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...

Author: K. Nagayama

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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-

N e w Y o r k - O x f o r d - Paris - S h a n n o n -

Tokyo

ELSEVIER

SCIENCE

B.V.

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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 << "c•. Otherwise the adsorption and the surface tension would be affected by the diffusion supply of surfactant molecules from the bulk of solution toward the expanding interface. The diffusion transport tends to reduce the increase of surface tension upon dilatation, thus apparently rendering the interface less elastic and more fluid. To describe the variation of the surface tension after an initial dilatation one is to solve the diffusion equation using an appropriate initial condition (see Section 1.3.1 for details). In such a case the Gibbs elasticity, Ec~, enters the theoretical expressions through this initial condition, which corresponds to an "instantaneous" dilatation of the interface (that is At << ~-,~), see e.g. Ref. [57]. This "instantaneous" dilatation decreases the adsorptions Fi and the subsurface concentrations ci, of the species (the subsurface is presumed to be always in equilibrium with the surface), but the bulk concentrations c;= remain unaffected [58,59]. This initially created difference between ci, and c;= further triggers the diffusion process. Now, let us try to extend this approach to the case of ionic surfactants. In the case of solution of an ionic surfactant, a non-uniform diffuse electric double layer (EDL) is formed in a vicinity of the interface; this is the major difference with the case of nonionic surfactant. The main question is whether or not the electric field in the EDL should be affected by the initial "instantaneous" dilatation of the interface. This problem has been examined in Ref. [60] and it has been established that a variation of the electric field during the initial dilatation leads to theoretical results devoid of sense. This is due to the following two facts" (i) The speed of propagation of the electric signals is much greater than the characteristic rate of diffusion. (2) Even a small initial variation in the surface charge density ,o, immediately gives rise to an electric potential, which is linearly increasing with the distance from the interface (potential of a planar wall). Thus a small initial perturbation of the interface would

Chapter 1

36

immediately affect the ions in the whole solution; of course, such an initial condition is physically unacceptable. In reality, a linearly growing electric field could not appear in the ionic solution, because a variation of the surface charge density would be immediately suppressed by exchange of counterions, which are abundant in the subsurface layer of the solution (see Fig. 1.4). The theoretical equations suggest the same: to have a mathematically meaningful initial condition for the diffusion problem, the initial dilatation must be carried out at constant surface charge density p, (p, = const, means also ~ , = const., see Eq. 1.66). Thus we can conclude that the initial "instantaneous" interfacial dilatation, which is related to the definition of Gibbs elasticity of a soluble ionic surfactant, must be carried out at Ps = const. From Eq. (1.19) one obtains

(dcy)p~ - (da. )o~ + (dad)p,

(1.87)

We recall that cr~ and cyd are, respectively, the contributions of the adsorption and diffusion layers to the total interfacial tension, or. An interfacial dilatation at constant p.,. and O.3 does not alter the diffuse part of the EDL, and consequently, ( d c r a ) p . , - 0 . Since, or,,-or 0 - k T J , the expressions for J in Table 1.1 show that or,, depends only on F~ at constant temperature. Then the definition of Gibbs elasticity of nonionic adsorption layers, Eq. (1.45), can be extended to ionic adsorption layers in the following way:

Ea--r,

~

.p,--rl~-~

r

(1.88)

The dependence of o" on FI for nonionic surfactants is the same as the dependence of oi, on F1 for ionic surfactants, see the surface tension isotherms in Table 1.1. Then Eqs. (1.45) and (1.88) show that the expressions for Ec, in Table 1.2 are valid for both nonionic and ionic surfactants. The effect of the surface electric potential on the Gibbs elasticity Eo of an ionic adsorption monolayer is implicit, through the equilibrium surfactant adsorption FI, which depends on the electric properties of the interface. To illustrate this let us consider the case of Langmuir isotherm; combining expressions from Tables 1.1 and 1.2 we obtain E c - F k T K a ~ , . Further, using Eqs. (1.49) and (1.81) we derive

E G - F~kTa,~ (KI e - * ' + K2a2~ )

(for Langmuir isotherm)

(1.89)

Planar Fluid Interfaces

37

Equation (1.89) visualizes the effect of salt on Ec: when the salt concentration increases, a2oo also increases, but the (dimensionless) surface potential ~s decreases; then Eq. (1.89) predicts an increase of Ec with the salt concentration. Note also that the values of Eo, calculated from the fits, like that in Fig. 1.6, depend on the type of the used adsorption isotherm; for example, the Frumkin isotherm gives values of E(;, which are systematically larger than those given by the van der Waals isotherm. The latter is preferable for fluid interfaces insofar as it corresponds to the model of non-localized adsorption. The definition of Gibbs elasticity given by Eq. (1.88) corresponds to an "instantaneous" (At << "tb) dilatation of the adsorption layer (that contributes to 0%) without affecting the diffuse layer and od. This will cause an initial change in the subsurface concentrations c;, of the species, which will further trigger a diffusion transport of components across a changing electric double layer. Thus we reach again the subject of the adsorption kinetics, which is considered in the next section.

1.3.

KINETICS OF SURFACTANT ADSORPTION

When a colloidal particle approaches an interface from the bulk of solution, or when an attached particle is moving throughout the interface, the surfactant adsorption layer is locally disturbed (expanded, compressed, sheared). The surfactant solution has the property to damp the disturbances by diffusion of surfactant molecules from the bulk to the interface (or in the opposite direction). If the particle motion is slow enough (compared with the relaxation time of surface tension "tb) the interface will behave as a two-dimensional fluid and surface elastic effects will not arise. On the contrary, if the characteristic time of the process of particle motion is comparable with or smaller than re,, the motion of the particle will be accompanied by surface elastic effects and adsorption dynamics. The criterion, showing when the latter effects would appear, is related to the relaxation time of the surface tension r~. Our attention in the present section will be focused on the theoretical results about "tb obtained for various types of surfactant adsorption, as follows: (i) adsorption under diffusion control, (ii) adsorption under electro-diffusion control, (iii) adsorption under barrier (kinetic) control,

38

Chapter 1

(iv) adsorption from micellar solutions, (v) adsorption from protein solutions. Our purpose is to give a brief review and related references in the context of the subject of this book; detailed information about the variety of experimental methods and theoretical approaches can be found elsewhere [58-66].

1.3.1. ADSORPTIONUNDERDIFFUSIONCONTROL Insofar as we are interested mainly in the relaxation time zc~, we will restrict our considerations to a physical situation, in which the interface is instantaneously expanded at the initial moment t = 0 and then (for t > 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 ;

erfc(x) = -[I

$11

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<(;/( I:".,kT))' ~

(for Langmuir isothcrm)

( I . 100)

Equation ( I , IOO) visualizes the vcry strong dcpendcnce of the rtlaxalion liirle on thc sut-lratant concentration

L'I;

i n gcnciA,

tn

can vai-y with inany order5 of riiagnitlrde as

;I

function of c .

Equation ( I , 100) shows dso that high Gibbs elasticily corrcsponds to short rclaxation time, and vice vci-sa.

41

Planar Fluid Interfaces

As a quantitative example let us take typical values of the parameters: K~ = 15 m3/mol, 1/F~o= 40/~2, D~ - 5.5 x 10-6 cm2/s and T=298 K. Then substituting c1= 6.5 x 10-6 M in Eqs. (1.45a) and (1.100) we calculate Ec -- 1.0 mN/m and ~'~r= 5 s. In the same way, for Cl= 6.5 x 10.4 M we calculate Ea ~ 100 mN/m and ~'cr= 5 x 10-4 s. As already mentioned, to directly measure the Gibbs elasticity Ec, or to precisely investigate the dynamics of surface tension, one needs an experimental method, whose characteristic time is smaller compared to ~'cr. Equation (1.100) and the above numerical example show that when the surfactant concentration is higher, the experimental method should be faster. Various experimental methods are available, whose operational time scales cover different time intervals. Methods with a shorter characteristic operational time are the oscillating jet method [71-73], the oscillating bubble method [74-77], the fast-formed drop technique [78,79], the surface wave techniques [80-83] and the maximum bubble pressure method [84-88]. Methods of longer characteristic operational time are the inclined plate method [89, 90] and the dropweight techniques [91-93]; see Ref. [64] for a detailed review.

1.3.2.

ADSORPTION UNDER ELECTRO-DIFFUSION CONTROL

Let us consider a solution of an ionic surfactant and salt; for simplicity we assume that the counterions due to the surfactant and salt are the same (an example is SDS and NaC1, both of them releasing Na + counterions; the coions are DS- and C1-). The adsorption of surfactant at the interface creates surface charge, which is increasing in the course of the adsorption process. The charged interface repels the new-coming surfactant molecules, but attracts the conversely charged counterions; some of them bind to the surfactant headgroups thus decreasing the surface charge density and favoring the adsorption of new surfactant molecules. The transport of the i-th ionic species, with valency Zi and diffusion coefficient Di , under the influence of electrical potential gt, is described by the set of electro-diffusion equations [58,59,94]:

at

-

az

+

kT

c/

-~z

(z > O, t > O)

i = 1,2,3

(1 101)

Chapter 1

42

The indices i = 1, 2 and 3 denote the surfactant ion, the counterion and the coion, respectively; ci is the bulk concentrations of the i-th ion which depends on time t and the distance z to the interface. The second term in the parentheses in Eq. (1.101), the so called "electromigration" term, accounts for the effect of the electric field on diffusion. The electric potential ~ is related to the bulk charge density through the known Poisson equation,

O 2~ _ Oz ~

47ce [Z,c, + Z~c~ + Z~c 3 ] , e - -

(1 102)

Now we have two adsorbing species" the surfactant ions and the counterions; the colons are not ,...,

expected to adsorb at the interface: F'3 = 0; on the other hand, 1-'3 - A 3 ve 0, see Eq. (1.69). Then the generalization of Eq. (1.91) is

d F i _ D ( Ol Ci Zie o~~ ) ~ + -c dt ~z kT ' o~z

(z =0, t > 0)

i = 1,2.

(1.103)

Note that the supply of surfactant ions to the interface is promoted by the gradient of concentration, Vcl, but it is opposed by the gradient of electric field, V~. The two effects compensate each other in such a way, that the effect of gc~ is slightly predominant (otherwise, there would not be surfactant adsorption). For the conversely charged counterions these tendencies have the opposite direction with a predominant effect of V ~. It is not an easy task to solve the electro-diffusion problem based on Eqs. (1.101)-(1.103). Dukhin et al.

[95-98] have developed a quasi-equilibrium model assuming that the

characteristic diffusion time is much greater than the time of formation of the electrical double layer, and consequently, the electro-diffusion problem is reduced to a mixed barrier-diffusion controlled problem. Bonfillon and Langevin [99] investigated the case of small periodic surface corrugations. MacLeod and Radke [94] obtained numerical solutions of the electro-diffusion problem without taking into account the effect of counterion binding, i.e. the formation of a Stern layer. Analytical results for the long-time asymptotics of adsorption and surface tension have been obtained in Refs. [60,70,100] without making simplifications of the physical model. Assuming

43

Planar Fluid Interfaces

small deviations from equilibrium the adsorption isotherm is linearized and a counterpart of Eq. (1.92) is obtained:

AF i

(t) --r i (t)--

i~

El, e

where Ac. ( t ) - c i , ( t ) - q , ~ ) ,

eA C I s (f)-]-

G~ C2sL

e

1104)

-"

"

i =1,2, are the perturbations in the subsurface concentrations of

surfactant ions and counterions. As usual, the subscript "s" denotes subsurface concentration and the subscript "e" refers to the equilibrium state. We recall that in the case of ionic surfactant two types of adsorptions can be introduced: Fi, which is mostly due to the surfactant ions and counterions immobilized at the interface, and F i - F i + A i which includes also a contribution from the whole diffuse EDL, see Eq. (1.69). Equation (1.104) expresses a local equilibrium between surface and subsurface; such an equation cannot be written for F i , because the latter quantity has a non-local, integral character. The result for the long-time asymptotics of the adsorption relaxation, derived on the basis of Eqs. (1.101)-(1.104), is [60,70] a F , (t) _ E ( t ) - r,,~ _ r ~ r , t AF' i (0) - F; (0) - Fi,e

(t >> 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) << r,

A = (1 + c-wr,.m+ c,-'

for large initial perturbation:

r,(O)= 0

The above algebraic equations enable one to calculate the relaxation times of surfactant and counterion adsorption.

71 and T?

, using Eq. ( I . 106). From Eqs. (1.74), ( 1 .81), ( I 3 2 ) and the

adsorption isotherms in Table I . 1 (with u r s=cJ one can deduce relatively simple expressions for the coefficients g,, [60]:

4;

J

(1.108)

: -

'I

-

kTT,

Using the expressions for Ec; for the various isotherms in Table 1.2, one can easily calculate

J,, and all coefficients g,l from Eqs. ( I . 107)-( I . 108). The result for the long-time asymptotics of the surface tension relaxation is [60,70]:

( t >> T ~ )

for [urge initial perturbation ( I ,109b)

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

w = 2 tanh-

@\.
~

J'l

where, as usual,

2

( 1 . 1 10)

is the equilibrium value of the dimensionless surface potential, cf. Eq.

(1.54) and the relaxation times

51

and

zl are given by Eq. (1,106). It should be noted that

usually A = I and, therefore, the values of T ~ zz , and z, are not so sensitive to the magnitude of

45

Planar Fluid Interfaces

the initial perturbation: small or large. In this respect zl, ~= and ~,~ can be considered as general kinetic properties of the adsorption monolayer from an ionic surfactant [70]. Let us now discuss the similarities and dissimilarities of the adsorption relaxation under diffusion and electro-diffusion control, which correspond to the cases of nonionic and ionic surfactants, respectively. In both cases AFt(t) and Ao-(t) tend to zero proportionally to 1/x/t, cf. Eqs. (1.98), (1.105) and (1.109). Hence, from the fact that the plot of Ao-(t) vs. 1/4~ is linear it is impossible to determine whether the adsorption is under diffusion or electro-diffusion control. The difference is that the slope of this plot depends on the concentration of added salt in the case of electro-diffusion control. Another difference is that for nonionic surfactants the relaxation time is the same for adsorption and surface tension, see Eq. (1.105), whereas for ionic surfactants these relaxation times are different:

vG~ "c~ ~ ~2. The latter difference

originates from the presence of diffuse electric double layer, whose relatively slow relaxation affects stronger ~-,~ than T~ a n d s . To visualize the difference between ~,~, "El and ~h, and to examine their dependence on the concentration of the dissolved salt, below we consider an illustrative example. The values of the parameters/~, F~, K~ and K2 , determined from the fits of the data in Fig. 1.6 (see Section 1.2.5) can be used to calculate the values of all parameters entering Eq. (1.106) and (1.110). Figure 1.9 shows the relaxation times of surface tension, surfactant adsorption and counterion adsorption, zo, L , and z-=, respectively, calculated in this way in Ref. [70]. They are plotted as functions of the surfactant concentration, c1=, for a solution, which does not contain NaCI" c 3 -

0. First of all, one notices the wide range of variation of the relaxation

times, which is 3 - 4 orders of magnitude. In particular, the relaxation time of surface tension, T~, drops from 0.1 s down to 1•

-5 s. Next, one sees that systematically z=<~'l < z ~ the

difference between these three relaxation times can be greater than one order of magnitude for the lower surfactant concentrations. Thus one can conclude that the terms with ~.~.e in Eq. (1.110), which lead to a difference between L and ~-o, play an important vole, especially for the lower surfactant concentrations. Figure 1.9 demonstrates that the approximation v~ =z'~, which is widely used in literature, is applicable only for the higher surfactant concentrations,

46

Chapter 1

1 0 -1

"~ ,...,

"-,, .

10-2

",.\ \

E

._ tO

..

10-a ~2

X

rr

.....-

"

10-4

10-5 2

a

4

5

6

7

10 0

1.5

2

3

4

5

6

7

TSDS concentration (mM)

Fig. 1.9. Relaxation times of surface tension, z,~, of surfactant adsorption, "el, and of counterion adsorption, "c2, calculated in Ref. [70] for surfactant solutions without added salt by means of Eqs. (1.106) and (1.110) as functions of surfactant (TSDS) concentration, c~oo, using parameters values determined from the best fit of the data in Fig. 1.6; a large initial perturbation is assumed. for which "t'o-->'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~ << F= both the Frumkin and Langmuir expressions in Table 1.4 reduce to the Henry expression. Substituting the expressions for Q from Table 1.4 into Eq. (1.111) and integrating one can derive explicit expressions for the relaxation of surfactant adsorption:

49

Planar Fluid Interfaces

Table 1.4. Expressions for the total rate of reversible surfactant adsorption, Q, corresponding to six different kinetic models [110]. Total rate of reversible adsorption

Type of adsorption isotherm

Q - raas (c I ,F 1) - rdes (1-'1)

Henry

Q = K a d s Cl -- gdesF1

Freundlich

Q -- gads Kin-1 c~n - KdesFl

Langmuir

Q = Kad~C, 1 - ~F1 / - gdesFl

Frumkin

F1

exp(- 2/3 F l

/

Volmer Q - KadsC1 -- KdesF1

F~

exp(

roo - r ,

~r~

F1 -r

/

1

)

van der Waals Q-KadsCl-KdesF1 F~F~-F1 exp( F1F~- F 1

2flF1)kT

O'(t)-O'e = Fl(t)-Fl'e = exp(t--~)

a(o)-a~

F, (o)- F,,~

(1.113)

k,-r~

Equation (1.113) holds for Ao(t) = o(t) - oe only in the case of small deviations from equilibrium, whereas there is not such a restriction concerning

AFt(t) = F~(t) - Fl,e" the

relaxation time in Eq. (1.113) is given by the expressions To" _

(Kde s

T o. =

/

)-1

Kde s +

(Henry and Freundlich) KadsCl

F

/'

(Langmuir)

(1.114)

(1.115)

Equation (1.113) predicts that the perturbation of surface tension, Ao-(t) = o(t) - oe, relaxes exponentially. This is an important difference with the cases of adsorption under diffusion and electro-diffusion control, for which Ao'(t) oo 1/47 ,cf. Eqs. (1.98), (1.105) and (1.109). Thus a

50

Chapter 1

test whether or not the adsorption occurs under purely barrier control is to plot data for ln[Ao-(t)] vs. t and to check if the plot complies with a straight line. When the rate of transfer of surfactant molecules from the subsurface to the surface is not-tooslow compared with the rate of diffusion, one deals with the more general case of mixed barrier-diffusion control [110]. In such a case, the "kinetic" boundary condition (1.111) is used

in conjunction with the "diffusion" boundary condition (1.91). Numerical analysis of this process has been performed on the basis of the Henry and Langmuir models [113], and the Frumkin model [111]. Analytical solution of the mixed (diffusion-barrier) problem has been published in Ref. [59] for the case of the Henry isotherm: F 1( t ) - Fj (0)

rl, e - r l (0)

= f(~:,b2)- FU:,b~),

F('c,b) -

1-exp(b 2r)erfc(b ~r}-)

(1.116)

2b x/a 2 - i

where z = Kdest is dimensionless time, b~,2 = ~ + (0~ 2_ 1)~/2 are dimensionless parameters with o~

-

Kads/(4D 1Kdes )1/2 being a dimensionless diffusion-kinetic ratio; the complementary error

function erfc(x) is defined by Eq. (1.95). Equation (1.116) is valid not only for a > 1, but also in the case c~ < 1 (fast diffusion). Despite the fact that in the latter case b~ and b2 become complex numbers, Eq. (1.116) gives real values of Fl(t). In the limit c~ --+ 0 (complete barrier control) Eq. (1.116) reduces to Eqs. (1.113)-(1.114). In fact, Eq. (1.116) describes the transition from diffusion to adsorption control: for a > 1 / ~ -

the diffusion control is

predominant and AF(t) ~: 1/~/7 for t--+~; on the other hand, for o~ < 1 / ~

the barrier control is

predominant and AF(t) decays exponentially for t-->~,. One can estimate the characteristic time of relaxation under mixed diffusion-barrier control by using the following combined expression: ~'o = ~

1 (

o~+

01r -1 11/2)2

(1.117)

For o~ << 1 Eq. (1.117) reduces to the result for barrier control, Eq. (1.114), whereas for o~ >> 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 << 2"m. Note, that Eq. (1.134) is applicable only for small perturbations. An approximate analytical approach, which is applicable for both small and large deviations from equilibrium, is developed in Ref. [ 134].

1.3.5.

ADSORPTION FROM SOLUTIONS OF PROTEINS

The kinetics of adsorption of proteins and other macromolecules is a complex process, in which several steps, some of them occurring simultaneously, can be conceptually identified [136,137]: (i) transport of the native protein to the interface by diffusion; (ii) adsorptiondesorption from the phase boundary, which can happen under barrier control, see section 1.3.3; (iii) changes in the molecular structure (denaturation) including unfolding and spreading of the molecules over the interface; (iv) rearrangement of some structural groups of the adsorbed protein molecules. Many results accumulated in this field can be found in Refs. [63,138-140]. As an example let us consider a process of protein adsorption including stages (i) and (iii). We denote by F~ and 1-'2 the adsorptions of the native and the denatured protein, respectively. The changes of F1 and F2 due to the denaturation process are equal by magnitude but have opposite signs. Then the interfacial mass balances for the two modifications of protein at the interface take the form [ 141 ]:

--" ~

~ --.

q

C; O

.0

~

0

9

~

"

r

~

~

~,

~

~

.,-

-,

~

~ ~ --.

~ ~"

-

0

=

-.

~ ~

~.

o_

O

=r

~

ON

~

~"

,-i

~

~

'-~ =~"

:

0

:::::1

0

~

_0 , .

~-<. ~

~

~ m"

~"

~ I~

r~

~L

~"

E

B

9

=r

O

0

~ ~ ..

O

~

--"

m

~

~-

t,a

~

4~

~,

F

III

I

~

4~

~ ~

O

9

~

I

'-~

~

II

I>

~

._.

+

~

o

. e~

~~ II

-

~

-,.

4~

'o' II

~

~

~.

.

o

~..,. ~

=..

9

~- . m

9

~

~-~

~

-a

~.

=;" ~

"

~

~ ;~

I

I .-

-

I

I

L/I

Planar Fluid Interfaces

57

order of micrometers because of the formation of a diffuse electric double layer in a vicinity of the phase boundary. In the latter case the interfacial tension (and the bending moment as well) can be expressed as a sum of a double-layer and a non-double-layer contribution, Eq. (1.19). The thermodynamics of fluid interfaces describes how the composition in the bulk of solution determines the composition at the phase boundary and the interfacial tension. Various surfactant adsorption isotherms can be used to process experimental data; the most popular of them are listed in Table 1.1, where the respective surface tension isotherms are also shown. In the case of solutions of ionic surfactants two types of adsorptions can be introduced: F~, which represents a surface excesses of component "i" with respect to the uniform bulk solution, and Fi representing a surface excess with respect to the non-uniform diffuse electric double layer, see Eq. (1.69). Correspondingly, the Gibbs adsorption equation for a charged interface can be expressed in two equivalent forms, Eqs. (1.68) and (1.70). Not only surfactant ions, but also counterions, do adsorb at the interface; the counterionic adsorption can be described by the Stern isotherm, Eq. (1.82), which is thermodynamically compatible with the surfactant adsorption isotherms (Table 1.1) if only Eq. (1.81) is satisfied. The occupancy of the Stern layer by adsorbed counterions could rise up to 70 - 80 % (Fig. 1.8) and should not be neglected. The double-layer contribution to the interfacial tension depends on the valence of the electrolyte, see Table 1.3. The value of the Gibbs elasticity, Ec, of an adsorption layer determines whether the interface behaves as a two-dimensional fluid or as an elastic body. This rheological behavior can strongly influence the attachment of a particle to an interface, as well as the capillary forces between attached particles. Definitions for Gibbs elasticity of adsorption layers from non-ionic and ionic surfactants are presented and discussed, see Eqs. (1.45), (1.88) and Table 1.2. For soluble surfactants the effect of the Gibbs elasticity can be suppressed by the diffusive supply of surfactant to an expanding interface; the diffusion tends to render the interface more fluid. Thus one can estimate whether or not the effect of Gibbs elasticity will show up by a comparison of the characteristic adsorption relaxation time with the characteristic time of the specific process. Expressions for the relaxation time of surface tension, v•, are presented for the cases of adsorption kinetics under diffusion control: Eq. (1.94); electro-diffusion control:

58

Chapter 1

Eq. ( 1.110); barrier control: Eqs. (1.114)-(1.115), mixed diffusion-barrier control: Eq. (1.117), adsorption from micellar solutions, Eq. (1.134), and adsorption from protein solutions, see Eq. (1.140). The quantities introduced in Chapter 1, and the relationships between them, are important for the theoretical description of the particle-interface interaction and the particleparticle capillary forces, as this will be seen in the next chapters.

1.5.

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I.

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2.

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3.

J.S. Rowlinson, B. Widom, Molecular Theory of Capillarity, Clarendon Press, Oxford, 1982.

.

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~

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J.W. Gibbs, The Scientific Papers of J.W. Gibbs, vol. 1, Dover, New York, 1961. G. Bakker, Kapillarit~it und Oberfl~ichenspannung, in: Handbuch der Experimentalphysik, Band 6, Akademische Verlagsgesellschaft, Leipzig, 1928. F.P. Buff, J. Chem. Phys. 19 (1951) 1591. I.W. Plesner, O. Platz, J. Chem. Phys. 48 (1968) 5361. C.A. Croxton, R.P. Ferrier, (1971), J. Phys. C: Solid State Phys. 4 (1971) 2433. A.I. Rusanov, E.N. Brodskaya, Kolloidn. Zh. 39 (1977) 646.

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Planar Fluid Interfaces

59

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60

Chapter 1

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Planar Fluid Interfaces

61

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62 92. P. Joos, J. van Hunsel, Colloid Polym. Sci. 267 (1989) 1026. 93. V.B. Fainerman, R. Miller, Colloids Surf. A. 97 (1995) 255. 94. C. MacLeod, C. Radke, Langmuir 10 (1994) 3555. 95. S. Dukhin, R. Miller, G. Kretzschmar, Colloid Polym. Sci. 261 (1983) 335. 96. S. Dukhin, R. Miller, G. Kretzschmar, Colloid Polym. Sci. 263 (1985) 420 97. S. Dukhin, R. Miller, Colloid Polym. Sci. 269 (1991) 923. 98. S. Dukhin, R. Miller, Colloid Polym. Sci. 272 (1994) 548. 99. A. Bonfillon, D. Langevin, Langmuir 10 (1994) 2965. 100.P.M. Vlahovska, K.D. Danov, A. Mehreteab, G, Broze, J. Colloid Interface Sci. 192 (1997) 194. 101.W.N. Bond, H.O. Puls, Phil. Mag. 24 (1937) 864.

102.K.S.G. Doss, Koll. Z. 84 (1938) 138. 103. S. Ross, Am. Chem. SOC.67 (1945) 990. 104.C.M. Blair, J. Chem. Phys. 16 (1948) 113. 105. A.F.H. Ward, Surface Chemistry, London, 1949. 106. R.S. Hansen, T. Wallace, J. Phys. Chem. 63 (19.59) 108.5.

107. D.G. Dervichian. Koll. Z. 146 (1956) 96. 108. J.F. Baret, J. Phys. Chem. 72 (1968) 275.5.

109. J.F. Baret, J. Chem. Phys. 65 (1968) 89.5. 110. J.F. Baret, J. Colloid Interface Sci. 30 (1969) I. I 1 1 . R.P. Borwankar, D.T. Wasan, Chem. Eng. Sci. 38 (1983) 1637. 1 12. L.K. Filippov, J. Colloid Interface Sci. I64 ( 1994) 47 I .

1 13. R. Miller, G. Kretzschmar, Colloid Polym. Sci. 258 ( 1 980) 8.5.

114. C. Tsonopoulos, J. Newman, J.M. Prausnitz, Chem. Eng. Sci. 26 (1971) 817

115. A. Yousef, B.J. McCoy, J. Colloid Interface Sci. 94 (1983) 497. 116. Z. Adamczyk, and J. Petlocki, J. Colloid Interface Sci. 118 (1987) 20.

I 17. Z. Adamczyk, J. Colloid Interface Sci. 120 (1987) 477.

I 18. J. Balbaert, P. Joos, Colloids Surf. 23 ( 1987) 259.

I 19. V.V. Kalinin, C.J. Radke, Colloids Surf. A, 1 14 (1996) 337.

Planar Fluid Interfaces

63

120. E.A.G. Aniansson, S.N. Wall, J. Phys. Chem. 78 (1974) 1024. 121. E.A.G. Aniansson, S.N. Wall, J. Phys. Chem. 79 (1975) 857. 122. E.A.G. Aniansson, S.N. Wall, in: E. Wyn-Jones (Ed.) Chemical and Biological Applications of Relaxation Spectrometry, Reidel, Dordrecht, 1975, p. 223. 123. E.A.G. Aniansson, S.N. Wall, M. Almgren, H. Hoffmann, I. Kielmann, W. Ulbricht, R. Zana, J. Lang, C. Tondre, J. Phys. Chem. 80 (1976) 905. 124. C. D. Dushkin, I. B. Ivanov, Colloids Surf. 60 (1991) 213. 125. C. D. Dushkin, I. B. Ivanov, P. A. Kralchevsky, Colloids Surf. 60 (1991) 235. 126. B. A. Noskov, Kolloidn Zh. 52 (1990) 509. 127. B. A. Noskov, Kolloidn Zh. 52 (1990) 796. 128. D. H. McQueen, and J. J. Hermans, J. Colloid Interface Sci. 39 (1972) 389. 129. J. Lucassen, J. Chem. Soc., Faraday Trans. 172 (1976) 76. 130. R. Miller, Colloid Polym. Sci. 259 (1981) 1124. 131. R. H. Weinheimer, D. F. Evans, and E. L. Cussler, J. Colloid Interface Sci. 80 (1981) 357. 132. D.E. Evans, S. Mukherjee, D.J. Mitchell, B.W. Ninham, J. Colloid Interface Sci. 93 (1983) 184. 133. P. Joos, L. van Hunsel, Colloids Surf. 33 (1988) 99. 134. K.D. Danov, P.M. Vlahovska, T. Horozov, C.D. Dushkin, P.A. Kralchevsky, A. Mehreteab, G. Broze, J. Colloid Interface Sci. 183 (1996) 223. 135. G.C. Kresheck, E. Hamori, G. Davenport, and H.A. Scheraga, J. Am. Chem. Soc. 88 (1966) 264. 136. E. Tornberg, J. Colloid Interface Sci. 64 (1978) 3. 137. J.D. Aptel, A. Carroy, P. Dejardin, E. Pefferkorn, P. Schaaf, A. Schmitt, R. Varoqui, J.C. Voegel, in: J.L. Brash and T.A. Horbett (Eds.) Proteins at Interfaces: Physicochemical and Biochemical Studies, ACS Publications, Washington, DC, 1987. 138. J.D. Andrade (Ed.) Surface and Inteffacial Aspects of Biomedical Polymers, Vol. 2: Protein Adsorption, Plenum Press, New York, 1985. 139. J.L. Brash and T.A. Horbett (Eds.) Proteins at Interfaces: Physicochemical and Biochemical Studies, ACS Publications, Washington, DC, 1987. 140. R. Miller, V.B. Fainerman, A.V. Makievski, J. Kr~gel, D.O. Gfigoriev, V.N. Kazakov, O.V. Sinyachenko, Adv. Colloid Interface Sci. 86 (2000) 39. 141. G. Semen, G. Geeraerts, P. Joos, Colloids Surf. 68 (1992) 219.

64

CHAPTER 2 INTERFACES OF MODERATE CURVATURE: THEORY OF CAPILLARITY

This chapter gives a brief presentation of the conventional theory of capillarity, which is based on the Laplace and Young equations, and neglects such effects as interfacial bending moment and curvature elastic moduli (the latter effects are subject of the next Chapter 3). The Laplace equation is derived by a force balance per unit area of a curved interface, as well as by means of a variational method. Various forms of Laplace equation are presented depending on the symmetry of the phase boundaries. Special attention is paid to the physically and practically important case of axisymmetric interfaces. Equations are given, which describe the shape of sessile and pendant drops, of the fluid interface around a vertical cylinder, floating solid or fluid particle, hole in a wetting film, capillary "bridges", Plateau borders in foams, the profile of the free surface of a fluid particle or biological cell pressed between two plates, etc. The values of the contact angles subtended between three intersecting surfaces are determined by the force balance at the contact line, which is given by the Young and Neumann equations. It is demonstrated that these equations (likewise the Laplace equation) can be derived by variation of the thermodynamic potential. The rule how to calculate the net force exerted on a particle at an interface is discussed. Linear excess energy (line tension) can be ascribed to a contact line. When the contact line is curved, the line tension gives a contribution to the Young and Neumann equations. The presence of line tension effect is indicated by dependence of the contact angle on the curvature of the contact line. The contact angles can vary also due to the phenomenon hysteresis, which is considered in relation to the line tension effect. The chapter represents a basis for most of the subsequent chapters since the subjects of first importance in this book are the shapes of the menisci around attached particles, the shapes of fluid particles approaching an interface, the balances of forces exerted on particles at interfaces, and various kinds of capillary forces.

Interfaces of Moderate Curvature." Theor3, of CapillariO,

2.1.

THE LAPLACE EQUATION OF CAPILLARITY

2.1.1.

LAPLACE EQUATION FOR SPHERICAL INTERFACE

65

Let us consider a spherical interface between two fluid phases (spherical liquid drop or gas bubble). In Fig. 2.1 Pl and P2 denote the inner and outer pressure, respectively; R is the radius of the spherical dividing surface (defined as the surface of tension) and o- is the interfacial tension. Let us make the balance of all forces exerted on a small segment of the dividing surface situated around its intersection point with the z-axis (Fig. 2.1) and corresponding to a central angle 0. The area of this segment is A(O) = 2rtR2(1 - cos0) = 71;R202

(0 << 1)

(2.1)

(0 << 1)

(2.2)

The length of the circumference encircling this segment is L(0) = 2rtRsin 0 = 2rtR 0

Then the balance of the forces acting on the segment, resolved along the z-axis, reads: P1A(O) = P2 A(O) + (o"sin0)L(O)

(2.3)

The left-hand side of Eq. (2.3) represents the force directed upwards, whereas the right-hand side expresses the forces acting downwards (Fig. 2.1); these forces should counterbalance each other for an equilibrium interface. Substituting Eqs. (2.1) and (2.2) into Eq. (2.3), and carrying out the transition 0--->0, one obtains the Laplace equation of capillarity for a spherical interface

[1]: 2o

~ = P ~ -P2 R

(2.4)

Equation (2.4) shows that the pressure exhibits a jump, Pc = PI - P2

(2.5)

across a spherical interface; Pc is called capillary pressure, or Laplace pressure. In the limit R---~0 (planar interface) Eq. (2.4) yields P~ - P2, as this must be for a flat dividing surface, see

Chapter 2

66

i Rsin0..

.......... ~- . . . . . . . ,~,,

P~

'~c~sinO

~O ~ J R

~sinO,

-

|

!

\

/ \

/ N

/ \

%

l

Fig. 2.1. Balance of forces exerted on a segment of spherical interface or membrane of tension cr and radius R; the segment is encompassed by the circumference of radius Rsin0, where 0 is a central angle; P~ and P2 denote the inner and outer pressure, respectively. Section 1.1. The above purely hydrostatic derivation of the Laplace equation reveals its physical meaning: it expresses the normal force balance per unit area of the interface. Below we proceed with the derivation of the form of Laplace equation for an arbitrarily curved interface.

2.1.2.

GENERAL FORM OF LAPLACE EQUATION

Derivation by minimization of the grand potential. Let

us consider a two-phase fluid

system confined in a box of volume V, see Fig. 2.2. The volumes of the two phases are V~ and V2 ; we have Vz + V2 = V. We assume also that the chemical potentials of all components in the system are kept constant. Then the equilibrium state of the system corresponds to a minimum of the grand thermodynamic potential, s [2-4]:

f 2 - - I P l d V - IP2dV+crA v~

(2.6)

v2

where A is the area of the interface; the pressures P~ and P2 depend on the vertical coordinate z due to the effect of gravity:

Interfaces o f Moderate Curvature: Theoo, of CapillariO,

67

phase 2 9

V1

phase 1 0

x

Fig. 2.2. Sketch of a two-phase system composed of phases 1 and 2, which occupy volumes V~ and V2, respectively; z = u(x,y) is the equation of the phase boundary.

PI(Z) =

Plo-

(2.7)

P2(z) = P 2 0 - p 2 g z ,

plgz,

PI0 and P20 are constants, p~ and P2 are the mass densities of the two neighboring fluids, and g is the acceleration due to gravity. Let z =

u(x,y)

to be the equation describing the shape of the

interface. Then the area of the interface is A-

f dx ~x/ 1-1- U2x + U2 , ~0

Ux - -

, au ax

U,. ----

A0 is the projection of the interface on the coordinate plane

(2.8)

au oN

xy.

In addition, one derives

u(x,y)

b

(2.9) v~

Ao

o

v~

Ao

u( x, y )

where z = 0 and z - b are the lower and the upper side of the box (Fig. 2.2). The substitution of Eqs. (2.8) and (2.9) into Eq. (2.6) yields

f2 - ~dxdyL(u(x, y),ux(x, y),u, (x, y))

(2.10)

Ao

where

L(u,ux,u,)-- P,(z)&- P2(z)&+~ l+u~ +u ~ 0

u

(2.11)

68

Chapter 2

Equations (2.10) and (2.11) show that the grand potential f2 depends as a functional on the interfacial shape u(x,y). Then the necessary condition for minimum of f2 is given by the known Euler equation [5,6]" cgL

o~ cgL

cg u

O x O ux

O cgL ~ ~ = 0

(2.12)

O y cg u Y

Differentiating Eq. (2.11) one obtains cgL

~9u = -P1(u) + P2 (u)

(2.13)

Next, differentiating Eq. (2.1 l) one can derive O cgL 03 o~L ~ ~ + ~ ~ = 2 H o

Ox cgux

(2.14)

O y OUy

where we have used the notation

2H -VII 9I

Vn u

(2.15)

I

~/1 + IgllU] 2

VII

9 ~e

X

0

(2.16)

=--Ox+ey 0 y

Here VII is the two-dimensional gradient operator in the plane xy; H defined by Eq. (2.15) is a basic quantity in differential geometry, which is termed mean curvature of the surface [5,7,8]. Note that Eq. (2.15) is expressed in a covariant form and can be specified for any type of curvilinear coordinates in the plane xy (not only Cartesian ones). Substituting Eqs. (2.13) and (2.14) into Eq. (2.12) we obtain a general form of Laplace equation of capillarity [ 1]" 2Her = P2(u) - Pl(U)

(Laplace equation)

(2.17)

When the pressures Pl and P2 are dependent on the position in space, as it is in Eq. (2.7), their values at the interface enter the Laplace equation; in such a case the capillary pressure, P c -

Pl(u)

-

P2(u), varies throughout the interface.

Interfaces of Moderate Curvature." Theory of Capillarity

69

Various forms of Laplace equation. The mean curvature can be expressed through the two principle radii of curvature of the surface, R1 and R2 [5,7]"

. . . . +~ 2 R~ R 2

(2.18)

Combining Eqs. (2.17) and (2.18) one obtains another popular form of Laplace equation [9]:

O" ~ + ~ R1 R2

= P l ( u ) - P2(u)

(Laplace equation)

(2.19)

For a spherical interface the two principal radii of curvature are equal, R1 = R2 = R, and then Eq. (2.19) reduces to Eq. (2.4). The original form of Eq. (2.17), published by Laplace in 1805, can be obtained if the right-hand side of Eq. (2.15) is expressed in Cartesian coordinates and the differentiation is carried out [1 ]" + Uy Uxx --

(1

UxUy +

+ Ux U)9,

2+//2)3/2

+Ux

= [P2(u) - Pl(u)]/c~

(2.20)

y

Here uxx, Uxy and Uyy denote the respective second derivatives of u(x,y). One sees that in general the Laplace equation, Eq. (2.20), is a second order non-linear partial differential equation for determining the shape of the fluid phase boundary, u(x,y). The way we derived Eq. (2.20) shows that its solution, u(x,y), minimizes the grand thermodynamic potential, ~2, and consequently, corresponds to the state of mechanical equilibrium of the system. For interfaces of rotational or translational symmetry Eq. (2.20) reduces to an ordinary differential equation (see below), which is much easier to solve. If the curved interface in Fig. 2.2 has translational symmetry along the y-axis, i.e. z = u(x), then Uy = 0, Uxy= Uyy- 0, and Eq. (2.20) reduces to: u .~:~. 2 )3/2

= (P2 - Pl)/O"

(translational symmetry)

(2.21)

l+u x

If the curved interface has rotational symmetry around the z-axis (axial symmetry), then it is convenient to introduce polar coordinates (r, q3) in the plane xy. Due to the axial symmetry the

70

Chapter 2

equation of the interface has the form z = u(r). Then introducing polar coordinates in Eq. (2.15) one can bring Eq. (2.17) into the form [10]"

l d l r u r l = ( P 22) - P1/2 l)/cYO rdr +u r

(rotational symmetry)

(2.22)

where Ur -- du/dr. Sometimes it is more convenient to work in terms of the inverse function of z = u(r), that is r = r(z). In such a case Eq. (2.22) can be transformed in an equivalent form [10,11]: ~, 1 192 - Pl (1 + re2) 3/2 + r ( l + r z2) 1/2 = ~ ' 0 . .

dr q - dz '

d 2r ~ - dz-"

(2.23)

Two equivalent parametric forms of Laplace equation are often used for analytical and numerical calculations [ 10,11 ]: dsinq~ dr

t

sinq~ r

=

Pc G

,

dz tan(p = _ + ~ dr

(2.24)

(the angle q9can be defined with both positive or negative sign) and d(p ds

=

Pc o

sinq9 , r

dr ~ = cosq~, ds

dz ds

= sinrp

(2.25)

Here q) is the meniscus running slope angle and s is the arc length along the generatrix of the meniscus z = z(r); Pc is the capillary pressure defined by Eq. (2.7)" the sign of Pc is to be specified for every given interface. Equations (2.25) represent a set of three equations for determining the functions (,0(s), r(s) and z(s), which is especially convenient for numerical integration [11]" note that Eq. (2.24) may create numerical problems at the points with tan(p = _+~,, like the points on the "equator" of the fluid particle in Fig. 2.3. The Laplace equation can be generalized to account for such effects as the interfacial bending elasticity and shearing tension" such a generalization is important for interfaces and membranes of low tension and high curvature and can be used to describe the configurations of red blood cells, see Chapters 3 and 4.

Interfaces of Moderate Curvature: Theoo' of CapillariO'

71

Z.

phase 3

| w(r)

....... I ..... "

[

\

l

R ..........

3u(r)

o

Fig. 2.3. Cross-section of a light fluid particle (bubble or droplet) from Phase l, which is attached to the boundary between Phases 2 and 3. The equations of the boundaries between phases 1-2, 13 and 2-3 are denoted by u(r), v(r) and w(r), respectively; (Pc, 0 and gtc are slope angles of the respective phase boundaries at the contact line, which intersects the plane of the drawing in the point (rc,zc); ~ r ) is a running slope angle; R is "equatorial" radius and Rs is the curvature radius of the surface v(r), which can be a thin film of Phase 2, intervening between Phases 1 and 3.

2.2.

AXISYMMETRIC FLUID INTERFACES

Very often the boundaries between two fluid phases (the capillary menisci) have rotational (axial) symmetry. An example is the fluid particle (drop or bubble) attached below an interface, I

lhr

II

er Fig. 2.4. Menisci formed by the liquid around two vertical coaxial cylinders of radii R1 and R2: (I) Meniscus meeting the axis of revolution; (II) Meniscus decaying at infinity; (III) Meniscus confined between the two cylinders; hr is the capillary rise in the inner cylinder; hc and gtc are the elevation and the slope angle of Meniscus II at the contact line r = R2.

Chapter 2

72

which is depicted in Fig. 2.3: all interfaces, u(r), v(r) and w(r), have axial symmetry. In general, there are three types of axially symmetric menisci corresponding to the three regions denoted in Fig. 2.4: (I) Meniscus meeting the axis of revolution, (ll) Meniscus decaying at infinity, and (Ill) Meniscus confined between two cylinders, 0
2.2.1.

MENISCUS MEETING THE AXIS OF REVOLUTION

The interfaces u(r) and v(r) in Fig. 2.3 belong to this type of menisci, as well as the interfaces of floating lenses and any kind of sessile or pendant drops/bubbles. Such a meniscus is a part of a sphere when the effect of gravity is negligible, that is when the capillary (or Bond) number is small:

~c - gb2AP ~or << 1

(2.26)

Here, as usual, g is the acceleration due to gravity, Ap is the magnitude of the difference in the mass densities of the two fluids and b is a characteristic radius of the meniscus curvature. For example, if Eq. (2.26) is satisfied with b = Rl, see Region I in Fig. 2.4, the capillary rise, hr, of the liquid in the inner cylinder is determined by means of the equation [ 13]

hr = (20"cosO0/(Ap g RI)

(2.27)

a is a contact angle which can be both acute and obtuse, depending on whether the inner surface of the cylinder is hydrophilic or hydrophobic; in the case o~ > 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. In this case the solution of Eq. (2.28) can be obtained as truncated asymptotic expansions with respect to the powers of/3c, see Ref. [ 14]:

r((P----~) =sinq~+eb/3c

[i3

+/3c2 7 + T c ~

( ~ c o t ~(P-+sin2(p --21sin(p]

2in2 ,asian/ l/ q~-7

~

sinq~-

z((p) = l + c o s q g + s f l c I3 sin2q)+72 lnsin (p 2 -7'

(2.31) l+-~cot2~

(1 + cos(p)]

cot

(15~

~

(2.32)

For /3c--+0 Eqs. (2.31) and (2.32) reduce to Eq. (2.30). Equations (2.31) and (2.32) are applicable with a good accuracy for 15~ (p < 180 ~ see Refs. [14,15] for details. In the case when 0~

q~ < 15 ~ instead of Eqs. (2.31) and (2.32) one can use the following asymptotic

expansions [14,15]:

74

Chapter 2

r = P-

_z

b

27p 7 + 108eflcP 5 + 144/~2P 3 -8e,~i~P 2 54(3p 4 + 2e[3 p2) + O(/3 c )

= 2 _ 1 ( p P + 52 e 13c l n ~P + O ( f 1 2 ) 2e

(2.33)

( 0 o < (p <

15 o)

8 l((p+ ~~02 ..l_~/~c ) ' see Ref.

where e is the Napier number, ln(e) = 1, and p --~

(2.34)

[15] for more

details. Other approximate solutions of Laplace equation can be found in Refs. [ 11,16]. For example, if the meniscus slope is small, Ur2 << 1, Eq. (2.22) reduces to a linear differential equation of Bessel type, whose solution reads z(r) = u(r) = 2[I0(qr) - 1]](bq2),

q =_(Ap g/(y)l/2,

Ur2

<<1

(2.35)

where I0(x) is the modified Bessel function of the first kind and zeroth order [5,17,18]. (iii) The gravity deformation is considerable and tic is not a small parameter. In this case one can integrate numerically the Laplace equation in its parametric presentation, Eq. (2.25). Despite the fact that in the presence of gravitational deformation the Laplace equation has no exact analytical solution, it is curious to note that there is an exact formula for the volume of a sessile or pendant drop (bubble), irrespective of the magnitude of the gravitational deformation. For example, the volume Vl of the lower part of the drop (bubble) in Fig. 2.3, that is comprised between the planes z = 0 and z = z~, is 27r

Zc

r(z)

zc

V1 - ~dooldz I r d r = ~ I d z r 2 ( z ) o o o o

Zc

- Jrzcr~-2~Idzzr(z) o

dr dz

(2.36)

Here co denotes the azimuthal angle in the plane xy, r= r(z) is the generatrix of the interfacial profile, r~ is the radius of the contact line (Fig. 2.3); at the last step in Eq. (2.36) we have integrated by parts. On the other hand, Eq. (2.28) can be expressed in the form d 2 - - ( r s i n qg) = - r dr b

2 +e q zr

(2.37)

Interfaces of Moderate Curvature: Theory of Capillarity

75

where q is defined by Eq. (2.35). Next, one multiplies Eq. (2.37) by dr/dz and integrates for 0 < z < Zc; in view of Eq. (2.29) the result reads: 2 Zc dr rc +eq 2~dzzr(z)~

rcsintp c -

b

o

(2.38)

&

The integral in Eqs. (2.36) and (2.38) cannot be solved analytically; however, this integral can be eliminated between these two equations to obtain

W 1 - I t , z c /.2 q.. ~

c

- q

eq2

(2.39)

sin (Pc

where qgc is the value of the slope angle q9 at z = Zc, see Fig. 2.3. Similar expression can be obtained also for the volume V2 of the upper part of the fluid particle in Fig. 2.3, that part situated above the plane z = Zc.

2.2.2.

MENISCUS DECAYING AT INFINITY

Examples are the outer meniscus, z = w(r) in Fig. 2.3 and the meniscus in the outer Region II in Fig. 2.4. In this case the action of gravity cannot be neglected insofar as the gravity keeps the interface flat far from the contact line. Let z = 0 be the equation of the aforementioned flat interface. Then P10

= P20

in Eq. (2.7) and the capillary pressure is Pc = Apgz; the Laplace

equation (2.22) for the function z(r) takes the form: Zrr

0 + z2)

•r 3/2 -I-

/2

rO + Z2r~

=q

2

d2z

& Z,

Z r ~-~

dr

,

Zrr

dr 2

(2.40)

Equation (2.40) has no closed analytical solution. The region far from the contact line has always small slope, Zr2 << 1. In this region Eq. (2.40) can be linearized and reduces to a modified Bessel equation; in analogy with Eq. (2.35) one derives

z(r) = A K0 (qr)

(Zr2 << 1)

(2.41 )

where A is a constant of integration and K0(x) is the modified Bessel function of the second kind and zeroth order [5,17,18]. The numerical integration of Eq. (2.40) can be carried out by

Chapter2

76

using Eq. (2.41) as a boundary condition, together with

Zr=-qAKl(qr)

for some appropriately

fixed r >> q-l [ 11 ]" the constant A is to be determined from the boundary condition at r = Re. Approximate analytical solutions of the problem are available [14,19-21]. For the case, when the radius of the contact line R2 (Fig. 2.4) is much smaller than the characteristic capillary length, q-l, Derjaguin [ 19] has derived an asymptotic formula for the elevation of the contact line at the outer surface of the cylinder r = R2:

(qR2)2 <<

hc = -R2 sin gtc ln[qR2 ye (1 + cosg4.)/4],

1

(2.42)

Here gtc is the meniscus slope angle at the contact line (Fig. 2.4), q is defined by Eq. (2.35) and Te = 1.781 072 418... is the Euler constant [17]. Note that Derjaguin's formula (2.42) is valid not only in the case 0 < ~c < 90 ~ but also in the case 90 ~ < gtc < 180 ~ in which the meniscus has

10

z

........................../ ....................

Fig. 2.5. Sketch of a circular hole in a liquid wetting film of thickness hc on a solid substrate; gtc is the contact angle and r0 is the radius of the "neck" of the meniscus. a neck, see e.g. Fig. 2.5. If the condition

(qR2)2 <<

1 is satisfied, the constant A in Eq. (12.41)

can be determined [ 19,14]"

z(r)- R2 sin l/tc K0(qr)

zr2 << 1,

(qR2)2 <<

1

(2.43)

Particles floating on a fluid interface of entrapped in a liquid film usually create small interfacial deformations, for which Eqs. (2.42) and (2.43) are applicable. These equations will be used below in this book to describe quantitatively the lateral capillary forces due to the overlap of the interfacial deformations (menisci) formed around such particles.

Interfaces of Moderate Curvature: Theory of Capillarity

If the condition

Zr 2 <<

77

1 is not fulfilled close to the cylinder r = R2, in this region one can use an

alternative "inner" asymptotic expression [ 14],

h incEarccoh/ r / accoh/ ' /1 R 2 sin I/tc

sin ~c

(qR2)2 << 1

(2.44)

In Eq. (2.44) r _> R2 and h,. is given by Eq. (2.42); arccoshx = ln[x+(x 2 - 1)1/2]; Eq. (2.44) has meaning for z > 0 . For large values of r Eq. (2.44) predicts negative values of z(r), which indicates that this asymptotic formula is out of the region of its validity. Higher order correction terms in Eqs. (2.42)-(2.44) are derived in Refs. [14,20]. If the thickness hc of the liquid film in Fig. 2.5 is small enough one can use (qhc) 2 as a small parameter and to obtain asymptotic expansions for the meniscus profile. In this way O'Brien [21] has derived an expression for the radius of the "neck", r0 (the meniscus has a vertical tangent at r = r0): r0 = - he{ In[ -~ q r0 7e cot(lffc/2)] }-l

(qhc )2 << 1,

90 ~ < ~c < 180 ~

(2.45)

For given hc and ~. the radius of the "neck" (or "pinhole") r0 can be calculated from Eq. (2.45) by iterations. For ~c = 90 ~ the radii of the neck and contact line coincide, r0 - R2, and in this special case Eqs. (2.42) and (2.45) coincide. For gtc--9 180 ~ the asymptotic formula (2.45) is not valid and another procedure of calculations is suggested in Ref. [21]; alternatively, Eq. (2.25) with Pc. = Ap g~ can be integrated numerically.

2.2.3. MENISCUS CONFINED BETWEEN TWO CYLINDERS, 0 < R1 < r < R2
This type of capillary menisci include the various capillary bridges (Fig. 2.6a), the channels (borders) in foams and some emulsions, as well as the borders around the model thin liquid films in the experimental cells of Scheludko [22-24] and Mysels [25]; such is the configuration of fluid particles or biological cells pressed between two surfaces, see Fig. 2.6b and Refs. [26,27]. The shape of these menisci have been first investigated by Plateau [28] and Gibbs [29] in connection to the borders in the foams

78

Chapter 2

If the gravitational deformation of the meniscus cannot be neglected, the interfacial shape can be determined by numerical integration of Eq. (2.25), or by iterative procedure [30]. If the meniscus deformation caused by gravity is negligible, analytical solution can be found as described below. Equation (2.24) can be presented in the form d

(rsin (p)- 2k, r,

k, = P~ - P2 2a

dr

TZ

(2.46)

. ,,i I

,,

0

o

Pt

(a)

(b)

Fig. 2.6. Capillary bridges between two parallel plates: (a) concave bridge, (b) convex bridge; rc is the radius of the three-phase contact line; ro is the radius of the section with the plane of symmetry; P1 and P2 are the pressures inside and outside the bridge, respectively. The pressures in phases 1 and 2, P1 and P2, and the coordinate of the point with vertical tangent, r0, are shown in Fig. 2.6. To determine the shape of the menisci depicted in Figs. 2.6a and 2.6b we integrate Eq. (2.46) from r0 to r to derive sin (p - k~ r + r~ r

- k I r0)

(2.47)

New,t, using Eq. (2.47)and the identity tan(p= • sin(p/(1- sin2(p) 1/2 one obtains dz kl @2 _ ~ )+ r0 d---~= +[(r 2 _ 4 ) ( r , 2_r~)l/21k, l,

1 - k I r0

r, -

(2.48)

k,

Equation (2.48) describes curves, which after Plateau [28,31,10] are called "nodoid" and "unduloid," see Fig. 2.7. One sees that the nodoid (unlike the unduloid) has points with horizontal tangent, where dz/dr = 0. With the help of Eq. (2.48) one can deduce that the

Interfaces of Moderate Curvature: Theory of Capillarity

79

meniscus generatrix is a part of nodoid for klro ~ ( - ~ , 0)u(1, +~), whereas the meniscus generatrix is a part of unduloid for kl r0 ~ (0, 1).

Z ~

Z ~

(nodoid)

(unduloid)

I I

I

R1

R2

1

_

0

(a)

R1

R2

(b)

Fig. 2.7. Typical shape of the curves of Plateau: (a) nodoid, (b) unduloid. The curves are confined between two cylinders of radii R1 and R2.

In the special case klro= 1 the meniscus is spherical. In the other special case, klr0=0, the meniscus has the shape of catenoid, i.e.

1, Qualitatively, the catenoid looks like the meniscus depicted in Fig. 2.6a; it corresponds to zero capillary pressure and zero mean curvature of the interface (Pc= 0 and H = 0). In this case H = 0 because the two principle curvatures have equal magnitude, but opposite sign, at each point of the catenoid. This is the reason why it is called also "pseudosphere". The meniscus has a "neck" (Fig. 2.6a) when k~ro ~ ( - ~ , 1/2); in particular, the generatrix is nodoid for klro~ (-~o, 0), catenoid for klr0=0, and unduloid for klro~ (0, 1/2). For the configuration depicted in Fig. 2.6a one has rl > r0 (in Fig. 2.7 Rl = r0, R2 = rl) and Eq. (2.48) can be integrated to yield

Chapter 2

80

z ( r ) : +_ r0F(q)l,q, )+ rlsgnk I E(q~,,ql ) - r r ,

where sngx denotes the sign of x, q l = ( 1 - r o Z / r l 2 ) 1/2 , sin(pl=ql-l(1-ro2/r2)l/2; F(q),q) and E(O, q) are the standard symbols for elliptic integrals, respectively, of the first and the second kind [5,17,18]:

F(q~, q) -

d_lff , 41_ q2 sin 2~

r

E(O,q)

f - 0

,/l - q2 sin 2 ~ d ~

(2.51)

A convenient numerical method for computation of F(O, q) and E(q~,q) is the method of the arithmetic-geometric mean, see Ref. [ 18], Chapter 17.6. The meniscus has a "haunch" (Fig. 2.6b) when klro 9 (1/2, +oo); in particular, the generatrix is unduloid for klro 9 (1/2, 1), circumference for klro= 1, and nodoid for klro 9 (1,+oo). For the configuration depicted in Fig. 2.6b one has r0 > r~ (in Fig. 2.7 R~ = r~, R2-" r0) and Eq. (2.48) can be integrated to yield z ( r ) - _+[(r0 - 1/k I )F(q~2, q2 ) - r0 E(q~2,q2 )],

(r1 < r < r0 )

(2.52)

where q2 = (l-r1 2 /ro 2 ) 1/2, sin~ = q2 -1(1 - r 2/ro 2 )1/2 More about capillary bridges can be found in Chapter 11. Equation (2.52) has been also used [26,27] to describe the shape of drops and blood cells entrapped in liquid films.

2.3.

F O R C E BALANCE AT A THREE-PHASE-CONTACT LINE

2.3.1.

EQUATION OF YOUNG

As discussed in Section 2.1.2, the Laplace equation is a differential equation, which needs boundary conditions to obtain an uniquely defined solution for the shape of the interface. In Section 2.1.1 we demonstrated that the Laplace equation can be derived as a necessary condition for minimum of the grand thermodynamic potential f2 by using variations in the meniscus shape u(x,y) at fixed boundaries. In addition, the boundary conditions for Laplace equation can be derived in a similar way by using variations in the meniscus shape at mobile boundaries, see e.g. [8,32-34].

Interfaces of Moderate Curvature: Theory of Capillarity

81

V2

(2)

A2

8u!.

8u, A1

(1) v

0

Xl

Y

Fig. 2.8. Sketch of a two-phase system composed of phases 1 and 2 occupying volumes Vi and V2 separated by an interface of equation z = u(x); A1 and A2 are the contact areas of the respective phases with the right-hand side wall; a is contact angle; 5u(x) and 8u~ represent variation in the shape of the interface and in the position of the contact line, respectively.

Let us consider again a two-phase system closed in a box of volume V = V~ + V2, see Fig. 2.8. For the sake of simplicity, let us assume that the meniscus has a translational symmetry along the y-axis; then the meniscus profile is z = u(x). Let the right-hand-side wall of the box (Fig. 2.8) be a vertical solid plate situated at x = Xl. W e consider variations, 8u(x), of the meniscus profile for movable contact line at the vertical wall at x = Xl. The grand potential f2 can be expressed in the form

- - _ [ P 1 d V - f P2dV+(ya + O',sal + (Y2sa2 vj

(2.53)

where A~ and A2 are the contact areas of the vertical wall with phase 1 and 2, respectively (Fig. 2.8); 0-,,3 and O2s can be interpreted as surface excess densities of f~ for the boundaries s o l i d / p h a s e 1 and s o l i d / p h a s e 2. Since the meniscus has translational symmetry, one can write

xl

u(x)

0

0

Xl

b

0

u(x)

(2.54) Vl

Vl

82

Chapter 2

where l is the length of the meniscus along the y-axis; z = 0 and z = b are the lower and the upper side of the box (Fig. 2.8). The area of the boundary between phases 1 and 2 is x,

~/ l+u 2

a-lldx 0

c)u u~--

(2.55) c?x

'

Let the variation 8u(x) of the meniscus profile is accompanied by a variation 8Ul of the position of the contact line at the vertical wall at x

= Xl.

Then 8A~ =

-

~n 2 =

1~b/1 and in view of Eqs.

(2.53)-(2.55) the first variation of the grand potential f~ can be expressed in the form: Xl

,Sf~ -lI(,SL)dx +(~,, - ~2, )hSu~

(2.56)

0

where

Pl(z)dz-

L(u, U x ) - -

0

(2.57)

P2(z)dz+o" l+u2x u

Differentiating Eq. (2.57) one obtains aL = c?L au + c?L - - d((~u) - -

Ou

(2.58)

c?ux dx

Next, we integrate by parts the last term in Eq. (2.58)"

oaUx

ax

x=xI

ZLi-y

au,-

audx

(2.59)

Combining Eqs. (2.56)-(2.59) one derives

~ ~'~ -- I

--~R -- -'~X "-~Rx

~ u dx

k l ~ u x x=xl -t--O'ls- O'2s l ~ U 1

(2.60)

We already know that the meniscus profile satisfies the Laplace equation, Eq. (2.21), which is equivalent to the equation o~__L_L_ d | @0=. (_,_9 1| _, ' )

9u

dx to.

)

Hence, the integral term in Eq. (2.60)is equal

to zero. Furthermore, the necessary condition for minimum is ~5U2 = 0, and 8uj is an independent and arbitrary variation. Then the term multiplying 5u~ in Eq. (2.60) must be equal to zero:

Interfaces of Moderate Curvature: Theory of Capillarity

83

[9~-~uL ) +c~,-c~,-0

(2.61)

x= x I

Using Eq. (2.57) and the identity ux = tan(p, where (p is the running slope angle, one can derive: c? L

0- u x

0- tan (p

c?ux

41+u 2

~/1 + tan2 (p

Finally, in Eq. (2.62) we set x

= Xl

= 0-sin (p

(2.62)

and take into account that (sin(p)x=x I - c o s ~ , where c~ is

the contact angle (Fig. 2.8)" then Eq. (2.61) transforms into the Young [35] equation: 0-2, = 0-1, + 0- cosa

(Young equation)

(2.63)

An equivalent equation has been obtained by Dupr6 [36] in his analysis of the work of adhesion. We derived Eq. (2.63) for the special case of meniscus of translational symmetry, however, this equation has the same form for arbitrarily shaped meniscus and solid surface, see e.g. Ref. [8]. The above derivation of Young equation, which is similar to the derivation of Laplace equation in Section 2.1.1, implies that the Young equation expresses the balance of forces per unit length of a three-phase-contact line, likewise the Laplace equation expresses the force balance

per unit area of a phase boundary. According to Gibbs [29] 05, and 0-2, can be interpreted as surface tensions (forces per unit area): O"1sand 0"2, oppose every increase of the "wet" area (without any deformation of the solid) in the same way as 0- opposes every dilatation of the interface between fluids 1 and 2; Gibbs termed 0-I, and 0"2, "superficial tensions". Equation (2.63), presented in the form coso~ = (0-2s- 0-~,)/0-, shows that the contact angle o~ can be expressed in terms of the three surface tensions, which in their own turn are determined by the excess molecular interactions in the zone of the phase boundaries (Section 1.1.1). Therefore, the magnitude of the equilibrium contact angle is determined by the intermolecular forces. It is worth noting that ~5~ = 0 is only a necessary, but not sufficient, condition for a minimum of the grand potential f~. This condition is fulfilled also in the case of maximum and inflection

84

Chapter 2

point of f2. Quite surprisingly, Eriksson & Ljunggren [37] established that for gas bubbles attached to a hydrophobic wall the Young equation can correspond to a maximum of f2, i.e. to a state of unstable equilibrium. Further, in Ref. [38] the conditions for stable or unstable attachment of a fluid particle (drop, bubble) to a solid wall have been examined. The force interpretation of the Young equation is illustrated in Fig. 2.9a. One sees that Eq. (2.63) represents the horizontal projection of the vectorial force balance 13' + O'ls + O'2s + fR ----0

(2.64)

where the underlined sigma's denote vectors, each of them being tangential to the respective interface and normal to the contact line; fR is the vector of the bearing reaction, which is normal to the surface of the solid substrate and counterbalances the vertical projection of the meniscus surface tension" fR = cr sino~

(2.65)

Equation (2.65) can be useful in the analysis of the forces applied to a particle, which is attached to an interface (see below). Note that the thermodynamic interpretations of or, o'~s and crzs as surface excess densities of the grand thermodynamic potential f2 are completely compatible with their mechanical interpretation as forces per unit length (Fig. 2.9a) insofar as boundaries of a liquid phase are considered and deformations of the solid walls are neglected. These two interpretations (as energy per unit area and force per unit length) are mutually complementary, in spite of the fact that some authors favor the thermodynamic and other ones - the mechanical interpretation. As already mentioned, Gibbs [29] interpreted O~s and crzs as tensions opposing the increase of the wet area on the solid surface without any deformation of the solid. On the other hand, differences between energy per unit area and force per unit length appear if deformations of the solid surface take place, see e.g. Refs. [39-43]. Then the force per unit length becomes a two-dimensional tensor, whereas the energy per unit area remains a

scalar, and consequently, these two quantities cannot be equal" the trace of the tensor (the surface tension of the solid) is related to the excess surface energy per unit area by means of the Shuttleworth equation [39-43]. Deformations of the solid phase are out of the conventional capillary hydrostatics and thermodynamics, and will not be considered here.

hzterfaces of Moderate Curvature: Theory of Capillarity

85

V

fluid 2

N eum ann triangle

l

/

sinc

(3)

w(r)

(Yv

!A (2)

~

1

u(r)

(a)

], ~J~,

(b)

Fig. 2.9. Force interpretation of (a) Young equation (2.64) and (b) the Neumann triangle (2.66); fR is the bearing reaction of the solid substrate, which counterbalances the normal projection of the surface tension, o'sina; a and t3 are contact angles.

2.3.2.

TRIANGLEOF NEUMANN

Three-phase-contact line can be formed also when all the three phases are line of intersection of the menisci

u(r), v(r)

and

w(r)

fluid:

such is the

in Fig. 2.3 (drop or bubble attached to an

interface) and Fig. 2.9b (liquid lens at a fluid phase boundary). In this case one can prove (again by a variational method, see e.g. Refs. [32,33]) that the force balance per unit length of the contact line is given by the Neumann [44] vectorial triangle: ff~, + ~v + _~w = 0,

(Neumann triangle)

(2.66)

see Fig. 2.9b and Refs. [3,31,45,46]. In this case there are two independent contact angles: a and 13. Applying the known cosine theorem to the triangle in Fig. 2.9b one obtains: 0.w 2 ---- 0.u 2 + 0.v 2 +

2o;,0-v c o s o ~ ,

0-v2 "- 0.u 2 q- 0.w 2 -F

20.,,0.w cosfl,

(2.67)

From Eq. (2.67) one determines [3,46-48] 0.2

coso~ -

2

2

w-0., -o'v, 20.,,0.v

2

2

2

cosfl - 0-v - 0 - , -0-w 20.,0. w

(2.68)

86

Chapter2

Equations (2.68) relate the contact angles o~ and/3 to the three interfacial tensions. Hence, we again arrive to the conclusion that the magnitude of the contact angles is determined by the intermolecular forces (which determine also the values of the sigma's) and, moreover, the contact angles do not depend on applied external fields such as gravitational or centrifugal. Note however, that if a line tension is present at the contact line (Section 2.3.3), the contact angle becomes (in principle) dependent on applied external fields as proven by Ivanov et al. [32]. It must be emphasized that the above conclusions related to the physical nature of the contact angles are valid only for the true, i.e. correctly defined, contact angles. Experimentally, by means of some optical method the shapes of the interfaces at some distance from the contact line are usually determined, and then by extrapolating them until they intersect the contact angle is found, for review see e.g. Refs. [30,49,50]. The first problem encountered when this procedure is carried out in practice is that the minimum distance from the contact line, at which one can still obtain experimental information about the interfacial shape, is limited by the magnification of the microscope [51]. The second problem is related to the transition region in a narrow vicinity of the contact line in which the interactions between the three neighboring phases affect the interfacial tensions and the shape of the interfaces. Although the width of the transition region (it was estimated to 1 lain in Ref. [52]) is smaller than the resolution of the optical methods, if some of the experimental points happen to lie there, this will again affect the extrapolation procedure, and hence - the value of the macroscopic contact angle. The only way to avoid these two errors is [32]: (i) to make sure that all experimental points used lie outside of the transition region and (ii) to carry out the extrapolation of the surface in such a way that its shape at all points satisfies the Laplace equation with the macroscopic value of the surface tension.

Interfaces of Moderate Curvature: Theor)' of Capillarity

87

2.3.3. THE EFFECT OF LINE TENSION

As discussed above, the true thermodynamic contact angle is subtended between the

extrapolated surfaces, which obey the Laplace equation. On the other hand, the interactions between the phases (the disjoining pressure effects) in a vicinity of the three-phase-contact line may lead to a deviation of the real interfacial shape from the extrapolated surface and to a variation of the interfacial tensions in this narrow region. In such a case, to make the idealized system (composed of uniform bulk phases and interfaces obeying the Laplace equation at constant surface tension) equivalent to the real system, some excesses of the thermodynamic parameters can be ascribed to the three-phase contact line. The excess linear density of the grand thermodynamic potential f2 has the meaning of a line tension, likewise the excess surface density of f2 is identified as the surface tension, see e.g. Ref. [53]. Furthermore, the line tension can be interpreted as a force, directed tangentially to the contact line, just like the tension of a stretched fiber or string [54]. For the first time the concept of line tension was formulated by Gibbs [29] in his theory of capillarity. The thermodynamic theory of systems with line tension was developed by Buff and Saltsburg [45], and Boruvka and Neumann [53]. The theory of line tension for systems containing

thin liquid film was worked out in Refs. [52,55,56]. Following Veselovsky and

Pertsov [54], let us consider a small circular portion (arc) of a curved contact line with length s, curvature radius rc and line tension K', which is depicted in Fig. 2.10. The considered portion of the contact line is situated symmetrically with respect to the x-axis, which is chosen to pass through the center of curvature O. Due to the action of line tension, a force of magnitude • and direction tangential to the contact line is exerted at both ends of the considered arc (Fig. 2.10). Then one obtains s = 20 rc,

IfxI = 2resin0

(2.69)

where fx is the sum of the x-projections of the two forces due to the line tension. Then it turns out that the existence of line tension leads to the appearance of a force

o'~ - l i m ~ o--,0 s

=-

rc

(2.70)

Chapter 2

88

X

t

x s

t

i

i I x

C

t

~\

/

x

i \ x

I

C

1//

r

O Fig. 2.10. The line tension ~, acting tangentially to a small portion (arc) of a curved contact line, leads to the appearance of a force per unit length o-~ = tc/rc, which is directed toward the center of curvature of the arc, point O; rc is the curvature radius of the considered elementary arc of the contact line.

acting per unit length of the contact line. In each point of the contact line this force, ~ ,

is

directed toward the center of curvature (toward the center of the circumference, if the contact line is circular). The above derivation is valid for any smooth curved line, insofar as every small portion of such a line can be approximated with an arc of circumference [7]. If the effect of line tension is included into the Young equation, Eqs. (2.63) and (2.64) acquire the more general form [57]:

0"2,` - 0"1., + 0" c o s a + - r

(2.71 )

(3"+O'ls+(Y2s+O'tr +[R =0

(2.72)

In this case the contact line belongs to a solid surface and the force interpretation of ~; and ~

is

similar to the interpretation of _~s and O'2s given by Gibbs (see above); in particular, tr can be interpreted as a force, which opposes the extension of the perimeter of the wet zone in every process of variation of the wet area without any deformation in the solid substrate.

Interfaces of Moderate Curvature: Theory of Capillarity

N eum ann quadrangle

(~./4-~(~,; ~

89

i t

hl

,,2

I

!

,

P i m

' !~ 2re ,I

(3)

,

(Jv

i

I

(2) ~ / ( y ~ ,

(a)

(b)

Fig. 2.11. Balance of forces per unit length of the contact line (a) around a floating lens (b) around a spot of tension ?'1 and thickness hI in a liquid film of tension T2 and thickness h2; cry:is the contribution of line tension. Likewise, the Neumann triangle, Eq. (2.66) transforms into a "Neumann quadrangle" defined by the following vectorial equation [32,48,58-60] ~, + ~v + ~w + ~,,-= O,

(2.73)

see Fig. 2.11a. Note that Eq. (2.73) can be derived from the thermodynamic condition for minimum of the thermodynamic potential [32,33]. To do that the capillary system is considered as being built up from bulk (three-dimensional) phases, surface (two-dimensional) phases and line (one-dimensional) phases [53,60]. Then the grand potential is expressed in the form: f2 - - Z P Vi + Z CrmA,, + Z tcnL,, i

m

(2.74)

n

where the indices "i", "m" and "n" numerate the bulk, surface and line phases" as usual, P denotes pressure, V - volume, 0 - surface tension, A - area, to- line tension and L - length. Next, let us consider a circular spot of radius rc, thickness h~ and tension y~ formed in a foam (or emulsion) film of thickness h2 and tension ?~, see Fig. 2.1 lb. Such spots are typical for the transition from primary to secondary film (from common black to Newton black film) in foams

90

Chapter 2

[25,61,62], as well as for the multiple step-wise transitions in the thickness of liquid films containing surfactant micelles or other spherical colloid particles [63,64]. The force balance per unit length of the line encircling the spot is (Fig. 2.1 l b): K"

- - =)'2 -71 rc

(2.75)

Equation (2.75) is obviously a linear counterpart of the Laplace equation (2.4). One could estimate the magnitude of tr for the spot in Fig. 2.11 b in the following way [64]. The periphery of the spot can be approximated with a step of total height Ah

= h2 - hi.

The excess energy due

to the formation of the spot is tr = o-Ah, where o" is the surface tension of the film. Thus with typical values o" = 36 mN/m and Ah = 5 nm one calculates tr = 1.8 x 10-l~ N. It should be noted that this estimate is pertinent to the system depicted in Fig. 2.11 b, and it is certainly irrelevant for other systems and/or configurations. What concerns the sign of tc, it must be positive if the shape of the spot is circular. Indeed, at tr > 0 each deviation from circular shape (at constant spot area) would lead to an increase of the length, L, of the contact line encircling the spot and of the line energy, teL; therefore, the spot will spontaneously acquire a circular shape, which minimizes the line energy. Such circular spots are observed also in adsorption monolayers of insoluble molecules, like phospholipids, which exhibit coexisting domains of different twodimensional phases [65-67]. In some cases, however, the boundaries between such coexisting domains are highly irregular and unstable, which can be attributed (at least in part) to the action of negative line tension. Note, that in (physically) two-dimensional systems, like the film in Fig, 2.1 l b, there is no other force, but a positive line tension, which tends to keep the shape of the contact line circular. That is the reason why the line tension is an effect of primary importance for such "two-dimensional" systems, in which the contact line separates two surface phases (Fig. 2.1 l b). Quite different is the case of a

three-phase-contact line, see Fig. 2.1 l a and Eq. (2.73). In order

to minimize the surface area (and energy) the fluid interfaces acquire axisymmetric shape and their lines of intersection (the three-phase-contact lines) are usually circumferences. Then even a negative line tension cannot disturb the regular shape of the contact line, which is preserved by the surface tensions in the Young equation (2.71) and Neumann quadrangle (2.73). The

htterfaces of Moderate Curvature: Theor3' of Capillarity

91

(J2sl 7 . ( y s i n o~

(2) 61

Fig. 2.12. Balance of forces per unit length of the contact line of a small solid sphere attached to the planar interface between the fluid phases 1 and 2; o" is the interfacial tension, o'l.,,and o'2.,,are the two solid-fluid tensions, o'~ is the line tension effect, fR is the bearing reaction of the solid particle: fR + o'~sina = o'sino~. accumulated results for three-phase systems show that the line-tension term turns out to be only a small correction, which can be (and is usually) neglected in Eqs. (2.71)-(2.73), see Section 2.3.4. As an example for application of the Young equation (2.72) let us consider a small spherical particle attached to the interface between two fluid phases, Fig. 2.12. We presume that the weight of the particle is small and the particle does not create any deformation of the fluid interface [68,69]. All forces taking part in Eq. (2.72) are depicted in Fig. 2.12, including the normal projection, osino:, of the interfacial tension.

Its tangential projection, ocoso~, is

counterbalanced by O2s + O"tcCOS~- t~ls in accordance with the Young equation. Then one could conclude (erroneously!) that o s i n a and/or oscsino~, which have non-zero projections along the z-axis (Fig. 2.12), give rise to a force acting on the particle along the normal to the fluid interface. If such a force were really operative, it would create a deformation of the fluid interface around the particle, which would be in contradiction with the experimental observations. Then a question arises: how to calculate correctly the net force exerted on a fluid particle attached to an interface at equilibrium?

92

Chapter 2

The rule (called sometimes the principle of Stevin), stems from the classical mechanics and it is the following: The net force exerted on a particle originates only from phases, which are o u t e r with respect to the given particle: the pressure of an outer bulk phase, the surface tension

of an outer surface phase and the line tension of an outer line phase. (For example, if a particle is hanging on a fiber, then the tension of the fiber has to be considered as the line tension of an "outer line phase".) In our case (the particle in Fig. 2.12) the o u t e r forces are the pressures in the two neighboring fluid phases 1 and 2, and the surface tension, o, of the boundary between them. The integral effect of their action gives a zero net force for the configuration depicted in Fig. 2.12 due to its symmetry. Then there is no force acting along the normal to the interface and the latter will not undergo a deformation in a vicinity of the particle. (Such a deformation would appear if the particle weight and the buoyancy force were not negligible.) On the other hand, the solid-fluid tensions, o~, and O'2s, the tension o'~-due to line tension, and the bearing reaction of the solid, fR, cannot be considered as outer forces. However, O'~s, Cr2s and o'~ also affect the equilibrium position of the particle at the interface insofar as they (together with o) determine the value of the contact angle o~, see Eq. (2.71).Additional information can be found in Chapter 5 below, where balances of forces experienced by particles attached to the boundary between two fluids are considered.

2.3.4. HYSTERESIS OF CONTACT ANGLE AND LINE TENSION

The experimental determination of line tension is often based on the fact, that the presence of a ~rc term in Eqs. (2.71) and (2.73) leads (in principle) to a dependence of the contact angle o~ on the radius of the contact line rc (o', o'ls and O2s are presumably constants), see Refs. [70-79]. However, there is another phenomenon, the hysteresis of contact angle, which also leads to variation of the contact angle, see e.g. Ref. [80]. Both phenomena may have a similar physical origin [75]. The fact that a hysteresis of contact angle takes place with liquid menisci on a s o l i d substrate has been known for a long time [81,82]. It is an experimental fact that a range of stable contact angles can be measured on a real solid surface. The highest of them is termed "advancing", and

Interfaces of Moderate Curvature: Theory of CapillariO,

93

the lowest one -"receding" contact angle. The difference between the advancing and receding angles is called "the range of hysteresis", or shortly, "hysteresis" [83,84]. The widely accepted qualitative explanation of this phenomenon is that the hysteresis is caused by the presence of surface roughness and chemical heterogeneity of the real solid surfaces [75, 85-96]. From this viewpoint, the Young equation is believed to be valid only for an ideal solid surface, which is molecularly smooth, chemically homogeneous, rigid and insoluble [84]. However, hysteresis of contact angle can be observed even on an ideal solid surface if a thin liquid film is formed in front of an advancing meniscus, or left behind a receding meniscus; this was proven theoretically by Martynov et al. [97], see also Refs. [98,99]. In this case the hysteresis is due to the action of an adhesive surface force within the thin film, which opposes the detachment of the film surfaces and facilitates their attachment. Such forces are present (and hysteresis is observed) not only in wetting films on a solid substrate, but also in free foam and emulsion films stabilized by usual surfactants [100-102] or by proteins [99]. It turns out that, as a rule, one observes hysteresis of contact angle and only with some special systems hysteresis is completely missing. Such special systems can be liquid lenses on a fluid interface [30, 103-107] or thin films without strong adhesive forces [108]. The occurrence of hysteresis is different for a completely fluid three-phase-contact line and for a three-phase contact involving one solid phase: In the former case at complete equilibrium (immobile contact line) an equilibrium contact angle is established [99-102]; in contrast, in the latter case (in the presence of solid phase) it is practically impossible to figure out which angle, could be identified as the equilibrium one within the range between the receding and advancing angles. Coming back to the line tension issue, in Fig. 2.13 we demonstrate, that in some cases the line tension could be a manifestation of the hysteresis of contact angle. Let us assume that for some value of the contact angle, c~ = o~1, the Young equation (2.63) is satisfied (Fig. 2.13a). Due to the hysteresis another metastable contact angle, a2, exists (~2 > ~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

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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.

1 1 I . T.D. Gurkov, P.A. Kralchevsky, J. Disp. Sci. Technol. 18 (1997) 609. 1 12. P. Tarazona, G. Navascues, Physica A 1 1.5 (1982) 490. 1 13. G. Navascues, L. Mederos, Surf. Technol. 17 (1 982) 79.

I 14. T. Kolarov, Z.M. Zorin, Colloid J. USSR 42 (1980) 899. I 1.5. N.D. Denkov, D.N. Petsev, K.D. Danov, J. Colloid Interface Sci., 176 (1995) 189. 116. J.A. Wallace, S. Schurch, J. Colloid Interface Sci. 124 (1988) 4.52. 1 17. J.A. Wallace, S. Schurch, Colloids Surf. 43 (1990) 207. 118. J. Gaydos, A.W. Neumann, J. Colloid Interface Sci. 120 (1987) 76

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'

_(<) ~ c3D

H - - 2kcD

(3.9)

where Bo = -4kc Ho

(3.1 O)

is an expression for the bending moment of a planar interface in the framework of the Helfrich model. In general, Eqs. (3.9) can be interpreted as truncated power expansions of B and O for small deviations from planar interface (small H and D; quadratic and higher-order terms neglected). On the other hand, the original Helfrich formula, Eq. (3.7), has been postulated to give the leading term in the power expansion of wf in the case when H is close to H0. If the values of B0 and k c are known for a planar interface (H = 0) and then H0 is formally calculated from Eq. (3.10), there is no guarantee that the obtained result for H0 could be physically interpreted as a spontaneous curvature, i.e. that for H = H0 the free energy of flexural deformation wf has a minimum. In other words, if Eq. (3.8) is integrated, along with Eq. (3.9), in general there is no guarantee that the integration constant will be equal to 2 k c H g , cf. Eq. (3.7).

3.1.3.

FUNDAMENTAL THERMODYNAMICEQUATION OF A CURVED INTERFACE

Let us consider an elementary interfacial parcel of area dA. The Gibbs excesses of internal energy, entropy and number of molecules from the/-the species, corresponding to this parcel, are dU,-

uL,.dA,

dS,-

s, dA,

d~,-

co, dA,

d N 7 - FidA,

(3.11)

Chapter 3

110

where u s, s s, cos and

F i are

surface densities of the respective quantities. Further, let the area dA

of the elementary parcel increases with (3.12)

5 ( d A ) - 5 ( d A ) dA - 5o~ dA dA

due to the occurrence of some thermodynamic process. The Gibbs approach to nonuniform interfaces consists in application of the fundamental equation of an uniform interface locally, i.e. to every element dA of the nonuniform interface [1 ]. (Of course, this approach is more general and applicable to any nonuniform phases, not necessarily curved interfaces.) Thus, in the course of a thermodynamic process the variations of the parameters in Eq. (3.11) obey the fundamental equation k

8 ( d U s) : T(~(dS s ) + E l t i d ( d N

s) + 8W s

(3.13)

i=1

where T and ]-/i a r e the temperature and chemical potential, and 6Ws is the mechanical work of deformation of the surface element dA. In keeping with Eq. (3.1) one can write [30, 31 ]: (SW , = (Sws dA = (y (Sa + ~ (S fl + B (SH + | 6 D ) dA

(3.14)

With the help of Eqs. (3.11) and (3.12) one obtains =

+ ,. 6

)da

(3.15)

Similar expressions can be deduced also for dS, and dNi ~ . In this way the fundamental equation (3.13) can be transformed to read [29-31 ] k

aus - T(Sss + ~__,,uiaFi + ( 7 - c o , ~ o ~ + ~ [ 3 + B a H + |

(3.16)

i=1

where k

co, - u , - Ts, -~_~ It i F i

(13.17)

i=1

is the surface excess of the grand thermodynamic potential. Differentiation of Eq. (3.17) and substitution of the result for ~u, in Eq. (3.16) yields

Surface Bending Moment and Curvature Elastic Moduli

111

k

~ co - - s , ~ T - ~__ F ~~ t.t i + O, - co , )~ a + ~ (~ ,13 + B (~H + | ~ D

(3.18)

i=1

If the interface can be treated as a t w o - d i m e n s i o n a l f l u i d , then the interfacial tension equals the density of the surface excess grand thermodynamic potential [1] ? ' - co,

(3.19)

Equation (3.19) can be obtained in the following way. Imagine an elementary interfacial parcel of area dA, whose boundary is a contour permeable for the transport of all components. An imaginary process of expansion of this contour at constant intensive parameters (#T = 0, ~#i = 0, ~13 = 0, ~H = 0, ~D = 0) leads to enlargement of the parcel owing to the transfer of molecules across the permeable boundary line. In the course of this process matter with the same intensive properties, and especially with the same co,, is added to the considered parcel. Then, setting &o, = 0 in Eq. (3.18) we arrive at Eq. (3.19). If the interface represents such a two-dimensional fluid, then Eq. (3.19) is valid irrespective of whether components, which are insoluble in the bulk phases, are present at the interface. In this case, from Eqs. (3.18) and (3.19) we obtain a generalizes version of the Gibbs adsorption equation: k

~y--s,.~T-~_Fi~l.t

, +~fl

+ B~H

+|

(3.20)

i=1

cf. Eq. (1.35). It is worthwhile noting that the sign of B and | is a matter of convention. It is determined by the choice of the direction of the running unit normal to the interface, n. From the differential geometry it is known that the surface tensor of curvature, b, can be defined as the surface gradient of n [36]: b = -V~n

(3.21)

The latter equation is known also as the Weingarten's formula [37]. Since the principal curvatures, cl and cx, are the eigenvalues of b, then in view of Eqs. (3.3) and (3.21) an inversion of the direction of n will lead to a change of the sign of H and D. Moreover, ~wf in

112

Chapter 3

Eq. (3.8) must not depend on the choice of direction of n; consequently, the inversion of the direction of n leads also to an inversion of the sign of B and |

From Eq. (3.8) it could be

realized that a positive B tends to bend the interface around the inner phase, that is the phase for which n is an outer normal. The latter rule is general, i.e. it is independent of the choice of the direction of n.

3.2.

THERMODYNAMICS OF SPHERICAL INTERFACES

3.2.1.

DEPENDENCE OF THE BENDING MOMENT ON THE CHOICE OF DIVIDING SURFACE

Spherical fluid interfaces are often observed due to the fact that the spherical shape corresponds to minimal surface energy if the gravitational deformation is negligible. After Gibbs [1] an interface can be modeled as a mathematical dividing surface separating two bulk phases. As noticed in Chapter 1, in reality there is a narrow transitional zone between the two phases, whose thickness could be from few angstroms to dozens of angstroms. For that reason, a problem arises about the exact definition of the position of the dividing surface. From symmetry considerations it follows that in the case of spherical interface the dividing surface must be a sphere. However, an additional condition must be imposed to uniquely define the radius of this sphere. It can be proven [7,8] that for an arbitrary definition of the spherical dividing surface the following two equations hold:

d w , = 7dc~ +

a

+

~a

-~a d a

- P~ -PI~

(3.22)

(3.23~

Here a is the radius of the dividing surface, PI and PII are the pressures, respectively, inside and outside the spherical drop (bubble, vesicle); [O~/Oa] is a formal derivative of )'with respect to the radius a; here and hereafter the brackets symbolize formal derivatives, which correspond to an imaginary variation of the choice of a at fixed physical state of the system. The comparison of Eqs. (3.22) and (3.1) (the latter for ~fl= 0, ~D = 0 and a = - l / H ) yields [29-31]:

Surface Bending Moment and Curvature Elastic Moduli

B - a 2 [ ~ l-~-a '

113

(3.24)

that is the formal derivative turns out to be proportional to the bending moment B. One way to uniquely define the dividing surface is to impose the additional condition the formal derivative of y to be always equal to zero:

- 0

(3.25)

a -a x

This special dividing surface, introduced by Gibbs [1], is called the surface of tension, cf. Eq. (1.13); here its radius is denoted by a,. For this dividing surface Eq. (3.23) reduces to the common capillary equation of Laplace: = PI - Pn

(3.26)

a s

Eliminating P I - Pn between Eqs. (3.23) and (3.26) we obtain

dy

2 +-ydx x

2-

0

(3.27)

where we have introduced the notation

x-

a/a,,

y-y/y,

(3.28)

The solution of Eq. (3.27), satisfying the boundary condition y(1) = 1, reads 1

y ( x ) - 3x 2

2

+ -x 3

(3.29)

The latter equation, first derived by Kondo [7], describes the dependence of the interfacial tension yon the choice of the dividing surface at fixed physical state of the system. From Eqs. (3.24), (3.28) and (3.29) one can deduce a similar dependence for the bending moment B [29]: B -

B

2ya

=

x 3 -1 2x 3 + 1

(3.30)

Chapter 3

114

y,B

4

3 . ~ .... ".......... i .... i ..... i .... i ..... i ..... i..... i ..... ..... i ..... i .... i ..... i ..... '..... i..... '..... i ..... i .... i ..... i .... i ..... i ..... i..... i..... ..... i..... i .... 3 2.5 2 1.5 1

i

,.

0.5

:

:

" .....................

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

0

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

.--~---~------~-

......

- 1

-1.5

......

~

.

.

O.

~

i

" .....................

" ......................

" ....

-

-o.5 ...................-~--~..............i...... B = i

B/(2ya)

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

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

.

.

.

.

.

0.5

.

0.9

'

1.3

.

.

.

. . 1.7

.

.

. . 2.1

.

.

.

2.5

2.9

x = a/as

Fig. 3.2. Dependence of the thermodynamic interfacial tension, 7, and the dimensionless bending moment, B , on the choice of the dividing surface (of radius a) for a fixed physical state of the system. Equations (3.29) and (3.30) are illustrated graphically in Fig. 3.2. In accordance with Eq. (3.25), the surface of tension corresponds to the minimum of the curve 7 vs. a. Moreover, the dimensionless bending moment, B , which takes part in the Laplace equation, 2y (1 + B ) = PI - e l i ,

( 3 . 3 1 )

a

cf. Eq. (3.23), (3.24) and (3.30), turns out to be a function of a with bounded variation (Fig. 3.2). In addition, one could verify that the dependence B vs a,

B

-

2(a2asl a

--#Ysas

,

(3.32)

as

which stems from Eqs. (3.28)-(3.30), satisfies the differential equation [31]

E

-~a

- 27,

(3.33)

which is analogous to Eq. (3.24). The surface of tension turns out to be convenient in many cases, because it simplifies the shape of the Laplace equation, which determines the shape of interfaces in the capillary hydrostatics,

Surface Bending Moment and Curvature Elastic Moduli

115

see Chapter 2. On the other hand, as noticed by C. Miller [17], in the case of low interfacial tension (critical emulsions, microemulsions, lipid vesicles, biomembranes) the mathematical surface of tension is situated away from the physical transition zone between the two phases; see also eq. (3.62) below. In such a case, which is equivalent to the presence of essential contribution from the interfacial bending moment, it is appropriate to use the equimolecular dividing surface.

3.2.2.

EQUIMOLECULAR DIVIDING SURFACE AND TOLMAN LENGTH

As mentioned above, an additional condition must be imposed to define uniquely the radius of the dividing surface. Instead of setting the formal derivative [O~'/Oa] equal to zero, as it is for the surface of tension, one could require the formal derivative to be equal to the physical derivative, viz.

(3.34) 7aa ~=av -- ~av

where a v is the radius of the dividing surface defined by Eq. (3.34), termed the e q u i m o l e c u l a r dividing surface [2,3,6,8]; see also Section 1.2.2 above. The partial derivative 0~4/Oav is related to the physical dependence of ~'v on av for fixed values of the other thermodynamic parameters of state. Different possible choices of the latter parameters correspond to different definitions of av. The distance (~ = a v -

(3.35)

as

between the surface of tension and the equimolecular dividing surface is called the Tolman length. Tolman [2] has derived the equation

,0/120 )

(3.36)

a v

Y0 -

lim ?'v, a v ---~oo

~0 = lim a v ----)oo

(3.37)

116

Chapter 3

which expresses in first approximation the physical dependence of Yv on the interfacial curvature. Setting a = av in Eq. (3.32) and expanding in series for &lay << 1 we obtain [38]:

2ro, o +... The next terms in the expansion (3.38) can be found only if the dependence ~ = g)(av) is known. From Eqs. (3.36) and (3.38) we deduce B0

7'v - 7o

- + ....

B 0 - B,,la~ - 2)'o~ 0

(3.39)

a v

The last equation allows estimates for the magnitude of Bo to be made. For example, for the surface of liquid argon at temperature -188.85~

we have ~'0 = 13.45 raN/m, & = 3.6 A, see

Ref. [39], and then from Eq. (3.39) we compute B0 = 9.7 x 10 -12 N. Equation (3.39) demonstrates the connection between the Tolman length, &, and the bending moment B0, which in its own turn is proportional to the Helfrich's spontaneous curvature H0, see Eq. (3.10); all these quantities are related to the curvature dependence of the surface tension. Another relationship between the Tolman length and the parameters of the equimolecular dividing surface can be obtained in the following way. From Eqs. (3.28) and (3.29) one can deduce

Vav - s/2 -

+

- 3

3a v

/

(3.40)

Likewise, from Eqs. (3.26), (3.30) and (3.31) we obtain 27' v av

+

Bv 2

av

=

27' s

(3.41)

as

The elimination of ~/as between Eqs. (3.40) and (3.41), along with Eq. (3.35), yields the sought for expression for ~ [31 ]"

- a~. 1 -

)'v-By/av ?'v +5i Bv / av

(3.42)

Surface Bending Moment and Curvature Elastic Moduli

117

Equation (3.42) shows that the sign of ~ is connected with the sign of Bv: indeed, one could verify that Eq. (3.42) yields g)v > 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

r
(3.46) r>a

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 << 1, from Eqs. (3.78) - (3.80) one calculates B~/ =

4.2 x 10 -11 N, k el = 1.4 x 10-20 J and k~Yt = - 0 . 7

x

10-20 J.

Since, B~t , kf / and k~Y~ are proportional to (AV) 2 their sign does not depend on the sign of the surface potential AV. For oil-water interface Bo t and Bo w have the same sign: both of them tend to bend the interface around the oil phase. In contrast, for air-water interface B~l and Bo w have the opposite signs: B~l bends around the gas phase, while B~w bends around the water phase (the phase of higher Hamaker constant). Equations ( 3 . 7 8 ) - (3.80) show also that B~l, k~land k~Y/ depend on the choice of the Gibbs dividing surface through the distances Ii and 12. Moreover, if d

-

12-11

> 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 << 1.

(3.95)

cf. Eq. (3.10); here r is the radius of the flat film; the multiplier 2 accounts for the fact that the film has two surfaces. The increment of W T , which is due to the formation of a flat film, is A W f = WT (O) - Wu ( H ) - - 21r r2 Bo H + O ( H 2 ) - 21r r 2 Bo / a + . . .

(3.96)

The bending moment can be estimated as follows: I B0l = I Bo w + B~tl -- 5 • 10-1IN, see the numerical values given above in this chapter. Using the latter value together with k T = 4 . 1 x l 0 -21 J (room temperature) and a tentative value r - - a ~ 2 0 ,

from Eq. (3.96) we

estimate

Consequently, for emulsion drops of radii 10 --6cm < a < 10 -4cm the contribution of the interfacial bending moment to the energy of interaction between two drops will be 2 k T < I A W f f < 200 k T . In other words, we arrive at the conclusion that the interfacial bending

moment B0 could be important for the interactions between sub-gin, and even lure-sized, drops [63]. This conclusion is intriguing insofar as it is usually believed that the bending energy is essential only for objects of very high interfacial curvature, like the nuclei of a new fluid phase and the droplets in microemulsions. It should be also noted that B0 > 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" for biomembranes at body temperature [9, 51 ]. Thus one could seek the membrane profile as a solution of the equation [50]

0" ~ + dr

4.4.2.

r

- AP+--cos r

dr

rcos

dr

-~r

(rsin0)

]t

(4 110)

BOUNDARY CONDITIONS AND SHAPE COMPUTATION

To find the solution of Eq. (4.110), along with Eq. (4.105), one needs 4 boundary conditions. The following boundary conditions have been used in Ref. [50] to find the shape of erythrocytes attached to a glass substrate (Fig. 4.4): (i-ii) z = 0 and 0 = 0 at r = 0, i.e. at the apex of the membrane (the point where the membrane intersects the z-axis); (iii) the membrane curvature varies smoothly in a vicinity of the membrane apex; (iv) 0 = 0t, for r = rf (Oh - contact angle; r f - radius of the adhesive film); (v) the total area of the membrane, At, is known; this condition was used in Ref. [50] to determine the unknown material parameter (4.111)

~ = 0"1kc

To solve Eq. (4.110) it is convenient to introduce the auxiliary function

F =

dsin0 -

dr

t-

sin0

2

r

b

,

b -

20"

,

(4.112)

AP

where b is constant if the effect of gravity on the membrane shape is negligible. Then Eq. (4.110) acquires the form

General Curved Interfaces and Biomembranes

165

c

6 E

,-

4

o

3

.-"O L O

/

5

\

O

.~

'...v, ,..~

2

>

'// 7:;

..,~

tD

1 .

-~.

.

.

-4

.

.

-3

//..... .

.

-2

.

.

-1

.

.

0

.

.

1

.

.

2

.

.

3

.

4

5

Horizontal coordinate (gm)

Fig. 4.4. Shape of an erythrocyte adherent to a glass substrate, determined in Ref. [50]; the zone of the flat adhesion film is in the lower part of the graph. (a) For osmolarity 143 mOsm of the hypotonic solution the non-adherent part of the membrane is spherical; the shape for (b) osmolarity 153 mOsm and (c) 156 mOsm is reconstructed from experimental data by solving Eq. (4.110) for kc= 1.8 x 1 0 -19 J and for fixed membrane area AT = 188 gm 2. cos0 d (

r

dr

r cos0

dF) dr

- AF

(4.113)

It is convenient to use as an independent variable the length of the generatrix of the membrane profile, s, whose differential is related to the differentials of the cylindrical coordinates (r,z) as follows"

dr - cos0 ds;

dz - sin 0 ds

(4.114)

The introduction of s as a variable of integration helps to avoid divergence in the procedure of numerical integration at the "equator" of the erythrocyte, where cos0 = 0. The differential of the membrane area, A, is simply related to ds:

dA = 2fords Combining Eqs. (4.112) and (4.114) one obtains

(4.115)

Chapter 4

166

dO - -

ds

=

2

sin0

b

r

(4.116)

~ F(s)

Likewise, from (4.113) and (4.114) one derives

d2F I cosO I dF ~ = ~ F - - ds 2 r ds

(4.117)

Equations (4.114)-(4.117) form a set of 5 equations for determining the 5 unknown functions

r(s), z(s), O(s), F(s) and A(s). In particular, the functions r(s) and z(s) determine the profile of the axisymmetric membrane in a parametric form. Following Ref. [50], to determine the profile of the adherent erythrocyte (Fig. 4.4) one starts the numerical integration of Eqs. (4.114)-(4.117) from the apex of the membrane surface, i.e. from the upper point of the profiles in Fig. 4.4, where the generatrix intersects the axis of revolution. The boundary conditions are: r(s=0) = 0;

z(s=0) = 0;

O(s = 0) - 0

and

A(s=0) = 0

(4.118)

b is assumed to be a known parameter; in Ref. [50] it has been determined from the experimental data. Finally, the two boundary conditions for Eq. (4.117) are:

F(s=0) = q

and

dF -- = 0 ds

for s = 0

(4.119)

The latter boundary condition removes a divergence in Eq. (4.117) for s = 0 (r = 0). On the other hand, q is a unknown parameter, which is to be determined, together with the other unknown parameter ~,, from the area of the free (non-adherent) portion of the membrane:

AF = A T - ~ rj2

(4.120)

where At is the total area of the membrane, assumed constant; rr is assumed to be known from the experiment. The two unknown parameters, q and 2, are to be determined from the following two conditions: r - rl

and

0 - 0h

for

A = AF

(4.121 )

General Curved Interfaces and Biomembranes

167

,/" A

E

v

=I.

2

f,,'"

,,,,, ,

/

b

.,L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-

a,.

t~

e.m

"~ O O

o

0 -1

.~ x._

-2

>

-3

-,..i

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

t...-

'\

/

\

/ ",

/ -,

-4 -6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Horizontal coordinate (gm)

Fig. 4.5. Shape of a closed membrane calculated in Ref. [50] by means of Eq. (4.110)" (a) spherocyte; (b) discocyte corresponding to cr= 1.8x 10 -4 mN/m and AP = 0.036 Pa; (c) discocyte corresponding to o" = 3.6 x 10 .4 mN/m and AP = 0.072 Pa; k C and Ar are the same as in Fig. 4.4. To obtain the profiles of adherent cells shown in Fig. 4.4 one can start the integration from the membrane apex, s =0, with tentative values of the unknown parameters, q and ~,. Further, the integration continues until the point with A = AF is reached. Then we check whether Eq. (4.121) is satisfied. If not, we assign new values of q and X and start the integration again from the apex s = 0 . This continues until we find such values of q and )~, which lead to fulfillment of Eq. (4.121). These values can be automatically determined by numerical minimization of the function * ( q ' / ~ ) - [r(AF " q ' & ) - r[ ]2 + [O(A F . q,/~) - 0 ~ ]2

(4.122)

with respect of q and X. Thus one determines the values of ~ and q. The curves in Fig. 4.5 illustrate what would be the shape of the same erythrocyte (of the same fixed area A T - 188 ~lm2 as in Fig. 4.4) if it were not attached to the substrate. In other words, Fig. 4.5 shows the shape of a free (non-attached) erythrocyte of area AT at various values of the membrane tension o" and the transmembrane pressure difference AP, specified in the figure caption. The curves in Fig. 4.5 are obtained in Ref. [50] by numerical integration of Eqs. (4.114)-(4.117). As before, the integration starts from the apex of the profile, i.e. from the

Chapter 4

168

intersection point of the generatrix with the axis of revolution, where we set s = 0. The numerical integration starts with a tentative value of q in Eq. (4.119). The value of q is determined from the condition that when the integration reaches the membrane "equator", where cos0= 0, the membrane area must be A = At~2. Since the membrane profile is symmetric with respect to the "equatorial" plane, in this case we carry out the numerical integration above the equator, and then we obtain the profile below the equator as a mirror image. Of course, one can continue the numerical integration after crossing the equator and the same profile will be obtained; however, due to accumulation of error from the numerical procedure this could decrease the accuracy of the calculated profile for the higher values of s (below the equator). Such calculations indicate that the generalized Laplace equation, Eq. (4.110), or the equivalent set of equations (4.114)-(4.117), has at least two types of solutions for a free (non-adherent) erythrocyte of fixed area and given membrane tension or. The first type is a "discocyte" like those in Fig. 4.5. The second type looks like an oblong ellipsoid of revolution, which resembles the shape of a red blood cell penetrating along a narrow capillary (blood vessel).

4.5.

M I C R O M E C H A N I C A L EXPRESSIONS FOR THE SURFACE PROPERTIES

In the Section 4.3.2 we derived the formula for the work of interfacial deformation, Eq. (4.72), starting from a purely phenomenological expression, Eq. (4.73); as a result we obtained relationships

between

the

surface

thermodynamical

and

mechanical

properties,

Eqs.

(4.79)-(4.80). In the present section we will derive again Eq. (4.72), however this time by taking excesses with respect to the bulk phases; this alternative approach provides

micromechanical expressions for the surface properties of an arbitrarily curved interface, such as generalizations of Eqs. (3.69) and (3.70).

4.5.1. SURFACE TENSIONS, MOMENTS AND CURVATURE ELASTIC MODULI Basic relationships. Our purpose here is to extend the hydrostatic approach to the theory of a spherical phase boundary from Section 3.2.3 to the general case of an arbitrarily

General Curved Interfaces and Biomembranes

169

curved interface. We follow Refs. [23,26]. As discussed in Chapter 1, in a vicinity of a real fluid interface the pressure P represents a tensorial quantity, which depends on the position in the transition zone between the two phases. After Gibbs [1 ], let us define an idealized (model) system, which is composed of two homogeneous and isotropic fluid phases, I and II, separated by a mathematical dividing surface. If R(ul,u 2) is the running position vector of a point from the dividing surface, we can present the position-vector of a given point in space in the form: r - R ( u l , u 2) + 2 n ( u l , u 2)

(4.123)

As before, n is the running unit normal to the interface; 2~is the distance from the given point in space to the dividing surface. The pressure in the idealized system can be expressed in the form

P - PU,

P = P~O(-&)+ Pn0(&)

(4.124)

where {~ 0(A,) -

for~<0; for & > 0,

(4.125)

is the step-wise function of Heaviside; P~ and Pn are the pressures in the bulk of the respective two neighboring phases, while U is the unit tensor in space. Having in mind Eq. (1.22), one can express the work of interfacial deformation, produced during a time interval St, [52]:

f~w= dA = - I ( P - P ) A

9( ~ t )

dV

(4.126)

V

As before, &v, is the work of interfacial deformation per unit area, d A - a l / Z d u l d u

2

is a

surface element, and W is the spatial rate-of-strain tensor (the strain tensor is ~ = W~t): 1 [ V w -k- (Vw) T]

(4.127)

Here w is the velocity field in the bulk phases and the superscript "T" symbolizes conjugation. For slow quasistatic processes one can use the kinematic formula of Eliassen [21 ]" w-

v-,~(V~v).n

(4.128)

where the surface velocity v is defined by Eq. (4.18). Equation (4.128) corresponds to such a

170

Chapter 4

deformation, for which every material point is fixed at a point of the curvilinear coordinate network (u I ,u 2,/~ ), which moves together with the deforming dividing surface. In particular, every layer of material points, corresponding to ~ = const., remains parallel to the dividing surface in the course of deformation. The integration in the left-hand side of Eq. (4.126) is carried out over an arbitrarily chosen parcel A from the dividing surface, while the integration in the right-hand side - over the corresponding cylindrical volume depicted in Fig. 4.6. The lateral surface of this volume is perpendicular to the dividing surface. The vectorial surface element of the lateral surface can be expressed in the form [21 ]: (4.129)

ds = v . L d/~ dl

where dl is a linear element on the contour C, and v is an outer unit normal to C, see Fig. 4.6; L = (1-2)~H)U s +~b

(4.130)

in Eq. (4.129) is a surface tensor, cf. Eqs. (4.14) and (4.17). The volume element d V can be expressed in the form [53]: X = ( 1 - / ~ H ) 2 - ~ 2 D 2= 1 - 2M-/+/~2K

dV=zdAd),,

(4.131)

A substitution of w from Eq. (4.128) into Eq. (4.127) yields [23]

i ! !

\

; '1 x-o i

A--~. 1

I

Fig. 4.6. An imaginary cylinder, whose lateral surface is perpendicular to the dividing surface between phases I and II, and whose bases are parallel to it; n and v (nA_v) are running unit normals to the dividing surface and to the contour C, respectively; moreover, v is tangential to the dividing surface.

General Curved Interfaces and Biomembranes

171

- a u a v [(1 - 2)~H)(wu, v + wv, ~ ) + ~,(wu,c~bGv + wv,c~b ~ ) - 2(buy - ~ Kauv )v ~n)]/22"

(4.132)

Next, with the help of Eqs. (4.17), (4.19), (4.21), (4.128), (4.132) and the identities (4.61)-(4.62) one can prove that [23] (U s "~F)U s + ( q ' ~ ) q - ( Z U s a + 2 " q ~ - Z / ~ L I ( t - 2 ~ q . L D ) / 2 "

(4.133)

Since the surface parcel A has been arbitrarily chosen, from Eqs. (4.126) and Eq. (4.131), we get: ws - - ~ ( P ~

9~ ) ~ t 2" d/~,

ps = p _ ~

(4.134)

A further simplification of Eq. (4.134) can be achieved if some specific information about the process of deformation is available. To specify the process one can assume that the surface rate-of-strain tensor can be expressed in the form [23]: 9 - 8 9 s "~F)U s + 89

~)q

(4.135)

The latter equation implies that the tensor ~F has diagonal form in the basis of the principle curvatures. In other words, the deformation of the dividing surface is related to the deformation in the adjacent phases. Using Eq. (4.135) we obtain: ps . q j _ ~ ps . [ U s ( U s . qj) + q (q. qj)]

(4.136)

Equation (4.136) can be also obtained using the alternative assumption that the pressure tensor can be expressed in the form P

7(Us "P)U s + 8 9

(4.137)

From Eq. (4.137) one can deduce Eq. (4.136) without imposing any limitations on the form of the rate-of-strain tensor qJ. The expression (4.137), which has been introduced by Buff [54, 55], means that at every point two of the eigenvectors of the tensor P are directed along the lines of maximal and minimal curvature of the surface/~ = const., passing through this point, whereas the third eigenvector is always normal to the dividing surface. In other words, the orientation of the eigenvectors of P in a vicinity of the phase boundary is induced by the geometry of the

Chapter 4

172 interface, which is a reasonable presumption from a physical viewpoint.

Combining Eqs. (4.133), (4.134) and (4.136) one obtains again the known expression for the work of surface deformation, 8Ws = ?'(5o~+ ~(5~ + B3H + |

see Eq. (3.1), where [23, 26]"

7 - - l fUs .Pszd

(4.138)

~ - - 8 9 f q" pSzd/~

(4.139)

B-

(4.140)

f L " P~,d~, q#

)-i

j'(q. L)" PS/~d/~

|

(4.141)

/1,1

Equations (4.138)-(4.141)provide general micromechanical expressions for the interfacial tensions and moments of an arbitrarily curved interface. In the special case of a spherical interface of radius a Eqs. (4.138) and (4.140) reduce to known expressions first derived by Tolman [56]: these are Eq. (3.59) and the equation ae B - 2 j"(Pv - P)(1 + ~ / a)~dA

(4.142)

The special form of Eqs. (4.138)-(4.141) for a cylindrical surface can be found in Refs. [14,22].

Expressions for the spontaneous curvature and the elastic moduli k C and k c . From Eqs. (4.140)-(4.141) one can deduce micromechanical expressions for the bending moment of a planar interface, B0, and for the bending and torsion elastic moduli, k C and k c . We will use the simplifying assumption for "tangential isotropy", which has been introduced by Buff [53]" m

ps _ p§ Us + pN nn;

P§ - P T - P ;

From Eqs. (4.17) and (4.130) we obtain

P~v --PN - P

(4.143)

173

'1.L = ( I - M ) q + A D U ,

(4.143)

A substitution 01' Eys. (4.143)-(4.144) i n l o I%].

(4.141) yields

7

8 = 211 t;';t"dL

(4.145)

i l

The comparison of the lattcr equation with Eq. (3.9) gives "1

k, =-j,yi%a

(4.146)

4

Equation (4.146) ha\ been fir41 obtained by Helfrich [57].Further, from Eqs. (4.17) and (4.130) oiic can derive

I, = (I-AH)IJ, + A 13 q

(4.147)

Substituting Eqs. (4.143) and (4.147) inio Eq. (4.140) we get (4.148)

From the latter equation, in view of Eqs. (3.9), (3.10) and (4.146), we obtain (4.149)

(4.150)

As usual, H,, is the spontaneous curvature. Equations (4.149) and (4.150) have been first

obtained in a quite different way. respectively, by Helfrich [571 and Szleifer et al. [58]. Equations (3.136).(4.149)and (4.150) wcrc utilized by Lckkcrkcrkcr [59] for calculating the electrostatic components of B,, , k , and

Lc.,see Eqs. (,3.91)-(3.93).

Froin the above equations we can deduce also expressions for the coefficients CI and C?, see

Eqs. (3.5) and (3.h), in the framework of the simplifying assumption for tangential isotropy, Eq. (4.143). For example, combining Eqs. (4.135) and (4.148) with Eq. (3.6)one can derive

Chapter 4

174

C2 - - 1 0 _ _ i p r 2D ~j

h Cl - B - 2 C 2 H = 2 f P T ~ f l ,

22d2;

d]t

(4.151)

fl,l

Likewise, a combination of Eqs. (4.131), (4.138) and (4.143) yields x2

~ ' = - I ( 1 - 2 H ~ + K ~ Z ) P T ~ d2

(4.152)

21

Equations (4.151) and (4.152) are equivalent to the results by Markin et al. [30] (the latter ,-v

authors use the notation J = - 2 H and CI =-C1/2).

Expressions for the surface densities of extensive thermodynamic parameters. In Ref. [23] a micromechanical derivation of the fundamental thermodynamic equation, Eq. (3.16), was given, which provided expressions also for the adsorptions of the species, 1-'~, and the surface excesses of internal energy Us and entropy Ss, as follows ;~

r k - f(n k -gk)zdA,

,h

us - f ( u - g ) z d ~ , ,

,h

s,-

f(s--s)zdA

(4.153)

Here Z is defined by Eq. (4.131), n~, u and s are bulk densities of the k-th spacies, internal energy and entropy in the real system; the respective densities in the idealized system are -n

k

t 0 ( - X ) + n~IO (X),

-= l l k

g - uIO ( - X ) + uIIO (X),

-g - slO ( - X ) + snO (X)"

the superscripts 'T' and "II" denote properties of phases I and II. Note that the expression for F1, in Eq. (4.153) represents a generalization of Eq. (1.36) for an arbitrarily curved interface.

4.5.2.

TENSORS OF THE SURFACE STRESSES AND MOMENTS

General micromechanical expressions. Following Ref. [26] let us consider a sectorial strip AA, which is perpendicular to the interface and corresponds to a linear element dl from an arbitrarily chosen curve C on the dividing surface (Fig. 4.7). As usual, n and v (n_l_v) are, respectively, running unit normals to the dividing surface and the curve C. It is presumed that the ends of the sectorial strip ALl,, corresponding to ,%= &l and 2, = 2,2, are located in the volume

General Curved Interfaces and Biomembranes

175

2=22

11

/

Fig. 4.7. Sketch of a sectorial strip, which is perpendicular to the dividing surface. The strip corresponds to an element dl from the curve C on the dividing surface; n and v are running unit normals to the surface and the curve C, respectively; in addition, n _1_v.

of the two adjacent phases, i.e. far enough from the interface to have isotropic pressure tensor P. The force acting on the strip AA, in the real system is

I dsn.

(4.154)

P

AAs

In the idealized system the pressure tensor is assumed to be isotropic by definition, see Eq. (4.124); to compensate the difference with the real system, a surface stress tensor, if, is introduced. Hence, the force exerted on the strip AAs in the idealized system is

iidsn.P)-v.~dl

(4.155)

AAs

Demanding the f o r c e acting on the strip AA, to be the same in the real and idealized system, we obtain [ 14]: ~2

v

.ff - - v-

I dXL- P ' ,

P" - P - P ,

(4.156)

Xl

where we have used Eq. (4.129). Likewise, demanding the m o m e n t acting on the strip AA, to

176

Chapter 4

be the same in the idealized and the real system, we obtain [14]" '~2

v.N-v.~xR--v,

fdXL. P s xr

(4.157)

~,1

Using the arbitrariness of v and the identity a v x n - a U e u v ,

from Eqs. (4.123), (4.156) and

(4.157) one deduces [14]: ~2 P

_ _ ]dAL.p s

(4.158)

d A]

~2

N--

[d~L-P

s

(4.159)

Q~

In addition, with the help of the identities M - N-e_

and e - e - - U s

from Eq. (4.159) one

derives

M - [dAAL.ps

o

(4.160)

Us

Equations (4.158)-(4.160) represent the sought for micromechanical expressions for the tensors of the surface stresses and moments, see Fig. 4.1. It is worthwhile noting that in contrast with the tensor o-, the tensors M and N, defined by Eqs. (4.159) and (4.160), have no transversal components (components directed along n), which is consonant with Eqs. (4.28) and (4.46). In addition, Eq. (4.83) shows that the bending and torsion moments, B and O, are equal to the trace and the deviator of the tensor M: B-Us'M,

O-q'M

(4.161)

The substitution of M from (4.160) into (4.161) gives exactly Eqs. (4.140) and (4.141), which is an additional evidence for the selfconsistency of the theory presented here. Likewise, from Eq. (4.24) we obtain O" _ I U5

, . ,0 -

7/ _ 1 5 q ' o "

Combining Eqs. (4.158) and (4.162) one derives [ 14]"

(4.162)

General Curved Interfaces and Biomembranes

177

~2

o - _ - 7 [ d A L " ps __

(4.163)

1

~2

r / = - g 1 ] ' d 2 ( q . L ) . ps

(4.164)

One can verify that if the micromechanical expressions for 7, {, B, |

a and 77, Eqs.

(4.138)-(4.141) and (4.163)-(4.164), are substituted into Eqs. (4.81) and (4.82), the latter are identically satisfied; this is an additional test for selfconsistency of the theory. m

Special case o f tangentially isotropic tensor P: From the definition ps = p _

p it

follows that in such a case the tensor ps will be also tangentially isotropic, cf. Eq. (4.143), and 7 will be given by Eq. (4.152). From Eqs. (4.139), (4.163) and (4.164) we obtain ]t.2

(4.165)

- - I d/~ (1 - M-/)Pr ~

O-

)q ~2

- O,

71 - - D I d ~ P ~

- --y~ (BD + O H

)

(4.166)

hl

As mentioned at the end of Section 4.3.2, two alternative definitions of f l u i d interface are possible: (i) mechanical: the surface stress tensor, o', is isotropic, that is r/= 0. (ii) thermodynamical: no work is produced upon a deformation of surface shearing, i.e. { = 0. Equation (4.166) shows that the hypothesis for tangential isotropy, Eq. (4.143), is consistent with the thermodynamic definition for fluid interface, that is { = 0. On the other hand, from the tangential isotropy of the tensor P (and P ' ) does not follow isotropy of its surface excess, the tensor Or, cf. Eq. (4.158). In fact, the anisotropy of o- stems from the anisotropy of the curvature tensor b of the arbitrarily curved interface, see Eqs. (4.130) and (4.158). The fact that the hypothesis for tangential isotropy of the pressure tensor P is consistent with the thermodynamic definition of surface fluidity should not be considered as an argument in

Chapter 4

178

favor of the latter definition. One should have in mind that, in general, the statistical mechanics predicts a non-isotropic pressure tensor P in a vicinity of an arbitrarily curved interface; see the review by Kuni and Rusanov [60].

4.6.

SUMMARY

In mechanics the stresses and moments acting in an interface or biomembrane can be taken into account by assigning tensors of the surface stresses, ~, and moments, M, to the phase boundary, see Fig. 4.1. Three equations determine the shape and deformation of a curved interface or biomembrane: they correspond to the three projections of the vectorial local balance of the linear momentum, see Eqs. (4.37), (4.39) and (4.40). Additional useful information is provided by the vectorial local balance of the angular momentum, see Eqs. (4.43)-(4.44), which imply that the tensor M is symmetric, and that its divergence is related to the transverse shear stress resultants of ~, see Eqs. (4.28) and (4.49). The normal projection of the surface linear momentum balance has the meaning of a generalized Laplace equation, which contains a contribution from the interfacial moments, see Eq. (4.51). Alternatively, variational calculus can be applied to derive the equations governing the interfacial/membrane shape by minimization of a functional - " t h e thermodynamic approach". The surface/membrane tension depends on the local curvature of the surface and should not be treated as a constant Lagrange multiplier. The correct minimization procedure is considered in detail in Section 4.3.1. In the theoreti6al derivations one should take into account also the work of surface shearing, even in the case offluid interface/membrane; see the discussions after Eqs. (4.83) and (4.166). Thus it turns out that the generalized Laplace equation can be derived in two alternative ways: mechanical and thermodynamical, cf. Eqs. (4.50)-(4.51) and Eqs. (4.70)-(4.71). This fact provides a test for a given surface mechanical (theological) model: if a model is selfconsistent, the two alternative approaches must give the same result. The connection between them has the form of relationships between the mechanical and thermodynamical surface tensions and moments, see Eqs. (4.79)-(4.82). Different, but equivalent, forms of the generalized Laplace equation are considered and discussed, see Section 4.3.3. In fact, the mechanical and

General Curved Interfaces and Biomembranes

179

thermodynamical approaches are mutually complementary parts of the same formalism. The general theoretical equations can give quantitative predictions if only theological constitutive relations are specified, which characterize a given interface (biomembrane) as an elastic, viscous or visco-elastic two-dimensional continuum. The most popular constitutive relations for the tensors of the surface stresses ~ and moments M are Eqs. (4.86) and (4.96), which stem from the models of Boussinesq-Scriven and Helfrich. The latter leads to a specific form of the generalized Laplace equation, which is convenient to use in applications, such as determination of the axisymmetric shapes of biological cells, see Eq. (4.103) and the whole Section 4.4. In particular, the axial symmetry reduces the generalized Laplace equation to a system of ordinary differential equations, for which a convenient method of integration is proposed (Section 4.4.2). Finally, micromechanical expressions for the surface tensions and moments are derived in terms of combinations from the components of the pressure tensor, see Eqs. (4.138)-(4.141) and (4.158)-(4.164). As corollaries from the latter general equations one can deduce theoretical expressions for calculation of the bending and torsion elastic moduli, kc and k C, and the spontaneous curvature, H0, see Eqs. (4.146), (4.149) and (4.150).

4.7.

REFERENCES

1. J.W. Gibbs, The Scientific Papers of J.W. Gibbs, Vol. 1, Dover, New York, 1961. 2. L. Boruvka, A.W. Neumann, J. Chem. Phys. 66 (1977) 5464. 3. J.G. Kirkwood, I. Oppenheim, Chemical Thermodynamics, McGraw-Hill, New York, 1961. 4. G. Kirchhoff, Crelles J. 40 (1850) 51. 5. A.E.H. Love, Phil. Trans. Roy. Soc. London A 179 (1888) 491. 6. P.M. Naghdi, The Theory of Shells and Plates, in: Handbuch der Physik, S. Fltigge, Ed., Vol. Via/2, Springer, Berlin, 1972. 7. W. Helfrich, Z. Naturforsch. 28c (1973) 693. 8. W. Helfrich, Z. Naturforsch. 29c (1974) 510.

180

Chapter 4

9. E.A. Evans, R. Skalak, Mechanics and Thermodynamics of Biomembranes, CRC Press, Boca Raton, Florida, 1979. 10. E. Cosserat, F. Cosserat, Theorie des corps deformables, Hermann, Paris, 1909. 11. A.G. Petrov, The Lyotropic State of Matter: Molecular Physics and Living Matter Physics, Gordon & Breach Sci. Publishers, Amsterdam, 1999. 12. Ya. S. Podstrigach, Yu. Z. Povstenko, Introduction to Mechanics of Surface Phenomena in Deformable Solids, Naukova Dumka, Kiev, 1985 (in Russian). 13. S. Ljunggren, J.C. Eriksson, P.A. Kralchevsky, J. Colloid Interface Sci. 161 (1993) 133. 14. P.A. Kralchevsky, J.C. Eriksson, S. Ljunggren, Adv. Colloid Interface Sci. 48 (1994) 19. 15. J. Serrin, Mathematical Principles of Classical Fluid Mechanics, in: S. Fltigge (Ed.), Handbuch der Physik, Vol. VIIF1, Springer, Berlin, 1959. 16. L.D. Landau, E.M. Lifshitz, Theory of Elasticity, Pergamon Press, Oxford, 1970. 17. L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, 1984. 18. C.E. Weatherburn, Differential Geometry of Three Dimensions, Cambridge University Press, Cambridge, 1939. 19. A.J. McConnell, Application of Tensor Analysis, Dover, New York, 1957. 20. R. Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics, Prentice-Hall, Englewood Cliffs, 1962. 21. J.D. Eliassen, Interfacial Mechanics, Ph.D. Thesis (1963), University of Minnesota, Dept. Chemical Engineering; University Microfilms, Ann Arbour, 1983. 22. S. Ljunggren, J.C. Eriksson, P.A. Kralchevsky, J. Colloid Interface Sci. 191 (1997) 424. 23. P.A. Kralchevsky. J. Colloid Interface Sci. 137 (1990) 217. 24. T.D. Gurkov, P.A. Kralchevsky, Colloids Surf. 47 (1990) 45. 25. L. Brand, Vector and Tensor Analysis, Wiley, New York, 1947. 26. I.B. Ivanov, P.A. Kralchevsky, in: I.B. Ivanov (Ed.) Thin Liquid Films, Marcel Dekker, New York, 1988, p. 91. 27. 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. 28. J.T. Davies, E. K. Rideal, Interfacial Phenomena, Academic Press, New York, 1963. 29. A.W. Adamson, A.P. Gast, Physical Chemistry of Surfaces, Wiley, New York, 1997. 30. V.S. Markin, M.M. Kozlov, S.I. Leikin, J. Chem. Soc. Faraday Trans. 2, 84 (1988) 1149.

General Curved Interfaces and Biomembranes

181

31. J.C. Melrose, Ind. Eng. Chem. 60 (1968) 53. 32. Ou-Yang Zhong-can, W. Helfrich, Phys. Rev. Lett. 59 (1987) 2486. 33. Ou-Yang Zhong-can, W. Helfrich, Phys. Rev. A 39 (1989) 5280. 34. Ou-Yang Zhong-can, Phys. Rev. A 41 (1990) 4517. 35. M. Reiner, Rheology, in: S. Flfigge (Ed.), Handbuch der Physik, Vol. VI, Springer, Berlin, 1958. 36. E.D. Shchukin, A.V. Pertsov, E.A. Amelina, Colloid Chemistry, Moscow Univ. Press, Moscow, 1982 (in Russian). 37. M.J. Boussinesq, Ann. Chim. Phys. 29 (1913) 349. 38. L.E. Scriven, Chem. Eng. Sci. 12 (1960) 98. 39. N. Dan, P. Pincus, S.A. Safran, Langmuir 9 (1993) 2768. 40. H. Funaki, Japan J. Physiol. 5 (1955) 81-92. 41. P. Canham, J. Theor. Biol. 26 (1970) 61. 42. H.J. Deuling, W. Helfrich, Biophys. J. 16 (1976) 861. 43. H.J. Deuling, W. Helfrich, J. Physique 37 (1976) 1335. 44. P.R. Zarda, S. Chien, R. Skalak, J. Biomech. 10 (1977) 211. 45. H. Vayo, Canadian J. Physiol. Pharmacol. 61 (1983) 646. 46. S. Svetina, B. Zek~, Eur. Biophys. J. 17 (1989) 101. 47. U. Seifert, K. Berndl, R. Lipowsky, Phys. Rev. A, 44 (1991) 1182. 48. B. L.-S. Mui, H.-G. Dobereiner, T.D. Madden, P.R. Cullis, Biophys. J. 69 (1995) 930. 49. A.G. Volkov, D.W. Deamer, D.L. Tanelian, V.S. Markin, Liquid Interfaces in Chemistry and Biology, John Wiley & Sons, New York, 1998. 50. K.D. Tachev, J.K. Angarska, K.D. Danov, P.A. Kralchevsky, Colloids Surf. B: Biointerfaces 19 (2000) 61. 51. E.A. Evans, R.M. Hochmuth, in: A. Kleinzeller & F. Bronner (Eds.), Current Topics in Membranes and Transport, Vol. 10, Academic Press, New Work, 1978; p. 1. 52. A.I. Rusanov, Phase Equilibria and Surface Phenomena, Khimia, Leningrad, 1967 (in Russian); Phasengleichgewichte und Grenzfl~chenerscheinungen, Akademie Verlag, Berlin, 1978. 53. F.P. Buff, J. Chem. Phys. 25 (1956) 146.

182

Chapter 4

54. F.P. Buff, Discuss. Faraday Soc. 30 (1960) 52. 55. F.P. Buff, The Theory of Capillarity, in: S. Flfigge (Ed.), Handbuch der Physik, Vol. X, Springer, Berlin, 1960. 56. R.C. Tolman, J. Chem. Phys. 16 (1948) 758. 57. W. Helfrich, in: Physics of Defects (R. Balian et al., Eds.), North Holland Publ. Co., Amsterdam, 1981; pp. 716- 755. 58. I. Szleifer, D. Kramer, A. Ben Shaul, D. Roux, W. M. Gelbart, Phys. Rev. Lett.60 (1988) 1966. 59. H.N.W. Lekkerkerker, Physica A 167 (1990) 384. 60. F.M. Kuni, A.I. Rusanov, in: The Modern Theory of Capillarity, A.I. Rusanov & F.C. Goodrich (Eds.), Akademie Verlag, Berlin, 1980.

183

CHAPTER 5 LIQUID FILMS AND INTERACTIONS BETWEEN PARTICLE AND SURFACE The collision of a colloid particle with an interface, or with another particle, is accompanied by the formation of a thin liquid film. The particle(s) will stick or rebound depending on whether repulsive or attractive forces prevail in the liquid film. In the case of an equilibrium liquid film the repulsive forces dominate the disjoining pressure, which is counterbalanced by the action of transversal tension, the latter being dominated by the attractive forces in the transition zone film-meniscus. The Derjaguin approximation allows one to calculate the force across a film of uneven thickness if the interaction energy per unit area of a plane-parallel film is known. Next we consider interactions of different physical origin. Expressions for the van der Waals interaction between surfaces of various shape are presented. Hypotheses about the nature of the long-range hydrophobic surface force are discussed. Special attention is paid to the electrostatic surface force which is due to the overlap of the electric double layers formed at the charged surfaces of an aqueous film. The effects of excluded volume per ion and ionic correlations lead to the appearance of a hydration repulsion and an ion-correlation attraction. The presence of fine colloidal particles in a liquid film gives rise to an oscillatory structural force which could stabilize the film or cause its step-wise thinning (stratification). At low volume fractions of the fine particles the oscillatory force degenerates into the depletion attraction, which has a destabilizing effect. The overlap of "brushes" from adsorbed polymeric molecules produces a steric interaction. The configurational confinement of thermally excited surface modes engenders repulsive undulation and protrusion forces. Finally, the collisions of emulsion drops are accompanied with deformations, i.e. deviations from the spherical shape. They cause extension of the drop surface area and change in the surface curvature, which lead to dilatational

and

bending

contributions

to

the

overall

interaction

energy.

The

total

particle-surface (or particle-particle) interaction energy is a superposition of contributions from all operative surface interactions. In addition, hydrodynamic interactions, due to the viscous friction in a liquid film, are considered in the next Chapter 6.

Chapter 5

184

5.1.

MECHANICAL BALANCES AND THERMODYNAMIC RELATIONSHIPS

5.1.1. INTRODUCTION A necessary step in the process of interaction of a colloidal particle (solid bead, liquid drop or gas bubble) with an interface is the formation of a liquid film (Fig. 5.1). For example, a liquid film of uniform thickness can be formed when a fluid particle approaches a solid surface, see Fig. 5.1 a. The shape of such a film is circular; the radius of the contact line at its periphery is denoted by rc. From a geometric (and hydrodynamic) viewpoint a liquid film is termed thin when its thickness h is relatively small, viz. h/rc << 1. From a physical viewpoint a liquid film is called thin if its thickness is sufficiently small that the molecular interactions between the two adjacent phases across the film are not negligible; as a rule this happens for h _< 100 nm [1]. These molecular interactions across the film are often termed surface forces [2,3]. The surface force per unit area of the film is called disjoining pressure, H, [4]. In general, the disjoining pressure depends on the film thickness, H = H(h). Since H is an excess pressure in the thin liquid film with respect to the bulk liquid, H vanishes in a thick film, that is I-I--->0 for h---~,x,. The disjoining pressure can be both repulsive (H > 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- << 1)

(5.11)

O-

In addition, the combination of Eqs. (5.6) and (5.10) yields

"t= G [ 1 - ( l + f / o - ) z ~ / Z ~ ( - 2 f o - ) l / 2

Equations (5.10) and (5.11) show that the interaction free energy must be negative, that is f < 0; otherwise equilibrium attachment of a particle to the interface (Fig. 5.2) is impossible. The condition f < 0 is often satisfied because at long distances the integrand 1-I(h) in Eq. (5.9) is negative, which in a final reckoning can give a negative f For aqueous films the disjoining pressure is often a superposition of electrostatic repulsion and van der Waals attraction [ 16,17]: H(h) = C exp(-tch)

AH 6~h3

(5.12)

Here C is a constant, tc is the Debye screening parameter and AH is the Hamaker constant; see Section 5.2 for details. A typical shape of the H(h) dependence, determined from Eq. (5.12), is shown in Fig. 5.3; the portion of the curve on the left of the primary minimum is due to the short-range Born repulsion, which is not accounted for in Eq. (5.12). One sees that the equation II(h) = Pc may have three roots corresponding to three possible equilibrium states the liquid

Liquid Films and Interactions between Particle and Surface

/

189

electrostatic barrier

rI=P

primary minimum Fig. 5.3. A typical disjoining pressure isotherm, 1-I vs. h, predicted by Eq. (5.12). The intersection points of the curve I-I(h) with the horizontal line I-I = Pc correspond to equilibrium states of the film: Points 1 and 2 - stable primary and secondary films; Point 3 - unstable equilibrium state. film, see Eq. (5.3). Point 1 in Fig. 5.3 corresponds to a film, which is stabilized by the double layer repulsion; sometimes such a film is called the primary film or common black film. Point 3 corresponds to unstable equilibrium and cannot be observed experimentally. Point 2 corresponds to a very thin film, which is stabilized by the short range repulsion; such a film is called the secondary film or Newton black film. Transitions from common to Newton black films are often observed with foam and emulsion films [ 18-21]. As an example, let us assume that the state of the film in Fig. 5.2 corresponds to Point 1 in Fig. 5.3. Then obviously II(h~) = Pc > 0, i.e. the disjoining pressure is repulsive and keeps the two film surfaces at an equilibrium distance h~ apart (film of uniform thickness is formed). On the other hand, the attractive surface forces (the zone of the "secondary minimum" in Fig. 5.3) prevail in the integral in Eq. (5.9). In such case we have f(h~) < 0 and consequently, the contact angle a does exists, see Eq. (5.10), and the transversal tension ~-is a real positive quantity, see

Chapter 5

190

•••sjn ~

Plateau border

p,

Pin i .....

h_l_2__ _~, .r t Pin .Pt+ H-r

-h/2

ro

"i Fig. 5.4. Schematic presentation of the detailed and membrane models of a thin liquid film: on the leftand right-hand side, respectively. Eq. (5.11). Note that in Fig. 5.2 H and v have the opposite directions; indeed, as seen from Fig. 5.3, and Eqs. (5.9) and (5.11), their values are determined by the predominant repulsion (for H) and attraction (for v). The fact the directions of H and "r are opposite has a crucial importance for the existence of equilibrium state of an attached particle at an interface. To demonstrate that let us consider the total balance of the forces exerted on the fluid particle in Fig. 5.2. If the particle is small (negligible effect of gravity), then the integral of Pl over the surface of the fluid particle in Fig. 5.2 is equal to zero. Then the total balance of the forces exerted on the particle reads [22,23] /D'c 2 171 ---- 21fro ~"

(5.13)

i.e. the disjoining pressure H, multiplied by the film area, must be equal to the transversal tension ~, multiplied by the length of the contact line. Thus it turns out that the fluid particle sticks to the solid surface at its contact line (at the film periphery) where the long-range attraction (accounted for by r) prevails; on the other hand, the repulsion predominates inside the film, where H = Po > 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 << 1 Eq. (5.57) reduces to Uvw (ho)=

A~I Y = - 2n'RI R~2 An 12 (1 + y)x R 1 + R 2 12g'ho

(5.59)

Equation (5.59) can be also derived by substituting fvw(h) from Eq. (5.56) into Derjaguin approximation (5.51). It is worthwhile noting, that the logarithmic term in Eq. (5.57) can be neglected only if x << 1. For example, even when x = 5 x 10-3, the contribution of the

Liquid Filmsand InteractionsbetweenParticleand Surface

205

logarithmic term amounts to about 10% of the result (for y = 1); consequently, for larger values of x this term must be retained [44]. For the configuration

sphere - wall,

which is depicted in Fig. 5.1b, an expression for the

interaction energy can be obtained setting R1 ~ o,, and R2 = R in Eqs. (5.57) and (5.58):

U vw(h~

AI4 ( ~2R 2R +~+21n

--~-

ho

ho

2R + h o

)

(5.60)

2R + h o

Alternatively, substituting fvw(h) from Eq. (5.56) into the Derjaguin approximated formula (5.49) one derives Uvw (ho)_

A/_/ 2R 12 ho

(5.61)

which coincides with the leading term in Eq. (5.60) for Next, let us consider the configuration

ho/(2R) << 1.

truncated sphere - wall, which

is depicted in Fig. 5.1 a.

An expression for the interaction energy in this case has been derived by Danov et al. [37]:

Uvw (ho ) -

-AH(2R12 / h~ ~re2 2R ho + h---7+ ~ + 21n l + ho 1+ ho

where l - R + (R 2 -

2re2 ]

(5.62)

lh

re{)1/2. Alternatively, substituting fvw(h) from Eq. (5.56) into Derjaguin

approximated formula (5.50) one obtains Uvw (ho )= - - AH(2R_ ~ --~o +

h-~ore2)"

(5.63)

Obviously, the latter approximate expression contains the two leading terms in the right-hand side of Eq. (5.62) for ho--->0. In the case of film between

two identical deformed emulsion droplets,

like those depicted in

Fig. 3.5 with a = R, r = rc and h = ho, the respective droplet-droplet interaction energy can be expressed in the form [37]

AHIR-;-- +-7~re2 +-73 + 21n(__~) - 12re2 Roj h Uvw (ho ) = --72IZLno h~ 4

(ho, re << R)

(5.64)

Chapter 5

206

Equation (5.64) represents a truncated series expansion; the exact formula, which is rather long, can be found in Ref. [37]. Expressions for Uvw for other geometrical configurations are also available [52]. Further, we consider expressions for calculating the Hamaker constant An, which enters Eqs. (5.56)-(5.64). For that purpose two approaches have been developed: the microscopic theory due to Hamaker [51] and the macroscopic theory due to Lifshitz [53].

Microscopic theory: its basic assumption is that the van der Waals interaction is pairwise additive, and consequently, the total interaction energy between two bodies can be obtained by interaction over all couples of constituent molecules. Thus, for the interaction between two semiinfinite phases, composed from components i and j, across a plane-parallel gap of vacuum, one obtains Eq. (5.56) with AH = Aij, where Aij is expressed as follows

A 0 --~2piPjO~ij

(5.65)

p/and pj are the densities of the respective phases and o~ijis a molecular parameter defined by Eq. (5.55). Usually, the dimension of Pi and pj is expressed in molecules per cm 3, and then AH and A 0 have a dimension of energy. For a plane-parallel film from component 3 between two semiinfinite phases from components 1 and 2 the microscopic approach gives again Eq. (5.56), but this time the compound Hamaker constant is determined by the expression [44]

Au -

A132 =

A33 +

AI2 -

A13 -

A23

(5.66)

Here A 0 (ij = 1,2,3) is determined by Eq. (5.65). If the film is "filled" with vacuum, then/93 = 0 and Eq. (5.66) reduces to AH = A~2, as it could be expected. If the Hamaker constants of the symmetric films, viz. Aii and Ajj, are known, one can estimate Aij (i:/:j) by using the approximation of Hamaker

Aij=(A~ Ajj

(5.67)

If components 1 and 2 are identical, AH is positive. Therefore, the van der Waals interaction between identical bodies is attractive across any medium. Besides, two dense bodies (even if nonidentical) will attract each other when placed in medium 3 of low density (gas, vacuum).

Liquid Films and Interactions between Particle and Surface

I

3L

2L

207

hi2

!

1L

1R

2R

3R

4R

hll h21

"1 I ,..|

Fig. 5.8. Sketch of two multilayered bodies interacting across a medium 0; the layers are counted from the central film 0 outward to the left (L) and right (R). On the other hand, if the phase in the middle (component 3) has an intermediate Hamaker constant between those of bodies 1 and 2 (say All < A33 < A22), then the compound Hamaker constant AH can be negative and the van der Waals disjoining pressure can be r e p u l s i v e (positive). Such is the case of an aqueous film between mercury and gas [54], or liquid hydrocarbon film on alumina [55] and quartz [56]. It is worthwhile noting that the liquid helium climbs up the walls of containers because of the repulsive van der Waals force across the wetting helium film [3, 57, 58]. Equation (5.66) can be generalized for multilayered films. For example, two surfactant adsorption monolayers (or lipid bilayers) interacting across water film can b e modeled as a multilayered structure: one layer for the headgroup region, other layer for the hydrocarbon tails, another layer for the aqueous core of the film, etc.). There is a general formula for the interaction between two such multilayered structures (Fig. 5.8) stemming from the microscopic approach [52]: fvw - - ~_~ NL ~-~ NR a ( i , j)2 , i=l j=l 12rth o

a ( i , j ) - a i,j - ai,

, - a i - j + Ai- ,j ~

(5.68)

where NL and NR denote the number of layers on the left and on the r i g h t from the central layer, the latter denoted by index "0" - see Fig. 5.8 for the notation; Ai,j (= A O) is defined by Eq. (5.65). Equation (5.68) reduces to Eq. (5.56) for N L - NR - 1 and hi1 = h.

208

Chapter 5

Macroscopic theory: An alternative approach to the calculation of the Hamaker constant A/_/in condensed phases is provided by the Lifshitz theory [53, 57], which is not limited by the assumption for pairwise additivity of the van der Waals interaction, see also Refs. [2, 3,52]. The Lifshitz theory treats each phase as a continuous medium characterized by a given uniform dielectric permittivity, which is dependent on the frequency, v, of the propagating electromagnetic waves. A good knowledge of quantum field theory is required to understand the Lifshitz theory of the van der Waals interaction between macroscopic bodies. Nevertheless, the final results of this theory can be represented in a form convenient for application. For the symmetric configuration of two identical phases i interacting across a medium j the macroscopic theory provides the expression [3]

AH =-Aiji -A~ v=~ +A/~/v.>~

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) << 1

(5.89)

where C is a constant independent of h; as usual, ~" is the Debye screening parameter. The constant C was determined by Verwey and Overbeek [17] in the following way. Let us consider a film of two identical surfaces and let us deal with a solution of symmetric

electrolyte: Zl = -Z2 - Z. Combining the Boltzmann equation (5.85) with Eq. (5.80), and expanding in series, one obtains

[

I-Ie~ = 2nokT cosh

kT

1 ~ nokT

- 1

kT

(5.90)

2

where gtm - gt(0) is the potential in the middle of the film (Fig. 5.9b), which is assumed to be small: ~_

/

<< 1

(5.91)

Note that we have set gt = 0 in the bulk of solution, see Eq. (5.85); hence small gtm means a

weak overlap of the two double electric layers in the middle of the film. In such case one can use the superposition approximation,

I//m =

2~1, that

is the potential in the middle of the film is

equal to two times the potential at a distance h/2 from a single surface, see Fig. 5.9b for the notation. Since gtl is also a small quantity, with the help of Eq. (1.65) one obtains gtm -- 2gtl -- 8tanh Zegt,

4kT

exp -

(5.92)

---2-

where g~ is the value of the electric potential at the surface of the film. The substitution of Eq. (5.92) into Eq. (5.90) yields [17]

1-Ie, (h)

64nokT(tanhZellts12 exp(-x'h) 4kT

for exp(-tch) << 1

(5.93)

By integration of Eq. (5.93) one can derive expressions for the free energy (per unit area) due to the electrostatic interaction, fd(h), as well as the interaction energy between two bodies, Uel(h0),

216

Chapter 5

with the help of Eqs. (5.9) and (5.49)-(5.52). It is interesting to note, that when gts is large enough, the hyperbolic tangent in Eq. (5.93) is identically 1 and I-Iej (as well as fel and Ue0 becomes independent of the surface potential (or charge). Equation (5.93) can be generalized for the case of 2:1 electrolyte (divalent counterion) and 1:2

electrolyte (divalent coion) [94]: Vi:j

II~l=432n(2)kT tanh--4-

exp(-r..h)

(5.94)

where n(2) is the concentration of the divalent ions, the subscript "i:j" takes value "2:1" or "1:2", and

v21:lnI,/,+2ep///1

,2 lnE(2exp(/+l),1

(5.95)

Equation (5.93) can be generalized also for the case of two non-identically charged interfaces of surface potentials g61 and gts2 for Z:Z electrolytes [2]

1-Ie'(h)= 64n~

7~ - tanh('Z]egts~ '

k-l,2

(5.96)

Equations (5.93)-(5.96) are valid for both low and high surface potentials, if only exp(-r,,h) << 1. The comparison of these equations with Eq. (5.89) allow one to determine the parameter C for each specific system. In addition, the expression for the total interaction energy U = Uvw + Ue~ can be used to predict the critical electrolyte concentration for coagulation in a colloid, see e.g. Ref. [3,44]. Likewise, one can determine the critical concentration of electrolyte which is needed for colloid particles to adhere to an interface with the same sign of the surface charge.

5.2.5. REPULSIVE HYDRATION FORCE

The DLVO theory predicts that the height of the electrostatic barrier (see Fig. 5.3) decreases with the increase of the electrolyte concentration in solution. In other words, the added electrolyte suppresses the electrostatic repulsion. In contrast, the experiment [95-98] shows that sometimes at higher electrolyte concentrations (above c.a. 10-3 M) a strong repulsive force is

Liquid Films and Interactions between Particle and Surface

/

-

217

-

,~'~'--0 1 M

'~,__.,,

10

Vs = -128.4 mV electrolyte" KC1 T = 298 K v = 1.2x 10-27m 3

-2

--"~.~\\

10

M

, 103

t

E

Z

f ,

III

f

,,

.

\

"... -...

\

...~... 0

,o-4 ...

\ 9

0.1

-. \

10

"-. 20

30

40

50

h [nm]

Fig. 5.10. (a) Theoretical dependence of F/R - 2z0~on the film thickness h for various concentrations of KC1, denoted in the curves. For all curves the surface potential is gts = -128 mV, the temperature is 298 K and the excluded volume per ion is v = 1.2 x 10-27 m3; results from Ref. [100l. detected, which completely dominates the effect of the van der Waals attraction at short distances (h < 10 nm), see Fig. 5.10. This repulsive interaction is called the hydration force. It appears as a deviation from the DLVO theory for short distances between two molecularly smooth electrically charged surfaces. {Note that sometimes other, different effects are also termed "hydration force", see Ref. [99] for review. } Experimentally the existence of hydration repulsive force was established by Israelachvili et al. [95,96] and Pashley [97,98] who examined the validity of DLVO-theory at small film thickness in experiments with films from aqueous electrolyte solutions confined between two mica surfaces. At electrolyte concentrations below 10 -4 M (KNO3 or KCI) they observed the typical DLVO maximum, However, at electrolyte concentrations higher than 10 -3 M they did not observe the expected DLVO maximum; instead a strong short range repulsion was detected; cf. Fig. 5.10. Empirically, the hydration force appears to follow an exponential law

[31:

Chapter 5

218

fhyar = f0 exp(-h/2o) where, as usual, h is the film thickness; the decay length is 2o -- 0 . 6 - 1 . 1

(5.97) nm for 1:1

electrolytes; the pre-exponential factor, f0, depends on the specific surface but is usually about 3 - 30 mJ/m 2. The hydration force stabilizes thin liquid films and dispersions preventing coagulation in the primary minimum (that between points 2 and 3 in Fig. 5.3). In historical plan, the hydration repulsion has been attributed to various effects: solvent polarization and H-bonding [101], image charges [102], non-local electrostatic effects [103], existence of a layer of lower dielectric constant, e, in a vicinity of the interface [ 104, 105]. It seems, however, that the main contribution to the hydration repulsion between two charged interfaces originates from the finite size of the hydrated counterions confined into a narrow subsurface potential well [ 100]. (The latter effect is not taken into account by the DLVO theory, which deals with point ions.) Indeed, in accordance with Eq. (1.65), at high electrolyte concentration (large to) and not too low surface potential t/rs, a narrow potential well is formed in a vicinity of the surface, where the concentration of the counterions is expected to be much higher than its bulk value. At such high subsurface concentrations (i) the volume exclusion effect, due to the finite ionic size, becomes considerable and (ii) the counterion binding (the occupancy of the Stern layer) will be greater, see Fig. 1.4. The formed dense subsurface layers from hydrated counterions prevent two similar surfaces from adhesion upon a close contact. This is probably the explanation of the experimental results of Healy et al. [106], who found that even high electrolyte concentrations cannot cause coagulation of amphoteric latex particles due to binding of strongly hydrated Li + ions (of higher effective volume) at the particle surfaces. If the Li + ions are replaced by weakly hydrated Cs + ions (of smaller effective volume), the hydration repulsion becomes negligible, compared with the van der Waals attraction, and the particles coagulate as predicted by the DLVO-theory. The effect of the volume excluded by the counterions becomes important in relatively thin films, insofar as the aforementioned potential well is located in a close vicinity of the film surfaces. In Ref. [100] this effect was taken into account by means of the Bikerman equation [107, 108]:

219

L i q u i d Films a n d Interactions b e t w e e n Particle a n d Surface

1-v~_~nk(z) n i(z)

-

1--v~nko

U i -~ -

ni~

Z i e llt ~

(5.98)

kT

k

Here z is the distance to the charged surface, ni and

kT units)

energy (in

Ui are the

number density and the potential

of the i-th ion in the double electric layer; nio is the value of

ni in the

bulk

solution; the summation is carried out over all ionic species; v has the meaning of an average excluded volume per counterion; the theoretical estimates [100] show that v is approximately equal to 8 times the volume of the hydrated counterion. The volume exclusion effect leads to a modification of the Poisson equation (5.86); it is now presented in the form

,E d2111 ZZien~expUi ~.

4:re dz 2

i

n i =--

--= p ( Z ) ;

i

where

p(z)

ni~

1+ v Z

1+ vZ n; expUi

nkO

(5.99)

k

denotes the charge density in the electric double layer. For v = 0 Eq. (5.99) reduces

to the expression used in the conventional DLVO theory. Taking into account the definition of

Ui, one

can numerically solve Eq. (5.99). Next, the total electrostatic disjoining pressure can be

calculated by means of the expression [328]

Fl+v~_.n]exp(-Zkelltm/ kT)1

I11m

H:~t = -

IPmdl]l-o

- ~ l n [.

~

1--~vv~ ~k

(5.100)

where the subscript "m" denotes values of the respective variables at the midplane of the film. Finally, the non-DLVO

hydration

force can be determined as an excess over the conventional

DLVO electrostatic disjoining pressure: Hhydr

1-[ tot 1-[ DLVO --'= ~ e l -- ~ e l

(5.101)

DLVO is defined by Eq. (5.80), which can be deduced from Eq. (5.100) for v -+ 0. The where H el effect of v :~ 0 leads to a larger value of gtm, which contributes to a positive (repulsive) Hhydr. Similar, but quantitatively much smaller, is the effect of the lowering of the dielectric constant,

Chapter 5

220

e, in a vicinity of the interface [! 00]. The quantitative predictions of Eqs. (5.99)-(5.101) are found to agree well with experimental data of Pashley [97,98], Claesson et al. [109] and Horn et al. [110]. In Fig. 5.10 results from theoretical calculations for F/R - 2rtf vs. h are presented; here F is the force measured by the surface force apparatus between two crossed cylinders of radius R; as usual, f is the total surface free energy per unit area, see Eq. (5.9). The dependence of hydration repulsion on the concentration of electrolyte, KC1, is investigated. All theoretical curves are calculated for v = 1.2 x 10 -27 m 3 (8 times the volume of the hydrated K + ion), AH = 2.2 x 10 -20 J and Vs = -128.4 mV; the boundary condition of constant surface potential is used. In Fig. 5.10 for

Ce, = 5 x 10-5 and 10-4 M a typical DLVO maximum is observed. However, for Cel = 10-3, 10-2 and 10 -1 M maximum is not seen, but instead, the short range hydration repulsion appears. These predictions agree with the experimental findings. Note that the increased electrolyte concentration increases the hydration repulsion, but suppresses the long-range double layer repulsion.

5.2.6.

ION-CORRELATION SURFACE FORCE

The positions of the ions in an electrolyte solution are correlated in such a way that a counterion a t m o s p h e r e appears around each ion thus screening its Coulomb potential. The

latter effect has been taken into account in the theory of strong electrolytes by Debye and Htickel [111, 112], which explains why the activities of the ions in solution are smaller than their concentrations, see Refs. [ 113, 114] for details. The energy of formation of the counterion atmospheres gives a contribution to the free energy of the system called correlation energy [ 115]. The correlation energy provides a contribution to the osmotic pressure of the electrolyte solution, which can be expressed in the form [111, 112] k

I-losm - k T ~ n i i=1

kT~r 3

241r

(5.102)

The first term in the right-hand side of the Eq. (5.102) corresponds to an ideal solution, whereas the seconds term takes into account the effect of electrostatic interactions between the ions. The expression for Hemin the DLVO-theory, Eq. (5.80), obviously corresponds to an ideal

Liquid Films and Interactions between Particle and Surface

221

solution, that is to the first term in Eq. (5.102), the contribution of the ionic correlations being neglected. In the case of overlap of two electric double layers, formed at the surfaces of two bodies interacting across an aqueous phase, the effect of the ionic correlations also gives a contribution, FIcor, to the net disjoining pressure, as pointed out by Guldbrand et al. [116]. 1-Icor can be interpreted as a surface excess of the last term in Eq. (5.102). In other words, the ionic correlation force originates from the fact that the counterion atmosphere of a given ion in a thin film is different from that in the bulk of the solution. There are two reasons for this difference: (i) the ionic concentration in the film differs from that in the bulk and (ii) the counterion atmospheres are affected (deformed) due to the neighborhood of the film surfaces. Both numerical [116-118] and analytical [119,120] methods have been developed for calculating the

ion-correlation component

of disjoining pressure, I-Icor. Attard et al. [119]

derived the following asymptotic formula, which is applicable to the case of symmetric (Z:Z) electrolyte and sufficiently thick films [exp(-tch) << 1]:

I-Icor-

ZcorI-Ie 1

+O(e -~h )

Z2e2~ (ln2+ 21c) ~87r162

Aco r -

(5.103)

Here 1-Ie~is the conventional DLVO electrostatic disjoining pressure, see Eq. (5.80), and

Ic--~1 ( l + J ) l n 2 + J

2-2s 3+ s 2s(2s 2 -1) 2

2S 2 - - 3

-

-- l ) 3 '

Ps is the surface charge

J)ln(s+s 2) ~/sZ-lIl+J+4(2s2-

212

- 1)-3 larctan ~S-ls_ +1

s

s - 1+ 2~ePs ekT~:

~

2S

1(12

density, i.e. the net surface electric charge per unit area.

The theory [117-120] predicts that for films of identically charged surfaces Ilcor is (Acor < 0) and corresponds to

attraction, which

negative

can be comparable by magnitude with I-Ivw. In

the case of 1:1 electrolyte FIcor is usually a small correction to 1-Iel. In the case of 2:2 electrolyte, however, the situation can be quite different: for electrolyte concentrations above a certain

critical

value the ion-correlation attraction could become greater than the double layer

Chapter 5

222

10o A 2 area per charge 1:1

--.. .......

-2 -3

(b

~-4 ~

-7 10 -5

.

. 10 -4 .

.

3

.

:

3

. 10. -3

1 0,.2

,

,0,_1 1

,

, 10 0

electrolyte concentration [M]

Fig. 5.11. Theoretical dependence of 1-Icor/I-Iel on the electrolyte concentration for 1:1, 2:2 and 3:3 electrolytes calculated by means of Eq. (5.103); for all curves the area per surface charge is l e/ps I = 100/~2; after Ref. [ 121 ]. repulsion. In other words, in the presence of bivalent and multivalent counterions Hcor could become the predominant surface force. To illustrate the theoretical predictions, in Fig. 5.11 we present numerical data computed by means of Eq. (5.103). At constant Ps the coefficient Acor, multiplying Flel in Eq. (5.103), is independent of the film thickness h. In other words, for exp(-tch) << 1, the ratio Hcor/Hel is independent of h. In Fig. 5.11 we plot Hcor/Flel vs. the electrolyte concentration for 1:1, 2:2 and 3:3 electrolytes; we have used the value le/psl = 100 ~2 for the area per surface charge. The "critical" electrolyte concentration corresponds to the intersection points of the curves with the horizontal line a t - 1 in Fig. 5.11; for electrolyte concentrations above the critical one the calculated ionic correlation attraction becomes greater by magnitude than the double layer repulsion. One sees in the figure that for a 2:2 electrolyte the critical concentration is about 1 mM, whereas for a 3:3 electrolyte it is below 10-4 M. In the case of secondary thin liquid films, stabilized by ionic surfactant (h - h2 in Fig. 5.3), the measured contact angle is considerably larger than the theoretical value predicted if only van der Waals attraction is taken into account [122]. The experimentally detected additional

Liquid Films and Interactions between Particle and Surface

223

attraction in these very thin films (h = 5 nm) can be attributed to short range ionic correlation effects [123] as well as to the discreteness of the surface charge [2, 124, 125]. Short-range net attractive ion-correlation forces have been measured by Marra [126, 127] and Kjellander et al. [ 128, 129] between highly charged anionic bilayer surfaces in CaCI2 solutions. These forces are believed to be responsible for the strong adhesion of some surfaces (clay and bilayer membranes) in the presence of divalent counterions [ 128, 130]. On the other hand, Kohonen et al. [131 ] measured a monotonic repulsion between two mica surfaces in 4.8 x 10-3 M solution of MgSO4; the lack of attractive surface force in the latter experiments could be attributed, at least in part, to the presence of a strong hydration repulsion due to the Mg 2+ ions. Additional work is necessary to verify the theoretical predictions and to clarify the physical significance of the ion-correlation surface force. DLVO, corresponds to a In summary, the conventional electrostatic disjoining pressure, 1-Ie~- He~ mean-field model, i.e. ideal solution of point ions in the electric field of the double layer. The hydration and ionic-correlation components of disjoining pressure, Hhydr and 1-Icor, represent "superstructures" over the conventional DLVO model of the double-layer forces. In particular, Flhydr takes into account the effect of the ionic finite volume. In addition, Ilcor, accounts for the non-ideality of the electrolyte solutions, which is caused by the long-range electric forces

between the ions. The total surface force, due to the overlap of electric double layers, is equal to the sum of the aforementioned three contributions: -I el t~ = 1-Iel + l-lhydr + I-Icor

(5.104)

Note that in view of Eq. (5.89) and (5.103) one obtains I-Id + I-Icor= (1 + Acor)FIel = (1 + Acor)Cexp(-tch) = C exp(-tch) where C is a "renormalized" pre-exponential factor. In practice C is determined from the experimental fits and it is often identified with the pre-exponential factor in Eq. (5.93). Thus an apparent (lower) value of the surface potential ~ is determined neglecting the effect of the ionic correlations. Of course, this would be correct if IAco~l << 1. It turns out that the contribution of the ionic correlations can be detected if only an independently determined value

224

Chapter 5

of ~ is available. However, in the case of strong ionic correlations one could have 1 + Acor < 0, that is Hcor/Hel < - 1 in Fig. 5.11; in such a case the net interaction between similar surfaces would become attractive and the effect of I-Icor could not be misinterpreted as He~ at lower surface potential.

5.2.7.

OSCILLATOR Y STRUCTURAL AND DEPLETION FORCES

Oscillatory structural forces are observed in two cases: (i) In very thin liquid films (h < 5 nm) between two molecularly smooth solid surfaces; in this case the period of oscillations is of the order of the diameter of the solvent molecules. These, so called solvation forces [3, 40], could be important for the short-range interactions between solid particles in dispersions. (ii) In thin liquid films containing colloidal particles (including surfactant micelles, protein globules, latex beads); in this case the period of the oscillatory force is close to the diameter of the colloid particles, see Fig. 5.12. At higher particle concentrations these colloid structural

forces stabilize the liquid films and colloids [132-135]. At lower particle concentrations the structural forces degenerate into the so called depletion attraction, which is found to destabilize various dispersions [ 136-138]. In all cases, the oscillations decay with the increase of the film thickness; in the experiment one rarely detects more than 8-9 oscillations.

Physical origin of the oscillatory force. The oscillatory structural force appears when monodisperse spherical (in some cases ellipsoidal or cylindrical) particles are confined between the two surfaces of a thin film. Even one "hard wall" can induce ordering among the neighboring molecules. The oscillatory structural force is a result of overlap of the structured zones formed at two approaching surfaces, see Fig. 5.13 and Refs. [3,139-141 ]. A wall can induce structuring in the neighboring fluid only if the magnitude of the surface roughness is negligible compared with the particle diameter, d. If surface irregularities are present (say a rough solid surface), the oscillations are smeared out and oscillatory structural force does not appear. If the film surfaces are fluid, the role of the surface roughness is played

Liquid Films and Interactions between Particle and Surface

100

.

.

.

.

i

.

.

.

.

i

.

.

.

.

90

i

"

.

.

.

.

i

.

.

.

225

.

i

.

.

.

,

.

.

.

.

i

.

.

.

.

|

,

,

"

~:~i--'_~+, .g;~.~~, ..~%_i+"-.-',,~:_-4_:k~_~. , .

80 E

.

% :,S %~.;~i~-.,~'%~,.[+~ "!-',../;~.~"water

70

e-

o"

ra ffl

50

z

40

o ~

1-

~x

)o

ao 20 10 .

.

.

.

.

.

.

.

20

.

.

.

.

.

.

.

40

.

.

.

.

,

60

. . . .

80

,

100

. . . .

. . . . . .

120

140

160

TIME, seconds

Fig. 5.12. Experimental curve: thickness of an emulsion film, h, vs. time; the step-wise thinning of the film is clearly seen. The film is formed from micellar aqueous solution of the ionic surfactant sodium nonylphenol-polyoxyethylene-25 sulfate with 0.1 M NaC1; the height of a step is close to the micelle hydrodynamic diameter. The steps represent metastable states corresponding to different number micelle layers inside the film, see the inset; data from Marinova et al. [149]. by the interfacial fluctuation capillary waves, whose amplitude ( 1 - 5 / k ) is comparable with the diameter of the solvent molecules. Structural forces in foam or emulsion films appear if the diameter of the colloidal particles is much larger than the amplitude of the surface corrugations. Surfactant micelles can play the role of such particles; in fact the manifestation of colloid structural forces was first observed with foam films formed from micellar surfactant solutions. Johnott [142] and Perrin [143] observed that the thickness of foam films decreases with several step-wise transitions. This phenomenon was called "stratification". Bruil and Lyklema [144] and Friberg et al. [145] studied systematically the effect of ionic surfactant and electrolyte on the occurrence of the step-wise transitions. Keuskamp and Lyklema [146] suggested that some oscillatory interaction between the film surfaces must be responsible for the observed phenomenon. Kruglyakov et al. [147, 148] and Marinova et al. [149] observed stratification with emulsion films, see Fig. 5.12. Stepwise structuring of colloidal particles has been observed also in wetting films (with one solid surface) [150].

Chapter 5

226 a

b

c

d

h=O h=d

e

f

h=2d

g

h=3d

(a) I-Iosc

b A d il/5 ~/ /

Repulsive f [-[=Pc Attractive

I

1

I

2

I

3

I

4

I

5

h/d

(b) Fig. 5.13. (a) From right to the left: consecutive stages of thinning of a liquid film containing spherical particles of diameter d. (b) Schematic plot of the oscillatory-structural component of disjoining pressure, FI.... vs. the film thickness h. The metastable states of the film (the steps in Fig. 5.12) correspond to the intersection points of the oscillatory curve with the horizontal line FI = Pc, see Eq. (5.3). The stable branches of the oscillatory curve are those with ~I-I/Oh < 0; see Ref. [3] for details.

As a first guess, it has been suggested [ 148, 151 ] that a possible explanation of the phenomenon can be the formation of surfactant lamella liquid-crystal structure inside the film. Such lamellar micelles are observed to form in surfactant solutions, however, at concentrations much higher than those used in the experiments with stratifying films. The latter fact makes the explanation with lamella liquid crystal irrelevant. Nikolov et al. [41,132-135] observed stratification not only with micellar surfactant solutions but also with suspensions of latex particles of micellar size. The step-wise changes in the film thickness were approximately equal to the diameter of the spherical particles, contained in the foam film. The observed multiple step-wise decrease of the film thickness (see Fig. 5.12) was attributed to the layer-by-layer thinning of a colloidcrystal-like structure from spherical particles inside the film, which is manifested by the appearance of an oscillatory structural force [133]. The metastable states of the film (the steps) correspond to the roots of the equation 1-I(h)= Pc for the stable oscillatory branches with

Liquid Films and Interactions between Particle and Surface

227

OH~Oh <0; in Fig. 5.13 there are three such roots; cf. Figs. 5.3 and 5.13; Pc is the applied capillary pressure. The mechanism of stratification was studied theoretically in Ref. [ 152], where the appearance and expansion of black spots in horizontal stratifying films was described as a process of condensation of vacancies in a colloid crystal of ordered particles within the film. This mechanism was confirmed by subsequent experimental studies with casein submicelles and silica particles [153,154]. Additional studies with vertical liquid films containing latex particles indicated that the packing of the structured particles is hexagonal [ 155]. The stable branches of the oscillatory disjoining pressure isotherm were experimentally detected for films from micellar solutions by Bergeron and Radke [156]. Oscillatory structural forces due to micelles and microemulsion droplets were directly measured by means of a surface force balance [157, 158]. Static and dynamic light scattering methods were also applied to investigate the micelle structuring in stratifying films [ 159]. Theoretical expressions for the oscillatory forces. As already mentioned, the period of the oscillations is close to the particle diameter. In this respect the structural forces are appropriately called the "volume exclusion forces" by Henderson [160], who derived an explicit (though rather complex) analytical formula for calculating these forces. Modeling by means of the integral equations of statistical mechanics [161-164] and numerical simulations [165-167] of the oscillatory force of the step-wise film thinning are also available. A convenient semiempirical formula for the oscillatory structural component of disjoining pressure was proposed [168]

Hosc (h)= P0

~d 1

exp d.d2

(5.~05)

d2

= - Po

for O
where d is the diameter of the hard spheres, dl and d2 are the period and the decay length of the oscillations which are related to the particle volume fraction, qg, as follows [168]

d

+ 0.237 Aq9 + 0.633.A(p. 2"( )

d2 0.4866 . . . . d Aq)

0.420

(5.106)

Chapter 5

228

1

2

3

4

5

h/d

Fig. 5.14. Plot of the dimensionless oscillatory disjoining pressure, 1-loscd3/kT, vs. the dimensionless film thickness h/d for volume fraction q9= 0.357 of the particles in the bulk suspension. The solid curve is calculated from Eq. (5.105), the dotted curve - from the theory by Henderson [160], the dashed curve is from Ref. [162] and the x-points - from Ref. [165]; after Ref. [168].

Here Aq9 = qgmax- q9 with (Pmaxbeing the value of q9 at close packing: qgmax= *c/(3-~ ) = 0.74. P0 is the particle osmotic pressure determined by means of the Carnahan-Starling formula [ 169]

Po - n k T

1+ q~ + q92 -q9 3 ( 1 - q~)3 '

6q9 n - red3,

(5.107)

where n is the particle number density. For h < d, when the particles are expelled from the slit into the neighboring bulk suspension, Eq. (5.105) describes the so called depletion attraction, see the first minimum in Fig. 5.13. On the other hand, for h > d the structural disjoining pressure oscillates around P0, defined by Eq. 5.107. As seen in Fig. 5.14, the quantitative predictions of Eq. (5.105) compare well with the Henderson theory [160] as well as with numerical results Kjellander and Sarman [162] and Karlstr6m [165]. It is interesting to note that in oscillatory regime the concentration dependence of Ilosc is dominated by the decay length de in the exponent, cf. Eq. (5.106). Roughly speaking, for a

Liquid Films and Interactions between Particle and Surface

229

given distance h the oscillatory disjoining pressure Hosc increases five times when tp is increased with 10%, see Ref. [ 168]. The contribution of the oscillatory structural forces to the interaction free energy per unit area of the film can be obtained by integrating Hosc in accordance with Eq. (5.9): oo

fosc(h)- II-Iosc (h')dh'- F(h~

for h _>d

h

=F(d)-Po(d-h ),

for

F(h)- P~

ONhNd

[ d2

(5.108)

[ dl

If the colloidal particles are not real hard spheres, then their effective hard-core diameter can be estimated from the formula [134] (5.109)

d= [3 fl2/(4~)] 1/3

Where/32 is the second virial coefficient in the virial expansion of the osmotic pressure due to the particles,

Posm/(nkT)= 1 + flzn/2 +

...; ,62 can be determined by static light scattering [170].

In the case of electrically charged particles the effective diameter can be estimated from the expression [ 132] d

= dH +

2U 1

(5.110)

where dn is the hydrodynamic diameter of the colloid particles, which can be determined by dynamic light scattering [ 171 ].

Depletion interaction. With

the decrease of particle (micelle) volume fraction tp the

amplitude of the oscillations decreases and the oscillatory structural force degenerates into the depletion interaction (only the first minimum in the oscillatory curve in Fig. 5.13b remains). The latter interaction manifested itself in the experiments by Bondy [172], who observed coagulation of rubber latex in presence of polymer molecules in the disperse medium. In the case of plane-parallel films the depletion component of disjoining pressure is

Chapter 5

230 ...

I

...

....

%

m

..

J"

%

I

!

I

I i

., ..,

_

h0

I I

I \

/ %

%

j

Fig. 5.15. Schematic presentation of the overlap of the depletion zones around two larger particles of diameter dL separated at a distance h0. The centers of the smaller particles (of diameter d) cannot come closer to the larger particles than the circumferences denoted with dashed line.

]-I dep ( h ) -

-

Fidep(h)= 0

Po

for hd

(5.111)

which is a special case of Eq. (5.105) for small d2, see Ref. [3] for details. Evans and Needham [173] to measured the depletion energy of two interacting bilayer surfaces in a concentrated Dextran solution; their results confirm the validity of Eq. (5.111). Asakura and Oosawa [ 136, 137] published a theory, which attributed the observed interparticle attraction to the overlap of the depletion layers at the surfaces of two approaching larger colloid particles, see Fig. 5.15. The centers of the smaller particles, of diameter, d, cannot approach the surface of a larger particle (of diameter dD at a distance shorter than d/2, which is the thickness of the depletion layer. When the two depletion layers overlap (Fig. 5.15) some volume between the large particles becomes inaccessible for the smaller particles. This gives rise to an osmotic pressure, which tends to suck out the solvent between the larger particles thus forcing them against each other. The total depletion force experienced by one of the larger particles is [136, 137] Foep - - kTnS(h o )

(5.112)

Liquid Films and Interactions between Particle and Surface

231

where the effective depletion area is S (h0 ) - ~ ( 2 d L + d + h 0 )(d - h 0 ) for 0 < h 0
(5.113)

Here h0 is the shortest distance between the surfaces of the larger particles and n is the number density of the smaller particles. By integrating Eq. (5.112) one can derive an expression for the depletion interaction energy between the two larger particles, gdep(h0).

For dL >> d this

expression reads U d e p ( h o ) / k T ~ ---j3 q) --~(d d L - ho

)2 ,

O < ho < d

(5.114)

where q~ = 7cnd3/6 is the volume fraction of the small particles. The maximum value of Uaep at h0 = 0 is gdep(O)/kr -~ -3(pdL/(2d).

-7.5 kT.

For example, if dL/d = 50 and (p = 0.1, then gdep(0) =

This depletion attraction turns out to be large enough to cause flocculation in

dispersions and attachment of particles to surfaces [42, 138,174-177], as well as attraction between lipid bilayers [ 178, 179].

5.2.8.

STERIC INTERACTION DUE TO ADSORBED MOLECULAR CHAINS

The adsorption of polymeric molecules at an interface may lead to the appearance of polymeric brushes, see Fig. 5.16. The polymers could be attached to the surface by chemical bonding

(chemisorption, anchoring) of some groups. Alternatively, a polymeric coverage of the surface can be achieved by physical adsorption of nonionic surfactants, whose hydrophilic moieties represent water-soluble polymers (typically- polyoxyethylenes). When two surfaces covered with polymer brushes approach each other, the overlap of the brushes gives rise to a strong osmotic repulsion, which protects the surfaces from adhesion and the colloidal dispersions from coagulation [3, 42, 180-182]. This steric repulsion plays a role similar to that of the electrostatic repulsion with respect to colloid stability. Thickness of a separate polymeric brush. Obviously, the range of action of the steric

repulsion is determined by the thickness of the brush, L. The latter can be defined as the meansquare end-to-end distance of the hydrophilic portion of the chain. If a chain, composed of N

232

Chapter 5

L

Brush

Fig. 5.16. Polymeric chains adsorbed at an interface form a "brush" of thickness L. The overlap of the brushes formed at the two surfaces of a thin liquid film gives rise to a steric interaction.

segments, were completely extended, then L would be equal to lN, where 1 is the length of a polymeric segment. However, due to the Brownian motion L < lN. For an isolated anchored chain in an ideal (theta) solvent L can be estimated as [42] L -- L 0 - l ~ -

(5.115)

The solvent-polymer interactions may essentially influence L, and therefrom - the steric interaction. The osmotic pressure of either dilute or concentrated polymer solutions can be expressed in the form [183] Posm = 1 ~ --nv 1 + ln2 w nkT N 2 3

q-...

(5.116)

Here n is the number segment density, v and w account for the pair and triplet interactions between segments. In fact v and w are counterparts of the second and third virial coefficients in the theory of non-ideal gases [ 114]. v and w can be calculated if information about the polymer chain and the solvent is available [42] w 1/2

--

~ m / N A,

v - wl/2 (1 - 2Z),

(5.117)

where ~ (m3/kg) is the specific volume per segment, m (kg/mol) is the molecular weight per segment, NA is the Avogadro number and Z is the Flory parameter [ 114]: c Z ---~(ueP

+Uss - 2 U e s )

(5.118)

c is the number of the closest neighbors of a molecule, Ua8 stands for the energy of interaction

Liquid Films and Interactions between Particle and Surface

233

between molecules type "A" and "B" (A,B = P,S; "P" = polymeric segment, "S" = solvent molecule). The parameter v can be zero (see Eq. 5.117) for some special temperature, called the

theta temperature. The solvent at theta temperature is known as theta solvent or ideal solvent. At the theta temperature the intermolecular (intersegment) attraction and repulsion in polymer solutions are exactly counterbalanced. In a good solvent, however, the repulsion prevails over the attraction and v > 0. In contrast, in a poor solvent the intersegment attraction prevails, so v<0. In a good solvent L > L0, whereas in a poor solvent L < L0. In addition, L depends on the surface concentration, F, of the adsorbed chains, i.e. L is different for an isolated molecule and for a brush. If the segments repel each other, larger 1-" leads to greater L. The mean field approach [42, 184], applied to polymer solutions, provides the following equation for calculating L:

~3

/

1+

-

9

--v

(5.119)

6

where L, F and g are the dimensionless values of L, F and v defined as follows:

L,=L/(I~ t

I'=FN~-ww/1,

~'=vFN 3/2/l

(5.120)

For an isolated adsorbed molecule (1-'= 0) in an ideal solvent (~ = 0) Eq. (5.119) predicts L = 1, that is L = L0.

Interaction between two overlapping polymeric brushes. As already mentioned, the major source of the steric repulsion between brushes (Fig. 5.16) is the increased osmotic pressure in the zone of overlap. However, two other factors tend to reduce the osmotic repulsion. (i) In a poor solvent the segments of the chain molecules attract each other; hence the overlap of the two approaching layers of polymer molecules will be accompanied with the appearance of more intersegment contacts which will decrease the free energy of the system. The latter effect could sometimes prevail over the osmotic repulsion in the case of small overlap (two brushes just touching each other). However, in the case of larger overlap (smaller h) the osmotic repulsion becomes predominant, see Fig. 5.17. (ii) Due to the repulsive

234

Chapter 5

.5--

"

I I

I

I

I

I

I

I

/ J

C k

i i

.0--

"

.5--

-

d,i

15~ t"q

ts

II

0.0

~I

~rl

~D O

-0.5 -

-1.0

0

1

2

3

I

I

I

4

5

6

Fig. 5.17. Experimental data for the steric interaction obtained by surface force apparatus. Plot of F/R - 2rtf vs. h for two surfaces covered with adsorption monolayers from the nonionic surfactant C12(EO)6 for ~ various temperatures. The appearance of - minima indicates that with the increase of temperature the water becomes a poor solvent for the polyoxyethylene chains. (From Ref. 190.) 7

Distance h (nm)

interactions with the chains of neighboring molecules in the brush each polymeric chain is subjected to an extension (L > L0, see Eq. 5.119), which produces an extra elastic stress. This elastic stress can be partially released when two such monolayers are pressed against each other. As a result, the free energy of the system decreases, which is equivalent to an effective attractive contribution to the net steric surface force, which will be briefly termed elastic attraction.

Dolan and Edwards [185] calculated the steric interaction free energy per unit area, f~t, for two polymeric adsorption monolayers in an ideal s o l v e n t as a function of the film thickness, h,:

-

t-~--~5-

for h < Lo4~J"

(5.121)

Liquid Films and Interactions between Particle and Surface

235

3h2/

fst(h)=4FkTexp - 2---~0

for 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 << 2,, we have I'Iprotr oc h -l, that is I-[protr is divergent for h--> 0. Integrating Eq. (5.125) in accordance with Eq. (5.9) one obtains a relatively compact expression for the free energy of the protrusion interaction per unit area: cx~

fprotr(h) = IIIprotr (]~)d/~ = -FkTln[1 - (1 + h/,a,)exp(-h/,a,)]

(5.126)

h

5.2.10. FORCES DUE TO DEFORMATION OF LIQUID DROPS

Effect of the interfacial dilatation.

In the course of collision of a liquid drop with a

solid surface (Fig. 5.19a) or with another drop (Fig. 5.19b) interfacial deformations may happen. We assume that before the collision the fluid particle is sphere of radius R. When the surface-to-surface distance h is sufficiently small, a flattening (a film of radius rc) could appear in the zone of contact. This deviation from the spherical shape causes a dilatation of the surface of the fluid particle; the respective increase of the surface energy can be deduced as follows. For small dilatation, o~- dA/A, the surface tension can be expanded in series: o" = o'0 + Eccz + -.. where o'0 is the surface tension of the non-deformed drop, A and Ec denote area and Gibbs elasticity, see Eq. (1.145). Then the work of dilatation per unit area is

238

Chapter 5

ht

"

(b

(a)

Fig. 5.19. Deformation of interacting emulsion drops: (a) drop colliding with a solid surface; (b) central collision of two drops; R is the radius of the spherical part of the drop; rc is the radius of the contact line at the boundary film-meniscus.

a Wdil : I (GO "k- E c a ) d a

o

AA E c ( A A ] 2 = (7o ~ +

Ao T

Z-

(5.127)

where A0 is the area if the nondeformed (spherical) drop and AA = A - A0 is the increase of its area upon deformation. The total energy of surface dilatation is [37]

Udil "" Ao Wdil

,

=

j O'oAA + -~ AoE G

(5.128)

where j = 1,2 for the system depicted in Fig. 5.19a and 5.19b, respectively. Further, the surface area and the volume of a deformed drop are A =/lTR 2 (2 + 2 coso~ + sin 2 o~)

(5.129)

V - ~ R 3 ( 2 + 2coso~ +sin 2 o~cos~) 3

(5.130)

where sina = rJR, see Fig. 5.19 for the notation. We consider small deformations at fixed volume of the drop; then using series expansion in Eq. (5.130) for e = sin2o~ << 1 and fixed V one can derive

Liquid Films and Interactions between Particle and Surface

R 2 - R~ 1 + - - + - . . 8

,

239

R0 -

(5.131)

Substituting Eq. (5.131) into Eq. (5.129) and expanding again in series for small e one obtains

a-ao + =4 Rg

-ao +ao

(ao =4 Ro )

(5.132)

Finally, combining Eqs. (5.128) and (5.132) one deduces [37] 9

Udi , -

4

rC

/rj

r,8

cro--RT + ~ - ~ E c - - ~ + . . . ,

(rJR) 2 << 1

(5.133)

Calculations for typical emulsion systems show that the condition (rflR) 2 << 1 is always satisfied and Eq. (5.127) holds with a good precision. The contribution of Ec to U~il, that is the last term in Eq. (5.133) is usually an effect of higher order and can be neglected. However, for microemulsion drops the surface tension is rather low, (3" << 1 raN/m, and then the term with E c in Eq. (5.133) may become significant. On the other hand, if there is no adsorbed surfactant on the drop surface, then Ec = 0. In all c a s e s Udil > 0, i.e. the interfacial dilatation gives rise to an effective repulsion between the two droplets. Equation (5.133) predicts that Udil strongly increases with the rise of the film radius rc.

Effect of interfacial bending. The flattening of the drop surface in the zone of contact (Fig. 5.19) is accompanied by change in the interfacial bending energy, Ubend. In Section 3.3.2 we have considered in detail this effect; in agreement with Eq. (3.96) the energy of interfacial bending is [39]: Ubend --

-j~rcZBo/R,

(rc/R) 2 << 1

(5.134)

where B0 is the interfacial bending moment, and j is the same as in Eq. (5.128). Note that B0=-4kcH0, where/4o is the spontaneous curvature and kc is the interfacial bending elastic modulus. As discussed in Section 3.3.2, for oil-in-water (O/W) emulsions Ubend > 0 and, consequently, the interfacial bending is energetically unfavorable. However, for water-in-oil (W/O) emulsions Ubend< 0, which favors the flattening [39].

Chapter 5

240 5.3.

SUMMARY

The act of collision of a colloid particle with an interface (or with another particle) is accompanied by the formation of a thin liquid film between the two approaching surfaces. If attractive forces prevail in the liquid film, the latter becomes unstable, breaks and the particle "enters" the interface (or coalesces with the other particle). Conversely, if the repulsive forces are predominant, the particle will rebound from the interface and there will be no attachment/ coagulation. A third possibility is the attractive and repulsive forces to counterbalance each other; in such case an equilibrium film is formed between the particle and the interface (the other particle). The balance of all forces exerted on an attached particle is considered, see Fig. 5.2. It turns out that at equilibrium the repulsive forces dominate the disjoining pressure I-I, which is counterbalanced by the action of transversal tension T, the latter being dominated by the attractive forces in the transition zone film-meniscus, see Eq. (5.13). Thermodynamic relationships of the latter quantities with the contact angle are derived. Next we consider the Derjaguin approximation, which allows one to calculate the interaction across a film of nonuniform thickness if the interaction energy per unit area of a plane-parallel film is known, see Eq. (5.46). Further we consider interactions of different physical origin in thin liquid films. Expressions for the van der Waals interaction between particles / interfaces of various shapes are presented, Eqs. (5.57)-(5.64). Equations for calculating the Hamaker constant are reviewed, see Eqs. (5.65)-(5.78). Hypotheses about the nature of the long-range hydrophobic surface force and its physical significance are discussed in Section 5.2.3. Special attention is paid to the electrostatic

surface force which is due to the overlap of the electric double layers formed at the charged surfaces of an aqueous film, Section 5.2.4. A more detailed theory of the interactions across films from electrolyte solutions should take into account also the hydration repulsion (Section 5.2.5) and ion-correlation attraction (Section 5.2.6). The presence of fine colloidal particles (surfactant micelles, protein globules, etc.) in a liquid film gives rise to an oscillatory structural

force, which could stabilize the film or cause its step-wise thinning (stratification), Section 5.2.7. At low volume fractions of the fine particles the oscillatory force degenerates to the

depletion attraction, which leads to particle

attachment

and flocculation;

see

Eqs.

Liquid Films and Interactions between Particle and Surface

241

(5.111)-(5.114). Adsorbed polymeric molecules form "brushes" at the film surfaces; the overlap of such "brushes" in the course of film thinning produce a steric interaction. The latter can be repulsive or attractive depending on whether the water is good or poor solvent for the polymeric chains, see Section 5.2.8. Similar steric-osmotic effects appears when configurational confinement of thermally excited surface modes takes place in a thin liquid film. The bending mode of surface fluctuations gives rise to the undulation force, whereas the discrete protrusions of adsorbed amphiphilic molecules lead to the appearance of a short-range protrusion force, both of them being repulsive, see Section 5.2.9. Finally, the collisions of

emulsion drops are accompanied with deformation, i.e. deviation from the spherical shape, see Fig. 5.19. This causes extension of the drop surface area and change in the surface curvature, which lead to dilatational and bending contributions to the overall interaction energy, see Eqs. (5.133) and (5.134). The total particle-surface (or particle-particle) interaction energy is a superposition of contributions from all operative surface interactions, see Eq. (5.53). Another important contribution to the particle-surface and particle-particle interactions stems from the viscous friction in the thin liquid films. This hydrodynamic interaction is considered in the next Chapter 6.

5.4.

1.

REFERENCES

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248

CHAPTER 6 PARTICLES AT INTERFACES: DEFORMATIONS AND HYDRODYNAMIC INTERACTIONS Here we consider some aspects of the interaction of colloidal particles with a phase boundary, which involve deformations of a fluid interface and/or hydrodynamic flows. First, we discuss the energy changes accompanying the collision of a fluid particle (emulsion drop of gas bubble) with an interface or another particle. If the interaction is governed by the surface dilatation and the DLVO forces, the energy of the system may exhibit a minimum, which corresponds to the formation of a floc of two attached fluid particles. If oscillatory-structural forces are operative, then the energy surface exhibits a series of minima separated by barriers, whose physical importance is discussed. The radius of the liquid film formed between a fluid particle and an interface can be determined by means of force balance considerations. For small contact angles the film radius is proportional to the squared radius of the particle. Next we consider the hydrodynamic interactions of a colloidal particle with an interface (or another particle), which are due to flows in the viscous liquid medium. The theory relates the velocity of mutual approach of the two surfaces with the driving force. The respective relationships depend on the shape of the particle, its deformability and surface mobility. The gradual 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; the comparison of theory and experiment is discussed. Finally, we consider the factors and mechanisms for detachment of an oil drop from a solid surface in relation to the process of washing. The destabilization of the oil-water interface and of the three-phase contact line are known as, respectively, "emulsification" and "rolling-up" mechanisms of drop removal. Some surfactants are able to produce penetration of aqueous films between oil and solid, which is a purely physicochemical "disjoining film" mechanism for dr~,p detachment. Attention is paid to the detachment of oil drops from the orifice of a pore in a.latioll to the methods of emulsification by ceramic and glass membranes.

Particles at Interfaces: Deformations and Hydrodynamic Interactions

6.1.

DEFORMATION OF FLUID PARTICLES APPROACHING AN INTERFACE

6.1. l.

THERMODYNAMIC ASPECTS OF PARTICLE DEFORMATION

249

As demonstrated in Section 5.2.10, the deformation of a droplet at fixed volume leads to an expansion of its surface area, Eq. (5.133). In addition, the flattening of the droplet surfaces in the zone of their contact is accompanied with a variation of the interfacial bending energy of the droplets, Eq. (5.134). Last but not least, the formation of a thin liquid film between the two drops much enhances the role of the surface forces, such as the van der Waals attraction, electrostatic repulsion, oscillatory structural forces, steric interactions, etc., see Section 5.2. In Ref. [ 1] it was demonstrated that the energy of interaction between two fluid particles (drops or bubbles) calculated for the model shape of truncated spheres (Fig. 5.19) quantitatively agrees very well with the energy calculated by means of the "real profile", i.e. by accounting for the transition zone between the flat film and the spherical portions of the drop surfaces. Therefore, below we will use the configuration of truncated spheres. Equation (5.50) with h0 - h reads:

Sf(h)dh +zrf S(h)

oo

U(h, rc)-(2zR/j)

(6.1)

h

where j = 1,2 for the systems depicted in Fig. 5.19a and 5.19b, respectively. One sees that the energy of interaction between two deformed fluid particles, U, depends on two geometrical parameters, the film thickness, h, and the film radius, rc. However, it is natural to present the interaction energy as a function of a single parameter, which can be the distance z between the droplets' mass centers, i.e. U =

U(z). In the rigorous approach to this problem, the dependence

of the interaction energy on the distance z is characterized by the potential of the mean force,

Umf(Z) = -kTlng(z), where, as usual, k is the Boltzmann constant, T is temperature, and g(z) is the pair (radial) correlation function, see Ref. [2]. The latter function is determined by statistical averaging over all possible droplet configurations (of various h and rc) corresponding to a given z:

Chapter 6

250

)1/4 g(z)-1.103

~R2cr

2kT

1

-R ~exp{-U[h(rc'z)'rcl/kT}drc

(6.2)

Here R and • are the radius and the interfacial tension of the fluid particle; h(rc,z) represents the geometrical relation between h and rc for a given z and fixed drop volume. To calculate UmC(Z) one needs to know the function U = U(h, rc), which may contain contributions due to the various effects mentioned in the beginning of this section. As an illustration, let us consider the function U(h, rc) in a typical case, in which the interaction energy between two identical emulsion drops (Fig. 5.19b) is determined by the van der Waals attraction, the electrostatic repulsion and the interfacial dilatation:

U(h, rc) - Uvw + Uel + Udil

(6.3)

Here Uvw and Uaii are determined by Eqs. (5.64) and (5.133). To obtain an expression for Uej one can substitute

I-[el(h) from

Eq. (5.93) into Eq. (5.9), and then the calculated f e l - into Eq.

(6.1) withj = 2; the result reads [1-3]

rc 64 n~

~"

4kT

E

exp(-tch)rc2+

9

28 2e

ekT

2n~

(6.4)

where U~ is the Debye screening length, no is the concentration of a symmetric Z:Z electrolyte, e denotes the dielectric permittivity; ~ is the surface potential of the particle. Figure 6.1 shows a contour plot of U(h, rc), calculated by means of Eq. (6.3) with parameter values R= 1 gm, gts= 100 mV, if= 1 mN/m, no = 0.1 M and Hamaker constant AH = 2 x 10-2o J; the term with the Gibbs elasticity E~ in Eq. (5.133) is neglected. The minimum of the potential surface U(h,r,:) corresponds to an equilibrium doublet of two attached drops with a film formed between them; the thickness and the radius of this film will be denoted by depth of the minimum in Fig. 6.1a is

g(heq, rc,eq) = -60kZ.

heq

and

rc,eq. The

Hence, the equilibrium doublet

should be rather stable. The numerical computations [2] show that the radius of the equilibrium

film rc,eq,and the area of attachment, increases with the rise of both electrolyte concentration no and drop radius R.

Particles at Interfaces." Deformations and Hydrodynamic Interactions

0.01 ~

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

100

.

o

251

,

1

..,

,

75

% ~

5o

i

O.007U

25

0.006

.

0.1 0.02

0.04

0.06

0.08

.

.

.

0.2

0.3

0.4

0.5

no (M)

O. 1

rc,/R (a)

(b)

Fig. 6.1. (a) Contour plot of the total drop-drop interaction energy, U(h, rc) = Uvw + Ue~ + Udil for various values of h/R and rc/R, see Fig. 5.19b. The parameter values are: R = 1 ~m, ~ =100 mV, cro = 1 mN/m, no = 0.1 M, AH = 2 • 10-20 J. The distance between two neighboring contours equals 2 kT; the minimum of the potential surface is U(heq,rc,eq) = -60 kT. (b) Plot of AUeq vs. electrolyte concentration no for AH = 1 • 10-20 J and three values of the drop radius: R = 0.5, 1.0 and 2.0 ~tm for the dashed, continuous and dotted line, respectively [4,5]. Let U(h*,0) be the minimum value of U along the ordinate axis rc = 0 in Fig. 6.1 a; the points on this axis correspond to two spherical (non-deformed) drops. Figure 6. l b shows the calculated dependence of AUeq - U(heq, rc,eq) - U(h*,O) on the bulk electrolyte concentration no for three different values of the drop radius R. In fact, AUeq characterizes the gain of energy due to the transition from two interacting spherical drops to two deformed drops (Fig. 5.19b). This energy gain is due to the interactions of the two drops across the formed film; see the term rcr~Zf(h) in Eq. (6.1); note that at equilibrium f(h) < 0, cf. Eq. (5.10). Figure 6. l b shows that the effect of deformation, characterized by AUeq, strongly increases with the rise of no and R; this can be attributed to suppression of the electrostatic repulsion and enlargement of the contact area.

Effect of the oscillatory structural force. Very often the fluid dispersions contain small colloidal particles (such as surfactant micelles or protein globules) in the continuous phase. As described in Section 5.2.7, the presence of these small particles gives rise to an oscillatory structural force, which affects the stability of foam and emulsion films as well as the flocculation processes in various colloids. At higher particle concentrations (volume fractions

Chapter 6

252

0.014

000000000 00000 00000000000000

9 ~ .................

0.012

0

0.01

o. oo8

i

-

o . ~ .............

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

o o o o o o o o o o o o d o

0.006

. . . . . . . . . . .

0.004 0

0.02

0.04

0.06

JO

0.08

rc/R Fig 6.2. Contour plot of the energy, U(h,rc) = Uvw + Ue~ + Udil 4- Uosc, between two oil drops of radius R = 2 gm in the presence of ionic micelles in water. The parameters correspond to a micellar solution of SNP-25S, see Fig. 5.12: d = 9.8 nm, q~ = 0.38, o" = 7.5 mN/m, AI-I = 5 x 10-21 J, gts =-135 mV, no = 25 mM, tr-1 = 1.91 nm. The points on the contour plot denote three local minima: U/kT = - 406; -140, and -37, corresponding to film containing 0, 1 and 2 micellar layers, respectively [4,5]. above c.a. 15 %) the structural forces stabilize the liquid films and emulsions. At lower particle concentrations the structural forces degenerate into the depletion attraction, which is found to have a destabilizing effect. To quantify the contribution of the oscillatory forces, a respective term, Uosc, is to be included in the expression for the interaction energy:

U(h,rc) = Uvw + Uel + Udil-k- Uosc

(6.5)

cf. Eq. (6.3). Uosc can be calculated by a substitution offosc(h) from Eq. (5.108) into Eq. (6.1); the other terms in the right-hand side of Eq. (6.5) are determined as explained above. Figure 6.2 shows a contour plot of U (h, rc), which is similar to Fig. 6.1a, but computed by means of Eq. (6.5). The oscillatory term Uosc leads to the appearance of several local minima separated by "mountain ranges". If the particle volume fraction is smaller than c.a. 10% in the continuous phase, the height of the taller "range" is smaller than kT, and it cannot prevent the flocculation of two droplets in the deep "depletion" minimum - the deepest minimum (down right in Fig. 6.2). On the other hand, at higher micellar volume fraction these "ranges" become

Particles at Interfaces." Deformations and Hydrodynamic Interactions

E E

-r a_

180

140

00

D O

'

::........ ~ .

120

::

~

;

' 22.3 rnM

C, ............. i-O ...... ~ ....... ! O . " ~ ........... .... " : ................. i ..................... [ ..... . ...... ]~...

~

.

~;~"~-]-i~

253

.

.

&-,Ak

i ....... 9 k

::-A--" i . . . . i k "

jet

.

.

.

. . . .

.-

,

] .

.

33. .... A--~--A---{ .

I .... l l

:: . 6 7 m M .-

.......

i .<> ~

80---'

~

2o

(-9

0

"I'Lu

.

.....

0

I' .

.

.

.....

.

.

.

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

60

120

.

.

.

.

.... ------i ...............

180

.......... - . . . : .................... . . . . . . . . i......... . . . . . .

240

300

360

420

T I M E , rain

Fig. 6.3. Plot of the height of the water column, separated below a 20% styrene-in-water emulsion, as a function of time. The curves correspond to different surfactant (SNP-25S) concentrations, denoted in the figure, all of them above the CMC [5,6]. taller than kT and act like barriers against the closer approach and flocculation of the two fluid particles. In such case the oscillatory structural forces have a stabilizing effect, which could be a possible explanation of the experimental data shown in Fig. 6.3. The three curves in Fig. 6.3 correspond to three oil-in-water emulsions containing different concentrations of sodium nonylphenol polyoxyethylene-25 sulfate (SNP25S) in the aqueous phase, viz. 22.3, 33.5 and 67 mM, all of them much above (from 80 to 240 times) the critical micellization concentration, CMC = 0.28 mM. The height of the column of the aqueous phase, below the emulsion cream, is plotted in Fig. 6.3 as a function of time. The cream represents oil drops concentrated below the upper surface of the emulsion owing to the buoyancy force. The initial slope of the curves shows that the rate of water separation diminishes as the surfactant concentration increases. In addition, the more concentrated system finally produces a more

loosely packed cream (note the positions of the plateaus, Fig. 6.3), possibly due to hampered flocculation. One can attribute the observed effects to the oscillatory structural forces, which impede the flocculation of the emulsion drops and thus decelerate the separation driven by the Archimedes force; see Refs. [5-7] for more details. In conclusion, we have to mention that for each specified system an estimate should be done to

254

Chapter 6

reveal which of the terms in Eq. (5.53) are predominant, and which of them can be neglected; see also Eqs. (6.3) and (6.5). Similar approach can be applied to describe the multi-droplet interactions in flocs, because in most cases the interaction energy is pair-wise additive. Application of this approach to the description of the kinetics of simultaneous flocculation and coalescence in emulsions can be found in Ref. [8]. 6.1.2.

DEPENDENCE OF THE FILM AREA ON THE SIZE OF DROP~BUBBLE

In Figs. 6.1 a and 6.2 the equilibrium position of two attached fluid particles was determined as a minimum of the energy surface U(h,rc). Alternatively, it is possible to determine the radius, rc, of the equilibrium (or quasi-equilibrium) film with the help of macroscopic force balances. As an example, let us consider a fluid particle from phase 1, which approaches the boundary between the phases 2 and 3 driven by the buoyancy force; the drop is immersed in the heavier phase 3, see Fig. 6.4. For example, the fluid particle could be an oil drop or air bubble in water. For sufficiently short distance between the fluid particle and the interface a liquid film of uniform thickness h is formed; as a rule, h decreases slowly with time due to a viscous outflow of liquid, until eventually an equilibrium film is formed [9]. The pressures acting on the film surfaces are shown schematically in Fig. 6.4. At equilibrium the net force exerted on the particle should be equal to zero; this yields the following force balance, which is equivalent to Eq. (5.14): ~r

- 2lvr c "c cos0 + F b

(6.6)

P2 + Pf is the pressure inside the liquid film; ~"is the transversal tension; s i n 0 - rc]Rf where Rf is the curvature radius of the film surface; F a is the buoyancy (Archimedes) force, which is equal to the integral of the outer pressure, 4

3

F b - e~. ~ds P2(Z) - 7 7 C R o A p g "

Pz(Z)

- P2(0)

-

Apgz, over the surface of the particle:

(6.7)

at the last step, the Gauss-Ostrogradsky theorem has been applied; z is the vertical coordinate; Ap is the difference between the mass densities of phases 3 and 1; g is the acceleration due to gravity; R0 is the radius of the nondeformed (spherical) fluid particle. The transversal tension ~" accounts for the interaction in the transition zone film-meniscus and is related to the contact

Particlesat Interfaces:Deformationsand HydrodynamicInteractions

255

Phase 2 P2 P2

G2

,,,~

P2 R

41

Phase 3 Phase 1

Fig. 6.4. Sketch of a fluid particle from phase 1, which is attached to the boundary between phases 2 and 3; o-I and o'2 are the surface tensions of the boundaries 1/3 and 2/3; z"is transversal tension. The pressure balances at the two film surfaces illustrate the derivation of Eqs. (6.9) and (6.10). angle o~ by Eq. (5.6), viz. "c=a~ sina;

PT is

the excess pressure (with respect to the bulk

pressure P2) exerted on the film surface. In general

PT can

be presented as a sum of a viscous

and a disjoining pressure term:

Pf= Pvisc + H

(6.8)

In thick films the disjoining pressure Fl is negligible, and consequently

Pf-- Pvisc; in

thin quasi-equilibrium films the rate of thinning is low, Pvisc = 0 and then

Pf =

contrast, in

YI. However, at

any thickness the Laplace equation is satisfied for the lower and upper film surfaces (Fig. 6.4): 2o', = PI -(P2 +

Pr )- AP- PT

(6.9)

Rf 2~

=(Pz+PT)-Pz-Pu

(6.10)

Rf In Eqs. (6.9) and (6.10) the surface tensions of the liquid film are set approximately equal to the surface tensions of the boundaries between the respective bulk phases, o"f-- o'l and o"f = o'2, see Fig. 5.5. The pressure difference, across the drop surface, AP - P~ - P2, is 2al = AP R

(6.11)

Chapter 6

256

The three equations (6.9), (6.10) and (6.11) form a system for determining the three parameters,

RT, Pf and AP. The resulting expression for PT reads

~-

PU = mR,

where

_

20"10" 2

o" - cy~~+ cy2

(16.12)

Next, we substitute Eqs. (6.7) and (6.12) into Eq. (6.6) to obtain O" 27crczcosO- 74~R3A/9 g = 0 lrr~Z---t~-

(16.13)

we have set R = R0, which is a good approximation in the case of small deformations, that is

(rflR) 4 << 1" see Eq. (5.131). The solution of the quadratic equation (6.13) for rc reads

rc -

R f-- c o s 0

~

+

R2

R4 T_2 COS20 -k- 4Apg ~ 3~

/1/2 (6.14)

(the other root is physically meaningless). For small contact angles (z or sino~ << 1), Eq. (6.14) reduces to the simpler expression

r c - A R 2,

A = J~ 4Ap -~ g

(6.15)

If phases 1 and 2 are identicalfluids, then crl = 0"2 = o" and Eq. (6.12) yields ~-= o. In contrast, if phase 2 is solid and we deal with the configuration in Fig. 5.19a,

one may set o5 = or,

0"2--~ o% and then Eq. (6.12) yields ~-= 2o. Versions of Eq. (6.15) with ~ - - 2 o " have been derived by Derjaguin and Kussakov [10] and Allan et al. [11]. Expression equivalent to Eq. (6.15) follows from the theory of sessile drops/bubbles for zero contact angle measured across the outer phase (o~= 0); see Ref. [12] and Eq. (31)in Ref. [13]. Equation (6.15) holds irrespective of whether the pressure in the film, Pf, is dominated by the viscous pressure Pvi~ or disjoining pressure H. This equation has been derived neglecting the terms with the transversal tension z. The effect of z could show up for small drops (bubbles) for which the contribution of the buoyancy force (the term ~: R4 in Eq. 6.14) is vanishing. Hence, a deviation from the dependence rc = AR 2 for small drops could be interpreted as an effect of v, see Eq. (6.14), and could serve for determination of the contact angle, o~-- arcsin(v/G).

Particles at Interfaces: Deformations and Hydrodynamic Interactions

257

250.0

"E~ ,..? 'E

200.0

150.0 100.0

50.0

0,0

0.0

~

I

0.1

,

I

0.2

,

I

0.3

,

I

0.4

0.5

Squared drop radius, R2 (ram 2) Fig. 6.5. Experimental dependence of the film radius rc on the "equatorial" radius R of xylene oil drops situated below the phase boundary water-xylene, see Fig. 6.4. The straight line is the best fit with linear regression [ 14]. In Fig. 6.5 we present experimental data of Basheva [14] for small xylene drops; the system is similar to that in Fig. 6.4 where phases 1 and 2 are of xylene. Phase 3 is a 0.01 M solution of sodium dodecyl sulfate (SDS), preequilibrated with xylene. The data are plotted as rc vs. R 2 in accordance with Eq. (6.15). The density difference between water and xylene is A p = 0.12 g/cm3; the interfacial tension x y l e n e - aqueous solution is c r - 5.1 mN/m. With the latter parameter values from Eq. (6.15) one calculates A = 5.54 cm -l" on the other hand, the best linear fit of the data in Fig. 6.5 gives a slope A = 5.52 cm -l. Apparently, there is an excellent agreement between Eq. (6.15) and the experiment. One may check that the experimental data for rc and R, published in tables in Ref. [ 15], also agree well with Eq. (6.15). Taking square of Eq. (6.15) and using the expression for the buoyancy force, F b _ ~4~R3 A p g , one obtains [10]:

r~ = RFb_

(6.16)

]to" A generalization of Eq. (6.16) was obtained by Ivanov et al. [16] for two fluid particles (drops,

Chapter 6

258

bubbles) of different radii, R1 and R2, pressed against each other by an external force F: r~2 = R F , 27r~

where

-R - 2R1R2 R~ + R 2

(6.17)

R and ~- can be interpreted as mean diameter and surface tension, see also Eq. (6.12). If one of the two drops is a semiinfinite liquid phase, then R2 ----) co, R~ = R, R = 2R and Eq. (6.17) reduces to Eq. (6.16). The latter two equations have found applications in the studies of the hydrodynamic interactions of emulsion drops (see Fig. 6.6 below).

6.2.

HYDRODYNAMIC INTERACTIONS

The motion of a colloidal particle toward an interface (or another particle) is always affected by the viscous drag force. The strongest viscous dissipation of energy happens in the narrow zone of collision, where the two surfaces approach close to each other. The friction, accompanying the expulsion of the liquid from the collision zone, can cause a local deformation of fluid particles (gas bubbles, emulsion droplets or lipid vesicles), see e.g. Ref. [9]. The present section is devoted to such hydrodynamic interactions which are related to the viscous friction. We review the most useful theoretical expressions. One could find additional information in the comprehensive treatises on hydrodynamic interactions, Refs. [5, 9, 17-22].

Stokes regime of particle motion. At comparatively large surface-to-surface separations a spherical particle, moving under the action of a total driving force F, will obey the known Stokes equation for the velocity, see e.g. Ref. [23], F Vst - - ~

6jrr/R

,

(6.18)

where 77 is the dynamic viscosity of the liquid medium, and R is the radius of the particle. Equation (6.18) is obeyed not only by solid beads, but also by small (spherical) drops and bubbles in the presence of surfactant dissolved in the liquid medium. The role of surfactant, even at comparatively low concentrations, is to render the surface of the fluid particle tangentially immobile owing to the formation of a dense adsorption monolayer.

Particles at Interfaces: Deformations and Hydrodynamic Interactions

6.2.1.

259

TAYLOR REGIME OF PARTICLE APPROACH

At shorter distance between a spherical particle and an interface (or another particle) the hydrodynamic interaction between them becomes significant. This results in a dependence of the velocity of mutual approach on the surface-to-surface distance, h. For h/R << 1 the latter dependence is given by the Taylor formula [24] for the velocity:

2hF VTa =

3/rr/R 2

-

Vst

4h R

.

(6.19)

Since h/R << 1, it follows that VTa << Vst, i.e. the velocity of mutual approach is considerably decreased by the hydrodynamic interactions. Note that Eq. (6.19) is valid for two identical spheres of radius R. This equation can be generalized for two spheres of different radii, R1 and Re [18]"

2hF

(6.20)

WTa -- 3//:/'/~-2

where R is defined by Eq. (6.17). In the limit of two identical spheres (R~ = R2 = R) one has R = R, whereas for the interaction of a spherical particle with a planar interface (R~ = R, R2--->~) 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 ~ << 1) one has gTa cx: h and ~-,~--->~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.

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K.D. Danov, D.N. Petsev, N.D. Denkov, R. Borwankar, J. Chem. Phys. 99 (1993) 7179.

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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.

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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 ~

::<: ::: : ;~..; .. ..."" .~ ....... ~. ",

Isin~lsi: a~7 < OI i m m e r s i o n forces exist

flotation forces disappear (e)

for R < 10 ~ m

0

(f)

even for R ~ 10 n m

Q

Fig. 7.1. Capillary forces of flotation (a,c,e) and immersion (b,d,f) type: (a) attraction between two similar floating particles; (b) attraction between two similar particles immersed in a liquid film on a substrate; (c) repulsion between a light and a heavy floating particle; (d) repulsion between a hydrophilic and a hydrophobic particle; (e) small floating particles do not deform the interface and do not interact; (f) small particles confined within a liquid film experience capillary interaction because they deform the film surfaces due to the effects of wetting [21 ]. Fig. 7. l b illustrates the other case in which force of capillary attraction appears: the particles (instead of being freely floating) are partially immersed (confined) into a liquid layer [14, 18-21 ]. The deformation of the liquid surface in this case is related to the wetting properties of the particle surface, i.e. to the position of the contact line and the magnitude of the contact angle, rather than to gravity. To distinguish between the lateral forces in the case of floating particles and in the case of particles immersed in a liquid film, the former are called capillary flotation forces and the latter - capillary immersion forces [20,21]. These two kinds of force exhibit similar dependence on

290

Chapter7

the interparticle separation but very different dependencies on the particle radius and the surface tension of the liquid. The flotation and immersion forces can be both attractive (Fig. 7.1a and 7.1b) and repulsive (Fig. 7.1c and 7.1d). This is determined by the signs of the meniscus slope angles ~ and ~2 at the two contact lines: the capillary force is attractive when sin ~tl sin ~2 > 0 and repulsive when sin ~ sin ~2 < 0. In the case of flotation forces ~t > 0 for light particles (including bubbles) and ~ < 0 for heavy particles. In the case of immersion forces between particles protruding from an aqueous layer ~ > 0 for hydrophilic particles and ~ < 0 for hydrophobic particles. When ~t = 0 there is no meniscus deformation and, hence, there is no capillary interaction between the particles. This can happen when the weight of the particles is too small to create a significant surface deformation, Fig. 7.1e. The immersion force appears not only between particles in wetting films (Fig. 7.1b,d), but also in symmetric fluid films (Fig. 7.1f). Capillary immersion forces appear also between partially immersed bodies like vertical plates, vertical cylinders (rods), etc. Nicolson [3] derived an approximated analytical expression for the capillary force between two floating bubbles. Calculations about the capillary force per unit length of two infinite parallel horizontal floating cylinders were carried out by Gifford and Scriven [ 15] and by Fortes [ 16]. In this simplest configuration the meniscus has a translational symmetry and the Laplace equation, describing the interfacial profile, acquires a relatively simple form in Cartesian coordinates [7,15,16]. Chan et al. [17] derived analytical expressions for floating horizontal cylinders and spheres using the Nicolson's superposition approximation and confirmed the validity of this approximation by a comparison with the exact numerical results for cylinders obtained by Gifford and Scriven [15]. The aforementioned studies [3,15-17] deal with floating particles, i.e. with flotation forces driven by the particle weight. For the first time the capillary forces between two vertical cylinders and between two spheres partially immersed in a liquid layer have been theoretically studied in Ref. [14]. A general expression for the interaction energy has been used [14], which includes contributions from the energy of particle wetting, the gravitational energy and the energy of increase of the meniscus area due to the deformation caused by the particles; this expression is valid for both floating and confined particles. Expressions and numerical results

Lateral Capillary Forces between Partially Immersed Bodies

291

for the energy and force of interaction have been obtained for the case of small slope of the deformed meniscus; this case has a physical and practical importance because it corresponds to the usual experimental situation with small particles. The theory has been extended also to particles entrapped in thin films, for which the disjoining pressure effect, rather than gravity, keeps the non-deformed surface planar [14]. A new moment in Ref. [14] is the analytical approach to solving the Laplace equation: instead of using the approximation about a mere superposition of the known axisymmetric profiles around two separate particles, the linearized Laplace equation has been solved directly in bipolar coordinates. Thus one can impose the correct boundary conditions (constancy of the contact angle in agreement with the Young equation) at the particle contact lines. Thus a more rigorous theoretical description of the force at small interparticle separations is achieved, which is not accessible to the superposition approximation. Solutions for the meniscus profile in bipolar coordinates have been obtained in Ref. [18] for other configurations: vertical cylinder - vertical wall, and particle - vertical wall. A different, force approach to the calculation of the lateral capillary interactions has been applied to obtain

both analytical and numerical results. It has been established that the force exerted on the particle and the wall have equal magnitudes and opposite signs, as required by the third Newton's law; this is a check of the validity of the derived analytical expressions, which are subject to some approximations (small particle, small meniscus slope). The theory developed in Refs. [14] and [18] was further extended in Ref. [19] in the following two aspects. First, the energy approach and the force approach have been simultaneously applied to the same object (vertical cylinders and particles in a liquid film). The two approaches were found to give numerically coinciding results, although their equivalence had not been proven analytically there. Second, an analytical solution of Laplace equation in bipolar coordinates was obtained for the case of two dissimilar particles: vertical cylinders and/or spheres confined in a film. Attractive and repulsive capillary forces were obtained depending on the sings of the meniscus slopes at the contact lines of the two particles [ 19]. The theory of capillary forces between small floating particles of different size was extended in Ref. [20] on the basis of the results for the meniscus profile from Ref. [19]. The energy

292

Chapter 7

approach was applied to calculate the capillary interaction. Appropriate analytical expressions have been derived and numerical results for various configurations were obtained. The superposition approximation of Nicolson [3] was derived as an asymptotic case of the general expression for the interaction free energy, and thus the validity of this approximation was analytically proven. It was noticed that in a wide range of distances the capillary forces obey a power law, which resembles the Coulomb's law of electricity. Following this analogy "capillary charges" of the particles have been introduced. The physical nature and the magnitude of the lateral capillary forces between

floating and

confined particles have been compared in Ref. [20] and the differences between them have been explicitly analyzed. It has been established that the energy of capillary interaction between floating particles becomes negligible (smaller than the thermal energy kT) for particles smaller than 5-10 lam. On the other hand, when particles of the same size are partially immersed into a liquid film (instead of being freely floating), the energy of capillary interaction is much larger, and it can be much greater than kT even for particles of nanometer size. This analysis has been extended in Ref. [21] where the capillary forces in other configurations have been described theoretically; these are (i) two particles in a symmetric liquid film with account for the disjoining pressure effect, and (ii) two particles of fixed contact lines (rather than fixed contact angles). It has been established that the interaction at fixed contact angle is stronger than that at fixed contact line. Using the apparatus of the variational calculus the equivalence of the energy and force approaches to the capillary interactions has been analytically proven in Ref. [21 ] for the case of two vertical semi-immersed cylinders. As noticed Ref. [ 18], the meniscus between a vertical cylinder (or particle) and a wall has the same shape as the meniscus between two identical particles, each of them being the image of the other one with respect to the wall. For that reason the capillary interaction between the particle and the wall is the same as between the particle and its mirror image. In this respect there is analogy with the image forces in electrostatics. This idea has been applied and developed in Refs. [22] and [23], in which the capillary image forces

between particles

floating over an inclined meniscus in a vicinity of a wall have been theoretically and experimentally investigated.

Lateral CapillaryForces between PartiallyImmersedBodies

293

In Ref. [24] the theory of capillary forces has been extended to describe the interaction between particles attached to a spherical interface, film or membrane. In contrast with the planar interface (or film) the spherical interface has a restricted area and "infinite" interparticle separations are not possible. These geometrical differences can affect the trend of the lateral capillary force between identical particles: for spherical film it can be sometimes nonmonotonic: repulsive at long distances and attractive at short distances. On the other hand, in the case of planar geometry the capillary force between identical particles is always monotonic attraction. In Ref. [25] the theory of the lateral capillary forces was extended to describe the interaction between inclusions in phospholipid membranes; for that purpose a special mechanical model accounting for the elastic properties of the lipid bilayer was developed. A general conclusion from all studies on capillary immersion forces is that they are strong enough to produce aggregation and ordering of micrometer and sub-micrometer particles [14,18-25]. This fact could explain numerous experimental evidences about the formation of two-dimensional particle arrays in liquid films [26-45] and phospholipid membranes [46-48]. The problem about the capillary interaction between horizontal floating cylinders was reexamined by Allain and Cloitre [49,50], who used the linear superposition approximation and alternatively, a more rigorous expressions for the free energy of the cylinders; they calculated the capillary force for both light and heavy cylinders (for both small and large Bond numbers). It should be noted that the lateral capillary forces are distinct from the popular capillary bridge

forces, which form contacts between particles in the soil, pastes, and which are operative in some experiments with the atomic force microscope (AFM) [51-56]. The capillary bridge forces act normally to the plane of the contact line on the particle surface, while the lateral capillary forces are directed (almost) tangentially to the plane of the contact line. Theory about another kind of capillary force, which can be operative between particles of

irregular wetting perimeter has been proposed by Lucassen [57]. The irregular contact line induces respective irregular deformations in the surrounding liquid surface, even if the weight of the particle is negligible. The overlap of the deformations around such two particles also gives rise to a lateral capillary force. For the time being only a single theoretical study, Ref. [57], of this kind of force is available.

294

Chapter 7

The present chapter is devoted to the capillary immersion forces. First we derive an asymptotic expression for the immersion forces at not-too-small separations and consider the comparison of this expression with the experiment. Next we present an appropriate solution of Laplace equation in bipolar coordinates and obtain more general expressions for the capillary immersion forces using the energy and force approaches. The following configurations are described theoretically: two semi-immersed vertical cylinders, two spherical particles, vertical cylinder and sphere, vertical cylinder (or sphere) and vertical wall. The boundary conditions for fixed contact angle and fixed contact line are considered. The next Chapter 8 is devoted to the lateral capillary forces between two floating particles and between a floating particle and a wall (capillary image forces); applications of the theory of flotation forces to the measurement of the surface drag coefficient of small particles and the surface shear viscosity of surfactant adsorption monolayers are described. Chapter 9 presents the theory of the lateral capillary forces between particles bound to a spherical interface or thin film. An extension of the theory of the lateral capillary forces to the interactions between inclusions (membrane proteins) in lipid bilayers (biomembranes) is considered in Chapter 10. 7.1.2.

LINEARIZED LAPLACE EQUATION FOR SLIGHTLYDEFORMED LIQUID INTERFACESAND FILMS

Let z = ~(x,y) be the equation of the deformed fluid interface. The interfacial shape obeys the Laplace equation of capillarity, see Eqs. (2.15)-(2.17):

v,,

/vti /

c) 0 V n = e~ -~--+ e y ax

-[Pii(o

- Pi(O]/0-

(7.1)

(7.2)

Here, as usual, VII is the two-dimensional gradient operator in the plane xy. Note that Eq. (7.1) is expressed in a covariant form and can be specified for any type of curvilinear coordinates in the plane xv (not only Cartesian ones). The pressures P~ and PII on the two sides of the interface can dependent on ~" because of the effects of hydrostatic pressure and disjoining pressure, see below.

Lateral Capillary Forces between Partially Immersed Bodies

295

Z

T ~(r) //

ho

/1

Fig. 7.2. Colloidal sphere partially immersed in a liquid layer on a substrate; ~'(r) describes the shape of the meniscus formed around the sphere; P~ and PI~ are the pressures inside the liquid layer and in the upper fluid phase; ho is the thickness of the non-disturbed liquid layer; the latter is kept plane-parallel by the gravity, when the layer is thick, and by a repulsive disjoining pressure when the film is thin. As an example, let us consider a spherical particle which is entrapped into a wetting liquid film, Fig. 7.2. The upper surface of the liquid film is planar far from the particle; this plane is chosen to be the level z = 0 of the coordinate system. The thickness of the plane-parallel liquid film far from the particle is h0. The pressure inside and outside the film (in phases I and II) can be expressed in the form [58,59,21 ]: PI(O = go) _ p l g ~ ' + rI(ho+r

Pii(~) = plli( o ) _ PIIg~,,

IViiO2 << 1

(7.3)

Here, as before, g is the acceleration due to gravity, p~ and/91I are the mass densities in phases I and II, ~o) and ~0) are the pressures in the respective phases at the level z = 0; I-I is the disjoining pressure, which depends on the local thickness of the wetting film. The terms and

PigS,

P~ig~, express the hydrostatic pressure effect, which is predominant in thick films, i.e. for

h0 >> 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

=

=

<0

(7.4)

h=h0

see e.g. Ref. [60]. Expanding the disjoining pressure term in Eq. (7.3) in series one obtains n ( h 0 + O = n(h0) + 1-I' ~"+ ...

(7.5)

Usually the slope of the meniscus around particles, like that depicted in Fig. 7.2, is small enough and the approximation IVnO2 << 1 can be applied. Then combining Eqs. (7.1)-(7.5) one obtains a linearized form of Laplace equation [14]" V2 2 II~" -- q

~',

q

2

=

Apg O"

-l-I"

nt - ~

(Ap =/9i-

PlI, IVIIO 2 << 1)

(7.6)

O"

Note that 1-I' < 0. The disjoining pressure effect is negligible when the film is thick enough to have - F I ' (h0) << Ap g. In the latter case the upper film surface behaves as a single interface (it does not "feel" the lower film surface). The quantity q-~ is a characteristic capillary length, which determines the range of action of the lateral capillary forces. In thick films FI' is negligible and q-1 is of the order of millimeters, e.g. q-1 = 2.7 mm for water-air interface. However, in thin films I-I' is predominant and q-1 can be of the order of 10-100 nm, see Ref. [21 ]. In other words, the asymptotic expressions for q2 are:

7.1.3.

q 2 -Apg/cr

(in thick film)

q 2 --1-I ' / cr

(in thin films)

(7.7)

I M M E R S I O N FORCE: THEORETICAL EXPRESSION IN SUPERPOSITION A P P R O X I M A T I O N

Following Ref. [64] let us consider a couple of vertical cylinders, each of them being immersed partially in phase I, and partially in phase II. For each of these cylinders in isolation (Fig. 7.3) the shape of the surrounding capillary meniscus can be obtained by solving Eq. (7.6). The latter equation, written in cylindrical coordinates, reduces to the modified Bessel equation, whose solution (for small meniscus slope) has the form ~'k(r) = r~ sin gtk K0(qr) = Qk K0(qr),

(~: = ~,e)

(7.8)

Lateral Capillary Forces between Partially Immersed Bodies

297

AZ

9

O'1,1~1

r

r~

vl

Fig. 7.3. A vertical cylinder (rod) of radius r, creates a convex meniscus on an otherwise horizontal fluid interface of tension ~ the boundaries of the cylinder with the phases I and II have solidfluid surface tensions o'~,~and os,H;a, is the three-phase contact angle; ~V~is the meniscus slope at the particle contact line. see Eq. (2.43), where r~ is the contact line radius and ~V~ is the meniscus slope angle at the contact line, and is the so called "capillary charge" [20,21 ]; K0 is the Macdonald function of Qk - rk sin ~k

(k = 1,2)

(7.9)

zero order, see Refs. [61-63]. The contact angle a~ at the three phase contact line of the k-th cylinder (k = 1,2) obeys the Young equation: O'k,II-- O"k,I-" O " C O S O ~ k

"-

osin~t~ =

crQ~/r~

(k = 1,2)

(7.10)

cf. Fig. 7.3 and Eq. (2.2). Here o'~,i and o'k,i~ are the superficial tensions of fluids I and II, respectively, with solid 'k', see Section 2.3.1. The two cylinders are assumed immobile in vertical direction. Let us assume that cylinder 1 is fixed at the z-axis (Fig. 7.3) and let us consider a process in which the vertical cylinder 2 is moved in horizontal direction from infinity to some finite distance L (L >>

r~,r2). At a distance L the level of the liquid meniscus created by cylinder 1 is

~'1(L), see Fig. 7.3, and consequently, the elevation of the liquid around cylinder 2 rises with ~'I(L). Thus the surface area of cylinder 2 wet by phase I increases, whereas the area wet by phase II decreases. As a result, the energy of wetting of cylinder 2 will change with [64]

Chapter 7

298 AWw = - 2:rer2~'l(L) (cr2,ii- O'2,i) = - 2:rccrQ1Q2Ko(qL)

(7.11)

where at the last step Eqs. (7.8)-(7.10) have been used. Finally, identifying the capillary force with the derivative of the wetting energy, F = - dAWw/dL, we differentiate Eq. (7.11) using the identity [61,62] dKo(x)/dx - - K l (x)

(7.12)

and thus we obtain an (approximate) expression for the capillary immersion force [21,64]:

F =-2JrcrQ1QzqKl(qL ),

rk << L

(7.13)

Similar approximate expression for the flotation capillary force has been obtained long ago by Nicolson [3]; see also Refs. [17] and [20]. A more rigorous expression for F is given by Eq. (7.86) below. The above derivation of Eq. (7.13) makes use of the approximations ri << L, IVn~ 2 << 1. In particular, we have implicitly made use of the assumption, that for L >> ri the elevation of the liquid at cylinder 2 is equal to the superposition of the elevation at the isolated cylinder 2 plus the elevation ~'l(L) created by cylinder 1 at a distance L. The latter assumption is known as the superposition approximation; it can be obtained as an asymptotic case of the more rigorous solution, see Eq. (7.89) below. In the case of spherical particles the variation in the position of the contact line on the particle surface is accompanied with a variation of the contact line radii, rl and r2, and of the slope angles gt~ and gt2 ; these effects are taken into account in Refs. [ 14,18-20,24], see Section 7.3.2 for details. In spite of being approximate, the derivation of Eq. (7.13), clearly demonstrates the physical origin of the immersion force: the latter is (approximately) equal to the derivative of the

wetting energy Ww, see Eq. (7.11); similarly one can obtain (Section 8.1.1) that the flotation force can be approximated with the derivative of the gravitational energy Wg of a floating particle, which as a final result gives again Eq. (7.13). Using the identity Kl(X) = 1/x for x << 1 [61,62,65], one derives the asymptotic form of Eq. (7.13) for qL << 1 (q-1 = 2.7 mm for water), F - -2Jrcr Qj Q2 L

rk << L << q-l,

(7.14)

which looks like a two-dimensional analogue of Coulomb's law of electrostatics. The latter fact explains the name "capillary charge" of QI or Q2 [20,21].

Lateral Capillary Forces between Partially Immersed Bodies

299

~

Fig. 7.4. Sketch of the experimental set up used in Ref. [67] to measure the capillary immersion force between two vertical cylinders, '1' and '2'; '3' is a glass needle, which transfers the horizontal force exerted on cylinder '1' to a piezo-resistive sensor '4'. Thus the force, converted into electric signal, is measured as a function of the distance L. 7.1.4. MEASUREMENTS OF CAPILLARY IMMERSION FORCES Measurement of lateral capillary force (of the immersion type) has been carried out by Camoin et al. [66] with millimeter-sized polystyrene spheres attached to the tip of rod-like holders. By means of a sensitive electro-mechanical balance it has been established that the force is attractive and decays (approximately) exponentially, which corresponds to the long-distance asymptotics of Eq. (7.13), see e.g. Refs. [61, 65]:

F = -~:~Q1 Q2

exp(-qL) 1 +

~

(qL > 2)

(7.15)

A detailed comparison of the experimental results from Ref. [62] with the theory is not possible, because data for the surface tension, contact angle and the contact line radius are not given in that paper. Capillary immersion forces between two vertical cylinders, and between a vertical cylinder and a wall, were measured by means of a piezo-transduser balance [67], see Fig. 7.4. One of the cylinders ('1' in Fig. 7.4) is connected by a thin glass needle to a piezoresistive sensor; thus

Chapter 7

300

12 10

F qoQ1Qz

~

'

i

3

qL Fig. 7.5. Force F of capillary attraction between two hydrophilic vertical cylinders measured in Ref. [67] by means of the piezo-transducer balance sketched in Fig. 7.4; F is plotted vs. the distance L between the axes of the cylinders; the parameters values are q-1 = 2.72 mm, o"= 72.4 mN/m, Ql = 0.370 mm, Q2 = 0.315 mm. The solid line is calculated by means of Eq. (7.13)" no adjustable parameters.

the sensor can detect the pressure caused by the needle, which is in fact the horizontal component of the force exerted on the vertical cylinder 1. The other cylinder 2 can be moved during the experiment in order to change the distance L between the bodies. Figure 7.5 presents the dimensionless attractive capillary force F/(qcrQ1Q2) vs. the dimensionless distance qL measured in Ref. [67]. The liquid is pure water, o-= 72.4 mN/m, q-1 = 2.72 mm; the two cylinders are hydrophilic, so gtl = gt2 = 90~ the radii of the cylinders are rl = 370 Bm and 1"2 = 315 Bm. The solid curve in Fig. 7.5 is drawn by means of Eq. (7.13) without using any adjustable parameters. One sees that Eq. (7.13) agrees well with the experiment except in the region of small distances, where the asymptotic formula (7.13), derived under the assumptions for small meniscus slope and long distances, is no longer valid. The experimental data in Fig. 7.5 correspond to attraction between two similar (hydrophilic) rods. On the other hand, repulsive capillary force have been detected between two dissimilar

Lateral Capillam" Forces between Partialh, Immersed Bodies

301

F x 105 IN] 0.0

J

-0.1 -0.2 -0.3 -0.4 -0.5

9

.7 I

- 0 " 6 . 2 ' 140 ' 0.6

,

I

0.B

~

10

1.

~

i

1.2

qL

,

J

1.4

~

i

16

~

i

1.B

J

I

2.0

Fig. 7.6. Force F of capillary repulsion between hydrophilic (a~ = 0 ~ r l = 370 lam) and hydrophobic (a2 = 99 ~ r2 = 315 gm) vertical cylinders measured in Ref. [67] by means of the balance sketched in Fig. 7.4; F is plotted vs. the distance L between the axes of the cylinders; the parameters values are q-~ = 2.72 mm, o = 72.4 mN/m. The circles and triangles are results from two separate runs. The solid line is calculated by means of Eq. (7.13); no adjustable parameters. rods, a hydrophilic and a hydrophobic one, see Fig. 7.6. At long distances the experimental data agree very well with Eq. (7.13)" see the solid curve in Fig. 7.6, which is drawn without using any adjustable parameter. For short distances the data do not comply with Eq. (7.13), which is a manifestation of non-linear effects. Systematic measurements of capillary immersion force between partially immersed bodies of various shape (two vertical cylinders, cylinder and sphere, two spheres, sphere and vertical wall) were carried out in Refs. [68-70] by means of a torsion micro-balance, see Fig. 7.7. The latter in principle somewhat resembles the balance used by H. Cavendish to determine the gravitational constant in 1798 [71], but is much smaller. The interaction force for two couples of ,~ertical cylinders and/or spheres (Fig. 7.7) was measured by counterbalancing the moment created by the two couples of forces with the torsion moment of a fine platinum wire, whose diameter was 10 btm and 25 /am in different experiments. The angle of torsion, (p, was measured by reflection of a laser beam from a mirror attached to the anchor of the balance, see Fig. 7.7. Figure 7.8 shows data from Ref. [68] for the capillary force between two identical

Chapter 7

302

f

Fig. 7.7. Sketch of a torsion balance, used in Refs. [68-70] to measure the capillary attraction between two pairs of small, partially immersed, glass spheres (1-1' and 2-2') attached to holders. The immersed part of the holders is shown dashed. One of the particles in each pair (these are particles 1 and 2) is connected to the central anchor 3, which is suspended on a platinum wire 4; the angle of torsion is measured by reflection of a light beam from the mirror 5. 10

.4

r c= 50/~m r c=165/~m 9 r c=365/~m 9

10 .5

.

9

-

9

~

. -

10

-

-

-6

1,,--,-I

Z Ig

9

10 .7

10 -8

10 .9

J

0.0

,J

I

I

0.2

z

I

I

I

0.4

=

I

i

I

I

0.6

I

t

I

0.8

i

l

i

I

1.0

J

I

i

1.2

L [cm] Fig. 7.8. Plot of the force of capillary attraction F vs. the distance L between the axes of two identical vertical cylinders of radius rc. The force is measured in Refs. [68] by means of the torsion balance shown in Fig. 7.7; the three curves correspond to rc = 50, 165 and 365 ~tm. The solid lines are drawn by means of Eq. (7.13); no adjustable parameters.

Lateral Capilla 9 Forces between Partially Immersed Bodies

vertical cylinders for r~ =

r2 =

303

50, 165 and 365 btm; the solid lines in Fig. 7.8 are calculated by

means of Eq. (7.13) without using any adjustable parameter. It is seen that the theory and experiment agree well in the range of validity of the theoretical expressions. At shorter distances between the two interacting bodies, at which the linearized theory is not accurate, deviations from Eq. (7.13) are experimentally detected [69], as it could be expected.

7.1.5. The

ENERGY AND FORCE APPROACHES TO THE LATERAL CAPILLARY INTERACTIONS

energy approach to

the lateral capillary interactions (both immersion and flotation) is

based on an expression for the grand thermodynamic potential of a system of N particles attached to the interface between phases 1 and 2, which can be written in the form [ 14,20,21]:

f~(rl .....

(7.16)

rN)---- Wg -t- Ww -k- Wm +" const.

N

Wg= ZmkgZ~ C,- Z k=l

f PvdV

(7.17)

Y=I,I1 Vy

N

W-s

s

Wm - o ' ~ ,

(7.18)

k=l Y=I.II

where rl, r2 ..... ru are the position vectors of the particle mass centers and m~ (k = 1,2 ..... N) are the masses of the particles, ~

is the projection of rk along the vertical, Pv and Vy (Y = I,II) are

pressure and volume of the fluid phases I and II; cr is the surface tension of the interface (the meniscus) between fluid phases I and II; AA is the difference between the area of this meniscus and the area of its projection on the plane i.e. levels off to a plane at infinity); the boundary between particle

'k'

xy (AA is finite

Akv and

and phase

even if the meniscus has infinite area,

Oer are area and the surface free energy density of

'Y'; the

additive constant in Eq. (7.16) does not

depend on rl, r2 ..... ru. Wg, Ww and Wm are respectively the gravitational, wetting and meniscus contribution to the grand potential f2. Then the lateral capillary force between particles 1 and 2 is determined by differentiation:

-

=

----,

r,2

r12 =

Irl -

r2]

(7.19)

304

Chapter 7

z

Fig. 7.9. Illustration of the origin of capillary force between two spheres partially immersed in a liquid film: The net horizontal force F (1) exerted on particle 1 is a sum of the surface tension vector integrated along the contact line LI and of the pressure distribution integrated throughout the particle surface S~ (the same for particle 2), see Eqs. (7.21)-(7.23). When the distance between two particles varies, the shape of the meniscus between phases I and II (and consequently Wm) alters; during the same variation the areas of the particle surfaces wet by phases I and II also vary, which leads to a change in Ww; last but not least, the change in the meniscus shape is accompanied by changes in the positions of the mass centers of particles and fluid phases, which gives rise to a variation in their gravitational energy accounted for by Wg. Equations (7.16)-(7.19) are applicable also to thin films; one should take into account the fact that in such a case the meniscus surface tension depends on the local thickness of the film,

cr = o'(g'), so that [21,72] d~" =

FI

(thin films)

(7.20)

where, as usual, H is the disjoining pressure. In other words, the disjoining pressure effect is "hidden" in the meniscus energy term, Wm, in Eq. (7.16). The explicit form of Eqs. (7.14) and (7.15), and the relative importance of Wg, Ww and Win, depend on the specific configuration of the system. For example, in the case of flotation force is dominated by Wg, whereas in the case of immersion force ~ is dominated by W,~. This leads to different expressions for ~ corresponding to different physical configurations.

Lateral Capillary Forces between Partially Immersed Bodies

305

In the force approach, which is different but equivalent to the above energy approach, the lateral capillary force exerted on each of the interacting particles is calculated by integrating the meniscus interfacial tension r along the contact line and the hydrostatic pressure P throughout the particle surface [ 18-21 ]" F (k) = F(~~

F (kp) ,

k = 1,2 .....

(7.21)

k = 1,:2.....

(7.22)

k = 1,2 .....

(7.23)

where the contribution of interfacial tension is F(kO) - UII. ~ d l m ~ L~

and the contribution of the hydrostatic pressure is

F(~) -

UH" ~ d s ( - n P ) St

Here UH is the unit operator (tensor) of the horizontal plane xy; in Eqs. (7.22) and (7.23) this operator projects the respective vectorial integrals onto the xy-plane; L~ denotes the three phase contact line on the particle surface (Fig. 7.9) and dl is a linear element; the vector of surface tension ~ = mcr exerted per unit length of the contact line on the particle surface, is simultaneously normal to the contact line and tangential to the meniscus, and has magnitude equal to the surface tension or" m is a unit vector; S~ denotes the particle surface with outer unit running normal n; ds is a scalar surface element; the vector ' - n ' has the direction of the outer pressure exerted on the surface of each particle. In Refs. [ 18,19,21 ] it has been proven, that the integral expressions (7.21)-(7.23) are compatible with the Newton's third law, i.e. F (~) =

-

F (2),

as it must be. Note that the interfacial bending moment can also contribute to the lateral capillary force, see Ref. [25] and Chapter 10 below, although this contribution is expected to be important only for interfaces and membranes of low tension or. As an example, let us consider two particles entrapped in a liquid film on a substrate, see Fig. 7.9. If the contact lines L~ and

L2 were

horizontal, the integrals in Eqs. (7.22) and (7.23) would

be equal to zero because of the symmetry of the force distributions. However, due to the overlap of the interfacial perturbations created by each particle, the contact lines are slightly

306

Chapter 7

inclined, which is enough to break the symmetry of the force distribution and to give rise to a non-zero net (integral) force exerted on each of the two particles, F ~ and F ~2) in Fig. 7.9. The existence of inclination of the contact line can be clearly seen in Fig. 7.10, which represents three photographs of thin vertical hydrophobic glass rods partially immersed in water; the photographs have been taken Velev et al. [67] with the experimental set up sketched in Fig. 7.4. One sees that the contact line on an isolated rod is horizontal (Fig. 7.10a); when two such rods approach each other inclination of the contact line appears (Fig. 7.10b) and grows with the decrease of the distance between the rods (Fig. 7.10c). Let us imagine now that the upper part of the rods shown in Fig. 7.10 is hydrophobic, whereas the lower part is hydrophilic. In such a case the three-phase contact line can stick to the horizontal boundary between the hydrophobic and hydrophilic regions and the contact line will remain immobile and horizontal (no inclination!) when the two rods approach each other. Nevertheless, in such a case a lateral force of capillary attraction will also appear [21 ] because of the contact angle hysteresis: the meniscus slope varies along the circular contact line of each rod. The meniscus slope is the smallest in the zone between the two vertical cylinders (rods); then the integration in Eq. (7.22) yields again an attractive net force, see Ref. [21] and Section 7.3.4 for more details. It is worth noting that for small particles, rl,r2

<<

q-l, the contribution of the pressure to the

capillary force is negligible, IF(kP)I << IF(k~

for rl,r2 << q-l,

(7.24)

see Refs. [ 19,20] and Section 7.4 below. As established by Allain and Cloitre [49], the pressure contribution can prevail for ~'(rk) >> q-1 (k = 1,2), i.e. for large Bond numbers" however, this is not the case with colloidal particles, for which Eq. (7.24) is satisfied. It is not obvious that the energy and force approaches, based on Eqs. (7.16)-(7.19) and (7.21)-(7.23), respectively, are equivalent. Numerical coincidence of the results provided by these two approaches has been established in Refs. [19,20]. Analytical proof of the equivalence of the two approaches has been given in Ref. [21] for the case of two vertical cylinders; Eqs. (7.21)-(7.23) are derived by a differentiation of f2, see Eqs. (7.16) and (7.19).

Lateral Capillary Forces between Partially Immersed Bodies

307

(a)

(b) S'! ,,

(c) Fig. 7.10. Photographs, taken by Velev et al. [67], of two partially immersed vertical hydrophilic glass rods of radii rl = 315 lam and r2 - 370 lam. Note that the inclination of the three-phase contact lines on the rods increases when the distance between them decreases.

308

Chapter 7

7.2.

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

7.2.1.

SOLUTION OF THE LINEARIZED LAPLACE EQUATION IN BIPOLAR COORDINATES

When the Young equation holds and the three-phase contact angle is constant, the appearance of a small inclination of the contact line gives rise to the lateral capillary force, see Fig. 7.9. The simple superposition approximation is too rough to provide a quantitative estimate of this fine inclination. Indeed, the meniscus shape in superposition approximation does not satisfy the boundary condition for the constancy of the contact angle at the particle surface. A quantitative description can be obtained by solving the linearized Laplace equation, Eq. (7.6), in bipolar (bicylindrical) coordinates (v, co) in the plane xy, see e.g. Ref. [63]

x

a sinh z

-

coshz-cosco '

Y

--T1 --~ T ~ T2,

-

a sin co cosh z - cos co

(7.25)

--7~ < CO< Z

(7.26)

The elementary lengths along the T- and c~-lines of the respective orthogonal curvilinear coordinate network are [63] =

dl, - x/g,, dz,

dlo) - ~/go)o) d o ,

grT - go)o)

a

2

(coshz_cosco)2

(7.27)

where gTT and go)o)are components of the metric tensor. In Fig. 7.11 the circumferences Cl and C2, of radii rl and r2, represent the projections of the contact lines L~ and L2 on two interacting particles onto the plane xy (see e.g. Fig. 7.9). In the case of two vertical rods CI and C2 will be exactly circumferences; in the case of two spheres (Fig. 7.9) the contours C1 and C2 will slightly deviate from the circular shape, but this deviation is small for small particles, that is for (qrk) 2 << 1, and can be neglected [14]. The x-axis in Fig. 7.11 is chosen to pass through the centers of the two circumferences. The coordinate origin is determined in such a way that the tangents OA~ and OA2 to have equal lengths, a; in fact this is the geometrical meaning of parameter a in Eq. (7.25). From the two rectangular triangles in Fig. 7.11, O 0 1 A 1 and 002A2, one obtains Sk 2 -- a 2 = Fk2,

(k = 1,2).

(7.28)

Lateral Capillary Forces between Partially Immersed Bodies

Z'=

A1

309

A2

~

e

-""~2

c,

!

',0 2

,',~

S1

,._.

Fig. 7.11. Introduction of bipolar coordinates in the plane xy, see Eq. (7.25): the x-axis passes through the centers O1 and 02 of the contact line projections C1 and C2; the coordinate origin O is located in such a way that the two tangents, OA~ and OA2, have equal length a. The two circumferences in Fig. 7.11 correspond to fixed values of the parameter 7:, 7: = -7:1 and 7: = 7:2, where 7:1 and r2 are related to the geometrical parameters as follows: coshT:/, = s d r ~ ,

sinhT:~ = a/rk

(k = 1,2).

(7.29)

A substitution of Eqs. (7.29) into Eq. (7.28) yields the known identity cosh27:~- sinh27:k = 1. The r-lines are a family of circumferences in the plane x y determined by the equation [63]:

X2 4- ( y -

a cothz') 2 = a2/sinh2~"

(7.30)

Since sinhr~ - a/r~ one realizes that the two circumferences 7: = -7:~ and 7: = 7:2 have really radii rl and r2, see Fig. 7.11. The parameter a is related to the distance L = Sl + s2 by means of the expression [ 19]: a 2 = [L 2 - (rl

+/'2)2] [L 2 -

(rl - r2)2]/(2L) 2

(7.31)

One sees that a = L for L---~oo, and a---~0 at close contact, L--~(r~ + r2). In bipolar coordinates Eq. (7.6) takes the form [63,14,19]:

(cosh,T _ cos C0)2(c)2~ q c)2~ ) 3,c--5-

~

=(qa)

2

~(~,m)

(7.32)

310

Chapter 7

For small particles and not too large interparticle separations one has (qa) 2 << 1, and then Eq. (7.32) contains a small parameter. In such a case, following the method of the matched asymptotic expansions [73] one can consider an inner and an outer region" inner region (close to the particles)" (cosh~'- cosco) 2 >> (qa) 2

(7.33)

outer region (far from the particles): ( c o s h ~ - cosco) 2 << (qa) 2

(7.34)

We seek the solution of Eq. (7.32) for two vertical cylinders, like those depicted in Fig. 7.12. (In Section 7.3 it will be demonstrated that the results for vertical cylinders can be extended to describe the case of spherical particles.) The meniscus slope at the cylinders is determined by the slope angles I/tk = rt/2 - O~k,

k = 1,2,

(7.35)

where al and o~2 are the contact angles. Consequently, the following boundary conditions must be satisfied at the two contact lines [ 19]" 0---~-=(-1)k ~/g~sinvk

for r = (--1)k~'k

(k = 1,2)

(7.36)

z 2F 1

i

]

I.,

2r 2

,d

I

i i i i i

_j. . . . . . .

Lll I

X

cl i i i S 1

[

S2

Fig. 7.12. Schematic view of the capillary meniscus around two partially immersed vertical cylinders of radii rl and r~;_ al and a,~_are contact an,,les~ ,, gtl and ~2 are meniscus slope angles at the respective contact lines LI and L2, whose horizontal projections are denoted by CI and C2.

Lateral Capillary Forces between Partially Immersed Bodies

311

Other boundary condition is the meniscus to level off far from the cylinders: lim ~" - 0,

r

r----)oo

~

X2 + y-

(7.37)

In Refs. [14] and [ 19] the method of the matched asymptotic expansions [73] was applied and the solution was found in the form of a compound expansion:

~-= Cn + C u t _ (~ut)in

(7.38)

where ~,ut= (Q1 + Qz)Ko(qr)

(Q, - r, sin l/t,, k = 1,2)

(7.39)

(~,out)in ._-(Q1 4- Q2) ln(Teqr/2)

Te = 1.78 10724 18...

(7.40)

r

_ I r (%',0~) for --q;1 ~'u~O

[ ~2 (x,r

(7.41) for O<x<x 2

Here 7e is the Euler-Masceroni number, see e.g. Ref. [61] and the functions ~'1 and ~'2 are defined as follows [19]:

oo ~', (~, o))= Co + Q, ln(2 cosh27- 2 coso)) + Z c,~,~)cosh n[27- (-1)'27k] cos no)

(7.42)

n=l

where the coefficients are given by the expressions

oo

A - Z 1 sinh n(r I - 272) ,,=1 n sinh n(27 1 + 272)

Co = (Q1 - Q2)A - (Q1 + Q2) ln(Teqr),

C(~k)

_ _ 2(Qk-Qj

)

n

sinh n27j

j , k = 1,2" j ~ k ,

n = 1,2,3 ....

(7.43)

(7.44)

sinh n(T 1 --}-T 2 )

Eqs. (7.39)-(7.44) describe the meniscus profile around two cylinders of different radii supposedly the condition (qa) 2 << 1 is satisfied. Such is the case of colloid-sized particles, which represents a physical and practical interest. For two identical cylinders QI = Q2 = Q,

si = s2 - s,

rl - r2 = re,

271 = ~'2 = rc,

(7.45)

312

Chapter 7

and in such a case Eqs. (7.41)-(7.42) for Cn considerably simplifies [14,19]"

Cn(~, (_O)-" Q [ln(2 coshz'- 2 coso))- 2 ln(yeqr)]

(7.46)

The above expressions serve as a basis for the quantitative description of the lateral capillary forces between cylindrical and spherical particles (see below). First we will obtain some useful auxiliary expressions for the mean elevation and the shape of the contact line.

7.2.2.

MEAN CAPILLARY ELEVATION OF THE PARTICLE CONTACT LINE

As already mentioned, the two contact lines, ~-= -T~ and ~"= r2, are not perfectly horizontal, that is ~n depends on o)along the contact line. The deviation from horizontality is small for small particles (thin cylinders) [14]. The mean elevation of the contact lines above the horizontal interface far from the cylinders is [14,19]:

k= 1,2.

hk _1__~__ ~dl~in((_l)k72k,(O )

(7.47)

ZI~k ~k Using Eqs. (7.27) and (7.41)-(7.44) one can solve the integral in Eq. (7.47) to obtain [19]: hk -Qk {~'k + 21n[1- exp(-2~'~ )]}- (Q1 +Q2)ln()'eqa ) + ( Q t - Q 2 ) A - ( - I ) k ~[ 2 exp(-n'tk)sinhnz'j ,,--1 n

(j :/: k, j,k = 1,2)

(7.48)

s i n h n(z" l + ~2 )

In accordance with Eq. (7.29) zk can be expressed in the form: "rk - l n [ ~a + l a---3-+ 2 1/ , rk rk"

k = 1,,.'~"

(7.49)

a and A are given by Eqs. (7.31) and (7.43), respectively. The value of hk can be both positive and negative. For two identical cylinders Eq. (7.45) holds and h~ = h2 = he; in this special case Eq. (7.48) considerably simplifies [14]: hC - Q r C+ 2In l - e x p ( - 2 r )~, ?'~qa ]j

(identical cylinders~ "

(7.50~ "

Lateral Capillary Forces between Partially Immersed Bodies

313

Note that both Eqs. (7.48) and (7.50) are derived under the assumption, that (qa) 2 << l, which means that both the cylinder radii, r~ and r2, and the distance between the cylinders, L, is small compared to the capillary length, q-1. The restriction qL << 1 can be overcome applying again the method of the matched asymptotic expansions, as follows. First of all, we note that in the limit of infinite separation between the two cylinders, L---~oo, the limiting value hkoo of hk can be calculated by using the Derjaguin formula [74] for an isolated cylinder:

h~oo - Qk In

4 -- Qk In ~ , 2 ~Zeqrk (1 + COS~k ) 7eqrk

k = 1,2;

(qrk) 2 << 1.

(7.51)

At the last step we have used the fact that in the considered case of small meniscus slope we have sinZ~k << 1, which implies cos~tk -- 1. In the case of two identical cylinders using Eqs. (7.29) and (7.45) one can represent Eq. (7.50) in the following form [22]"

h~n - Qln

2 2 + Qln , ~'eqrc ~'eq(S + a)

(qa) 2 << 1,

(7.52)

without using any approximations. The subscript "in" means that we consider Eq. (7.52) as a limiting expression for hc in the "inner region" of relatively short interparticle distances, for which (qa) 2 << 1. In the complementary "outer region" one can use the superposition approximation of Nicolson (3), see Section 7.1.3, to derive

h~

hcoo + QKo(2qs),

(qa) z > 1.

(7.53)

where hcoo can be calculated from Eq. (7.5 l) with r~ = re. For small values of the argument the K0 function can be expressed in the form [61,65]"

K0(x)-ln

2

+O(xlnx)

x << 1.

(7.54)

]/e x Taking into account Eqs. (7.51)-(7.54) one can obtain the leading term in the compound expansion for hc [22]"

314

Chapter 7

(qrc) 2 << 1.

hc = hc= + QKo(q(s + a)),

(7.55)

One can check that for short distances, q(s + a) << l, Eq. (7.55) reduces to the "inner expansion", Eq. (7.52), whereas for long distances one has a = s and Eq. (7.55) reduces to Eq. (7.53). It turns out that Eq. (7.55) predicts the capillary elevation hc with a good accuracy for the whole range of distances between the two cylinders, from close contact up to infinite separations. In the case of two dissimilar cylinders, like those depicted in Fig. 7.12, one can obtain a generalization of Eq. (7.55) by using Eq. (7.48) and asymptotic expansions proposed in Ref. [ 19]: h~ = hk= + Qj K0(q(sk + a)),

j 4: k, j , k = 1,2;

(qr~) 2 << 1.

(7.56)

Equation (7.56) is subject to the additional condition (rk/sk) 4 << 1, which is violated for close distances between the two cylinders. That is the reason why the usage of Eq. (7.48) is recommended for close distances between the two cylinders and Eq. (7.56) can be applied for all other distances. In Section 7.3 we make use of Eqs. (7.55) and (7.56) to quantify the capillary interaction by means of the energy approach.

7.2.3.

EXPRESSIONS FOR THE SHAPE OF THE CONTACT LINE

Let us begin with the case of two identical vertical cylinders. (As already mentioned, in Section 7.3 it will be demonstrated that the results for vertical cylinders can be extended to describe the case of spherical particles.) In view of Eqs. (7.29) and (7.45) one can write "Cc= ln(a/rc + ~ a 2 /re2 +1 ) - ln(s/rc

+ ~/s 2 /re2 -1 ) = ln[(a + s)/rc]

(7.57)

By means of the last equation one can bring Eq. (7.46) for ~"= "Ccinto the form [22]:

~,~n = (~.in)r

- Qln ='re

2 ~'e qrc

+ Q ln

2

(qa) 2 << 1.

(7.58)

)/e 2qa 2/( s - rc cos CO)

~'e.in(co) describes the shape of the three-phase contact line at the surface of each cylinder in the "inner region" corresponding to relatively shorter distance between the cylinders, for which (qa) 2 << 1. In the complementary case of large separations, (qa) 2 _> l, one can use the

Lateral Capillary Forces between Partially ImmersedBodies

315

superposition approximation representing the meniscus shape around a couple of particles as a sum of the deformations, created by two isolated particles. The meniscus around a single axisymmetric particle is described by the Derjaguin equation, z(r) = QKo(qr), see Section 2.2.2. Thus one obtains [22]: ~'c~

= QKo(qrr) + QKo(qrl),

(qa) 2 > 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 << 1, one can write a - ~/s 2 - re? = S and to rewrite Eq. (7.60) in the form rt2 = (x + a) 2 +

y2,

rr2 = ( x - a) 2 + y 2 .

(7.61)

Next, for the particle contact line, (x,y) ~ C, we substitute x and y from Eq. (7.25) into Eq. (7.61) and rearrange the result using Eqs. (7.29) and (7.45); thus we obtain [22]: rl2 =

2a2(s+a)

r2 =

s - rc cos o9'

2a2(s-a)

(7.62)

s - rCcos co

To find the inner asymptotics of the "outer expansion" in Eq. (7.59) we carry out a transition

a--+O (qrl, qrr --+ 0); then from Eqs. (7.54) and (7.59) we obtain [22] (7.63)

(~'~ In = O(ln 2~+~zeqrr In ~/eqrl2 I

The substitution of Eq. (7.62) into Eq. (7.63) after some algebra gives exactly Eq. (7.58), without any approximations, i.e. (~c~ I n =-~.~n. Finally, in keeping with Eq. (7.54) we return back to K0 function in Eq. (7.58) to obtain a "compound" expression for ~'c((O), which for small separations reduces to ~'cin, Eq. (7.58), and for large separations yields ~'c~ Eq. (7.59) [22]:

~c(m)-hc=+QKo(2qa2) s - rCcoso)

(qrc)2 << 1

IVn~]2<< 1

(7.64)

316

Chapter 7

where he= is the Derjaguin's [74] expression for the elevation of the contact line for an isolated axisymmetric particle: he= -- Q In ~

2

(7.65)

~'eqrc

cf. Eq. (7.51). Equation (7.64) can be applied for any interparticle distances, characterized by the parameter a, see Eq. (7.31). For a--->0 Eq. (7.64) predicts ~'c(o9)-->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 << 1, is violated for such short distances. Equation (7.64) can be generalized to describe the shape of the contact lines, ~'c,l(r ~'c,2(o9), on two cylinders of

different

and

radii, rl and r2. Applying to Eq. (7.41)-(7.44) the

asymptotic procedure described in the Appendix of Ref. [19] one can derive

(c'~(c~176 (j :/: k, j,k =

I s k -2qa2r~ cosco / '

(qr~)2<< 1,

IV,I~2 << 1,

(rk/Sk)4<
(7.66)

1,2), where hk= is given by Eq. (7.51). The validity of Eq. (7.66) is limited by one

additional condition, (rk/Sk) 4 << 1, as compared with Eq. (7.64); if the latter condition is violated at short interparticle distances, the usage of the more rigorous expressions, Eq. (7.41)-(7.44), is recommended. In Section 7.4 below we make use of Eqs. (7.64) and (7.66) to quantify the capillary interaction by means of the force approach.

7.3.

ENERGY APPROACH TO THE LATERAL CAPILLARY INTERACTIONS

7.3. l.

CAPILLARY IMMERSION FORCE BETWEEN TWO VERTICAL CYLINDERS

We begin with the case of two partially immersed vertical cylinders, Fig. 7.12. According to Eq. (7.16) the free energy (the grand thermodynamic potential) of the system can be expressed as a sum of three terms, Wg, Ww and Wm, which are the contributions of the gravitational energy, the wetting of the cylinder surfaces and the meniscus surface energy, respectively. Below we will separately consider these three contributions.

Lateral Capillary. Forces between Partially Immersed Bodies

317

The meniscus surface energy can be expressed as Wm = o'AA, see Eq. (7.18). Its contribution to the capillary interaction energy between two cylinders can be written in the form AWm = O"(Zk4 - AAoo)

(7.67a)

where AAoo is the value of AA at infinite distance between the two cylinders. The difference AA between the area of the meniscus and the area of its projection on the plane xy can be expressed in the form [14]

AA - I [(1 + IVii~'121/2 - 1]ds

(7.67b)

Sm

where Vn is the gradient operator in the plane xv defined by Eq. (7.2), ds = dxdy is the surface element and the integration is carried out over the projection, Sin, of the meniscus on the plane

xy. If the meniscus slope is small, the square root in Eq. (7.67b) can be expanded in series ~=1

(IVllO2 << l)

I(VlI~'). (Vli~)ds

(7.68)

Sm

With the help of the linearized Laplace equation, V~2~" - q2 ~., one derives [14]"

2

(VlI~)'(VII~)--VlI-(~'VlI~)--~711~--Vll

.(~7 ~ )

II . --

q2~,2

(7.69)

Further, we substitute Eq. (7.69) into Eq. (7.68), and in view of Eq. (7.67a) and the definition q2._ A/9 g]ry we obtain [14] A Wm "- O" ( I c - Icoo) - m D g(/v -/voo)

(7.70)

where we have introduced the notation (7.71)

Ic ~ 1 IdSVlI .(~'Vll~ ) Sm

iv---

y Vm

Sm

lf s 2 0

(7.72)

Sm

Vm is the volume comprised between the meniscus surface z = ~(x,y) and its projection Sm on the plane xy; Icoo and Ivooare the limiting values of Ic and Iv for infinite distance (L---~oo) between

Chapter 7

318

the axes of the two vertical cylinders. Equation (7.71) can be rearranged using the Green theorem [ 14,75]: I C - 8 9~

~d/fi.(~Vxi~')

(7.73)

k=l,2 Ck

Here, as usual, the contours Ck (k = 1,2) are the projections of the contact lines Lk on the plane xy; the contour Ck is oriented clock-wise and fi is its running unit normal directed inward; to

obtain Eq. (7.73) it has been also used that the integrand ~;7H~"vanishes at infinity. For two identical cylinders one can write ~

= ~2 = ~c, and then from the boundary condition for

constant contact angle, Eq. (7.36), it follows fi.Vii~. =(_l)k

1 ~ =sin~c =const. grr ~

for v=(-1)k't'c

(7.74)

In such a case Eq. (7.73) reduces to I~ =sin 1VC~ d l ~ - 2zcrch ~ sin I/t~= 2~Qh c"

hC -

Cl

1

~dl~

(7.75)

CI

cf. Eqs. (7.9), (7.45) and (7.47). In this way the mean elevation of the contact line he enters the expression for the energy of capillary interaction. Using Eq. (7.75) one can represent the meniscus surface energy, Eq. (7.70), in the form AWm

=

(17.76)

27ccr Q(hc - hcoo) - A p g(Iv - Ivoo)

The gravitational potential energy, Wg, given by Eq. (7.17) varies only because the

shape of the interface between phases I and II changes when the distance between the two cylinders is altered; the mass centers of the cylinders are not supposed to change their positions, Zk(c). Since the interface is flat far from the cylinders, the hydrostatic pressures in the two neighboring phases can be expressed in the form Pv = Po - pvgz, where P0 is the pressure at the level z - 0, see Fig. 7.12. Then Eq. (7.17) reduces to

w~

const.- ~

,[PYdV - c o n s t . + A p g ~[zdV= const. + ApgI~ "

r=I,II Vv

gm

(17.77)

Lateral Capillary Forces between Partially Immersed Bodies

319

here, as usual, Ap - P r - P.; we have used Eq. (7.72) and the fact that the total volume of the system, V~ + Vii, is constant. Then the contribution of the gravitational potential energy to the capillary interaction energy becomes [14] AWg = Ap g(Iv - lw)

(7.78)

Summing up Eqs. (7.76) and (7.78) one obtains (7.79)

AWm + AWg = 27ro- Q(hc - hcoo)

Since Q and ( h e - he=) have the same sign, then the combined contribution of the meniscus surface energy and the gravitational potential energy, AWw + AWg, is always positive, i.e. it corresponds to repulsion. This is related to the fact that the capillary rise hc increases when the cylinders come closer; simultaneously the deviation of the meniscus from planarity increases, which is energetically unfavorable. The energy of wetting, given by Eq. (7.18), can be expressed as follows N

Ww - 2

Z CrkrAkr = 2(cri -

o'ii)

k=l Y=I,II

~ dl~ + const. = 4rtrc(o5 - oi,)hc + const.

(7.80)

Cl

cf. Eq. (7.75); here we have used the fact that for identical cylinders crkr = err (k = 1,2; Y = I,II). The contribution of the energy of wetting to the capillary interaction energy is AWw - Ww- Ww= = 4rtrc(cri-

oii)(hc- hcoo)

(7.81)

AWw is defined in such a way that AWw ---~0 for large distances, L---~o,,. We assume that the Young equation (see Section 2.3.1) holds, on -

o-~ = o- c o s o ~ = cr s i n g t c ,

(7.82)

and then Eq. (7.81) becomes AWw = -4rtcr rc (he - h,.oo) sin ~. = -4rto" Q (he - hcoo)

(7.83)

As mentioned above, Q, and ( h e - he,o) have the same sign, then the contribution of the wetting energy, AWw, is always negative, i.e. it corresponds to attraction. The comparison between Eqs. (7.79) and (7.83) shows that AWw is two times larger by magnitude than AWm + AWg;

320

Chapter 7

consequently, the work of wetting AWw determines the sign and the trend of the total capillary interaction energy [14]: Af2 = AWw + kWm + AWg = -2r~ry Q (hc - hcoo)

(7.84)

Thus the intuitive assumption that the interaction energy can be identified with a half of the work of wetting (wetting of one of the two cylinders), which was used in Section 7.1.3 to obtain Eq. (7.11), turns out to be correct. The substitution of Eq. (7.55) into Eq. (7.84) gives the dependence of the interaction energy Af2 on the distance L between the axes of the two identical vertical cylinders: (qrc) 2 << 1.

AE2 =-2rtcr Q2Ko(q(s + a))

(7.85)

Note that a = ~/s 2 - rc2 and s = L/2 - re. Differentiating Eq. (7.85) one obtains the capillary immersion force: F --

dAf~ _ _2;go_Q2 a + S q K 1(q(s + a)) dL 2a

(qrc) 2 << 1.

(7.86)

For (rJs) z << 1 one has s ~. a = L/2 and then Eqs. (7.85) and (7.86) reduce to their long-distance asymptotic forms, which can be obtained by means of the superposition approximation, see Section 7.1.3 above: A ~ =-2rtcr Q2Ko(qL),

F =-2trey QZqKI(qL)

[(qrc) 2 << 1, (rc/S) 2 << 1].

(7.87)

If the two cylinders have different radii, r~ and r2, see Fig. 7.12, Eq. (7.84) can be generalized following the same scheme of derivation [19]: A ~ = - rtcr Z Qk (h~- hkoo),

(Q/, = rk sin lff~)

(7.88)

k=l,2

The dependence of A~ vs. distance L can be obtained substituting the respective expression for hk, Eq. (7.48) or Eq. (7.56), into Eq. (7.88). For example, the combination of Eqs. (7.56) and (7.88) yields A~--~cy

QIQ2 Z K0 (q(sk + a)), k=l,2

[(qr/r 2 << 1, (rl,./s) 4 << 1],

(7.89)

Lateral Capillary Forces between Partially Immersed Bodies

1012

321

Cylinder 1"

Cylinder 2:

~p,=5 ~

W~= 5 ~

r~ = 0.8 k t m

10-.3

.L

.....

I----'1

r, = 8 / t m .

10 TM

.

.

.

.

.

.,. .

.

.

.

,... .

.

.

.

.

.

.

.

.

!.

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

r2 = 0.8 ~ m

1015

r~ = 0.08 ktm 1016 ,

0

I

100

~

I

200

~

I

300

,

I

,

400

I

500

,

I

,

600

l

700

,

800

b [pm] Fig. 7.13. Calculated energy of capillary attraction A~ vs. the distance b between the surfaces of two vertical cylinders. The meniscus slope angles are ~ - ~2 = 5~ the radius of the first cylinder is rl = 0.8 lam; the curves correspond to three values of the radius r2 of the second cylinder, which are given in the figure [19]. which reduces to Eq. (7.11) in the limiting case of long distances (Sk = a = L/2). Note that the dependence.of a on L is given by Eq. (7.31) and that sk = x/r2 + a 2 As an illustration Fig. 7.13 represents the dependence of Af2 on the surface-to-surface separation b - L -

rl - r2 between two vertical cylinders of different radii, rl and r2, but of

equal contact angles. Af2 is calculated from Eqs. (7.48) and (7.88) [19]. As could be expected AF2 is negative (corresponds to attraction) and IAf~I decreases with the increase of the separation b. As seen in Fig. 7.13, IAs

is much larger than the thermal energy kT = 4 x 10-21 J.

The same is true for spherical particles (instead of cylinders) - see below.

7.3.2. CAPILLARY IMMERSION FORCE BETWEEN TWO SPHERICAL PARTICLES Now our system is a flat horizontal solid surface covered with a liquid layer of thickness 10. Following Ref. [19] we consider two spheres of radii Rl and R2 which protrude from the liquid layer (Fig. 7.14), i.e.

Chapter 7

322

L,

21,

1

2r

~,1

h ,k

x

:,/ /

//

U/////////

///~/,.~

y sI

J"l

S2

Fig. 7.14. Sketch of the capillary meniscus around two spherical particles of radii R~ and R2, which are immersed partially in a wetting liquid film, whose thickness far from the particles is uniform and equal to/0; h~ and h2 denote the capillary elevation of the meniscus at the respective particle contact line; ak and Nk is contact angle and meniscus slope angle, respectively (k = 1,2). l0 < min(2Ri,2R2);

(7.90)

Our considerations are restricted to small particles and small meniscus slopes at the contact lines: (qRk) << 1,

sin2~tk << 1,

k = 1,2.

(17.91)

In this case the projections of the contact lines in the plane xy can be treated approximately as circumferences of radii rl and r2, see Ref. [ 14] for details. Instead of Eq. (7.35) now we have Nk = arcsin(r~/Rk) - ak,

k = 1,2;

(17.92)

see Fig. 7.14. Let us mention in advance that when the two particles are small, i.e. (qRk) << 1, they create small meniscus slope, that is the second condition in Eq. (7.91), sinZN~ << 1, is automatically satisfied, irrespective of the values of the contact angles al and a2. In the cases of two spheres (Fig. 7.14), unlike the case of two vertical cylinders, the radius of the contact line r~ and the slope angle ~k (k = 1,2) vary with the interparticle distance, L Sl + s2. This is due to the fact that the increase of the wet area of each particle in Fig. 7.14 is accompanied with a shrinkage of the contact line. The latter fact has to be accounted for in the

Lateral Capillary Forces between Partially Immersed Bodies

323

expression for the meniscus surface energy, Eq. (7.76), which can be generalized to the case of two different particles as follows [19]:

Z ( Q k h k -- Qk hk= - r,2 + r~=) - Apg(Iv - Iw)

AWm = r

(7.93)

k=l,2

As before, the subscript "co" denotes the value of the respective quantity at infinite interparticle separation, L---~,,o. The term r2 - rk2 accounts for the fact that the meniscus area alters when the radius of the contact line varies; the integral Iv is given again by Eq. (7.72). Since the two particles (Fig. 7o14) move only in horizontal direction, their gravitational potential energy does not change with L. However, the gravitational energy of the two fluid phases varies because of the dependence of the meniscus shape on L. The generalization of Eq. (7.78) to the case of two different spherical particles reads [ 19]: AWg = Apg(lv - I w -

(7.94)

Alp)

where A/p is a small correction accounting for the gravitational potential energy of the liquid displaced by a portion of the particle volume, denoted by Vpl and gp2 in Fig. 7.14:

(7.95) k-l,2

Vpk

Vp~(~)

Using geometrical considerations one can derive an explicit expression for A/k [19]:

Alk = _~{ 89(h _ hkZ ) (2lo ( 2 Rk _ lo ) _ hZ _ hkZ )+ 47( Rk _ lo ) (h3 _ hkoo )_ rZ hk2 + rk~ 2 hk~ 2

(7.95a)

It turns out that the contribution of Alp is always negligible for small particles, that is for

(qrk) 2 << 1. Again by using geometrical considerations one can derive that the areas of particle 'k' wet by phases I and II are respectively AkI =

2rtRl<(h~ + lo)

and

AklI

=

2rtR~[2R~- (h~ +/o)]

Then in view of Eq. (7.18) and the Young equation,

O ' k , I I - O"k,I = O"

(k = 1,2)

(7.96)

COSt[Zk, one obtains a

counterpart of Eq. (7.83) for spherical particles: AWw = -2rtcr ~ R k (h~ - hk~)cosO~k k=l,2

(7.97)

324

Chapter 7

A summation of Eqs. (7.93), (7.94) and (7.97) finally yields an expression for the energy of capillary interaction between the two partially immersed spheres [ 14,19]: 2

Af2= -/rcrZ{2Rk (h k - h k ~ ) c o s a k -Qkh~ + Q ~ h ~ +r2-rk2}-ApgAlp

(7.98)

k=l

Further, the capillary immersion force between the two particles can be calculated by differentiation of Eq. (7.98): F = -dAf2/dL. The numerical calculations show that the wetting term, R~(hk - h~=)coso~, in Eq. (7.98) is predominant and determines the sign and the magnitude of the attractive capillary immersion force. Since COSO~kis the largest for contact angle o~k = 0 ~ completely "hydrophilic" particles experience a relatively strong capillary

immersion force when captured in a liquid film. This is understandable, because hydrophilic particles also deform the film surface(s) when the film thickness l0 is smaller than the particle diameter 2Rk, as it is in Fig. 7.14. Note however, that hydrophilic particles (of density higher than that of water) cannot experience capillary flotation force, see Fig. 7.1a, because they cannot float on the aqueous surface, but instead they sink into the water phase. To compute the dependence of Af~ vs. L one has to first calculate the values of some geometric parameters. The following procedure of calculations has been proposed in Refs. [ 14,19]: The input geometrical parameters are the distance between the centers of the two spheres, L, the thickness of the layer far from the particles, 10, the particle radii Rk and contact angles o~k, (k = 1,2). From the equation of the particle spherical surface one calculates the contact radius rkoo of an isolated particle for a given value of the mean elevation hk= :

rkoo(hkoo)= [(/0 + hko~)(2Rk- lo- hkoo)] 1/2

(17.99)

Next from Eq. (7.92) one determines ~tk=(hkoo)= arcsin(rk=(hk=)/R~) - o:~. The calculated values of r~ooand ~k= are finally substituted into the Derjaguin equation (7.51)

h~= =-rkoo(h~=) sin~koo(hk=) ln[Teqr~=(h~=)/2], which is solved numerically to determine h,= (as well as r,=, ~kooand Q,= = r, oosin~tkoo).

(7.100)

Lateral Capillary Forces between Partially Immersed Bodies

325

Further, hk, r~ and ~t, are determined in the following way. From Eq. (7.99) (with r~, hk instead of r, oo, h,oo) one calculates rk(h,) for each given hk. Then in view of Eq. (7.92) one calculates ~,(h,) = arcsin(r,(hk)/Rk)- a,. The obtained values of rk and ~tk are finally substituted into Eq.

(7.48) or Eq. (7.56), which along with Eqs.(7.9), (7.31) and (7.49) determines hk as a function of L, rk, and ~,: hi, = ~/,(L; rl(hl), r2(h2), IKl(hl), I//2(h2)),

k = 1,2.

(7.101)

Equation (7.101) for k = 1,2 represents a set of two equations for determining h~ and h2 for each given interparticle distance L. In Ref. [ 19] h~ and h2 have been determined by numerical minimization of the function G(hl, h2) = Z

[hk - ~k(L; rl(hl), rz(h2), I//l(hl), Ipr2(h2))]2

(7.102)

k=l,2

In view ofEq. (7.101)the minimum value of G(hl, h2) is zero. The couple (hi*, h2*) satisfying the equation G(hl, h2) = 0 is the sought for solution for hi(L) and hz(L), which is to be further substituted in Eq. (7.98) to calculate the interaction energy Af2(L). To find (hi*, h2*) in Ref. [19] hi and h2 have been varied within the limits -10 < hi, < 2Rk- 10, k = 1,2, using the method of Hooke and Jeeves [76]. In the case of two identical spherical particles the calculation procedure is similar but simpler: having in mind that a = (s 2 - rc2) 1/2 and s = L/2, Eq. (7.55) [or Eq. (7.50) along with Eq. (7.57)] provide an equation for determining hc: hc = ~(L; rc(hc), ~c(hc)),

(7.103)

where the functions rc(hc) and ~c(hc) are determined in the same way as for the case of two different particles, viz. from Eq. (7.99) with rc, hc, and R instead of rkoo, hkoo and Rk, and from the relationship ~c(hc) = arcsin(rc(hc)/R) - a. Further, Af2 can be calculated either from Eq. (7.84), or from Eq. (7.85) with Q = rc sin~tc is to be computed with the values of r,. and ~c obtained for each L by solving Eq. (7.103). As an illustration in Fig. 7.15 we present the calculated capillary interaction energy Af2 as a function of the distance L between the centers of two identical hydrophilic spherical particles of radius R = 0.8 ~tm and contact angle a = 0, which are confined in a wetting film (see Fig. 7.1 b).

Chapter 7

326

.0 ..........................................................................................

~"

10 = 1 . 2 # m

-5.0

x

=

-10.0

aq-1=__00.2 ~ cm

. /~m

R = 0.8/tm -15.0 0

, , , , . . . . . . . . . . . 200 400 600

, , , , 800 1000

L/2R Fig. 7.15. Theoretical dependence of the capillary interaction energy A ~ vs. L/2R, calculated in Ref. [14] for two identical spheres of radius R -- 0.8 lam separated at a center-to-center distance L. The spheres are hydrophilic ( a = 0 ~ and are partially immersed in a liquid layer, whose thickness far from the particles is uniform and equal to /0; the two curves correspond to l0 = 0.4 and 1.2 lam; the capillary length is q-~ = 0.2 cm. T h e thickness of the liquid layer far from the particles is taken to be lo = 1.2 and 0.4 lam; the capillary length is q-~ = 0.2 cm. F o r lo = 1.2 ~tm the particles create a relatively small d e f o r m a t i o n of the u p p e r surface of the w e t t i n g film, w h e r e a s for 1,, = 0.4 lam the d e f o r m a t i o n is greater. C o r r e s p o n d i n g l y , the m a g n i t u d e o f Af2 is greater for lo = 0.4 lam, see Fig. 7.15.

Z

I_.,

2/"1

~1

L. 2r2 ~l

rm,

P"I

S1

"r

S2

Fig. 7.16. Sketch of a vertical cylinder of radius rl and sphere of radius R2, which are partially immersed in a liquid layer, whose thickness far from the particles is uniform and equal to/0; r2 and h2 are the radius of the particle contact line and its capillary elevation; a~ and ~k are contact angle and meniscus slope angle, respectively (k = 1,2).

Lateral Capillary Forces between Partially Immersed Bodies

327

For both values of 1,, in Fig. 7.15 the interaction energy A~ is much greater than the thermal energy k T -- 4 x 10-21 J. As expected, Af~ is negative and corresponds to attraction, which turns out to be rather long-ranged: even at distance L / 2 R = 1000, Af2 is considerably larger than k T [14]. Such a long-range attraction would lead to two-dimensional disorder-order phase transition and formation of ordered clusters or larger domains of particles depending on the experimental conditions; in fact, this has been observed experimentally [11,12,27-29]; see Chapter 13.

7.3.3. CAPILLARY IMMERSION FORCE BETWEEN SPHERICAL PARTICLE AND VERTICAL CYLINDER

The method described above can be directly applied to calculate the capillary interaction between a vertical cylinder and a partially immersed sphere [ 19]. The system is depicted in Fig. 7.16. The geometrical parameters belonging to the cylinder and the sphere are denoted by indices 1 and 2, respectively. In particular, r~ and R2 denote the radii of the cylinder and the sphere; ~zk and ~ (k = 1,2) are the respective contact and meniscus slope angles, see Fig. 7.16; r2 and L have the same meaning as in the previous section; a can be calculated from Eq. (7.31). Again we will make use of the assumptions for small particles, (qr/,) 2 << 1, and small meniscus slope, sinZ~k << 1. Then following the procedures of derivation of the interaction energy for two cylinders, Eq. (7.88), and for two spheres, Eq. (7.98), one can obtain the following expression for the energy of capillary interaction between a vertical cylinder and a sphere [ 19]: Af~ = -rtcr [ ( h , - h l ~ ) r l s i n gt~ - Q~h 2 + Q2~h2~ + 2R2(h2-h2~)cos % + r22 - r 2 ] - A p g A l 2

(7.104)

where A/2 is defined by Eq. (7.95a) for k = 2. The parameters r2=, gt2= and h2~ can be determined from Eqs. (7.99) and (7.100) for k = 2. hi= can be calculated directly from Eq. (7.51). hi and h2 can be calculated using Eqs. (7.101) and (7.102); simultaneously r2 and gt2 are determined; the numerical procedure is simpler than that for two spheres because of the constancy of some parameters: for the cylinder rl = const and gfi = rd2 - o~ = const. As an illustration Fig. 7.17 presents the calculated dependence of the capillary interaction energy Af2 on the distance L between a vertical cylinder of radius r~ = 0.5 ~tm and a sphere of contact angle c~2 = 10~ the values of the other parameters are denoted in the figure. The three

328

Chapter 7

0.0 R2 = 1 , u m

R2 =_2.8 m . . . . . .

-1.0 ,.,.... . . . . . . . . . . . . . . . . . . . . . . .

RI---" 3 _,um. . . . .

r""-I

'"'

-2.0

....,

.,,

X /

-3.0

1o= 0.5 p m a = 40 mN/m

-

i

/ -4.0

Cylinder 1"

Sphere 2:

~01 = 5 ~

a2 = 10 ~

r, = 0.5 p m -5.0

.

0

.

.

.

I

50

.

.

.

.

I

.

100

.

.

.

I

150

.

.

.

.

200

L [,um] Fig. 7.17. Theoretical dependence of the capillary interaction energy A~ vs. the distance L = s~ + s2, calculated in Ref. [19] for the configuration of cylinder and sphere depicted in Fig. 7.16. The three curves correspond to various values of the particle radius R2 denoted in the figure; the values of the other parameters are r~ = 0.5 ~m, ~ = 5 ~ a2 = 10~ l0 = 0.5 lam, and o"= 40 mN/m. curves correspond to three values of the particle radius R2. One sees that A ~ is negative (corresponds to attraction) and IA~21 increases with the increase of particle radius R2 at fixed thickness l0 of the wetting film. Again IA~I is much larger than the thermal energy kT; that is the capillary force prevails over the Brownian force exerted on the particle.

7.3.4.

CAPILLARY INTERACTIONS AT FIXED ELEVATION OF THE CONTACT LINE

All cases considered in the previous sections of this chapter correspond to the boundary condition of fixed contact angle at the particle surface. In particular, the obtained solution of the Laplace equation, Eqs. (7.38)-(7.44), satisfies the boundary condition for constant contact angle: Eq. (7.36) along with Eq. (7.35) for cylinders, or with Eq. (7.92) for spheres. In the present section, following Ref. [21], we consider another physical situation: fixed meniscus position (instead of fixed meniscus slope) at the contact line. As shown schematically in Fig. 7.18a,b this can happen when the contact line is located at some edge at the particle surface. Other possibility is the contact line to be attached to the boundary between hydrophilic and hydrophobic domains of the surface, as sketched in Fig. 7.18c; similar is the configuration

LateralCapillaryForcesbetweenPartiallyImmersedBodies

329

of two membrane proteins incorporated in a lipid bilayer, which is considered in details in Chapter 10 below. Note that in the case of fixed position of the contact line, the variation in the meniscus shape due to a change in the distance L is accompanied by a

hysteresis of

the contact

angle (see Section 2.3.4), i.e. by a variation of the meniscus slope angle (rather than meniscus elevation) at the contact line. Z

~

1

-

.

.

.

.

~o~ I ~ __ _1_ _ i I ' ~ / / / / / / /I / . i ////..,,//.//////.,///.,,,//..,, L

t -T- . . . . . . . . .

i

!

i

t

.....

I !i

t1'

ooy..

/

t

!

~t. . . . . . . .

o y/...,

i[

I. I~(x,y)

;

.--l---

I i

.

L

Iz

yc ~ 2.

.

.

.

" ..........

i~

[~(x,y) " ....

.... ] i l

i I

~

~---

i

L

,,.I

--i

Fig. 7.18. Examples for capillary interaction at fixed elevation hcooof the contact line: (a) two cylinders or disks immersed in a liquid layer; (b) two vertical cylinders whose lower bases are attached to a liquid surface; (c) two particles in an emulsion f i l m - the contact lines are attached to the boundaries between the hydrophilic and hydrophobic domains on the particle surface (shown with different shadowing). Since the contact lines are immobilized, the energy of wetting does not contribute to the capillary interaction unlike the case of mobile contact lines shown in Figs. 7.12, 7.14 and 7.16.

Chapter 7

330

Let us consider the meniscus around two identical axisymmetric bodies like these depicted in Fig. 7.18. Note that the exact geometry of the bodies (spheres, cylinders, etc.) is not important insofar as the contact line is circular and fixed at the surface of the respective axisymmetric bodies. The derivation of the expression for the capillary interaction energy Af~ follows exactly Eqs. (7.67)-(7.79) with the only difference that Eq. (7.74) does not hold and Eq. (7.75) takes the alternative form

Ic - h~ooC, ~ dl(II.

VIIi)

-

~ d l ( f i . V '~~') 2a'rchco~ sin T c 9 sin T c - ~ c1 c,

(7.105)

(~" = hcoo at the contact line). For that reason, instead of Eq. (7.79) one obtains [21 ] A~2 = AWm + AWg = 2~o'rchcoo[sinTc(L) - sin~c~o]

(fixed elevation)

(7.106)

We have taken into account the fact that there is no change in the energy of wetting, AWw - 0, when the contact line is fixed, cf. Eq. (7.16). [In the case of symmetric film, Fig. 7.18c, there are two deformed interfaces and consequently the interaction energy is twice Af2 as given by Eq. (7.106).] In spite of the fact that Eq. (7.106) does not include a direct contribution from the work of wetting, AWw, the interaction energy Af~ is again connected to the special wetting properties of the particle surface due to the fixed position of the contact line, which bring about meniscus deformations and give rise to a non-zero contribution from the meniscus surface energy and the gravitational energy, AWm + AWg. To find the profile of the capillary meniscus we will use again bipolar coordinates (z, co) in the plane xy, see Eq. (7.25). The projections of the two contact lines on the plane xy are two circumferences z = -+Zc of radius rc, see Eq. (7.57) where a = (s 2 - rc2) 1/2 and s = L/2. The mathematical description of the capillary interaction at fixed position (elevation) of the contact line demands to find a solution of Laplace equation, which satisfies the following (inner) boundary condition ~'(z=+zc, o)) = he= = const

(fixed elevation of the contact line)

(7.107)

The other (the outer) boundary condition is the meniscus to level off at infinity, that is Eq. (7.37) to be satisfied. We seek a solution of Eq. (7.32) satisfying the aforementioned two

331

Lateral Capillary Forces between Partially Immersed Bodies

boundary conditions. In Ref. [21] inner and outer asymptotic regions are considered, cf. Eqs. (7.33)-(7.34), and a compound solution, ~'= ~n + ~ut_ (~ut)in, is obtained, in which

= 2coshn~" exp(_nL)cosnco ~n= hcoo+ Acln(2cosh~r- 2cosco) - ~7 + Ac ~,=~ncos---hnr---~ r -- (X2 "+-y2)1/2

~ut = 2AcK0(qr),

(~.out)in _-- -2Acln(yeqr/2)

(7.108)

(7.109) (7.110)

Here, as usual, ]'e = 1.781072418... (lnye = 0.577...) is the constant of Euler-Masceroni [61 ] and the parameter Ac is defined by the following expression [21 ]:

Ac = h~=I T~ - 2 ln(yeqa) - ~~ 2 exp(-nT~ ) 1-1 ,=l n cosh n~"C

(7.111)

Note that the last two terms in the brackets in Eq. (7.111) are logarithmically divergent for a-+0, that is for small distances between the two bodies; however, these two divergent terms cancel each other. To prove that we first notice that for a--+0 we have vc-+a/rc << 1 and in such a case the sum in Eq. (7.111) can be exchanged with an integral as follows [21 ]: oo exp(_nz. C)

--->2

,,=1 nc~

i ~ dx + -In 4Yea 2,rc x(e x + 1) n'rc

(7.112)

At the last step integration by part has been used along with the identity [77] i eXlnx d x - l l n (

(e x q- 1) 2

zr ) -2- ~ 2Ye

(7.113)

Then the terms containing lna in Eqs. (7.111) and (7.112) cancel each other for a--+0 and, consequently, Ac remains finite in the same limit. Next, introducing bipolar coordinates in Eq. (7.105) and substituting there Eq. (7.108) one can derive

sinTc(L)=

,

do ~

=re

rc

(7.114)

332

Chapter 7

In the other limit, (qa) 2 _> 1, one can find an expression for Uec(L) using the superposition approximation. In the framework of this approximation the elevation hc of the contact line on each of two vertical cylinders can be presented in the form

hc = hc= + Ahc +

(

~)h~ / (sinq~-sinq~c~) ~ sin W c rc

(7.115)

Here hc= is the elevation for L---~,,oand Ahc = rc sin~g~ooK0(qL)

(7.116)

is the elevation created by a single cylinder at a distance L from its axis (the distance at which the second cylinder is situated); the last term in Eq. (7.115) accounts for the change in hc due to the change in ~c. Differentiating the Derjaguin's formula, Eq. (7.65), with Q = rc sin~c, one obtains ~hc

~)sin W c

I rc

- rc In

2 7eqrr

(7.117)

Now we impose the boundary condition Eq. (7.107), which requires hc = hcoo, and then the combination of Eqs. (7.115)-(7.117) yields [21 ]

sinWc(L) = sinWcoo 1 - In

2

K 0 (qL)

(7.118)

Yeqrc

Equation (7.118) holds when the distance L between the two bodies is large enough and the second term in the brackets is small, i.e.

e(L) -

/ /' In

2

K0(qL ) << 1.

(7.119)

7eqrc

Equation (7.118) implies that ~Pc(L) < ~Pcooand consequently, the interaction energy A~, given by Eq. (7.106), is negative and corresponds to attraction. The two asymptotics, Eq. (7.114) for short distances and Eq. (7.118) for long distances can be matched applying the standard procedure [see Eq. (7.38)] to the function 1/sin~c(L); the resulting compound expression reads [21]:

Lateral Capillary Forces between Partially Immersed Bodies

sinWc(L) -

rc + K 0 ( q L ) _ l n

2

333

r

(qrc) 2 << 1.

(7.120)

Teq L

To achieve better accuracy in computations, it is recommended to use Eq. (7.118) when e(L) << 1, and to use Eq. (7.120) in all other cases. The energies of capillary interaction corresponding to the two different boundary conditions, fixed slope and fixed elevation, are compared in Fig. 7.19. In the case of fixed slope the energy is calculated from Eq. (7.85), whereas in the case of fixed elevation - from Eqs. (7.106) and (7.120). To have a basis for comparison the parameters values denoted in Fig. 7.19 are taken to be the same for the two cases. As seen in the figure, in both cases Af2 is negative and can have a magnitude of the order of 10-12 J. In other words, in both cases the interaction energy is much greater than the thermal energy k T and corresponds to attraction. The new moment is that the interaction at constant slope is stronger than the interaction at constant elevation [21 ].

0.0 constant elevation -0.5

o

.-

-1.0

Aa_

r

. . . . . . . . . . o. . . . . . .

X

/Aft21

-slope--

rc- 10/tm

/

-1.5

q-1 = 2.7 m m

/

a = 72.7 m N / m

!

/ -2.0

.

hero = 5.0 ktm .

.

.

I

5

.

.

.

.

I

10

.

.

.

.

I

.

.

15

.

.

I

20

,

,

,._a_.l

~

25

i

i

t

30

L/2r c

Fig. 7.19. Calculated energy of capillary attraction Af~ vs. L/2rc, where L is the distance between the axes of symmetry of the particles (see Fig. 7.18) and rc is the radius of the contact line. Af~l and A~22 correspond to the cases of constant slope, Eq. (7.88), and constant elevation, Eq. (7.106). The values of the parameters We,o, rc, q-l, cr and h
334

Chapter 7

7.4.

FORCE APPROACH TO THE LATERAL CAPILLARY INTERACTIONS

7.4.1.

CAPILLARY IMMERSION FORCE BETWEEN TWO CYLINDERS OR TWO SPHERES

As already noted in Section 7.1.5, the lateral capillary force exerted on each of two interacting particles is a sum of the net forces due to the interfacial tension and hydrostatic pressure: F (k~ = F(k'r)+ F (kp) (k = 1,2), see Eqs. (7.21)-(7.23). F (k~ is calculated by integration of the meniscus

interfacial tension cr along the contact line, while F (kp) is determined by the integral of the hydrostatic pressure P throughout the particle surface. Our purpose below is following Ref. [19] to obtain explicit analytical expressions for F (~~ and F (kv~, and to compare the numerical results obtained by means of the alternative force and energy approaches. First, let us calculate the capillary force F (~ for each of the two partially immersed vertical cylinders depicted in Fig. 7.12. It is convenient to make a special choice of the coordinate system, see Fig. 7.20. The z-axis coincides with the axis of the considered cylinder. The plane xy, as usual, coincides with the horizontal fluid interface far from the cylinders. The x-axis is

directed from the cylinder of consideration toward the other cylinder. The symmetry of the system implies that the y- components of F (~~ and F (~) are equal to zero. For that reason our task is reduced to the calculation of F~k~ = e~.F (k~

and

F(xkp) = ex. F (kp)

(7.121)

where e~ is the unit vector of the x-axis. Note that due to the specific choice of the coordinate system, the positive (negative) value of the projection F! k) corresponds to attraction (repulsion) between the two cylinders. Force F! ~~ due to the interfacial tension.

Let z = ~'(r

be the equation of the contact

line with r being the azimuthal angle in the plane xy, see Fig. 7.20. The position vector of a point belonging to the contact line is R(cp) = e~ rk coscp + ey rk sincp + ez ~'(r

(7.122)

where, as usual, rk is the radius of the contact line. The linear element along the contact line is dl = Z dcp,

Z -IdR/d~

= [rk2 + (d~]dfp)2] 1/2 ,

(7.123)

Lateral Capillary Forces between Partially Immersed Bodies

335

Z

O"

Ii,

x rk Fig. 7.20. Sketch of the capillary meniscus around one of the two cylinders depicted in Fig. 7.12; rk is the radius of the cylinder, ak is contact angle; z = ~'(qg) is the equation of the three-phase contact line with running unit tangent t; n is a running unit normal to the cylinder surface; b = t x n is a unit binormal; ~ is the vector of surface tension, which is perpendicular to t, but tangential to the meniscus surface. The slope of the contact line is exaggerated. The vector of the running unit tangent to the contact line is

t -

1!

= - - e~ r k sin q~ + e ~r k cos q0 + e z ~ X. dq~ X

/

(7.124)

The vector of the outer running unit normal to the cylindrical surface is (Fig. 7.20) n = e~ cosq9 + ey sinq9

(7.125)

At each point of the contact line (see the point M in Fig. 7.20) one can define the vector of the unit binormal as follows" b=t•

(7.126)

The vector of the interfacial tension ~, exerted at the contact line, is simultaneously tangential to the meniscus surface and normal to the contact line; hence cy belongs to the plane f o r m e d by the vectors n and b" = cr (b sin ~t~ + n cos ~t~),

(7.127)

336

Chapter 7

see Fig. 7.20. Next, we substitute Eqs. (7.124)-(7.126) into Eq. (7.127) to derive

r

ex'rd =( ~s (c~176 i n q g s - 1i 2'ndcp~ e / -

(7.128)

Combining eqs. (7.22). (7.121), (7.123) and (7.128) one obtains [19]

F(xe~ ~dl ~x =-rto'sinl/rk Lk

~od~

sin(pdtp+AF (k~

(7.129)

x

where we have introduced the notation 2

/',Fx"

=

o-cos~,~, ~Xcos~dq, o

=~ r~0

cos@dq9

At the last step we have used the condition for small meniscus slope,

(7.130)

[ V l l ~ 2 <<

1, and have

expanded in series the square root in Eq. (7.123). Further, it is convenient to introduce parametrization of the contact line in terms of the angle co of the bipolar coordinate system ('r, co), see Eq. (7.25). By means of some geometrical considerations one can find the connection between co and rp [ 19]: cos@=

1 - cosh ~-k cos co cosh'r k -cosco

,

0<@
rc > c o > 0

(7.131)

where "ck is the value of the bipolar coordinate ~ at the contact line, see Eq. (7.49). With the help of Eq. (7.131) one can bring Eqs. (7.129) and (7.130) into the form [19]:

Fff ~ = 2rta sin Ck sinh rk

= --

i d~ sin codm + AFff ~ o do) cosh "tk - cos o3

(1 - cosh 1:k costa)do3

(7.132)

(7.133)

a 0

In fact Eqs. (7.132)-(7.133) are the final equations for calculating F} k~ one has to substitute ~'(co) = ~'c,k(co) from Eq. (7.66) and then to carry out numerically the integration in Eqs. (7.132)-(7.133). For not too large distances between the cylinders, for which (qa) 2 << 1, it

Lateral Capillary Forces between Partially Immersed Bodies

337

is better to substitute ~'(co) = ~'k(~'~, o9) from Eq. (7.42); then the integration in Eq. (7.132) can be carried out analytically [19]:

F~k~ = 2rtcrsingtk Qke -~ + 2(Q 1 - Q 2 ) ( - 1 ) k sinh'c k ~ n=l

1

E

+ AF~ k)

(7.134)

with E (~ - exp[-2n('rl + 7:2)]. However, the integral in Eq. (7.133) cannot be solved analytically and has to be calculated numerically. The calculations show that for small particles, (qr~) 2 << 1, the quadratic term AF~ k) is only a small correction in comparison with F~k~ and it is not a great loss of accuracy if the term AFt}k~ in Eqs. (7.132) and (7.134) is neglected. For large interparticle separations from Eq. (7.134) one can obtain a simple asymptotic formula, see Ref. [ 19] for details: F~~~ -- 2rto" sin gtl sin gt2 rl r2 , L

(r~/L) 2 << 1,

Force F (kp~ due to the hydrostatic pressure.

(qa) 2 << 1.

(7.135)

To calculate F~kp~ one can identify the

integration surface Sk in Eq. (7.23) with the part of the cylindrical surface comprised between the horizontal planes z = za and z = zb, see Fig. (7.20). The hydrostatic pressure can be expressed in the form PI for

P-

f

Za<_ Z <-- ~ ( ( p )

(7.136)

Pn for ~'((p)_< z _< z b

where PY = P o - pvgz, Y = I, II is the hydrostatic pressure in the respective phase and P0 is the pressure at level z = 0; as before, ~'((p) is the equation of the contact line. Combining Eqs. (7.23), (7.121), (7.125) and (7.136) one derives [19]

d(pre cosq) d z P -

F(kP~x - -It

where Ap = P i equivalent form

Za

A p g r k ~2(q))cosq0dq)

(7.137)

0

Pn. Using Eqs. (7.29) and (7.131) one can express Eq. (7.137) into the

Chapter 7

338

F(xkp~ - Apga ! 1-cosh~'~ coso) ~.2do) (cosh ~'~ - -

(7.138)

C O S (.0) 2

To calculate F} tp) from Eq. (7.138) one is tosubstitute

~" = ~c,k(OO) from Eq. (7.66)

[or alternatively ~"= (k(V~, CO)from Eq. (7.42)] and then to carry out numerically the integration. The series expansion for long distances leads to the following asymptotic expression for F(xkp) [19]F} kp) -- 2rto" (qrk) 2 hk rj sin N L1 '

(rk]L) 2 << 1,

(qa) 2 << 1.

(7.139)

j,k = 1,2;j 4: k. The comparison of Eqs. (7.135) and (7.139) shows that the ratio F(xkP)/F)k~ _(qr k )z

hk

(7.140)

rk sin~k is a small quantity for the case of small particles, that is for (qrg) 2 << 1; indeed, the Derjaguin formula, Eq. (7.51), shows that hJ(rk sin gtk) = ln(2/yeqr~) and the latter logarithm is a quantity of the order of 1 up to 10. Thus one may conclude that for small particles, (qrk) 2 << 1, F~!kp) is much smaller than Fx(k~ , and therefore in a first approximation F~kp) can be neglected.

sSSSS~

Zb

// t

Za

l l

o

.0fp

"-

dsc~S |

Fig. 7.21. Cross section of a spherical particle: the pressure P is directed normally to the spherical element ds,, whose projection on the vertical cylindrical surface is denoted by dsc.

Lateral Capillary Forces between Partially Immersed Bodies

339

Application to Spherical Particle. In Section 7.3.2 we demonstrated that the expression

for the meniscus profile around two vertical cylinders, ~'('t', o9), can be applied to approximately calculate the capillary interaction between two spheres confined in a liquid film. Similarly, when the deviation of the contact line from the horizontal position is not too large, one can combine one of Eqs. (7.42), (7.46), (7.64) or (7.66) with Eqs. (7.132) and (7.138) to calculate Fx(k~ and ~kp). It should be taken into account that for spherical particles rk and ~k depend on the distance L; the later dependence can be obtained by applying the numerical procedure based on Eqs. (7.101)-(7.102). The applicability of Eq. (7.138) to spherical particles needs additional discussion. Let us consider an element dss from the surface of a sphere, whose projection on the cylindrical surface (see Fig. 7.21) is dsc = (dss)cosO. Then the horizontal projection of the force exerted on the spherical element dss is Pcos0 dss = Pdsc. Note also that z~ and Zb in Fig. 7.21 are the same as in Fig. 7.20. Hence the integration over the spherical belt can be replaced by an integration over the portion of the cylindrical surface comprised between the planes z = Za and z = Zb. In other words, Eq. (7.137), and its corollary (7.138), can be used also in the case of spherical particle.

1.6

5.0 r~ = G = 1 # m o = 40 mN/m

1.2

~

~, ~'

=

r~ = G = 1 p m o -- 40 mN/m

2.5

~P2= 5~

i,-.,a

0.8

~:~

2--5 ~

0.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

O.4

~

t" ~p,= -~p~=-5 -2.5

0.0

(a)

-0.4 ................................................................................................................................................. 0 25 75 50 100

L ~m]

o

t I

(b)

I 1 -5.0

.

0

.

.

.

'

25

.

.

.

.

J

.

.

50

.

.

t

75

.

.

.

.

! 00

L [~m]

Fig. 7.22. Calculated in Ref. [19] plots of capillary force vs. distance L between two semi-immersed vertical cylinders of equal radii, r~ = r2 = 1 lam at various contact angles" (a) contribution of the hydrostatic pressure, ~kp), calculated from Eq. (7.138)" (b) contribution of surface tension, ~ ~ calculated from Eq. (7.134).

Chapter 7

340

Numerical results and discussion. Figure 7.22 shows plots of the calculated F~!k~ and F~kp) vs. the distance L between the axes of two vertical cylinders of radii r~ = r2 = 1 gm. F~k~ is calculated from Eq. (7.134) while F~kp) is calculated by means of Eqs. (7.138) and (7.46). The other parameters values are o" = 40 mN/m and Igtll = Igt21 = 5 ~ Figure 7.22 illustrates the fact that the lateral capillary forces can be either attractive or repulsive depending on the sign of the angles gtl and gt2, cf. Fig. 7.1. The numerical results for F~kp) and F~k~ shown in Fig. 7.22a and 7.22b confirm the conclusion drawn from Eq. (7.140) that for small particles, (qr~) 2 << 1, the force F! ~p~ due to the hydrostatic pressure is much smaller than the force Fx(k~ due to the interfacial tension. For the numerical example shown in Fig. 7.22 F~kp~ is with 5 orders of magnitude smaller than F! ~~ . Table 7.1 contains numerical data calculated for two vertical cylinders of different radii ( r l 10 gm and r2 = 30 gm) and different contact angles (gtl = 10 ~ and gt2 = 1~ The capillary immersion force F)k~=F(x~~ F) kp~ is calculated by means of Eqs. (7.133), (7.134) and (7.138). If the approximations used to derive the latter equations are correct one should obtain FJ~= F~ 2~ (the third Newton's law), irrespective of the fact that the two cylinders have different radii and contact angles. The data in Table 7.1 for F) ~) and F~2) really confirm the validity of the employed approximations. Moreover, the force and energy approaches must be equivalent, that is F~,~ =F)2)= d(Af~)

(7.141)

dL In other words the differentiation of Eq. (7.88) [or Eq. (7.89)], expressing AlL should give the same values of the force as the integral expressions, Eqs. (7.133), (7.134) and (7.138), obtained by means of the force approach. The numerical data in Table 7.1 confirm that Eq. (7.141) is satisfied with a very good accuracy, irrespective of the differences in the procedures and the approximations used to calculate

F! l~ and F (z) in the force approach and d(Af~)/dL in the

energy approach. Note, for example, that in the energy approach we have worked in terms of the average elevation of the contact line, h~, just as if the contact lines were horizontal, see

Lateral Capillary Forces between Partially Immersed Bodies

341

Eq. (7.88), whereas in the force approach the inclination of the contact line, d(,/dq9, plays a central role; indeed Eqs. (7.129), (7.130) and (7.137) give zero force if d~/dq9 is set zero.

Table 7.1. Comparison of the calculated values of Fxr , Fx(2) and d(A~2)/dL for r~ = 10 ~tm, r2 = 30 ~tm, ~ = 10 ~ and ~2 = 1~ for various values of the distance L between the axes of two vertical cylinders, partially immersed in a liquid of surface tension a = 40 mN/m.

s = L/2 [~m]

F~') [N]

F (2) [N]

d(Af~)/dL [N]

50

1.482 x 10 -8

1.485 x 10 -8

1.483 x 10 -8

100

3.054 x 10 -9

3.061 x 10 - 9

3.054 x 10 -9

150

1.737 x 10 -9

1.742 x 10 -9

1.737 x 10 -9

200

1.231 x 10 -9

1.234 x 10 -9

1.231 x 10 -9

250

9.590 x 10 -1~

9.612 x 10 -l~

9.588 x 10 -l~

300

7.876 x 10 -l~

7.893 x 10 -l~

7.875 x 10 -l~

350

6.692 x 10 -l~

6.706 x 10 -l~

6.691 x 10 -l~

400

5.822 x 10 -l~

5.834 x 10 -l~

5.821 x 10 -~~

450

5.155 x 10 -l~

5.165 x 10 -l~

5.154 x 10 -~~

500

4.626 x 10 -1~

4.635 x 10 -1~

4.625 x 10 -l~

The data in Table 7.1 confirm numerically the equivalence of the force and energy approaches to the calculation of the lateral capillary forces. One can find an analytical proof of this equivalence in Ref. [21 ] for the case of two vertical cylinders.

7. 4.2.

ASYMPTOTIC EXPRESSION FOR THE CAPILLARY FORCE BETWEEN TWO PARTICLES

In Section 7.1.3 we derived the asymptotic formula F -- - 2 r t a Q1Q2qKl(qL) for the capillary force by using the energy approach, see Eq. (7.13). Our purpose here is to demonstrate that the

force approach yields the same asymptotic formula. Our starting point is Eq. (7.66), which describes the shape of the contact line on a vertical cylinder (and in first approximation - on a spherical particle as well). W e expand Eq. (7.66) in series for r~/sk << 1, which means that we seek the shape of the contact line at relatively long distances between the two particles (then sl = s2 ~- a = L/2):

342

Chapter 7

~k(co) = hkoo + Qj[Ko(qL) - 2 r k q K l ( q L ) costa+ ...]

(j,k = 1,2;j 4: k)

(7.142)

(j,k= 1,2; j :g: k)

(7.143)

Differentiating Eq. (7.142) one obtains ~=2rkQjqKl(qL)sino)

do)

Since for small particles, that is for (qr~) 2 << 1, the terms zSawx(e)and F~ kp~ represent only small corrections in the expression for the capillary force, then the force is given with a good accuracy by the integral term in Eq. (7.132):

Fx
i d(~ sin r 0 d o cosh z-e - cos o)

(7.144)

Next, we substitute Eq. (7.143) into Eq. (7.144) to obtain the sought for asymptotic formula [64]: F x = 2oQiQ2qKl(qL)sinh'c2

i

sin r

----

2

-,~ cosh ~-~ - cos co

= 27ccyQ, Q 2 q K , ( q L )

(7.145)

(the choice of the coordinate system in Fig. 7.20 implies that Fx > 0 corresponds to attraction). At the last step we have used the fact that Qk - rk sin gt~, the identity [ 19]

si___n_2co/o)

i = 2n: exp(-~" 2 ) -,~ cosh z-2 - cos o)

(7.146)

and the approximation 2sinh'c2 exp(-~'2) = 1

for

z'2 > 2

(r2[$2 << 1)

(7.147)

see also Eq. (7.29). As could be expected, the derived asymptotic expression for the lateral capillary force, Eq. (7.145), is identical to Eq. (7.13), both of them corresponding to the boundary condition of fixed contact angle. Note that during the derivation of Eq. (7.145) it was not necessary to specify whether the capillary force is of flotation or immersion type, or whether we deal with a single interface or with a thin liquid film. We have used only the integral expression for the capillary force, Eq. (7.144), which is valid in all aforementioned cases, as well as Eq. (7.66) for the shape of the contact line. The latter equation accounts /'or the overlap of the interfacial

343

Lateral Capillary Forces between Partially Immersed Bodies

l z

t,

2r

~1

0

"Y//Y/Y/Ill // -.~

Y/Ill ,

v

s

Fig. 7.23. Sketch of the capillary meniscus around a spherical particle, which is situated at a distance s from a vertical wall. The particle is confined in a liquid film whose thickness is uniform and equal to l0 far from the particle and the wall; R2 and r2 are the radii of the particle and its contact line; al and a2 are three-phase contact angles; W~and W2are meniscus slope angles; h2 is the capillary elevation of the contact line at the particle surface. deformations created by the two particles (cylinders) irrespective of the origin of the deformation: weight of the particle or capillary rise (wetting). Consequently, the above derivation of the expression for the capillary interaction by means of the force approach once again confirms the general conclusion that all kind of lateral capillary forces are due to the overlap of perturbations in the interfacial shape created by attached bodies. Note that Eq. (7.145) is an approximate asymptotic formula, which is valid for comparatively long distances between the particles (L >> r~,r2). For not-too-long distances the more accurate analytical expressions for the capillary force from Sections 7.3 and 7.4.1 have to be used.

7. 4.3.

CAPILLARY IMMERSION FORCE BETWEEN SPHERICAL PARTICLE AND WALL

In this section following Ref. [ 18] we consider another configuration: a planar vertical wall and a planar horizontal substrate covered with a liquid layer, which has thickness equal to l0 far from the wall. Our aim is to determine the lateral capillary force between the wall and a sphere, which is partially immersed in the liquid film, see Fig. 7.23. As usual, the coordinate plane x y is chosen to coincide with the horizontal upper surface of the liquid layer far from the sphere and the wall. In addition, the x-axis is oriented perpendicular to the vertical wall. The geometric parameters related to the wall are denoted by subscript 1, whereas those related to the particleby subscript 2, see Fig. 7.23 for the notation.

Chapter 7

344

Following an approach analogous to that from Section 7.2.1 one can find a compound asymptotic solution for the shape of the meniscus in Fig. 7.23 assuming that the meniscus slope is small, i.e. IVn~I2 << 1. The compound solution, obtained in terms of the bipolar coordinates, Eq. (7.25), reads [18]: ~'(z-, (o)=q-lcotale-qX+Q2{2Ko(qr)+ln[(cosh1:-cosoo)r2/(2a2)]

},

x_>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, the contribution of the hydrostatic pressure to the capillary force is found to be negligible (Eq. (7.140) and Fig. 7.21) and one can calculate the capillary force from Eq. (7.132), which represents a contribution from the interfacial tension. The latter expression, can be employed also to calculate the capillary immersion force between particle and wall, see Section 7.4.3. A test of the theoretical expressions for the capillary force, stemming from the alternative energy and force approaches, show that they are in a very good numerical agreement (Table 7.1). In conclusion, the capillary immersion forces can appear in a variety of systems with characteristic particle size from 1 cm down to 2 nm (see Fig. 8.3 below); in all cases the lateral capillary interaction has a similar origin (overlap of interfacial deformations created by the particles) and is subject to a unified theoretical treatment.

7.6.

REFERENCES

1. D.F. Gerson, J.E. Zaijc, M.D. Ouchi, in: M. Tomlinson (Ed.) "Chemistry for Energy", ACS Symposium Series, Vol. 90, p.66, Amer. Chem. Soc., Washington DC, 1979. 2.

J.D. Henry, M.E. Prudich, K.R. Vaidyanathan, Sep. Purif. Methods 8 (1979) 81.

3.

M.M. Nicolson, Proc. Cambridge Philos. Soc. 45 (1949) 288.

4.

K. Hinsch, J. Colloid Interface Sci. 92 (1983) 243.

5.

C. Allain, B. Jouhier, J. Phys. Lett. 44 (1983) L421.

348

Chapter 7

6. C. Allain, M. Cloitre, in: R. Jullien et al. (Eds.), Springer Proceedings in Physics, Vol. 32, Springer Verlag, Berlin, 1988, p.146. 7. B.V. Derjaguin, V.M. Starov, Colloid J. USSR Engl. Trans. 39 (1977) 383. 8. H. Yoshimura, S. Endo, M. Matsumoto, K. Nagayama, Y. Kagawa, J. Biochem. 106 (1989) 958. 9.

H. Yoshimura, M. Matsumoto, S. Endo, K. Nagayama, Ultramicroscopy 32 (1990) 265.

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Lateral Capillary. Forces between Partially Immersed Bodies

349

29. M. Yamaki, J. Higo, K. Nagayama, Langmuir 11 (1995) 2975. 30. K. Nagayama, S. Takeda, S. Endo, H. Yoshimura, Jap. J. Appl. Phys. 34 (1995) 3947. 31. C.A. Johnson, A.M. Lenhoff, J. Colloid Interface Sci. 179 (1996) 587. 32. M. Sasaki, K. Hane, J. Appl. Phys. 80 (1996) 5427. 33. N.D. Denkov, H. Yoshimura, K. Nagayama, Phys. Rev. Lett. 76 (1996) 2354. 34. N.D. Denkov, H. Yoshimura, K. Nagayama, Ultramicroscopy 65 (1996) 147. 35. F. Burmeister, C. Sch~ifle, T. Matthes, M. Bohmisch, J. Boneberg, P. Leiderer, Langmuir 13 (1997) 2983. 36. H. Du, P. Chen, F. Liu, F.-D. Meng, T.-J. Li, X.-Y. Tang, Materials Chem. Phys. 51 (1997) 277. 37. S. Rakers, L.F. Chi, H. Fuchs, Langmuir 13 (1997) 7121 38. S. Matsushita, T. Miwa, A. Fujishima, Langmuir 13 (1997) 2582. 39. J. Boneberg, F. Burmeister, C. Schafle, P. Leiderer, D. Reim, A. Fery, S. Herminghaus, Langmuir 13 (1997) 7080. 40. P.C. Ohara, J.R. Heath, W.M. Gelbart, Angew. Chem. Int. Ed. Engl. 36 (1997) 1078. 41. N. Bowden, A. Terfort, J. Carbeck, G.M. Whitesides, Science 276 (1997) 233. 42. S.V. Kukhtetskii, L.P. Mikhailenko, Doklady Academii Nauk 357 (1997) 616. 43. H. Shibata, H. Yin, T Emi, Philos. Trans. Roy. Soc. London A 356 (1998) 957. 44. F. Burmeister, C. Sch~ifle, B. Keilhofer, C. Bechinger, J. Boneberg, P. Leiderer, Adv. Mater. 10 (1998) 495. 45. K.P. Velikov, F. Durst, O.D. Velev, Langmuir 14 (1998) 1148. 46. H. Aranda-Espinoza, A. Berman, N. Dan, P. Pincus, S. Safran, B iophys. J. 71 (1996) 648. 47. M.Ge, J.H. Freed, Biophys. J. 76 (1999) 264. 48. T. Gil, J.H. Ipsen, O.G. Mouritsen, M.C. Sabra, M.M. Sperotto, M. Zuckermann, "Theoretical Analysis of Protein Organization in Lipid Membranes", BBA - Reviews on Biomembranes, Vol. 1376 (3), Elsevier, Amsterdam, 1998; pp. 245-266. 49. C. Allain, M Cloitre, J. Colloid Interface Sci. 157 (1993) 261. 50. C. Allain, M Cloitre, J. Colloid Interface Sci. 157 (1993) 269. 51. C.M. Mate, V.J. Novotny, J. Chem. Phys. 94 (1991) 8420. 52. M.L. Forcada, M.M. Jakas, A. Gras-Marti, J. Chem. Phys. 95 (1991) 706. 53. A. Marmur, Langmuir 9 (1993) 1922. 54. G. Debregeas, F. Brochard-Wyart, J. Colloid Interface Sci. 190 (1997) 134. 55. O.P. Behrend, F. Oulevey, D. Gourdon, E. Dupas, A.J. Kulik, G. Gremaud, N.A. Burnham, Applied Physics A 66 (1998) $219.

350

Chapter 7

56. H. Suzuki, S. Mashiko, Applied Physics A 66 (1998) S1271. 57. J. Lucassen, Colloids Surf. 65 (1992) 131. 58. B.V. Derjaguin, Kolloidn. Zh. 17 (1955) 207. 59. I.B. Ivanov, P.A. Kralchevsky, in: I.B. Ivanov (Ed.) Thin Liquid Films, Marcel Dekker, New York, 1988, p. 49. 60. P.A. Kralchevsky, K.D. Danov, N.D. Denkov, Chemical Physics of Colloid Systems and Interfaces, in: K.S. Birdi (Ed.) Handbook of Surface and Colloid Chemistry, CRC Press, Boca Raton, 1997. 61. E. Janke, F. Emde, F. L6sch, Tables of Higher Functions, McGraw-Hill, New York, 1960. 62. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. 63. G.A. Korn, T.M. Korn, Mathematical Handbook, McGraw-Hill, New York, 1968. 64. P.A. Kralchevsky, K. Nagayama, Adv. Colloid Interface Sci. 85 (2000) 145. 65. H.B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan Co., New York, 1961. 66. C. Camoin, J.F. Roussell, R. Faure, R. Blanc, Europhys. Lett. 3 (1987) 449. 67. O.D. Velev, N.D. Denkov, V.N. Paunov, P.A. Kralchevsky, K. Nagayama, Langmuir 9 (1993) 3702. 68. C.D. Dushkin, P.A. Kralchevsky, H. Yoshimura, K. Nagayama, Phys. Rev. Lett. 75 (1995) 3454, 69. C.D. Dushkin, P.A. Kralchevsky, V.N. Paunov, H. Yoshimura, K. Nagayama, Langmuir 12 (1996) 641. 70. P.A. Kralchevsky, C.D. Dushkin, V.N. Paunov, N.D. Denkov, K. Nagayama, Prog. Colloid Polymer Sci. 98 (1995) 12. 71. About the measurement of the gravitational constant by H. Cavendish see Rose et al., Phys. Rev. Lett. 23 (1969) 655. 72. P.A. Kralchevsky, I.B. Ivanov, J. Colloid Interface Sci. 137 (1990) 234. 73. A.H. Nayfeh, Perturbation Methods, Wiley, New York, 1973. 74~ B.V. Derjaguin, Dokl. Akad. Nauk SSSR 51 (1946) 517. 75. A.J. McConnell, Application of Tensor Analysis, Dover, New York, 1957. 76. R. Hooke, T.A. Jeeves, J. Assoc. Comp. Mach. 8 (1961) 212. 77. A.P. Prudnikov, Y.A. Brychkov, O.I. Marichev, Integrals and Series, Nauka, Moscow, 1981 ; in Russian.

351

CHAPTER 8 LATERAL CAPILLARY FORCES BETWEEN FLOATING PARTICLES This chapter contains theoretical and experimental results about the lateral capillary interaction between two floating particles, and between a floating particle and a vertical wall. The origin of this "flotation" force is the overlap of the interfacial deformations created by the separate floating particles. The difference between the "flotation" and "immersion" forces is manifested through the distinct physical origin of the respective "capillary charge" Q, which results in a different dependence of Q on the interfacial tension cr and particle radius R. In some aspect these two kinds of capillary interactions 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). Theoretical expressions for calculating the capillary charges and interaction energy of two floating particles are obtained. Numerical results for the dependence of the interaction energy on the interparticle distance, particle radius, density and contact angle are presented and discussed. In all cases the dependence of force vs. distance is a monotonic attraction or repulsion depending on whether the particles are similar or dissimilar. A single particle floating in a vicinity of a vertical wall experiences the action of a "capillary image force". The latter can be formally considered as an interaction between the particle and its mirror image. This force can be attractive or repulsive depending on whether the contact angle or contact line is fixed at the wall. The presence of an inclined meniscus in a neighborhood of the wall may lead to a non-monotonic dependence of the interaction energy on the particle-wall separation. This dependence is derived by means of both energy and force approaches. A convenient asymptotic formula for the capillary force is obtained, which compares very well with the output of the more accurate theory. The derived expressions are in a very good agreement with experimental data for the equilibrium position of floating particles. The obtained theoretical results have been applied to determine the surface drag coefficient of floating particles and the surface shear viscosity of surfactant adsorption monolayers.

352

Chapter 8

8.1.

INTERACTIONBETWEEN TWO FLOATING PARTICLES

8. ]. ].

FLOTATION FORCE: THEORETICAL EXPRESSION IN SUPERPOSITION APPROXIMATION

Similarly to the derivation of the asymptotic formula for the i m m e r s i o n force (see Section 7.1.3 above), one can apply the superposition approximation to derive an expression for the f l o t a t i o n force, which is valid for not-too-small distances between the particles. Following Nicolson [1] let us consider a floating spherical particle of mass mk, which creates an interracial deformation, see Fig. 8.1. Equation (7.8) for k = 1,2, which describes the meniscus profile ~'k(r) around each of the two particles in isolation, holds again. The geometrical configuration and the meaning of the parameters is illustrated in Fig. 8.1 for k = 1. The force due to gravity (weight + buoyancy), Fg(k), which is exerted on the k-th particle (k = 1,2) is counterbalanced by the vertically resolved surface tension force, acting per unit length of the three-phase contact line: Fg(k) = 2Jrcr r~ sin gtk = 27ro- Q/,

(k = 1,2)

(8.1)

see Fig. 8.1 and Eq. (2.2). Here Fg(~) = mk g - Fb with Fb being the buoyancy (Archimedes) force.; expression for Fb for floating particles can be found in Ref. [2]; see Eq. (8.8) below.

z

t

PII

~

,

K

2

L

j

....... ,cysin~Ox .

.

.

.

~ Fg(l ) Fig. 8.1. A heavy spherical particle creates a concave meniscus on an otherwise horizontal fluid interface of tension ~, Fg(l) is the net force due to gravity (a combination of particle weight and buoyancy); gtl is the meniscus slope at the particle contact line of radius rl.

Lateral Capillary Forces Between Floating Particles

353

Let us consider particle 2 situated at a horizontal distance L from particle 1. After Nicolson [ 1] we assume that due to the existence of a meniscus created by particle 1, the mass-center of particle 2 is situated at a distance ~'I(L) below the horizontal plane z = 0, see point 2 in Fig. 8.1. The work carried out by the gravitational force to bring particle 2 from level z - 0 (infinite interparticle separation) down to level z = - ( l ( L ) is [1,3] (8.2)

AWg - - - Fg(2)(I(L) - " - 2IrGQ1Q2Ko(qL)

where at the last step Eqs. (7.8) and (8.1) have been used. Having in mind that

F - - dAWg ~ ; dL

~dK o=(x) dx

- K l (x)

(8.3)

one obtains that (at not-too-small distances) the flotation force, likewise the immersion force, obeys the asymptotic law [1,3]

F = -2rccy Q1Q2qKI(qL),

rk << L.

(8.4)

The above derivation of Eq. (8.4) makes use of several approximations - see the discussion after Eq. (7.13) above. Expressions, which are more accurate than Eq. (8.4) can be found in Section 8.1.4. As already noticed, the equation F = -2trey Q~Q2qKI(qL) expresses the derivative of wetting energy in the case of immersion force, and the derivative of gravitational energy in the case of

flotation force. The fact that the results are formally identical stems from the usage of the same expression for the meniscus shape, viz. Eq. (7.8). Indeed, the meniscus shape (the solution of Laplace equation) "feels" the interacting bodies only through the boundary condition that the meniscus slope is ~k at r = rk, irrespective of the physical irrespective of the physical cause of the interfacial deformation (the capillary rise in Fig. 7.3. or the particle weight in Fig. 8.1). As a result, we have F ~ Kl(qL) for both immersion and flotation lateral capillary forces; however, the coefficient of proportionality, Q1Q2 depends on the type of the capillary force, as demonstrated below.

Chapter 8

354

-

ir

i

l

.

|

ir~i E r2 i ~ ,~----~i ~ ................ "~" .... ~..................... , ..... "7i;....................

,

~ r2i

7 ..................

"

i

|

(a)

(b)

Fig. 8.2. Sketch of two floating particles of radii R1 and R2 separated at a center-to-center distance L. (a) Two light particles. (b) Two heavy particles, rl and r2 are the radii of the contact lines, a~ and a2 are three-phase contact angles, ~ and ~t2 are meniscus slope angles; h~ and h2 are the elevations of the contact lines with respect to the level z = 0 of the non-disturbed horizontal interface far from the floating particles.

8.1.2.

"CAPILLARY CHARGE" OF FLOATING PARTICLES

First of all, let us specify the sign of the capillary charge, Qk - rk sin ~tk,

(8.5)

which is a matter of convention. W e will use the convention that the angle ~k is positive for

light particles (Fig. 8.2a) and negative for heavy particles (Fig. 8.2b). Correspondingly, Qk and hk are positive for light particles and negative for heavy particles. The p a r a m e t e r b~ (k = 1,2) in Fig. 8.2 denotes the depth of i m m e r s i o n of the respective particle in the lower fluid (phase I). The v o l u m e of the part of the sphere which is i m m e r s e d in the lower fluid is

Vt (~) = lr,bkZ(R~ - b~/3),

k = 1,2,

(8.6)

supposedly the contact line is horizontal. The radius of the contact line can be also expressed in terms of b~ : rk = [ ( 2 R k - b~)bk] 1/2,

k = 1,2.

(8.7)

As shown in Ref. [2], the gravitational force (particle weight + buoyancy force) exerted on a floating particle is

Fg(k) = g[(Pl - Pk)VI (k) + (PlI- pk)Vu (k) --/~P ~rkZhk]

(8.8)

Lateral Capillary Forces Between Floating Particles

355

where VuCk) denotes the upper part of particle volume, that immersed in phase H; as before p~ and Pl~ denote the mass densities of the fluid phases I and II, whereas Pk is the mass density of the k-th particle; Ap - p~ - p.. Note that Eq. (8.8) can be used not only for solid spheres, but also for fluid particles (drops, bubbles), like that in Fig. 2.3, which are composed of two axisymmetric segments, but are not spherical as a whole. In the special case of spherical particle of radius Rk and volume Vs r one can write Vu (k) = Vs (k) - VI (k),

4 z R k3 Vs (k) = -~

(8.9)

(q2 = Ap g/or)

(8.10)

Then Eq. (8.8) can be represented in the form [4] Fg(k) = Cr q2(Vl (k) - O k V s (k) - TcrkZhk),

where D k = Pk -- PII

(8.11)

P I -- PII

Combining Eqs. (8.1), (8.5), (8.6) and (8.10) one obtains an expression for the capillary charge

[4]: Qk

= 21q 2 [bk2 (Rk - b k / 3 ) - - - ~ 4D k R k 3 _ rZhk ]

(8.12)

By means of geometrical considerations from Fig. 8.2 one can deduce I/tk - a r c c o s ( b k - l l - a k

t gk

)

(k = 1,2)

(8.13)

Equations (7.56), (8.5), (8.7), (8.12) and (8.13) form a set of 10 equations for determining the 10 unknown parameters: Qk, hk, rk, ~k and bk (k = 1,2) for every value of the distance L between the two floating particles. It turns out [3,4] that for small particles the capillary charge Qk is a rather weak function of the distance L. For that reason one can obtain a convenient approximate formula for Qk. With that end in view we first notice that the lower portion of particle volume can be expressed in the form 1 3 Vt(k) = 7TCRk (2 + 3cos0 k -COS 3 Ok 1

Ok = arcsin(rk/Rk) - ak + ~k.

(8.14)

356

Chapter 8

Combining Eqs. (8.10) and (8.14) one derives Fg(k )

1 2 R k3 ( 2 - a D k +3cos0k - c o s 3 0 k ) = -~rccrq

(qRk << 1)

(18.15)

Here we have used the fact that for qR~ << 1 the term rtr~2h~ in Eq. (8.10) can be neglected. In addition, for qR~ << 1 one can use the approximation 0h = a~. Then the combination of Eqs. (8.1) and (8.15) yields [3,4] 0/, -- Qk~ - ~ q 2Rk3(2_4Dk

+3COSO~k_COS30~k)[I+O(qR~)]

(8.16)

Equation (8.16) allows one to calculate the capillary charge Qk directly from the particle radius Rg and contact angle o~. The capillary charge of a floating particle Qk is a quantity of bounded variation. Indeed, from Eqs. (8.5) and (8.14) one can deduce that Qk = Rk sin(~k + Nk) sinNk. The condition for extremum of Qk reads 0 - dQ~ = Rk [cos(a k +l/t~ ) sinl/t k + sin(a k +l/t~ )cosl/t k ] - R k sin(o~k + 21/tk ) dNk

(8.17)

Equation (8.17) has two roots of physical importance: Nk = - ak/2 and Nk = ( r t - ak)/2; they determine the minimum and maximum values of Qk [4,5]" - Rk sinZ(ak/2) <_ Qk <-RkCOS2(O~k/2)

(8.18)

For Qk < -R~, sin2(a,/2) the particle would sink into the lower phase I; on the other hand, for Qk > Rk COS2(ak/2) the particle would enter the upper phase II. In particular, if the contact angle is o~k = 0, then Eq. (8.18) implies Q~, > 0, that is only light particles can attach to the interface (heavy particles with Qk < 0 would sink into phase I).

8.1.3.

COMPARISONBETWEEN THE CAPILLARYFLOTATIONAND IMMERSION FORCES

Equations (7.13) and (8.4), and their asymptotic form for qL << 1, F - -2//~O"QIQ2 L

r~ << L << q- 1

(8.19)

(q-~ = 2.7 mm for water), show that the immersion and flotation forces exhibit the same functional dependence on the interparticle distance L. On the other hand, the "capillary

Lateral Capillary Forces Between Floating Particles

357

101~ - o=40omN.m-1 _ lO~

~2"-./ /

L=2R

/

PI = 1.0 g . c m -3

lO6 -

/

=6o

_41~.g~ /

./~,o~O~

. _ = o

/

~,

/-,

lO4 102 lO~

.......

)-10

.......

10-9

, ...........

10-8

, ....

10-7

.......

10~

-,,r

: ..........

104

; ..........

10~

10-3

Rim] Fig. 8.3. Energy of capillary attraction AW, in kT units, plotted vs. the radius R of two similar particles separated at a center-to-center distance L = 2R. If AW > kT, the capillary attraction is stronger than the Brownian force and can cause a two-dimensional aggregation of the particles [4,6]. charges", Q1 and Q2, can be very different for these two kinds of capillary force. To demonstrate that let us consider the case of two identical particles, for which R1 - R2 = R and ~zl = oh = a; then using Eqs. (8.5) and (8.16) one can derive [4,6,7] F ~: (R 6 / o ) K 1(eL)

for flotation force

F ~ ~ R 2 K l (qL)

for immersion force

(8.20)

Consequently, the flotation force decreases, while the immersion force increases, when the interfacial tension o increases. Besides, the flotation force decreases much stronger with the decrease of particle radius R than the immersion force. Thus Fflotation is negligible for R < 5 - 1 0 lam, whereas Fimmersion can be significant even for R = 2 nm [4,6]. This is illustrated in Fig. 8.3, where the two types of capillary interaction are compared, with respect to their energy AW(L) = ~; F ( L ' ) d L ' , for a wide range of particle sizes. The values of the parameters used are: particle mass density pp = 2 g/cm 3, density difference between the two fluids Ap = 1 g/cm 3, surface tension a -- 40 mN/m, contact angle ~ = 60 ~ interparticle distance L - 2R, and

358

Chapter8

thickness of the non-disturbed planar film l0 = R. Protein molecules of nanometer size can be considered as "particles" insofar as they are much larger than the solvent (water) molecules. For example, the radius of a water molecule is about 0.12 nm; then a protein of radius R > 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 << 1

~

sin2gtk<< 1

(k = 1,2)

(18.29)

Combining the latter relationships with Eqs. (7.51) and (8.16) one can derive [4]" IQ~I/Rk - (qRk) 2 << 1,

IhffRkl ,- (qRk)2 Iln (qRk)l << 1,

(k = 1,2)

(18.3o)

As demonstrated in Appendix 8A Eqs. (8.29) and (8.30) imply [ l + O ( q 2R k2 )1-7(qrk 1 2 dhk 2R2k )1 -dbk - ~ - = - r k -dllt~ - ~ - - -- - ~ -dQk ) --~--[1 + O ( q

(18.31)

A differentiation of Eqs. (8.23), (8.27) and (8.28), along with Eqs. (8.7) and (8.31) yields [4]:

dAWw

~=-7c6 dL dAW m

dL

~.,(qr k

)~

dh~

(8.32)

R k cosa k --~[1 + O ( q 2 R k )1

k=l,2

dhk

2

2

= nc~ ~ [ O ~ + (q~):n~ coso~ 1--~- [1 + O(q n~ )1 k=l,2

(8.33)

Lateral Capillary Forces Between Floating Particles

361

dAWg dh k R2 = -~cr ~ 2Qk [1 + O ( q 2 )] dL --ffff k k=l,2

(8.34)

Finally, in view of Eq. (8.26) we sum up Eqs. (8.32)-(8.34) to obtain an expression for the capillary flotation force between small particles [4]: F-

dAf~ . . . . dL

7rty Z Q k

dhk

2 2

[l+O(q R k)]

(8.35)

k=l,2

The form of Eqs. (8.32)-(8.35) calls for some discussion. First, one sees that the contributions of the wetting, meniscus surface and gravitational energies have comparable magnitudes. However, when deriving the expression for F the derivative d(AWw)/dL is canceled by a part of d( AWm )/dL, and the result represents a half of the gravitational contribution d( AWg )/dL. Thus, in a final reckoning, F turns out to be (approximately) equal to the half of the gravitational contribution, just as assumed by Nicolson long ago [ 1], see Section 8.1.1. Equation (8.31) shows that the dependence of Q~ on L is rather weak; then the integration of Eq. (8.35) can be formally carried out at constant Qk [4]: 2

2

Af~ =-Jrt7 ~_,(Qkhk -Qk~hk~o)[1 + O ( q R~ )]

(8.36)

k=l,2

The substitution of some of the expressions for hk derived in Section 7.2.2 in Eqs. (8.35) and (8.36) allow one to calculate the dependencies F(L) and Af2(L). More details about the procedure of calculation are given in the next section 8.1.5. In particular, from Eqs. (8.35) and (8.36), along with Eq. (7.56), one recovers the asymptotic expressions

8.1.5.

F = - 2rtcr Q1QzqKI(qL)[ 1 + O(qZRk2)],

rk << L,

(8.37)

Af2 = - 2rtcr QIQzKo(qL)[ 1 + O(qZRk2)],

r~ << L.

(8.38)

NUMERICAL RESULTS AND DISCUSSION

Procedure of calculations. The derived equations enable one to calculate the energy

and force of capillary interaction between two floating particles, AU2 and F. The input parameters are usually the interfacial tension o', the capillary length q-l, the density ratio Dk, the contact angle ak, the particle radius Rk (k = 1,2), and the interparticle distance L.

362

Chapter 8

The remaining 10 parameters, Qk, h,, r,, gtk and bk (k = 1,2) can be determined for every value of the distance L between the two floating particles using the set of Eqs. (7.48), (8.5), (8.7), (8.12) and (8.13). For longer distances between the particles (L >> r,) the usage of Eq. (7.56) is recommended instead of Eq. (7.48). For Rk < 850 lam one can use a much faster iteration procedure proposed in Ref. [4]. Since for small particles the meniscus slope is also small, sin2~, << 1, cf. Eq. (8.29), one can write (8.39)

r, = R, sin(a, + Vt,) -- R~ cosc~k sin~k + Rk sin~k

In view of Eq. (8.5) one can substitute s i n ~ -- Qflrk and then Eq. (8.39) transforms into a quadratic equation for rk, whose solution reads i [Rksin~k+ ( Rk2 sin20r + 4QkRkcosO~k)l/2] r~ = -5

(18.40)

Next, from Eq. (8.3 l) one obtains

Q(,,+,~ __Q~n) ,

__h~n) - 7 (1q r k ) 2(h(n+l) , k )

(k = 1,2)

(18.41) In)

where Q~n+~) and ~, o (') are two consecutive approximations for Q,, as well as .~, h(n+' and h,

are

two consecutive approximations for hk; n = 0, 1, 2 . . . . . The iteration procedure has the following steps [4]" (i) As a zeroth-order approximation one can use rk(0~ - r,oo,

t3(0~ ~k - Q~. . . .

h(~ k -hkoo ,

(8.42) 9

where Qkoo is determined from Eq. (8.16), r~,oo is then calculated from Eq. (8.40) and hkoo is determined from Eq. (7.51). (ii) h (n+~) is calculated from Eq. (7.48) [or Eq (7.56) for L >> 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 << 1; note that L is plotted in log scale along the abscissa of Fig. 8.5. In addition, in Fig. 8.5 one sees that for such small particles (R = 10 lam) the energy of capillary attraction due to the flotation force is relatively small, not larger than several kT, and increases markedly with the rise of the particle density &. For the couple of particles, whose density (Pk = 1.05 g/cm 3) is close to that of the lower fluid phase (Pi = 1 g/cm 3) A ~ is smaller than the thermal energy kT, and in this case the capillary

attraction between the particles is negligible. Figure 8.6 illustrates the fact that for two identical particles the dependence of the energy of capillary attraction, A~, on the particle mass density, &, is a non-monotonic one. Qualitatively, this can be easily anticipated, because for light and heavy particles one has the configurations depicted in Figs. 8.2a and 8.2b, respectively, corresponding to positive and negative meniscus slope angle ~k. Then for some intermediate value of the particle density &* the slope angle becomes ~ = 0, which means that there is no interfacial deformation and capillary attraction; the latter situation corresponds to the upper points (with A ~ = 0) of the curves in Fig. 8.6.

Lateral Capillary Forces Between Floating Particles

365 0.0

0.0 O~

ACI/kT

-5.0

G=70 mN.m" R=I0.0 lam

L=2R -7.5

AOIkT

"'"'",,,,.,,

-2.5

3

",,

'k~ ~

pl,=O

\

\,\

...... O=/u ml'~.m ' R=10.0 !am

', '\,

~176 Pl=l-0g 'cm3

-5.0

~

',',c~ = 9 0 ~ 'k " k

"'",_ ~ P k ",, "",,,

L=2R -10.0

= 1.3 g . e m - a

"...pk= 1.5 g.ern-a "............................

0C=60o p,=l .Og.cm 3 Pll =0

-10.0

. . . . . . . . . 0.O 0.5

cx- 150 ~ ' .... \ .... 1.0 1.5

\ ~"

i

2.0

- 15.0 0

3~0

p~ [g.em -3]

6'0

9'0

12 0

r

15 0

18 0

c~ [deg ]

Fig. 8.6. Plot of the capillary interaction energy Fig. 8.7. Plot of the capillary interaction energy A~2/kT vs. particle mass density Pk Af2/kT vs. contact angle a calculated in calculated in Ref. [4] for three different Ref. [4] for three different values of the values of the contact angle a of two mass density Ok of two identical spherical identical spherical particles of radius R particles of radius R = 10 gm separated at 10 gm separated at a distance L = 2R. a distance L = 2R. Since for gtk = 0 one has Qk = 0, we can estimate the value of Pk* from Eqs. (8.11) and (8.16)" P~ - Pn

_

1

(2 + 3 c o s a k -

(8.45)

cos 3ak)

PI - PlI

In particular, for ak = 90 ~ Pi = 1 g/cm 3 and PlI

"-

0 Eq. (8.45) predicts Pk* = 0.5 g/cm 3 which

exactly corresponds to the maximum of the curve with a = 90 ~ in Fig. 8.6. Figure 8.7 shows plots of A~/kT vs. a, calculated in Ref. [4], which illustrate the dependence of the capillary interaction energy on the contact angle; the three curves correspond to three different values of the particle mass density Pk (two identical particles, o~ - ak). The values of the other parameters in Fig. 8.7 are the same as in Fig. 8.6. One sees in Fig. 8.7 that the dependence of All on a is the most pronounced in the interval 45 ~ < a < 135 ~ whereas outside this interval Al-~ is almost constant. This behavior can be attributed to the fact that the capillary charge Q~ depends on cosock, see Eq. (8.16), and cosa~ varies strongly in the interval 45 ~ < cx < 135 ~ while it is almost constant (coso~ = +1 ) outside this interval. Figure 8.8 shows plots of AQ/kT vs. o', calculated in Ref. [4], which illustrate the dependence of the capillary interaction energy on the value of the interfacial tension (case of two identical

Chapter 8

366

Afl/kT

0

AD./kT

-10 -20

4000

' Pk : 1.20 g.cm-a

3000

-30

.............. "......

-40 -50 -60 -70

/ ../

20

3'0

'

4'0

'

5~0 '

6'0

'

7J0

",,,

",,,. ....."--.,.,.

-,4.

L=2RR~=R'=R= 15 gm

2000

o~ =o~z=c~=60 ~ pi=l.0g.cm 3 p.=0

/,'" / !pk = 1.5o g.cm-3

R 1 = 40 # m

1000

'

0

................. L~

Rl = 2 0 p,m

2;0

80

~;0

7;0

iooo

L [/zm]

(y [ m N . m l ]

Fig. 8.8. Plot of the capillary interaction energy A~/kT vs. interfacial tension o" calculated in Ref. [4] for three different values of the mass density Pk of two identical spherical particles of radius R = 15 gm and contact angle ak = 60 ~ separated at a distance L = 2R.

RI= 30/zm

CY=35 mN.m -~ R2=20 g m 91=2.0 g.cm 3, p2=0 0~=80 ~ 0~2=15~ 91=l'0g "cm3, Pll=0

Fig. 8.9. Plot of the energy of capillary repulsion zkQ/kT vs. distance L between a heavy particle of radius R1 and a light particle (bubble) of radius R2 = 20 gm; the curves are obtained in Ref. [4] for three different values of R~.

particles). One sees that the lower the interfacial tension or, the greater the magnitude of the interaction energy IAtll. To understand this behavior one first notices that q2 oc l/or and then Eq. (8.16) gives Q, o~ 1/o-. Then in accordance with Eq. (8.38) one can write All o~ cr Qk2 o< l/or, which explains the trend of the curves in Fig. 8.8. From a physical viewpoint this means that when the interfacial tension (3"is smaller, a floating heavy (or light) particle creates a larger surface deformation (sin gq o~ Qk), which gives rise to a stronger capillary attraction if the menisci formed around two such particles overlap. This behavior of the dependence A ~ vs. (3has been first theoretically established by Chan et al. [3] for the case of flotation forces. Note, however, that in the case of immersion force the trend is exactly the opposite: At) grows with the increase of o', i.e. At) o~ o', see Eq. (8.20) and Ref. [6]. Up to here, all considered numerical examples (Figs. 8.4-8.8) correspond to the case of two identical particles, which experience attractive capillary force. The derived equations enable one to obtain theoretical results also for the case of two non-identical particles, which may experience repulsive capillary force. As an example, in Fig. 8.9 we give the plot of Afl/kT vs. L for a couple of dissimilar particles: a heavy particle of mass density Pl = 2 g/cm 3 and a hollow sphere (bubble) of mass density P2 = 0; the values of the other parameters are shown in the

367

Lateral Capillary Forces Between Floating Particles

figure. The configuration resembles that in Fig. 7. l c and one could expect that the interaction between the two dissimilar particles will be repulsive. The three curves in Fig. 8.9, calculated in Ref. [4] for three values of the radius of the heavy particle, Re = 20, 30 and 40 pro, really correspond to AU2 > 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~ << L,

(8.46)

where 10 for fixed contact angle at the wall; -

for fixed contact line at the wall.

(8.47)

Such capillary interactions between a floating particle and a vertical wall, which are equivalent to the interaction between the particle and its mirror image, are termed "capillary image forces" in Ref. [9]. In fact, they resemble the electrostatic image forces, appearing when an electric charge imbedded in a medium of dielectric permittivity el is located in the neighborhood of a boundary with a second medium of permittivity e2. The electrostatic image force is proportional to the difference (el - e2), see e.g. [11,12], and consequently, this force can be repulsive or attractive depending on whether el > 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 ~ <90~ ~'~(x) is the non-disturbed meniscus of the wall and ~o(x,y) is the interfacial perturbation caused by the particle. (b) Typical dependence of the capillary interaction energy Af~ vs. distance s corresponding to the above configuration of floating particle and wall [9]. along the inclined meniscus. We will restrict our considerations to the case of small meniscus slope, sin21//1 << 1,

(I/./1

=

90 ~ - oh)

(8.48)

(otherwise the gravitational force will be predominant and the effect of the lateral capillary force becomes negligible). Equation (8.48) corresponds to the nontrivial case, in which the total force exerted on the particle is an interplay of the gravitational effect (particle weight + buoyancy force) and the effect of the capillary image force. Prior to any quantitative considerations one can anticipate qualitatively the trend of the particle-wall interaction: In Fig. 8.11 a the contact angle oh < 90 ~ is assumed constant, but the contact line on the wall is mobile. In such a case (like in Fig. 8.10a) the capillary image force is attractive, whereas the gravitational force will push the particle away from the wall. If the particle is large enough, the capillary image force will prevail at short distances, whereas the gravitational force will prevail

Lateral Capillary Forces Between Floating Particles

(a)

371

~" L~'--"

s

A~

.i

S*

0 (b)

1

Fig. 8.12. (a) Sketch of a heavy particle of radius R2 floating at a distance s from a vertical wall; the three-phase contact line is fixed at the edge of the wall, which is situated at the level z - H below the horizontal liquid surface z=0 far from the wall. ~'l(x) is the non-disturbed meniscus of the wall and ~o(x,y) is the interfacial perturbation caused by the particle. (b) Typical dependence of the capillary interaction energy Af~ vs. distance s corresponding to the above configuration of floating particle and wall [9]. at long distances. Therefore, one can expect that at some distance s = s* these two forces will counterbalance each other and the energy of the particle Af~ will exhibit a maximum when plotted vs. the distance s, see Fig. 8.11 b. If the slope of the meniscus at the wall has the opposite sign, i.e. o~ > 90 ~ and ~, < O, then the plot of A ~ vs. s for a heavy particle (Q2 < O) will be a monotonically increasing curve, because in this case both the gravitational and the capillary image force will have a tendency to bring the particle closer to the wall. From a physical (including experimental) viewpoint more interesting is the case presented in Fig. 8.12: the position of the contact line is fixed at the edge of a plate or to the boundary between hydrophilic and hydrophobic regions on the wall. As mentioned above, in this case the capillary image force is repulsive. Let us denote by H the z-coordinate of the contact line on the wall; the choice of the coordinate system is shown in Fig. 8.12a, where H < 0. In such a case, the gravitational force will tend to bring a floating heavy particle (with Q2 < 0) closer to the wall. Then at some separation s = s* the gravitational and the capillary image forces can

Chapter 8

372

counterbalance each other and the particle will have a stable equilibrium

position,

corresponding to a minimum of the energy Af2 (Fig. 8.12b). The dependence of Af2 vs. s is similar if H > 0 and the particle is light (Q2 > 0). The existence of a minimum of Af2 at s = s* allows one to determine experimentally the dependence of s* on H and thus to verify the theoretical predictions, see Section 8.2.7 below.

Shape of the contact line on the particle surface.

As in Chapter 7, to quantify the

capillary interaction we have to solve the Laplace equation and to determine the meniscus shape. Since we assume small meniscus slope, the Laplace equation can be linearized, see

((x,y)

Eq. (7.6), and the solution z -

can be formally expressed as a sum of the meniscus (l(x)

formed at the wall in the absence of a floating particle, and the deformation

(o(x,y)

created by

the particle in the absence of inclined meniscus at the wall (gtl = 0, see Fig. 8.10) [9]:

~(x,y) = (o(x,y) + (~(x). Both

~o(x,y) and

(8.49)

~'l(x) satisfy the linearized Laplace equation of capillarity:

o~2(0 o~ 2~'o 2( O~X'-'-'-'-~nt- 0y 2 ' -- q 0,

d2~'l = q2~-

(8.50)

dx 2

The solution for (l(X), which levels off at infinity, has the form: (l(X) = q-1 tan~rl exp(-qx)

(fixed contact angle ctl = 90 ~ - ~1)

(8.51)

(l(x) = H exp(-qx)

(fixed contact line on the wall)

(8.52)

On the other hand, consequently,

(o(x,y)

(o(x,y)

corresponds to the configurations depicted in Fig. 8.10, and

is determined by Eqs. (7.38)-(7.44) with Q1 = Q2 in the case of fixed

contact angle and with Q~ = - Q 2 in the case of fixed contact line. Further, in the case of fixed contact

angle

the shape of the contact line on the particle surface, (0(co), is determined by

Eq. (7.64), that is

~o (co) = h:= + Q2Ko ( 2 q a 2

s - r 2 cos co

I,

(fixed contact angle al)

In Eq. (8.53) o9 is bipolar coordinate, cf. Eq. (7.25); in addition,

(8.53)

Lateral Capillary Forces Between Floating Particles

h2oo = Q 2 In ~ 2

,

373

a --- (s 2 -

r22) 1/2.

(8.54)

~'e q r2

The validity of Eq. (8.53) is limited to small particles, (qr2) 2 << 1, and small meniscus slope, IVii~'0l2 << 1, which is the case considered here. Next, our purpose is to derive a counterpart of Eq. (8.53) for the case of fixed contact line at the wall. As in Section 7.2.2 we will use the method of the matched ("outer" and "inner") asymptotic expansions [13]. For large particle-wall separations, (qa) 2 > 1, one can use the superposition approximation representing the meniscus shape around a couple of particles (the particle and its mirror image) as a sum of the deformations, created by two isolated particles. Since the mirror image now has the opposite capillary charge (see Fig. 8.10b) the superposition approximation yields Eq. (7.59), but with " - " instead of "+" [9]: ~"0~

"-

(qa) 2 _> 1

QzKo(qrr) - QzKo(qr;),

(8.55)

where the superscript "out" means that Eq. (8.55) is valid in the outer asymptotic region of nottoo-small interparticle separations, (qa) 2 > 1, in which the superposition approximation can be applied; r; and rr are expressed again by Eq. (7.62). Taking into account the known asymptotic formula [ 14-1 6] 2 K0(x)-- l n ~ ~'ex

for x << 1

(8.56)

and using Eq. (7.62) one obtains the "inner" limit of Eq. (8.55) for a-+O (r;, rr --~ 0): (~"0~

in "-

Qeln(r;/rr) = Qzln[(a + s)]r2] = Q2372

(8.57)

Here 372is the value of the bipolar coordinate 37at the contact line, see Eq. (7.57). On the other hand, analytical expression corresponding to the inner asymptotic region, (qa) 2 << 1, can be obtained setting 37= 372, Ql = -Q2 and 37~= 372 in Eq. (7.42) [9]: ~"0in(0-)) -"

Q2 ~'2 (co),

(qa) 2 << 1,

(8.58)

where ~2 37~(o3) - 372+ ~l= n tanh(n372) exp(-nT2) cos (no))

(8.59)

374

Chapter 8

For large separations ~ --->~o, r'2 ---~r2, and then with the help of Eqs. (8.57) and (8.58) one may check that (~'oin)~ = (~'o~ in,

(8.60)

as required by the method of the matched asymptotic expansions [13]. Equation (8.58) can be represented in the equivalent form

~"0in(co) .

Q2In 2. Q2 2 . . In yeqr2 ?'eq?2 (C0)'

r2 (re) - r2 exp,"2 ((o)

(8.6 !)

The compound solution, which is valid in both the "inner" and "outer" region reads [9]" (fixed contact line at the wall)

~'0(co) = h2= - K0(q g2 if.o) )

(8.62)

where h2oo is defined by Eq. (8.54) and 72fro) is defined by Eqs. (8.59) and (8.61). Having in mind Eq. (8.56) one sees that for small distances, (qa) 2 << 1, Eq. (8.62) reduces to the "inner" expression Eq. (8.61), whereas for longer distances, (qa) 2 _> 1, Eq. (8.62) asymptotically tends to the "outer" expression Eq. (8.55). Equations (8.53) and (8.62) will be used in Section 8.2.5 to quantify the particle-wall interaction in the framework of the force approach.

8.2.3.

ELEVATION OF THE CONTACT LINE ON THE SURFACE OF THE FLOATING PARTICLE

Here we continue the theoretical description of the configurations of particle and wall depicted in Figs. 8.11 and 8.12. In Ref. [9] it has been established that the capillary charge of the floating particle is given by the expression Q2 = 89

-b2/3)-4DzR~

-rZh2]

(8.63)

which is in fact Eq. (8.12) for k = 2, but the capillary elevation h2 now has contributions from both the particle meniscus ~'0 and the meniscus on the wall ~'l, see Figs. 8.11 and 8.12, and Eq. (8.49): h2 = h20 + h2!

(8.64)

Lateral Capillary Forces Between Floating Particles

375

Below we specify the expressions for h2 in the two alternative cases of fixed contact angle a~ and fixed elevation H at the wall. Fixed contact angle at the wall. As already discussed, in this case the shape of the

meniscus ~'0 is identical to that around two similar floating particles of capillary charge Q2, see Fig. 8.10a. Consequently, h20 can be expressed by means of Eq. (7.55); in addition, h21 can be determined from Eq. (8.51): h21 = ~'l(S). Thus in view of Eq. (8.64) one obtains [9] h2 = q-i tangtl exp(-qs) + h2~ + QzKo(q(s + a))

(fixed contact angle cq);

(8.65)

see also Eq. (8.54). Fixed contact line at the wall. In this case h21 can be determined from Eq. (8.52):

h21 =

~l(S)

-"

(8.66)

Hexp(-qs)

To determine h20 we will employ again the method of the matched asymptotic expansions. The value of h20 in the inner asymptotic region of short particle-wall separations can be obtained by integration of Eqs. (8.58)-(8.59) in accordance with Eq. (7.47):

l

h~n~- - ~ 2

~ondl--Q2"c 2

c2

(qa) 2 << 1,

(8.67)

"t'2= ln[(a + s)/r2].

(8.68)

where

f2

=

T2 +

~2 n~l;= tanh(n~'2)exp(-2nz'2),

On the other hand, in the outer asymptotic region of longer particle-wall separations h20 can be expressed by means of Eq. (7.53) with Q =-Q2: h Out

20 - h2= __ Q2Ko(2qs)

(qa) 2 ~>1 ,

(8.69)

(qa) 2 << 1

(8.70)

Equation (8.67) can be rewritten in the form h~ - Q 2 l n - - ~ 2 - Q2 In 2 ~eqr2 ~'eqr2 expf 2

The compound expression can be formally obtained if the logarithms in Eq.(8.70) are exchanged with K0-functions in agreement with Eq. (8.56) [9]:

Chapter 8

376

(8.71)

h20 = h2~ - Q2Ko(qr2 expf2)

see also Eq. (8.54). For short particle-wall separations Eq. (8.7 l) reduces to Eq. (8.70), whereas for longer separations f2--)2"2 and Eq. (8.71) transforms into Eq. (8.69). Finally, substituting Eqs. (8.66) and (8o71) into Eq. (8.64) one obtains the sought-for expression for the capillary elevation of the contact line on the particle surface: h2 = Hexp(-qs) + h2~ - Q 2 K o (qr 2 expf 2)

8.2.4.

(fixed elevation at the wall)

(8.72)

ENERGY OF CAPILLARY INTERACTION

In accordance with Eq. (7.16) the energy and the force of capillary interaction can be presented in the form A~'-~ "- ~"~(s) - ~'-~(c~) = AWw q- AWm q" AWg,

F =-d(A~)/ds,

(8.73)

where the additive constant in the expression for Af2 is determined in such a way that Af2---~0 for s--->,,,,; the last three terms in the expression for Af2 represent the wetting, meniscus and gravitational contributions, respectively. Following the derivation of Eq. (8.23) one can deduce [9]: AWw = - 2 ~

[Rzb2 cosa2 + ~Qzq -1 tan N1 e x p ( - q a ) ] - Ww=,

(8.74)

where X is defined by Eq. (8.47) and the constant Ww= is defined in such a way that AWw---~0 for s---~,~; the two terms in the brackets represent, respectively, contributions from the wetting of the particle and the wall. Further, in Ref. [9] it is proven that the sum of the gravitational and the meniscus surface energy can be expressed in the form (8.75)

AWm + mWg = m w m q- mWg

where AWg is defined by a counterpart of Eq. (8.28),

mWg" = -/rcr{2O2h2 - q2[ 88163 ~

2 +(402R3-R2b22 +~lb32)(R2-b2)]}-ff'~,oo

(8.76)

and A W m "- ~G[Q2h2 - ( - 1 ) X Q 2 ~ l ( s ) - r2-

7(qr2

Al/l~m=

(8.77)

Lateral Capillary Forces Between Floating Particles

377

Here ~ is defined by Eq. (8.47); ~'l(x) is to be calculated from Eq. (8.51) or (8.52) depending on whether the contact angle or contact line is fixed at the wall; likewise, h2 is to be calculated from either Eq. (8.65) or Eq. (8.72); the constant AWm= is defined in such a way that AWm -->0 for s--->,,,,. Equations (8.73)-(8.77) determine the dependence AF~ = A~(s). A convenient procedure of numerical calculations is described in Section 8.2.6 below. A relatively accurate and simple asymptotic formula for the force F experienced by the floating particle can be derived in the following way [9]. Equation (8.31) for k = 2 can be presented in the form: dh2 dill 2 r2 dr2 dQ 2 =-r 2-= ~ = _ _ _ [1 +O(q2R~)]--~(qr2)2---;--[l+O(q ds ds R 2 - b 2 ds ds

db 2

2

2

R 2 )],

(8.78)

see also Eq. (8A.3) in Appendix 8A. The differentiation of Eq. (8.74), along with Eqs. (8.77), gives:

[

I

dAWw )2 dh2 qr 2 ds =Tccy - ( q r 2 R 2 coso~2 --~s + ~, 2Q2 +

dh21

tanl/tle

qsI oq2R2 [1 +

)]

(8.79)

Next, differentiating Eq. (8.77) along with Eq. (8.78), and taking into account that b 2 - R 2 ~R2cosa2 and d~l(s)/ds =-q~'l(S), see Eqs. (8.51)-(8.52), one obtains" dAWm =7c~ ([Q2 - l ( q r 2 ) 2 h 2 +(qr2)2R2coso~2 ] dh2 + ds ds (-1)Z q~'(s)+q[qr2~(s)]2

(8.8o)

x[l +O(q2R2)]

The differentiation of Eq. (8.76) gives an expression analogous to Eq. (8.34)" ....,

dAWg

= -Jvcr ~ 2Q2 k=l,2

dh2

7-s

2

2

(8.81)

[1 + O(q R 2 )]

Since our purpose is to obtain the long distance asymptotics, that for s >> r2, we notice that in this limit both Eq. (8.65) and Eq. (8.72) can be presented in the form h2 =

~'l(S)

-1"

h2~ + Q2Ko(2qs),

Then in view of Eqs. (8.51)-(8.52) one obtains

s >> r2,

(8.82)

Chapter 8

378

dh 2

ds

-q~'l(S) - 2qQ2Kl(2qs),

s >> r2,

(8.83)

Finally, in accordance with Eq. (8.73) we sum up Eqs. (8.79)-(8.81) and substitute Eq. (8.83) in the result; after some algebra we obtain [9]:

F(s)~--rccrq [2Q2~l(S)+(qr2~, (s))2-2(-1)~Q2Kl(2qs)](l+O(q2R2))2

(s>> r2)

(8.84)

~'l(s) is to be substituted from Eq. (8.51) or (8.52) depending on the boundary condition on the wall; ~ is defined by Eq. (8.47). The range of validity of Eq. (8.84) is verified in Fig. 8.15 below. The meaning of the three terms in Eq. (8.84) is the following. First we notice that the gravitational force exerted on the particle is Fg -- 2rtr2cysingt2 = 2/1;0"Q2, cf. Eq. (8.1). In addition, the slope of the interface is characterized by sin ~ s ) = tan ~ s ) = d~'l (s)/ds = -q~l (s).

(8.85)

Then one obtains-2tier Q2 q~l(s) = Fg sin~s). Hence, the first term in the brackets in Eq. (8.84) expresses the effect of the gravitational force, Fg sin gt(s), which tends to "slide" the particle along the inclined meniscus. The second term in the brackets in Eq. (8.84),--g(~q(qr2~l(S)) 2, is proportional to rtr22, that is to the area encircled by the contact line. This term takes into account the pressure jump across the interface. The respective force can be estimated multiplying the area ~r22 by the hydrostatic pressure A~gh 2 and by sin~tt(s) ----q~l(s) to take a projection along the tangent to the meniscus. Taking into account the fact that h2 = ~l(S) and Apg = crq2, one o b t a i n s (rr,ri2)(Apgh2)sin Ill(s ) =

-Tr,CYq(qr2~l(S))2. This term is always negative, i.e. it always corresponds to an effective particle-wall attraction [9]. The third term in the brackets in Eq. (8.84), 27rcr(-1) a qQ2 K, (2qs), expresses the contribution of the capillary image force, see Eq. (8.37), which is attractive in the case of fixed contact angle at the wall (Fig. 8.10a), but repulsive in the case of fixed contact line at the wall (Fig. 8.10b). Equation (8.84) has found applications for the interpretation of experimental data about the measurement of the surface drag coefficient of floating particles and surface shear viscosity of

Lateral Capillary Forces Between Floating Particles

379

surfactant adsorption monolayers, see Section 8.2.7 below. Note that Eq. (8.84) can be integrated at fixed Q2 to obtain an approximate expression for the interaction energy [9]"

A~(s)=-/I:Cr

[ 2Q2 r

1

~'l(S)) 2 - ( - 1

),~ Q22 KI (2qs)lO+O(q2

R22));

(s >> 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 << 1) (8.102)

Since F ~- Fx = Fx {m + Fx~p), one sums up Eqs. (8.100) and (8.102) and as a result one arrives again at Eq. (8.84). In other words the energy and the force approaches give the same asymptotic expression for the capillary force, as it should be expected.

8.2.6.

NUMERICAL PREDICTIONS OF THE THEORY AND DISCUSSION

The procedure of numerical calculations, proposed in Ref. [9], is the following: (i) The input parameters are the mass densities of the phases, PI, P~I and/92, the particle radius and contact angle, R2 and a2, and the distance s between the particle and wall. Then q = (Apg]cy) 1/2 and from Eq. (8.11) one calculates Oz. In addition, the parameters gq and H are known in the case of fixed contact angle or line, respectively, see Fig. 8.10. Note that ~ and H can be both positive and negative; for example, in Fig. 8.11 gq is positive, whereas in Fig 8.12 H is negative. (ii) Equations (8.5), (8.63), (8.65) and the two geometrical relationships, be = R2[1 + cos(o~2 + Ipr2)],

r2 = [(2R2 - bz)b2] 1/2

(8.103)

form a set of five equations for determining the five unknown variables Qe, ~2, he, r2 and b2. Note that Eq. (8.72) must be used instead of Eq. (8.65) when the contact line (rather than the contact angle) is fixed at the wall. To solve the aforementioned system of five equations one may use the following procedure of iterations [9]: ( 1) As a zeroth-order approximation one may use r2 t~ = R2 sino~2,

1//2 (0) ----- 0

and Q2 (~ = 0;

Lateral Capillary Forces Between Floating Particles

383

(2) Q2 (k+l) = r2~k)sin~t2~k), k = 0, 1,2 .... 9cf. Eq. (8.5)" (3) h2(k+l) is calculated from the values of r2 (k) and Qz(k): in the case of fixed contact angle

Eq. (8.65) is used; in the case of fixed contact line Eq. (8.72) is used; (4) b2 ~k+l) and r2(k+l) are calculated from the value of ~2 ~) using Eq. (8.103);

(5) ~2 ~+~) is calculated from Eq. (8.63), along with Eq. (8.5), substituting the values of h2 ~k), b2~k) and rz(k); (6) If I1 - Qz(k)/Q2(k+I)I <

g

the iteration process stops; otherwise the iterations continue from

step (2). Here e is a small parameter determining the accuracy of the numerical calculations; for example, one could set e = 10-8. (iii) The limiting (for s--->oo) values Q2oo, ~2oo, h2oo, r2oo and b2oo are determined from the same set of five equations, in which Eq. (8.65) or (8.72) for h2 is exchanged with Eq. (8.54) for h2oo. (iv) Using Eq. (8.73), along with Eqs. (8.74)-(8.77) one calculates the energy A~2 and force F of capillary interaction by means of the energy approach. (v) Alternatively, one can use the force approach, i.e. Eq. (8.99) along with Eqs. (8.51) and (8.53) in case of fixed contact angle, or along with Eqs. (8.52) and (8.62) in case of fixed contact line; thus the force F can be calculated for each given distance s, and then the energy A~2 can be obtained by means of numerical integration of the function F(s). In Ref. [9] it was established that the numerical procedures based on the energy and force approaches

give very close results;

small differences may originate from the used

approximations for small particle and small meniscus slope.

Numerical results and discussion. As illustrative examples here we present numerical results obtained in Ref. [9]. The curves in Figs. 8.14a and 8.14b are calculated for the same particle (R2 -- 500 Jam, a2 = 70 ~ floating at air-water interface ( a = 72.4 mN/m, Ap = 1.0

g/cm3), but the boundary conditions at the wall are different: fixed contact angle c~ = 90 ~ - ~l (Fig. 8.14a) and fixed contact line (Fig. 8.14b). The solid lines are calculated using the procedure described above, whereas the dashed lines are obtained by means of the approximate expression Eq. (8.86), where ~'l(s) is determined from Eqs. (8.51) or (8.52). One sees that the

Chapter 8

384 ~1=1 ~ (fixed c o n t a c t a n g l e )

ACzlkTx 10 "11

9(1)=13.5 g/cm 3

2.0

13(2)= 8.9 g/cm 3

2.o

1.o i

1.0

0.0

0.0

-1.0 -2.0 (a)

1,

0.0

I ] I

G 72.4 9 mNlm ~o - 1.0 g/cmS R2 = 500 Jlnl (z= = 70=

i ! i l i

i

.

.

.

s.o

.

.

.

.

.

.

.

.

10.0

s/R2 .

.

.

.

.

.

1 s.o

H = - 5 0 ~m (fixed c o n t a c t line)

AC~IkTx 10 "11

G = 72.4 mNlm ~o - 1.0 glcmS _.~" i .

a2=70 =

-1.o -2.0 20.0

( b )

0.0

9(3)= 5.4 g/cm 3 '

'

"

s.o

10.0

s/R2

"

i s.o

-

20.0

Fig. 8.14. Plots of the interaction energy Af~ vs. separation s between a floating particle and a wall: (a) in the case of fixed contact angle at the wall (al = 90 ~ - gtl = const.) and (b) in the case of fixed contact line at the wall (H = const). The different curves, calculated in Ref. [9], correspond to different values of particle mass density. The solid and dashed lines are obtained using Eqs. (8.73) and (8.86) respectively. approximate

expression

(8.86)

is rather accurate,

except for very short particle-wall

separations. The different curves in Fig. 8.14, as well as in Fig. 8.15, are calculated for three different values of the particle mass density, corresponding to the densities of mercury, copper and titanium. The most important difference between the curves in Fig. 8.14a and 8.14b is that a m a x i m u m of Af~ (state of unstable equilibrium) exists in the case of fixed contact angle (cf.

Fig. 8.11), whereas a m i n i m u m of Af~ (state of stable equilibrium) is present in the case of fixed contact line (cf. Fig. 8.12). The existence of such a state of stable equilibrium at some distance s = s* has been established experimentally in Ref. [18], see Section 8.2.7 below. Figure 8.15 shows the curves of the force F vs. distance s, corresponding to the curves A ~ vs. s in Fig. 8.14. The solid lines are calculated by means of the more rigorous set of equations, Eqs. (8.73)-(8.77), whereas the dashed lines are obtained by means of the approximate Eq. (8.84). One sees that the latter approximate formula gives amazingly good numerical results. Qualitatively the plots of F vs. s in Fig. 8.15 look similar to the plots of A ~ vs. s in Fig. 8.14; the points in which F = 0 correspond to the maxima or minima of the Aft(s) curves. In the latter points the capillary image force, that is the term Fimage -- 2~o'(-1 )~qQ22 Kl (2qs)

(8.104)

in Eq. (8.84) is exactly counterbalanced by the gravitational force exerted on the particle

Lateral Capillary Forces Between Floating Particles

385

~1=1~ (fixed contact angle)

F[N] x 107

2.0

:

1.0

i

.2

3

1_

P(1)=13.5 g/cm3 p(2)= 8.9 g/cm 3

F[N]

p(3)= 5.4g/cm 3

-1

(a)

Li

~/

/

o/ i / ! !

lill

"2"~

s.o . . . .

"= = 500.m

....

I I I ~, !t ~

(~=72.4mN/m Zko = 1.0 g/cm8 R2 = 500 pm

It 1~ 1j~

0,0 .............................................................

z~o=l.og/cms

l~176176

2.0

1.0

o.o 4---/i .............................................

H=-50 l~m (fixed contact line)

x 10 7

2r

(b)

-1.o

3

"2"00.0 . . . .

5'.0 . . . .

2 1 1 I~.0' S"--2//'f '1 5.0 . . . .

20.0

Fig. 8.15. Plots of the interaction force F vs. separation s between a floating particle and a wall: (a) in the case of fixed contact angle at the wall (a~ - 90 ~ - ~ = const.) and (b) in the case of fixed contact line at the wall (H = const). The different curves, calculated in Ref. [9], correspond to different values of particle mass density. The solid and dashed lines are obtained using Eqs. (8.73) and (8.84) respectively. (weight + buoyancy force), this is the term Fgravity ~ -rtcrq[2Q2~](s) + (qrz~l(s)) 2]

(8.105)

in Eq. (8.84), i.e. F = Fimage -t- Fgravity -" 0. It should be noted that the plot of F vs. s is not always non-monotonic as it is in Fig. 8.15. Indeed, if the contact angle o:~ in Fig. 8.11 were larger than 90 ~ or the contact line elevation H in Fig. 8.12 were positive, then Fimag e and Fgravity would have the same sign and the net dependence F(s) would be monotonic. To examine the dependence of the force F on the particle contact angle ~2 in Fig. 8.16 we plot the dependence of F vs. a2 for a fixed particle-wall separation: s = 5R2; the other parameter values are the same as in Figs 8.14 and 8.15. One sees that the dependence of F vs. o~2 is rather weak in both cases: fixed contact angle (Fig. 8.16a) and fixed contact line (Fig. 8.16b). This fact implies that measurements of the capillary force, F, or of the equilibrium distance particlewall, s*, cannot be used as a method for determining contact angles of small particles. On the other hand, it turns out that the theoretical results for F give the possibility to determine the drag coefficient of particles sliding down an inclined meniscus and then to extract the coefficient of surface shear viscosity from the data, see Section 8.2.7.

Chapter8

386 F[N]

• 10 7

, F [ N ] • 10 7

~1=1 ~ (fixed contact angle)

1.0

H=-50 ~tm (fixed contact line)

3.0 s ,, 5/12

9(2)= 8.9 g/cm 3 0.0 .....................................................................

-1.0

9(1)=13.5 g/era 3

1.0 s = 51:12

-2.0 ('a)

R~ = 500 lun (~ = 72.4 mN/m ~o = 1.0 g/cma

2.0

"

~

~

90)=13-5 g/cm3

R2 = 500 pm a = 72.4 mN/m Z~o = 1.0 g/craB

"3"~ . . . . 8'0 . . . . e'o. . . . 9 0 ' " i 2 0 " a2 [deg]

oo

i~O"

ieo

(b)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-i;i;;=--6.~i-~c;;;~

-1.0 0 . . . . a'o. . . . do . . . . 9 ' 0 i20" a2 [deg]

i50''

i60

Fig. 8.16. Capillary force F as a function of the particle contact angle a2 for a fixed value of the particle-wall separation, s = 5R2: (a) in the case of fixed contact angle at the wall (a] = 90 ~ ~t~ = const.) and (b) in the case of fixed contact line at the wall (H = const). The two curves, calculated in Ref. [9], correspond to particle mass density p(~) = 13.5 g/cm 3 and /9(2) = 8.9 g/cm3o

8.2.7. EXPERIMENTALMEASUREMENTSWITHFLOATINGPARTICLES The configuration with repulsive capillary image force, which is depicted in Fig. 8.12, is realized experimentally in Refs. [18] and [19] as shown in Fig. 8.17. In these experiments the "wall" is a hydrophobic Teflon barrier, whose position along the vertical can be precisely varied and adjusted. The total lateral capillary force exerted on the particle depicted in Fig. 8.17 can be described by the asymptotic expression Eq. (8.84) in which ~'~(s) is substituted from Eq. (8.52):

F(s) =-~(yqI2Q2He-qS +(r2qHe-qS~-2Q2KI(2qs)I(I+O(q2R2))

(8.106)

As before, H characterizes the position of the contact line on the wall with respect to the nondisturbed horizontal liquid surface far from the vertical plate (Fig. 8.17) and s is the particlewall distance.

Equilibrium measurements. In keeping with the sign-convention accepted in the present chapter, both H and Q2 in Eq. (8.106) should be negative quantities. Then the first two terms in the brackets in Eq. (8.106) are positive, whereas the third one is negative. Therefore, as

Lateral Capillary Forces Between Floating Particles

387

1

Fig. 8.17. Experimental set up for studying capillary interactions [ 18], for measurement of surface drag coefficient [19], and surface shear viscosity of surfactant adsorption monolayers [23]. Particle 1 is floating at a distance s from a hydrophobic plate 2, whose lower edge is located at a distance H below the level of the non-disturbed horizontal liquid surface; H can be varied by means of the micrometric table 3 and screw 4. discussed in the previous section, for each given H there will be a distance s = s*, corresponding to an equilibrium position of the particle, for which F(s*) = 0. In other words, one can expect that a particle floating in a vicinity of the vertical wall (Fig. 8.17) will stay at an equilibrium distance s* from the wall. The measurements carried out in Ref. [18] show that really this is the experimental situation. Varying H one can change the distance s*. Figure 8.18 shows experimental points for H vs. s* measured with a hydrophobized copper bead floating on the surface of pure water. The radius of the bead is R2 = 700 + 15 lam and its contact angle with pure water is o~2

-

-

100 ~ The accuracy and the reproducibility of the measurement are

about +2 Bm for H and +20 lam for s*. The theoretical curve (the dashed line in Fig. 8.18) is drawn with R2 = 711 lam which agrees well with the optically measured radius of the bead; the more rigorous expression Eq. (8.99) is used in the calculations, instead of the asymptotic formula Eq. (8.106). One sees in Fig. 8.18 that the agreement between theory and experiment is very good. Figure 8.19 shows similar data as Fig. 8.18, however this time the floating particle is a mercury (Hg) drop. Since the mercury is liquid, the shape of the Hg drop is composed of two spherical

Chapter 8

388

300 o

250 -

\

\

n

~-

200 -

& zz

150 -

'Ib

100 500

0

!

I

I

I

I

1000

2000

3000

4000

5000

s*~m] Fig. 8.18. Experimental data (o) from Ref. [18] for the dependence of H on s* for a hydrophobized copper sphere, see the text for the notation. The line represents the theoretical dependence calculated by setting F = 0 in Eq. (8.99); experimental values of the contact angle a2 = 100 ~ and sphere radius R2 = 711 lam are used.

200 180

160 140

1

120 1 O0

,,, \ o, m \ ,

8O 60 40 -

""......... "41...,...i... "..-..11....111 9

2O

0o

I

1000

,

I

~

2000

I

3000

,

I

4000

m

I

:5000

s" ~m] Fig. 8.19. Experimental data from Ref. [18] for the dependence of H on s* for two mercury drops of different size. The lines represent theoretical fits calculated by means of Eq. (8.99) using the experimental values of the radii Rw = 440 gm (the lower curve) and Rw = 512 gm (the upper curve); the three-phase contact angle mercury-water-air is a 2 - 86 ~

Lateral Capillary Forces Between Floating Particles

389

segments of radii Rw and Ra, corresponding to the Hg-water and Hg-air regions. The values of Rw, Ra and of the contact angle a2 have been determined in Ref. [18] by means of optical measurements. Then the theoretical plot of H vs. s* has been calculated using Eq. (8.99) (see the dashed lines in Fig. 8.19) and compared with the experimental data (the symbols in Fig. 8.19). Again, there is an excellent agreement between theory and experiment. It should be noted that with the exception of an unknown (instrumental) additive constant in the experimental data for H, the theoretical curves in Fig. 8.19 are drawn without using any other adjustable parameter [ 18]. D y n a m i c m e a s u r e m e n t s . It was demonstrated in Ref. [ 19] that if the capillary force F is

calculated by means of Eq. (8.106), and the particle velocity, k, is measured in dynamic experiments, then one can determine the drag force, Fa" Ira = 67~r/R2f d ~

F d = F - mg,

(8.107)

where R2, m and ;/ are the particle radius, mass and acceleration, r/is the viscosity of the liquid and fd is the drag coefficient. If the particle were entirely immersed in the bulk liquid, Fd would 0.6 0.5

/

/ ra0

,-~

. ~.,,.~

IB

0.4 0.3

O

0.2 9IH1= 0.0914 cm 9 1/41 = 0.1102

0.1 /

~176

cm

91/41= 0.1220 cm ,

014

,

t

,

,.

12

t

,.6 I

t

capillary force x 10 3 [dyn]

Fig. 8.20. Experimental data from Ref. [19] for the velocity ~ of a "sliding" glass sphere (Fig. 8.17) plotted vs. the capillary force F calculated from Eq. (8.106); the sphere is floating on the surface of pure water. The slope of the dashed line, drawn in accordance with Eq. (8.107) (the inertial term m g neglected), gives drag coefficient fj = 0.68. The data are obtained in 3 runs corresponding to 3 fixed values of H denoted in the figure.

Chapter 8

390

be given by the Stokes formula, Fd = 6~r/R2 k, and fd would be equal to 1. In the considered case, fd differs from unity because the particle is attached to the interface and protrudes from the underlying liquid phase. In Fig. 8.20 the experimentally measured velocity k of a particle approaching the wall is plotted vs. the capillary force F(s), calculated by means of Eq. (8.106) for the experimental values of the distance s. The particle is a glass sphere of radius R2 = 229 lam and contact angle o~2 = 48.7 ~ at the surface of pure water, o" = 72 mN/m. The straight line in Fig. 8.20 corresponds to Fd = F(s), i.e. drag force equal to the capillary force" from the slope of the straight line one determines fd = 0.68. The slight deviation of the data from the straight line for the largest F(s) is not a discrepancy between theory and experiment: this deviation is due to the inertia term mg in Eq. (8.107), which is not negligible for shorter particle-wall distances, for which the approximation Fd ~ F(s) is not good enough. The hydrodynamic theory by Brenner and Leal [20,21 ], and Danov et al. [22], predicts that the drag coefficient fd of a particle attached to a planar fluid interface is a function only of the of the viscosities of the two fluids and of the three-phase contact angle, o:2. The experiments in

0.6 9 pure waterfd = 0.67, R2 = 222/~m 9 SDS fd = 1.60, R2 = 216/tm 9 HTAB f~ - 1.61, R2= 259/2m

0.5 r"--'l

-~

0.4

,~

0.3

/ A/

0.6 t

,1

O

0.2

,,t,

x

0.4 .~

/

0.3

/

.4 r .,t.-~'" 0.2 ~,~I"

0.1

....

o.t ~ u u ~ ~ ' . ~ 0

O. .0

~

I

0.4

J

I

0.8

~

0"0

i

1

1.2

i

2

t[s] i

3

4

1.6

2.0

capillary force x 10 3 [dyn] Fig. 8.21. Experimental data from Ref. [23] for the velocity k of a glass sphere plotted vs. the capillary force F calculated from Eq. (8.106). The slope of each experimental line gives the value of the drag coefficient, fj. Data about the type of the solution, the determined fj and the particle radius R: are given in the figure. The inset shows the experimental plot of k vs. time, which becomes linear if plotted as k vs. F.

Lateral Capilla~ Forces Between Floating Particles

391

Ref. [19] with particles on air-water interface give fd varying between 0.68 and 0.54 for particle contact angle a2 varying from 49 ~ to 82 ~ (the less the depth of particle immersion, the less the drag coefficient fd); these experimentally obtained data for fd are in a very good quantitative agreement with the hydrodynamic theory of the drag coefficient [22]. If the floating particle is heavy enough, it creates a considerable deformation of the surrounding liquid surface; this deformation travels together with the particle thus increasing fd several times [ 19]; for the time being there is no quantitative hydrodynamic theory of the latter effect. The addition of surfactant in the solution strongly increases fd. The latter effect has been used in Ref. [23] to measure the surface viscosity of adsorption monolayers from low molecular weight surfactants exhibiting fast kinetics of adsorption. For these surfactants the surface viscosity is too low to be accessible to the conventional experimental methods, like the deep-

channel surface viscometer [24-30], or the disk viscometer [31-37]. Fortunately, the motion of a sphere of radius 200-300 lain along a slightly inclined meniscus turns out to be sensitive to the friction within the adsorption layer (thick no more than 2 rim) of surfactants such as sodium dodecyl sulfate (SDS) and hexadecyl-trimethyl-ammonium-bromide (HTAB). This fact is utilized by the sliding-particle method for measurement of surface viscosity, which has been proposed in Ref. [23]. In Fig. 8.21 we present experimental data from Ref. [23] for the velocity of a particle (Fig. 8.17) plotted vs. the capillary force F(s), calculated by means of Eq. (8.106). One sees in Fig. 8.21 that again the data for ~ vs. F(s) comply with straight lines, whereas the plots of ~ vs. time t are non-linear (see the inset in Fig. 8.21). From the slopes of the straight lines in Fig. 8.21 the drag coefficient fd has been determined; the obtained values of fd are also shown in Fig. 8.21. Note, that the addition of surfactant increases fd from 0.66 (for pure water) up to 1.6. This effect, converted in terms of surface viscosity, gives respectively r/s = 1.5 and 2.0

x

10 -6

kg/s for the surface viscosity of dense SDS and HTAB adsorption

monolayers [23]. These results compare well with values of r/s obtained by means of knife-edge surface viscometers [38-43]. If the kinetics of surfactant adsorption is not fast enough to damp the surface elastic effects, the drag coefficient fd can be influenced not only by the surface viscosity, but also by the surface (Gibbs) elasticity (see Chapter 1 about the definitions of and theoretical expressions for the

392

Chapter 8

Gibbs elasticity and adsorption relaxation times). The complete theoretical dynamic problem, involving the effects of surface viscosity, surface elasticity and dynamics of surfactant adsorption has not yet been solved.

8.3.

SUMMARY

In this chapter we presented theoretical and experimental results about the lateral capillary interaction between two floating particles, and between a floating particle and a vertical wall. The origin of this "flotation" force, F, is the overlap of the interfacial deformations created by the separate floating particles. In this aspect the "floatation" force is similar to the "immersion" force (see Chapter 7) and it is described by the same asymptotic formula, Eq. (8.4). The difference between the "flotation" and "immersion" forces is manifested through the different physical origin of the respective "capillary charge" Q, which results in a different dependence of Q on the interfacial tension o and particle radius R. Thus in the case of "flotation" force F o~ R6/G, whereas in the case of "immersion" force F ,,~ R=o-, see Eq. (8.20). The strong

decrease of the "flotation" force with the diminishing of particle radius R leads to the consequence that this force becomes negligibly small for R < 5 ~tm, i.e. for Brownian particles, for which the "immersion" force is still rather powerful, see Fig. 8.3. In some aspect these two kinds of capillary interactions 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). The capillary charge of a floating particle can be calculated by means of Eq. (8.12) or by means of the more convenient, but approximate, Eq. (8.16). The energy of capillary interaction between floating particles is composed of contributions from the gravitational potential energy of particles and fluid phases, from the wetting of the particle surface and from the deformation of the fluid interface caused by the particles, see Eq. (8.26). In the asymptotic case of small deformations and long interparticle separations the sum of the aforementioned three contributions turns out to be equal to the half of the particle gravitational potential energy, see Eqs. (8.32)-(8.34); the latter finding validates the usage of the popular superposition approximation, Section 8.1.1. A convenient numerical procedure for a precise calculation of

Lateral Capillary Forces Between Floating Particles

393

the lateral flotation force is described, Section 8.1.5. Numerical data about the dependence of the interaction energy Af~ on the interparticle distance L, particle radius R~, density p~ and contact angle o~, are presented and discussed. In all cases the capillary interaction between floating particles, as a function of L, is a monotonic attraction or repulsion depending on the sign of the product of the capillary charges: QIQ2> 0

or

Q1Q2

<

0, see Figs. 8.5-8.9.

Next, we considered the case of a single particle floating in a vicinity of a vertical wall. Such a particle experiences the action of a capillary image force. The latter can be formally considered as interaction between the particle and its mirror image (with respect to the wall), which can be attraction or repulsion depending on whether the contact angle (Fig. 8.10a) or contact line (Fig. 8.10b) is fixed at the wall. If an inclined meniscus is formed in a neighborhood of the wall, then the interplay of the gravitational and capillary image forces can lead to a nonmonotonic dependence of the interaction energy on the particle-wall distance, see Figs. 8.11,

8.12, 8.14 and 8.15. Analytical expressions for calculating the flotation interaction are obtained by means of both energy and force approaches, see Sections 8.2.4 and 8.2.5. A convenient asymptotic formula, Eq. (8.84), for the capillary force is obtained, which compares very well with the output of the more accurate theory, see Figs. 8.14 and 8.15. The derived expressions are in a very good agreement with experimental data for the equilibrium position of floating particles, Figs. 8.18 and 8.19. The obtained theoretical results have been applied to determine experimentally the drag coefficient of floating particles and the surface shear viscosity of surfactant adsorption monolayers, see Figs. 8.20 and 8.21. Finally, we should note that in this chapter our attention was focused on the capillary interactions of floating spherical particles. In Refs. [44-47] one can find theoretical expressions and numerical results for two floating parallel horizontal cylinders.

394

8.4.

Chapter 8

REFERENCES

1. M.M. Nicolson, Proc. Cambridge Philos. Soc. 45 (1949) 288. 2. I.B~Ivanov, P.A. Kralchevsky, A.D. Nikolov, J. Colloid Interface Sci. 112 (1986) 97. 3. D.Y.C. Chan, J.D. Henry, L.R. White, J. Colloid Interface Sci. 79 (1981) 410. 4.

V.N. Paunov, P.A. Kralchevsky, N.D. Denkov, K. Nagayama, J. Colloid Interface Sci. 157 (1993) 100.

5. E.D. Shchukin, A.V. Pertsov, E.A. Amelina, Colloid Chemistry, Moscow Univ. Press, Moscow, 1982 [in Russian]. 6. P.A. Kralchevsky, K. Nagayama, Langmuir 10 (1994) 23. 7.

P.A. Kralchevsky, K.D. Danov, N.D. Denkov, Chemical Physics of Colloid Systems and Interfaces, in: K.S. Birdi (Ed.) Handbook of Surface and Colloid Chemistry, CRC Press, Boca Raton, 1997.

8. P.A. Kralchevsky, K. Nagayama, Adv. Colloid Interface Sci. 85 (2000) 145. 9. P.A. Kralchevsky, V.N. Paunov, N.D. Denkov, K. Nagayama, J. Colloid Interface Sci. 167 (1994) 47. 10. P.A. Kralchevsky, N.D. Denkov, V.N. Paunov, O.D. Velev, I.B. Ivanov, H. Yoshimura, K. Nagayama, J. Phys.: Condens. Matter 6 (1994) A395. 11. L.D~ Landau, E.M. Lifshitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1984. 12. J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, London, 1992. 13. A.H. Nayfeh, Perturbation Methods, Wiley, New York, 1973. 14. E. Janke, F. Emde, F. L6sch, Tables of Higher Functions, McGraw-Hill, New York, 1960. 15. H.B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan Co., New York, 1961. 16. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. 17. V.N. Paunov, P.A. Kralchevsky, N.D. Denkov, I.B. Ivanov, K. Nagayama, Colloids Surf. 67 (1992) 138. 18. O.D. Velev, N.D. Denkov, P~ Interface Sci. 167 (1994) 66.

Kralchevsky, V.N. Paunov, K. Nagayama, J. Colloid

19. J.T. Petkov, N.D. Denkov, K.D. Danov, O.D. Velev, R. Aust, F. Durst, J. Colloid Interface Sci. 172 (1995) 147. 20. H. Brenner, L.G. Leal, J. Colloid Interface Sci. 65 (1978) 191.

Lateral Capillary Forces Between Floating Particles

395

21. H. Brenner, L.G. Leal, Jo Colloid Interface Sci. 88 (1982) 136. 22. K.D. Danov, R. Aust, F. Durst, U. Lange, J. Colloid Interface Sci. 175 (1995) 36. 23. J.T. Petkov, K.D. Danov, N.D. Denkov, R. Aust, F. Durst, Langmuir 12 (1996) 2650. 24. W.D. Harkins, R. J. Meyers, Nature 140 (1937) 465. 25. D.G. Dervichian, M. Joly, J. Phys. Radium 10 (1939) 375. 26. J.T. Davies, Proc. 2nd Int. Congr. Surf. Act. 1 (1957) 220. 27. R.J. Mannheimer, R.S. Schechter, J. Colloid Interface Sci. 32 (1970) 195. 28. A.J. Pintar, A.B. Israel, D.T. Wasan, J. Colloid Interface Sci. 37 (1971) 52. 29. D.T. Wasan, V. Mohan, Interfacial theological properties of fluid interfaces containing surfactants, in: D.O. Shah and R.S. Schechter (Eds.) Improved Oil Recovery by Surfactant and Polymer Flooding, Academic Press, New York, 1977, p. 161. 30. D.A. Edwards, H. Brenner, D.T. Wasan, Interfacial Transport Processes and Rheology, Butterworth-Heinemann, Boston, 1991. 31. F.C. Goodrich, A.K. Chatterjee, J. Colloid Interface Sci. 34 (1970) 36. 32. PoB. Briley, A.R. Deemer, J.C. Slattery, J. Colloid Interface Sci. 56 (1976) 1. 33. R. Shail, J. Engng. Math. 12 (1978) 59. 34. S.G. Oh, J.C. Slattery, J. Colloid Interface Sci. 67 (1978) 516. 35. A.M. Davis, M.E. O'Neill, Int. J. Multiphase Flow 5 (1979) 413. 36. R. Shail, D.K. Gooden, Int. J. Multiphase Flow 7 (1981) 245. 37. R. Miller, R. Wustneck, J. Kr~igel, G. Kretzschmar, Colloids Surf. A 111 (1996) 75. 38. AoG. Brown, W.C. Thuman, J.W. McBain, J. Colloid Sci. 8 (1953) 491. 39. N. Lifshutz, M.G. Hedge, J.C. Slattery, J. Colloid Interface Sci. 37 (1971) 73. 40. F.C. Goodrich, L.H. Allen, J. Colloid Interface Sci. 40 (1972) 329. 41. F~

Goodrich, L.H. Allen, A.M. Poskanzer, J. Colloid Interface Sci. 52 (1975) 201.

42. A.Mo Poskanzer, F.C. Goodrich, J. Colloid Interface Sci. 52 (1975) 213. 43. A.M. Poskanzer, F.C. Goodrich, J. Phys. Chem. 79 (1975) 2122. 44. W.A. Gifford, L.E. Scriven, Chem. Eng. Sci. 26 (1971) 287. 45. M.Ao Fortes, Can. J. Chem. 60 (1982) 2889. 46. C. Allain, M Cloitre, J. Colloid Interface Sci. 157 (1993) 261. 47. C. Allain, M Cloitre, J. Colloid Interface Sci. 157 (1993) 269.

396

CHAPTER 9

CAPILLARY FORCES BETWEEN PARTICLES BOUND TO A SPHERICAL INTERFACE

This chapter contains theoretical results about the lateral capillary interaction between two particles bound to a spherical fluid interface, liquid film, lipid vesicle or membrane. The capillary forces in this case can be only of "immersion" type. The origin of the interfacial deformation and capillary force can be the entrapment of the particles in the liquid film between two phase boundaries, or the presence of applied stresses due to outer bodies. The stability of a liquid film is provided by a repulsive disjoining pressure, which determines the capillary length q-1 and the range of the particle-particle interaction. The calculation of the capillary force is affected by the specificity of the spherical geometry. The spherical bipolar coordinates represent the natural set of coordinates for the mathematical description of the considered system. They reduce the integration domain to a rectangle and make easier the numerical solution of the Laplace equation. Two types of boundary conditions, fixed contact angle and fixed contact line, can be applied. Coupled with the spherical geometry of the interface they lead to qualitatively different dependencies of the capillary force on distance. The magnitude of the capillary interaction energy can be again of the order of 10-100 kT for small sub-micrometer particles. In such a case, the capillary attraction prevails over the thermal motion and can bring about particle aggregation and ordering in the spherical film. In this respect, the physical situation is the same for spherical and planar films, if only the particles are subjected to the action of the lateral immersion force. The study of a film with one deformable surface, described in this chapter, is the first step toward the investigation of more complicated systems with two deformable surfaces such as a spherical emulsion film or a spherical lipid vesicle containing inclusions.

Capillary Forces between Particles Bound to a Spherical hlterface

9.1.

397

ORIGIN OF THE "CAPILLARY CHARGE" IN THE CASE OF SPHERICAL INTERFACE

Spherical interfaces and membranes can be observed frequently in nature, especially in various emulsion and biological systems [1-3]. As a rule, the droplets in an emulsion are polydisperse in size, and consequently, the liquid films intervening between two attached emulsion drops have in general spherical shape [4]. It is worthwhile noting that some emulsions exist in the form of globular liquid films, which can be of W1/O/W2 or O1/W/O2 type ( O = o i l , W = water), see e.g. Ref~ [5]. If small colloidal particles are bound to such spherical interfaces (thin films, liposomes, membranes, etCo) they may experience the action of lateral capillary forces. The spherical geometry provides some specific conditions, which differ from those with planar interfaces or plane-parallel thin films. For example, in the case of closed spherical thin film it is important that the volume of the liquid layer is finite. In addition, the capillary force between two diametrically opposed particles, confined in a spherical film, is zero irrespective of the range of the interaction determined by the characteristic capillary length

q-1.

As already discussed, the particles attached to an interface (thin film, membrane) interact through the overlap of the perturbations in the interfacial shape created by them. This is true also when the non-disturbed interface is spherical; in this case any deviation from the spherical shape has to be considered as an interfacial perturbation, which gives rise to the particle "capillary charge", see Section 7.1.3 above. The effect of gravity is negligible in the case of spherical interfaces (otherwise the latter will be deformed), and consequently, it is not expected the particle weight to cause any significant interfacial deformation. Then a question arises: which can be the origin of the interfacial perturbations in this case? Let us consider an example depicted in Fig. 9.1a: a solid spherical particle attached to the surface of a spherical emulsion drop of radius R0. Such a configuration is typical for the Picketing emulsions which are stabilized by the adsorption of solid particles and have a considerable importance for the practice [6-10]. The depth of immersion of the particle into the drop phase, and the radius of the three-phase contact line, rc, is determined by the value of the contact angle a (Fig. 9.1 a). The pressure within the drop,

PI, is larger than

the outside pressure

398

Chapter 9 z

Oc

(a)

(b)

Fig. 9.1. (a) Spherical particle attached to the surface of an emulsion drop of radius R0; a is the three phase contact angle; rc is the contact line radius; PI and PH are the pressures inside and outside the drop. (b) Particle of radius Rp entrapped between the two lipid bilayers composing a spherical vesicle of radius R0; ~" is the running thickness of the gap (filled with water) between the two detached bilayers. P~ because of the curvature of the drop surface. The force pushing the particle outside the drop (along the z-axis) is Fout = grc2pi ;

(9.1)

on the other hand, the force pushing the particle inside the drop is due to the outer pressure and the drop surface tension resolved along the z-axis (Fig. 9.1 a): Fin = 7r,rc2pii +

2rtrccrsin0c

(9.2)

Here 0c is a central angle: sin0c= rJRo. At equilibrium one must have Fin-" Fout; then combining Eqs. (9.1) and (9.2) one obtains the Laplace equation P I - P I I -" 2cr/Ro which is identically satisfied for a spherical interface. Thus we arrive at the conclusion that the force balance Fin -" Fout is fulfilled for a spherical interface. The same conclusion can be reached in a different way. The configuration of a spherical particle attached to an emulsion drop must have rotational symmetry. It is known [11] that for an axisymmetric surface intersecting the axis of revolution the Laplace equation, Eq. (2.24), has a single solution: sphere (gravity deformation negligible). If a second particle is attached to the drop surface it can acquire the same configuration as that in Fig. 9. l a; only the radius of the spherical surface will slightly increase due two the volume of the drop phase displaced by the

Capillao' Forces between Particles Bound to a Spherical bTterface

399

P2

Fig. 9.2. Sketch of two solid particles entrapped into a spherical film which intervenes between two emulsion drops of different size; Pj and P2 denote the pressures into the two drops and P3 is the pressure in the continuous phase. second particle. In other words the force balance Fin

= Fout

is fulfilled for each separate particle

and the drop surface remains spherical. Moreover, if there is no deviation from the spherical shape, then lateral capillary force between the particles c a n n o t appear. Hence, if aggregation of particles attached to the surface of such emulsion drop is observed, it should be attributed to other kind of forces. After the last 'negative' example, let us consider another example, in which both deformation and lateral capillary forces do appear. Pouligny and co-authors [12-14] have studied the sequence of phenomena which occur when a solid latex microsphere is brought in contact with an isolated giant spherical phospholipid vesicle. They observed a spontaneous attachment (adhesion) of latex particles to the vesicle, which is accompanied by complete or partial wetting (wrapping) of the particle by lipid bilayer(s). In fact, the membrane of such a vesicle can be composed of two or more lipid bilayers. As an example, in Fig. 9.1b we present a configuration of a membrane consisting of two lipid bilayers; the particle is captured between the two bilayers. The observations show that such two captured particles experience a long range attractive force [ 15]. There are experimental indications that in a vicinity of the particle the two lipid bilayers are detached (Fig. 9. l b) and a gap filled with water is formed between them [15]. The latter configuration resembles that depicted in Fig. 7. l f, and consequently, the observed long range attraction could be attributed to the capillary immersion force [15]. Similar configurations can appear also around particles, which are confined in the spherical film intervening between two attached emulsion droplets (Fig. 9.2), or in the globular emulsion films like those studied in Ref. [5]. In these cases the interfacial deformations are related to the confinement of the particles within the film.

Chapter 9

400

(a)

cell interior

(b)

cell / interior

/

Fig. 9.3. Deformations in the membrane of a living cell due to (a) a microfilament pulling an inclusion inward and (b) a microtubule pushing an inclusion outward. Looking for an example in biology, we could note that the cytoskeleton of a living cell is a framework composed of interconnected microtubules and filaments, which resembles a "tensegrity" architectural system composed of long struts joined with cables, see Refs. [ 16,17]. Moreover, inside the cell a gossamer network of contractile microfilaments pulls the cell's membrane toward the nucleus in the core [17]. In the points where the microfilaments are attached to the membrane, concave "dimples" will be formed, see Fig. 9.3a. On the other hand, at the points where microtubules (the "struts") touch the membrane, the latter will acquire a "pimple"-like shape, see Fig. 9.3b. Being deformations in the cell membrane, these "dimples" and "pimples" will experience lateral capillary forces, both attractive and repulsive, which can be employed to create a more adequate mechanical model of a living cell and, hopefully, to explain the regular "geodesic forms" which appear in some biological structures [17]. Other example can be a lipid bilayer (vesicle) containing incorporated membrane proteins, around which some local variation in the bilayer thickness can be created. The latter is due to the mismatch in the thickness of the hydrophobic zones of the protein and the bilayer. The overlap of such deformations can give rise to a membrane-mediated protein-protein interaction [18]. A peculiarity of this system, which is considered in Chapter 10 below, is that the hydrocarbon core of the lipid bilayer exhibits some elastic properties and cannot be treated as a simple fluid [ 19,20].

Capilla O, Forces between Particles Bound to a Spherical Interface

401

Coming back to simpler systems, in which lateral capillary forces can be operative, we should mention a configuration of two particles (Fig. 9.4b), which are confined in a liquid film wetting a bigger spherical solid particle. The problem about the capillary forces experienced by such two particles has been solved in Ref. [21]. The developed theoretical approach, which is applicable (with possible modifications) also to the other systems mentioned above, is described in the rest of the present chapter.

9.2.

INTERFACIAL SHAPE AROUND INCLUSIONS IN A SPHERICAL FILM

9.2. l.

LINEARIZATION OF LAPLACE EQUATION FOR SMALL DEVIATIONS FROM SPHERICAL SHAPE

Figure 9.4 shows schematically a spherical solid substrate (I) of radius Rs covered with a liquid film (F) intervening between the substrate and the outer fluid phase (II). The film contains two identical entrapped particles which deform the outer film surface. The non-disturbed spherical liquid film can have a stable equilibrium thickness h0 only due to the action of some repulsive forces (positive disjoining pressure) between the two film surfaces. For that reason a thin film,

Z~

zT

m

sO

(a)

n

(b)

"

[" "

/

(II)

Fig. 9.4. (a) Two 'cork-shaped' particles and (b) two spherical particles of radius Rp protruding from a liquid layer on a solid substrate of radius Rs ; the angles 0. and 0c characterize the particle positions and size; rc is the contact line radius; hc is the elevation of the contact line above the level of the reference sphere of radius R0; a is the three-phase contact angle; in both (a) and (b) the meniscus slope angle ~c is subtended between the normal to the segment ON and the tangent to the meniscus [21 ].

402

Chapter 9

i.e. a film for which the effect of the disjoining pressure H is not negligible, is considered here. Below we restrict our considerations to film thickness and particle size much smaller than Rs. An auxiliary system is depicted in Fig. 9.4a, in which each of the two particles have the special shape of a part of slender cone with vertex in the center of the substrate. In Ref. [21 ] it has been demonstrated, that the consideration of such cork-shaped particles is useful for the subsequent treatment of the more realistic system with two spherical particles depicted in Fig. 9.4b. We will first present the results for cork-shaped particles, which will be further extended to spherical particles in Section 9.3.3 below. The deviation of the outer film surface (Fig. 9.4) from the spherical shape is caused by the capillary rise of the liquid along the particle surface to form an equilibrium three-phase contact angle o~. For given radius of the substrate, film volume and particle shape there is one special value ~ of the contact angle (Fig. 9.4b), which corresponds to spherical shape of the outer film surface ( ~ = re/2 for the configuration in Fig. 9.4a). The radius of this sphere is denoted by R0; below it will be termed the reference sphere [21] and the interfacial deformations created by the trapped particles (for a 4: ~ ) will be accounted with respect to this spherical surface. The radial coordinate of a point of the deformed film surface can be presented in the form r = R0 + ~'(0,(p)

(9.3)

where 0 and (p are standard polar and azimuthal angles on the reference sphere r = R0 and ~'(0,(p) expresses the interfacial deformation due to the presence of the two particles. We assume small deformations, I~'/R01 << 1

and

]VIIi"]2 << 1,

(9.4)

where VII denotes surface gradient operator in the reference sphere. At static conditions the interfacial shape obeys the Laplace equation of capillarity, which in view of Eq. (9.4) can be linearized [22,23]: /2 PR + II(h) - Pn = if(h) Ro

2~" Rg

V,2br/

(9.5)

Capillao, Forces between Particles Bound to a Spherical Interface

403

Here Pn is the pressure in the outer fluid (II), PR is the reference pressure in the thin liquid film, FI is disjoining pressure and o" is the interfacial tension of the boundary film-phase II. If the liquid film contacts with a bulk liquid phase, as it is in Fig. 9.2, then the reference pressure can be identified with the pressure in this phase, that is PR - P3 for the system depicted in Fig. 9.2. On the other hand, if the film is closed as it is in Fig. 9.4, then PR can be determined from the condition for constancy of the volume of this film, see Eq. (9.18) below. For ~'--+0 Eq. (9.5) reduces to the Laplace equation for the surface of a spherical thin liquid film, see Eqs. (6.8)(6.10) and Ref. [24]. Both II and o" depend on the film thickness ho - Ro - Rs = const.

h = ho + ~,

(9.6)

Moreover, for a wetting film 1-I and cr are connected by means of the relationship [25] oo

(9.7)

o(h) = ooo + f H ( h ) d h , h

where o'oo is the surface tension of the bulk liquid phase (infinitely thick film), see Eq. (5.8) above. The integral term in Eq. (9.7) expresses the equilibrium work (per unit area) carried out by the surface forces to bring the two film surfaces from infinity to a finite separation h. For << h0, using Eqs. (9.6) and (9.7), one can expand FI(h) and o'(h) in series: n = no + n'r +

....

~=

~0- n0r

89F I ' ~ "2 + ....

(9.8)

where GO = O']h=ho ,

1-I0=

1-Ilh=ho ,

I 1 ' = (dIl/dh)lh=ho ,

(9.9)

Next, we substitute Eq. (9.9) into Eq. (9.5) and obtain the linearized Laplace equation in the form [21] 7 2II~"--q 2~" - 2AH, where the following notation has been introduced:

(9.10)

404

Chapter 9

2 q =

H'

2

2H o

cro

Ro

OoRo

1 -

(9.1 l)

1

1

R'

R

. . . .

Ro

1 -

--(PR

20"0

+

[ I 0 - PR)

(9.12)

Here q-1 has the meaning of capillary length, which determines the range of the interfacial deformation and of the lateral capillary force; note that for R0--->~ Eq. (9.11) reduces to the form of Eq. (7.7) for flat thin films. On the other hand, if disjoining pressure is missing (as it is for the systems depicted in Fig. 9.3) then q2 = -2/R0; in such a case q will be an imaginary number and the Laplace equation, Eq. (9.10), will have oscillatory solutions. Following Ref. [21 ], we will assume that the effect of disjoining pressure is predominant (this guarantees the stability of the films in Figs. 9.2 or 9.4), and we work with real values of the parameter q. Indeed, for stable films H' < 0 , see e.g. Ref. [26]; we assume that IH'l is large enough to have

q2>0. In Eq. (9.12) AH stands for the change in the mean curvature of the film surface due to the deformation caused by the two entrapped particles (Fig. 9.4); R can be interpreted as the outer radius of an imaginary spherical layer of thickness h0, whose internal pressure is equal to the pressure inside the perturbed film [21 ].

9.2.2~

" C A P I L L A R Y C H A R G E " AND R E F E R E N C E P R E S S U R E

AH and PR can be determined from the physical condition that the volume of the liquid film does not change, i.e. the liquid within the film is incompressible and the phase boundaries are closed for the exchange of molecules with the neighboring phases [21]. In such a case the integral

RO% r

Vm - IIdO dq) sinO Idrr2 - Ids ~ + ~ + ~ , So R~ 3R2 So Ro

(9.13)

expressing the change in the film volume due to the surface deformation, must be equal to zero; the integration is taken over the surface domain So representing the radial projection of the deformed film surface on the reference sphere; ds is a surface element. For the system with cork-shaped particles (Fig. 9.4a) Vm = 0 is a rigorous relationship, whereas for the system with

Capillao' Forces between Particles Bound to a Spherical Interface

405

spherical particles Vm = 0 is an approximate expression because the small volumes shown shaded in Fig. 9.4b are neglected. Linearizing the integrand in Eq. (9.13) and substituting ~" from Eq. (9.10) one derives [21 ] 0 = Vm ~ ~ d s r - q-2 fUs(V~I~" So So

__ 2Z~)

(9.14)

Further, from Eq. (9.14) one obtains

8J~R~

2 ~ I d s 2 A H - I d s V i , . (V ,i~") - E ~dlrl. vii ~ , So So k=l Ck

(ro/Ro) 2 <<1

(9.15)

At the last step we have used the Green-Gauss-Ostrogradsky theorem, see e.g. Refs. [27-29]; Ck (k = 1,2) are two circular contours representing the orthogonal (radial) projections of the two particle contact lines onto the reference sphere of radius R0; dl is a linear element; fi is a unit running normal to Ck directed inwards. The linear integral yields ~dlfi . V n~ - 2n'ro tanN c ,

(9.16)

ro - Ro sinOc

Ck where tan Nc is the average meniscus slope at the contact line, see Fig. 9.4, and ro is the radius of the circumference C~. In the case of fixed contact angle a, for the system depicted in Fig. 9.4a Nc = n/2 - o~ = const, and one can directly write fi.Vn~" = tan Nc. Finally, combining Eqs. (9.15) and (9.16) one obtains the sought-for expression for AH [21 ]: A H = Q/(2Ro2),

Q - ro sin ~c = r0 tan ~c

(sin2~c << 1)

(9.17)

Here, as usual, Q denotes the "capillary charge" of the entrapped particles, cf. Eqs. (7.9) and (7.14). Finally, from Eqs. (9.12) and (9.17) one determines the reference pressure inside the closed film [21 ]" PR = P I I - FI0 + 2o'0/Ro- ooQ]Ro 2

(9.18)

Since AH is constant (independent of the surface coordinates 0 and q~) it is convenient to present Eq. (9.10) in the simpler form

V2"" ,~ II~" - q-f,

(9.19)

( - ( ~ + 2q-ZAH)/Ro

(9.20)

where

Chapter 9

406

9.2.3.

INTRODUCTION

OF SPHERICAL BIPOLAR COORDINATES

To integrate conveniently Eq. (9.19) special bipolar coordinates on a sphere have been invented in Ref. [21]. The connection between the Cartesian coordinates (x,y,z) and these curvilinear coordinates (r,m, T) are" -1.0

(a)

1 . ~ y o

i ~ !

x

~

-10/

1.0 / 1.o

/ /0 z -1.0

/

/

iI

(b)

-Yl

-'gc

Fig. 9.5. (a) Bipolar coordinate lines on the unit sphere: the lines z" = const, are analogous to the parallels, while the lines co = const, connecting the two "poles" resemble meridians, cf. Eq. (9.21). (b) The parametrization of the reference sphere, r = Ro in Fig. 9.4, by means of spherical bipolar coordinates reduces the integration domain of Eq. (9.19) to a rectangle; after Ref. [21].

Capillary Forces between Particles Bound to a Spherical lntelface

x=

r~/~ 2 - l s i n h z A cosh z - cos co

,

y=

r~, 2 -lsin ~ ,~ cosh z - cos co

407

,

z-

r(cosh z - ~ cos co) /~ cosh "r - cos co

(9.21)

The surfaces r = const, are spheres; the lines co = const. (on each sphere r = const.) are circumferences which are counterparts of the meridians; the lines z = const, are circumferences

-

counterparts of the parallels of latitude, see Fig. 9.5a. In each point on the sphere the respective z-line and co-line are orthogonal to each other. These bipolar coordinates on a sphere induce orthogonal biconical coordinates in space, which in fact are defined by Eq. (9.21). A detailed description of these coordinates can be found in Ref. [21], including expressions for the components of the metric tensor and Christoffel symbols, for various differential operators, components of the rate-of-strain tensor and the respective form of the Navier-Stokes equation. In the special case considered here we will use the spherical bipolar coordinates (co, 7:) for a parametrization of the reference sphere r = R0 = const. In terms of these coordinates the surface gradient operator acquires the form [21 ]

(9.22)

where a

Z -

,

a- R0(~2

1)1/2

(9.23)

~, cosh z - cos m e~o and eT are the unit vectors of the local surface basis. With the help of the spherical bipolar coordinates we bring Eq. (9.19) in the form [21]

(ik:oshz'- cosco) 2

/

+ o~z2 ) = (qa) 2 ff (co, r)

(9.24)

where a is defined by Eq. (9.23); note that Eq. (9.24) much resembles Eq. (7.32). The parameter ,~, which depends on the distance between the two particles, is defined by means of the expression [21 ] - c~ COS0 a

(0, > 0c)

(9.25)

Chapter 9

408

Note that ,a, can vary in the interval 1 < ,a, < oo; in particular, ,~ = 1 when the two particles in Fig. 9.4a touch each other, while ,~ = ~ when the particles are diametrically opposed. The contact lines on the two particles correspond to "t - +'cc, where [21 ]" "t'c - arctanh[(cos20c - COS20a)l/2/sinOa]

(Oc--< O, < re/2)

(9.26)

Thus the integration domain of Eq. (9.24) acquires the simple form of a rectangle, see Fig. 9.5b: -rt __ co _<+rt,

-~c < -c < +rc

(9.27)

The boundary condition of fixed contact angle (stemming from the Young equation) for the system depicted in Fig. 9.4a implies fi'Vii~'= (-1)~tanVc

at contour C~

(k = 1, 2)

(9.28)

fi has the same meaning as in Eq. (9.15)" since C~ represents a line of fixed r-coordinate ('r = +To), then fi = +e~ at the contour Ck. Thus with the help of Eqs. (9.20), (9.22) and (9.23) the boundary condition (9.28) can be expressed in the form [21]

(

cgff/,

= =r,.

sinl/ccx/& 2 - 1

(fixed contact angle)

(9.29)

,~ cosh ~"c - cosCO

The boundary condition of fixed contact line has the same physical meaning as in Sections 7.3.4 and 8.2.1: the position of the contact line is fixed at some edge on the particle surface or at the boundary between hydrophilic and hydrophobic domains. In this case ~"= hc = const, at the contact line, see Fig. 9.4. The linear elements along the co-lines and "r-lines in spherical bipolar coordinates are

dG = zd(.o,

dlr = z d r

(9.30)

Taking into account Eqs. (9.16), (9.17), (9.20), (9.22) and (9.30), the boundary condition ~"= hc can be expressed in terms of ~" [21]"

= ~

+

(fixed contact line)

(9.31)

Capillary Forces between Particles Bound to a Spherical Interface

409

Note that when the position of the contact line is fixed on the surface of an axisymmetric particle, the exact shape of the particle (cork-like or spherical) is not important insofar as in all cases the contact line is a circumference of a given fixed radius. In this aspect Eq. (9.31) is applicable for both types of particles depicted in Fig. 9.4.

9.2.4.

PROCEDURE OF CALCULATION AND NUMERICAL RESULTS

To obtain numerical results about the interfacial shape, and further to apply them to the calculation of the lateral capillary force (Section 9.3 below), we first have to specify the input parameters. It is convenient to scale all parameters using the reference radius R0 and the interfacial tension if0; then in the computations one can work only in terms of dimensionless parameters and variables. Thus the input parameters characterizing the film are ho/Ro and qRo. The input parameters related to the confined particles are Oa, Oc and hc/Ro (or o0 when the contact line (or the contact angle) on the particle is fixed. Next, ~, and ~:c are calculated by means of Eqs. (9.25) and (9.26). In the considered case of spherical geometry the distance between the two particles can be characterized by the length of the shortest arc connecting the axes of the two particles: L = 20aRo. In the numerical calculations it is convenient to use the dimensionless distance (20c/rt < L < 1)

L - L/(rtRo) = 20a/rt

(9.32)

Note that the maximum value L = 1 corresponds to a configuration of two diametrically opposed particles. As already mentioned, the usage of spherical bipolar coordinates transforms the domain of integration of Eq. (9.24) into a rectangle in the (co,~:)-plane, which is bounded by the lines (_o= _+rt and ~:= -+~:c,see Fig. 9.5b. Due to the symmetry it is enough to find the meniscus profile ~" (oo,r) only in a quarter of this rectangle, say in the quadrant

0
0 < ~:< ~:c. The

symmetry implies the following additional boundary conditions at the inner boundaries of the quadrant:

0

0

Chapter 9

410

The rectangular shape of the integration domain considerably simplifies and accelerates the numerical solution of Eq. (9.24) obeying the boundary conditions (9.29) and (9.33), or alternatively, (9.31) and (9.33). In Ref. [21] the conventional finite difference scheme of second order was used for discretization of the boundary problem. In this way for each node of the integration network one linear algebraic equation was obtained. The resulting set of equations was solved by means of the Gauss-Seidel iterative method [30,31] combined with successive over-relaxation (SOR) and the Chebyshev acceleration technique [32]. When the boundary condition for

fixed contact line,

Eq. (9.31), is used, the numerical

procedure can be simplified if one seeks a solution in the form [21] ~" (c0,v) = A f(co, v),

A = const.

(9.34)

The constant A is determined as follows. The function f(co, v) apparently satisfies Eq. (9.24) and the boundary conditions (9.33) withfinstead of ~". Then one imposes the requirement ~]T--r, = A

or

f[r=r~ = 1

(fixed contact line)

(9.35)

Combining Eqs. (9.31) and (9.35) one determines the constant A :

A=

he 1 ~dco(Of R 0 1--]~] '2R 2 d0 /O~V =~,

(9.36)

One has to first solve (numerically) the boundary problem for f(co, v), then to calculate A from Eq. (9.36) and finally to determine ~"(co,v) from Eq. (9.34). When the boundary condition

for fixed contact angle,

Eq. (9.29), is used, the average elevation

of the contact line, hc, is not constant, but varies with the distance L between the two particles. From the calculated meniscus shape one can determine hc by means of the equation [21]

hc'" hc -Ro=q

~cdl--~4~2--1 i

~(co'Tc)dco -

rcsmOc 0 'a'c~

sin I//c sin Oc

q2R2

where ~'c(co)expresses the position of the contact line on the particle surface:

(9.37)

Capillary Forces between Particles Bound to a Spherical Interface

411

0.010 c'q II

qR0=l .... qRo=2 qR0=3

0.008 _

o 0.006

-

x

~ x

.. ~.

* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.004 0.002 0

I

I

0.25

0.50

. . . . .

I

,

0.75

1.00

L Fig. 9.6. Plot of the dimensionless average capillary elevation, hc/Ro, vs. the dimensionless distance L between two "cork-shaped" particles (Fig. 9.4a) calculated in Ref. [21] for three different values of the parameter qRo at fixed values of the angles Oc = 2 ~ and ~c = 3 ~

~'c(co) - ~'(co,'cc)

(9.38)

As an illustrative example in Fig. 9.6 we show the dependence of the dimensionless elevation h C on the dimensionless interparticle distance L for three different values of qRo for corkshaped particles (Fig. 9.4a). The values of the other parameters, 0c = 2 ~ and ~c = 3 ~ are the same for the three curves in Fig. 9.6. The increase of h c with the decrease of L is due to the increasing overlap of the menisci created by the two particles. Since q-1 is the characteristic size (length) of the meniscus zone, at fixed L the overlap is greater (and consequently hc is larger) when q-J is greater, i.e. qRo is smaller, see Fig. 9.6. The overlap of the menisci around the two particles has also another consequence: in general the contact line is inclined, i.e. it does not lie in a plane perpendicular to the particle axis. This effect is similar to that with the two vertical cylinders in Fig. 7.10. For small particles (re << R0) this inclination turns out to be very small. Indeed, it can be characterized by the angle r/c defined as follows [21 ] tan r/c - ~c (7c)- ~c (0) = ~'c (Jr).~- ~c (0) 2r~. 2(1 + he)sin0 c

rc = (Ro + hc)sin0c

(9.39)

412

Chapter 9

2.0 "-"eJ:} 1.5 0 1.0

~ ,,,,

0.5

0c= 1o Oc = 2~ .... 0c = 30 ,..

0

0.1

0.2

0.3

0.4

L Fig. 9.7. Plot of the inclination angle of contact line, r/c, vs. the dimensionless interparticle distance L calculated in Ref. [21] for three different values of the angle Oc determining the size of a 'cork-shaped' particle (Fig. 9.4a); the values of the parameters qRo = 5 and gtc = 5 ~ are fixed. The dependence of the contact line inclination r/c on the dimensionless interparticle separation L is illustrated in Fig. 9.7 for three different values of Oc for c o r k - s h a p e d particles (Fig. 9.4a). The values of the other parameters, qR o = 5 and gtc = 5 ~ are the same for the three curves in Fig. 9.7. For two diametrically opposed particles the inclination disappears because of the symmetry of the system, that is r/c = 0 for L = 1, see Fig. 9.7. As seen in the figure r/c is small even for short separations L . This fact will be utilized in Section 9.3.3 below in the procedure for calculating the lateral capillary force experienced by spherical particles.

9.3.

CALCULATION OF THE LATERAL CAPILLARY FORCE

In Fig. 9.4 the particles are pressed against the solid substrate by the meniscus surface tension; the resultant of the force applied on each particle along the normal to the substrate is counterbalanced by the bearing reaction. Our aim below is to calculate the tangential component of the capillary force Ft, which represents the lateral capillary force acting between the two particles. Let e~ be the unit vector of the particle axis. We denote by e, a vector, which is perpendicular to e,, and belongs to the plane xz, see Fig. 9.4a. Then in view of Eqs. (7.21)-(7.23) the contributions of surface tension and pressure to the lateral capillary force are Ft ~G'

et . ~dlmr C

,

F, (p' - e, . ~ d s ( - n P ) , S

(9.40)

Capillary Forces between Particles Bound to a Spherical Interface

Ft = Fff r) + Ft (p)

413

(9.41)

The integration in Eq. (9.40) is carried out along the three-phase contact line C and throughout the particle surface S. As usual, n is the running outer unit normal to the particle surface and In is the unit vector in the direction of the surface tension, see Fig. 9.4a; m is simultaneously perpendicular to the contact line and tangential to the liquid interface. Following Ref. [21] below we calculate Ft for the cases of fixed contact line and contact angle separately.

9.3.1.

BOUNDARY CONDITION OF FIXED CONTACT LINE

This boundary condition reads (see Fig. 9.4): ~"= hc = const.

(at the contact line)

(9.42)

In such a case the contact line is a circumference, which is perpendicular to the vector e,, i.e. to the particle axis. Then the symmetry of the system implies Ft (r~ = 0. On the other hand, Ft (~ is not zero insofar as the meniscus slope

tan ~

(9.43)

= Z

=T c

varies along the contact line (this is a manifestation of contact angle hysteresis); Z is given by Eq. (9.23). The unit vectors of the biconical coordinates, eo), er and er (see Fig. 9.5a), form a local basis in each point of the contact line. Then the unit vector in the direction of surface tension can be expressed in the form m = - ercos Vt- er sin

(9.44)

Next, we introduce polar coordinates (p, q~), with running unit vectors ep and % , in the plane of the contact line. Then for the points of the contact line one can write e~ = - %cos0c + eu sin0c,

er= ep sinOc + e~, cosOc.

Taking into account that erep = cos0 and erea et. m = (cos0c c o s g t - sin0~ sin gt)cos~

=

(9.45)

0 we combine Eqs. (9.44) and (9.45) to obtain (9.46)

414

Chapter 9

The assumption for small meniscus slope, Eq. (9.4), used by us implies that angle gt is small; then in view of Eq. (9.43) we can write (9.47) Z

=To

Recalling that F,(p) = 0, from Eqs. (9.40), (9.46) and (9.47) one obtains

F t = - aorc odOC~

--~ ~

+ - - ~Or sin0cz

c~

1

(9.48)

r:~

where rc is given by Eq. (9.39) and higher order terms are neglected. The running angles 0 and co provide two alternative parametrizations of the contact line, connected as follows [21 ]"

dO do)

~coshv c cosco- 1 COS ~ :

,~ cosh z'c - cos co

,

~

(,,~2 c o s h 2 Tc _ 1 ) 1 / 2 :

(9.49)

~ cosh zc - cos co

A substitution of Eq. (9.49) into (9.48), in view of Eqs. (9.20) and (9.23), finally yields [21]"

Ft __O.oRo f dco(~coshzc cosco_ l ) 0

1

c)( 2

(~2 _ 1)1/2 -~Z

2sin0 c 3~" + 2, cosh "Co- cos co 03Z"

(9.50) T----Tc

Having solved numerically the boundary problem in the case of fixed contact line, see Eqs. (9.34)-(9.36), one can further substitute the result for the computed function ~"(co, v) into Eq. (9.50) to calculate the lateral capillary force by means of numerical integration. Results are shown in Section 9.3.4 below.

9.3.2.

BOUNDARY CONDITION OF FIXED CONTACT ANGLE

The main difference with the previous Section 9.3.1 is that the contact line in general does not lie in a plane perpendicular to the particle axis, see Fig. 9.8. As a result Ft (r) is no longer zero and the interfacial tension o can vary along the contact line in accordance with Eq. (9.8). Thus from Eqs. (9.8) and (9.40) one obtains [21]

Ft (~) - ~dl(e t 9m)(o" o -H0~" ~. +...) C

(9.51)

Capillary Forces between Particles Bound to a Spherical Interface

415

Fig. 9.8. The right-hand-side cork-shaped particle in Fig. 9.4a sketched in an enlarged scale: t is a unit vector tangential to the three-phase contact line; n is an outer unit normal to the particle surface; b = t x n is a binormal; In is the unit vector in the direction of the surface tension, which is tangential to the liquid meniscus and normal to the contact line; o~ is the three-phase contact angle.

where ~'~.is defined by Eq. (9.38). First we derive expressions for calculating Ft in the case of cork-shaped particles; the case of spherical particles is considered in Section 9.3.3 below. The unit vector in direction of the surface tension at the contact line is rn = n cos gtc + b sin gtc,

b-txn,

(9.52)

where, as usual, n is the outer unit normal to the particle surface, t is the running unit tangent to the contact line; b is a binormal and gtc = rt/2 - o~ = const., see Fig. 9.8. The unit vectors et, ey and ea form a basis, which can be used to express n: n = et cos0c cosq~ + ey cos0c s i n 0 - e, sin0c

(9.53)

Let us denote by R the running position vector of a point from the contact line; then 10R

t=--~, t C cgq~

R - [ R 0 + ~'c (~)]er

(9.54)

where "9

.

)

to- = (R0 + ~'c)2 sln-0c + (d~'c/d0)-,

er = et sin0c cos~ + e,, sin0c sinq~ + e, cos0c

(9.55)

By means of Eqs. (9.52)-(9.55) one can prove that

e,. m - -

Id~.

., d(IRIsinO) ~ I sin~tc cos ~ 0~. sin 0 -~ sin - 0~. + cos 0,. cos ~t~.c o s 0 L dO dO tC

Further we substitute Eq. (9.56) into Eq. (9.51) to obtain [21]:

(9.56)

Chapter9

416

Ft(~) --2O'oCOS0c(cos0csinv,. + sin0,.) i d 0 -dr ~sin0 0

O_0( d~.c + cos0, dOcos0 7o( dO

where higher-order terms have been neglected. To calculate

2 _ 2Horo~-C

(9.57)

Ft(p) we integrate the pressure

throughout the lateral surface of the cork-shaped particle (Fig. 9.8); thus using Eqs. (9.40) and (9.53) we get

2 /

Ft(p) =-cos0c Idq}sin0ccosq~ PR Srdr+Pn [rdr 0

r1

(9.58)

Ro+~c

where rl and r2 are boundaries of integration satisfying the relationship rl <- ~'c(q})-< r2 for any ~; the exact choice of rl and r2 is not important, because they drop out from the final expression for F//'). Indeed, integrating in Eq. (9.58), substituting PR from Eq. (9.18), and neglecting higher order terms one obtains [21] 7r

F/p) = 2(Ho -

2ao/Ro)rocosO,: ~d 0 ~c cosO

(9.59)

0

Next, we integrate by parts in Eq. (9.59), and combine the result with Eqs. (9.41) and (9.57)"

!/.;/2

!.;

Ft=-2o'0cos0c(cos0csinvc- sin0c) d O - ~

sinO+--r0 cos0 c dO ~, dO

cosO

(19.60)

Note that the terms with H0 in Eqs. (9.57) and (9.59) cancel each other. The numerical procedure from Section 9.2.4 gives ~'(co,v)" therefore it is more convenient to rearrange Eq. (9.60) using co instead of 0 as integration variable; by means of Eqs. (9.20) and (9.49) one obtains [21]" /~t - -2 (cosO,. sin~c - sina.)fdco d~' sin coc~ ~/2"2-1 0 d o ~,coshv,.-cosco 2

(9.61) x/X2 - lo

/ do)

Capillary Forces between Particles Bound to a Spherical Interface

where ~'c - ~"(c~

417

and F t denotes the dimensionless lateral capillary force:

,.,.,

F t - F,/(~yoRo)

(9.62)

To calculate the capillary force Ft one has to first determine ~"(co,~-) solving numerically Eq. (9.24) and then to use Eq. (9.50) or (9.61) depending on whether the case of fixed contact line or angle, respectively, is considered.

Note that in the case of fixed contact angle Eq. (9.61) contains the derivative 0~"/0c0 and consequently the capillary force Ft stems from the inclination of the contact line. In contrast, when the contact line is fixed at the particle surface, Eq. (9.50) contains only the derivative 0~"/0~ and then Ft originates from the variation of meniscus slope along the contact line.

9.3.3.

CALCULATION PROCEDURE FOR CAPILLARY FORCE BETWEEN SPHERICAL PARTICLES

As already mentioned, if the boundary condition for fixed contact line, Eq. (9.42) is used, then the capillary force Ft is calculated from Eq. (9.50) in the same way for spherical particles (Fig. 9.4b) and cork-shaped particles (Fig. 9.4a). Indeed, in both cases the contact line is an immobile circumference perpendicular to the particle-substrate axis. In other words, if the contact line is fixed, then F, is identical for spherical and cork-shaped particles having the same radius of the contact line. In contrast, if the boundary condition for fixed contact angle, Eq. (9.29), is imposed, then the contact line moves along the particle surface when the interparticle separatio'n L varies. In addition, when the two particles approach each other (i) the capillary elevation of the contact line hc defined by Eq. (9.37) increases (Fig. 9.6), and (ii) the inclination angle rh. defined by Eq. (9.39) also increases. As demonstrated in Fig. 9.7 for not too small values of the separation L the inclination angle is rather small (tanr/c << 1) and the contact line is almost perpendicular to the particle-substrate axis. This fact allows one to utilize the following approximate two-step procedure, which is analogous to that used in Section 7.3.2 for the case of a planar interface. At the f i r s t step, as a zeroth approximation one assumes that the contact line lies in a plane perpendicular to the axis determined by angle 0, (Fig. 9.4b). In such a case the contact line is a circumference of radius rc elevated at a distance hc from the reference sphere of radius R0.

418

Chapter 9

At the second step one formally replaces the sphere by a cork shaped particle having the same values of 0c, Nc and he, as well as of re, 0,, ~'c, etc. Next, one solves numerically Eq. (9.24), along with the boundary conditions (9.29) and (9.33) and substitutes the result for ~"(6o, r) in Eq. (9.61) to calculate the capillary force Ft. The value of Ft thus obtained, which is accurate for the cork-shaped particle, gives a first approximation for the capillary force exerted on the spherical particle. Note that for different values of the distance L between two spherical particles the contact radius rc and the capillary elevation hc (Fig. 9.6) are different, and consequently, using the above procedure we replace the sphere with different cork-shaped particles (with different 0c). This reflects the shrinking of the contact line on the spherical surface with the increase of he.. The mathematical background of the above procedure is the following [21]. The radius of the contact line re. can be related to the radius of the reference sphere R0 and the particle radius Rl, (Fig. 9.4b)" m

rc = R0(1 + hc singr

= R/, sin(o~ + 0c + g4.),

(9.63)

We have introduced the notation h C = hc/(Ro sin gto)" for the numerical calculations it is important that hC is insensitive to the value of gtc. The length of the segment OM in Fig. 9.4b can be expressed in a similar way: m

IOMI = R0 (1 + hc singtc)COS0c = R0 - h0 + R/, [ 1 + cos(a + 0c + I/tc)],

(9.64)

The angle 13- o~ + 0c + gtc is also shown in Fig. 9.4b. From Eqs. (9.63) and (9.64) one eliminates angle 13 and obtains" Nc(0e.) = arcsin ( ( Y - 1)/hc),

(9.65)

Y - (1 - ho/Ro + Rp/Ro)cosO~. + [(Rp/Ro) 2 - (1 - ho/Ro + Rp/Ro) 2 sin20c] 1/2

(9.66)

where

Now we can formulate a procedure of calculations, which is based on the above equations [21]" The input parameters are R1,/Ro, ho/Ro, qRo, o~ and 0~,. Further, 0,. and ~. are calculated as follows.

Capillary

Forces

between

Particles

Bound

to a Spherical

419

Interface

(b)

(a) 0

..

0

;.:.

.... . . . . . . . . . . . . 9

s,

-1 ~

~

-3

/ !

-5

0

I

/

/

/

/

~

0.25

0.75

/

-2 / /

I

AW

--2oo k-T

t

1

/

1,00

-100

/

R_/h,,=l p u . . . . Rv'h o ~- 1.5 - - Rr o = 2

0.50

-

J

2~

,/

I

--

,,.,,''

.."

-2

2,... ;.,.- a . . - - , -

-4

0

-

0.25

0.50

0.75

-300

1.00

Fig. 9.9. Dimensionless (a) force Ft and (b) energy AW of capillary interaction plotted against the dimensionless separation /~ between two spherical particles entrapped in a spherical liquid film (Fig. 9.4b). The three different curves, calculated in Ref. [21 ], correspond to three values of the parameter Rp/ho; the values of the other parameters are fixed: qRo = 1 and o~ = 60 ~ The right-hand-side scale of AW/kT shows the value of the capillary interaction energy for the special case of R0 = 1 lam, T = 298 K and a0 = 30 mN/m. (i) One chooses an initial guess for 0c and Nc. (ii) ~'(CO,T) is obtained by numerical integration of Eq. (9.24), along with the boundary conditions (9.29) and (9.33), as described in Section 9.2.4. (iii) Next, h c = h~./singtc is determined by numerical calculation of the integral in Eq. (9.37). (iv) A new value of Nc is calculated from Eqs. (9.65)-(9.66). (v) A new value of 0c is obtained by solving numerically Eq. (9.63). The next iteration repeats from point (ii) with the new values of gtc and0c until convergence is achieved. (vi) With the obtained final parameter values and ~"(co, r) one calculates the dimensionless capillary force F t from Eq. (9.61) for various 0,, that is for various values of the dimensionless interparticle separation L - 20,/re; F, thus calculated corresponds to the boundary condition of fixed contact angle. (vii) If the boundary condition for fixed contact line is used, then hc and 0c are input parameters (instead of o~ and Rp/Ro) and steps (i)-(vi) are not necessary. In this case Eq. (9.24) is integrated

420

Chapter 9

numerically using the boundary conditions (9.33)-(9.36)" next, the capillary force is calculated from Eq. (9.50). (viii) Finally, one can determine the energy of lateral capillary interaction AWby integration: "-" ~

AW(L)

1

(9.67) fioRd where AW is the dimensionless interaction energy. The additive constant in the energy is determined in such a way, that A W - 0 for two diametrically opposed particles, i.e. for L = 1.

9.3.4.

NUMERICAL RESULTS FOR THE FORCE AND ENERGY OF CAPILLARY INTERACTION

Using the procedure described in Section 9.3.3 the force and energy of capillary interaction between two spherical particles entrapped into a spherical liquid film (Fig. 9.4b) have been calculated [21]. Results for the case of fixed contact angle (o~ = 60 ~ are shown in Figs. 9.9a and 9.9b, where the dimensionless force and energy, F, and AW [see Eqs. (9.62) and (9.67)], are plotted against the dimensionless interparticle separation L = L/(rcRo); the range of variation of L is 20Jrt
At a given R0, the size of

the particles is determined by the angle 0,., which for the curves in Figs. 9.10 and 9.11 takes

C a p i l l a r y F o r c e s b e t w e e n P a r t i c l e s B o u n d to a S p h e r i c a l I n t e r f a c e

0.0

421

.. 9.r162..:.:-":""" *.,-" . . . . .

.'J . . ' f

."/ ~,

-25

-0.5

x

-50

~

AW kT

-1.0

0c = 1o

-75

. . . . . 0,:=2" - - - 0c=3 o -100 i

-

t

1

9

,

I

1.5 0

1

l

1

0.25

0.5

,.,.o

L Fig. 9.10. Case of fixed contact angle: dimensionless energy Aft" of capillary interaction plotted against the dimensionless separation i

between two cork-shaped particles entrapped in a spherical

liquid film (Fig. 9.4a). The three curves, calculated in Ref. [21], correspond to three values of O denoted in the figure" the other parameters are gt. = 5 ~ and qR,, = 5. The right-hand-side scale of A W / k T is as in Fig. 9.9.

./...2...-7..-7. ---, .,., 0.0

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

. ." r.i?. : __ . : . .~. . ".'44".~. . ~ . 7." ~'_~_ . . . . . . . . . .:" i /

-10

tm

AW

-0.2

kT

//

<3 -0.4

-20

...........Oc:,O

/

!

l

.....

0c =2~

i

i

t

--"

0c=3"

-30

! |

0.0

I

I

I

[

I

I

I

I

0.5

Fig. 9.11. Case of fixed contact line: dimensionless energy

1.0

AW

of capillary interaction

vs.

dimensionless separation L between two cork-shaped particles entrapped in a spherical liquid film (Fig. 9.4a). The three curves are calculated in Ref. [21] with the same values of 0~. as in Fig. 9.10" for each curve h J R o is fixed and equal to the respective values of h ~ / R o (0.00389, 0.00585 and 0.00719) for the curves in Fig. 9.10. The right-hand-side scale of A W / k T is as in Fig. 9.9.

Chapter 9

422

values 1~ 2 ~ and 3 ~ i.e. the particles are small, rc << R0. One sees again that the dimensional interaction energy AW can be of the order of (10-100)kT. From a physical viewpoint

AW/kT >> 1 means that the capillary attraction prevails over the thermal motion and can bring about particle aggregation and ordering in the spherical film. Note that the parameters values in Figs. 9. l0 and 9. l 1 are chosen in such a way, that the shape of the fluid interfaces to be identical in the state of zero energy, i.e. for two diametrically opposed particles. This provides a basis for quantitative comparison of the plots of AW vs. L in these two figures, calculated by using the two alternative boundary conditions. The curves in Fig. 9.10 (as well as in Fig. 9.9) are calculated assuming fixed contact angle; one sees that the interaction energy AW is always negative, i.e. corresponds to attraction. On the other hand, the curves in Fig. 9.11 are calculated assuming fixed contact line. In the latter case the interaction energy changes its sign at comparatively large interparticle distances: attractive at short distances becomes repulsive at large separations. The fact that that the interaction energy can change sign in the case of fixed contact line, but the energy is always negative in the case of fixed contact angle, is discussed in Ref. [21]. It is concluded that the non-monotonic behavior of the capillary interaction energy (Fig. 9.11) is a non-trivial effect stemming from the spherical geometry of the film coupled with the boundary condition of fixed contact line; such an effect is difficult to anticipate by physical insight. Note that in the case of planar geometry the capillary force between identical particles is always monotonic attraction.

9.4.

SUMMARY

The fluid interfaces acquire spherical shape when the gravitational deformations are negligible. Hence, lateral capillary forces of "flotation" type (Chapter 8), which are due to the particle weight, do not appear between particles attached to a spherical interface, liquid film or membrane. The capillary forces in this case can be only of "immersion" type (Chapter 7). In such a case the origin of the interracial deformation and the capillary force is the entrapment of particles in the membrane of a spherical multilayered liposome (Fig. 9.1b), as well as in "opened" (Fig. 9.2) and "closed" (Fig. 9.4) liquid films. Interfacial (membrane) deformations

Capillao, Forces between Particles Bound to a Spherical blterface

423

and lateral capillary interactions can originate also from stresses due to outer bodies, like the microfilament in Fig. 9.3a and the microtubule in Fig. 9.3b. The calculation of the capillary force between particles trapped in spherical films is affected by the specificity of the spherical geometry. For example, the condition for constancy of the volume of the liquid in a closed spherical film (Fig. 9.4) leads to a connection between the particle "capillary charge" Q and the pressure PR within the film, see Eq. (9.18). The stability of such a film is provided by the repulsive disjoining pressure 1-I exerted on its surfaces. The disjoining pressure effect determines the capillary length q-i, see Eq. (9.11), and consequently, the range of the lateral capillary forces. If disjoining pressure is missing (as it is in Fig. 9.3) then q is an imaginary number and the Laplace equation, Eq. (9.19), has oscillatory solutions. In our numerical solutions we have assumed that the effect of disjoining pressure is predominant (this guarantees the stability of the films in Figs. 9.2 or 9.4), and we work with real values of the parameter q. The spherical bipolar coordinates, Eq. (9.21), represent the natural set of coordinates for the mathematical description of the considered system: two axisymmetric particles entrapped into a spherical liquid film. Thus the integration domain is reduced to a rectangle (Fig. 9.5) and the numerical solution of the Laplace equation is made easier. The two types of boundary conditions, fixed contact angle, Eq. (9.29), or fixed contact line, Eq. (9.31), lead to two different expressions for the lateral capillary force, Eqs. (9.61) and (9.50), respectively. The calculation of the capillary interaction between two spherical particles is more complicated in the case of fixed contact angle due to the mobility of the contact line; to solve the problem in Section 9.3.3 we have employed auxiliary cork-shaped particles. The interaction energy is always negative (attractive) in the case of fixed contact angle (Figs. 9.9 and 9.10). On the other hand, it turns out that the energy can change sign in the case of fixed contact line (attractive at short distances but repulsive at long distances, Fig. 9.11). This non-monotonic behavior of the capillary interaction energy is a non-trivial effect stemming from the specificity of the spherical geometry coupled with the boundary condition of fixed contact line; such an effect does not exist in the case of planar geometry, for which the capillary force between identical particles is always monotonic attraction.

424

Chapter 9

The magnitude of the capillary interaction energy can be of the order of 10-100 kK see Figs. 9.9-9.11, for sub-micrometer (Brownian) particles. In such a case, the capillary attraction prevails over the thermal motion and can bring about particle aggregation and ordering in the spherical film. In this respect, the physical situation is the same for spherical and planar films, if only the particles are subjected to the action of the lateral immersion force.

9.5.

REFERENCES

1. J. Sj6blom (Ed.), Emulsions and Emulsion Stability, M. Dekker, New York, 1996. 2.

S. Hyde, S. Anderson, K. Larsson, Z. B lum, T. Landh, S. Lidin, B.W. Ninham, The Language of Shape, Elsevier, Amsterdam, 1997.

3.

A.G. Volkov, D.W. Deamer, D.L. Tanelian, V.S. Markin, Liquid Interfaces in Chemistry and Biology, Wiley, New York, 1998. S. Hartland, Coalescence in Dense-Packed Dispersions, in: I.B. Ivanov (Ed.) Thin Liquid Films, M. Dekker, New York, 1988; p. 663.

5.

H. Wangqi, K.D. Papadopoulos, Colloids Surf. A 125 (1997) 181.

6.

S.U. Picketing, J. Chem. Soc. 91 (1907) 2001.

7.

Th. F. Tadros, B. Vincent, in: P. Becher (Ed.) Encyclopedia of Emulsion Technology, Vol. 1, Marcel Dekker, New York, 1983, p. 129.

8.

N.D. Denkov, I.B. Ivanov, P.A. Kralchevsky, J. Colloid Interface Sci. 150 (1992) 589.

9.

N. Yan, J.H. Masliyah, Colloids Surf. A 96 (1995) 229 and 243.

10. B.R. Midmore, Colloids Surf. A 132 (1998) 257. 11. H.M. Princen, The Equilibrium Shape of Interfaces, Drops, and Bubbles, in: E. Matijevic, (Ed.) Surface and Colloid Science, Vol. 2, Wiley, New York, 1969, p. 1. 12. C. Dietrich, M. Angelova, B. Pouligny, J. Phys. II France 7 (I 997) 1651. 13. K. Velikov, C. Dietrich, A. Hadjiisky, K.D. Danov, B. Pouligny, Europhys. Lett. 40, (1997) 405. 14. K. Velikov, K.D. Danov, M. Angelova, C. Dietrich, B. Pouligny, Colloids Surf. A, 149 (1998) 245.

Capillary Forces between Particles Bound to a Spherical Interface

425

15. K.D. Danov, B. Pouligny, M.I. Angelova, P.A. Kralchevsky, "Strong Capillary Attraction between Spherical Inclusions in a Multilayered Lipid Membrane", in: The Proceedings of the International Conference on Colloid and Surface Science, Tokyo, November 2000; Studies in Surface Science and Catalysis, Elsevier, Amsterdam, 2000. 16. D.E. Ingber, Ann. Rev. Physiol. 59 (1997) 575. 17. D.E. Ingber, Scientific American, January 1998, p. 30. 18. J.N. Israelachvili, Biochim. Biophys. Acta 469 (1977) 221. 19. A.G. Petrov, I. Bivas, Prog. Surface Sci. 16 (1984) 389. 20. P.A. Kralchevsky, V.N. Paunov, N.D. Denkov, K. Nagayama, J. Chem. Soc. Faraday Trans. 91 (1995) 3415. 21. P.A. Kralchevsky, V.N. Paunov, K. Nagayama, J. Fluid. Mech. 299 (1995) 105. 22. L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, 1984. 23. P.A. Kralchevsky, I.B. Ivanov, J. Colloid Interface Sci. 137 (1990) 234. 24. I.B. Ivanov, P.A. Kralchevsky, Mechanics and thermodynamics of curved thin films, in: I.B. Ivanov (Ed.) Thin Liquid Films, M. Dekker, New York, 1988; p. 49. 25. I.B. Ivanov, B.V. Toshev, Colloid Polymer Sci. 253 (1975) 593. 26. B.V. Derjaguin, N.V. Churaev, V.M. Muller, V.M., Surface Forces, Plenum Press: Consultants Bureau, New York, 1987. 27. C.E. Weatherburn, Differential Geometry of Three Dimensions, Cambridge University Press, Cambridge, 1939. 28. L. Brand, Vector and Tensor Analysis, Wiley, 1947. 29. A.J. McConnell, Application of Tensor Analysis, Dover, New York, 1957. 30. G.A. Korn, T.M. Korn, Mathematical Handbook, McGraw-Hill, New York, 1968. 31. A. Constantinides, Applied Numerical Methods with Personal Computers, McGraw-Hill, New York, 1987. 32. R.W. Hockney, J.W. Eastwood, Computer Simulation Using Particles, McGraw-Hill, New York, 1981.

426

CHAPTER 10 MECHANICS OF LIPID MEMBRANES AND INTERACTION BETWEEN INCLUSIONS

A bilayer lipid membrane cannot be simply modeled as a thin liquid film because the hydrocarbon-chain interior of the membrane exhibits elastic behavior when its thickness is varied. The hybrid mechanical behavior of a lipid bilayer (neither liquid nor bulk elastic body) can be described by means of a mechanical model, which treats the membrane as a special elastic film (the hydrocarbon chain interior) sandwiched between two Gibbs dividing surfaces (the surface polar-headgroup layers of the membrane). The latter "sandwich" model involves mechanical parameters such as the shear elastic modulus of the hydrocarbon chain interior, the bilayer surface tension, stretching (Gibbs) elasticity, surface bending moment and curvature elastic moduli. A mechanical analysis of the bilayer deformations enables one to derive expressions for the total stretching, bending and torsion (Gaussian) moduli of the membrane as a whole in terms of the aforementioned mechanical parameters of the model. Inclusions in a lipid membrane (like membrane proteins) cause deformations in the bilayer surfaces accompanied by displacements in the membrane hydrocarbon interior. The presented mechanical model provides a set of differential equations, which could have both monotonic and oscillatory solutions for the membrane profile. In the case of not-too-low membrane surface tension the shape of the membrane surfaces is governed by an analogue of the Laplace equation of capillarity. The theory of the lateral capillary forces, presented in Chapter 7, is extended and applied to describe the interactions between two inclusions in a lipid membrane. The range of the obtained attractive force turns out to be of the order of several inclusion radii. The magnitude of interaction is estimated to be sufficient to bring about aggregation of the inclusions. The theoretical predictions are consonant with the experimental observations, although more reliable data about the membrane mechanical parameters are needed to achieve an actually quantitative comparison. The presented mechanical model of lipid membranes can be helpful for the theoretical description and interpretation of various processes involving bilayer deformations and interactions between protein inclusions.

Mechanics of Lipid Membranes and Interaction between Inclusions

10.1.

427

D E F O R M A T I O N S IN A LIPID MEMBRANE DUE TO THE PRESENCE OF INCLUSIONS

The integral (transmembrane) proteins, having a relatively rigid structure, create some deformations in the surrounding lipid membrane (Fig. 10.1). The overlap of such two deformed zones is expected to give rise to a membrane (lipid) mediated protein-protein interaction, as it is for particles confined into a liquid film (Chapter 7) or floating on a liquid interface (Chapter 8). Actually, the experiment shows that some integral proteins can form twodimensional ordered arrays in native membranes [1-6], which can be attributed to interactions of this kind. Although the physical basis of the interactions between inclusions in liquid films and in lipid membranes is similar (overlap of deformations), the formalism developed in Chapters 7 and 8 for liquid interfaces and films cannot be directly applied to the case of lipid membranes, because of their more complicated mechanical properties. Some elastic properties of the membrane interior and surfaces should be taken into account to develop an adequate mechanical model of a lipid bilayer. For review on membrane properties see Refs. [6-8].

L (a)

(b)

(c) Fig. 10.1. The hydrophobic thickness of an inclusion (transmembrane protein) can be (a) greater or (b) smaller than the thickness of the non-disturbed phospholipid bilayer. The overlap of the deformations around two similar inclusions gives rise to attraction between them. In the case of dissimilar inclusions (c) repulsion is expected.

428

Chapter 10

A variety of theoretical models have been proposed to describe the properties of phospholipid bilayers [9-18]. Generally, two types of models can be distinguished. The first type includes molecular models explicitly accounting for the lipid-lipid interactions (of van der Waals,

electrostatic and steric origin) as well as for the configurational entropy of the hydrocarbon chains [11,13]. Similar are the mean-field models based on calculation of the energy and entropy of a given lipid molecule by averaging over all possible chain configurations of its neighbors [9,10,12]. These models give predictions about the elastic constants, spontaneous curvature, chain order parameter, etc. Two essential conclusions can be drawn from these studies: (i) the CH2-group density and ordering are rather uniform across the membrane [10,12,13] and (ii) the lipid hydrocarbon tails strongly interact with each other and create significant mechanical stresses in the membrane interior. The second type of phenomenological models [11,14-18] allows one to calculate the variation of the membrane shape, energy, etc., around a given non-perturbed state using some phenomenological relationships and parameters, such as interfacial tension, stretching and bending elastic moduli, spontaneous curvature of the constituent lipid monolayers, elasticity of the hydrocarbon core, and others. An approach to the lipid-mediated interactions between the membrane proteins, based on the calculation of the chain interaction energy and entropy, has been developed in several works [9,19-21]. This theoretical approach is related to the experimental findings that proteins incorporated in membranes perturb the neighboring lipid molecules [22-24], and especially affect the fluidity of their hydrocarbon chains. However, experiments performed by means of ESR and NMR methods demonstrated that the degree of ordering and fluidity of the hydrocarbon chains of lipid molecules bound to membrane proteins is not very different from those of unbound molecules [25-27], in contract with the initial hypotheses [9,19]. Other idea about the origin of the membrane-mediated interactions between inclusions stems from the experiments by Chen and Hubbell [28], who have observed aggregation of the transmembrane protein rhodopsin in cases in which there has been a mismatch between the width of the hydrophobic belt of the protein and the thickness of the hydrophobic interior of the lipid bilayer, see Fig. 10. lb. These results imply that the perturbation of the bilayer thickness in

Mechanics of Lipid Membranes and Interaction between Inclusions

429

a vicinity of an incorporated protein may give rise to protein-protein attraction. This effect was studied both experimentally [29-34] and theoretically [35-41]; see Ref. [6] for a recent review. Lewis and Engelman [30] showed that bacteriorhodopsin forms aggregates in the membrane of vesicles (prepared from lipids of different chain length) only when the mismatch is greater than 1 nm for thinner bilayers (Fig. 10.1a) and 0.4 nm for thicker bilayers (Fig. 10.1b). Likewise, protein aggregation at considerable hydrophobic mismatch was detected with other natural proteins [29,31,32,34] and synthesized polypeptides [31]. As noted above, if two different protein inclusions create mismatches of the opposite sign, see Fig. 10.1c, then repulsion between the inclusions is expected [42]. Dan et al. [38] proposed a relatively simple phenomenological description, which models a lipid bilayer as two identical tensionless surfaces separated at a distance equal to the local bilayer hydrophobic thickness; see also Refs. [39-41]. Each of these surfaces models a tensionless monolayer, which possesses its own bending elasticity (stiffness) and spontaneous curvature. At a first glance this model, related to the theory of the smectic liquid crystals [43], completely disregards the elastic properties of the membrane interior, however, a closer inspection (see Appendix 10A) shows that the latter effect is implicitly accounted for by the surface free energy. The model by Dan et al. gives predictions about the membrane shape and the interaction between two protein inclusions; generally both the calculated shapes and interaction turn out to be non-monotonic (oscillatory) [38,40]. The comparison between theory and experiment can be realized through the radial correlation function of the protein distribution throughout the membrane, which is simultaneously liable to measurement and calculation, once the interaction potential is available [40]. Unfortunately, for the time being it is not possible to achieve an actually quantitative comparison between theory and experiment because (i) there are no appropriate experimental sets of data and (ii) the values of the model parameters are not certainly known. The conclusions that the lipid hydrocarbon tails in a bilayer are subjected to substantial mechanical stresses [10,11,13,44] does not contradict the experimental fact that usually the total tension of a lipid bilayer, Yb, is rather low (}'~, << 1 mN/m). The low value of ?'/, is due to the fact that considerable negative (compressing) stresses, localized at the membrane-water interfaces, are almost completely counterbalanced by strong positive (stretching) stresses in the

430

Chapter 10

hydrocarbon-chain region, which are even able to cause conformational changes in the imbedded proteins [44]. One could expect an impact of this internal non-uniform stress distribution (neglected in Refs. [38-40]) on the lipid mediated protein-protein interaction. This was a part of our motivation to propose in Ref. [45] an appropriate "sandwich" model of the lipid membrane, which is presented in the next Section 10.2, and further applied to describe the bilayer shape (Section 10.3) and the lateral force between two inclusions (Section 10.4).

10.2.

"SANDWICH" MODEL OF A LIPID BILAYER

10.2.1. DEFINITION OF THE MODEL; STRESS BALANCES IN A LIPID BILAYER AT EQUILIBRIUM

A lipid bilayer drawn to scale is shown in Fig. 10.2. One can distinguish a hydrophobic hydrocarbon chain region sandwiched between two hydrophilic regions of the lipid polar headgroups. Recent studies [6, 46] indicate that the headgroup region can be more voluminous than it is depicted in Fig. 10.2. It is generally accepted that from a mechanical viewpoint a lipid bilayer behaves as a t w o - d i m e n s i o n a l viscous f l u i d at body temperature. This two-dimensional fluidity is manifested when inclusions are moving throughout the membrane [47]. On the other hand, the bilayer exhibits elastic properties in processes of dilatation or bending, both of them being accompanied by extension or compression of the hydrocarbon chains of the lipids.

~-- polar head-group hydrocarbon/water interface

hydrocarbon chain region

Fig. 10.2. A lipid (lecithin) bilayer drawn to scale; after Ref. [50].

Mechanics of Lipid Membranes and Interaction between Inclusions

431

In other words, a bilayer can exhibit different theological behavior (fluid, elastic or hybrid) depending on the mode of deformation. This is not surprising, because a bilayer is neither a three-dimensional, nor a two-dimensional continuum and the hydrocarbon region is neither an isotropic liquid, nor a solid. The generally accepted theoretical model treats a thin liquid film as a liquid layer confined between two Gibbs dividing surfaces, whose interaction is accounted for by an excess disjoining pressure (surface force per unit area) [48-50]. In Ref. [45] the thin-film approach has been extended to phospholipid bilayers (membranes). The hydrocarbon interior of the bilayer has been mechanically described as an elastic medium (rather than as a liquid), which is sandwiched between two Gibbs dividing surfaces modeling the two head-group regions, cf. Fig. 10.2. As already mentioned, in this chapter we present the description of the stresses in a lipid membrane provided by this s a n d w i c h m o d e l , as well as its application to quantify the interaction between two inclusions (like those depicted in Fig. 10.1). Every deformation represents a change in the state of the bilayer with respect to some initial r e f e r e n c e state. Following Ref. [45] we will consider the reference state to be a plane-parallel

bilayer, formed as a result of self-assembly of lipid molecules. Due to the specific state of their hydrocarbon chains built into the bilayer, some internal stresses exist in the chain region [10,11,13,44]. They can be modeled by stresses in an elastic medium. In analogy with the micromechanical approach to the interfaces and thin films, see Eq. (1.4), one can express the pressure tensor in the chain region in the form P=Pr(exe/, +e,e,)+PNe~e Z ,

(10.1)

see Fig. 10.3 for the notation. As in the case of thin liquid films the bilayer (film) surface tension o can be defined as an excess with respect to the tangential component of the pressure tensor [51 ]: 1 ,l ,/ 2

o - - f[P~'(z)-Pr]dz0

oo .f [ p(cal

(z)-Poldz

(10.2)

h/2

where h is the thickness of the hydrocarbon chain region, p7eaJ is the tangential component of

Chapter 10

432

Po

Ip~+rI ~, 7b

h

cI

/

ol~

+

ex

Fig. 10.3. Force balances in the reference state of a lipid bilayer modeled as an elastic medium; PN and Pr are the normal and tangential components of the pressure tensor exerted on an element of the medium (the inset). Normal balance: the outer pressure P0 is counterbalanced by the sum of PN and the disjoining pressure FI. Tangential balance: the total bilayer tension ~4, is a superposition by the compressive surface tension o-at the membrane surfaces and the repulsive effect of the stresses in the bilayer interior, ( P r - Po)h, where h is the thickness of the hydrophobic core; ex, e,, and e: are the unit vectors of the coordinate axes. the real pressure tensor and P0 is the pressure in the bulk of the aqueous phase; PN and PT are attributes of the model, which can be determined as follows [45]. At equilibrium the force balance per unit area of the membrane surface is (see Fig. 10.3)

Po = PN + I-I

(10.3)

where, as usual, FI is the disjoining pressure accounting for the excess molecular interactions across the film [48-51]. Since the hydrocarbon chain region is a non-polar medium, one can expect that the value of FI will be determined mostly by the attractive van der Waals (dispersion) surface force, i.e. FI =

AH 6~rh 3

(10.4)

where An is the Hamaker constant see Chapter 5 and Refs. [49,50,52]. For a hydrocarbon film in water AH ~- 10 -2o J [50]" then with h - 3 nm one obtains I-I = - 2 x 104 Pa. Using the value P0-- 105 Pa (the atmospheric pressure), from Eq. (10.3) one estimates PN -~ 1.2 X 105 Pa [45].

Mechanics of Lipid Membranes and Interaction between Inclusions

433

In addition to the normal force balance, Eq. (10.3), we consider also the tangential force balance. As seen in Fig. 10.3, the stresses acting in lateral direction are related to 0" and PT. The total tension 7b characterizes the bilayer as a membrane of zero thickness intervening between two aqueous phases of pressure P0. Then in view of Fig. 10.3 one can write [45] 7h = 20" + ( P o - Pr)h

(10.5)

Equation (10.5) is in fact a form of the Rusanov equation, which has been originally derived for a thin liquid film [53]. The substitution of Eq. (10.2) into (10.5) yields 4-oo

'}"b ---- I [ P; eal (Z)

- P0]dz

(10.6)

_oo

Equation (10.6) shows that 7~ does not depend on the parameters of the model, such as the film thickness h. On the other hand, 0" and PT depend on the specific definition of the model. However, once this definition has been made, all parameters have well defined values for a given state of the real system. Such a situation is typical for the thermodynamics of thin liquid films [51 ]. One possibility to specify the model is to identify the bilayer surface tension 0" with the interfacial tension of a lipid adsorption monolayer on a hydrocarbon-water interface at the same temperature, composition of the aqueous phase and area per lipid molecule as for the bilayer [45]. Indeed, the surface tension of the bilayer 0" is expected to depend mostly on the interactions in the polar head-group region (Fig. 10.2) and to be insensitive to the state of the hydrocarbon interior. Actually, the experiments with dense lipid monolayers on hydrocarbonwater interface show that the interfacial tension does not depend on the length of the hydrocarbon chains of the lipids at varying area per molecule [54]. Therefore, identifying the surface tension of the bilayer, 0", with the interfacial tension of the respective monolayer at an oil-water interface, one subtracts the contribution from the interactions in the head-group region from the total membrane tension, 7b, and consequently, attributes the excess stresses in the chain region to the tangential stress PT: (Po - PT )h = 0, - 20"

(10.7)

434

Chapter 10

see Eq. (10.5). In other words, having once 0" defined, Eq. (10.7) determines the other parameter of the model, PT. Often the lipid membranes are flaccid and their tension ~4, is rather low: 7b << 2o'; this is usually quoted as the tension free state [15]. Equation (10.7) gives a simple interpretation of the tension free state (yb--+0), viz. the lateral compressing force (per unit length) in the two head-group regions of the bilayer, 20- (which is mostly due to the tendency to decrease the area of hydrocarbon-water contact), is counterbalanced by the lateral

repulsive stress in the chain region, ( P T - Po)h [44,45]. As a numerical example let us take the experimental value 0- = 14 mN/m from Ref. [54] for area 67/~2 per molecule of 1,2-distearoyl lecithin monolayer at the heptane-water interface. For a flaccid lipid bilayer

PT "~ Po + 20-/h

(?'b << 20")

Then taking again P0 -- 105 Pa and h = 3 rim, one calculates PT = 9.4

x 10 6

(10.8) Pa [45].

An inclusion (like these depicted in Fig. 10.1) generates deformations (strains) and creates additional stresses in the surrounding portion of the lipid membrane. As mentioned earlier, the overlap of the deformations caused by two inclusions is expected to give rise to a lateral force, which is analogous to the capillary immersion force described in Chapter 7. The difference with the common immersion force stems from the complicated theology of the lipid membrane: neither liquid film nor elastic plate. A natural approach to the mechanics of such complex body is to use different constitutive relations (connecting stress and strain) for the different independent modes of deformation. The displacement vector u can be expressed as a sum of components due to the independent modes [45]: U = Ustretching q" Ubending 4- Usqueezing

(10.9)

In the cases of uniform stretching and bending the bilayer behaves as an incompressible elastic body. However, in the case of a "squeezing" (peristaltic) deformation, which consists in a variation of bilayer thickness at immobile bilayer midplane (like it is in Fig. 10.1), the twodimensional fluidity of the lipid membrane shows up, see Section 10.3 below. In the next two Sections, 10.2.2 and 10.2.3, following Ref. [45] we will consider briefly the modes of uniform

Mechanics of Lipid Membranes and Interaction between Inclusions

435

Fig. 10.4. The uniform stretching of a lipid bilayer results in a change of the shape of the hydrocarbon portion of each molecule (sketched as rectangle) at constant volume; after Ref. [45].

stretching and bending. Further, in Section 10.3, we pay special attention to the squeezing mode, which is related to the lateral interactions between membrane proteins. 10.2.2. STRETCHING MODE OF DEFORMATION AND STRETCHING ELASTIC MODULUS

Deformation of uniform stretching is presented schematically in Fig. 10.4. Each rectangle symbolizes the hydrocarbon portion of a lipid molecule, which is considered as an

incompressible elastic body; the shadowed circles symbolize the headgroups. Lateral slip between the neighboring "rectangles" is allowed, which reflects the two-dimensional fluidity of the bilayer. The stretching results in a change of the shape of the hydrocarbon portion of each molecule at constant volume (Fig. 10.4). This deformation is not accompanied with a lateral slip between the neighboring lipid molecules. Therefore, in this case the two-dimensional fluidity does not show up and the hydrocarbon interior of the membrane can be treated as an elastic medium, see e.g. Ref. [55]. Let us denote by

,~kPN

and APT the changes in PN and PT, which are due to the stretching of the

bilayer. Following Chapter 5 of the book by Landau and Lifshitz [55] for a deformation of uniform stretching one can derive 72xx---. 72yy= - A P T ;

(10.10)

72zz--- APN ,

where ~ij (i,j = x, y, z) are components of the stress tensor (the stress equals the negative pressure by definition). The components of the strain tensor are defined as follows [55]:

z ( OuJ

o~ui l '

Uij = --2, c)xi +oqxj

(Xl-X,

x2-y,

X3-Z)

(10.11)

Chapter 10

436

where b/i (i = x, y, z) is a component of the displacement vector u. For an elastic body the connections between strain and stress are given by Eq. (4.8) in Ref. [55], which in view of our Eq. (10.10) acquires the form [45]: ///. . . . .

....

1

( t e N q- APT) -

1

(teN -- APt)

(10.12)

9K e 1

1

uxx= uy~,=-9K---~(APN+ APt) + - ~ (APN- APt)

(10.13)

Here Ke is the elastic compressibility modulus and/~ is the coefficient of shear elasticity. For a liquid-like medium Ke >> X, i.e. such a medium can be regarded as being incompressible. In addition, the terms (APN- APt) and (APN + APt) have the same order of magnitude (it turns out that IAPrl >> IAPNl, see Ref. [45] for more details). Hence the terms proportional to 1/K~ in Eqs. (10.12)-(10.13) can be neglected: 1 blxr = b l y y - ----2 blzz = ~

1 ( t e N -- APT)

(10.14)

The components of the strain tensor determined by Eq. (10.14) obey the incompressibility condition V.u = 0, which in view of Eq. (10.11) takes the form uxx + uyv + u~: = 0. Further, let A and AA be the area of the bilayer and its change in the course of deformation. Then the relative dilatation (stretching) of the bilayer is [45] (10.15)

=-- A A ] A = Uxx + uyy

Likewise, the relative change in the thickness is

Ah/h = u~: = -o~,

(10.16)

where we have used the incompressibility condition. The stretching elastic modulus of the lipid bilayer (membrane) Ks is defined by the relationship [15]

Ayh= K~a

(10.17)

where A~'~,is the increment of the membrane tension ~'~,due to the dilatation o~. To obtain an expression for Ks in the framework of the "sandwich" model we differentiate Eq. (10.5):

(lO.18)

AN, = 2Act + ( P 0 - Pr)Ah -hAPr The differentiation of Eqs. (10.3) and (10.4) yields t e N = - n I - I (10.16). Then from Eqs. (10.14) and (10.15) one obtains

-"

3HAh/h =-3IIo~, see Eq.

Mechanics of Lipid Membranes and Interaction between Inclusions

Z ~ T "- Z ~ N -- 3 , ~ = - 3 ( F I

437

+ ,~)o~

(10.19)

Further, in linear approximation one can write Ao = Ec, c~

(10.20)

where Ec is the Gibbs elasticity of a lipid monolayer, cf. Eq. (1.45) and Table 1.2. Finally, in Eq. (10.18) we substitute A0- from Eq. (10.20), (P0 - Pr) from Eq. (10.8), Ah from Eq. (10.16) and APT from Eq. (10.19); as a result we obtain Eq. (10.17) in which Ks is determined by the following expression [45]: Ks = 20- + 2Ec; + 3I-Ih + 3,~h

(stretching elastic modulus)

(10.21 )

The term 20- + 2Ec, in Eq. (10.21) accounts for the elastic response of the two membrane surfaces, whereas the term 3Flh + 3,~h represents a correction related to the internal hydrocarbon chain region. Using the numerical values given after Eq. (10.4) one estimates 31-Ih = 0.18 raN/m, which is much smaller than the total value of Ks, which can be of the order of 100 mN/m. Femandez-Puente et al. [56] estimated that the elasticity of the chain region contributes about 20 mN/m to the total stretching modulus Ks. If one can identify this contribution with the last term in Eq. (10.21), that is 3Xh = 20 raN/m, then with h = 3.6 nm one calculates ,~ ~ 2

x 10 6

Pa [45].

It should be noted that Eq. (10.17) describes the direct dilatation of the membrane. However, the real flaccid lipid membranes are corrugated by thermally excited undulations (capillary waves), and most frequently the average (projected)

area of the membrane

can be

experimentally measured, see e.g. Ref. [57]. The projected area is predicted to increase through superposition of two dilatory effects: (i) suppression of the thermal undulations and (ii) direct elastic stretch of the molecular surface area; the corresponding generalized form of Eq. (10.17) reads [58]: o~ =

kT

ln(1 + AA~'b/4rck,) + zX~,/Ks

(10.17a)

87ckt

where A is the membrane area, o~ is the dilatation of the projected membrane area and kt is the total bending elastic modulus of the membrane as a whole (see the next Section 10.2.3). The experiment [57] confirms that for smaller dilatation (say o~ < 0.02) ln(A~4,) is a linear function

438

Chapter 10

of o~ as predicted by Eq. (10.17a), whereas for larger deformations (say 0.02 < ot < 0.05) AN, grows linearly with o~ in agreement with Eq. (10.17)" for ~z > 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~-=~.=<. h e

(Or

= (a~'/aco)o,=o = ( a C / a ( o ) o , = ~ -

0

(10.98)

In Ref. [45] the conventional second-order finite-difference scheme [75,76] has been used for discretization of the boundary problem. In this way Eq. (10.97) is presented as a system of linear equations, which can be solved by means of one of the standard methods. In Ref. [45] the Gauss-Seidel iterative method has been combined with successive over-relaxation (SOR) and Chebyshev acceleration technique, see e.g. Refs. [75-77].

454

Chapter 10

Having calculated ~', one obtains u~ from Eq. (10.58). In view of Eqs. (10.61) and (10.86) to determine the lateral projection of the displacement vector Un(o9,T) from the calculated profile ~'(o9,z) one is to first find the auxiliary function f(o9, T). The symmetry implies that f(o~, z) is an even function of both o9 and z. Moreover, since f obeys Eq. (10.87), it can be expressed as a Fourier cosine expansion [45]" c,o

f(og, T) = ~ E n coshnv cosno9

(10.99)

n=l

The coefficients En are determined by substituting Eqs. (10.86) and (10,99) into the boundary condition (Og/Oz) T:r = 0, which follows from Eq. (10.64); thus one derives [45]

En =

2 do9 cos no9 nrchq 2 sinh n z c _~

(10.100) :r~

The functions ~'(o9,z) and u(og, z') thus obtained represent the complete solution of the problem about the bilayer deformation. Below we proceed with the calculation of the force between two inclusions, which is due to the overlap of the deformed zones of the membrane around each inclusion.

10.4.

L A T E R A L I N T E R A C T I O N B E T W E E N T W O I D E N T I C A L INCLUSIONS

10.4.1. DIRECT CALCULATION OF THE FORCE

The force approach (Section 7.1.5) can be applied to calculate the lateral capillary force, F, between two identical cylindrical inclusions, like those depicted in Figs. 10.1 and 10.6 [45]: F - 2U H .~dl (m.~)

(10.101)

C

Here C denotes the contact line; the multiplier 2 accounts for the presence of two identical contact lines (upper and lower) on each inclusion. A difference between Eqs. (10.101) and (7.22) is that the membrane surface tension is a tensor, ~, rather than a scalar, see Eq. (10.68). Since the lateral stresses due to the deformation are zero, ~j = 0, (i, j = x, y), see Eqs. (10.54) and (10.63), in the present case there is no contribution analogous to F (kp) in Eq. (7.23). In view of Eqs. (10.71) and (10.78), the two terms in Eq. (10.68) give rise to contributions, F ('r) and

Mechanics of Lipid Membranes and Interaction between Inclusions

455

F (B), originating from the scalar surface tension and bending moment, respectively; thus Eq. (10.101) can be represented in the form [45]" F = F (a) + F (m

(10.102)

F(G)= 2~d/(U n -m)o"

(10.103)

C

F (m=

2(kcq= - Bo/h) ~dl(m.

Vs()(Uii

.n)

(10.104)

c

As in Eq. (10.65), here Vs is a gradient operator in the curved surface

~(x,y).

The latter

equations show that the interfacial bending moment can also give a contribution to the lateral capillary force: this conclusion has a more general validity, i.e. it holds for any interface, not only for lipid membranes. However, it is to be expected, that F <m can be comparable by magnitude with F (m only for interfaces of low tension. Geometrical considerations (see Fig. 10.6) yield Uii.m = fi cos Ip'c,

Un.n = fi sin ~c

(10.105)

Note that the slope angle gtc varies along the contact line due to the overlap of the deformations created by the two inclusions. With the help of Eq. (10.66) one obtains cosgtc = e~. n = (1 + IVii~'12)-1/2

for T - ~'c

(10.106)

sin ~c = fi" n = -(fi-Vn~(1 + IVII~]2) -1/2 = - m - V ~ "

for ~'= ~c

(10.107)

Combining Eqs. (10.103)-(10.107) one derives [45]

F (G>=

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

<1 -6.0

-8.0 I

lo = 3 . 0 n m 2 = 2 x 1 0 6 N m -2

f . ,~.,,~symptotic r formula

o"o = 35 m N . m -1 Bo = -3.2x10 -11 N

-10.0 1.0

I

I

I

2.0

I

3.0

I

4.0

L/2rc Fig. 10.9. Comparison of curves for A~/kT, vs. L/2rc calculated in Ref. [45] by means of Eq. (10.125) and the asymptotic Eq. (10.129); the bilayer thickness is h = 1.55 nm; the other parameters are as in Fig. 10.7. m e m b r a n e proteins. One sees in the figure that for small m i s m a t c h e s (around h/lo - l) the energy of lateral attraction A ~ is smaller than kT, i.e. negligible. H o w e v e r , IAf~l rises strongly when the mismatch increases in both directions. The increase of IA~I with the rise of the surface tension of the non-perturbed bilayer, o'0, is also illustrated in Fig. 10.8. W e recall again

Mechanics of Lipid Membranes and htteraction between Inclusions

463

that o0 is a parameter of the "sandwich" model utilized here and it is to be distinguished from the total membrane tension, ?'b, which is usually rather low (N << 1 raN/m). By definition (see Section 10.2.1) o'0 is the interfacial tension of a lipid adsorption monolayer at oil-water interface for the same area per headgroup as in the bilayer. In the tension free state (?'b = 0) the bilayer surface tension cY0 is completely counterbalanced by the stresses in the hydrocarbonchain region, see Eq. (10.8). It should be noted that Figs. 10.7 and 10.8 give an illustration of the magnitude and range of the lateral protein-protein interaction energy, rather than a quantitative comparison between theory and experiment. An actual comparison demands to know the real values of the various parameters (/l,o0, etc.) which can depend on temperature and the nature of the lipid, the latter being different in the different experiments [30]. On the other hand, the comparison between the two versions of the theory, the force approach, Eqs. (10.117)-(10.118), and the energy approach, Eqs. (10.126)-(10.127), give coinciding numerical results for Af~ as expected [45]. These equations have been used to calculate the curves in Figs. 10.7 and 10.8. In Fig. 10.9 we compare plots of Af~ vs. L/2rc calculated by means of the approximate Eq. (10.129) (the broken line) and by the more rigorous Eqs. (10.126)-(10.127) (the continuous line). This numerical text shows that the asymptotic formula (10.129) provides a good accuracy [45]. In Appendix 10A we describe the connections between the "sandwich" model presented in this chapter and the phenomenological model of a lipid membrane used by Dan et al. [38,40], which is based on a postulated expression for the free energy per molecule of a curved monolayer as a constituent part of bilayer lipid membrane.

10.6.

SUMMARY

Although at body temperature a bilayer lipid membrane behaves as a two-dimensional liquid, mechanically it cannot be simply described as a thin liquid film. The reason is that the hydrocarbon-chain interior of the membrane exhibits elastic behavior if the thickness of the bilayer is varied. This hybrid behavior of the lipid bilayers (neither liquid nor solid) can be

464

Chapter 10

described by means of a mechanical model, which treats the membrane as a special elastic film (the hydrocarbon chain interior) sandwiched between two Gibbs dividing surfaces (accounting for the polar headgroup regions on the membrane surfaces). The latter "sandwich" model involves mechanical parameters such as the shear elastic modulus of the hydrocarbon chain interior 2, and the properties of the two bilayer surfaces: surface tension o, stretching (Gibbs) elasticity Ec,, surface bending moment B0 (proportional to the spontaneous curvature), surface bending and torsion elastic moduli, kc and k C. All these parameters can be determined in experiments with lipid monolayers and bilayers. In the framework of the "sandwich" model the tension-free state of a bilayer is attributed to the counterbalancing of compressing stresses in

the membrane surfaces by stretching stresses in the membrane interior, see Eq. (10.7). A mechanical analysis of the bilayer deformations enables one to derive expressions for the total stretching, bending and torsion moduli of the membrane as a whole, Ks, kt and k t, in terms of the aforementioned mechanical parameters of the model, see Eqs. (10.21 ), (10.49) and (10.50). Inclusions (like proteins) in a lipid membrane cause deformations in its surfaces accompanied by displacements in the membrane hydrocarbon interior, see Figs. 10.1 and 10.6. The resulting stresses are described by an appropriate constitutive relation, Eq. (10.54), and the interfacial stress balance, Eq. (10.65). The model provides a set of differential equations for determining the displacement vector, u, and the interfacial shape, z = ~(x,y). The linearized problem can be reduced to solving a fourth-order differential equation for ~(x,y), Eq. (10.79). The latter could have monotonic and oscillatory solutions for the membrane profile depending on the values of the involved physical parameters. The case of monotonic solutions (corresponding to not-toolow membrane surface tension) is investigated in details; in this case the shape of the membrane surfaces is governed by a second order differential equation, Eq. (10.85), which is analogous to the Laplace equation of capillarity. The account for the compressing stresses at the bilayer surfaces and the elastic stresses in the bilayer interior leads to the appearance of the bilayer surface tension 05 and the chain shear elasticity ~ in the expression for the characteristic capillary length q-~, see Eq. (10.84). The theory of the capillary immersion forces, presented in Chapter 7, can be extended and applied to describe the interactions between two inclusions in a lipid membrane; the derived expressions for the interaction force and energy are Eqs. (10.117) and (10.125). The range of

Mechanics of Lipid Membranes and Interaction between Inclusions

465

the resulting attractive interaction turns out to be of the order of several inclusion radii. The magnitude of the interaction energy Af~ is proportional to the mismatch hc between the hydrophobic zone of the inclusion and the hydrocarbon core of the bilayer. For not-too-small hc the magnitude of A~ can be from 2 to 8 k T at close contact of the inclusions (Figs. 10.7-10.9); this is sufficient to bring about their aggregation. An asymptotic formula for Af~ is derived, Eq. (10.129), which compares well with the numerical calculations; this formula makes the calculation of the interaction energy much easier at the cost of some approximations. The theoretical predictions qualitatively agree with the experimental observations of Lewis and Engelman [30], although more reliable data for the membrane mechanical parameters are needed to achieve a real quantitative comparison between theory and experiment. The parameters of the "sandwich" model can be related to the parameters of the phenomenological model by Dan et al. [38], which is also designed to describe the interactions between inclusions in lipid membranes, see Appendix 10A. The presented mechanical model of lipid membranes can be helpful for the theoretical description of various processes involving bilayer deformations and for the interpretation of experimental data about the interactions and aggregation of protein inclusions.

10.7.

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466

Chapter 10

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469

CHAPTER 11 CAPILLARY BRIDGES AND CAPILLARY-BRIDGE FORCES

The importance of capillary bridges has been recognized for many systems and phenomena like consolidation of granules and soils, wetting of powders, capillary condensation and bridging in the atomic-force-microscope measurements, etc. The capillary bridge force is oriented normally to the plane of the three-phase contact line and consists of contributions from the capillary pressure and surface tension. The toroid (circle) approximation can be applied to quantify the shape of capillary bridges and the capillary-bridge force. More reliable results can be obtained using the exact profile of the capillary bridge, which is determined by the Plateau sequence of shapes: (1) nodoid with "neck", (2) catenoid, (3) unduloid with "neck", (4) cylinder, (5) unduloid with "haunch", (6) sphere and (7) nodoid with "haunch". For the shapes (1-5) the capillary-bridge force is attractive, it is zero for sphere (6) and repulsive for the nodoid (7). Equations connecting the radius of the neck/haunch with the contact angle and radius are derived. The procedures for shape calculations are outlined for the cases of bridges between a plane and an axisymmetric body, and between two parallel planes. In the asymptotic case, in which the contact angle belongs to the range 70 ~ < (Pc < 90 ~ the elliptic integrals reduce to elementary algebraic functions and the capillary bridge can be described in terms of the toroid approximation. Some upper geometrical and physical stability limits for the length of the capillary bridges are considered; the latter can be established by analysis of diagrams of volume vs. pressure. Attention is paid also to the thermodynamics of nucleation of capillary bridges between two solid surfaces. Two plane-parallel plates are considered as an example. The treatment is similar for

liquid-in-gas bridges between two hydrophilic plates and for gas-in-

liquid bridges between two hydrophobic plates. Nucleation of capillary bridges is possible when the distance between the plates is smaller than a certain limiting value. Equations for calculating the work of nucleation and the size of the critical (and/or equilibrium) nucleus are presented.

Chapter I 1

470

11.1.

ROLE OF THE CAPILLARY BRIDGES IN VARIOUS PROCESSES AND PHENOMENA

McFarlane and Tabor [1] studied experimentally the adhesion of spherical beads to a flat plate. They established that in clean dry air the adhesion was negligible. In humid atmosphere, however, marked adhesion was observed, particularly with hydrophilic glass surfaces. At saturated humidity the adhesion was the same as that observed if a small drop of water was placed between the surfaces [1], see Fig. 11.1. Similar results were obtained in the earlier experimental studies by Budgett [2] and Stone [3]. The formation of a liquid bridge between two solid surfaces can lead to the appearance of attractive (adhesive) force between them owing to the decreased pressure inside the liquid bridge and the direct action of the surface tension force exerted around the annulus of the meniscus. We should say from the very beginning that in some cases the force due to capillary bridge can be also repulsive, see Eq. (11.15) below. In all cases this force is perpendicular to the planes of the three-phase contact lines (circumferences) on the solid surfaces, in contrast with the lateral capillary forces considered in Chapters 7-10. The importance of the capillary bridges has been recognized in many experimental and practical systems [4]. For example, the effect of capillary bridges is essential for the assessment of the water saturation in soils and the adhesive forces in any moist unconsolidated porous media [5-7]; for the dispersion of pigments and wetting of powders [8]; for the adhesion of dust and powder to surfaces [9]; for flocculation of particles in three-phase slurries [10]; for liquidphase sintering of fine metal and polymer particles [11,12]; for obtaining of films from latex and silica particles [13-15]; for calculation of the capillary evaporation and condensation in various porous media [16-19]; for estimation of the retention of water in hydrocarbon reservoirs [20], and for granule consolidation [21,22]. The action of capillary-bridge force is often detected in the experiments with atomic force microscopy [23-25]. The capillary-bridge force is also one of the major candidates for explanation of the attractive hydrophobic surface force [26-31], see Section 5.2.3 above. Pioneering studies (both experimental and theoretical) of capillary bridges have been undertaken by Plateau, who classified the shapes of the capillary bridges (the surfaces of

Capillary Bridges and Capillar3,-Bridge Forces

471

diameter Fig. I 1.1. Liquid bridge between a spherical particle and a planar solid surface. constant mean curvature) and investigated their stability [32-34]. The study of the instability of cylindrical fluid interfaces by Plateau was further extended by Rayleigh, who considered also jets of viscous fluid [35-38]. The shapes of capillary bridges between two solid spheres and between sphere and plate were experimentally investigated by McFarlane and Tabor [1], Cross and Picknett [39], Mason et al. [40-41 ], Erie et al. [42]. Exact solutions of the Laplace equation of capillarity for the respective bridges have been obtained in the works by Fisher [7], Melrose [43], Erie [42], Orr et al. [4]. In some cases appropriate simpler approximate solutions can be applied [4, 41,44-46]. All these studies deal with capillary bridges between two solids. Capillary bridges can appear also between solid and fluid phases. Taylor and Michael [47] studied (both theoretically and experimentally) the formation and stability of holes in a sheet of liquid, see Fig. 2.5. Forcada et al. [48] examined theoretically the appearance of a capillary bridge, which "jumps" from a liquid film to wet the tip of the atomic force microscope. Debregeas and Brochard-Wyart [49] investigated experimentally the nucleation and growth of liquid bridges between a horizontal liquid interface and a horizontal solid plate at a short distance apart. The interaction between solid particle and gas bubble, studied by Ducker et al. [28] and Fielden et al. [50], also leads to the formation of a capillary-bridge-type meniscus. Capillary bridges between two fluid phases are found to have crucial importance for the process of antifoaming by dispersed oil drops by Ross [51 ], Garrett [52, 53], Aveyard et al. [54-56], and Denkov et al. [57,58]. When an oil droplet bridges between the surfaces of an aqueous film,

472

Chapter 11

two scenarios of film destruction are proposed: (i) dewetting of the drop would create film rupture [52]; (ii) the oil bridge could have a unstable configuration and film rupturing could happen at the center of the expanding destabilized bridge [53]. The latter mechanism has been recorded experimentally with the help of a high-speed video camera [57] and the results have been interpreted in terms of the theory of capillary-bridge stability [58]. More about the bridging between two fluid phases and the antifoaming action can be found in Chapter 14 below. Many works have been devoted to the problem of capillary-bridge stabili~; we give a brief review in Section 11.3.4 below. Comprehensive review articles about the progress in this field have been published by Michael [59] and Lowry and Steen [60]. Everywhere in this chapter we consider relatively small bridges and neglect the gravitational deformation of the meniscus.

11.2.

DEFINITION AND MAGNITUDE OF THE CAPILLARY-BRIDGE FORCE

11.2. I. DEFINITION

Let us consider an axisymmetric fluid capillary bridge formed between two solid bodies, say two parallel plates, or particle and plate (Fig. 11.1), or two particles (Fig. 11.2), or two circular rings [47]. The presence of capillary bridge will lead to interaction between the two bodies, which can be attractive or repulsive depending on the shape of the bridge (see below). As in the case of lateral capillary forces (see Chapters 7 and 8) the total capillary force F, is a sum of contributions from the surface tension cy and the meniscus capillary pressure, Pc " (11.1)

F,.= U ~ + F p)

Due to the axial symmetry F,. is directed along the z-axis (Fig. 11.2). To calculate the force exerted on the upper particle (Fig. 11.2) one can apply the classical Stevin approach: the upper part of the capillary bridge (above the plane - = 0) can be considered to be "frozen" and the zcomponents of the forces exerted on the system "frozen bridge + upper particle" to be calculated. Thus one obtains F ~ F~. = - rt(2roCY- ro2 Pc.),

9

-2~rot7 and FP~= rcro'Pc, and consequently,

Pc. = PI - P2

(11.2)

Capillary Bridges and Capillap3'-Bridge Forces

473

zt,: A r

\

Fig. 11.2. Sketch of the capillary bridge between two axisymmetric particles or bodies. P~ and P2 are the pressures inside and outside the bridge of length L; r0 and rc are the radii of the neck and the contact line; q~c and q~ are the values of the meniscus slope angle at the contact line and at an arbitrarily chosen section AB.

In Eq. (11.2) negative Fc corresponds to attraction between the two bodies, whereas positive Fc corresponds to repulsion. In spite of the fact that Eq. (11.2) has been obtained for the section across the neck (or the "haunch", see Fig. 2.6b) of the bridge, the total capillary force Fc is independent of the choice of the cross-section. To prove that one first represents the Laplace equation, Eq. (2.24), in the form tan(p - d z / d r

d(rsinq~)/dr = Pc r

(11.3)

and then integrates: olrsinq~- r0) = 7I Pc(r 2

r02)

(11.4)

Comparing Eqs. (11.2) and (11.4) one obtains the expression for the capillary force Fc corresponding to an arbitrary section of the bridge, say the section A B in Fig. 11.2: Fc = - rc(2ro sin(p - r 2 P~,)

(0 < (p < re)

(11.5)

Note that Eq. (11.5) can be directly obtained by making the force balance for the section A B , in the same way as we did for the section z = 0 (Fig. 11.2). Equation (11.4) guarantees that the result for Fc will be the same, irrespective of the choice of the cross-section. If r is chosen to be the radius rc of the contact line at the particle surface, then q~= (,0~.is the meniscus slope angle at the contact line:

474

Chapter 11

F,. = - rt(2rco" sinq~c- ~ Pc)

(11.5a)

Note that the surface tension term in Eq. (11.5) always leads to attraction between the two particles, whereas the capillary pressure term corresponds to repulsion for P,: > 0, and to attraction for Pc < 0 (in the latter case the bridge is nodoid-shaped with a neck, see Section 11.3.1 for details. In accordance with the Laplace equation, Eq. (2.19), the capillary pressure Pc, = Pl - P2 of an axisymmetric meniscus can be expressed as follows: (11.6)

Pc = cr(1/rm + 1/r,),

where rm and r, are, respectively, the meridional and azimuthal radii of curvature. In general, rm and ra vary from point to point and can have positive or negative sign. The sign convention followed in this chapter corresponds to positive rm and ra for a sphere. When the capillary bridge has small length L, but relatively large volume, then r, >> rm and Eq. (11.6) can be written in the form Pc = a ]rm

(r, >> rm)

(11.7)

Since Pc = constant (the gravity deformation negligible), then Eq. (11.7) gives rm = constant, that is the generatrix of the meniscus is a circle in this asymptotic case.

11.2.2. CAPILLARYBRIDGE IN TOROID (CIRCLE) APPROXIMATION

As shown in Chapter 2, in the case of capillary bridges the generatrix of the meniscus is usually an arc of nodoid or unduloid, which are mathematically expressed in terms of elliptic integrals, see Fig. 2.7 and Eqs. (2.50)-(2.52). In some special cases the meniscus can be catenoid, cylinder or sphere, see Section 11.3.1. For the sake of estimates, in the literature the generatrix is often approximated with an arc of circle, and correspondingly, the meniscus is described as a part of a toroid [1,4, 39, 41]. In the asymptotic case described by Eq. (11.7) this is the exact profile. When the toroid approximation is used, the meridional radius 1,, is uniquely determined by the boundary condition for fixed contact angle at the line of three-phase contact (the Young equation), see e.g. Eq. (11.10) below. On the other hand, in various works the toroid approximation is used with different definitions of the azimuthal curvature radius r, :

Capillary Bridges and Capillary-Bridge Forces

475

\ !

,O1 .... fP!

Ze

|

,Zm r

l ,s

}

Fig. 11.3. In toroid approximation the generatrix of the profile of the capillary bridge is approximated with a circle (with center at 02 in the figure). 0~ and 02 are the three-phase contact angles at the surfaces of the solid plate and the spherical particle of radius R, respectively.

r, = rJsinq~c

(maximum possible value of r~)

(11.Sa)

ru = rc

(an intermediate value of ra)

(11.Sb)

r~ = ro = r c - rm(1 - sin(pc)

(minimum possible value of r~,)

(1 1.8c)

where, as before, r0 is the radius of the "neck" and q~c is the meniscus slope angle at the contact line, which is a circumference of radius re. Actually, for a given capillary bridge, r,, varies from rJsinq0c at the contact line down to r0 at the "neck". Equation (11 .Sa) was used by Orr et al. [4] to derive approximate expression for the capillary-bridge force between a spherical particle and a planar plate for arbitrary values of the contact angles. Equation (11.8c) was utilized by Clark et al. [41 ] for a similar system in the special case of zero contact angle (hydrophilic particle and plate). Below we demonstrate that for 70 ~ < q~c < 90 ~ Eq. (11.Sb) compares well with the result for the capillary pressure Pc of a symmetric nodoid-shaped bridge with "neck", see Eq. (1 1.54). Let us consider the application of the toroid (circle) approximation to the capillary bridge between a spherical particle of radius R and a planar plate, for arbitrary values of the contact angles, denoted by 01 and 02, see Fig. 11.3. The contact radius and meniscus slope at the p a r t i c l e contact line are denoted by rc and q0c, respectively; zc is the distance between the plane

of the particle contact line and the planar solid surface. For each of the two circumferences in Fig. 11.3 one can obtain an expression for z~:

Chapter 11

476

R(1 - cos ~) = zc - z,,, - rm cosqgc,

am = rm cos01

(1 1.9)

From Eq. (1 1.9) one can determine the meridional radius of curvature [4]: rm = -R( 1 - cos IV)(cos01 - cosq)c) -1

(toroid approximation)

(11.10)

In addition, we notice that (Pc = ze - (gt + 02) and re. = Rsin gt, see Fig. 1 1.3. Then substituting Eqs. (11.6), (l l . 8a) and (1 1.10) into Eq. ( l l . 5 a ) one determines the capillary-bridge force in toroid approximation: G = -zeRo" {sin(gt+ 02)singt+ (1 + cos gt) [cos 01 + cos(gt+ 02)]}

(11.11a)

In the same way, but using Eq. (1 1.8b), instead of Eq. (11.8a), one derives an alternative expression: F~. = -zeRo" { [2sin(gt+ 02) - 1]singt+ (1 + COSI//)[COS01 4- COS(I//4- 02)]}

(ll.llb)

A third version of the expression for F~. can be obtained combining Eq. ( l l . 8 c ) with Eqs. (1 1.2), (1 1.6) and (1 1.10): Fc =-zeo-

where a

-

1 + cos ~t asingt

COS01 4-

[a s i n l / t - ( 1 - c o s l / t ) ( l - sinl/t)](a + 1 - cos~)

COS(Ip r 4- 0 2 ) .

(ll.llc)

In spite of the different form of Eqs. (1 1.1 la)-(1 1.1 lc) all of

them give the same asymptotics for gt--+0, Fc -- -2zeo- R(cos01 + cos02) = 4zeo- R cos 01 + 02 cos 2

1

-

-

2

0n

-

(gt << 1)

(1 1.12)

which corresponds to the limiting case of a very small (thin and fiat) bridge in the form of ring around the touching point of the sphere and plane [4]. Such a "pendular ring" can be formed between a hydrophilic sphere and a plane owing to a local condensation of water, which gives rise to a strong adhesion [1,61]. However, if the wetting is sufficiently imperfect that 01 + 02 > ze, then F,. has positive sign and corresponds to repulsion. For 01 = ze/2 Eq. (1 1.12) reduces to a formula reported by Cross and Picknett [39]. In the special case of hydrophilic surfaces, 01 = 02 = 0, Eq. (1 1.1 1c) yields the formula derived by Haynes, see Ref. [41 ]: F,. = -2zeo-R(2singt+ c o s g t - 1)/singt

(1 1.13)

Capillary Bridges and Capillai3'-Bridge Forces

477

For two equal non-zero contact angles, 01 = 02= 0, Eq. (1 1.12) reduces to the formula of McFarlane & Tabor [ 1] F,. = -4~;o-Rcos0. If both the sphere and plane are hydrophilic (0 = 0), then F,. = - 4 ~ R c ~ . It is really astonishing that a tiny microscopic pendular ring, localized in the narrow contact zone sphere-plane, can create a force equal to twice the surface tension, 2o', multiplied by the equatorial length of the sphere, 2KR. However, this force is not due to the direct contribution of the surface tension, / ~ ) =-2rcr,.cy sinq~.. For ~--> 0 the thickness of the gap h = R ( 1 - c o s ~ ) - + 0 ,

and in view of Eq. (11.10) rm--->0. In such a case, the term l/rm

dominates the capillary pressure, P~, and the total capillary bridge force Fc, see Eqs. (11.5a) and (11.6). Therefore, for small pendular rings (~t--->0) Eq. (11.7) holds and the toroid approximation can be applied with a good precision. As already mentioned, other case, in which the toroid approximation works accurately, is that of symmetric nodoid-shaped bridges for 70~ q~c<90 ~ see Eq. (11.54) below. However, to achieve really accurate and reliable numerical results it is preferable to work with the rigorous expressions for the capillary bridge shape, given in the next Section 11.3, and to calculate the capillary bridge force using Eq. (11.2), or its equivalent forms (11.5) and (11.5a).

11.3.

GEOMETRICAL AND PHYSICAL PROPERTIES OF CAPILLARY BRIDGES

11.3.1. TYPES OF CAPILLARY BRIDGES AND EXPRESSIONS FOR THEIR SHAPE

Let us define the dimensionless capillary pressure p - P c r o / ( 2 G ) - klro

(1 1.14)

Here kl stands for the mean curvature of the capillary meniscus. The sequence of meniscus shapes, observed when p is increased, has been classified by Plateau [34], see Section 2.2.3 and Table 11.1. The capillary pressure Pc can be both positive and negative; in general - ~

Chapter 11

478

Table 11.1. Types of capillary bridges depending on dimensionless capillary pressure p --Pcr0/(2o'); kl -- p/ro; ro is the radius of the "neck" or "haunch"; r~. is contact line radius; L,n, Lnh, Lun, Luh, L*, Lc and Ls are upper stability limits; see Eq. (1 1.16) and Fig. 1 1.5 for the notation. bridges with "neck" p=0

-~
bridges with "haunch" 0
1/2

nodoid

catenoid

unduloid

Lnn, see

L*/r,, = 1.3255

Lun, see

gq. (11.59)

L*/2ro = 1.200

gq. (11.56)

Notation"

p = 1/2 cylinder

Lc/2ro = rc

1/2 < p < unduloid

Notation"

< +c~

nodoid Lnh, see

Lsl2ro = 1

gq. (l 1.60) q2 = (1 - pl2/po 2)112,

sin% = q2-1(1 - p21p02)l12

e = +1 for unduloid

(Po < P -< P~)

Ik, lz(p) = +-{p,E(0l,ql) + cpoF(01,q,)

1
sphere

Luh, see

sin01 = qj-l(l - po21p 2) 1/2

Shape:

p=l

Eq. (1 1.57)

ql = (1 - po2/pl 2)112,

e = -1 for nodoid,

1

r=ro

e = -1 for nodoid,

e = +1 for unduloid

Shape:

<

(Pl

P < Po)

kiz(p) = -+-[poE(02,q2) + ( 1 - p o ) F ( 0 2 , q 2 ) ]

_[ (p: _ po2)(pi 2_ p2)]l/e/p }

r

r0

A(r) = 2 r t ; d r r ( 1

Area:

+

Zr2) 1/2

A

A(r) = 2re I d r r ( l

Area:

ro

+

Zr2) 1/2

r

2rtro z 2~

k2

27z"

{pIE(OI'ql)--[(p2--po2)(p'2--p2)]'/2/P}

k,-

Jb

r

Volume:

V( r) = rc ; dr r 2 dzldr =

V( r) = - rc I dr r 2 dz/dr =

Volume"

ro

r

V __

a:Pl I/3 E(01,q,)- Po2F(01,ql) 31k~l

(~ + p2)[(p2--po2)(p,2--p2)]ll2/(Ppl) [ /~- (2po 2 + 2pl 2 + 3s

Running slope:

tan(0 = dz/dr

COS2(0= (p2-po2)(p12-p2)/p2

Po E(q}2,q2)

= 7[ro 2 Z

217

3k 3 [fl po E(02,q2)- pop, 2F(r

+ p[(f_pl2)(po2_f) ]1/2[ ,8 - (2po 2 + 2pl 2 + 3epopl)

Running slope" cos 2(0=

tan(0 = dz/dr

(p2-pl2)(po2-f )/p2

Capillao', Bridges and Capillao'-Bridge Forces

479

With tile help of Eq. (11.4) one can bring Eq. (11.2) into the form

F,. = -2rcr0o" ( 1 - p)

(11.15)

For a spherical capillary bridge p = 1 and Eq. (11.15) gives zero capillary bridge force, F, = 0. The latter means that for a spherical bridge the repulsive capillary pressure force, F ~p) = rcro2p,, exactly counterbalances the attractive contribution of the surface tension, U ~) = -2~r0o", cf. Eq. (11.2). The fact that if F, = 0, then the capillary bridge must be a zone of sphere, has been established by Mason and Clark [62]. Note also, that the nodoid with "haunch" (1 < p < oo) is the only type of capillary bridge, for which the total capillary-bridge force is repulsive, F, > 0. On the other hand, the nodoid with "neck" is the only type of capillary bridge, for which the capillary pressure is negative, p < 0. To describe the shape of the capillary bridges we will use the same notation as in Section 2.2.3" in addition, we introduce the following dimensionless variables: t9= Ikllr,

/9o= Ikllro,

Pl = Ikllrl,

pc = Ikllrc

(11.16)

In view of Eq. (2.28) one has Pl = I1 - po sign(p)l

(11.17)

where sign(p) denotes the sign of p. Then Eq. (2.48), which governs the shape of the capillary bridge, can be represented in the form 2

tan(.p -

dz PoP~ + sp = + dr - ](p2 _ pg )(p2 _ p2 )

(11.18)

Here, as before, q~ is the running meniscus slope angle, see Fig. 11.2, and the parameter e = • is defined in Table 11.1. In addition, Po < P < Pl

for a bridge with "neck",

(11.19a)

PJ < P < Po

for a bridge with "haunch".

(11.19b)

The integration of Eq. (11.18), in view of Eq. (11.16), yields the expressions for the generatrix of the meniscus profile, z(p), which are given in Table 11.1" in fact, these expressions are

Chapter 11

480

equivalent to Eqs. (2.50) and (2.52). The elliptic integrals of the first and second kind, F(~, q) and E(0, q), are defined by Eq. (2.51). In the special case of catenoid the meniscus shape is determined by Eq. (2.49). For the catenoid the meridional and azimuthal curvature radii are equal by magnitude, I rm I = I r, I, as it is for a sphere; however, rm = - r , . For that reason the surface obtained by a rotation of a catenoid is sometimes termed "pseudosphere". For a sphere the deviatoric curvature D -

1/r~, -

1/rm is constant, D = 0, whereas for a pseudosphere

D = ro/r 2, i.e. it is not zero and varies from point to point. Table 11.1 contains also expressions for the area A(r) and volume V(r) derived with the help of Eq. (11.18). A(r) and V(r) represent, respectively, the meniscus area and the volume of a portion of the bridge confined between the cross-sections of radii r and r0, the latter being the section across the neck/haunch. With the help of the expressions for A(r) and V(r) given in Table 11.1 one can determine the portions of bridge surface area and volume, A(rx, r,,,) and

V(rx, rv) comprised between every two sections of radii rx and r,,, A(rx, r s) = IA(r,) -T- a(r,,)l,

V(rx, r,,)= I V(rx) -T- V(rv)l

(11.20)

where the signs " - " and "+" stand, respectively, for sections situated on the same and the opposite side(s) with respect to the section r = r0 (the neck/haunch).

l l. 3.2. RELATIONS BETWEEN THE GEOMETRICAL PARAMETERS First, let us note that

122 <-- PoPl

for nodoid with "neck"

(11.21)

192 > PoPl

for nodoid with "haunch"

(1 1.22)

inflection point for all unduloids

(11.23)

p2

= PoPl

In addition, from the definition of g (Table 11.1 ) and from Eq. (11.17) it follows (Pl + ,~P0)2 - 1

(11.24)

Then using the identity (cos(o) -2 - 1 + tan2(o from Eq. (11.18) one can express the cosine of the running slope angle: cos2(p = [(p2 _ po2)(pl2 _ p2)]/p2

(11.25)

Capillary Bridges and Capillary-Bridge Forces

481

Equation (11.25), which holds for both nodoids and unduloids, can be represented also in the form (00/)1) 2 - p2(p02 + 112) + p2COS2q) + 0 4 = 0,

(11.26)

which can be considered as a quadratic equation for determining/)2 if Po and (p are given: p2 = Zl [b _+ (b 2 _ 4po2p12)l/2]

(11.27)

b - po 2 + pl 2 - cos2(p = sin2(p- 28pOPl

(11.28)

The fact that Eq. (11.27) contains two roots

for/92 deserves a special attention.

In the case of nodoid (P0 - Pl -- +1) Eq. (11.27) has always two positive roots for p2, which in view of Eqs. (11.21) and (11.22) correspond to the two types of nodoids:

p2 = PoPl + 71 {sin2q)m [(sin2(p+ 4poPl)sin2q)]l/2]}

(nodoid with 'neck")

(11.29)

p2 = PoPl + -51 {sin2q~+ [(sin2(p + 4poPl)sin2(p ]1/2] }

(nodoid with "haunch")

(1 ~.3o)

In the case of unduloid (P0 + Pl = 1) Eq. (11.27) has real roots for/32 only when 0 < P0 < sin 2 q~ 2

(for unduloid with "neck", q~< rt/2)

(11.31)

sin 2 q~ < P0 < 1 2

(for unduloid with "haunch", q~> re/2)

(11.32)

To understand the meaning of the two roots in the case of unduloids, one can express Eq. (11.27) in the form p2= Popl + 71 { sin2(p-4popl -_J- [(sin2(0

4poPl)sin2(p]l/2]}

(all unduloids)

(11.33)

Then in view of Eq. (11.23) the two roots in Eq. (11.33) are the radial coordinates of the two points with the same slope angle q~, situated on the left and right from the inflection point of the unduloid at p2= POP1. Another possible problem, appearing when the boundary conditions for the meniscus shape are imposed, is to determine the radius of the neck/haunch, P0, for given p and q~. With that end in view we use Eq. (11.24) to represent Eq. (11.26) in the form

482

C h a p t e r 11

bl 2 --

2gp2u + (/94 -/92 sin2(P) = 0,

u - PoPl

(11.34)

Solving Eq. (11.34) for nodoid-shaped bridge one obtains u = p2 ___/9 sin(P, where the roots with "+" and " - " correspond to meniscus with "neck" and "haunch", respectively. Then using the fact that for nodoid p~ = P0 -+ 1, and consequently, u = p02 _+P0, one derives Po = 71 {[1 + 4p(p + sin(P)] 1/2 - 1 }

(nodoid with "neck")

(11.35)

1 P0 = -~ {[1 + 4 p ( p - sin(P)] 1/2 + 1 }

(nodoid with "haunch",/9 > sin@

(11.36)

Solving Eq. (11.34) for unduloid-shaped bridge one obtains u = -/92 -t-- p sin(p (the other root must be disregarded). Since for nodoid u = P0 - P02, one finally derives 1 P0 = 7{1 - [ 1 -4p(sin(P-p)] 1/2}

(unduloid with "neck", p < sin(P),

(11.37)

P0 = 71 { 1 + [1 - 4p(sin(P- p)]l/2}

(unduloid with "haunch", p < sin@

(11.38)

i and -~ < P0 < 1 respectively, which in view of Equations (11.37) and (11.38) yield 0 < P0 < 7, Table 11.1 determines the type of the bridge (for unduloids P0 = P).

Capillary bridge formed between axisymmetric surface and a plane. To illustrate the application of the above equations let us consider the capillary bridge formed between a curved axisymmetric surface of equation z = h(r) and a plane. The equation z = h(r) may represent a sphere (see Fig. 11.3), or the shape of the cantilever of the atomic force microscope, see Refs. [23-25,46]. To specify the problem we assume that the contact angles 01 and 0~ are known (cf. e.g. Fig. 11.3), and that the capillary pressure Pc is negative, i.e. we deal with a nodoidshaped bridge with neck. Such bridges can be spontaneously formed by capillary condensation of water between two hydrophilic surfaces at atmospheric humidity lower than 100%: alternatively, such bridges can be spontaneously formed by capillary cavitation of vapor-filled bridges between two hydrophobic surfaces at temperatures lower than the boiling point of the aqueous phase; see Section 11.4 for more details. The meniscus slope angle at the curved surface is (Pc" = 02 -t-

arctan(dh/dr)

(11.39)

Capillary Bridges and Capillary-Bridge Forces

483

Then for a given (dimensionless) contact radius on the curved solid surface, Pc. = Ikllrc, using Eq. (11.35) one calculates the radius of the neck, Po = 71 { [ 1 + 4pc(pc + sin(pc)] 1/2 - 1 },

(1 1.40)

of the nodoid-shaped surface; the real bridge meniscus represents a part (zone) of this surface. With the value of P0 thus obtained from Eq. (11.29) one determines the (dimensionless) contact radius p,. at the planar solid surface: ~2 _(kl~.)2 =PoPl + 71 {sin201 -[(sin201 +4pOPl)Sin201]l/2]}

(Pl = P 0 + 1)

(11.41)

Further, the length of the capillary bridge (the distance between the planes of the contact lines on the curved and planar solid surfaces) can be expressed in the form

h(rc) = Iz(pc) +- z( fie- )1

(11.42)

where the function z(p) is given in Table 11.1; the sign is "+" or " - " depending on whether the neck appears on the real meniscus, or on its extrapolation; as already mentioned, h(r) is a known function. Having in mind Eqs. (11.40) and (11.41) one concludes that Eq. (11.42) relates two parameters: Pc and k~, or equivalently, rc = Pc./Ikll and Pc = 2ok~. If one of them (the contact-line radius re., or the capillary pressure Pc.) is known, then we can determine the other one by solving numerically Eq. (11.42). Similar procedure can be applied to the case of unduloid with neck; in the latter case one has to use Eqs. (11.37) and (11.33) instead of eqs. (11.40) and (11.41).

11.3.3.

SYMMETRIC NODOID-SHAPED BRIDGE WITH NECK

In this subsection we consider another example of physical importance: nodoid-shaped bridge with neck, formed between two identical parallel planar solid surfaces separated at a distance h (Fig. 11.4). As mentioned earlier, when the liquid phase is water, an aqueous bridge can be formed between two hydrophilic solid surfaces, or alternatively, a vapor-filled bridge can be formed between two hydrophobic solid surfaces. In both cases we will denote by (p~ the meniscus slope angle at the contact line, which represents also the three-phase contact angle

Chapter 11

484

hl2

.

0

"

Fig. 11.4. Capillary bridge formed between two plane-parallel solid surfaces separated at a distance h; ro and r. are the radii of the neck and the contact line; qg. is the contact angle measured across the phase of the bridge.

measured across the phase of the bridge (q~, < 90~

As before, p, = I k~lr, is the dimensionless

radius of the contact line. With the help of Table l l.1 and Eq. (11.25) the equation for the length of the bridge, h = 2z(p,), can be represented in the form (11.43)

Ikll h/2 + plE(q~l,ql) = p0F(~l,ql) + COSq), where ql - (1 - po2/p12) 1/2,

sinO1 - (1 - po2/pc2)l/2/ql,

Pl = Po + 1.

(11.44)

In view of Eqs. (11.40) and (11.44) one can use Eq. (11.43) to calculate p, if kl is known, or vice versa, to determine kl (and the capillary pressure P, = 2O'kl) if r, = p c / l k l l

is given. To

achieve that one has to solve Eq. (11.43) numerically; the stable numerical method of the "arithmetic-geometric mean" can be applied to calculate the elliptic integrals, see Ref. [63] Chapter 17.6 therein. Depending on the parameters values Eq. (11.43) could have one or two solutions. Long ago Plateau [34] established that two solutions are possible for a nodoid-shaped bridge with neck, formed between two parallel circular discs, at given volume, radius r, and separation h; see also Refs. [59,60]. Experimentally, one of the solutions corresponds to a stable bridge, but loses its stability when it coincides with the other solution, in which case the fluid separates axisymmetrically forming cap-shaped drops onto the two discs [34,59].

Capillar3, Bridges and Capillar),-Bridge Forces

485

Let us now investigate analytically the case 70 ~ < q~ < 90 ~ in which the elliptic integrals can be asymptotically expressed in terms of algebraic functions. (Note that in the case of hydrophobic plates gas-filled bridge 70 ~ < q~c < 90 ~ corresponds to 90 ~ < 0 < 110 ~ where 0 is the contact angle measured across water,

see Fig. 11.4.) In this range of angles, which is often

experimentally observed, one has sinqgc = 1 -//2/2,

/7 = COS(pc,

//2 << 1

(11.45)

The substitution of the latter expression in Eq. (11.40), after expansion in series, yields Po = P c - Pc//2/[2(1 + 2pc)] + 0(//4)

(11.46)

Then one derives (1

2 -~ - Po/P,.")l/2

=

//(1 + 2pc) -1/2 + 0(//3)

(11.47)

With the help of Eqs. (11.17) and (11.46) one obtains: ( 1 - po21pl 2) 1/2 _-- ( 1 4- 2pc)1/21( 1 + Pc)+ 0(//2)

(11.48)

Next, the combination of Eqs. (11.47) and (11.48) yields" sin01(pc ) - (1 - po2/pc:) 1/2 (1 - po2]p12) -1/2 - 1I (1 + pc)/(1 + 2p~.) + 0(//3)

(11.49)

One sees that sinq~l(p~.) = 0(7"/) is a small quantity. Then the elliptic integrals can be expanded in series:

dc~

sinq~f

F(4h,ql)

J0 .,h-q,

sinqh + O(sin3q~l) -- E(01,ql)

(11.50)

Eqs. (11.49) and (11.50) lead to F(01,ql ) -- E(01,ql ) -- sin01 - 7/(1 + pc)/(1 + 2pc) + 0(//3)

(11.51)

Finally, the substitution of Eq. (11.51) into Eq. (11.43) yields a simple relation between the thickness of the gap, h, and the dimensionless radius of the contact line Pc.: h ~-

2cos q~c

p~.

Ikll

1 +2pc.

(0 < Pc" < oo" 70 ~ < q)c < 90 ~

(11.52)

486

Chapter 11

Note that the above expansions for small 7/ are uniformly valid for 0 < A < oo and kl < 0. For pc---)0 (and kl < 0) Eq. (11.52) gives h ---~0, as it could be expected. In the other limit, p,.--)oo Eq. (11.52) reduces to hma• = c~ Ikll

(70 ~ < r < 90 ~

(11.53)

where hmax denotes the maximum possible length of the bridge (the maximum width of the gap between the plates), for given values of the capillary pressure, Pc. = 2O'kl, and the contact angle r

the latter being measured across the bridge phase.

It is worth noting that f o r f i x e d h and pc--> 0 Eq. (11.52) gives kl -->0, but the ratio rc = p c / I k l l -->h/(2cosq~c) tends

to a non-zero

constant.

One

can

obtain

smaller

values

of

re.

(i.e. r~. < h/2cosq~,.) only using unduloid (rather than nodoid) with "neck". It should be also noted, that Eq. (11.52) can be presented in the alternative form

-Pc = a 9

(2c~

1/ h

rC

(70 ~ < (Pc < 90 ~

(11.54)

The comparison between Eqs. (11.54) and (11.6) shows that the meridional curvature radius is constant: r m - -h/(2cos(p,,) = const. In fact, Eq. (11.54) represents the result, which would be obtained if the "toroid" ("circle") approximation were directly applied to express the capillary pressure using the definition (11.8b) for the azimuthal curvature radius r,. Hence, it turns out that the toroid approximation can be applied with a good precision to n o d o i d - s h a p e d bridges with neck, if the contact angle belongs to the interval 70 ~ < q~c < 90 ~ Similar expansion for small 7/2 can be applied also to the case of u n d u l o i d with neck; as a starting point one is to use Eq. (11.37) with p = Pc, instead of Eq. (11.40). In first approximation one obtains again Eq. (11.54), that is the toroid approximation.

11.3.4. GEOMETRICAL AND PHYSICAL LIMITS FOR THE LENGTH OF A CAPILLARY BRIDGE

Although derived for a special type of capillary bridge, Equation (11.53) demonstrates the existence of limits for the length of a capillary bridge. The nodoid and unduloid are periodical curves along the z-axis, see Figs. 2.7 and 11.5. Moreover, the nodoid has self-intersection

Capillary Bridges and Capillar?'-Bridge Forces

Y

f

<'',

487

r

Lun

Luh

Luh

[

f

r

~-

Fig. 11.5. Geometrical upper limits for the stability of capillary bridges: (a) distances between two closest points with vertical tangents for an unduloid with "neck", Lun, and with "haunch", Luh; (b) distances between two closest points with horizontal tangents for a nodoid with "neck", Lnn, and with "haunch", Lnh. points. Unduloid-shaped bridges of length greater than the period of unduloid as a rule are physically unstable, see Refs. [59,60]; hence the period of unduloid gives an upper limit for the stability of the respective bridges (the longer bridges are unstable, whereas the shorter bridges could be stable or unstable). The distance (along the z-axis) between two consecutive points with horizontal tangent (slope angle q~ = 0) on the nodoid (Fig. l l.5b) can also serve as an upper limit for the stability of nodoid-shaped bridges. Below we provide expressions for these upper stability limits, which could be helpful for estimates. In Fig. 11.5a the geometrical limits for the stability of the unduloids with "neck" and "haunch" are denoted by Lun and Luh, respectively. Each of them can be identified with 2z(p~), where the function z(p) is given in Table 11.1. In addition, for p = p l

one has sinai = sinO2 = 1;

consequently, ~ = ~2 = ~/2, and the elliptic integrals in Table 11.1 reduce to the respective total elliptic integrals [63-66]: K(q) -- F(•/2, q),

E(q) - E(rc/2, q),

(11.55)

Then with the help of Eqs. (11.14), (11.16), (11.17) and the expressions for z(p) in Table 11.1 one obtains

Chapter 11

488

1-p

Lun(p)/(2ro) = K(ql) +

E(ql),

ql =

K(q2),

q2 =

p 1-p

Luh(p)/(2ro) - E(q2) +

41-2p l-p ~/2p-I

P

1 for 0 < p < -5

(11.56)

1 for 5 < P < 1

(11.57)

p

For p--+ 71 one has ql - q2 - 0" in addition, K(0) - E(0) - rt/2. Then from Eqs. (11.56) and (11.57) one obtains Lc = lim Lun(P)= lim Luh(p)=2gro p--~l / 2 p---~l/ 2

(1 ~.58)

Indeed, the upper limit for the stability of a cylindrical (p = 21 ) capillary bridge of radius r0 is Lc = 2rtr0. This critical value was given first by Beer [67] and obtained in the studies by Plateau [34]. Lc= 2rtr0 is the limit of stability of the cylindrical bridge against axisymmetric perturbations at fixed (controlled) volume of the bridge. In the case of pressure control the limit of stability appears at two times shorter length: L = rtr0, see e.g. Refs. [59,60]. For p ~ 1 (spherical bridge, Table 11.1 ) Eq. (11.57) yields Luh ----)2r0; indeed, from geometrical viewpoint the larger possible diameter of a spherical bridge is the diameter of the sphere, 2r0. The geometrical limits for the length of a nodoid-shaped bridge (see Fig. 11.5b) can be determined in a similar way:

Lnn =F(q}lg'qlg )+ ~/21 p l + l 2r 0

I pl

I pl+lE(d)lg,qlg I pl

Lnh = E((Peg ,q2g ) - p - 1F(~2g ,q2g ) 2r0

)

-~ < p < 0

(11.59)

0 < p < oo

(11.60)

P

where qlg - (21pl + 1)1/2(1 + Ipl)-l" sin~lg - [(1 + Ipl)/(2lpl + 1)] 1/2", q2g - (2p - 1)l/2p -l

,

sinq~2g - [p/(2p - 1)] I/2" the fact that the nodoid has horizontal tangent at p = pg - (PoPl

and

)1/2 has

been used. For p---~l Eq. (11.60) yields Lnh---) 2r0, i.e. we arrive again to the result for a sphere, see above. For p + 0 Eqs. (11.56) and (11.59) give divergent values for Lun and Lnn; this result can be attributed to the fact that there are no geometrical limitations for the length of a catenoid. On the other hand, there are physical limitations for the length of a catenoid stemming from the

Capillary Bridges and Capillary-Bridge Forces

489

boundary conditions for the Laplace equation. Plateau [34] produced a catenoid in stable equilibrium by suspending oil on two circular rings and adjusting the volume of the oil so that the interface across the rings was planar. He found that the catenoid thus produced was at the limit of its stability when the distance apart of the rings L to the diameter 2rc reached a value approximately 0.663. He recognized also that for L/2re < 0.663 there is an alternative catenoid solution not observable in the experiments, and that the limit of stability is reached when the two solutions coincide; see Ref. [59] for more information. To elucidate this point one can use the equation of the catenoid, Eq. (2.49), to obtain the connection between the length of the bridge, L, and the contact radius re: ( 11.61)

rc/ro = cosh(L/2ro)

Introducing variables x - L/2ro and a - L/2rc one transforms Eq. (11.61) to read x = a cosh x

(11.62)

When a is small enough the straight line y~(x) = x has two intersection points with the curve y2(x) = a cosh x ; they represent the two roots of Eq. (11.62) corresponding to the two catenoids recognized by Plateau. For larger values of a Eq. (11.62) has no solution. For some intermediate critical value a = a* the line y~(x) is tangential to the curve y2(x) and the two roots coincide; from the condition for identical tangents, y~'(x*) = y2'(x*), one obtains a*sinh x* = 1. The combination of the last result with Eq. (11.62) yields a transcendental equation for x* 9 x* = coth x*

~

x* - L*/2ro = 1.1996786...

(11.63)

Finally, one recovers the result of Plateau" L*/2rc - a* = (sinh x*) -l - 0.6627434... = 0.663. The latter value determines the maximum length, L*, of the catenoid formed between two identical circular rings of a given radius rc. S t a b i l i ~ o f c a p i l l a r y bridge menisci. As mentioned above, the parameters Lun, Luh, Lc,

Lnn, Lnh and L* calculated above serve as upper limits for the length of the bridges: the longer bridges are unstable, but the shorter bridges could be stable or unstable, depending on the specific conditions. In general, the bridges are more stable when the volume of the bridge and the position of the three-phase contact line are fixed. The bridges are less stable when the pressure, rather than the volume, is fixed; see Refs. [59, 60] for a detailed review.

490

Chapter 11

In general, two methods are used to investigate the stability of capillary bridges. As demonstrated in Section 2.1.2 the Laplace equation of capillarity corresponds to an extremum of the grand thermodynamic potential f~. A solution of Laplace equation describes a stable or unstable equilibrium meniscus depending on whether it corresponds, respectively, to a minimum or maximum of ~. Then the sign of the second variation of ~ is an indication for stability or instability. This approach was applied to capillary bridges by Howe [68], and utilized by many authors [69-71 ]. The second method for determining the stability of capillary menisci arose from the analysis of the behavior of pendant and sessile drops and bubbles used in the methods for surface tension measurements. These observations revealed that the stability limits always lie at turning points in the plots of volume against pressure (PV-diagrams) [60]. The turning points in volume are the points on the PV-diagram at which the volume has local minimum or maximum; they represent stability limits in the case of fixed (controlled) volume. Likewise, the turning points in pressure are the points on the PV-diagram at which the pressure has local minimum or maximum; they represent stability limits in the case of fixed (controlled) pressure. Classical example for turning points in pressure are the local extrema of pressure in the PVdiagram predicted by the well-known van der Waals equation of state for temperatures below the critical one; these turning points separate the unstable region from the region of (meta)stable gas or liquid, see e.g. Ref. [72]. Another example for turning point in pressure is observed with the known "maximum bubble pressure method" for measurement of dynamic surface tension [73-75]; the transition from stability to instability occurs when the bubble reaches hemispherical shape and maximum pressure [76]. Note, however, if the same system is under volume control (say liquid drop of controlled volume instead of bubble) the turning point in pressure is no more a stability limit: stable states beyond hemisphere can be realized. The idea that stability changes for drops always occur at turning points was put forward as a proposition for meniscus stability analysis by Padday & Pitt [77] and Boucher & Evans [78]. In addition, bifurcation points on the PV-diagrams are also recognized as stability limits, especially for relatively long capillary bridges at volume control (the bridges under pressure control are less stable and cannot survive until the appearance of bifurcation points), see Refs. [60,79] for details.

Capillary Bridges and Capillary-Bridge Forces

V Vcyll

491

I

I I I I I

E I

D 0 -0.5

A !

0

0.5

1

1.5

Fig. 11.6. Plot of the volume of a liquid bridge, V, scaled with the volume of the cylindrical bridge, Vcy~, against the dimensionless pressure difference across the meniscus surface, Pc , see Eq. (11.64)" the bridge is symmetric like that in Fig. 11.4; V and ~ vary at fixed contact line radius rc and fixed length L of the bridge {after Ref. [60] }. For example, the stability limit L = rtr0 for cylindrical bridge under pressure control is a turning point in pressure, whereas the stability limit L = 2rtr0 for cylinder under volume control is a bifurcation point [59,60]. Figure 11.6 represents a sketch of a typical PV-diagram for relatively short liquid capillary bridges, L/rc < L*/rc = 1.3255, which are symmetric with respect to the plane of the "neck"/ "haunch", like it is in Fig. 2.6. Moreover, it is assumed that the contact radius, re, is fixed, but the slope angle at the contact line, qgc, can vary, cf. Fig. 11.4. The dimensionless pressure in Fig. 11.6 is defined as follows:

Pc = Pc r,:/(20") For catenoid and cylinder Pc = P = 0 and -~, 1 respectively, but in general Pc :/: P

(11.64) cf. Eqs.

(11.14) and (11.64) (p is not suitable to be plotted on a PV-diagram, because it reflects not only variations of the pressure Pc, but also changes in the neck radius r0). Each point on the curve in Fig. 11.6 corresponds to a capillary bridge in mechanical equilibrium, which could be stable or I cf. unstable. In particular, the section AB corresponds to unduloid-shaped bridges (0 < p < 5,

Table 11.1) with a very thin neck, which are unstable [60]. At the point A the neck radius r0 becomes zero and the bridge splits on two pieces of spheres. Point B is a turning point in volume; it represents a boundary between stable bridges of controlled volume (on the left) and

492

Chapter 11

the unstable bridges (on the right). Likewise, point D is a turning point in pressure: the whole line DEFGH corresponds to stable bridges under pressure (or volume) control, whereas the section DA corresponds to bridges unstable under pressure control. The section BD represents bridges with neck, which are stable under volume control, but unstable under pressure control. The points C and E, at which Po = P = 0, represent the two catenoids, determined by the roots of Eq. (11.62). The catenoid bridge E (that of greater volume) is stable in both the regimes of fixed volume and pressure, whereas the catenoid bridge C (that of smaller volume) is stable only in the regime of fixed volume [60]. When L/rc = L*/rc = 1.3255 points C and E merge with point D, and for L/rc > 1.3255 there are no equilibrium catenoid (and nodoid with neck) bridges. The section EF corresponds to stable unduloid-shaped

bridges with "neck"

1 ( 0 < p < 7), the section FG represents stable unduloid-shaped bridges with "haunch"

(-5l < p < 1), and finally, the section GH corresponds to stable nodoid-shaped bridges with "haunch" (p > 1, cf. Table 11.1). The equilibrium bridges at the points F and G have the shape of cylinder and truncated sphere, respectively. At the point H one has

L/ro -- Lnh/ro,

see

Eq. (11.60), which is the limit of stability of the menisci with "haunch" [60]. For L/rc = ~ the point F (cylindrical bridge) coincides with the point D (turning point in pressure) and the cylindrical bridge loses its stability under regime of pressure control. For L/rc > ~ the regions with stable bridges on the PV-diagrams become more narrow, and the PV-

diagrams (representing mostly unstable states) become more complicated, see the review by Lowry and Steen [60].

11.4.

NUCLEATIONOF CAPILLARY B R I D G E S

11.4.1. THERMODYNAMIC BASIS

The thermodynamics of nucleation (creation of a new phase from a supersaturated mother phase), stems from the works of Gibbs [80] and Volmer.[81], and describes the formation and growth of small clusters from the new phase (drops, bubbles, crystals) in a process of phase transition like condensation of vapors, cavitation (formation of bubbles) in boiling liquids, precipitation of a solute from solution, etc. [82-86]. As a driving force of nucleation the

Capillar)' Bridges and Capillar)'-Bridge Forces

493

increased chemical potential of the molecules in the mother phase with respect to the new phase is recognized. The nuclei of the new phase in the processes of homogeneous (bulk) condensation and cavitation are spherical drops and bubbles. On the other hand, in the case of

heterogeneous nucleation (nucleation on a surface) the nuclei have the shape of truncated spheres. In both cases the presence of a convex spherical liquid interface of large curvature (small nucleus) leads to a large value of the pressure inside the nucleus, which in its own turn increases the molecular chemical potential with respect to its value in a large phase of planar interface. That is the reason why a necessary condition for such nuclei to appear is the initial phase to be "supersaturated": in the case of condensation the humidity must be slightly above 100 %; in the case of cavitation (boiling) the equilibrium vapor pressure of the liquid to be higher than the applied outer pressure. The nodoid-shaped capillary bridges (as well as the concave spherical menisci in capillaries) have negative capillary pressure and provide quite different conditions for nucleation. Formation of nuclei with such interfacial shape makes possible the condensation to occur at humidity markedly below 100% and the cavitation to happen when the equilibrium vapor pressure of the liquid is considerably lower than the outer pressure. The effect of concave menisci on nucleation was first established in the phenomenon capillary condensation, which appears as a hysteresis of adsorption in porous solids [17, 86-88]. The experiments of McFarlane and Tabor [ 1] on the adhesion of spherical beads to glass plate in humid atmosphere give an example for nucleation of liquid capillary bridges. The formation of nodoid-shaped cavities between two solid surfaces was examined by Yushchenko et al. [89] and Parker et al. [90]. As already mentioned, the formation of such cavities is proposed as an explanation of the attractive hydrophobic surface force [26-31 ]. As a physical example let us consider the nucleation of nodoid-shaped capillary bridges in the narrow gap between two parallel plates (Fig. 11.4). The theoretical treatment is the same for

liquid bridges between two hydrophilic solid surfaces and for gas bridges between two hydrophobic solid surfaces. In other words, the approach can be applied to both capillary condensation and capillary cavitation. The work of formation of a nucleus can be expressed in the form [82, 83, 85]:

Chapter 11

494 W(r,.) = A t a -

2 A c a coscpc - v(n)[p(n)(rc)

- P] + N(n)[l./(n)(rc) - ,/./]

(11.65)

which represents the difference between the free energies of the system in the states with and without nucleus. The meaning of the symbols and terms in Eq. (11.65) is the following. As usual, o is the surface tension, AI denotes the area of the liquid meniscus and ( 11.66)

Ac = rcrc2

is the area encircled by each of the two contact lines of radius rc (Fig. 11.4). We will use rc as a parameter which identifies the capillary bridge, just as in the theory of homogeneous nucleation the drop/bubble radius is used to identify the spherical nuclei. The first and the second terms in the right-hand side of Eq. (11.65) represent the work of formation of new phase boundaries liquid/gas and solid/fluid, respectively; the third term is the mechanical work related to the change in the pressure inside the nucleus, P(~, in comparison with the pressure P in the ambient mother phase; ],/(n)(r) and/l are chemical potentials of the molecules in the nucleus and in the ambient mother phase; Ar

and g (n) are the number of molecules in the nucleus and its

volume. We assume that the mechanical equilibrium has been attained (the Laplace and Young equations are satisfied); however, the bridges could be out of chemical equilibrium. In particular, the multiplier-a coscpc in Eq. (11.65) stems from the Young equation. As in Section 11.3.3 angle q3c is the three-phase contact angle measured across the bridge phase (in the case of gas-in-liquid bridge the complementary angle, 0=re-(Pc, is traditionally called 'the contact angle'). Depending on whether we deal with gas (vapor) or liquid bridges, the following expressions are to be substituted: ]v(n)[]./(n)(rc) - 1./] -- P(n)(rc)l, An) ln[P(n)(rc)/Po] N(n)[l./(n)(rc) - 1./] = (v(n)/gm){

[P(n)(rc)-P]Vm - k T l n ( P ' / P o ) }

(gas bridges)

(11.67)

(liquid bridges)

(11.68)

As before, p(n) is the pressure in the nucleus (in the capillary bridge); P0 is the equilibrium vapor pressure of a planar liquid surface at that temperature; P' is the vapor pressure in the gas phase surrounding a liquid-bridge nucleus; Vm is the volume per molecule in the liquid phase. To obtain Eq. (11.67) we have used the expression for the chemical potential of the vapors inside the nucleus, fl(n)(rc) = ]-/0 + k T lnp(n)(r~.), and that for the vapors which are in equilibrium

Capillar 3, Bridges and Capillar3,-Bridge Forces

495

with the mother phase,/1 = Y0 + k T lnP0; /to is a standard chemical potential; the ideal gas equation, Pm)(rc)lr n~ = N(")kT, has been also employed. To obtain Eq. (11.68) we have used the expressions p = ~to + k T l n P '

and /t(n)(rc) = /t0 4

k T lnp(V)(rc), where the equilibrium vapor pressure of the concave liquid bridge, p(v), is given

by the Gibbs-Thomson equation p(v) = Po exp(P~.Vm/kT) = Po exp{ [P(n)(rc) - P ] V m / k T }

(11.69)

Note that quantities like P, Po, P', P, 1/,11,are independent of re. Let us now investigate the dependence of nucleation work W on rc. Using the Gibbs-Duhem equation for the phase of the bridge, one can write: -Vm)dp(n)/drc + l~n)d/t(n)/drc = 0

(11.70)

Differentiating Eq. (11.65), along with Eqs. (11.66) and (11.70) one can derive dA 1 dV (n) dN(n_____s) drc = cr drc - 4rcrcocosq0~. + [P - P(n)(rc)] d r c + [p(n)(r~.) - / t ] drc

dW

(11.71)

The meniscus area and the volume of the bridge can be expressed in the form hi2

hi2

A, = 2rt f d z

r(z)(1 + r, z)l/2

l/n) = rt Idz rZ(z)

-h/2

(11.72)

-h/2

where r(z) expresses the meniscus profile in cylindrical coordinates and r~ - Or~Oz. Note, that in general the meniscus profile depends on rc through the boundary condition at the contact line; in other words r = r(z,r,.). Differentiating under the sign of the integrals in Eq. (11.72) and using integration by parts one can prove that (7"

dA l

+ [ P - P(n)(r~.)]

dr. -- 2re

d V (n )

=

dr c d z r - -ar

-h/2

1

o~r~. r(1 + rT. )l/e

.....

r..

P

(1 + r~ )3/'~. + ~ o -

P

+ 4rcr,.o'cos(p,.

(11.73)

where r:: = i02r/~)z 2. The expression in the brackets must be equal to zero because the meniscus shape r(z) obeys the Laplace equation, Eq. (2.23). Then substituting Eq. (11.73) into Eq. (11.71) one obtains

Chapter 11

496

dN (n) dW = [/l(n)(rc) _/1] ~ dr c dry.

Ill 1.74)

In the theory of nucleation [81-86] the critical nucleus is defined as the nucleus with maximum

W. Then for the critical nucleus dW/drc = 0 and from Eq. (11.74) one obtains

(111.75)

]./(n)(rc*) = ]./

that is the critical nucleus of contact radius rc = rc* is in chemical equilibrium with the ambient mother phase. In view of Eqs. (11.67) and (11.68) this yields

P(n)(rc*) = Po

(gas bridge)

(11.76)

P(n)(rc*) = P - (kT/Vm)ln(Po/P')

(liquid bridge)

(11.77)

P is to be identified with the atmospheric pressure. Below the boiling temperature of the liquid (say at room temperature) one has P(n)(rc*) = P0 < P for a gas bridge; likewise, the capillary condensation usually takes place at humidity below 100 %, i.e. P'/Po < 1 and then Eq. (11.77) gives also P(n)(rc*) < P. As mentioned earlier, a negative pressure difference, P(n)(rc* ) - P < O, can be attained only in capillary bridges with generatrix nodoid (-co < p < 0, see Table 11.1).

11.4.2. CRITICAL NUCLEUS AND EQUILIBRIUM BRIDGE The dependence W = W(rc) can be calculated in the following way. For given values of Pc., h and (Pc, from Eqs. (11.40) and (11.43) one calculates P0 and kl. The substitution of the results in the expressions for the area and volume of nodoid-shaped bridge with neck (Table 11.1) give

Agrc) and l,xn)(rc), where rc = pc/Ikll. Further, in view of Eq. (11.14) the capillary pressure is [P-P{n)(rc)] = 20"lkll. Finally, a substitution of the results in Eqs. (11.65)-(11.68) gives W(r~.). For the smallest values of r~: the (non-equilibrium) bridges are unduloids with neck, for which W(rc) can be calculated in a similar way, but Eq. (11.37) with p = Pc must be used instead of Eq. (11.40). Physical interest represent situations, in which h < hmax, where hm~x denotes the maximum possible length of the bridge, see Fig. 11.4. For 70 ~ < (Pc < 90 ~ one can estimate hmax with the help of Eq. (11.53). For example, let us consider the case of nucleation of vapor-filled bridges

Capillary Bridges and Capillary-Bridge Forces

497

Table 11.2. Values of the maximum possible length, h . . . . of a vapor-filled equilibrium capillary bridge between two hydrophobic plates in water as a function of the contact angle 0 (Fig. 11.4) at temperature 20~ as predicted by Eq. (11.78). 94 ~

98 ~

102 ~

106 ~

110 ~

hmax (nm)

103

205

306

405

503

in water at temperature 20~

In this case the surface tension is c y - 72.75 mN/m and the

O - r e - q~c

90 ~

equilibrium vapor pressure of water is P0 = 2337 Pa, which is only 2.3 % of the normal atmospheric pressure at the sea level, P = 101 325 Pa. Equation (11.53) acquires the form hmax -- 20"c~ P - Po

(70 ~ < q~c < 90 ~

(11.78)

Table 11.2. contains numerical results for hmax calculated with the above parameters values. One sees that hmax markedly increases when the contact angle 0 of the hydrophobic plate increases beyond 90 ~ It is worthwhile noting that hmax is obtained from Eq. (11.52) for p c - - ~ , that is the limit for a large bridge, whose azimuthal curvature radius is much greater than the meridional one, r, >> rm. For such bridges rm =-h/(2cos(pc) and using Eq. (11.7) one obtains that the length of the bridge is given again by Eq. (11.78) for every (Pc e [0 ~ 90 ~ (that is 90 ~ < 0 < 180~

In this way Table 11.2 can be extended for 0 > 110~ then the maximum possible

value of the gap width, corresponding to 0 - 180 ~ is hmax - 1470 nm. The large equilibrium bridges (for which r, >> rm) can be considered as a result of the process of nucleation, which begins with a fluctuational formation of much smaller capillary bridges. In the theory of nucleation the small nuclei are generally out of chemical equilibrium with the mother phase, and this is the reason why they could spontaneously grow owing to the addition of new molecules to the nucleus. A typical picture, originating from the theory of homogeneous condensation (and cavitation), is that a critical drop (bubble) exists, corresponding to a maximum of the work of nucleation W, i.e. to a state of unstable equilibrium, see e.g. Refs. [85,86]. If a molecule is added to the critical nucleus it begins to grow spontaneously; on the contrary, if a molecule is detached from the critical nucleus, it spontaneously diminishes and disappears. For example, in the case of

Chapter 11

498 homogeneous condensation of water vapors at 0~

and P'/Po = 4 the radius of the critical

droplet was calculated to be about 0.85 nm [91,86]. Let us consider now the case of nucleation (condensation or cavitation) of capillary bridges in a narrow gap. The condition for extremum (maximum or minimum) of W determines uniquely the pressure inside the (critical or equilibrium) nucleus, p(n), see Eqs. (11.76) and (11.77), and the value of the parameter k~ - [p(n) _ P]/(2cr). Then the number of the extrema of W is equal to the number of roots of Eq. (11.43) for the respective value of k~ and for the given thickness of the gap, h, and contact angle qgc. A maximum of W for some value of rc can be interpreted as existence of a critical nucleus in a state of unstable equilibrium with the ambient mother phase. A local minimum of W(rc) corresponds to a capillary bridge in state of stable equilibrium. Additional information about the nucleation of liquid and gas capillary bridges can be found in Refs. [90, 92].

11.5.

SUMMARY

The role of capillary bridges has been recognized to be important for many systems and phenomena such as adhesion of particles (dust, powders) to solid surfaces, consolidation of granules and porous media, dispersion of pigments and wetting of powders, obtaining of latex films, antifoaming, capillary condensation, bridging force in experiments with atomic force microscope (AFM), attraction between hydrophobic surfaces, etc. The capillary bridge force is oriented normally to the plane of the three-phase contact line and its magnitude is determined by the contributions of the capillary pressure and the normally resolved surface tension force, see Eq. (11.5). The simplest way to quantify the shape of the capillary bridges and the capillary-bridge force is to use the toroid (circle) approximation (Section 11.2.2). Like every approximation it has some limits of validity; moreover, there is an ambiguity in the definition of the azimuthal radius of curvature, which results in different expressions for the capillary bridge force, see Eqs. (11.8) and (11.11). For that reason more reliable results can be obtained using the exact profile of the capillary bridge, which is determined by the Plateau sequence of shapes: (1) nodoid with "neck", (2) catenoid, (3) unduloid with "neck", (4) cylinder, (5) unduloid with "haunch", (6) sphere and (7) nodoid with

Capilla~. Bridges and Capillary-Bridge Forces

499

"haunch". The capillary-bridge force is attractive for the shapes (1-5), zero for sphere (6) and repulsive for the nodoid (7). Expressions for the bridge shape, area and volume are given in Table ll.1. In addition, equations connecting the radius of the bridge neck/haunch with the contact angle and radius are derived in Section 11.3.2. For the reader's convenience the procedures for shape calculations are outlined for the cases of bridges between plane and axisymmetric body, and between two parallel planes, see Eqs. (11.39)-(11.44). It is demonstrated that in the asymptotic case, in which the contact angle belongs to the range 70 ~ < q~c < 90 ~ (or 90 ~ < 0 < 110 ~ for hydrophobic plates), the elliptic integrals reduce to elementary algebraic functions and the capillary bridge can be described in terms of the toroid approximation, see Eqs. (11.52)-(11.54). Some upper "geometrical" stability limits for the length of the capillary bridges are related to the distances between the points with horizontal and vertical tangents of the nodoid and unduloid, see Eqs. (11.56)-(11.69). The limits for the length of a catenoid-shaped bridge are connected with the possibility to satisfy the boundary conditions with this special shape, see Eq. (11.63). The real "physical" limits for the bridge stability can be established by analysis of diagrams of volume vs. pressure, see Fig. 11.6 and its interpretation. Finally, we consider the thermodynamics of nucleation of capillary bridges between two solid surfaces. Two plane-parallel plates are considered as an example. The treatment is similar for liquid bridges between two hydrophilic plates and for gas bridges between two hydrophobic plates; in both cases the work of nucleation is determined by Eq. (11.65). Nucleation of capillary bridges is possible when the distance between the plates is smaller than a certain limiting value hmax, see Eq. (11.78) and Table 11.2. Equations for calculating the work of nucleation and the size of the critical (and/or equilibrium) nucleus are presented. 11.6.

REFERENCES

1. J.S. McFarlane, D. Tabor, Proc. R. Soc. London A, 202 (1950) 224. 2.

H.M. Budgett, Proc. R. Soc. London A, 86 (1912) 25.

3.

W. Stone, Phil. Mag. 9 (1930) 610.

4.

F.M. Orr, L.E. Scriven, A.P. Rivas, J. Fluid Mech. 67 (1975) 723.

5.

W.B. Haines, J. Agric. Sci. 15 (1925) 529.

500

Chapter 11

6.

W.B. Haines, J. Agric. Sci. 17 (1927) 264.

7.

R.A. Fisher, J. Agric. Sci. 16 (1926) 492.

8.

P.C. Carman, J. Phys. Chem. 57 (1953) 56.

9.

A.D. Zimon, Adhesion of Dust and Powder, Plenum, London, 1969.

10. J. Woodrow, H. Chilton, R.I. Hawes, J. Nucl. Energy B, Reactor Tech. 2 (1961) 229. 11. W.D. Kingery, J. Appl. Phys. 30 (1959) 301. 12. R.B. Heady, J.W. Cahn, Met. Trans. 1 (1970) 185. 13. D.P. Sheetz, J. Polymer Sci. 9 (1965) 3759. 14. J.W. Vanderhoff, H.L. Tarkowski, M.C. Jenkins, E.B. Bradford, J. Macromolec. Chem. 1 (1966) 361. 15. A.S. Dimitrov, T. Miwa, K. Nagayama, Langmuir 15 (1999) 5257. 16. R. Defay, I. Prigogine, in: Surface Tension and Adsorption, Wiley, New York, 1966; p. 217. 17. D.H. Everett, in: E.A. Flood (Ed.) The Solid-Gas Interface, Vol. 2, M. Dekker, New York, 1967; p. 1055. 18. J.C. Melrose, J. Colloid Interface Sci. 38 (1972) 312. 19. M. Iwamatsu, K. Horii, J. Colloid Interface Sci. 182 (1996) 400. 20. N.R. Morrow, J. Can. Pet. Tech. 10 (1971) 38. 21. S.M. Iveson, J.D. Lister, Powder Technol. 99 (1998) 234. 22. S.M. Iveson, J.D. Lister, Powder Technol. 99 (1998) 243. 23. C.M. Mate, V.J. Novotny, J. Chem. Phys. 94 (1991) 8420. 24. O.P. Behrend, F. Oulevey, D. Gourdon, E. Dupas, A.J. Kulik, G. Gremaud, N.A. Burnham, Applied Physics A, 66 (1998) $219. 25. H. Suzuki, S. Mashiko, Applied Physics A, 66 (1998) S1271. 26. H.K. Christenson, J. Fang, J.N. Israelachvili, Phys. Rev. B, 39 (1989) 11750. 27. V.S.J. Carig, B.W. Ninham, R.M. Pashley, J. Phys. Chem. 97 (1993) 10192. 28. W.A. Ducker, Z. Xu, J.N. Israelachvili, Langmuir 10 (1994) 3279. 29. J.C. Eriksson, S. Ljunggren, Langmuir 11 (1995) 2325. 30. V.V. Yaminsky, Colloids Surf. A, 129-130 (1997) 415. 31. V.V. Yaminsky, Langmuir 13 (1997) 2. 32. J. Plateau, Experimental and Theoretical Researches on the Figures of Equilibrium of a Liquid Mass Withdrawn from the Action of Gravity, in: The Annual Report of the Smithsonian Institution, Washington D.C., 1863; pp. 207-285. 33. J. Plateau, The Figures of Equilibrium of a Liquid Mass, in: The Annual Report of the Smithsonian Institution, Washington D.C., 1864; pp. 338-369. 34. J. Plateau, Statique Exp6rimentale et Th6oretique des Liquides Soumis aux Seules Forces Mol6culaires, Gauthier-Villars, Paris, 1873.

Capillao, Bridges and Capillao'-Bridge Forces

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35. Lord Rayleigh, Proc. London Math. Soc. l0 (1879) 4. 36. Lord Rayleigh, Proc. R. Soc. London 29 (1879) 71. 37. Lord Rayleigh, Philos. Mag. 34 (1892) 145. 38. Lord Rayleigh, Philos. Mag. 34 (1892) 177. 39. N.L. Cross R.G. Picknett, Particle Adhesion in the Presence of a Liquid Film, in: H.R. Johnson and D.H. Litter (Ed.) The Mechanism of Corrosion by Fuel Impurities, Butterworths, London, 1963; p. 383. 40. G. Mason, W.C. Clark, Chem. Eng. Sci. 20 (1965) 859. 41. W.C. Clark, J.M. Haynes, G. Mason, Chem. Eng. Sci. 23 (1968) 810. 42. M.A. Erle, D.C. Dyson, N.R. Morrow, AIChE J. 17 (1971 ) 115. 43. J.C. Melrose, AIChE J. 12 (1966) 986. 44. W. Rose, J. Appl. Phys. 29 (1958) 687. 45. N.L. Cross, R.G. Picknett, Trans. Faraday Soc. 59 (1963) 846. 46. A. Marmur, Langmuir 9 (1993) 1922. 47. G.I. Taylor, D.H. Michael, J. Fluid Mech. 58 (1973) 625. 48. M.L. Forcada, M.M. Jakas, A. Gras-Marti, J. Chem. Phys. 95 (1991) 706. 49. G. Debregeas, F. Brochard-Wyart, J. Colloid Interface Sci. 190 (1997) 134. 50. M.L. Fielden, R.A. Hayes, J. Ralston, Langmuir 12 (1996) 3721. 51. S. Ross, J. Phys. Colloid Chem. 54 (1950) 429. 52. P.R. Garrett, J. Colloid Interface Sci. 76 (1980) 587. 53. P.R. Garrett, in: P.R. Garrett (Ed.) Defoaming: Theory and Industrial Applications, M. Dekker, New York, 1993; Chapter 1. 54. R. Aveyard, P. Cooper, P.D.I. Fletcher, C.E. Rutherford, Langmuir 9 (1993) 604. 55. R. Aveyard, B.P. Binks, P.D.I. Fletcher, T.G. Peck, C.E. Rutherford, Adv. Colloid Interface Sci. 48 (1994) 93. 56. R. Aveyard, J.H. Clint, J. Chem. Soc. Faraday Trans. 93 (1997) 1397. 57. N.D. Denkov, P. Cooper, J.-Y. Martin, Langmuir 15 (1999) 8514. 58. N.D. Denkov, Langmuir 15 (1999) 8530. 59. D.H. Michael, Ann. Rev. Fluid Mech. 13 (1981) 189. 60. B.J. Lowry, P.H. Steen, Proc. R. Soc. London A, 449 (1995) 411. 61. J.N. Israelachvili, Intermolecular and Surface Forces, Academic Press, New York, 1992. 62. G. Mason, W.C. Clark, Brit. Chem. Engng. l0 (1965) 327. 63. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. 64. E. Janke, F. Erode, F. L6sch, Tables of Higher Functions, McGraw-Hill, New York, 1960. 65. H.B. Dwight, Tables of Integrals and Other Mathematical Data, Macmillan Co., New York, 1961.

502

Chapter 11

66. G.A. Korn, T.M. Korn, Mathematical Handbook, McGraw-Hill, New York, 1968. 67. A. Beer, Ann. d. Phys. u. Chem. 96 (1855) 1; ibid. p. 210. 68. W. Howe, Ph. D. Dissertation, Friedrich-Wilhelms UniversitS.t zu Berlin, 1887. 69. A.D. Myshkis, V.G. Babskii, N.D. Kopachevskii, L.A. Slobozhanin, A.D. Tyuptsov, Lowgravity Fluid Mechanics, Springer-Verlag, New York, 1987. 70. L.A. Slobozhanin, J.M. Perales, Phys. Fluids A 5 (1993) 1305. 71. J.C. Eriksson, S. Ljunggren, Langmuir 11 (1995) 2325. 72. T.L. Hill, An Introduction to Statistical Thermodynamics, Addison-Wesley, Reading, MA, 1962. 73. K.J. Mysels, Colloids Surf. 43 (1990) 241. 74. T.S. Horozov, C.D. Dushkin, K.D. Danov, L.N. Arnaudov, O.D. Velev, A. Mehreteab, G. Broze, Colloids Surf. A, 113 (1996) 117. 75. S.S. Dukhin, G. Kretzschmar, R. Miller, Dynamics of Adsorption at Liquid Interfaces, Elsevier, Amsterdam, 1995. 76. G.F.C. Searle, in: Experimental Physics, Cambridge Univ. Press, Cambridge, 1934; p.128. 77. J.F. Padday, A.R. Pitt, Phil. Trans. R. Soc. Lond. A 275 (1973) 489. 78. E.A. Boucher, M.J.B. Evans, Proc. R. Soc. Lond. A 346 (1975) 349. 79. T.I. Vogel, SIAM J. Appl. Math. 49 (1989) 1009. 80. J.W. Gibbs, The Scientific Papers of J.W. Gibbs, vol. 1, Dover, New York, 1961. 81. M. Volmer, Kinetik der Phasenbildung, Edwards Brothers, Ann Arbor, Michigan, 1945. 82. A.I. Rusanov, Phase Equilibria and Surface Phenomena, Khimia, Leningrad, 1967 (in Russian); Phasengleichgewichte und Grenzfl~ichenerscheinungen, Akademie Verlag, Berlin, 1978. 83. V.P. Skripov, Metastable Liquid, Moscow, 1972 (in Russian). 84. F.F. Abraham, Homogeneous Nucleation Theory, Academic Press, New York, 1974. 85. E.D. Shchukin, A.V. Pertsov, E.A. Amelina, Colloid Chemistry, Moscow Univ. Press, Moscow, 1982 (in Russian). 86. A.W. Adamson, A.P. Gast, Physical Chemistry of Surfaces, 6th Edition, WileyInterscience, New York, 1997. 87. L.H. Cohan, J. Am. Chem. Soc. 60 (1938) 433. 88. J.C.P. Broekhoff, B.G. Linsen, in: B.G. Linsen (Ed.) Physical and Chemical Aspects of Adsorbents and Catalysts, Academic Press, New York, 1970; p. 1. 89. V.S. Yushchenko, V.V. Yaminsky, E.D. Shchukin, J. Colloid Interface Sci. 96 (1983) 307. 90. J.L. Parker, P.M. Claesson, P. Attard, J. Phys. Chem. 98 (1994) 8468. 91. R.S. Bradley, Q. Rev. (London) 5 (1951) 315. 92. P. Attard, Langmuir 16 (2000) 4455.

503

CHAPTER 12 CAPILLARY FORCES BETWEEN PARTICLES OF IRREGULAR CONTACT LINE Solid particles attached to a fluid interface often exhibit irregular wetting perimeter. The latter induces interfacial deformations, which are theoretically found to engender a strong lateral capillary force between the particles. For the time being this force is not investigated experimentally, although there are some indications about its existence. A quantitative theoretical description of such capillary interactions can be achieved for irregularities of a given characteristic amplitude and period, which can be approximated with a model sinusoidal contact line. The latter disturbs the smoothness of the fluid interface creating a wave-like profile. The overlap of the disturbances produced by two particles gives rise to the capillary interaction, which becomes significant for distances comparable with, or smaller than, the wavelength of the contact-line undulation. If the phase shift between the two sinusoidal contact lines is not too large, the interaction is always attractive at long distances between the particles. On the other hand, at short distances the interaction is always repulsive. Only in the special case of coinciding wavelengths and amplitudes the particles attract each other at all distances. The interaction energy exhibits a minimum at some interparticle separation. The depth of the minimum could be greater than the thermal energy kT even if the contact-line undulations have nanometer amplitude. The latter fact implies that this type of capillary force could be significant even for colloid nano-particles and protein macromolecules. Any deformation of a particulate adsorption monolayer, either by dilatation or by shear, which takes the particles out of the potential minimum, is resisted. As a result, the particle monolayer exhibits dilatational and shear elasticity. These elastic properties can be estimated if the average value and the dispersion of the amplitude of the contact-line undulations are known.

Chapter 12

504

.~

+" #

,+

,.,

+

(a)

t

+ '+

+

+,

,++'+u

++ " l ' + + Al

+

'r

/m+_,.++:+~.:+,r

+ .... ' ' +

,+,

++'+"+++ ,++-an

+

(b) Fig. 12.1. Electron micrographs: ordered monolayers made of gm-sized crystals (a) from hexahedral silver bromide, (b) from octahedral silver bromide. The monolayers were initially formed at the water-air interface, then transferred to a mica support and observed by a scanning electron microscope by Heki and Inoue [2] (reprinted with permission).

Capillary Forces between Particles of lrregular Contact Line

12.1.

505

SURFACE CORRUGATIONS AND INTERACTION BETWEEN TWO PARTICLES

12.1.1. INTERFACIAL DEFORMATION DUE TO IRREGULAR CONTACT LINE

As discussed in Chapter 8, if the weight of a floating particle is smaller than 5-10 lain, it is not able to create any physically significant interfacial deformation; therefore the lateral flotation force is negligible for such particles. This conclusion holds for spherical particles of regular (smooth) wetting perimeter. Lucassen [1] pointed out that even submicrometer floating particles can create interfacial deformation if the wetting perimeter is not fully located in the plane of the surface, i.e. there are some irregularities of the contact line. Often this is the practical situation with solid particles having some surface roughness and/or inhomogeneity. In the case of crystal particles the contact line can be attached to some jagged edge on the particle surface. Qualitatively, it is to be expected that capillary interaction will appear as soon as the deformed zones around two such particles overlap. Capillary interaction of this type seems to be rather universal and significant, and it is astonishing that for the time being the only study in this area is the work by Lucassen [1]. At least in part, this can be attributed to the impossibility to obtain a quantitative theoretical description, or reproducible experimental data, for particles of completely irregular periphery. Lucassen has overcome this difficulty considering irregularities of a given characteristic amplitude and period, and approximating them with a model sinusoidal contact line. This treatment provides a complete theoretical description of the meniscus shape and the interparticle force and gives a physical insight about the sign and magnitude of the interaction energy in more complicated cases. In the present chapter we follow the approach from Ref. [1]. Before the theoretical considerations, let us mention some experimental facts, which could be related to the action of the aforementioned capillary force. Heki and Inoue [2] observed the formation of 2-dimensional ordered arrays of lain-sized crystals of silver bromide floating at the air-water interface, see Fig. 12.1. Arrays from hexahedral (cubic) and octahedral crystals were produced, which were separated by areas of bare interface. The latter fact implies that some attractive force has collected the floating microcrystals, which could be the Lucassen's lateral

506

Chapter 12

Fig. 12.2. Copolymer latex particles made of polystyrene + 2-hydroxyethyl-metacrylate (PS/HEMA): Secondary electron images of the surface of a colloid crystal obtained by ultra highresolution-field-emission scanning electron microscope. From Cardoso et al [3], 9 American Chemical Society; with permission. capillary force. Other possible explanation could be the attraction due to van der Waals and hydrophobic forces, or the simultaneous action of several kinds of forces. A contact line of approximately sinusoidal shape can also appear when particles with "undulated" surface are attached to a fluid phase boundary. The spontaneous formation of such particles was observed in the experiments by Cardoso et al. [3], who synthesized latex particles by batch surfactant-free emulsion copolymerization of styrene and 2-hydroxyethyl-metacrylate (PS/HEMA particles), see Fig. 12.2. The properties of 3-dimensional colloid crystals from such particles have been studied by means of electron-microscopic techniques [3]. The model particles used in the theoretical approach of Lucassen, shown in Fig. 12.3, much resemble the cubic microcrystals of silver bromide in Fig. 12.1 a. It is assumed that a sinusoidal contact line creates small interfacial deformations, q~" << 1, where q-~ is the capillary length.

Capillary Forces between Particles of lrregular Contact Line

507

zT X

" /

; .,-

" ,;

9 ....

g

; z

Fig. 12.3. Sketch of a solid particle with sinusoidal contact line located in a plane, which is perpendicular to the x-axis. The undulation of the contact line creates interfacial corrugations, which decay with the increase of the distance x from the particle. In such case the right-hand side of the linearized Laplace equation (7.6) can be neglected and one obtains [ 1] {?2r "

a~

- -

o~x----T+-~ 0

(q~" << 1)

(12.1)

As known, the general solution of Eq. (12. l) can be presented in the form

~(x,y) =

~ ( A k e -z~ + B~ e ~)(C, sin ky + D, cos ky)

(12.2)

k

The constants A,, B,, C~ and D~, are to be determined from the boundary conditions. For example, if the contact line in Fig. 12.3 has equation z

=

~'1 COS kly, then the meniscus shape

exhibits oscillations, which decay with the distance x from the contact line: ~(x,y)

---- ~'1

Here ~'~ and / ~ undulations.

exp(-klx) c o s 2g/kl

kly

(12.3)

characterize the amplitude and the wavelength of the contact-line

The interfacial corrugations caused by the sinusoidal

contact line decay

exponentially with the distance x from the edge of the particle. The surface deformation and the capillary interaction are physically significant at distances of the order of, or shorter than, the wavelength/~ = 2rck~-~ , see below. Next, after Lucassen [ 1], we consider two solid particles floating on a fluid interface (Fig. 12.4) assuming that their sinusoidal edges are parallel to each other and to the y-axis. The coordinate

Chapter 12

508 !=

L

Fig. 12.4. Sketch of two model particles with sinusoidal contact lines belonging to two parallel vertical planes separated at a distance L; ,~l and A2 are the wavelengths of oscillation of the respective contact lines.

X

origin is chosen in the middle between the two particles; in this way their edges are situated at

x = -L/2 and x = +L/2. The equations of the respective contact lines are taken in the form z = (1 cos kly and z = ~'2 cos(kzy + (p), where (p represents a phase difference. Then in view of Eq. (12.2) the shape of the meniscus between the particles is [1]

((x,y)

=

(1 s i n h k l ( L / 2 - x ) sinhkz(L/2+x) cos kl y + (2 cos(k2 y + (p) sinh k 1L sinh kzL

(12.4)

Equation (12.4) corresponds to the general case of two different amplitudes, ~'l, ~'2, and wave numbers, kl, k2, in the presence of phase shift (p. On the other hand, the validity of Eq. (12.4) is limited to contact lines lying in two parallel vertical planes (Fig. 12.4).

12.1.2.

ENERGY AND FORCE OF CAPILLARY INTERACTION

As mentioned above, we consider small particles and neglect the gravitational effects. Moreover, the contact lines are fixed to the respective sinusoidal edges (Fig. 12.4), and therefore the energy of wetting does not vary with the interparticle distance L. Then the only nontrivial contribution to the grand thermodynamic potential, Eq. (7.16), originates from the increased area of the corrugated meniscus [1 ]: Y0

L

n= ~AA = K dy dx 2 y,, __L 2

+

(112.5)

2

cf. Eq. (7.68). The substitution of Eq. (12.4) into Eq. (12.5) in general gives an explicit analytical expression for f2. Below we will restrict our considerations to the simpler case of

Capillao, Forces between Particles of Irregular Contact Line

509

equal wave numbers, k2 = kl, and integer number of wave-lengths along the y-axis,

k~yo =

2rm;

n = 1,2,3 ..... In this case the interaction energy is [ 1]: Af~ - f~(L) - f~(oo) =

E

1

2

= -g rtno (~-2 + ( 2 ) coth

klL + tanh klL 2 -2)-(2~l~2c~176

2

2

The asymptotic form of Eq. (12.6) for long distances,

klL >>

(12.6)

1, reads:

A~')(L) = -/rno" [(2~'1~'2 cosq))e -klL - (~'~ + ~,2) e-2klL + O(e-3klL)] Equation (12.7) shows that

at long distances

(12.7)

the interaction is attractive (Af~ < 0) if (p < 90 ~

but it is repulsive (Af2 > 0) if q) _> 90 ~ On the other hand,

at short distances, klL

<< 1,

Eq. (12.6) is dominated by the hyperbolic cotangent: 1 1 A~'~(L)- ~/17/70" [((~ 4" ~.,2 _ 2(1(2 COS(p) ~_.7__7 _ (~'2 q-~22 ) q- O(klL)] Kl*-,

(12.8)

Equation (12.8) implies that at short distances the capillary interaction is always repulsive (Af2 > 0). An exception is the very special case, in which simultaneously ~'1 = ~'2, q9 = 0 and the terms with hyperbolic cotangents cancel each other; in this case the interaction is attractive for all distances. The above analysis implies that for q9 < 90 ~ A~(L) has a minimum at a certain distance Lmin, which can be found from Eq. (12.7) [1 ]:

Lmin k11

= - - arc cosh

/ 2(1(2 cos(p /

(12.9)

This minimum, corresponding to equilibrium interparticle separation, exists for - 9 0 ~ < q~< 90 ~ but it disappears for larger phase shift, 90 ~ < (o<270 ~ Special cases of Eq. (12.9) are 1 Lmin- -7- [ln(~'l/~'2)]

for q~=0

(12.10)

for ~1 "- (2

(12.11)

kl 1 /l+sinrp') Lmin- ~-1 In ~ c o s q~ )

Chapter12

510 10

,

,

,

,

0.3

,

8

I

'

~

i

'

0.2

"X 6 x

< 4

0.1

,- 0.0

2 cD

o 0

O

"= -0.1 i

~" cD

-2

~D

-0.2 -4

-6

0

100

200

300

400

-0.3

0

100

200

300

Distance L [nm]

Distance L [nan]

(a)

(b)

400

Fig. 12.5. Plots of (a) energy Af2 and (b) force Fx vs. distance L, calculated by means of Eqs. (12.6) and (12.12) for /~ = 1 lain, n = 1, cr = 40 mN/m and ~'j = ~'2 = 60 rim; the separate curves correspond to the following values of the phase-shift angle: (A) q9 = 0, (B) (p = 15 ~ (C) q9= 35 ~ (D) (p = 60 ~ and (E) q~= 90 ~ The force between the two particles can be deduced by differentiation of Eq. (12.6) [1 ]"

Fx-- Om~-26~Z --~]Oll~Ykl[~2

+l ~ s - i2~1~2 n hc~2 (- ~k ( l+c~-sh~ ~2 L +/ <2~1~2 L2 / 2)7 c~

(12.12)

The dependence of Af2 on the phase angle (p leads to the appearance of a force component also along the y-axis [ 1]: FY=

k~_l_q9 ~ l d-

2 rtncrkl~'l~'2 sin(p ( coth

klL2- tanh klL2/

(12.13)

Fy is a force which tends to make the particles slide with respect to one another in a direction parallel to the y-axis. The action of this force will eventually bring the particles in a position without phase shift, i.e. with q9 = 0. In the latter case the expression for the interaction energy, Eq. (12.6), reduces to 1 Af~ = 7~:no" [(~, +

~:)2tanh(klL/2)+(~,- ~2)2coth(klL/2)-2(~'~ +~-2) ] 9

(12.14)

Capillary Forces between Particles of Irregular Contact Line

511

The minimum value of Af~ can be obtained substituting Eq. (12.10) into Eq. (12.14). Let us specify that ~'2 < ~'l. Then one obtains

tanh(klgmin/2) = ~ l - ~ 2 = [coth(k~gmin/2)]-I ~'1 q-~'2

((,0= 0, ~2 <-- ~1)

(12.15)

A~'-'~min= A~'-~(Lmin) = -g?/O" ~2

((p-" 0, ~'2 < ~,)

(12.16)

Let us check whether the depth of the minimum, [Af~min], is significant compared with the energy of the thermal motion of a Brownian particle, which is approximately equal to kT. The value of the amplitude ~'2, which provides A~min = kT, is ~'2 = (kT/rcncY)l/2; taking surface tension cy=40mN/m, temperature 25~

and n =1 (the length of the particle equals 1 wave-

length) one calculates ~'2 = 1.8 A. This result is really astonishing: it turns out that even edge irregularities of angstrom amplitude may lead to physically significant attraction between the particles! However, in the angstrom scale the fluid interfaces are no longer smooth: they are naturally corrugated by thermally excited fluctuation capillary waves, whose amplitude is typically between 3 and 6 A [4]. Then one can expect that the effect of the undulated particle contact lines becomes significant when the amplitude of the undulations is greater than this stochastic noise, that is for nanometer and larger amplitudes. In any case, one may expect that such capillary interactions could be important even for large adsorbed macromolecules of irregular shape, like the proteins. Figure 12.5a shows the dependence of the interaction energy Af~ on the distance L calculated by means of Eq. (12.6) for four different values of the phase angle (p; for all curves n = 1,

-2rc/kl = 1 btm and the amplitudes of the two sinusoidal contact lines are equal: ~'1 = ~'2 = 60 nm.

As noted above, the interaction is monotonic attraction at all distances only if

simultaneously p = 0 and ~'l = ~'2. For 0 < q0<90 ~ the curves exhibit minima at L = Lmin, see Eq. (12.11) and Fig. 12.5a.

For 90~

~ the interaction is monotonic repulsion. The

magnitude of interaction at short separations (L < 500 nm) is much larger than the thermal energy kT. For example, for the curve with (p = 0 the magnitude of the energy IAf~l is equal to 1.1 x 105 kT at close contact (L = 0), and it drops down to 1 kT at a separation L = 2.0 lam = 2X.

Chapter 12

512

1

0.3 0.2 •

0.1

o

o 0.0 ~o s9 o

T

I

ATTRACTI ~ t ON.I.

-0.1

o

= -0.2 -0.3

0

1

100

200

300

400

Distance L [nm] Fig. 12.6. Plots of the lateral capillary force F, vs. distance L, calculated by means of Eq. (12.12) for Z = 1 gm, n = 1, o-= 40 mN/m and q9 = 0; the separate curves correspond to (A) ~'1 = 60 nm, ~'2 = 60 nm; (B) ~', = 56 nm, ~'2 = 64 nm; (C) ~'1 = 50 nm, ~'2 = 70 nm; (D) ~'l = 40 nm, ~'2 = 80 nm. Figure 12.5b presents the dependence of force Fx on distance L calculated by means of Eq. (12.12) for the same values of the parameters as in Fig. 12.5a. The points with Fx = 0 correspond to the equilibrium separation between two particles. One sees again that the interaction becomes more repulsive with the rise of the phase shift qz Figure 12.6 illustrates the tendency the interaction to become more repulsive when the difference between the amplitudes ~] and ~'2 increases. For example, a difference ~] - ~'2 = 5 nm produces the same effect as a phase shift q~ = 10 ~ In Fig. 12.6 we have fixed q~ = 0, and consequently, the interaction is attractive at long distances for every values of ~'l and ~'2.

12.2.

ELASTIC PROPERTIES OF PARTICULATE ADSORPTION MONOLAYERS

The fact that the capillary interaction energy Af2(L) has a m i n i m u m leads to the appearance of elastic behavior of a fluid interface covered by "rough-edged" particles (i.e. particles with corrugated contact line). As noted by Lucassen [1], any deformation of this surface, either by dilatation or by shear, will take the particles out of their equilibrium positions and will,

Capillar3,Forces between Particlesof Irregular ContactLine

513

therefore, be resisted. As a consequence, the particulate monolayer will exhibit dilatational and shear elastic properties.

12.2.1. SURFACEDILATATIONALELASTICITY Let us consider a fluid surface covered with square particles of edge length y0 = ~ = 2rt/k~. It is natural to assume that the particles spontaneously adjust their lateral positions so that Fy = 0 and consequently, r = 0. Let us denote the area per particle in the monolayer with a. Then the surface dilatational elasticity of the particulate monolayer can be defined in the same way as for surfactant molecules: dO'ap

EG = a ~

(12.17)

da

cf. Eq. (1.45); here O'ap is the apparent surface tension of the particulate monolayer. For our rectangular particles of edge-length 2, separated at edge-to-edge distance L one has a = (,~ + L) 2. Then da = 2(2 + L)dL and Eq. (12.17) acquires the form Ec

_

-4-(~ + L) dO'ap 1

-

dL

1,~

=-y

dO'ap

dL

1 dEx 2--~

= -

( L < < ,,q,)

(12.18)

where at the last step we have identified the variation of the apparent surface tension with the increment of F~ per unit length, i.e. drYap = IdFxl/)u. Setting r = 0 in Eq. (12.12) and differentiating one obtains

dFx '~cyk~[~ - ; 2 ~ c~ -(~, dL = - - 8 ' sinh 2 (klL/ 2)

q-~2Y

tanh(klL/2) ) c-~sh y ~ I L / 2 )

(12.19)

As already mentioned, we consider small deviations from the equilibrium position of the particles. Then setting L = Lmi~ in Eq. (12.19) and using Eqs. (12.15) and (12.18) we derive

dE~ =-2rto- k(~'2~'2

dc

( L - Cmin)

(12.20)

;7

Let us denote by ~'0 and A~', respectively, the average value and the dispersion of the amplitude of the contact-line undulations for the particles from the adsorption monolayer; we will assume

514

Chapter 12

A~" << ~'0. Then combining Eqs. (12.18) and (12.20) we obtain an estimate for the surface

dilatational elasticity: Ec," = 2/r30"~"3 ~2A~.

(12.21)

where the identity kl = 2rt/2, has been used. Note that the surface dilatational elasticity Ec is sensitive to both ~'0 and A~'. With values o" = 40 raN/m, 2, = 1 gm, ~'0 = 60 nm and A~" = 6 nm Eq. (12.21) gives Eo = 89 mN/m; this is a considerable value, which is comparable with the surface elasticity of surfactant adsorption monolayers. The divergence of Ea for k ~ ' + 0 is related to the fact that in the limit A~'---~0 the particles come into direct contact (Lmin = 0). Indeed, with A~'= ~'l- ~'2 Eq. (12.10) yields

(m~ << ~'0)

Lmin = A~"/~'o

12.2.2.

(12.22)

SURFACE SHEAR ELASTICITY

Let us consider again an equilibrium monolayer of square "rough-edged" particles having sidelength X. A small shear deformation along the y-axis (Fig. 12.4) is accompanied by the appearance of a shear stress: Auy Tsh = Esh - ~

where

k______k q~/ 1 ~ Esh Lmin

(12.23)

Esh is the surface shear elasticity, Auy = co/k~ is a small relative particle displacement

along the y-axis, and Ax = Lmin is the equilibrium separation between the edges of the two particles along the x-axis; see e.g. Ref. [5]. For small displacements (small co) one can estimate the surface shear stress Z'shwith the help of Eq. (12.13):

"Csh--Fv/'~= 2ruyk~(o~2 coth "

-tanh 2

(12.24) 2

Combining Eqs. (12.23) and (12.24), along with Eqs. (12.10), (12.15) and (12.22), after some transformations we obtain the following expression for the surface shear elasticiU: Esh = 271;20"(~0//~)2

(12.25)

Capillary Forces between Particles of Irregular Contact Line

515

where the meaning of ~'0 and X is the same as in Eq. (12.21). The comparison of Eqs. (12.21) and (12.25) shows that, unlike the dilatational elasticity Ec,, the shear elasticity Esh does not depend on the dispersion A~" of the amplitudes of the particle edge roughness. With o = 40 mN/m and ~'0/~,= 0.1 Eq. (12.25) yields Esh -- 8 mN/m. Equation (12.25) has been derived for the case of small shear displacements, for which we can substitute sin(p-- r in Eq. (12.13); i.e. we have dealt with small deviations from the state of

minimum energy. For larger shear displacements the interaction energy Af~ passes through a maximum (barrier) at q0 = Jr, which should be overcome by the particles in order to have a viscous shear flow in the interface [6]. The height of this barrier, Urn, can be estimated by means of Eq. (12.6)

Urn--A~'-~(~--~)-A~"~((p-0)---no"~'l ~"2 (coth k

k 2' L - tanh k'L 2

I

(12.26)

,/

The coverage of the fluid interface with square particles of edge ,~ is o~ - ,~2/(~ + L):. Using the latter relationship and k~ = 2jr/X one obtains

k,L = Jr(o~-'/2 - 1)

(12.27)

Using Eqs. (12.26), (12.27) and the values o: = 0.8, ~'l = ~'2 = 60 nm and o" = 40 mN/m one calculates Um -- 2.7 x 105

kT. It seems that such a high energy barrier will prevent any surface

shear flow (the surface viscosity is r/s ~ exp(Um/kT), see Ref. [6]), and consequently, the particulate monolayer will behave as an elastic two-dimensional continuum. Note however, that the expression for the height of the barrier Eq. (12.26), is obtained assuming a

translational slip along the y-axis. However, in reality each particle is free to rotate around its axis, and this fact certainly increases the interracial fluidity. The estimate of the surface

shear

viscosity based on account for the particle rotation is out of the applicability range of the present simple model with square particles. 12.3.

SUMMARY

The contact line of a solid particle attached to a fluid interface may have irregular shape due to surface roughness or inhomogeneity. Such an irregular contact line corrugates the surrounding fluid interface. The overlap of the corrugations around two particles gives rise to a lateral capillary force between them. A quantitative theoretical description of this effect was given by

Chapter 12

516

Lucassen [1] who considered irregularities of a given characteristic amplitude and period, and approximated them with a model sinusoidal contact line, Fig. 12.3 and Eq. (12.3). Assuming that the sinusoidal contact lines of the two interacting particles belong to two parallel vertical planes, one can derive an analytical expression for the meniscus shape, Eq. (12.4). In general, the corrugations increase the interfacial area. When the approaching of two particles causes decreasing of the interfacial area, the interparticle force is attractive; in the opposite case (increasing of the area) the particles repel each other. The latter effects are quantified by Eq. (12.6). If the phase shift q3 between the two sinusoidal contact lines is not too large (0 < q3< 90 ~ the interaction is always attractive at long distances between the particles, see Eq. (12.7). On the other hand, at short distances the interaction is always repulsive, see Eq. (12.8) and Figs. 12.5 and 12.6. The only exclusion (monotonic attraction at all distances) is the case, in which there is no phase shift (q3 = 0), and moreover, the amplitudes are identical (~'l = ~'2). The interaction energy exhibits a minimum at some interparticle separation Lmin, given by Eqs. (12.9)-(12.11). The depth of the minimum turns out to be greater than the thermal energy k T even for contact-line undulations of nanometer amplitude, see Eq. (12.16). This striking fact implies that the considered type of capillary force could be significant even between nanoparticles and protein macromolecules, and that its physical importance should not be neglected. Any deformation of a particulate adsorption monolayer, either by dilatation or by shear, which takes the particles out of their equilibrium positions, is resisted. As a result the particle monolayer exhibits dilatational and shear elasticity, which can be estimated by means of Eqs. (12.21) and (12.25) knowing the average value, ~'0, and the dispersion, A~', of the amplitude of the contact-line undulations. 12.4.

1. 2. 3. 4. 5. 6.

REFERENCES

J. Lucassen, Colloids Surf. 65 (1992) 131. T. Heki, N. Inoue, Forma 4 (1989) 55. A.H. Cardoso, C.A.P. Leite, F. Galembeck, Langmuir 15 (1999) 4447. J.S. Rowlinson, B. Widom, Molecular Theory of Capillarity, Clarendon Press, Oxford, 1982. E.S. Basheva, A.D. Nikolov, P.A. Kralchevsky, I.B. Ivanov, D.T. Wasan, in: K.L. Mittal and D.O. Shah (Ed.) Surfactants in Solution, Vol. 11, Plenum Press, New York, 1991. J. Frenkel, Kinetic Theory of Liquids, Clarendon Press, Oxford, 1946.

517

CHAPTER 13 TWO-DIMENSIONAL CRYSTALLIZATION OF PARTICULATES AND PROTEINS

First we give an overview of the methods for producing ordered 2D arrays from colloid particles and proteins in relation to their physical mechanisms and driving forces. From this viewpoint the following methods can be distinguished: evaporation of liquid suspension films; ordering owing to Kirkwood-Alder transition at high particle concentrations; self-assembly of colloidal particles at a single fluid interface; ordering of particles forced by the action of applied external electric, magnetic or optical fields; structuring in insoluble particle monolayers in a Langmuir trough, and ordering due to the binding of particles at structured adsorption sites. The major part of this chapter is devoted to the method for obtaining particle and protein 2D arrays in evaporating liquid suspension films; this method is closer to the subject of the present book and to the research fields of the authors. The respective mechanism of 2D crystallization is based on the fact that the decrease of the film thickness (because of evaporation) forces the particles to enter and/or deform the liquid interface, which automatically "switches on" the strong attraction due to the capillary immersion force. The latter collects the particles into ordered aggregates, which further grow owing to an evaporation-driven influx of new particles. Such evaporating films can be formed by spreading of protein solution in a mercury trough, which has been applied to crystallize dozens of proteins. Special attention is paid to the occurrence and advantages of 2D array formation over a liquid substrate: fluorinated oil and mercury. The particle size separation during 2D crystallization and the methods for obtaining large 2D-crystalline coatings on solid substrates are described. We consider also the formation of 2D arrays in free foam films and their observation by electron cryomicroscopy,

which

ensures

an excellent

structure

preservation of delicate vesicles or molecular complexes. Finally we review the various applications

of particulate

2D

arrays

in

optics,

optoelectronics,

nano-lithography,

microcontact printing, in fabrication of nanostructured surfaces for catalytic films and solar cells, as well as the usage of protein 2D crystals for immunosensors and extremely isoporous ultrafiltration membranes, for creation of bioelectronic and biophotonic devices, etc.

518

13.1.

Chapter 13

METHODS FOR OBTAINING 2 D ARRAYS FROM MICROSCOPIC PARTICLES

In 1959 R. Feynman attracted the attention of the scientists at a huge unexplored field for a future development of research and technology: that is to manipulate matter on very small scale and to create nanodevices and nanotechnologies, see e.g. Ref. [1]. Since that time much has been done in this field, but an enormous amount still remains to be done. The achievements and the perspectives are summarized in several recent books on nanotechnology [2],

microfabrication

[3],

biomolecular

electronics

[4-6].

A

necessary

step

of

a

microfabrication is to order macromolecules or particles into structures and to interconnect them [7]. Some proteins can form three-dimensional (3D) and two-dimensional (2D) crystals due to specific

molecular

interactions

and

molecular

recognition

[8-12].

In

other

cases

macromolecules or colloidal particles can form ordered 3D and 2D structures, called colloid crystals, under the action of nonspecific physical interactions [13,14]. The term "ordered array" is also used as a synonym of "colloidal crystal" [15]. Our attention in the present chapter will be focused on the methods for obtaining ordered 2D arrays from protein molecules and colloidal particles. In Section 13.8 we review their practical applications.

13.1.1. FORMATION OF PARTICLE 2 D ARRAYS IN EVAPORATING LIQUID FILMS

In his work on determining the Avogadro number Perrin [16] measured the size of sub-lam particles by forming a 2D array of the particles on a solid substrate. He used an aqueous suspension containing monodisperse spheres of gomme-gutte. Some amount of this suspension was deposited on a glass substrate. After the water evaporated, Perrin observed and photographed particle 2D arrays. Much work in the field of colloidal crystals were carried out with monodisperse spherical latex particles. They were first obtained by emulsion polymerization and examined with the help of electron microscopy by Backus and Williams [17]. Alfrey et al. [18] studied the optical properties of colloid crystals obtained by drying of aqueous latex suspensions; 14 different monodisperse samples, having particle diameters from 165 to 986 nm, were used. The diffraction of light from some of the crystalline samples was shown to be that for a two-

Two-Dimensional Co,stallization of Particulates and Proteins

519

!

~"

~;~.',i~.

,.

,~

9

Fig. ]3.]. Electron micrograph of a two-dimensional crystal of the protein ferritin, obtained by Yoshimura et al. [32] by means of the mercury trough method. dimensional grating. It was proposed to use the latex crystals as diffraction gratings []8] and as standards for calibration of electronic microscopes []7,]9]. Since then many authors have obtained ordered 2D arrays of latex spheres by evaporation from suspension layers, however it is not easy to obtain large mono-crysta]lino domains without a special control of the conditions, see Rof. [20] for details. In ]945 Price otal. [2]] evaporated an aqueous suspension of a purified plant virus (southern bean mosaic virus, bushy stunt virus and tobacco necrosis virus) deposited on a collodion-

covered screen used for electron microscopy. The electron micrographs sowed that each of these viruses had formed 2D (and sometimes 3D) crystals over the substrate. In a similar way crystals from other plant viruses were obtained [22,23]. In 1974 Home and Pasquali Ronchetti [24] formulated the negative staining-carbon film method for preparing 2D arrays of viruses and proteins. As a necessary step this method involves spreading and drying of suspension on a solid substrate. In this way, 2D crystals of various viruses [25,26] and proteins [27-30] have been obtained. Yoshimura et al. [31,32] spread protein suspension on the surface of mercury in a special mercury trough. After the evaporation of water large ordered protein 2D arrays were obtained (Fig. 13.1). This method was applied to crystallize a dozen of proteins, see Section 13.2.

520

Chapter 13

The mechanism of 2D crystallization in evaporating films from suspensions on solid substrate has been investigated in details by Denkov et al. [33,34] in experiments with latex spheres in a specially designed experimental cell. The direct observations of the dynamics of array formation reveals that the nucleation and growth of the 2D colloid crystals is governed by an interplay of the lateral immersion force (Chapter 7) and an evaporation-driven convective flow, see Section 13.3.1 for details. Similar experimental setup has been used by Dushkin et al. [35,36] to study the kinetics of 2D-array growth and the color phenomena observed with ordered multilayers from sub-lain latex spheres on solid substrate (Section 13.3.2). Subsequent experiments confirmed that the mechanism of ordering, established by Denkov et al. [33,34] for wetting films on solid substrate, is operative also in the case of 2D crystallization on liquid substrates: fluorinated oil [37] and mercury [38], see Section 13.4 below. Yamaki et al. [39] observed experimentally and simulated by computer a phenomenon of size separation of sub-lam particles (latex spheres, viruses, proteins) during 2D crystallization; this is an additional evidence for the action of the lateral immersion force (see Section 13.5). Large 2D-crystalline coatings from latex particles have been obtained by Dimitrov and Nagayama [40,41] by pulling up a plate out of a suspension with a controlled speed. An wetting film is deposited after the receding meniscus, in which the colloidal particles are ordered under the combined action of the capillary forces and the evaporation-driven convective flux. The deposition of a wetting film containing colloidal particles, and subsequent formation of large uniform 2D arrays, is possible also by deposition with a plate (playing the role of a brush) [42] or by continuous supply of the suspension with the help of an extruder [43], see Section 13.6. Ordered 2D arrays can be obtained not only in wetting films, but also in free foam films [4447]. To observe the arrays from sub-lain particles in such a foam film, the latter can be quickly frozen and examined by electron microscopy [44-46]; in this way arrays from delicate "soft" particles, like lipid-protein vesicles or surfactant micelles, could be observed (Section 13.7). It should be noted that large and well ordered particle monolayers can be formed under controlled conditions, which give the possibility to adjust the rate of receding of the liquid

Two-Dimensional Crystallization of Particulates and Proteins

521

meniscus equal to the rate of growth of the ordered array due to the supply of particles by the evaporation-driven convective flux [33-45]. In the absence of such a control it is possible to find only small ordered clusters or separate domains [48-50]. If simply a sessile drop (having the shape shown in Fig. 2.13) of the suspension is evaporated one observes only a narrow monolayered zone in the feet of a multilayered hill of particles, which has a ring-shape; in other words, the particles contained in the droplet are deposited in a vicinity of the receding contact line of the drying drop, see Section 13.3.1. The formation of such ring-shaped deposits by drying sessile drops has been examined by Deegan et al. [51 ]. In a similar way Maenosono et al. [52] obtained and investigated rings of semiconductor nanoparticles. Ring-shaped structures from particles can be created also by an expanding contact line: if a thinning suspension film is unstable and breaks, the contact line of the growing hole collects the particles into an annular ring-like array [53]. The simplest way to obtain a uniform monolayered domain, instead of multilayered ring, is to place a suspension drop on a substrate and then to smear it with a glass rod, before the drying [54]. A more elaborated way is to apply the spin-coating technique (spreading of a droplet over a rotating substrate) [50,55-57], which is currently used in industry to make polymeric films. Varying the particle concentration and spin speed, one could reproducibly fabricate monolayers and bilayers of hexagonal packing [58]. Another relatively simple method was introduced by Micheletto [59]: a drop of the colloid suspension is placed on a glass plate, which is then tilted at an appropriate angle. Thus the array formation starts at the upper edge of the drop and proceeds downwards. This method was applied by Burmeister et al. [60], who afterwards stabilized the monolayer by vacuum deposition of a metal or by thermal annealing. Further, the substrate was slowly dipped into water and the stabilized ordered particle monolayer gradually floated off as a whole onto the water surface. The floating array was finally transferred onto the desired surface [60]. Ordered particle monolayers of good quality can be produced if a drop of suspension is spread on a solid substrate encircled with a Teflon ring, as demonstrated by Denkov et al. [33]. Thus a slightly concave meniscus is formed, which leads to initiation of the ordering in the middle of the encircled area (see Section 13.3.1). In this way D u e t

al. [61] obtained ordered

522

Chapter 13

monolayers from monodisperse copolymer latex spheres of diameter less than 100 nm. The authors varied the concentrations of electrolyte and surfactant (emulsifier) in order to improve the quality of the ordered arrays. Likewise, with the help of a Teflon ring Sasaki and Hane [62] assembled polystyrene latex particles (diameter 64, 137 and 330 nm) and gold particles (diameter 39 rim) into regular 2D arrays on a glass substrate. To improve the quality of the array the authors applied an ultrasonic radiation technique. The ultrasonic radiation (frequency 2.4 MHz) increases the temperature of water (from 18 to 30 ~

and controlling its power one

can adjust the evaporation. In addition, the treatment with ultrasonic waves manipulates the array in such a way that densely packed lattice is formed, whereas the appearance of voids and random assemblies is reduced [62]. As already noted, the method for obtaining 2D ordered particle arrays in evaporating liquid films can be applied also to form 2D ordered arrays of protein macromolecules. For example, Haggerty et al. [63] spread 10 lal droplets of aqueous protein solutions (lysozyme or chymotrypsinogen A) on freshly cleaved pyrolytic graphite. The samples were dried at room temperature for about 30 min and then examined by electron microscopy. Regular twodimensional arrays of protein molecules were seen [63]. Likewise, by evaporation of the solvent from suspension films, ordered 2D arrays of inorganic particles on graphite [641] and amidine latex on mica [65] have been assembled. As mentioned above, the ordering of particles in evaporating films is promoted by the attractive capillary immersion forces and the directional influx of particles driven by the evaporation (see Section 13.3 for details). On the other hand, it should be noted that in some cases particle ordering in liquid films is observed even when the latter two factors are missing. Such cases are considered in the next Section 13.1.2.

13.1.2. PARTICLE ORDERING DUE TO A KIRKWOOD-ALDER TYPE PHASE TRANSITION

Three-dimensional and two-dimensional colloid crystals can be formed also in the absence of attractive forces between the particles, if only the particle volume fraction is high enough. First, this type of disorder-order phase transition has been theoretically predicted by Kirkwood et al. [66,67] and then confirmed by computer simulations by Alder and Wainwright [67,68]

Two-Dimensional Crystallization of Particulates and Proteins

523

for a system of hard disks and hard spheres. It is usually termed the Kirkwood-Alder phase transition. The freezing point and the melting point for 3D systems were respectively estimated to be at particle volume fractions 0.50 and 0.55 [70]. Subsequent computer experiments showed that the same kind of first-order phase transition is observed for "soft" spheres, repelling each other electrostatically (the inverse power potentials) [71 ]. The essential feature of the Kirkwood-Alder phase transition is that it occurs between particles interacting only via repulsive forces. Such a phase transition was observed by Hachisu et al. [13] in suspensions of monodisperse latex particles and attributed by Wadati and Toda [72] to a Kirkwood-Alder transition. The existence of this phase transition in the bulk of latex suspensions was confirmed by many succeeding experimental works [73-77]. When the particles are situated in a vicinity of a planar interface (wall), or in a liquid film sandwiched between two interfaces, the particle structuring is facilitated by the ordering effect of the wall [78-82]. This is the same effect, which leads to the appearance of stratification with foam films [83] and gives rise to the oscillatory surface forces detected by the surface force apparatus [84], see Section 5.2.7. Pieranski et al. [85] investigated experimentally a layer of deionized latex suspension confined between two glass surfaces: a flat plate and a sphere. Ordering of the latex particles has been observed. In subsequent studies Pieranski et al. [86,87] examined a deionized latex suspension (particle diameter 1.1 gm) contained in the wedge-shaped gap between two planar glass plates. It was observed that with the increase of the gap thickness a consequence of ordered monolayer, bilayer, trilayer, etc. had been formed in which the particles had alternatively changing triangular and square packing as follows: 1 A - 2 ~ - 2 A - 3 0 - 3 A .... (the cipher denotes the number of layers and the symbol indicates the type of packing). Similar experiments were carried out later by Van Winkle and Murray [88] with smaller polystyrene latex spheres (diameter 305 nm). The suspension was subjected to deionization, so 3D arrays were present also in the bulk. For gap thickness h = 1 gm the authors detected formation of a hexagonally ordered particle monolayer. As h increased, the evolution from two- to three-dimensional crystals was observed as a series of structural transitions, which

Chapter 13

524

resembled that established by Pieranski et al. [86,87]. A melting transition was also detected

[89]. Since the Kirkwood-Alder phase transition occurs in the absence of attraction between the particles, it cannot be classified as a self-assembly process. The volume restriction effect forces the particles to form ordered 3D and 2D arrays in this case. Such conditions of volume restriction are created when a colloidal dispersion, driven by the capillary action, fills the micro-channels of specially designed micromolds [90]. This micromolding method was used to prepare patterned arrays of polystyrene microspheres and other microscopic structures [90,91].

13.1.3.

SELF-ASSEMBLY OF PARTICLES FLOATING ON A LIQUID INTERFACE

As described in details in Chapter 8, the weight of floating colloidal particles, of diameter bigger than 5-10 gm, is large enough to deform the fluid interface and to bring about interparticle attraction and agglomeration due to the lateral flotation force. Bowden et al. [92] prepared regular arrays of topologically complex, millimeter-scale objects, floating on the interface between perfluorodecalin and water. Ordered patterns of different topology were obtained by making hydrophilic or hydrophobic specified edges of the particles. For particles smaller than 5 ~tm the lateral flotation force is usually negligible; an estimate can be made with the help of Eq. (8.2). Then if self-assembly of such small floating particles is observed, it can be attributed either to the van der Waals attractive surface force [93], or to the Lucassen's [94] capillary force between particles with undulated contact line (Chapter 12) for details. The energy Wvw of van der Waals attraction between two identical spherical particles of radius R, can be estimated by means of the Derjaguin approximation (Section 5.1.4): Wvw =

AnR

(~3.1)

12h where An is the Hamaker constant, and h is the surface-to-surface distance between the particles. Taking typical values, An = 5 x 10 -20 J and R = 1 lain, one estimates that Wvw becomes equal to the energy of the thermal motion,

kT, at distance h = 1.02 lain. In other

words, the van der Waals attraction can be significant for such comparatively large Brownian

Two-Dimensional C 9

of Particulates and Proteins

525

particles, and its range can be of the order of the particle radius. Note that the value AH = 5 • 10-2o J is applicable to latex particles interacting across air. If the particles interact across water, AH is about 10 times smaller [84], and then the effect of the van der Waals forces is significant only at close contact. Hence, the latter effect is expected to be more pronounced for floating hydrophobic particles, whose close contact is realized across the air phase. In many cases the surface of the colloidal particles is not smooth, but it has some wave-like folds and/or edges. For example, some samples of latex "spheres" have corrugated surfaces. When attached to a fluid interface such particles exhibit undulated contact lines, and the Lucassen's capillary forces are expected to be operative. The latter are physically significant at distances equal to, or smaller than, the wavelength of the undulations, see Chapter 12. Pieranski [14] examined by optical microscope a layer of latex suspension deposited on a glass substrate. The diameter of the polystyrene spheres was 245 nm. The thickness of the layer was decreased from a few microns to zero. For layer thickness about 300 nm only one layer of particles remained. When the thickness of the water layer was decreased rapidly, the particles were transferred from the bulk onto the water-air surface and were observed to form there ordered or disordered arrays, respectively, for higher and lower particle concentration. The observed uniform distribution of the particles indicated the existence of electrostatic repulsion. In the ordered arrays the interparticle distance was > 1 ~tm, which was considerably greater than the particle diameter (245 nm). The observed ordering could be attributed (i) to a two-dimensional Kirkwood-Alder transition or (ii) to a combination of the electrostatic repulsion and the powerful attraction due to the capillary immersion force (Chapter 7)" the latter could appear if during the structuring the particles have touched the substrate, as it is in Fig. 7.lb. Evidence for clustering of silica particles (300 nm in diameter) at air-water interface was reported by Hurd and Schaefer [96]. The ionic strength of the electrolyte dissolved in the aqueous phase was high enough to limit the range of action of the electrostatic repulsive forces below 1 nm. Under these conditions irreversible dendritic aggregation was observed with particles apparently in direct contact. The bonding was attributed to the van der Waals forces.

526

Chapter 13

Similar experiments with particles at air-water interface were carried out by Onoda [97]. He used larger monodisperse polystyrene latex spheres with size from 1 to 15 lain in diameter. The ionic strength of the aqueous solution was 0.02 M. With the smaller particles (1 lain) little clustering was evident. With the 2 gm particles formation and disassembly of transient 2D aggregates was noticed. With the larger particles, of diameter 5, 10 and 15 lain, irreversible clustering into 2D aggregates was observed. By means of the DLVO theory the depth of the secondary minimum is estimated to be 0.2, 1, 5 and 10 kT (with Hamaker constant An = 5 • 10 -21 J) for particles of diameter 1, 2, 5 and 10 gin, respectively, which agrees well with the experimental evidence. Although the weight of a single particle is small and cannot create interfacial deformation, the weight of larger particle clusters can bend the interface and give rise to capillary flotation forces between the 2D aggregates [97]. Kondo et al. [98] investigated the 2D aggregation of monodisperse spherical silica particles on the surface of benzene. The experiments included two stages: (i) formation of particle clusters at the air-liquid interface under the action of the van der Waals forces; (ii) formation of ordered domains during the drying of the 2D particle structures on a mica substrate. At the second stage case the capillary immersion forces were operative. Heki and Inoue [99] observed the formation of 2D ordered arrays of lain-sized crystals of silver bromide floating at air-water interface - see Fig. 12.1. Such ordering of microcrystals was aimed at improving the quality of photographic materials. Initially the microcrystals, which are hydrophobic, have been dispersed in water with the help of gelatin. The ionic strength of the solution was low to prevent coagulation in the bulk. Next, the gelatin was removed by washing with distilled water and the microcrystals emerge at the water-air interface. There they form 2D structures, separated by areas of bare interface. As discussed in Chapter 12, the latter fact suggests that some attractive force has collected the floating microcrystals. This could be the Lucassen's [94] lateral capillary force, the van der Waals and hydrophobic interactions, or their combination. On molten steel surfaces a strong long-range attraction, which can extend to tens of micrometers, was clearly observed to operate between floating solid particles and to collect them into clusters [100,101]. Such a long-range interaction cannot be attributed to the van der

Two-Dimensional Cr3,stallization of Particulates and Proteins

527

Waals force [100]. Later estimates [101] showed that most probably this interparticle attraction is a manifestation of the lateral flotation force (see Chapter 8).

13.1.4. FORMATION OF PARTICLE 2 D ARRAYS IN ELECTRIC, MAGNETIC AND OPTICAL FIELDS

The electrophoretic deposition method provides another tool for assembly of charged colloidal particles at the surface of an electrode. In the experiments of Giersig and Mulvaney [102,103] citrate- and alkanethiol-stabilized gold particles (samples of average diameter 3.5, 14.1 and 18.5 nm), as well as latex particles of diameter 440 rim, were electrophoretically deposited onto carbon-coated copper grids. A conventional dc power supply was used to generate the applied voltage. After deposition was complete, the copper grid was removed, while the electrodes were still polarized to prevent the occurrence of any desorption of colloid particles. The polarization was then switched off and the solvent in the remaining thin film was allowed to evaporate; finally, the coated grids were examined by electron microscope and ordered 2D domains were observed. The particle agglomeration into clusters evidenced that some attractive surface force must exist between the adsorbed gold particles [102]; it could be attributed to the van der Waals attraction since the Hamaker constant of gold across water is rather high: An = 3 x 10 -19 j. On the other hand, for the latex particles An is almost 100 times smaller; one possible explanation for their ordering is the occurrence of a Kirkwood-Alder transition in a dense population of electrophoretically deposited latex spheres. Alternatively, the observed ordering could be attributed to the action of the capillary immersion force (Fig. 7.1) during the drying of the deposited film. The electrophoretic deposition of latex and silica particles has been examined in Refs. [104108]. In some of these studies [105,108] a transparent electrode of conducting glass (indiumtin oxide, or tin oxide) was used to observe directly the process of particle assembly. The deposited particles were seen to spontaneously assemble into clusters, which further grow into ordered domains. In other words, there is a clear evidence about the existence of a lateral attraction between the deposited latex particles [105,108]. It cannot be the van der Waals force (the Hamaker constant is too small), neither it can be the lateral immersion force (the particles do not protrude from a fluid interface). In Ref. [105] it is hypothesized that this attractive interaction is caused by electro-hydrodynamic effects arising from the charge

528

Chapter" 13

accumulation near the electrode due to the passage of ionic current. In our opinion, the charged dielectric latex particles certainly disturb the electric double layer formed in a vicinity of the electrode; the overlap of the disturbances around two deposited particles could produce interaction between them. This effect calls for additional investigation. Not only electric, but also magnetic field was used to produce 2D arrays of colloidal particles. Hexagonally packed 2D arrays and chain-like clusters of polystyrene latex particles ("magnetic holes") were formed when a magnetic field was applied to ferrofluids such as Fe304, iron or cobalt, and the pattern was numerically simulated by using the Monte Carlo method [ 109,110]. Dimitrov et al. [111 ] investigated experimentally 2D arrays of polystyrene beads, containing a ferrite core of Fe304 and y-Fe203, formed at air-water and glass-water interfaces. Takahashi et al. [112] developed a theoretical model describing the forces acting on the particles. The computer simulations reproduced the observed hexagonally aligned 2D patterns, including the interparticle distance and the time of array formation. It was established that the major role is played by the magnetic force and the monopole electrostatic repulsion between two particles; on the other hand, the contribution of the lateral capillary forces and the electrostatic dipolar repulsion were found to be negligible for this experimental system [ 112]. Finally, it was found experimentally that intensive optical (electromagnetic) fields can cause ordering of microscopic dielectric objects into 2D arrays [ 113,114]. Two principles of particle organization have been proposed [114]. Firstly, the dielectric particles are ordered in direct response to the externally applied standing wave optical fields; this is the same mechanism, which provides trapping (levitation) of dielectric objects into gradients of optical fields, as established by Ashkin [115] and others [116]. Secondly, the external optical fields can produce interactions between dielectric particles (due to oscillating optically induced dipoles), which can also result in the creation of complex structures [113,114]. In this way "optical crystallization" of polystyrene spheres (3.4 lain in diameter) was experimentally achieved with the help of an argon ion laser delivering power up to 10 W at wavelength 514.5 nm.

Two-Dimensional Crystallization of Particulates and Proteins

529

13.1.5. 2 D ARRAYS OBTAINED BY ADSORPTION AND/OR LANGMUIR-BLODGETT METHOD

Particles or macromolecules can adsorb spontaneously at a given interface. When the energy gain accompanying the adsorption is large enough, the adsorption layer becomes dense and the particles acquire hexagonal packing, i.e. a 2D array is formed. On the other hand, if the obtained adsorption monolayer is not dense, it can be compressed by the movable barrier of a Langmuir trough, which is suitable for handling monolayers from insoluble (or slightly soluble) molecules or colloid particles. The latter method was used by Goodwin et al. [117] to form 2D ordered arrays of latex particles, which were further transferred onto glass slides using the Langmuir-Blodgett approach, see e.g. Ref. [118]. This method was applied also by Fendler and coworkers to prepare particulate mono- and multilayers from surfactant stabilized nanosized magnetite crystallites [119], from cadmium sulfide clusters [120], from silver, platinum and palladium nanocrystallytes [121,122], and from ferroelectric lead zirconium titanate particles [123]. The same method can be applied to form 2D arrays of proteins, see e.g. Refs. [ 118, 124, 125]. The above experimental approach can be extended to particulates or macromolecules which do not adsorb at the interface itself, but which can bind to some appropriate pre-spread adsorption monolayer. Fendler et al. have used this version of the method to grow various nanocrystallites below adsorption monolayers; for review see Refs.

[126,127]. Lipid

adsorption monolayers and liposomes are found to be suitable substrates for the growth of two-dimensional crystals of proteins by adsorption from solutions [128-133]. Muratsugu et al. [134-137]

have

developed

a

method

for

preparation

of

2D

arrays

of

proteins

(immunoglobulins) on a plasma-polymerized allylamine film covered on a silver plate; such arrays find biomedical application as immunoassays or immunosensors. Yoshimura et al. [ 138] achieved formation of well ordered 2D arrays of proteins by injection of protein solution into an aqueous subphase that had a higher density and surface tension than the protein solution (solutions of 2 % glucose, 0.15 M NaC1 and 10 mM CdSO4 at pH = 5.7 gave the best results). The buoyancy made the injected drop of protein solution to rise and to spread quickly on the liquid-air interface. Firstly, the proteins that reached the interface unfolded instantaneously and formed a continuous film. Secondly, new-coming protein

530

Chapter 13

molecules adsorbed below the latter continuous film and formed ordered 2D arrays of intact proteins [ 138]. The next sections of this chapter are devoted to a more detailed description of the methods and mechanisms for production of 2D arrays of proteins and particulates in liquid films under the action of the lateral capillary forces and evaporation-driven flow of the fluid.

13.2.

2D

CRYSTALLIZATION OF PROTEINS ON THE SURFACE OF MERCURY

13.2.1. THE MERCURY TROUGH METHOD

3D protein crystals are used to establish the tertiary structure of these macromolecules; for that purpose hundreds of proteins have been crystallized [139-145]. In addition, 2D protein crystals are used for electron microscope crystallography [146-148]. When large 2D crystals of good quality are obtained, the tertiary structure of the protein can be determined by image analysis of the electron micrographs [149-152]. Moreover, as mentioned in the beginning of this chapter, 2D arrays of proteins and other nanoparticles represent considerable practical interest for design of nano-technological devices, which are a possible step towards a future high technology at the macromolecular level. Yoshimura et al. [32] developed a method for 2D crystallization of proteins by spreading a drop of concentrated protein solution on the surface of mercury. There are three reasons to chose mercury as a substrate: (i) its surface is perfectly flat because mercury is liquid (no surface roughness); (ii) the protein suspensions spontaneously spread on the surface of mercury to form a very thin and stable wetting film, in which the two-dimensional crystallization of the protein is accomplished; (iii) in contrast with the methods using localized adsorption [128-133] the protein molecules are not immobilized in adsorption cites; thus they can adjust their location and orientation, which results in the formation of larger well-ordered domains (Fig. 13.1). Two-dimensional crystals of many proteins have been obtained using this method, see Table 13.1. The 2D-crystallization of proteins is carried out in a mercury trough with a special design [32], see Fig. 13.2. It provides all necessary conditions to create and maintain clean mercury surface, which turns out to be of crucial importance. First of all, the mercury trough is closed

Two-Dimensional Co;stalIization of Particulates and Proteins

Beam spot o: ellipsometry

531 SamplingI point l

120

Fig. 13.2. Schematic view of the mercury trough: with the help of the mobile barriers "A" and "B" the mercury surface can be compressed, expanded and cleaned. The surface properties are monitored by ellipsometer. By means of a syringe a drop of protein solution is loaded, and after its spreading sample of the formed film is taken by pressing gently a carbon coated copper grid to the mercury surface. in a chamber filled with pure oxygen (99.99%). The two rotating barriers (Fig. 13.2) are used to create a new mercury surface and to collect the possible interfacial contamination in a narrow sector; the contaminated surface is removed by ejection of the interfacial layer with the help of a Teflon capillary connected to a vacuum pump. The interfacial cleanness is controlled by measurement of the mercury surface tension by means of the Wilhelmi plate method; an amalgamated platinum plate is used for that purpose [32]. In addition, the cleanness is controlled by means of ellipsometric measurements, which give also the thickness of the spread protein layer [32,153]. The composition of the atmosphere in the chamber with the mercury trough is also important. Experiments with helium, nitrogen and oxygen have been carried out [32]. It turns out that the 2D crystallization of proteins is successful only in the presence of oxygen atmosphere, which provides a uniform spreading of the protein solution on mercury. This is attributed to physical adsorption of oxygen molecules on the mercury surface, which makes it more hydrophilic [154]. The humidity of the atmosphere in the chamber also turned out to be important. In a humid O2 atmosphere (relative humidity below 50 %) the protein formed an amorphous monolayer rather than an ordered crystal [154]. The best 2D crystals were obtained if the atmosphere before the spreading was 100 % dry oxygen, without water vapors. The effect of

Chapter 13

532

r

,~,

i~.,,: r: : ' ,,~" " ~. ,,

,:<,,,, o, 2, .: '

i :i :i

:,

,'~

'

'

(a)

(b)

Fig. 13.3. Electron micrograph of 2D arrays of ferritin obtained by Yamaki et al [153] in a mercury trough by spreading of (a) 2btL and (b) 10 laL suspension drops, each of them containing the same amount (0.4 mg) ferritin in the presence of glycerol. The insets show first order diffraction patterns. humidity on the 2D crystallization is related to two factors, both of them decelerating the spreading and drying of the protein solution" (i) adsorption of water on mercury decreases the spreading coefficient [118]" (ii) the rate of evaporation decreases in humid atmosphere.

13.2.2. EXPERIMENTAL PROCEDURE AND RESULTS

In the experiments [153] a sector of the mercury surface of area 430 cm 2 was used setting the angle between the two radial barriers (Fig. 13.2) equal to 300 ~ In each experiment a droplet of protein solution, of volume between 2 and 10 microliters, was placed on the mercury surface. The spreading of the droplet is very fast; it takes less than 1 s. To improve the spreading 0 . 1 1 M NaCl was added to the protein solution. Addition of either glucose or glycerol improves the spreading more than the addition of NaC1 [153]. Moreover, the glucose or glycerol protects the protein molecules from denaturation upon contact with the mercury surface. After 2 rain. of spreading the dried protein monolayer was transferred to a carbon film

Two-Dimensional Co'stallization of Particulates and Proteins

533

(supported by a copper grid) by gently attaching it to the mercury surface. If necessary, the specimen could be stained with 1% uranyl acetate. Finally, the samples were analyzed by transmission electron microscopy (TEM). Figure 13.3 shows two micrographs of 2D arrays of the protein ferritin spread in the presence of glycerol. Some samples (Fig. 13.3a) show almost perfect hexagonal 2D crystals. Other samples (Fig. 13.3b) show the formation of "2D foam", that consists of dispersed empty zones surrounded by interconnected domains of ordered protein monolayer. The absence of separate protein molecules dispersed in the empty zones evidences for the existence of some attractive force between the protein molecules, which tends to keep them together. To reveal the nature of this force experiments with various nanoparticles have been carried out in the mercury trough [39], see Section 13.5 for more details. Two-dimensional crystallization of many proteins has been attempted in the mercury trough. One-third of the trials led to obtaining of satisfactorily ordered protein 2D arrays; they are listed in Table 13.1, where the molecular weight of the protein, the symmetry of the obtained 2D crystal and the experimental diffraction order are also listed. The 2D crystals denoted by asterisk (*) are formed by using the alternative glucose-solution-method [138]. The proteins are not highly symmetric molecules; consequently, the crystal symmetry is often P1 (Table 13.1). On the other hand, the hexagonal lattice is a natural form of closely packed globular particles and it is actually the lattice observed with various protein samples tested until now [ 155]. Micrographs, like Fig. 13.3b, reveal that the close packing is not a result of high protein density, but more likely of the attractive force between the protein molecules. The successful results obtained with the mercury trough stimulated further research on the mechanism, driving force and kinetics of the two-dimensional crystallization in liquid films. The major results are described in Sections 13.3 - 13.7. It is worthwhile noting that, except mercury, there is another metal, gallium, which is liquid at body temperature (more precisely, above 29.8~

The results of the first attempts for

crystallization of ferritin on gallium surface were recently reported [156]. Other liquid substrate attempted for protein 2D-crystallization is perfluorinated oil, see Ref. [37] and Section 13.4.1 below.

534

Chapter 13

Table 13.1. Two-dimensional (2D) crystals of proteins obtained by means of the mercury trough method; 2D crystals obtained using the glucose-solution method [138] are denoted by asterisk (*). Protein

Size

Crystal

Diffraction

(kDa)

symmetry

order

Chaperonin (from thermophilic bacterium)

420

P1

Holo-ferritin (horse)

400

P3

Apo-ferritin (horse)

400

P3

[158]

Apo-ferritin (human)

400

P3

[159]

H+-ATPase (TF1 - PS3)

340

P1 (P3)

[31,32]

H+-ATPase (TF1 - Sulfolobus)

400

PI (P3)

[151]

o~3133(PS3)

280

P3

[31]

LDH (thermophilic bacterium)

300

Proteasomes (thermophilic bacterium)

300 ?

3*

[138]

C-reactive protein (human)

118

3*

[16~]

Myosin S 1 (rabbit skeletal)

100

[16o1

Troponin (rabbit skeletal)

68

[ 1601

Tropomyosin (rabbit skeletal)

64

[ 16(31

Streptoavidin

60

Cytocrome b562

20

Reference

[157] 6; 7*

[32,153]

[ 160]

P2

[16(31

[~ 6{3]

Methallothionetin (rabbit liver)

[160]

H+-ATPase F0-F1 (PS3)

500

Pl(P3)

[1551

LP ring, flagella motor (salmonella)

1300

P3(P6)

[162]

Na +, K+-ATPase (dog kidney)

120

[~55]

Cytocrome P450

54

[155]

Two-Dimensional Crystallization of Particulates and Proteins

13.3.

DYNAMICS OF

2D

535

C R Y S T A L L I Z A T I O N IN E V A P O R A T I N G L I Q U I D F I L M S

13.3.1. MECHANISM OF TWO-DIMENSIONAL CRYSTALLIZATION

Many methods for obtaining two-dimensional ordered arrays of proteins and colloidal particles involve a deposition and/or spreading of a suspension droplet over a substrate, followed by evaporation of the solvent (usually water). Examples are the mercury trough method described in Section 13.2, the method used to crystallize viruses [21-23]; the negative staining-carbon film procedure [24-30], the formation of 2D arrays on graphite and mica [63-65], ordering of latex, silica and other colloidal particles on various substrates [59-62, 163-167]. Many of the works on 2D array formation describe the applied procedure and the final result of ordering. However, the mechanism and stages of the process of ordering demand a special investigation, which can reveal the forces and factors governing the 2D array formation. Such a study has been carried out by Denkov et al. [33,34], who undertook direct microscopic observations of the 2D crystallization of lain-sized latex spheres. The experimental cell used in this study is depicted in Fig. 13.4. In the experiments a drop of latex suspension spreads over the accessible area of a hydrophilic glass substrate encircled by a Teflon ring of inner diameter 14 mm. The inner wall of the ring is cut slantwise (the inset in Fig. 13.4) to ensure the formation of slightly concave liquid meniscus, which favors the formation of larger 2D ordered

9

~

..

"~ ....

,

,

,

4

,,

,

.

~

,,

.

.

"~

/. .........................

1

Fig. 13.4. Sketch of the experimental cell used by Denkov et al. [33] to produce particle 2D arrays in a liquid layer with concave surface (1) over a glass substrate (2) encircled by a hydrophobic (Teflon) ring (3), which is fixed with the help of a brass plate (4) and screws (5) to the microscope table (6); (7) is microscope objective.

Chapter 13

536

9~ oo OooOi 2R > h

(a)

2R > h

(b)

Fig. 13.5. Schematic presentation of the process of 2D ordering of suspension particles in an evaporating liquid layer on a solid substrate: (a) Brownian motion of the particles in a thick layer (b) after the particle tops protrude from the liquid layer, lateral capillary forces appear and cause aggregation of the particles.

Fig. 13.6. Photographs taken by Denkov et al. [33] of two consecutive stages of the process of 2D array formation from latex particles (1.7/.tm in diameter): (a) Brownian motion of the particles in a 10 gm thick aqueous layer; (b) growth of a hexagonal array --- the tracks of particles rnoving toward the ordered phase are seen. arrays. The events happening in the experimental cell were observed from below by means of an optical microscope. The stages of the 2D array formation, as observed in Refs. [33,34], are described below. At the initial stage of the experiments the liquid layer has a thickness of about 100 lam. The microscopic observations show that the latex particles (of diameter 1.70 ~tm) are involved in intensive Brownian motion, see Fig. 13.5a and 13.6a. As the layer thickness gradually decreases owing to the water evaporation, the particle concentration increases, the particles come closer and closer and often collide with each other. However, no aggregation or irreversible particle attachment to the glass surface is noticed.

Two-Dimensional Cr?,stallization of Particulates and Proteins

.....

537

:i:,~,:i;ii~:!9:!!iiI~:~

.....~

',

Fig. 13.7. The boundary between an ordered particle monolayer (the dark zone) and a "lake" (water layer free of particles - the bright zone); every transition from dark to bright interference band corresponds to a 102 nm difference in the water layer thickness; photograph from Ref. [33].

Fig. 13.8. A hexagonal monolayer (bright, lower right) and a hexagonal bilayer (dark, upper left) from ordered 1.7 lam latex spheres; particles packed in a square lattice are seen in the transition zone; photograph from Ref. [33].

When the layer thickness becomes c.a. 10 gm, one can see the appearance of concentric Newton interference rings in the central zone of the wetting film, where the thickness is the smallest. From the number and brightness of the respective rings one can estimate the local thickness of the wetting film (monochromatic light of wavelength/~ = 546 nm has been used) [33,34]. As the thickness continues to decrease, the latex particles remain in Brownian motion. Occasionally one can observe that the tops of few larger particles (having diameter greater than the average for the suspension) protrude from the aqueous film; no movement of these particles is noticed after that. Such particles are found eventually to create defects in the 2D crystal. When the film thickness in the central zone of the cell becomes approximately equal to the particle diameter, one observes the formation of a ring-shaped narrow zone (nucleus) of closely packed particles over the middle of the glass substrate, Fig. 13.5b. The ordered region is surrounded by a thicker liquid film, in which the volume fraction of the particles is lower than 10 % [33,34]. This is the beginning of growth of a two-dimensional crystal, which suddenly changes the pattern of the particle motion. The particles in the thicker layer rush towards the ordered zone and upon reaching the boundary of the array they are trapped in it (Fig. 13.6b). Thus the front of the 2D crystal advances with time in a radial direction, from the center of the

Chapter 13

538

substrate toward the ring wall. Inside the ordered array of hexagonal packing sometimes one can observe "lakes" representing regions free of particles, where the glass substrate is covered only by an aqueous layer. Counting the number of the interference fringes in a vicinity of the shore of such a lake (Fig. 13.7) one can estimate that the thickness of the water layer at the boundary of the 2D array is slightly below the particle diameter (1.7 lam) [33,34]. When the radius of the ordered domain becomes about 3-4 mm and approaches the boundary of the experimental cell, one often observes a transition from monolayer to bilayer (Fig. 13.8). Usually at the boundary between hexagonal monolayer and bilayer one observes small domains of particles packed in square lattice (Fig. 13.8). In some experiments multilayers have been obtained with the following sequence of layers: 1 A - 2 0 - 2 A - 3 ~ - 3 A .... [33,36]; here the ciphers correspond to the number of layers and the symbols mean hexagonal (A) or tetragonal (~1) packing of the particles. This order exactly coincides with the phase diagram calculated and observed by Pieranski et al. [86,87] for colloid particles confined in a narrow wedgeshaped gap between two solid plates. The experimental study of the influence of various factors on the occurrence of the twodimensional crystallization is helpful for revealing the driving forces behind the observed events. In Ref. [33] the effect of particle concentration was examined.

Although the

concentration was varied over one order of magnitude (from 0.25 to 2.5 wt %), no substantial difference in the occurrence of the 2D crystallization was established. At the lower concentrations a "2D-foam" structure, i.e. large zones free of particles formed amidst interconnected bands of ordered particles, were often observed (Fig.

13.9). Note the

resemblance between Figs. 13.9 and 13.3b. On the other hand, when the particle concentration was higher, larger areas were covered by bilayers. Other factor examined in Ref. [33] is the electrolyte concentration, which affects the electrostatic interactions between the negatively charged latex particles. By electrophoretic measurements it was found that the addition of 5 x 10-4 M BaCI2 in the suspension alters the particle ~'-potential f r o m - 1 0 6 t o - 5 3 inV. As a result, at the stage of Brownian motion (Fig. 13.6a) a pronounced tendency for formation of transient aggregates (of 2-5 particles) was observed, which could be attributed to the screened electrostatic repulsion. The aggregation

Two-Dimensional Crystallization of Particulates and Proteins

539

ii!iiiiiiiiiii Fig. 13.9. A dried "2D-foam" structure of ordered latex particles (diameter 1.7 ~m) obtained in the presence of 0.008 M SDS at a low particle concentration; photograph from Ref. [33].

Fig. 13.10. Photograph from Ref. [34] of the growth of a 2D array from latex spheres in the presence of glucose, which decreases the rates of water evaporation and particle motion: the tracks are shorter as compared to those in Fig. 13.6b.

was reversible and the process of ordering followed the same pattern as in the absence of BaC12. Similar was the effect of addition of 0.01 M NaC1. On the other hand high electrolyte concentration (say BaC12 above 2 x 10-3 M) brings about coagulation in the bulk of the latex suspension and 2D crystallization is not observed. It has been established [33] that the water evaporation rate is an important factor for the 2D array formation. The evaporation rate can be changed by varying the volume of the air space above the liquid layer, or by creation of a vertical temperature gradient [37]. For example, the reduction of the volume of the gas space from 250 to 1 cm 3 resulted in a 10-times decrease of the rate of all processes, including the speed of the directional motion of the particles (Fig. 13.6b) and the rate of array growth. Moreover, this decrease of the volume caused formation of larger ordered domains and larger areas covered with bilayer. A further increase of humidity of the air in the cell leads to complete stopping of the process of ordering and even to disintegration of the ordered clusters and restoration of the chaotic particle motion [33]. Note that the evaporation rate can be slowed down also by the addition of anionic (SDS) or cationic (HTAB) surfactant, which form a dense adsorption layer on the liquid surface, which

540

Chapter 13

decelerates the evaporation. The addition of cationic surfactant has the disadvantage to create irreversible attachment of the negatively charged latex particles to the negatively charged glass substrate, which eventually causes many defects in the obtained 2D array [33]. In other experiments glucose was added to the suspension [34]; it also decreases the kinetics of 2D array growth (Fig. 13.10) because of reduction of the evaporation rate and increase of the viscosity of the aqueous phase.

The shape of the surface of the liquid layer also influences the occurrence and the result of the 2D crystallization. In the experimental cell depicted in Fig. 13.4 a slightly concave meniscus is formed and the 2D crystallization starts from the central (thinner) part of the liquid layer. This geometry leads to the formation of large ordered particle monolayers or bilayers, see Fig. 13.11. On the other hand, if a drop of the latex suspension is placed on the same glass plate, but without surrounding Teflon ring, the drop spreads over a certain area and forms a convex meniscus, which meets the glass surface at a contact angle of a few degrees. In this case the thinnest zone of the liquid layer is at the periphery of the drop, where the growth of particle array follows the shrinking of the contact line of the drying drop. In this case a directional motion of latex particles from the center of the drop toward its periphery is observed, which results in the formation of a thick multilayer of particles, see Fig. 13.12. In the central part only a small amount of particles remains and forms small clusters. Large and well ordered arrays were not obtained with convex drops [33]. Similar mechanism of 2D crystallization was established irrespective of the particle si~e: latex particles of diameters 1700 and 814 nm [33], latex particles of diameter 144 and 55 nm [35,36,38], latex particles of diameter 95 and 22 rim, virus and protein of diameters 30 and 12 nm, respectively [39]. First of all, the fact that the addition of electrolyte strongly suppresses the electrostatic interactions without substantial changing of the ordering process, shows that the observed 2D crystallization can be attributed neither to the DLVO surface forces (responsible for the bulk coagulation) [84,95], nor to a Kirkwood-Alder phase transition in concentrated suspensions (a volume restriction effect, see Section 13.1.2). Similarly, the change of the particle concentration does not affect the onset of ordering, whereas with 3D crystals this is the major factor governing the phase transitions. In all experiments [33-38] the

Two-Dimensional Crystallization Of Particulates and Protein.s"

541

Fig. 13.11. A transition from a dried ordered Fig. 13.12. Periphery of a latex suspension drop monolayer to area free of particles (1.7 drying on a glass plate without a ~m in diameter); photograph from Ref. surrounding ring; a thick multilayer of [33]. particle deposits is formed (lower dark zone); photograph from Ref. [33]. 2D array formation starts when the thickness of the water layer becomes approximately equal to the particle diameter and the crystal grows through a directional motion of particles toward the ordered regions. A coexistence of ordered domains and regions free of particles, with a sharp boundary between them, is often observed (Figs. 13.9 and 13.11); the latter fact cannot be explained with the action of repulsive forces alone. Considering the experimental facts the following two-stage mechanism of 2D crystallization was proposed [33,34]: 1) At the first stage, immediately after the protrusion of the particle tops from the liquid layer, the attractive lateral immersion forces collect particles into a "nucleus" of the ordered phase. The immersion force is significant even for nm-sized particles, see Chapter 7 and Fig. 8.3. 2) Once the nucleus is formed, the second stage of crystal growth starts through directional motion of particles toward the ordered array. It is caused by the hydrophilic nature of the surface of the particles: the level of the liquid in the nucleus and in the growing ordered 2D array must be high enough to wet the predominant area of the particle surface and to ensure the formation of a small contact angle particle-water-air, see Fig. 13.13. Therefore, the water

542

Chapter

13

Evaporation

Water surface

/

/

,

|

|

i

|

:

:

1

,.

:

, / / / /

,"

.,,

Fig. 13.13. Schematic presentation of the particle assembly process" the water evaporated from an aggregate of hydrophilic particles is compensated by the influx of water from the surrounding thicker liquid layer; the flux brings new particles to the growing 2D array. evaporated from the 2D array must be compensated by the influx of water from the surrounding thicker liquid layer (Fig. 13.13) in order to keep wet the particles in the ordered domain. This brings about an intensive water flow toward the ordered domains, which carries along suspended particles. Upon reaching the boundary of the array the "newcomers" remain attached captured by the capillary attraction (the immersion force). The above two-stage mechanism agrees well with numerous experimental data, see Refs. [3339]. It provides a quantitative theoretical description of the kinetics of 2D crystal growth, which compares well with the experiment" see the next Section 13.2.3.

13.3.2.

KINETICS

OF TWO-DIMENSIONAL

CRYSTALLIZATION

IN CONVECTIVE

REGIME

An important feature of the mechanism of 2D crystal growth described in Section 13.2.2 is the presence of evaporation-driven convective flow, which carries the suspended particles towards the nucleus [33,34]. This convective regime provides a rapid growth and good particle ordering, which makes a big contrast with the diffusion limited crystallization [168,169]. The latter is characterized by a relatively low rate and frequent formation of random dendrite (fractal) structures [170,171], like that observed in the processes of ice-growth on a cold glass or formation of snowflakes in the air. As discussed in the previous section, the kinetics of convective growth depends on the evaporation rate and the shape of the liquid meniscus (concave or convex), which in turn depend on the specific construction of the experimental cell.

Two-Dimensional Cr~'stallization of Particulates and Proteins

r

ryJ~mdriwcall

?

543

light source

L rc

R

r

Fig. 13.14. Sketch of the experimental cell with paraffin wall used by Dushkin et al. [35] to produce 2D arrays of sub-~m polystyrene latex spheres. The processes in the cell are recorded by a videomicroscope system. The evaporation flux j,~ drives convection fluxes of water, jw, and particles, jp, toward the growing array. To study the kinetics of 2D crystallization Dushkin et al. [35] used an experimental cell depicted in Fig. 13.14. It was made by piercing a circular hole of diameter 2 mm across a paraffin block, which was then sealed to a hydrophilic transparent plate (glass or mica) representing the substrate for 2D crystal growth. The sealing was achieved simply by a local melting of the paraffin surface (at about 70~

In the experiments the cell was loaded with 1

microliter aqueous suspension of polystyrene latex particles of diameter 144 nm (+2 nm) at particle volume fraction 0 = 0.001. The upper side of the cell was kept open allowing the water to evaporate at constant temperature 20~

(+2~

and relative humidity 30 % (+3 %). The

crystal growth was recorded by microscope and video-camera. The observed consequence of events is similar to that described in the previous section. Owing to the water evaporation the liquid layer in the cell thins; at a certain moment a plane parallel liquid film is formed in the central zone, which can be distinguished by the appearance of interference rings around its periphery (Fig. 13.15a). The ordering starts from the center of the cell, where the film is the thinnest and the protrusion of the particle tops from the liquid first happens. A "2D-foam" structure with many empty areas is observed in the central zone, see Fig 13.15b, which is taken about 1 rain. after the beginning of the experiment; the boundary (of elliptic shape) between the array and the surrounding thicker liquid layer is also seen in Fig. 13.15b. The area occupied by the particle array expands with time. Fig. 13.15c shows the final result, after evaporation of the whole amount of water. Because of some instabilities the central

Chapter 13

544

,,

Fig. 13.15. Consecutive stages of 2D array growth video-recorded by Dushkin et al. [35]: (a) interference fringes (light wavelength 540 rim) from a thinning liquid film prior to the array formation; (b) initial stage of growth of 2D array of latex particles (diameter 144 rim); the array has "2D-foam" structure with empty places; (c) final view of the whole area (3.14 mm 2) covered with radial domains of particle monolayer, separated by multilayered ridges and/or empty zones. The bars correspond to 200 lain.

T w o - D i m e n s i o n a l Crystallization o f Particulates and Proteins

545

zone is surrounded by alternating multilayer and monolayer rings. During the formation of such a multilayered ring one observes a decrease of the speed of the boundary between the array and the thicker liquid layer. The reasons for this instability are discussed at the end of this section. The area between the multilayered rings is occupied by a hexagonally packed particle monolayer like that in Fig. 13.11" this was established by means of electron microscopy [35]. To describe quantitatively the kinetics of growth, let us introduce the fluxes of water molecules and colloidal particles at the boundary of the growing 2D array" jw = CwVw,

jp = CpVp= Cpfl Vw

(13.2)

Here Vw and Vp are the velocities of water and particles at the boundary array-meniscus; the coefficient/~ accounts for the fact that the velocity of the particles convected by the flow could be somewhat different from the hydrodynamic velocity of water, Vp - / 5 Vw" Cw and Cp denote the concentrations of water and particles (number of molecules or particles per unit volume). The number of particles joining the array per unit time is [35]

dNp -- jpAh dt

(13.3)

where Ah is the cross sectional area of the periphery of the crystal of thickness h, and t is time. The increase of the number of the attached particles can be related to the increase of the volume of the array [35]"

d(Ah) dNp = ~a ~

(13.4)

vo

where Vp is the volume of a colloidal particle, A and ~, are the area of the array and the volume fraction of the particles within it. Further, we notice, that the water flux jw through the periphery of the array is driven by the evaporation flux je of water molecules per unit area of the evaporating surface, Ae, of the array [35]: j w = j e a~

(13.5)

Ah

The combination of Eqs. (13.2) and (13.5) yields

Cp Cp jpAh- fl ~ A h j w - j6 ~A~je Cw

Cw

(13.6)

Chapter 13

546

Next, substituting Eqs. (13.4) and (13.6) into the mass balance equation (13.3) one obtains the basic differential equation describing the growth of the ordered area A(t) [35]:

dA = fl Vp C________Ae pp je dt (?.hc w

(13.7)

Here we have used the fact that between two step-wise transitions the thickness of the array remains constant, h = const. Further, to obtain quantitative results one has to additionally specify the system. For the experimental cell, used by Dushkin et al. [35] the additional conditions are: (i) During the process of growth the boundary of the array, the contact line array-meniscus, is (approximately) a circle of radius re(t). (ii) The convection at the boundary array-meniscus is stationary, that is the parameters Cw, Cp and ,/3, as well as the rate of evaporation j~, can be (approximately) considered as being independent of time. (iii) Owing to the fast evaporation, the central part of the array has already dried; consequently, evaporation takes place from a circular ring of width b (wet array) situated in a close vicinity of the contact line; in stationary regime b is assumed independent of time [35]. (iv) The defects of the ordered monolayer, such as empty areas and multilayers, cover relatively small area and can be neglected in a first approximation. The additional conditions (i) and (iii) imply

A = rcr c,2

Ae = 2rcr~.b,

(b << rc)

(113.8)

The substitution of Eq. (13.8) into Eq. (13.7) yields a simple equation for rc(t):

dr c dt

= vc,

where

vc =

fl jeVpcpb (P.cwh

=

fl jeVwO b

(13.9)

q). (1-Q)h

Here we have expressed the particle and water numerical concentrations, Cp and Cw, through the volume fractions of the particles in the suspension q~" Vw = 30 ~3 is the volume per water molecule; vc has the meaning of a velocity of advance of the contact line, or a linear rate of

Two-Dimensiona I Crysta lliza tion of Pa rticu la tes and Proteins

547

3

~

2

0

0

5

10

15

20

t (min) Fig. 13.16. Experimental data for the area, A, of a growing 2D vs. the elapsed time, t, corresponding to 4 different runs (the different symbols). The solid lines are drawn with the help of Eq. (13.10) by the least squares method; results from Ref. [35]. growth of the ordered array. In stationary regime vc is constant and from Eqs. (13.8) and (13.9) one obtains [35] re-

+ re0

+

A0-

Equations (13.10) predict that in stationary regime the radius of the array rc increases linearly with time, and the area A of the 2D crystal grows as a quadratic function of time; rco and A0 are, respectively, radius and area of the ordered domain formed at the initial stage of 2D crystallization, before the establishment of stationary regime. Figure 13.16 shows experimental data of Dushkin et al. [35] for A(t) in the case of particle diameter 144 rim; the data agrees well with Eq. (13.9). From the best fits values of vc from 6.8 lam/s (Curve 4) to 10.7 gm/s (Curve 1) have been calculated. The evaporation flux je = 6.12 x 1017 cm -2 s-1 was measured in an independent experiment, and then the value of the width of the wet zone was estimated to be b = 40 gin, which seems reasonable [35]. The theoretical model agrees well also with similar data for particles of diameter 55 gin. Coming back to the multilayers, whose formation has been neglected in the previous theoretical consideration, we should mention that depending on their thickness (number of layers) they exhibit different colors when observed in reflected or transmitted polychromatic light [36]. The most pronounced color effect was observed with multilayers deposited on a gold-coated glass

548

Chapter 13

substrate, see Table 13.2. The colors of the layers from transparent latex spheres are due to light interference at plane parallel films of different thickness. By measuring the wavelengths, at which the reflectance from two neighboring multilayers coincide, it has been established that the particle array has hexagonal packing, which is in agreement with the electronic microscope pictures; see Ref. [36] for details. The same interference method has been applied to establish that the spherical micelles in stratifying foam films are also hexagonally packed [172]. Table 13.2. Interference colors (in reflected light) exhibited by dried multilayers of 55 nm polystyrene latex spheres deposited on gold-coated glass; data from Ref. [36]. number of

thickness

color in

number of

thickness

color in

layers, k

hk ( r i m )

reflected light

layers, k

hk (nm)

reflected light

1

47

ochre

7

277

magenta

2

85

brown

8

316

blue-purplish

3

124

navy blue

9

354

green

4

162

sky blue

10

392

yellow-green

5

201

yellow

11

431

orange

6

239

orange

12

469

red

Finally, we give an idea about the origin of the instability, which causes the appearance of alternatively changing rings of particle monolayers and multilayers (Fig. 13.15c). First, notice that evaporation takes place not only from the drying 2D array, but also from the surrounding liquid layer. Let us denote the volume of the latter by V: R

V= 2rt f d r r z(r)

(13.11)

Here z(r) is the shape of the generatrix of the axisymmetric meniscus encircling the array; R is the radius of the inner cylindrical wall of the experimental cell, see Fig. 13.14. The profile z(r) can be determined by integration of the Laplace equation of capillarity, Eq. (2.24) with (p- 0: r sin0= r,. sin0c + kl (r 2 - re2),

kl =- Pc/2(Y

(13.12)

Two-Dimensional Crystallization of Particulates and Proteins

549

For small meniscus slope, as it is in Fig. 13.14, with the help of Eq. (13.12) one obtains & -

-

dr

= tan0 = sin0 - kit + r,, (sin0c - klr,.)/r

(13.13)

Integrating Eq. (13.13) one determines z(r), and further, substituting Eq. (13.11) one obtains analytical expression for V - V(rc). For our considerations it is important that the evaporation from the encircling meniscus provides an expression for the velocity of the contact line,

VcZ

dr C dr,. dV ~ dt z ~dV~ dt

-

~

dr C 2 dV Vwjezr(R -r~2)

(b << re)

(13.14)

at the last step we have approximated the evaporating area with /17(R2 - rc2), which is a good approximation for small meniscus slope. Equation (13.14) expresses the rate of receding of the liquid meniscus, whereas Eq, (13.9) represents the rate of advance of the ordered array owing to the supply of new particles. For a steadily growing 2D array these two rates must coincide. The latter physical requirement determines the local thickness, h, of the particle array: insofar as Eqs. (13.9) and (13.14) must give the same Vc, one obtains

re0, ( 1 - 0 )

- 7~-rc

(R 2 - re2)h

One can check that dV/drc is not very sensitive to the variation of re. With the help of Eq. (13.15) the instability in the 2D crystal growth can be interpreted in the following way: (i) With the growth of the ordered array the term (R 2 -r~ 2) decreases, see the denominator in Eq. (13.15), which can be compensated by increase of h, i.e. by the formation of a multilayer. (ii) As experimentally observed [35], when a multilayer is formed the velocity of advance of the contact line decreases, which will certainly result in a shrinking of the width b of the wet 2D array zone because of the advancing front of the dried array. (iii) Since b and h enter Eq. (13.15) as a ratio (b/h), the decrease of b can be compensated by a decrease of h, that is again a monolayer (and even intermediate zone free of particles) can be formed. The monolayer grows and after a certain time the events are repeated starting from step (i) above.

550

Chapter 13

The above three-step mechanism can be applied to quantify the consecutive formation of monolayers and multilayers (Fig. 13.15c); an adequate description requires to take into account the increase of the particle volume fraction, i.e. ~ - @t), and the variation of the width b(t) of the wet zone.

13.4.

LIQUID SUBSTRATES FOR 2 D ARRAY FORMATION

13.4.1. FLUORINATED OIL AS A SUBSTRATE FOR TWO-DIMENSIONAL CRYSTALLIZATION

The possibility for application of fluorinated oil (F-oil) as a liquid substrate for 2D array formation was studied in Ref. [37]. Perfluoromethyldecalin (PFMD) was used, which possesses some of the appropriate properties of the mercury substrate (molecularly smooth and tangentially mobile surface, cf. Section 13.2.1) as well as some additional advantages: (i) it is chemically inert and hazardless [173-175]; (ii) it allows the merging and rearrangement of already ordered domains into larger ones; (iii) the formed 2D structures can be gently deposited onto another surface after evaporation of all F-oil and (iv) it is difficult to contaminate the fluorocarbon surface insofar as the common surfactants adsorb poorly at the fluorocarbonwater interface [ 176]. The physicochemical properties of PFMD are: molecular mass 512.1 Daltons, mass density 1.94 g/cm 3 at 25~

boiling temperature 141~

melting temperature -10~

9insoluble in water

or liquid hydrocarbons; refractive index 1.315; surface tension 19.2 mN/m" interfacial tension against pure water 53.4 mN/m.

___1

L__ r

i

4 r....ll.'//.///..//,,I/7"//./////

1/

//I

Fig. 13.17. Experimental cell used in Ref. [37] to produce 2D arrays of particles on F-oil substrate: l- glass plate, 2- Teflon ring, 3- glass container for control of temperature and evaporation, 4- micro-syringe for control of the oil level and the meniscus shape.

Two-Dimensiona I Co,sta llization of Pa rticu lates and Proteins

~ C ~ t _~_ Capillary ....... ffrces

551

Water evaporation

TTTTT

/ __~ ......

F-oil 7 Solid "// / substrate

//

//

/

I / So'i~d/ substrate

"

.,

j/

(a)

//

F-oil evaporation

,,

/

..,,

/

(b)

Fig. 13.18. (a) The 2D array nucleation and growth over F-oil substrate are performed under the combined action of attractive lateral capillary forces and an evaporation driven supply of particles; (b) the formed 2D array is automatically deposited on a solid substrate after the consecutive evaporation of the water and F-oil. The experiments on two-dimensional crystallization have been carried out in a cell depicted in Fig. 13.17. The special cross section ensures attachment of the meniscus of the aqueous layer to the concave corner of the Teflon ring. The bottom of the experimental cell is transparent and allows observations of the processes on the F-oil surface from below, through the F-oil phase. It is possible to control the rate of evaporation from the aqueous layer (and the rate of 2D array growth) varying the temperature of the water circulating through the glass container at the top of the cell. In addition, via sucking or injection of F-oil by a microsyringe one can control the thickness of the aqueous layer spread over the F-oil [37]. To spread drops of aqueous latex suspensions or protein solutions over the fluorinated oil appropriate fluorinated surfactants were used [37]. Best results were obtained with a mixture of the surfactant perfluorononylpolyoxyethylene, C9F19CH2CH(OH)CH2(OCH2CH2)9OCH3, and the fluorinated alcohol CsFI7(CH2)zOH as a cosurfactant. This mixture of surfactant and cosurfactant provided hydrophilization of the F-oil surface and spreading of aqueous layers which were very stable and did not rupture during the experiments with latex particles and ferritin suspensions.

552

Chapter 13

.

9

4

);)4,.

~

)""(,

: " (

~ "~

' " " "(

~"

),..~,.),.~t

"

"(

~.4

'

Fig. 13.19. Large, well ordered domains of 1.7 lam polystyrene latex spheres (1.7 gm in diameter) obtained over F-oil substrate: (a) optical microscope view; (b) scanning electron microscope view; micrographs taken by Lazarov et al. [37]. The mechanism of two-dimensional crystallization over F-oil substrate resembles that over a solid substrate (Section 13.3). Again the ordering started when the water film thickness becomes about the particle diameter, which was 1.7 lam for the used latex spheres [37]. A difference with the case of solid substrate is that as a first step small groups of particles are formed in the aqueous film. Next one observes that these groups (2D aggregates) attract each other, come closer and merge into larger ordered aggregates. The interfacial deformations around each aggregate are visible due to the formation of interference fringes in the surrounding aqueous layer of uneven thickness. Consequently, one can be sure that the observed attraction and coalescence of particle aggregates is due to the action of the lateral immersion forces (Chapter 7). At the first stages of the ordering process a "2D foam" (cf. Figs. 13.3b and 13.9) is formed. During the later stages one observes the formation of a ring-shaped homogeneous particle monolayer, which grows owing to the evaporation-driven convective supply of particles (Fig. 13.18a). After the evaporation of the water from the F-oil surface, the oil itself starts to evaporate. Thus the formed particle structure gradually approaches the glass plate at the bottom of the cell. After the evaporation of the oil the particle array is transferred to the solid substrate (Fig. 13.18b). In order to prepare samples for electron microscopy a specimen grid was put at the bottom of the cell before loading it with F-oil and particle suspension. After drying, the formed

553

Two-Dimensional Crystallization of Particulates and Proteins

84

e~,ee

. . . .

.... ,

i~..,

--;s

,.

.

.

.

.

.

.

.

: :9: .

":-...-+i--+;v-!

Fig. 13.20. Photograph from Ref. [37] of two overlapping ordered monolayers from latex spheres; the apparent "superlattice" (down right) is a result of the moir6 optic effect.

Fig. 13.21. Illustration of the moir6 optic effect due to the relative rotation of two identical lattices; the dependence of the "superlattice" constant on the magnitude of rotation can be seen [37].

structures were directly deposited on the specimen grid and were studied by electron microscopy. Fig. 13.19 shows large well ordered domains of latex particles formed on F-oil substrate. Two additional differences between the ordering on a glass (solid) and F-oil (liquid) substrate deserve to be mentioned. As noted in Sections 13.3, in some experiments with glass substrate multilayers have been obtained with the sequence of layers: 1 A - 2 [ ] - 2 A - 3 0 - 3 A .... established by Pieranski et al. [86,87] for colloid particles confined in a narrow wedge-shaped gap between two solid plates. In contrast, on the F-oil substrate only hexagonally (A) packed multilayers, 1A-2A-3A .... , are formed; tetragonal (O) zones are not observed at the boundary between two hexagonal layers. This can be attributed to the "soft" oil-water interface which can bend in the transition zone between the multilayers [37].

554

Chapter 13

If the rate of water evaporation is high, the approaching ordered domains (which are almost dry) can tuck one underneath another upon collision instead of merging and rearranging [37]. Then if the axes of symmetry of the two hexagonally-packed domains are rotated with respect to each other at an angle not divisible by 60 ~ they form a peculiar "superlattice". The latter exhibits an interesting optical property termed the "moir6 effect" [177]. One observes regions of hexagonal packing of unit-cell constant larger than that of the particle monolayer (Fig. 13.20). The relation of the "moir6 effect" to the relative rotation of the axes of the two 2D lattices is illustrated in Fig. 13.21. In conclusion, the results with F-oil substrate [37] confirm that the capillary forces and the convective particle flux are the main factors governing the 2D array formation. The quality of the array can be improved by control of the evaporation rate and the meniscus shape.

13.4.2. M E R C U R Y AS A SUBSTRATE FOR TWO-DIMENSIONAL CRYSTALLIZATION

The processes in the mercury trough (Section 13.2) happen too fast and one can examine only the final result of the two-dimensional crystallization. To investigate its mechanism, Dimitrov et al. [38] carried out separate experiments with lain-sized latex particles and recorded the

r

v

4

2

ff

1

3 Fig. 13.22. Experimental cell used in Ref. [38] to obtain ordered 2D arrays of colloid particles on the surface of mercury: (1) mercury, (2) bronze cover, (3) container for mercury, (4) syringe needle, (5) aqueous suspension of latex particles, (6) plastic ring representing the wall of the crystallization cell.

Two-Dimensional Co, stallization of Particulates and Proteins

~,~v11P'~

-

~

(a)

l ( c )

555

",

0

~

"

~

(b)

"

o

Fig. 13.23. Consecutive stages of the ordering of 1.7 gm latex spheres on the surface of mercury: (a) in the beginning (t = 0) the particles associate in 2D clusters; (b) at t = 1 min one sees an increased number of clusters; (c) at t = 4 min the coalescence of clusters has led to the formation of a large aggregate (nucleus), which begins to grow due to a directional influx of new particles; (d) a large ordered domain as a final result of the 2D crystallization; the length of the bar is 20 gin; micrographs taken by Dimitrov et al. [38]. dynamics of the process. For that purpose a special experimental cell was constructed (Fig. 13.22). Latex suspension layer was spread over a part of the mercury surface restricted by a plastic ring of inner diameter 1 cm. The plastic ring had thickness 1 mm and separated the bronze cover from the mercury substrate. The electric potential difference between the mercury and the bronze cover could be varied and its effect on the 2D ordering was examined. Before each experiment the mercury surface was treated with 0.01 M solution of sodium dodecyl sulfate, which ensured hydrophilization and spreading of the latex suspension droplet. The latex particles bear a negative surface charge. When a potential equal to - 2 0 0 mV, or more negative, is applied to the mercury, it repels the latex particles in the spread thick layer of

suspension [38]. This electrostatic "buoyancy force" brings the particles on the upper (waterair) interface, where they are further observed to attract each other and to form hexagonally packed domains. At the upper interface the particles have a configuration like that in Fig. 8.2a;

Chapter 13

556

hence, the attraction between them can be attributed to a counterpart of the lateral flotation force, but driven by the electrostatic upthrust rather than by the usual Archimedes force, which is due to gravity. The hexagonal particle arrays thus formed were not very stable and were destroyed by convective fluxes in the suspension layer [38]. On the other hand, if a relatively high positive electric potential (> +100 mV) is applied to the mercury, then the latex particles stick to its surface, which does not facilitate the 2D ordering. It was found that the best conditions for 2D crystallization are provided when the potential of the mercury is low and positive, between +3 and +10 mV [38]. In this potential range uniform aqueous suspension film was formed on the mercury. The nucleation and growth of 2D particle arrays was observed in films of thickness slightly smaller than the particle diameter; the latter was 1.70 + 0.05 gm for the particles in Fig. 13.23. First one sees (Fig. 13.23a) that the particles attract each other and associate into small clusters under the action of the lateral immersion force (see Chapter 7). After 1 rain. the particle packing into clusters becomes denser (Fig. 13.23b); on the other hand, zones free of particles appear. After 4 rain. the larger particle assemblies are observed, which are formed by coalescence of smaller clusters (Fig. 13.23c); the evaporation from the assembly triggers a slight directional particle motion towards the array. Finally, a large two-dimensional ordered array was formed; it consists of hexagonally-packed domains (Fig. 13.23d). The boundary between two neighboring domains can be removed (to form a larger domain) by a consecutive increase and decrease of the humidity of the air above the array, which causes a partial "melting" and subsequent "re-crystallization" of the 2D array; this procedure was termed "annealing" [38]. The experiments on mercury surface reported in Ref. [38] convincingly demonstrated the role of the lateral immersion forces in the process of two-dimensional aggregation and ordering of colloidal particles confined in a liquid film.

13.5.

SIZE SEPARATION OF COLLOIDAL PARTICLES DURING 2D CRYSTALLIZATION

Consider an aqueous layer containing a mixture of larger and smaller colloidal particles. When the thickness of the layer decreases owing to water evaporation, the tops of the larger particles first protrude from the liquid layer, and the overlap of the menisci formed around these particle gives rise to a strong capillary attraction (Fig. 7. l b), which is expected to cause aggregation of

Two-Dimensional Co'stallization of Particulates and Proteins

557

the larger particles. Later, with the advance of water evaporation, the tops of the smaller particles will also protrude from the aqueous layer and then they will also aggregate driven by the lateral immersion force. As a result, the larger and the smaller particles are expected to form separate aggregates. Such a size separation (segregation) of particles during 2D crystallization can serve as an indicator for the involvement of the lateral immersion force into the process of ordering. In Ref. [33] observations of the formation of 2D array from a mixed suspension, containing particles of diameter 1.70 and 0.81 gm, are reported. It has been observed, that first the larger particles form ordered clusters, and at a later stage these clusters are encircled by ordered array of the smaller particles. Yamaki et al. [39] carried out experiments on formation of 2D arrays of latex particles in the mercury trough, see Section 13.2.1. The average size of the particles was 55 nm and their polydispersity is characterized with a standard deviation +3.4 nm. In the resulting arrays (see Fig. 13.24a) ordered clusters of the largest particles are surrounded by 2D array of the smaller particles; the particle size decreases with the increase of the distance from the center of the cluster. As discussed above, this effect can be attributed to the fact, that the largest particles first protrude from the thinning liquid layer and are grouped together by the lateral immersion force to form a 2D aggregate. The evaporation from this aggregate brings about a convective influx of water. The flow carries along suspended particles; only those of them, whose tops protrude from the liquid layer, can be captured by the aggregate owing to the capillary attraction. Since the thickness of the aqueous layer decreases with time (due to the evaporation), smaller and smaller particles are able to join the aggregate; the latter explains the decrease of the particle size with the increase of the distance from the center of the cluster. This phenomenon could be further developed into a method for size-fractionating of colloidal particles, including proteins and viruses [39]. Yamaki et al. [39] performed experiments in the mercury trough also with mixtures of larger and smaller particles. Examples for the observed size separation are shown in Figs. 13.24 b-d. As a rule, the larger particles are collected in the center of the aggregate and are surrounded by an ordered array of the smaller particles. In Fig. 13.24b the larger, 144 nm in diameter, latex

Chapter 13

558

~"

"

200

(a)

I1,1 Jl

nm

(b)

q

q

r

"77~:e4:::,:, g;~+Ea,,:,::;t~'..

Ic) Fig. 13.24. Effect of size separation during 2D crystallization: transmission electron microscope images taken by Yamaki et al. [39] from samples of 2D arrays formed in a mercury trough: (a) from polydisperse latex spheres of diameter 55 _+ 3.4 rim; (b) from a homogenized suspension mixture containing latex particles 55 + 3.4 nm and 144 + 2.0 nm in diameter; (c) from a mixture containing the protein ferritin and polio virus of diameters 12 nm and 30 nm, respectively; (d) from a suspension mixture containing latex particles 95 _+ 2.0 and 144 _+2.0 nm in diameter.

Two-Dimensional Co'stallization of Particulates and Proteins

559

Fig. 13.25. A result from the computer simulation of 2D size separation in a mixture of larger and smaller particles (diameter ratio 2:1) in the presence of evaporation and capillary immersion force [39]. spheres (in the center) are separated from the smaller latex particles of diameter 55 nm. Similar result was obtained with a mixture of latex particles of diameter 22 nm and ferritin (protein) molecules of diameter 12 nm [39]. In Fig. 13.24c the polio viruses of diameter 30 nm are grouped in a small cluster in the center, surrounded by the separated ferritin molecules. The separation of the larger from the smaller particles is not good when there is a relatively strong attraction between the particles of the two different species. The theory of the double-layer interactions [95] predicts that colloid particles of different surface potential could experience electrostatic attraction at short distances, even when the surface potentials have the same sign. An effect of this type was observed with a mixture of polystyrene latex particles of diameters 144 and 95 nm [39], see Fig. 13.24d. To obtain a better understanding of the observed size separation, computer simulations were also carried out by means of the Monte Carlo method [39]. The behavior of the particles in a mixture of 37 large and 84 small particles was simulated in the presence of lateral capillary force and convective flow. The computer simulations reproduced a size separation (Fig. 13.25), which is very similar to the experimental results.

560

Chapter 13

It is interesting to note that almost simultaneously with the work by Yamaki et al. [39] another work reporting a similar phenomenon appeared. Ohara et al [178] studied the formation of 2D crystals (opals) from suspensions of polydisperse rim-sized gold particles in hexane. The resulting structures were detected by means of transmission electron microscopy (TEM). In some of the experiments a drop of such suspension was evaporated directly on an amorphous carbon-coated TEM grid. In other experiment a layer of the gold-in-hexane suspension was evaporated over water in a 10 cm diameter Petri dish and a sample for TEM was taken after the evaporation of the hexane. In both cases it was found that hexagonally-packed domains of the largest particles (5-6 nm diameter) were formed at the center, which were surrounded in radial direction by successively smaller particles. Ohara et al. [178] hypothesized that the size separation of the particles is due to the size dependence of the van der Waals attraction between the gold particles. Computer simulations of the process of ordering was carried out by means of the Monte Carlo method in conjunction with the Hamaker formula for the van der Waals interaction between two spherical particles of different radii. The computer simulation reproduced a size separation, which is very similar to the experimental results [ 178]. Then a question arises: which is the cause for the particle size separation - the van der Waals surface force or the lateral capillary force? For the system studied by Ohara et al. [ 178] perhaps both of them are simultaneously operative and the size segregation happens under their combined action. On the other hand, the proteins, viruses and latex particles have a very low Hamaker constant across water: An = 5-8 x 10-2~ J for two protein molecules across water [ 179], whereas AH = 3.1 x 10-19 J for two gold particles across dodecane [ 178]. For that reason, it is unlikely the van der Waals forces to play an essential role in the ordering and size separation observed by Yamaki et al. [39]. Moreover, for particles of radius greater than 10 nm the energy of the capillary immersion interaction becomes larger than 100 kT (see Fig. 8.3), which is certainly much greater than the energy of the van der Waals attraction between such particles, except might be at close contact. The effect of size separation under the action of the lateral immersion force and convective flux was confirmed both experimentally (polydisperse latex particles of average size 80 and 220

Two-Dimensional Crystallization of Particulates and Proteins

561

nm) and by computer simulations by Rakers et al. [180]. The 2D arrays were formed on a silicon substrate, whose temperature could be varied. At lower temperatures (2~

a more

pronounced effect of ordering and size segregation was observed as compared with the result at higher temperature (20~

The temperature affects the 2D crystallization through the rate of

evaporation. At higher temperature the evaporation is stronger and the growth of the ordered aggregates is dominated by the convective flux, rather than by the lateral capillary force. As a result, the size selection carried out by the capillary force is suppressed. In contrast, at low evaporation rates (low temperatures) the uptake of particles by the aggregates is dominated by the capillary force, which brings about a size segregation of the particles.

13.6.

METHODS FOR OBTAINING LARGE 2D-CRYSTALLINE COATINGS

13.6.1.

WITHDRAWAL OF A PLATE FROM SUSPENSION

For examination of 2D arrays of sub-gm particles and protein macromolecules by electron microscopy relatively small (microscopic) samples are sufficient. On the other hand, the usage of 2D arrays in industrial applications may demand the formation of larger ordered areas (1 cm 2 and larger) [40,41]. As discussed in Section 13.3, the two-dimensional crystallization in a cylindrical cell (Figs. 13.4 and 13.14) gives ordered domains of area not larger than 1 mm 2 due to the appearance of instabilities in the crystal growth (consecutive formation of monolayers and multilayers). In principle, this problem could be overcome by adjustment of the rate of receding of the capillary meniscus in such a way, that a particle monolayer to be formed. In other words, the velocity of the contact line array/meniscus is to be determined by Eq. (13.9) with h = 2Rp, Vc = fl jegw ~9 b , 2Rp0a (1 - q})

(for monolayer)

(13.16)

where Rp is the particle radius. A possible way to realize receding of the meniscus, with a velocity vc given by Eq. (13.16), is to withdraw a hydrophilic plate from a suspension with a constant speed. This was realized by Dimitrov and Nagayama [40] as described below, and cmsized 2D arrays of colloidal particles were obtained on hydrophilic glass plates.

Chapter 13

562 Substrate Z

""

b

u

~ Suspension 9 0 9

Fig. 13.26. The withdrawal of a plate from a suspension carries along a liquid film containing particles; the evaporation of the solvent (water) from that film gives rise to a convective influx of water jw and particles jp; the latter stick to the periphery of the growing array arrested by the capillary immersion force. In the beginning of each experiment the plate (in vertical position) is immersed partially into a suspension of monodisperse latex spheres. The water rises along the hydrophilic plate to form a very small (practically zero) contact angle. The wedge-shaped meniscus just below the contact line contains latex particles. Because of water evaporation from the suspension, the level of the liquid slowly decreases, which results in a deposition of latex particles on the vertical glass substrate in the zone of the receding contact line. In fact, this is a known phenomenon: always when water contacts with wettable surfaces and evaporates, the substances dissolved in the water accumulate in the wetting film in the vicinity of the three-phase contact line [40,59]. An initial deposit of latex particles is thus obtained; then a stepper motor is switched on and the glass plate is pulled up out of the suspension with an appropriate rate in the range from 0.1 to 30 ~tm/s. The particle array deposited on the plate is observed by microscope and the rate of plate withdrawal, vc, is adjusted to obtain a monolayer in a stationary regime of continuous 2D array growth (Fig. 13.26). Monodisperse suspensions of particle radii ranging from 79 nm up to 2.1 ~tm have been used at fixed particle volume fraction: r

0.01. It has been established

experimentally [40] that vc ~: 1]Rp, exactly as predicted by Eq. (13.16). If v~ is decreased two times by the stepper motor, one observes formation of a particle bilayer.

Two-Dimensional Crystallization of Particulates and Proteins

563

Fig. 13.27. Photographs from Ref. [40] of the region around the border of growing 2D array on a vertical plate, which is being pulled up from an aqueous suspension, which contains particles of diameter (a) 814 nm and (b) 953 nm. The particles in the liquid film below the array are dragged upwards by the evaporation driven water flow (Fig. 13.26); particles moving during the exposition period look fuzzy; the arrow in (b) indicates one of the larger particles immobilized in the thicker water film below the boundary of the array. As already discussed in Section 13.3, the growth of particle 2D array in the experimental cells depicted in Figs. 13.4 and 13.14 happens through an evaporation-driven convective flow of water, which brings along suspended particles toward the boundary of the 2D array; once reaching that boundary the newcomer-particles are captured by the lateral capillary force and firmly attached to the growing array. Neither the convective flow, nor the capillary interaction is dependent on the gravitational force, so one could expect that the mechanism of growth is not dependent on the orientation of the plate: horizontal (Figs. 13.4, 13.14 and 13.17) or vertical (Fig. 13.26). By means of optical microscope it was observed that in stationary regime of the plate withdrawal the width of the wet zone is b ~ 2 mm [40]. Similar value of b was obtained from Eq. (13.16) using experimentally measured values of Vc and je. The microscopic observations showed also the existence of directional motion of particles, dragged by the water flow toward the boundary of the growing array, see Fig. 13.27. Some bigger particles are immobilized before reaching the array and subsequently they cause defects in the 2D crystal, cf. Fig. 13.27 and 13.6b. All these similarities between the 2D array formation on a horizontal and vertical substrate imply that the underlying physical mechanism of growth is essentially the same. The

564

Chapter 13

advantage of the technique with vertical plate is that the rate of withdrawal can be relatively easily controlled to obtain large ordered particle monolayers in a continuous regime of production [40]. The formed dried ordered monolayers exhibit uniform coloring due to interference of light, cf. Table 13.2. Moreover, the 2D arrays from particles, whose diameter is greater than the wavelength of the visible light, exhibit brilliant (opalescent) diffraction colors, which depend on the particle size and the orientation of the 2D crystal domains. The brilliant coloring can be enhanced by coating the particulate monolayers with silver or gold (up to 10 nm thick); the coating stabilizes mechanically the arrays, which are fragile before the metal coating. The usage of these ordered arrays as diffraction gratings is discussed in Ref. [40,41].

13.6.2. DEPOSITION OF ORDERED COATINGS WITH A "BRUSH"

As discussed above, larger domains of ordered 2D arrays can be obtained if the speed of meniscus receding, vc, is appropriately related to the evaporation flux je and the width of the wet array, b, see Eq. (13.16). An experimental method, which allows a control of Vc and je was developed in Ref. [42] and successfully applied to obtain large 2D crystals of the protein holoferritin. The basic part of the experimental setup is shown in Fig. 13.28. The substrate is a silicon wafer (20 x 20 x 1 mm), which is fixed to a movable horizontal stage. A vertical platinum plate (20 x 20 x 2 mm) plays the role of a "brush". In the experiments 10 microliters protein solution is introduced at the corner between the horizontal silicon wafer and the vertical Pt-plate. Then by means of a motor the stage with the silicon wafer begins to move horizontally with a constant velocity of 2 lam/s. After the receding meniscus, which is attached to the vertical Pt-plate, a thin wetting film is deposited on the silicone substrate. Large and ordered two-dimensional arrays of protein are formed in the drying film: this was established by scanning electron microscope [42]. It was established that better two-dimensional arrays of holoferritin are formed when the humidity of the atmosphere above the deposited film is higher and the rate of evaporation is lower [42]. Moreover, it was found experimentally, that good hexagonal ordering is obtained when the pH is sufficiently higher than the isoelectric point of the holoferritin (at pH = 4.8),

Two-Dimensional Crystallization of Particulates and Proteins

Protein s o l u t i o n ~ ~ Pt plate (P)

SteS) /

565

m~,

Latex particles fsuspension

I

Fig. 13.28. Deposition of a film from protein or Fig. 13.29. Deposition of a film from latexparticulate suspension over a moving particle suspension over a moving horizontal horizontal substrate with the help of an substrate with continuous supply of immobile vertical plate: sketch of the suspension through a channel of width 7 mm experimental setup used in Ref. [42]. between two parallel plates: experimental setup used in Ref. [43]. and respectively, the protein is negatively charged. The latter effect could be attributed to the electrostatic repulsion between the negatively charged protein and the silicon substrate, which prevents irreversible protein-to-silicon attachment and promotes the arrangement of the molecules into a regular hexagonal lattice. A method, which allows a continuous supply and deposition of the suspension was developed in by Matsushita et al. [43]. The suspension was supplied by means of an extruder (Fig. 13.29), which was constructed from two parallel glass plates (25 • 25 mm); the width of the gap between them was 0.7 mm. After injection of suspension in the gap of the extruder, the glass substrate was translated horizontally with a constant speed, which was fixed at 5 lam/s in order to form a monolayer of latex particles. In the beginning an wetting film from the suspension is deposited. During the drying of this film the latex particles are ordered into a hexagonal lattice by the attractive lateral capillary forces [43]. In the specific experiment a mixture of fluorescent and non-fluorescent latex particles of equal diameters, 1.00 _+0.05 lain was used. The quality of the obtained 2D ordered arrays was found to increase with the increase of the particle charge, which can be attributed to the enhanced particle-substrate and particle-particle repulsion. In a subsequent experimental study Matsushita et al. [181] applied their technique to create large

566

Chapter 13

2D arrays of ordered SiO2 particles, which were further used as a matrix for fabrication of mesoporous structured TiO2 films exhibiting photocatalytic activity.

13.7.

2D CRYSTALLIZATION OF PARTICLES IN FREE FOAM FILMS

13.7.1.

ARRAYS FROM MICROMETER-SIZED PARTICLES IN FOAM FILMS

In the previous sections of this chapter we considered two-dimensional structuring of particles in wetting films formed on substrates. Particle structuring can be realized also in free liquid films formed between two gas phases (foam films). Experimental studies on particle structuring in such films revealed the features and advantages of this approach to the obtaining of 2D arrays [44-47]. In Ref. [47] the foam films were formed from a suspension of latex particles having diameter 7 ~tm and bearing a negative surface potential (about - 7 0 mV). The processes of particle agglomeration and ordering were observed by optical microscopy. To form the films, the liquid was gradually ejected from a cylindrical capillary in the experimental cell invented by Scheludko & Exerowa [182,183], see Fig. 13.30. During the measurements the experimental cell was kept in a closed environment saturated with water vapor, which prevented the evaporation from the film surfaces. After the formation of a liquid film, its thickness progressively decreases because of a continuous outflow of water from the film towards the encircling Plateau border. It was established that the nature of the dissolved surface-active agent, used to stabilize the thin film, has a crucial impact on the observed phenomena [47]. In some experiments the films were stabilized by the addition of 0.016 M sodium dodecyl sulfate (SDS), which is an anionic surfactant. Experimentally it was observed that although the thick films contained latex particles, in the course of film thinning all particles were pushed out of the film area [47], see Fig. 13.31a. Therefore, particle ordering was not detected in the foam film. The main reason for this negative result is the convective outflow of water from the film, which is not essentially resisted by the adsorption monolayers from SDS at the film surfaces insofar as SDS exhibits a low surface viscosity upon adsorption [184]. Moreover, both the latex particles and the film surfaces bear a negative surface charge, and consequently, repel eachother.

Two-Dimensional Crystallization of Particulates and Proteins

~

Air phase

3pquid"lm

Microscope objective

I

'

567

~'~,..OO

lateau border ~

Latex

Ok, 9

, 9

Excesswater phase ' " ."

(a)

NO

Capillary force

: ..

is

(b)

| Pressure control system

t

Fig. 13.30. Sketch of the experimental setup used Fig. in Ref. [47] to produce ordered 2D arrays from negatively charged spherical colloid particles in a freely suspended foam film.

0(~

(c)

13.31. Schematic description of the experimental results obtained in Ref. [47]: (a) with anionic surfactant (SDS) all particles are expelled from the film; (b) with protein (BSA) some particles are entrapped in the film and experience longranged attraction; (c) with cationic surfactant (DTAB, HTAB) large and wellordered arrays of particles can be formed.

In another cycle of experiments, to increase the surface viscosity the films were stabilized with the addition of 0.1 wt % protein, bovine serum albumin (BSA), instead of SDS [47]. In this case it was possible to entrap latex particles within the foam film. The captured particles tended to form small 2D aggregates; the latter were observed to attract each other from distances, which could reach up to 100 lam (Fig. 13.31b). This attraction can be attributed to the lateral immersion force, see Fig. 7.1f. At higher particle concentration the formation of array having "2D-foam" structure was observed. On many occasions the entrapped particles caused rupture of the otherwise stable film [47]. Best 2D arrays of negatively charged latex spheres were formed in films stabilized by a cationic surfactant,

like dodecyl-trimethyl-ammonium bromide (DTAB) and hexadecyl-

trimethyl-ammonium bromide (HTAB). In this case latex particles adhered to the sutfactant

568

Chapter 13

adsorption monolayers at the air-water surfaces. As a result, the particle mobility was much decreased and interfacial deformations appeared when the particles bridged the two film surfaces; the latter caused a strong interparticle capillary attraction (Fig. 7.1f). At a higher surfactant concentration (0.03 M DTAB) bulk aggregation of the latex particles was observed, which was undesirable. Best hexagonal 2D arrays were observed for an intermediate surfactant concentration, viz. 0.006 M DTAB. The ordering induced by the lateral capillary force started as a ring-shaped zone at the periphery of the circular film (Fig. 13.31c). By a series of consecutive increasing and decreasing of the pressure in the Plateau border (shrinking and expansion of the contact line) it was possible to rearrange the ring-shaped array into a large compact 2D array with excellent hexagonal packing, which covered the whole film area [47]. Part of the difficulties encountered in Ref. [47] are due to the fact that the cell with the foam film has been kept in a closed environment with saturated aqueous vapor, which has prevented the evaporation. Therefore, the predominant convective flux was directed outwards and pushed the particles out of the film; the only factor promoting the 2D aggregation in this case was the lateral immersion force. On the contrary, if evaporation from the foam film is present, it causes a radial water influx, from the Plateau border towards the center of the film. As discussed in section 13.3.1 the influx brings new particles within the film and in combination with the capillary force leads to a 2D crystal-growth. The latter mechanism (combined convective flux and capillary force) has been employed by Denkov et al. [44-46] to obtain two-dimensional crystals of sub-lain particles in free foam films; see the next section.

13.7.2.

ARRAYS FROM SUB-MICROMETER PARTICLES STUDIED BY ELECTRON CRYOMICROSCOPY

The 2D arrays of sub-lam particles cannot be detected by usual optical microscopy. For that reason a special experimental technique was developed in Refs. [44,45] to accomplish observation of structures in free foam films by means of electron microscopy. At a given stage of the experimental procedure the foam film was frozen by a quick immersion into a cooling liquid, which was 1:1 mixture of ethane and propane at -190~

Such a sudden cooling led to a

vitrification of the aqueous films, that is formation of amorphous ice. (The appearance of any

569

Two-Dimensiona I Co, sta lliza tion of Pa rticu la tes and Prote ins

3 2

___ ~1///

/--

(b)

Fig. 13.32. (a) Photograph of the disk-shaped holder made of copper, which was used by Denkov et al. [44-46] to produce and cryo-vitrify 2D arrays in free foam films. The foam film is created in the central hole (the dark circle in the center of the holder), which is first loaded with a droplet of particle suspension; a glass capillary attached to the holder (on the right) is used to suck out liquid from the loaded droplet through a narrow slit. (b) Thus a free foam film (1) is formed, which is encircled by a meniscus border (2) attached to the inner edge of the holder (3). ice crystals would damage the structures formed within the foam film.) In these experiments a free foam film of relatively large diameter (= 100 gm) was created in the central hole of a diskshaped copper holder (Fig. 13.32a); the latter was 60 lam thick and had an outer diameter 3 mm. The central hole was connected by a narrow channel with the periphery of the holder. During the film formation and the subsequent thinning the tip of a glass capillary was attached to the outer end of the channel. The capillary was connected to a microsyringe, which supplied the investigated suspension for creating a film in the hole (Fig. 13.32b), and allowed for the precise control of the film diameter. The thickness of the foam film was determined by means of an optical interferometric method with accuracy of about +1 nm [44,45]. The formation of 2D arrays in free foam films is based on the same mechanism as the 2D crystallization in wetting films (Section 13.3.1), i.e. on the combined action of capillary forces and evaporation-driven convective flux, see Fig. 13.33. Initially, a relatively small film (diameter = 30 gm) is created and then the contact line encircling the film is gradually expanded up to diameter 100-150 lam. In this way, an ordered monolayer is formed within the film if the particle concentration in the bulk suspension (about 10 vol%), the rate of film expansion (5-101am/s), and the evaporation rate are appropriately chosen. If the optimal

Chapter 13

570 Evaporation

.

.

.

.

.

.

.

.

.

.

.

.

.

Liquid film Meniscus region Fig. 13.33. Sketch of the process of particle ordering in an evaporating free foam film. A flux of particles directed from the meniscus region toward the film appears as a result of the water evaporation from the film region. The particles-newcomers are arrested at the periphery of the growing array by the attractive capillary immersion force. regime is violated, the quality of the formed array worsens. For example, at lower particle concentration and/or at a faster expansion of the contact line one obtains films deprived of particles. On the other hand, a higher particle concentration and enhanced evaporation result in thick films containing bilayers and multilayers [45]. After the gradual expansion of the contact line and the (hoped for) production of 2D array within the film, the glass capillary is detached from the holder with the film (Fig. 13.32a) and the latter is plunged for vitrification into the 1:1 liquid mixture of ethane and propane cooled by liquid nitrogen. Then the vitrified film is transferred into the electron microscope by means of a cryo-transfer holder. The vitrified foam films are rather fragile and any mechanical disturbance easily brakes them. For that reason the safe transfer of the frozen films into the electronic microscope is the most delicate step in these experiments. With this procedure 30% of the films thicker than 40 nm have been safely transferred into the microscope. In several experiments even films of thickness 30 nm have been transferred intact. It is expected that by means of appropriate automation the observation of films as thin as 20 nm will become attainable [44,45]. Using this method ordered monolayer of polystyrene latex spheres (66 and 144 nm in diameter) have been obtained [44,45]. It is curious to note, that the vitrified aqueous films containing ordered monolayer of latex particles is much more fragile than a film of the same thickness but without latex particles. As a rule, the latex-containing frozen films brake along some of the

Two-Dimensional Crystallization of Particulates and Proteins

571

Fig. 13.34. Ordered monolayer of bacteriorhodopsin vesicles formed within a free liquid film of thickness 59 nm; the determined average radius of the vesicles is R0 = 18.5 nm with a standard deviation AR = _+ 1.1 nm, which is due to their polydispersity; the inset shows an ordered bilayer of vesicles; micrographs taken by Denkov et. al. [45] with the help of cryovitrification of the foam film and electron microscopy. axes of the hexagonal array [45]. In other experiments [44-46], vesicles (diameter -- 39 rim) made of the membrane protein bacteriorhodopsin and lipids were ordered into large hexagonal monolayers or bilayers, see Fig. 13.34. The structure of these fine vesicles cannot be investigated by the negative staining method because they are easily deformed and even totally destroyed by the staining agent. For that reason the electron cryomicroscopy is an appropriate method to investigate such soft and delicate particles. By controlling the rate of evaporation from the film (Fig. 13.33) large ordered domains (thousands of particles) have been formed. The averaged value of the vesicle radius from many studied samples was R0 = 18.5 nm with a standard deviation AR = 1.1 nm, which represents a polydispersity of 6 %. Analyzing the electronic micrographs, such as Fig. 13.34, it is possible to determine the bending elasticity of the vesicle membrane

[46].

Indeed, these vesicles

are formed

spontaneously: purple membrane is subjected to the action of surfactant (octylthioglucoside), which dissolves selectively lipids from one of the two membrane constituent lipid monolayers. This gives rise to a spontaneous curvature, whose radius can be identified with the

572

Chapter 13

experimental mean value R0. Then with the help of the fluctuation theory [185] one can derive a Gaussian size distribution of the vesicles [46]:

C(R) = C(Ro) exp - - ~

R0

,

4~kt R~

(13.17)

where C(R) is the concentration of vesicles of radius R and kt is the total bending elastic modulus of the membrane. Then from the experimental value of the standard deviation AR, determined from Fig. 13.34, Denkov et al. [46] calculated kt = 0.87 x 10-19 J. (For lipid bilayers kt is usually of the order of 10-19 J, see e.g. Eq. (10.52) above.)

In conclusion, free foam films combined with the described vitrification cryo-technique can serve as a method for obtaining and investigation of 2D arrays from colloidal particles. This method ensures the excellent structure preservation of delicate hydrated molecular complexes and fine vesicles.

13.8.

APPLICATION OF 2 D ARRAYS FROM COLLOID PARTICLES AND PROTEINS

13.8.1. APPLICATION OF COLLOID 2D ARRAYS IN OPTICS AND OPTOELECTRONICS

The first studies of colloid crystals from latex particles revealed that they exhibit color diffraction patterns and it was suggested these crystalline arrays to be used as spectroscopic

diffraction gratings [18]; see Ref. [186] about more recent results. It should be noted that the precious opals, mined mostly in Hungary and South Australia, show brilliant colors due to the same diffraction effect. The colors in the opals are caused by a regular array of silica particles, whose diameter and spacing determine the color range of an opal. Similar hexagonal arrays of particles determine the coloring of the wings of the Morpho butterfly [ 187, 40, 41 ]. Moreover, ordered multilayers from transparent spherical particles exhibit interference colors [36], see Table 13.2. It is interesting to note that this effect has been utilized in the eye of the night moth, which is covered with a 2D array of 150 nm particles. This ordered monolayer allows the moth eye to remove light reflections making the most of the light available in the dark night. This example suggests application of coatings from particulate arrays for production of antireflection optical surfaces and elements.

Two-Dimensional Crystallization of Particulates and Proteins

573

Hayashi et al. [188] demonstrated that the separate particles from a monolayer of lam-sized latex spheres produce imaging of an object in a fashion similar to that of a compound eye. It is interesting that these small particles are fulfilling finely a lens function in a microfield beyond the range of the unassisted eye [188]. The usefulness of ordered arrays from latex spheres as a standard for determining the magnification and shadow-casting angle in electron microscopy was reported by Backus and Williams [17] and discussed by Gerould [19]. Such standards can be also serviceable for the test of novel observation schemes, like that reported in Ref. [ 189], aimed at improvement of the contrast and the point resolution of the electron microscopy. In addition, colloidal nanostructures can be used as test samples for optical nearfield microscopy [190,191]. A promising application of ordered arrays from semiconductor nanocrystallites is to utilize them as a photo- and electro-luminescent material [192-196]. This application stems from the fact that some fundamental properties of the solids are characterized by length scales on the order of 1 to 20 nm; if a crystal has at least one of its dimensions smaller than the length scale of the same property, then that property is "confined" and becomes dependent on the size and shape of what is now called a "quantum crystal" [ 196]. As a result, such quantum crystals may exhibit fundamentally new properties. For example, sufficiently small (~ 2 nm) silicon crystallites can become a very efficient photo- and electro-luminescent material, despite the fact that the macroscopic Si crystals are not a useful optoelectronic light source [196]. For device-oriented applications such nanocrystallites have to be assembled into superlattice structures termed also quantum dot lattices [193, 197, 198].

13.8.2. NANO-LITHOGRAPHY, MICROCONTACT PRINTING, NANOSTRUCTURED SURFACES

As already mentioned, technological applications demand fabrication of small surface structures approaching dimensions of a few nanometers. The widely used photo-lithographic techniques have some limitations [199, 200]. A new, complementary technique has been proposed, which employs 2D arrays of sub-lam colloid spheres as masks for etching or vacuum deposition. Such masks are suitable in applications, in which a periodic arrangement of the structures is required. The latter technique has been termed "natural lithography" [55] or

574

Chapter 13

Fig. 13.35. Schematic description of the colloid-monolayer lithography (after Burmeister et al., Ref. 199): (a) droplet of suspension is spread and dried on a given substrate to obtain an ordered particle monolayer; (b) metal is deposited onto the particle monolayer; (c) regularly arranged metal structures are found on the substrate after removal of the particles. "nanosphere lithography" [58]. Its principle is illustrated in Fig. 13.35. First an ordered 2D array of particles is formed on a substrate by evaporation of a suspension film (see Section 13.3). Then metal is deposited using the 2D array as a deposition mask. For example, Fischer and Zingsheim [201] formed an array from sub-lam polystyrene spheres on a glass substrate. Then a vacuum deposition of platinum led to the formation of triangular platinum spots on the substrate, which are located in the gaps (interstices) among the hexagonally packed polystyrene spheres. The spheres were finally removed by sonification in benzene. Likewise, Deckman and Dunsmir [55] formed a hexagonal array of 400 nm polystyrene spheres and evaporated silver to form triangular silver posts on the substrate; next, the spheres were dissolved in methylene chloride. Alternatively, these authors used a monolayer of latex particles as an etching mask. A hexagonal 2D array from the polystyrene spheres was assembled on a silicon wafer. Then the silicon was etched with a 500-eV ion beam produced from CF4 gas discharge. Finally, the polystyrene particles were removed from the apex of the

Two-Dimensiona I Crysta lliza tion of Pa rticu la tes and Prote ins

575

silicon posts with a 500-eV oxygen ion beam. Morphology of the individual posts can be altered by changing the reactivity of the ion beam and/or varying the endpoint of the etching process [55,57]. Using hexagonal polystyrene monolayers as lithographic masks Boneberg et al. [202] found out two other types of structures, in addition to the aforementioned triangular posts. These are hillocks (nano-dots) representing residual material from detached latex particles, and nanorings found around the original location of the polystyrene spheres on the substrate. The hillocks appear as a consequence of the strong adhesion of the spheres to the surface, while the rings originate from organic solutes in the aqueous suspension, which are deposited after the evaporation of water. It was suggested to use these structures as fluorescent dye rings of submicron size [202]. Burmeister et al. [199] investigated the possibility to induce crystal symmetries other than simply hexagonal. To achieve this a prestructured substrate could be used. By means of optical lithography silicon grids were fabricated as substrates. A periodic series of parallel trenches were created on the silicon surface; depending on the period of this grid, hexagonal or quadratic packing of deposited particles was obtained. Xia et al. [200] described several methods that generate patterned relief structures for casting the elastomeric stamps needed in m i c r o c o n t a c t printing. The stamps, made from polydimethylsiloxane (PDMS), can be used in combination with selective wet etching to generate patterns in thin films of silver and gold. The schematic procedure for preparation of stamps with the help of 2D array from polystyrene spheres (diameter = 200 nm) is shown in Fig. 13.36. First the 2D array is formed by evaporation from a latex suspension film (see Section 13.3). Then the particle array is covered with PDMS; finally, two types of stamps can be prepared (Fig. 13.36), which produce patterns complementary to each other [200]. Semiconducting materials having surface pores of size from tens to hundreds of nanometers, which are called n a n o s t r u c t u r e d or m e s o s t r u c t u r e d m a t e r i a l s , have recently found various technological applications. Some examples are photo-electrochemical solar cells [203-205], high-performance photocatalytic films [206], sensors [207] and electrocatalysts [208-210].

576

Chapter

Assemble polymer beads

{ ~ ~ ~S~

...........

olystyrene beads

,,.'j .,. f

Cast PDMS

, y....r

1

PDMS (Type I stamp)

~ !i ...........

TiO "

1

~

/'f

r

~Annealing

CUI'X'X'X3 S

.....f

Anatase ~ microcrystal ~

PDMS

~

.,.'J

Two-dimensional array of SiO 2

Cover glass

~ s i ~

]

/..J

13

.,.-~"~ Ni substrate

PDMS

OZxExEXExD (Type II stamp)

Fig. 13.36. Scheme of the procedure used by Xia et al. [200] for casting PDMS stamps from 2D array of polystyrene microspheres assembled on a planar Si substrate.

~s

~

TiO~ film on the SiO 2 two:dimensional array by a spray-pyrolysis teclmxque

" Sample was up-side~; ~;<.•215215 down on Ni substrate

~HF treatment Nanostructured TiO2 surface

Fig. 13.37. Scheme of the procedure used by Matsushita et al. [213] for fabrication of nanostructured TiO2 surfaces with the help of ordered 2D arrays of SiO2 spheres.

Mesoporous TiO2 materials, which exhibit photocatalytic properties, have been obtained by replication of 3D latex particle arrays produced via ultracentrifugation [211 ] or filtration [212]. Matsushita et al. [213] produced a new nanostructured TiO2 surface by using a 2D array-based template and examined its photocatalytic activity; see also Ref. [181]. Their procedure for surface fabrication is schematically presented in Fig. 13.37. First, a 2D array from SiO2 particles (530 nm in diameter) is created on a glass plate by means of the method of continuous deposition depicted in Fig. 13.29, Section 13.6.2. Second, the 2D array is covered with a micrometer-thick TiO2 film, which is deposited by a spray-pyrolysis technique. Third, the TiO2 film was fixed to a Ni substrate by epoxy for preservation of the obtained fragile structure. Next, the glass plate and the SiO2 particles were dissolved by treatment with HF solution, which does not dissolve TiO2, epoxy or Ni. The photocatalytic activity of the produced mesostructured surface was tested by means of photoreduction of Ag + to produce silver particles [ 181].

Two-Dimensional Crystallization of Particulates and Proteins

577

13.8.3. PROTEIN 2D ARRAYS IN APPLICATIONS

As already mentioned, one of the major application of protein 2D arrays is in electron microscope crystallography for determining the tertiary structure of the protein molecules

[146-152, 214]. A biomedical application, which may find a wide usage, is to employ 2D arrays as immunosensors and immunoassays [ 134-137]. Immunosensors represent a particular type of biosensor that are based on the recognition

properties of antibodies (usually globular proteins). By means of the protein-engineering techniques (hybridization and cloning) it is possible to produce antibodies capable of recognizing and binding almost any type of molecule or antigen; for more details see Refs. [215,216]. As an example, in Fig. 13.38 we present a sketch of an optical immunosensor. It can be based on the total internal reflection fluorescence (TIRF) [217, 218] or on the surface plasmon resonance (SPR) [219]. In the TIRF-based immunosensors a 2D array of antibodies is deposited on the surface of an optic fiber. After the binding of fluorescent antigens to adsorbed antibodies, one can register the fluorescence excited by the evanescent-light wave, which is located in a close vicinity of the solid substrate. The experimental studies have indicated picomolar

to

micromolar

detection

limits

of

this

method

[215,218].

Gravimetric

immunosensors, based on surface acoustic wave, have been also developed; they employ a protein 2D array adherent to the surface of a quartz piezocrystal [220].

Optical Immunosensor ~" ~: Aqueous~solutions~ Antibo% ~

~r

Fluorescent +// antigen ~"

Guided optics

4

Data acquisition & analysis

Fig. 13.38. Sketch of an optical immunosensor based on TIRF technique; after Ref. [215].

578

Chapter 13

Several applications of natural ordered protein 2D arrays, present in the membranes of numerous kinds of bacteria, have been proposed. These most commonly observed bacterial cell surface structures, termed S-layers by Sleytr [221 ], are composed from a single protein species; they completely cover the bacterial cell surface and exhibit either oblique, square or hexagonal lattice symmetry. The interaction, which assembles the protein molecules into a S-layer, is due to hydrogen bonding. Consequently, a complete disintegration of S-layers into their constituent subunits can be achieved by a treatment with H-bond breaking agent, such as guanidinium hydrochloride. After removal of the disintegrating agent (e.g. by dialysis), the protein subunits are able to recrystalize into 2D arrays either in the bulk of solution or on interfaces [223]. Most of the S-layers are 5-15 nm thick and possess identical pores of size from 2 to 6 nm. This determines their potential application as isoporous ultrafiltration membranes, which exhibit very sharp molecular exclusion limits. To produce S-layer ultrafiltration membranes, the respective protein 2D arrays are deposited on the surface of commercial microfiltration membranes of pore size < 0.5 lain [222, 223]. S-layers can be applied also as supports for Langmuir-Blodgett films and covalent attachment of macromolecules, or as carriers of artificial antigens for vaccine development [222]. Finally, we arrive at the most challenging application of 2D arrays, the molecular electronics, which is still in its early age. During the last decades, the technology revolution in microelectronics was based on the fact that the density of device components, packed into a single integrated circuit, had grown exponentially with the passage of time. Extrapolating this tendency one could conclude that the device size would eventually reach the atomic scale. However, the experts believe that the latter is impossible to accomplish because of limitations imposed by quantum and thermal effects [224-226]. The most advanced machinery, the living organisms, operate with functional elements having molecular dimensions. Accordingly, the accessible limit of miniaturization seems to lie in the employment of structural units and principles of architecture used by the living organisms. The performance of an animal or human brain, in comparison with the most advanced silicon based computer, is a living proof that the approach of molecular electronics is justified [226].

Two-Dimensional Crystallization of Pa rticulates and Proteins

579

Protein 1 protein 2

Accepto.r,r

Electrode

j/

Donor

Electrode

Fig. 13.39. Scheme of a protein molecular diode; after Ref. [231]. In his fine review Hong [226] notes that instead of being viewed as revolutionary technology threatening to replace conventional microelectronics, molecular electronics is to be more accurately viewed as complementary at the present stage of development. The biomaterials are mechanically delicate and chemically unstable. They easily denature and lose their functions. The living organisms operate on the principle of continuous self-repair and self-assembly, which is not attainable by the present technologies. A reasonable approach is to advance stepby-step with the development of biomolecular devices and prototypes, which at the first time may seem as clumsy hybrids between the devices of silicon-based microelectronics and the appliances of living organisms. Proteins are the major building blocks of many functional units in the biological cells and represent important materials for building biomolecular electronic devices. For example, Aviram [227] proposed a molecular rectifier, in which different functional parts of a single protein molecule play the role of an electron donor, an insulator and an electron acceptor. A variety of designs of molecular electronic devices have subsequently appeared, exploiting either the electronic or the photonic properties of proteins [4-6, 228-231]. For example, Fig. 13.39 shows a sketch of a molecular diode (rectifier). It is composed of two protein monolayers, which possess different electronic energy levels at their electron transfer sites. Each protein monolayer is electronically linked to the surface of a supporting electrode. Thus

580

Chapter 13

the electron flux between the pair of electrodes is rectified by the sandwiched protein bilayer [231]. One of the first biophotonic achievements is the development of the "Biochrom" films, which are thin-film-optical-memory devices involving the stable protein bacteriorhodopsin (bR), which is embedded into an appropriate polymeric matrix [232, 233]. These films are able to record optical information upon light exposure, to store it permanently or reversibly; they exhibit high spatial resolution, photosensitivity and cyclicity [226]. Mutated bR films have been used as holographic media [234, 235]. Miyasaka et al. [236] deposited a bR containing film on a SnO2 conductive electrode by the Langmuir-Blodgett method. Then the bR film was put into contact with a layer of aqueous electrolyte gel and with a second gold electrode. Thus an electrochemical sandwich-type photocell was constructed, with a junction structure SnOz/bR/electrolyte/Au. Irradiated with visible

light, this photocell

produced

rectified photocurrent

with unique

differential

responsivity to light intensity. Further, an artificial photoreceptor, comprising a pixel network of such bR photocells, was fabricated [236]. To ensure better orientation of the bR molecules into light-sensing photoelectric devices, antibodies were also involved in the composition of the photocells [237]. Hartley et al. [238] determined the surface potential of two-dimensional bR crystals at various pH and ionic strengths by using a silica sphere, attached to the cantilever of an atomic force microscope (AFM). A review on the photovoltaic effects in bR-containing and other biomembranes can be found in Ref. [239].

13.9.

SUMMARY

The attachment of particles to an interface, accompanied with the action of lateral interparticle forces, often leads to the formation of ordered two-dimensional arrays (2D colloid crystals) from particulates or protein macromolecules. Several methods for producing such 2D arrays can be distinguished. The most widely used approach is to evaporate the solvent from a film formed from a suspension, which contains colloidal particles or macromolecules (Section 13.1.1). The decrease of the film thickness (due to the evaporation) forces the particles to enter the liquid interface, which automatically "switches on" the powerful attraction due to the capillary

Two-Dimensional Crystallization of Particulates and Proteins

581

immersion force. The latter collects the particles into ordered aggregates, which further grow owing to an evaporation-driven convective influx of new particles. This method has been applied by many authors to obtain 2D crystals from proteins, viruses and various colloids. Particle structuring is possible even in the absence of any attractive force. At sufficiently high concentration the Kirkwood-Alder phase transition occurs with both 2D and 3D systems of particles experiencing only electrostatic and/or hard-core repulsion (Section 13.1.2). Colloidal particles, which are attached to a single interface (instead of being confined into a liquid film), are not subjected to the action of the capillary immersion force, but they can also form ordered aggregates assembled by either the capillary flotation force, van der Waals attraction or Lucassen's capillary force (Section 13.1.3). Applied external electric, magnetic or optical fields can bring colloid particles to an interface and can force them to form ordered 2D arrays (Section 13.1.4). Dense monolayers of adsorbing particles acquire hexagonal packing due to the area-exclusion effect and/or attractive interparticle forces. Interfacial monolayers from insoluble molecules or particulates can be compressed to form ordered 2D arrays with the help of a Langmuir trough and can be further transferred onto a solid substrate by means of the Langmuir-Blodgett technique (Section 13.1.5). The chapter is devoted mostly to the practical methods for preparation of particle and protein 2D arrays in evaporating liquid films. The method of the mercury trough has been applied to crystallize dozens of proteins (Section 13.2). Experiments revealing the mechanism and kinetics of 2D crystallization in evaporating films are considered in details (Section 13.3). The occurrence and advantages of 2D array formation over a liquid substrate (fluorinated oil or mercury) are described in Section 13.4. During the process of 2D crystallization a size separation of the particles occurs, which is an evidence about the action of capillary immersion forces (Section 13.5). Methods for obtaining large 2D-crystalline coatings (plate withdrawal and "brush"/extruder methods) are described in Section 13.6. Ordered particle 2D arrays can be produced also in free foam films. To observe the arrays by electron microscopy such foam films are frozen (vitrified) by a quick immersion into a cooling liquid; this electron cryotechnique ensures an excellent structure preservation of delicate vesicles or hydrated molecular complexes (Section 13.7).

582

Chapter 13

The 2D arrays from colloidal particles find various applications in optics and optoelectronics (Section 13.8.1), in nano-lithography and microcontact printing, in the development of methods for fabrication of nanostructured and mesostructured surfaces for catalytic films and solar cells (Section 13.8.2). Protein 2D arrays are used for investigating molecular structure by electron microscope crystallography, for preparation of immunosensors and immunoassays, for fabrication of extremely isoporous ultrafiltration membranes, for creation of bioelectronic and biophotonic devices, etc. (Section 13.8.3).

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591

CHAPTER 14 EFFECT OF OIL DROPS AND PARTICULATES ON THE STABILITY OF FOAMS

In some cases the foaminess is desirable, while in other cases it is not wanted. The attachment of ~m-sized oil droplets and/or solid particles to fluid interfaces has a foam-destabilizing effect, which 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. Direct observations show that when the foam decay is slow, the antifoam particles are expelled from the foam films into the Plateau borders, where the particles may enter the air-water interface and destroy the neighboring foam cell(s). In contrast, when the particles exhibit a fast antifoaming action, they are observed to directly break the foam films, which thin much faster than the Plateau borders. Three different mechanisms

of antifoaming

action have been established:

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. Criteria for spontaneous occurrence of entering, spreading and bridging have been proposed. The experiment shows that the key determinant for antifoaming action is the stability of the asymmetric particle-water-air film. Repulsive interactions in this film may create a high barrier to drop entry. The major thermodynamic factors, which stabilize the asymmetric film, are the disjoining-pressure barriers due to the double layer repulsion, steric polymer-chain interaction, and oscillatory structural forces. In addition, there are kinetic stabilizing factors, such as the surface elasticity and viscosity, which damp the instabilities in the liquid films. On the other hand, a 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. Solid particulates of irregular shape, adsorbed at the oil-water interface, have a "piercing effect" on the asymmetric oil-water-air films. The evaporation from a foam can also help for overcoming the disjoiningpressure barrier(s). The accumulated knowledge about the mechanisms of foam destruction enables one to give definite predictions and prescriptions concerning the foam stability after a careful examination of the factors operative in each specific case.

Chapter 14

592

14.1.

FOAM-BREAKING ACTION OF MICROSCOPIC PARTICLES

14.1.1.

CONTROL OF FOAM STABILITY; ANTIFOAMING VS. DEFOAMING

Foams have many applications in industry and in every-day life. In some cases the formation of foams is desirable (under a certain control); such are the applications in personal-care and house-hold detergency, in fire-fighting, ore flotation, foods and drinks. On the other hand, in many cases the spontaneous formation of foams is not wanted insofar as it hampers the efficient operation of industrial processes such as paper pulp processing, paper making, polymer, sugar and food processing, textile dyeing and scouring, wastewater treatment, fermentation in pharmaceutical, food and chemical technologies, phosphoric acid production, gas-oil and distillation separation processes in petroleum industry [1-6]. It has been established that microscopic oil droplets or solid particles [7-11], and their combination [4-6, 12-16], may exhibit a foam-destructive effect. For example, small oil droplets dispersed in shampoos provide hair conditioning; special surfactant compositions have been invented to protect the foams from the destructive action of the droplets [17]. In other cases, very active antifoams are intentionally added to suppress the development of unwanted foamability during industrial processes [18]. In both cases one could achieve the desired effect utilizing the knowledge about the foam-breaking action of microscopic oil drops and particulates. Often the investigators distinguish between defoaming and antifoaming as two different methods for foam destabilization [5, 6]. In defoaming first the foam is formed and then the foam-breaking agent is added onto the foam. In antifoaming first the foam-inhibiting agent (antifoam) is dispersed in the foaming solution, and then foam is produced, which is less stable and less voluminous because of the action of the agent.

Defoamers added on the top of an existing foam (Fig. 14.1 a) begin to break the foam films one after another. Water-insoluble alcohols (like octanol) are good defoamers but ineffective as antifoams [6,19]. A possible mechanism of action of oily defoamer droplets is that they spread on one of the two surfaces of the foam films thus creating asymmetric (water-oil) films, which exhibit a kinetic instability during the process of oil spreading. The mechanism of defoaming action of hydrophobic solid particles could be attributed to the fact that they quickly adsorb

Effect of Oil Drops and Particulates on the Stability of Foams

593

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Fig. 14.1. (a) Defoamer particles are added on the top of existing foam; they break the foam films one after another. (b) Antifoam particles are contained in the solution before the production of foam; they are drawn into the newly created loam, which can be obtained by circulation of solution, as depicted in the figure. surfactant molecules from the water-air interfaces in the foam thus creating destructive local chemical shocks [5, 12]. Note that the defoamer particles approach the air-water interfaces from the side of the air phase. Antifoam particles, initially contained in the solution (Fig. 14.1b), approach the air-water interfaces from the side of the water phase. Sometimes (as in the case of shampoos) their antifoaming action is an undesirable side effect; in other cases they are added on purpose. As already mentioned, three types of particles are known to exhibit antifoaming action [5, 6]: (i) nonpolar oils, which can be silicone or organic, including nonionic surfactants above their cloud point and some fatty esters; (ii) hydrophobic solid particles: hydrophobized silica, microcrystalline waxes, hydrophobic polymers, etc.; particles of irregular shape might have a strong antifoaming effect; (iii) mixtures of nonpolar oils and hydrophobic solid particles: in combination their foambreaking activity increases synergistically. As mentioned above, in the present chapter our attention is focused on the mechanism of action

594

Chapter 14

of fluid and solid particles, which exhibit antifoaming performance. The latter is related to the attachment of the particles to the surfaces of the foam films and/or the adjacent Plateau borders.

14.1.2. STUDIES WITH SEPARATE FOAM FILMS

Before considering the mechanisms of antifoaming action (Section 14.2), it is helpful to present some research methods and illustrative examples. The process of foam generation involves the formation of liquid films and the drainage of liquid out of the films, along the Plateau borders [20,21]. The antifoaming particles migrate throughout the foam driven by the flow of water. Model experiments with separate films can provide information about the stages of film thinning and particle migration [22,23]. A useful tool for investigating separate horizontal liquid films is the Scheludko cell [24,25], whose construction is shown in Fig. 14.2. Within such a cell one could form either foam films (gaswater-gas), emulsion films (oil-water-oil), or asymmetric (oil-water-gas) films. It allows one to measure the lifetime of the separate foam films, the variation of film thickness, contact angle and capillary pressure during the process of film drainage. Higher capillary pressures can be achieved with the experimental cell of Mysels [26], in which the ejection of liquid is realized by means of a porous glass plate. In particular, the film thickness can be measured by means of the micro-interferometric method, see e.g. Refs. [27-30].

Fig. 14.2. Sketch of the experimental cell constructed by Scheludko and Exerowa [24,25]. First the cylindrical glass cell is filled with Fluid 1; next it is immersed in Fluid 2; then a portion of Fluid 1 is ejected from the cell through the orifice in the glass wall. Thus in the central part of the cell a liquid film is formed, which is encircled with a Plateau border.

595

Effect of Oil Drops and Particulates on the Stabilio, of Foams

I rectangular frame

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@ light source Fig. 14.3. Sketch of an experimental cell for measurements with vertical films formed by pulling up a glass frame out of the surfactant solution. One can use a simple rectangular frame (up-right) or the "three-leg" frame constructed by Koczo and Racz [31 ]. The formed vertical films are observed with a horizontal microscope connected to CCD camera and video recorder. The area of the foam films has a significant effect on their stability. The films formed in the Scheludko cell are relatively small (less than a millimeter in diameter), while the films in the real foams can be larger. Moreover, the films in real foams are in contact with Plateau borders of triangular cross section, whose area decreases with time due to the water drainage out of the foam. The oil droplets can be trapped in the narrowing Plateau borders with subsequent drop entry at the surface and destruction of the neighboring foam films. The latter processes cannot be modeled in the Scheludko cell, where the Plateau border is not similar to that in the real foams. Vertical foam films of size several centimeters can be formed by pulling pull up a frame out of the surfactant solution [31,17]. A special frame with three "legs" was proposed in Ref. 31; it allows one to form simultaneously 3 vertical films, subtending angles of 120 ~ with each other and forming a vertical Plateau border, see Fig. 14.3. This configuration allows one

Chapter 14

596

( Stages of Thinning of Foam Film with Particles'"1

(a) Approach of two concave menisci

(b) Dimple formation

(c) Drainage of a thick film

Drainage of a planar thin film

~'----:7---7--~ (t) Stratification of thin film

Thin black film

Fig. 14.4. Schematic presentation of the main evolution stages of a foam film, which is formed in a Scheludko cell from surfactant solution containing gm-sized oil drops; after Refs. [ 17, 18]. to study the behavior of large foam films and the effect of oil drops on the stability of Plateau borders. The "three-leg" frame enables one to obtain information about the rate of film drainage, film lifetime, period of time needed for formation of black film, critical thickness of rupture (or equilibrium film thickness), etc. [17]. Aronson [32] observed that during the process of film thinning the antifoam particles quickly move out of the foam films and enter the Plateau borders, where they are carried along by the flow of the outgoing water. This was confirmed in the experiments of Wasan et al. [23] and Koczo et al. [33] with separate foam films, both horizontal and vertical. Figure 14.4 illustrates the consecutive stages of thinning of a horizontal liquid film initially containing oil droplets, as observed by Denkov et al. [17,18] in experiments with Scheludko cell. In the beginning a thick liquid layer is formed between the two approaching concave menisci (Fig. 14.4a). Next, due to the hydrodynamic interaction between the two liquid surfaces a thicker lens-shaped zone appears in the middle of the film, which is usually termed "dimple formation" [34], see Fig. 14.4b. Particles are observed both in the initial thick layer

Effect of Oil Drops and Particulates on the Stability of Foams

597

Fig. 14.5. Consecutive video-frames from Ref. [17] taken in light reflected from a thinning foam film, which is produced in a Scheludko cell from 0.1 M solution of the anionic surfactant sodium dodecyl-trioxyethylene-sulfate (SDP3S); additional non-amphiphilic electrolyte of ionic strength 0.066 M and 0.1 wt% silicone-oil droplets of average size 5 lam are also present. (a) Channel-like pattern formed after the outflow of the dimple; average film thickness h 550 nm. (b) Pattern at thickness 100 nm: small emulsion drops are seen (down-left and upright). (c) Appearance of stratification spots: four levels of step-wise thinning are seen; the number of micelle layers is denoted. (d) After the last step-wise transition, a black film is formed, which does not contain surfactant micelles. The scaling bar corresponds to 100 gm. and in the dimple. The latter is a unstable formation which soon disappears leaving some transient channel-like patterns (Fig. 14.5a). After that, an almost plane parallel film is formed, which initially may contain some oil drops (Figs. 14.4c and 14.5b), but they are soon expelled into the adjacent Plateau border driven by the water outflow from the thinning film (Fig. 14.4d). If the volume fraction of the surfactant micelles in the solution is high enough, one can observe step-wise thinning (stratification of the film), see Figs. 14.4e and 14.5c,d. Each of the steps (which appear as spots of different darkness) represents a liquid film containing a

Chapter 14

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6000

s

Fig. 14.6. Effect of oil drops on the decay of a foam column: experimental data from Ref. [17] for the foam volume vs. time. The foam has been produced in a setup like that in Fig. 14.1b from solution containing 0.1 M blend of 80 % anionic surfactant, SDP3S, and 20 % amphoteric surfactant, lauryl-amide-propyl-betaine. Stage I: drainage of water out of the foam and formation of foam cells; Stage H: narrowing of the Plateau borders at constant foam w~lume; Stage III: decrease of the foam volume due to destruction of foam cells by oil drops located in the Plateau borders; Stage IV: the antifoaming action of the oil drops is completed and the foam volume remains constant. Stages III and IV appear on the curve which corresponds to solution containing 0.1 wt% added silicone-oil emulsion of average drop-size 15 Jam. given number of micelle layers: from 4 down to 0 in Figs. 14.5c,d. Finally, a thin black film is formed, which contains neither antifoam particles nor micelles (Figs. 14.4f and 14.5d). Based on similar observations Koczo et al. [33] suggested that the rupture of foam cells does not happen by direct breaking of the foam films by the particles (at the stage depicted in Fig. 14.4c), instead, rupturing occurs when the particles become trapped in the thinning Plateau borders. This is typical for systems, in which the particles cause a slow foam decay (from 5 rain to hours). Figure 14.6 gives a typical example for such a system [17]: the initial decrease of the foam volume is due to the drainage of liquid out of the foam, Stage I in the figure; Stage II corresponds to slow narrowing of the Plateau borders at constant foam volume; further decrease of the foam volume, due to destruction of foam cells (Stage III), is observed if only a

Effect of Oil Drops and Particulates on the Stability of Foams

#

599

ir~

Fig. 14.7. Photograph of foam cells from Ref. [17]. Many oil drops entrapped in the Plateau borders can be seen. The experimental system is the same as in Fig. 14.6. sufficient amount of silicone-oil emulsion is present in the solution; finally, the antifoaming action of the oil drops is completed and the foam volume levels off at a certain non-zero value, Stage IV in Fig. 14.6. A microscope view of the same foam at Stage II shows the presence of silicone oil drops entrapped in the Plateau borders, see Fig. 14.7, where a train of many captured oil drops is seen. Observation with a higher magnification reveals that the drops are certainly strongly compressed, since the walls of the Plateau border have acquired a wavy shape [ 17]. A different consequence of events is observed in systems with f a s t antifoaming action (from seconds to minutes). For example, Denkov et al. [18] formed foam from solution of the surfactant sodium dioctyl-sulfosuccinate (AOT) in the presence of antifoaming oil emulsion: poly-dimethyl-siloxane (PDMS) containing 4.2 wt% hydrophobized silica particles. A quick destruction of the foam is observed with this system. The investigation of separate films formed in the Scheludko cell show that before the expulsion of the oil drops out of the thinning film, some of them can "bridge" the two film surfaces, which appears as a typical interference pattern, called by the authors the "fish eye", see Fig. 14.8. Soon after the appearance of such "fish eyes" the foam film ruptures. With the help of a high-speed video-camera it was

600

Chapter 14

oo

.......~

16

'594

~

~:

Fig. 14.8. Two video-frames taken by Denkov et al. [18] from a foam film formed in a Scheludko cell and observed in reflected light; interference fringes corresponding to zones of equal film thickness are seen. The foam films contain oil capillary bridges (shown by arrows), which look like "fish eyes" since they produce local concavities on the film surfaces. The films are made from AOT solution with 0.01 wt% added PDMS-emulsion; the latter contains 4.2 wt% hydrophobized silica particles established that the "fish eye" indicates the formation of an oil capillary bridge with "neck" (see Chapter 11). Under certain conditions such a bridge becomes unstable" it spontaneously stretches and ruptures [18, 35]" for more details - see below. In summary, the experiments show that the foam-breaking action of small oil drops can be located either in the foam films (Fig. 14.8) or in the Plateau borders (Figs. 14.6 and 14.7), depending on the specific system. It seems that the former situation is typical for systems with

fast foam decay (less than a minute), whereas the latter situation is characteristic for slowly decaying foams (with lifetime from minutes to hours).

14.1.3.

HYDRODYNAMICS OF DRAINAGE OF FOAM FILMS

A comprehensive review on hydrodynamics of drainage of liquid films can be found in Ref. [34]. At higher surfactant concentrations the liquid surface is occupied by a dense surfactant adsorption monolayer and it can be treated as

tangentially immobile. At lower

surfactant concentrations the hydrodynamic drag, due to the outflow of water, may create gradients of surfactant adsorption and surface tension, i.e. the

effect of Marangoni takes place.

In any case the fluid interfaces are deformable and their shape can change during the process of

Effect of Oil Drops and Particulates on the Stability of Foams

601

thinning. The quantitative theoretical description requires a rather complicated mathematical treatment to be used [34,36], which is out of the scope of the present chapter. It is instructive to consider a simpler case, which allows analytical solution. Imagine two planeparallel and tangentially immobile film surfaces, which approach each other with a given velocity u. Let us assume that the two film surfaces are circular disks (of radius R) as it can be with the liquid films formed in the Scheludko cell, Fig. 14.2. The drainage of the liquid out of the gap (of width h) between such two disks is a classical problem solved by Reynolds [37,38]. If the film is thin (h/R << 1) and the velocity of drainage is small (small Reynolds number), the Navier-Stokes equation, connecting the pressure P inside the film and the velocity v of the draining water, acquires the following simpler form, known as the lubrication approximation [37,38]: 6~2Vr

~

m

mm ~

6~Z 2

1 OP

o3P

Or'

Oz

77

(h/R << 1)

=0

(14.1)

Here 7"/ is the dynamic viscosity of the aqueous phase; the flow is assumed to have axial symmetry; r and z are the common cylindrical coordinates (along the radial and axial directions). Integrating Eq. (14.1), along with the boundary conditions at the film surfaces, V r [z=O -- O,

Vr

Iz=h= 0

(tangential immobility),

(14.2)

one obtains the distribution of the radial velocity across the film [37,38]: vr

"-

lOP

(Z2 -- hz)

(14.3)

20 o3r which is related to the radial distribution of the pressure inside the draining film, P(r). Further, we use the continuity equation for an incompressible fluid [38]"

CgV_____Lz. = CgZ

1 c~(rVr) r Or

(14.4)

For v~ one can impose the following boundary conditions" vz Iz=0= O,

vz Iz=h = - u

(14.5)

602

Chapter 14

i.e. the lower surface is immobile, whereas the fluid at the upper film surface moves downwards with a given velocity u. Next we substitute Eq. (14.3) into Eq. (14.4) and integrate between z = 0 and z = h using Eq. (14.5); the result reads [38]: r

Or Tr

- -12flu

r

(14.6)

Finally, we integrate Eq. (14.6), along with the boundary conditions P(r=-0) < ~,

P(r=R) = PPB,

(14.7)

to get the radial distribution of the pressure inside the film [37,38]: 3flu

P ( r ) = PPB + - - ~ (R 2 - r2)

(14.8)

Here R is the radius of the foam film and PPB is the pressure in the neighboring Plateau border. Equation (14.8) shows that the hydrodynamic pressure P(r) in the center of the film (r = 0) is the highest. Similar is the situation when the deformability of the film surfaces is taken into account [34]. The highest pressure in the central part of a draining film is the reason fi~r the appearance of a "dimple", see Fig. 14.4b. The above equations will be used in Section 14.3.3 to discuss the effect of evaporation on the breakage of foam films.

14.2.

MECHANISMS OF FOAM-BREAKING ACTION OF OIL DROPS AND PARTICLES

14.2.1. SCHEME OF THE CONSECUTIVE STAGES

As mentioned earlier, knowing the mechanism of antifoaming action of colloid particles one can control the stability of foams, which could be regulated in the two opposite directions: stabilization or destabilization. Many combinations of foam-stabilizing surfactants and foambreaking particles have been investigated.

Different authors have proposed different

mechanisms of antifoaming [4, 6, 18, 39-47]. Although there is no single universal mechanism, the accumulated experimental evidence implies the existence of several possible scenarios, which are summarized in Fig. 14.9. The transition from state A to state B (Fig. 14.9) corresponds to drop (particle) entry at the surface of the liquid film. In the case of oil drop it is possible a molecularly thin film of oil to

603

Effect of Oil Drops and Particulates on the Stability of Foams

MECHANISMS OF ANTIFOAMING ACTION Particles in thePlateau Border (slow antifoaming)

Particles in the Film (fast antifoaming)

(A)

@ particleentry1

=_

__~

rupture of asymmetric film

/

(D

(B)

(v)

(G)

monolayer 1 spreading

(c)

-,ql--

(E) spreading of thick layer

bridging "Q

"

ldewetting

(D) spreading mechanism

(K)

bridging-dewetting mechanism

I stretching (L)

--)@

bridgins mechanism

Fig. 14.9. Stages and transitions involved in the mechanisms of antifoaming action. The antifoam particle is schematically depicted as a sphere, but it can acquire a lens-shape if it is an oil drop. Stage A: particle in a foam film. Stage B: entry of a particle at the film surface. Stage C: spreading of a molecularly-thin oil film from an oil droplet. Stage D: spreading of a thick oil film (for positive spreading coefficient, S > 0), which leads to destabilization and rupture of the foam film. Stage E: bridging of the two film surfaces by an antifoam particle. Stage F: antifoam particle located in a Plateau border. Stage G: particle entry at one of the surfaces of the Plateau border. Stage H: particle entry at a second surface of the Plateau border. Stage K: dewetting of the formed "bridge" and rupture of the foam film. Stage L: Stretching of an oil bridge, which ruptures at its thinnest part and breaks the whole foam film.

604

Chapter 14

spread over (or among) the hydrocarbon tails of the adsorbed surfactant molecules: this is the transition B---~C in Fig. 14.9. The presence of molecular spreading can be established by measuring whether or not the surface tension exhibits a decrease after a small amount of oil is dropped on the surface of the surfactant solution. The propagation of the surface tension gradient, which accompanies the molecular spreading of oil, drives a water flow, which can locally accelerate the thinning of the foam film and can cause its destabilization and rupture. The molecular spreading (B---~C) could be followed by a spreading of a thicker oil layer over the surface of the foam film (C---~D). The latter leads to strong instabilities, manifested as irregular changes in the film thickness, which finally lead to rupturing of the film. The stages C and D (Fig. 14.9), followed by film rupture, represent the spreading mechanism of antifoaming action, proposed by Ross and McBain [39]; this mechanism has been observed and/or discussed by many authors [5,6,40-43]; for more details see Section 14.2.3 below. The drop, lens or particle entry at one of the film surfaces could be followed by a second entry at the other film surface due to the thinning of the foam film, i.e. bridging could happen. This is the transition B--~E in Fig. 14.9. The bridging could be accelerated by the spreading of oil monolayer, C---~E. In the case of hydrophobic particle the bridging can be followed by dewetting, step E----~K in Fig. 14.9, which eventually leads to breakage of the foam film. The sequence of stages E and K represents the bridging-dewetting mechanism, proposed by Garrett [44] and examined in many succeeding studies [4,15,23,45-47], see Section 14.2.4. In the case of bridging by oil drops another scenario is also possible and experimentally observed [18,35]. An oil drop, bridging the surfaces of a foam film, is not spherical; it acquires the shape of a capillary bridge (see Chapter 11). A capillary bridge with "neck", formed in a foam film, is usually unstable: the oil bridge begins to expand in lateral direction and to thin in its central part (stage L in Fig. 14.9), which soon leads to perforation of the bridge and rupture of the foam film. The sequence of stages E and L represents the bridging-stretching mechanism [18,35]. Note that oil capillary bridges with "haunch" (see Fig. 2.6) appear to be stable and do not break the foam film (unless oil spreading is also present). A criterion for determining whether or not the formation of an oil capillary bridge will cause rupturing of a foam film has been formulated by Garrett [45, 4] in terms of the so called bridging coefficient, see Eq. (14.11)

Effect of Oil Drops and Particulates on the Stability of Foams

605

below. In Refs. [18,35] it has been established that the action of the investigated strong silicone antifoam follows the route A---~B---~E---~L; see Section 14.2.5. As pointed out in Section 14.1.2, in many cases the antifoaming particles (solid particulates or oil drops) can "peacefully" leave the foam film with the draining water, without producing any destabilizing effect (A---~F in Fig. 14.9). Thus the particles are accumulated in the Plateau borders of the foam (stage F); see also Fig. 14.7. These particles could even have a transient foam-stabilizing action insofar as they hinder the water from leaving the foam through the Plateau borders [6]. In this way the drainage of water can be decelerated, but it cannot be prevented. As a result, the cross section of the Plateau borders progressively decreases, which brings about immobilization and pressing of the particles against the walls of the channel. This eventually leads to entry of some particle at one of the surfaces: see the transition F-+G in Fig. 14.9; the latter takes much longer time compared to the transition A---~B. As mentioned earlier, the transition A---~B (entry in foam films) is pertinent to f a s t antifoaming, whereas the transition F---~G (entry in Plateau borders) is typical for slow antifoaming. If the particle is a drop of oil, which can spread over the surface of the Plateau border and the neighboring films (transition G---~C in Fig. 14.9), then the film rupturing can occur following the spreading mechanism (C---~D). In fact, A--+F---~G-~C~D is found [17] to be the route for destruction of shampootype foams by silicone oil droplets, which are added as a hair-conditioning agent, see Section 14.2.3 for details. An alternative scenario is the transition G---~H to occur (Fig. 14.9), i.e. the particle confined in the Plateau border to enter two of its three surfaces, see Ref. [6]. In this way the particle actually enters the periphery of the foam film, to whom the two surfaces belong. The latter is equivalent to an act of bridging, which may lead to rupture of the foam film by means of the bridging-dewetting or bridging-stretching mechanisms (E--~K or E---~L in Fig. 14.9). If the particle is an oil droplet, after the first entry (stage G in Fig. 14.9) it acquires the shape of a lens. Then the bridging ( G ~ H ) can be realized as a coalescence of two oil lenses located at two different surfaces of the Plateau border. Such a mechanism of bridging has been observed by Wang et al. [48]. A necessary condition for this coalescence to happen is the oil-water-oil film, formed between two lenses upon contact, to be unstable. This can be achieved by the

606

Chapter 14

addition of small silica crystallites, which spontaneously occupy the oil-water interface of the lenses and promote the breakage of the oil-water-oil film [48]. After the lens coalescence, the formed oil bridge leads to rupturing of the foam cell; the exact route has not been detected; it could be the bridging-stretching mechanism (E---)L). Knowing the diversity of combinations between foam-stabilizing surfactants and foamdestabilizing particles, one could not rule out the possibility for the existence of stages, transitions or mechanisms different from those mentioned above and illustrated in Fig. 14.9.

14.2.2. ENTERING, SPREADING AND BRIDGING COEFFICIENTS

Having in mind the variety of scenarios for the occurrence of the antifoaming action, it would be helpful if some criteria can allow one to foresee which is the most probable mechanism for a given system. All mechanisms presented in Fig. 14.9 involve the particle entry at the air-water interface as a necessary stage. Robinson and Woods [49] proposed a criterion for drop entry in terms of the surface tensions of the air-water, oil-water and oil-air interfaces, (JAW, (jOW and (JOA, respectively: E > 0,

E

= (JAW "k- (jOW -- (JOA

(E - entering coefficient)

(14.9)

A sufficient condition for spontaneous spreading of oil over the air-water interface was formulated by Harkins [50]: S > 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

<0.

(14.15)

As usual, h denotes the film thickness, and PA is the pressure difference applied across the surface of the film. For example, if one of the film surfaces represents a liquid-gas interface one can write [62] PA =

Pg - Pl + 2CYlg/Rf

(14.16)

where PI and Pe is the pressure in the bulk liquid and gas phases, respectively; RU is the radius of curvature of the film surface, and o'tg is its surface tension (see Chapter 5). For a flat film (Rf-a r

one has simply

P A = Pg-

Pl. Note that for oil droplets captured in foams the

asymmetric oil-water-air films are curved and the term 2C~ig/Rfin Eq. (14.16) must be taken into account. The condition FI

=

PA means that at equilibrium the disjoining pressure 1-I

counterbalances the pressure difference

PA

applied across the film surface. In addition, the

condition OH~Oh < 0 guarantees that the equilibrium is stable (rather than unstable).

618

Chapter 14 (a) Electrostatic DLVO Barrier

(b) Multiple Barriers Due to Micelles FI

9

(1) 1--[rnax _

_

_

1111 ,

Fig. 14.17. Typical plots of disjoining pressure FI vs. film thickness h; PA is the pressure difference applied across the film surface; the equilibrium states of the liquid film correspond to the points in which I-I = PA. (a) DLVO-type disjoining pressure isotherm 1-I(h); the points at h = hi and h2 correspond to primary and secondary films, respectively; I--[maxis the height of a barrier due to the electrostatic repulsion between the film surfaces. (b) The presence of surfactant micelles (or other monodisperse colloidal particles) give rise to an oscillatory structural force between the two surfaces of a liquid film; the disjoining pressure isotherm 1-I(h) exhibits multiple decaying oscillations; the stable equilibrium films with thickness h0, h~, h2 and h3 correspond to films containing 0, 1, 2 and 3 layers of micelles, respectively. As an illustration, Fig. 14.17a shows a typical DLVO-type disjoining pressure isotherm I-I(h), see Chapter 5 for more details. There are two points, h = hi and h = h2, at which the condition for stable equilibrium, Eq. (14.16), is satisfied. In particular, h = hi corresponds to the so called

primary film, which is stabilized by the electrostatic (double layer) repulsion. The addition of electrolyte to the solution may lead to a decrease of the height of the electrostatic barrier, 1-Imax [61,63]; at high electrolyte concentration it is possible to have Hmax < PA ; then primary film does not exist. The equilibrium state at h = h2 (Fig. 14.17a) corresponds to a very thin secondary film, which is stabilized by the short-range Born repulsion. The secondary film represents a bilayer of two adjacent surfactant monolayers; its thickness is usually about 5 nm (slightly greater than the doubled length of the surfactant molecule) [64]. Secondary films are observed in stable "dry foams" formed after the drainage of most of the water out of the foam.

Effect of Oil Drops and Particulates on the StabiliO, of Foams

619

The situation is more complicated when the aqueous solution contains surfactant micelles, which is a very common situation in practice. In such a case the disjoining pressure isotherm 1-I(h) can exhibit multiple decaying oscillations, whose period is close to the diameter of the micelles (Fig. 14.17b); see Section 5.2.7 above. The condition for equilibrium liquid film, Eq. (14.15), can be satisfied in several points, denoted by h0, hi, h2 and h3 in Fig. 14.17b; the corresponding films contain 0, 1, 2 and 3 layers of micelles, respectively. The transitions between these multiple equilibrium states represent the phenomenon stratification, see Refs. [30,65-76]. The presence of disjoining pressure barriers, which are due either to the electrostatic repulsion (Fig. 14.17a), or to the oscillatory structural forces (Fig. 14.17b), has a stabilizing effect on liquid films and foams. The existence of a stable equilibrium state (cf. Fig. 14.17) does not guarantee that a draining liquid film can "safely" reach this state. Indeed, the hydrodynamic instabilities, accompanying the drainage of the water, could rupture the film before it has reached its thermodynamic equilibrium state [77]. There are several kinetic stabilizing factors, which suppress the hydrodynamic instabilities and decelerate the drainage of the film, thus increasing its life-time. Such a factor is the Gibbs (surface) elasticity of the surfactant adsorption monolayers; it tends to eliminate the gradients in adsorption and surface tension (the Marangoni effect) and damps the fluctuation capillary waves. At high surfactant concentrations the Gibbs elasticity is also high and it renders the interface tangentially immobile, see e.g. [78]. The surface viscosity also impedes the drainage of water out of the films because of the dissipation of a part of the kinetic energy of the flow within the surfactant adsorption monolayers. The surfactant adsorption relaxation time (see Chapter 1) is another important kinetic factor. In the process of foam formation a new water-air interface is created. If the adsorption relaxation time is small enough, a dense adsorption monolayer will cover the newly formed interface and will protect the gas bubbles against coalescence upon collision. In the opposite case (slow adsorption kinetics) the bubbles can merge upon collision and the volume of the created foam (if any) will be relatively small.

620

Chapter 14

14.3.2. MECHANISMS OF FILM RUPTURE

The role of the foam stabilizing (or destabilizing) factors can be understood if the mechanism of film rupture is known. Several different mechanisms of rupture of liquid films have been proposed, which are briefly described below.

(a) Fluctuation Capillary Waves in relatively thick films; de Vries (1958)

(b) Nucleation of Pores in thin bilayer films; Derjaguin & Gutop (1962)

Illllll

\

/ .. Pore

(c) Transport of Solute across the Film Marangoni instability; Ivanov et al. (1987) Phase I

.

" . "

9

Film

9

9

9

9

oi

.

si0n. .

transport

Phase II

Fig. 14.18. Mechanisms of rupturing of liquid films. (a) Fluctuation-wave mechanism: the film rupture is a result of growth of capillary waves enhanced by attractive surface forces. (b) Porenucleation mechanism: it is expected to be operative in very thin films, representing two attached monolayers of amphiphilic molecules. (c) Solute-transport mechanism: if a solute is transferred across the two surfaces of the liquid film due to chemical-potential gradients, it may provoke Marangoni instability and fihn rupture.

The most popular is the capillary-wave mechanism proposed by de Vries [79] and developed in subsequent

studies

[80-85,36,78],

see Fig.

14.18a. The conventional version of this

mechanism is developed for the case of monotonic attraction between the two surfaces of the liquid film (say, van der Waals attraction). Thermally excited fluctuation capillary waves are always present at the film surfaces. With the decrease of the average film thickness, h, the attractive disjoining pressure enhances the amplitude of some modes of the fluctuation waves. At a given critical value of the film thickness, her, corrugations on the two opposite film surfaces can touch each other and then the film will break [36]. The critical thickness of a film having area ~R 2 can be estimated using the following relationship [81,78]"

Effect of Oil Drops and Particulates on the Stability of Foams

cr

--~

-> jl2 - 5.783...

621

(14.17)

where jl is the first zero of the Bessel function J0; as usual, cy denotes surface tension. The relationship (14.17) can be satisfied only for positive OH~Oh. In the special case of van der Waals interaction one can substitute OH~Oh = Au/(2h4), where AH is the Hamaker constant; then Eq. (14.17) shows that the critical thickness increases with the increase of the film radius R, i.e. the films of larger area break easier (at a greater thickness) than the films of smaller area. An estimate for the critical thickness of the film between two bubbles of radius a is available [78]:

crlaR2AJI7 288~20-2

(14.18)

Equation (14.18) is found to agree well with experimental data for hcr vs. R. The effect of kinetic factors, such as the surface viscosity, elasticity and diffusivity are taken into account in Refs. [36, 78, 83-85]. The mechanism of film rupture by nucleation of pores has been proposed by Derjaguin and Gutop [86] to explain the braking of very thin films, built up from two attached monolayers of amphiphilic molecules. Such are the secondary foam films and the bilayer lipid membranes. This mechanism was further developed by Derjaguin and Prokhorov [61,87, 88], Kashchiev and Exerowa [89-91], Chizmadzhev et al. [92-94], Kabalnov and Wennerstr6m [95]. The formation of a nucleus of a pore (Fig. 14.18b) is favored by the decrease of the surface energy, but it is opposed by the edge energy of the pore periphery. The edge energy can be described (macroscopically) as a line tension [87-91], or (micromechanically) as an effect of the spontaneous curvature and bending elasticity of the amphiphilic monolayer [95]. For small nuclei the edge energy is predominant, whereas for larger nuclei the surface energy gets the upper hand. Consequently, the energy of pore nucleation exhibits a maximum at a given critical pore size; the larger pores spontaneously grow and break the film, while the smaller

pores shrink and disappear. A third mechanism of liquid-film rupture is observed when there is a transport of solute across the film, see Fig. 14.18c. This mechanism, investigated experimentally and theoretically by

Chapter 14

622

2r

@000 ,0000 Fig. 14.19. The formation of spots of lower thickness in stratifying liquid films could be attributed to condensation of vacancies in the colloid-crystal-like structure of surfactant micelles formed inside the liquid film [68]; r denotes the spot radius. Ivanov et al. [96-98], was observed with emulsion systems (transfer of alcohols, acetic acid and acetone across liquid films), but it could appear also in some asymmetric oil-water-air films. The diffusion transport of some solute across the film leads to the development of Marangoni instability, which manifests itself as a forced growth of capillary waves at the film surfaces and eventual film rupture. Note that Marangoni instability can be caused by both mass and heat transfer [98-101 ]. In the case of oils, which are soluble in water, the instability could be caused by surface tension gradients due to the diffusion transport of oil to the water-air interface [ 102]. A fourth mechanism of film rupture is the barrier mechanism. It is directly related to the physical interpretation of the equilibrium states in Fig. 14.17. For example, let us consider an electrostatically stabilized film of thickness hi (Fig. 14.17a). Some processes in the system may lead to the increase of the applied capillary pressure PA. For example, if the height of a foam column increases from 1 cm to 10 cm, the hydrostatic effect increases the capillary pressure in the upper part of the foam column from 98 Pa up to 980 Pa. Thus PA could become greater than the height of the barrier, I-[max, which would cause either film rupture, or transition to the stable state of secondary film at h = h2 (Fig. 14.17a). The latter two possibilities could be realized with different probability, say 80 % of the films may rupture and 20 % of the films may safely reach the next equilibrium state of secondary film, see e.g. Ref. [103]. The increase of the adsorption density stabilizes the secondary films. In addition, the decrease of I-[max decreases the probability the film to rupture after the barrier is overcome. Indeed, the overcoming of the barrier is accompanied with a violent release of mechanical energy accumulated during the increase of PA. If the barrier is high enough, the released energy could break the liquid film. On the other hand, if the barrier is not too high, the film could survive the transition.

Effect of Oil Drops and Particulates on the Stability of Foams

623

14.3.3. OVERCOMING THE BARRIER TO DROP ENTRY

In reality, the overcoming of the barrier can be facilitated by various factors. Most frequently the transition happens trough the formation and expansion of spots of lower thickness within the film (Fig. 2.1 l b), rather than by a sudden decrease of the thickness of the whole film. Physically this is accomplished by a nucleation of spots of sub-lain size, which resembles a transition with a "tunnel effect", rather than a real overcoming of the barrier. A theoretical model of spot formation in stratifying films by condensation of vacancies in the structure of ordered micelles (see Figs. 14.17b and 14.19) has been developed in Ref. [68]. The nucleation of spots makes the transitions less violent and decreases the probability for film rupturing. The increase of the applied capillary pressure PA facilitates the spot formation and the transition to state with lower film thickness; this has been established by Bergeron and Radke [71], who experimentally obtained portions of the stable branches of the oscillatory disjoining-pressure curve (Fig. 14.17b). Other factors could also facilitate the overcoming of the disjoining-pressure barrier(s). For example, oil droplets contained within a foam migrate driven by the flow of the outgoing water. The motion of such a droplet could disturb the uniformity of the surfactant adsorption monolayers due to the hydrodynamic friction with the surfaces of the foam films or Plateau borders (Fig. 14.20a). The resulting nonuniform surfactant adsorption may have a destabilizing effect on the asymmetric oil-water-air films and could promote drop entry; this effect has not yet been modeled theoretically. The experiment shows that the presence of small solid crystallites at the oil-water interface (Fig. 14.20b) facilitates the entry of an oil drop at the air-water interface [4,52, 103-106]. Aveyard et al. [15] carried out experiments with separate foam films and observed that oil lenses at the film surface act together with hydrophobic solid to rupture the films. Wasan et al. [23] studied the effect

of

hydrophobic silica particles on the stability of asymmetric

oil-water-air film formed at the tip of a glass capillary of inner diameter 2 0 0 - 300 ~tm. They established the existence of a critical (threshold) concentration of the solid particles, below which their antifoaming effect strongly decreases.

624

Chapter 14

(a) Non-equilibrium Film Surfaces

"-_-:-7}-_-Z-X-}-_}P-_-_

(b) Piercing Effect of Solid Particles

..... --.v_v_v___v_-

_-X-;-~X-;-X-X-r

_-X-X-:-}~-X-X-

Air

Air

Fig. 14.20. Factors which promote overcoming of the disjoining-pressure barrier to oil-drop entry. (a) The motion of a migrating oil droplet, driven by the flow of the outgoing water, disturbs the uniformity of the surfactant adsorption monolayers due to the viscous friction. (b) Small solid crystallites, adsorbed at the oil-water interface, facilitate the rupturing of the asymmetric oil-water-air films. Wang et al. [48] found out that hydrophobic silica particles concentrate in the oil-water interface. These authors observed that couples of oil lenses, attached to different surfaces of the Plateau border, merge upon contact to form a unstable oily bridge, whose rupture breaks the film/border. It seems that in this case the role of the solid particles is to rupture the s y m m e t r i c oil-water-oil film between the two lenses and to effectuate the bridging. Denkov et al. [18,106] found out that the rupture of foam films by means of the bridgingstretching mechanism leads to a separation of the solid silica particles from the silicon oil droplets. Thus with the advance of the antifoaming process the initial, silica-containing oil drops, which exhibit a strong antifoaming action due to a synergistic effect, are transformed into two different populations of particles, silica-free and silica-enriched, both of them having a poor antifoaming performance. This is one of the possible mechanisms of antifoam deactivation (exhaustion) [18, 105, 106]. A powerful factor, which can bring about overcoming of the disjoining-pressure barrier, is the e v a p o r a t i o n of water from the foam. It is known that a foam decays faster if it is exposed to

atmosphere of lower humidity. The evaporation of water from the upper surface of a foam film is compensated by the influx of water from the neighboring Plateau border. The resulting viscous flow leads to a strong decrease of the local hydrodynamic pressure inside the film, which can cause overcoming of the barrier and film rupture. To estimate the latter effect let us consider a plane-parallel film, whose surfaces are circular disks of radius R. Equations (14.1) -

Effect of Oil Drops and Particulates on the Stabili O' of Foams

625

(14.4) are valid also in the present case. Water is supposed to evaporate only from the upper film surface (at z = h); then instead of Eq. (14.5) we have the following boundary condition: v z Iz=0 = 0,

Vz lz=h = Vwje

(14.19)

Here je is the number of water molecules evaporating per unit time from unit area of the upper film surface; Vw is the volume per water molecule in the aqueous phase; h is the width of the gap in which the fluid flow takes place. The hydrodynamic pressure of the liquid in the film, Pl, is given by Eq. (14.8) with u = - Vwje :

Pl(r) = PPB -

3rlVw L (R 2 r 2) h3 -

(0 < r < R) - -

(14.20)

As before, r is the radial coordinate (r = 0 in the center of the film). The substitution of Eq. (14.20) into Eq. (14.16) yields

PA = Pg - PeR +

20"lg

+

37"/VwJe h3 (R 2 - r 2)

(14.21 )

Rf

The last term in Eq. (14.21) represents a viscous contribution to the capillary pressure difference applied across the film surface. The disjoining-pressure barrier (Fig. 14.17.a) will be overcome if 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

1. W. Gerhartz (Ed.), Ullmann's Encyclopedia of Industrial Chemistry, 5th ed., VCH Publishers, New York, 1988, pp. 466-490. 2. N.P. Ghildyal, B.K. Lonsane, N.G. Karanth, Adv. Appl. Microbiol. 33 (1988) 173. 3. J.I. Kroshwitz, M. Howe-Grant (Eds.), Kirk-Othmer Encyclopedia of Chemical Technology, Vol. 7, Wiley-Interscience, New York, 1993, pp. 430-447. 4.

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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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.

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37. O. Reynolds, Phil. Trans. Roy. Soc. (London) A177 (1886) 157. 38. L. D. Landau, E. M. Lifshitz, Fluid Mechanics, Pergamon Press, Oxford, 1984. 39. S. Ross, J.W. McBain, Ind. Eng. Chem. 36 (1944) 570. 40. S. Ross, J. Phys. Colloid Chem. 59 (1950) 429. 41. R.E. Pattle, J. Soc. Chem. Ind. 69 (1950) 363. 42. W.E. Ewers, K.L. Sutherland, Aust. J. Sci. Res. 5 (1952) 697. 43. L.T. Shearer, W.W. Akers, J. Phys. Chem. 62 (1958) 1264, 1269. 44. P.R. Garrett, J. Colloid Interface Sci. 69 (1979) 107. 45. P.R. Garrett, J. Colloid Interface Sci. 76 (1980) 587. 46. A. Dippenaar, Int. J. Mineral Process. 9 (1982) 1. 47. G.C. Frye, J.C. Berg, J. Colloid Interf. Sci. 127 (1989) 222; ibid. 130 (1989) 54. 48. G. Wang, R. Pelton, A. Hrymak, N. Shawafaty, Y.M. Heng, Langmuir 15 (1999) 2202. 49. J.V. Robinson, W.W. Woods, J. Soc. Chem. Ind. 67 (1948) 361. 50. W.D. Harkins, J. Chem. Phys. 9 (1941) 552. 51. A.W. Adamson, A.P. Gast, Physical Chemistry of Surfaces, 6th Ed., Wiley-Interscience, New York, 1997. 52. V. Bergeron, P. Cooper, C. Fischer, J. Giermanska-Kahn, D. Langevin, A. Pouchelon, Colloids Surf. A, 122 (1997) 103. 53. S. Ross, G. Young, Ind. Eng. Chem. 43 (1951) 2520. 54. S. Ross, A.F. Highes, M.L. Kennedy, A.R. Mardonian, J. Phys. Chem. 57 (1953) 684. 55. S. Ross, Chem. Eng. Progr. 63 (1967) 41. 56. P.M. Kruglyakov, T.T. Kotova, Doklady Akad. Nauk SSSR 188 (1969) 865. 57. R.S. Bhute, J. Sci. Ind. Res. 30 (1971) 241. 58. J.V. Povich, Am. Inst. Chem. Eng. J. 25 (1975) 1016. 59. S. Ross, G.M. Nishioka, J. Colloid Interface Sci. 65 (1978) 216. 60. R.M. Hill, S.P. Christiano, Antifoaming Agents, in J.C. Salamone (Ed.) The Polymeric Materials Encyclopedia, CRC Press, Boca Raton FL, 1996. 61. B.V. Derjaguin, Theory of Stability of Colloids and Thin Liquid Films, Plenum Press: Consultants Bureau, New York, 1989. 62. I.B. Ivanov, P.A. Kralchevsky, Mechanics and Thermodynamics of Curved Thin Liquid Films, in: I.B.Ivanov (Ed.) Thin Liquid Films, M. Dekker, New York, 1988, p. 49.

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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 << 1, the deviation of the contact line from the horizontal position is a higher order effect [2] and it can be neglected. In other words, the contact lines in Fig. 8.2 are assumed to be horizontal. Then using geometrical considerations one obtains bk = Re[1 + cos(ak + gte)],

re = Re sin(c~e + gte).

(8A. 1)

Differentiating the above expression for be and then substituting the above expression for re one derives

db~ _ d~z~ - -r k dE dE

(8A.2)

where we have taken into account that the particle radius Rk is independent of the interparticle distance L. Next, we differentiate the relationship between bk and rk given by Eq. (8.7):

dbe =

aL

~

d_2,.

(8A.3)

8~ -b~ dL

Using the definition Qk = r~ sinNk, and Eqs. (8A.2)-(8A.3) one further obtains

dQ~ =/r~ cosN~ + sinl/t k/dr~ _ _db~ [1 + O(q2R~)] dE \ bk - Rk ) dE dE

(8A.4)

At the last step we have made use also of Eqs. (8.16) and (8.29). An alternative expression for

dQffdL can be obtained differentiating Eq. (8.12) along with Eq. (8.7): dQe = _}2q2( 2 dbk _ 2here drk 2d_~h) dE rk dL d L - r ~ dL

(8A.5)

Finally, into Eq. (8A.5) we substitute drffdL from Eq. (8A.3) and then we substitute dbffdL from Eq. (8A.4) to derive

dQk -4 2 2 dhk 2 2 dE - -q rk --~-[1 +O(q Re)] Combining Eqs. (8A.2), (8A.4) and (8A.6) one obtains the sought-for Eq. (8.31).

(8A.6)

Appendices

636

REFERENCES: APPENDIX 8A V.N. Paunov, P.A. Kralchevsky, N.D. Denkov, K. Nagayama, J. Colloid Interface Sci. 157 (1993) 100. 2.

D.Y.C. Chan, J.D. Henry, L.R. White, J. Colloid Interface Sci. 79 (1981) 410.

APPENDIX 1 0 A :

CONNECTIONS BETWEEN TWO MODELS OF LIPID MEMBRANES

Here our purpose is to compare the "sandwich" model described and applied in Chapter 10 with the model developed by Dan et al. [1-3]. Our attention will be focused on the relationships between the parameters entering the two models. The model by Dan et al. is based on a postulated expression for the free energy per molecule of the lipid membrane [1 ]: f(~', Z ) = f 0 ( Z ) + ~Z) d2~

----gdx

+

Kd(~_,,)(d2~12 ~ dx 2 _

( 10A. 1)

where X is the surface area per lipid molecule; tc and Kd are parameters whose physical meaning will be discussed below. In Ref. [1] translational symmetry (along the y-axis) has been assumed, and consequently, the membrane profile is described by the function ~" = ~'(x). As in Fig. 10.6, ~"denotes variation in the membrane thickness which is caused by the presence of inclusion(s) in the membrane. Equation (10A.1) corresponds to an one-dimensional model in which the inclusion is a fiat wall perpendicular to the planar midsurface of the membrane. This one-dimensional model exhibits features that are qualitatively similar to those obtained by taking the inclusion to be a cylindrical object [3]. Note also that Eq. (10A.I) corresponds to the linearized theory for small deformations. On substituting Eqs. (10.32) and

ar

= 0 into Eq. (10.41), representing the Helfrich [4]

constitutive relation, one derives

Wf-"

-~1kc(d 2~/ dx 2 -

2H0) 2

(10A.2)

Having in mind that f is energy per molecule and wf is energy per unit area one can compare the coefficients before (d2~/dx2) 2 in Eqs. (10A.1) and (10A.2) to obtain [5]

Appendices

637

kc = 2Kd/'2;

(10A.3)

Equation (10A.3) relates the Helfrich bending elastic modulus kc (of a lipid monolayer) with the bending stiffness Kd in Refs. [1-3]. In the model of Dan et al. [1] the change of the monolayer free energy (per unit length along the y-axis) due to the presence of inclusions is expressed in the form

F.:

3oc[dxlf~ L2--U4"2 1r

( ~ -tr

dx 2

gd (d2~"/21 +--i-tdU ,

b-

h/2

(10A.4)

where L is the distance between the two inclusions, b is a half of the thickness of the lipid bilayer, and the prime denotes 3/3E. By variational minimization of Eq. (10A.4) one can derive the following equation for ~'(x) [3, 5]:

2/

2~"

__ K.t

b

-Z dx 2

2Kd d4~" _ f_0"5~.

Z dx 4

b2

(10A.5)

'

The comparison of Eq. (10A.5) with Eq. (10.79), in view of Eq. (10A.3), yields

~= f~

'

~o - Cro+ Bo/h = ~

~

/

(101.6)

Disregarding the small term with II', Eq. (10A.6) shows that the effect of chain shear elasticity is connected with f0 in the model of Dan et al. [1]. Besides, in the latter model the surface tension effect is missing; indeed there is no term (yo(d~/dx) 2 in Eq. (10A.4). Therefore, to make a correct comparison we should set o'0 = 0 in Eq. (10A.6), which leads to [5]

Bo=-Z 3B~ cgZ - 4 ( - - ~ - Ztr /

~

B0 = 4~: Z

(101.7)

Having in mind that B0 = -4kcH0 [6], from Eqs. (10A.3) and (10A.7) one obtains the following expression for the spontaneous curvature H0:

Ho =- ~r

89lr

(10A.8)

In Refs. [1,3] K/Kd is termed the spontaneous curvature. Note that for symmetric bilayer lipid membranes the spontaneous curvature of the bilayer as a whole is zero, but the spontaneous

Appendices

638

curvature of each constituent monolayer (irrespective of its definition as H0 or tr

in general

is not zero. In Ref. [1] expressions forf0(Z), h-'(Z) and Kd(Z) in terms of molecular parameters are provided" the substitution of these expressions into Eqs. (10A.3), (10A.6) and (10A.7) yield [5]:

~= 3al kT/Z 2,

kc= azb3kT(1 + 12g')/(222),

Bo =-8a3vZ g kT/Z 4

(10A.9)

where the I I ' term in the expression for 2 is neglected; al = art2/24, a2 = a3 = a71;2/32, a is a molecular length scale, v is volume per amphiphile; this molecular approach is originally developed for nonionic amphiphiles such as (CzH4)n(OCHzCH2)mOH;

g" = ( n - m ) ( m + n )

characterizes the asymmetry of the amphiphile; see Ref. [1] for details. Taking typical parameter values, a = 1 rim, b = 1.5 rim, Z - 0.70 nm 2, kT = 4 x 10 -21 J, v = b]~, from Eq. (10A.9) one calculates ~ = 9.8 x 106 Pa and kc = 4.2 x 10-21 J, which are close to the values of the respective parameters (~, = 2-3 x 106 Pa and kc = 4 x 10-21 J) used by us in Chapter 10. Further, in view of Eq. (10A.7) and (10A.9) one obtains

B o = Z(OBo/~Z) = 32a3v 2g kT/Z 4

(10A. 10)

In Ref. [1] the following values of the parameter g have been used: g = 0,

0.025,

0.1

and

-0.1

(10A.11)

The corresponding values of B0 and B~, calculated from Eqs. (10A.9)-(10A. 11), are Bo = (0,

-0.11,

-0.45

and

+0.45) x 10 -ll N

(10A.12)

B 0 = (0,

0.44,

1.8

and

-1.8) x 10 -~1N

(10A.13) P

The latter values are to be compared with the values B0 = 7 x 10-ll N and B 0

_.

- 3 . 2 x 10-1~ N

used by us in Chapter 10, which are estimated in Ref. [5] from the experimental Hamaker constant and AV potential. From the values of B0 in Eq. (10A.12) one can calculate the respective values of the spontaneous curvature H0 = - Bo/4kc of a lipid monolayer; with kc = 4.2 x 10 -21J one obtains: H o I = ~,

15 nm,

3.7 nm

and

-3.7 nm

(10A. 14)

Appendices

639

In conclusion, the main difference between the model by Dan et al. [1-3] and the "sandwich" model described in Chapter 10 is that the latter accounts for the structure of the bilayer membrane (headgroup and chain regions, see Fig. 10.2), whereas the former treats the membrane just as a couple of attached lipid monolayers obeying a Helfrich constitutive relation, see Eq. (10A.1). Despite this difference, the parameters of the two models can be related, see Eqs. (10A.6)-(10A.8). An exception is the surface tension effect, which is neglected in Refs. [ 1-3, 7], i.e. o'0 is set zero. The importance of the surface tension o'0 for the theoretical description of self-assembled structures, such as lamellar bilayers and micelles built up from amphiphilic molecules, deserves a special discussion. Israelachvili, Mitchell and Ninham [8] have proposed a unified treatment of the various self-assembled structures, which is physically realistic and widely accepted; see also Ref. [9]. In this approach the type of the self-assembled structure, formed by a given amphiphile, is determined by the dimensionless ratio of the following three parameters: optimal area per molecule a0, volume of the molecular hydrocarbon moiety v, and "critical chain length" lc [8, 9]: v/(ao lc) < -~1

spherical micelles

for

cylindrical micelles

for

89< v/(ao Ic) < 71

spherical bilayers

for

71 < v/(ao lc) < 1

planar bilayers

for

v/(ao lc) = 1

The optimal area per molecule can be estimated as a0

=

(Kw/O'0)1/2, where Kw is a parameter

related to the interaction constant in the van der Waals equation of state of an amphiphilic adsorption monolayer [9]. The latter equation is found to describe well the state of phospholipid adsorption monolayers [10]. Consequently, the surface tension (surface pressure) contributions turn out to be of primary importance for the theoretical description of all selfassembled structures, including the planar bilayers. In contrast, the effect of curvature elasticity seems to be a higher order correction, see e.g. Ref. [9], Section 17.9 therein. From the viewpoint of the unified treatment of the various self-assembled structures there should not be drastic differences between the theoretical description of micelles and bilayers, and therefore

640

Appendices

one could expect that the surface tension effects (o'0, Kw) are significant not only for micelles, but also for bilayers and lamellar structures.

REFERENCES: APPENDIX I OA

1. N. Dan, P. Pincus, S.A. Safran, Langmuir 9 (1993) 2768. 2.

N. Dan, A. Berman, P. Pincus, S.A. Safran, J. Phys. II France 4 (1994) 1713.

3.

H. Aranda-Espinoza, A. Berman, N. Dan, P. Pincus, S. Safran, Biophysical J. 71 (1996) 648.

4.

W. Helfrich, Z. Naturforsch. 28C (1973) 693.

5. P.A. Kralchevsky, V.N. Paunov, N.D. Denkov, K. Nagayama, J. Chem. Soc. Faraday Trans. 91 (1995) 3415. 6.

P.A. Kralchevsky, J.C. Eriksson, S. Ljunggren, Adv. Colloid Interface Sci. 48 (1994) 19; Eq. (3.29) therein.

7.

P. Helfrich, W. Jacobson, Biophys. J. 57 (1990) 1075.

8. J.N. Israelachvili, D.J. Mitchell, B.W. Ninham, J. Chem. Soc. Faraday Trans. I, 72 (1976) 1525. 9.

J.N. Israelachvili, Intermolecular and Surface Forces, 2nd Ed., Academic Press, London, 1992.

10. T.D. Gurkov, P.A. Kralchevsky, K. Nagayama, Colloid Polymer Sci. 274 (1996) 227.

641

INDEX

Activities of ionic species, 20 Activity coefficient, 21 Adhesion of latex particles to vesicle, 399 of spherical beads to a flat plate, 470 Adsorption flux of surfactants, 52 Adsorption from ionic surfactant solutions, 25 from micellar surfactant solutions, 53 from nonionic surfactant solutions, 15 from solution of proteins, 55 Adsorption isotherms, 16, 49 of Freundlich, 49 of Frumkin, 16, 49 of Henry, 49 of Langmuir, 16, 49 of van der Waals, 16, 49 of Volmer, 16, 49 Advancing contact angles, 269, 271 Alexander-de Gennes theory, 235 Angular momentum, 449 surface balance, 449 Antifoaming, 471,592, 611,613, 615 Antifoaming action, 591,599, 603, 605,624 fast, 599 slow, 605, 615 bridging-dewetting mechanism, 603, 613 bridging-stretching mechanism, 603 mechanisms, 603 of dispersed oil drops, 471 of oil drops and particulates, 591 of solid crystallites, 623 spreading mechanism, 603 Antifoam deactivation (exhaustion), 624 Antifoam particles, 593, 596 Apo-ferritin, 534 Arrays from microscopic particles, 518,566 methods for obtaining, 518 Asymmetric films - stability, 617

Asymmetric (oil-water-gas) films, 594 Atomic force microscopy, 470 Bacteriorhodopsin, 429, 460, 580 Bacteriorhodopsin vesicles, 571 Bakker equation, 4, 10 Balance of the angular momentum, 149,449 Balance of the linear momentum, 147 Barrier mechanism of film rupture, 622 Barrier to drop entry, 623 Bearing reaction, 84 Bending elastic modulus, 109, 123-128, 161, 163, 173,428, 438-444, 572 Bending moment, 6, 105, 108, 112, 122, 123125, 129-130, 156, 440, 442 contribution of the diffuse layer, 129 contribution of the Stern layer, 129 contribution of the steric interaction, 124 contribution of van der Waals interaction, 124 contribution to the interaction between two emulsion drops, 130 dependence on choice of dividing surface, 112 electrostatic components, 125 in linear approximation, 442 Betaine, 609 B iconical coordinates, 407 Biological cells, axisymmetric shapes, 162 Biomembranes, 106, 115, 137 Biomolecular devices, 579 Bipolar (bicylindrical) coordinates, 308,453 Born repulsion, 618 Bovine serum albumin, 567 Bridging coefficient, 607, 609 Brush/extruder methods, 565 Capillary attraction, 302, 542 Capillary bridges, 77-78, 211,469, 474, 477, 478,482, 486, 490, 493,497, 604, 615

642

Index

between hydrophobic surfaces, 482, 497 bridges with neck, 604, 615 expressions for their shape, 477 gas-in-liquid, 211,469 liquid-in-gas, 469 limits for the length, 486 maximum length, 497 nucleation of capillary bridges, 493 radius of the neck, 483 stability limits, 490 toroid (circle) approximation, 474 types of capillary bridges, 478 Capillary-bridge force, 469, 472 Capillary cavitation, 493 Capillary charge, 297, 354, 356, 397,405 of floating particles, 354, 356 of particles in a spherical film, 397,405 Capillary condensation, 493 Capillary elevation, 312 Capillary flotation force, 356, 360 asymptotic expression, 360 Capillary force between two inclusions in lipid membrane, 457 Capillary force between two spheres partially immersed in a liquid, 304, 324 Capillary forces, 288-289, 341, 351, 381,396, 414, 417, 503, 505-506, 524 asymptotic expression, 341, 381 between particles bound to a spherical interface, 396 between particles in spherical film, 414 between particles of irregular contact line, 503 between particles with "undulated" surface, 506 between spherical particles, 417 flotation force, 289, 351 immersion force, 289 Lucassen's force, 505, 524 types of capillary forces, 288 Capillary image force, 351,367, 370-371 particle-wall interaction, 367

attractive, 370 repulsive, 371 Capillary immersion force between spherical particle and vertical cylinder, 327 Capillary immersion force between spherical particle and wall, 343 Capillary immersion forces, 298-299, 316, 321, 356, 434, 541,559, 568 asymptotic form, 298 between vertical cylinders, 316 between two spherical particles, 321 measurement, 299 Capillary interaction, 303,305, 316, 334, 358, 364-366, 506 energy approach, 303, 316 force approach, 305,334 dependence of the distance, 364 force, 508 energy, 508 Capillary interactions at fixed elevation of the contact line, 328 Capillary interaction energy, 320, 358, 364366, 376, 419 dependence on contact angle, 365 dependence on interfacial tension, 366 Capillary length, 296 Capillary menisci, 71,490 stability, 490 Capillary meniscus around two axisymmetric bodies, 308 Capillary repulsion, 366 Capillary rise, 72 Capillary waves, 161 Capillary-wave mechanism of liquid film rupture, 620 Carnahan-Starling formula, 228 Catenoid, 79, 478 Catenoid bridge - maximum length, 489 Cavitation, 493 Cell-monolayer adhesive energy, 197 Cell-water-air film, 196 Chemical potential, 12

Index

643

Codazzi equation, 160 Coefficient of shear elasticity, 439 Coions, 8 Colloid 2D arrays in optics and optoelectronics, 572 Colloid crystal, 522 Colloid structural forces, 224

Defoaming, 592 Deformation of fluid particles, 249 Dendritic aggregation, 525 Depletion attraction, 224, 228, 229

Colloidal particles- size-fractionation, 557 Colloid-monolayer lithography, 574 Color effects observed with multilayers, 547 Common black film, 89, 189 Compound Hamaker constant, 207 Condensation, 493 Contact angles, 86, 187 Contact angles of liquid films, 197 Contact line, 314, 374 expressions for the shape, 314 on the surface of the floating particle, 374 Cosserat continuum, 139 Counterion adsorption (binding), 8, 28, 52, 31, 218 Critical nucleus, 496 Critical thickness of rupture of a liquid film, 262, 621 Cryo-vitrification, 571 Crystallization of ferritin on gallium, 533 Crystallization of particles in foam films, 566 Crystallization of proteins, 530 Curvature, 107, 440-441 deviatoric, 107, 441 Gaussian, 108,440 mean, 107,440 Curvature deviatoric tensor, 143 Curvature elastic moduli, 105, 125, 162, 168

Derjaguin's formula, 76, 313 Detachment of an oil drop, 279 Detachment of oil drops, 280 edge-cut mechanism, 279 from a solid surface, 268 physicochemical factors, 280 Detergency, 268, 592 Deviatoric curvature, 108 Dewetting, 472 Dielectric permittivity, 9 Diffraction colors, 564 Diffraction gratings, 572 Diffuse electric double layer, 8, 22, 127 contribution to the interfacial tension, 10 Diffusion coefficient, 39 Dilatation, 11,107,437,449 interfacial, 11 of a bilayer surface, 449 of a surface element, 107 of flaccid lipid membranes, 437 Diminishing bubbles, 96 Dimple in a liquid film, 596 Dippenaar - experimental cell, 613 Disjoining film mechanism, 280-281 Disjoining pressure, 184, 186, 194, 201,432 thermodynamic definition, 194

Curved interface, 109 Cylindrical bridge- stability limit, 492

Disjoining pressure isotherm, 189

Cylindrical inclusions in a lipid membrane, 453 Debye screening parameter, 24, 215,250 Debye-Hfickel theory, 21,220 Definitions of fluid interface, 156

Depletion minimum, 252 Deposition of a film from suspension, 565 Derjaguin approximation, 197

Disjoining pressure barrier, 625 Dispersion interaction, 203, 208 Dividing surface, 12, 112, 126 equimolecular, 14 DLVO barrier, 618 DLVO theory, 219 Double emulsions, 131

Index

644 Drag coefficient, 389-390 Drag force, 389 Drainage of foam films - hydrodynamics, 600 Drop detachment, 270 disjoining film mechanism, 280 emulsification mechanism, 270 integral criterion, 269 local criterion, 274 rolling-up mechanism, 270 Drop-drop interaction energy, 251 Drops exposed to shear flow, 268 Drop-weight techniques, 41 Dynamics of 2D crystallization, 535 Dynamics of surfactant adsorption see Kinetics of surfactant adsorption Elastic bodies, 142 Elastic compressibility modulus, 436 Elastic modulus, 109 bending, 109 torsion, 109 Gaussian, 109 Elastic properties of a membrane, 427 Electric charge density, 22 Electric double layer, 8, 20 Electrochemical potential, 20 Electro-diffusion equations, 41 Electromagnetic retardation effect, 209, 267 Electromigration, 42 Electron cryomicroscopy, 568 Electrostatic disjoining pressure, 212 Electrostatic interactions, 123 Electrostatic surface force, 212, 250 Ellipsometric measurements, 531 Elliptic integrals, 80, 484 Emulsion films, 225, 594 Emulsions, 115, 129, 239, 253 critical, 115 interaction between drops, 129 microemulsions, 115, 130-131 oil-in-water, 239, 253

water-in-oil, 239 Energy of capillary attraction, 321 Energy of interaction between a particle and a surface, 203 Energy of interaction between two inclusions in lipid membrane, 459, 461 Enhanced oil recovery, 268, 282 Entering (entry) coefficient, 607, 609 Entering of an oil drop at the air-water interface, 607 Equimolecular dividing surface, 4, 115 Erythrocyte membrane, 163 Erythrocyte, 165, 167 adherent to a glass substrate, 165 free (non-attached), 167 Euler-Masceroni constant, 331 Evaporating liquid films, 602, 624-625 effect on foam destruction, 624 Fast formed drop technique, 41 Ferritin, 533, 551,558 Film radius, 257 Film rupture - mechanisms, 620 Film trapping technique, 196 Films from latex and silica particles, 470 Flexural deformation, 440 Floating particles - experimental measurements, 386 Floating particle - at air-water interface, 383 Flory parameter, 232 Flotation force, 352, 356, 524 superposition approximation, 352 Fluctuation capillary waves, 235, 620 Fluctuation forces, 235 Fluid interface - definitions, 177 Fluid particles of completely mobile surfaces, 263 Fluid particles with partially mobile surfaces, 263 Fluorinated oil - as a substrate for twodimensional crystallization, 550 Fluorocarbon-water interface, 550

Index

Foam column - stages of decay, 598 Foam films, 266 thinning, 597 Foam-breaking action of oil drops and particles, 602 Foams - stability, 591 Forces due to deformation of liquid drops, 237 Free energy, 30, 439 of surfactant adsorption, 30 of lipid bilayers, 439 per unit area of a thin liquid film, 188 Fundamental thermodynamic equation, 109 Gallium- substrate in 2D crystallization, 533 Gas bridges between two solids, 493 Gas bubbles, 264 Gas-in-liquid bridge, 494 Gaussian (torsion) elasticity see Torsion (Gaussian) elastic modulus General curved interfaces - thermodynamical and mechanical approaches, 138-139 Generalized Laplace equation, 151, 157, 161162, 448,450 versions, 157 deviation by minimization of the free energy, 151 for bilayer surfaces, 448,450 in parametric form, 162 Geodesic forms in biological structures, 400 Gibbs adsorption equation, 12-13, 25, 111,633 Gibbs-Duhem equation, 192, 495 Gibbs local approach, 110 Gibbs (surface) elasticity, 19, 33, 36, 237, 264, 391,437, 443, 633 for ionic surfactants, 33 of a lipid monolayer, 437 Gold-coated glass, 547 Gold particles, 560 Good solvent, 233 Gouy equation, 23, 34 Gouy plane, 8 Green theorem, 148

645 Hamaker approximation, 206, 209 Hamaker constant, 124, 188, 204, 206, 208, 210, 432 microscopic theory, 206 macroscopic theory, 208 screened by electrolyte, 210 Helfrich surfaces, 441 Helfrich constitutive relation, 108 Hole in a wetting film, 76 Holes in a sheet of liquid, 471 Homogeneous nucleation, 494 Humidity, 531,543 Hydration force, 216 Hydrodynamic drag, 269 Hydrodynamic interactions, 248, 258 Hydrogen bonds, 211 Hydrophobic plates- connected by vapor-filled capillary bridge, 497 Hydrophobic surface force, 211 Hydrostatic equilibrium condition, 2 Hysteresis of contact angle, 92, 95, 271,306, 329 Ideal solvent, 233 Immersion capillary force - see Capillary immersion force Inclined plate method, 41 Inclusions in a lipid membrane, 426 Induction interaction, 203,208, 210 Instability in the 2D crystal growth, 549 Integral proteins, 427 Interfaces: spherical, 108 cylindrical, 108 Interfacial bending, 239 Interfacial dilatation, 237, 250 Interfacial (surface) tension, 113-114 mechanical, 119, 156 thermodynamical, 119, 156 Interference colors, 548 Interference fringes, 196 Interferometric method, 569

646

Index

Inversion thickness, 260 Ion correlation component of disjoining pressure, 221 Ion correlation surface force, 220 Ionic strength, 21 Kinematics of a curved surface, 142 Kinetics of surfactant adsorption, 37 under barrier control, 38 under diffusion control, 38 under electro-diffusion control, 41 under mixed barrier-electrodiffusion control, 51 under mixed diffusion-barrier control, 50 Kinetics of 2D-array growth Kinetics of two-dimensional crystallization, 542 Kirchhoff - Love, 149-150 first three equations, 149 second three equations, 150 Kirkwood-Alder phase transition, 522, 524-525 Kronecker symbol, 9 Langmuir isotherm, 28 Langmuir-Blodgett films, 578 Langmuir-Blodgett method, 529, 580 Laplace equation of capillarity, 65, 68, 75, 114, 137,294, 308, 398, 401,407 for small deviations from spherical shape, 401 for biological membranes, 162 in bipolar coordinates, 308 in spherical bipolar coordinates, 407 linearized, 294, 308 Lateral capillary forcessee Capillary forces Latex particles, 506, 522, 536, 551 amidine, 522 from copolymer, 506 polystyrene, 522 Latex suspensions, 518 Leukemic Jurkat cell, 196

Life time of a doublet from two emulsion drops, 262 Lifshitz theory, 208 Ligand-receptor interaction, 197 Line tension, 87-88, 92, 95, 194 in Young equation, 88 Linear momentum, 2, 447 balance 2 interfacial balance, 447 Lipid adsorption monolayer, 433 Lipid bilayers, 106, 231,399, 430, 432, 435, 438,444, 458 force balances, 432 grand thermodynamic potential, 458 bending mode of deformation, 438 squeezing (peristaltic) mode of deformation, 444 stretching mode of deformation, 435 Lipid membranes, 426, 636 mechanics, 426 free energy per molecule, 636 Lipid vesicles, 115 Liquid bridgesee Capillary bridges Liquid crystals, 106, 139 Liquid films, 183, 201, 261,289, 397 globular, 397 Liquid-bridge- nucleus, 494 Liquid-film: rupture due to transport of solute, 621 Lithographic masks from polystyrene monolayers, 575 Long-ranged attraction, 567 Lubrication approximation, 601 Lucassen's capillary force, 505 Magnetic field used to produce 2D particulate arrays, 528 Marangoni effect, 20, 264 Marangoni instability, 622 Matched asymptotic expansions, 310-311 Maximum bubble pressure method, 41,490

Index

647

Mean curvature, 143

Navier-Stokes equation, 142, 162, 407

Mechanical surface tension -dependence on dilatation and curvature, 442

Neck of capillary bridge, 475

Membrane deformations caused by inclusions, 444 Membrane emulsification, 268 Membrane of a living cell, 400

two-dimensional, 162 Necking, 271,276 Negative staining-carbon film method, 519 Neumann triangle, 85,607 Neumann-Young equation, 194, 197

Membrane proteins, 400, 426, 445,460 energy of interaction, 460

generalization, 197 Newton black film, 89, 189

Meniscus between two cylinders, 77

Nodoid, 79, 478

Meniscus decaying at infinity, 75

Nodoid with "haunch", 480 Nodoid with "neck", 480

Mercury - as a substrate for two-dimensional crystallization, 554 Mercury trough method, 519, 530 Mesostructured materials, 575 Micelles (in surfactant solutions), 53,225, 639 spherical, 639 cylindrical, 639 Microcontact printing, 575 Micromechanical expressions, 121,172, 174 for the adsorption of the species, 174 for the curvature elastic moduli, 172 for the entropy, 174 for the interfacial tensions, 172 for the interfacial moments, 172 for the internal energy, 174 for the spontaneous curvature, 172 for the surface properties, 168 for the tensors of surface stresses, 174 for the tensors of surface moments, 174 Mobility factor, 266 Moir6 optic effect, 553 Molecular electronics, 578 Monte-Carlo method, 559 Mysels experimental cell, 594

Nodoid-shaped bridge, 475, 483,496 with neck, 483,496 Nodoid-shaped cavities between two solids in water, 493 Nonionic surfactants, 15,234 Nucleation and growth of 2D colloid crystals, 520 Nucleation and growth of 2D particle array, 556 Nucleation of capillary bridges, 469, 492 Nucleation of pores in thin films, 620 Nuclei of a new fluid phase, 130 Oil detachment- mechanism, 268 in detergency, 282 Oil drops protruding from pores, 276 Opals, 572 Optical field- producing ordering of microscopic dielectric objects, 528 Optimal area per molecule - in self-assembled structures, 639 Ordered monolayers of lam-sized crystals, 504 Orientation interaction, 203,208, 210 Oscillating bubble method, 41

Nanocrystallites, 529

Oscillating jet method, 41

Nano-lithography, 573

Oscillatory disjoining pressure, 228

Nanostructured surfaces, 573

Oscillatory structural forces, 224, 251

Nanostructured TiO2 surface, 576 Navier equation, 142

Plateau- classification of meniscus shapes, 477 Particle 2D array, 518, 527

648

Index

in evaporating liquid films, 518 in electric, magnetic & optical fields, 527 Particle array, 543 Particles in a narrow wedge-shaped gap, 538 Particles with corrugated contact line, 512 Particle-wall interaction, 368 force approach, 379 Particulate monolayer- elastic properties, 512 Pascal law, 2 Pendant drops, 72 Pendular ring, 476 Phospholipid bilayer, 427, 431 Photocatalytic films, 575 Pickering emulsions, 397 Plant viruses, 519 Plateau borders, 595,599 Poisson equation, 42, 219 modified, 219 Poisson-Boltzmann equation 22 Polymeric brush, 231,233 thickness, 231 interactions, 233 Polystyrene latex spheres, 552 Poor solvent, 233 Pressure tensor, 2 Principal curvatures, 111, 107, 162, 198 Protein 2D-crystallization, 530 Protein 2D arrays in applications, 577 Protein adsorption, 55, 196 Protein aggregation, 460 Protein-engineering, 577 Protein molecular diode, 579 Protein molecules, 287 Protein-protein interaction, 430, 463 lipid mediated, 430 energy, 463 Protrusion force, 235-236 PV-diagrams, 490 Radiotracer method, 3 l Rate of surfactant adsorption, 48

Rate-of-strain tensor, 14 l, 144, 407 Receding contact angle, 269, 271 Receding contact line, 521 Receding meniscus, 96 Relaxation time of counterion adsorption, 44, 46 of surface tension, 35, 39, 46 of surfactant adsorption, 44, 46 Retentive capillary force, 269 Reynolds formula, 261 Reynolds number, 276 Rheological constitutive relations for interfaces, 158, 160 Rheological models, 141 Rheology - interfacial, 108 Rhodopsin, 428 Rolling-up mechanism, 273 Rupture of foam cells, 598 Rusanov equation, 191,433 Saddle-shape deformation, 108 Sandwich model of a lipid bilayer, 430, 438, 463, 636 Scheludko cell, 594, 599 Screening in electrolyte solutions, 210 Self-assembly of floating particles, 524 Self assembled structures, 639 Semiconductor nanoparticles, 521,573 Sessile drops, 72 Shampoo-type system, 608 Shape computation for biomembranes, 164 Shear deformation, 107 Shear elasticity, 436 Shear- surface rate, 144 Shearing tension, 440 Silicone oil, 624 Size separation of colloidal particles, 556 S-layers, 578 Solvation forces, 224 Spherical bipolar coordinates, 406 Spherical fluid interface, 396

Index

649

Spherical interfaces, thermodynamics, 112 Spin-coating technique, 521 Spontaneous curvature, 123, 163,428,441, 571,637 of lipid monolayers, 428 Spots in horizontal stratifying films, 227, 622

Surface strain tensor, 106 Surface stress tensor, 448 of Boussinesq-Scriven, 158 constitutive relation, 158

Spread protein layer, 531

Surface tension, 4, 6, 117, 168 diffuse layer contribution, 25 mechanical, 119, 156 thermodynamical, 119, 156 of a micellar solution, 54 Surface unit tensor, 143 Surface viscometer, 391 deep-channel, 391 sliding-particle method, 391 knife-edge viscometers, 391 Surface viscosity, 158, 392 dilatational, shear, 158 transversal, 159 Surface wave techniques, 41

Spreading coefficient, 609 Spreading mechanism, 611 Stability of liquid films, 617 Step-wise thinning of foam films, 225,597 Steric interaction due to adsorbed molecular chains, 231 Steric interaction, 123, 231 Stern isotherm, 27, 52 Stern layer, 8, 25, 32, 127, 218 Stokes equation, 258 Strain tensor, 141,435 Stratification of liquid films, 225,597,619, 622 Stress tensor, 435 Stretching elastic modulus of a lipid bilayer, 428,436 Superficial tensions, 83 Superposition approximation of Nicolson, 292, 298 Superposition approximation for capillary image force, 373 for capillary flotation force, 352 for capillary immersion force, 296 Surface balance of the angular momentum, 149 Surface balance of the linear momentum, 147 Surface elasticity see Gibbs (surface) elasticity Surface elasticity of particulate monolayer dilatational, 513 shear, 514

Surface stretching elasticity, 444 Surface tension isotherm, 17

Surfactant adsorptionsee Adsorption from ... Surfactant micelles, 225,597 Tangential isotropy assumption, 172 Taylor formula, 259 Tensegrity, 400 Tension free state, 434 Tensionless monolayer, 429 Tensor of curvature, 111 in linear approximation, 441 Tensors of surface stresses and moments, 145 Tensor of the surface moments, 160 Thermal undulations, 437 Thermodynamic equilibrium, 13

Surface electric potential, 32

Thermodynamic surface tension, 440

Surface force apparatus, 201

Thermodynamics of nucleation, 492

Surface forces, 184 Surface local basis, 143

Thermodynamics of thin liquid films, 191 Theta solvent, 233

Surface moments, 108

Theta temperature, 233

Surface shear viscosity, 385

Thickness of a liquid film-

hzdex

650 critical and transitional, 265 Tolman length, 115

between two spheres, 204 Vertical films, 595

Tolman term, 442

Vesicles, 106

Toroid approximation, 476 Torsion (Gaussian) elastic modulus, 109, 123128, 161,163, 173,438,443,444 Torsion elastic modulus of lipid bilayer, 443 Torsion micro-balance, 301 Torsion moment, 108, 123, 156, 440 Torsion of a surface element, 107 Transitional thickness of a liquid film, 265 Transmembrane protein, 428 Transversal tension, 186, 195,254 Trapping (levitation) of dielectric objects into gradients of optical fields, 528 Turning points in volume and in pressure, 490-492 Two-dimensional arrays of holoferritin, 564 Two-dimensional colloid crystal, 519, 537 Two-dimensional crystallization, 517, 535, 551 of particulates and proteins, 517 mechanism, 535 over fluorinated oil, 551 Two-dimensional fluid, 111 Ultrasonic radiation technique, 522 Unduloid, 79, 478,480, 496 inflection point, 480 with neck, 496 Unduloid-shaped bridges, 491 Undulation forces, 235-236 Unstable bridges, 616 van der Waals force (attraction), 123, 188 van der Waals disjoining pressure (surface force), 203 repulsive, 207 between multilayered films, 207 between sphere - wall, 205 between truncated sphere- wall, 205 between two identical deformed emulsion droplets, 205

giant spherical phospholipid, 399 Viscous body (fluid), 142 Vitrified film, 570 Volta (AV) potential, 33, 125,443 Volume exclusion effect, 218 Volume exclusion forces, 227 Washing of fabrics, 268 Washing, 282 Weingarten's formula, 111 Wetting film, 187, 225 Wetting of powders, 470 Wilhelmi plate method, 531 Withdrawal of a plate from suspension, 561 Work of interfacial deformation, 106 Work of nucleation, 497 Work of surface deformation, 154 Work of surface deformation, 442 Young equation, 80, 83, 9 l, 270, 297 Zeta potential, 33 Zwitterionic surfactant, 125

651

NOTATION ai

activity of a solute, Eq. (1.46)

aioo

activity of a solute in the bulk of solution

ais

activity of a solute at the subsurface, Eq. ( 1.71)

ally

surface metric tensor, Eq. (4.15)

a~,

vector of the local surface basis, Eq. (4.13)

A

area of a surface

AH

Hamaker constant, see Section 5.2.2

b, buy

curvature tensor and its components, Eq. (3.21) and (4.17)

B

bending moment of an interface or membrane, Eq. (3.1)

B

dimensionless bending moment, Eq. (3.30)

Bo

bending moment of a flat interface, Eqs. (1.11) and (3.9) surface rate-of-strain tensor, Eq. (4.19)

D

deviatoric curvature, Eq. (3.3)

Di

bulk diffusion coefficient of component i, Eq. (1.101)

Ds

surface diffusion coefficient

e

Napier number, e = 2.7182818...

e

charge of the electron

ei

unit basis vector of the Cartesian coordinates

E6

surface (Gibbs) elasticity, Eq. (1.45)

E(r

elliptic integral of the second kind, Eq. (2.51)

F(0,q)

elliptic integral of the first kind, Eq. (2.51)

f

surface density of the interaction free energy, see Eq. (5.9)

f

acceleration due to a body force, Eq. (4.1)

F

force

g

acceleration due to gravity

h

thickness of a film or membrane; surface-to-surface distance

H

mean curvature, Eq. (3.3)

H0

spontaneous curvature, Eqs. (3.7) and (3.10)

I

ionic strength, Eq. (1.52)

[ln(e) = 1]

Notation

652 J

integral in the expression for the surface tension, Eqs. (1.43) - (1.44)

k

Boltzmann's constant

kc

bending elastic modulus of an interface or membrane, Eq. (3.7)

kc

torsion (Gaussian)elastic modulus, Eq. (3.7)

kt, k t

total bending and torsion elastic moduli of a lipid bilayer, Eqs. (10.49)-(10.50)

Ks

stretching elastic modulus of a membrane, Eq. (10.17)

K

Gaussian curvature, Eq. (3.4)

K~(x)

modified function of Bessel of the second kind and order n (n = 0, 1.... )

1

length of a line

L

distance between two particles or cylinders, Eq. (7.11)

L

tensor defined by Eq. (4.130)

M, M~v

symmetric tensor of the surface moments and its components, Fig. 4.1

ni

number density of the i-th component running unit normal to a surface

N, Nuv

tensor of the surface moments and its components, Fig. 4.1

Ni

number of the molecules of the i-th component

P

pressure tensor

PT, PN

components of the pressure tensor, Eq. (1.4)

P

isotropic pressure in the bulk of a fluid phase

Pc

capillary pressure inverse capillary length, Eq. (2.35)

q, q,v

curvature deviatoric tensor and its components, Eq. (4.14)

Qk

"capillary charge" of the particle "k" attached to an interface, Eq. (7.9) radius vector of a point in space radius of curvature of a contact line

r0

radius of the neck of a capillary bridge, Fig. 2.6

R

radius of a colloid particle, Fig. 5.19

R

mean radius of two fluid particles, Eq. (6.17) distance between particle and wall, Fig. 8.10 excess surface density of entropy, Eq. (3.11)

Notation

S

entropy, Eq. (1.28)

t

time

T

temperature

T

total stress tensor, Eq. (10.67) excess surface density of internal energy, Eq. (3.11)

Us 1

U ,tt

653

2

surface curvilinear coordinates, Section 4.2.2

U

displacement vector

U

internal energy, Eq. (1.28), or interaction energy between particles, Eq. (5.43)

U

unit tensor (idemfactor) in space

U~

unit tensor (idemfactor) in a surface, Eq. (4.14)

UII

unit tensor (idemfactor) in the plane xy

v

velocity

V

volume

VRe

Reynolds velocity of thinning of a liquid film, Eq. (6.25)

VTa

Taylor velocity of approach of two particles, Eq. (6.20)

w

mechanical work per unit area of a surface

W

mechanical work

x, y, z

Cartesian coordinates

Zi

scaled ionic valence, Eq. (1.71)

L

ionic valence

t~

dilatational deformation of a surface, Eqs. (3.2) and (4.22)

t2

rate of surface dilatation, Eq. (4.21)

fi

shearing deformation of a surface, Eq. (3.2) and (4.22)

fi

rate of surface shearing, Eq. (4.21)

?

thermodynamic surface tension; tension of a liquid film constant of Euler-Masceroni (~ = 1.781072418...)

Y+

activity coefficient, Eq. (1.50)

Fi

adsorption of component i

F=

maximum (saturation) adsorption Tolman length, Eq. (3.35)

Notation

654 ~l, 52

thickness of adsorption layer, Eqs. (1.83) and (1.84) symbol of Kronecker

8

dielectric permittivity

((r)

deformation in the shape of an interface, Fig. 7.2 mechanical shearing tension, Eq. (4.24)

7/d, 7/s

dilatational and shear surface viscosity, Eq. (4.86)

0

torsion moment of an interface or membrane, Eq. (3.1) Debye screening parameter, Eq. (1.64) see eq. (1.56) line tension, Eq. (2.74)

J./i

chemical potential of component i thermodynamical shearing tension, Eq. (3.1) Archimedes number (3.1415927...)

FI

disjoining pressure mass density

Pe

bulk electric charge density

Ps

surface electric charge density

o

(mechanical) surface tension

o

mean surface tension of two fluid particles, Eq. (6.12)

o'~, o'd

contributions of adsorption and diffuse layer to the surface tension, Eq. (1.19)

(I)

dimensionless electric potential, Eq. (1.54) surface dimensionless potential, Eq. (1.60)

V

electric potential

~s

surface electric potential bipolar coordinate, Eqs. (7.25) and (9.21)

ms

excess surface density of the grand thermodynamic potential, Eq. (3.11) grand thermodynamic potential gradient operator in space

Vs

surface gradient operator, Eq. (4.13)

VII

gradient operator in the plane xy, Eq. (2.16)