STAROV Wetting and Spreading Dynamics

WETTING AND SPREADING DYNAMICS

© 2007 by Taylor & Francis Group, LLC

DANIEL BLANKSCHTEIN

Department of ChemicalEngineeringMassachusetts Institute of TechnologyCambridge, Massachusetts

S. KARABORNI

Shell International PetroleumCompany LimitedLondon, England

LISA B. QUENCER

The Dow Chemical CompanyMidland, Michigan

JOHN F. SCAMEHORN

Institute for Applied SurfactantResearchUniversity of OklahomaNorman, Oklahoma

P. SOMASUNDARAN

Henry Krumb School of MinesColumbia UniversityNew York, New York

ERIC W. KALER

Department of ChemicalEngineeringUniversity of DelawareNewark, Delaware

CLARENCE MILLER

Chemical and Biomolecular Engineering DepartmentRice UniversityHouston, Texas

DON RUBINGH

The Procter & Gamble CompanyCincinnati, Ohio

BEREND SMIT

Shell International Oil Products B.V.Amsterdam, The Netherlands

JOHN TEXTER

Strider Research CorporationRochester, New York

SURFACTANT SCIENCE SERIES

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ARTHUR T. HUBBARD

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138. Wetting and Spreading Dynamics, Victor M. Starov, Manuel G. Velarde, and Clayton J. Radke


WETTING ANDSPREADING DYNAMICS

Victor M. StarovLoughborough University

Loughborough, U.K.

Manuel G. VelardeInstituto Pluridisciplinar

Madrid, Spain

Clayton J. RadkeUniversity of California at Berkeley

Berkeley, California, U.S.A.

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Starov, V. M.Wetting and spreading dynamics / Victor Starov, Manuel Velarde, and Clayton

Radke.p. cm. -- (Surfactant science ; 138)

Includes bibliographical references and index.ISBN-13: 978-1-57444-540-4 (alk. paper) 1. Wetting. 2. Surface (Chemistry) I. Velarde, Manuel G. (Manuel García) II. Radke, Clayton. III. Title. IV. Series.

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Contents

Preface .............................................................................................................. xviiAcknowledgments ..............................................................................................xxi

Chapter 1 Surface Forces and the Equilibrium of Liquids on Solid Substrates.........................................................................................1

Introduction ...........................................................................................................11.1 Wetting and Young’s Equation ....................................................................21.2 Surface Forces and Disjoining Pressure ...................................................11

Components of the Disjoining Pressure ...................................................13Molecular or Dispersion Component............................................13The Electrostatic Component of the Disjoining Pressure ............19Structural Component of the Disjoining Pressure ........................21

1.3 Static Hysteresis of Contact Angle ...........................................................23Static Hysteresis of Contact Angles from Microscopic Point of View: Surface Forces ................................................................28

References ...........................................................................................................30

Chapter 2 Equilibrium Wetting Phenomena ..................................................31

Introduction .........................................................................................................312.1 Thin Liquid Films on Flat Solid Substrates .............................................31

Equilibrium Droplets on the Solid Substrate under Oversaturation (Pe < 0) ..........................................................................................36

Flat Films at the Equilibrium with Menisci (Pe > 0) ...............................38S-Shaped Isotherms of Disjoining Pressure in the Special Case S– < S+.... 40

2.2 Nonflat Equilibrium Liquid Shapes on Flat Surfaces...............................41General Consideration ...............................................................................42Microdrops: The Case Pe > 0....................................................................47Microscopic Equilibrium Periodic Films..................................................49Microscopic Equilibrium Depressions on β-Films...................................54

2.3 Equilibrium Contact Angle of Menisci and Drops: Liquid Shape in the Transition Zone from the Bulk Liquid to the Flat Films in Front.....56Equilibrium of Liquid in a Flat Capillary: Partial Wetting Case .............57Meniscus in a Flat Capillary .....................................................................60Meniscus in a Flat Capillary: Profile of the Transition Zone ..................63Partial Wetting: Macroscopic Liquid Drops .............................................65Profile of the Transition Zone in the Case of Droplets............................71Axisymmetric Drops .................................................................................71


Meniscus in a Cylindrical Capillary .........................................................72Appendix 1 ................................................................................................73

2.4 Profile of the Transition Zone between a Wetting Film and the Meniscus of the Bulk Liquid in the Case of Complete Wetting..............74

2.5 Thickness of Wetting Films on Rough Solid Substrates ..........................812.6 Wetting Films on Locally Heterogeneous Surfaces: Hydrophilic

Surface with Hydrophobic Inclusions.......................................................902.7 Thickness and Stability of Liquid Films on Nonplanar Surfaces ..........1002.8 Pressure on Wetting Perimeter and Deformation of Soft Solids............1062.9 Deformation of Fluid Particles in the Contact Zone ..............................113

Two Identical Cylindrical Drops or Bubbles..........................................115Interaction of Cylindrical Droplets of Different Radii...........................119Shape of a Liquid Interlayer between Interacting Droplets: Critical

Radius ..........................................................................................1232.10 Line Tension ............................................................................................130

Comparison with Experimental Data and Discussion ............................1422.11 Capillary Interaction between Solid Bodies ...........................................144

Appendix 2 ..............................................................................................152Equilibrium Liquid Shape Close to a Vertical Plate...................152

2.12 Liquid Profiles on Curved Interfaces, Effective Disjoining Pressure. Equilibrium Contact Angles of Droplets on Outer/Inner Cylindrical Surfaces and Menisci inside Cylindrical Capillary ................................154Liquid Profiles on Curved Surface: Derivation of Governing

Equations .....................................................................................154Equilibrium Contact Angle of a Droplet on an Outer Surface of

Cylindrical Capillaries.................................................................159Equilibrium Contact Angle of a Meniscus inside Cylindrical

Capillaries ....................................................................................161References .........................................................................................................163

Chapter 3 Kinetics of Wetting......................................................................165

Introduction .......................................................................................................1653.1 Spreading of Droplets of Nonvolatile Liquids over Flat Solid

Substrates: Qualitative Consideration .....................................................174Capillary Regime of Spreading...............................................................179Similarity Solution of Equation 3.18 and Equation 3.19 .......................181Gravitational Spreading...........................................................................186Similarity Solution ..................................................................................187Spreading of Very Thin Droplets ............................................................190

3.2 The Spreading of Liquid Drops over Dry Surfaces: Influence of Surface Forces.....................................................................197Case n = 2 ...............................................................................................205Case n = 3 ...............................................................................................205


Comparison with Experiments ................................................................209Conclusions..............................................................................................211Appendix 1 ..............................................................................................211Appendix 2 ..............................................................................................213Appendix 3 ..............................................................................................214Appendix 4 ..............................................................................................216

3.3 Spreading of Drops over a Surface Covered with a Thin Layer of the Same Liquid ............................................................................................217

3.4 Quasi-Steady-State Approach to the Kinetics of Spreading...................2253.5 Dynamic Advancing Contact Angle and the Form of the Moving

Meniscus in Flat Capillaries in the Case of Complete Wetting .............235Appendix 5 ..............................................................................................242

3.6 Motion of Long Drops in Thin Capillaries in the Case of Complete Wetting.....................................................................................................245Appendix 6 ..............................................................................................255

3.7 Coating of a Liquid Film on a Moving Thin Cylindrical Fiber.............259Statement of the Problem........................................................................260Derivation of the Equation for the Liquid–Liquid Interface Profile ......262Immobile Meniscus .................................................................................264Matching of Asymptotic Solutions in Zones I and II (Figure 3.17)......265Equilibrium Case (Ca = 0)......................................................................267Numerical Results ...................................................................................269

3.8 Blow-Off Method for Investigation of Boundary Viscosity of Volatile Liquids .....................................................................................................270Boundary Viscosity..................................................................................270Theory of the Method .............................................................................271

Experimental Part ........................................................................284Conclusions..............................................................................................287

3.9 Combined Heat and Mass Transfer in Tapered Capillaries with Bubbles under the Action of a Temperature Gradient............................287Cylindrical Capillaries.............................................................................292Tapered Capillaries ..................................................................................293

3.10 Static Hysteresis of Contact Angle .........................................................296Equilibrium Contact Angles ....................................................................297Static Hysteresis of the Contact Angle of Menisci ................................301Static Hysteresis Contact Angles of Drops.............................................308Conclusions..............................................................................................312

References .........................................................................................................312

Chapter 4 Spreading over Porous Substrates...............................................315

Introduction .......................................................................................................3154.1 Spreading of Liquid Drops over Saturated Porous Layers ....................315

Theory......................................................................................................316


Liquid inside the Drop (0 < z < h(t,r)) .......................................316Inside the Porous Layer beneath the Drop (–D < z < 0, 0 < r < L) ...............................................................318Materials and Methods ................................................................325Results and Discussion. Experimental Determination of Effective Lubrication Coefficient ω........................................327

4.2 Spreading of Liquid Drops over Dry Porous Layers: Complete Wetting Case............................................................................................331Theory......................................................................................................332

Inside the Porous Layer outside the Drop (–D < z < 0, L < r < l) ................................................................338Experimental Part ........................................................................343Independent Determination of Kppc ............................................344Results and Discussion................................................................345

Appendix 1 ..............................................................................................3514.3 Spreading of Liquid Drops over Thick Porous Substrates: Complete

Wetting Case............................................................................................354Theory......................................................................................................355

Inside the Porous Substrate .........................................................358Experimental Part ........................................................................358Results and Discussion................................................................360Spreading of Silicone Oil Drops of Different Viscosity over Identical Glass Filters..................................................................363Spreading of Silicone Oil Drops over Filters with Similar Properties but Made of Different Materials................................364Spreading of Silicone Oil Drops with the Same Viscosity (η = 5P) over Glass Filters with Different Porosity and Average Pore Size .......................................................................366Conclusions..................................................................................368

4.4 Spreading of Liquid Drops from a Liquid Source .................................369Theory......................................................................................................370Experimental Set-Up and Results ...........................................................374

Materials and Methods ................................................................374Results and Discussion................................................................376Conclusions..................................................................................379

Appendix 2 ..............................................................................................379Capillary Regime, Complete Wetting .........................................380Gravitational Regime, Complete Wetting ...................................384Partial Wetting .............................................................................387

References .........................................................................................................388

Chapter 5 Dynamics of Wetting or Spreading in the Presence of Surfactants ...................................................................................389

Introduction .......................................................................................................390


5.1 Spreading of Aqueous Surfactant Solutions over Porous Layers...........390Experimental Methods and Materials [1] ...............................................391

Spreading on Porous Substrates (Figure 4.4) .............................391Measurement of Static Advancing and Receding Contact Angles on Nonporous Substrates ................................................391Results and Discussion................................................................393Advancing and Hydrodynamic Receding Contact Angles on Porous Nitrocellulose Membranes.........................................398Static Hysteresis of the Contact Angle of SDS Solution Drops on Smooth Nonporous Nitrocellulose Substrate.........................400Conclusions..................................................................................403

5.2 Spontaneous Capillary Imbibition of Surfactant Solutions into Hydrophobic Capillaries..........................................................................403Theory......................................................................................................406

Concentration below CMC..........................................................410Concentration above CMC..........................................................413Spontaneous Capillary Rise in Hydrophobic Capillaries ...........417

Appendix 1 ..............................................................................................4195.3 Capillary Imbibition of Surfactant Solutions in Porous Media and

Thin Capillaries: Partial Wetting Case....................................................421Theory......................................................................................................422

Concentration below CMC..........................................................424Concentration above CMC..........................................................432Experimental Part ........................................................................434Results and Discussions ..............................................................435

5.4 Spreading of Surfactant Solutions over Hydrophobic Substrates ..........436Theory......................................................................................................437

Experiment: Materials .................................................................442Monitoring Method .....................................................................442Results and Discussion................................................................443

5.5 Spreading of Non-Newtonian Liquids over Solid Substrates ................445Governing Equation for the Evolution of the Profile of the Spreading

Drop .............................................................................................446Gravitational Regime of Spreading.........................................................452Capillary Regime of Spreading...............................................................455Discussion................................................................................................459

5.6 Spreading of an Insoluble Surfactant over Thin Viscose Liquid Layers ......................................................................................................460Theory and Relation to Experiment........................................................462

The First Spreading Stage...........................................................465The Second Spreading Stage ......................................................470Experimental Results...................................................................473

Appendix 2 ..............................................................................................475Derivation of Governing Equations for Time Evolution of Both Film Thickness and Surfactant Surface Concentration .....475


Appendix 3 ..............................................................................................476Influence of Capillary Forces during Initial Stage of Spreading .....................................................................................476

Appendix 4 ..............................................................................................478Derivation of Boundary Condition at the Moving Shock Front.............................................................................................478

Appendix 5 ..............................................................................................479Matching of Asymptotic Solutions at the Moving Shock Front.............................................................................................479

Appendix 6 ..............................................................................................480Solution of the Governing Equations for the Second Stage of Spreading.................................................................................480

5.7 Spreading of Aqueous Droplets Induced by Overturning of Amphiphilic Molecules or Their Fragments in the Surface Layer of an Initially Hydrophobic Substrate.....................................................481Theory and Derivation of Basic Equations.............................................482Boundary Conditions...............................................................................487

Solution of the Problem ..............................................................493Comparison between Theory and Experimental Data............................497

References .........................................................................................................499

Conclusions .......................................................................................................501Frequently Used Equations...............................................................................502

Navier–Stokes Equations.........................................................................502Navier-Stokes Equations in the Case of Two-Dimensional Flow..........504Capillary Pressure....................................................................................505

List of Main Symbols Used..............................................................................505Greek........................................................................................................505Latin.........................................................................................................506Subscripts.................................................................................................506


Preface

This book is for anyone who has recently started to be interested in, or is alreadyinvolved in, research or applications of wetting and spreading, i.e., for newcomersand practitioners alike. Its contents are not a comprehensive and critical reviewof the existing research literature. Needless to say, it rather reflects the authors’recent scientific interests and understanding. The authors presume that the readerusing this book has some knowledge in thermodynamics, fluid mechanics, andtransport phenomena. Yet the book has been written in an almost self-containedmanner, and it should be possible for a graduate student, scientist, or engineerwith a reasonable background in differential equations to follow it. Although invarious parts we have used the phrase “it can be shown …” or the like, the authorshave tried to go as deep into the details of derivation of results as required tomake the book useful.

The term wetting commonly refers to the displacement of air from a solidsurface. Throughout this book we shall be discussing wetting and spreadingfeatures of liquids, which partially (the most important example being water andaqueous solutions) or completely (oils) wet the solids or other liquids.

Wetting water films occur everywhere, even in the driest deserts or in thesauna and bathtub, although you might not see them with the naked eye becausethey are too thin or because they seem to disappear too quickly. Water is essentialfor life. It may very well be that without water, life would have not have startedon Earth. In fact no life seems possible without fluids! Life, as we know it startedin a little “pond,” the “primordial soup” leading to the first replicating bio-relatedamino acids.

In the processes of wetting or spreading, three phases — air, liquid, andsolids — meet along a line, which is referred to as a three-phase contact line.Recall the spreading drop and the drop edge, which is the three-phase contactline. In the vicinity of a three-phase contact line, the thickness of the dropletbecomes very thin and, even more, virtually tends to zero. In a thin water layer,new very special surface forces come into play. These forces are well known incolloid science: forces in thin layers between interfaces of neighbor particles,droplets, and bubbles in suspensions and emulsions. Understanding of the impor-tance of surface forces in colloid science has resulted in substantial progress inthis area. In fact, it is the reason why colloid science is referred to nowadays ascolloid and interface science.

Surface forces of the same nature act in thin liquid layers in the vicinity ofthe three-phase contact lines in the course of wetting and spreading. Surprisingly,the importance of surface forces has been much less recognized in wetting andspreading than it deserves. In Chapter 1 through Chapter 3 we will try to convince


the reader that virtually all wetting and spreading phenomena are determined bythe surface forces acting in a tiny vicinity of the three-phase contact line.

Water is, indeed, a strange liquid. For example, if you place a glass bottlefull of pure water (H2O) in the deep freezer, the bottle will break as water increasesin volume while solidifying as ice, an anomalous property relative to other liquids.Life (fish) in frozen lakes would not be possible without the anomalous behaviorof water around 4˚C.

We shall see that a property of water relative to “surface” forces is key tounderstanding its wetting and spreading features. We will also find that surfaceforces (frequently also referred to as disjoining pressure) have a very peculiarshape, in the case of water and aqueous solutions. This fact is critical for theexistence of our life in a way which is yet to be understood.

Wetting and spreading are dramatically affected by SURFace ACTtiveAgeNTS (in short, surfactants). Their molecules have a hydrophilic head (ionicor nonionic) with affinity for water and a hydrophobic tail (a hydrocarbon group),which is repelled by an aqueous phase. Fatty acids, alcohols, and some proteins(natural polymers), and washing liquids, powders, and detergents all act as sur-factants. It is the reason why the kinetics of wetting and spreading of surfactantsolutions is under investigation in this book.

On the other hand a number of solid substrates — printing materials, textiles,hairs — when in contact with liquids are porous in different degrees. In spite ofmuch experimental and practical experience in the area, only a limited number ofpublications are available in the literature that deals with fundamental aspects of thephenomenon. We show in this book that spreading kinetics over porous substratesdiffers substantially as compared with spreading over nonporous substrates.

Aiming at a logical progression in the problems treated with discussion ateach level, building albeit not rigidly, upon the material that came earlier, thebook can be divided into two parts: Chapter 1 to Chapter 3 form one part, andChapter 4 and Chapter 5 constitute the other. Chapter 1 is key to the former inthat its reading is a must for the understanding of Chapter 2 and Chapter 3. Toa large extent Chapter 4 and Chapter 5 can be read independently from thepreceding chapters, yet they are tied to each other and to the previous three.

Chapter 1 introduces surface forces and a detailed critical analysis of thecurrent understanding of Young’s equation, the building block in most wettingand spreading research and in a number of publications. The surface forces arealso frequently referred to in the literature as colloidal forces and disjoiningpressure. All these terms are used as equivalents in this book, following appro-priate clarification of concepts, terminology, and origins. Colloidal forces act inthin liquid films and layers when thickness goes down to about 10–5 cm = 0.1μm = 102 nm. Below this thickness the surface forces or disjoining pressurebecome so increasingly powerful that they dominate all other forces (for example,capillary forces and gravity). Accordingly, surface forces determine the wettingproperties of liquids in contact with solid substrates. One purpose of Chapter 1through Chapter 3 is to show that progress in the area of equilibrium and dynamicsof wetting demands due consideration of surface forces action in the vicinity of


the three-phase contact line. Chapter 2 and Chapter 3 look sequentially at theequilibrium and kinetics or dynamics of wetting, showing that the action ofsurface forces determines all equilibrium and kinetics features of liquids in contactwith solids. Note that Chapter 3 cannot be read and understood without readingthe introduction to the chapter.

Colloidal forces or disjoining pressure are well known and widely used incolloid science to account for equilibrium and dynamics of colloidal suspensionsand emulsions. The current theory behind colloidal forces between colloidalparticles, drops, and bubbles is the DLVO theory, an acronym made after thenames of Derjaguin (B.V.), Landau (L.D.), Verwey (E.J.W.) and Overbeek(J.Th.G.). The same forces act in the vicinity of the three-phase contact line, andtheir action is as important in this case as it is in the case of colloids. Unfortu-nately, most authors currently ignore the action of colloidal forces when discuss-ing the equilibrium and dynamics of wetting. It is our belief that this has hamperedprogress in the area of wetting phenomena for decades.

Chapter 4 and Chapter 5 are devoted to a detailed discussion of some recent,albeit still fragmentary, developments regarding the kinetics of spreading overporous solid substrates, including the case of hydrophobic substrates in thepresence of surfactants. Noteworthy are some new and universal spreading lawsin the case of spreading over thin porous layers discussed in Chapter 4. Somearguments and theory in Chapter 5 are experiment-discussion oriented and heu-ristic or semiempirical in approach (Section 5.4 and Section 5.5) and should bejudged accordingly. To our understanding, little is well established about spread-ing over hydrophobic substrates in the presence of surfactants. Our treatment ofthe spontaneous adsorption of surfactant molecules on a bare hydrophobic sub-strate ahead of the moving liquid front, making an initially hydrophobic substratepartially hydrophilic, allows a good description of a number of phenomena. Yetan understanding of the actual mechanism of transfer of surfactant molecules ina vicinity of the three-phase contact line will require considerable theoretical andexperimental efforts. We close the book with a few comments and warnings ina chapter of conclusions.

Victor M. Starov Loughborough University, Leicestershire, United Kingdom

Manuel G. VelardeInstituto Pluridisciplinar, Universidad Complutense, Madrid, Spain

Clayton J. RadkeUniversity of California at Berkeley


Acknowledgments

In 1974 Victor M. Starov met Prof. Nikolay V. Churaev, the beginning of acollaboration that has continued for more than 30 years and for which authorStarov would like to express very special thanks. Churaev involved Starov in theinvestigation of wetting and spreading phenomena in the former Surface ForcesDepartment, Moscow Institute of Physical Chemistry (MIPCh), Russian Academyof Sciences. This collaboration soon included a number of other colleaguesfrom MIPCh; appreciation is extended to these, especially professors Boris V.Derjaguin, Georgy A. Martynov, Vladimir D. Sobolev, and Zinoviy M. Zorin.

In 1981 Starov took the position of head of the Department of AppliedMathematics, Moscow University of Food Industry. He organized a weekly sem-inar there, where virtually all problems presented in this book were either solved,initiated, or at least discussed. These seminars were carried on until the Soviet Unioncollapsed. Author Starov would like to thank all members of the seminar butespecially professors Anatoly N. Filippov and Vasily V. Kalinin, and Drs. YuryE. Solomentsev, Vladimir I. Ivanov, Sergey I. Vasin, and Vjacheslav G. Zhdanov.

In 1987, the University of Sofia celebrated its centennial. This book’s firsttwo authors, Victor M. Starov and Manuel G. Velarde, were honored by beingchosen by Prof. Ivan B. Ivanov to be centennial lecturers at his university. Beyondbeing an honor, this was a lucky event in their lives. Both knew of Ivanov forquite some time but had not met him earlier nor had they worked together in thesame field, although both had common interests in the interfacial phenomena.While in Sofia, hearing each other lecturing and discussing science “and beyond,”they felt that it would be interesting to work together one day, particularly inexploring the consequences of surface tension and surface tension gradients, thelatter of which, e.g., creates flow or alters an existing one (the Marangoni effect).

In 1991 Starov was able to visit with Manuel G. Velarde at the InstitutoPluridisciplinar of the Universidad Complutense, Madrid, Spain. Both were for-tunate once more in being visited by Dr. Alain de Ryck, a young French scientistand brilliant experimentalist. He produced experiments where both Starov andVelarde were able to observe the striking role of the Marangoni effect in thespreading of a surfactant droplet over the thin aqueous layer. Later, the scientificrelationship between the first two authors of this book was strengthened by thevisit of Prof. Vladimir D. Sobolev, MIPCh, an outstanding scientist who wentbeyond being a highly skilled experimentalist. His work cemented the earliermentioned scientific relationship and collaboration between Starov and Velarde.It was further enhanced when the former moved from Moscow to the ChemicalEngineering Department, Loughborough University, United Kingdom, in 1999.There, Sobolev also worked with both Starov and Velarde, and this was the


beginning of numerous Loughborough–Madrid exchanges involving also severalyounger colleagues: Drs. Serguei R. Kosvintsev, Serguei A. Zhdanov, and AndreL. Zuev.

Then in 2001, the first two authors of this book jointly organized a summerschool on wetting and spreading dynamics and related phenomena at El Escorial,Madrid, under the sponsorship of the Universidad Complutense Summer Pro-gramme. Economic support also came from the European Union (under theICOPAC Network), the European Space Agency (ESA), Fuchs Iberica, L’Oreal,Inescop, and Unilever, Spain. Among the prestigious speakers from Bulgaria,France, Germany, Israel, the United States, and Spain was one of the invitedlecturers, the third author of this book, Clayton J. Radke. We decided not toproduce proceedings of that school, but soon after, the three future coauthors ofthis book started thinking of writing a joint monograph. Indeed, the present bookis the result of our concern about the lack of systematized knowledge on wettingand spreading dynamics, i.e., the lack of a monograph for the use of basic andapplied scientists, applied mathematicians, chemists, and engineers.

Two other schools are also worth mentioning. One on complex fluids, wetting,and spreading-related topics, coordinated by Velarde, took place in 1999 atLa Rabida, Huelva, Spain. The other course, much more focused on spreadingproblems, coordinated by Starov, was scheduled in 2003 at CISM (InternationalCenter for Mechanical Sciences) in Udine, Italy. There are proceedings of thelatter (“Fluid mechanics of surfactant and polymer solutions,” edited by Starovand Ivanov; Springer Verlag, 2004)) but not of the former. In the past few yearsseveral other workshops, discussion meetings, and international conferences tookplace in Madrid and Loughborough on the subject.

The authors would like to express their gratitude to Nadezda V. Starova.Without her energy, endless patience, kindness, and expertise, this book mostsurely would have never been finished. We are also happy to thank Maria-JesusMartin (Madrid) for her help in the preparation of the manuscript.

We wish to express our gratitude to the coauthors of our joint publications:Nikolay N. Churaev, Boris V. Derjaguin (deceased), Ivan B. Ivanov, VladimirI. Ivanov, Vasiliy V. Kalinin, Olga A. Kiseleva, Serguei R. Kosvintsev, Georgy A.Martynov, David Quere, Alain de Ryck, Ramon G. Rubio, Victor M. Rudoy,Vladimir D. Sobolev, Serguei A. Zhdanov, Pavel P. Zolotarev, and Zinoviy M.Zorin.

We also would like to recognize the following colleagues, fruitful discussionswith whom stimulated our research: Anne-Marie Cazabat, Pierre-Gilles deGennes, Benoit Goyeau, George (Bud) Homsy, Dominique Langevin, FranciscoMonroy, Alex T. Nikolov, Francisco Ortega, Len Pismen, Yves Pomeau, UweThiele, and Darsh T. Wasan.

Preparation of the manuscript was supported by a grant from the RoyalSociety, United Kingdom, which we would like to acknowledge. We wish toparticularly acknowledge the support by Prof. John Enderby. The final revisionof the manuscript was done while Manuel G. Velarde was Del Amo FoundationVisiting Professor with the Department of Mechanical Engineering and Environ-


mental Sciences of the University of California at Santa Barbara. This waspossible thanks to the hospitality of Prof. George M. Homsy.

Last but not least, we acknowledge the support for the research leading tothis book which came from the Engineering and Physical Sciences ResearchCouncil, United Kingdom (Grants EP/D077869 and EP/C528557), and from theMinisterio de Educacion y Ciencia, Spain (Grants MAT2003-01517, BQU2003-01556, and VEVES).


1

1 Surface Forces and the Equilibrium of Liquids on Solid Substrates

INTRODUCTION

In this chapter, we shall give a brief account of the theory and experimentalevidence of the action of surface forces, i.e., forces needed to account for phe-nomena occurring near surfaces, very thin layers, corners, borders, contact lines,etc. All forces do originate at the microscopic level, but we shall look at thephenomenological, macroscopic manifestations of those forces. In particular, weshall emphasize the role of the so-called disjoining pressure. Such terminologyis a bit misleading because, in a number of cases, action would be just theopposite: conjoining pressure (attraction). However, current use or historicalreasons lead us to maintain the term disjoining pressure, whatever the particularsituation might be.

The disjoining pressure acts in the vicinity of the three-phase contact line,and its action becomes dominant, e.g., as a liquid profile approaches a solidsubstrate, or with colloidal particles or drops. In the study of wetting and spread-ing processes, its importance seems less common than in colloid science, in spiteof the same nature of the forces and the same level of necessity.

The relationship between the disjoining pressure and the thickness of a liquidfilm is frequently referred to as disjoining pressure isotherm because it is generallymeasured at a given temperature. It is noteworthy that for water and aqueoussolutions, the disjoining pressure isotherm has an S-shape, hence alerting us toinstability, metastability, and bistability (in the spirit of van der Waals and Max-well description of thermodynamic equilibrium phases). Our life is very muchtuned to the properties of water (as carbon is also linked to life). To what extentdoes the S-shape of the disjoining pressure isotherm of water affect life? This isan interesting question to be answered. However, we do not address this problemin this book.

We shall start with a discussion about a well-known and much used Young’sequation in spreading and wetting dynamics. We advise reading the original paperby Young, as it is a masterpiece of scientific literature. Yet we hope to convincethe reader of the ill-founded thermodynamic support of the (historical) standardform of such relationship. We shall argue and prove that the thermodynamically


2 Wetting and Spreading Dynamics

sound Young’s equation, which is frequently referred to as the Derjaguin–Frumkinequation, is only possible if due account is given to the disjoining pressure. Weshall see that the disjoining pressure action either in the case of complete orpartial wetting always leads to the formation of a thin liquid layer in the vicinityof the three-phase contact line. The latter results in a microscopic flow that isdetermined by both the disjoining pressure action and the topography of thesurface (roughness, heterogeneity, chemical, or otherwise). As a result, never isthere a real three-phase contact line but only an apparent macroscopic contactline.

Then, we shall describe and comment upon the three most used componentsof the disjoining pressure. Finally, we shall consider at the heuristic level thestatic contact angle hysteresis when, for example, a drop spreads on a smoothand homogeneous solid substrate. We shall show that microscopic flow in thevicinity of the apparent three-phase contact line is unavoidable. The complicationintroduced by such microscopic flow seems responsible for the present lack of asound theory of the kinetics of spreading in the case of partial wetting, in contrastto the case of complete wetting, where the theory is well developed and leads toa quite good agreement with most, if not all, experimental observations.

1.1 WETTING AND YOUNG’S EQUATION

Why do droplets of different liquids deposited on identical solid substrates behaveso differently? Why do identical droplets — for example, aqueous droplets depos-ited on different substrates — behave so differently?

When we attempt to make a uniform layer of mercury on a glass surface, wefind it impossible. Each time we try, the mercury layer will immediately form adroplet, which is a spherical cap with the contact angle bigger than π/2(Figure 1.1). Note, the contact angle is always measured inside the liquid phase(Figure 1.1 to Figure 1.3). However, it is easy to make an oil layer (hexane ordecane) on the same glass surface; for this purpose an oil droplet can be depositedon the same glass substrate, and it will spread out completely (Figure1.3). In thiscase, the contact angle decreases with time down to a zero value.

Now let us try the same procedure with an ordinary tap water droplet. Anaqueous droplet deposited on the same glass substrate spreads out only partiallydown to some contact angle, θ, which is in between 0 and π/2 (Figure 1.2). Thatis, an aqueous droplet on a glass surface behaves in a way that is intermediatebetween the behavior of the mercury and oil.

These three cases (Figure 1.1, Figure 1.2, and Figure 1.3) are referred to as:nonwetting, partial wetting, and complete wetting, respectively.

FIGURE 1.1 Nonwetting case: contact angle is bigger than π/2.

θ


Surface Forces and the Equilibrium of Liquids on Solid Substrates 3

Now let us try to make a water layer on a Teflon surface. We will be unableto do this, exactly in the same way as we were unable to in the case of mercuryon a glass surface. That is, the same aqueous droplet can spread out partially ona glass substrate and does not spread at all on a Teflon substrate. The inabilityto spread on the Teflon surface indicates that the wetting or nonwetting is not aproperty of the liquid but rather a property of the liquid–solid pair.

In broader terms, complete wetting, partial wetting, and nonwetting behaviorare determined by the nature of both the liquid and the solid substrate.

Let us consider a picture presented in Figure 1.4. Let us say that the three-phase contact line is the line where three phases: liquid, solid, and vapor meet.Consideration of forces in the tangential direction at the three-phase contact lineresults in the well-known Young’s equation, which connects three interfacialtensions, γsl, γsv, and γ with the value of the equilibrium contact angle, θNY

(Figure 1.4), where γsl, γsv, and γ are solid–liquid, solid–vapor, and liquid–vaporinterfacial tensions, respectively:

cos θNY = (γsv – γsl)/γ (1.1)

FIGURE 1.2 Partial wetting case: the contact angle is in between 0 and π/2.

FIGURE 1.3 Complete wetting case: the droplet spreads out completely, and only thedynamic contact angle can be measured, which tends to become zero over time.

FIGURE 1.4 Interfacial tensions at the three-phase contact line. R is the radius of thedroplet base, ℜ is the radius of the droplet. The droplet is small enough, and the gravityaction can be neglected.

θ

θ(t)

R

ℜ

θ

γ

γsv γsl



Note, we marked the equilibrium contact angle in Equation 1.1 as θNY, andwe see in the following section that there is a good reason for that.

According to Figure 1.4, the complete wetting case corresponds to the casewhen all forces cannot be compensated in the tangential direction at any contactangle, that is, if γsv > γsl + γ. Partial wetting case, according to Equation 1.1,corresponds to 0 < cos θNY < 1, and, last, the nonwetting case corresponds to1 < cos θNY < 0. That is, Equation 1.1 reduces complete wettability, partialwettability, and nonwettability cases to the determination of three interfacialtensions, γsl, γsv, and γ. It looks like everything is very easy and straightforward.However, as we see in the following section, the situation is far more complexthan that.

Let us try to deduce Equation 1.1 using a rigorous theoretical procedure basedon the consideration of the excess free energy of the system presented inFigure 1.4. Let us assume that the excess free energy of the small droplet (neglect-ing the gravity action) is as follows:

(1.2)

where S is the area of the liquid–air interface, Pe = Pa – Pl is the excess pressureinside the liquid, Pa is the pressure in the ambient air, Pl is the pressure insidethe liquid, and R is the radius of the drop base. The last term on the right-handside of Equation 1.2 gives the difference between the energy of the part of thebare surface covered by the liquid drop as compared with the energy of the samesolid surface without the droplet.

Note that the excess pressure, Pe, is negative in the case of liquid droplets(concave liquid–air interface) and positive in the case of meniscus in partially orcompletely wetted capillaries (convex liquid–air interface).

Let h(r) be the unknown profile of the liquid droplet; then the excess freeenergy, as given by Equation 1.2, can be rewritten as

(1.3)

Now we use one of the most fundamental principles: any profile, h(r), in thelatter expression, should give the minimum value of the excess free energy as inEquation 1.2. Details of the procedure are given in the next chapter (see Section 2.2).

Under equilibrium conditions, the excess free energy should reach its mini-mum value. The mathematical expressions for this requirement are the followingconditions: (1) the first variation of the free energy, δΦ, should be zero, (2) thesecond variation, , should be positive, and (3) the transversality condition atthe drop perimeter at the three-phase contact line — that is, at r = R — shouldbe satisfied. In Section 2.2, these conditions are discussed in more detail, and it

Φ = + + −γ π γ γS PV Re sl sv2( ),

Φ = + ′ + + −( )∫2 1 2

0

π γ γr h P h dre sl sv

R

δ2Φ



is shown that actually one extra condition should be fulfilled. However, at themoment we will ignore this extra condition, because it is easy to check that thiscondition is always satisfied in the case of the excess free energy given byEquation 1.3. Conditions 1 and 2 are actually identical to those for a minimumof regular functions. Condition 3 is usually forgotten and deduced using a differentconsideration.

Condition 1 results in the Euler equation, which gives an equation for thedrop profile:

,

where

or

(1.4)

Solution of the latter equation is a part of the sphere of radius 2γ/Pe (Figure 1.4).The second condition gives:

,

or

,

which is always satisfied. The latter means that Equation 1.4 really gives aminimum value to the excess free energy in Equation 1.3.

Now the third, transversality condition is as follows:

∂∂

− ∂∂ ′

=f

h

d

dr

f

h0

f r h P he sl sv= + ′ + + −

γ γ γ1 2 ,

γr

ddr

rh

h

Pe′

+ ′( )

=

12

12

∂∂ ′

>2

20

f

h

γ

10

23 2

+ ′( )>

h/

f hfh

r R

− ′ ∂∂ ′

=

=

0



or

Taking into account that we conclude from the previous equation

(1.5)

Figure 1.4 shows that

Substitution of the latter expression in Equation 1.5 results in Equation 1.1.To summarize: application of the rigorous mathematical procedure to excess

free energy given by Equation 1.3 results in:

1. A spherical profile of the droplet with a radius of the curvature

, (1.6)

2. The Young’s equation (Equation 1.1) for the equilibrium contact angleθNY.

The equation for the equilibrium contact angle shows that the derivation ofYoung’s equation (Equation 1.1) is based on a firm theoretical basis if we adoptthe expression for the free energy, Equation 1.3. The free energy equation consid-eration means that Young’s equation (Equation 1.1) is valid only when the adoptedexpression for the excess free energy (Equation 1.3) is valid. Consideration of thinfilms on curved surfaces was undertaken by I. Ivanov and P. Kralchevsky in [9].

Let us ask ourselves a question: How many equilibrium states can a thermo-dynamic system have? The answer is well-known: either one or, in some specialcases, two or even more states that are separated from each other by potentialbarriers. According to condition 1 and condition 2, we get an infinite and con-tinuous set of equilibrium states, which are not separated from each other bypotential barriers. Young’s equation does not specify the equilibrium volume ofthe droplet, V, or the excess pressure inside the drop, Pe, which can be any negativevalue. Both volume of the droplet and the excess pressure can be arbitrary. Thelatter means that the volume of the droplet is not specified; a droplet of anyvolume can be at the equilibrium.

r h P h hr h

he sl svγ γ γ γ

11

2

2+ ′ + + −

− ′ ′

+ ′

=

=r R

0.

hr R=

= 0,

γ γ γ1

02+ ′

+ −

=

=hsl sv

r R

′ = −=

hr R NYtan .θ

RPe

= − 2γ



That means Young’s equation (Equation 1.1) is in drastic contradiction withthermodynamics. Why is it so? Where is there a mistake? Definitely not in thederivation. That means we should go back to basics. Is the expression for theexcess free energy (Equation 1.3) correct?

At equilibrium, the following three equilibrium considerations should hold:

1. Liquid in the droplet must be in equilibrium with its own vapor.2. Liquid in the droplet must be in equilibrium with the solid.3. Vapor must be in equilibrium with the solid substrate.

Step by step, in the following section, we show that none of these threeequilibriums are taken into account by the expression for the excess free energy(Equation 1.3).

The first requirement in the list above results in the equality of chemicalpotentials of the liquid molecules in vapor and inside the droplet. This results inthe following expression of the excess pressure, Pe:

, (1.7)

where is the molar volume of liquid, is the pressure of the saturated vaporat the temperature T, R is the gas constant (do not confuse with the radius of thedrop base), p is the vapor pressure that is in equilibrium with the liquid droplet.Equation 1.7 determines the unique equilibrium excess pressure Pe and, hence,according to Equation 1.6, the unique radius of the droplet, ℜ.

We remind the reader now that the excess pressure inside the drop, Pe, shouldbe negative (pressure inside the droplet is bigger than the pressure in the ambientair). That means that the right-hand side in Equation 1.7 should also be negative,but negativity is possible only if p > ps, that is, the droplets can be at equilibriumonly with oversaturated vapor. It is really troublesome because the equilibrationprocess goes on for hours, and it is necessary to keep oversaturated vapor overa solid substrate under investigation for hours. To the best of our knowledge,nobody can do that, which would mean that it is difficult to experimentallyinvestigate equilibrium droplets on the solid substrate. There is a plethora ofinvestigations published on the literature of the equilibrium contact angles ofdroplets on solid substrate. The previous consideration shows that the contactangles measured are mostly not in equilibrium. It is a different story when contactangles were really measured. Only in Chapter 3 will we be ready to clarify thesubject completely (see Section 3.10).

Unfortunately, this is not the end of the troubles encountered with Young’sequation (Equation 1.1) because now we should consider requirements of theequilibriums 2 and 3. Let us assume that we can create, at least theoretically, anoversaturated vapor over the solid substrate and wait long enough until theequilibrium is reached. Now the liquid molecules in the vapor are at equilibrium

PRT

v

p

pem

s= ln

νm ps



with the liquid molecules in the droplet. Note that the solid–liquid interfacialtension, γsl, differs from the solid–vapor interfacial tension, γsv. If they are notdifferent, then according to Equation 1.1, the contact angle is equal to 90o (anintermediate case between partial wetting and nonwetting). In the case of partialwetting or complete wetting, γsl < γsv. The latter expression means that thepresence of liquid on the solid substrate results in lower surface tension ascompared with the surface tension of the bare solid surface, γsv. Now back to ourtheoretical case of the liquid droplet on the solid substrate in equilibrium withthe oversaturated vapor. We should now take into account the equilibrium betweenthe liquid vapor and the solid surface; it is unavoidable as the liquid moleculeadsorption on the solid substrate and the presence of liquid molecules on thesurface changes the initial surface tension. This means that the liquid moleculesfrom the vapor must adsorb on the solid substrate outside the liquid droplet underconsideration. The latter consideration results in the formation of an adsorptionliquid film on the surface and a new interfacial tension, γhv, where h is thethickness of the adsorbed layer.

It may be said that it does not make sense to talk about a monolayer, or inthe best case, several layers of the adsorbed liquid molecules on the solid sub-strate, as the influence on the macroscopic droplet will be negligible. Let usconsider a simple but important example of the presence of a single monolayerdrastically changing the wetting property. Take a microscope glass cover and putan aqueous droplet on this surface. The droplet will form a contact angle, whichconsiderably depends on the type of the glass, and in some special case (whichwe consider now), it will be as small as 10˚. Now let us place a monolayer ofoil on the glass surface (reminder: a monolayer is a layer with thickness of1 molecule). Now again, let us place a water droplet on a new glass surfacecovered by a monolayer of oil. The droplet will form a contact angle that is higherthan 90˚. That is, the presence of only one tiny monolayer changed partial wettingto nonwetting.

Now, back to the droplets on the solid surface at equilibrium with the over-saturated vapor. As we now understand, the adsorption of vapor on the solidsubstrate is very important, and instead of the interfacial tension of the bare solidsurface, γsv, we should use γhv. The latter interfacial tension is to be investigatedin Chapter 2 (Section 2.1).

The previous consideration shows that to investigate equilibrium liquid drop-lets, the following procedure should be followed:

• The solid substrate under investigation should be kept in the atmo-sphere of the oversaturated vapor until equilibrium adsorption of vaporon the solid substrate is reached, and a new interfacial tension γhv shouldbe measured.

• Then, the droplet, which has a size that should be in equilibrium withthe oversaturated vapor, should be deposited and kept until the equi-librium is reached.



• Young’s equation (Equation 1.1) should now be rewritten as

. (1.8)

The characteristic time scale of the latter processes depends on the liquidvolatibility and viscosity and is, in general, of hours of magnitude. No such kindof experiment has ever been attempted in the atmosphere of an oversaturated vaporto the best of the author’s knowledge. This would mean that equilibrium liquiddroplets of volatile liquids probably have never been observed experimentally.

It is obvious from the same reasons as given before that the thickness of theadsorbed layer, h, depends on the vapor pressure in the ambient air; that is, γhv

is a function of the pressure in the ambient air and, hence, according to Equation1.8, the contact angle changes with vapor pressure. Is this dependency strong orweak? The answer will be given in Chapter 2 (Section 2.1 and Section 2.3).

Is this the end of the problems with Young’s equation (Equation 1.1)? Unfor-tunately not, because we still did not consider the last, but not the least, require-ment of the equilibrium (3). In Figure 1.5, an equilibrium liquid droplet ispresented in contact with an equilibrium-adsorbed liquid film on the solid surface.What happens in the vicinity of the line where they meet?

Is the situation presented in Figure 1.5 possible? The answer is obvious: suchsharp transition from the liquid droplet to the liquid film is impossible. On theline shown by the arrow, the capillary pressure will be infinite. Hence, it shouldbe a smooth transition from the flat equilibrium liquid film on the solid surfacesto the spherical droplet, as shown in Figure 1.6, where this smooth transition isshown.

Let us call this region, where transition from a flat film to the droplet takesplace, a transition zone. The presence of the transition zone leads us into muchbigger problems than before, because pure capillary forces cannot keep the liquidin this zone in equilibrium; the liquid profile is concave (hence, the capillarypressure under the liquid surface is higher than in the ambient air) to the rightfrom the arrow in Figure 1.6, and the liquid profile is convex (hence, the capillary

FIGURE 1.5 Cross section of an equilibrium liquid droplet (at oversaturation) in contactwith an equilibrium-adsorbed liquid film on the solid substrate. What happens on the line(shown by an arrow) where they meet?

cos θ γ γγe

hv sl= −

θ



pressure under the liquid surface is lower than in the ambient air) to the left fromthe arrow in Figure 1.6. We come back to this paradox a bit later. Now we onlyremark that the consideration of this paradox was one of the motivations to replacethe name colloid science with a new name, colloid and interface science. How-ever, for a moment let us forget about the transition zone.

The preceding consideration shows that Young’s equation can probably beused only in the case of nonvolatile liquids, as we have too many problems withvolatile liquids. Can a liquid really be a nonvolatile one? Usually, low volatibilitymeans liquids with big molecules that have high viscosity and a correspondinghigher characteristic time scale of equilibration process with the oversaturatedvapor. In spite of that, let us assume that the liquid is nonvolatile. In the case ofpartial wetting, as we have already seen, liquid droplets cannot be in equilibriumwith a bare solid surface. There should always be at equilibrium an adsorptionlayer of the liquid molecules on the solid substrate in front of the droplet on thebare solid surface. If the liquid is volatile, then this layer is created by means ofevaporation and adsorption. However, if the liquid is nonvolatile, the same layershould be created by means of flow from the droplet edge onto the solid substrate.As a result, at equilibrium the solid substrate is covered by an equilibrium liquidlayer of thickness, h. The thickness of the equilibrium liquid film, h, is determined(as we see in the following section), by the potential of action of surface forces.The characteristic time scale of this process is hours, because it is determined bythe flow in the thinnest part in the vicinity of the apparent three-phase contactline, where the viscose resistance is very high. During these hours, evaporationof the liquid from the droplet cannot be ignored, and we go back to the problemof volatibility.

Let us assume, however, that the equilibrium film, after all, forms in front ofthe liquid droplet, and we have waited enough for the equilibrium. However, nowwe have again the following three interfacial tensions: γ, γsl, and γvh, which areliquid–vapor interfacial tension, solid–liquid interfacial tension and solid sub-strate, covered with the liquid film of thickness h–vapor interfacial tensions. Wecan refer back to the same problem as in the case of volatile liquid. We can neithermeasure the interfacial tension, γvh, nor use it in Equation 1.8. However, there isan answer, and the answer will be given in Section 2.1.

FIGURE 1.6 Transition zone from the flat equilibrium liquid film on a solid surface tothe liquid droplet. The arrow shows the point to the right where the liquid profile is concaveand to the left where the profile is convex.



This would mean that even in the case of nonvolatile liquids, the applicabilityof Young’s equation (Equation 1.1) still remains questionable.

In view of the preceding features, from now on in this book we shall be usingapparent three-phase contact line because there is no such line at the microscopicscale.

1.2 SURFACE FORCES AND DISJOINING PRESSURE

The presence of adsorbed liquid layers on a solid substrate is a result of the actionof some special forces, referred to as surface forces.

Let us go back to Figure 1.6 and consider the transition zone between thedroplet and the flat liquid films in front of it. It looks like the profile presentedin Figure 1.6 cannot be in equilibrium because capillary pressure should changethe sign inside the transition zone, and it is in contradiction with the requirementof the constancy of the capillary pressure everywhere inside the droplet. Someadditional forces are missing. The mentioned problem was under considerationby a number of scientists for more than a century. Their efforts resulted inconsiderable reconsideration of the nature of wetting phenomena.

A new class of phenomena has been introduced [1]: surface phenomena,which are determined by the special forces acting in thin liquid films or layersin the vicinity of the apparent three-phase contact line.

Surface forces are well-known and are widely used in colloid and interfacescience. They determine the stability and behavior of colloidal suspensions andemulsions. In the case of emulsions/suspensions, their properties and behavior(stability, instability, rheology, interactions, and so on) are completely determinedby surface forces acting between colloidal particles or droplets. This theory iswidely referred to as the DLVO theory [1] after the names of four scientists whodeveloped the theory: Derjaguin, Landau, Vervey, and Overbeek. No doubt thatall colloidal particles have a rough surface and, in a number of cases, evenchemically inhomogeneous surfaces (living cells, for example). Roughness andinhomogeneity of colloidal particles can modify substantial surface forces: theirnature, magnitude, and range of action. However, the roughness and inhomoge-neity of the surface of the colloidal particles does not influence the main phe-nomenon; all their interactions and properties are determined by the action of thesurface forces [2].

There is something unconventional about wetting studies as compared withanalogous studies in colloid and interface science. It is widely (and erroneously)accepted that roughness and nonuniformity of the solid substrate in contact withliquids can in itself explain wetting features, without consideration of the surfaceforces acting in a vicinity of the apparent three-phase contact line. As a result,the influence of surface forces on the kinetics of wetting and spreading is muchless recognized than in the study of colloidal suspensions and emulsions, in spiteof the same nature of surface forces.

It has been established that the range of action of surface forces is usuallyof the order of 0.1 µm [1]. Note that in the vicinity of the apparent three-phase



contact line, r = R (Figure 1.4), the liquid profile, h(r), tends to be of zerothickness. This thickness means that close to the three-phase contact line, surfaceforces come into play and their influence cannot be ignored.

A manifestation of the action of surface forces is the disjoining pressure. Toexplain the nature of the disjoining pressure, let us consider the interaction oftwo thick, plain, and parallel surfaces divided by a thin liquid layer of thicknessh (aqueous electrolyte solution, for example). The surfaces are not necessarily ofthe same nature as two important examples show: (1) one is air, one is a liquidfilm, and one is solid support, and (2) both surfaces are air, and one is a liquid film.Example 1 is referred to as a liquid film on a solid support and models the liquidlayer in the vicinity of the three-phase contact line, Example 2 is referred to asa free liquid film. There is a range of experimental methods to measure theinteraction forces between these two surfaces as a function of the thickness, h(gravity action is already taken into account) (Figure 1.7) [1,3,4].

If h is bigger than ≅10–5 cm = 0.1 µm, then the interaction force is equal tozero. However, if h < 10–5 cm, then an interaction force appears. This force candepend on the thickness, h, in a very peculiar way. The interaction forces dividedby the surface area of the plate has a dimension of pressure and is referred to asthe disjoining pressure [1]. Note that this term is a bit misleading, because thementioned force can be both disjoining (repulsion between surfaces) and con-joining (attraction between surfaces).

Now we discuss the physical phenomena behind the existence of surfaceforces. Let us consider a liquid–air interface. It is obvious that the physicalproperties of the very first layer on the interface are substantially different fromthe properties of the liquid (in bulk) far from the interface. What can we sayabout the properties of the second, third, and other layers? It is understandablethat the physical properties do not change by jumping from the very first layeron the interface to the subsequent layers, but the change proceeds in a continuousway. This continuous change results in the formation of a special layer, whichwe refer to as the boundary layer, where all properties differ from correspondingbulk properties. Do not confuse the introduced boundary layer with a boundarylayer in hydrodynamics; they have nothing do to with each other.

Such boundary layers exist in proximity to any interface: solid–liquid,liquid–liquid, or liquid–air. In the vicinity of the apparent three-phase contact

FIGURE 1.7 Measurement of interaction between two thick plates 1 and 2, possibly madeof different materials, with a thin layer 3 in between.

2

3

1

h



line (Figure 1.8), these boundary layers overlap. The overlapping of boundarylayers is the physical phenomenon that results in existence of surface forces. Thesurface force per unit area has a dimension of pressure and is referred to asdisjoining pressure, as we have already mentioned in the preceding section. Letthe thickness of the boundary layers be δ. In the vicinity of the three-phase contactline, the thickness of a droplet, h, is small enough, that is, h ~ δ, and henceboundary layers overlap (Figure 1.8), which results in the creation of disjoiningpressure. The above mentioned characteristic scale, δ ~ 10–5 cm, determines thecharacteristic thickness where disjoining pressure acts. This thickness is referredto as the range of disjoining (or surface forces) action, ts.

The main conclusion: the pressure in thin layers close to the three-phasecontact line is different from the pressure in the bulk liquid, and it depends onthe thickness of the layer, h, and varies with the thickness, h.

In the following, we briefly review the physical phenomena that result in theformation of the above mentioned surface forces and disjoining pressure.

COMPONENTS OF THE DISJOINING PRESSURE

Several physical phenomena have been identified for the appearance of the dis-joining pressure. Here, we consider only three of them.

Molecular or Dispersion Component

Let us start with the most investigated molecular or dispersion component ofsurface forces. Note that in a number of cases, this component is the weakestamong all the other components considered in the following section. Surprisingly,this component is used more frequently than others.

It is well known that at relatively large distances (but still in the range ofangstroms, that is, 108 cm) all neutral molecules interact with each other, and theenergy of this interaction is proportional to const/r 6, where r is the distancebetween molecules. This is apparent by examining two surfaces made of different

FIGURE 1.8 The liquid profile in the vicinity of the apparent three-phase contact line:(1) bulk liquid, where boundary layers do not overlap, (2) boundary layer in the vicinityliquid–air and liquid–solid interfaces, (3) a region where boundary layers overlap, and (4)flat thin equilibrium film. The latter two are the regions where disjoining pressure acts.

2

21

3 4



materials placed inside an aqueous electrolyte solution at a distance, h, from eachother (Figure 1.7).

Calculation of the molecular contribution to disjoining pressure, Πm, has beenapproached in two ways: from the approximation of interactions as a pairwiseadditive, and from a field theory of many-body interactions in condensed matter.The simpler and, historically, earlier approach followed a theory based on sum-ming individual London–van der Waals interactions between molecules pair-by-pair, undertaken by Hamaker [1].

The more sophisticated, modern theory of Πm was developed (see review [1])based on the consideration of a fluctuating electromagnetic field. In the following,we give an expression for the molecular component of the disjoining pressure,Πm, for a film of uniform thickness, h, between two semiinfinite phases in vacuum(for simplicity). The expression is [1]:

where c is the speed of light, s1 ≡ (ε1 – 1 + p2)1/2, s2 ≡ (ε2 – 1 + p2)1/2, and the dielectric constants ε1, ε2 are functions of imaginary frequency ω ≡ iξ, given by:

,

where is the imaginary component of the dielectric constant. In the limitingcase of film thickness h, small in comparison with the characteristic wavelength,λ, of the adsorption spectra of the bodies, the molecular component of disjoiningpressure is inversely proportional to the cube of film thickness [1]:

(1.9)

In the limiting case of h, large in comparison to λ on the other hand, disjoiningpressure turns out to be inversely proportional to the fourth power of film thick-ness [1]:

Πm N

kT

cp

s p s p

s p s p

p=+( ) +( )−( ) −( )π

ξ3

2 3 1 2

1 2

2exp(

ξξN

N

hc

) −

−∞

=

∞

∫∑ 1

1

10

++( ) +( )−( )

s p s p

s p s

1 1 2 2

3 3 2

ε εε −−( )

−

−

p

p hc

dpN

εξ

2

1

21exp

ε ξπ

ω ε ωω ξ

ω( )( )

i d= + ′′+

∞

∫12

2 2

0

′′ε ω( )

Πmh

i i

i=

− − +

�8

1 1

12 3

1 2

1π

ε ξ ε ξ

ε ξ

( ) ( )

( ) + =

∞

∫ ε ξξ

20

31( ).

id

A

hH



where s10 ≡ (ε10 – 1 + p2)1/2, s20 ≡ (ε20 – 1 + p2)1/2, and ε10, ε20 are the electrostaticvalues of the dielectric constants, i.e., the values of the dielectric constant at ξ = 0.

There are corresponding expressions for the molecular component of thedisjoining pressure of films of nonpolar liquids. Those expressions are presentedin Reference 1. However, the functional dependency

remains valid. In the following section, we use only the expression derived fromEquation 1.9 for the molecular component because the contribution of the dis-joining pressure at “big” film or layer thickness at h > λ is relatively small ascompared with the first part at h < λ.

For a sufficiently long time it was believed that the Lifshitz theory of van derWaals forces was but an elegant formalism, as the necessary dielectric constantsacross the entire frequency range could not readily be determined. Then Parsegianand Ninham discovered a technique for calculating those properties to an adequateapproximation from dielectric data (see the review [1]). Precise measurementsof Πm both in-thin and thin-liquid films are in good agreement with the theorypredictions [1]. However, the latter theory does not apply to films so thin as tohave dielectric properties that vary with thickness.

Using the first historically approximate direct summation of all molecularinteraction in the system, we obtain the following expression for the molecularor dispersion components of the disjoining pressure:

, (1.10)

where AH is referred to as the Hamaker constant, after the scientist who carriedout these calculations around a half-century ago [1]. The Hamaker constant, AH,

Πm

c

h

x

p

s p s p

s p=

+( ) +( )−( )

∞∞

∫∫�32 2 4

3

2

10

10 20

10π[

ss pex

20

11−( ) −

−]

[++( ) +( )−(

s p s p

s p

10 10 20 20

10 10

ε εε )) −( ) −

=−

s pe dpdx

B

hx

20 20

14

1ε

] ,

Πm

H

h

A

hh

B

hh

( )

,

,

=<

>

3

4

λ

λ

Πm = − A

hH

6 3πA A A A AH = + − −33 12 13 23,



depends on the properties of the phases 1, 2, and 3 through the Hamaker constants,Aij, of phases i and j. Equation 1.10 shows that the Hamaker constant can beeither positive (attraction) or negative (repulsion). Note that the functional depen-dency of the molecular component of the disjoining pressure, according to Equa-tion 1.10, coincides with the exact Equation 1.9. However, the precise value ofthe Hamaker constant, according to direct summation in Equation 1.10, can becompletely wrong. This is the reason why a number of approximations have beendeveloped to precisely calculate the Hamaker constant [1].

In the case of oil droplets on the glass surface, when the dispersion componentis the only component of the disjoining pressure acting in thin films, the dispersioninteraction is repulsive, i.e., the Hamaker constant is negative. In the following,we mostly consider the latter situation (thin liquid films on solid substrates) wherethe Hamaker constant is negative. For this purpose, we rewrite Equation 1.10 as

(1.11)

and just that constant, A, is referred to as the Hamaker constant. Note that thepositive Hamaker constant, A, now indicates a repulsion, and the negative constantindicates an attraction.

The characteristic value of the Hamaker constant is A ~ 10–14 erg (oil filmson glass, quartz, or mica surfaces). This value of the Hamaker constant showsthat when the liquid layer is at a thickness of h ~ 10–7 cm, the dispersion componentof the disjoining pressure is Πm ~ 10–14/10–21 = 107 dyn/cm2. Let us consider a smalloil droplet of a radius ℜ ~ 0.1 cm on a solid substrate (Figure 1.4); the surfacetension of oils is about γ ≅ 30 dyn/cm. The capillary pressure inside the sphericalpart of the droplet is

This value shows that in the vicinity of the three-phase contact line, the capillarypressure is much smaller than the disjoining pressure. Let us assume for a momentthat the droplet shape remains spherical until the contact with the solid substrate.However, as we have already seen in the preceding section, the capillary pressureis much smaller than the disjoining pressure and cannot counterbalance thedisjoining pressure. This means that the disjoining pressure action substantiallydistorts the spherical shape of droplets in the vicinity of the three-phase contactline. Droplets cannot retain their spherical shape up to the contact line. See further

Before further discussing the next electrostatic component of disjoining pres-sure, a few words should be said about the electrical double layer.

ΠmHA

hA

A= = −3 6

,π

2 2 300 1

6 102 2γℜ

⋅ = ⋅~.

.dyn cm/


consideration of the profile of liquid droplets in Section 2.3.


Electrical Double LayersNeutral molecules of many salts, acids, and alkalis dissociate into ions (cationsand anions) in water, forming aqueous electrolyte solutions. For example, NaCldissociates with the formation of a cation Na+ and an anion Cl–. Even more, ifwe assume water completely pure without any salts, acids, and so on, then thewater molecule itself, H2O, also dissociates according to the following dissocia-tion reaction: H2O ↔ H+ + OH–.

That is, even in pure water, both cations, H+, and anions, OH–, are alwayspresent. Note that the two ions H+ and OH– play the most important role inkinetics of wetting and spreading of aqueous solutions.

The total charge of cations is completely counterbalanced by the total chargeof anions in the bulk of the liquid. These ions are called free ions. All free ionscan be transferred both by means of convection (by the flow of water) and bydiffusion, if a gradient of concentration of any ion is imposed. Ions also can betransferred under the action of the gradient of electric potential (electromigration),either imposed or spontaneous.

In the aqueous electrolyte solutions, the majority of solid surfaces acquire acharge. Before mentioning the mechanism of formation of this charge, let usemphasize that these charges are mostly fixed rigidly on the solid surface andcan usually be moved only with the solids. There are two main mechanisms offormation of the charge of the solid surface in aqueous electrolyte solutions: thedissociation of surface groups (briefly discussed in the following section) and theunequal adsorption of different types of ions.

A considerable number of solid surfaces have the following type of surfacegroups on the solid–liquid interface R-OH, where R- is the group that is rigidlyconnected to the solid. The –OH groups can dissociate in aqueous solutions,which results in the formation of negatively charged groups, R-O– returning theH+ ion into the solution. According to this mechanism or a similar one, many ofthe solid surfaces (actually majority) in aqueous solutions acquire a negativesurface charge. It is obvious that this charge strongly depends on the pH ofsolution, i.e., depending on the concentration of H+ ions in the volume of solution;pH = log cH

+, where cH+ is the concentration of H+ in mol/l. Note that pH = 7

corresponds to the neutral solution, pH < 7 is an acidic solution, and pH > 7corresponds to an alkaline solution.

In all processes, the free and bound ions behave in different ways: free ionscan freely be moved, but the bound ions only move with the solid surface. Letus consider the distribution of ions in the close vicinity of a negatively chargedsurface in contact with aqueous electrolyte solution, for example, NaCl. NaCldissociates as NaCl → Na+ + Cl–. The electroneutrality condition requires equalconcentrations of cations and anions in the bulk solution, far from the chargedsurface. However, close to the charged surface, according to Coulomb’s law, thefree cations Na+ are attracted by the negatively charged solid surface, and thenegatively charged ions Cl– are repulsed from the same surface. As a result, theconcentration of cations is higher near the surface, and the concentration of anions



is lower than the corresponding concentration in the bulk solution. We recall theprocess of diffusion. The basic task of diffusion is to destroy all nonuniformitiesin the distribution of ions. In this case, the diffusion will attempt to make anexact opposite, in comparison with Coulomb’s interaction, to decrease the con-centration of cations near the surface and to increase the concentration of anions.As a result of these two opposed trends near the negatively charged surface, alayer of finite thickness is created in which the concentration of cations reachesits maximum near the surface and monotonically decreases into the depths of thesolution to its bulk value, whereas the concentration of anions monotonicallygrows from its minimum value near the surface to its bulk value in the depths ofthe solution. This layer, where the concentration of cations and anions differ fromtheir bulk values, is referred to as a diffusive part of the electrical double layer.The characteristic thickness of the diffusive part of electrical double layer is theDebye length, Rd. The characteristic value of the Debye length is

where the electrolyte concentration, C, should be expressed in mol/l. This expres-sion shows that the higher the electrolyte concentration, the thinner is the elec-trical double layer. For example, at C = 10–4 mol/l, Rd = 3·10–6 cm (which isconsidered as a large thickness), whereas at C = 10–2 mol/l, Rd = 3·10–7 cm (whichis considered as a very small thickness). The electrical double layer is formedfrom two parts; the first part is the charged surface (usually negatively charged)with immobile ions, whereas the second part is the diffusive part. The electricalpotential of the charged solid surface is referred to as the zeta potential (ζ). Acharacteristic value of the ζ potential is equal to RT/F = 25 mV, where R is theuniversal gas constant, T is the absolute temperature in °K, and F is the Faradayconstant.

The difference in mobility of free mobile ions in the diffusive part of electricaldouble layer and on the charged surface determines the electrokinetic phenomena,which are totally determined by properties of electrical double layer.

Electrokinetic PhenomenaCurrently, a number of electrokinetic phenomena have been discovered and inves-tigated. Only one of them is briefly discussed as follows: the streaming potential.Let us consider the flow of an electrolyte solution in a capillary with negativelycharged walls (for example, a glass or quartz capillary). In the initial state, thefeed solution and the receiving solution have equal concentrations of electrolyte.The electrolyte solution starts to flow after a pressure difference is applied toboth sides of the capillary. This flow involves mobile cations in a electrical doublelayer near the solid negatively charged walls of the capillary into a convectivemotion, which is an electric current. As a result of the convective electric current,the concentration of cations increases in the receiving solution and the excesspositive charges accumulate there. These excess charges cause the appearance of

RC

d = ⋅ −3 10 8

cm,



an electric potential difference between the entrance and the end of the capillary,which generates the electric current in the direction opposite to the direction ofthe flow. This electric current destroys the emerging surplus of cations in theoutflowing solution. Electric potential difference appearing between the ends ofcapillary, in this case, is called a streaming potential. Let us note that the totalelectric current in the system is equal to zero, i.e., there is no electric current inthe system, in spite of an electric potential difference between the ends of thecapillary.

The Electrostatic Component of the Disjoining Pressure

Now we shall continue the examination of the next component of the disjoiningpressure, the electrostatic component.

Let us return to the examination of two charged surfaces (not necessarily ofthe same nature) in aqueous electrolyte solutions (Figure 1.9a and Figure 1.9b).The surfaces are assumed to have equal charges or opposite charges, i.e., thereare electrical double layers near each of them. The sign of the charge of thediffusive part of each electrical double layer is opposite to the sign of the chargeof the corresponding surface. If the width of clearance between surfacesis , the electrical double layers of surfaces do not overlap (Figure 1.9a),and there is no electrostatic interaction of surfaces. However, if the thickness ofthe clearance, h, is comparable with the thickness of the electrical double layer,Rd, then electrical double layers overlap, and it results in an interaction between

(a)

(b)

FIGURE 1.9 (a) ζ1 and ζ2 are negative. Distance between two negatively charged surfaces,h, is bigger than the thickness of the Debye layers, Rd. Electrical double layers do notoverlap, and there is no electrostatic interaction between these surfaces; ζ1 and ζ2 areelectrical potentials of charged surfaces. (b) ζ1 and ζ2 are negative. Distance between twonegatively charged surfaces, h, is smaller or comparable with the thickness of the electricaldouble layer, Rd. Electrical double layers of both surfaces overlap, which results in aninteraction that is repulsion, in the case under consideration.

h Rd>>

Rd

h

ζ1

ζ2

+

+ + + + + + + + +

+++++++++

– – – – – – – – – –

––––––––– –

h

–––––––––

– – – – – – – – –

++++ ++

++

++++

++++

+

+ζ1

ζ2



the surfaces. If the surfaces are equally charged, their diffusive layers are equallycharged as well, i.e., the repulsion appears as a result of their overlapping (theelectrostatic component of the disjoining pressure is positive in this case).

If the surfaces have opposite charges, an attraction would ensue as a resultof the overlapping of opposite charges. The electrostatic component of the dis-joining pressure is negative in this case (Figure 1.10a and Figure 1.10b).

There are a number of approximate expressions for the electrostatic compo-nent of the disjoining pressure [1]. For example, in the case of low ζ potentialsof both surfaces, the following relation is valid [1]:

, (1.12)

where ε is the dielectric constant of water and 1/κ = Rd, respectively. ζ potentialis considered to be low if the corresponding dimensionless potential

.

(a)

(b)

FIGURE 1.10 (a) ζ1 > 0 and ζ2 < 0. Distance between two surfaces, baring the oppositecharges, h, is bigger than the thickness of the Debye layers, Rd. Electrical double layersdo not overlap, and there is no electrostatic interaction between these surfaces. ζ1 and ζ2

are electrical potentials of charged surfaces. (a) ζ1 > 0 and ζ2 < 0. Distance between twosurfaces with opposite charges, h, is smaller or comparable with the thickness of theelectrical double layer, Rd. Electrical double layers overlap, which results in an interactionthat is attraction, in the case under consideration.

Rd

h

ζ1

ζ2

+++++

+ + + + + + + + + +

++++ –––––

– – ––– – – – –

–––– –

h

+

+++++++++

+ + + + + + + +ζ1

ζ2

– –– – – – – – –

–––

––

––

––

––

Πe hh

h( )

cosh

sinh=

− +( )εκπ

ζ ζ κ ζ ζ

κ

21 2 1

222

28

2

F

RT

ζ < 1



Note that in the case of oppositely charged surfaces and at relatively smalldistances, the following expression for the electrostatic component of the disjoin-ing pressure is valid [1]:

, (1.13)

which is always attraction. It is necessary to be very careful with the latterexpression because in this case, the attraction can change to repulsion at a pointbeyond certain small critical distances [1].

Equation 1.12 and Equation 1.13 show that the disjoining pressure does notvanish even in cases when only one of the two surfaces is charged (for example,ζ1 = 0). The physical reason for this phenomenon is the deformation of theelectrical double layer, if the distance between the surfaces is smaller than theDebye radius.

The theory for the calculation of the disjoining pressure based on the twoindicated components, i.e., dispersion, Πm(h), and electrostatic, Πe(h), is referredto as the DLVO theory. According to the DLVO theory, the total disjoiningpressure is a sum of the two components, i.e., Π(h) = Πm(h) + Πe(h). The DLVOtheory made possible the explanation of a range of experimental data on thestability of colloidal suspensions/emulsions as well as the static and the kineticsof wetting. However, it has been understood later that only these two componentsare insufficient for explaining the phenomena in thin liquid films, layers and incolloidal dispersions. There is a requirement of a third important component ofdisjoining pressure, which becomes equally important in aqueous electrolytesolutions.

Structural Component of the Disjoining Pressure

This component of disjoining pressure is caused by the orientation of watermolecules in a vicinity of aqueous solution–solid interface or aqueous solution–airinterface. Keep in mind that all water molecules can be modeled as an electricdipole.

In the vicinity of a negatively charged interface, a positive part of waterdipoles is attracted to the surface. That is, the negative part of dipoles are directedoppositely and the next set of water dipoles is facing a negatively charged partof dipoles, which in its turn, results in the orientation of the next layer of dipolesand so on. However, thermal fluctuations try to destroy this orientation (Figure1.11).

As a result of these two opposite trends, there is a formation of a finite layer,where the structure of water dipoles differs from the completely random bulkstructure. This layer is frequently referred to as the hydration layer. If we nowhave two interfaces with hydration layers close to each of them (or even one ofthem), then at a close separation, comparable with the thickness of the hydration

Πe hh

( ) = −−( )ε

πς ς

81 2

2

2



layer, these surfaces “feel each other,” that is, hydration layers overlap. Thisoverlapping results either in attraction or repulsion of these two surfaces.

Unfortunately, until now, there is no firm theoretical background on thestructural component of the disjoining pressure, and we are unable to deducetheoretically those cases in which the structure formation results in an attractionand those in which it results in a repulsion. As a consequence, only a semiem-pirical equation exists, which gives a dependence of the structural component ofdisjoining pressure on the thickness of the liquid film [1]:

, (1.14)

where K and λ are constants. There is a clear physical meaning of the parameter1/λ, which is the correlation length of water molecules in aqueous solutions. Thisparameter further gives 1/λ ~ 10–15 Å, which is the characteristic thickness ofthe hydration layer.

However, we are still far from a complete understanding of the preexponentialfactor K, which can be extracted on the current stage only from experimentalmeasurements of the disjoining pressure.

Currently, it is assumed [1] that the disjoining pressure of thin aqueous filmsis equal to the sum of the three components

. (1.15)

In Figure 1.12, the dependences of the disjoining pressure on the thicknessof a flat liquid film are presented for the cases of the complete wetting (curve 1that corresponds to a dispersion or molecular component of disjoining pressure,Πm(h)) and partial wetting (curve 2 that corresponds to a sum of all three compo-nents of the disjoining pressure, according to Equation 1.15). Disjoining pressurepresented by curve 1 in Figure 1.12 corresponds to a case of complete wetting, for

FIGURE 1.11 Formation of a hydration layer of water dipoles in the vicinity of a nega-tively charged interface. The darker part of water dipoles is positively charged, whereasthe lighter part is negatively charged.

Hydration layer

ΠShh Ke( ) = −λ

Π Π Π Πh h h hm e s( ) = ( ) + ( ) + ( )



example, oil droplets on glass substrate, whereas curve 2 corresponds to the caseof partial wetting, for example, aqueous electrolyte solutions on glass substrates.

In Reference 1 to 4, a number of experimental data on measurement ofdisjoining pressure are presented. The dependency (Equation 1.11) has beenfirmly confirmed in the case of oil thin films on glass, quartz, and metal surfaces,which corresponds to the case of complete wetting. In Figure 1.13, experimentaldata and calculations according to Equation 1.15 for aqueous thin films arepresented.

In Reference 1, all necessary details concerning experimental data presentedin Figure 1.13 are given.

1.3 STATIC HYSTERESIS OF CONTACT ANGLE

The previous consideration shows that the situation with Young’s equation (Equa-tion 1.1) is far more difficult than it is usually assumed. This equation is supposedto describe the equilibrium contact angle. We explained in Section 1.1 that thelatter equation does not comply with any of the three requirements of the equi-librium: liquid–vapor equilibrium, liquid–solid equilibrium, and vapor–solidequilibrium.

However, there is a phenomenon that is far more important than the previousones from a practical point of view. It is called the static hysteresis of contactangle.

The derivation of Equation 1.1 and further considerations show that the givenequation (or its modifications) determines only one unique equilibrium contact

FIGURE 1.12 Types of isotherms of disjoining pressure, which are under considerationbelow: (1) complete wetting, observed for oil films on quartz, glass, metal surfaces [1];(2) partial wetting, observed for aqueous films on quartz, glass, metal surfaces [1];(3) nonwetting case.

1

2

3

h

Π



FIGURE 1.13 Calculated and experimentally-measured isotherms of disjoining pressure,Π(h), of the films of water on a quartz surface at concentration of KCl: C = 10–5 mol/l,pH = 7, and dimensionless ς potential of the quartz surface equals to 6 [1]. (a) Within theregion of large thicknesses: dimensionless ς potential of the film–air interface equals to2.2 (curve 1), 1 (curve 2), and 0 (curve 3); (b) within the region of small thicknesses:dimensionless ς potential of the film–air interface equals to 2.2 (curve 1). The structuralcomponent, ΠS(h), of the disjoining pressure isotherm and electrostatic component, Πe(h),are indicated by curves 2 and 3, respectively. Curves 4 in both part (a) and (b) are calculatedaccording to Equation 1.13.

20001000

1 2 3

–2

0

2

44

βα

h, Å

Π-1

0–3, d

yn/c

m2

(a)

(b)

–1

–2

0 4 200 400

2

3

1t0

1

2

Π . 1

0–5, d

yn/c

m2

h, Å



angle. Static hysteresis of contact angle results in an infinite number of equilib-rium contact angles of the drop on the solid surface, not the unique contact angle,θe, but the whole range of contact angles, θr < θe < θa, where θr and θa are thecorresponding static receding and advancing contact angles.

The meaning of static advancing and receding contact angles is apparentwhen we consider a liquid droplet on a horizontal substrate that is slowly beingpumped through an orifice in the solid substrate (Figure 1.14). Let us assumethat, in some way, an initial contact angle of the droplet was equal to theequilibrium one. When we carefully and slowly pump the liquid through an orificein the center, the contact angle will grow. However, the radius of the drop basewill not change until a critical value of the contact angle, θa, is reached. Furtherpumping will result in spreading of the drop.

If we start from the same equilibrium contact angle and then pump out theliquid through the same orifice, the contact angle will decrease further, but thedroplet will not shrink until the critical contact angle, θr, is reached. After that,the droplet will start to recede.

For example, in the case of water droplets on a smooth homogeneous glasssurface that is specially treated for purity, θr ~ 0°–5°, whereas θa is in the rangeof 40°–60°.

It is usually believed that the static hysteresis of contact angle is determinedby the surface roughness and/or heterogeneity (Figure 1.15).

Figure 1.15b presents the magnified vicinity of the three-phase contact lineof the same droplet as in Figure 1.15a. This picture gives a qualitative explanationof the phenomenon of the static hysteresis of contact angle, which is widelyadopted in the literature. The static hysteresis of contact angle is connected withmultiple equilibrium positions on the drop edge on a rough surface. No doubt

FIGURE 1.14 Schematic presentation of a liquid droplet on a horizontal solid substrate,which is slowly pumped through the liquid source in the drop center. R is the radius ofthe drop base; θ is the contact angle; (1) liquid drop, (2) solid substrate with a small orificein the center, (3) liquid source (syringe).

3

R1θ

2



that a roughness and/or a chemical heterogeneity of the solid substrate contributesubstantially to the contact angle hysteresis.

As already mentioned, the static hysteresis of the contact angle is usuallyrelated to the heterogeneity of the surface, either geometric (roughness) [6,7] orchemical [8]. In this case, it is assumed that at each point of the surface theequilibrium value of the contact angle of that point is established, depending onlyon the local properties of the substrate. As a result, a whole series of localthermodynamic equilibrium states can be realized, corresponding to a certaininterval of values of the angle. The maximum value corresponds to the value ofthe advancing contact angle, θa, and the minimum value corresponds to thereceding contact angle, θr.

According to such a model, the dependency of contact angle on velocity ofmotion should be as presented in Figure 1.16.

There is no doubt that heterogeneity affects the wetting process. However,heterogeneity of the surface is apparently not the sole reason for hysteresis ofthe contact angle. This follows from the fact that not all predictions made on the

FIGURE 1.15 (a) Droplet on a solid substrate with a small roughness, which is invisibleto a naked eye, (b) magnification of the apparent three-phase contact line, (c) magnificationon the apparent three-phase contact line with the rough surface covered by the liquid filmthat flows out from the droplet. An arrow shows the zone where a microscopic motionoccurs.

θef

(a)

θefθi

Solid

(b)

θef

θ

(c)

Solid



basis of this theory have turned out to be true [10,11]. Besides, hysteresis hasbeen observed in cases of quite smooth and uniform surfaces [12–17]. Further,the static hysteresis of contact angle is present even on surfaces that are definitelyfree liquid films [17–18].

Now we recall that in the vicinity of the apparent three-phase contact line,surface forces (disjoining pressure) disturb the liquid profile substantially, andthe picture presented in Figure 1.15b is impossible. Immediately after the dropletis deposited, the disjoining pressure comes into play. This pressure results in acoverage of the substrate in the vicinity of the apparent three-phase contact lineby a thin liquid film. This would mean that the liquid edge is always in contactwith an already wetted, rough solid substrate. A more realistic picture is depictedin Figure 1.15c, which describes the situation more adequately in the vicinity ofthe apparent three-phase contact line. Equilibrium and hysteresis contact angleson rough surfaces have never been considered from this point of view before andare the subject of future investigations.

These considerations would suggest that the picture presented in Figure 1.16cannot be realized either on smooth or rough substrate. This is the reason whywe consider the static hysteresis on completely smooth substrates.

In earlier studies [5], a completely new concept of hysteresis of contact angleon smooth homogeneous substrates has been suggested. This mechanism will bediscussed in Section 3.10. In the following, we give a qualitative description ofthis phenomenon.

The picture presented in Figure 1.16 is in contradiction with the thermo-dynamics, which requires a unique equilibrium contact angle, θe, on smoothhomogeneous substrates. The latter means that at any contact angle, θ, in therange θr < θ < θa and different from the equilibrium one, the liquid droplet cannotbe at the equilibrium but in the state of a very slow “microscopic” motion. Moredetailed observations and theoretical considerations show (see Section 3.10) thatat any contact angle different from the equilibrium one, θe, the liquid droplet isin a state of slow microscopic motion, which is located in the tiny vicinity of theapparent three-phase contact line. The microscopic motion abruptly becomes“macroscopic” after the critical contact angles θa or θr are reached.

This observation shows that the dependency presented in Figure 1.16 shouldbe replaced by a more complicated but realistic dependency as shown in Figure 1.17.

FIGURE 1.16 Dependency of the contact angle on the velocity of advancing (v > 0) orreceding (v < 0) meniscus.

θa

θrv



The presence of the contact angle hysteresis indicates that the actual equilib-rium contact angle is very difficult to obtain experimentally even if we neglectthe equilibrium with vapor and solid substrate.

Static Hysteresis of Contact Angles from Microscopic Point of View: Surface Forces

At this stage, we are capable of explaining the nature of the hysteresis of contactangles via the S-shape of the isotherm of disjoining pressure (curve 2 in Figure1.12) in the case of partial wetting. More details are given in Section 3.10.

First of all, we recall what the hysteresis of contact angle in capillaries means.Let us consider a meniscus in the case of partial wetting in a capillary (Figure 1.18aand Figure 1.18b). Note that the capillary is in contact with a reservoir wherethe pressure, Pa – Pe, in the reservoir is lower than the atmospheric pressure, Pa.

If we increase the pressure under the meniscus, then the meniscus does notmove but changes its curvature to compensate for the excess pressure and, as aconsequence, the contact angle increases accordingly. The meniscus does notmove until a critical pressure and critical contact angle, θa, are reached. Afterfurther increase in pressure, the meniscus starts to advance. A similar phenomenontakes place if we decrease the pressure under the meniscus; it does not recedeuntil a critical pressure and corresponding critical contact angle, θr, are reached.This indicates that in the whole range of contact angles, θr < θ < θa, the meniscusdoes not move macroscopically. It is obvious that on the smooth homogeneoussolid substrate only one contact angle corresponds to the equilibrium position,and all the rest do not. Based on that idea, in Figure 1.17 we present a dependencyof the contact angle on the velocity of motion, which shows that all contact angles,θ, in the range, θr < θ < θa, correspond to a slow microscopic advancing orreceding of the meniscus. This microscopic motion abruptly changes to macro-scopic as soon as θr or θa are reached.

Explanation of the dependence presented in Figure 1.18 is based on theS-shaped isotherm of disjoining pressure in the case of partial wetting. This shape

FIGURE 1.17 At any deviation from the equilibrium contact angle θe, the liquid drop isin the state of a slow microscopic motion, which abruptly transforms into a state ofmacroscopic motion after critical contact angles θa or θr are reached.

θa

θr

θe

v



determines a very special shape of the transition zone in the case of the equilib-rium meniscus (see Section 2.3). In the case of pressure increases behind themeniscus (Figure 1.18a), a detailed consideration (Section 3.10) of the transitionzone indicates that close to a “dangerous” point marked in Figure 1.18a, the slopeof the profile becomes steeper with increasing pressure. In the range of very thinfilms (region 3 in Figure 1.18a), there is a zone of flow. Viscous resistance in thisregion is very high, hence the very slow advancement of the meniscus. After acertain critical pressure behind the meniscus is reached, the slope at the dangerouspoint reaches π/2, and the flow proceeds stepwise, occupying the region of thickfilms. Thus, the fast “caterpillar” motion begins, as shown in Figure1.18a.

In the case of pressure decreases behind the meniscus, the event proceedsaccording to Figure 1.18b. Again, up to a certain critical pressure, the slope inthe transition zone close to the point marked dangerous becomes more and moreflat. In the range of very thin films (region 3 in Figure 1.18b), there is a zone offlow. Viscous resistance in this region again is very high. This is why the recedingof the meniscus proceeds in a very slow manner. After the attainment of thecritical pressure behind the meniscus, the profile in the vicinity of the dangerouspoint shows a discontinuous behavior, which is obviously impossible. That meansthe meniscus will start to slide along a thick β-film, moving relatively fast andleaving behind the thick β-film. The latter phenomenon (the presence of a thickβ-film behind the receding meniscus of aqueous solutions in quartz capillaries)has been discovered experimentally (see discussion in Section 3.10). This dis-covery supports our arguments explaining static contact angle hysteresis onsmooth homogeneous substrates.

FIGURE 1.18 Hysteresis of contact angle in capillaries in the case of partial wetting (S-shaped isotherm of disjoining pressure). (a) Advancing contact angle. (1) a sphericalmeniscus of radius ρa, (2) transition zone with a point dangerous marked (see explanationin the text), (3) zone of flow, (4) flat films. Close to the marked point, a dashed line showsthe profile of the transition zone just after the contact angle reaches the critical value θa,which indicates a beginning of the caterpillar motion. (b) Receding contact angle. (1) aspherical meniscus of radius ρr < ρa, (2) transition zone with a point dangerous marked(see explanation in the text), (3) zone of flow, (4) flat films. Close to the marked point, adashed line shows the profile of the transition zone just after the contact angle reaches thecritical value θa.

2 24

334

1 1ρa ρr

θa θr

(b)(a)



REFERENCES

1. Deryaguin, B.V., Churaev, N.V., and Muller, V.M., Surface Forces, ConsultantsBureau, Plenum Press, New York, 1987.

2. Russel, W.B., Saville, D.A., and Schowalter, W.R., Colloidal Dispersions, Cam-bridge University Press, Cambridge, U.K., 1999.

3. Exerowa, D. and Kruglyakov, P., Foam and Foam Films: Theory, Experiment,Application, Elsevier, New York, 1998.

4. Israelashvili, J.N., Intermolecular and Surface Forces, Academic Press, London,1991.

5. Starov, V.M., Adv. Colloid Interface Sci., 39, 147, 1992.6. Wenzel, R., Ind. Eng. Chem., 28, 988, 1936.7. Deryagin, B.V., Dokl. Akad. Nauk SSSR [in Russian], 51, 357, 1946.8. Johnson, R.E. and Dettre, R.H., Surface and Colloid Science, Vol. 2, Wiley, New

York, 1969, p. 85.9. Ivanov, I.B. and Kralchevsky, P.A., In “Thin liquid films. Fundmentals and Appli-

cations.” Ivanov, I.B. (ed.). Surfactant Science Series, Marcel Dekker Inc., NewYork and Basel, v. 29 (1988).

10. Schwartz, A.M., Racier, C.A., and Huey, E., Adv. Chem. Ser., 43, 250, 1964.11. Neumann, A.W., Renzow, D., Renmuth, H., and Richter, I.E., Fortsch. Ber. Kol-

loide Polym., 55, 49, 1971.12. Holland, L., The Properties of Glass Surfaces, London, 1964, p. 364.13. Zorin, Z.M., Sobolev, V.D., and Churaev, N.V., Surface Forces in Thin Films and

Disperse Systems [in Russian], Nauka, Moscow, 1972, p. 214.14. Romanov, E.A., Kokorev, D.T., and Churaev, N.V., Int. J. Heat Mass Transfer,

16, 549, 1973.15. Neumann, A.W., Z. Phys. Chem. (Frankfurt), 41, 339, 1964.16. Zheleznyi, B.V., Dokl. Akad. Nauk SSSR [in Russian], 207, 647, 1972.17. Platikanov, D., Nedyalkov, M., and Petkova, V. Advances in Colloid and Interface

Science, Vol. 100–102, 2003, pp. 185–203.18. Petkova, V., Platikanov, D., and Nedyalkov, M., Adv. Colloid and Interface Sci.,

104, 37, 2003.


31

2 Equilibrium Wetting Phenomena

INTRODUCTION

In this chapter, we shall discuss equilibrium liquid shapes on solid substrates,which demands equilibrium of liquid–vapor, liquid–solid, and vapor–solid. Notalways do authors take into account all the three equilibria. The vapor–solidequilibrium determines the presence of adsorbed liquid layers on solid surfacesfor both complete and partial wetting. Even at this moment, we and everythingaround are covered by a thin water film. The thickness of aqueous films dependson the humidity in the room, and the adsorption is exactly at equilibrium withthe surrounding humidity, no matter how low or high.

The presence of liquid layers on solid substrates is determined by the actionof surface forces (the disjoining pressure), which was discussed in Chapter 1.The disjoining pressure isotherm is normally dealt with because it is usuallymeasured at a constant temperature. In the case of water and aqueous solutions,the disjoining pressure is S-shaped. Water and aqueous solutions are crucial forlife. Does the peculiar shape of the disjoining pressure isotherm of water andaqueous solutions in some way determine our existence?

It is well known that all properties of water and aqueous solutions are vitallyimportant for life. This means that the peculiar shape of disjoining pressureisotherms of water and aqueous solutions, in some unknown way, determines theexistence of our life. At the moment we do not know how the process works.

Further in this chapter, we investigate the influence of the combined actionof disjoining pressure and capillary forces on the equilibrium shapes of liquidson solid substrates.

2.1 THIN LIQUID FILMS ON FLAT SOLID SUBSTRATES

In this section, we shall consider the properties and stability of liquid films onsolid substrates under partial or complete wetting conditions. We shall accountfor the disjoining pressure action alternatively with the action of surface forces.As discussed in Chapter 1, the adsorption of liquid on solid substrates is amanifestation of the action of surface forces. But before we start, let us recallthat partially or completely wetted solid surfaces, at equilibrium, are alwayscovered by a liquid film that is at equilibrium with the vapor pressure, p, of thesurrounding air. The free energy of such a solid covered substrate is lower thanthe free energy of the corresponding bare solid substrate. Hence, in all cases here



to be considered, there is no real three-phase contact line at equilibrium becausethe whole solid surface is covered by a flat equilibrium liquid film (on occasionwe shall mention the apparent contact line).

In this section, we consider properties and stability of liquid films in the caseof both partial and complete wetting. The two terms that are equally used are:disjoining pressure action and the action of surface forces.

As we already discussed in Chapter 1, the adsorption of liquid on solidsubstrates is a manifestation the action of surface forces. This means that thelatter forces must be taken into account if we are to consider equilibrium statesof liquid films on solid substrates. We also noted that, in all cases under consid-eration, there is no real three-phase contact line at the equilibrium because thewhole solid surface is covered by flat equilibrium liquid film.

The excess free energy per unit area of a flat equilibrium liquid film ofthickness he on a solid substrate at equilibrium with the vapor in the surroundingair is:

(2.1)

where S is the surface covered by the liquid film, and fD (he) is the potential ofsurface forces; γ, γsl, and γsv are liquid–air, solid–liquid, and liquid–vapor inter-facial tensions, respectively; the excess pressure Pe = Pa – Pl, where Pl is thepressure inside the liquid film, and Pa is the pressure in the ambient air. Notethat, according to the spontaneous adsorption of liquid molecules in partial orcomplete wetting cases, the latter excess free energy should be negative; otherwisethe liquid molecules would not adsorb at all.

Owing to the equilibrium of the liquid film with the vapor, the excess pressure,Pe, cannot be left as an arbitrary constant; it is determined by the equality ofchemical potentials of liquid molecules in the film and in the vapor. This require-ment results in the well-known Kelvin’s equation:

, (2.2)

where R, T, and vm are the universal gas constants, the absolute temperature, andthe liquid molar volume, respectively; ps and p correspond to the pressures of thesaturated vapor and the vapor at which the liquid film is at equilibrium. The latterexpression shows that the excess pressure, Pe, cannot be fixed arbitrarily but isdetermined by the vapor pressure in the ambient air, p. It must be noted thatEquation 2.2 expresses the equality of chemical potentials of water molecules invapor and liquid phases.

The excess free energy, according to Equation 2.1, is a function of the variable,he, which is the thickness of the equilibrium film. Hence, the usual conditions ofthermodynamic equilibrium should hold, which give a minimum value to theexcess free energy. Those conditions are:

Φ/S P h f he e D e sl sv= + + + −γ γ γ( ) ,

PRT

v

p

pem

s= ln


Equilibrium Wetting Phenomena 33

.

The first requirement results in

, (2.3)

and the second requirement yields

, (2.4)

where

is referred to as the disjoining pressure [1]. The disjoining pressure, Π(h), is thephysical property that can be experimentally measured. That is, the considerationthat follows is based on the consideration of the disjoining pressure. Using theprevious definition, we can rewrite the excess free energy fD(h) as:

.

Equation 2.3 determines the thickness of the equilibrium liquid film, he, viadisjoining pressure isotherm. Equation 2.4 gives the well-known stability condi-tion of flat equilibrium liquid films [1].

According to the stability condition (2.4), all flat equilibrium films are stablein the case of complete wetting (curve 1, Figure 2.1), and only films in the rangeof thickness 0 to tmin (these films are referred to in the following section as α-films,which are absolutely stable) and at h > tmax (the latter films are referred to asβ-films, and it is shown that they are metastable) in the case of partial wetting(curve 2 in Figure 2.1) are stable. Hence, only those α- and β-films can exist asflat films.

Note again that the S-shaped disjoining pressure isotherms (curve 2 inFigure 2.1) are characteristic shapes in the case of water and aqueous solutions.All properties of water and aqueous solutions are vitally important for life. Thelatter means that the peculiar shape of disjoining pressure of water and aqueoussolutions presented in Figure 2.1 in some way determines the existence of life.At the moment we do not know how, but the peculiar shape of curve 2 inFigure 2.1 does tell us something about the process.

d

dh

d

dhe e

Φ Φ= >0 02

2,

P he e= Π( )

d h

dhe

e

Π( ) < 0

Π( )( )

hdf h

dhD= −

f h h dhD

h

( ) ( )=∞

∫ Π



We can rewrite the expression for the excess free energy of the film Equation2.1 using the disjoining pressure in the following way:

(2.5)

The latter expression gives the excess free energy via a measurable physicaldependency, Π(h), which is the disjoining pressure isotherm.

Now we can rewrite expression (2.5), of the excess free energy of thin liquidfilms as

(2.6)

where

(2.7)

is the “interfacial tension” (actually the excess free energy) of the solid substratecovered with the liquid film of thickness he.

The preceding expression determines the unknown value of γsvhe in Young’s

equation 1.8 in Chapter 1, Section 1.1:

FIGURE 2.1 Two types of isotherms of disjoining pressure, which are under considerationbelow: 1 — complete wetting, 2 — partial wetting. Isotherms for partial wetting areobserved for water films on almost all surfaces, for example, on quartz, glass, and metalsurfaces [2,3]. Isotherms of type 1 are observed in a number of cases of complete wetting,for example, at oil films on quartz, glass, and metal surfaces [4].

2

1

tmin tmax h

S+

S–t0

–Πmin

Πmax

Π

Φ Π/S P h h dhe e

h

sl sv

e

= + + + −∞

∫γ γ γ( ) .

Φ/S svh sve= −γ γ ,

γ γ γsvh e e

h

sle

e

P h h dh= + + +∞

∫ Π( )



. (2.8)

Combination of Equation 2.7 and Equation 2.8 results in

. (2.9)

The latter equation is the well-known Derjaguin–Frumkin equation for theequilibrium contact angle, which has been deduced using a different thermo-dynamic consideration [1]. Equation 2.9 is a very important equation and isdeduced in Section 2.3 in a different way.

Because –1 < cos θe < 1, using Equation 2.9 we conclude that the integralon the right-hand side should be negative. This requirement is satisfied in thecase of partial wetting (see curve 2 in Figure 2.1).

. (2.10)

The latter inequality is satisfied if

, (see Figure 2.1). (2.11)

In the case of complete wetting, the right hand side in Equation 2.9 is alwayspositive, that is, equilibrium droplets cannot exist on the solid substrate eitherunder oversaturation or undersaturation; they spread out completely and evapo-rate. However, equilibrium menisci (at undersaturation) can exist in capillaries.That is, the behavior of droplets and menisci in the case of complete wetting iscompletely different.

Using Equation 2.9, we can rewrite the expression for the excess free energyof a flat liquid film in Equation 2.1 as

(2.12)

According to spontaneous adsorption of liquid molecules on solid substrates,in the case of partial and complete wetting, the right-hand side value correspond-ing to the excess free energy (Equation 2.12) is negative, and hence,

cos θ γ γγe

svh sl= −

cos

( )

θ

γ

γe

e e

h

P h h dh

e=

+ +∞

∫ Π

Π( )h dhhe

∞

∫ < 0

S S− +>

Φ/S e sl sv e NY= + − = −( )γ θ γ γ γ θ θcos cos cos .



. (2.13)

That is, even if γsl and γsv are measured, the contact angle according to theoriginal Young’s equation,

, (2.14)

is smaller than the real equilibrium contact angle.Frequently, a contact angle determined according to Equation 2.14 is identi-

fied with a static advancing contact angle, θa. It is obvious that the static advancingcontact angle, θa, is bigger than the equilibrium contact angle, θa > θe. If we nowcompare this inequality with Equation 2.13, we can conclude that there is nojustification for the identification of θNY and θa because θNY < θe < θa.

EQUILIBRIUM DROPLETS ON THE SOLID SUBSTRATE UNDER OVERSATURATION (Pe < 0)

As we already noticed, the excess pressure, Pe, is negative at oversaturationaccording to Equation 2.2. The equilibrium film or films are determined accordingto Equation 2.3 at both undersaturation and oversaturation.

Figure 2.2 shows that, in the case of complete wetting, there are no flatequilibrium films on solid substrates under oversaturation because the line, Pe < 0,does not intersect (curve 1 in Figure 2.2). Hence, there are also no equilibriumdroplets on completely wettable solids at oversaturation; they are in the surround-ing air.

FIGURE 2.2 Two equilibrium flat films on solid substrates under oversaturation: stablefilm of thickness he and unstable film of thickness hu.

cos cos ,θ θ θ θe NY e NY− < >0

cos θ γ γγNY

sv sl= −

t0 tmin

tmax h

huhe

–Πmin

Π

Pe



However, in the case of partial wetting, Equation 2.3 has two solutions (Figure2.2). According to the stability condition of flat films in Equation 2.4, one ofthem corresponds to the stable equilibrium film of thickness he, and the secondone corresponds to the unstable film of thickness hu (Figure 2.2). This wouldsuggest that equilibrium droplets in the case of partial wetting are “sitting” onthe stable equilibrium film of thickness he.

However, even in the case of partial wetting, equilibrium droplets can existon the solid substrate only in a limited interval of oversaturation, which isdetermined by 0 < Pe < –Πmin (Figure 2.2) or using Equation 2.2 in the followingrange of oversaturated pressure, p, over the solid substrate

. (2.15)

If Πmin is in the range 106–107 dyn/cm2, then the latter inequality takes thefollowing form:

;

that is, the equilibrium droplets in the case of partial wetting exist only in a verylimited interval of oversaturation on the solid substrates. Beyond this interval, athigher oversaturation, neither equilibrium liquid films nor droplets exist on thesolid substrate as in the case of complete wetting. Probably, the critical oversat-uration pcr

,

determined from Equation 2.15, corresponds to the beginning of homogeneousnucleation, and at higher oversaturations, homogeneous nucleation is more favorable.

Let ℜ be the radius of the equilibrium droplet. According to the definitionof the capillary pressure,

Hence, the radius of equilibrium drops is

1 < <

p

p

v

RTs

mexp minΠ

1 1 1 001 1 01< < + ≈ −p

p

v

RTs

mΠmin . .

p

p

v

RTcr

s

m=

exp minΠ

Pe = −ℜ2γ

.

ℜ =−2γPe

.



In the above-mentioned narrow interval of oversaturation, the radius of the equi-librium drops changes from infinity at p → ps to

at p = pcr. If Πmin ≈ 106 dyn/cm2 and γ ≈ 72 dyn, then

that is, the critical size is out of the range of the action of surface forces and thedroplet size is sufficiently big. However, if Πmin ≈ 107 dyn/cm2, then

and the whole droplet is in the range of the action of surface forces. In this case,the drop is so small that it does not have anywhere (even on the very top) aspherical part that is undisturbed by surface forces.

FLAT FILMS AT THE EQUILIBRIUM WITH MENISCI (Pe > 0)

Equation 2.3 and Figure 2.3 show that, in the case of complete wetting, there isonly one equilibrium flat film, hc, which is stable according to the stabilitycondition (2.4).

FIGURE 2.3 Disjoining pressure isotherm in the case of complete wetting (1), and partialwetting (2). In thick capillaries (H > γ/Πmax), there are three solutions of Equation 2.3.

ℜ =cr

2γΠmin

ℜ ≈ =cr

144

101 44

6. µm.

ℜ ≈ = =cr

144

100 144 1440

7. µm A

�

2

1

hu hc hhe tminPe

hβ

Πmax

Π



In the case of partial wetting, (Figure 2.3) and Equation 2.3 show differentsolutions in the case of Pe > Πmax and Pe < Πmax. If Pe > Πmax, Equation 2.3 hasonly one solution that is stable (according to the stability condition 2.4) and isreferred to as α-film. In the second case, Pe < Πmax (Figure 2.3) and Equation 2.3show three solutions, one of which corresponds to the stable equilibrium α-filmwith thickness, he. The second solution of Equation 2.3, hu, is unstable accordingto the stability condition in Equation 2.4, and the third solution, hβ, is stable againaccording to the same stability condition in Equation 2.4. The latter films arereferred to as β-films. Note that the thickness of an equilibrium film in the caseof complete wetting, hc, is bigger than the thickness of β- film, hβ, in the case ofpartial wetting.

Let us compare the excess free energy of flat α- and β-films, he and hβ.According to the definition, this difference is equal to

. (2.16)

The difference (hβ – he) is always positive (Figure 2.3) in the case of partialwetting, S > S–, according to Equation 2.11. Hence, the integral on the right-handside of Equation 2.16 is negative. Hence, the excess free energy of β-films ishigher than the excess free energy of α-films. This means that β-films are lessstable than α-films, and that is why β-films are referred to as metastable films,and α-films as absolutely stable films.

It is necessary to make additional comments on α-films and β-films in thecase of partial wetting. If we increase the vapor pressure over partially wettablesurfaces from p = 0 to the saturation pressure, ps, then we can observe theformation of only α-films on the solid substrate. The thickness of these filmschanges correspondingly (according to Equation 2.3 and Figure 2. 3) from zeroat p = 0 to t0 ≈ 70 Å [1]. However, β-films cannot be obtained in the course ofthe adsorption process; they can be obtained only by decreasing the thickness ofvery thick films down to the equilibrium thickness of the β-film. This is whyα-films are referred to as adsorption films (because they can be obtained in thecourse of adsorption), and β-films are referred to as wetting films.

Let ρ be the radius of the curvature of a meniscus in a flat capillary (a meniscusbetween two parallel plates). According to the definition of the capillary pressure,

Let us introduce

∆ Φ Φ Παβ β α β

β

= − = −( ) − ∫( ) ( ) ( )h h S P h h h dhe e

he

h

Pe = γρ

.

ρ γmax

max

=Π



(Figure 2.3), and consider Pe > Πmax (Figure 2.3). We define a capillary as a “thin”capillary if ρ < ρmax. In such capillaries, only thin α-films can be at equilibriumwith the meniscus, and equilibrium β-films do not exist in such thin capillaries.However, if the capillary is “thick,” that is, ρ > ρmax, then in such capillaries,both α- and β-films can be at equilibrium with the meniscus. However, β-filmsare metastable.

If we adopt γ ~ 70 dyn/cm and Πmax ~ 104 dyn/cm2 for estimations, then ρmax ~7⋅10–3 cm.

S-SHAPED ISOTHERMS OF DISJOINING PRESSURE IN THE SPECIAL CASE S– < S+

Let us consider the case when the disjoining pressure isotherm is S-shaped as inFigure 2.3, curve 2. However, let us assume that

,

that is, S– < S+ (Figure 2. 3). In this case, from Equation 2.16 we conclude:

The latter means that at low Pe (or high humidity), β-films are more stable thanα-films. It is easy to check using Equation 2.16 that is an increasing functionof Pe because

Hence, can become positive at some value of Pe, and after that, thick β-filmsbecome less stable than thin α-films. This instability occurs if

In this case, if Pe increases from zero (where thick β-films are more stable thanthin α-films), it reaches a critical value Pcr, such as < 0 at 0 < Pe < Pcr, and

> 0 at Pcr < Pe < Πmax. This would indicate that in the range 0 < Pe < Πcr,thick β-films are more stable than thin α-films; however, at Pcr < Pe < Πmax,

α-films become more stable than β-films. This consideration shows that a cyclepresented in Figure 2.4 with a spontaneous and reversible transition from α-films

Π( )h dhhe

>∞

∫ 0

∆αβ PS S

e = = − <− +00.

∆αβ

d P

dPe

e

∆αβ( ).> 0

∆αβ

∆ Παβ( ) .max

PPe

e = > 0

∆αβ∆αβ



to β-films (along DA) should take place with a decrease in Pe (along CD). Also,a spontaneous reversible transition from β-films to α-films (along BC) shouldtake place with an increase in Pe (along AB). Such spontaneous reversible tran-sitions have been discovered by Exerowa et al. [5–8]. In their experiments, thedisjoining pressure isotherm was S-shaped, but the minimum value of the dis-joining pressure isotherm was positive, which means that the condition

,

that is, S– < S+, was satisfied.

2.2 NONFLAT EQUILIBRIUM LIQUID SHAPES ON FLAT SURFACES

In thin flat liquid films (oil and aqueous thin films, thin films of aqueous elec-trolyte and surfactant solutions, and both free films and films on solid substrates),the disjoining pressure acts alone and determines their thickness. However, if thefilm surface is curved or uneven, both the disjoining and the capillary pressuresact together. In the case of partial wetting, their simultaneous action is expectedto yield nonflat equilibrium shapes. For instance, due to the S-shaped disjoiningpressure isotherm, microdrops, microdepressions, and equilibrium periodic filmscould exist on flat solid substrates. We shall establish a criteria for both existenceand stability of such nonflat equilibrium liquid shapes. On the other hand, we

FIGURE 2.4 S-shaped disjoining pressure isotherm with S– < S+. Reversible transitionfrom α- to β-films along DA at Pe < Pcr and from β- to α-films along BC at Pe > Pcr.

tmaxtmin

Π

Πmax

−Πmin

PcrD

C B

A

h

Π( )h dhhe

>∞

∫ 0



shall see that a transition from thick films to thinner films can proceed transitorilyvia nonflat states with microdepressions and wavy shapes, both of which can be

mentioned nonflat equilibrium shapes on flat solid substrates are to be discoveredexperimentally.

The equilibrium contact angle of either an equilibrium drop or a meniscus inthe capillary can be expressed via disjoining pressure isotherm (see Section 2.3,Figure 2.3):

(2.17)

where Pe is the excess pressure (positive in the case of the meniscus and negativein the case of drops), he is the equilibrium of an absolutely stable α-film (Figure2.3), H is the radius of the capillary in the case of meniscus and the maximumheight in the case of drops.

Equation 2.17 shows that the partial wetting case corresponds to S– > S+, thatis, S-shaped isotherm 2 in Figure 2.3. Equation 2.17 also shows that the equilib-rium contact angle is completely determined by the shape of the disjoiningpressure in the case of molecular smooth substrates. No doubt that the surfaceroughness influences the apparent value of the contact angle. However, it isobvious that the roughness cannot result in a transition from the nonwetting tothe partial wetting case or from the partial wetting to the complete wetting case.That is why, in this section, only molecularly smooth solid substrates are underconsideration. The influence of roughness and chemical heterogeneity is consid-ered in Section 2.4 and Section 2.5.

The main idea of this section is to show that the simultaneous action of thecapillary pressure and S-shaped disjoining pressure isotherm results in the for-mation of nonflat equilibrium liquid shapes even on smooth homogeneous solidsubstrates. Again, we should emphasize that the shape is specific for water andaqueous solutions and hence is vitally important for life. However, we are stillcompletely unaware of the way in which it is important.

Consideration of the equilibrium nonflat liquid layers allows the suggestionof a new scenario of rupture of thick metastable β-films and their transition toabsolutely stable α-films (see the following section).

GENERAL CONSIDERATION

The excess free energy, Φ, of a liquid layer, drop, or meniscus on a solid substratecan be expressed in the following way:

(2.18)

cos

( )

( )θγ

γe

he

ehe

h dh

hH

h dh=

+

−≈ + ≈ −

∞

∞∫∫

11

11

11

Π

Π SS S− +−γ

,

Φ Φ Φ= + + −γS PVe D ref ,


more stable than flat films in some range of hydrostatic pressures [30]. All


where S, V, and ΦD are excesses of the vapor–liquid interfacial area, the excessvolume, and the excess energy associated with the action of surface forces,respectively; γ is the liquid–vapor interfacial tension; Pe is the excess pressure(see Section 2.1); Φref is the excess free energy of a reference state (see thefollowing section). The gravity action is neglected in Equation 2.18. The excesspressure, Pe, is introduced as in Section 1.2 as

, (2.19)

where Pa is the pressure in the ambient air, and Pl is the pressure inside the liquid.The latter pressure is referred to as the hydrostatic pressure inside the liquid.This means that Pe > 0 in the case of a meniscus, and Pe < 0 in the case of adrop. Pe is uniquely determined by the ambient vapor pressure p, according toEquation 2.2 (Section 2.1). Equation 2.2 shows that Pe > 0 corresponds to anundersaturation, whereas Pe < 0 corresponds to an oversaturation. That is, meniscican be at equilibrium at undersaturation, and drops can be at equilibrium atoversaturation.

In the following text, only two-dimensional equilibrium systems are underconsideration for the sake of simplicity. In this case, the excess free energy inEquation 2.18 can be rewritten as:

(2.20)

Integration, in the preceding equation, is taking over the whole space occupiedby the system. Note that the excess free energy in Equation 2.20 is selected as anexcess over the energy of a reference state, which is the state of the same flat surfacecovered by a stable equilibrium α-film. A selection of any reference state resultsin an additive constant in the expression 2.20. However, the reference state isimportant during the consideration of the liquid profiles in the vicinity of theapparent three-phase contact line.

Any liquid profile, h(x), which gives the minimum value to the excess freeenergy, Φ, according to Equation 2.20, describes an equilibrium liquid configu-ration. For the existence of the minimum of the excess free energy 2.20, thefollowing four conditions should be satisfied:

1. δΦ = 0

2.

where

P P Pe a l= −

Φ Π Π= + ′ −( ) + −( ) + −

∞ ∞

∫ ∫γ 1 12h P h h h dh he e

h he

( ) ( )

∫ dx.

∂∂ ′

>2

20

f

h

f h P h h h dh he e

h he

= + ′ −( ) + −( ) + −∞ ∞

∫ ∫γ 1 12 Π Π( ) ( )



3. Solution of Jacoby’s equation, u(x), should not vanish at any position,x, inside the region under consideration except for boundaries of theregion of integration in Equation 2.20.

4. The transversality condition at the apparent three-phase contact lineshould be satisfied. It provides the condition of a smooth transitionfrom a nonflat liquid profile to a flat equilibrium film in front. Thetransversality condition reads

where B is the position of the three-phase contact line.

The preceding condition can be rewritten using the aforementioned definitionof f as:

This condition shows that the three-phase contact line should be determinedat the intersection of the liquid profile with the equilibrium liquid film of thicknesshe, and not at the intersection with the solid substrate as usually assumed. Thisfurther results in

or

which is obviously satisfied only at

or (2.21)

This transversality condition is discussed in Section 2.3. There it is shownthat the condition actually means

f hfh

B

− ′ ∂∂ ′

= 0,

γ γ1 1

1

22

+ ′ −( ) + −( ) + − − ′+ ′

∞

∫h P h h h dh hh

he e

h

Π Π( ) ( )22

0he B

∞

∫

=

1 11

022

2+ ′ −( ) − ′

+ ′

=hh

h B

1

11

2+ ′

=h B

,

′( ) =hB

0 ′ =h he( ) 0



, (2.22)

and the meaning of x → ∞ is clarified, which is, “tends to the end of the transitionzone.”

The first condition (1) results in the well-known equation

, (2.23)

which should be referred to as Derjaguin’s equation because Derjaguin was thefirst to introduce it [1]. The first term on the left-hand side of Equation 2.23corresponds to the capillary pressure, and the second term represents the disjoin-ing pressure action. If the thickness of the liquid is out of the range of thedisjoining pressure action, then Equation 2.23 describes either a flat (Pe = 0)liquid surface, a spherical drop profile (at Pe < 0), or a spherical meniscus profile(at Pe > 0).

The second condition (2) is always satisfied because

.

The third condition (3) results in Jacoby’s equation:

(2.24)

Direct differentiation of Equation 2.23 results in

.

Comparison of Equation 2.23 and Equation 2.24 shows that the solution ofJacoby’s equation is:

. (2.25)

Hence, if h′(x) does not vanish anywhere inside the system under consider-ation, then the system is stable; however, if h′(x0) = 0 and x0 is different from theends of the system under consideration (that is, inside the range of integration inEquation 2.20), then the system is unstable.

′ → → ∞h x0,

γ ′′+ ′

+ =h

hh Pe

( )( )

/1 2 3 2Π

∂∂ ′

=+ ′

>2

2 2 3 210

f

h h

γ( ) /

d

dx

u

h

d h

dhu

γ ′+ ′

+ =( )

( )/1

02 3 2

Π

d

dx

h

h

d h

dh

dh

dx

γ ′′+ ′

+ =( )

( )/1

02 3 2

Π

u const h= ⋅ ′



The second-order differential Equation 2.23 can be integrated once, whichgives:

(2.26)

where C is an integration constant to be determined. The important observationis that the right-hand side of the preceding equation should always be positive.

In the case of a meniscus in a flat capillary, the integration constant, C, isdetermined from the following condition: at the capillary center, h′(H) = –∞,which gives C = PeH, where H is the half-width of the capillary (see Section 2.3).In the case of equilibrium droplets, the constant should be selected using adifferent condition at the droplet apex, h = H: h′(H) = 0 (see Section 2.3), whichresults in C = γ + PeH. An alternate way of selection of the integration constant,C, is by using the transversality condition (2.21).

The integration constant, C, in this section is selected individually accordingto boundary conditions in each case under consideration.

In the case of equilibrium liquid drops and menisci (see Section 2.3), theyare supposed to be always at equilibrium with flat films with which they are incontact with in the front. Only the capillary pressure acts inside the sphericalparts of drops or menisci, and only the disjoining pressure acts inside thin flatfilms. However, there is a transition zone between the bulk liquid (drops ormenisci) and the thin flat film in front of them. In this transition zone, both thecapillary pressure and the disjoining pressure act simultaneously (see Section 2.3for more details). A profile of the transition zone between a meniscus in a flatcapillary and a thin α-film in front of it, in the case of partial wetting, is presentedin Figure 2.5. It shows that the liquid profile is not always concave but changesits curvature inside the transition zone. Just this peculiar liquid shape in thetransition zone determines the static hysteresis of contact angle (see Chapter 3)

FIGURE 2.5 Partial wetting. Magnification of the liquid profile inside the transition zonein “thick capillaries.” S-shaped disjoining pressure isotherm (left side, a) and the liquidprofile in the transition zone (right side, b).

1

1 2+ ′=

− −∞

∫h

C P h h dhe

h

Π( )

,γ

bahh

hu

he

Pe xxminxmax xuxβ

hβ

tmax

tmin

Π

θe



and a number of other equilibrium and nonequilibrium macroscopic liquid prop-erties on solid substrates (Chapter 2 and Chapter 3).

In the transition zone (Figure 2.5), all thicknesses are presented from verythick (outside the range of the disjoining pressure action) to thin α-films. Thismeans that the stability condition of flat films (Equation 2.4, Section 2.1) cannotbe used any more because this condition is valid only in the case of flat films.The more sophisticated Jacoby’s condition (3) should be used instead, whichshows that the transition zone is stable if h′(x) does not vanish anywhere insidethe transition zone.

A peculiar shape of the transition zone, where both the capillary pressure andthe disjoining pressure are equally important, provides an idea to consider solu-tions of Equation 2.26 in the case of the S-shaped isotherm and to see if thisequation has other stable solutions different from flat liquid films of a constantthickness. Each of these solutions (if any) corresponds to a nonflat liquid layer,whose stability should be checked using Jacoby’s condition (3). In the followingtext, we show that such nonflat equilibrium liquid shapes can exist.

MICRODROPS: THE CASE Pe > 0

In the following text, we consider the possibility of existence of microdrops, thatis, drops with an apex in the range of influence of the disjoining pressure. In thiscase, the drop does not have a spherical part even at the drop apex because itsshape is distorted everywhere by the disjoining pressure action.

The liquid profile, h(x), that is to be determined, is obtained by the integrationfrom Equation 2.23 with an integration constant, C, as described by Equation 2.26.

The transversality condition (2.21) at h = he gives h′(he) = 0, which meansthe drop edge approaches the equilibrium film of thickness he on the solid surfaceat zero microscopic contact angle. This condition allows the determination of theintegration constant in Equation 2.26 as

.

Hence, the drop profile is described by the following equation:

(2.27)

where

C P h h dhe e

he

= + +∞

∫γ Π( )

′ = −−( )

−hL h

γ

γ

2

21

( ),

L h P h h h dhe e

he

h

( ) ( ) ( ) .= − − ∫Π



The expression under the square root in Equation 2.27 should be positive, thatis, the following condition should be satisfied:

(2.28)

where the first equality corresponds to the zero derivative, and the second onecorresponds to the infinite derivative.

Let h+ be the apex of the microdrop. The upper part of Figure 2.6 shows theS-shaped dependence of the disjoining pressure isotherm, Π(h), whereas the lowerpart of Figure 2.6 shows the curve L(h) that has a value of maximum or minimumthickness, which are solutions of Pe = Π(h). At the apex of the drop, when h =h+, the first derivative should be zero, that is, h′(h+) = 0, or from Equation 2.27,

L(h+) = 0 . (2.29)

The origin is placed at the center of the drop. In the following text, we consideronly the situation that corresponds to the formation of microdrops at undersatu-ration, that is, at Pe > 0. Equilibrium macrodrops at oversaturation, that is, atPe < 0, are considered in the Section 2.3.

At 0 < Pe < Πmax, the equation Pe = Π(h) has three roots (Figure 2.6), thesmallest of which corresponds to the equilibrium flat α-film of thickness, he. For

FIGURE 2.6 Determination of the microdrop apex. Upper part: S-shaped disjoining pres-sure isotherm, lower part: L(h). L(h+) = 0 determines the drop apex, h+.

h

h

huhe

Pe

0

0

L

h+

hβ

Π

Πmax

–Πmin

0 ≤ ≤L h( ) ,γ



the existence of microdrops, the following conditions should be satisfied: h″ < 0at h = h+, and h″ > 0 as h → he (Figure 2.7); hence, the following inequalityshould be satisfied: hu < h+ < hβ. At the drop apex h′(h+) = 0, and hence, accordingto Equation 2.29,

(2.30)

and the solution of this equation, h+, should be located in the following range:hu < h+ < hβ (see the lower part in Figure 2.6).

The left-hand side in Equation 2.30 is positive, and so should be the right-hand side. Hence, a sufficient condition for the existence of equilibrium micro-drops is as follows: S– < S+, that is, the disjoining pressure isotherm should beS-shaped but the equilibrium contact angle should be equal to zero.

This means that the equilibrium microdrops do not exist either in the case ofpartial wetting, when S– > S+, or in the case of a “regular” complete wetting,when the disjoining pressure decreases in a monotonous way, as for example,

.

However, conditions for the existence of equilibrium microdrops are satisfied inthe experiments mentioned already by Exerowa et al. [5–7].

MICROSCOPIC EQUILIBRIUM PERIODIC FILMS

In this section, we consider the possibility of existence of equilibrium periodicliquid films that are situated completely in the range of the disjoining pressureaction (partial wetting, S-shaped disjoining pressure isotherm). Undersaturationis under consideration, that is, Pe > 0.

FIGURE 2.7 Profile of an equilibrium microdrop. Note that the apex of the microdrop isin the range of the disjoining pressure action, that is, the drop does not have any sphericalpart (even at the drop apex).

x

he

h+

P h h h dhe e

h

h

e

( ) ( ) ,+ − =+

∫Π

Π( )hA

h=

3



Let h+ and h be the maximum and minimum heights of an equilibrium periodicfilm (Figure 2.8). Derivatives should be zero at h = h– and h = h+ or h′(h–) =h′(h+) = 0. Using the second of these two conditions in Equation 2.26, we candetermine the integration constant C as

.

Hence, the profile of the equilibrium periodic film is described by the followingequation:

, (2.31)

where

(2.32)

The following condition should be satisfied to have the positiveexpression under the square root in Equation 2.31.

The origin is placed at the position of the maximum height (Figure 2.8) andEquation 2.31 is written for a half period of the periodic film from x = 0, whichcorresponds to the position of the maximum, to x = x–, which corresponds to theposition of the minimum height (Figure 2.8). Notice that h–, h+, and x– are to bedetermined.

As we must have h′(h–) = 0 at the position of the minimum height of the film,it follows from Equation 2.31 that

(2.33)

FIGURE 2.8 Equilibrium periodic film. h+ denotes maximum thickness, and h– denotesminimum thickness; x– is the length of the half-period of the film.

x–

xh–

h+h

0

C P h h dhe

h

= + ++

∞

+

∫γ Π( )

′ = −−( )

−+

hL h

γ

γ

2

21

( )

L h P h h h dhe

h

h

+ += − − ++

∫( ) ( ) ( ) .Π

0 ≤ ≤+L h( ) γ

P h h h dhe

h

h

( ) ( ) .+ −− =−

+

∫Π



The latter condition relates the two unknown thicknesses h– and h+:

h– = h– (h+).

Equation 2.23 can be rewritten as

.

The latter equation and Figure 2.8 show that near the minimum height, h = h–,the liquid profile is convex, h″ > 0, that is, Pe > Π(h+); similarly, near themaximum height, h = h+, h″ < 0 if Pe < Π(h+).

At Pe > 0, for every pressure Pe there exists either no solution at all of Equation2.33, or there exists an interval of values h+ min < h+ < h+ max, where h+ min, h+ max

are determined by the following conditions h–(h+ min) = he, h–(h+ max) = hu.In the following text, we give a method for determining the unique value of

h+, that is, the value that is actually realized at the equilibrium.The excess free energy of a half period, x– (Figure 2.8), of the periodic film

is given by the same relation (2.20) where we, however, omit the additive constantdetermined by the reference state. The latter, as we see in the following text, isunimportant. From Equation 2.31, we can express

.

After substitution of the latter expression into Equation 2.20, we arrive at

(2.34)

The latter expression includes only one undetermined parameter, h+, becauseh– is expressed via h+ according to Equation 2.33. Only shapes with the minimumvalue of the excess free energy, Equation 2.34, can be realized, i.e., the unknownh+ should be determined using the following conditions:

(2.35)

′′ =+ ′( )

−( )hh

P he

13 2/

( )γ

Π

dxdh

L h

= −

−( )−

+

γ

γ

2

21

( )

Φ

Π

=

+ ′ + +

− −

∞

+

−

+ ∫∫

γ

γ

γ

1

1

2

2

2

h h dh P h

L h

h

e

h

h ( )

( )

ddh.

∂∂

= ∂∂

>+ +

Φ Φh h

0 02

2, .



As the volume of the half period per unit length of the periodic film is

conditions (2.35) are identical to the usual thermodynamic conditions of equilibrium:

.

Conditions 2.35 completely determine the equilibrium shape of the periodic film.The procedure suggested in Equation 2.35 (minimization of the excess free

energy) is consistent with Euler’s Equation (Equation 2.23), which minimizes thesame excess free energy. Computer calculations shown in the following textindicate that there is a unique h+ value satisfying conditions (2.35). These con-ditions prove the thermodynamic stability of periodic films.

For calculations of the dependence of the excess free energy on h+, accordingto Equation 2.34, a disjoining pressure isotherm should be selected. It is selectedfor the calculation as follows (Figure 2.9):

This choice approximately corresponds to aqueous films on quartz surface[1] and, according to Equation 2.17, gives:

FIGURE 2.9 Isotherm of disjoining pressure used for calculations of the excess freeenergy of periodic films.

t0ts

hhβh+

h−tmin

tmaxt2

Pe

−Πmin

Πmax

Π

0

V hdx V hx

x

= = +

−

+

∫ ( ),

∂∂

= ∂∂

>Φ ΦV V

0 02

2,

t t t t06 6

2610 1 5 10 2 10= = ⋅ = ⋅ =− − −cm cm cm, . , ,min max 33 10

5 10 10

6

6 6

⋅

= ⋅ = =

−

−

cm

cm

,

, ,max mints Π Πdyn/cm2 110 727 dyn/cm dyn/cm2, γ =



Using the adopted disjoining pressure isotherm (Figure 2.9), the dependenceof the excess free energy on h+ is calculated according to Equation 2.34. Theexcess pressure, Pe, varied from 0 to

Two calculated dependences are shown in Figure 2.10. Each of these plots has asharp minimum value. The minimum value determines the unique h+ value, whichis realized at the equilibrium.

According to Equation 2.34 and using the isotherm of disjoining pressurepresented in Figure 2.9, the calculated dependences are:

1. Pe = 0.7⋅106 dyn/cm2. 2. Pe = 0.2⋅106 dyn/cm2.

Let us compare the excess free energy of the corresponding β-film, Φβ, ofthe same length as a half period of the periodic film,

with the energy of a half period of the periodic film. The comparison is presentedin Table 2.1, which shows that at Pe < 0.6 · 10–6 dyn/cm2, the excess free energy

FIGURE 2.10 The excess free energy of periodic films, Φ(h+), calculated according toEquation 2.34, using the isotherm of disjoining pressure presented in Figure 2.9. (1) Pe =0.7⋅106 dyn/cm2, (2) Pe = 0.2⋅106 dyn/cm2.

43210

20

30

40

50

Φ . 1

04 , erg

5

12

h+ . 106, cm

cos . .θγe

S S≈ − − ≈− +1 0 94

Πmax .= 106 dyn/cm2

Φ Πβ βγβ

= + +

∞

−∫ ( ) ,h dh P h xh

e



of (β-films is lower than the corresponding energy of the periodic films (that is,(β-films are more stable); however, at Pe > 0.6 · 10–6 dyn/cm2, and the free energyof the periodic film becomes lower.

This would mean that close to the maximum value of the disjoining pressureisotherm, Πmax, periodic films are more stable than β-films, that is, the periodicfilms are a transitional state before rupture of β-films.

It was previously mentioned that periodic films exist only in the case of partialwetting, that is, if S– > S+. Periodic films are to be experimentally discovered.

The case p/ps > 1, when Pe < 0, can be treated similarly. It is possible toshow that in this case, the maximum thickness of periodic films can be outsidethe range of the disjoining pressure action, that is, periodic films in this case areactually a periodic array of drops.

MICROSCOPIC EQUILIBRIUM DEPRESSIONS ON ββββ-FILMS

In this section, an existence of equilibrium depressions on the surface of thickβ-films is considered (Figure 2.11).

A minimum thickness of a depression is marked as h– (Figure 2.11). Thederivative should be zero at the top of the depression, that is, h′(hβ) = 0. Usingthe preceding condition, an integration constant in Equation 2.26 can be deter-mined as follows:

.

After which Equation 2.26 can be rewritten as

TABLE 2.1Calculated Excess Free Energy of the Periodic Film and ββββ-Film

Pe⋅⋅⋅⋅106 dyn/cm2

F⋅⋅⋅⋅106 dyn (the periodic film)

Fββββ⋅⋅⋅⋅106 dyn(ββββ-film)

0.1 1130 11150.2 1212 12000.3 1303 12900.4 1299 12900.5 1446 14380.6 1633 16350.7 1727 17300.8 2227 22300.9 2250 2265

C P h h dhe

h e

= + +∞

∫γ β

β

Π( )



(2.36)

where

The right-hand side in Equation 2.36 should be positive, and it gives the followingrestrictions: 0 ≤ L– (h) ≤ γ.

The derivative should be zero at the bottom of the depression, that is, h′(h–) =0. This condition gives an equation for the determination of h:

(2.37)

The procedure of determination of h– is shown in Figure 2.12.

FIGURE 2.11 Schematic presentation of an equilibrium depression on the β-film withthickness hβ. h denotes the minimum thickness of the depression.

FIGURE 2.12 Determining the minimum thickness of the depression, h–, S-shaped dis-joining pressure isotherm (the upper part), and function L–(h) (the lower part).

hβ h–

1

1 2+ ′= − −

h

L hγγ

( ),

L h P h h h dhe

h

h

− = − − + ∫( ) ( ) ( ) .β

β

Π

L h P h h h dhe

h

h

− − −

−

= − − + =∫( ) ( ) ( ) .β

β

Π 0

h

h

hβhu

he

Pe

Π

Πmax

−Πmin

0

0

L–

h–



It is possible to show that

1. h– always exists in the case of partial wetting, that is, if S– > S+.2. Equilibrium depressions have lower excess free energy than the cor-

responding flat β-films above certain critical value of Pe.

Now we can suggest a new scenario of the transition from thick β-films to thinα-films in the case of partial wetting. At low values of Pe, β-films are more stablethan the equilibrium depressions or periodic films. However, above some criticalvalue of Pe, β-films have higher energy as compared to the equilibrium depres-sions. That means isolated depressions develop on the β-film. At further increasesof Pe, their excess free energy exceeds the corresponding value of a periodic film,and a transition from isolated depressions to a periodic film takes place. As wementioned previously, there is a critical value of Pe above which periodic filmscannot exist any more. This results in a transition from the periodic film to theα-film, with the microdrops sitting on it. This transition can be described as “arupture.” Residual microdrops cannot be at equilibrium with the α-film in thecase of partial wetting and gradually disappear by evaporation and/or hydrodynamicflow.

To summarize: we have shown that in the case of S-shaped disjoining pressureisotherm microdrops, microdepressions and equilibrium periodic films are pos-sible on flat solid substrates. Criteria have been provided for both existence andstability of these nonflat equilibrium liquid shapes. It has been suggested thattransition from thick films to thinner films goes via intermediate nonflat stateslike microdepressions or periodic films, which are more stable than flat films insome hydrostatic pressure ranges. Flat liquid films are unstable in the regionbetween α- and β-films and hence cannot be experimentally observed. However,the predicted nonflat stable liquid layers (microdepressions and periodic films)are located in this unstable region. Accordingly, experimental measurements ofprofiles of these nonflat layers open the possibility of determining the disjoiningpressure isotherm in the unstable region.

2.3 EQUILIBRIUM CONTACT ANGLE OF MENISCI AND DROPS: LIQUID SHAPE IN THE TRANSITION ZONE FROM THE BULK LIQUID TO THE FLAT FILMS IN FRONT

In this section, we shall show that the disjoining pressure action determines thepeculiar shape of liquid inside the transition zone from the bulk to the flat liquidfilms ahead for both menisci and drops. Thus, the equilibrium contact angle isdetermined using the disjoining pressure isotherm.

In Figure 1.12, three types of disjoining pressure isotherms are presented,which correspond to three different situations presented in Figure 1.1. Disjoining



pressure isotherms can be directly measured, not in the whole range of filmthickness, as we already discussed in Section 2.1 but only in the regions whereflat films are stable (see the stability condition of flat films in Equation 2.4). Thiswould indicate that the experimental measurements of the disjoining pressure canbe undertaken only in the case of flat and absolutely stable α-films and metastableβ-films. In this section, only partial wetting cases are under consideration. Com-plete wetting cases are under consideration in Section 2.4.

There is no doubt that the surface roughness influences the apparent value ofthe contact angle. However, it is obvious that the roughness cannot result in atransition from nonwetting to partial wetting or from partial wetting to completewetting. That is why only molecularly smooth solid substrates are under consid-eration in the following section. Consideration of equilibrium liquid states onrough substrates, when both capillary forces and surface forces are taken intoaccount, is a challenging subject to be developed in the future.

EQUILIBRIUM OF LIQUID IN A FLAT CAPILLARY: PARTIAL WETTING CASE

The excess free energy, Φ, of a liquid layer, drop, or meniscus on a solid substratecan be expressed by Equation 2.20 (see Section 2.2).

Equation 2.2 and Equation 2.19 show that the case Pe > 0 corresponds to thecase of menisci or other nonflat liquid shapes (see Section 2.2) at equilibriumwith undersaturated vapor; and the case Pe < 0 corresponds to the case of dropsor other nonflat liquid shapes (see Section 2.2) at equilibrium with oversaturatedvapor.

The difference between a volatile and a nonvolatile liquid determines onlythe path and the rate of a transition to the equilibrium state but not the equilibriumstate itself. In the following section, only the equilibrium states are under con-sideration, and hence it is not specified in this chapter whether the liquid is volatileor nonvolatile.

As already mentioned in Chapter 1, all solid surfaces in contact with a volatileor nonvolatile liquid at equilibrium are covered by a thin liquid film. The thicknessof this equilibrium film is determined by the action of surface forces (disjoiningpressure isotherm). That is, the choice of the reference state is uniquely deter-mined in order to consider the vicinity of the three-phase contact line at theequilibrium state of a bulk liquid in contact with a solid substrate; the referencestate is the state of solid substrate covered with the equilibrium liquid film. Thatis why a reference state that has a plane parallel film with the lowest possibleequilibrium thickness (that is, α-films introduced in Section 2.1), which corre-sponds to the vapor pressure p in the ambient air, is selected. In this section, two-dimensional equilibrium menisci in a flat chamber with a half-width H or two-dimensional equilibrium liquid drops are considered for simplicity. Extension ofthe derivation, in the following text, to axial symmetry is briefly discussed at theend of this section.



According to this selection, Equation 2.20 can be rewritten as

(2.38)

where he is the thickness of the equilibrium plane parallel α-film, and fD(h) is thedensity of the energy of surface forces. Two substantial simplifications are adoptedin the expressions for the free energy in Equation 2.38:

1. The density of the energy of surface forces, fD(h), depends only on thefilm thickness, h, and is independent of the derivatives of the filmthickness.

2. The interfacial tension retains its bulk value, γ. The first assumptionmeans that only profiles with low slope can be described using suchapproximation. The only attempt to take into account a dependency ofthe surface forces, fD(h), on the first derivative of the liquid profile ofdispersion forces has been undertaken in Reference 8. However, thecalculations in Reference 8 are based on a direct summation of molec-ular forces. These forces are well known to be of nonadditive nature[1]. Probably, this was the reason why a controversial nonzero equi-librium contact angle has been predicted in the case of complete wet-ting [8]. That is why consideration of surface forces in the case ofnonflat profiles remains a challenge, and we use this assumption (1)even in cases where it is not rigorously valid.

Actually, the two assumptions, (1) and (2), are strongly interconnected. If thedensity of energy of surface forces, fD(h, h′), depends on the derivative of thefilm profile, h′, then the tangential stress on the surface of the liquid is unbalanced.However, if we adopt both assumptions (1) and (2), at least from this point ofview, we do not have any contradictions; constant interfacial tension results inzero tangential stress under equilibrium conditions.

Let us briefly discuss what happens if the density of energy of surface forces,fD(h, h′), depends on the derivative of the film profile, h′. In this case, Equation2.38 takes the following form:

At equilibrium, the first condition (1) (Section 2.2) must be satisfied, whichresults in

.

F h P h h f h f h dxe e D D e= + ′ −

+ −( ) + −{ }∫ γ 1 12 ( ) ( ) ,,

Φ = + ′ −

+ −( ) + ′ −{ }γ 1 12h P h h f h h f he e D D e( , ) ( )∫∫ dx.

γ ′′+ ′

− ∂∂

+′

′′ +′

′h

h

f

h

d f

dhh

d f

dhdhD D D

( ) /1 2 3 2

2

2

2

hh Pe=



Let us introduce the following functions:

;

then the latter equation takes the following form:

.

This means that (1) The effective surface tension, , depends onboth thickness, h, and the slope, h′. (2) The effective disjoining pressure nowdepends on both the thickness and slope values as the effective surface tension.The consequences of such dependences, as well as the physical meaning of theseeffective values are to be understood. This is the reason why we use the approx-imation adopted in Equation 2.38, that is, the density of the energy of surfaceforces, fD(h), depends only on the film thickness, h, and is independent of thederivatives of the film thickness.

Integration, as seen in Equation 2.38, is taking over the whole space occupiedby the flat meniscus.

Any liquid profile, h(x), which gives the minimum value to the excess freeenergy, Φ, according to Equation 2.23, describes an equilibrium liquid configu-ration on the planar surface. For the existence of the minimum value, the fourconditions introduced in Section 2.2 should be satisfied (see conditions (1)–(4)and the discussion there).

The first requirement (1) shows that the liquid profile gives minimum ormaximum to the excess free energy, Φ, whereas two other requirements, (2) and(3), prove that the profile provides a minimum value to the excess free energyΦ. It is necessary to note that both requirements (2) and (3) must be satisfied;only in this case the excess free energy (Equation 2.38) has a minimum value.

The requirement (1) results in the Euler’s equation (Equation 2.23), whichfor the first time has been suggested by Derjaguin [1] and should be referred toas Derjaguin’s equation, where disjoining pressure is introduced as

.

If the requirement (3) is not satisfied, then the solution of Equation 2.23 doesnot provide a stable solution.

Condition (3) shows that Equation 2.23 can be integrated once, which resultsin Equation 2.26. Note that the right-hand side of Equation 2.26 should alwaysbe positive.

a h hd f

dhh h h

f

h

dD D( , ) , ( , )/

′ =′

+ ′( ) ′ = − ∂∂

+2

22

3 21 Π

22 f

dhdhhD

′′

γ + ′( ) ′′

+ ′+ ′ =

a h h h

hh h Pe

( , )

( )( , )

/1 2 3 2Π

γ + ′a h h( , )

Π( )( )

hdf h

dhD= −



MENISCUS IN A FLAT CAPILLARY

In the case of a meniscus in a flat capillary, the integration constant, C, isdetermined from the condition at the capillary center:

(2.39)

which gives

where H is the half-width of the capillary. Using this constant, Equation 2.26 canbe rewritten as

(2.40)

This equation describes an equilibrium profile of the meniscus in flat capillaries.Let us consider the solution of Equation 2.23 in more detail. This equation

determines the liquid profile in three different regions:

(1) A spherical meniscus, which is not disturbed by the action of surfaceforces. That is, the disjoining pressure action can be neglected, and wearrive at a regular Laplace equation:

. (2.41)

(2) In the case of a flat liquid film in front of the meniscus,

. (2.42)

(3) At a transition zone in between, both the capillary force and the dis-joining pressure are equally important.

Note that Equation 2.42 coincides with Equation 2.3, but we keep this equa-tion for convenience. In the following text, we consider only “macroscopic cap-illaries.” In these capillaries, the radius, H, is much bigger than the range of actionof surface forces. Let the radius of the action of surface forces be ts ≈ 10–5 cm =1000 Å = 0.1 µ = 100 nm, that is, at h > ts: Π(h) = 0. In this case, Equation2.23 can be rewritten at h > ts as Equation 2.41 with boundary conditions

′ = −∞h H( ) ,

C P H h dhe

H

= +∞

∫Π( ) ,

1

1 2+ ′=

−( ) − ∫h

P H h h dhe

h

H

Π( )

.γ

γ ′′+ ′

=h

hPe

( ) /1 2 3 2

Π( )h Pe=



(2.43)

and Equation 2.39. Solution of Equation 2.41 with boundary conditions accordingto Equation 2.39 and Equation 2.43 gives a spherical profile

, (2.44)

that is, a spherical meniscus of radius

.

The preceding equation describes an idealized profile of a spherical meniscus.Intersection of this profile with the thin equilibrium film of thickness he definesthe apparent three-phase contact line and the macroscopic equilibrium contactangle θe (Figure 2.13). A simple geometrical consideration shows that

. (2.45)

At h = he: h′2 = 0, and we conclude from Equation 2.45 and Equation 2.40:

, (2.46)

FIGURE 2.13 Profile of a meniscus in a flat capillary. 1 — a spherical part of the meniscusof curvature ρe, 2 — transition zone between the spherical meniscus and flat films in front,3 — flat equilibrium liquid film of thickness he. Further in the text, the liquid profile insidethe transition zone will be considered in more detail.

h H( )0 =

( )H hP

xPe e

− + −

=

2

2 2γ γ

ρ γe

eP=

PHe

e= γ θcos

1 =

−( ) − ∫γ θ

γ

cos( )e

e

h

H

HH h h dh

e

Π

x

2H

he32

1

ρeθe



which allows the determination of the contact angle via disjoining pressureisotherm as

(2.47)

The preceding equation, for the first time, has been deduced by Derjaguinand Frumkin [1] and named after these two scientists as the Derjagun–Frumkinequation. Note that this equation was deduced using completely different argu-ments from those previously mentioned and for many years was considered asindependent of Equation 2.23. The derivation given in the preceding section showsthat Derjaguin–Frumkin Equation 2.47 is a direct consequence of Equation 2.23.Also note that the same equation for the contact angle has been deduced inSection 2.1 in a different way.

Equation 2.47 can be approximately rewritten (Figure 2.1) as:

. (2.48)

The preceding equation shows that cos θe < 1 only if S– > S+ (Figure 2.1).Now, at last, we can precisely define the term partial wetting: (1) S-shaped

disjoining pressure isotherm (curve 2 on Figure 2.1) and (2) S– > S+.Let us consider the case when Pe < Πmax,

,

or H > Hcr, where

.

We refer to such capillaries as thick capillaries. In the case of aqueous solutions,γ ~ 70 dyn/cm, erg, and hence, . Otherwise,the capillary is referred to as a thin capillary, that is, the capillary is thin if itsthickness H is in the range ts << H < Hcr, where ts is the radius of the disjoiningpressure action. According to the definition of thin capillaries, these capillariesare still big enough when compared with the radius of the action of surface forces,ts. If the capillary radius is compared with the radius of action of surface forces, ts,

cos

( )

( ) ,θγ

γeh

H

eh

h dh

hH

h dhe

e

=

+

−≈ +

∫∫∞

11

11

1Π

Π at hh t He s< << .

cos θγe

S S≈ − −− +1

γ θcosmaxH

< Π

Hcr ~max

γΠ

Πmax ~ 105 Hcr ~ ~7 10 104 3⋅ − −cm cm



then such a capillary should be referred to as a microscopic capillary. Only thin,macroscopic capillaries are under consideration in this book.

In the case of thick macroscopic capillaries, Equation 2.42 has three solutions,one of which corresponds to the stable equilibrium α-film with thickness he. Theexcess free energy of α-films is equal to zero, according to our choice in Equation2.38. The second solution of Equation 2.42 in this case, hu, is unstable accordingto the stability condition (2.4, Section 2.1), and the third solution, hβ, is β-film,which is also stable according to the same stability condition (2.4, Section 2.1).It has been shown in Section 2.1 that β-films have higher excess free energy ascompared with α-films, that is, β-films are less stable and eventually rupture tothinner and absolutely stable α-films.

However, in thin capillaries, Equation 2.42 has only one solution (not shownin Figure 2.3), which is an absolutely stable α-film.

MENISCUS IN A FLAT CAPILLARY: PROFILE OF THE TRANSITION ZONE

Let us estimate the length of the transition zone, L. Inside the transition zone,the capillary pressure and disjoining pressure are of the same order of magnitudeof

.

According to Equation 2.23, inside the transition zone, the capillary pressure canbe estimated as

or .

From the latter estimation, we conclude that

. (2.49)

In the case of he ~ 10–6 cm, H ~ 0.01 cm, and the latter estimation gives L ~1 µm. Note that the same estimation of the length of the transition zone is alsovalid in the case of droplets.

Now let us rewrite Equation 2.23 in the following form:

. (2.50)

PHe ~γ

γ γ′′

+ ′

h

h

h

Le

1 2 2~

γ γh

L He

2~

L h He~

′′ = + ′ −h h P he

11 2 3 2

γ( ) [ ( )]/ Π



The preceding equation shows that the sign of the second derivative is deter-mined by the difference Pe – Π(h). In the case of thick capillaries, that is, H >Hcr, Figure 2.5 shows that

h″ > 0, in the following range of thickness: hβ < h < H; the profile isconcave.

h″ <, at hu < h < hβ; the profile is convex.h″ > 0, at he < h < hu; the profile is concave again.

This would mean that the profile of the liquid inside the transition zone doesnot remain concave all the way through the transition zone, but it changes itscurvature in two inflection points: (Figure 2.5). The mag-nification of the liquid profile inside the transition zone is schematically shownin Figure 2.5.

Now an important question arises: flat thin films in the range of thicknessfrom tmin to tmax are unstable according to the stability condition (2.4, Section2.1). We would like to emphasize that the aforementioned condition is the stabilitycondition of flat films. As already discussed in Section 2.2 and in this section,the stability condition (3) of Section 2.2 of nonflat liquid layers is completelydifferent, and according to Equation 2.25, the condition is satisfied inside thetransition zone (see Figure 2.5 where h′ is positive everywhere). Nobody shouldexpect any convergence of the two stability conditions (2.4, Section 2.1) of flatfilms and (condition 3, Section 2.2) of nonflat films; they are completely different.A qualitative physical explanation of the stability of the transition zone insidethe “dangerous” range of thickness from tmin to tmax is as follows. The extent ofthe dangerous region from xmax to xmin (Figure 2.5) is small enough, that is, anyfluctuation inside this dangerous region is dampened by the neighboring stableregions from both sides (Figure 2.5).

The liquid profile inside the transition zone in the case of thin capillaries,that is, H < Hcr , is much simpler (Figure 2.14), and it does not have any inflectionpoints as in the case of thick capillaries. Here, according to Equation 2.50, the liquidprofile is always concave. The stability of the liquid shape inside the dangerousregion of thickness from tmin to tmax is proven in precisely the same way as in thecase of thick capillaries.

Note that in all cases under consideration, there is no real three-phase contactline at the equilibrium because the whole solid surface is covered by a flatequilibrium liquid film. This is the reason why we refer to it as an apparentcontact line. The transversality condition (4 of Section 2.2) at the apparent three-phase contact line results in Equation 2.21 (Section 2.2):

(2.51)

Let us consider the latter condition in more detail in Appendix 1. Thisconsideration shows that in general cases (except for a very special model iso-therm of disjoining pressure), the transition from nonflat transition zone to flat

h x h h x hu u( ) , ( )β β= =

′ =h he( ) .0



equilibrium films goes very smoothly and asymptotically as (the sense ofthe latter statement is clarified in Appendix 1). That is, in a general case, thereis no final point where the transition zone ends, but it approaches the flat filmasymptotically.

PARTIAL WETTING: MACROSCOPIC LIQUID DROPS

We have to remind ourselves that Pe < 0 in this part because the liquid drops canbe at equilibrium with oversaturated vapor only. A macroscopic drop means thatthe drop apex, H, is outside the range of surface forces (or disjoining pressure)action. Microscopic drops, that is, drops with the apex in the range of thedisjoining pressure action, are considered in Section 2.2.

The equilibrium films are determined according to Equation 2.42. Note thatin the case of complete wetting, there are no equilibrium flat films on solidsubstrates because the line Pe < 0 does not intersect (curve 1 in Figure 2.1).Hence, there are no equilibrium droplets on completely wettable solids underoversaturation.

However, in the case of partial wetting, Equation 2.42 has two solutions(Figure 2.17, left-hand side). According to the stability condition of flat films(condition 4, Section 2.2), one of them corresponds to the stable equilibrium filmof thickness he, and the second one corresponds to the unstable film of thicknesshu (Figure 2.17, left-hand side). The latter means that equilibrium droplets in thecase of partial wetting are sitting on the stable equilibrium film of thickness he.

However, even in the case of partial wetting, equilibrium droplets can existon the solid substrate only in a limited interval of oversaturation, which isdetermined by the following inequality: or using Equation 2.2 inthe following range of oversaturated pressure over the solid substrate,

. (2.52)

FIGURE 2.14 Magnification of the transition zone in the case of partial wetting in “thincapillaries.”

x

b

θe

hha

ΠPe

he

hu

tmin

tmax

hβ

x → ∞

0 > > −Pe Πmin ,

1 < <

p

p

v

RTs

mexp minΠ



If Πmin is in the range 106–107 dyn/cm2, then the latter inequality takes thefollowing form

,

that is, the equilibrium droplets exist only in a very limited interval of oversatu-ration. Beyond this interval, at higher oversaturation, neither equilibrium liquidfilms nor droplets exist on the solid substrate as in the case of complete wetting.Probably, the critical oversaturation pcr:

determined using Equation 2.52 corresponds to the beginning of homogeneousnucleation, and values below this critical limit would indicate that a heterogeneousnucleation is more favorable.

The radius of curvature of an equilibrium drop is

In the aforementioned narrow interval of oversaturation, the radii of the equilib-rium drops change from infinity at to

If Πmin ≈ 106 dyn/cm2 and γ ≈ 72 dyn, then

that is, the critical size is out of the range of the action of surface forces. However,if Πmin ≈ 107 dyn/cm2, then

and the whole droplet is in the range of the action of surface forces. That is, inthe latter case, the drop is so small that it does not have anywhere (even at thevery apex) a spherical part undisturbed by the action of the disjoining pressure.

1 1 1 001 1 01< < + ≈ −p

p

v

RTs

mΠmin . .

p

p

v

RTcr

s

m=

exp minΠ

ℜ =−e

eP2γ

.

p ps→

ℜ =cr

2γΠmin

.

ℜ ≈ =cr

144

101 44

6. µm,

ℜ ≈ = =cr

144

100 144 1440

7. µm Å,



In the following text, we consider only two-dimensional drops for simplicity.Three-dimensional axisymmetric drops and menisci in cylindrical capillaries arebriefly considered towards the end of this section.

On the drop apex, H, the derivative vanishes: . Using the lattercondition and Equation 2.26, we arrive at the following integration constant

.

In this case, Equation 2.26 transforms as follows:

. (2.53)

The preceding equation describes the profile of equilibrium liquid drop onflat solid substrate at Pe < 0.

As in the case of a meniscus, the whole profile of a droplet can be subdividedinto three parts:

1. A spherical part of the drop2. A transition zone where both capillary pressure and disjoining pressure

are equally important3. A region of flat equilibrium liquid film in front of the drop

Outside the range of the disjoining pressure action, we can neglect the actionof surface forces in Equation 2.53, and these equations describe the profile of aspherical drop:

. (2.54)

The preceding equation describes an idealized profile of a spherical droplet.Intersection of this profile with the thin equilibrium film of thickness he definesthe apparent three-phase contact line and the macroscopic equilibrium contactangle h′(he) = –tgθe. Substitution of this expression into Equation 2.54 results in

.

′ =h H( ) 0

C P H h dhe

H

= + +∞

∫γ Π( )

1

1 2+ ′=

+ − − ∫h

P H h h dhe

h

H

γ

γ

( ) ( )Π

1

1 2+ ′= + −

h

P H heγγ( )

PHe

e= − −γ θ( cos )1



Casting this expression into Equation 2.53 at h = he results in the followingdefinition of the contact angle in the case of drops on a flat substrate:

(2.55)

which is similar to Equation 2.47 in the case of the meniscus. At he << H,expressions for the equilibrium contact angles of the menisci (2.47) and droplets(2.55) coincide. However, it is necessary to note that expressions for the equilib-rium contact angle in the case of menisci (Equation 2.47) and drops (Equation2.55) are still different; integration in these expressions, even in the case of thickcapillaries and big drops, ts << H, starts from different values of he. In the caseof drops, this value is always bigger than that of the menisci. This results indifferent values of equilibrium contact angles in the two cases.

Let us rewrite Equation 2.53 for the drop profile in the identical form, usingthe transversality condition at h = he (h′(he) = 0). This results in

(2.56)

where

(2.57)

The expression under the square root in Equation 2.56 should be positive,that is the following condition should be satisfied:

(2.58)

where the first equality corresponds to the zero derivative, and the second equalitycorresponds to the infinite derivative. Note that beyond the radius of the actionof surface forces, ts, that is, at h > ts: L(h) = Pe (h – he) becomes a straight line(lower part in Figure 2.15).

Now, it is important to mention that the droplet is in equilibrium with thevapor in the surrounding air. This means that the droplet volume cannot be fixed,

cos

( )

( ) ,θγ

γeh

H

eh

h dh

hH

h dhe

e

= +−

≈ +∫

∫∞

1

1

11

1Π

Π at tt Hs << ,

′ = −−( )

−hL h

γ

γ

2

21

( ),

L h P h h h dhe e

he

h

( ) ( ) ( ) .= − − ∫Π

0 ≤ ≤L h( ) ,γ



and hence the droplet height, H, cannot be fixed. That is, the drop height, H, isto be determined in the following section.

The upper part of Figure 2.15 shows the S-shaped dependence of the disjoin-ing pressure isotherm, Π(h), whereas the lower part of Figure 2.15 shows thedependency of L(h), which has maximum or minimum thickness which aresolutions of Pe = Π(h). At the apex of the drop, when h = H, the first derivativeshould be zero, that is, h′(H) = 0 or from Equation 2.58:

L(H) = 0. (2.59)

In this section, macrodrops, that is, drops that have their apex outside therange of the disjoining pressure action (partial wetting), are considered. In thiscase, max L(h) = L(hu) > 0 and L(h) → –∞ as h → +∞ (see Figure 2.15, lowerpart). Therefore, Equation 2.59 always has a solution hu < H < ∞ (the lower partof Figure 2.15).

If the value of H lies outside the range of the influence of the disjoiningpressure, then it becomes possible to determine the equilibrium macroscopiccontact angle of the drop. Depending on the value of the radius of the curvaturein the central part of the drop, ℜ, three different possibilities can occur (Figure2.16a–Figure 2.16c):

FIGURE 2.15 Determining the droplet height, H. Upper part: disjoining pressure iso-therm; lower part: shape of function L(h) (see definition given by Equation 2.58).

h

h H

ts hu he

ts hu he

Pe

Π

Lmax

L

γ

–Πmin



1. ℜ > H he, which corresponds to the contact angle 0 < θe < π/2, thepartial wetting case.

2. ℜ < H he but 2ℜ > H – he, which corresponds to the macroscopiccontact angle π/2 < θe < π, the nonwetting case.

3. If 2ℜ < H – he, despite the apex of the drop being outside the rangeof influence of the disjoining pressure, it is impossible to determine themacroscopic contact angle, as there is no intersection of the circle ofradius ℜ with the solid surface (Figure 2.16c). The last case can bereferred to as the complete nonwetting case and can be referred to asθe > π, similar to the case of complete wetting, when cos θe > 1 (seeSection 2.4).

It is interesting to note that probably cases 2 and 3 (Figure 2.16) have neverbeen observed experimentally. It means that either such disjoining pressure iso-therms do not exist in nature or such cases are yet to be discovered.

It is possible to check (using Equation 2.55) whether in partial wetting θe(Pe),dependence increases with decreasing Pe, i.e., the drop elevates itself above thesolid surface as Pe decreases, and at Pe = –Πmin, the drop separates itself from the

FIGURE 2.16 Determining the equilibrium contact angle of droplets. (a) partial wetting,(b) nonwetting, and (c) complete nonwetting.

(c)

(a)

(b)

x

H

x

θe

H

he

he

xθe

H

he

�

�

�



solid surface and goes into the surrounding air. It corresponds to a transition froma heterogeneous to a homogeneous nucleation.

PROFILE OF THE TRANSITION ZONE IN THE CASE OF DROPLETS

Equation 2.50 and Figure 2.17 (left-hand side) show that, in the case of droplets,there is only one inflection point on the drop profile at h(xu) = hu. Hence, thedrop profile inside the transition zone is shown in Figure 2.17 (right-hand side).

Deviations from the spherical profile start immediately as the surface forcesstep into action at h < ts. Note that those deviations are in the opposite directionas compared to the drop profile (Figure 2.5 and Figure 2.14).

AXISYMMETRIC DROPS

In this case, expression for the excess free energy takes the following form:

which gives the following equation for the liquid profile of an axisymmetric drop:

, (2.60)

where

,

FIGURE 2.17 The drop profile inside the transition zone.

Φ = + ′ −

+ −( ) + −{ }2 1 12π γr h P h h f h f he e D D e( ) ( )∫∫ dr,

γr

d

dr

rh

hh Pe

′

+ ′

+ =

1 2Π( )

Pe = −ℜ2γ

xs xu x

θe

Pe

he

hu

t

h

Π–Πmin



which results in

for drops. Note the multiplier 2 in the latter expressions.Unfortunately, Equation 2.60 cannot be integrated as is done in the case of

two-dimensional menisci and drops. However, the latter equation can be rewrittenas

. (2.61)

The first term on the left-hand side of the preceding equation is due to thefirst curvature (similar to the case of the two-dimensional menisci or drops inEquation 2.23), and the second term is due to the second curvature, which isshown in the following text to be small as compared to the first term. Thecharacteristic length of the transition region, L, is given by Equation 2.49:

. The latter expression shows that L << H. Let us estimate the ratioof the second term to the first term on the left-hand side of Equation 2.61:

The latter estimation shows that the second term on the left-hand side ofEquation 2.61 is small as compared to the first term and can be neglected in thetransition region. After that, Equation 2.61 can be integrated, and in a similarmanner as Equation 2.23, Equation 2.26 can be recovered. Outside the region ofthe action surface forces, Equation 2.60 can be easily solved. This solution is the“outer solution,” whereas the solutions obtained in the previously mentionedmethod all are “inner solutions.” The matching of these two asymptotic solutionsgives the real profile in the case of axial symmetry (see this procedure in the caseof complete wetting in Section 2.4).

MENISCUS IN A CYLINDRICAL CAPILLARY

In this case, the expression for the excess free energy is as follows:

PHe

e= − −2 1γ θ( cos )

γ γ′′

+ ′( )+ ′

+ ′+ =h

h r

h

hh Pe

1 123 2 2/

( )Π

L h He~

γ γr

h

h

h

h

hrh

h L

Hh L

LH

′+ ′

′′

+ ′( )′′′

=1 1

2 23 2 2/

~ ~ ~/

/

hhH

e << 1.

Φ = −( ) + ′ − −( )

{

+ −

∫ 2 1 2πγ

π

H h h H h

P H

e

e hh H h H f h f h dxe D D e( ) − −( )

+ − }2 22π ( ) ( )



where H is the radius of the cylindrical capillary. The aforementioned procedurealso results in

.

Note that the disjoining pressure in this case is

,

which is different from the disjoining pressure of flat films Π(h). We shall discussthis further in Section 2.7.

APPENDIX 1

Let us assume that the transition zone profile does not tend asymptotically to theequilibrium thickness he but meets the film at the final point x = x0. In this case,in the vicinity of this point, we approximate the disjoining pressure isotherm bya linear dependency , where is a positivevalue; he is a stable flat liquid film, and the derivative of the disjoining pressureshould be negative and . The liquid profile in this region has a lowslope, which means Equation 2.23 can be rewritten as

.

Solution of the latter equation is

, (A1.1)

where

,

and C1 and C2 are two integration constants.At x = x0, according to Equation 2.21, the following two boundary conditions

should be satisfied:

γ γ′′

+ ′( )+

− + ′+

−=h

h H h h

H

H hh Pe

1

1

123 2 2/

( )Π

H

H hh

−Π( )

Π Π( ) ( ) ( )h h a h he e≈ − − a he= − ′Π ( )

Π( )h Pe e=

γ ′′ + − =h a h he( ) 0

h x h C x C xe( ) exp( ) exp( )= + + −1 2α α

αγ

= a

h x h

h x

e( ) .

( ) .

0

0 0

=

′ =



Equation A1.1 and the two boundary conditions result in the following systemof algebraic equations for the determination of the integration constants C1 and C2:

.

.

The only solution of the aforementioned system is C1 = C2 = 0, which isobviously a contradiction. Hence, the only possibility is that C1 = 0, and the liquidprofile has the following form if h → he:

.

That is, the liquid profile in the transition zone tends asymptotically to theequilibrium thickness he and does not meet the equilibrium flat film in any finalpoint x0.

Note that in the case when , at h < t0, our assumption on linearizationof the disjoining pressure isotherm is not valid anymore, and the only specialcase here is when the transition zone profile meets the equilibrium flat liquid filmat the final point x0. In this case, the upper limit of the integration in Equation 2.1should be replaced by x0.

Note, that

gives a new scale of the transition zone, which is 1/α. It is possible to checkwhether the new scales and the previous one given by Equation 2.49 are of thesame order of magnitude. Indeed,

2.4 PROFILE OF THE TRANSITION ZONE BETWEEN A WETTING FILM AND THE MENISCUS OF THE BULK LIQUID IN THE CASE OF COMPLETE WETTING

The profile of a liquid in the transition zone between a capillary meniscus and awetting film has been calculated for two types of disjoining pressure isotherms

C x C x1 0 2 0 0exp( ) exp( )α α+ − =

C x C x1 0 2 0 0exp( ) exp( )α α− − =

h x h C xe( ) exp( )= + −2 α

Π( )h = ∞

αγ

= a

1α

γ γ γ γγ

=′

= = =Π Π( )

~( )

~ .h h h

hP

hH

h H Le e e

e

e

ee/ /



(both in the case of complete wetting). As discussed in Section 2.3, wetting filmsahead of the meniscus are separated from the capillary meniscus by a transitionzone where the surface forces and the capillary forces act simultaneously. As themeasurements of equilibrium contact angles and surface curvature of bulk liquidsshould be carried out outside the transition zone, its size and profile are of interest.Moreover, the shape of the liquid profile in the transition zone supplies informa-tion concerning the disjoining pressure isotherm of liquid films on a given solidsubstrate.

In the following text, we consider a transition zone under equilibrium con-ditions between a capillary meniscus between two parallel plates and wettingfilms in front (Figure 2.18). The width of the capillary, 2H, is assumed to bemuch larger than the thickness of the equilibrium flat film, he. In the case underconsideration, the thickness of the liquid layer, h(x), is a function of a single coor-dinate, x, directed along the capillary surface. It was already shown in Section 2.2that the meniscus profile comes to the flat liquid film at zero contact angle,according to the transversality condition (2.21), and the condition for that followsfrom Equation 2.26 as:

,

where the equilibrium thickness, he, is determined as

.

FIGURE 2.18 Complete wetting case. Schematic representation of a circular capillarymeniscus (1), transition zone (2), wetting films (3) in a flat capillary. Continuation of aspherical meniscus (broken line) does not intersect either the solid walls of the capillaryor the thin liquid film of thickness he in front of the meniscus. The radius of the curvatureof the meniscus, Pe, is smaller than the half-width H.

γ = − −∞

∫P H h h dhe e

he

( ) ( )Π

Π( )h Pe e=

x

2H

23

0

1 Pe

he

h

h∗



In the case of complete wetting and disjoining pressure isotherms of the typeΠ(h) = A/hn, Equation 2.21 and Equation 2.26 result in

,

and

,

where n is the exponent in the expression for the isotherm, γ is the surface tensionof the bulk liquid, and H is the half-width of the capillary. That is, we have twoequations with two unknown values, Pe and he. The preceding two equationsdetermine the equilibrium pressure, Pe, via the thickness of the equilibrium flatfilm, he. This results in

(2.62)

Note that the equilibrium pressure is equal to

according to Equation 2.62. This means that the radius of the curvature of themeniscus (in the case of the complete wetting) and the contin-uation of the spherical meniscus does not intersect either the capillary walls orthe flat liquid films in front of the meniscus (Figure 2.18).

According to Section 2.2 and Section 2.3, the profile of the meniscus, thetransition zone, and the flat wetting films in front are described by Equation 2.23,which in the case under consideration becomes

(2.63)

where h(x) is the local liquid profile, A/hn is the local disjoining pressure isotherm.Inside the transition zone (Figure 2.18), the liquid profile has a very low slope,

γ = − −− −P H h

A

n he e

en

( )( )1 1

A

hP

en e=

PH

n

nh

H hh

n

nh he

e

e e=−

−

=−

=−

>∗

γ γ

11

, .*

PP H he

e

= =− ∗

γ γ,

P H h He = − <* ,

γ ′′

+ ′( )+ =h

h

A

hP

n e

1 23 2/

,



that is, (dh/dx)2 << 1 is satisfied, and the disjoining pressure isotherm, A/hn, offlat films can be safely used at each point of the liquid layer of varying thickness.As previously mentioned, we assume that the surface tension of the film is thesame as that of the bulk liquid.

The following consideration is carried out for the case of complete wetting.As the liquid profile inside the transition zone satisfies the condition

(dh/dx)2 << 1, the low slope approximation can be used. Introducing the dimen-sionless variables, ξ = h/he and y = [x – (H – h*)]/l, where l is a length scale tobe determined along x. It is shown in the following text that y is the local variableinside the transition zone. Using these notations in Equation 2.63, we arrive at

(2.64)

where ξ = ξ(y), and the length scale is selected as

(2.65)

Note that the latter selection is in excellent agreement with our previousestimation of the length of the transition zone in Section 2.3 (Equation 2.49).The thickness of the equilibrium flat film, he, is determined as before from thecondition A/hn

e = Pe.According to Jacoby’s condition, the dependency ξ(y) is a monotonic one.

Therefore, as the independent variable, y, does not appear explicitly in Equation2.64, we can introduce a new unknown function, ξ′ = function(ξ). Taking intoaccount that ξ′(1) = 0, Equation 2.64 can be rewritten as

(2.66)

The preceding equation is solved for the most important cases: n = 3 andn = 2. Films that obey the Π = A/h3 law (that is, n = 3) correspond to the casewhere the disjoining pressure of the film is determined by dispersion forces [1].For nonpolar liquids on solid dielectrics, we can adopt A = 10–14 erg [1] and γ =30 dyn/cm.

In particular, for n = 3, it follows from Equation 2.66 that

′′ + =ξξ1

1n

,

l h H he= −( )* .

′ = − −( ) +−

−

−ξ ξ

ξ2 1

21

11

1( ).

n n

′ = − − +ξ ξξ

ξ12 1.



Upon integration, the solution of the latter equation is

(2.67)

where C is the integration constant.If in Equation 2.67, that is, in the region where the transition zone

and the meniscus meet each other, we conclude that

(2.68)

In order to determine C, we consider the meniscus shape corresponding tolarge h values for which it can be assumed in Equation 2.63 that the disjoiningpressure can be neglected, which results in

Integration of this equation with the boundary condition, h(0) = H, resultsin the following solution for the spherical meniscus:

.

Using the same local variables previously mentioned for the transition zone,ξ(y) and y << 1, in the latter equation, we conclude that

(2.69)

Comparison of Equations 2.68 and Equation 2.69 results in C = 0. That is,Equation 2.67 can now be rewritten as

(2.70)

2 11

3

2 1 3

2 1 3ξ

ξξ

+ ++ −+ +

= − +( )ln ,y C

ξ( )y >> 1

ξ( )( )

.yy C≈ + 2

2

γ γ′′

+ ′( )

= =−

h

h

PH he

12

3 2/*

.

h x H H h x H h( ) * *= − −( ) − − −( )( )2 2

ξ( ) .yn

n

y≈−

+1 2

2

2 11

3

2 1 3

2 1 3ξ

ξξ

+ ++ −+ +

= −ln .y



An “ideal profile,” , is as shown in the following function:

(2.71)

The “real profile,” calculated according to Equation 2.70, and the ideal profile,according to Equation 2.71 at n = 3, are presented in Figure 2.19. It shows thatthe extent of the transition zone can be roughly estimated from y = 1.2 to y =1.7. That is, the total length of the transition zone, L, in dimensional units is

Figure 2.19 shows that the maximum deviation of the real profile from idealprofile is at y = 0, that is, at the position corresponding to minimum value ofcontinuation of the spherical meniscus, and the maximum deviation is around2.5 he. Figure 2.19 also shows that the deviation of the real profile from the idealprofile is roughly symmetrical from both sides, pertaining to the position of themaximum deviation at y = 0.

FIGURE 2.19 Real profile (1) inside the transition zone calculated according to Equation2.70 and the ideal profile (2,3) according to Equation 2.71 at n = 3.

ξi y( )

ξi y

n

n

yy

y

( ), .

,

= −+ <

>

1 20

1 0

2

L l h H he= = −( )2 9 4 3. . .*

–2 –1 0 1 2

y

3

1

1.0

22.0

2.5

3.0

0.5

ξ

1.5



The isotherm Π = A/h2 corresponds to thick β films of water, which arestabilized by the ionic–electrostatic component of the disjoining pressure [1]. Inthis case, integration of Equation 2.66 at n = 2 results in

, (2.72)

where the integration constant is equal to zero, which is concluded in preciselythe same way as in the case, n = 3.

Table 2.2 shows that although the absolute thickness of the transition zone, L,decreases with a decrease in the capillary radius, H, the relative transition zonethickness, L/H, increases with a decrease in H. The L values vary within a rangefrom 37 ηm for thick films (he ~ 500 Å) to 0.2 ηm for thin films (he ~ 30 Å).

In the case n = 2, calculations made using Equation 2.72 and adopting A =2⋅10–7 dyn, γ = 72 dyn/cm (Table 2.3), and the ideal profile (according to Equation2.71 at n = 2) are presented in Figure 2.20. The figure shows that the extent ofthe transition zone can be roughly estimated from y = –2.4 to y = 1.9. That is,the total length of the transition zone, L, in dimensional units is

It also shows that the maximum deviation of the real profile from the ideal profileis at y = 0, that is, at the position that corresponds to the minimum value of the

TABLE 2.2Characteristics of the Transition Zone for the Case of the ΠΠΠΠ = A/h3 Isotherm

H, cm 0.3 0.2 0.1 5.10–2 10–2 10–3 10–4

he, Å 445 405 322 256 150 70 32L, cm 3.7·10–3 2.88·10–3 1.81·10–3 1.14·10–3 3.92·10–4 0.85·10–4 1.81·10–5

TABLE 2.3Characteristics of the Transition Zone for the Case of the ΠΠΠΠ = A/h2 Isotherm

H, cm 1 0.5 0.2 0.1 10–2 10–3 10–4

he, Å 5250 3730 2640 1660 525 166 53L, cm 3.62·10–2 2.1·10–2 115·10–2 6.44·10–3 1.15·10–3 2.04·10–4 3.64·10–5

ξξ

ξ+

−

+= −1

2

1

1 2ln

y

L l h H he= = −( )4 3 4 3. . .*



continuation of the spherical meniscus, and the maximum deviation is around2.5he. However, now the deviation is bigger from the meniscus side than fromthe flat film side. Figure 2.20 also shows that the deviation of the real profilefrom the ideal profile is not symmetrical from both sides, from the position ofthe maximum deviation at y = 0, but decreases more rapidly from the flat liquidfilm side and is extended more into the depth from the meniscus side. Thisbehavior is different from the case n = 3. The thickness of the L regions is largerat n = 2 than at n = 3 at equal H values; the maximum L values are about 362 µmfor thick film in wide slots (H = 1 cm, he ~ 0.525 µm) and 0.36 µm for thin films(H = 1 µm, he = 53 Å).

Thus, the transition zone is very much extended for low capillary meniscuspressures and thick liquid films. The radius of the meniscus curvature must bestudied outside the transition zone, i.e., at a distance bigger than L, from theapparent three-phase contact line.

2.5 THICKNESS OF WETTING FILMS ON ROUGH SOLID SUBSTRATES

Let us now consider the case of thin equilibrium liquid films on rough solidsubstrates when there is complete wetting. In such a case, it is possible to measureonly the mean thickness of the film,

–h. It appears that the mean thickness,

–h, of

wetted films on rough solid surfaces is bigger than the corresponding thicknessof a flat film, he, on a smooth substrate. It also appears that the mean thickness,

–h,

approaches the value, he, at high and low film thicknesses when the latter are

FIGURE 2.20 Real profile (1) inside the transition zone calculated according to Equation2.72 and the ideal profile (2,3) according to Equation 2.71 at n = 2.

–3 –1–2 0 1 2

y

3

1

2

ξ

2

4

5

6

7

1

3



smaller or bigger in relation to the characteristic scale of roughness of the solidsubstrate, α. For

–h >> α, the effect of the roughness is made negligible, and the

surface of the film at the interface with the gas becomes practically smooth.When

–h << α, the film copies the surface of the substrate, maintaining a constant

value of the film per unit area of surface. Hence, the maximum deviation of thethickness of liquid films on a rough substrate from the corresponding thicknesson a flat substrate should be expected when

–h ≈ α.

When a wetting film of uniform thickness covers a curved surface, its equi-librium with the vapor of the same liquid is determined by Equation 2.23 inSection 2.2, which can be rewritten as

, (2.73)

where K is the capillary pressure due to the local curvature of the surface of thefilm; γ is the surface tension; Π(h) is the disjoining pressure, which is a functionof the local thickness of the film h; R is the gas constant; T is the temperature;vm is the molar volume of the liquid; p is the equilibrium pressure of the vaporabove the film; and ps is the pressure of the saturated vapor of the liquid.

Let us make a further examination for nonpolar one-component liquids (com-plete wetting), where the disjoining pressure isotherm, Π(h), is determined onlyby the dispersion forces. The isotherm of the disjoining pressure of a flat film inthis case has the same form as in the previous Section 2.4.

(2.74)

where n = 3 for small and n = 4 for large thicknesses of the film. The Hamakerconstant A is characterized on the basis of the spectral characteristics of the filmand the substrate [9].

In the general case, the disjoining pressure in thin films on a curved substrate,, is different from the corresponding disjoining pressure in films on a flat

substrate, Π(h). Thus, for example, for films at the internal surface of a capillaryof radius, r, the disjoining pressure, , (in the approximation h << r) is givenby [10]

, (2.75)

where Π(h) is the disjoining pressure of a flat film of the same thickness on aflat substrate;

γK hRT

v

p

pP

m

se+ = =Π( ) ln

Π( ) ,h A hn= /

Πr h( )

Πr

Π Π

∆ ∆ ∆ ∆

∆ ∆r h

h

r

d

= +

⋅ +( )

⋅

∞

∞

∫

∫( ) 1

2

32

0

21 32 21

32

0

2

ξ

11dξ



,

is the dielectric permeability, which is a function of the angular frequencyξ, taken at the imaginary axis [11]. The subscripts 1, 2, and 3 relate to the gas,the film, and the solid substrate, respectively. At h/r → 0, Πr → Π. Quantitativeevaluations for films of decane on the surface of quartz show that the contributionof the second term on the right-hand side of Equation 2.75 is relatively small.Thus, with h/r ≤ 0.2, the difference between Πr and Π does not exceed 2.5%.

Thus, under the condition of very small curvature, h/r, the isotherm (Equation2.74) of the disjoining pressure of flat films can be used with a sufficient precision.

Real surfaces, as a rule, have a roughness. In this case, the local thickness ofthe film is a function of the coordinate, and a mean value of the thickness of thefilm,

–h, should be used. The problem is how significantly the mean thickness,

–h,

differs from the thickness of a flat film on an ideally smooth surface of the samenature, and how these differences affect the roughness (or topology) of the surface.

Let us consider a simplified model of one-dimensional roughness (Figure2.21), where the profile of the surface is a function of one coordinate x. Let Hs

(x) be the equation of the surface of the substrate, and H(x) be the equation ofthe surface of the film, forming a boundary with the gas. The local thickness ofthe film is determined as

. (2.76)

Using the latter notations, Equation 2.73 takes the following form

(2.77)

FIGURE 2.21 Calculations of the thickness of wetting films on a rough cylindrical sur-face. Hs(x) is the profile of the solid substrate, H(x) is the liquid profile, and h(x) = H(x) –Hs(x) is the film thickness.

∆iki k

i k

= −+

ε εε ε

ε ξ( )i

h x H x H xs( ) ( ) ( )= −

γ ′′

+ ′

+ =H

H

A

hP

n e

1 23 2

( ),

/

x0

H

H(x)

h(x) Hs(x)



where H′ and H″ are the first and the second derivatives of H(x), and

is determined by the vapor pressure in the ambient air.On a plane substrate, the liquid curvature is zero and Equation 2.77 gives

. (2.78)

Substitution of Equation 2.78 into Equation 2.77 results in

(2.79)

Let us introduce the average values of any function, ϕ, of a random variableas follows:

,

where λ is a characteristic scale of surface roughness in the direction x; theoverbar means averaged over random substrate.

Let us integrate both sides of Equation 2.79 from 0 to X. We assume that thesurface is statistically homogeneous, that is, there is no preferable positive ornegative curvature of the liquid film. We can subdivide the whole interval ofintegration form 0 to X into a big number, N, of subintervals of a small length,λ = X/N. After that,

where θ is the local slope of the liquid profile. Hence, in the case of random andstatistically homogeneous roughness, the average value of the left-hand side ofEquation 2.79 vanishes, and the average of the right-hand side results in

PRTv

ppe

m

s=

ln

A

hP

en e=

ddx

H

H

A

h

A

hnen

γ ′′

+ ′

= − −

1 2

1 2( )

./

ϕ ϕ ϕ λ= ≈ >>→∞ ∫ ∫lim ( ) ( ) ,

X

X X

Xx dx

Xx dx X

1 1

0 0

1 1 1

0 0

1

1

XKdx

XKdx

Xx

X

x

x

i

N

i

i

i

= = −∫ ∫∑+

=+sin ( ) sinθ θθ

λθ

( )

sin ( )

x

Nx

i

i

N

i

( )

= −

=

+

∑0

11 1 11 1

00N

xi

i

N

i

N

sin ( ) sin s_____

θλ

θ==∑∑

= − iin ,

, , , ,

_____

θ

=

= ⋅ =

0

0 1 2x i ii ∆ …… =N x XN, ,



(2.80)

Equation 2.80 can be rewritten in the following form:

,

that is, the measured average disjoining pressure on a rough substrate coincideswith the disjoining pressure on a corresponding flat substrate. Hence the latterexpression can be rewritten as

(2.81)

Let us recall a well-known theorem from the probability theory. Let usconsider a concave function, ϕ, of random variable, h. Then

that is, the average of the concave function is bigger than the function of theaverage. The disjoining pressure isotherm of the type under consideration is aconcave function of h because the second derivative is positive:

Application of the aforementioned theorem results in

Comparison of this inequality and Equation 2.81 results in

10

0X

A

h

A

hdx

nen

X

−

=∫ .

1

0X

A

hdx P

n

X

e∫ =

A

hP

n e= .

ϕ ϕ( ) ( ),______

h h>

A

h

n n A

hn n

′′

= − >−( )

.1

02

A

h

A

hn n> .

A

h

A

hen n

> ,



and hence,

(2.82)

The preceding inequality shows that, on a rough solid substrate, the measuredaverage thickness of the liquid film is bigger than the equilibrium thickness ofthe film on a corresponding flat substrate.

In the following text, we investigate how the thickness of the film on a roughsubstrate influences the disjoining pressure measurements. For that purpose, weconsider a model rough surface of the following kind:

Hs(x) = α cos kx (2.83)

that is, a periodic roughness with an amplitude α. This allows us to restrict ourconsideration to x in the following range:

.

In the case of the model roughness (Equation 2.83), we consider only halfof the period. Hence, the following boundary conditions are satisfied:

, (2.84)

and

, (2.85)

where X = π/k is the half period.To simplify the calculation shown in the following text, we assume that δ =

α · k << 1. Then, the model roughness has a low slope, and the solution ofEquation 2.77 can be expanded in a series in terms of the small parameter

(2.86)

where Hi(x) ~ δi, i = 1, 2, …. We limit ourselves to the first two terms of theexpansion, as the nonlinearity of the curvature is of the third order of smallness.Using these notations, we can write

h he> .

− < <π πk

xk

′ =H ( )0 0

′ =H X( ) 0

H h H He= + + + …1 2 , ,

A

h

A

h

nA

hH H H

n n A

hH

nen

en s

en

= − + + …−( ) + ++ +1 1 2 2

1

2

( )11 2

2+ + …−( ) + …H Hs .



Substitution of the preceding expression into integral (Equation 2.80) andcollecting only the terms of the first and second order pertaining to the smallparameter δ, we conclude that, in the first order,

, (2.87)

and collecting the two terms of the second order,

.

The latter equation can be rewritten as

. (2.88)

Let us now determine the average thickness as

According to Equation 2.87, the first term on the right-hand side of thepreceding equation is equal to zero, and the second term is given by Equation 2.88.Hence, the latter equation can now be rewritten as

. (2.89)

The integral on the right-hand side of the preceding equation is alwayspositive. Hence, we come to the same conclusion as in the general case (see theinequality Equation 2.82 previously mentioned): the average film thickness on arough substrate is always bigger than the corresponding film thickness on a flatsubstrate.

Equation 2.89 shows that we have to determine only the first function, H1(x),to calculate the second correction to the average thickness,

–h.

H H dxs

X

1

0

0−( ) =∫

− + + −( ) =+ +∫ ∫nA

hH dx

n n A

hH H dx

en

X

en s

X

1 2

0

2 1

2

0

10

( )

H dxn

hH H dx

X

es

X

2

0

1

2

0

1∫ ∫= + −( )( )

h hX

H H H dx hX

H H dxe s

o

X

e s

o

X

= + + + …−( ) ≈ + −( ) +∫ ∫1 11 2 1

112X

H dxo

X

∫ .

h hn

XhH H dxe

es

X

= + + −( )∫( )11

2

0



Substituting the expansion (2.86) into Equation 2.77 and collecting the termsof an identical order, we obtain

, (2.90)

which shows that

is a constant that is equal to the thickness of the equilibrium film on a flat substrate,he. The next equation is

(2.91)

with the periodic boundary conditions

(2.92)

Introducing the notation

instead of Equation 2.91 we arrive at

(2.93)

Solution of Equation 2.93 with the boundary conditions given by Equation2.92 is

. (2.94)

For a crest of the sinusoid, from Equation 2.94 we get

,

A

HP

n e0

=

HA

Pe

n

0

1

=

/

γ ′′− ⋅ −( ) =+HnA

hH H

en s1 1 1 0

′( ) = ′

=H Hk1 10 0π

.

anA

hen

21

= +γ,

′′− + ⋅ ⋅ =H a H a kx12

12 0α cos .

Ha kx

a k1

2

2 2= ⋅ ⋅

+α cos

hk

hk

a ke

π α

= −+

2

2 2



and for a trough,

,

i.e., the film is thicker in a depression than on a convexity.Using Equation 2.94, we conclude:

.

Substitution of this expression into Equation 2.89 results in the following secondapproximation of the mean thickness of the film:

. (2.95)

The preceding equation shows that tends to zero, at both

and

The difference in the thickness of the films on a rough and smoothsubstrate goes via a maximum deviation at some value of Pe max, which can bedetermined by direct differentiation of the second term on the right-hand side ofEquation 2.95. This results in

. (2.96)

At–h >> α, the effect of the roughness is damped, and the surface of the film

at the interface with the gas becomes practically smooth. At–h << α, the film

copies the surface of the substrate. Hence, the maximum deviation of the thickness

h hk

a ke0

2

2 2( ) = ++

α

H Hk kx

a ks1

2

2 2− = − ⋅ ⋅

+α cos

h hk P n

AnP

Ak

ee

n

n en

n

= ++( )

+

+

α γ

γ

2 4 1

11 1

12

1

2

/

//

/

2

∆ = −h he

Pppe

s

→ ∞ →

or 0

Pppe

s

→ →

0 1or .

∆ = −h he

Pk

n nAe

n n

nmax

/

/

( )=

+

⋅

+( )+( )γ 2

1

1 1

2 1



of liquid films on a rough substrate from the corresponding thickness on a flatsubstrate should be expected at

–h ≈ α .

The approximate form of the function–h (Pe) is shown in Figure 2.22 (curve 2).

The straight line (1) illustrates the disjoining pressure isotherm of flat filmsaccording to Equation 2.74. The maximum deviations of the isotherm (curve 2)from the isotherm (line 1) correspond to the values of Pe max.

Because polished surfaces, as a rule, have grooves left by the solid grains ofthe polishing pastes, these surfaces are expected to have a qualitative picturepresented in Figure 2.22. This was experimentally observed in Reference 12 forwetting films of tetradecane on the polished surfaces of steel, where the qualitativepicture presented in Figure 2.22 was experimentally observed in the case ofdisjoining pressure isotherm for complete wetting, n = 3 (Figure 2.23).

2.6 WETTING FILMS ON LOCALLY HETEROGENEOUS SURFACES: HYDROPHILIC SURFACE WITH HYDROPHOBIC INCLUSIONS

Liquid films on heterogeneous solids, that is, solids where solid hydrophilicsurfaces include hydrophobic spots, are considered in this section [31,32].Changes in the profile of a wetting film over the spots and its eventual breakdownare expected. The combined action of the disjoining pressure and capillary forcesshould allow the prediction of the critical width of the hydrophobic spot on suchheterogeneous substrates before wetting film breakdown. Needless to say, thiscritical size is supposed to depend on the parameters of the disjoining pressureisotherms of the hydrophilic and hydrophobic parts and the relative vapor pressurein the surrounding medium. Let us take two different thin liquid film disjoiningpressure isotherms for the hydrophilic and hydrophobic parts of the substrate.

FIGURE 2.22 Disjoining pressure isotherm Pe(he) = A/hn. Sketch of deviations of themeasured average liquid film thickness,

–h, from the predicted film thickness on a flat

surface, he. (1) thickness of the film on a flat substrate; (2) average film thickness on arough substrate.

21

lg Pe max lg Pe

lg hmax

lg h



Complete wetting is assumed for the hydrophilic part and partial wetting for thehydrophobic or less hydrophilic spot. The equilibrium profiles should be calcu-lated according to the spot size and the value θe of the equilibrium contact angleof the hydrophobic spot. It is shown in the following text that the critical widthof a hydrophobic spot decreases by an order of magnitude with an increase inthe contact angle of the more hydrophobic spot from 10˚ to 90˚.

Let us consider a flat hydrophilic surface covered by a sufficiently thickwetting film in the presence of a hydrophobic strip of width 2L (Figure 2.24a).The origin of the x-axis corresponds to the middle of the strip.

As a reference system, a state was chosen in which each part of the surfaceis covered by a corresponding equilibrium film and an interaction between thefilms is absent. The equilibrium thickness of each of the films, He and he, isdetermined by the corresponding disjoining pressure isotherms, Π(H) and π(h),respectively:

, (2.97)

where is the excess pressure of the film as compared with a bulk liquid at thesame temperature T, R is the gas constant, is the molar volume of the liquid,and p and ps are the equilibrium and saturated vapor pressure, respectively.

Between two idealized states of the films as shown in Figure 2.24a, a transitionzone is formed, the possible shape of which is shown in Figure 2.24b, andFigure 2.24c. Consideration of the variations in the free energy of the system(similar to Section 2.1) results in the following set of differential equations

FIGURE 2.23 Experimental data on average film thickness of tetradecane on steel,obtained by continuous thinning of the film (open symbols) and subsequent thickening(closed symbols). Solid line according to the disjoining pressure isotherm Π = A/h3 [12].

lg h

lg Π

2.6

2.5

2.4

2.3

2.2

2.7 2.9 3.1 3.3 3.5

Π H h PRT

v

p

pe e em

s( ) = ( ) = = ⋅π ln

Pe

vm



enabling calculation of the profile of the transition zone for hydrophobic andhydrophilic regions of the surface, respectively:

(2.98)

where γ is the surface tension of the liquid; h′, H ′ and h″, H″ are the first andthe second derivatives of the film thickness over x within the zones 0 < x < Land x > L, respectively.

FIGURE 2.24 Schematic representation of a wetting film on a solid surface containing ahydrophobic spot at –L < x < L. (a) Reference system: stepwise film profile, equilibriumfilm thickness He and he, on the corresponding hydrophilic and hydrophobic parts, if eachpart is unbounded. (b) Depression formation over a hydrophobic spot when the width Lof a hydrophobic part is smaller than critical value Lc. (c) Rupture of a wetting film if thewidth of the hydrophobic part L > Lc, and formation of a thin film on the hydrophobic part.

x

x

x

H

He

He

He

he

x0he

h

H

H

he

h

h

h(x)H(x)

a

b

c

L

L

0

L0

0

γ π′′ + ( ) = < <h h P x Le , ,at 0

γ ′′ + ( ) = >H H P x LeΠ , ,at



The conditions according to Equation 2.98 are applicable in the case of lowslope profiles, when (h′) << 1 and (H ′) << 1. The first terms on the left-hand sideof Equation 2.98 determine the local capillary pressure, and the second termsdetermine the local disjoining pressure. The pressure sets the chemical poten-tial of the system and the relative vapor pressure according to Equation 2.97. Inthe case of flat films, the capillary terms are equal to zero.

Let us now formulate the boundary conditions that are used for solvingEquation 2.98. The first condition characterizes the equilibrium state of the thickhydrophilic film, far from the hydrophobic spot:

at x → ∞. (2.99)

The second condition reflects the symmetry of the system

(2.100)

To match the profiles at x = L, the following two conditions of continuity ofthe liquid profile should be used:

(2.101)

and

(2.102)

As already discussed in Section 2.2, not all solutions of Equation 2.98 withboundary conditions (2.99 through Equation 2.102) describe the stable equilib-rium profiles, but only those that satisfy Jacoby’s condition (3) (Section 2.2).This condition in application to the previously discussed problem has the follow-ing form:

at 0 < x < L;

at x > L, (2.103)

with the following boundary condition:

,

and continuity at x = L, where U(x) and V(x) are the Jacoby functions. Therequirement of the stability of the solution is as follows: the solution of Equation

Pe

H x He( ) →

′( ) =h 0 0.

h L H L( ) = ( );

′( ) = ′( )h L H L .

γ π′( )′ + ′( ) =U h U 0,

γ ′( )′ + ′( ) =V H VΠ 0,

U 0 0( ) =



2.103 does not vanish anywhere except at the point x = 0 and x = ∞. The solutionis given as follows:

After differentiating the conditions given by Equation 2.98 over x, we obtain:

at

at . (2.104)

Comparing the given equations with Equation 2.103, we conclude that U(x) =const·h′(x) and V(x) = const·H′(x). That means, the profiles h(x) and H(x) must behavein a monotonous way inside the corresponding zones, as shown in Figure 2.24band Figure 2.24c. Nonmonotonous behavior results in the loss of stability.

Further calculations are made using simplified expressions for isotherms ofdisjoining pressure (Figure 2.25) consisting of linear parts:

(2.105)

(2.106)

FIGURE 2.25 Simplified forms of disjoining pressure isotherms of the films formed onthe hydrophilic surface, Π(h) (curve I) and on its hydrophobic part, π(h) (curve 2).

γ π′′( )′ + ′( ) ′ =h h h 0, 0 < <x L;

γ ′′( )′ + ′( ) ′ =H H HΠ 0, x L>

π h

h h

a h t h h t

h t

e

e s

s

( ) =

∞ <

− < <

>

, ,

( ), ,

,0

Π hA t H h t

h t

s

s

( ) =− < <

>

( ), ,

, .

0

0

1

ts Hh

2

Hehe

Pe

P2

–Pmin

–Π

+Π



The thickness, ts, characterizes the range of action of surface forces and isselected identically for both the disjoining pressure isotherms. The equilibriumthickness, he, is independent of the vapor pressure in the surrounding air, p,according to adopted isotherm, Equation 2.105. The film thickness, He, accordingto Equation 2.106, depends on the vapor pressure in the surrounding medium.The parameters

and

determine the slope of the isotherms. For the films on hydrophilic surface (curve1, Figure 2.25), the isotherm ranges from

(when and He = ts) to Π = P2,

when He = 0. At some definite value of the pressure, Pe (between 0 and P2), thefilm thickness equals He (Figure 2.25). Such a form of the isotherm correspondsto the case only when the repulsion forces (Π > 0) act in the film and completewetting takes place.

Equilibrium films on a hydrophobic surface (curve 2 in Figure 2.25) havesmaller thickness he, which is adopted to be independent of relative vapor pressurein the range of Pe higher than –Pmin. At h > he, attractive forces act in the films(Π < 0), which makes the films unstable in this region of thickness. Stable filmsin the system are considered only at undersaturation, that is, at

and Pe > 0.The less hydrophilic (hydrophobic) spot may be characterized by the value

of the contact angle, θe, which is a droplet of the liquid form on the hydrophobicsubstrate. The contact angle, θe, is calculated on the basis of the equation deducedin Section 2.1 (Equation 2.9) using the disjoining pressure isotherm, π(h), of thefilms on the hydrophobic substrate:

. (2.107)

Substituting the model disjoining pressure isotherm, Equation 2.105, intoEquation 2.107, we obtain:

AP

ts

= 2 aP

t hs e

=−min

Π = 0p

ps

= 1

p

ps

< 1

cos θγ γ

πγ

πe e e

h

t

h

P h h dh h dh

e

s

e

= + + ⋅ ( ) ≈ + ⋅ ( )∫11 1

11

tts

∫



. (2.108)

In the framework of the adopted model, we can characterize the state of thehydrophobic surface by the contact angle, θe, that is calculated using Equation2.108. This equation includes parameters a, ts, and he of the isotherm, as well asthe surface tension γ of the liquid.

We would like to analyze two possible situations that are schematically shownin Figure 2.24b and Figure 2.24c. In the first case, the transition zones betweenhydrophilic and hydrophobic parts overlap, and the film thickness in the middle(at x = 0) is higher than the equilibrium film thickness of the hydrophobic spot,he. In the second case, in the middle of a wider hydrophobic spot, the filmthickness is equal to the equilibrium value, he, and deviation from this thicknessstarts only at x > x0.

According to the transversality requirement discussed in Section 2.2 (condi-tion 4), the condition h′(x0) = 0 holds at x = x0.

Let us consider the first case, Figure 2.24b. Equation 2.98, which determinesthe film profile, takes the following form using the disjoining pressure isothermsgiven by Equation 2.105 and Equation 2.106:

(2.109)

(2.110)

The solution of Equation 2.110 that satisfies the boundary condition (2.99) is:

, (2.111)

where is an integration constant. Solution of Equation 2.109 is different forthe first and the second cases (Figure 2.24b and Figure 2.24c, respectively). Inthe first case (Figure 2.24b), the solution that satisfies the symmetry condition(2.100) has the form

. (2.112)

The integration constants, C1 and C2, should be determined using boundaryconditions (Equation 2.101 and Equation 2.102) at x = L, at the border between

cos θγe

s ea t h= −

−( )1

2

2

γ ′′ + −( ) =h a h t Ps e ,

γ ′′ + −( ) =H A t H Ps e.

H H C x LA

e= + − −( )

2

1 2

exp/

γ

C2

h tP

aC x

ae= + + ⋅ ⋅

⋅

1 cosγ



the hydrophilic and hydrophobic zones. It follows from Jacoby’s conditions thatthe profile within the hydrophobic zone (between x = –L and x = L) is stablewhen the following restriction is satisfied:

In this case, the solutions of Equation 2.110 vanish only at the point x = 0. Thisrestriction gives an estimation of the critical width of the hydrophobic spot whenthe stability condition is violated. Therefore, this condition gives the critical sizeof the hydrophobic spot:

(2.113)

From the two boundary conditions (Equation 2.101 and Equation 2.102), andEquation 2.111 and Equation 2.112, two algebraic equations are obtained, whichare used for the determination of the unknown constants C1 and C2:

, (2.114)

(2.115)

whereTaking into account Equation 2.113, the constant β can range between 0 and

π …, and hence, sin β > 0. Therefore, parameters C1 and C2 have the identicalsign. Substitution of the expression for C2 from Equation 2.114 into Equation2.115 results in

(2.116)

The profile of the liquid within the hydrophobic zone must have monoto-nously increasing thickness at the point x between 0 and L (Figure 2.24b), whichmeans that the value of the parameter must be negative. However, the latteris possible only when

.

This leads to a stronger restriction of the critical width of the hydrophobic zoneas compared with Equation 2.113: L < (γ /a)1/2 arctan (A/a)1/2. Owing to

L a/γ π( ) <1 2/

L a< ⋅( )π γ /1 2/

.

CP

A

P

aCe e

2 1− = + ⋅cos β

C A a C2

1 2

1/( ) =/

sin ,β

β γ= ( )L a/1 2/

.

C P A a a Ae1

1 21 1= − ( ) + ( ) − ( )

⋅ / / /cos sin/

β β.

C1

cos sinβ β> a

A



arctan

a more accurate definition of the critical width of the hydrophobic zone whenthe wetting film ruptures is:

(2.117)

Further analysis shows that a stronger limitation of the critical L values exists,which follows from the condition that film thickness at cannot be lowerthan he, where he, is the equilibrium film thickness at the center of the hydrophobicspot at any given pressure, Pe, and corresponding value of the vapor pressure,p/ps, in the surrounding media. From Equation 2.112 for the film profile andEquation 2.116 for the parameter at x = 0, the expression for the thickness ofthe film in the center, h0 is:

. (2.118)

Hence, the preceding equation and the condition finally determine thecritical length of the hydrophobic spot, Lc:

(2.119)

where βc = Lc (γ/a)1/2.As distinct from the previous approximations (Equation 2.113 and Equation

2.117), the critical width of the hydrophobic zone, Lc, according to Equation 2.119depends on the parameters of both isotherms, a and A, as well as on the relativevapor pressure in the surrounding media that is characterized by the pressure Pe.

The profiles of a transition zone beyond the hydrophobic spot, at x > L, iscalculated using Equation 2.111 and Equation 2.115. At x = L, the two profiles arematched according to boundary conditions (Equation 2.101 and Equation 2.102).

Let us find the critical width of the hydrophobic zone using a simplifieddefinition given by Equation 2.117. In this case, the value of may be calculateddependent on the degree of surface hydrophobicity that is characterized by thecontact angle. Substituting the expression for from Equation 2.108into Equation 2.117, we conclude

.2 (2.120)

A a/( ) <1 2

2

/,

π

L a= ( )γ π/

1 2

2

/.

x = 0

C1

h tPa

P A a a Aee0

1 21 1= − − ( ) + ( ) − ( )/ / /cos sin

/β β

h he0 =

cos sin/β βc ce

ee ea A t

Pa

hPA

Pa

− ( ) = − −

+

/ 1 2

,

Lc

1−( )cos θe

L t h tc s e e s e= −( ) −( ) ≈ ⋅ −π θ θ2 2 1 1 1 11 2

cos . cos/ (( )1 2/



The results of calculations of the dependence of Lc/ts on the contact angle θe

are shown in Figure 2.26. These results show that the values of the critical width,Lc, decrease with an increasing contact angle and fall sharply at θe > 30˚. Supposethe range of action of surface forces, ts, is of the order of 10–6 cm; we may thenconclude that the critical width of a hydrophobic spot decreases from Lc ≈ 10–5 cmat θe = 10˚ to Lc ≈ 10–6 cm at θe = 180˚. Note that the Equation 2.119 shows that at

,

wetting film thickness, He, approaches its highest value ts, and the critical widthtends to decrease.

The prediction of the theory is in line with experimental investigations ofwetting film stability on heterogeneous methylated glass surfaces [13]. Filmrupturing is sensitive to the contact angle values at θe < 45˚. At larger values ofcontact angles, the effect is practically not dependent on the degree of hydropho-bicity of the hydrophobic spot.

In a similar way, the length (the extension of a part of the hydrophobicspot covered with an equilibrium film with the thickness he) may be calculated(Figure 2.24c):

(2.121)

FIGURE 2.26 Calculated according to Equation 2.24, dependence of the critical widthof a hydrophobic spot, Lc, at which wetting film ruptures, on the value of contact angleθ that characterizes the hydrophobic spot on the surface.

θπ2π/3π/30

5

10

L c/t s

p

ps

→ 1

x0

x P ae0

1 2= − ( ) ( )β ε γ /

/,



where the function is determined from the following equation:

(2.122)

Therefore, the solutions obtained open the possibility of studying the criticalsize of a hydrophobic spot, Lc, dependent on the parameters of disjoining pressureisotherms. When the width, L, of a hydrophobic spot is smaller than Lc, adepression cavity is formed over the spot (Figure 2.24b), and this is reflected onthe equilibrium liquid profile, h(x), which describes a profile of a relatively thickfilm even over the hydrophobic spot. However, at L ≥ Lc, the thick wetting filmruptures, and a part of the hydrophobic surface becomes almost “dry.” Furtherincrease in the dimension of the hydrophobic spot leads to the expansion of adry part of the surface (Figure 2.24c).

In view of the preceding features, we can say that the presence of morehydrophobic spots on smooth hydrophilic substrates results in the formation ofdepressions where the film thickness is lower than the thickness on the rest ofthe substrate. Accordingly, the presence of more hydrophobic spots results in alower mean thickness of the film relative to the thickness on a uniform hydrophilicsubstrate. Thus the surface roughness and the presence of hydrophobic spots on thesurface influence the mean thickness of the equilibrium film in opposite ways:the presence of roughness results in an increase in the mean film thickness (Section2.5), whereas the presence of hydrophobic inclusions leads to the contrary.

2.7 THICKNESS AND STABILITY OF LIQUID FILMS ON NONPLANAR SURFACES

In this section, conditions are determined about the stability of liquid films oncylindrical and spherical solid substrates. We shall see that stability is determinedby an effective disjoining pressure isotherm, Πeff (h), which differs from thecorresponding disjoining pressure isotherm of (flat) liquid films on flat (planar)solid substrates. The effective disjoining pressure on curved surfaces is consideredin more detail in Section 2.12. An analysis is given of the different types ofisotherms Πeff (h) relating the film thickness h to the total change in pressure inthe film relative to the bulk phase of the same liquid.

When a liquid film covers a planar solid surface, its equilibrium with thevapor is determined by Equation 2.23 (see Section 2.2), which takes into accountthe simultaneous action of both the capillary pressure and the disjoining pressurein these liquid layers.

In the following text, we consider thin liquid layers on curved surfaces, forexample, the inner and outer surfaces of cylinders. Let a be the inner or outerradius of the cylinder. If the film thickness, h, is much smaller than the cylinderradius, (h << a), we can use, as a first approximation, the isotherm of thedisjoining pressure, Π(h), of a planar layer. However, in the general case, the

ε Pe( )

cos sin/

ε ε− ( ) = ( ) + ( ) + ( )a A P A P a t P ae e e/ / / /1 2

−− he .



disjoining pressure in thin liquid films on a curved substrate, Πa, is different fromthe corresponding disjoining pressure on a flat surface, Π. In Section 2.5, wealready gave an estimation of the distortion of dispersion forces in this case. Ithas been shown that at h/a ≤ 0.2, the difference between Πa and Π does notexceed 2.5%. Capillary effects, as we see in the following text, have a consider-ably more pronounced influence on the thickness and stability of films on non-planar surfaces. Hence, in subsequent calculations, we use the disjoining pressureisotherm, Π(h), of flat liquid films, the same that is used in Equation 2.23(Section 2.2).

Let us first examine equilibrium films on the convex surface of a cylinderwith a radius, a. The conditions of the equilibrium have the form given byEquation 2.249 and Equation 2.251 (see Section 2.12 for the derivation), wherethe effective isotherm of disjoining pressure is given by Equation 2.245, whichis rewritten as

, (2.123)

where γ is the surface tension, Πeff (h) is the effective disjoining pressure isothermof the curved film, and the excess pressure Pe is given by Equation 2.2 inSection 2.1. According to Equation 2.123, the Πeff (h) dependency intersects theaxis of thickness at Π(h) = γ /a. Consequently, when p/ps = 1, the film on a curvedsurface has a finite thickness h0 that is determined by the specific form of thedisjoining pressure isotherm, Π(h). Note that according to Equation 2.123, theeffective disjoining pressure isotherm depends on the radius of the cylinder, a.

For planar films in Section 2.1, we deduced the condition of stability of flatfilms on flat solid substrate:

.

Applying the similar method to a curved film and using the effective disjoiningisotherm Πeff (h), we conclude (see Section 2.12) that the stability condition ofliquid films of uniform thickness on the outer cylindrical surface reads

. (2.124)

In comparison with the stability condition , the condition (2.124)depends on the vapor pressure in the surrounding air; in the case of undersatu-ration (Pe > 0), the latter condition is less restrictive than on a planar substrate,whereas at oversaturation, the condition is more restrictive, as it makes films ona convex surface less stable than on a flat surface.

Π Πeff ha

a hh

a( ) ( )=

+−

γ

d

dhh

Π Π= ′ <( ) 0

′ <Πeff eh( ) 0

′ <Π ( )h 0



It is interesting to note that in the case of complete wetting,

,

the equilibrium adsorption does not take place under oversaturation on a flatsurface. However, according to effective disjoining pressure given by Equation2.123, adsorption is possible on an outer cylindrical surface at oversaturations,in the range from

p/ps = 1 to

However, adsorption films are stable only in the region of thicknesses where thestability condition (2.124) is satisfied (see Section 2.12).

We now calculate the critical thickness h* for the isotherm corresponding tocomplete wetting A/h3, using the stability condition (2.124). We conclude that

. (2.125)

For films of decane on a cylindrical quartz surface, assuming that A = 1.6⋅10–13

erg [14] and γ = 23 dyn/cm, we find, using Equation 2.125, that for a = 10–3,10–4, and 10–5 cm, h* = 1190, 376, and 119 Å, respectively. However, the loss ofstability occurs only at Πeff < 0 (and hence when p/ps > 1), i.e., in the region ofoversaturation. When p/ps < 1, the films remain stable, but their thickness (incontrast to planar films) do not tend to infinity as p/ps → 1 but rather toward alimiting value

.

Let us now consider the more complex case of partial wetting, when theisotherm for planar films, Π(h), intersects the axis of thickness (see Chapter 1).The form of such an S-shaped isotherm is shown schematically in Figure 2.27(curve 1). Curves 2–4 are possible variants of the effective disjoining pressure,Πeff (h), in accordance with Equation 2.123. The values Πeff (h) > 0 correspond

Π( )hA

h=

3

p pv

aRTsm/ =

>exp .γ

1

ha A

*

/

≈

3 2

1 4

γ

hAa

0

1 3

=

γ

/



here to the region of undersaturation (p/ps ≤ 1), and values Π(h) < 0 correspondto the region of oversaturation (p/ps > 1).

As in Section 2.2, stable films with thicknesses h < tmin are referred to asα-films, and metastable films with thicknesses h > tmax as β-films (Figure 2.27).According to Figure 2.27 (curves 2,3), as the radius of the cylinder, a, decreases,then both the region of metastable films on curved substrates and the region ofstable α-films shrink.

A new phenomenon appears: a finite extent of the region of stable β-films(Figure 2.27, curve 2). For the Πeff (h) isotherm shown by curve 2 (Figure 2.27),the region of thicknesses, where β-films are stable, is bounded by thicknesses fromt1 to t2. In the case of even smaller radius of the cylinder, the region of existenceof stable β-films at p/ps < 1 disappears completely (curve 3 in Figure 2.27).

Figure 2.27 shows that a decrease in radius of cylinders, a, leads to a decreasein adsorption (at undersaturation, p/ps ≤ 1). The latter phenomenon has nothingto do with the nature of the cylinders and depends only on the geometric features,i.e., the curvature of the surface.

Let us now analyze the effective disjoining pressure isotherms, Πeff (h), on aconcave surface, for example, on the inner surface of a cylindrical capillary, thatis, inside the cylindrical capillary of radius a. The equation for the equilibriumof liquid and vapor in this case has the following form (see Section 2.12):

. (2.126)

FIGURE 2.27 Effective disjoining pressure, Πeff (h), for films on convex surface. Upperpart: 1 — disjoining pressure isotherm of flat liquid films on flat solid substrate, Π(h);2–3 effective disjoining pressure isotherms, Πeff (h), at different radiis of the outer cylin-drical surface, a: a2 > a3.

Π(h)Πeff (h)

0h

3

2

1

tmax

tmint1

t2

a

a hh

ah Peff e−

+

= ( ) =Π Π( )γ



The corresponding stability condition becomes

(2.127)

Here, in contrast to convex surfaces, the film thickness, h, is evidently limitedby the value of a. However, long before h approaches the inner radius, a, of thecapillary, it becomes necessary to account for the influence of overlapping fieldsof surface forces of all sections of the capillary surface. For slit pores, thecorresponding evaluations have been made [15]. We limit ourselves in this section,as previously, to an analysis of the solution for rather large values of a, when thecondition h << a is fulfilled.

Figure 2.28 schematically shows the isotherms of a planar film Π(h) (curve 1)and the effective disjoining pressure isotherms, Πeff (h), (curves 2 and 3) corre-sponding to the case of an inner surface of a cylindrical capillary.

With decreasing capillary radius, a, the region of stable state of α-films isnarrowed, but their thickness is bigger than the corresponding thickness of filmson a planar surface. The appearance of a lower limit of film stability, t1, and anupper limit, t2, also contributes to the existence of a narrowed region of β-filmsstability. For the disjoining pressure isotherm shown as curve 2 (Figure 2.28), β-films can exist only in the interval of film thicknesses from t1 to t2, correspondingto a certain interval of p/ps in the region of undersaturation.

In narrower cylindrical pores (curve 3, Figure 2.28), only thin α-films arestable, and β-films disappear completely.

These conclusions are supported by published experimental data. The exist-ence of an upper limit of stability for β-films in cylindrical capillaries wasexperimentally discovered [16]. It has also been experimentally observed [17]that in glass cylindrical capillaries with radius a > 0.4 µm, thin α-films are formedwith thickness h ≈ 50–60 Å. However, in thinner capillaries with a = 0.2–0.3 µm(with a corresponding reduction of p/ps), thicker β-films appear with thickness

FIGURE 2.28 Effective disjoining pressure, Πeff (h), for films on the inner surface of acylinder of radius a (concave surfaces). 1 — disjoining pressure isotherm of flat liquidfilms on flat solid substrate, Π(h); 2–3 effective disjoining pressure isotherms, Πeff (h), atdifferent curvature of cylindrical capillaries, a: a2 > a3.

Π(h)Πeff (h)

h1

2

3

0 t1 t2

′ <Πeff eh( ) .0



h ≈ 300–400 Å. Both the experimental observations correspond to the isothermΠeff (h) shown by curve 2 in Figure 2.28.

To conclude this section, we examine the case of complete wetting, when theisotherm Π(h) of a planar film can be represented by A/h3.

According to Equation 2.127, the critical thickness h* can be calculated whenthe film of a uniform thickness loses its stability. Based on Equation 2.126 andEquation 2.127, we can conclude that h* = (3Aa2/γ)1/4. At h ≥ h*, the films on theinner capillary surface lose stability, and the liquid changes into a more stablestate, forming a capillary condensate. For a ~ 10–4 cm, A ~ 10–14 erg, and γ ~30 dyn/cm, we obtain a thickness h* of the order of 10–6 cm. Thus, the conditionh << a is fulfilled over the entire interval of film thicknesses that is physicallyrealizable in a capillary. Even with a ~ 10–6 cm, the values of h* are no greaterthan 10–7 cm.

As an example, we show in Figure 2.29 curves plotted on the basis of Equation2.126 for the thickness of a decane film, h, on the inner surface of quartzcapillaries, as a function of the relative vapor pressure, p/ps. For this purpose, inEquation 2.126 we use a disjoining pressure isotherm as given by A/h3, and Pe

is replaced by its values from Equation 2.2, which results in

. (2.128)

FIGURE 2.29 Adsorption isotherms, h (p/ps), for films of decane in quartz capillarieswith radius a = ∝ (1), 10–5 cm (2), and 10–6 cm (3).

a

a h

A

h a

RT

v

p

pm

s

−+

=3

γln

h, Å

40

20

0

1

3

p/ps

2

0.6 0.8 1



In the calculations shown, it is assumed (the same as previously) that A =8.5⋅10–15 erg, γ = 23 dyn/cm, vm = 195 cm3/mole, and T = 293˚K.

Curve 1 in Figure 2.29 is the isotherm h(p/ps) at a = ∞, i.e., for a planarsurface. Curves 2 and 3 were plotted for capillaries with radii a = 10–5 and 10–6 cm,respectively. Whereas on the planar surface, h → ∞ when p/ps → 1 (curve 1), wefind that, in the capillaries, the isotherm breaks off at p/ps = 0.98 (curve 2) andp/ps = 0.79 (curve 3). Breakoff of the isotherms corresponds to loss of filmstability, in accordance with Equation 2.127. Note that in the entire region ofphysically realizable film thicknesses, h ≤ h*, the condition h << a is fulfilled.That is, capillary condensation starts much earlier than when the capillary is filledwith the liquid.

Even though the thickness of adsorbed films with equal values of p/ps isbigger in a capillary with a smaller capillary radius, the region of their existenceis curtailed quite substantially. When h > h*, the films lose stability, and thecapillary is filled with condensate.

The properties of films on curved surfaces that have been examined in thissection, i.e., film stability and thickness, should be taken into account to inves-tigate processes of polymolecular adsorption in fine porous solids, the surface ofwhich have convex and concave portions. These properties should also be takeninto account to study adsorbate transfer processes occurring in these solids. Toconclude, analogous calculations can be carried out for any form of the disjoiningpressure isotherms, Π(h) (or isotherms of adsorption on a planar surface), includ-ing isotherms obtained experimentally. We have selected the isotherm A/h3, whichcorresponds to adsorbate or adsorbent interaction solely due to dispersion forces,only in order to illustrate the solutions obtained.

2.8 PRESSURE ON WETTING PERIMETER AND DEFORMATION OF SOFT SOLIDS

We shall continue the consideration of the simultaneous action of capillary anddisjoining stresses applied on thin liquid layers. On that basis, the distribution ofnormal pressure on a solid substrate in the vicinity of the apparent three-phasecontact line is going to be determined. In the general case, the substrate issubjected to both tensile and compressive stresses. The extent of the zone inwhich normal pressure acts corresponds to the extent of the transition zonebetween the bulk part of the liquid under the drop or meniscus and the equilibriumflat films in front [33]. We shall be concerned with the influence of the verticalcomponents of capillary and surface forces acting on a solid support in the vicinityof the apparent three-phase contact line. For a rigid inelastic substrate, the verticalcomponent of those forces can be ignored, as its action does not lead to shapechange. For an elastic solid substrate, the change in equilibrium conditions underthe influence of surface deformation is known.

Within the framework of the model of an absolutely rigid substrate, thevertical component of those forces can be ignored, as its action does not lead to



any change in the shape of the substrate. For an elastic solid substrate, the changein equilibrium conditions under the influence of surface deformation was inves-tigated [18,19]. It has been assumed that the force, F, acts in a certain zone withina thickness δ, corresponding to the thickness of the surface layer of the liquidwhere the pressure is anisotropic. The thickness of the zone, δ, within the frame-work of this theory, remained undetermined.

Through an examination of the transition zone between the bulk liquid andthe film covering the substrate surface (Figure 2.30), we can relate the force onthe substrate to the disjoining pressure, Π(h), and to the action of capillary forces.This approach is based on an analysis of the distribution of the disjoining pressurewithin the limits of the transition zone; the shape and extent of this zone havebeen investigated in Section 2.3 and Section 2.4.

Our subsequent analysis of the problem of pressure on a wetting perimeteris performed within the framework of the theory, which as previously mentioned,includes a simultaneous action of the capillary and disjoining pressure in theliquid layers presented in the previous sections of this chapter. It was shown inSection 2.3 that the equilibrium contact angle, θe, can be expressed via thedisjoining pressure isotherm, Π(h), by Equation 2.47.

The vertical component of the forces acting on the wetting line, F = γ sinθe,does not vanish, when θe > 0, i.e., in the case of partial wetting. This corresponds,in accordance with Section 2.1 through Section 2.3, to an S-shaped disjoiningpressure isotherm, Π(h), entering the region of negative values in a certain intervalof film thicknesses (see Figure 2.1).

FIGURE 2.30 Profile of transition zone h(x) between bulk liquid and flat wetting film(1), disjoining pressure isotherms Π(h) (2,3), and profile of normal forces acting onsubstrate (Equation 2.131) in the case when a model disjoining pressure isotherm (2) isadopted; xNY is the position where the vertical force is exerted.

x

x

xNY x0 0

0

4

Π −Π

θe

he

h3 h

h2

ts

t0

2 3

00

t1

h1

f(x)

x_

h

F

F+

F–

x+

1



In the following text, examine the transition zone between a flat meniscusand a film in front at θe > 0 (curve 1, Figure 2.30), that is, in the case of partialwetting. The bulk liquid here forms a wedge, changing over, as x → ∞, to a flatfilm with a thickness he. With this model, we can limit ourselves to a one-dimensional solution of the problem without any loss of generality of the analysis.

The coordinate origin x = 0 is taken as a point of the profile lying beyondthe limits of action of surface forces (Figure 2.30). The profile of the transitionzone, h(x), can be obtained by solving Equation 2.23 deduced in Section 2.2,which includes the combined action of capillary and disjoining pressure:

(2.129)

where h′ = dh/dx; h″ = dh 2/dx2; and Pe is the excess pressure of the drop ormeniscus. In the region of the flat equilibrium film, h″ = 0 and Π(he) = Pe. Inthe bulk part of the liquid, beyond the limits of action of surface forces, Π = 0and Pe = ±γ /ℜ, where ℜ is the radius of surface curvature of the drop or concavemeniscus. For a planar wedge (Figure 2.30), ℜ = ∞ and Pe = 0.

The resultant forces in the vertical direction caused by the pressure on thesolid substrate,

(2.130)

is calculated in the following text based on Equation 2.129. Replacing the dis-joining pressure, Π(h), in Equation 2.130 by its expression from Equation 2.129,and keeping in mind that Pe = 0, we obtain

(2.131)

In the derivation of the latter expression, we have used the boundary condi-tions h′(∞) = 0 and h′ (0) = tanθe.

Thus, integration of the local disjoining pressure leads to the same value ofthe total vertical force caused by the pressure on the substrate as Young’s equationdoes. However, in contrast to Young’s equation, the mentioned forces are notexerted on a particular point but distributed over the whole region where thedisjoining pressure acts, that is, over the transition zone. Note that the actual

γ ′′

+ ′( )

+ ( ) =h

h

h Pe

12

3 2/,Π

F h dx= ( )∞

∫Π · ,0

Fddx

h

h

dx= − ′

+ ′( )

=γγ

12

1 2/

′′( )+ ′( )

=⋅

+= ⋅

h

h

e

e

eee

0

1 0 12

γ θ

θγ θ

tan

tansin

00

∞

∫ .



equilibrium contact angle, θe, is different from that calculated according toYoung’s equation, θNY (see the discussion in Section 2.1).

Now let us consider how the pressure on the substrate is distributed insidethe transition zone, i.e., in the vicinity of the apparent three-phase contact line.Figure 2.30 presents the transition region that is subject to the action of pressureon the solid substrate. This region is determined by the difference between thecoordinates of the points at which Π(h) = 0 (large h) and Π(he) = 0. The lengthscale of the transition zone has been estimated in Section 2.3 and calculated inthe case of complete wetting in Section 2.4.

We now perform quantitative evaluations of the extent of this region in thecase of partial wetting, using a model disjoining pressure isotherm made up oflinear sections (curve 2, Figure 2.30). Such a simplification, while retaining thebasic properties of the real isotherm (curve 3, Figure 2.30), enables us to obtainan analytical solution of the problem of the pressure distribution inside thetransition zone. The equation of the model isotherm used has the following form:

(2.132)

where ts corresponds to the finite radius of the action of the surface forces; t0 =he is the thickness of the equilibrium film with Π(t0) = 0; and the parameters aand b characterize the slopes of the linear sections of the disjoining pressureisotherm in the region of thicknesses from 0 to t1 and from t1 to ts, respectively.The coordinate origin x = 0 is selected at h = ts.

The profile of the transition zone corresponding to the model disjoiningpressure isotherm (Equation 2.132) is depicted by curve 1 in Figure 2.30. Wesubdivide the whole profile of the liquid into three sections: h1(x) for the regionof thicknesses from he = t0 to h = t1, where the liquid surface is concave; h2(x)for the region of thicknesses from t1 to ts, where the surface is convex; and h3(x)for the region of the wedge, where h > ts and the surface is flat.

For contact angles that are not too large, we can assume that (h′)2 << 1. Then,for each of these three zones, Equation 2.129 can be rewritten in the followingform:

(2.133)

Π h

a t h h t

b t h t h ts( ) =

−( ) < <

−( ) ≤ ≤

0 1

2 1

0

0

,

,

,,

,

h ts≥

γ

γ

γ

⋅ ′′+ −( ) =

⋅ ′′+ −( ) =

′′ =

h a t h

h b t h

h

s

1 0 1

2 2

3

0

0

0

,

,

..



The profile of the liquid in these three regions should be linked at the points(0, t2) and (x0, t1) by the conditions of equality of thicknesses and of the derivatives,h′: h1(x0) = h2(x0) = t1 and h1′ (x0) = h2′ (x0); h2(0) = h3(0) = ts, and h2′ (0) = h3′ (0).The following boundary condition should be used for a smooth transition fromthe bulk liquid to the flat equilibrium film: h1(x) → t0 as x → ∞ (Section 2.3,Appendix 1). The solution of the Equation 2.133, with the preceding boundaryconditions, has the following form:

(2.134)

where

Using the solutions obtained from Equation 2.134, we can find the distributionof force on the substrate: f(x) = Π(x) = Π [h(x)]. The profile f(x) correspondingto the model isotherm in Equation 2.132 is shown in Figure 2.30 as curve 4. Atthe point x = x0, the force f changes its sign abruptly in the same way as thedisjoining pressure Π (curve 2). In the case of real isotherms (curve 3), the transitionfrom compressive to tensile stresses is not that sharp; however, the qualitativeform of the profile of the local force f(x) is retained.

It is important to emphasize that the solid substrate is not subjected to atensile force only (i.e., force directed upward) as had been assumed previously.Young’s equation gives only the net value of the force F; but, as can be seenfrom Figure 2.30, both tensile forces (F+ > 0) and compressive forces (F < 0)contribute to the resultant force.

The point of application of the resultant force F can be found from theequation

,

where the value of F is determined using Equation 2.130. After substituting thevalues of Π[h(x)] from Equation 2.132 and Equation 2.134, and further integra-tion, we obtain

h x t Ca

C CC C

xa

a b

1 0 11 2

1 2

1 2

( ) = + +−

−γ

γ

/ /

exp

( ) = − −( )

,

sinh/

h x tb

C C xb

s2 12

22

1 2γ

γ

( ) = + −( )

,

,/

h x t x C Cs3 12

22

1 2

Ca

t t Cb

t ts1 1 0 2 1= −( ) = −( )γ γ, .

x F x x dxNY = ( ) ( )∞

∫10

/ · ·Π



According to the preceding expression, the value of xNY is close but does notcoincide with the predicted position based on Young’s equation, that is, with theposition of the line of contact between the wedge and film in the absence of atransition zone (see Figure 2.30). Also note that the position of the applicationof the force, xNY, (Figure 2.30) is located at the intersection of the continuationof the bulk liquid profile with continuation of the equilibrium film of thicknesshe and not with the solid substrate. The positions of the maximum and minimumforces, f+(x) and f(x), are shifted from the position xNY into the depth of the bulkliquid.

We now find the values of the resultants separately for the tensile (F+) andthe compressive (F–) forces

(2.135)

Summation of the latter expressions for F+ and F– gives the total resultantforce value, F = γ.sinθe. This follows from Equation 2.47 after substituting theequation of the disjoining pressure isotherm (Equation 2.132), which gives

(2.136)

The latter expression as given by Equation 2.133 is used for relatively smallcontact angles. That is, the following relation is used: θe ~ tanθe.

For the points of application of the resultant forces, F+ and F–, we obtain

(2.137)

respectively.

x t tNY s e= −( )0 tan .θ

F f x dx C

F f x dx C

+ +

∞

− −

∞

= ( ) ⋅ = − ⋅

= ( ) ⋅ = ⋅ −

∫

∫

0

1

0

1 1 1

γ

γ

,

−− ( )

C C2 1

2/

cos / · .θe C C= − ( ) −( )1 1 2 12

22

xb

C C C C

C C+ = ⋅

+( ) −( ) −{ }− − ( )

γ ln 1 2 1 2

2 1

2

1

2 1 1 /

<

= ⋅ + +−

−

x

xa

ab

C CC C

0

1 2

1 2

112

,

lnγ

> x0 ,



The approach that we use in the preceding expression can also be utilized inthe case of a concave meniscus in a capillary, or in the case of drops in partialwetting, that is, forming a finite contact angle, θe, with the equilibrium films ona solid substrate. In Figure 2.31, we qualitatively show the form of distributionof the excess forces f(x), normal to the substrate for a meniscus in a flat capillarywith a width of 2H (Figure 2.31a, curve 1, Pe > 0) and for a drop (Figure 2.31b,curve 2, Pe < 0). In the two latter cases, we took into account the fact that, inthe bulk part of the meniscus or drop, the substrate is subjected to an excesspressure Pe = const (excess in comparison with the pressure in the surroundinggas phase). An equal hydrodynamic pressure acts at the equilibrium state in thetransition zone and in the flat film as well. Therefore, the values of f(x) must bedetermined as the difference

f(x) = Π(h) –Pe. (2.138)

In the bulk part of the meniscus or drop, Π(h) = 0, and hence, f(x) = Pe. Ina flat film, Π(he) = Pe, and hence, f(x) = 0.

From Equation 2.129 and the definition given by Equation 2.138, we notethat the value of the local force f(x) = Π(h) – Pe is determined by the localcurvature of the liquid surface: f(x) = γ · K(x), where K(x) is the local curvature

FIGURE 2.31 Profiles of transition zone, isotherms Π(h), and profiles of normal forcesin the case of partial wetting for (a) a concave meniscus (curve 1) and (b) a drop (curve2). (c) For a meniscus, in the case of complete wetting (curve 3). Deformation of thesubstrate is shown to be proportional to the local force applied.

H

P

θePe1

h

he

Pe 0 –Π+Π

he

h

Pe0 –Π+Π

�

2 θe Pe

PeH

P

3he

he

he

Pe Pe0 0+Π

45he

h h

a

b

c



of the liquid surface. The local curvature, K(x), depends on the shape of the liquidin the transition zone, h(x), which in turn is determined by the combined actionof capillary forces and disjoining pressure.

In conclusion, we examine the case of complete wetting where the meniscusdoes not intersect the plane of the substrate and does not form a contact anglewith the substrate. As was shown previously in Section 2.4, as in the case ofpartial wetting, a transition zone is formed between the spherical part of themeniscus and the flat equilibrium films in front. The distribution of excess forcef(x) is shown for this case by the curve 3 in Figure 2.31c. This plot shows thatfor Π(h) isotherms that decrease monotonically and lie entirely in the positiveregion, Π > 0 (curve 4), the pressure on the substrate f(x) also decreases mono-tonically from the value Pe under the bulk meniscus to 0 as h → he. The extentof the zone of action f(x) coincides with the extent of the transition zone, L,which has already been evaluated in Section 2.3 as L ~ , where H is thehalf-width of the flat capillary, and he is the equilibrium thickness of the film infront. Thus, even in the case of complete wetting, when Young’s equation givesF = 0, a certain pressure f(x) acts on the substrate beyond the limits of the bulkpart of the spherical meniscus. This force may also be of a sign-alternatingcharacter if the isotherm (still corresponding to the complete wetting) intersectsthe vertical line P = Pe (curve 5 in Figure 2.31c). We would like to emphasizethat the consideration based on the combined action of the disjoining pressureand capillary pressure inside the transition zone is not only more precise thanYoung’s equation but also gives a far broader picture of the events in the vicinityof the apparent three-phase contact line.

The same approach can be used for the calculation of normal forces in the caseof nonflat liquid layers that were discussed in Section 2.2. All those nonflat layersare located inside the range of action of the surface forces (that is, h(x) < ts);there is no bulk part of the nonflat layer. The pressure on the substrate is deter-mined by the values of f(x) = γK(x), where the curvature K varies with the radialcoordinate x in all parts of the nonflat layer.

The next step should be the determination of the profile of a substratedeformed under the influence of the distributed force f(x). In Figure 2.31, weadopted for simplicity that the deformation of the solid substrate is proportionalto the local force. This assumption is definitely an oversimplification and shouldbe replaced by a more realistic hypothesis.

Note that we have taken into account only the normal component of forceacting on a solid substrate. Adding the effect of the tangential component of theforce inside the transition zone represents a challenging problem.

2.9 DEFORMATION OF FLUID PARTICLES IN THE CONTACT ZONE

The hydrostatic pressure in thin liquid films intervening between two drops orbubbles differs from the pressure inside the drops or bubbles. This difference is

heH



caused by the action of both capillary and surface forces. The manifestation ofthe action of surface forces is the disjoining pressure, which has a special S-shapedform in the case of partial wetting (aqueous thin films and thin films of aqueouselectrolyte and surfactant solutions). Disjoining pressure solely acts in thin flatliquid films and determines their thickness. If the film surface is curved, thenboth the disjoining and the capillary pressures act simultaneously. A theory isdeveloped in the following text enabling one to calculate the shape of the liquidinterlayer between emulsion droplets or between gas bubbles of different radiiunder equilibrium conditions, taking into account both the local disjoining pres-sure of the interlayer and the local curvature of its surfaces [33–36].

The model of solid nondeforming particles is frequently used when carryingout an analysis of the forces acting between colloidal particles. However, realdroplets or bubbles and even soft solid particles within the contact zone candeform, which can change the conditions of their equilibrium. In this case,interaction is not limited only to the zone of a flat contact but is extended ontothe surrounding parts within the range of action of surface forces. For elasticsolid particles, such a problem was discussed [20–21]. Here, the case of dropletsor bubbles is considered (e.g., emulsions, gas bubbles in a liquid), where shapechanges very easily under the influence of surface forces (Figure 2.32).

We now take into account both the finite thickness of a liquid interlayerbetween droplets or bubbles and the variation in thickness in the transition zonebetween the interlayer and the equilibrium bulk liquid phase. It was alreadymentioned that there is a problem in using the approach of thickness-dependentinterfacial tension: if we try to use the Navier–Stokes equation for description offlow or equilibrium in thin liquid films, a thickness-dependent surface tensionresults in an unbalanced tangential stress on the surface of thin films. This is thereason why this particular approach is not used in this book.

In the following text, we use the same approach as in the previous sectionsof this chapter, which takes into account the interlayer thickness and the effect ofthe transition zone between the thin interlayer and the bulk liquid. This effect isequivalent to the line tension that is considered in Section 2.10. A low slope andconstant surface tension approximations are used. Then, as was shown earlier inSection 2.1 through Section 2.3, it is possible to use the equation taking intoaccount both the disjoining pressure and the capillary pressure in the interlayer.

FIGURE 2.32 Two identical drops or bubbles (1) at equilibrium in a surrounding liquid,(2); h(x) is a half thickness of the liquid film between two drops or bubbles.

1

1

22

Pl h(x)x

Pd



TWO IDENTICAL CYLINDRICAL DROPS OR BUBBLES

First, let us consider a case of two identical cylindrical drops or bubbles (Figure2.33). It is assumed for simplicity that the radius of action of surface forces islimited to a certain distance, ts. Beyond this distance, the surface of dropletsretains a constant curvature radius, R, and is not disturbed by surface forces. Theinteracting droplets are considered as being surrounded by a liquid with constantpressure, Pl.

The thickness of the liquid interlayer, 2h(x), varies from 2h0 on the axis ofsymmetry at x = 0 to 2h = ts at x = x0 (Figure 2.33).

For the drop profiles not disturbed by the surface forces, we use the desig-nation hs at y < B, and Hs for y > B (Figure 2.33). The liquid in the droplets isassumed to be incompressible, which leads to the condition of the constancy ofthe volume of droplets per unit length,

where the superscript 0 marks an undisturbed isolated droplet prior to contact.Note that, in general, the system of two droplets is thermodynamically unsta-

ble with regard to coalescence. Therefore, the following calculations give theconditions of the metastable equilibrium of droplets separated by a thin interlayerof the surrounding liquid.

The excess free energy of the two cylindrical drops (per unit length ofcylindrical droplets), Φ, is equal to Φ = γS + ΦD + PeV , where S is the total

FIGURE 2.33 The equilibrium profile of interlayer, h(x), between two cylindrical dropletsof the same radius, R.

V H h dx R consts s

R

= −( ) ⋅ = =∫4 0 0 0 2

0

0

π ( ) ,

2B 2h0

h(x)

2x0

Hs(x)

hs(x)

xts

PlR

R

ϕϕ

θe



interfacial area per unit length, γ is the interfacial tension, ΦD is the excess freeenergy per unit length determined by the surface forces action, Pe is the excesspressure, and V is the volume per unit length. In the case of cylindrical drops orbubbles S, FD, and V are as follows:

,

(2.139)

where h(x), hs (x), Hs (x), x0, and radius, R are determined in Figure 2.33; ts isthe radius of the action of surface forces.

Substitution of the latter expression into the excess free energy results in

A variation of the excess free energy with respect to h(x), hs (x), Hs (x), andthe two values x0 and R results in the following equations:

(2.140)

where the last two equations give the equation of a circle of radius R. The latterimmediately determines the unknown excess pressure as

S H dx h dx h dxs

R

s

x

R x

D

= + ′ + + ′ + + ′∫ ∫ ∫4 1 4 1 4 12

0

2

0

2

0

0

Φ ==

∞

∫∫220

0

Π( ) ,h dh dxh

x

and

V H h dx H h dx R consts s

x

R

s

x

= −( ) ⋅ + −( ) = =∫ ∫4 4

0 0

0

02π ,

Φ = + ′ + + ′ + + ′

∫ ∫ ∫4 1 4 1 4 12

0

2

0

2

0

0

γ H dx h dx h dxs

R

s

x

R x

+

+ −

∞

∫∫2

4

20

0

Π( )h dh dx

P H

h

x

e s hh dx H h dxs

x

R

s

x

( ) ⋅ + −( )

∫ ∫0 0

0

4 .

γ

γ

·

·

/′′ + ′ + ( ) = −

′′ + ′

−h h h P

h h

e

S S

1 2

1

23 2

2

Π

= −

′′ + ′ =

−

−

3 2

23 2

1

/

/·

P

H H P

e

S S eγ



After that, the first part of Equation 2.140 takes the following form:

(2.141)

where the first term on the left-hand side is due to the capillary pressure, and thesecond term is determined by the local value of the disjoining pressure, Π(2h).

The boundary conditions for Equation 2.141 are as follows:

(2.142)

(2.143)

(2.144)

We remind ourselves that hs describes the profile of the droplet at 2hs > ts.If the droplets are located at a distance, 2h0 > ts, then Π(2h) = 0; and in this

case, the solution of Equation 2.141 gives the profile hs(x), which corresponds tothe circular form of the cross section of nondeformed droplets that do not interactwith one another. However, at 2h0 < ts, the interlayer is under the effect of bothsurface forces, whose contribution is determined by the term Π(2h) and thecapillary forces, depending on the local curvature of the interlayer surfaces.

Equation 2.141 with boundary conditions (2.142 through 2.144) and with thecondition of constancy of volume (2.139) provide a solution to the problem; itenables us to determine the profile of an interlayer, h(x), between the droplets atthe known isotherm of disjoining pressure, Π(2h).

An example of the profile of the droplets in the transition zone, in the caseof the S-shaped disjoining pressure isotherm, is shown in Figure 2.34. Note that,in general, the thickness in the central part between drops is different from theequilibrium thickness he, and this is the reason why it is referred to in Figure 2.33as h0. The latter two thicknesses coincide if the central part between the two dropsis flat.

The solution thus obtained may be verified in the following way: at theequilibrium state, the total force of interaction of droplets per unit length, F,should be equal to zero. According to Section 2.8, this force

(2.145)

PRe = − γ

.

γ γ⋅ ′′ + ′ + ( ) =−

h h hR

1 223 2/

,Π

h x h x h x h x h x ts s s0 0 0 0 0 2( ) = ( ) ′( ) = ′ ( ) ( ) =, , ,/

h R H R h R H Rs s S S( ) = ( ) ′ ( ) = − ′ ( ) = ∞; ,

′( ) = ′ ( ) =h HS0 0 0.

F h dx

x

= ( )∫2 20

0

Π · .



Substituting the expression for Π(2h) from Equation 2.140 and Equation 2.141into Equation 2.145 and then carrying out integration, we obtain:

(2.146)

In view of the boundary condition (2.142), at the point x = x0, both valuesof h(x) = hs(x), and their derivatives, h′ = h′s, are equal. This enables one to expressh′(x0) through the central angle ϕ (Figure 2.33) and the values x0 and R:

(2.147)

Substituting the latter expression for h′(x0) into Equation 2.146, we obtainF = 0, as should be at the equilibrium. Thus, it should be expected that theconditions of equilibrium given by Equation 2.141, and by the boundary conditions(2.142 through 2.144), correspond to zero interaction forces between the droplets.

It should be noted that the angle ϕ has a value that is very close to that ofthe contact angle θe, to be determined at the point of intersection of the contin-uation of the undisturbed profile of a droplet with axis x (Figure 2.33: ho > he orwith the continuation of the equilibrium film (Figure 2.34: h0 = he). The valuesof θe and ϕ practically coincide when the interlayer thickness is small as comparedwith R and when x0 is not too small. This enables us to use Equation 2.147 forthe calculation of the contact angles.

Thus, derived values of x0, B, θe, and the droplet profile, h(x), give the fullsolution of the problem, where the distance between the centers of droplets, B,(Figure 2.33) is:

FIGURE 2.34 Partial wetting. S-shaped disjoining pressure isotherm (left side) and theliquid profile in the transition zone (right side). Magnification of the upper part of thetransition zone between the drop or bubble and the thin liquid interlayer. 1 — real deformedprofile of the drop or bubble, 2 — ideal spherical profile, when the influence of disjoiningpressure has been ignored, 3 — thin liquid interlayer of thickness 2he. x0 should be replacedby r0 in the case of a spherical drop/bubble.

θe

x0

3 3

1 2

Pe

2 he

2 tmin

2 tmax

2 ts

h

Π

F xh x

h x

= −′( )

+ ′( )

21

00

0

2γ .

′( ) = = −( )−h x x R x0 0

202

1 2tan .

/ϕ



(2.148)

INTERACTION OF CYLINDRICAL DROPLETS OF DIFFERENT RADII

Let us now consider a more complicated case of interaction of cylindrical dropletsof different radii, R2 > R1 (Figure 2.35a).

Applying the same method of minimization of the excess free energy as inthe preceding section, we obtain the following equations:

(2.149)

(2.150)

where h1(x) and h2(x) are measured from an arbitrary plane that is perpendicularto the axis of symmetry, and t(x) = h1(x) – h2(x), is the thickness of the interlayer.Equation 2.149 and Equation 2.150 enable the determination of two profiles, h1(x)and h2(x). Equation 2.149 and Equation 2.150 along with the boundary conditions

(2.151)

FIGURE 2.35 The equilibrium profile of interlayer t(x) = h1.(x) – h2 (x) between twodroplets of different radii, R1 and R2, in the general case (a); and in the simplified case(b), when the transition region is neglected.

B t R x t xsf s= + −( ) = + ( )2 2202

1 2

0

/tan ./ ϕ

γ γ· ,/

′′ + ′( )

+ ( ) =−

h h t R1 1

23 2

11 Π /

γ γ⋅ ′′ + ′( )

− ( ) = −

−

h h t R2 2

3 2

212

/

,Π /

′( ) = ′ ( ) = ( ) =h h t x ts1 2 00 0 0; ;

r R1

ϕ

ϕ1ϕ1

ϕ2ϕ2x

H2s(x)

h1(x)h2(x)h2s(x)

h1s(x)

h1s(x)

B

R2 R2

R1

y

2x02x0

ttsft(x)

(a) (b)



(2.152)

and with the conditions of constancy of volumes, give a solution to the problem.After the addition of Equation 2.149 and Equation 2.150, we obtain

(2.153)

which means that the curvature of the whole interlayer equalizes the capillarypressure drop between the droplets.

In a similar way, we may solve the same problem for two droplets of differentcomposition with different interfacial tensions, γ1 and γ2, of the first and thesecond droplet, respectively. In this case, even for the droplets of the same radius,the interlayer on the whole proves to be curved owing to the appearance of acapillary pressure drop, ∆Pk = (γ1 – γ2)/R.

In the case of a not strongly curved interlayer between droplets, the term (h′)2

may be neglected as compared with 1. From Equation 2.149, Equation 2.150,and Equation 2.153, taking into account that t(x) = h1(x) – h2(x) and t″(x) = h″1(x)–h″2 (x), we obtain:

(2.154)

This equation should be subjected to the following boundary conditions:

(2.155)

and coupled with the conditions of constancy of volumes, which determine theunknown radii R1 and R2. The latter conditions and Equation 2.154 and Equation2.155 enable the calculation of t(x), determining the variable thickness of theinterlayer.

Thereafter, on substituting the known dependence Π[t(x)] = Π(x) into Equa-tion 2.150, it is possible to obtain from

(2.156)

the profile h2(x) of the lower surface of the interlayer. The boundary conditionsfor Equation 2.156 are as follows:

′( ) = = −( )−h x x R x1 0 1 0 1

202

1 2tan ;

/ϕ

′ ( ) = = −( )−h x x R x2 0 2 0 2

202

1 2tan ,

/ϕ

γ γ/ /R R Pk1 2( ) − ( ) = ∆ ,

γ γ⋅ ′′ + ⋅ ( ) = +

t tR R

21 1

1 2

Π .

′( ) = ( ) = ′( ) = −( ) +−

t t x t t x x R x Rs0 0 0 0 0 12

02

1 2

22; ;

/−−( )

−x0

21 2/

γ γ· ′′ − ( ) = −h x R2 2Π /



(2.157)

It has been taken into account in the preceding expression that angle isdetermined from Equation 2.152. The sum, h2(x) + t(x), gives the profile of theupper surface of the interlayer.

Calculation of the interaction of cylindrical droplets of different radii (R2 >R1) can be simplified if we assume that the interlayer is of a constant thickness.This means that the effect of a transition zone is neglected, which is justifiedonly at

In this case, the curvature of each surface of the interlayer is constant (Figure2.35b), and Equation 2.149 and Equation 2.150 may be rewritten in the followingway:

(2.158)

(2.159)

This system of equations enables one to determine two unknown values: t,the interlayer thickness; and r, the radius of curvature of its surface on the sideof the smaller droplet. By summing up and subtracting the terms in Equation2.158 and Equation 2.159, we obtain (at r >> t):

(2.160)

(2.161)

If the shape of disjoining pressure isotherm, Π(t), is known, then Equation2.161 determines the equilibrium thickness t = const of the curved interlayer.

At R2 >> R1, Equation 2.160 and Equation 2.161 result in:

(2.160’)

(2.161’)

It should be noted that Equation 2.160 and Equation 2.161 may be derivedby another method using the concept of disjoining pressure:

(2.162)

′ ( ) = ( ) = ′ ( ) =h h x R h x2 2 0 2 2 0 20 0; cos ; tan .ϕ ϕ

ϕ2

x ts0 >> .

γ γ/ /r t R( ) + ( ) =Π 1,

γ γ/ /r t t R+( ) − ( ) = −Π 2.

r R R R R~ ,2 1 2 2 1−( )

Π t R r R R R R( ) − +( )~ ~ · .γ γ γ/ /1 2 1 1 22

r R~ ,2 1

Π t R( ) ~ .γ /2 1

Π = −P Pd l ,



where Pd is the pressure under the interlayer surface, and Pl is the pressure inthe bulk phase in which the interlayer is at equilibrium.

On the side of a smaller droplet, we have Pd = Pl + (γ /R1) – (γ /r); on the sideof a larger droplet, Pd = Pl + (γ /R2) + γ(r + (t). As the disjoining pressure doesnot depend on the side of the interlayer from which it is determined, then fromEquation 2.162 it follows that:

(2.163)

which coincides with Equation 2.158 through Equation 2.161.However, determination of r and t does not completely solve the problem

because the position of the center of the interlayer curvature remains unknown.Its position can be determined by minimizing the value of the free energy of thesystem

(2.164)

and by taking into account the condition of the constancy of the volume ofdroplets:

(2.165)

where R10 and R20 are the radii of the undisturbed droplets (atIt is possible to express all the values via x0: x0 = R1 sinϕ1; x0 = R2 sinϕ2;

and x0 = r sinϕ. Then the condition ∂Φ/∂x0 = 0 is used to determine the value ofx0, corresponding to the equilibrium position of the droplets. In carrying out theaforementioned procedure, the values of t and r can be expressed through R1, R2,and γ, in accordance with Equation 2.160 and Equation 2.161.

In conclusion, let us consider the equilibrium conditions for the most commoncase of two spherical droplets of different radii and compositions. Using thepreviously mentioned method of minimization of the excess free energy, weobtain:

(2.166)

Π = ( ) − ( ) = ( ) + +( ) γ γ γ γR r R r t1 2 ,

Φ Π= ⋅ −( ) + ⋅ −( ) + + ( )

∞

∫2 2 2 22 2 1 1γ π ϕ γ π ϕ ϕ γ ξ ξR R r dt

R r12

1 121

22

12

2π ϕ ϕ ϕ ϕ π− +

+ −

=sin sin · RR const

R r

102

22

2 221

22

12

=

− +

− −

,

sin siπ ϕ ϕ ϕ nn ,2 202ϕ π

= ⋅ =R const

h tsf0 > ).

γ11

1

23 2

1

1

21

1 1

′′

+ ′( )

+ ′

+ ′( )

h

h

h

r h/ //

,2

1

1

2

+ ( ) =Π tRγ



, (2.167)

where r is now the radial coordinate.The boundary conditions for Equation 2.166 and Equation 2.167 are given

by Equation 2.151 through Equation 2.152, where x0 should be replaced by r0.In this case, conditions of the constancy of the volume of droplets can be writtenas:

At (h′)2 << 1, Equation 2.166 and Equation 2.167 can be simplified. Summingup these equations would give:

(2.168)

where t(r) = h1(r) – h2(r).Solution of Equation 2.168 describes the variations in the thickness, t(r), of

the interlayer between the droplets. The distance between the centers of thedroplets can be obtained using the same methods as are used in the case ofcylindrical droplets.

Thus, the use of the isotherms of disjoining pressure of thin interlayers enablesus to solve the problem of the equilibrium of droplets in contact with one anotherand to calculate the shape of the deformed droplets and of the interlayer.

SHAPE OF A LIQUID INTERLAYER BETWEEN INTERACTING DROPLETS: CRITICAL RADIUS

As seen in the previous section, the shape of a liquid interlayer between interactingdroplets depends on their size, interfacial tensions, and the shape of the disjoiningpressure. The calculations presented in the following text describe the shape ofan interlayer between droplets under equilibrium conditions until its possiblerupture.

γ 22

2

23 2

2

2

21

1 1

′′

+ ′( )

+ ′

+ ′( )

h

h

h

r h/ //2

2

2

2

− ( ) = −Π tRγ

V H h rdr H h rdrs s

r

R

s

r

1 1 1

0

1

1

0

0

2 243

= −( ) ⋅ + −( ) =∫ ∫π π ππ

π π

R const

V H h rdr H hs s

r

R

s

103

2 2 2

0

2

12 2

=

= −( ) ⋅ + −∫ (( ) = =∫0

0

2034

3

r

rdr R constπ

′′ + ′ + +

⋅ ( ) = +

ttr

tR R

1 12

1 1

1 2 1 2γ γΠ ,



In carrying out further calculations, we assume that the interlayer thickness,h(r), varies within the contact region but not very sharply, that is, the approxi-mation (h′)2 << 1 can be used. This enables the usage of the isotherm of disjoiningpressure of flat interlayers, Π(h), in equations of equilibrium, as well as to simplifythe expression for the local curvature of the interlayer surfaces.

Let us consider the interactions of two identical spherical droplets (Figure 2.37).In this case, the equation of the interlayer profile, h(r), can be derived by solvingEquation 2.168:

(2.169)

where h′ = dh/dr; h″ = d2h/dr2; γ is the interfacial tension; and R is the radius ofdroplets.

For carrying out quantitative calculations, we use the model isotherm ofdisjoining pressure, ΠΠΠΠ (h), in the following form (Figure 2.36a):

(2.170)

Such an isotherm (Figure 2.36a) provides a possibility for obtaining ananalytical solution of the problem, and at the same time, it possesses the mainproperties of real isotherms (Figure 2.36b) corresponding to the attraction ofdroplets at large separations, i.e., t0 < h < ts, and to their repulsion at smalldistances, at h < t0. Here, the thickness ts corresponds to the radius of action ofsurface forces. Beyond its limits, at h > ts, the interaction forces vanish, that is,Π = 0. Parameter a = (Π1 + Π min)/ts determines the slope of the isotherm.

Solution of Equation 2.169 together with the isotherm as given by (Equation2.170) has the following form:

FIGURE 2.36 The disjoining pressure isotherm, Π(h), as used in calculations.

Π

−Π −Πmin

Π1

0t0

ts h

(a) (b)

Π

−Π

0h

γ γ2 4

2′′ + ′

+ ( ) = =hh

h R PeΠ ,

Π hh t

a t h h t

s

s

( ) =≥

− ≤ ≤

0

00

,

( ),.



(2.171)

where z = r(2a/γ)1/2, and I0 is Bessel’s function of an imaginary variable. Theconstant A is determined from the boundary condition, h(r0) = ts, where r0 is theradius of the zone of deformation (Figure 2.37), and z0 = r0(2a/γ)1/2

(2.172)

Substituting Equation 2.172 into Equation 2.171, we obtain an equationdetermining the profile of the interlayer between droplets:

(2.173)

Accordingly, the minimum distance between the surfaces of droplets is equalto

(2.174)

In Equation 2.173 and Equation 2.174, the value of r0 still remains to bedetermined. For this purpose, let us use the condition (2.145):

(2.175)

Substituting into Equation 2.175 the equation of isotherm (Equation 2.170)and replacing h(r) by its expression from Equation 2.143, we obtain the followingexpression:

(2.176)

In the case when (sufficiently large drops), the integral in Equation 2.176is equal to In this case, Equation 2.176 results in the following expres-sion for the radius of the contact zone:

(2.177)

h r t aR A I z( ) = − ( ) + ( )0 02γ · ,

A t t aR I zs= − + ( ) ( )0 0 02γ .

h r taR

t taR

I z

I zs( ) = − + − +

⋅( )

( )0 00

0 0

2 2γ γ.

h h taR

t taR I z

s0 0 00 0

02 2 1= ( ) = − + − +

⋅ ( )γ γ

.

F h r dr

r

= ( ) ⋅ ⋅ =∫Π 2 00

0

π .

raR

t taR

rI z

I zdsf

r

02

00

0 00

4 20

= − +

( )( )∫γ

γ· rr.

z0 5≥r a0

1 22( ) ./γ /

r aR t t

as

0

1 2 02

21= ( ) −( )

+

γ/

.



Using Equation 2.177, the relative extension of the contact zone, where theeffect of the surface forces is pronounced, can be expressed as:

(2.178)

At or R >> 2γ/a(ts – t0) the ratio r0/R tends to the value

which is independent of the radius of the drop, R. This would indicate the geometricsimilarity of all large fluid droplets that are deformed by the contact interaction.

For the droplets of small radius, R, one may use another approximation forI0(z), which is valid at z < 1: I0(z) = 1 + (z2/4). In this case, integration in Equation2.176 yields:

(2.179)

Let us compare the preceding expression with the solution for solid spheres.Their equilibrium state (Figure 2.37b) can also be characterized by the interactionregion of radius r0, within which the surface forces exert their effect. The profileof the solid sphere can be represented as

FIGURE 2.37 The schematic representation of the contact interaction of the fluid (a,c)and the solid (b) particles. In the case (c), the thickness in the center coincides with theequilibrium thickness he.

rR

at t

aRs0 1 2 022

1= ⋅( ) − +

γγ

/.

1 20/ )/aR t ts<< −( γ

t ta

s −( )

0

1 2

2·

γ

rR t t

aR t ts

s02 0

0

2

1 2=

−( )− ( ) ⋅ −( )/ γ

.

h h R r R= + ⋅ − − ( )

0

22 1 1 / .

(a) (b) (c)

R

R

2r0

2h0

h(x)

2B θe

R

R

2r0

ts 2h0 θe

R

R

2r0



At r0 << R, its approximate form may be used:

(2.179′)

The minimum distance, h0, between the solid particles can be determinedfrom the same condition (2.145) and Derjaguin’s known approximation [1]:

(2.180)

Replacing the lower integration limit by ts and using the model disjoiningpressure isotherm of Equation 2.170 (after integration in Equation 2.180), weobtain the following expression: h0 = 2t0 – t1. On substituting this expression intoEquation 2.179 and taking into account that h(r0 ) = ts, we get:

(2.181)

The same value of r0 is also obtained by solving Equation 2.179 for smallfluid droplets. This means that very small emulsion droplets and gas bubblespractically behave as solid particles; they do not deform in the contact zone.

Let us call the critical size of fluid droplets R*, such that at R << R*; theirinteraction does not differ from that of solid spheres. Comparing Equation 2.179and Equation 2.181, we note that these coincide under the condition that thesecond term in the denominator of Equation 2.179 is much smaller than unity.This condition allows the determination of R* as

(2.182)

The distance between the centers of the droplets can be determined usingEquation 2.148 and simple geometrical considerations:

. (2.183)

Accordingly, the contact angle, θe, can also be determined at the point ofintersection between the nondeformed surface of a droplet and the r-axis (Figure2.37a):

(2.184)

h h r R≅ + ( )02 .

F h R h dh

h

0

0

( ) = ( )∞∫π · · .Π

r R t ts02

02= −( ).

Rt t as

*

·.=

−( )2

0

γ

B t R rs= + −( )2 202

1 2/

cos ./

θe sB R t R r R= = ( ) + − ( )

/ / /2 2 1 0

21 2



This expression holds only for relatively large droplets. As has already beenpointed out, small droplets (R < R*) are practically nondeformable, and theirundisturbed, circular profile does not intersect the r-axis. In the case of smalldroplets, similar to that of complete wetting (see Section 2.4), the contact angleis absent.

Let us now numerically calculate the profiles of fluid droplets in the contactzone using Equation 2.173, and compare them with two other known models:the model of solid nondeformable particles (Figure 2.37b), and the model of aflat interlayer between similar droplets (Figure 2.37c). In the latter model, theeffect of a transition zone between the surrounding bulk medium and the flatportion of the interlayer is not taken into account. In the case of flat interlayers,let us use the equations of equilibrium of thin flat films (Equation 2.47 in Section2.3) which is modified for the case of two droplets or bubbles:

, (2.185)

where θe is the equilibrium contact angle determined at the point of intersectionof the continuation of the nondeformed part of a sphere and the surface of theequilibrium flat film. In that case, the thickness of the flat interlayer, 2he, isdetermined using the disjoining pressure isotherm, Π(h), as Π = Pe, where

is the capillary pressure drop at the spherical interface. The contactarea of droplets at he << R is equal to , where r0 is the radiusof the contact zone.

Substituting Equation 2.170 of the disjoining pressure isotherm into Equation2.185, we obtain the following expression:

(2.186)

The calculations were carried out for the model disjoining pressure isotherm,Π(h), (Equation 2.170) preset by the following parameters: γ = 30 dyn/cm; ts =3 · 10–6 cm; t0 = 2 · 10–6 cm; Π3 = Π(0) = 3 · 106 dyn/cm2, and Πmin = –Π (ts) =–1.5 · 106 dyn/cm2, which gives a = 1.5 · 1012 dyn/cm3. According to Equation2.185, the adopted values correspond to the contact angle θe = 9˚.

The left-hand part of the plots in Figure 2.38 relates to the spherical dropletshaving radius R = 10–3 cm, and the right-hand part to R = 10–4 cm. As appearsfrom the left-hand part, the large-sized droplets deform considerably, thus forminga practically flat contact zone. Profiles 1 and 2 are close to each other, and thetransition zone occupies rather a small region immediately near the contactperimeter. Now, deviations from the profile of solid particles are very large(curve 3). Thus, in the case of R >> R*, the conditions of equilibrium of thedroplets may be described with in the framework of the theory of flat interlayers,i.e., on the basis of Equation 2.185.

2 1 22

⋅ −( ) = ( ) ⋅ + ⋅∞

∫γ θcos e e e

h

h dh P h

e

Π

P Re = 2γ /π π θr R e0

2 2 2= ⋅ sin

cos ·θ γ γe sa t t aR R= − ( ) −( ) − ( )

+ (1 4 2 10

2 2/ / / )) − ( ) · .t aR0 2γ /



A decrease in the size of droplets (the right-hand side of the graph in Figure2.38) makes the profiles of solid particles (curve 3) and droplets (curve 1)approach one another. In the central part of the contact region, the flat interlayerregion reduces, whereas the differences from the profile of the flat interlayers(curve 2) increases. A further decrease in the droplet radius causes a still greaterapproach of the profiles of the fluid and the solid particles to one another and adisappearance of the flat portion of the interlayer. As already shown, at R << R*,the known solutions for the interaction of solid spheres may be used.

In this connection, it is important to evaluate the values of R*. This allowsthe determination of the regions of applicability of different solutions — namely,flat interlayers for R >> R*; solid particles for R << R*; and finally, the approachdeveloped for the intermediate values of R.

As appears from Equation 2.182, the critical radius R* depends on the param-eters of the disjoining pressure isotherm (a, ts, and t0) and the interface tension.Under otherwise equal conditions, a decrease in the interface tension reduces thevalues of R*, thus limiting the region of the applicability of the theory of solidspheres.

Let us evaluate R*, assuming γ = 50 dyn/ cm, and (ts –t0) = 10–6 cm. Thevalues of parameter a will be varied, which, in accordance with Equation 2.186,is equivalent to a change in the contact angle θe (omitting small terms):

(2.187)

FIGURE 2.38 The profiles, h(r/R), of the contact zone of spherical droplets, R = 10–3 cm(to the left) and the R = 10–4 cm (to the right), calculated while taking into account thetransition zone (1), in the approximation of a flat interlayer (2), and solid particles (3).

R = 10–3 R = 10–4

3

22

31

1

0.1

0.2

r0r0

rR

h, nm20 10 0 10 20

cos · .θ γe sa t t= − ( ) −( )1 4 0

2/



The calculations show that the values of R* decrease as θe increases. Thus,at small values of θe close to zero, R* = 10–3 cm; at θe = 5–6˚, R* = 10–4 cm; atθe = 20–30˚, R* = 5 × 10–6 cm; and at θe = 90˚, R* = 5 × 10–7 cm. In this manner,an increase in the contact angle, corresponding to the enhancement of the dropletinteraction, causes a decrease in the critical radius R*. Now, this means that inthe case of a strong interparticle interaction, the approach of solid spheresbecomes less applicable; yet, in the case of weakly interacting droplets, theparticular approach may be used even for relatively large-sized droplets. Thesolutions obtained allow the evaluation of R* and help to choose the correspondingequations for the calculation of the equilibrium shape of the droplets within thecontact zone.

2.10 LINE TENSION

The presence of the transition zone between a drop or a bubble and thin liquidinterlayers can be described in terms of line tension, τ, a concept first introducedby Gibbs (see for example [22]). In the case of surface tension, the transitionzone between the liquid and vapor is replaced by a plane of tension with excesssurface energy, γ. By analogy, the transition zone between a drop or a bubble andthe thin liquid interlayer may be replaced by a three-phase contact line with anexcess linear energy, τ. In contrast to surface tension defined always as positive,the value of the line tension may be positive and negative. When positive, itcontracts the wetting perimeter, whereas the perimeter expands if the line tensionis negative [33–36].

A number of attempts have been made to improve Young’s equation and tomake it more theoretically justified. The most important of them is the introduc-tion of line tension, τ. In Section 2.3, it has been shown that the drop profilecannot keep its spherical shape up to the three-phase contact line. It has beenshown in the same section that the action of surface forces results in a substantialdeviation of the drop shape, in the vicinity of the three-phase contact line, froma spherical shape. It results in the formation of a transition zone where theinfluence of the disjoining pressure is important and cannot be ignored. However,if we still want to consider a spherical droplet, then the existence of the transitionregion can be effectively taken into account by replacing the whole transitionregion by an additional free energy located on the three-phase contact line. Thisconsideration, in a way, is similar to the introduction of an interfacial tension. Ifthe line tension is taken into account, then the excess free energy of the dropleton the solid substrate, Φ, should be written as:

, (2.188)

where r0 is the radius of the droplet base. Note that, according to Section 2.1,we used the interfacial tension of the solid substrate covered by an equilibriumliquid film, , but not the corresponding interfacial tension of a bare solid

Φ = + + − +γ π γ γ π τS P V r re sl svh02

02( )

γ svh



substrate, . In Section 2.1, we explained that the latter surface tension cannotbe used at the equilibrium conditions, and these two interfacial tensions can differconsiderably. Using the expression for the droplet profile, hid(r), the latter equationcan be rewritten as

, (2.189)

where τ is line tension, and hid(r) is the idealized droplet profile: it has a sphericalpart up to the intersection with the surface of the equilibrium flat film. The firsttwo conditions of the minimum value of the excess free energy result in theidentical equation for the spherical drop profile (41), Section 2.3. However, thetransversality condition (4) in Section 2.2 takes the following form:

(2.190)

If we now introduce a new real equilibrium contact angle that takes intoaccount the line tension as θe and use the previous definition of the contact angle,which is referred to now as θe∞ according to Equation 2.47, then Equation 2.190can be rewritten as

. (2.191)

Note that the derivative in the preceding equation is usually neglected withoutany justification. Neglecting the derivative of the line tension in Equation 2.191results in:

. (2.192)

If the line tension is negative, then the influence of line tension results in abigger contact angle as compared with predictions according to Equation 2.192.

In the general case, Equation 2.191 can be rewritten as

, (2.193)

where m and “plus” or “minus” depends on the system geometry: m = 1 and aminus sign corresponds to the drop on the solid substrate, m = 2 and a plus sign

γ sv

F r h P h dri d sl svh

r

= + ′( ) + ⋅ + −

+∫2 1 2

2

0

0

π γ γ γ πrr0τ

γ γ γ τ τ1

02

00

0+ ′

+ −

+ + =

=h

d

drrsl svh

r r

.

γ θ θ τ τcos cose e

d

dr r−( ) + + =∞

0 0

0

cos cosθ θ τγe e r

= −∞0

mr re eγ θ θ τ τ⋅ −( ) = ± + ∂

∂

∞cos cos

0 0



correspond to the two identical drops or bubbles in contact (Figure 2.37); θe∞refer to the contact angle in the case of big cylindrical drops (see the followingsection). In the case of liquid drops on the flat solid substrate, the positive valuesof τ cause an increase in the values of contact angles θe, whereas the negativevalue results in their decrease. In the case of flat films in contact with a concavemeniscus, the influence of the line tension τ, is inversed because the minus signshould be used now in Equation 2.193.

For water and aqueous electrolyte solutions, the line tension values are in therange of 10–6–10–5 dyn [23] and below. Thus, the terms on the right-hand side ofEquation 2.193 become noticeable at r0 ≤ 10–4 cm.

It is important to recall that the aforementioned value of the line tension canbe used only at equilibrium. It is impossible to experimentally measure theequilibrium liquid droplets on solid substrates because they should be at equilib-rium with oversaturated vapor, as was explained in Section 2.3. This would meanthat usually everything in Equation 2.193 is either far from equilibrium or undera quasi-equilibrium condition (as caused by the hysteresis of contact angle —see Section 3.10). In that case, the value of line tension can be many orders ofmagnitude higher than 10–6–10–5 dyn. However, this line tension should bereferred to as dynamic line tension. To the best of our knowledge, there has notbeen any attempt as yet to introduce or investigate the dynamic line tension.

The values of the line tension, τ, for drops on solid substrates has beencalculated as a difference between the values of , and is calcu-lated in two different ways: (1) neglecting the transition zone, (2) by taking itinto account. As the line tension arises due to the existence of the transition zone,it is clear that this difference is just associated with the additional terms on theright-hand side of Equation 2.192.

An expression for τ was obtained in the case of a model isotherm of disjoiningpressure [23], and the line has been estimated as dyn and negative.De Feijter and Vrij [24] have considered the transition zone between a Newtonblack film (a different name for α-films) and bulk liquid. According to theirestimations, the line tension value is also negative and has the same order ofmagnitude. Kolarov and Zorin [25] have measured the line tension value. Theyhave used Sheludko’s cell for measurements of properties of free liquid films.An aqueous solution of 0.1% NaCl with sodium dodecyl sulphate (SDS) 0.05%concentration (carboxymethylcellulose or CMC = 0.2%) has been used. Theyhave calculated the line tension for this system using Young’s equation (Equation2.193). The value of line tension has been found to be dyn. That is, itis negative and in good agreement with the theoretical predictions [23].

However, Platikanov et al. [26] have carried out experimental measurementsof line tension dependency on salt concentration and presented this experimentalevidence of line tension sign change. In the following text, we present a theorythat is modified as compared with the presented theory [23], and it explains theexperimentally discovered sign change in the line tension.

γ θ⋅cos e γ θ⋅ ∞cos e

10 105 6− −÷

− ⋅ −1 7 10 6.



Let us consider two big (R >> R*) identical drops or bubbles in contact(Figure 2.37) under equilibrium conditions. It has been explained in Section 2.3that it is important to properly select the reference state. This reference state isintroduced as a flat equilibrium liquid film of thickness 2he. The reason for thesame was presented in Section 2.1 and Section 2.3. From a mathematical pointof view, it means addition or subtraction of a constant to or from the excess freeenergy. The latter constant does not influence the final equation, which describesthe profile in the transition zone. However, as already seen in Section 2.3, thischoice is essential for transversality conditions at the apparent three-phase contactline. In the following text, we see that this choice is also important for thedefinition of the line tension. This means that the choice of the reference state isvery important, and we use the same choice of the reference state as in Section2.1, which is the uniform flat equilibrium film of thickness 2he, where he is thehalf-thickness of the equilibrium film. Using this choice, the excess free energy,Φ, of the system (curve 1 in Figure 2.34) has the following form:

(2.194)

where h(x) is the half-thickness of the liquid layer; he is the half-thickness of theflat equilibrium thin liquid film; Pe = 2γ/R is the excess pressure; Π(h) is thedisjoining pressure; and h(R) = H is the position of the end of the drop. Thelower limit of integration corresponds to the end of the transition zone where h =he. We can use infinity as the upper limit of integration instead of R because, atthis stage, we are not interested in the upper part of the drop or bubble. Note thataccording to the definition given by Equation 2.162 in Section 2.9, the excesspressure, Pe, is positive.

Under the equilibrium condition, the system is at the minimum free energystate, that is, conditions (1) through (4) should be satisfied (Section 2.2). Thisresults in the equation for the determination of the liquid profile in the transitionzone:

(2.195)

The transversality condition (4) results in (see Appendix 1)

(2.196)

Φ Π Π= + ′ − + − +∞ ∞

∫ ∫2 2 1 1 22

2 2

π γr h h dh h dh Ph h

e

e

( ) ( ) ( ) ⋅⋅ −

∫ ( ) ,h h dre

R

0

γr

ddr

rh

h

h P⋅ ′′

+ ′( )

+ ( ) =

1

22

12

Π ee.

′ →h 0,



at the end of the transition zone, which means a smooth transition from thetransition zone to the flat thin film.

Let us introduce an ideal profile of the liquid interlayer in the transition zone,2hid, which is a spherical part up to the intersection with the equilibrium liquidinterlayer of thickness 2he (Curve 2 in Figure 2.34). The excess free energy ofsuch an ideal profile differs from the exact excess free energy given by Equation2.194 because the presence of the transition zone is ignored in the case of theideal profile. This means that the line tension, τ, should be introduced to com-pensate the difference:

(2.197)

Under equilibrium conditions, the same equilibrium conditions (1–4) shouldbe satisfied, which gives an equation for the ideal liquid profile in the transitionzone

(2.198)

and the transversality condition in the case of ideal liquid profile with excess freeenergy given by Equation 2.197 is as follows:

(2.199)

where

Substitution of the latter expression into the condition (2.199) results in thefollowing equation at r = r0 and h = he:

.

Φ Π= + ′( ) −

− ( ) + ⋅∞

∫2 2 1 1 22

2

π γr h h dh P hi d

h

e i

e

dd e

r

h dr r−( )

+∞

∫0

2 0π τ

γr

ddr

rh

h

Pid

id

′

+ ′( )

=

12

12

,

− − ′ ∂∂ ′

+ ==

f hf

h

d rdrid id

id

id r r0

0

0

0( )

,τ

f r h h dh P hid id

h

id

e

= + ′ −

− + ⋅ −∞

∫2 1 1 22

2

γ Π ( ) ( hhe) .

2 1 12

1

2

2

2

2γ γ+ −( ) − −

′

+ ′

∞

∫h h dhh

hid

h

id

ide

Π ( )

− +

=

=r r

d

dr r0

0 0

0τ τ



After a rearrangement, the latter condition becomes:

, (2.200)

where

. (2.201)

Equation 2.200 coincides with Equation 2.193 if we select m = 2 and acorresponding plus sign. Equation 2.201 gives the expression for the contact angleof a big cylindrical drop (see the following text).

Excess free energy given by Equation 2.194 and Equation 2.197 should beequal. The latter gives the following definition of the line tension, τ:

(2.202)

The preceding equation presents an exact definition of the line tension, τ, incontrast to Equation 2.200, where the value of the line tension is unknown. InEquation 2.202, the real liquid profile, h(r), is the solution of Equation 2.195,and the ideal liquid profile, hid(r), is the solution of Equation 2.198.

Dependency of the line tension on the radius, r0, has been investigated [23]in the case of a model disjoining pressure isotherm. Here, we focus on the absolutevalue of the line tension and a possible comparison with experimental data.

For this purpose, let us consider the line tension in the simplest possible case:contact of two identical cylindrical drops or bubbles. In this case, the correspond-ing excess free energies given by Equation 2.194 and Equation 2.197 take thefollowing form:

(2.203)

20 0

γ θ θ τ τ⋅ −( ) = + ∂∂

∞cos cose e r r

2 22

γ θ γcos ( )e

h

h dh

e

∞

∞

= + ∫ Π

r h h dh P h hi d

h

i d e

e

2 1 1 22

2

γ + ′( ) −

− ( ) + ⋅ −(∞

∫ Π ))

+ =

+ ′ −( ) +

∞

∫ dr r

r h

r0

0

22 1 1

τ

γ (Π hh dh h dh P h h drh h

e

e

) ( ) ( )2 2

2

∞ ∞

∫ ∫− + ⋅ −

Π ..0

∞

∫

Φ Π Π= + ′ −( ) + − + ⋅∞ ∞

∫ ∫2 1 1 22

2 2

γ h h dh h dh P hh h

e

e

( ) ( ) ( −−

∞

∫ h dxe) ,0



and

, (2.204)

where Φ is an excess free energy per unit length.From Equation 2.203, we conclude that

. (2.205)

Equation 2.205 describes the whole range of the liquid profile, including thelower bulk part of the drop or bubble, the thin flat liquid interlayer in front of it,and the transition zone in between. The boundary conditions for Equation 2.205are

, (2.206)

(2.207)

We can integrate Equation 2.205 using the boundary condition (2.206), whichyields:

, (2.208)

where

.

Equation 2.208 can now be rewritten as:

. (2.209)

Φ Π= + ′( ) −

− ( ) + ⋅ −∞

∫2 1 1 22

2

γ h h dh P h hi d

h

e i d

e

ee

x

dx( )

+∞

∫0

τ

γ ⋅ ′′

+ ′( )

+ ( ) =h

h

h P

1

22

3

2

Π

h R H h R( ) = ′ ( ) = −∞,

h h x

h x

e

h he

→ → ∞

′ = → ∞=

,

, .0

γ

12

1

2+ ′( )

=

h

L he ( )

L h P H h h dhe e

h

( ) = ⋅ −( ) −∞

∫ Π ( )2

′ = − −hL he

γ 2

21

( )



The left-hand side of Equation 2.208 is always positive and less than γ. Thatmeans, the same should be true for the right-hand side of Equation 2.208:

(2.210)

where andCondition (2.207) results in:

(2.211)

The capillary pressure can be expressed as before:

, (2.212)

where is the radius of the curvature of the cylindrical drop. Simple geometricalconsiderations show that

. (2.213)

With the help of the preceding condition, we can conclude that

. (2.214)

Using Equation 2.211 and Equation 2.214, we conclude:

, (2.215)

(2.216)

0 ≤ ( ) ≤L h γ,

L h h he( ) ,= =γ if L h h H( ) , .= =0 if

P H h h dhe e

he

⋅ −( ) = +∞

∫γ Π ( ) .2

PRe = γ

R

RH

e

=cos θ

PHe

e= ⋅γ θcos

cos

( )

θγ

eh

e

h dh

h

H

e=

+ ⋅

−

∞

∫11

2

1

2

Π

γ θ⋅ = ( )cos eeH

hΠ 2



The two latter equations are modifications of our previous consideration in thecase of drops/menisci (see Section 2.3). If he /H << 1, the contact angle inEquation 2.215 is referred to as θe∞ and coincides with that given by Equation2.201.

In the case of partial wetting, the contact angle is in the following range:

,

or 0 < cos θe < 1. Using condition (2.210) the latter inequality can be rewritten as

Hence, the integral,

,

should be negative in the case of partial wetting.In the case of the ideal profile, we should use Equation 2.204, which results in

. (2.217)

The boundary conditions for Equation 2.217 are

(2.218)

Equation 2.217 can be integrated using boundary conditions (2.218), whichyields:

(2.219)

where

02

≤ ≤θ πe

− < ( ) < −⋅

∞

∫γγ

Π h dhh

Hhe

e

2

.

Π h dhhe

( )∞

∫2

γ ⋅ ′′

+ ′( )

=h

h

Pi d

i d

e

12

3

2

h R H h Ri d i d( ) = ′ ( ) = −∞, .

′ = − −hL

i did

γ 2

21,

L h P H hid id id( ) = −( ).



Line tension, τ, can be expressed using Equation 2.203 and Equation 2.204 as

Using Equation 2.209 and Equation 2.219, we can rewrite the precedingequation as

(2.220)

Using Equation 2.209 and Equation 2.219, we can switch from the integrationover x to integration over the thickness, h. The latter transformation of Equation2.220 gives

(2.221)

Note that the integration on the right-hand side is over h from he to infinity,which does not depend on the profile (real or ideal profile) but only on theintegration limits.

Equation 2.221 can be rewritten as

(2.222)

In the case of h << H inside the whole transition zone, expressions for L(h)and Lid (h) can be rewritten as

τ γ= + ′ −( ) + − + ⋅ −∞ ∞

∫ ∫2 1 1 22

2 2

h h dh h dh P hh he

Π Π( ) ( ) ( hh dx

h

e

i d

)

− + ′( ) −

∞

∫0

22 1 1γ

− ( ) + ⋅ −( )

∞

∫ Π h dh P h h dxh

i d e

e2

2xx0

∞

∫

τ γ γ= + ′ −

− + ′( ) −∞

∫2 1 2 12

0

2h L h dx h L hi d id( ) ( iid

x

dx) .

∞

∫0

τ γ γ= − − −( )∞

∫2 2 2 2 2L h L h dhid

he

( ) ( ) .

τγ

=

− −

−

∞∞

∫∫2

2

1

2

2

2

P H h h dh h dhhh

( ) ( ) ( )Π Π

LL h L hdh

idhe

2 2 2 21( ) ( ).

/ /γ γ+ −

∞

∫

L h h h h dh

L h

e

h

id

( ) cos ( ) , ( ) ( )

( )

= −( ) =∞

∞

∫γ θ ε εγ1

2

Π

== ∞γ θcos e



Using the latter expression, Equation 2.222 can be rewritten as

(2.223)

In the case of small contact angles, ε(h) << 1, and the latter equation takesthe following form:

(2.224)

It is possible to show that

in the case under consideration. Let the mean value of the latter expression be ω,where 0.5 < ω < 1; then Equation 2.224 takes the following form:

(2.225)

Equation 2.201 can be rewritten as , or

(2.226)

Combining the preceding expression and Equation 2.225 results in

.

(2.227)

τ γθ

θ ε ε

θ ε= −

+ + −∞

∞

∞

2 2

1 12

2

sincos ( ) ( )

cos ( )e

e

e

h h

h εεθ

2

2

( )

sin

.h

dh

e

he

∞

∞

∫

τ γθ

εθ ε

θ

=+ +∞ ∞

∞

∞

∫4

1 12

2

tan( )

cos ( )

sin

.e e

e

h

h

hdh

e

0 51

1 12

1

2

.cos ( )

sin

<+ +

<∞

∞

θ εθe

e

h

τ ωθ

ωθ

=

≈∞

∞∞

∞∫∫4 4

22tan

( ) (e

hhe

h dh dh

e

Π Π hh dh dhhhe

) .22

∞∞

∫∫

cos ( )θ εe eh∞ = +1

sin ( ) ( ) .θ θ εγe e e

h

h h dh

e

∞ ∞

∞

≈ = − = − ∫2 21

2

Π

τ ω

γ

=

−

∞

∞∞

∫∫∫2

1

2

22Π

Π

( )

( )

h dh

h dh dh

h

hh

e

e

==

−

∞

∞∞

∫∫∫2

2

22

ω γ

Π

Π

( )

( )

h dh

h dh dh

h

hh

e

e



We can compare the experimental data by Platikanov et al. [26] to the theorypredictions according to Equation 2.227 (at ω = 1).

We have used only the dispersion and electrostatic components of the dis-joining pressure. The following expressions for the different components of thedisjoining pressure are used:

for the dispersion component:

, (2.228)

where A ≈ 10–14 dyn·cm is the Hamaker constant and for the electrostatic component:

(2.229)

where R, T, F,

,

are the concentration of the electrolyte, the universal gas constant, temperaturein °K, Faraday number, dimensionless zeta potential of the film surfaces, and theinverse Debye length, respectively; εw is the dielectric constant of water. Hence,the total disjoining pressure is

(2.230)

Unfortunately, the disjoining pressure in the form given by Equation 2.230does not allow any equilibrium liquid films at low thickness (α-films). To over-come this problem, we introduce a cutoff thickness, t* (Figure 2.39).

According to this choice, the equilibrium thickness 2he does not depend onthe pressure inside the drops or bubbles and is always equal to t*.

Platikanov et al. [26] have determined both contact angle and line tension onthe NaCl concentration in the range of concentrations 0.2–0.45 mol/l. These twodependencies are discussed in the following section.

Πd

A

h= −

3

Πel c R T h= ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅64 4tanh( ) exp( ),ψ κ/

ψ ψ= F

RT

κ πε

= 8 2F c

RTw

Π( ) tanh( ) exp( ).hA

hc R T h= − + ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅

364 4ψ κ/



COMPARISON WITH EXPERIMENTAL DATA AND DISCUSSION

Details of the experimental measurement and system under consideration aregiven in Reference 26. Line tension of the free liquid film between two bubbleswas investigated in the range of NaCl concentration 0.2–0.45 mol/l. The film andbubble surfaces were stabilized by surfactants [26]. The zeta potential of the filmsurface was ψ = 17 mV or according to Reference 26.

The cutoff thickness, t*, was used as a fitting parameter. We used experimentalvalues of the contact angle from Reference 26 on salt concentration to determinethe cutoff thickness t*, according to Equation 2.226. A reasonable agreementbetween experimental dependency of the contact angle on the salt concentrationand the calculated one according to Equation 2.226 has been attained. The fitteddependency of the contact angle on the salt concentration was much weaker thanthe original experimental data [26]. However, we tried to compare our calculationof line tension according to Equation 2.227 and the corresponding experimentaldata of line tension from Reference 26, using the already calculated cutoff thick-ness, t*. The calculated dependency of line tension on the electrolyte concentrationis shown in Figure 2.40.

The following conclusion can be made, based on the consideration of thedependency presented in Figure 2.40:

• The line tension dependency on the salt concentration is in a qualitativeagreement with experimental dependency in Reference 26, that is, linetension goes from positive to negative values with an increase in theelectrolyte concentration.

• The absolute values of the calculated line tension were found consid-erably different from the corresponding experimental values; the cal-culated electrolyte concentration (0.022 mol/l) at which line tension

FIGURE 2.39 Model disjoining pressure isotherm according to Equation 2.230 withcutoff thickness t*.

ht∗

Π

ψ = 0 68.



switches from positive to negative values does not match the experi-mental value (0.36 mol/l).

• The calculated line tension remains almost constant in the range ofelectrolyte concentrations above 0.1 mol/l used in Reference 26.

• The line tension decreases much faster with the electrolyte concentra-tion (Figure 2.40) than experimental values [26].

The discrepancy between the measured and calculated line tension depen-dencies can be caused by one or both of the following reasons:

• Only the dispersion and electrostatic components of disjoining pressurehave been used to compare with the experimental data. It looks likethese two components are not enough to adequately describe the behav-ior of thin liquid films and the transition region in the system underconsideration. The influence of both the structural (caused by theorientation of water dipoles in the vicinity of free film surfaces) andsteric (caused by the direct interaction of the head of the surfactantmolecules on the film surfaces) components cannot be ignored. Thetheory of these components of disjoining pressure is to be developed.

• According to the definition of line tension (2.222), it is determined bythe equilibrium liquid profile in the transition region from the thin flatliquid interlayer to the bulk surface of bubbles. Note that in the caseof partial wetting, which is under consideration, the static hysteresisof contact angle (see Chapter 3, Section 3.10) is unavoidable. The latterphenomenon can substantially influence the comparison.

In Reference 26, considerable efforts have been made to reach the equilibriumstate. We would like to emphasize again: if the liquid profile in the transitionregion from a thin liquid interlayer to the bulk drop or bubble interface is not at

FIGURE 2.40 Calculated dependency of line tension in electrolyte concentration.

Electrolyte concentration, mol/l

Line

tens

ion

τ ∗ 10

7 , dyn

10

8

6

4

2

0

–2

0.1 0.2 0.3 0.4 0.5



equilibrium, then values of line tensions can differ considerably from the theo-retically predicted equilibrium values.

In the case of quasi-equilibrium (no macroscopic motion but possible micro-scopic motion inside the transition zone, see Chapter 3, Section 3.10), it isnecessary to introduce a new dynamic line tension. We believe that this is achallenging area that requires further research.

2.11 CAPILLARY INTERACTION BETWEEN SOLID BODIES

In this section, we shall consider the capillary interaction between two solid platespartly immersed in liquid, which can be, one or both, completely wetted, partiallywetted, or nonwetted by the liquid. We shall derive an expression for the forceof interaction between them at large separation distances. We shall show that ifone of the plates is wetted and the other is not, then a critical separation existssuch that below this separation, the plates attract each other, whereas there is arepulsion at larger separations. Similarly, we shall see that there is a critical angleof relative inclination that also delineates the regions of attraction and repulsion

advancing or static receding, depending on the way the system is placed.First we calculate the force of interaction between the plates at large separa-

tions, L >> a, where is the capillary length, γ is the surface tension,ρ is the density of the liquid, and g is the gravity acceleration. We consider onlythe case of two partially (or complete) wettable plates, that is, θ1 < π/2, θ2 < π/2,as the case of two nonwettable plates, θ1 > π/2, θ2 > π/2, can be treated in asimilar manner.

Note that in the case of partial or complete wetting, the height of the liquidbetween the plates is higher than the height outside the plates, that is, h1 > H1

and h2 > H2 (Figure 2.41).

FIGURE 2.41 Liquid profile between two partially wettable plates that are partiallyimmersed into the liquid. Different contact angles, θ1 ≠ θ2.

a g= γ ρ/

21

h1

h2 H2

H1

xm xL0

θ1

θ1 θ2θ2


[37]. We shall not specify the kind of contact angle as it can be either static


We use the principle of the frozen state [27] to calculate the force of inter-action between the plates. We imagine that the liquid between plates 1 and 2(Figure 2.41) has solidified above the level of the free liquid surface. We dividethe resulting solid body by a plane parallel to plates 1 and 2, passing throughx = xm, where

.

Displacement of plate 2 through dL while plate 1 is fixed, changes the excessfree energy of the system, Φ, by

where l is the width of the plates. As dΦ = FdL, where F is the force of attractionbetween the plates, we get

. (2.231)

According to the previous consideration, the shape of a liquid surface platesin the gravity field can be described by the following equation:

or

Close to the position xm, the liquid profile has a low slope, that is, h′2 << 1,and hence, in the vicinity of this position, the shape of the liquid surface can bedescribed by the linearized equation when close to x = xm,

(2.232)

The condition

dh

dxx xm=

= 0

dgl

h dLmΦ = ρ2

2 ,

Fg

lhm= ρ2

2

γ ρ′′

+ ′( )=h

hgh

1 23 2/

,

′′

+ ′( )=h

h

h

a1 23 2 2/

.

′′ =h h a/ 2,

h x c x a c xp L x a( ) = −( ) + − −( )( )1 2exp ./

dh

dxx xm=

= 0



results in

(2.233)

where hm = h(xm).The exact solution for the shape of a liquid surface close to an isolated plate

is deduced in Appendix 2:

(A2.8)

Using the latter expression at x >> a, we get where

Substituting ci = c∞(θi), i = 1,2 from the preceding equation into Equation2.232 and Equation 2.233 gives an expression for the force of interaction betweenthe two plates at L >> a:

(2.234)

In the case of two identical plates (θ1 = θ2 = θ), this expression results in

(2.234’)

and in the case of complete wetting (θ = 0),

(2.234’’)

h c c L am2

1 24= −( )exp ,/

xa

h

a=

− +( ) + −

+ +( )2

2 1 1 14

2 1

2

2

ln

sin

sin

θ

θ 11 14

2 1 22

2

2

2

2

− −

+ + − −

h

a

ah

asinθ

h x c x a( ) ( )exp( ),≈ −∞ θ /

c a∞ = − ++ +

( )sin

sin.θ θ

θ4

2 1

2 1

F ga l=− ++ +

− ++ +

322 1

2 1

2 1

2 12 1

1

2ρθθ

θsin

sin·

sin

sinn,/

θ2

e L a−

F ga l e L a= − ++ +

−322 1

2 12ρ θ

θsin

sin/

F ga le L a= −5 5 2. /ρ



Equation 2.234’ and Equation 2.234’’ show that two partially wettable platesattract each other, and the force of attraction between these plates decays expo-nentially at L >> a.

We can similarly calculate the force of interaction between any two bodiesfor which the surface between the bodies, along which the height of capillaryrise of the liquid has a minimum value, is a plane. Examples are: two identicalspherical or cylindrical particles. In Reference 28, the profile of a liquid close topartially immersed cylindrical plates has been calculated. Using this expressionfor the shape of the surface of the meniscus at a vertical cylinder of radius R atlarge separations, we get the following expression:

(2.235)

where K0 is a cylindrical function. The latter expression shows that the forcebetween two partially wetted cylinders is also that of attraction, and this forcedecays faster compared to the case of the two plates.

In the following text is an expression deduced in Appendix 2 for the height ofcapillary rise of the meniscus at the plates on the free liquid side at θ = θi, i = 1, 2:

. (A2.5)

We now consider the interaction between wettable and nonwettable plates.Let there be an isolated plate 1 with contact angle θ1 < π/2. At some distancesLk, we draw an intersecting plate 2 (Figure 2.42).

FIGURE 2.42 Capillary interaction between wettable and nonwettable plates.

F ga R K x d x L a

ga

L a

= ( ) −

≈

∞

∫4 42 202 2 2 2

2

3 2

ρ θ

π ρ

cos/

/

/

22 2R

L aL a

cos·exp ,

θ/

/−{ }

H ai i= −( )2 1 sin θ

θ2

θ2θ1 θ1

0 xLk

1 2



The angle between the surface of the unperturbed meniscus and the inter-secting plate θ2 > πI2 (Figure 2.42). If we replace the intersecting plate by a realplate with contact angle θ2, the surface of the meniscus between the plates willobviously be unchanged; the meniscus at the second plate on the side of the freeliquid surface will sink to a height H2, according to Equation A2.5. Thus, thereare no forces acting along the x-axis at plates l and 2. We denote the distance atwhich the angle between the intersecting plate and the meniscus is equal to thewetting angle θ2 as Lk (θ1, θ2). It is easy to check whether Lk(θ1, θ2) exists only if

θ1 + θ2 < π, θ1 < , θ2 > . (2.236)

If θ1 + θ2 → π , Lk → 0, and if θ2 → π/2, Lk → ∞. If the condition θ1 + θ2 < πholds, we can easily determine Lk(θ1, θ2) by substituting

into the equation for the unperturbed meniscus at plate 1 (A2.8), which results in

(2.237)

The angle between the meniscus and the intersecting plate when L < Lk, isgreater than θ2, and a consequent displacement of plate 2 from the position L =Lk toward L < Lk, causes the liquid to rise between the plates and thus sets upan attraction between the two plates. Similarly, a displacement of plate 2 fromthe position L = Lk toward L > Lk causes the liquid to fall between the plates andconsequently sets up a repulsion. Thus, L = Lk(θ1, θ2) delineates two regions ofinteraction between the plates: attraction when L < Lk, and repulsion when L >Lk, i.e., L = Lk is a state of unstable equilibrium.

Important conclusion: according to condition (2.236), any completely wetta-ble plate on a liquid surface is attracted to a nonwettable plate at sufficientlysmall distances.

At separations L >> a (when L > Lk), between plates, one of which is partiallywetted whereas the other is not, the liquid surface between them must intersectat the level h = 0.

Using the principle of the frozen state, the force of interaction between theplates in this case is

π2

π2

H a2 22 1= −( )sin θ

La

k θ θθ θ

1 2

1 2

2

2 1 2 1

2 1, ln

sin sin

sin( ) =

− +( ) + +( )+ + θθ θ

θ θ

1 2

1 2

2 1

2 1 1

( ) − +( )+ + − +( )

sin

sin sin .a



where α = arctan h′(x0) and h(x0) = 0. Close to x = x0, the liquid profile, h(x),as in the preceding case becomes when

and for the interaction force, F, we get the following expression:

.

As before, by expressing c1, c2 from the equations for unperturbed surfaces, wefinally get

. (2.238)

Note that the latter force is a repulsive force, which is completely differentfrom the case of two partially wettable plates, according to Equation 2.234.

Now let us place a nonwettable plate 2 at an arbitrary position L > Lk at anangle α(L) > 0 (Figure 2.43), choosing α(L) so that the meniscus at the pointx = L – ∆ would not be disturbed by the presence of plate 2.

FIGURE 2.43 Critical inclination of plates.

F l= − −γ α( cos ),1

h c x a c L x a= − − − −1 2exp( ) exp( ( ) ),/ /

cos expα =+ ′ ( )

≈ − ′ ( ) = −1

11

12

12

20

20

1 22

2

h xh x

c c

aL/aa( ),

Flc c

ae L a= − −2 1 2

2

γ /

F l= −− +

+ +⋅

− +

+ +32

2 1

2 1

2 1

2 1

1

1

2γθ

θ

θ

θ

sin

sin

sin

sin 22

e L a− /

2

θ2

θ2α

L – ∆x

1



Figure 2.43 shows that the angle α(L) has to be selected as

(2.239)

where ∆, according to Figure 2.43, is given by

(2.240)

At L > Lk and L >> a, h′ → 0, and hence, according to Equation 2.239,

(2.241)

Thus, there are no forces acting on plate 1 when plate 2 is positioned at anangle αk; this applies equally to plate 2. Inclination of plate 2 at an angle smallerthan α(L) causes the liquid to rise between the plates, i.e., sets up an attractionbetween them, but inclination at an angle bigger than α(L) causes the liquid tofall between plates, i.e., sets up a repulsion between the two plates. Thus, theangle α(L) delineates the regions of attraction and repulsion between the plates,defining a state of unstable equilibrium.

Equation 2.239 and Equation 2.240 can be used for numerical calculation ofthe unknown wetting angle of plate 2 at all separations, whereas Equation 2.241can be used for large separations only.

In conclusion, we note that positioning two nonwettable plates at an angle β =θ2 – π /2 from the vertical (Figure 2.44a) will constrain the liquid to a horizontalposition between them, i.e., the liquid between the plates will form a convexmeniscus when the inclination angle is smaller than β, and a concave meniscuswhen the inclination angle is bigger than βk. That is, in the latter case, the liquidwill rise between the plates, although it does not wet either plate.

Similarly, positioning wettable plates at an angle δ = π/2 – θ1 (Figure 2.44b)will constrain the liquid to a horizontal surface between them, i.e., when theinclination angle is bigger than δ, the liquid will sink between wettable plates,forming a convex meniscus, although it wets both plates.

The latter consideration along with Figure 2.44a and Figure 2.44b exemplifythe fact that capillary imbibition of a liquid into a porous body significantlydepends on the angle of opening of the pores and its change along the pore axis,i.e., the pore distribution function with respect to a derivative of the radius [29].

α π θL h L( ) = − − ′ −( )2 arctan ,∆

∆ ∆= ( ) + − ′ −( )

cotanα π

L a h L2 12

sin arctan

= ( ) ++ ′ −( )

.

cotanα L ah L

2 11

1 2 ∆

α π θ( ) .∞ = − 2



The liquid cannot move beyond those places where the meniscus in the porebecomes flat, i.e, where the effective pore radius is equal to infinity. In particular,the latter phenomenon is one of the reasons of the capillary hysteresis in porousbodies and the presence of trapped air.

In summary, we have shown that the capillary attraction or repulsion betweensolid bodies depends not only on their wetting features but also on their separationand on the mutual angle of relative inclination.

FIGURE 2.44 (a) Capillary rise between nonwettable plates. (b) Capillary interactionbetween wettable plates at inclination.

θ2θ2

θ2θ2

β

x

(a)

θ1

θ1θ1

(b)

x

γ

θ1



APPENDIX 2

Equilibrium Liquid Shape Close to a Vertical Plate

Let us consider a vertical plate partially immersed in a liquid of density, ρ, andsurface tension, γ.

In this case, only the capillary and gravity forces act, and the equation thatdescribes the liquid profile is as follows:

, (A2.1)

where g is the gravity acceleration. Equation A2.1 is the differential equation ofthe second order; hence, the two boundary conditions should be specified. Theseconditions are

(A2.2)

(A2.3)

Condition (A2.3) is used as

. (A2.3’)

Using the capillary length

,

Equation A2.1 can be rewritten as

. (A2.1’)

Multiplication of the latter equation by and integration with x results in

,

γ ρ′′

+ ′( )=h

hgh

1 23 2/

′ = −h an( ) cot ,0 θ

h x x( ) , .→ → ∞0

′ ==

hh 0

0

ag

= γρ

′′

+ ′( )=h

h

h

a1 23 2 2/

′h

1

1 221 2

2

2+ ′( )

= −h

Ch

a/



where C is an integration constant. Using the boundary condition (A2.3’), weconclude that C = 1 and

. (A2.4)

The left-hand side of the preceding equation is positive and so should be the

right-hand side, which results in the restriction . We further see that

H = corresponds to the maximum possible elevation (Figure 2.45) in the

case of complete wetting, that is, at θ = 0.

Using the boundary condition (A2.2), we conclude from Equation A2.4 that

, (A2.5)

which gives H = in the case of complete wetting.From Equation A2.4, we conclude:

, (A2.6)

with boundary condition

(A2.7)

FIGURE 2.45 Liquid profile close to the vertical wall. θ is the contact angle, H is themaximum elevation.

x

θ

H

h(x)

1

11

221 2

2

2+ ′( )

= −h

h

a/

h a≤ 2

a 2

H a= −2 1( sin )θ

a 2

′ = −−

−

hh

h

a

ah

a

22

2 12

2

2

2

2

h H a( ) ( sin ).0 2 1= = − θ



Integration of Equation A2.6 with the boundary condition (A2.7) results in

,

or

. (A2.8)

Equation A2.8 gives an implicit dependence of the liquid profile h on x.

2.12 LIQUID PROFILES ON CURVED INTERFACES, EFFECTIVE DISJOINING PRESSURE. EQUILIBRIUM CONTACT ANGLES OF DROPLETS ON OUTER/INNER CYLINDRICAL SURFACES AND MENISCI INSIDE CYLINDRICAL CAPILLARY

In this section, we shall deduce effective disjoining pressure isotherms for liquidfilms of uniform thickness on inner and outer cylindrical surfaces and on thesurface of spherical particles. This effective disjoining pressure is expected todepend on the surface curvature. From its expression, we shall be able to calculatethe equilibrium contact angles of drops on the outer surface of a cylinder andmenisci inside cylindrical capillaries. We shall see that the contact angle is almostindependent of liquid geometry. However, there are differences in the expressionsfor equilibrium contact angles according to geometry, in view of the differencein thickness of films of uniform thickness with which the bulk liquid (drops ormenisci) is at equilibrium. The latter thickness determines the lower limit of theintegral in the expression for the equilibrium contact angle.

LIQUID PROFILES ON CURVED SURFACE: DERIVATION OF GOVERNING EQUATIONS

Excess free energy, Φ, of the liquid droplet on an outer surface of a cylindricalcapillary of radius a is as follows:

− =−

−∫x

a

h

a

hh

a

dhH

h 12

14

2

2

2

2

xa

h

a=

− +( ) + −

+ +(12

2 1 1 14

2 1

2

2

ln

sin

sin

θ

θ )) − −

+ + − −

1 14

2 1 22

2

2

2

2

h

a

h

asinθ



where, as before, we selected a reference state as the outer surface of the cylin-drical capillary covered by a the equilibrium liquid film of the thickness he; x isin the direction parallel to the cylinder axis (Figure 2.46).

The condition of equilibrium according to Section 2.2 (condition (1)) resultsin the following equation describing the liquid profile on the surface of thecylindrical capillary:

. (2.243)

Note that the latter equation is different from Equation 2.23 in Section 2.2not only due to the presence of the second curvature,

,

but also due to a difference in the definition of the disjoining pressure, which isnow

instead of in Equation 2.23, Section 2.2. This difference results in substan-tial consequences as shown in Section 2.7.

FIGURE 2.46 Cross section of an axisymmetric liquid droplet on the outer surface of acylinder of radius a. H is the maximum height of the droplet, he is the thickness of anequilibrium film of uniform thickness.

Φ = +( ) + ′ − +( )

+ + −( )2 1 2 2 2πγ πa h h a h P a h ae e ( ) −− + −( )

{

+ −

∞

∞

∫

∫

( )

( )

a h a

a h dh

e

h

2 2

0

2π Π Π(( )h dh dxhe

∞

∫

(2.242)

γ γ′′

+ ′( )−

+( ) + ′+

+=h

h a h h

a

a hh Pe

1 123 2 2/

( )Π

−+( ) + ′

γ

a h h1 2

a

a hh

+Π( )

Π( )h

ahe θe H

x



Note that the excess pressure, Pe, is determined by the vapor pressure in thesurrounding air and given by Equation 2.2 (Section 2.1). In the case of dropletson the cylindrical capillary, as in the case of droplets on flat solid substrates, theequilibrium is possible only at oversaturation, that is, at Pe < 0.

For the equilibrium film of uniform thickness, he, we conclude from Equation2.243:

(2.244)

Let us introduce the effective disjoining pressure as

(2.245)

In the following text, we show that the introduced effective disjoining pressureprovides the correct stability condition. For that purpose, we consider the excessfree energy per unit length of the capillary, Φe, of the equilibrium film of a uniformthickness, he, which is,

.

(2.246)

According to the requirements of equilibrium, the following conditions shouldbe satisfied:

, (2.247)

. (2.248)

The first condition (2.247) of the equilibrium results in

.

The latter equation can be rewritten using the definition of the effectivedisjoining pressure given by Equation 2.245 as

−+( ) +

+=γ

a h

a

a hh P

e ee eΠ( ) .

Π Πeff ha h

a

a hh( ) ( ).= −

+( ) ++

γ

Φ Πe e e ea h P a h a a h dh= + + +( ) −

+ +2 22 2πγ π π( ) ( ) 22π γ γa sl sv

he

( )−∞

∫

d

dhe

e

Φ = 0

d

dhe

e

2

20

Φ >

γ + + − =P a h a he e e( ) ( )Π 0



(2.249)

The second condition (2.248) gives

. (2.250)

Let us check whether the effective disjoining pressure isotherm that is intro-duced according to Equation 2.245 satisfies the stability condition given byEquation 2.250. Indeed:

.

Substituting the expression for Π(he) from Equation 2.249, we obtain:

according to condition (2.250).Hence, we conclude

, (2.251)

which means that the effective disjoining pressure introduced according to Equa-tion 2.245 will possess all the necessary properties according to Equation 2.249and Equation 2.251. Just this effective disjoining pressure isotherm is used inSection 2.7 for the investigation of stability of uniform liquid films on cylindricalsurfaces.

In the case of a uniform film on a spherical particle of radius, a, we get anexpression for the excess free energy similar to the one given by Equation 2.244:

(2.252)

Πeff e eh P( ) =

P a he e− ′ >Π ( ) 0

d

dh

a

a hh

a

a

a hheff

ee

ee

e

ΠΠ Π= −

+( )−

+

+′

2( ) (

γ))

d

dh

a

a h aP

a h

a aeff

ee

ee

Π= −

+( )+ +

−

2

γ γ +++

′

=+

′ − <

a

a hh

a ha h P

ee

ee e

Π

Π

( )

( )1

0

d

dheff

e

Π< 0

Φ

Π

e e e ea h P a h a

a

= + + +( ) −

+

443

4

2 3 3

2

πγ π

π

( )

(( ) ( ).h dh a sl sv

he

+ −∞

∫ 4 2π γ γ



Let us introduce the effective disjoining pressure isotherm in this case as

. (2.253)

The same procedure as shown in the preceding section results in two equi-librium conditions (2.249) and (2.251). This means that the effective disjoiningpressure isotherm defined according to Equation 2.253 really describes the sta-bility of uniform films on spherical particles.

In the case of liquid layers inside the inner part of the capillary of radius a,we have the following expression for the excess free energy:

which is similar to the expression for the excess free energy on the outer cylin-drical surface Equation 2.242 (Figure 2.47).

Exactly in the same way as in the case of the outer cylindrical surface, wededuce the following equation for the liquid profile:

. (2.255)

Note again that the resulting equation (Equation 2.255) is different from boththe corresponding Equation 2.243 (liquid on the outer cylindrical surface) andEquation 2.23 in Section 2.2 (for a flat surface).

FIGURE 2.47 Profile of a meniscus in a cylindrical capillary of radius a. 1 — a sphericalpart of the meniscus of curvature Pe, 2 — transition zone between the spherical meniscusand flat films in front, 3 — flat equilibrium liquid film of thickness he.

Π Πeff ha

a hh

a h( ) ( )=

+( )−

+

2

2

2γ

Φ = −( ) + ′ − −( )

+ − −( )2 1 2 2 2πγ πa h h a h P a a he e ( ) −− − −( )

{

+

∞

∫ a a h

a h d

e2 2

0

2

( )

( )π Π hh h dh dxh he

−

∞ ∞

∫ ∫ Π( ) , (2.254)

γ γ′′

+ ′( )+

−( ) + ′+

−=h

h a h h

a

a hh Pe

1 123 2 2/

( )Π

x

2a

he 3 2 1

Peθe



Let us introduce the effective disjoining isotherm in the latter case as

. (2.256)

The corresponding expression for the excess free energy per unit length of auniform film on the inner cylindrical surface is

(2.257)

The latter expression and the definition given by Equation 2.256 result in theconditions (2.249) and (2.251), which describe the stability of a film of uniformthickness on the inner surface of a cylindrical capillary.

EQUILIBRIUM CONTACT ANGLE OF A DROPLET ON AN OUTER SURFACE OF CYLINDRICAL CAPILLARIES

The droplet profile is described by Equation 2.243. Let H be the maximum heightof the droplet in the center, that is, . Let us introduce a new unknownfunction

in this equation and integrate Equation 2.243, which results in

, (2.258)

where condition is taken into account. If we neglect the disjoiningpressure on the right-hand side of Equation 2.258, we get the “outer solution,”which describes the drop profile not distorted by the disjoining pressure action:

. (2.259)

Π Πeff ha

a hh

a h( ) ( )=

−+

−( )γ

Φ Πe e e ea h P a a h a h dh= − + − −( )

+ +2 22 2πγ π π( ) ( ) 22π γ γa sl sv

he

( )−∞

∫

h H( )0 =

uh

=+ ′

1

1 2

1

11

2 2

2

2 2

+ ′= +

+ − −

−

∞

∫h

PH

aH ahh

a h dhe

h

Π( )

(γ aa h+ )

′ =h H( ) 0

1

11

2 2

2

2 2

+ ′= +

+ − −

+h

PH

aH ahh

a h

e

γ ( )



If we continue the outer solution to the intersection with the surface of thecylinder, we get h′(0) = –tan θe, where θe is the equilibrium contact angle to bedetermined. Using this condition in the outer solution obtained from Equation2.259, we conclude:

,

or

. (2.260)

The preceding expression shows that the equilibrium droplets on the outersurface of a cylinder can be at equilibrium only at oversaturation as droplets ona flat substrate.

From the whole of Equation 2.258, we conclude that the local profile tendsasymptotically to the film of the uniform thickness, he. Therefore, locally, theprofile satisfies the condition . Using this condition, we conclude fromEquation 2.258:

,

or

, (2.261)

where the equilibrium thickness of the uniform film is determined from thefollowing equation:

. (2.262)

Substitution of Equation 2.260 into Equation 2.261 results in the followingequation for the determination of the equilibrium contact angle,

cos θγe

ePH

aH

a= +

+

1

2

2

Pa

H aHe

e=−( )

+<

cos θ γ1 2

20

2

′ =h he( ) 0

02 2

2 2

=

+ − −

−

+

∞

∫PH

aH ahh

a h dh

a h

e ee

h

e

e

Π( )

(γ ))

PH

aH ahh

a h dhe ee

he

2 2

2 20+ − −

− =

∞

∫ Π( )

Π Πeff ee e

e eha h

a

a hh P( ) ( )= −

+( ) ++

=γ



. (2.263)

If we omit the small terms,

,

in Equation 2.263, we arrive at

. (2.264)

This form of the preceding equation is identical to Equation 2.47 (meniscusin a flat capillary) and Equation 2.55 (droplet on a flat substrate) deduced inSection 2.3. However, there are substantial differences between Equation 2.47,Equation 2.55, and Equation 2.264: the lower limit of integration in these equa-tions, which corresponds to the thickness of the uniform film, is substantiallydifferent in each of them.

EQUILIBRIUM CONTACT ANGLE OF A MENISCUS INSIDE CYLINDRICAL CAPILLARIES

In this case, the meniscus profile is described by Equation 2.255. We introducea new unknown function,

as in the case of a droplet on a cylindrical surface. After an integration, Equation2.255 takes the form

, (2.265)

where we already take into account the condition in the center of the capillary,. If we neglect the disjoining pressure in the latter equation, we arrive at

cos ( )θγe

e e hah h

H aH

h dh

e

= +− +

+

∞

∫11

12

2

12

2

Π

2

2

2

2

ah h

H aHe e+

+

cos ( )θγe

h

h dh

e

≈ +∞

∫11 Π

uh

=+ ′

1

1 2

1

1

2

2

2

+ ′=

−( ) −

−

∞

∫h

Pa h a h dh

a h

e

h

Π( )

( )γ

′ = −∞h a( )



, (2.266)

which describes the spherical meniscus profile. Continuation of this profile to theintersection with the capillary surface results in a manner similar to the previouscase of droplets:

,

or

, (2.267)

as expected.Now, from the whole of Equation 2.265, we conclude that the local profile

tends asymptotically to the film of uniform thickness, he. That is, locally, theprofile satisfies the condition h′(he) = 0. Using this condition and Equation 2.267,we conclude from Equation 2.265 that

, (2.268)

where the thickness of the uniform film is determined from

. (2.269)

If we omit small terms such as

in Equation 2.268, we arrive at the same functional dependency of cosθe as earlier(Equation 2.264, Equation 2.47, and Equation 2.55 in Section 2.3).

Let us insist on the significant difference (discussed in Section 2.3) betweenthe equilibrium of drops and menisci: in the case of drops (no difference with a

1

1 22+ ′=

−( )h

P a he

γ

cos θγe

eP a=2

Pae

e= 2γ θcos

cos ( )θγe

ee h

h

ah

a

h dh

e

=−

+

−

∞

∫1

1

1

1

12

Π

Π Πeff ee

ee

eha

a hh

a hP( ) ( )=

−+

−( ) =γ

h

ae << 1



flat surface or outer cylindrical surface), the external supersaturated vapor pres-sure in the ambient air may be arbitrary inside the narrow limits determined inSection 2.3, Equation 2.52. The drop size adjusts to the imposed pressure atequilibrium. The situation is very different in the case of an equilibrium meniscuson a capillary (no difference between a flat chamber or a cylindrical capillary)Here, there is only a single vapor pressure allowing the equilibrium of themeniscus. At all other vapor pressures, the meniscus cannot be at equilibrium.

REFERENCES

1. Derjaguin, B. V., Churaev, N. V., and Muller, V. M. Surface Forces, Plenum Press,New York, 1987.

2. Deryagin, B. V., Starov, V. M., and Churaev, N. V. Colloid J., Russian Academyof Sciences, 38, 789 (1976).

3. Deryagin, B. V. and Churaev, N. V. Dokl. Akad. Nauk USSR, 207, 572 (1972).4. Deryagin, B. V. and Zorin, Z. M. Sov. J. Phys. Chem., 29, 1755 (1955).5. Exerowa, D. Adv. Colloid Interface Sci, 96, 75 (2002).6. Exerowa, D. and Kruglyakov, P. M. Foam and Foam Films, Elsevier, 1998.7. Cohen, R., Exerowa, D., Kolarov, T., Yamanaka, T., and Muller, V. Colloids &

Surfaces, 65, 201 (1992).8. Miller, C. A. and Neogi, P. Interfacial Phenomena, New York, Marcel Dekker,

1985; Miller, C.A. and Ruckenstein, E. J. Colloid Interface. Sci., 48, 368 (1974).9. Churaev, N. V. Colloid J., Russian Academy of Sciences, 34, 988 (1972); 36, 318

(1974).10. Parsegian, V.A. Mol. Phys., 37, 1503 (1974).11. Dzyaloshinskii, I. E., Lifshitz, E. M., and. Pitaevskii, L. P. Zh. Eksp. Teor. Fiz.,

37, 229 (1959).12. Shishin,V. A., Zorin, Z. M., and Churaev, N. V. Colloid J., Russian Academy of

Sciences, 39, 520 (1977).13. Slavchov, R., Radoev, B., and Stöckelhuber, K.W. Colloids & Surfaces A: Phys-

icochemical and Engineering Aspects, 261 (1-3), 135, (2005). Mahnke, J., Schulze,H. J., Stöckelhuber, K. W., and Radoev, B. Colloids & Surfaces A: Physicochem-ical and Engineering Aspects, 157 (1–3), 1–9 (1999); Mahnke, J., Schulze, H. J.,Stöckelhuber, K. W., and Radoev, B. Ibid. 11–20.

14. Churaev, N. V., Colloid J., Russian Academy of Sciences, 36, 318 (1974).15. Derjaguin, B.V., Starov, V.M., and Churaev, N.V. Colloid J., Russian Academy of

Sciences, 37(2), 219, (1975).16. Us’yarov, O. G. “Research in the Field of Stability of Disperse Systems and

Wetting Films” [in Russian]. Doctor of Sciences Thesis. SanktPetersburg (Lenin-grad) University, Department of Colloid Science, (1976).

17. Zorin, Z. M. and Sobolev, V. D. In: “Research on Surface Forces,” 3, ConsultantsBureau. New York, p. 29 (1971).

18. Rusanov, A. I. Colloid J., Russian Academy of Sciences, 37 (4), 678 (1975).19. Lester, G. R. J. Colloid Sci., 16, No. 4, 315 (1961).20. Derjaguin, B. V., Muller, V. M., and Toporov, Yu. P. Colloid J., Russian Academy

of Sciences, 37, 455 (1975); 37, 1066 (1975).



21. Muller, V. M. and Yushchenko, Y. S. Colloid J., Russian Academy of Sciences,42, 500 (1980).

22. Rowlinson, J. F. and Widom, B. Molecular Theory of Capillarity, Clarendon,Oxford, 1984.

23. Starov, V. M. and Churaev, N. V. Colloid J., Russian Academy of Sciences, 42,703 (1980).

24. de Feijter, J. A. and Vrij, A. J. Electroanal. Chem., 37, 9 (1972).25. Kolarov, T. and Zorin, Z. M. Colloid Polym. Sci., 257, 1292 (1979).26. Platikanov, D, Nedyalkov, M., and Nasteva, V. J. Colloid Interface Sci., 75, 620

(1980).27. Stevin, S. De Beghinselen der Weeghconst. The Elements of the Art of Weighing,

Francois van Raphelinghen, Leyden (1586).28. Deryagin, B. V. Dokl. Akad. Nauk SSSR, 51, 517 (1946).29. Eremeev, G. G. and Starov, V. M. J. Phys. Chem. of the USSR, (in Russian),

47(11), 2921 (1973).30. Starov, V. J. Colloid Interface Sci., 269, 432 (2003).31. Starov, V. M. and Churaev, N.V. Colloid J., Russian Academy of Sciences, 60 (6),

770 (1998).32. Starov, V. M. and Churaev, N. V. Colloids and Surfaces A: Physicochemical &

Engineering Aspects, 156, 243-248 (1999).33. Starov, V. M., Churaev, N. V., and Derjaguin, B. V. Colloid J., Russian Academy

of Sciences, 44(5), 770 (1982).34. Starov, V. M. and Churaev, N. V. Colloid J., Russian Academy of Sciences, 45(5),

852 (1983).35. Churaev, N. V. and Starov, V. M. J. Colloid Interface Sci., 103(2) 301 (1985).36. V.Starov. In Emulsions: structure, stability and interactions, Edited by D. N.

Petsev. Elsevier Academic Press, Amsterdam-Boston-Heidelberg-London-NewYork-Oxford-Paris-San Diego-San Francisco-Singapore-Sydney-Tokyo, 183–214(2004).

37. Derjaguin, B. V. and Starov, V. M. Colloid J., Russian Academy of Sciences,39(3), 383 (1977).


165

3 Kinetics of Wetting

INTRODUCTION

In Chapter 2, the importance of surface forces in the vicinity of the three-phasecontact line has been shown and investigated in the case of equilibrium.

It is obvious that the same forces are equally important in the kinetics ofspreading. In this chapter we consider separately two cases that are drasticallydifferent: the complete wetting that we begin with and then the partial wetting.The discussion in this chapter shows that, in the case of spreading, the wholedrop profile should be divided into a number of regions, which are briefly dis-cussed below.

Complete wetting. We should decide which dimensionless parameters areimportant in the case of spreading, and also which drops can be considered as“small drops” and which of them are “big drops.” The latter is determined bygravity action.

Let us consider the simplest possible example of spreading of two-dimen-sional (cylindrical) droplets (Figure 3.1a). The capillary regime is an initial stageof spreading of small drops, whereas the gravitational regime is the final stageof spreading of small drops or the regime of spreading of big drops. Transitionfrom the capillary regime of spreading to the gravitational regime takes place atthe moment tc, when

R(tc) ~ ,

where R(t) is the radius of the drop base at time t, which is referred to below asthe radius of spreading; γ and ρ are the interfacial tension and density of theliquid, respectively; g is the gravity acceleration.

In the case of water, (γ = 72.5 dyn/cm, ρ = 1 g/cm3), and hence, a ~ 0.27 cm.The next two important parameters are the Reynolds number and the capillary

number. The Reynolds number characterizes the importance of inertial forces ascompared with viscose forces. To deduce the relevant expression for the Reynoldsnumber, let us consider the spreading of a two-dimensional droplet over a solidsurface (gravity action is neglected). In this case, the Navier–Stokes equationwith the incompressibility condition takes the following form:

ag

= γρ

ρ ηuux

vuy

px

u

x

u

y

∂∂

+ ∂∂

= − ∂∂

+ ∂∂

+ ∂∂

2

2

2

2



where is the velocity vector, and the gravity action is neglected. Let U*

and v* be scales of the velocity components in the tangential and the verticaldirections, respectively. Using the incompressibility condition we conclude that

,

or

.

FIGURE 3.1 (a) Spreading of liquid droplet over flat solid substrate. R(t) – radius ofspreading, which is the position of the apparent three-phase contact line; θ(t) – dynamiccontact angle; 1 – vicinity of the apparent three-phase contact line. (b) A magnificationof the vicinity of the moving apparent three-phase contact line in the case of completewetting: (1) spherical part of the drop, which forms a dynamic contact angle, θ, with thesolid substrate; (2) a region where a spherical shape is distorted by the hydrodynamicforce; (3) a region where disjoining pressure comes into play and become increasinglyimportant towards the end of the region 3; and (4) a region where a macroscopic descriptionis not valid any more and surface diffusion takes place.

43

2

1

θ

b

a

R(t)x1

H

θ(t)

y

ρ ηuvx

vvy

py

v

x

v

y

∂∂

+ ∂∂

= − ∂∂

+ ∂∂

+ ∂∂

2

2

2

2

∂∂

+ ∂∂

=ux

vy

0

�v u v= ( , )

Ur

vh

*

*

*

*

=

v Uhr

* **

*

,= =ε ε


Kinetics of Wetting 167

If the droplet has a low slope, then ε << 1, and hence, the velocity scale in thevertical direction is much smaller than the velocity scale in the tangential direc-tion. Now, using the first Navier–Stokes equation, we can estimate

,

and hence, all derivatives in the low-slope approximation in the x direction canbe neglected as compared with derivatives in the axial direction y. Now we canestimate the Reynolds number:

or

(3.1)

The latter expression shows that the Reynolds number under the low slopeapproximation is proportional to ε2. Hence, during the initial stage of spreading,when ε ~ 1, the Reynolds number is not small, but as soon as the low slopeapproximation is valid, Re becomes small even if

is not small enough. This means that, during the short initial stage of spreading,both the low-slope approximation and low Reynolds number approximations arenot valid. However, we are interested only in the main part of the spreadingprocess when the short initial stage is over. In the following text, we see that theRe number should be calculated only in the close vicinity of the moving contactline, where the low slope approximation is valid (see the following text), becausein the main part of the spreading droplet the liquid is moving much slower thanit does close to the edges. Hence, the inertial terms in Navier–Stokes equationscan be safely omitted and only Stokes equations should be used instead:

ρ ρ ρ η ε η ηuux

vuy

Ur

u

x

u

y

∂∂

∂∂

∂∂

∂∂

<< ∂~ ~ , ~*

*

2 2

22

2

2

2uu

y

u

y

U

h∂∂∂2

2

2 2, ~ *

*

η η

Re ~ ~

*

*

*

*

* *

*

ρ

η

ρ

ηρ

ηε ρu

uxu

y

UrU

h

U hr

∂∂

∂∂

= =2

2

2

2

22 UU r* *

η

Re *= ε ρη

2 Ur

ρηUr*

02

2

2

2= − ∂

∂+ ∂

∂+ ∂

∂

px

u

x

u

yη



.

The tangential stress on the free drop surface at is

Under the low slope approximation, ε << 1, we can easily check that h′ << 1.Using this estimation, the aforementioned condition can be rewritten as

Using the previous estimations we conclude:

This estimation shows that, under the low slope approximation, the tangentialstress on the free liquid interface is

.

This boundary condition is used below in Chapter 3.The capillary number,

,

characterizes the relative influence of the viscose forces as compared with thecapillary force. Let us estimate possible values of the Ca. Let us consider an oildroplet with r* ~ 0.1 cm, γ ~ 30 dyn/cm and η ~ 10–2 P. Let the droplet moveforward on the distance equal to its radius over 1 sec, which can be consideredas a very high velocity of spreading. This gives the following estimation: Ca ~3⋅10–5 << 1. That means, we should expect Ca to be even less than 10–5 over theduration of spreading. According to the previous estimation, we assume belowthat both the capillary and Reynolds numbers are very small except for a very

0

0

2

2

2

2= − ∂

∂+ ∂

∂+ ∂

∂

∂∂

+ ∂∂

=

py

v

x

v

y

ux

vy

η

y h x= ( )

η ′ ′ ∂∂

+ ∂∂

+ ′ ∂

∂− ∂

h huy

vx

hux

v2 22∂∂

+ ′ ∂∂

+ ∂∂

=y

huy

vx

0

η ∂∂

+ ∂∂

=uy

vx

0

∂∂

∂∂

= << ∂∂

uy

Uh

vx

Ur

Uh

uy

~ ; ~* * *

ε ε2

η ∂∂

=uy

0

CaU= ηγ



short initial stage of spreading. We estimate the duration of the initial stage ofspreading in Chapter 4 (Section 4.1) immediately after the drop is deposited onthe solid substrate.

Let us consider the consequence of the smallness of the capillary number,Ca << 1, using the simplest possible example of spreading of two-dimensional(cylindrical) droplets (Figure 3.1a). Let the length scales in both x and y directionin the main part of the spreading drop (Figure 3.1a) be r*, then the pressure insidethe main part of the droplet has the order of magnitude of the capillary pressure,that is

.

Using the incompressibility condition, we immediately conclude that velocitiesin both directions, u and v, have the same order of magnitude U*. Let us introducethe following dimensionless variables, which are marked by an over-bar

.

Using these variables, the Stokes equations can be rewritten as

We already concluded that Ca << 1; that means the right-hand side of bothforegoing equations is very small. Hence, these equations can be rewritten as

which means that the pressure remains constant inside the main part of thespreading droplet. If we now write down the normal stress balance on the mainpart of the spreading droplet, we get

.

pr

~*

γ

ppr

xxr

yyr

uu

Uv

vU

= = = = =γ /

, , , ,* * * * *

∂∂

= ∂∂

+ ∂∂

∂∂

= ∂∂

+ ∂

px

Cau

x

u

y

py

Cav

x

2

2

2

2

2

2

2

�vv

y∂

2.

∂∂

=

∂∂ =

px

py

0

0

,

,�

phh

Cah

huy

vx

= ′′+ ′

+ −+ ′( ) − ′ ∂

∂+ ∂

∂

( ) /1

2

12 3 2 2 − ∂

∂− ′ ∂

∂

vy

hux

2



Using the condition Ca << 1, we conclude from the latter equation that

,

even in the case when the droplet profile does not have the low slope approxi-mation, that is, even if that is not small. This shows that the spreadingdroplet retains its spherical shape over the main part of the droplet. Note that theradius of the droplet base, R(t), changes over time, and this change results in aquasi-steady-state change of the droplet profile.

In the case of moving meniscus in a capillary, a similar estimation shows thatCa << 1 results in a spherical shape of the meniscus in the main part of thecapillary.

Once again, the smallness of the Ca means that the surface tension is muchmore powerful over the most part of the droplet/meniscus, and hence, the droplet/meniscus has a spherical shape everywhere except for a vicinity of the apparentthree-phase contact line. The size of this region, l*, is estimated in this chapterin Section 3.2. It will be shown that the following inequality is satisfied:

.

Hence,

is a small parameter inside the vicinity of the moving contact line. The lattermeans that the curvature of the liquid interface inside the vicinity of the movingcontact line can be estimated as

This provides a very important conclusion: the low slope approximation isvalid inside the vicinity of the moving contact line even if the drop profile is notvery low sloped, that is, even if is not small. Hence, we can always usethe low slope approximation inside the vicinity of the moving contact line exceptin the case when the slope is close to π/2 (see Section 3.10).

Let us estimate a possible range of capillary numbers. If Ca ~ 1, then, in thecase of water, we conclude that

.

phh

const= ′′+ ′

=( ) /1 2 3 2

′h 2 1~ ,

h l r* * *<< <<

δ = <<hl

*

*

1

γγ γ

′′

+ ′( ) + ′

=h

h

h

l

h

lh

11

23 2

2

2

22

3 2/

*

*

*

*

/~

hh

l

h

h

l

*

*/

*

*

.2

2 23 2 2

1+ ′( )≈

δγ

′h 2 1~

CaU U P

~ ~ ~* *ηγ

⋅ −1072

12

dyn/cm



This results in U* ~ 72 m/sec. The velocity here is so high that it probably canbe achieved only under very special conditions. In the case of r* ~ 0.1–1 cm, thevelocity results in Re ~ 7.2·104 – 7.2·105, which is a turbulent flow, beyond thescope of this book. This signifies that the possible range of Ca is between 0 and 1.Low Ca << 1 denotes a relatively low rate of spreading, whereas Ca ~ 1 denotesa very high velocity of motion. In its turn, the case Ca << 1 includes a highcapillary number limit (see Section 3.5 through Section 3.7). Therefore, the caseof low capillary numbers, Ca << 1, and intermediate capillary numbers, Ca ~ 1,should be considered in a completely different way. The situation is similar tothe case of the Reynolds number: consideration of flows at low Reynolds numbersis very much different from that at high Reynolds numbers.

Now we are ready to consider in more detail the vicinity of the moving contactline (region 1 in Figure 3.1a), which is magnified in Figure 3.1b.

The whole vicinity of the three-phase contact line can be subdivided into fourregions (Figure 3.1b). Region 1 is a spherical meniscus in the main part of thespreading droplet. This region is included to show the dynamic contact angle,θ(t), which is defined at the intersection of the tangent to the spherical part ofthe droplet with the solid substrate. The dynamic contact angle is unknown andshould be determined by matching all regions presented in Figure 3.1b. Insidethe next region, 2, the spherical shape is distorted by the hydrodynamic flow.This region is followed by region 3, where disjoining pressure comes into play.Over region 3, the disjoining pressure action becomes increasingly important ascompared to the capillary force. Toward the end of region 3, the disjoiningpressure overcomes the capillary force and becomes the only driving force of thespreading process. Region 3 is followed by region 4, where a macroscopicdescription of the spreading process becomes impossible because the character-istic scale in the vertical direction is of the order of the molecular size. We referto region 4 as the region of surface diffusion.

The picture of a spreading drop profile in a vicinity of the three-phase contactline, presented in Figure 3.1b, has been understood only recently. A number ofsimplified physical mechanisms have been introduced previously, based on asimplification of the above picture.

For a long time the so-called singularity on a three-phase contact line [1] hasbeen considered as a major problem in the consideration of the kinetics ofspreading. We explain in the following text the source of this singularity and whyit is removed by the disjoining pressure acting in a vicinity of the apparent three-phase contact line.

It is easy to see that the viscose stress in the tangential direction close to thethree-phase contact line (Figure 3.1b) is

,

as h → 0. This means that the drop cannot spread out because the friction forceat the moving front becomes infinite. The way to overcome this problem has been

η η∂∂

→ ∞vy

Uh

r ~ *



suggested in Reference 2. The concept is as follows: the very first layer of theliquid molecules on the liquid–solid interface is attached to the solid substrateby a force of adhesion. However, the adhesion force is finite, not infinite. If thetangential stress has become large enough, then the first layer of the liquidmolecules will be swept away by the tangential stress. The result is a “slippagevelocity,” which is introduced as follows:

,

where α is a proportionality coefficient. That is, the slippage velocity is propor-tional to the applied shear stress on the solid substrate. This definition can berewritten as

, (3.2)

where λ has a dimension of length and can be referred to as a slippage length.However, it turns out [3] that λ ~ 10–6 cm, which is located just in the rangewhere surface forces are the most powerful. Therefore, Condition 3.2 cannot beused as a macroscopic condition because the thickness is in the range of surfaceforces action. Hence, a modified consideration in this region, which takes intoaccount surface forces action, should be used instead. Note that, in the case ofcomplete wetting, the disjoining pressure is equal to

as h → 0, that is, even faster than the tangential stress. Hence, the disjoiningpressure is the driving force of spreading in a vicinity of the three-phase contactline. Let us estimate the thickness at which the disjoining pressure overcomesthe increasing tangential stress:

,

or

.

That is, the lower the velocity of spreading, U, the higher the thickness below whichthe disjoining pressure overcomes the tangential stress. If, as before, we adopt at

η α∂∂

= [ ]

==

vy

vr

y

r y0

0

∂∂

=

== =

vy

vr

y

r

y0 0λλ η

α,

Π( )hAh

= → ∞3

ηUh

Ah

* < 3

hAU

ht<

=η *

/1 2



the initial stage of spreading a very high spreading velocity, U* ~ 10–1 cm/sec,A ~ 10–14 erg, η ~ 10–2 P, then ht ~ 0.3⋅10–6 cm, that is, in the range where thedisjoining pressure is the most powerful.

A simple way to overcome the problem of “singularity on the moving contactline” has been suggested in Reference 4: a cutting length was introduced in avicinity of the three-phase moving contact line. However, the introduction ofcutting length is similar to the introduction of slippage velocity.

A simplifying approach has been suggested in Reference 5. According to thisapproach, the hydrodynamic flow in region 2 and region 3 is ignored, as well asthe disjoining pressure action in region 3 (Figure 3.1b). According to this approach,a spherical meniscus is followed directly by region 4, where surface diffusiontakes place. This approach results in the following equation for the velocity ofspreading:

that is, the velocity of spreading is proportional to the difference between cosθe,where θe is the equilibrium contact angle (θe is a fitting parameter in the theory[5]) and cosθ(t), where θ(t) is the instantaneous dynamic contact angle. In thecase of complete wetting, that is at cosθe = 1, the latter equation results in

It is well established that, in the case of complete wetting (see Section 3.1and Section 3.2), the law is

Comparison of these last two equations shows that the approach suggestedin Reference 5 does not agree with well-established theoretical predictions.

The next approach we mention was tried long ago and is based on theconsideration of dynamic surface tension in the vicinity of the apparent movingthree-phase contact line. It has been assumed that the surface tension of “thefresh interface” (which appears close to the apparent three-phase contact line) ishigher than the surface tension behind the apparent three-phase contact line on“the old interface.” This surface difference could be the driving force behindspreading. However, both experimental investigations [6] and theoretical estima-tions [7] showed that the relaxation time of the surface tension on a fresh liq-uid–air interface of pure liquids is too small, and hence, cannot influence thespreading process, which proceeds on much larger time scales. However, recently,an attempt has been made to revive the same idea of a high surface tension on afresh liquid–air interface [8]. The approach suggested in Reference 8 also com-pletely ignores the disjoining pressure action in the vicinity of the moving three-phase contact line. This approach was criticized in Reference 7.

�R const te= ⋅ − cos cos ( ) ,θ θ

�R const t= ⋅θ2( ).

�R const t= ⋅θ3( ).



Surface diffusion (region 4 in Figure 3.1) results in an effective slippage [9].The first attempt to introduce surface slippage based on the consideration of surfacediffusion was undertaken in Reference 9. This approach is to be developed further.

3.1 SPREADING OF DROPLETS OF NONVOLATILE LIQUIDS OVER FLAT SOLID SUBSTRATES: QUALITATIVE CONSIDERATION

In this section we shall discuss the viscous spreading of drops over solid surfaceswhen there is complete wetting, and spreading proceeds completely down to themolecular level. We shall disregard the influence of surface forces in the vicinityof the apparent three-phase contact line. Thus, we know there is a singularity atthe moving three-phase contact line (as shown in the Introduction to this chapter).In spite of that, we shall find the correct time dependence of spreading (spreadinglaws in the case of capillary and gravitational spreading). Subsequently, we shallconsider the spreading of microdrops, that is, very small drops, entirely subjectedto the action of the surface forces. As expected, we shall see that the disjoiningpressure action removes the singularity on the moving three-phase contact line.Also to be expected is that, giving due consideration to the disjoining pressure, theliquid profile does not end at any particular point but vanishes asymptotically.Hence, only an apparent moving front can be identified.

Considerable experimental and theoretical material dealing with the spreadingof droplets over a solid, nondeformable, dry surface has been accumulated fordecades [10–13]. In the following text we present a derivation of equations thatdescribe the kinetics of spreading in different situations.

Note that all considerations below are undertaken in the case of completewetting. Partial wetting is considered in Section 3.10, which shows that the partialwetting case is far more complicated compared to complete wetting. This explainswhy the spreading law in the case of partial wetting is to be deduced.

Let us consider an axisymmetric spreading of liquid droplet over dry solidsubstrate (Figure 3.1). Let R(t) be the radius of spreading, θ(t) the dynamic contactangle, and h(t,r) the unknown liquid. Our objective is to deduce the spreadinglaw, R(t) (Figure 3.1). As discussed in the Introduction to this chapter, the Rey-nolds number is small over the main duration of the spreading process; hence,Re << 1. We also assume that the low slope profile approximation

is valid, which means the scale in the vertical direction, h*, is much smaller thanthe scale in the radial direction, r*.

The cylindrical coordinate system (r,ϕ, z) should be used in the case ofaxisymmetric spreading. Because of symmetry, vϕ = 0, and all unknown valuesare independent of the angle ϕ.

ε ~∂∂

<<hr

1



Let U* and v* be the scales of the velocity components in the tangential andvertical directions, respectively. Using the incompressibility condition, we con-clude that

, or . (3.3)

Hence, the velocity scale in the vertical direction is much smaller than thevelocity scale in the tangential direction. This means:

.

Using the same small parameter ε, we can conclude that all derivatives in theradial direction, r, are much smaller than derivatives in the vertical direction z:

, (3.4)

where f(r,z) is any function.Now we can easily conclude from the Stokes equation for the vertical velocity

component that

;

hence, the pressure depends only on the radial coordinate, r, that is p = p(r) andremains constant through the cross section of the spreading drop. Therefore thepressure can be presented as

p = pa –γK – Π(h) + ρgh,

where pa is the pressure in the ambient air, γK is the capillary pressure, K is themean curvature of the interface (negative in the case of droplets of a flat solidsubstrate and positive in the case of menisci in capillaries), g is the gravityacceleration, and ρ is the liquid density, Π(h) is the isotherm of the disjoiningpressure.

In the low slope case (see Equation 3.3) we conclude

.

Ur

vh

*

*

*

*

= v U Uhr

* **

*

,= << = <<ε ε 1

v vz r<<

∂∂

<< ∂∂

fr

fz

∂∂

=pz

0

Kr r

rhr

= ∂∂

∂∂

1



Hence,

. (3.5)

After that the only equation left is that for the radial velocity component, vr,which now can be written as

, (3.6)

with nonslip boundary conditions at z = 0:

vr (t,z) = 0, (3.7)

and no-tangential stress condition on the liquid–air interface surface:

. (3.8)

To deduce this condition we should take into account both Equation 3.3 andEquation 3.4; after that, all terms, except for the one shown in Equation 3.8,disappear (see Introduction to this chapter).

Integration of Equation 3.6 with boundary condition (3.7) and (3.8) resultsin the following expression for the velocity profile:

. (3.9)

This equation allows calculation of the flow rate, Q:

. (3.10)

The conservation of mass reads:

.

p pr r

rhr

h gha= − ∂∂

∂∂

− +γ ρ1 Π( )

∂∂

= ∂∂

pr

v

zrη

2

2

η ∂∂

= =vz

z h t rr 0 at ( , )

vpr

h zz

r = − ∂∂

−

1

2

2

η

Q rv dz r hprr

h

= = − ∂∂∫2

23

0

3π πη

2 0πrht

Qr

∂∂

+ ∂∂

=



Substitution of Equation 3.10 into the preceding one results in

, (3.11)

which is referred to in the following text as the equation of spreading.Nonvolatability of the liquid results in the conservation of the total liquid

volume, V:

, (3.12)

where R(t) is the radius of the drop base, which is frequently referred to as theradius of spreading. Finally, we have two equations (3.11 and 3.12) with twounknowns: the liquid profile, h(t,r), and the radius of spreading, R(t). Note thatthe pressure, p(t,r), in Equation 3.9 is specified according to Equation 3.5.

All the thicknesses of a spreading droplet should be divided into the followingparts:

• Big thickness, h ≥ 10–5 cm ~ ts; the effects of the disjoining pressuremay be neglected. According to Equation 3.5, the hydrodynamic pres-sure within this region is equal to

. (3.13)

Let us introduce the following dimensionless variables, which aremarked with an overbar:

.

Note that all dimensionless values have the same order of magnitudeof 1. Using the dimensionless variable we conclude from Equation 3.13that

. (3.14)

This expression includes two dimensionless parameters:

.

∂∂

= ∂∂

∂∂

ht r r

r hpr

13

3

η

20

π r h dr V const

R t( )

∫ = =

p pr r

rhr

gha= − ∂∂

∂∂

+γ ρ1

rrr

hhh

r rr h hh= = = =* *

* *, ,or

p ph

r r rr

hr

gh ha= − ∂∂

∂∂

+γ ρ*

**2

1

γ ρh

rgh*

**2

and



The first one gives the intensity of capillary forces, and the second onethe intensity of the gravitational force.

If capillary forces prevail, then the capillary regime of spreadingtakes place; that is, if

,

where a is the capillary length.If gravity prevails, then the gravitational regime of spreading takes

place; that is, if

In the case of water, the capillary length is equal to a ≈ 0.27 cm. Thecapillary regime is an initial stage of spreading of small drops, whereasthe gravitational regime is the final stage of spreading of small dropsor the regime of spreading of big drops. The transition from the cap-illary regime of spreading to the gravitational regime should take placeat the moment tc, when Rtc) ~ a, where R(t) is the radius of the dropbase at time t.

• Intermediate thickness 10–6 cm ≤ h ≤ 10–5 cm. In this region both thecapillary and the disjoining pressures act simultaneously, i.e., thehydrodynamic pressure takes on the form

. (3.15)

• Small thickness h ≤ 10–6 cm. Here, the value of the capillary pressureis negligible in comparison to the disjoining pressure:

. (3.16)

Accordingly, the following stages of the spreading of a drop can be identified:

1. Inertial, when both Re and Ca are not small. We will not consider thisstage in this book; however, an estimation of the duration of this periodis given in Chapter 4, Section 4.1.

2. Gravitational, in the case of big droplets.

γ ρ γρ

h

rgh r

ga*

** *2

>> << =or

γ ρ γρ

h

rgh r

ga*

** * .

2<< >> =or

p pr r

rh t r

rha= − ∂

∂∂ ( )

∂

− ( )γ ,Π

p p ha= − ( )Π



3. In the case of small droplets (which is considered in the followingtext), the gravitational stage is preceded by the capillary stage (3).

4. Disjoining pressure action, when the complete droplet is in the rangeof disjoining pressure action.

Only during the first stage does the spreading process proceed without regardto the form of the isotherm of the disjoining pressure, and all other stages aredetermined to different degrees by the disjoining pressure.

The apparent macroscopic wetting perimeter r0(t) (Figure 3.2) and the truemicroscopic wetting perimeter, R(t), are different because of the formation of theprecursor film, caused by the action of both hydrodynamic and surface forces.In the vicinity of the point r0(t) there is a transitional region, where both thecapillary and disjoining pressures act. In the following text we consider so-calledsimilarity solutions of the equation of spreading (3.11) in the case of capillaryand gravitation regimes of spreading.

CAPILLARY REGIME OF SPREADING

Small drops with an initial characteristic size smaller than the capillary length,a, are considered here. That means, after a short inertial period, the capillaryregime of spreading begins. In this case the pressure is given by

,

and Equation 3.11 takes the following form:

FIGURE 3.2 Spreading of a spherical droplet. At h > h1, the spherical droplet profile isnot distorted by the hydrodynamic flow; ts < h1 is the radius of action of the disjoiningpressure; r0(t) is the macroscopic wetting perimeter (the apparent three-phase contact line);R(t) is the true microscopic wetting perimeter; θ(t) is the dynamic contact angle; and H(t)is the drop apex.

r

ℜ

θ(t)

H(t)

R(t)h1ts

r0(t)0

Z

p pr r

rhra= − ∂

∂∂∂

γ 1



, (3.17)

with conservation law (3.12) and the boundary conditions by virtue of the sym-metry at the center of the droplet:

. (3.17’)

First, let us introduce dimensionless values using the following characteristicscales h*, r*, and t*. From Equation 3.17 we conclude that

.

Hence,

.

In the same way, from Equation 3.12,

.

Hence,

.

In Section 3.2 we will see that the estimation of the characteristic time scale isunrealistically small. However, for a moment we ignore this because, as we will seein the following text, the spreading problem cannot be solved precisely in this way.

Now, Equation 3.17 and Equation 3.12 can be rewritten as

, (3.18)

. (3.19)

∂∂

= − ∂∂

∂∂

∂∂

∂∂

ht r r

r hr r r

rhr

γη3

13

∂∂

= ∂∂

== =

hr

hrr r0

3

30

0

ht

ht

h

r r rr h

r r rr

hr

*

*

*

*

∂∂

= − ∂∂

∂∂

∂∂

∂∂

γη

4

43

3

1 1

ht

h

rt

r

h*

*

*

**

*

*

,= =γη

ηγ

4

4

4

33

3or

2 2

0

πr h r h dr V

R t

* *

( )

∫ =

22

3 222

3 10

3π

ππ ηγ

r h V hV

rt

r

V* * *

**

*, ,( )= = =or

∂∂

= − ∂∂

∂∂

∂∂

∂∂

ht r r

r hr r r

rhr

1 13

r h dr

R t

0

1

( )

∫ =



SIMILARITY SOLUTION OF EQUATION 3.18 AND EQUATION 3.19

Let ξ be a new variable, which is a combination of . The dimensionlessthickness, , should depend on this new single variable. Such a solution isreferred to as a similarity solution.

Let , where is a new unknown function. From Equation 3.19we conclude that

,

or

. (3.20)

This equation must be independent of . This gives two conditions:

, (3.21)

where λc is an unknown constant. Equation 3.18 and Equation 3.19 do not includeany dimensionless parameters; it means that λc ~ 1. As we see in the followingexample, the latter constant cannot be determined without consideration of thedisjoining pressure, and it will be determined only in Section 3.2.

The second condition, which immediately follows from Equation 3.20, is

. (3.22)

According to Equation 3.21, the spreading law is known if the function,, is determined.

Similarity solution (3.22) should be substituted in Equation 3.18.Let us calculate

r tandh t r( , )

ξ = rf t( ) f t( )

rf tf t

h drf tf t

R t

( )( )

( )( )

( )1 1

10

=∫

ξ ξh

f td

R t f t

2

0

1( )

( ) ( )

=∫t

R t f t const R tf tc

c( ) ( ) , ( )( )

= = =λ λ

h A f t= ( ) ( )ξ 2

R t( )f t( )

∂∂

= ′ ′ + ′

= ′

ht

A rf t f t A f t f t

A

( ) ( ) ( ) ( ) ( ) ( )ξ ξ2 2

(( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

ξ ξrf t f t f t A f t f t

f t

′ + ′

= ′

2

ff t A A( ) ( ) ( )′ + ξ ξ ξ2



Using the latter equation and

,

we conclude from Equation 3.18

. (3.23)

This equation should not depend on time but on variable ξ; hence,

,

or

. (3.24)

The solution of this equation is

,

where C is an integration constant.From this equation and Equation 3.21,

.

Using the initial condition R(0) = R0, where R0 is different from the initialradius of the droplet but equal to the radius of the droplet in the end of the inertialperiod of spreading (see further discussion in Chapter 4, Section 4.1). Therefore,we should choose the scale of the radius of the droplet as follows: .

According to this choice, . Hence,

.

rf t

= ξ( )

f t f t A A

f t f t

( ) ( ) ( ) ( )

( ) ( )

′ ′ + =

−

ξ ξ ξ2

8 4 11 13

ξ ξξ

ξ ξ ξξ

ξd

dA

dd

dd

dAd

f t f t f t f t( ) ( ) ( ) ( )′ = − 8 4

′ = −f t f t( ) ( )11

f tt C

( ).

=+( )

1

100 1

R t t Cc( ).

= +( ) λ 100 1

r R* = 0

R( )0 1=

Cc

= 110 10λ



It is reasonable to assume (it will be justified partially in Chapter 4, Section 4.1),that the constant is equal to the duration of the inertial stage of spreading. Hence,we adopt

,

where tin is the duration of the inertial stage of spreading. Using this notation wecan write the spreading law as

.

Now back to dimensional variables:

Finally,

. (3.25)

Using Equation 3.25 and the definition (3.22) we conclude that

,

and

,

where the dynamic contact angle is determined as

.

Ctt

tinin= =

*

R t t tc in( ) ( )( ). .= +λ 100 1 0 1

R tR

t tt

R t

cin

c

( )( )

( )(

.

*

.

0

0 1

0 1

0

10

10

= +

=

λ

λ ..

.

.

.)( )

1

30 1

30 1

0 1

3 2π

γη( )

+ =V

t t in 00 653

0 1

0 1

. ( ) .

.

.

λ γηc in

Vt t

+

R tV

t tc in( ) . ( )

.

.=

+0 65

30 1

0 1λ γη

H tV

t tin( ) ≈

+ −ηγ

2 1 5

1 5

/

/( )

θ ηγ

tV

t tin( ) ≈

+ −3

3

1 10

0 3

/

,( )

θ( ) ~( )( )

tH tR t



The spreading rate according to Equation 3.25 is

.

If we express time in this equation via the dynamic contact angle θ(t), wearrive at

,

which is the well-known and well-verified Tanner’s law of dynamics of spreadingin the case of complete spreading. This law has already been mentioned in theIntroduction to this chapter.

As we already commented, the unknown constant, λc, is close to 1.According to Equation 3.25, the exponent 0.1 and R(t)/V 0.3 are independent

of the drop volume. These two conclusions have been well confirmed in the caseof complete wetting by numerous experimental data.

Similar calculations can be carried out for the dynamics of spreading of aone-dimensional droplet (cylinder), which results in

.

However, there is one very substantial drawback to the obtained solution (seethe following text).

Let us go back to the Equation 3.23, which takes the following form afterselection (3.24):

.

The equation can be transformed as follows:

UdR t

dtV

t tc in= =

+ −( )

. ( )

.

.0 0653

0 1

0 9λ γη

U C~ .0 065 3λ γη

θ

R t V t tin( ) ≈

+γη

3

1 7

1 7

/

/( )

′ + =

A Ad

dA

dd

dd

dAd

( ) ( )ξ ξ ξξ ξ

ξξ ξ ξ

ξξ

21 13

′ + =

A Ad

dA

dd

dd

dAd

( ) ( )ξ ξ ξ ξξ

ξξ ξ ξ

ξξ

2 321

′ =A

dd

Ad

dd

d( )ξ ξ

ξξ

ξ ξ2 3 1

ξξξ

ξdAd



The preceding equation shows that either A(ξ) = 0, or

.

This equation describes the spreading droplet profile up to the point ofintersection with the axis ξ; that is, to the point where A(ξ0) = 0. Hence, thefollowing problem should be solved:

. (3.25′)

The equation should be solved with two additional conditions

,

which determine the unknown constants a and b. Note that the first condition issimply the conservation law (3.21). Hence, we can now determine . Itcould be the complete solution to the problem of the capillary regime of spreading.

Unfortunately, no such solution of the above problem exists: at any choiceof constants a and b, the solution behaves as shown in Figure 3.3: the similaritydrop profile never intersects the axis (curve 1, calculated according to the abovesystem). Curve 2 in Figure 3.3 presents the parabolic profile

for comparison. This is the manifestation of the so-called “singularity” at themoving apparent three-phase contact line.

A Ad

dd

ddAd

A Ad

d

( )ξ ξ ξξ ξ ξ

ξξ

ξξ

2 3 1

2

=

− 110

ξ ξξ

ξd

ddAd

= .

ξξ ξ ξ

ξξ

−

=A

dd

dd

dAd

2 10

ξξ ξA

AA A

A a

A

A b

2 2

0

0 0

0

= ′′′ + ′′ − ′

=

′ =

′′ = −

( )

( )

( )

ξ ξ ξ ξξ

A d A( ) , ( )= =∫ 1 00

0

0

where

λ ξC = 0

ab−2

2ξ



This contradiction is resolved in Section 3.2. Surprisingly, the spreading law(3.25) remains almost untouched except for the determination of the unknownconstant λc.

GRAVITATIONAL SPREADING

In this case the radius of spreading is bigger than the capillary length. Hence,according to Equation 3.14, we can omit the capillary pressure. This results inthe following expression for the pressure inside the drop:

.

Substitution of this expression into the equation of spreading (3.11) results in

, (3.26)

with the same conservation law (3.12) and the boundary condition

.

Let us introduce the following dimensionless values using characteristicscales h*, a, and t*. The end of the capillary stage of spreading is when the radiusof spreading reaches a. It is the reason why the latter length is selected as thecharacteristic length scale. From Equation 3.26 we conclude:

.

FIGURE 3.3 Dependency of a similarity profile of the spreading droplet, A(ξ): it neverintersects the axis. (1) calculated according to Equation 3.25′; (2) the spherical (parabolicprofile).

21

A

a

c∗

ξ∗ξ

p p gha= + ρ

∂∂

= ∂∂

∂∂

ht

gr r

r hhr

ρη3

3

∂∂

==

hr r 0

0

ht

ht

gh

a r rr h

hr

*

*

*∂∂

= ∂∂

∂∂

ρη

4

23

3

1



Hence,

.

Now, from conservation law (3.12) we conclude:

.

Hence,

.

Using these notations we can conclude from Equation 3.26 and conservation law(3.12):

, (3.27)

. (3.28)

SIMILARITY SOLUTION

Let ξ be a new variable, which is a combination of r and t as in the previous caseof capillary spreading; h(t,r) should depend on this new single variable. Suchsolution is referred to as a similarity solution.

Let ξ = r f(t), where f(t) is a new unknown function. Substituting the latterdefinitions into Equation 3.28, we find that

.

This equation must be independent of t; this gives two conditions:

, (3.29)

ht

gh

at

a

gh*

*

**

*

,= =ρη

ηρ

4

2

2

33

3or

2 2

0

πa h r h dr V

R t

*

( )

=∫

22

3 222

3 8

3π

ππ η

ρa h V h

V

at

a

gV* *,

*,

( )= = =or

∂∂

= ∂∂

∂∂

ht r r

r hhr

1 3

r h dr

R t

=∫ 10

( )

ξ ξh

f td

R t f t

2

0

1( )

( ) ( )

=∫

R t f t const R tf tg

g( ) ( ) , ( )( )

= = =λλ



and

, (3.30)

where λg should be close to 1. Equation 3.29 shows that the spreading law, R(t),is known if the unknown function, f(t), is determined.

Similarity solution (3.30) should be substituted in Equation 3.27. Calculationssimilar to the previous case results in an identical expression for the time deriv-ative. Using the definition

we conclude from Equation 3.27 that

. (3.31)

Equation 3.31 should not depend on time but on variable ξ only; hence,

(3.32)

or

Note the “minus” in the latter equation because the radius of spreading, R(t),should be an increasing function of time. Solution of Equation 3.32 is

,

where C is an integration constant.From the latter equation and Equation 3.29 we conclude:

. (3.33)

The initial condition can be specified as R(0) = a or in dimensionless unitsas . Using the latter initial condition we conclude from Equation 3.33:

h A f t= ( ) ( )ξ 2

rf t

= ξ( )

f t f t A A f t f td

dA( ) ( ) ( ) ( ) ( ) ( )′ ′ + =ξ ξ ξ

ξ ξξ2

18 2 33 dAdξ

f t f t f t f t( ) ( ) ( ) ( )′ = − 8 2

′ = −f t f t( ) ( ).9

f tt C

( )/

=+( )

1

81 8

R t t Cg( )/

= +( ) λ 81 8

R( )0 1=



,

which we can refer to as the duration of the capillary stage of spreading, tc.Therefore. Equation 3.33 can be rewritten as

–R(t)– = (λg81/8) –(t + –tc)1/8. After we

go back to dimensional variables, the equation becomes

,

or

, (3.34)

where λg is close to 1. Exponent 1/8 agrees very well with a number of experi-mental observations.

Now we can consider the shape of the spreading droplet over the duration ofthe gravitational regime of spreading. Equation 3.31, taking into account Equation3.32, now takes the following form:

.

After simple rearrangements (similar to those in the capillary case), we arriveat

.

Solution of this equation is

, (3.35)

where the integration constant ξ0 should be determined from the conservation law

,

Cg

= 18 8λ

R ta

t ttg

c( )( )/

*

/

= +

λ 81 8

1 8

R tgV

t tg c( ) . ( )

/

/=

+0 57

31 8

1 8λ ρη

′ + = −

A Ad

dA

dAd

( ) ( )ξ ξ ξξ ξ

ξξ

21 3

′ = −AAξ

2

A g( )/

/ξ λ ξ=

−( )3

2

1 32 2 1 3

ξ ξ ξλ

A d

g

( ) =∫ 10



which results in λg ≈ 1.37, which is close to 1 as predicted earlier. Using thisvalue, we can rewrite Equation 3.34 as

. (3.36)

That is, in the case of gravitational spreading, the spreading law can bedetermined completely according to Equation 3.36.

However, we still have one substantial problem in the case of gravitationalspreading. The profile of the droplet in this case is shown in Figure 5.25 (thecase n = 1). This figure shows that the low slope approximation used in ourconsideration is severely violated in a vicinity of the edge of the spreading droplet.

In Figure 3.4 the time evolution of radius of spreading is schematicallypresented in log–log coordinates. Capillary and gravitational regimes are shownby line 1 (according to Equation 3.25, with λc still unknown) and line 2 (accordingto Equation 3.36), respectively.

The capillary regime of spreading switches to the gravitational regime ofspreading at the moment marked as tc, when the radius of spreading reaches thevalue of the capillary length, a. The real time evolution is shown by curve 3 inFigure 3.4. The experimental evidence of such dependency was obtained inReference 15, where a transition from the capillary regime of spreading to thegravitation regime has been demonstrated similar to that shown in Figure 3.4.

SPREADING OF VERY THIN DROPLETS

In the earlier part of this section we showed that pure capillary spreading resultsin inconsistency of the mathematical treatment in the vicinity of the apparentmoving contact line of the spreading droplet. This inconsistency is usuallyreferred to as a singularity at the three-phase contact line. As we already men-tioned in the Introduction to this chapter, this inconsistency is the result of

FIGURE 3.4 Time evolution of the radius of spreading in log–log coordinate system (3).(1) capillary spreading; (2) gravitational regime of spreading.

�n tc

�n R

�n a

�n t

3

2

1

R tgV

t tc( ) . ( )

/

/=

+0 78

31 8

1 8ρη



neglecting the disjoining pressure action in the vicinity of the moving apparentthree-phase contact line. Remember that as the drop thickness tends to zero, thedisjoining pressure overcomes the capillary pressure and dominates the spreadingprocess.

This is why we consider here the spreading of the microdroplet, whose apexis located in the range of action of the surface forces. In the following text, weneglect the effect of capillary pressure. The hydrodynamic pressure in the liquidis described by Equation 3.16.

Thus, the case under consideration refers to the final stage of the spreadingof droplets, which completely wets the substrate. The equation describing thedroplet profile, h(t, r), of a spreading droplet has the following form:

. (3.37)

This is obtained by a substitution of Equation 3.16 into the equation of spreading(3.11). The boundary conditions for second-order differential equation (3.37) areas follows:

, (3.38)

, (3.39)

and the conservation law (3.12). Note that Equation 3.37 has the form of anonlinear equation of thermal conductivity. As we know, an equation of thermalconductivity often (for example, in the linear case) results in an infinite rate ofpropagation of perturbations. Therefore, features similar to that of Equation 3.37should be expected in the case under consideration.

Let us first consider the simplest form of the isotherm of the disjoiningpressure, Π(h), in the case of complete wetting:

(3.40)

In this case, Equation 3.37 takes the form

. (3.41)

∂∂

= − ∂∂

′ ∂∂

ht r r

r h hhr

13

3

ηΠ ( )

∂∂

= =hr

r0 0,

h t,∞( ) = 0

Π hb t h h t

h t

s s

s

( ) =−( ) ≤ ≤

≥

,

,.

0

0

∂∂

= ∂∂

∂∂

ht

br r

r hhr3

3

η



Introducing dimensionless values as before in the equation, we conclude:

.

Using these notations we can conclude from Equation 3.41 and conservation law(3.12) that

, (3.42)

. (3.43)

Surprisingly, Equation 3.42 and Equation 3.43 are identical to Equation 3.27and Equation 3.28 in the case of gravitational spreading. Hence, we can use thealready deduced similarity solution (3.35) for the droplet profile with λg ≈ 1.37.After that, the spreading law becomes, as in (3.36),

, (3.44)

where t0 is the time when this stage of spreading starts. The only differencebetween the expression from (3.36) and (3.44) is that ρg in Equation 3.36 isreplaced by b in Equation 3.44. At any rate, the disjoining pressure action, evenin the simplest possible form (3.40), removed the artificial singularity on themoving apparent contact line.

Let us consider an isotherm of the disjoining pressure of the form

, (3.45)

which, as we already know, is relevant in the case of complete wetting (forexample, spreading of oils over glass and metals).

Equation 3.37 now becomes

. (3.46)

Introducing dimensionless values using the following characteristic scales h*,r*, t* in Equation 3.46, we conclude:

22

3 222

3 8

3π

ππ η

r h V hV

rt

r

bV* * *

**

*, ,( )= = =or

∂∂

= ∂∂

∂∂

ht r r

r hhr

1 3

r h dr

R t

=∫ 10

( )

R tbV

t t( ) . ( )

/

/=

+0 78

31 8

01 8

η

Π h A hn( ) = /

∂∂

= ∂∂

∂∂

−ht

nAr r

r hhr

n

32

η



.

Hence, we can choose

.

Using the conservation law (3.12) we conclude as before:

.

Now Equation 3.46 and Equation 3.12 can be rewritten as

, (3.47)

and (3.19).Let , where is a new unknown function. Substitution of the

foregoing definitions into Equation 3.19, we find that

.

This equation must be independent of t; this gives two conditions:

, (3.48)

and

, (3.49)

where λn should be close to 1 and depends on the exponent n of the disjoiningpressure isotherm. Equation 3.48 shows that the spreading law, R(t), is known ifthe unknown function, f(t), is determined.

ht

ht

nAh

r r rr h

hr

nn*

*

*

*

∂∂

= − ∂∂

∂∂

−−

3

22

3

1

η

tr

nAh n**

*

= −3 2

2

η

hV

rt

V

r

n

n n**

**

,( )

= =−

− −2

3

22

2

2 2 6πη

π

∂∂

= ∂∂

∂∂

−ht r r

r hhr

n1 2

ξ = rf t( ) f t( )

ξ ξh

f td

R t f t

2

0

1( )

( ) ( )

=∫

R t f t const R tf tn

n( ) ( ) , ( )( )

= = =λ λ

h A f t= ( ) ( )ξ 2



Similarity solution (3.49) should be substituted in Equation 3.47. Calculationssimilar to the previous cases result in an identical expression for the time deriv-ative. Using the definition

,

we conclude from Equation 3.47 that

(3.50)

Equation 3.50 should not depend on time but on variable ξ only, hence,

(3.51)

or

Note the minus in the latter equation because the radius of spreading, R(t),should be an increasing function of time. Now Equation 3.50 takes the followingform:

,

or, after rearrangement similar to the previous cases,

. (3.52)

Let us consider two cases, n = 3 and n = 2, in the Equations 3.51 and 3.52.In the case n = 2 we conclude that

and .

Taking into account Equation 3.48, we conclude that

, (3.53)

rf t

= ξ( )

f t f t A A f td

dAn( ) ( ) ( ) ( ) ( )′ ′ + = − −ξ ξ ξ

ξ ξξ2

18 2 2 nn dAdξ

.

f t f t f tn( ) ( ) ( )′ = − −8 2

′ = − −f t f tn( ) ( ).7 2

′ + = −

−A Ad

dA

dAd

n( ) ( )ξ ξ ξξ ξ

ξξ

21 2

ξξ

= − −AdAd

n1

′ = −f t f t( ) ( )3 ξξ

= − −AdAd

1

R t t t( ) = +( )λ2 22



and

, (3.54)

where is the dimensionless time of the beginning of this stage of spreading,and C is the integration constant to be determined from the conservation law.According to Equation 3.54, the droplet profile does not vanish anywhere buttends asymptotically to zero. That is, we can determine only the effective apparentthree-phase contact line as follows: According to conservation law,

;

hence, after integration.

.

If we select the apparent contact line as the point where the profile is closeto zero, that is,

,

then . In this case, C ≈ 0.9, and we conclude that

, (3.55)

.

Note that the spreading law (3.55) with exponent 0.5 is the fastest yet thatwe have found.

Let us emphasize again that there is no distinct three-phase contact line inthe case under consideration; the droplet profile tends to zero only asymptotically.This is the consequence of neglecting the surface diffusion in front of the movingcontact line. It is still an open problem as to how to connect the macroscopicdescription based on the consideration of the disjoining pressure with surfacediffusion, which is a microscopic description.

A C( ) expξ ξ= −

2

2

t2

ξ ξ ξλ

C dexp −

=∫

2

02

12

C 12

122

− −

=exp

λ

exp .−

=λss

20 1

λ2 2 10 2 15= ≈ln .

R t t t( ) .= +( )3 04 2

A( ) . expξ ξ= −

0 92

2



In the case n = 3, we conclude from Equation 3.51 and Equation 3.52 that

and .

The solution of the latter equations is

and ,

where is the time when this stage of spreading begins, and C is an integrationconstant. As in the previous case, there is no distinct three-phase contact linebecause the droplet profile tends to zero only asymptotically. We select an appar-ent contact line as follows from the conservation law:

,

which gives

.

The latter equation has the following solution:

or

.

According to the previous consideration, λ3 should be around 1. Hence, if weselect

,

then

, (3.56)

′ = −f t f t( ) ( ) ξξ

= − −AdAd

2

f t t t( ) exp (= − +( )3 AC

( )ξξ

=+

1

2

2

t3

ξ ξξ

λd

C +=∫ 2

02

13

ln 12

132

+

=λC

12

32

+ =λC

e,

λ32

21

Ce= −

Ce

~( )

.1

2 10 29

−=

R t t t( ) exp= +( )λ3 3



and

.

The latter Equation 3.56 gives the highest possible rate of spreading.

3.2 THE SPREADING OF LIQUID DROPS OVER DRY SURFACES: INFLUENCE OF SURFACE FORCES

In this section we consider the spreading of an axisymmetric liquid drop on aplane solid substrate in the case of complete wetting. Both capillary and disjoiningpressure are taken into account [17]. As we already concluded in Section 3.1 andthe Introduction to this chapter, neglecting of the disjoining pressure in the vicinityof the moving apparent three-phase contact line results in a contradiction. On theother hand, we already showed at the end of Section 3.1 that the disjoiningpressure action removes the singularity on the moving contact line.

A cylindrical coordinate system is used in the following text. The initial sizeof the droplet is assumed to be smaller than the capillary length

;

that is, the gravity is neglected.We already estimated in the Introduction to this chapter that

1. The capillary number, Ca ~ 10–5 << 1, that is, the main part of thespreading droplet remains the spherical shape.

2. A low slope approximation is valid in the vicinity of the moving contactline. However, we should still estimate the scale of the narrow zone,where the latter approximation is satisfied.

This means that the liquid profile in the main part of the spreading droplet is

, (3.57)

(see Figure 3.5). Using simple geometrical consideration, we conclude fromFigure 3.2 that H and ℜ are expressed via the dynamic contact and the radius ofspreading as

.

A( ).

ξξ

≈+

1

0 292

2

ag

= γρ

h r r H( ) ( )= ℜ − − ℜ −2 2

ℜ = = −RH

Rsin

,( cos )

sinθθ

θ1



If we assume that the liquid is mostly located in the spherical part of thedroplet, then the volume, V, is

, (3.58)

or

. (3.59)

In the case of small contact angles, the latter equation results in

. (3.60)

Equation 3.60 gives a good approximation of the right-hand side of Equation3.59 in the range of dynamic contact angles 0 < θ(t) < π/4.

Using the low slope approximation we already deduced the equation ofspreading, which describes the shape of the liquid profile (3.11). Now, thisequation is used only in the vicinity of the moving contact line and will bematched with the solution (3.57).

Inside the same vicinity of the moving contact line the pressure is given byexpression (3.15), which should be substituted into Equation 3.11. In the case ofcomplete wetting, which is discussed in this section, we adopt the disjoiningpressure isotherm, Π(h), as

, (3.61)

FIGURE 3.5 Spreading of axisymmetric droplet. (1) vicinity of the moving contact line.

R(t)

ℜ

r 1

H

θ(t)

z

V R= +

π θ θ6 2

32

3 2tan tan

R tV

t t( )

tan( )

tan( )

/

=

+

6 1

23

2

1 3

2π θ θ

1 3/

R tV

( )/

=

41 3

πθ

Π hAhn( ) =



with A > 0 being the Hamaker constant and n = 2 or 3. Substituting the latterequations into Equation 3.11 and Equation 3.15 results in

. (3.62)

The first and the second terms in brackets on the right-hand side of Equation3.62 describe the effects of capillarity and disjoining pressure, respectively. Thedrop thickness decreases from the center of the drop to its spreading edge. In thissection we are interested in the three regions of the spreading drop (Figure 3.1b).The spherical part of the spreading droplet (that is, the region from the center ofthe drop to a point where the film thickness is still large, such that in this regionboth viscous and disjoining pressure effects are negligible compared to capillaryeffects) is referred to as the outer region. Further outward radially, there is aregion where both capillary and viscous effects dominate (region 2 in Figure 3.1b).The next region is further outward, where the film is thin and disjoining pressureeffects are of comparable importance to viscous and capillary effects (see Figure3.1b). Region 2 and Region 3 are referred to as the inner region.

Equation 3.62 can be made dimensionless by introducing appropriate scalesfor thickness, radius, and time,

, (3.63)

where we use the same notations for dimensionless values as for dimensional:h → h/h*, r → r/r*, t = t/t*, ε = 3η r*

4/γ h*3t*, λ = nAr*

2/h*n+1. The scales, h*, r*, and

t* are determined in the following text. According to Equation 3.12, 2π r*2h* = V,

where V is the drop volume. Hence, from this equation, h* is determined as h* =V/(2πr*

2), and there are now only two unknown scales, r* and t*.The volume of the liquid drop is constant during spreading; that is, Equation

3.12 should be used.If the whole liquid profile satisfies the condition h′2 << 1, then Equation 3.63

describes the liquid profile in all regions. In this case the following symmetryconditions are valid at r = 0:

. (3.64)

The preceding case does not differ substantially from the more general case,when the dynamic contact angle is not sufficiently small. This is why only thelow slope approximation, which is valid over the whole droplet, is under consid-eration here.

∂∂

γη

∂∂

∂∂

∂∂

∂∂

ht r r

rhr r r

rhr

= −

31 13

−

+

nAh

hrnγ

∂∂1

− =

ε ∂

∂∂

∂∂

∂∂

∂∂∂

ht r r

r hr r r

rhr

1 13

−

+

λ ∂∂h

hrn 1

∂∂

∂∂

hr

hr

= =3

3 0



The initial solid surface is dry in front of the spreading droplet; that is:

. (3.65)

It is necessary to comment on the latter boundary condition. As we alreadyestablished in Section 3.1, the liquid profile may tend asymptotically to zero; forthis case the boundary condition will be specified in the following text.

It is shown in the following text that both dimensionless constants in Equation3.63 are small:

, (3.66)

and

. (3.67)

Condition (3.66) expresses the smallness of viscous forces, Fη, as comparedwith capillary forces, Fγ : Fη ∼ η∂2vr /∂z2 ~ η r*/t* h*

2, and

,

where z is the vertical coordinate and vr the radial component of velocity. Hence,Fη /Fγ ~ η r*

4/(γ t* h*3) = ε/3 << 1. This consideration shows that, according to its

physical meaning, ε is the modified capillary number, Ca. Condition (3.67)expresses the smallness of the disjoining pressure as compared with capillaryforces at large thickness.

Letting ε = λ = 0, we obtain the outer solution of Equation 3.63 in the regionof the spherical part of the droplet as

, (3.68)

where ξ = r f(t), and an unknown function f(t) is determined as follows. Equation3.68 determines the parabolic profile of the drop away from the drop edge. Theapparent contact line corresponds to the condition ξ = 1, i.e.,

, (3.69)

where r0 (t) is the macroscopic apparent radius of the spreading drop (Figure 3.2).In deriving Equation 3.68, the conservation law (3.12) was used (see Appendix 1).It was supposed that almost the whole volume of the spreading droplet is locatedin the spherical part.

h r R t→ →0, ( )at

ε << 1

λ << 1

Fr r r

rhr

h

rγ

∂∂

γ ∂∂

∂∂

γ~ ~

∗

∗3

h r t f t, ,( ) = ( ) −( ) < <4 1 0 12 2ξ ξ

r t f t0 1( ) = ( )



To derive the inner solution of Equation 3.63 in the region, where the dropprofile is distorted by the action hydrodynamic flow and disjoining pressure, weintroduce new variables:

, (3.70)

and

, (3.71)

where χ(t), ψ(µ), and h0(t) are new unknown functions. Letting χ << 1, h0 << 1and focusing only on the largest terms (see Appendix 1), from Equation 3.63 weobtain that

,

or

, (3.72)

where the overdot indicates the first derivative with respect to dimensionless time,t. For both sides of Equation 3.72 to be explicit functions of µ but not time, werequire

, (3.73)

. (3.74)

Equation 3.73 and Equation 3.74 are two ordinary differential equations forthree unknown functions f(t), χ(t), and h0 (t).

Integrating Equation 3.72 and using condition (3.64), which in a dimension-less form becomes ψ(µ) → 0, as η → ∞, we obtain

, (3.75)

where ′ and � indicate first and third derivatives in regard to µ, respectively. Thefirst and second terms in the brackets on the left-hand side of the equation aredue to the capillary and disjoining pressure effects, respectively.

µ ξ χ= −( ) ( )1 t

h h t= ( ) ( )0 ψ µ

− = − +ε

χψµ χ µ

ψ ψµ

λ χψ

h f

fdd

h f dd

dd f hn

0 04 4

43

3

3

2

20

1

�nn

dd+

1

ψµ

− = − + +ε χ ψ

µ µψ ψ

µλ χ

ψ

3

03 5

33

3

2

20

1 1

�fh f

dd

dd

dd f h

dn n

ψψµd

ε χ 3

03 5 1

�fh f

= −

λ χ 2

20

1 1f hn + =

ψ ψ ψψ

21

1′′′ − ′

=+n



The condition for matching the inner and outer solutions is

, (3.76)

where B is a positive constant to be determined. The matching condition (3.76)is the third equation, which is required to determine the three unknown functionsf(t), χ(t), and h0 (t). From Equation 3.73, Equation 3.74, and Equation 3.76 wefind (see Appendix 2)

, (3.77)

, (3.78)

, (3.79)

where E = 8/B.It follows from Equation 3.69 and Equation 3.77 that the dimensionless

macroscopic radius of the spreading drop r0 (t) has the following time-dependence:

. (3.80)

The constant B in Equation 3.77 and Equation 3.79 can be determined usingthe matching condition according to Equation 3.76:

. (3.81)

Unfortunately, Equation 3.75 (see Appendix 3) does not have the requiredasymptotic behavior at to satisfy the condition (3.81). This suggests thatin order for both the outer, Equation 3.68, and the inner, Equation 3.75, solutionsto be meaningful in some common range of variables ξ and µ, it is necessary toreplace the matching condition (3.81) with an approximate condition

dd

fh

Bψµ

χ µ= − = − = −∞82

0

,

f tE t( ) = +

−

0 5102

13

10

0 1

..

ε

χ λε

tE

E tn

n

n

( ) =

+

+( )+

−2 102

12 2

1

11 3

10

+−( )

n

n

15 1

h tE

E tn n

0

6

2

11 3

10

35 12 10

21( ) =

+

− −λε

(( )

r tE t

0

3

10

0 1

2102

1( ) = +

ε

.

dd

Bψµ

µ= − = −∞,

µ → −∞



, (3.82)

where µ* > 0 is an unknown constant. According to the physical meaning of thepoint µ* the requirement µ* >> 1 should be satisfied, which is checked in thefollowing text.

Let us find an expression for µ*. The outer solution, Equation 3.68, obtainedfrom Equation 3.63 at ε = λ = 0, becomes meaningless when h3 ∼ ε. Changing thevariable according to Equation 3.71 gives h0

3ψ*3 ∼ ε, where ψ* = ψ(–µ*) >> 1.

Substituting Equation 3.75 into this expression and omitting the time-dependentterm, we have

∼ ε.

The above-mentioned approximate condition is chosen from the latter equation,where ∼ is replaced by equality; that is,

,

and, at last, the necessary equation for is

. (3.83)

Note that the condition ψ* >> 1 should be satisfied.Now we are ready to calculate parameter ε. In terms of dimensional variables,

Equation 3.80 becomes

.

Differentiating the latter equation we conclude that

.

Based on the foregoing two equations, we choose and

.

dd

Bψµ

µ µ= − = − ∗,

26 23

1 3λ ψ/E n( ) −∗

26 23

1 3λ ψ ε/E n( ) =−∗

µ ∗

ψ ε λ∗−= ( )1 3 21

164/ /E n

r t rE t

t0

3

10

0 1

2102

1( ) = +

∗∗ε

.

d r t

d t

E r

tE t

t0

3

9

3

10

0 9

2102

1( ) = +

∗

∗ ∗

−

ε ε

.

r r∗ = 0 0 2( )/

r

t

d r

d t∗

∗= − ( )

0 500.



The two aforementioned equations determine the unknown scales, and ,and, hence, ε

. (3.84)

From definitions of ε, h*, r*, and Equation 3.84, we conclude

,

where is the initial macroscopic drop radius:

. (3.85)

We show in Appendix 3 that as

, (3.86)

. (3.87)

From Equation 3.82, Equation 3.83, Equation 3.86, and Equation 3.87 wededuce that

. (3.88)

From Equation 3.84 and Equation 3.88 we have an equation for parameter B:

. (3.89)

Now, from Equation 3.84, Equation 3.88, and Equation 3.89 we can determineB, µ*, and ε. Neglecting the time-dependent term in deriving Equation 3.83 andEquation 3.89 is justified, as discussed in Appendix 4, where the dependence ofB on time is shown to be weak.

Let us consider the solution of Equation 3.89 for the two particular cases ofdisjoining pressure isotherm mentioned earlier.

r∗ t ∗

ε = =E

B

3

10 32

1

2

tr B

V

r B

V∗ = =

3

21 45

30010 3

6 30010 3

3

π ηγ

ηγ

.

r00

r r00 0 0= ( )

µ → −∞

′≈ −ψ µ31 3 1 3/ /ln

ψ µ≈ 31 3 1 3/ /ln

ψ ελ

µ µ µ∗

−

∗ ∗ ∗=

= = =1 32

11

1 3 1 313/ / /ln ex

BB B

n

pp B 3 3/( )

12 3

2

11

1 3 2 3

λ BB B

n

= ( )−/ exp /



CASE n = 2

From Equation 3.89 we conclude in this case that

. (3.90)

Let us examine the magnitude of the parameter λ for typical values of h*, r*,A, and γ. For h* = 0.01 cm, r* = 0.05 cm, A = 10–7 dyn, and γ = 20 dyn/cm weconclude from λ definition: λ = 2.5·10–5 << 1. Substituting the value of λ intoEquation 3.90, we have B4 exp(B3/3) = 3.18·104. This gives B = 2.68. Substitutingthis value of B into Equation 3.84 and Equation 3.88, we find ε = 0.026 << 1,ψ* = 1648 >> 1, and µ* = 615 >> 1. As λ changes from 1.25·10–5 to 5·10–5, Band ε change from 2.76 to 2.60 and from 0.024 to 0.029, respectively. For thesame range of λ, ψ* and µ* change from 3025 to 904 and from 1096 to 348,respectively. This means that the approximate conditions (Equation 3.82 andEquation 3.83) are practically equivalent to the matching condition (Equation 3.81)over some range containing the point µ = –µ*. Also note that the values ofparameters satisfy the conditions mentioned before: ε << 1, λ << 1, µ* >> 1, andψ* >> 1.

CASE n = 3

From Equation 3.89 we conclude in this case that

. (3.91)

Let us estimate the magnitude of λ for typical values of h*, r*, A, and γ. Forh* = 0.01 cm, r* = 0.05 cm, A = 10–14 erg, and γ = 20 dyn/cm: λ = 3.75·10–10 <<1. Substituting the value of λ into Equation 3.91, we have B3 exp(B3/3) = 4.10·104.This gives B = 2.82. Substituting this value of B into Equation 3.84 and Equation3.88, we find ε = 0.022 << 1, ψ* = 5138 >> 1, and µ* = 1819 >> 1. As λ changesfrom 1.875·10–10 to 7.5·10–10, B and ε change from 2.86 to 2.79 and from 0.0213to 0.0231, respectively. For the same range of λ, ψ* and µ* change from 7073 to3735 and from 2472 to 1341, respectively. Similar to the previous case, this meansthat the approximate conditions (3.82, 3.83) are practically equivalent to thematching condition (3.81) over some range containing the match point µ = –µ*.Also note that the values of parameters satisfy the conditions mentioned earlier:ε << 1, λ << 1, µ* >> 1, and ψ* >> 1.

Using Equation 3.84 and Equation 3.85 we can simplify the equation for r0 (t)dependency as

. (3.92)

B B4 31 3

31

2exp

//( ) =

λ

B B3 3

1 33

1

2exp

//( ) =

λ

r t rt

t0 00

0 110

1( ) = +

∗

.



From Equation 3.68, Equation 3.77, and Equation 3.84 we can derive the dropletprofile as a function of radial position and time, h(r, t) = 4 f 2(t) (1 – r2 f 2(t)), or,in dimensional variables,

. (3.93)

The apex height of the drop is

, (3.94)

where is the initial apex height of the drop. The error of assigning isnegligible, as discussed in Appendix 2.

For a small angle, the advancing dynamic contact angle θ(t) can be derivedfrom Equation 3.93 as θ(t = –∂h/∂rR, at r = r0(t), or

. (3.95)

Assuming the spreading drop has the shape of a spherical cap, and the angleis small, another advancing dynamic contact angle, θRH(t), can be derived fromEquation 3.92 and Equation 3.94:

. (3.95’)

The right-hand site in Equation 3.95’ is identical to that in Equation 3.95 asit should be in the case of low slopes.

We now derive the relationship between the advancing dynamic contact angleand the capillary number Ca, which is defined as Ca = ηU/γ, where the spreadingspeed at the drop edge, U, is defined as

. (3.96)

From Equation 3.95 we conclude that

. (3.97)

h r t h t t r r t t,.( ) = +( ) − ( ) +(∗ ∗

−∗ ∗10 1 1 4 10 1

0 2 2 2/ / / ))

−0 2.

H t h t( ) = ( )0,

H t h t h t t h t t( ) = ( ) = +( ) = +( )∗ ∗−

∗−

0 10 1 10 10 2

00,.

/ /00 2.

h00 h h∗ = 00

θ t h r t t h r t t( ) = ( ) +( ) = ( )∗ ∗ ∗−

∗/ / / /10 1 2 100 3

00 00

.++( )−

10 3.

θRH t H t r t H t r t r( ) = ( ) = =−2 2 210 0 00tan ( ) ( ) ( ) ( )/ / /rr t t00

0 310 1( ) +( )∗

−/

.

U d r d t r t t t= = ( ) +( )∗ ∗−

0 00

0 910 1/ / /

.

θ t BCa

BCa

( ) = ( )=

2

82 64

1 3 1 3

1 3

/ /

/.

in radians

in degrrees( )



We can rewrite the power laws in (10 t/t* + 1) of Equation 3.92, Equation3.94, and Equation 3.95 to power laws in te, where te is the experimentallymeasured time. Then,

(3.98)

where K = 1.21·B0.3. Using definitions of h*, r*, t*, and Equation 3.85, we canreduce Equation 3.94 to

, (3.99)

and Equation 3.95 to

. (3.100)

Equation 3.98, Equation 3.99, and Equation 3.100 are the same as thosederived in Section 3.1, except for the prefactors, which could not be deduced inthat section.

From Equation 3.89 we know that is a function of λ:

and, in turn, depends on the parameters n and A of the disjoining pressureisotherm. For the n = 2 case, with typical values of h*, r*, A, and γ given above,we have K = 0.900, and Equation 3.98 becomes

. (3.101)

Similarly, for the n = 3 case, we have K = 0.887, and

. (3.102)

Let us now examine the influence of the constant A (Hamaker constant in then = 3 case) on K in the spreading laws (3.102) and (3.103). For the case,

r t V B t V Be e03 2 3

0 10 1 3 3640 3 1 21( ) = ( ) = (γ π η γ η/ /

.. . ))

= ( )

0 10 1

30 1

0 1

..

..

t

K V t

e

eγ η/

H t B V te e( ) = ( ) ( ) −2 3 4030 2

3 20 1

0 2. .

.η π γ/

θ η π γt B V te e( ) = ( ) ( ) −2 27 200030 3

3 30 1

0 3. .

./ in raddians( )

B

λ π γ= + + + +n Ar Vn n n n002 4 1 3 12/ ,

r t V te e03

0 10 10 900( ) = ( ).

..γ η/

r t V te e03

0 10 10 887( ) = ( ).

..γ η/

n = 2



the value of B changes from 2.76 to 2.60, and hence, K changes from 0.892 to0.909. For the case, the value of changes from 2.86 to 2.79, and therefore,K changes from 0.883 to 0.890. It is obvious that although the value of K issomewhat larger in the n = 2 case than in the n = 3 case, the difference is notsignificant between the two cases. Thus, K is only slightly dependent on n andthe constant A of the disjoining pressure isotherm in the case of complete wetting.

From Equation 3.92 and Equation 3.98 the wetted area predicted by ourtheory, St, is

We now briefly discuss the applicable conditions of the results:

1. The solid surface must be flat and smooth, and the liquid is Newtonianand nonvolatile.

2. The gravity and inertia effects must be negligible compared to capillaryeffects. That means the Bond number

and Weber number

,

where ρ is the liquid density.3. From the solution it follows that H → 0, as t/t* → ∞. However, as the

droplet apex is in the range of surface forces action, the influence ofdisjoining pressure is in the drop center, and even a weak volatility ofthe liquid becomes significant. The equilibrium film thickness, he, isdetermined by the vapor pressure in the surrounding air and, accordingto Chapter 2, is

This means that H → he, as t/t* → ∞, and hence, our results are valid only whenthe drop apex is much bigger than the final equilibrium film thickness, H >> he;

n = 3 B

S r t r t t Vt = = +( ) =∗π π γ π η02

000 2 0 2 3 210 1 640 3( ) . .

/ / BB t

K V t

e

e

30 2

0 2

2 30 2

0 2

( )= ( )

..

..γ η/

BogH

= <<ρ

γ

2

1

WeU H

= <<ρ

γ

2

1

A

hRT v p p

en m s= ( ) ( )/ /ln .



(4) as we only discuss the macroscopic aspect of the spreading problem, thecondition h(r,t) → 0, as r → ∞, is the macroscopic condition. Macroscopicdescription (including the disjoining pressure) is meaningless if the scale of thethickness is comparable with the molecular length scale, m. Thus, our results arevalid in the wetted area within the radius rm, where h(rm, t) > m.

In conclusion, we compare the calculated radius of time dependency accord-ing to Equation 3.92 and Equation 3.25 deduced in Section 3.1. We rewriteEquation 3.92 as

,

where tin = t*/10. This comparison results in λC = 1.34; that is, λC ~ 1, as predictedin Section 3.1.

COMPARISON WITH EXPERIMENTS

Chen [16,17] has reported a series of experiments on spreading drops of poly-dimethylsiloxane. His data allow us to compare the deduced Equation 3.92,Equation 3.94, Equation 3.95, and Equation 3.97. In those experiments the con-ditions described in the theory are satisfied. The experiments involve depositinga liquid drop on a glass surface and monitoring the silhouette of the spreadingdrop: for the drop radius, r0(t), the drop apex height, H(t), and the advancingdynamic contact angle, θ(t). All experiments are run at room temperature. Theerrors of measurement are within 0.001 cm, 0.0002 cm, and 0.75˚, respectively.These errors are estimated from three repeated measurements for each quantity.

The liquid used is a silicone liquid (a polydimethylsiloxane, Dow Corning200 fluid; Dow Corning Corporation, Midland, MI). It has a number-averagemolecular weight of 7500. At the room temperature of 22.5 to 24.0˚C, its viscosityranges from 1.98 to 1.93 P, its density from 0.970 to 0.969 g/cm3, and its surfacetension against air from 20.9 to 20.8 dyn/cm. The glass sample used is a soda-lime glass plate and a borosilicate microscope slide.

An example of the comparison is shown in Figure 3.6. The straight lines arethe results of least-square-fit to data in power laws. We first compare the least-square-fit results in power laws of

(3.103)

with the theory, according to Equation 3.92, Equation 3.94, and Equation 3.95.The values of M and N obtained from least-square-fit of experimental data showexcellent agreement with the theory prediction not only in exponents, N, but alsoin prefactors, M.

r tV

t tin0

3 0 10 1

0 89( ) ..

.=

+( )γη

r , H, and /0 θ θRH

NM t t= ⋅ +( )∗10 1 ,



Figure 3.7 shows a typical comparison between our theory predictions andexperimental data. These comparisons show that the prefactors predicted by thetheory agree well with the experimental values.

The least-square-fit results for data points of θ(Ca) dependency and for thesame number of data points of θRH(Ca) from all experiments results in

FIGURE 3.6 Experimental dependencies of radius of spreading, dynamic contact angle,and the drop apex height on time fitted according to Equation 3.103.

FIGURE 3.7 Comparison between theoretical prediction (according to Equation 3.92)and the same experimental data as in Figure 3.6. Comparison of all other dependences(contact angle, the drop apex) show the similar excellent agreement between theoreticalequations and experimental data.

0.1

0.05

0.03

0.02

0.01

0.005

0.003

0.0021 3 4 5 6 7 8

2

4

6

8

10

12

2

θ, d

eg.

r 0, H

cm

10 t/t∗ + 1

∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗

radius

apex

θθRH

∗

∗

∗∗

∗∗

∗∗ ∗

∗

0.15

0.145

0.14

0.135

0.13

0.125

0.12

0.115

0.113 4 5 6 7 8 9 10

10 t/t∗ + 112

r 0, c

m



, (3.104)

. (3.105)

The value of Ca ranges from 1.8·10–6 to 3.3·10–4 for experiments. Theprefactors and exponents in Equation 3.104 and Equation 3.105 are close to thosepredicted by Equation 3.97, which can be written as

. (3.106)

Ausserre et al. [18,19] have measured the time-dependence of the radius andadvancing dynamic contact angle for a number of spreading drops of polydime-thylsiloxane with different molecular weights. They found that the radius andcontact angle show a power law dependence on time with an exponent of 0.100 ±0.010 and 0.3 ± 0.015, respectively. These values agree with our predictions, 0.1and 0.3. For similar experiments, Tanner [20] found that the dynamic contactangle follows a power law in time with an exponent ranging from 0.317 to 0.335,which is close to our predicted value, 0.3.

CONCLUSIONS

The combined effect of viscosity, surface tension, and disjoining pressure isincluded in the theory for an axisymmetric, nonvolatile, Newtonian liquid dropspreading over a horizontal, dry, smooth, flat solid surface in the case of completewetting. It is shown that the disjoining pressure action removes the singularityon the moving three-phase contact line. The drop profile is calculated as a functionof time and radial position. The spreading radius, the apex height of the drop,and the advancing dynamic contact angle are found to follow different powerlaws in time. The dynamic contact angle is found to follow a power law incapillary number. Both the prefactors and exponents in the power laws are pre-dicted, and the predicted power laws agree with known experimental data.

APPENDIX 1

This appendix shows the derivation of Equation 3.68 and Equation 3.72.Let ε = λ = 0; Equation 3.63 becomes, in this case,

.

Integrating once with respect to r results in

.

θ = ⋅ ( )222 0 353Ca . in degrees

θRH Ca= ⋅ ( )242 0 353. in degrees

θ θ= = ⋅ ( )RH Ca238 0 333. in degrees

∂∂

∂∂

∂∂

∂∂

rrh

r r rr

hr

3 1

= 0

hr r r

rhr

Cr

3 11∂∂

∂∂

∂∂

=



As h is finite at r = 0, C1 must be 0. Integrating with respect to r three times,we have h = C2r 2/4 + C3 ln r +C4 . For the same reason, C3 must be 0. Recognizingh(r,t) = 0 at r = r0, we have h = (C2/4)(r2 – r 2

0 ). Now taking into accountconservation law (3.12) and defining r0 = 1/f(t), we arrive at Equation 3.68.


where the overhead dot and the superscript indicate the first derivative with respectto dimensionless time, t, and coordinate, µ, respectively.

As , hence,

In view of this we conclude that

In the following text we use the estimations

.

These estimations give

Hence, combining all previous estimations, we get the following:

(A1.1)

∂∂

= ( ) ( ) + ( ) ∂∂

= + ′−h

th t h t

dd t

h hrf� �

0 0 0 0ψ µ ψµ

µ ψ ψχ rrf −( )1

2

�χχ

�

��

h h

ff

0 0

1

ψ ψµ

χχ µχ

χ+ ′

+

−,

µ χ µ χ= → << <<rf 1 1 1and then /,

µχ

χ µχ µχ+

− ≈ − ≈1 ��

��

�ff

ff

ff

.

∂∂

= + ′ht

h hf

f�

�0 0ψ ψ

χ.

� �hht

fft0

0≈ ′ ≈ ≈, ,ψ ψµ

��

h

hf

f

h

hf

f

0

0

0

0

1ψ

ψχ

ψη

ψχ

ηχ′

≈ = << .

∂∂

= ′ht

hf

f0ψ χ

�.



Now, for µχ << 1, the right-hand side of Equation 3.63 can be simplified as

Now, from Equation A1.1 and Equation A1.2, we arrive at Equation 3.72.

APPENDIX 2

This appendix shows the derivation of Equation 3.77 through Equation 3.79 anddiscusses the negligible error in assigning H(0,0) = H* in the derivation.

From Equation 3.73 and Equation 3.76 we have

(A2.1)

Here, the overhead dot indicates the first derivative with respect to the dimen-sionless time, t. Integrating Equation A2.1 with respect to t once, we have

(A2.2)

To determine the integration constant, we make use of the following initialcondition at and t = 0: H(0,0) = H*‚ which in a dimensionless form is

, (A2.3)

where H(0,0) is the initial apex height of the drop. Substituting Equation A2.3into Equation 3.68, we find that

f(0) = 1/2. (A2.4)

1 131r r

rhr r r

rh

r h

h

rn

∂∂

∂∂

∂∂

∂∂

− ∂

∂+

λ

=+

hf d

d04

4

4

11χ µχ µ

11

113+( ) +

+( )

µχ ψ

µ µχ µµχ ψ

µd

d

d

d

d

d

−

+ +

λχψ

ψµ

2

20

1 1f h

d

dn n

= − + +hf d

d

d

d f h

d

dn n04

4

43

3

3

2

20

1 1χ µψ ψ

µλχ

ψψµµ

(A1.2)

�ff

E11

3

= −ε

.

f E t const− = +10 310 /ε .

r r0 0 0= ( )

h r t( , ) ,0 0 1 0 0= = =at and



Substituting Equation A2.4 into Equation A2.2, we find that the constant =210 and

,

which coincides with Equation 3.77. From Equation 3.74, Equation 3.76, andEquation 3.77 we can derive Equation 3.78 and, at last, from Equation 3.76,Equation 3.77, and Equation 3.78 we can derive Equation 3.79.

We now discuss the error in assigning H* = H(0,0) instead of using H* =V/(2πR2

*). In view of r* = r0(0)/2 and Equation 3.85, H* can be written as

. (A2.5)

If the drop shape is a spherical cap, then

, (A2.6)

where H00 = H(0,0) is the initial apex height of the drop, and r00 = r0(0) is theinitial macroscopic drop radius. From Equation A2.5 and Equation A2.6, weconclude that

. (A2.7)

From Equation A2.7, we estimate the error in assigning H* = H(0,0) is 0.75%when H00/r00 is 0.15. Thus, the error is negligibly small.

APPENDIX 3

This appendix shows the derivation for Equation 3.86 and Equation 3.87.As , Equation 3.75 becomes

(A3.1)

Let us define . According to the chain rule, we have

.

f t E t( ) = +( )−0 5 10 2 13 10

0 1.

./ ε

H V R V r∗ ∗= ( ) = ( )2 22002π π

V H r H= ( ) +( )π 00 002

0026 3/

H H r r H H H r* = ( ) +( ) = +(00 002

002

002

00 002

0023 3 1 3/ / ))

µ ψ→ −∞ → ∞,

ψ ψ2 1′′′ = .

ϕ ψ ψ ψ µ( ) = ′ = d d/

′′ = = = ⋅ = ′

′′′ = ′(ψ ψ µ ϕ µ ϕ ψ ψ µ ϕ ϕ

ψ ϕ ϕ

d d d d d d d d

d

2 2/ / / /

)) = ′ + ′ = ′′ + ′( )d d d d dµ ϕ µ ϕ ϕ ϕ µ ϕ ϕ ϕ ϕ/ / 2 2



Equation A3.1 now becomes

. (A3.2)

Let y = ln ϕ and applying the chain rule once again, we have

Substituting these results into Equation A2.2, we have

. (A3.3)

Assuming the solution to Equation A3.3 has the form , where D andG are constants to be determined,

.

If G = 2/3, then � � = �D�3y1/3 2/3 , at y . Hence, G = 2/3 cannotsatisfy Equation A3.3 at y . However, if G = 1/3, then

Hence, at , for G = 1/3, we have from Equation A3.3 that D = –31/3 and

. (A3.4)

From Equation A3.4 we have Equation 3.86 and .Introducing and integrating the above equation by parts, we conclude

that

ψ ϕ ϕ ϕ ϕ2 2 21′′ + ′( )( ) =

′ = = ⋅ =

′′ = =

ϕ ϕ ψ ϕ ψ ϕ ψ

ϕ ϕ ψ ψ

d d d dy dy d

d d dy d d

y/ / / /

/ /2 2 ϕϕ ψ ϕ ψ

ϕ ϕ

y yy

yyy

yy

y

dy d e dy

e e e

/( ) = ( )

= −( )

−

− − − == −( )−e yyy y

2 ϕ ϕ .

ϕ ϕ ϕ ϕ ϕy yy y2 2 1+ −( ) =

ϕ = DyG

ϕ ϕ ϕ ϕ

ϕ ϕ

y yyG

yG

G GD y

GD y

2 2 3 3 2

23

3 1

2 1+ = −( )=

−

−

ϕ ϕy2 → ∞ → ∞

→ ∞

ϕ ϕ ϕ ϕy yy D y y2 2 39 0+ = ( ) → → ∞, .at

y → ∞

ϕ ψ ψ= = ′ = − = −Dy y1 3 1 3 1 3 1 3 1 33 3/ / / / /ln

d dψ ψ µ/ ln / /1 3 1 33= −z = ln ψ

ln // / /− − −( ) = = + ( )∫ 1 3 1 3 1 3 1 3ψ ψd e z dz e z e zz z z −−

− − −

∫∫= + ( ) ≅

4 3

1 3 4 3 11 3

/

/ /ln ln ln

dz

dψ ψ ψ ψ ψ/ // ,3∫ ψ



as . Hence,

. (A3.5)

Solving Equation A3.5 in the limits , we conclude that

This expression coincides with Equation 3.87.

APPENDIX 4

This appendix discusses the weak dependence of B on dimensionless time, t.In deriving Equation 3.83 and Equation 3.89, we have neglected the time-

dependence term in . Here, we include the time-dependence term for an esti-mation. We determine match point as the point where the followingcondition is satisfied:

(A4.1)

where and, in view of Equation 3.79, Equation A4.1 can berewritten as

(A4.2)

From Equation 3.84 and Equation A4.2 we conclude that

where . Following the same derivation for Equation 3.89, wehave

.

Let us define ; then,

and .

ln ln ,/ /4 3 1 3ψ ψ ψ>> ′ → ∞at

ψ ψ µ/ ln / /1 3 1 33= −

µ ψ>> >>1 1and

ψ µ ψ µ µ ψ µ= = ( )( ) ≈3 3 3 31 3 1 3 1 3 1 3 1 3 1 3/ / / / / /ln ln ln lnn /1 3 µ

h03

µ µ= − ∗

h03 3ψ ε∗ = ,

ψ ψ µ∗ ∗= −( ) >> 1,

( ) ./ /2 10 2 16 2 3 1 3 10

9 5 13λ τ ε ψ ε/ /E En n−( ) −( )∗+( ) =

ψ ε λ ε∗

−( ) − −( )= ( ) +( ) =1 3 21 1 3 5 1 1 364 10 1// / //E t

n nEE

n2

1 164/ Λ( ) −( )/

,

Λ = +( )λ 10 13 5

t/

ΛB B Bn

21 1

1 3 2 32 3( ) = ( )− −( ) / exp /

F B B n Bn n( ) = = −( ) −( )1 2 1 31 3 2 3/ /Λ / exp

dF dB n B n B/ /= + −( )( )2 1 2 Λ dF dB dB dt d dt/ / /⋅ = − ( )Λ Λ2



As (dΛ/dt)/Λ = 6/(10t + 1), we have dB/dt = –6 B/[(10t + 1)(2n + (n – 1) B3)].For n = 3 and B ≈ 3, we conclude from the preceding equation that

Hence,

.

The latter equation proves that B is weakly dependent upon time and justifiesthe omission of the time-dependent term in deriving Equation 3.83 and Equation3.89.

3.3 SPREADING OF DROPS OVER A SURFACE COVERED WITH A THIN LAYER OF THE SAME LIQUID

In this section, a solution is obtained for the problem of viscous spreading of aliquid drop on a plane solid surface that has been prewetted with a film of thesame liquid. The film thickness is assumed thick enough, that is, the thicknessis bigger than the radius of disjoining pressure action. Appearing in the spreadingequation is a universal small parameter, which is independent of the nature ofthe liquid–substrate system and which is a characteristic of the viscous spreadingregime. By an expansion in terms of this small parameter, a solution has beenobtained for the problem, through which the profile of the spreading drop andthe velocity of motion of the drop boundary can be determined [11].

Let us examine a drop of a viscous liquid on a planar horizontal solid surfacecovered with a layer of the same liquid with thickness h0 (Figure 3.8). Thethickness h0 is assumed to be outside the range of disjoining pressure action.However, the droplet is still small enough to neglect the gravity action. As thespreading process is axisymmetric in the case under consideration, the height ofthe drop, h(r,t), is a function of both the distance, r, from the coordinate origin(which is located on the plane of the solid surface in the center of the drop)

FIGURE 3.8 Spreading of liquid droplets over the solid surface covered with a film ofthe same liquid of uniform thickness h0.

dB dt t/ = − +( )0 03 0 1. .

B t= − +( )3 0 03 0 1. ln .

L∗

h0r0(t)

r

h(r, t)

H(0, t)

0



(Figure 3.8), and the time, t. For a sufficiently low-sloped drop, for which grav-itational and inertial forces can be neglected, the equation describing the processof viscous spreading of the drop is deduced in Section 3.1 (Equation 3.17). Thisassumption means h* << r*, that is, the length scale in the vertical direction ismuch smaller than in the horizontal direction. In view of the symmetry of thedrop, the boundary conditions in the droplet center (Equation 3.17’) are satisfied.

As flow occurs only where the liquid surface is curved, so that there is noflow going out into the film to infinity, the excess volume of liquid above thefilm remains constant:

. (3.107)

Far from the droplet, its profile tends to that of the film, and hence

. (3.108)

In addition to the four boundary conditions (3.17’), (3.107), and (3.108),Equation 3.17, which is the fourth order partial differential equation, requires theassignment of an initial condition, which is formulated in the following text. Letus introduce the dimensionless quantities using the characteristic scales r*, h*, t*,and U* for the horizontal dimension, the height of the drop, the time, and thespreading velocity, respectively, with U* = r*/t* and V = 2πr*

2h*, the latter rela-tionship determining the horizontal scale of r* with a given drop volume andinitial height following from the conservation law (3.107) in the same way as inSections 3.1 and 3.2. Using the following dimensionless variables r → r/r*, t →t/t*, h→ h/h*, and h0 → h0/h*‚ keeping the same symbols for dimensionless valuesas for dimensional, we arrive at the equation of spreading:

, (3.109)

where the dimensionless parameter ε:

(3.110)

which, as before, represents the ratio of characteristic values of the force ofviscous friction in the drop Fη = ηU*/h*

2 and the horizontal component of thegradient of capillary pressure Fγ = γh*/r*

3. The conservation law (3.107) nowbecomes

2 0

0

π r h r t h drR V const,( ) − = =∞

∫

h r t h r( , ) ,= → ∞0

1 13

r rrh

r r rr

hr

∂∂

∂∂

∂∂

∂∂

= − ∂ε hh

t∂

ε ηγ

= 3 3

3

U rh* *

*



, (3.111)

and the condition (3.108) has an identical form.It is easy to check that

ε << 1. (3.112)

We also assume that the film thickness is much smaller than the characteristicsize of the droplet; that is,

h0 << 1. (3.113)

In solving this problem, we use the method of matching of asymptotic expan-sions, which we already used in Section 3.2. In view of the smallness of theparameter ε, the entire drop can be subdivided into two zones: the outer zone 0 ≤r < r0(t) that determines the macroscopic boundary of the drop, i.e., the coordinateat which h(r0(t),t) ≈ h0, and the inner zone r > r0(t) with the characteristic scale,L*, much smaller than the scale of the outer region (Figure 3.8): L* << r*. Thus,the integral in Equation 3.111 can be rewritten as follows:

These integrals can be estimated using dimensional variables as

.

This estimation shows that

;

hence, the entire volume of the drop, with the accuracy with which the problemis being solved, is actually concentrated in the outer zone; i.e., we can set

(3.114)

r h r t h dr,( ) − =∞

∫ 0

0

1

r h h dr r h h dr r h h dr

r t

−( ) = −( ) − −( )∞( )∞

∫∫∫ 0 0 0

000

0

.

r h h dr h r r h h dr h Lr

r

−( ) ≈ −( ) ≈∞

∫∫ 0 0 0

0 0

0

* * *and

r h h dr r h h drhh

Lr

r

r

−( ) −( ) ≈ <<∞

∫ ∫0 0

0

0

0

0

1*

*

*

r h h dr

r t

−( ) ≈( )

∫ 0

0

1

0

.



In the zeroth approximation we conclude from Equation 3.109, setting ε =0, we obtain the equation for the outer solution:

. (3.115)

The boundary condition (3.108) should now be written as

. (3.116)

We obtain the solution of the problem (3.115), (3.111), (3.114), and (3.116),in the following form:

, (3.117)

where

, (3.118)

and f(t) is a new unknown function of time, which is determined in the followingtext; 0 ≤ ξ < 1. It follows from Equation 3.117 that the drop has the parabolicshape; the apparent radius of the spreading drop corresponds to ξ = 1, or

. (3.119)

In order to obtain the inner solution of the problem (3.109), (3.108), whichis valid close to the boundary of the spreading drop, we introduce new scales forthe space variable and a new form of the unknown h(r, t):

, (3.120)

,

where χ(t) and ψ(µ) are new unknown functions subject to determination; µ isthe new local variable inside the inner zone; χ(t) << 1. Note, in this section, h0

= const, which is different from Section 3.2. In the following text we present aslightly modified way of deducing the inner equation for the droplet profile ascompared with that in the previous section. This method is more appropriate forthe problem under consideration. To do this we integrate Equation 3.109 withrespect to the variable r:

. (3.121)

∂∂

∂∂

∂∂

=r r r

rhr

10

h r t h0 0,( ) ≈

h r t f t h,( ) = ( ) −( ) +4 12 20ξ

ξ = ( )rf t

r t f t0 1( ) = ( )

ξ χ µ− = ( )1 t

h h= ( )0ψ µ

rhr r r

rhr

rht

dr

r

3

0

1∂∂

∂∂

∂∂

= − ∂∂∫ε



We now calculate the integral on the right side of Equation 3.121, using theouter solution, as the integration takes place mostly across this region:

Substituting the resulting expression, together with the transformation (3.120),into Equation 3.121, we obtain

. (3.122)

Equation 3.122 should include only the variable µ; the latter gives the fol-lowing requirement:

. (3.123)

Hence, Equation 3.122 can be rewritten as

. (3.124)

The matching of the inner (3.120) and outer (3.118) solutions is determinedby condition

. (3.125)

This means that the following condition must be fulfilled:

, (3.126)

where B can no longer be selected arbitrarily but is determined by the matchingcondition.

− ∂∂

= − ( )∂∂

= − ( ) ′ −( )ε ε ξ ξ ε ξrr

tdr

f t

h

td

f tff

2 224

2 1 −− ′

∫∫∫ 2 2

000

f rf d

r

ξ ξ ξξξ

== − ′′ −( ) = − ′′ −( ) = − ′8 1 2 42 2 4

3

4

ε ξ ξ ξ ε ξ ξ ε ξf

fd

f

f

f

f22

0

0

h h−( )∫ξ

.

ψ ψµ

ε χψ3

3

303

3

51

d

d h

t f t

f t= −

( ) ′( )( ) −( )

ε χh

f

f03

2

51

′ =

ψ ψµ

ψ33

3 1dd

= −

dd

f t t

hψµ

χ= −

( ) ( )8 2

0

8 20f t t h B const( ) ( ) = =χ



The two equations (3.123) and (3.126) determine the two unknown functionsf(t) and χ(t) that have been introduced in the preceding text. The expressions(3.125) and (3.126), together with the boundary condition (3.108), give thefollowing boundary conditions for Equation 3.124:

, (3.127)

. (3.128)

The problem (3.124), (3.127), and (3.128) is investigated in Section 3.5. Itis shown that the function ψ(µ) has the form of damped oscillations as µ → +∞,

. (3.129)

When we take conditions (3.123) and (3.126) into account, we obtain

. (3.130)

Now we can return to the question of the initial condition for Equation 3.109.The analysis that has been performed is valid only after the passage of a certaininitial period of time, tin, at which the spreading regime described in this sectionis established. As we already mentioned, the duration of the initial stage ofspreading when both Re and Ca numbers are not small is estimated in Section 4.1.The moment t = tin is taken as the initial moment of time. At this time, a parabolicprofile has been formed in the center of the drop; for the assignment of this profileat the initial moment, all that must be known (apart from the condition ofconstancy of volume and initial radius r0(0)) is the height at the center, h(0,0).Selecting h(0,0)-h0 = h* at the scale on the z axis, we obtain the missing initialcondition for Equation 3.109:

. (3.131)

This condition determines the initial condition for the function f(t) that is thesolution of Equation 3.130: f(0) = 1/2. This makes it possible to determine fromEquation 3.130 and Equation 3.126 the functions f(t) and χ(t):

, (3.132)

dd

Bψµ

= −

ψµ→∞

= 1

ψ µ µ µ( ) = + −( )

1 2

32

C exp sin/

B f

f81

3

11

′ = −ε

h h0 0 1 0,( ) = +

f tB

t( ) = +

−12

51

3

1 10

/ε/



. (3.133)

The expression for χ(t) gives the scale of the inner zone, which should besmall, that is, χ(t) << 1. It follows from Equation 3.133 that this condition issatisfied if

. (3.134)

This provides the required restriction on the duration of the spreading process.After that the droplet becomes too small and undistinguished from the film.

From Equation 3.119 and Equation 3.132 we find an expression for the radiusof the spreading drop:

,

or in dimensional variables,

. (3.135)

From these expressions we obtain the time dependence of the dynamic contactangle at the drop boundary as

.

Expressing the right side of the last equation in terms of the spreadingvelocity, U(t) = dr0/dt, we arrive at

. (3.136)

It is possible to check that the dimensional combination appearing in Equation3.110 for the parameter ε does not depend upon the selection of the initial momentof time. Indeed, as we already established previously, the characteristic values of

χ εtB

hB

t( ) = +

2

510 3

1 5

//

Bh

Bt t

h B25

1 12

50 3

1 5 5

05 2

/ ,/

ε ε+

<< <<or

r tB

t0 3

1 10

25

1( ) = +

/ε/

r t rB

hr

t0 3

3

4

0 1

25

31( ) = +

*

*

*

.γη

tghr

hr B

tr r

θ ε= − ∂∂

= +

=

−

0

51

3

3 10

*

*

/

/

tg BU

B Caθ ηγ

=

= ( )33

1 31 3

//



the quantities appearing in this combination vary with time in accordance with thelaws r* ≈ r0(t) ≈ t1/10, h* ≈ h(0, t) ≈ t –1/5, t* ≈ t, i.e., it remains constant with time:r*

4/t*h*3 = const.

Differentiating both sides of Equation 3.135 with respect to time, we findthat, at the initial moment, t = 0,

. (3.137)

As r0 (0) = 2r*, taking as the characteristic value of velocity

,

from Equation 3.137 we find the characteristic value of spreading time: t* =(6Bηr*

4/(γh*3)).

In Section 3.2 we showed that Equation 3.124 does not have any solutionssatisfying the matching condition (3.127); therefore, in place of the matching ofthe outer (3.118) and inner (3.120) solutions, we require, as in Section 3.2, thatthey must be patched at a certain point: µ = –µ*:dψ/dµ = –B.

We determine the patching point, µ*, in precisely the same way as in Section3.2. The outer solution (3.118) obtained from Equation 3.109 with ε = 0 losesits meaning if its left-hand side becomes of the same order of magnitude as theright-hand side, i.e., when h3 = ε. We set as the patching point h3 = ε or, in viewof (3.120), h0

3 ψ3(–µ*) = ε.Equation 3.124 as µ → –∞ has the asymptotic representation

(see Section 3.2 for details), which gives the following equation for determiningof the quantity B. All details are identical to those presented in Section 3.2. Thisresults in

exp and . (3.138)

If we now set h* ≈ 0.1 cm and h0 ≈ 10–5 cm, then h0 ≈ 10–4; for B and ε weobtain the following values: B ≈ 2.753, ε ≈ 0.024. Correspondingly, µ* ≈ 1048and ψ(–µ*) ≈ 2886.

The calculated values of B and ε are very close to those found in the previoussection for the spreading over a dry substrate. Therefore, in the case of complete

drdt B

h

r0

3

3

3

1

3= γ

η*

*

220U

drdt

rt*

*

*

= =

ψ ξ ξ ξ ψξ

ξ ψ µ µ( ) ≈ ≈ ( ) ≈3 3 33 1 3 3 1 3 3 1 3ln , ln ln/ / /dd

µµ ψµ

µ, ln /dd

≈ − 33 1 3

B3 B h3031 2( ) = / ε = 1 2 3/ B



wetting, the preexponential constant is almost insensitive to the conditions infront of the spreading droplet. We will see a further evidence of this insensitivityin Section 4.1.

3.4 QUASI-STEADY-STATE APPROACH TO THE KINETICS OF SPREADING

A simplified approach is suggested here to the solution of problems of liquiddrop spreading. The approach consists essentially of assuming constancy of the dropspreading velocity, U, at each fixed moment of time, t. This gives the possibilityof determining the relationship U = f(r0), where r0 is the radius of the drop base.As U = dr0/dt, this equation can be used to obtain an expression for the spreadingradius as a function of time. The applicability of the method has been demon-strated using examples of spreading on a solid substrate (with disjoining pressureacting in the vicinity of the apparent three-phase moving contact line), spreadingon a prewetted solid substrate, and spreading under the influence of gravity andunder the influence of applied temperature gradient [52].

In Section 3.1 through Section 3.3 a number of spreading problems have beensolved. In the case of spreading of small drops over dry solid substrate (Section3.2), the explicit expressions obtained for the drop radius and height and thedynamic contact angle as functions of time. These expressions do not includeany fitting parameters and depend solely on the hydrodynamic characteristics ofthe liquid, including the disjoining pressure isotherm, proved to be in goodagreement with experimental data. This solution was limited to the case ofcomplete wetting for disjoining pressure isotherms of the type Π(h) = A/hn, whereA > 0 and n ≥ 2. Similarity solution of gravitational regime of spreading andspreading of very small liquid drops were obtained in Section 3.1, and spreadingover prewetted solid substrate was solved in Section 3.3. In all cases, the solutionincluded more or less sophisticated mathematical treatment. From an analysis ofall of the relationships that have been found between the drop radius and time,we arrive at a hypothesis of a “quasi-steady-state nature” of the spreading process.According to this hypothesis, the characteristics of spreading are determined forthe most part only by the instantaneous value of the spreading velocity.

Let us first examine the problem of spreading of a low sloped drop of aviscous liquid on a horizontal surface covered with a layer of the same liquidwith a thickness h0 — that is, the same problem as in Section 3.3. The samenotations as in Section 3.3 are used here. Let h(r,t) be the equation of the dropprofile; r is the radial coordinate, and t is the time. We use the characteristicscales; h*, r*, and t*, respectively. The following relationships are satisfied: h* >>h0; r* = r0(0)/2, where r0(t) is the radius of drop spreading; h* << r*. Then, asshown in Section 3.3, in dimensionless variables h → h/h*, r →r/r*, t → t/t*, andh0 → h0/h*, the spreading process is described by the differential equation (3.109),with conservation law (3.107) and boundary conditions (Equation 3.17’) and(3.108).



As was shown in Section 3.3, the condition ε << 1 is usually met; i.e., viscousforces are small in comparison with capillary forces. Now, in Equation 3.109,we use a new quasi-steady-state variable:

, (3.139)

where r0(t) is the spreading radius. Let be the spreading velocity; theoverdot denotes the derivative with respect to time t. Using these notations andassuming that the liquid profile depends on the new variable only — that is, h =h(ξ) — we obtain from Equation 3.39 that

. (3.140)

Setting ε = 0 in the equation, we obtain, in the same way as in Section 3.3,the outer solution of the problem:

,

where C is the integration constant, determined from conservation law (3.107).That gives C = 4/r0

2, whereas the outer solution is

, (3.141)

which coincides with the outer solution deduced in Section 3.3 (Equation 3.117).Equation 3.141 describes a parabolic profile of the drop and is valid far from theapparent moving three-phase contact line.

In the vicinity of the moving apparent three-phase contact line, ξ = 0, wherethe profile of the outer solution (3.141) intersects the surface of the liquid film,we introduce the inner variable as before:

(3.142)

Using the new inner variable and retaining in Equation 3.140 only the leadingterms, we obtain

ξ = − ( )r r t0

U r t= �0( )

εξ ξ ξ

ξξ ξ ξ

ξξ

Udhd

dd

hd

dd

ddhd

=

1 13

h C r h= −( ) +1 202

0ξ /

hr

r h= −( ) +41

02

202

0ξ /

µ ξχ

χ

ψ µ

= − <<

= ( )

r

h h

0

0

1,

.



. (3.143)

The requirement of “self-similarity” with the “frozen” time t results in

(3.144)

Integration of Equation 3.144 with the condition yields Equa-tion 3.124.

The condition of “matching” the outer solution (3.141) and the inner solution(3.124) at a certain point µ = –µ* has the form similar to that deduced previously(3.127):

. (3.145)

Now, using the first equation (3.144) and (3.145), we obtain

. (3.146)

We now take into account the fact that U = dr0 /dt. Then, from Equation 3.146with the initial condition r0(0) = 2, we obtain

. (3.147)

The law of spreading (3.147) coincides with that obtained previously inSection 3.3 by a more rigorous method. However (see the following text), theconstant B that appears in the spreading law (3.147), determining the point ofmatching of the outer and inner solutions differs from that found in Section 3.3.We assume, as in Section 3.3, that at the point of matching, h3 = ε. In this case,the right and left sides of Equation 3.140 have identical orders. Using thiscondition we conclude that

. (3.148)

ε χ ψµ µ

ψ ψµ

Urh

dd

dd

dd

03 3

03

33

3=

ε χ

ψµ µ

ψ ψµ

Ur

h

dd

dd

d

d

03 3

03

33

3

1=

=

.

ψ µ→ → ∞1,

dd r h

B constψη

χη η=− = − = − =*

8

02

0

εUB

r=

−83

09

rB

t0 3

0 1

25

1= +

ε

,

h h03 3

03 3ψ µ ψ ε−( ) = =* *



Taking as the initial condition for the velocity U(0) = 2 as in Section 3.3, weconclude from Equation 3.148 that the relation is identical to relation (3.138) inthe previous section: ε = 1/(2B3). The solution of Equation 3.124 has the asymp-totic form (see Section 3.2)

(3.149)

Now, using Equation 3.145, Equation 3.148, and Equation 3.149, we concludethat

.

For the determination of the constant B, we can limit ourselves in Equation3.148 to the initial moment of time, i.e., we can set U = U(0) = 2. Then the valueof B satisfies the relation ship

. (3.150)

Equation 3.150 is very similar to Equation 3.138, which determines theconstant B. The latter equation was obtained by a rigorous method in Section3.3, but it has a right side that is exactly twice as large. Taking as an examplethe same values as in Section 3.3: h* = 10–1 cm, h0 = 10–5 cm, and, consequently,h0 = 10–4, we obtain from Equation 3.150 the following values: B ≈ 2.781 andε ≈ 0.023, whereas the more rigorous method gave B ≈ 2.753 and ε ≈ 0.024. Ascan be seen, the approximate method under consideration, which is based on aquasi-steady-state approach, gives results in this particular problem that are verylittle different from those obtained by the rigorous method.

Let us pass on now to the case of spreading of a liquid drop over a dry surfacein the case of complete wetting — that is, with the disjoining pressure in thefollowing form: Π(h) = A/hn (A > 0, n ≥ 2). In this case, as was shown in Section3.2, the spreading equation is given by Equation 3.63.

We now change over, the same way as done previously, to a quasi-steady-state variable (3.139). This is substituted into Equation 3.63, which gives

. (3.151)

ψ η η

ψη

η η

3 3

3

3

3

≈

≈ − → −∞

ln

ln , .dd

ψ ε µ η µ* * * *ln exp303 3 3 3 3 33= = = = ( )U h B B B/

B B h6 3031exp( ) = /

εξ ξ ξ

ξξ ξ ξ

ξξ

λU

dhd

dd

hd

dd

ddhd h

=

−1 13

nn

dhd+

1 ξ



With , we obtain the outer solution of Equation 3.151 similar toEquation 3.68, which is valid far from the moving three-phase contact line:

. (3.152)

In the vicinity of the point ξ = 0, which corresponds to the apparent three-phase contact line, we carry out a replacement of variables in Equation 3.142, inwhich h0 is now a new unknown function to be determined.

The requirement of self-similarity of Equation 3.151 results in the followingconditions:

(3.153)

and the equation itself for the inner solution ψ(µ) describing the drop surfaceprofile in the vicinity of the moving apparent three-phase contact line takes theform

or, after integration, considering that ,

. (3.154)

Equation 3.154 was investigated in detail in Section 3.2 (see also the end ofSection 3.1).

The condition of matching of the inner and outer solutions, as discussed, hasthe form (3.145), and it leads, after taking into account that U = dr0/dt, to theexpression (3.147) for the spreading radius r0. Expressions for the unknownfunctions h0(t) and χ(t), determining the scale of the zone of the inner solution,can be found easily using Equation 3.153 and Equation 3.147 (see Section 3.2for details).

We focus below on determining the constant B, which, according to Equation3.148, determines the point of matching of the solutions and, in view of (3.138),determines the magnitude of the small parameter ε.

ε λ= = 0

hr

r= −( )41

02

202ξ /

ε χ

λ χ

Ur

h

r

hn

03

03

02

01

1

1

=

=+

,

dd

dd

dd

ddn

ψµ µ

ψ ψµ ψ

ψµ

= −

+3

3

3 1

1

ψ µ→ → +∞0,

ψ ψµ ψ

ψµ

23

3 1

11

dd

ddn−

=+



Equation 3.154, as µ → –∞, has the previous asymptotic behavior (3.149).Hence, in the same way as previously, we deduce that

. (3.155)

Substituting into Equation 3.153 r0 = 2 and U = 2 (values corresponding tothe initial moment of spreading), we now obtain

.

After that, using Equation 3.155 and taking into account that ε = 1/B3, weobtain

. (3.156)

Equation 3.156 for the determination of the constant B differs from Equation3.89, obtained by the more rigorous method in Section 3.2 only in the absenceof the factor 1/ on the left-hand side. Comparison of numerical results obtainedusing the quasi-steady-state approach, Equation 3.156, with those obtained inSection 3.2, Equation 3.89, for the examples that were considered in Section 3.2,give almost identical numerical values for both n = 2 and n = 3 for both B and ε.

Thus, in this particular problem as well, the proposed quasi-steady-statemethod of investigation of spreading has led to results that are essentially notdifferent from those obtained previously by a more rigorous method.

Now let us apply the proposed method to the solution of the problem of liquid-drop spreading on a horizontal substrate under the influence of gravitational forces.Neglecting capillary forces and considering the spreading of a two-dimensional(cylindrical) drop (along the OX axis), we write the spreading equation in dimen-sionless form according to Section 3.1 as

, (3.157)

where ρ is the density of the liquid; g is the acceleration ofgravity. The conservation of volume, V, in this case has the form

, (3.158)

ψ ε*

/exp= ( ) = ( )U h B B

1 3

03 3/

h U Bn0

1 2 3 2− = ( ) =λ ε λ/

132

11 3

BB B Bnλ( ) = ( )−

− exp /

23

∂∂

= ∂∂

∂∂

ht x

hhx

β 3

β ρ η= gh t r* * * ;3 2

20

0

hdx V

x

=∫



where x0 is the spreading radius, and V is the volume per unit length of thespreading droplet. Selection

,

results in

, (3.159)

and x0(t) is now dimensionless radius of spreading.Now, performing the replacement of variables according to Equation 3.139,

we obtain from Equation 3.157:

,

or, after integration,

.

Hence,

.

Using the condition of conservation of volume, we obtain

.

Therefore, from this equation we conclude that

,

hVr

**

=2

hdx

x

=∫ 10

0

β ρ η= gV t r3 88* * ,

− =

Udhd

dd

hdhdξ

βξ ξ

3

hdhd

U2

ξ β= −

hU= −

31 3

1 3

βξ

/

/

134

31 3

04 3

0

0

= =

−∫ hd

Ux

x

ξβ

/

/

U x x= = ( )�0 043

43β



and the drop-spreading law has the following form:

. (3.160)

The shape of the drop surface is determined here by the equation

. (3.161)

The spreading law (3.160) differs from the self-similar solution obtained inSection 5.5 (see also Reference 22) only in the value of the constant coefficient:k = 1.411 instead of k = 1.316.

Now, let the drop spreading take place under the influence of a temperaturegradient only. We consider the spreading of a nonvolatile liquid droplet using alow slope approximation — that is, h* << r* — and only spreading of a two-dimensional droplet is considered.

The Navier–Stokes equation in the case under consideration takes the fol-lowing form:

.

These equations give

, (3.162)

with the following two boundary conditions: a nonslip condition on the solidsubstrate

vx(0) = 0, (3.163)

and the condition of applied tangential stress on the droplet surface, caused bythe surface tension gradient, which in turn is caused by the applied temperaturegradient:

. (3.164)

x k t ks

s

01

132081

1 316= ( ) =

≈β /

/

, .

h x x x= −( ) ( )4 301 3

04 3/ /

∂∂

= ∂∂

∂∂

=

px

vz

pz

xη2

2

0

p p x vpx

z C z Cx= = ∂∂

+ +( ),1

22

1 2η

η γ γ∂∂

= ∂∂

= ∂∂

==

vz x

ddT

Tx

x

z h

Λ



Note that, in the latter condition we assume, the thickness of the droplet issmall enough, to neglect the temperature variation through the vertical crosssection of the droplet. Hence, the temperature is equal to the temperature of thesolid support. Note that, usually,

;

that is, the liquid–air interfacial tension decreases with temperature. In the fol-lowing text we consider the spreading of the droplet from the cold side in thecenter of the droplet to the hotter part of the solid substrate under the action ofthe constant temperature gradient. It is also assumed that in the range of temper-ature under consideration, the derivative, dγ /dT, is a negative constant. Hence,under the above assumption, Λ < 0 and remains constant.

Using the latter two boundary conditions, we conclude from Equation 3.162that

. (3.165)

The governing equation results from the integration of the continuity equationis

. (3.166)

Substitution of expression for the velocity according to Equation 3.165 intothe governing equation (3.166) yields

, (3.167)

with the conservation law according to Equation 3.158. The pressure inside thespreading droplet is given by Equation 3.5. However, in the following text weconcentrate on the spreading under the action of the temperature gradient only.That is, we assume that

.

Here we consider the negative values of the constant Λ.

ddT

γ < 0

vpx

zzh zx = ∂

∂−

+12

2

η ηΛ

∂∂

= − ∂∂ ∫h

t xv dzx

h

0

∂∂

= ∂∂

∂∂

−

ht x

h px

h13 2

3 2

ηΛ

2hr

p*

** << Λ



In this case Equation 3.167 becomes

, (3.168)

where

.

After the replacement of the variable according to Equation 3.139, we obtain

, (3.169)

or, after integration,

. (3.170)

In the situation under consideration, the drop surface has an abrupt changeat the point x = x0 (ξ = 0) corresponding to the liquid propagation front. Followingthe method described in Chapter 5, Section 5.6, we obtain the condition that mustbe satisfied by the solution of Equation 3.168 at the moving front.

We integrate Equation 3.168 over x from x– = x0 – δ to x+ = x0 + δ, where δis a small value:

.

Let us calculate

.

Hence,

,

where are thickness of the droplet just in front and behindthe moving front.

∂∂

= ∂∂

ht

hx

α2

αη

=Λ h t

r* *

*2

Udhd

dhdxξ

α=2

Uh h C= +α 2

∂∂

= −( )−

+

∫ + −ht

dx h hx

x

α 2 2

ddt

hdx x h hhtdx

x

x

x

x

= −( ) + ∂∂

−

+

−

+

∫ ∫+ −�0

∂∂

= − + −−

+

−

+

∫ ∫ + −htdx

ddt

hdx x h hx

x

x

x

�0( )

h h x h h x+ + − −= =( ), ( )



Taking the limit in the equation at (i.e., δ → 0), we obtain

.

In front of the moving droplet the solid substrate is dry; that is, h+ = 0. Hence,from the preceding equation we conclude that

. (3.171)

Equation 3.171, upon substitution into Equation 3.170, gives C = 0, and thusthe drop has the form of a flat step (pancake) with a height

, (3.172)

and a radius of the moving front x0(t). Equation 3.172 shows that the thicknessof the flat drop is a function of time only.

From conservation law (3.159) we conclude now that

.

Taking into account and h = 1/x0, we finally get the spreadinglaw as

. (3.173)

Summarizing the results obtained in this work, we can say that the proposedapproximate quasi-steady-state method for solving spreading problems gives asatisfactory accuracy of solution in all of the cases examined in this section. Themethod itself is mathematically simple, and it offers a means for reducing aproblem of complex nonlinear partial differential equations to the solution ofordinary differential equations. The quasi-steady-state approach that we haveexamined in this section can be used for the investigation of more complexspreading problems.

3.5 DYNAMIC ADVANCING CONTACT ANGLE AND THE FORM OF THE MOVING MENISCUS IN FLAT CAPILLARIES IN THE CASE OF COMPLETE WETTING

In this section we shall analyze how both the disjoining pressure and the widthof the capillary influence the dynamic advancing contact angle of the meniscus

x x x x− +→ →0 0,

�x h h h h02 2

+ − + −−( ) = −( )α

�x h0 = −α

h U= /α

10

0

0

0

0

= = =∫ ∫−

hdx hd Ux

x

x

ξ α/

U x x= =�0 0α/

x t0 2= α



of a liquid, completely wetting the substrate. For simplicity we shall considerthat the meniscus moves from an equilibrium position in a flat capillary. Theeffect of both the disjoining pressure and the width of the capillary on the dynamiccontact angle is investigated, following Reference 53. The problem of the formof the advancing meniscus and the dynamic contact angles have been consideredearlier [23–25], neglecting the disjoining pressure of the thin layer of liquids andlimiting the discussion only to capillary forces. However, in the close vicinity ofthe moving apparent three-phase contact line, the thickness on the liquid is sothin that the influence of disjoining pressure becomes significant.

The thickness of the film on the walls of the capillary ahead of the advancingmeniscus was assumed to be arbitrary in Reference 23 to Reference 25. Thismeans that, up to the start of the flow, the system is not in equilibrium, and that,consequently, there is flow from the meniscus to the film or the contrary, notconnected with the dynamics of the meniscus. These flows were also neglectedin solution of the problems in Reference 23 to Reference 25.

As has been shown earlier (Chapter 2, Section 2.2), in a state of equilibrium(Figure 3.9a) the capillary pressure of the meniscus, Pe, is connected with theisotherm of the disjoining pressure of flat wetting films, Π(h), by the followingrelationship:

, (3.174)

where γ is the surface tension of the liquid, H is the half-width of the capillary,and he is the thickness of the equilibrium film on the solid surfaces correspondingto Π(he) = Pe.

The relative pressure of the vapor, p/ps, above the films and the meniscusrelated to the equilibrium pressure, Pe, according to Equation 1.7 in Chapter 1.

FIGURE 3.9 Schematic presentation of the profile of the meniscus in a flat capillary.(a) at equilibrium, (b) velocity of motion below the critical velocity; ξ1, the only minimumon the liquid profile; (c) velocity of motion is above the critical velocity, ξ1 and ξ2 arethickness of the first minimum and maximum on the liquid profile; (1) the sphericalmeniscus, (2) the transition zone, (3) flat equilibrium film.

2H

he

her r

he he

U1 U2

U = 0

1

(a) (b) (c)

2 3x x x

ξ1

θd

ξ1 ξ2re

θd

P dh H he

h

e e

e

= +

−( ) =∞

∫γ Π Π



In sufficiently wide capillaries (H >> he), the meniscus in a central region has aspherical shape with a constant radius of curvature re = γ/Pe.

Below we solve the problem for the case of complete wetting, that is, forisotherms of the disjoining pressure of the type

, (3.175)

where n ≥ 2, and A is the constant of the surface forces (Hamaker constant inthe case n = 3). In Figure 3.9, schemes illustrating the profiles of the meniscusin a flat capillary in a state of equilibrium (a) and with different rates of motion(b, c) are presented.

In Chapter 2, Section 2.4, we showed that the radius of curvature of anequilibrium meniscus re < H, and in this case it is equal to

.

The form of the equilibrium profile of the liquid in the transitional zonebetween the meniscus and the film for isotherms of the type (175) was investigatedin Chapter 2, Section 2.4.

Let us consider the motion of the advancing meniscus in a flat capillary(Figure 3.9b and Figure 3.9c) from a state of equilibrium (Figure 3.9a). The zoneof the flow in which the main hydrodynamic resistance is exerted takes in aregion of the thicknesses of the layer h(x) above the surface of the substrate onthe order of he. x-axis is directed along the capillary axis (Figure 3.9).

According to the Introduction to this chapter, the profile of the movingmeniscus (Figure 3.9b and Figure 3.9c) can be subdivided at small capillarynumber, Ca << 1, in two regions: the outer region, where the meniscus has aspherical shape of an unknown radius of curvature, r, which is the function ofthe velocity of motion (or the same, capillary number, Ca), and the inner regionin between the spherical meniscus and the initial equilibrium flat film. In thelatter region, according to our estimations in the Introduction to this chapter, thecurvature of the liquid profile is small and a low slope approximation can be used.

In the case under consideration, the equation of spreading (3.62) from Section3.2 should be rewritten as

,

with the boundary condition

.

Π = >A hn/ 0

rP

Hn

hhe

ee= = −

−

γ1

∂∂

γη

∂∂ γ

∂∂

ht x

hhx

nAh

hrn= − ∂

∂−

−3

32

h h xe→ → ∞,



At the steady-state motion we can introduce a new coordinate system movingwith the meniscus y = x – Ut, which results in

, (3.176)

with the boundary condition

. (3.177)

Integration of Equation 3.176 with boundary condition (3.177) yields

, (3.178)

where ′ means differentiation with y.Let us introduce the following dimensionless variables z = y/y*,

means differentiation with z,

and the only one dimensionless parameter, which characterizes the intensity ofthe disjoining pressure action:

. (3.179)

It is important to note that using these scales we get the following estimationof the derivative in the flow zone:

.

The low slope condition is really satisfied in the flow zone in a vicinity of theapparent moving three-phase contact line as we predicted in the Introduction tothis chapter. It should be remembered that in the case under consideration, Ca ~10–6; hence, (Ca)1/3 ~ 0.01 << 1.

Substituting the aforementioned dimensionless variables into Equation 3.178results in

Ud hdy

dd y

hdhdy

nAh

d hdyn= −

−

γη γ3

32

h h ye→ → ∞,

γη γh

hnAh

h U h hn e

3

13′′′ − ′

= −( )+

ξ = h he/ ;′ ′′′ξ ξand

y h Uh

Cae

e*

/

/( )= ( ) =γ η/3

3

1 3

1 3

αγ η

=( )

−( )nA P

U

ne

n n1 1

1 3 2 33

/ /

/ /

′ = <<hhy

Cae~ ( )*

/3 11 3



(3.180)

In Appendix 5 we show (see Equation A5.9) that Equation 3.180 has theproper asymptotic behavior at z → –∞. In this sense, the problem under consid-eration is completely mathematically correct: it is possible to match asymptoticexpansions in the outer region (meniscus) and inner region in a vicinity of themoving apparent three-phase contact line. In the following text we undertake thematching procedure numerically.

Before passing on to the results of a numerical calculation according toEquation 3.180, let us analyze the equation. Considering the behavior of thedimensionless thickness, ξ, far from the meniscus (i.e., at 1 – ξ << 1), introducingυ = ξ – 1 and linearizing Equation 3.180, we conclude that

. (3.181)

We seek the solution of Equation 3.181 in the form Substitutionof this expression into Equation 3.181 yields

. (3.182)

The discriminant of Equation 3.182, Q = (1/4) – (α/3)3, can be either positiveor negative. If Q > 0 (i.e., with α < αc ≅ 1.89), Equation 3.182 has one realpositive root and two conjugate complex roots, whose real parts are negative.This corresponds to the presence of damped waves ahead of the moving meniscus.Such a situation (Figure 3.9c) is possible with U > Uc (with A and H = const), orwith A < Ac (with U and H = const) or, finally, with H > Hc (with U and A =const). With a decrease in α (with α < αc), the imaginary part of the roots risesmonotonically, and the real part decreases, which corresponds to an increase inthe amplitude and a decrease in the length of the surface waves.

At Q < 0 (i.e., with α > αc), Equation 3.182 has two negative real roots: –λ1

and –λ2 (λ1 > λ > 0) and one positive real root, λ3 > 0. These roots, as a functionof α, have the following properties: λ1 < , λ2 < . At α → ∞, λ2 → andλ2 ≈ 1/α. It follows from this that at α > αc, the function ξ(x) has a single minimumat the point z ≅ (2/ ) ln , which corresponds to the situation illustrated inFigure 3.9b. Thus, profiles of this type, as in Figure 3.9b, are realized with α >αc, i.e., with rather large values of A, and/or at a small velocity of motion of themeniscus U, or in narrow capillaries.

With an increase in the velocity of the flow or a decrease in the value of Asuch that α becomes less than αc, wavy films are formed ahead of the movingmeniscus, as in Figure 3.9c. Thus, if the effect of surface forces is neglected,(A = 0) — that is, α = 0 — the solution of Equation 3.8 can only have wavyprofiles of the film in front of the moving meniscus, as has been obtained earlierin Reference 24. The effect of surface forces results (at α < αc) in damping of the

ξ ξ αξ ξ ξ3 2 1′′′ − ′ = −−n .

′′′ − ′ =υ αυ 0

υ λ= exp( ).x

λ αλ3 1 0− − =

α α α

α α



waves and to an increase in the wavelength (still at α < αc) ahead of the movingmeniscus. The latter consideration shows the qualitative effect of the influenceof disjoining pressure of thin layers of liquids on hydrodynamics of moving ofmenisci.

Figure 3.10a gives the numerically calculated values of H/r (calculatedaccording to Equation 3.180) as a function of the parameter (Ca)2/3 for differentvalues of the width of the capillaries H. The values of r are located along a sectionof constant curvature, adjacent to the zone of the flow, appearing with a rise inthe value of the thickness, h, but still in the region of the low sloped profile. Thetransition to this section corresponds to the condition d2h/dy2 = γ/r = const. AtH/r < 1, the values of H/r = cos θd, where θd is the dynamic contact angle (Figure3.9b, c). Calculations were made for the following parameters A = 10–7 dyn andn = 2, characteristic for the value of β films of water on the surface of quartz[26]. The corresponding value of γ was taken as equal to 72 dyn/cm, and theviscosity η = 0.01P.

Figure 3.10b gives similar results for another isotherm Π(h), characteristicfor wetting films of nonpolar liquids on a solid dielectric [26]: A = 10–4 erg, n =3, γ = 30 dyn/cm, η = 0.01P.

Figure 3.10 shows that the calculated dependences of cos on Ca (curves 1–3)are similar to those calculated using the approximate Friz equation [24]: tanθd ≅2.36 (curve 4). However, as distinct from the Friz calculations, our calcula-tions show a dependence of θd on the width of the capillary. This becomesparticularly noticeable in “narrow capillaries,” when H < 103cm, where the

FIGURE 3.10 Dependence of the ratio H/r on the capillary number, Ca = Uη/γ.(a) Disjoining pressure isotherm Π(h) = 10–7/h2, H = 10–2 cm, he = 360 Å (1); H = 1.25.10–3,he = 125 Å (2); H = 10–5 cm, he = 11 Å (3). (b) Disjoining pressure isotherm Π(h) =10–14/h3, H = 10–2 cm, he = 150 Å (1); H = 1.25.10–3 cm , he = 74 Å (2); H = 1.25.10–5 cm,he = 16 Å (3). The dotted line 4 relates to the Friz equation [24].

1.0

0.9

0.8 1

1.0

0.9

0.8

–4 –6 –8 –2 –3

4 2 3

H/r

H/r

�n (Ca)2/3

(a) (b) �n (Ca)2/3

2 1

3 4

θd

Ca



thickness of the surface film becomes less than 100 Å. In the general case, anincrease in H, as can be seen from Figure 3.10, leads to a decrease in cosθd, i.e.,to a rise in θd.

Figure 3.11 presents the dependence on Ca of the characteristic thicknessinside the flowing zone: ξ1 = h1/he is the point of the first minimum, and ξ2 =h2/he is the point of the first maximum. As can be seen from these curves, thevalues of ξ1 fall with an increase in the rate of motion of the meniscus. Underthese circumstances a lowering of the thickness of the film near the movingmeniscus is observed at lower rates of motion; the lower the rate, the wider thecapillary. Thus, in a capillary H = 10–2 cm, the thinning of the film near themeniscus starts with Ca > 10–6 (for water, at U > 10–2 cm/sec).

The relative height of the first maximum of ξ2, on the contrary, rises with anincrease in Ca, i.e., with a rise in the rate of motion of the meniscus. However,the appearance of convexity of the film is observed with considerably greatervelocity U than the appearance of concavity.

It is important to note that, with a further rise in the value of the velocity ofthe meniscus, U, the values of ξ1 and ξ2 are stabilized, i.e., the first wavepractically does not change its form; an increase in U is accompanied by a changein the form of the second, third, etc. waves, propagating ahead of the meniscus.Under these circumstances, the value of α, corresponding to a transition to awavy profile, found from the values of (3Ca)2/3 with ξ2 = 0 by extrapolation ofthe curve of ξ2 (Ca) in Figure 3.11b, were found to lie in the interval from 1.5 to2.0, i.e., they were close to the values of αc ≅ 1.89, obtained with a linearizedpreliminary analysis of Equation 3.180.

As can be seen from Figure 3.11, the form of the wavy film ahead of a movingmeniscus depends more strongly than cosθd on the width of the capillary, H, andthe thickness of the equilibrium film, he, in front of the moving meniscus. Anincrease in the value of the constant A, signifying an increase in the effect of thesurface forces, leads (with H = const) to a rise in ξ1 and to a lowering of ξ2, withidentical values of Ca. Thus, the effect of surface forces, as followed from the

FIGURE 3.11 Dependences of the thickness of the first minimum, ξ1 = h1/he (Figure3.9c), (a), and the first maximum, ξ2 = h2/he (Figure 3.9c), (b) on Ca, obtained for valuesof H = 10–2 (1), 10–3 (2), and 10–5 cm (3). The parameters of the isotherm: A = 10–5 dyn,n = 2.

1.0

0.9 3 2 2 1 3

1

0.8 –4

ξ1ξ2

–6 –8 –10 –4 –6

(a) (b)

1.00

1.02

1.04

�n (Ca)2/3/�



preliminary analysis of Equation 3.180, leads to damping of the waves and makesthe profile of the film of the moving meniscus smoother.

The calculations made offered the possibility of evaluating the effect ofsurface forces on the dynamic contact angles θd and the profile of the film aheadof the moving meniscus for isotherms of the disjoining pressure, correspondingto complete wetting.

APPENDIX 5

Asymptotic behavior of solution of at .

At sufficiently big thickness we can neglect the disjoining pressure action inEquation 3.180. Hence, we come to the following differential equation

, (A5.1)

with the following boundary conditions

y → +∞, x → –∞. (A5.2)

According to the boundary condition (A5.2), Equation A5.1 asymptoticallycan be written as

. (A5.3)

Let us introduce a new unknown function w(y) in Equation A5.3

. (A5.4)

Hence,

.

Substitution of this expression into Equation A5.3 results in

. (A5.5)

yd y

dxy3

3

31= − y → ∞

yd ydx

y33

3 1= −

yd y

dx2

3

31=

w ydydx

( ) =

12

2

dwdy

d yd x

d wd y w

d yd x

= = −2

2

2

2

3

3

1

2,

yd wd y w

22

2

1

2= −



Let us introduce a new variable in this equation:

. (A5.6)

Using the new variable in Equation A5.6, we arrive at

. (A5.7)

Solution of Equation A5.7 can be expressed via cylindrical functions. How-ever, for the investigation of the asymptotic behavior of Equation A5.2, we limitourselves by considering only the main power terms in asymptotic expansion:

. (A5.8)

Substitution of this expression into Equation A5.7 results in

.

Equalizing different exponents in the preceding equation, we conclude thatonly three options are available for the exponent k: (1) k = 1, (2) k = 1/4, (3) k =1/2.

At k = 1 we conclude from Equation A5.7: C = 1, hence, p = w. Substitutionof this expression into Equation A5.6 gives w′ = w, or w = 2C1y, where C1 > 0is the integration constant. The next step is the substitution into Equation A5.4,which results in y′ = . The solution of this equation is

, (A5.9)

where C2 is a new integration constant. Note that the constant C1 can be onlypositive. We use the asymptotic solution (A5.9) in the current section 3.5 and inthe next section 3.6.

The second solution (at k = 1/4) does not satisfy the requirement (A5.2).Let us consider now the last, third exponent k = 1/2 , which results according

to Equation A5.8 in

. (A5.10)

dwd

p w yξ

ξ= =( ), ln

pdpdw

pw

− = − 1

2

p Cwk=

C w Ckwwk k2 2 1

1 2

2−

−

= −/

4C1y

y C x C= +1 22( )

p w= −12

1 2/



Substitution of this expression into Equation A5.10 and Equation A5.6 results in

.

Casting the latter expression into Equation A5.4 yields:

(A5.11)

Integration of the equation results in

.

Integration by parts gives

.

At y → + ∞ the following inequality holds: . Using this theequation can be simplified as

. (A5.12)

Equation A5.12 provides a possibility for deducing an explicit asymptoticbehavior of function y(x):

.

Taking into account that , we arrive at

. (A5.13)

Direct differentiation of Equation A5.13 gives

. (A5.14)

We used this asymptotic behavior in Section 3.2.

w y= ( )12

32 3

ln/

dydx

y= −31 3 1 3/ /ln

dyy

xln /

/1 3

1 33∫ = −

dyy

yy

dyyln ln ln/ / /1 3 1 3 4 3

13∫ ∫= +

ln ln/ /4 3 1 3y y>>

yy

xln /

/1 3

1 33= −

y x y x x y= − = ( )3 3 31 3 1 3 1 3 1 3 1 3 1 3/ / / / / /ln ln ln

x y>> ln /1 3

y x x x= → − ∞31 3 1 3/ /ln ,

dydx

x x= → − ∞31 3 1 3/ /ln ,



3.6 MOTION OF LONG DROPS IN THIN CAPILLARIES IN THE CASE OF COMPLETE WETTING

By now the reader is expected to be familiar with the fact that for nonpolar liquidswe can use disjoining pressure isotherms Π(h) = A/hn, A > 0, n ≥ 2 (h is thethickness of the film). Such isotherms pertain to the case of complete wetting.In this section we consider the motion of long oil drops or air bubbles in thincapillaries [54–56].

Let us consider the motion of a long drop or bubble in a thin capillary (gravityaction is neglected) (Figure 3.12). Under the action of applied pressure differencep– – p+ > 0, the drop or bubble moves from left to right with velocity U to bedetermined as a function of the applied pressure difference. Note that the velocityU is different from the average Poiseulle velocity because the drag force in thesystem presented in Figure 3.12 is different from the drag force in the samecapillary completely filled with liquid 1.

We consider below a relatively slow motion, when the capillary number,

,

where η1 is the viscosity of liquid 1 in the capillary, and γ is an interfacial tension,which can be substantially different from the liquid–air interfacial tension. Thelatter interfacial tension can be used only in the case of the motion of an airbubble. In the following text we show that the viscosity of liquid 2 inside thedrop or bubble, η2, doe not play any significant role and can usually be omitted.However, the interfacial tension in the case of the liquid bubble is still veryimportant as well as its difference from the liquid–air interfacial tension.

FIGURE 3.12 Schematic presentation of a motion of a drop or bubble in a capillary underthe action of applied pressure difference p– – p+ > 0, that is, the motion from left to right.(1) liquid in a capillary of length L, (2) drop or bubble of length �. L– and L+ are parts ofthe capillary without the drop or bubble, F′F and AA′ are parts of the capillary wherePoiseulle flow takes place, E and B are positions of the end of spherical menisci (curvatureof meniscus EE′ is bigger than the curvature of the meniscus BB′). ED and CB are transitionzones from menisci to the region of DC of the film of constant thickness, h0.

CaU= <<η

γ1 1

F ′ E ′ B ′ A ′B A

xp+

L+

E D

210

h0

p–

CF

ι

L

L–

r



Let us estimate the velocity of the motion in the case Ca ~ 1: if η1 ≈ 10–2P,γ ≈ 72 dyn/cm, then the corresponding velocity U ≈ 7200 cm/sec = 72 m/sec.Such huge velocity can be achieved in a sufficiently thin capillary with radiusless than 0.1 cm only under very special conditions. That means we can safelyconsider Ca << 1.

As we already discussed in the Introduction to this chapter,

1. The advancing meniscus EE′ has a constant curvature up to the zoneof the flow, ED, where the low slope approximation is valid. Thecurvature of the advancing meniscus EE′ is a function of the capillarynumber Ca to be determined,

2. The same is valid for the receding meniscus BB′ and the zone of flowCB. It is obvious that the curvature of the receding meniscus BB′ issmaller than the curvature of the advancing meniscus EE′.

The author of reference 23 had studied the motion of a long drop or bubblein a capillary without allowance for the effect of disjoining pressure. The follow-ing dependence of film thickness, h0, on drop velocity U has been deduced:

, (3.183)

where R is the radius of the capillary. Relation (3.183) yields a zero film thicknessat Ca → 0. However, as we already understood in Chapter 2, this is impossiblebecause the film thickness should tend to the equilibrium thickness, he, at Ca →0. This equilibrium value of the film thickness should be found from the followingcondition (see Chapter 2) Π(he) = Pe, where Pe is the excess pressure in the drop.Hence, ignoring the action of disjoining pressure results in a wrong prediction ofthe film thickness at low capillary numbers. However, according to Equation 3.183,the film thickness increases unboundedly at high capillary numbers. If h0 > ts,where ts is the radius of surface forces action, then Equation 3.183 should becomevalid. Hence, at low capillary numbers we should expect a substantial deviationfrom the prediction according to Equation 3.183, and at high capillary numbers,Equation 3.183 should hold asymptotically. Our calculations (see Figure 3.13)confirm the suggested dependency of film thickness on capillary number.

In the present section, we obtain the main characteristics of the motion of adrop under an applied pressure gradient, taking into account the disjoining pres-sure action.

We examine the motion of a long drop of an immiscible fluid 2 with a length �(Figure 3.12) inside the cylindrical capillary of radius R filled with fluid 1. Themotion is axisymmetric, and we introduce a coordinate system connected withthe drop. The x-axis coincides with the axis of the capillary. Let � be the lengthof the drop, L+ and L– the lengths of the drop-free sections of the capillary, andL the overall length of the capillary, (L = L+ + L– + �). The pressure difference,p+ – p– is applied at the ends of the capillary, where p– > p+. We examine the

h R Ca0

2 31 337= ( ).

/



steady motion of the drop along the capillary in the positive direction of the x-axisat the velocity U to be determined.

As before, we restrict ourselves to the cases of low Reynolds and capillarynumbers. We also assume that the drop is long, that is, R/� << 1. Let η1 and η2

be the viscosities of fluids 1 and 2, respectively. At η1 ≈ 10–2 P, γ ≈ 30 dyn/cm,U ≈ 1 cm/sec, we have Ca ≈ 3·10–4 << 1. As the value of ∼ can betaken as the upper bound of the velocity of the drop, we will assume that thecondition Ca << 1 is always satisfied.

We will divide the flow field in the capillary into regions following (A′A andFF ′) those parts of the capillary not containing the drop, where Poiseuille flowis realized; AB and EF are spherical menisci at the ends of the drop; CD is theregion with a constant film thickness, h0, to be determined; and BC and DE aretransitional regions from the constant-thickness film to menisci.

In Appendix 6 we show that the flow in both transition zones, ED and CB,and the zone of the constant-film thickness, DC, almost coincides with Equation3.178 deduced in the Section 3.5. It is assumed that the viscosity ratio of liquids1 and 2 is of the order of 1, i.e.,

–η = (η2/η1) ~ 1. Equation 3.178, however, doesnot include the viscosity of the liquid inside the drop or bubble, η2. The proof ofthat is given in Appendix 6. Here, we give a qualitative explanation.

As we already showed in the Introduction to this chapter, in zone EB the lowslope approximation is valid. Hence, the equality of tangential stress at theliquid–liquid interface, h(x), has the following form

,

FIGURE 3.13 Dependency of the constant film thickness, h0, inside the zone DC (Figure3.12) on capillary number, Ca. (1) according to Reference 23, when the disjoining pressureaction was ignored, (2) disjoining pressure A/h3, (3) disjoining pressure A/h2.

5

10

3

1

1 2 3(3Ca)1/3 . 102

h 0/R

. 104

2

0

U 1 cm/sec

η η11

22∂

∂= ∂

∂vx

vxh x h x( ) ( )



where v1 = v2 ~ U are velocities at the liquid–liquid interface, h(x). Let us estimatethe ratio of the tangential stress from the drop or bubble side to the tangentialstress from the film side

.

This estimation shows that the flow inside the drop or bubble can be safelyignored.

Using the low slope approximation, the curvature of the interface, h(x), insidezone EB (Figure 3.12) can be written as

. (3.184)

Using this expression, we conclude from Equation 3.178 that

.

In this equation we neglect the small difference between the actual effectivedisjoining pressure,

,

where Π(h) is the disjoining pressure of the corresponding flat films. See thejustification in the following text.

In the flow zone EB (Figure 3.12), the thickness of the film is much smallerthan the capillary radius; hence,

,

and the preceding equation can be rewritten as

. (3.185)

η η η η22

11

2 1∂∂

∂∂

vx

vx

UR

Uh

h x h x( ) ( )

~ == <<η hR

1

K xd hd x R h

( ) = +−

2

2

1

hd h

d x R h

dhd x

h dhd x

Ca h33

3 2

13+

−+

′( )

= −

( )

Πγ

hh0( )

Π ΠefR

R hh=

−( )

( )R h RhR

R− = −

≈2 2

221

hd h

d x R

dhd x

h dhd x

Ca h h33

3 2 01

3+ +′( )

= −( )Π

γ



In the following text we consider only the case of complete wetting, wherewe use the following isotherms of disjoining pressure:

.

Let us introduce the following dimensionless values in Equation 3.185: z =x/x*, ξ = h/h0, where the characteristic scale x* is the characteristic dimension ofthe regions BC and DE. In this section, our choice of this scale is different fromthe choice in the previous section because we are going to concentrate on atransitional regime of flow from equilibrium to a relatively high velocity. That iswhy we select this scale equal to the length of the transition zone at the equilib-rium (see Chapter 2, Section 2.3, Equation 2.49): x* = . It is important tonotice that using these scales we get the estimation of the derivative in the flowzone,

,

and the low slope condition is really satisfied in the flow zone in a vicinity ofboth advancing and receding menisci.

The choice also shows that

,

and, hence, the second term on the left-hand side of Equation 3.185 can beomitted. Therefore, Equation 3.185 can be rewritten as

, (3.186)

where

.

The case β = 0 corresponds to the case of filtrative motion at a high velocity,examined in Reference 23, where the effect of disjoining pressure was ignored.

Equation 3.186 has the solution ξ ≡ 1 corresponding to a film of constantthickness in the zone CD. This solution should ensure matching of this zone withthe surfaces of the spherical menisci in regions BB′ and EE′ at the ends of the drop.

Π hAh

nn( ) = =, , ,2 3 4

heR

′ = = <<hhx

h

h R

hR

e e

e

e~*

1

′′′( )

= >> ′ =hh

h R h R Rh

h

R h R

hR h

e

e e

e

e

e

e

~ ~/3 2 3 2 2

1 1 1

RR3

ddz

ddzn

3

3 1 3

1 1ξ βξ

ξ ε ξξ

− = −+

βγ

ε= = ( )n R Ah

Ca R hn0

0

323, /



The matching conditions for the leading end of the drop have the form

(3.187)

while for the trailing end

(3.188)

Satisfaction of conditions (3.187) and (3.188) is assured by the existence ofasymptotic solutions of Equation 3.186 having the form (see Appendix 5):

, (3.189)

(the + sign denotes the leading meniscus, while the – sign denotes the trailingmeniscus). The parameters A± and B±, dependent on the capillary number, Ca,are determined in the following using the numerical solution of Equation 3.186.

Comparing Equation 3.189 with Equation 3.187 and Equation 3.188, weconclude that

(3.190)

Assuming in Equation 3.186 that ε = 0 (or Ca = 0), we obtained an equationfor the transitional zone between the film and the meniscus for the equilibriumcase, which was examined in Chapter 2, Section 2.4. In the case of completewetting, the meniscus has the radius where he is the thicknessof the equilibrium film determined from the condition

.

Comparing these relations with those in Equation 3.190, we conclude that

ddz

RR

z

z

2

2

1

ξ

ξ

→ → +∞

→ → −∞

+ ,

,

d

d z

R

Rz

z

2

2

1

ξ

ξ

→ → −∞

→ → +∞

− ,

, .

η ≈ ( ) + ( ) → ±∞±

±A Caz B Ca z

22 ,

R R A

hR A

A B

R A

A B

± ±

+

+ +

−

− −

=

=−( )

=−( )

/

0

1 1.

R R n n he e± = − −( ) ,/ 1

A

hR

en e= 2γ /



(3.191)

Figure 3.13 shows the dependence of film thickness h0 on the dimensionlessvelocity (3Ca)1/3, whereas Figure 3.14 shows the dependences of the parametersA±, B±, and the radii of the leading R+ and trailing R– menisci on dimensionlessvelocity (3Ca)1/3. In each case, we adopted the radius of the capillary and theinterfacial tension R = 10–2 cm, γ = 30 dyn/cm, respectively.

Let us study the solution of Equation 3.186 at ε << 1, having represented itin the form

. (3.192)

Using this representation we arrive at the following equation for the zerothapproximation:

. (3.193)

Having integrated Equation 3.193, we obtain

. (3.194)

FIGURE 3.14 Dependency of parameters A+ (curve 1), A– (curve 2), B+ (curve 3), B–

(curve 4), and the radii of the leading meniscus, R+ (curve 5) and the trailing meniscus,R– (curve 6) on the capillary number, Ca.

lim ( )

lim ( ) .

ε

ε

ε

ε

→

±

→

±

= +−

=−

0

0

11

1

An

nhR

Bn

n

e

ξ ξ ξ= +e 3 1

ddz

ddz

e

en

e3

3 1 0ξ β

ξξ− =+

ddz n

constRR

e

en

e

2

2

ξ βξ

+ = =

4321

1

2

2.793

1.01

1.0

0.995

2

6

1

3

B±

R± /R

A±

0.7

0

(3Ca)1/3 . 102



Equation 3.194 describes the profile of the transitional zone between the filmand the equilibrium meniscus and, as was shown in Chapter 2, Section 2.4, it hasthe asymptotic solution

∼ , (3.195)

where

. (3.196)

The equation of the first approximation for (3.186) using the representation(3.192) takes the following form:

. (3.197)

Having integrated the equation, we conclude that

, (3.198)

where C± = const.At z → –∞ for the leading meniscus and z → +∞ for the trailing meniscus,

we conclude that ξ1 = 0. Then, we conclude from Equation 3.198 that

. (3.199)

As the solution of equation (3.198) satisfies the asymptotic conditions

, (3.200)

for the leading meniscus and

, (3.201)

ηe A Be eξ2

2+

ARR

A A Bn

nB Be

ee e e e e= = = =

−= =+ − + −,

1

ξ ξ βξη

ξeen e

ddz

d

d z n3

21

21

11−

= −+

ddz n

C d zen

e

e

21

21

1 3

1ξ βξξ

ξξ

ξ

− = − −+

±

± ∞

∫

C d ze

e

±

∞

± ∞

= −∫ ξξ

13

∓

ξ 11 2

12= + → +∞

++A

z B z,

ξ 11 2

12= + → −∞

−−A

z B z,



for the trailing meniscus, we find from Equation 3.198 that

. (3.202)

Equation 3.190 can be rewritten in the following form:

or, in a first approximation with respect to the parameter ε:

. (3.203)

Hence,

. (3.204)

As was shown in Section 2.4,

. (3.205)

To within the leading terms in ε we obtain the following relation usingEquation 3.204 and Equation 3.205:

. (3.206)

The final asymptotic representation of the parameters A± and B± takes the form

, (3.207)

. (3.208)

The constants C± were found from the numerical solution of Equation 3.194.It follows from Equation 3.207 and Equation 3.208 that at ε << 1 the parameters

A C± ±=

h B RA

0 11±

±= −

h BRA

Ae

e1

12

±±

=

BRA

h AR Ch Ae e e e

112 2

±± ±

= =

An

n

h

Re

e= +−

11

BRh

Ce

1± ±=

A A An

n

h

RCe

e± ± ±= + = +−

+ε ε1 11

B B Bn

nR

hCe

e

± ± ±= + =−

+ε ε1



A± and B± are linear functions of ε (or on Ca). At ε = 0 (Ca = 0), we recover theequilibrium values. At high velocities (Figure 3.13), the values of the parametersB± are close to the values of the constants obtained in Reference 23 as expected:B± ~ 2.79, B– ~ –0.7.

It can be seen from Figure 3.12 that the thickness of the film in the regionCD can be considered equal to the equilibrium thickness he at (3Ca)1/3 ≤ 8·10–3

for a disjoining-pressure isotherm of the form Π(h) = A/h3, and at (3Ca)1/3 ≤ 10–2

for Π(h) = A/h2. In the case of high velocities, the plot of the relation h0(Ca)tends asymptotically to that given by Equation 3.183, which is valid at a suffi-ciently high velocity of the drop or bubble.

To close the problem, we need to establish the relationship between theapplied pressure gradient, ∆p = p– – p+, at the ends of the capillary and the velocityof the drop, U. This relation can be represented in the following form (seeAppendix 6 for details):

. (3.209)

With allowance for Equation 3.190, we conclude from Equation 3.209 that

. (3.210)

The plot of the velocity, U, according to of Equation 3.210, obtained usingthe previously numerically calculated dependencies, is shown in Figure 3.14,where we used R = 10–2 cm, γ = 30 dyn/cm,

–η = 0, L– + L+ = 4 cm, � = 1 cm,η1 = 10–2 P.

For small ε, Equation 3.210 takes the following form:

, (3.211)

or

. (3.212)

Equation 3.212 shows that at small ε, the velocity U of the drop or bubblemotion depends linearly on the applied pressure difference, ∆p.

∆ pR

R

R

R

RCa

L L lR

= − + + +

+ −

+ −24

γ η

∆ pR

A Ca A Ca CaL L l

R= ( ) − ( ) + + +

+ −

+ −24

γ η

∆ pR

C C CaL L l

R= − + + +

+ −

+ −24

γ ε η( )

∆ pR

CaRh

C CL L l

R=

− + + +

+ −

+ −23 4

0

32γ η

( )



Let us examine the motion of a sequence k of drops or bubbles in the capillary.Let �i, i = 1, 2, …, k be the lengths of individual drops. We assume that theinfluence of drops on one another can be neglected if the distance between theirends exceeds 2R. Then, for a chain of k drops, Equation 3.210 can be rewritten as

. (3.213)

Figure 3.15 and Figure 3.16 show the plot of the velocity of motion, U, ofthe chain of bubbles according to Equation 3.213 at

and L+ + L– = 1 cm, γ = 30 dyn/cm, R = 10–2 cm.In numerical calculations for the isotherm of disjoining pressure Π(h) = A/h3,

we used A = 10–14 J/m at n = 2 and A = 10–21 J ·m at n = 3.

APPENDIX 6

The low slope approximation is valid two flow zones ED and CB, and the zoneof the flat film DC (Figure 3.12). At a low Reynolds number and taking intoaccount that all unknown functions do not depend on the angle, we get thefollowing system of equation, which describes the flow in both liquid 1 and 2:

FIGURE 3.15 Dependence of velocity of motion of a single drop or bubble, U, on theapplied pressure difference, ∆p. (1) Π(h) ≡ 0; (2) Π(h) = A/h2, according to Equation 3.210.

U . 1

05 , m/se

c

0.5 0.3 0.1

1

2

3

1 2

Δp, Pa

∆ pR

k A A Ca

L L l

R

i

i

k

= −( ) ++ +

+ −

+ −

=∑

24 1γ

µ

k li

i

k

= ==∑10 9

1

, ,cm



, (A6.1)

, (A6.2)

and the continuity equation

.

This equation can be rewritten as

. (A6.3)

The low slope in the zone EB results in

.

Using that, we conclude from Equation A6.1 and Equation A6.2 that

(A6.4)

. (A6.5)

FIGURE 3.16 Dependence of velocity of motion of a chain of drops on the appliedpressure difference, ∆p: (1) Π(h) ≡ 0; (2) Π(h) = A/h3, according to Equation 3.213.

U . 1

05 , m/se

c

10

2

0

5

1 2

Δp, Pa20

012

2

2

2 2= − ∂

∂+ ∂

∂+ ∂

∂+ ∂

∂−

pr

v

r

v

x rvr

v

rr r r rη

012

2

2

2= − ∂

∂+ ∂

∂+ ∂

∂+ ∂

∂

px

v

r

v

x rvr

x x xη

∂∂

+ ∂∂

+ =vr

vx

vr

r x r 0

∂∂

+ ∂∂

=rvr

rvx

r x 0

v vx r

r x<< ∂∂

<< ∂∂

,

0 = − ∂∂pr

,

02

21= − ∂

∂+

∂

∂+

∂∂

px

vxr r

vxr

η



Equation A6.4 means that the pressure in both liquids 1 and 2 is a functionof the axial co-ordinate x only. Integrating Equation A6.3 over radius in liquids1 and 2, we conclude that

, (A6.6)

, (A6.7)

where vrh, vx

h are velocities on the liquid–liquid interface. The integral on the right-hand side of the preceding equations can be transformed as

,

or

.

Substitution of this expression into Equation A6.6, and taking into account that

,

results in

,

or

. (A6.8)

Similar transformations of Equation A6.7 results in

. (A6.9)

∂∂

+ − =−

∫ rvx

dr R h vx

R h

rh

0

0( )

∂∂

− − =−∫ rv

xdr R h vx

R h

R

rh( ) 0

∂∂

= − ∂∂

− + ∂∂

− −

∫ ∫xrv dr

hx

R h vrvx

drx

R h

xh x

R h

0 0

( )

∂∂

= ∂∂

+ ∂∂

−− −

∫ ∫rvx

drx

rv drhx

R h vx

R h

x

R h

xh

0 0

( )

∂∂

= ∂∂

+ht

vhx

vxh

rh

( )R hht x

rv dr

R h

x− ∂∂

= − ∂∂

−

∫0

∂ −∂

= ∂∂

−

∫( )R ht x

rv dr

R h

x

2

0

2

∂ −∂

= − ∂∂

−∫( )R h

t xrv dr

R h

R

x

2

2



Integration of Equation A6.4 and Equation A6.5 using (1) a nonslip boundarycondition on the capillary surface at r = R, (2) a symmetry condition in thecapillary center at r = 0, and (3) equality of velocities and tangential stresses atthe liquid–liquid interface at r = R – h, we can deduce expressions for axialvelocities, vx, in both liquids 1 and 2. Substitution of those expressions in EquationA6.8 and Equation A6.9 results in

where p1(x) and p2(x) are pressures in liquids 1 and 2, respectively; ′ means aderivation with x.

Subtracting Equation A6.11 from Equation A6.10 and integrating over xresults in

(A6.12)

where A(t) is an integration function, which can dependent only on time.We consider the steady motion of the drop with the velocity U. Changing

over to the variable z = x – U t in Equations A6.10 and A6.11, we conclude that

(A6.13)

∂∂

∂∂ η η

R h

t

x

p R h pR h R h

−( )=

′ −( )− ′ −( ) −( )

2

2

4

2

1

1

2

8 4

22 2

4

2 1

121−

−−( ) ′ − ′( )

−

R

R h p p hRη

ln

(A6.10)

∂∂

∂∂ η η

R h

t x

p R h pR h R

−( )=

′ −( )− ′ −( ) −

2

2

4

1

1

1

2 2

8 8

−−( ) ′ − ′( ) −( ) −

−R h p p R h R

R

2

2 1

1

2 2

8 2η−−( ) −

h

hR

21ln ,

(A6.11)

− ′ −( ) − ′ − −( )

+′

pR h

pR R h

p

2

2

4 1

1

4 4

2

8 8η η

−− ′( )−( ) −( ) −

= ( )pR h R h R A t

1

1

2 2 2

42

η,

− ′ −( ) − ′ − −( )

+′

pR h

pR R h2

2

2 1

1

2 2

8 4η η

pp pR h

hR

U2 1

1

2

21

− ′( )−( ) −

=η

ln .



(A6.14)

In the steady-state case being examined, the integration constant A should beindependent of time. The film has a constant thickness h0 far from the ends of thedrop; hence, from Equation A6.13 and Equation A6.14, we obtained the solutions

, (A6.15)

, (A6.16)

where we introduced–η = η1/η2.

Equation A6.16 gives the dependence of the flow rate in the capillary on thethickness of the film, h0, and the velocity of the drop, U.

Equation A6.13 to Equation A6.16 are used to obtain an equation for thethickness of a film of liquid under a drop. Limiting ourselves to the leading termsof the expansion in h/R << 1, we move from Equation A6.16 to

, (A6.17)

where

.

If we assume that–η ~ 1, then P = 1, and Equation A6.17 takes the form of

Equation 3.185.

3.7 COATING OF A LIQUID FILM ON A MOVING THIN CYLINDRICAL FIBER

In this section we shall study how the disjoining pressure affects the thin filmadhering on the surface of a fiber being pulled out of a liquid. The problem of coating

− ′ − −( )

+′ − ′( )

−( )pR R h

p pR h1

1

2 22

2 1

1

2

8 2η η

lnR hhR

R R h−( ) +

+− −( )

22 2

12

= − −( )22

A U R h .

′ = ′ = −− + − −( )

p pU

R h R R h1 2

2

02 2

0

2

8

2

η

η( )

AU R h R R h

R h R R=

− + − − − + − −2 2

04 4

04

04 2

( ) ( )

( ) (

η

η hh02)

hdKdx

hdhdx

Ca h h P30

13+ ′

= −γ

Π ( ) ( )

P

h hR

hhR

hR

hR

=+ − +

+

+

1 2 8 1 4

1

0 2 02η η η

η

+ − +

1 4 160 2 0

2η ηh hR

hhR



by liquid films of a moving support is frequently encountered in technologicalprocesses. The problem of coating by a liquid film of a flat vertical support pulledout of the liquid in a gravity field has been investigated earlier in Reference 27and Reference 28, and the film thickness vs. support velocity was deduced for arelatively high velocity of coating. In Section 3.6 we found that the effect of thedisjoining pressure is predominant at low-support velocities. The problem ofcoating from this point of view is considered in this section. The effect of thedisjoining pressure in the thin film deposited on the surface of a fiber taken outof a liquid is taken into consideration in this section.

Let us examine the problem of pulling out a thin cylindrical fiber of radius athrough an interface between two immiscible liquids with a constant velocity U.The schematic presentation of the experimental system is shown in Figure 3.17. Thefiber moves along its own axis perpendicular to the interface of the liquids. Weneglect the forces of gravity in comparison to viscous and capillary forces.

STATEMENT OF THE PROBLEM

Let us introduce a cylindrical system of coordinates whose x-axis coincides withthe axis of the fiber, and r-axis is perpendicular to it. In this system of coordinates,the fiber moves with velocity U in the negative direction of x-axis (Figure 3.17).It is assumed that liquid 1 wets completely the fiber surface.

Both the Reynolds number, Re = Ua/η, and the capillary number, Ca = ηU/γ,are small, Re << 1, Ca << 1, and the film thickness, h0, is much smaller than theradius of the fiber, a, h/a << 1, where η is the dynamic viscosity of the liquid 1,and γ is the liquid–liquid interfacial tension on the interface of liquids 1 and 2.

The field of flow is subdivided as in Section 3.6 into the following threezones (Figure 3.17):

FIGURE 3.17 System under consideration: moving cylinder of radius a with velocity Uthrough the interface between liquids 1 and 2. (I) zone of the meniscus, (II) the flow zone,(III) zone of the film of the uniform thickness, h0, to be determined as a function of Ca.

1

I II x

III U

a

r h0

2

0



Zone I, the steady-state meniscus of the liquid. The movement of the liquidin this zone can be neglected.

Zone II, the flow zone in between the film of constant thickness and themeniscus of the liquid.

Zone III, a film of constant thickness, h0, to be determined as a functionof the Ca.

According to the previous consideration in zones II and III, the low slopeapproximation is valid. Hence, the equations of flow taking into account theprevious assumptions in zones II and III can be written in the following form:

, (3.214)

. (3.215)

. (3.216)

The continuity equation is

. (3.217)

The boundary conditions are: a nonslip boundary condition on the fibersurface,

; (3.218)

absence of tangential stress on the liquid–liquid interface (see the discussion inSection 3.6),

; (3.219)

and normal stress jump on the same interface results in

, (3.220)

where u and v are the velocity along axes x and r, respectively; p1 and p2 are thepressures in liquids 1 and 2; K(x) is the curvature of the surface of the film; and

∂∂

=pr1 0

∂∂

= ∂∂

∂∂

px r r

rur

1 1η

p const2 =

∂∂

+ ∂∂

( ) =x

ur r

rv1

0

u U r a= − =,

η ∂∂

= = +ur

r a h0,

p p K xa

a hh r a h2 1− = ( ) +

+( ) = +γ Π ,



Π(h) is the isotherm of the disjoining pressure of flat films. Note the presence ofthe term

in the boundary condition (3.220) (see Chapter 2, Section 2.12 for the derivation).

DERIVATION OF THE EQUATION FOR THE LIQUID–LIQUID INTERFACE PROFILE

Integration of Equation 3.214 and Equation 3.215 with consideration of boundaryconditions (3.218) and (3.219) gives the following equation for the velocity ofthe liquid:

, (3.221)

where . The flow rate of the liquid in the liquid layer is:

. (3.222)

Substituting Equation 3.221 for Equation 3.222, we obtain

The condition of constancy of flow in any section of the film, Q = const,follows from the equation of continuity (3.217). For a film of constant thicknessho in zone III, we deduce from boundary condition (3.220) and Equation 3.216that . The latter allows rewriting Equation 3.223 as

. (3.224)

aa h+

u p r a a h r a U= ′( ) − − +( )

−12 2 2

4 2/ /η ln

dp dx p1 1/ = ′

Q urdra

a h

=+

∫2π

Q

a h a a h a h

/2

4 2 22 2

22 2

π =

+( ) −

− +( )( ) +( ) ln aa h a a

p

+( ) −( ) +

′ −

1

4

2

1/ η UU a h a+( ) −

2 2 2.

(3.223)

′ =p1 0

Q V a h a= − +( ) −

0

2 2 2



Neglecting terms higher than the first order in zones II and III with respectto h/a << 1, we obtain from Equation 3.223 and Equation 3.224 that

. (3.225)

Boundary condition (3.220) with consideration of (3.216) can be used toobtain the equation for the profile of the liquid–liquid profile in zones II and IIIfrom Equation 3.225:

. (3.226)

The curvature of the surface of the liquid–liquid interface in the cylindricalsystem of coordinates is

.

Let us introduce as before the following dimensionless variables z = x/� andH = h/h0, where � is the scale of the flow zone. We are interested in low andintermediate capillary numbers; that is, the thickness of the film, h0, does notdiffer very much from the equilibrium thickness, he. This allows us, as in Section3.6, to select this scale as � = . According to that choice,

.

Using the new variables, we can rewrite the expression for the curvature as

. (3.227)

Using the same dimensionless variables in Equation 3.226 in combinationwith the latter Equation 3.227, we obtain

. (3.228)

− ′ = −( )p h U h h13

03/ η

ddx

K xa

a hh U

h hh

γ η( ) ++

( )

= ( ) −Π 3 03

K x h h a h h( ) = ′′ + ′( ) − +( ) + ′( )

1 1 123 2

21 2/ /

ah0

h ha

0 0 1�

= <<

K z a H( ) = ( ) ′′ − 1 1/

ddz

Ha

ha

H

h H′′ − ++

( )

=1

1

30

0

γΠ CCa

ah

H

H0

3 2

3

1

−/



According to Section 3.6, the dependency of the film thickness, h0, on thecapillary number is given by Equation 3.183. Therefore, our choice of the scaleof the transition zone � = tends to L = as Ca tends to zero, that is,to the length of the transition zone at the equilibrium, and

at sufficiently high Ca. This corresponds to the choice adopted in Reference 23,which is valid only at sufficiently high Ca.

IMMOBILE MENISCUS

In zone 1, the curvature of the interface should be constant, as the movement ofthe liquid in this region can be neglect (see for details Introduction to this chapter).However, away from the fiber, the interface is flat, and the curvature consequentlyvanishes. It is evident that h ≈ a and the scale of this zone has the same order ofmagnitude, a. Hence, in this zone, h′2 ≈ 1. That is, from the expression for thecurvature we conclude that

. (3.229)

Integrating Equation 3.229 once results in

, (3.230)

where C is the integration constant. The solution of Equation 3.230 is

, (3.231)

where C1 is the new integration constant. This equation has a stationary point h =hs, h′s = 0. Then we can determine, using Equation 3.231, that

. (3.232)

Let us shift the origin for x-axis to the stationary point, i.e., we select C1 =0 and expand the solution (3.231) in the Taylor series, which results in

(3.233)

ah0 ahe

� = ⋅ ⋅ =ah a a Ca aCa02 3 1 3~ / /

′′ + ′( ) − +( ) + ′

=h h a h h1 1 1 023 2

2/

′ = +( ) −h C a h2 21

h C C x C a= ( ) + −1 1/ cosh ( )

C a hs= +( )1

h hx

a hs

s

= ++

+ ⋅ ⋅ ⋅2

2( )



MATCHING OF ASYMPTOTIC SOLUTIONS IN ZONES I AND II (FIGURE 3.17)

Now we should match solution of the stationary meniscus according to Equation3.233 with the solution of Equation 3.228, which describes the liquid profile inzones II and III. Let us examine the asymptotic solution of Equation 3.228 forz → +∞ (or H → ∞), then we obtain as in the previous case (Equation 3.228).As we already mentioned, this equation has the following asymptotic:

, (3.234)

where the origin is transferred to the stationary point selected earlier to get ridof the term proportional to z; A0 and B0 are some functions of the capillary numberCa determined from the solution of Equation 3.228 with the boundary conditionH → 1 for z → –∞ and from the matching conditions, which is obtained in thetext below.

Let us rewrite solution (3.233) using the inner variables in Equation 3.228:

. (3.235)

Comparing Equation 3.234 and Equation 3.235, and as these expansionsshould coincide, we obtain two matching conditions for the solutions of the filmprofile equations in zones I and II:

, (3.236)

, (3.237)

where both thickness h0 and hs are still unknown. The two equations can be solvedas follows:

, (3.238)

. (3.239)

The constants A0 and B0 and their dependency on the capillary number, Ca,can be determined only numerically. The results of the numerical calculationsare presented in Figure 3.18. Only the case of complete wetting is under consid-eration in this section; that is why the isotherm of the disjoining pressure usedin the problem was selected as

H A z B= +02

02/

Hh

h h a

zs

s

= ++

+ ⋅ ⋅ ⋅0

211 2/

h h Bs / 0 0=

1 1 0+( ) =h a As /

h a A A B0 0 0 01= −( )

h a A As = −( )1 0 0



, (3.240)

where A is the Hamaker constant. A + Ca → 0, h0 → he, and B0 → 3/2 (as weshow in the following text). Hence,

,

where he is the equilibrium thickness of the film.

FIGURE 3.18 Calculated parameters A0 (1), A* (2), and B0 (3) on capillary number, Ca.

1 3

–16 –14 –12 –10 –8 �n Ca

–16 –14 –12 –10 –8 �n Ca

0

1

2

3 B A

1.00

0.99

(1.5)

(2.79)

2

�n A

∗

4

3

2

1

0

–1

Π hAh

( ) = 3

Aha

ha

Cae

e0

1

132

132

0→+

≈ − →,



EQUILIBRIUM CASE (Ca = 0)

In the equilibrium case (Ca = 0), the film profile equation will have the followingform according to Equation 3.226:

.

Integrating once, we obtain

. (3.241)

The integration constant is equal to zero because the interface is flat awayfrom the fiber.

In zone III we obtain the equation for determining the film thickness fromEquation 3.241, assuming h = he, h′ = 0, and h″ = 0:

, (3.242)

or, using the disjoining pressure isotherm in the case of complete wetting (3.240),Equation 3.242 is rewritten as follows: γ /a = A/he

3. This equation results in

. (3.243)

Equation 3.241 in the case of complete wetting takes the following form:

. (3.244)

In zone III, the film profile according to Equation 3.244 becomes flat andcoincides with Equation 3.243; in zone I, where the disjoining pressure can beneglected, we obtain the meniscus profile, which is similar to that given byEquation 3.231. All terms of Equation 3.244 are important in zone II. IntegratingEquation 3.244 once we obtain

,

ddx

h

h a h h

aa h

h′′

+ ′( )−

+( ) + ′+

+ ( )

1

1

123 2 2/

Π

= 0

′′

+ ′( )−

+ + ′+

+ ( ) = =h

h a h h

aa h

h const1

1

10

23 2 2/

( )Π

γ /a he= ( )Π

hAa

e =

γ

1 3/

′′

+ ′( )−

+( ) + ′+

+( ) =h

h a h h

Aa

a h h1

1

10

23 2 2 3/ γ

1 12

22

+ ′ = −

+( )h CAa

ha he γ



where Ce is the integration constant. At Ca → 0: h → he and h′ → 0, hence, weconclude from the latter equation:

.

Hence, finally, the liquid profile is

. (3.245)

The matching conditions in solutions of Equation 3.245 in zone II and theequation for the steady-state meniscus in zone I can be written as follows:

, (3.246)

where and refer to zones I and II, respectively.The solution of Equation 3.229, which describes the film profile in zone I, is

, (3.247)

where C2 is the integration constant. Using Equation 3.245 and Equation 3.247,we find that, using the matching condition (3.247),

. (3.248)

At the stationary point, hI = hse, where h′I = 0, and we conclude from Equation3.247 and Equation 3.248 that

,

or

, (3.249)

which is in agreement with the conclusion obtained in Section 2.4 (see Figure 2.19).

C a hAa

he e

e

= + +

2 2γ

1 12 2

22 2

+ = + +

−

h a h

Aa

h

Aa

he

e

'γ γ

+( )a h

hII

hI

II I

h h→∞ →

′ = ′lim lim0

′hI ′hII

1 1 22+ ′ = +( )h C a hI I

C Ce2 =

h C a hAa

hse e

e

= − = +

2 22γ

h hse e= 1 5.



Hence, we obtain the following expression for coefficient Be in the equilibriumcase (see Equation 3.234):

. (3.250)

NUMERICAL RESULTS

The coefficients A* = (1 – A0)/B0 (3 Ca)2/3, A0, and B0 as functions of the capillarynumber according to Equation 3.237 and Equation 3.236 obtained from thenumerical calculation and are presented in Figure 3.18. The isotherm of thedisjoining pressure is defined by Equation 3.240, that is, the case of completewetting is under consideration. Figure 3.18 shows that in this case at Ca → 0,B → 3/2, which corresponds to the prediction (3.249). At the same time, A0 →1 – (3/2) He as shown above, and A* → ∞.

For relatively large capillary numbers, Ca, the film thickness h0 becomesthick enough so that the effect of the disjoining pressure can be neglected. At thesame time, A* → 0.643, B → 2.79 (these values were obtained in Reference 23)and A0 ≈ 1 – 0, 643 (3Ca)2/3 · 3/2.

The dimensionless thickness H0 of the film remaining on the fiber as a functionof the capillary number Ca (dimensionless velocity) is presented in Figure 3.19.At Ca → 0, the graph has a plateau, h0 → he. For large Ca

, (3.251)

as was first deduced in Reference 23 earlier.

FIGURE 3.19 Dimensionless film thickness H0 = h0/a on the capillary number, Ca. (1)without disjoining pressure, according to Equation 3.251 and (2) with consideration of thedisjoining pressure, according to Equation 3.238. Experimental point from Reference 29.

B BeCa

= =→

lim .0

0 1 5

H Ca02 31 33≈ ( ). /

�n Hε

�n Ca∗ �n Ca

�n H0

–9

–15

1

2

–14 –12 –11

–7



Let us determine the critical velocity, Ca*, characteristic of the transitionalvelocity from small to large capillary numbers as the point of intersection of thelines ln he = const and ln 1.33 + 2/3 ln Ca. In our case, lnCa* = 12.85. Thecritical velocity obtained from the experimental data in Reference 29 is equal toln Ca* ≈ –13.26, that is, close to the above theoretical prediction.

3.8 BLOW-OFF METHOD FOR INVESTIGATION OF BOUNDARY VISCOSITY OF VOLATILE LIQUIDS

A blow-off method allows determining the boundary viscosity as a function ofthe distance to a solid substrate. A theory is suggested, taking into account notonly the flow of a liquid film but also its evaporation, with gas being blownthrough a plane-parallel channel over the film. The theory allows one to find thedependence of the dynamic viscosity, η, on the distance to the substrate, h, withη being a continuous function of h. A procedure is outlined for calculation of theviscosity, η, on h dependency based on experimental data. The theory is appliedto the calculation of the boundary viscosity of hexadecane. It turns out that theviscosity in thin layers (40–200 Å) is lower than that in the bulk [35,57].

BOUNDARY VISCOSITY

The measurements of the boundary viscosity are of substantial interest. Attemptshave been made to apply the conventional methods adapted to the measurementof boundary viscosity. These methods may be divided into three groups:

• Methods based on measurement of the rate of liquid flow throughcapillaries [62]

• Methods based on the rotation of coaxial cylinders or discs• Methods based on measurement of the velocity of a falling sphere

These methods afford the possibility of determining only the mean value ofviscosity for a sufficiently thick layer.

The blow-off method [30] allows measuring the boundary viscosity. Themethod consists of the following: One of the walls of a channel formed by twoplane-parallel surfaces is coated with a layer of the tested liquid. If a current ofair (or an inert gas) is then passed through the channel, a flow is generated in thefilm due to a tangential force induced by the flowing gas. In this case the filmprofile becomes wedge shaped, the slope of this wedge being dependent on theviscosity of the liquid.

If the liquid viscosity over the whole distance to the wall is invariable, theslope remains constant until the wetting boundary is reached. In the case of anincreased viscosity close to the wall, the slope increases; in the case of a decreasedviscosity, the slope decreases. The viscosity is calculated from the slope at agiven point of the profile.

The blow-off method was used to examine the boundary viscosity of someorganic liquids [31]. It has been established that the profile of a film of very pure



vaseline oil is rectilinear. In this case, no anomaly of viscosity is observed, andvaseline oil preserves its bulk viscosity value up to the layer thickness of about10–7 cm. The viscosity of incompletely hydrated benzontron was found to decreasein the boundary layer, whereas in a layer of 300 Å, the viscosity of chloroderiv-atives of saturated hydrocarbons was found to increase abruptly (jumpwise). Anincrease in the viscosity of some organic liquids and some vinyl polymers andtheir solutions in films up to 50,000 Å thick as well as the instability of the filmsof the solutions of vinyl polymers at certain thicknesses were observed in Ref-erence 32 to Reference 34.

Up to now, a limitation of the blow-off method has been its inapplicabilityto the measurement of the boundary viscosity of volatile liquids. In this section,the blow-off method has been modified so as to be applicable to studies of theboundary viscosity of volatile liquids. It is necessary to take into account the factthat when gas is blown through a plane-parallel channel, the flow of a film ofliquid occurs (as in the usual variant of the blow-off method) simultaneously withits evaporation.

The theory of the blow-off method, allowing determining the boundary vis-cosity of volatile liquids, is developed in the following text.

THEORY OF THE METHOD

Let us consider the profile of a film at instant of time t, which is characterizedby the relationship

, (3.252)

where x is the distance from the wetting boundary to the point at the surface ofthe film area having local thickness h. An initial condition is adopted as follows:

. (3.253)

An increment, dx, per elementary increment of time, dt, is the sum of anincrement, dx1, resulting from the flow under the action of the tangential shearstress per unit area of the blow-off current, τ, and an increment, dx2, due to theeffect of evaporation. It is obvious that the first increment due to the gas flow is

, (3.254)

where η(h) is the local viscosity in the boundary layer at distance h from thesubstrate. Evaporation results in

, (3.255)

x x h t= ( ),

x h,0 0( ) =

dxdh

hdt

h

1

0

= ( ) ⋅∫τη

dx idt

2 =sinα



where i is the linear evaporation rate and sinα is the slope of the film surface ata given point.

With the exception of a very short initial period, the film has a very low slopeprofile, which allows one to substitute

for sinα.Summing Equation 3.254 and Equation 3.255 and transforming them, we

eventually obtain

. (3.256)

The general solution of partial differential Equation 3.256 is

. (3.257)

The initial condition (3.253) results in

. (3.258)

Consider consequences of the solution obtained. If there is no special bound-ary viscosity, i.e., η(h) = η∞ = const, then

. (3.259)

Thus, having experimentally determined the relationship x = x(t, h) at constantthickness, h, and having revealed its strictly parabolic form on time, one mayascertain that the viscosity has a constant value. Therefore, using the parametersof the parabolic relationship x(t,h), at any h = const ≠ 0, one may determine bothη∞ and the evaporation rate i.

It is more convenient to attain the same objective by determining from theexperiment the dependence of h on t for a given value x and by checking itshyperbolic form.

dxdh

−1

dxt

ixh

dhh

h

∂− ∂

∂= ( )∫τ

η0

x f h iti

dhdh

h

hh

= +( ) − ( )∫∫τη1

2

200

1

xi

dhdh

hdh

dhh

hhhh it

= ( ) − ( )∫∫∫+

τη η1

2

21

2

20000

11

∫∫ ∫∫

= × ( )

+τ

ηidh

dhh

h

h

h it

12

20

1

xt

h it= +( )∞

τη2

2



If one determines only the velocity at which the wetting boundary recedes(which is the simplest of all), then Equation 3.259 becomes

, (3.260)

which is of interest for its method, thus allowing one to determine the ratio i/η∞.From Equation 3.259 we conclude that

.

If η(h) = η∞ = const, the equation gives η∞ and i at any constant x.From Equation 3.258 we conclude that

, (3.261)

, (3.262)

. (3.263)

Using Equation 3.262 and Equation 3.263 as the basis, it is difficult to obtainprecise data on the viscosity of boundary layers as it is difficult to measure x ata given h as a function of t. If we confine ourselves to the case where h = const ≠0, then the problem is rendered somewhat easier. Yet, we have to determine(∂x/∂h)t within the time interval it < HN, where HN is the thickness of the boundarylayer, and where the viscosity is different from its bulk value.

Measuring the values of h as a function of t at the fixed coordinate x, it ispossible to derive the experimental dependence h(t) or, conversely, t(h). Let usexplore the possibility of obtaining η(h) using such an approach.

For convenience, subsequently we use the following notation:

, (3.264)

i xt2 2η τ∞

=

hx

tit= −∞η

τ 2

∂∂

= ( )

+

∫xt

dthh

h it

τη

0

∂∂

=+( )

2

2

1xt

ih it

h

τη

∂∂ ∂

=+( )

2 1xh t h it

τη

Cxi

h it h= ( ) = ( )τ

ϕ,



, (3.265)

.

It follows from Equation 3.258 that f(h) satisfies the following functionalequation:

. (3.266)

As the viscosity is always positive, the definition (3.265) gives for all thick-nesses h

.

According to the definition, ϕ(h) = it(h) does not vanish anywhere, except ath = ∞. Note that ϕ(h) has the dimension of length.

Using Equation 3.266 we determine the dependence η(h). We proceed accord-ing to the following procedure:

(i) Let us define a sequence of film thicknesses, Hm, m = 0, 1, 2, 3, …such that

Having determined the dependence at , we find the valueof i.

(ii) We show that the function f(h) can be determined at h ≥ HN where h =HN is a film thickness such that η(h) = η∞ = const for h ≥ HN, i.e.,

.

(iii) The function f(h) for h ≥ HN, constructed according to (ii), will beextended to include values h < HN up to h = 0, which allows one tofind the dependence of viscosity on thickness for all h > 0, because

.

f h dhdh

h

hh

( ) = ( )∫∫ 1

1

2

200

η

f f h0 0 0 0( ) = ′( ) = ≥,

f h h f h C+ ( ) = ( ) +ϕ

f h f h f h( ) > ′( ) > ′′( ) >0 0 0, ,

dhdh

hC m

h

H

H

m

m

12

20

0 1 21

1

η( ) = = …∫∫−

, , , , .

η h( ) h Hm=

′′( ) = =∞

f h const1

η

η hf h

( ) =′′( )1



(iv) We develop a method for determining the thickness of the bound-ary layer of liquid.

Everywhere in (i)–(iv), the viscosity is considered as a continuous func-tion of thickness h.

The dependency t(h) is considered to be determined experimentallywith an accuracy sufficient for further consideration.

We now turn to the realization of the program indicated in the precedingdiscussion.

(i) Let us introduce the following notations:

(3.267)

where m = 0, 1, ….

At h = 0, from Equation 3.267 we conclude that f(ϕ–1) = f(0) + C = C, orf(H0) = C. Let us assume that h = Hm

, (3.268)

and prove Equation 3.268 by induction. We assume that it is valid at m = l andshow then that Equation 3.268 holds identically for m = l + 1.

Equation 3.256 at h = Hl gives f (Hl + ϕl) = f(Hl) + c = (l + 1)C + C = (l + 2)·C, or f(Hl+1) = (l + 2) C.

This relation proves the validity of Equation 3.268. As Equation 3.268 isvalid at m = 0, 1, it is valid for any m.

It follows from definition (3.267) that , i.e., is an increasingsequence of thicknesses; hence, it has a finite or an infinite limit .Passing over to the limit in the relationship Hm+1 = Hm + ϕ(Hm), we obtain H =H + ϕ(H). Hence, if H < ∞, then ϕ(H) = 0; however, as has already been statedpreviously, this is impossible as ϕ(h) = it(h) and does not vanish anywhere, exceptat h = ∞.

In the following text, we use this notation:

.

H N

ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ− − − −= ( ) = ( ) … = + + … +1 0 1 1 00 , , , m m 11

1 0 1 1 00

( ) …

= = … = + + … +− − −

, ,

, , ,H H Hmϕ ϕ ϕ ϕmm

m m mH H

−

+ = +

1

1

,

,

…,

ϕ

f H m Cm( ) = +( ) ⋅1

H Hm m≤ +1 Hm

H Hm m= →∞lim

f H f f H f f H fm m m m m m( ) = ′( ) = ′ ′′( ) = ′′, , ,

′( ) = ′ ′′( ) = ′′ϕ ϕ ϕ ϕH Hm m m m,



By differentiating Equation 3.266 with respect to h we conclude that

. (3.269)

At h = 0, we obtain from Equation 3.269 the following expression:

. (3.270)

As f ′(h) > 0 at h > 0, then ϕ′–1 = –1. We show below that the point at whichϕ′ (h) = –1 is unique. Let us assume that ϕ′ (h0) = –1 when h0 > 0. In this case,from Equation 3.269 we have

,

i.e., which is a contradiction, becauseBearing in mind Equation 3.265, the definition ϕ(h) gives or

. (3.271)

We show below that derivatives of the function at satisfythe following relation:

. (3.272)

Equation 3.269 at yields

. (3.273)

Assuming that Equation 3.272 is valid at m = l, we show its validity at m =l – 1. From Equation 3.269 at we have

(3.274)

which proves the relation (3.272).

′ + ( ) ⋅ + ′( ) = ′( )f h h h f hϕ ϕ1

′( ) + ′( ) = ′( ) =−f fϕ ϕ0 11 0 0

′( ) = ′ + ( ) ⋅ + ′( ) =f h f h h h0 0 0 01 0ϕ ϕ

′ =f h( ) ,0 0 ′ >f h( ) .0 0it′ = −( ) ,0 1

it

= −′( )1

0

f h( ) h H Hm N= <

′ = ′ ⋅ + ′( )=

−

∏f fm N i

i m

N

11

ϕ

h H N= −1

′( ) = ′ ⋅ + ′( )− −f H fN N N1 11 ϕ

h Hl= −1

′ = ′ ⋅ + ′( ) = ′ ⋅ + ′( )

− −

=

−

∏f f fl l N i

i l

N

1 1

1

1 1ϕ ϕ ⋅ + ′( )

= ′ ⋅ + ′( )

⋅

−

= −

−

∏

1

11

1

ϕ

ϕ

l i

N i

i l

N

f 11 11

1

1

+ ′( ) = ′ ⋅ + ′( )−= −

−

∏ϕ ϕl N i

i l

N

f ,



Now we show by induction that the second derivative with respect to h ofthe function f(h) at satisfies the following relation:

(3.275)

By differentiating Equation 3.266 twice with respect to h we obtain

. (3.276)

At Equation 3.276 changes to Equation 3.275. Assuming Equation3.275 to hold at m = l, we show its validity at m = l + 1. From Equation 3.275at we get

It follows from Equation 3.275 that when h = 0,

, (3.277)

here being determined from Equation 3.272 at m = 0. By differentiatingEquation 3.266 three times with respect to h and putting h = 0, we obtain

.

H Hm N<

′′ = ′′ ⋅ + ′( ) + ′ ⋅ ′′ + ′ ⋅ ′− − + − −f f f fN m N i N m N m N k12

1ϕ ϕ ′′

× + ′

− +=

−

= −

−

∑∏ ϕ

ϕ

N k

k

m

i N m

N

1

0

21

1 NN i

i k

m

− −= +

−

( )∏ 1

1

1

.

′′ + ( ) ⋅ + ′( ) + ′ + ( ) ⋅ ′′f h h h f h hϕ ϕ ϕ ϕ12

hh h( ) = ′′( )ϕ

h HN= −1,

H N l− −1

′′ = ′′ ⋅ + ′( ) + ′ ⋅ ′′− − − − − − − −f f fN l N l N l N l N l1 1

2

11 ϕ ϕ

== ′′ ⋅ + ′( ) + ′ ⋅ ′′ + ′ ⋅ ′′− + − − −f f fN i N l N l N k N k12

1ϕ ϕ ϕ −− − −= +

−

=

−

= −

−

⋅ + ′( )

∏∑∏ 1 1

2

1

1

0

21

1 ϕN i

i k

l

k

l

i N l

N

× + ′( ) + ′ ⋅ ′′ = ′− − − − −1 1

2

1ϕ ϕN l N l N l Nf f ⋅⋅ + ′( ) + ′ ⋅ ′′

+ ′

− − −= − −

−

∏ 12

1

1

1

ϕ ϕi N l N l

i N l

N

f

fNN k N k N i

i k

l

k

l

− − − − −= +=

−

⋅ ′′ ⋅ + ′( )∏∑ ϕ ϕ1 1

2

10

1

1 .

′′( ) = ′( ) ⋅ ′′− −f f0 1 1ϕ ϕ

′ −f ( )ϕ 1

′′′( ) = ′( ) ⋅ ′′′− −f f0 1 1ϕ ϕ



but, on the other hand,

.

Consequently,

. (3.278)

Thus, the sign of is opposite to that of the third derivative, ath = 0. It is shown in the following text that, in the case η(h) = η∞ = const,

i.e., .

The relations (3.268), (3.272), and (3.275) allow determining the dependenceη(h) at thicknesses h = Hm

, (3.279)

whereas the relation (3.278) gives a criterion for the behavior of viscosity at small h.Thus, we have fully completed the first part of the program of determining

the viscosity as a function of h.

(ii) As at then at such h

, (3.280)

where A and B are integration constants, which are as yet unknown.

Let us set

;

then, at we derive from Equation 3.266 that

′′′( ) = −′( )

( ) = − ′( ) ⋅ ′′ ( )f f00

00 0

22

ηη

η

′( ) = −′′′( )

′( ) ⋅ ′′ ( )−

ηϕ

ϕ ϕ0

0

012f

′η ( )0 ′′′ϕ ( ),h

′′′( ) =ϕ 0 0, ′( ) = ′ =∞ϕ η0 0

η Hf H

mm

( ) =′′( )

1

′′( ) = =∞

f h const1

ηh HN≥ ,

f hh

Ah B( ) = + +∞

2

2η

λ ηN

H

dh hN

= ( )∫ /0

h HN≥ ,



(3.281)

On the other hand, from Equation 3.280 we find that

. (3.282)

Comparing Equation 3.281 and Equation 3.282, we infer that

. (3.283)

Thus, A specifies an integral deviation of η(h) from η∞ at h < HN. At η(h) ≡η∞ for all h, we have from Equation 3.283: λN = HN /η∞ and A = 0. At this point,we assume λN and HN to be known, and they are defined in (iv).

From the second part of equality (3.281) we get

; (3.284)

hence,

. (3.285)

The condition λN = const at h ≥ HN imposes some restriction on the functionϕ(h). From Equation 3.284 ϕ(h) is determined by the following relation:

. (3.286)

For we get from Equation 3.286 that

. (3.287)

f h h f h dhdh

h

h HN

N+ ( ) − ( ) = ( ) = + −

∞ϕ

ηλ

η12

2

1

=+( )

− + −

++

∞ ∞ ∞

∫∫∫ dh

h h H

h

hh

h

h

NN

1

0

22

1

2 2

ϕϕ

ϕη

λη

=ϕ C.

f h f hh h

A C+( ) − ( ) =+( ) − + ⋅ =

∞ ∞ϕ

ϕη η

ϕ2 2

2 2

AH

NN= −∞

λη

ϕη

ϕ λη

2

20

∞ ∞+ − −

− =N

Nh HC

λϕ

ϕη ηN

NCh

h h H= ( ) − ( ) − −∞ ∞2

ϕ λ η η λ ηh h H C h HN N N N( ) = ⋅ + −( ) + − ⋅ + −( )∞ ∞ ∞2

2

λ η ηN Nh H C⋅ + −( ) >>∞ ∞2

2 ,

ϕ ηλ η

hC

h HN N( ) =

+ −∞

∞



This expression at results in

. (3.288)

Multiplying Equation 3.259 by and dividing it by τ yields

,

or, Hence,

, (3.289)

. (3.290)

As has been pointed out previously, at

,

and, hence, Equation 3.286 transforms to Equation 3.289. From Equation 3.289we conclude that

These properties mean that the inverse to function, which is h(t),decreases monotonically and is concave.

h H N N>> − ⋅[ ]∞λ η

ϕ ηh

Ch

( ) = ∞

iη∞

2 22η

τ∞⋅ = + ( )x i

hit it

2 2 2η ϕ ϕ∞ = +C h .

ϕ ηh h C h h( ) = + − ≥∞2 2 0,

ϕ η ηhC

hh C( ) = ⋅ >>∇∞

∞, 2

η η λη

hH

NN( ) ≡ =∞∞

,

0 12

1

2

2

2

2

> ′( ) = − ++

> −

′′( ) =+( )

∞

∞

∞

ϕη

ϕ η

η

hh

h C

hC

h C

,

33 2

25 2

0

6

20

0

/

/

,

,

>

′′′( ) = −+( )

<

′′′( )

∞

∞

ϕ η

η

ϕ

hC h

h C

== 0.

ϕ( ) ( )h it h=



Using Equation 3.268, from Equation 3.280 at we obtain

. (3.291)

As a result, we find that

. (3.292)

Thus, Equation 3.280 determines at (with and assumedto be known constants).

(iii) Let us set

(3.293)

at and

(3.294)

at h ≥ HN. Let h now vary from HN–1 to HN. In this case, by definition,h + ϕ(h) varies from HN to HN+1. However, at h ≥ HN, the unknownfunction, f(h), is already defined and given by Equation 3.294. Hence,in accordance with Equation 3.266, we can determine f(h) at smallerthickness in the interval, HN–1 ≤ h ≤ HN, as

(3.295)

Let us show that f(h) constructed according to Equation 3.295 is continuousat h = HN. Indeed, fN–1(HN = fN (HN+1) – C, but, by definition of HN+1: fN (HN+1) =fN (HN) + C.

From the two last equalities we deduce that fN–1 (HN) = fN (HN). That is, theconstructed function is continuous at h = HN.

Let us assume now that we have already determined the function fm+1(h), i.e.,the value of f(h) in the interval of thickness Hm+1 ≤ h ≤ Hm+2. Now, let h vary

h H N=

HAH B N CN

N2

21

η∞+ + = +( )

B N CH

A HNN= +( ) − − ⋅

∞1

22

η

f h( ) h H N≥ λN H N

f h f hm( ) = ( )

H h Hm m≤ ≤ +1,

f h f hh

Ah BN( ) = ( ) = + +∞

2

2η,

f h f h h C

h hA h h

N N−

∞

( ) = + ( ) −

=+ ( ) + +

1

2

2

ϕ

ϕ

ηϕ(( ) + −B C.



from Hm to Hm+1. According to the definition (3.267), h = ϕ(h) varies within Hm+1

to Hm+2. In this range, however, f(h) has already been determined and is equal tofm+1(h).

According to Equation 3.266, we define

. (3.296)

By construction the function f(h) satisfies the functional Equation 3.266. Theuniqueness of the function obtained follows from the additional condition: at h ≥HN:

.

Equation 3.295 and Equation 3.296 result in

which coincides with Equation 3.268.Let us show that the deduced function is continuous at the joining thicknesses,

that is, We do it for the case of fN–2(h) as an examplebecause other joining thickness can be considered in the same way:

.

In the latter derivation we used the continuity of at the thickness h =HN. The two last equalities give

.

Similarly, assuming the continuity of we can show the continuity offm(h) at thickness h = Hm+1.

At all other thicknesses (different from the joining thicknesses), the continuityof f(h) follows from the continuity ϕ(h) and relation (3.296).

Differentiating relation (3.296) with respect to h yields

.

f h f h h Cm m( ) = + ( ) −+1 ϕ

′′( ) = =∞

f h const1

η

f H f H C f H N m C

N

m m m m N N( ) = ( ) − = = ( ) − −( ) ⋅

= +( ) ⋅

+ +1 1

1

�

CC N m C m C− −( ) ⋅ = +( ) ⋅1 ,

f H f Hm m m m( ) ( ).+ + +=1 1 1

f H f H C N C C N CN N N N− −( ) = ( ) − = +( ) ⋅ − = ⋅1 1 1 ,

f H f H C f H C N C C NN N N N N N− − −( ) = ( ) − = ( ) − = +( ) ⋅ − = ⋅2 1 1 1 CC

f h( )

f H f H N CN N N N− − − −( ) = ( ) = ⋅1 1 2 1

f hm+1( ),

′ ( ) = ′ + ( ) ⋅ + ′( ) >+f h f h h hm m 1 1 0ϕ ϕ



As 0 > ϕ′ > –1 and at h = Hm, we derive from the last equality

which coincides with Equation 3.272.Differentiating relation (3.296) twice with respect to h, we obtain

. (3.297)

From Equation 3.297 at h = Hm, it is possible likewise to derive formula(3.275) by induction.

Thus, the function f(h) obtained is a unique solution of Equation 3.266 andsatisfies all the previous relations (3.268), (3.272), and (3.275). At any value ofh, the viscosity is now given by

(3.298)

We now turn to the determination of the quantities

and HN, which are as yet unknown.Let N0 be some value of the number that we provisionally take as the N

sought. In this case, according to Equation 3.285, at we conclude that

. (3.299)

Let denote function (3.286) at i.e.,

,

at Determine as the smallest number at which

′ >f hN ( ) 0

′ ( ) = ′ ( ) + ′ = ′ ( ) ⋅ ++ + + +f H f H f Hm m m m m m m1 1 2 21 1ϕ ′′( ) ⋅ + ′( )

= ⋅ ⋅ ⋅ = ′ ( ) ⋅ + ′( )

+

=

−

∏

ϕ ϕ

ϕ

m m

N N i

i m

N

f H

1

1

1

1

,,

′′ ( ) = ′′ +( ) + ′( ) + ′ +( ) ⋅ ′′+ +f h f h f hm m m1

2

11ϕ ϕ ϕ ϕ

η hf h

H h Hm

m m( ) =′′( ) ≤ ≤ +1

1, .

ληN

Hdh

h

N

= ( )∫0

h H N= 0

λϕ

ϕηN

N

NC0

0

0

2= −

∞

ϕ( ) ( )N h0 λN0,

ϕ λ η η λ ηNN N N Nh h H C h H0

0 0 0 0

22( )

∞ ∞ ∞( ) = ⋅ + −( ) + − + −( )

h HN≥0. N N0 ,



, (3.300)

at allRelations (3.299) and (3.300) determine the values of the thickness of a

boundary layer and λN.

Experimental Part [35]

The experimental device used for studying the boundary viscosity of volatilesubstances was substantially the same as that for nonvolatile liquids (Figure 3.20).

The body of device 1, made of brass, has a rectangular cutout, which receivesa steel insert 3, with its upper surface polished to the finish quality 14-b and thensubjected to additional optical polishing. Electron microscope examinationrevealed that the surface of the insert was fairly smooth. There are separaterandomly oriented scratches with rounded-off edges. Their depth is so small thatshading failed to be obtained. Their horizontal size varies from 20–160 Å, andtheir number is small. In the horizontal scale, the main working surface exhibitsroughness of the size smaller than 80 Å; hence, their depth is of the order of40 Å. At the top, the instrument is covered with prism 5. A plane-parallel channelis formed between the base of the prism and the upper surface of the insert. Acurrent of gas (air or nitrogen) is passed through the channel to blow off theliquid being tested.

The thickness of the channel is adjusted by putting calibrated shims 4 betweeninsert 3 and the base 2. For measuring the pressure gradient, two narrow openings6 with connections to join a pressure gauge are provided in the channel (theconnections are not shown in Figure 3.20).

In the device, a blow-off current is jetted up by a water-jet pump installed inan exhaust cabinet. This is done to prevent the vapor of volatile liquids fromescaping into the air of laboratory rooms. The pressure differential is checkedagainst a liquid differential pressure gauge, using a reading microscope.

FIGURE 3.20 Schematic diagram of the experimental device for investigations of theboundary viscosity of liquids.

ϕ ϕϕ

h h

h

N( ) − ( )( ) ≤

( )0

0 01.

h HN≥0.

6

57 7

2

13

1

4



The accuracy in the determination of differential pressure is 1%. Such anaccuracy is ensured because an additional capillary tube is connected to the systemin series with the instrument channel, the pressure drop being measured at theends of the capillary tube.

The capillary tube is chosen such that the pressure drop in the channel equalto 0.25 mm H2O corresponds to the pressure drop of 200 mm H2O in the capillarytube. This facilitates measurement and ensures the measurement accuracyrequired. In the experiments, the pressure drop was equal to 0.25 mm of waterover the length of the working portion of the film, which was equal to 3 cm. Thiscorresponds to the shear stress of about τ = 0.4 dyn/cm2. Experiments were alsoconducted at other pressure differentials, namely, at 0.06, 0.125, and 0.5 mm ofwater.

A saturated hydrocarbon, hexadecane, was used as a volatile liquid to betested. Hexadecane of various degrees of purification was used: industrial gradesqualified as “chemically pure,” a grade that has been additionally purified onsilica gel using the displacement chromatography method, and a chemically puregrade that has been purified in a chromatography unit. The content of organicadmixtures in hexadecane, which has been qualified as chemically pure, did notexceed 0.5%; whereas that in the chromatographically purified grade was below0.3%. The admixtures in such amounts certainly could not affect the measurementresults.

As a result of testing hexadecane, interference bands were found to occur, asbefore, on the insert. The movement of the wetting boundary was recoded. Thegap width was chosen to be 1 mm to ensure complete evaporation. It is difficultto blow off substances of very high volatility. In this case, evaporation was soprevalent over flow that during an experiment the spacing between the interferencebands would remain extremely narrow and measurements would become difficult.

The experiment is carried out as follows: The insert 3 (Figure 3.20) is coatedwith a film of the liquid to be tested. Then, the insert coated with the film isplaced in the device, and the film thickness is measured in the course of blowingoff at different distances l from the wetting boundary (l = 10 mm, l = 20 mm,and l = 30 mm) and at different shear stresses (0.1, 0.2, and 0.4 dyne/cm2). Thethickness of films is measured by the ellipsometric method.

Figure 3.21 gives experimental curves representing a variation in the filmthickness vs. the blow-off time. The curves have been obtained at ∆p = 0.25 mmof water (τ = 0.4 dyne/cm2 and l = 2 cm). The curves that have been obtained atother pressure differentials differ from the given curves in their slope as anincrease or a decrease in the pressure differential is equivalent to a decrease oran increase in the blow-off time.

The different curves in Figure 3.21 correspond to the virtually identicalexperimental conditions. However, curves are shifted from each other along thetime axis. It may be supposed that this shift depends on the difference in depositionof the original film. The initial condition (3.253) means that the initial contactangle of the film is 90˚ = π/2. As the real initial state of the film being blown offis different from the aforementioned one, this determines certain scattering in the



effective blow-off times. At considerable blow-off times, the influence of the initialstate x(h) results only in a shift of experimental curves from each other, the curvesthemselves remaining similar.

In the case under consideration, the dependences η(h) calculated by the abovemethod prove to be practically the same. Certain minor differences are observedfor larger thicknesses. This is quite understandable because the influence of thedeposition of the initial film prior to blowing off can influence only the initialstage of the process (at high-enough thicknesses).

The experimental data were processed according to Equation 3.298, whichgives the dependence of the boundary viscosity on film thickness.

Figure 3.22 represents the deduced dependences of the boundary viscosityon film thickness. Figure 3.22 shows that in the thickness range from 50 to 200 Å,the viscosity of hexadecane is lower than the bulk viscosity of liquid.

Note, the same decreased value of hexadecane viscosity in the boundary layerwas obtained for two different pressure differences, ∆p = 0.25 mm, and ∆p =0.06, 0.125, and 0.5 mm of water when the evaporation rates differ. This showsthe independence of the procedure from evaporation rates. Measurements ofviscosity at distances l = 1 cm and l = 3 cm from the wetting boundary alsoyielded the same decreased viscosity values within the indicated thickness range.Therefore, the phenomenon of film spreading at the solid substrate (in the oppositedirection to the blow-off gas) can be neglected. Otherwise, the spreading phe-nomenon would be most pronounced at l = 1 cm.

FIGURE 3.21 Thickness of the hexadecane film versus the blow-off time, t.

Time t, 103 sec2.8

0

100

200

300

400

500

600

700

2.9 3.0 3.1

Film

thick

ness

h, Å

12

34

×

×

×

×

×

×××



A hexadecane molecule is composed of 14 methylene groups that interactwith both the substrate and each other, and of two end methyl groups. The numberof methyl groups being small, they do not play an important part. In the layerclosest to the wall, one may expect a horizontal orientation of molecules owingto their sufficient rigidity and length.

The observed decrease in the viscosity of hexadecane seems to be attributableto the horizontal orientation of molecules.

CONCLUSIONS

The theory thus developed allows us to determine the boundary viscosity as afunction of the distance to the substrate, η(h). The dependency η(h) is determinedaccording to Equation 3.298, using the extension procedure set forth in (iii).

In the case of constant viscosity, the thickness of a layer at a given pointvaries with time according to the hyperbolic law

.

3.9 COMBINED HEAT AND MASS TRANSFER IN TAPERED CAPILLARIES WITH BUBBLES UNDER THE ACTION OF A TEMPERATURE GRADIENT

Simultaneous flows of vapor and a liquid in a thin liquid film under the actionof both temperature and pressure gradients are investigated as a function of theradius and taper of capillaries for decane and hexane. The regions of the greatesteffect of film flow are established [58].

FIGURE 3.22 Calculated dependencies of the boundary viscosity of hexadecane on thefilm thickness. Curves from 1 to 4 corresponds to curves from 1 to 4 in Figure 3.21.

Visc

osity

η, P

0.01

0.02

0.03

Film thickness h, Å2001000 300 400 500

1 23

4

h tit x

t∞

∞( ) = − +⋅2η

τ



Combined heat and mass transfer in porous media is of great interest in anumber of areas. Porous space inside any porous material has a sophisticatedstructure and is difficult to model even using computer simulations. It is evenmore difficult to model a combined heat and mass transfer in such complicatedstructures. In the following text we investigate a mass transfer of oils in a modelporous system, which is a tapered capillary. The mass transfer takes place betweentwo menisci of oil at x = 0 and x = L under the action of an imposed constanttemperature gradient (Figure 3.23).

We assume that the characteristic scale in the axial direction is much biggerthan the capillary radius — that is, r(x) << L.

As a result of vapor adsorption and the liquid flow on the capillary walls, afilm of adsorbed liquid, h(x), forms on the walls. It is assumed that in each crosssection there is a local equilibrium between the vapor and the adsorbed liquid.According to Chapter 2, the hydrodynamic pressure in the adsorbed liquid film,Pl, is equal to

, (3.301)

where Pa is the pressure in the ambient air, and γ(T) and Π(h) are the liquid–vaporinterfacial tension, which is temperature dependent, and the disjoining pressureof the flat oil films, respectively.

A bubble of air separates two menisci of the wetting liquid having a differentcurvature. The surface of the capillary in between the two menisci is covered by

FIGURE 3.23 Combined heat and mass transfer in a tapered capillary of the length L.Radius of the capillary at cross section x is r(x) = r0 + x · tanα, imposed temperature

gradient is constant,

L

α

xL0

r0

TLPLPLPsL

rLh(x)

y

T0

P0P0Ps0

dT

dx

T T

LconstL=

−=0 .

P x PT x

r x h xr x

r x h xh xl a( )

( )

( ) ( )( )

( ) ( )(= − [ ]

−−

−γ

Π ))[ ]



a wetting film whose thickness, h(x), is a function of the coordinate x; boundaryvalues of the film thickness are h(0) = h0 and h(L) = hL.

The excess pressure in the liquid film as before is

. (3.302)

Note that now the excess pressure is a function of the position. Equality ofchemical potentials of the liquid molecules in the vapor phase and in the liquidfilms results in the same equation as in Chapter 2, which applied however locallyin each cross-section:

, (3.303)

where T(x), ps(T), and p(x) are the local temperature, the local pressure of thesaturated vapor, and the local vapor pressure, respectively; R is the gas constant,and vm is the molar volume of the liquid. In the following text we assume thatthe liquid under investigation is an oil, which completely wet the capillary walls.Disjoining pressure in this case is determined by dispersion forces only — that is,

where A is the Hamaker constant.The combination of Equation 3.301 and Equation 3.303 results in

(3.304)

See Chapter 2, Section 2.7 and Section 2.12 for explanations on why the modifieddisjoining pressure

should be used.Now we should deduce an equation for a flow in thin liquid films under the

action of both the pressure gradient and the surface tension gradient.We consider a flow in thin films both at low Reynolds number Re << 1 and

capillary numbers Ca << 1 because, in the following text, we consider only slow

P P Pa l= −

PRTv

p Tp xm

s= ln( )( )

Π( ) ,hA

h=

3

P x PT x

r x h xr x

r x h xl a( )

( )

( ) ( )( )

( ) ( )= −

−

−−

γ

=

A

h x

RT xv

p T x

p xm

s

3( )

( )ln

( )

( ).

r x

r x h x

A

h x

( )

( ) ( ) ( )− 3



steady-state flow. This means that both front and rear menisci (Figure 3.23) havean equilibrium profile, and we ignore any deviation from the equilibrium shapes.

Taking into account the condition r(x) << L, we can conclude as we didearlier in this chapter that (1) the velocity in the radial direction is much smallerthan the velocity in the axial direction, (2) the velocity in the axial direction, v,depends only on the radial position r, and (3) the pressure in the films dependsonly on the axial component x. After that, the remaining Stokes equation, takinginto account the definition (3.302), becomes

,

with nonslip condition on the capillary walls

,

and the condition on the free film surface

,

where ′ marks the derivative with respect to x. This condition shows the tangentialstress on the free film surface caused by the presence of the surface tensiongradient, which, in its turn, is caused by the temperature gradient.

Solution of the preceding equation with two boundary conditions and Equa-tion 3.304 results in the following expression for the mass flow rate, Q:

, (3.305)

where the following expressions for mass flow rates in vapor, Q1; in the thinliquid film under the action of the temperature gradient (thermocapillary flow),Q2; and under the action of the pressure gradient, Q3, are:

(3.306)

− ′ = +

P

d v

dr rdvdr

η2

2

1

v r x( ( )) = 0

η γ γdvdr

ddx

ddT

Tr r x h= −

= = ′( )

Q Q Q Q= + +1 2 3

Qr h D

RTp

Qr h d

dTT

1 1

2

2 2

2

= − − ′

= ′

λ π µ

λ π ρη

γ

( )



where D is the diffusion coefficient of the vapor; ρ, η, and µ are density, viscosityand mass of the molecule of the liquid, respectively. We introduced additionalcoefficients λ1, λ2, and λ3 . Each of these coefficients can be 1 or zero to “switch”on or off the corresponding part of the mass flow rate.

The dependency of the film thickness on x, h(x), is determined by Equation3.304.

The boundary conditions for the first Equation 3.305 are as follows:

(3.307)

where we took into account that the saturated pressure depends only on thetemperature, T.

Equation 3.305 was differentiated with respect to x, and after that, was solvednumerically using a quasi-linearization method [36] combined with the methodof iterations:

, (3.308)

where n is the number of iterations and ε = 0.01 is the given relative error. Evenin the case where condition (3.308) is satisfied, not less than 10 iterations weremade.

The calculations were made using mean values of ρ, η, and D inside thebubble length, L, corresponding to the mean temperature Tm = (T1 + T2)/2. It wasassumed that a constant temperature gradient is imposed: dT/dx = const, i.e., thatthe distribution of the temperature, T, was linear inside the bubble. The distributionof the vapor pressure, p(x), corresponding to the imposed value of dT/dx = const,and the value of the total mass flow rate, Q, were calculated. Tabulated values ofthe liquid–air interfacial tension, γ(T), and the saturated vapor pressure, ps(T), wereused in the calculations. The intermediate values were found by linear (for theinterfacial tension, γ) and logarithmic (for the saturated pressure, ps) interpolations.

Calculations were made for two nonpolar liquids with a different volatility:decane and hexane. The well-known equation of the molecular component of thedisjoining pressure, Π(h) = A/h3, where A ~ 10–13 erg [37], was used as theisotherm Π(h). The value of the Hamaker constant, A, was regarded as indepen-dent of the temperature [38]. Dependences of the total mass flow rate, Q, and of

Qr RTh

vpp

pp

TT

ppm

s

s

s3 3

323

= − ′ − ′ + ′

λ π ρη

ln

p p TT v

r RT

p

sm( ) ( ) exp

( )

( ) ( )0 0

2 0

0 0= [ ] − [ ]

γ

(( ) ( ) exp( )

( ) ( )L p T L

T L v

r L RT Ls

m= [ ] − [ ]

2γ

Q Q Qn n n−( ) <+1 ε



the individual components of the flow rates on the radius and the taper of thecapillaries were obtained for dT/dx = const from 1 to 100 deg/cm with a constantmean temperature Tm = 300˚K. The total mass flow rate was calculated usingEquation 3.305, setting λ1 = λ2 = λ3 = 1. The vapor component, Q1, correspondingonly to diffusion of the vapor (in the presence of liquid films on the solid surfaceresults in decreasing of the cross section of the capillary), was calculated withλ1 = 1 and λ2 = λ3 = 0. An analogous method was used to calculate thecomponents Q2 (with λ2 = 1 and λ1 = λ3 = 0) and Q3 (with λ3 = 1 and λ1 = λ2 =0), corresponding, respectively, to the thermocapillary flow of the film and thefilm flow under the action of the pressure gradient dP/dx, due to the gradients ofthe disjoining and capillary pressures (see Equation 3.304).

The instantaneous radius of the conical capillary is given by the followingequation:

, (3.309)

where α = [r(L)–r(0)]/L = ∆r/L is the angle of taper of the capillary. Note thatwe are using the small angle approximation.

For hexane, the following values were given: D = 0.075 cm2/sec; µ = 86.17g/mol; ρ = 0.6534 g/cm3; η = 2.95 · 10–3 P; v = 132 cm3/mol. For decane, thecorresponding values are: D = 0.046 cm2/sec; µ = 142.28 g/mol; ρ = 0.7245 g/cm3;η = 8.2 · 10–3 P, v = 196 cm3/mol. The ratio L/r varied from 10 (for large capillaryradius, r) to 1000 (for small capillary radius, r). The absolute values of L werefrom the following range: from 10–1 to 10–3 cm. In view of this choice, with hightemperature gradient, dT/dx = const, the temperature difference at the boundariesof a bubble was very small. This justified the possibility of using constant meanvalues of ρ,η, and D.

CYLINDRICAL CAPILLARIES

We consider first the effect of the radius of the capillaries on film flow. To thisend, the values of the ratio Q/Q1 were calculated for cylindrical capillaries ofequal radius r (Figure 3.24). At Q/Q1 = 1, the principal mechanism of masstransfer is diffusion of the vapor: Q1 >> Q2 + Q3. Figure 3.24 shows that, withan increase in the radius of the capillaries, the effect of film flow decreases.However, in narrow capillaries (r ∼ 10–6 cm), film flow is the principal mechanismof transfer. The flow of vapor, Q1 in such thin capillaries makes a contributionto the total flow, which is an order of magnitude less, although the thickness ofthe films still remains considerably less than the radius of the capillaries (h/r =0.14). The effect of film transfer is more pronounced because the less volatiledecane: curve 1 passes above curve 2 for hexane. The results of the calculationsdo not disclose a dependence of Q/Q1 on the value of the temperature gradient,dT/dx = const, which is explained by the smallness of the absolute values ofT1 – T2, leading to a practically linear dependence of all the flows on the temper-ature gradient, dT/dx = const.

r r x= +1 α



Flow in thin films in cylindrical capillaries is mainly determined by thermo-capillary flow. The pressure-driven flow, Q3, is directed to the opposite side ascompared with the flows of vapor, Q1, and thermocapillary flow, Q2, and is alwaysmuch less as compared with thermocapillary flow, Q2. This can be shown alsoby obtaining analytical dependences for the individual components of the flow.Thus, in a cylindrical capillary, the ratio Q3/Q2 is equal to

. (3.310)

At r → ∞, (Q3/Q2) → 0. Therefore, we evaluate the ratio (3.310) for thethinnest capillaries, setting A = 10–13 erg and γ = 30 dyn/cm. Calculations showthat, for r ≥ 10–6 cm, the contribution of the flow Q3 does not exceed 3% incomparison with the flow Q2.

TAPERED CAPILLARIES

The picture of the flows in conical (tapered) capillaries has quite different featuresas compared with the flow in cylindrical capillaries. With an increase in the angleof taper of the capillary, α, the contribution of the flow, Q3, due to the differencesin the capillary pressures of the menisci, rises and, at sufficiently large values of

FIGURE 3.24 (a) Calculating scheme, which was used both in the case of tapered (r(0) <r(L)) and cylindrical capillaries (r(0) = r(L)); (b) dependences of the ratio of the totalmass flow rate, Q, to the mass flow rate of vapor, Q1, on the radius of cylindrical capillariesfor decane (1) and hexane (2): r = 10–2 cm, L = 10–1 cm; r = 10–3 – 10–5, L = 10–2; r =10–6, L = 10–3.

10–4 10–60

4

8

12

2

2

1

r, cm

r(x)

Q/Q 1

h(x)a

10–20

4

1L

0 x

b

Q Q A r3 22

1 34 3/ / /≈ ( )γ

/



α, becomes dominating. As an example, Figure 3.25 shows dependences of Q/Q1

on α for r0 = 10–4 cm. As can be seen from Figure 3.25, at α ≤ 10–5, the ratio ofQ/Q1 remains constant and has exactly the same value as in a cylindrical capillaryof radius r = r(0). The dashed lines show the course of the dependences of (Q1 +Q2)/Q1 on α. The flow rates Q1 and Q2 depend only weakly on α. Thus, thedeviation of curves 1 and 2 downward with α ≥ 10–5 is connected only with theeffect of the flow Q3 directed toward the side of the narrowing of the capillary.At α ≥ 10–4, its effect becomes dominating, as a result of which the total flowrate, Q, changes sign.

At some value of the taper α = α*, the total flow rate vanishes, Q = 0. Hence,the flow of vapor, Q1, and the thermocapillary flow, Q2, equalize out in a directionopposite to the direction of the film flow, Q3. Such circulating flows are realized,for example, in heat pipes as well as in porous media completely saturated by aliquid. The local taper of a pore, corresponding to the condition Q = 0, dependson the temperature gradient and the radius of the capillaries. It can be found usingthe calculating procedure suggested in this section.

In porous bodies, where the capillaries have a variable radius, bubbles of air,displaced toward the hot side, can be held in expanded pores if the values of r,α , and the imposed temperature gradient, ∆T, are such that the condition Q = 0is satisfied. This hold-up means that the air- and, consequently, moisture-contentcannot vary with time, in spite of the existence of the imposed temperaturegradient, ∆T, as the flow under the action of the temperature gradient, ∆T, iscompensated by a reverse flow under the action of the difference arising in thecapillary pressures.

FIGURE 3.25 Dependences of Q/Q1 on the taper of the capillaries, α, for decane (1) andhexane (2): r0 = 10–4 cm, L = 10–2 cm, dT/dx = –1 deg/cm.

10–6 10–5 10–4 10–3

1 2

0

5

10

Q/Q

1

α∗

α

–10

–20



With a transition to thinner capillaries, the picture of the mass transfer isbecoming even more complicated. Figure 3.26 gives the results of calculationsfor r(0) = 10–6 cm with the same values of L and ∆T as in Figure 3.25. Whereas,at a small taper (α < 10–7), the dependences of Q/Q1 have qualitatively the sameshape as in Figure 3.25, in the region of larger values of α, the dependency ofQ/Q1 on α undergoes a discontinuity and changes the sign. The reasons for sucha course of the curves can be established by analyzing the dependences of theindividual components of the flow on α. It follows from the calculations that thethermocapillary flow, Q2, is relatively small and almost does not depend on α .The flow of vapor, Q1, directed at small angles α toward the cold side, starts tobe retarded with a rise in α, as a result of an increase in the pressure of the vaporabove the meniscus r(L). Note that in the calculations it was assumed that r(0) =const, r(L) = r(0)+αL, that is, it increases with α. At α = α0 = 5⋅10–7, thedifference ∆p, due to the different temperatures of the meniscuses, is compensatedby the Kelvin’s difference in the pressures ∆p, connected with the differentcurvature of the meniscuses. At α > 5⋅10–7, the flow of vapor changes direction:the Kelvin’s difference in the pressures of the vapor exceeds the thermal differ-ence. An increase in the taper, α (at L = const), leads to a sharp increase in therate of the back flow of vapor (Q1 < 0).

FIGURE 3.26 Dependences of Q/Q1 on the taper of the capillaries, α, for decane (1) andhexane (2): r = 10–6 cm, L = 10–2 cm, dT/dx = –1 deg/cm.

Q/Q 1

2

1

100

10–910–810–710–610–5

–50

50

0

–100

2

1

α0α∗ α



Distinct from the vapor flow, Q1, the pressure flow in thin liquid films, Q3 ,is always directed toward the side of the narrower part of the capillary, and Q3 <0. With an increase in the taper, α, the absolute values of Q3 rise. At α > 5⋅10–8,Q3 makes the principal contribution to the total flow; hence, here it can be assumedthat Q ∼ Q3. As the flow Q1 approaches zero, the ratio Q/Q1 = Q3/Q1 → ±∞,where Q/Q1 → −∞ at Q1 > 0 and Q3/Q1 → + ∞ at Q1 < 0, which explains theresults given in Figure 3.26.

As can be seen from Figure 3.26, at α > α0 , the values of Q/Q1 fall with anincrease in the value of α. This is explained by the decreasing contribution ofthe film flow, Q3 , with an increase in the mean radius of the capillary rm = r(0) +(α/2). In thin pores (r ∼ 10–6 cm) and for small bubbles (L = 10–2 cm), even asmall taper leads to a sharp increase in the rate of mass transfer due to the flowof the liquid films. In the region of values of the taper α ~ α0 , the flow Q3 ~ Qcan exceed the flow of vapor. At α = α0 , the flow in the liquid phase is the solemass transfer mechanism in the system. For the less volatile decane, the effectof film transfer is more strongly expressed (curves 1) than for hexane (curves 2)(Figure 3.26).

The smaller the radius of the capillary, the greater the role of the taper; eventhe absolute value of the taper is insignificant. Whereas at r(0) = 10–4 cm, theeffect of the taper starts to be appreciable at α > 10–5, with r = 10–6 cm, it isalready appreciable at α > 10–8.

3.10 STATIC HYSTERESIS OF CONTACT ANGLE

We have argued in Chapter 1, Section 1.3, that the hysteresis of the static contactangle is usually related to the heterogeneity of the surface, either geometric(roughness) or chemical. The underlying assumption was that at each point ofthe surface there is an equilibrium value of the contact angle, depending only onthe local properties of the substrate. As a result, a whole series of local thermo-dynamic equilibrium states can be realized, corresponding to a certain intervalof values of the contact angle. The maximum possible value corresponds to thevalue of the static advancing contact angle of wetting, θa, and the minimumcorresponds to the static receding contact angle, θr.

In this section we present a theory of static hysteresis of the contact angle onsolid homogeneous substrates developed in terms of quasi-equilibrium phenom-ena. The values of the static receding, θr , and static advancing, θa , contact anglesare calculated, based on the shape of disjoining pressure isotherm. It is shown thatall contact angles, θ, in the range θr < θ < θa, which are different from the uniqueequilibrium contact angle, θe, correspond to the state of slow microscopic advanc-ing at all contact angles, θ, in the range θe < θ < θa or receding motion at allcontact angles, θ, in the range θr < θ < θe, respectively. This microscopic motion,step-wise, becomes fast macroscopic advancing or receding motion after the con-tact angle reaches the critical values θ = θa or θr = θ, correspondingly [59].



There is no doubt that heterogeneity affects the wetting properties of anysolid substrate. However, heterogeneity of the surface is apparently not the solereason for static hysteresis of the contact angle. This follows from the fact thatnot all the predictions made on the basis of this theory have turned out to be true[39,40]. Besides that, static hysteresis of the contact angle has been observed incases of quite smooth and uniform surfaces [41–45]. Further, it is present evenon surfaces that are definitely molecularly smooth: free liquid films [60,61].

Evidently, only a single unique value of an equilibrium contact angle, θe, ispossible on a smooth, homogeneous surface. Hence, the static hysteresis contactangles θa ≠ θe, θr ≠ θe, and all contact angles in between, observed experimentallyon such surfaces, can correspond only to a nonequilibrium or quasi-equilibriumstate of the system.

However, if the relaxation time of the system is long, a local equilibrium thatis not in equilibrium with the surrounding medium can be established at themeniscus or droplet. It is shown in the following text that there are certain criticalvalues of contact angles beyond which such a local equilibrium is not possiblebecause the relaxation time becomes very small. We relate these critical valuesof the angles to static advancing, θa, and static receding, θr, contact angles. Thus,the discussion of the static hysteresis phenomenon in this section is based on theanalysis of nonequilibrium states of a system and conditions of violation of localequilibrium of menisci or drops.

EQUILIBRIUM CONTACT ANGLES

In Chapter 2, Section 2.3, we considered an equilibrium state of the wettingmeniscus of a liquid (0 ≤ θe < 90˚) in a capillary with a flat slit of breadth 2H >>he, where he is the thickness of the equilibrium wetting film covering the surfaceof the capillary (Figure 2.13). If H >> he, the liquid in the central part of the slitis far from the range of the surface forces action. Neglecting the effect of grav-itational forces, the radius of curvature of the surface of the meniscus in thecentral part of the slot, ρe, is constant.

We denote by h the distance along the normal between the substrate and thesurface of the liquid. Then h(x), as before, is an equation of the profile of the liquidlayer, where x is the coordinate in the direction of the plane of symmetry of thecapillary.

Between the meniscus of constant curvature, ρe, and the flat film of thickness,he = const, there is a transition zone. Only capillary forces act in the region ofthe unperturbed meniscus, and only surface forces act in the flat film. However,both these forces act simultaneously within the boundaries of the transition zone(see Chapter 2, Section 2.3, for details).

When applied to a layer of liquid h(x), the equilibrium condition correspondsto a constant hydrostatic pressure, which can be written as follows (see Chapter 2,Section 2.1 and Section 2.3):



, (3.311)

where the first term characterizes the contribution of the capillary pressure causedby the local surface curvature. The second term of Equation 3.311 characterizesthe contribution of the disjoining pressure (surface forces) acting from the sideof the substrate.

Because of the symmetry of the meniscus, it is sufficient to consider theequilibrium of a liquid layer h(x) over the thickness range from h = he at x = ∞to h = H at x = 0. We solved Equation 3.311 in Section 2.3. Solution of Equation3.311 includes three constants, namely, the excess pressure Pe = const, and twointegration constants. For determining these constants, we used in Section 2.3three boundary conditions, with two at the center of the meniscus:

, (3.312)

and the condition of conjugation of the meniscus with the flat film:

. (3.313)

We multiply both sides of Equation 3.311 by h′ and integrate it with respectto x from 0 to x. After that we arrived in Section 2.3 at the following solution,

, (3.314)

where

. (3.315)

By solving Equation 3.315 with respect to h′, we obtain

. (3.316)

From this follows in particular the condition of equilibrium of the meniscuswith the film (see Section 2.3).

γ ′′+ ′

+ =hh

h Pe( )

( )/1 2 3 2 Π

h H h x= ′ = −∞ =, , 0

h h h xe= ′ = → ∞, ,0

γ ψ1 2+ ′

=h

h Pe( , )

ψ h P P H h dhe e

h

,( ) = −( ) −∞

∫Π

′ = − −

h

h Pe

γψ

2

2

1 2

1( , )

/



Substituting the condition (3.313) in (3.316), we conclude:

. (3.317)

In Section 2.3, using this equation, we deduced the expression (2.47) for theequilibrium contact angle, θe. For each value of H, the unique values of Pe canbe found from Section 2.3.

When the whole isotherm of the disjoining pressure is within the regionΠ > 0, i.e., in the case where complete wetting is realized, the concept of thecontact angle cannot be introduced, as the radius of the meniscus, ρe, of thecircumference of the existent transition zone does not intersect the surface of thecapillary (see Chapter 2, Section 2.4, where the profile of the transition zone wascalculated analytically).

Figure 3.27a shows two types of isotherms (curves 1 and 2) when partialwetting is possible. The first one has two branches, α and β, corresponding to

FIGURE 3.27 (a) Shapes of the disjoining pressure isotherms (curves 1, 2) in the caseof partial wetting; (b) corresponding dependences of ψ(h, P) (curves 3–5), which determinethe condition of the local equilibrium of the meniscus. See explanation in the text.

γ ψ= ( ) = −( ) −∞

∫h P P H h dhe e e e

he

, Π

2

1

Pe < Πmax

Pe > Πmax

0

Πmax

Πmin

ψ

γ

β

α

Π

3

4

0 h1 hh2

h4

h3

hehu hβ

h

b

a

5



stable states of the film. Systems that possess isotherms of the second type (curve 2,Figure 3.27a) have large equilibrium contact angles, θe.

For example, isotherms of the first type are inherent to water films on quartzsurfaces [26]. Only α films are observed on the surface of the capillary ifPe > Πmax, where Πmax is the stability limit of β films (Figure 3.27a). This happensin sufficiently narrow capillaries. If Pe > Πmax, the formation of metastable β filmsis possible, as observed experimentally in Reference 26. According to the datain Reference 46, this condition is fulfilled, e.g., in the case of water in quartzcapillaries with a radius bigger than 0.25 µm.

For analyzing the equilibrium profile of a liquid in a flat capillary, we rewriteEquation 3.311 in the following form:

. (3.318)

Equation 3.318 shows that the sign of h″, and therefore the curvature of thesurface, is determined by the sign of the difference, Pe – Π(h). Note that Pe =const, but the value of Π(h) varies from Π(he) = Pe to Π = 0 at h → H. If Pe > Πmax,all the values of the difference are positive and the liquid profile inside thetransition zone is concave everywhere (h″ > 0). This condition is always fulfilledin the case of an isotherm of type 2. However, in sufficiently thick capillaries,Pe < Πmax in the case of disjoining pressure isotherms of the first type. Then, h″ <0 over the range of thickness from hu to hβ (Figure 3.27), and the surface of theliquid should possess a convex segment.

According to Equation 3.314, the following inequality should hold: ψ(h, Pe) ≥0. On the other hand, Equation 3.316 shows that the second inequality should besatisfied: ψ(h, Pe) ≤ γ. Hence, the function ψ(h, Pe) should be located within thefollowing limits:

. (3.319)

Note: ψ(h, Pe) = 0 means the infinite derivative at the corresponding thicknessh, which is possible only in the center of the capillary at h = H. The equalityψ(h, Pe) = γ means the zero derivative at the corresponding thickness h. This issatisfied at h = he, and, based on this condition, we deduced an expression (2.47)for the equilibrium contact angle in Chapter 2, Section 2.3.

As an example of the equilibrium function ψ(h, Pe), curve 3 is shown inFigure 3.27. The positions of the extrema of curve 3 can be found by differenti-ation of Equation 3.315 with respect to h, which results in

. (3.320)

′′ = + ′( )

− ( ) h h P he

11

23 2

γ

/

Π

0 ≤ ( ) ≤ψ γh Pe,

d h Pdh

P hee

ψ( , ) = − + ( )Π



Hence, as the liquid thickness varies between he < h < H, the positions ofthe maximum and minimum on curve 3 correspond to the thicknesses hu and hβ,at which the straight line Pe = const intersects the disjoining pressure isotherm 1and Pe = Π(h) (Figure 3.27).

STATIC HYSTERESIS OF THE CONTACT ANGLE OF MENISCI

We now assume that the pressure over the meniscus is changed by a value ∆Prelative to the value of its equilibrium pressure Pe (Figure 3.28b), whereas thefilm in front and the gas maintain the initial equilibrium state.

Further consideration shows that the initial state as well as the presence orabsence of the film in front (zone 3 in Figure 3.28b) does not influence ourconsideration. That is, the same consideration as below can be applied to thestatic hysteresis of contact angle on initially dry surface.

A transport process starts at once under the action of the pressure differencecreated, which can be separated provisionally into a rapid and a slow one. As theresistance of the liquid–gas interface to changes of its form is small at lowcapillary numbers, Ca << 1, the most rapid change is that of the curvature of

FIGURE 3.28 Schematics of the profile of the menisci in a flat capillary of the thickness2H. (a) equilibrium state, no flow; (b) a state of local equilibrium, when the flow is locatedin zone 2 ((1) the new quasi-equilibrium spherical menisci and quasi-equilibrium transitionzone, (3) the equilibrium liquid film, which cannot be at the equilibrium with zone 1).

2H

a

b

x

Pe

he

h

h(x)

ρe

θe

0

2H

x

P = Pe + ∆P

he

hch

h(x)

ρθ

0 2 31



the meniscus. As a result of this, a new quasi-equilibrium state of the meniscusis formed with the pressure drop:

, (3.321)

outside the boundaries of the zone of action of surface forces, where ρ = cosθ/His the radius of curvature of the main part of the meniscus, and θ is a new valueof the contact angle, corresponding to the local equilibrium state (Figure 3.28b),which cannot be equilibrium any more.

The new state of the whole system is not in equilibrium. The change in thecurvature of the meniscus causes a change in the vapor pressure over it, as resultof which the liquid starts to transfer by evaporation or condensation from thesurface of the meniscus on the surface of the equilibrium film in front (zone 3in Figure 3.28b). Besides, the increase in pressure inside the liquid causes it toflow to those parts of the film at which the initial equilibrium pressure Π(he) =Pe is still maintained. All of this remains valid when the sign of ∆P is opposite.However, in the latter case, the direction of the transfer processes is inverted.

In the following text we limit the discussion to liquids with low volatilitywhose rate of evaporation and condensation are small.

The main assumption is that the liquid flow from the quasi-equilibriummeniscus to the equilibrium film in front is very slow until some critical pressuredifference, ∆Pa (in the case of advancing meniscus) or ∆Pr (in the case of recedingmeniscus) is reached. These conditions may not exist in the case of completewetting, when the equilibrium film is sufficiently thick. However, it is known thatin the case of complete wetting there is no static hysteresis of the contact angle.

Static hysteresis is usually observed in cases of partial wetting (at 0 < θ <90˚), when the surface of the solid body is covered with substantially thinnerfilms, where the viscose resistance is very high.

At ∆P ≠ 0, i.e., in a nonequilibrium system, we subdivide the whole systeminto the following regions (Figure 3.28b): a region 1 with a state of quasi-equilibrium inside where the hydrodynamic pressure is constant everywhere andequals P = Pe + ∆P; a transport region 2 where a viscous flow of liquid occurs andin which the pressure gradually changes from the value P to Pe; and a region 3 ofa thin flat film, where the pressure equals the initial equilibrium Pe (Figure 3.28b).As the largest pressure drop in the transition region occurs in the thinnest part 2,it is evident that region 2 covers part of the transition region that immediatelyadjoins the equilibrium film he in region 3.

We write the conditions of quasi-equilibrium of the meniscus in region 1, inthe boundaries of which fluxes can be neglected and where the excess pressurecan be considered to be constant at all points and equal to P = Pe + ∆P = const.We assume that Equation 3.311 still describes the quasi-equilibrium of profile ofthe liquid, h(x), in region 1 in the absence of full thermodynamic equilibrium inthe whole system. That is, we adopt:

P P Ph

hconste= + = ′′

+ ′( )= =∆ γ γ

ρ1 2 3 2/



. (3.322)

The boundary conditions (3.312) at the center of the meniscus are maintained.Instead of (3.313), another condition should be used, as region 1 cannot now beconnected by the equilibrium or quasi-equilibrium liquid profile with the flatequilibrium film in front; there is a flow zone 2 in between (Figure 3.28b). Thisthird condition is P = const, as the value of the excess pressure P is now fixedindependently and is not determined by the thickness of the slit and the isothermof the disjoining pressure, as in the case of the equilibrium.

The region of solution of Equation 3.322 is limited from below by a certainthickness hc ≥ he, corresponding to the beginning of the flow zone (Figure 3.28b).The condition h′ = 0 is not fulfilled at h = hc, and a micro-contact angle, tanθ ≈θ = –h′(he), is formed here, the values of which can be found by solving Equation3.322.

As previously, by integrating Equation 3.322 once, we obtain once moreEquation 3.314 to Equation 3.316, but with the difference that here

, (3.323)

where the equilibrium pressure, Pe, is replaced by the new nonequilibrium pres-sure, P.

The region in which a solution of Equation 3.322 exists is determined by thereal values of h′, which exist only under the same conditions as in the case ofequilibrium:

. (3.324)

There is no solution if ψ > γ or ψ < 0: if any of these conditions is violated,then the boundary of the flow zone, hc, and the center of the meniscus cannot beconnected by a continuous profile. This implies that quasi-equilibrium becomesimpossible, i.e., the meniscus cannot be at rest and must start the motion. Weshow below that the violation of one of the conditions (3.324) determines anadvancing contact angle, θa, and the violation of the other condition, the recedingcontact angle, θr. We refer to these two angles as static advancing and recedingcontact angles, respectively.

First, we determine the value of the static advancing contact angle, θa. Forthe motion of the meniscus in front of the capillary, the pressure in the liquidphase must be increased, which diminishes the capillary pressure drop at themeniscus–gas interface. Consequently, in this case, P < Pe and ∆P < 0. Thismeans that curve ψ(h, P) (curve 4) is located below curve ψ(h, Pe) (curve 3) in

γ ′′+ ′

+ =hh

h P( )

( )/1 2 3 2 Π

ψ h P P H h dhh

,( ) = −( ) −∞

∫Π

0 ≤ ( ) ≤ψ γh P,



Figure 3.27. As P < Pe, the interval of the thickness between the positions of theminimum and the maximum broadens in the case of an isotherm of disjoiningpressure in the case of partial wetting (curve 1 in Figure 3.27). Under the conditionP < Πmax, h1 < hu and h2 > hβ (Figure 3.27).

When the absolute value of ∆P increases to a certain critical value ∆Pa, thecurve ψ(h, P) can touch, at its minimum, the h-axis, as shown by curve 4 inFigure 3.27b. This means that a point with a vertical tangent appears on the profileat h = h1, where h′ = –∞ (Figure 3.29a).

At |∆P| > |∆Pa| the curve ψ(h, P) intersects the h-axis, and the value of ψbecomes negative in a certain region of thickness. Vertical tangents appear at theupper and lower parts of the profile, which become discontinuous (dashed lines inFigure 3.29a). Note that in the region of the profile marked by the dashed line, thedifferent disjoining pressure acts, which is always the attraction between the iden-tical phases. This means that the dashed part of the profile is unstable, the profileloses stability, and a transfusion of liquid starts towards the front of the spreadingfilm. It should be noted that a similar mechanism was proposed previously byFrenkel [47] for explaining the static hysteresis of a sliding drop over an inclinedplane. We wish to point out that Equation 3.322 does not give the profile of theflowing film but only determines the limiting positions of the static profile beforethe flow has set in.

This consideration shows the rather complicated picture of flow after themeniscus starts to advance macroscopically. This is the reason why the theory ofspreading is well developed in the case of complete wetting and is still to bedeveloped in the case of partial wetting.

Although the mechanism of violation of the equilibrium is understood phys-ically, the value of the static advancing contact angle, θa, cannot be calculatedprecisely at present, as the point h = h1 belongs to a region in which the conditionh′2 << 1 is violated and the disjoining pressure, Π(h), of flat films cannot be used.

FIGURE 3.29 Determination of the critical profiles and corresponding static advancingand receding contact angles: (a) static advancing contact angle with a vertical tangent atthe thickness h1; (b) static receding contact angle with violation of the quasi-equilibriumcondition in the region of thick β-film, h4. See explanation in the text.

H H

x

h1

PaPr

ρa ρr

θa θr0

a bh

h4



We therefore limit ourselves to the estimation of values of the static advancingcontact angles, θa. At θ = θa and P = Pa, the function ψ = 0 at h = h1. Hence,we conclude from Equation 3.323 that

. (3.325)

Pressure Pa and contact angle θa are connected by the general relationship

. (3.326)


. (3.327)

After subtracting the expression derived for the equilibrium contact angle,cosθe, according to Equation 2.47 (Section 2.3), we find

. (3.328)

The second term of the right-hand side of the equation is negative and smallerthan one; hence:

. (3.329)

In the case of the isotherm of disjoining pressure of the second type (Figure3.27a, curve 2), violation of quasi-equilibrium occurs at Pa ≤ 0 because, in thiscase, ψ(h, Pa) becomes negative simultaneously for the whole range of largethicknesses, h. Hence, in this case, the limiting condition Pa = 0 corresponds tothe contact angle θa = 90˚. However, at partial wetting, the condition θe < θa isalways fulfilled. Values of the static advancing contact angle, θa, close to 90˚have been observed experimentally in a number of systems [39].

The most important result of the above consideration is that the disjoiningpressure isotherm determines not only the equilibrium values of the contact anglesbut also the static advancing contact angle, θa.

P H h h dha

h

−( ) = ( )∞

∫1

1

Π

P Ha a= γ θcos /

cos ( )θγ γa

h hhH

h dh h dh=−

( ) ≈∞ ∞

∫ ∫1

1

1

11 1

Π Π

cos cosθ θγe a

h

h

h dh

e

− ≈ + ( )∫11

1

Π

cos cosθ θ θ θe a a e− < ° > >1 90and



We now consider the solution for a receding contact angle, θr. In this case,the value ∆P > 0, as the curvature of the meniscus increases with decreasingpressure in the liquid phase. It follows from Equation 3.316 and Equation 3.323that, at P > Pe, the values of ψ(h, P) should be everywhere above the equilibriumcurve ψ(h, Pe). The shape of the function ψ(h, P) in this case is shown by curve 5in Figure 3.27. Violation of the conditions of quasi-equilibrium occurs in thiscase at ψ(h, P) = γ, i.e., upon an increase in ∆P to such a critical value ∆Pr, thatthe curve ψ(h, Pr) intersects the dashed line γ = const (Figure 3.27, curve 5).

If the capillary is sufficiently narrow and Pe > Πmax, an intersection can occuronly in the region of small thicknesses, e.g., at h = h3 (Figure 3.27), as in thiscase the curve ψ(h, P) possesses only one maximum.

A formal intersection of the curve ψ(h, P) with the straight line γ = constshould occur at an arbitrarily small difference between P and Pe, as the curveψ(h, P) already touches the straight line γ = const at the point h = he. However,it should be remembered that there is a solution in region 1 (Figure 3.28b), limitedfrom below by the value hc > he. Violation of the quasi-equilibrium may occuronly if h3 > hc > he. Therefore, ∆Pr in this case has a small but final value. WritingEquation 3.323 for ψ = γ, P = Pr, assuming h = h3 ≅ hc, and subtracting from itEquation 3.315 for the equilibrium state, we derive

(3.330)

As the α branch of the disjoining pressure isotherm is generally quite steep,thicknesses he and h3 do not differ much from one another. Hence, the integralon the right-hand side of the equation is positive. Bearing in mind that H >>h3 ≅ he, it means that Pr > Pe, and consequently, θr < θe, which is indeed thecase. In addition, it follows from this discussion that the value of θr should differonly a little from that of θe. This theoretical conclusion agrees with experimentaldata in the case of water on hydrophobized glass [49].

The critical profile of the meniscus before receding sets in has such a formthat a thickness with a horizontal tangent appears at h = h3. Upon further increasein pressure, the profile of the meniscus can no longer be combined by a continuousline with the point h = h3. Consequently, the meniscus should start to recede at∆P > ∆Pr.

Things look differently in the case of violation of the quasi-equilibrium withreceding meniscus in thicker capillaries, when Pe < Πmax. Under certain condi-tions, which are determined by the actual form of the disjoining pressure isotherm,Π(h), with increasing ∆P, the curve ψ(h,P) can touch the straight line γ = constwith the right-hand rising maximum (at h = h4) earlier than thickness h3 comes outof the transfer zone (h3 < hc) (Figure 3.27b, curve 5). Then, a point with a horizontaltangent h′ = 0 appears on the convex part of the meniscus (Figure 3.29b). Violationof the quasi-equilibrium occurs at the point of the profile at h = h4. Note, that,at this particular moment the stability condition C (Chapter 2, Section 2.2) is

P H h P H h h dhr e e

h

he

−( ) − −( ) = ( )∫3

3

Π .



violated. The quasi-equilibrium liquid profile becomes unstable and the flow hasto set in.

At P > Pr, the part of the profile indicated in Figure 3.29b by dashed linesstarts sliding. It can be seen from Figure 3.27 that thickness h4 belongs to the βpart of the disjoining pressure isotherm 1. Thus, when the meniscus is displacedfrom it, a thick metastable β film should remain behind. This phenomenon (thickβ-films behind the receding meniscus in the case of S-shaped aqueous isothermof disjoining pressure) was confirmed in a number of experimental investigations[49–51].

As the profile of the receding meniscus in the transition zone is low sloped,the value of the static receding contact angle, θr, in the case of sufficiently thickcapillaries (that is, at Pe < Πmax) can be determined quite rigorously. For thispurpose we use the following system of equations:

, (3.331)

, (3.332)

where the first equation was derived from expression (3.323), under the conditionψ = γ, for P = Pr and h = h4. The second equation results from the fact that h4

is an inflection point at which h″ = 0. It then follows from Equation 3.322 thatΠ(h4) = Pr.

For the solution of the systems of Equation 3.331 and Equation 3.332, wemust know the form of the isotherm Π(h). If we choose for the β branch of thedisjoining pressure isotherm a relationship in the following form Π = B/h2, asfor β-films of water [26], we obtain, after integration in Equation 3.331,

, (3.333)

where B is a constant of the surface forces [26]. Under the condition H >> h4,these expressions transform into h4 ≅ and Pr ≈ γ /H, respectively. Hence,θr ≈ 0. Thus, if thick β- films are formed behind the receding meniscus, the staticreceding contact angle, θr, should be close to zero. This, again, is in goodagreement with experimental observations in Reference 49 to Reference 51.

For isotherms of type two (Figure 3.27) or for narrow capillaries, when Pe >Πmax, calculation of θr is further complicated by the fact that thickness hc, corre-sponding to the beginning of the zone of flow, is not known. In order to determineit as a function of ∆P, the corresponding hydrodynamic equation must be solved.This is the objective of a future investigation.

P H h h dhr

h

−( ) − ( ) =∞

∫4

4

Π γ

P hr = ( )Π 4

hB H

BP

B

hr4

42

1 1= + −

=

γγ

and

BH γ⁄



It should be noted that the solution obtained is valid also for the case ofcontact and flow of two immiscible liquids. The only difference is that, insteadof the isotherm of a wetting film, Equation 3.322 includes in this case thedisjoining pressure isotherm of thin layers of wetting liquids Π(h) enclosedbetween a solid substrate and a nonwetting liquid. For the existence of partialwetting, the form of the disjoining pressure isotherm, Π(h), of the layers musthere also be of the same type as shown in Figure 3.27a (curves 1 or 2).

STATIC HYSTERESIS CONTACT ANGLES OF DROPS

A similar approach is used in the following example for drops of liquids thatform equilibrium contact angles θe with a flat substrate. For the sake of simplicity,we limit the discussion to drops of a cylindrical shape, where the liquid profiledepends on only one variable x. This permits the reduction of the problem to atwo-dimensional one, as in the case of slit capillaries.

In the case of an equilibrium drop for solving Equation 3.311 instead ofcondition (3.312), a different boundary condition should be used:

(3.334)

where H is the maximum height of the drop over the surface, and we assumethat H >> he. By integration of Equation 3.311, we obtain for the derivative ofthe liquid profile, h′, in the same expression (3.316) as in the case of a meniscus.However, the function ψ now has a different form:

(3.335)

Condition (3.313) for equilibrium of a drop with an equilibrium flat film ofthickness he, h′ → 0 at h → he is written now in the form ψ = γ. Using theseconditions, we derive from Equation 3.335 that

. (3.336)

The excess pressure Pe is negative in the case of a drop, and Pe = –γ/ℜe,where ℜe is the radius of curvature of the spherical part of the surface of thedrop. It follows from simple geometrical considerations that cosθe = (ℜe–H)/ℜe.By substituting these expressions in Equation 3.336, we derive the condition ofequilibrium of a drop with a film that is identical with Equation 3.55 derived

x h H h= = ′ =0 0, , ,

ψ γ ϕ

ϕ

h P h P

h P P H h dh

e e

e e

he

, ( , ),

( , ) .

( ) = −

= − −( ) +∞

∫Π

P H h dhe e

he

−( ) − =∞

∫Π 0



previously in Chapter 2, Section 2.3 and Equation 2.47 in the same section forthe case of the menisci. This is not surprising; as in the case of H >> he, theconditions of equilibrium should not depend on the sign of the curvature of thesurface beyond the limits of the zone of influence of surface forces.

By differentiating Equation 3.336 with respect to Pe and bearing in mind thatPe < 0, we obtain

. (3.337)

This equation shows that the maximum thickness of equilibrium drops, H,decreases with increasing supersaturation, when the value of Pe diminishes. How-ever, this decrease has certain limits, as drops can be at equilibrium with flatfilms only if Pe > Πmin. It can be seen from Figure 3.30, curve 1, that Πmin is thepressure corresponding to the minimum of the isotherm Π(h). At Pe < –Πmin thereis neither a film nor a drop on the surface at the equilibrium.

FIGURE 3.30 The disjoining pressure isotherm Π(h) (1) in the case of partial wetting,and the corresponding curves of the functions ϕ(h) (2–4) determining the conditions ofquasi-equilibrium of the drop. See explanations in the text.

∂∂

= −−( )

− ( ) = − − >HP

H h

P hH h

Pe

e

e

e

eΠ0

0

Pe

hu

1

α

Π

he

h3 h4 h

Pa < Pe

Pr > Pe

h

β

–Πmin

ϕγ

3

4 2

0



At a sufficiently steep dependency of the α-branch of the disjoining pressureisotherm, as is usually the case, an increase in oversaturation causes a rise in thevalue of the contact angle θe, in accordance with Equation 2.47 from Section 2.3.Thus, the decrease in the size of the equilibrium drop causes an increase in θe;i.e., the value of θe is a function of the height of the drop H. When Pe decreasesand approaches –Πmin, the drops whose dimensions diminish should be torn offthe surface to pass into the gaseous phase.

Maintaining the external conditions, we consider now nonequilibrium profilesof drops when their volume changes by charging or withdrawing liquid (seeFigure 1.14). Advancing contact angles are formed at P < Pe, and receding onesat P > Pe. Expression (3.324) is a condition for the existence of a solution for(3.322), which is found by the method described previously and is written for adrop in the following form:

, (3.338)

where now the function ϕ(h, P) is given by

. (3.339)

Examples of the form of the function ϕ(h, P) (curves 2–4) for the disjoiningpressure isotherm Π(h) of the same type as curve 1 are shown in Figure 3.30.The extremum values of ϕ(h, P) are found from the condition P = Π(h) just aspreviously, i.e., from the points of intersection of the isotherm with the straightline P = const.

Note that, in the case of drops, the form of function ϕ(h, P) according toEquation 3.335 and Equation 3.339 differs from that of the corresponding functionψ(h, P) for menisci in capillaries (Figure 3.27) because of the difference in theboundary conditions. It is shown in the following text that this fact causes differentexpressions for the advancing and receding contact angles. Thus, for the sameliquid–solid system, the values of the static hysteresis angles depend on theconfiguration of the surface of the liquid (meniscus or drop).

There is also a difference in the equilibrium contact angles calculated accord-ing to Equation 2.47 and Equation 2.55 in Section 2.3, as the equilibrium excesspressure, Pe, differs in sign for a concave meniscus and a convex drop.

It follows from Equation 3.336 that, for an equilibrium profile of a drop, i.e.,at P = Pe, the function ϕ(h, Pe) vanishes at h = he. On the other hand ϕ(h, Pe)vanishes also at h = H. As ϕ(h, Pe) > 0, the function ϕ(h, Pe) has a maximum atthe thickness h = hu (curve 2 in Figure 3.30).

With increased pressure, i.e., at P > Pe, when the curvature of the surface ofthe drop decreases, the curve ϕ(h, P) (curve 3 in Figure 3.30) is located below

γ ϕ≥ ( ) ≥h P, 0

ϕ h P P H h h dhh

,( ) = − −( ) + ( )∞

∫Π



the equilibrium curve 2 (Figure 3.30). Figure 3.30 shows that, in this case, thecondition for the existence of a solution is violated even at a small excess of Pover Pe. The situation is similar to that considered previously for the case ofreceding contact angles in a capillary. The perimeter of the drop starts to recedeunder the condition h3 ≥ hc, where hc is the thickness corresponding to thebeginning of the flow zone. Thus, in the case of a drop, θr ≤ θe just as in the caseof a capillary.

At lowered pressures, i.e., at P < Pe, when the surface of the drops becomesmore convex, curve ϕ(h, P) (curve 4 in Figure 3.30) is located above that ofequilibrium curve 2 (Figure 3.30). The condition of a quasi-equilibrium is vio-lated, and the perimeter of the drop starts to advance after the maximum of ϕ(h, P)reaches the dashed line γ = const (curve 4 in Figure 3.30). This conditioncorresponds to the appearance of a thickness with a vertical tangent h′ = ∞ onthe profile of the drop. Just as in a capillary, a flow from the drop to the filmcommences by the Frenkel’s “caterpillar” mechanism if ϕ(h, P) > γ.

We calculate the value of the advancing contact angle θa, using the conditionϕ(h, P) = γ:

. (3.340)

Keeping in mind that Pa = γ (cos θa – 1)/Ha, we conclude from this expressionthat

. (3.341)

Let us calculate the difference cosθe – cosθa. Using Equation 2.55 fromSection 2.3 for the equilibrium contact angle, θe, and equilibrium excess pressure,Pe, in the drop, and Equation 3.341, we obtain

. (3.342)

The sign of the second term on the right-hand side is determined by the actualform of the disjoining pressure isotherm, Π(h). If (cos θe – cos θa) ≥ 1, thencosθa ≤ 0 and θa ≥ 90˚ > θe. If (cos θe – cos θa) ≤ 1, then cos θe > cos θa andθa > θe. Thus, in the case of a drop, the advancing angle is always larger thanthe equilibrium angle.

− −( ) + ( ) =∞

∫P H h h dha

h

4

4

Π γ

cos ( )θγa a

h

P h h dh= +∞

∫41

4

Π

cos cos ( )θ θγe a e e a

h

h

P h P h h dh

e

− = − − +

∫1

14

4

Π



CONCLUSIONS

A theory of static hysteresis of the contact angle has been developed on the basisof the analysis of conditions of quasi-equilibrium of the system and their violation.The suggested theory agrees qualitatively with known experimental data. Formore rigorous quantitative calculations, a theory of disjoining pressure should bedeveloped, which is applicable to profiles of drops or menisci having sufficientlysteep part of the profiles. It is also necessary to determine the thickness of thefilm thickness, hc, that corresponds to the beginning of the flow zone.

The fundamental conclusion of this section is the relation of the mechanismof static hysteresis of contact angle of smooth homogeneous surfaces to the formof the disjoining pressure isotherm. It should be noted that static hysteresis ofthe nature considered appears not only on smooth homogeneous surfaces but alsoon heterogeneous ones. Thus, in actual cases, the possibility of the simultaneousappearance of static hysteresis phenomena of different natures must be taken intoconsideration.

In order to prove the concepts developed in this section of static hysteresisand equilibrium angles, one must combine for these systems the derivation of thedisjoining pressure isotherms, Π(h), which characterize the forces acting in thinlayers and films. It would also be of interest to observe the behavior of deformedmenisci in a capillary or a drop at θe < θ < θa and at θe > θ > θr. If statichysteresis is related to the conditions of quasi-equilibrium, a slow motion ofthe position of the apparent three-phase contact line wetting perimeter shouldbe observed. The latter slow motion should change dramatically upon attainingthe static advancing or receding static hysteresis contact angles, θa and θr,respectively.

The previous remarks and conclusion also apply when the solid surface aheadof the menisci or the drops is initially dry.

REFERENCES

1. Dussan, E.B., Annu. Rev. Fluid Mech., 11, 371, 1979.2. Greenspan, H.P., J. Fluid Mech., 84, 125, 1978.3. Hocking, L.M. and Rivers, A.D., J. Fluid Mech., 121, 425, 1982. 4. de Gennes, P., Rev. Mod. Phys., 57(3), 827, 1985.5. Blake, T.D. and Haynes, J.M., J. Colloid and Interface Sci., 30(3), 421, 1969.6. Kochurova, N.N. and Rusanov, A.I., J. Colloid Interface Sci., 81(2), 297, 1981.7. Eggers, J. and Evans, R., J. Colloid Interface Sci., 280, 537, 2004.8. Blake, T.D. and Shikhmurzaev, Y.D., J. Colloid Interface Sci., 253, 196, 2002.9. Neogi, P. and Miller, C., J. Colloid Interface Sci., 86(2), 525, 1982.

10. Marmur, A., Adv. Colloid Interface Sci., 19, 75, 1983.11. Kalinin, V.V. and Starov, V.M., Colloid J. (Russian Academy of Sciences), 48(5),

767, 1986.12. Summ, B.D. and Goryunov, Yu.V., Physicochemical Principles of Wetting and

Spreading [in Russian], Khimiya, Moscow, 1976.



13. Starov, V.M., Adv. Colloid and Interface Sci., 39, 147, 1992.14. Voinov, O.V., Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza (in Russian) No. 5, 76,

1974.15. Cazabat, A.M. and Cohen-Stuart, M.A., J. Phys. Chem., 90, 5849, 1986.16. Chen, J.D., J. Colloid Interface Sci., 122, 60, 1988.17. Starov, V.M., Kalinin, V., and Chen, J.-D., Adv. Colloid and Interface Sci., 50,

187, 1994.18. Ausserre, D., Picart, A.M., and Leger, L., Phys. Rev. Lett., 57, 2671, 1986.19. Leger, L., Erman, M., Guinet-Picart, A.M., Ausserre, D., Strazielle, C., Benattar,

J.J., Rieutord, F., Daillant, J., and Bosio, L., Rev. Phys. Appl., 23, 104, 1988.20. Tanner, L.H., J. Phys. D, 12, 1979, 1473.21. Nayfeh, A.-H., Perturbation Methods, Wiley, New York, 1973.22. Starov, V.M., Velarde, M.G., Tjatjushkin, A.N., and Zhdanov, S.A. J. Colloid

Interface Sci., 257, 284, 2003.23. Bretherton, F.P., J. Fluid Mech., 10, 166, 1961.24. Friz, G., Angew Z. Phys., 19, 374, 1965.25. Ludviksson, V. and Lightfoot, E.N., AIChE J., 14, 674, 1968.26. Deryaguin, B.V., Churaev, N.V., Muller, V.M., Surface Forces, Consultants

Bureau, Plenum Press, New York, 1987.27. Deryagin, B.V. and Levi, S.M., Film Coating Theory, Focal Press, London-New

York, 1960.28. Levich, V.G., Physicochemical Hydrodynamics, Prentice Hall, Englewood Cliffs,

NJ, 1962.29. Quéré, D. and Di Meglio, J.-M. Advances in Colloid and Interface Science, 48,

141, 1994.30. Derjaguin, B.V., Strakhovskij, G.M., and Malisheva, D.S., Acta Physicochim.

URSS, 19, 541, 1944.31. Karasev, V.V. and Derjaguin, B.V., Zh. Fiz. Khim., 33, 100, 1959.32. Derjaguin, B.V., Zakhavaeva, N.N., Andreev, S.V., and Khomutov, A.M., Colloid J.

(Russian Academy of Sciences), 24(3), 289, 1962.33. Derjaguin, B.V., Zakhavaeva, N.N., Andreev, S.V., Milovidov, A.A., and Khomu-

tov, A.M., Research in Surface Forces, Consultants Bureau, New York, 1963,p. 110.

34. Derjaguin, B.V. and Zakhavaeva, N.N., Issledovanie v oblasti vysokomolekulamykhsoedinenii, Moscow-Leningrad, AN SSSR, 223, 1949.

35. Dejaguin, B.V., Karasev, V.V., Starov, V.M., and Khromova, E.N., J. ColloidInterface Sci., 67(3), 465, 1978.

36. Bellman, R.E. and Kalaba, R.E., Quasilinearization and Nonlinear BoundaryValue Problems, American Elsevier, New York, 1965.

37. Churaev, N.V., Colloid J. (Russian Academy of Sciences), 36, 323, 1974.38. Dzyaloshinskii, I.E., Lifshitz, E.M., and Pitaevskii, L.P., Usp. Fiz. Nauk (in Rus-

sian), 73, 381, 1961.39. Schwartz, A.M., Racier, C.A., and Huey, E., Adv. Chem. Ser., 43, 250, 1964.40. Neumann, A.W., Renzow, D., Renmuth, H., and Richter, I.E., Fortsch. Ber. Kol-

loide Polym., 55, 49, 1971.41. Holland, L., The Properties of Glass Surfaces, Chapman & Hall, London, 1964,

p. 364.42. Zorin, Z.M., Sobolev, V.D., and Churaev, N.V., Surface Forces in Thin Films and

Disperse Systems [in Russian], Nauka, Moscow, 1972, p. 214.



43. Romanov, E.A., Kokorev, D.T., and Churaev, N.V., Int. J. Heat Mass Transfer, 16,549, 1973.

44. Neumann, A.W., Z. Phys. Chem. (Frankfurt), 41, 339, 1964.45. Zheleznyi, B.V., Dokl. Akad. Nauk SSSR, 207, 647, 1972.46. Zorin, Z.M., Novikova, A.V., Petrov, A.K., and Churaev, N.V., in Surface Forces

in Thin Films and Stability of Colloids [in Russian], Nauka, Moscow, 1974, p. 94.47. Frenkel, S.Ya., Zh. Eksp. Teor. Fiz. (in Russian), 18, 659, 1948.48. Herzberg, W.J. and Marian, J.E., J. Colloid Interface Sci., 33, 164, 1970.49. Zorin, Z.M. and Churaev, N.V., Cololid J. (Russian Academy of Sciences), 30,

371, 1968.50. Deryagin, B.V., Ershova, I.G., and Churaev, N.V., Dokl. Akad. Nauk SSSR, 182,

368, 1968.51. Viktorina, M.M., Deryagin, B.V., Ershova, I.G., and Churaev, N.V., Dokl. Akad.

Nauk SSSR, 200, 1306, 1971.52. Kalinin, V.V. and Starov, V.M., Colloid J. (Russian Academy of Sciences), 54(2),

214, 1992.53. Starov, V.M., Churaev, N.V., and Khvorostyanov, A.G., Colloid J. (Russian Acad-

emy of Sciences), 39(1), 176, 1977.54. Ivanov, V.I., Kalinin, V.V., and Starov, V.M., Colloid J. (Russian Academy of

Sciences), 53(1), 25, 1991.55. Ivanov, V.I., Kalinin, V.V., and Starov, V.M., Colloid J. (Russian Academy of

Sciences), 53(2), 218, 1991.56. Starov, V.M., Kalinin, V.V., and Ivanov, V.I., Colloids Surf. A: Physicochemical

Eng. Aspects, 91, 149, 1994.57. Derjaguin, B.V., Karasev, V.V., Starov, V.M., and Khromova, E.N., Colloid J.

(Russian Academy of Sciences), 39(4), 584, 1977.58. Kiseleva, O.A., Starov, V.M., and Churaev, N.V., Colloid J. (Russian Academy of

Sciences), 39(6), 1021, 1977.59. Martynov, G.A., Starov, V.M., and Churaev, N.V., Colloid J. (Russian Academy

of Sciences), 39(3), 406, 1977.60. Platikanov, D., Nedyalkov, M., and Petkova, V., Adv. Colloid Interface Sci.,

100–102, 185–203, 2003.61. Petkova, V., Platikanov, D., and Nedyalkov, M., Adv. Colloid Interface Sci., 104,

37, 2003.62. Ershov, A.P., Zorin, Z.M., and Starov, V.M., Measurements of liquid viscosities

in tapered or parabolic capillaries. J. Colloid Interface Sci., 216(1), 1–7, 1999.


315

4 Spreading over Porous Substrates

INTRODUCTION

In this chapter, we consider the kinetics of spreading over porous substrates. Thisphenomenon is widely used in the industry: printing, painting, imbibition intosoils, health care, and home care products, and so on. However, it is only recentlythat this process started to develop on a theoretical and not just on a purelyempirical basis.

We show in this chapter that the kinetics of spreading over porous substratesis substantially different from the corresponding kinetics of solid nonporoussubstrates. First (Section 4.1), we considered the kinetics of spreading over theporous substrate already saturated with the same liquid. This allows extractionof an important parameter, which we referred to as the effective lubricationcoefficient. This coefficient turns out to be very insensitive to the properties ofthe porous substrate. Its pattern allows us to use its average value for the consid-eration of the kinetics of spreading over thin porous layers (Section 4.2). Weshow that the kinetics of spreading over these layers is described by a universaldependency. The unusual finding is that, in the case of complete wetting, thehysteresis of contact angle is present at the spreading on porous substrates. Wecall this form of hysteresis hydrodynamic, and it is determined by the flow in theporous substrate.

However, in the case of spreading over thick porous substrate, we cannotprovide a theory on the current stage and restrict ourselves by summarizingexperimental results only.

4.1 SPREADING OF LIQUID DROPS OVER SATURATED POROUS LAYERS

In Chapter 3, we have considered the kinetics of spreading over smooth homo-geneous surfaces. We made it clear that the singularity at the three-phase contactline is removed by the action of surface forces (see Section 3.2). However, thevast majority of real solid surfaces are rough to a varying degree, and in manycases, surfaces are either porous or covered with a thin porous skin. These featuresaffect the spreading process. Brinkman’s equations [1,17] are frequently used forthe description of the flow in porous media and have a reasonable semi-empiricalbackground [2] with physically meaningful coefficients: an effective viscosityand a permeability coefficient. A new method of calculation of these coefficients,as functions of the porosity of the porous media, has been suggested in Reference 3.



We shall use them to study the spreading of liquid drops over thin porous substratesfilled with the same liquid, i.e., when the thickness of the porous substrate, ∆, isassumed to be much smaller than the drop height, H, (H* sets the scaleof the drop height). We shall follow the evolution of the liquid, both in the dropabove the porous layer and inside it.

Spreading of small liquid drops over thin porous layers saturated with thesame liquid is investigated in this section, from both theoretical and experimentalpoints of view. A theory is presented, which shows that spreading is governedby the same power law as in the case of spreading over dry solid substrate.Brinkman’s equations are used to model the liquid flow inside the porous sub-strate. An equation of the drop spreading is deduced, which shows that both aneffective lubrication and the liquid exchange between the drop and the poroussubstrates are equally important. Presence of these two phenomena removes thewell-known singularity at the moving three-phase contact line, because it resultsin an effective “slippage” velocity on a moving three-phase contact line. Matchingof the drop profile in the vicinity of the three-phase contact line, with the mainspherical part of the drop, makes it possible to calculate preexponential factorsin the spreading law via the permeability and effective viscosity of the liquid inthe porous layer. However, this dependency turns out to be very weak. Spreadingof silicone oil over different microfiltration membranes is carried out. Radii ofspreading on time experimental dependencies confirm the theory predictions.Experimentally found coefficients agree with theoretical predictions.

Spreading of liquids over solid surfaces is one of the fundamental processes,with a number of applications such as coating, printing, and painting. In theprevious chapter, we considered the spreading over smooth homogeneous sur-faces. It has been established that singularity at the three-phase contact line isremoved by the action of surface forces (see Section 3.2).

An attempt to use Brinkman’s equations for description of the flow insidethe porous layer coupled with the drop flow over the layer has been undertakenin Reference 4. In the following text, the same approach is applied to investigatethe spreading of liquid drops over thin porous substrates filled with the sameliquid. Brinkman’s equations are used for description of the liquid flow insidethe porous substrate [5].

THEORY

The kinetics of spreading of small liquid drops over thin porous layers saturatedwith the same liquid is investigated in this section. Theoretical consideration takesinto account the kinetics of liquid motion, both in the drop above the porous layerand inside the porous layer itself. Consideration of the flow inside the porouslayer is based on Brinkman’s equations.

Liquid inside the Drop (0 < z < h(t,r), Figure 4. 1)

Let us consider the spreading of an axisymmetric liquid drop over a thin porouslayer with thickness ∆, saturated with the same liquid. The thickness of the porous

∆ << ∗H


Spreading over Porous Substrates 317

layer is assumed much smaller than the drop height, that is, ∆ << H*, where H*

is the scale of the drop height. The drop profile is assumed to have a low slope(H*/L* << 1, where L* is the length scale of the drop base). Influence of thegravity is neglected (small drops, Bond number << 1, or the size of the drop issmaller than the capillary length, a). Therefore, only capillary forces are takeninto account.

In the case under consideration, the liquid motion inside the drop is describedby the following system of equations:

, (4.1)

, (4.2)

, (4.3)

and boundary conditions:

, (4.4)

, (4.5)

, (4.6)

where t is the time; r, z are radial and vertical coordinates, respectively; z > 0and –∆ < z < 0 correspond to the drop and the porous layer, respectively; z = 0is the drop–porous layer interface; p, v, u are the pressure, radial, and verticalvelocity components, respectively; v0, u0 are velocity components at thedrop–porous layer interface, which are determined below by coupling with theflow inside the porous layer; h(t,r) is the drop profile; γ is the liquid–air interfacialtension; and pa is the pressure in the ambient air.

Equation 4.1 and Equation 4.2 are Stokes equations, in the low slope case;Equation 4.3 is the incompressibility condition; Equation 4.5 shows the absenceof a tangential stress on the liquid–air interface; and Equation 4.6 presents thepressure jump on the same interface determined by capillary forces only.

∂∂

= ∂∂

p

r

v

zη

2

2 2

∂∂

=p

z0

10

r

rv

r

u

z

∂( )∂

+ ∂∂

=

v v u u z= = = +0 0 0, ,

∂∂

= =v

zz h t r0, ( , )

p pr r

rh

ra= − ∂∂

∂∂

γ



Integration of Equation 4.1 to Equation 4.3 with boundary conditions (4.4 to4.6) results in the following equation, which describes the evolution of the dropprofile:

. (4.7)

The liquid velocity components, v0, u0, on the drop–porous layer interfaceare calculated in the following text.

Inside the Porous Layer beneath the Drop (–∆∆∆∆ < z < 0, Figure 4. 1)

If the porous layer is not completely saturated, then the capillary pressure insidethe saturated part of the porous layer, pc, can be estimated as

,

where r* is the scale of capillary radius inside the porous layer. According toEquation 4.6, the capillary pressure inside the drop, p – pa can be estimated as

.

Therefore, the capillary pressure inside the spreading drop is substantially smaller(actually several orders of magnitude smaller) than the capillary pressure insidethe porous layer, in the case of noncomplete saturation. This means that the droppressure cannot disturb, in any way, the drop–porous layer interface in front ofthe spreading drop when the porous layer is completely saturated. Hence, thisinterface always coincides with the surface z = 0. It is worth mentioning thateverything is happening on time scales much bigger than the initial period con-sidered in Reference 6.

The liquid motion inside the porous layer with thickness ∆ is assumed toobey Brinkman’s equations. In this case, the liquid motion inside the porous layeris described by the following system of equations:

, (4.8)

∂∂

= − ∂∂

∂∂

∂∂

∂∂

+h

tu

r rr h

r r rr

h

rv h0 3 01

31γ

η

prc ≈ γ*

p ph

L

h

L L L rpa c− ≈ = << << ≈γ γ γ γ*

*

*

* * * *2

∂∂

= ∂∂

−p

r

v

z

v

Kpp

η2

2



, (4.9)

, (4.10)

and boundary conditions:

(4.11)

, (4.12)

(4.13)

, (4.14)

where ηp, Kp are the viscosity and the permeability of Brinkman’s medium,respectively. The boundary condition (4.12) corresponds to the absence of atangential stress on the lower boundary of the porous layer, which correspondsto experimental conditions (see the following text).

FIGURE 4.1 Spreading of liquid drop over saturated porous layer of thickness ∆. L(t) —macroscopic radius of the drop base; (1) spherical cap (outer region); (2) vicinity of thethree-phase contact line (inner region). Inflections point, A, separates inner and outerregions. Inside the outer region, liquid flows from the drop into the porous layer; insidethe inner region, the liquid flows from the porous layer onto the drop edge.

A

1

2ζ0

∆

L(t)

∂∂

=p

z0

10

r

rv

r

u

z

∂( )∂

+ ∂∂

=

v v u u z= = = −0 0 0, ,

∂∂

= = = −v

zu z0, ∆

η ηp

z z

vz

vz

∂∂

= ∂∂=− =+0 0

,

p pz z=− =+

=0 0



Let us introduce Brinkman’s radius, as

(4.15)

The solution of Equation 4.8 to Equation 4.10 with boundary conditions (4.11to 4.15) results in:

(4.16)

Substitution of the latter expressions into Equation 4.7 results in the followingequation, which describes the kinetics of spreading of the liquid drop over aporous substrate:

(4.17)

where α = η/ηp is the viscosity ratio. According to Reference 3, effective viscosity,ηp, is always higher than the liquid viscosity, η, that is, α < 1.

If, instead of Brinkman’s equation (Equation 4.8), a slip condition on theliquid–porous layer interface is used [7], then the vertical velocity component,u0, should be set to zero in Equation 4.7, and the following boundary conditionshould be adopted for the radial velocity component:

, (4.18)

where β is an empirical parameter;

is the velocity inside the porous substrate; and δ/β is a slip length. If boundary conditions (4.13) and (4.18) are compared, then it is easy to see

that the former condition can be directly obtained from condition (4.18) if weadopt

δ η= p pK

v hp

ru

r rr h

p p

0 2 01 2 1= − +

∂∂

= ∂∂η

δδ

δη

δcoth ,∆ 22 2+( ) ∂

∂

∆δ p

r

∂∂

=

− ∂∂

+ + +

ht

r rr h h h

γη

αδ

δ α δ αδ3

13 6 33 2 2 2coth

∆ ∆

∂∂

∂∂

∂∂

r r rr

hr

1,

∂∂

= −( )=

v

zv v

z

p

0

0βδ

vK p

rpp

p

= − ∂∂η



.

Combination of this expression and boundary condition (4.13) gives the followingvalue of the empirical coefficient, β, as

. (4.19)

This means that the slip length is equal to

.

Kp and ηp dependencies on porosity can be calculated according to Reference 3.However, if the slip condition is used, then the omitted contribution of a

vertical component, u0, gives the comparable contribution in the resulting equation(see the following text).

Equation 4.17 should be solved with the symmetry condition in the dropcenter,

, (4.20)

and conservation of the drop volume condition,

, (4.21)

where L(t) is the macroscopic radius of the drop base (Figure 4.1).Everywhere, at r < L(t), except for a narrow region, ξ, close to the three-

phase contact line, the following inequality holds: h >> δ. The size of this narrowregion close to the three-phase contact line is calculated in the following text.The same consideration as in Chapter 3 (Section 3.2, Reference 8) shows thatthe solution of Equation 4.17 can be presented as “outer” and “inner” solutions(Figure 4.1). The outer solution can be deduced in the following way: the left-hand side of Equation 4.17 should be set to zero and solved with boundaryconditions (4.20 and 4.21), and an additional new boundary condition,

h(t, L – ξ) ≈ 0. (4.22)

η ηδp

z

ppv

z

v v∂∂

=−

=−0

0

βα

= 1

αδ ηη

=K p

p

∂∂

= ∂∂

= =h

r

h

rr

3

30 0,

20

π rhdr V

L

=∫



In a similar manner, as in Chapter 3 (Section 3.2), the outer solution isdeduced in the following form:

(4.23)

Equation 4.23 shows that the drop profile retains the spherical shape over theduration of spreading (except for a very short initial stage). Note that the timedependency of the macroscopic position of the apparent three-phase contact line,L(t), is to be determined.

The drop slope at the macroscopic apparent three-phase contact line can befound from Equation 4.23 as:

, (4.24)

which is used below as a boundary condition for the inner solution.Inside the inner region (Figure 4.1), the solution can be represented in the

following form,

, (4.25)

where f is a new unknown function; ζ is a similarity variable; and χ(t) << L(t)is the scale of the inner region. This means that ξ ≈ χ(t). Substitution of thesolution in the form (4.25) into Equation 4.17 results in the following equationfor determination of f(ζ):

, (4.26)

where over-dot means the differentiation with respect to time, t, and small termsare neglected in the same way as in Section 3.2.

The latter equation should not depend on time. This gives two equations:

(4.27)

and

. (4.28)

h t rV

LL r r L t( , ) ( ), ( ).= − <2

42 2

π

∂∂

= −=

h

r

V

Lr L

43π

h t r fr L t

t( , ) ( ),

( )( )

= = −δ ζ ζχ

df

dL t

t

d

df f f

ζγη

δχ ζ

αδ

α α�( )( )

coth= + + +3

3 6 33

33 2∆ ∆∆

δ ζ

d f

d

3

3

�L tt

( )( )

= γη

δχ3

3

3

df

d

d

df f f

d f

dζ ζα

δα α

δ ζ= + + +

3 23

33 6 3coth

∆ ∆



The solution of Equation 4.28 should be matched with the outer solution (4.23).Matching of asymptotic solutions gives the following condition:

= – (4.29)

The latter condition should not depend on time, t, which gives

= –λ (4.30)

and

, (4.31)

where λ is a dimensionless constant (see the following text). This equation gives

. (4.32)

This means that the scale of the inner region (Figure 4.1) is proportional toδ and is very small as compared with the size of the drop base.

The combination of Equation 4.31 and Equation 4.27 gives the followingequation for the determination of the radius of spreading, L(t):

(4.33)

The solution of this equation with the initial condition L(0) = L0, where L0

is the initial drop radius, is

, (4.34)

where

is the time scale of the spreading process.

δχ ζ ζ( )t

dfd →−∞

δχ ζ ζ( )t

dfd →−∞

dfdζ ζ→−∞

λ χπ δ

= 43

V t

L t

( )

( )

χ δ π λ( )

( )( )t

L t

VL t= <<

3

4

L t L tV9

3

34

( ) ( )� =

γη πλ

L t Lt

( ).

= +

0

0 1

1τ

τ ηγ

πλ=

310 4

0 03

3L L

V



Now back to the determination of the parameter λ. Integration of Equation 4.28and setting an integration constant zero (because of conservation of the dropvolume and vanishing of the drop profile in front of the spreading drop) gives:

(4.35)

This equation should be solved with the boundary conditions

, (4.36)

where ζ0 corresponds to the inner variable to the edge ξ0 in Figure 4.1, andboundary condition (4.30). We have seen already in Chapter 3 (see Section 3.2)that this condition cannot be satisfied because Equation 4.35 does not have aproper asymptotic behavior. An approximate method (“patching” of asymptoticsolutions) has been suggested in Section 3.2, which allows an approximate deter-mination of parameter λ. Now, the unknown constant λ can be calculated in thesame approximate way, as it has been done in Section 3.2. Estimations in thefollowing text show, however, that it is not worthwhile.

The second of the two boundary conditions (4.36) refers to a zero microcon-tact angle on the microscopic drop boundary.

Equation 4.24 gives the following value of an apparent dynamic contact angle,θ, (tanθ ≈ θ):

, (4.37)

or

. (4.38)


, (4.39)

where

d f

d

f

f f f

3

33 23 6 3ζ α

δα α

δ

=+ + +coth

∆ ∆

fdf

d( )

( )ς ςζ0

0 0= =

θπ

= 43

V

L

LV=

41 3

πθ

/

dL

dt= ω γθ

µ

3

ωλ

= 1

3 3



is referred to in the following text as an effective lubrication coefficient. If theliquid spreads over the solid substrate, this effective lubrication is determined bythe action of surface forces in the vicinity of the three-phase contact line (Section3.2). In this case, the effective lubrication coefficient has been calculated inSection 3.2, and its value is 1.36*10–2. It is reasonable to expect the ω value tobe higher than the one in the case under consideration. Spreading of the liquid overa prewetted solid substrate has been considered in Section 3.3 (see also Reference9). The effective lubrication coefficient in this case has been calculated as1.6*10–2. The latter shows:

1. An effective lubrication coefficient is not very sensitive to experimentalconditions. That is, we have chosen not to try to theoretically calculatethough the procedure of its approximate determination is very similarto those presented in Section 3.2.

2. It is reasonable to expect the values of the effective lubrication coefficientin between these two values. Experimentally determined values of thiscoefficient agree with our estimations reasonably well (see Table 4.1).

Materials and Methods [5]

Silicone oils SO50 (viscosity 0.55 P), SO100 (viscosity 1.18 P), and SO500(viscosity 5.58 P) purchased from PROLABO are used in these spreading exper-iments. The cellulose nitrate membrane filters purchased from Sartorius (type113) with an average pore size of 0.2 and 3 µm, respectively, are used as porouslayers. All membrane samples used are plane circular plates with a radius of25 mm and thickness from 0.0130 to 0.0138 cm. The porosity of the membranesranges between 0.65 and 0.87. Prior to spreading experiments, membranes weredried for 3–5 h at 95˚C and then stored in dry atmosphere.

TABLE 4.1Calculated Effective Lubrication Coefficient

Membrane Pore Size, µµµµm L0, cm ηηηη, P ττττ, s V, cm3 ωωωω

0.2 0.176 5.58 0.333 0.0039 0.017 ± 0.0040.2 0.193 0.55 0.0592 0.0040 0.018 ± 0.0040.2 0.150 1.18 0.00163 0.003 0.014 ± 0.0050.2 0.150 1.18 0.0026 0.0034 0.014 ± 0.0030.2 0.165 1.18 0.0086 0.003 0.016 ± 0.0053 0.119 1.18 0.0546 0.0055 0.015 ± 0.0093 0.250 1.18 0.461 0.005 0.016 ± 0.0093 0.253 0.55 0.609 0.005 0.018 ± 0.008

Optical glass 0.226 0.55 0.0378 0.0068 0.012 ± 0.003Optical glass 0.113 1.18 0.0103 0.002 0.010 ± 0.005



Figure 4.2 shows the sample chamber for monitoring the spreading drop overporous layers and dynamic contact angles. A porous wafer 1 (Figure 4.2) is placedin a thermostated and hermetically closed chamber 2 with a fixed humidity andtemperature. The chamber was made of brass to prevent temperature and humidityfluctuations. In the chamber walls, several channels were drilled to enable pump-ing of a thermostating liquid. The chamber is equipped with a fan. The temper-ature is monitored by a thermocouple. Droplets of liquid 3 are placed onto thewafer by a dosator 4 (Figure 4.2). The volume of drops is set by the diameter ofthe separable capillary of the dosator.

The chamber is also equipped with optical glass windows for observation ofboth the spreading drop shape and size (side view and view from above). TwoCCD cameras and tape recorders are used for storing the sequences of spreading.Different colors of monochromatic light are used for side views and viewing fromabove to eliminate spurious illumination on images. The optical circuit for view-ing from above (illuminator 7 as well as the camera 5) is equipped with interfer-ential light filters 8,9 with a wavelength 520 nm. The side view circuit (theilluminator 13 and camera 10) are equipped with filters 12 and 14 with a wave-length 640 nm. Such an arrangement suppresses the illumination of a CCD camera2 by the diffused light from the membrane and hence increases the precision ofmeasurements. The automatic processing of images is carried out using the (imageprocessor) Scion Image. The time discretization in processing ranges from 0.1 to1 sec in different experiments. The size of pixel on the image corresponds to0.0125 mm.

The experiments are organized in the following order:

A membrane under investigation is placed in the chamber; a big drop ofoil is deposited. The volume of the drop, V, is selected as ,where Rm is the radius of the membrane sample. This choice correspondsto the complete saturation of the membrane by a tested liquid.

FIGURE 4.2 Experimental set-up for monitoring the time evolution of droplets. (1) wafer;(2) sample chamber; (3) tested drop; (4) dosator; (5), (10) CCD cameras; (6), (11) VCRs;(7), (13) illuminators; (8), (9), (12), (14) interferential light filters (with wave length 520nm [8], [9], and with wavelength 640 nm [12], [14]).

42

1312

78

14 10 1116

1

3

2CCD

1CCD

15

9 5 6

V m Rm= π ∆ 2



After the imbibition process is completed (100–500 sec, depending on theliquid viscosity), the next tiny drop of the same liquid is deposited onthe saturated porous layer, and the spreading of this drop is monitored.Volumes of drops are measured by the direct evaluation of video images.The precision of measurement is around 0.0001–0.0005 cm3.

Results and Discussion. Experimental Determination of Effective Lubrication Coefficient ωωωω

According to the experimental observation [5], in all spreading experiments, dropsretain the spherical shape, and no disturbances or instabilities are detected. Exper-imental data are fitted using the following dependency

(4.40)

It is necessary to comment on the adopted fitting procedure. If experimentally determined values of the exponent, n, are taken for review

[10], it is easy to see that in most cases this exponent is higher (sometimesconsiderably) than 0.1. In the following text, we present a possible explanationof this phenomenon that we encountered in our experiments.

In all our experiments (probably in a number of others, too), drops have beenplaced on the solid substrate using a syringe. That is, the drops actually fall froma certain height. This means that during a very short initial stage of spreading,both inertia and a relaxation of the drop shape could not be ignored. This meansthat during the initial stage of spreading, both the Reynolds number, Re, and thecapillary number, Ca, are not small, and the capillary regime of spreading is notapplicable. Inertia spreading has been considered in Reference 11, where thefollowing spreading law has been predicted:

, (4.41)

where

and ρ is the liquid density. The latter equation shows that during the inertiaspreading regime, the drop spreads much faster than during the capillary regimeof spreading. Derivation in Reference 11 is applicable only if the Reynoldsnumber is high enough. Let us estimate the Reynolds number, which is

,

L L t n= +0 1( / )τ

L t v t( ) = ∞

vL∞ =

24

0

1 2γ

ρ

/

Re( )= ∞v H t ρ

η



where H(t) is the maximum drop height. According to Reference 11,

during the inertia period of spreading. Condition Re >> 1 gives:

or

. (4.42)

The time, tRe, values relevant for our experiments are calculated below. Letus estimate the Reynolds number during the capillary spreading stage. Equation4.39 gives the velocity of spreading, which should be used for calculation of theReynolds number. Simple rearrangement gives

.

The latter estimation should be compared with Equation 4.42, which gives

or t ~ 10tRe ~ 0.1 sec.This means that the capillary regime of spreading takes place only at t > 10 tRe. Equation 4.39 is used in the following text to determine the condition for the

fulfillment of the second requirement, Ca << 1. This equation can be rewrittenas . According to our experimental condition ω ≈ 10–2, θ ≈ 0.5. Thisgives the following estimation of the capillary number: Ca ~ 10–3. This means,

,

or t ~ 103 tCa ~ 1 sec, where

.

H tV

L t( )

( )=

2

V L

t

ρη

ργ0

1 2

2241

1

>>/

t t tV L<< =

Re Re

/ /

,ρ

ηρ

γ

1 2

0

1 4

24

Re = ≈ −ωγθη

3

2210

L

t

tRe

≈ −2

210

Ca = ωθ3

U L

t

ηγ

ηγ

≈ ≈ −10 3

tL

Ca = ηγ



It is necessary to emphasize that tRe is reversibly proportional to η1/2 (decreaseswith an increase in viscosity), but tCa is proportional to the viscosity (that is,increases with an increase in viscosity). This corresponds well to our experimentalobservations.

In any experimental observation, only a limited number of experimentalvalues of L(t) dependency are measured. If some of these measurements are takenat the initial regime of spreading, then a higher value results for the fitted exponentthan 0.1.

For example, let us take the experimental curve in the case of spreading ofSO50 over a saturated porous layer (Figure 4.3a and Figure 4.3b). In Figure 4.3a,this dependency is presented in a log–log coordinate system. It is easy to see thatthe whole spreading process consists of two stages corresponding to two differentpower laws. During the first stage, the inertia/shape relaxation cannot beneglected, whereas in the second stage, capillary spreading (exponent close to0.1) takes place. In Figure 4.3b, the fitting results for this particular spreadingexperiment are presented. The broken line corresponds to the fitting procedurewhen all experimental points are taken into account. In this case, the fittedexponent is higher than 0.1 (0.13 ± 0.01). However, if we do not take into accountthe first three points, located within the initial stage of spreading, then the fittedexponent becomes 0.11 ± 0.01, that is, much closer to 0.1. Figure 4.3a showsthat the initial stage of spreading continues approximately around 0.1 sec, whichagrees reasonably well with the previously mentioned estimations.

The following procedure for the definition of the parameters L0 and τ wasadopted [5]. First, the points, which correspond to the capillary stage of spreading,are selected using the presentation of experimental points in a log–log coordinatesystem. After this, the fitting procedure using Equation 4.40 is carried out usingonly the experimental points, which correspond to the capillary stage of spreading.This procedure gives the values of L0 and τ in each run.

After the experimental definition of L0 and τ, the value of ω is calculated inthe following manner. Equation 4.34 can be rewritten as

. (4.43)

Comparison of Equation 4.43 and Equation 4.40 gives

. (4.44)

Determined values of ω, as well as other experimental parameters, are pre-sented in Table 4.1. The last two rows of Table 4.1 consist of the results of thespreading over a dry glass (microscope optical glass). The data presented in Table4.1 show that the effective lubrication coefficient (1) is higher in the case of

L LV

Lt= +

0

3 3

010

0 1

1 104π

γη

ω

.

ω πτ

µγ

=

4

110

3

010

3

L

V



spreading over saturated porous substrate than in the case of “dry spreading,”(2) experimentally determined values of the effective lubrication coefficient, ω,agree well with the aforementioned theoretical estimations. However, precisionof experimental determination of this parameter does not allow us to extract moreinformation about effective viscosity of the porous substrate.

FIGURE 4.3 (a) Radius of the drop base on time in log–log coordinates. SO50 drop,volume 0.0039 cm3, on a porous membrane with average pore size 0.2 µm. (b) Radius ofthe drop base on time. SO50 drop with volume 0.0039 cm3, on a porous membrane withaverage pore size 0.2 µm. Broken line: fitted using all experimental points, solid line:fitted using only points that correspond to the capillary stage of spreading. Fitted param-eters are given in the insert.

1.0 0.5 –0.5 –1.0 –1.5 –2.0 –1.60

–1.55

–1.50

–1.45

–1.40

–1.35

–1.30

–1.25

–1.20

0.0

�n L

�n t

2.52.01.51.00.50.00.19

0.20

0.21

0.22

0.23

0.24

0.25

0.26

0.27

0.28

0.29

L, c

m

t, s

Data: 1–12 point

Data: 3–12 point

= 0.00002

= 09689

Model: L0 (1 + t/τ)

n

L0

L0

n

Model: L0 (1 + t/τ)

n

τ

n

τ

0.06

0.13

0.18

0.20

0.130.11

±0.01±0.08

±0.01

±0.03

±0.07±0.01

Chi 2/DoF

˘

= 52229 E-6Chi 2/DoF

˘

R 2

˘

= 098274R 2

˘



4.2 SPREADING OF LIQUID DROPS OVER DRY POROUS LAYERS: COMPLETE WETTING CASE

In this section, we take up the problem treated in the previous section but nowwe shall consider the drop spreading over a dry porous layer. We shall make useof the lubrication approximation, and we shall restrict consideration to the caseof complete wetting. Spreading of small liquid drops over thin dry porous layersis investigated in the following text [12]. Drop motion over a porous layer iscaused by an interplay of two processes: (1) the spreading of the drop over alreadysaturated parts of the porous layer, which results in an expanding of the dropbase, (2) the imbibition of the liquid from the drop into the porous substrate,which results in a shrinkage of the drop base and an expansion of the wettedregion inside the porous layer. As a result of these two competing processes, theradius of the drop goes through a maximum value over time. A system of twodifferential equations is derived to describe the evolution, with time of radii, ofboth the drop base and the wetted region inside the porous layer. This systemincludes two parameters; one accounts for the effective lubrication coefficient ofthe liquid over the wetted porous substrate, and the other is a combination ofpermeability and effective capillary pressure inside the porous layer. Two addi-tional experiments are used for an independent determination of these two param-eters. This system of differential equations does not include any fitting parameterafter these two parameters are determined. Experiments were carried out on thespreading of silicone oil drops over various dry microfiltration membranes (per-meable in both normal and tangential directions). The time evolution of the radiiof both the drop base and the wetted region inside the porous layer are monitored.All experimental data fell on two universal curves, if appropriate scales are usedwith a plot of the dimensionless radii of the drop base and of the wetted regioninside the porous layer on dimensionless time. The predicted theoretical relation-ships are the two universal curves accounting for the experimental data quitesatisfactorily. According to our theory prediction (1) the dynamic contact angledependence on the same dimensionless time (as before) should be a universalfunction, (2) the dynamic contact angle should change rapidly over an initialshort stage of spreading and remain a constant value over the duration of the restof the spreading process. The constancy of the contact angle at this stage hasnothing to do with hysteresis of the contact angle; there is no hysteresis in oursystem. These conclusions again are in good agreement with our experimentalobservations [12].

It has been shown in Section 4.1 that the presence of roughness or a poroussublayer changes the spreading conditions. In the same section, the spreading ofsmall liquid drops over thin porous layers saturated with the same liquid has beeninvestigated. Instead of the “slippage conditions,” Brinkman’s equations havebeen used in Section 4.1 for the description of the liquid flow inside the poroussubstrate.

In the present section, we take up the same problem, as is the case when adrop spreads over a dry porous layer. The problem is treated in the following text



under the lubrication theory approximation and in the case of complete wetting.Spreading of big drops (but still small enough to neglect the gravity action) overthin porous layers is considered in the following text.

THEORY

The kinetics of liquid motion both in the drop above the porous layer and insidethe porous layer itself are taken into account here. The thickness of the porouslayer, ∆, is assumed to be much smaller than the drop height, that is, ∆ << h*,where h* is the scale of the drop height. The drop profile is assumed to have alow slope (h*/L* << 1, where L* is the scale of the drop base) and the influenceof the gravity is neglected (small drops, Bond number , where ρ, g,and γ are the liquid density, gravity acceleration, and the liquid–air interfacialtension, respectively). That is, only capillary forces are taken into account.

Under such assumptions, a system of two differential equations is obtainedto describe the evolution with time of the radii of both the drop base, L(t), andthe wetted region inside the porous layer, l(t), (Figure 4.4). Further assumptionsmade are justified in the Appendix 1.

As in Section 4.2, the profile of axisymmetric drops spreading over the poroussubstrate (whether dry or saturated with the same liquid) is governed by thefollowing equation:

, (4.45)

where h(t,r) is the profile of the spreading drop; t and r are the time and the radialcoordinate, respectively; z > 0 corresponds to the drop; –∆ < z < 0 correspondto the porous layer; z = 0 is the drop–porous layer interface (Figure 4.4); v, u are

FIGURE 4.4 Cross section of the axisymmetric spreading drop over initially dry thinporous substrate with thickness: ∆; (1) liquid drop; (2) wetted region inside the poroussubstrate; (3) dry region inside the porous substrate; L(t) — radius of the drop base; l(t) —radius of the wetted area inside the porous substrate; ∆ — thickness of porous substrate;r, z — coordinate system; h(t, r) — profile of the spreading drop.

ρ γgL*2 1/ <<

∂∂

= − ∂∂

∂∂

∂∂

∂∂

+h

tu

r rr h

r r rr

h

rv h0 3 01

31γ

η

θ1

03223

L(t)

l(t)

r

z

h(t, r)∆



the radial and vertical velocity components, respectively; v0, u0 are velocitycomponents at the drop–porous layer interface; η is the liquid viscosity. Theliquid velocity components, v0, u0, on the drop–porous layer interface are calcu-lated by matching the flow in the drop with the flow inside the porous layer.

The porous layer is assumed to be very thin, and the time for saturation inthe vertical direction can be neglected, relative to other time scales of the process.Let us calculate the time required for a complete saturation of the porous layerin the vertical direction. According to Darcy’s equation

,

where Kp and pc are the permeability of the porous layer and the effective capillarypressure, respectively; z is the position of the liquid front inside the porous layer.The solution of this equation results in: , where t∆ is the time ofthe complete saturation for the porous layer in the vertical direction and hence,

.

The consideration in the following text is restricted to t > t∆. Estimations showthat t∆ is less than t0, which is the duration of the initial stage of spreading (seeSection 4.1 for details and an estimation of t0). The capillary spreading regime(the only one considered here) is not applicable at t < t0. Thus, we must considerthose instances such that t > max(t0, t∆), when both the initial stage is over, andthe porous layer is completely saturated in the vertical direction. Accordingly,the porous layer beneath the spreading drop (0 < r < L(t)) is always assumed tobe completely saturated.

The capillary pressure inside the porous layer, pc, can be estimated as

,

where r* is the scale of capillary radii inside the porous layer. The capillarypressure inside the drop, p – pa, can be estimated as

.

This means that the capillary pressure inside the pores of the porous layer is ofseveral orders of magnitude higher than the capillary pressure in the drop itself.

uK p

zz u

dzdt

p c= − < < =η

, ;∆ 0

∆ ∆2 2= K p tp c /η

tK pp c

∆∆=

2

2η

prc ≈ 2γ*

p ph

L

h

L L L rpa c− ≈ = << << ≈γ γ γ γ*

*

*

* * * *2



The boundary conditions for Equation 4.45 are as follows:

symmetry condition in the drop center,

, (4.46)

conservation of drop volume,

. (4.47)

The drop volume changes over time because of the imbibition of the liquidinto the porous layer, which means

, (4.48)

where V0 is the initial volume of the drop; m is the porosity of the porous layer;and is the radius of the wetted area inside the porous layer. The wetted regionis a cylinder with radius l(t) and the height ∆. l(t) is referred to here as the radiusof the wetted region inside the porous layer.

Let be the time instant when the drop is completely sucked by the poroussubstrate , where lmax

is the maximum radius of thewetted region in the porous layer. The preceding equation gives

. (4.49)

lmax is used in the following text to scale the radius of the wetted region inthe porous layer, l(t). It is easy to check that the previous equation results inlmax > L* in our case.


. (4.50)

Everywhere at (r < L(t)) except for a narrow region, ξ, close to the movingthree-phase contact line, we have h >> ∆, and the liquid motion inside the porouslayer under the drop can be neglected both in the vertical and horizontal directions(see the Appendix 1 for details). The size of this narrow region close to the moving

∂∂

= ∂∂

= =h

r

h

rr

3

30 0,

20

π rhdr V t

L

=∫ ( )

V t V m l t( ) ( )= −02π ∆

l t( )

tmax

V t V m l( )max max= = −0 02π ∆

lV

mmax

/

=

0

1 2

π ∆

2 02

0

π πrhdr V m l t

L

= −∫ ∆ ( )



three-phase contact line, where suction of the liquid from the drop into the poroussubstrate takes place, is estimated in the Appendix 1.

The latter means that Equation 4.45 can be rewritten as

. (4.51)

Equation 4.51 should be solved with boundary conditions (4.46 and 4.50)and leads to

h(t, L – ξ) ≈ 0. (4.52)

Following arguments developed in Chapter 3 (Section 3.2), the solution ofEquation 4.51 can be obtained using outer and inner solutions. The outer solutioncan be deduced in the following way: the left-hand side of Equation 4.51 shouldbe set to zero. After integration of the resulting equation with boundary conditions(4.46, 4.50, and 4.52) the outer solution becomes:

. (4.53)

Equation 4.53 shows that the drop surface profile remains spherical duringthe spreading process, except for a short initial stage when the porous layer isnot saturated and a final stage, when condition ∆ << h is violated everywhereover the whole profile of the drop.

Equation 4.53 gives the following value of the dynamic contact angle, θ,(tanθ ≈ θ):

, (4.54)

or else

. (4.55)

The drop motion is a superposition of two motions: (1) the spreading of thedrop over the already saturated part of the porous layer, which results in anexpansion of the drop base, and (2) a shrinkage of the drop base caused by theimbibition into the porous layer. Hence, we can write the following equation:

∂∂

= − ∂∂

∂∂

∂∂

∂∂

h

t r rrh

r r rr

h

r

γη3

1 13

< −, ( )r L t ξ

h t rV

LL r r L t( , ) ( ), ( )= − < −2

42 2

πξ

θπ

= 43

V

L

LV=

41 3

πθ

/



, (4.56)

where v+, v– are unknown velocities of the expansion and the shrinkage of thedrop base, respectively.

Let us take the time derivatives of both sides of Equation 4.55. It gives:

. (4.57)

Over the whole duration of the spreading over the porous layer, both thecontact angle and the drop volume can only decrease with time. Accordingly, thefirst term on the right-hand side of Equation 4.57 is positive and the second oneis negative.

Comparison of the two latter equations yields

(4.58)

There are two substantially different characteristic time scales in our problem:, where tη* and tmax are time scales of the viscous spreading and the

imbibition into the porous layer, respectively;

is a smallness parameter (around 0.08 under our experimental conditions, see thefollowing text). Both the time scales are calculated here. Then we have L = L(Tη,Tp) [13], where Tη is a fast time of the viscous spreading, and Tp is a slow timeof the imbibition into the porous substrate. The time derivative of L(Tη, Tp) is

. (4.59)

Comparison of Equation 4.56, Equation 4.58, and Equation 4.59 shows that

dL

dtv v= −+ −

dL

dt

V d

dt V= −

+

13

4 13

44

1 3

2

1 3

πθθ

π θ

/ /ddV

dt

vV d

dt

vV

+

−

= −

>

= −

13

40

13

4

4

1 3

2

πθθ

π θ

/

11 3

0/

.dV

dt>

t tη* max<<

λ η= <<t

t*

max

1

dL

dt

L

T

L

Tp

= ∂∂

+ ∂∂η

λ

vL

T

V d

dtv

L

Tp+ −= ∂

∂= −

= − ∂∂

= −η πθ

θ λ13

44

1 3/

,113

42

1 3

π θV

dV

dt

/



The smallness of

means that in the case under consideration, two processes actually functionindependently: the spreading of the drop over the saturated part of the porouslayer, and the shrinkage of the drop base caused by the imbibition of the liquidfrom the drop into the porous layer.

The decrease of the drop volume, V, with time is determined solely by theimbibition into the porous substrate and hence the drop volume, V, only dependson the slow time scale.

According to the previous consideration, the whole spreading process can besubdivided into two stages:

(1) A first fast stage, when the imbibition into the porous substrate can beneglected, and the drop spreads with approximately constant volume.This stage goes in the same way as the process of spreading oversaturated porous layer, and the arguments developed in Section 4.1 canbe used here again.

(2) A second slow stage, when the spreading process is almost alreadyover, and the evolution is determined by the imbibition into the poroussubstrate.

During the first stage, Equation 4.59 from Section 4.1 can be rewritten as

, (4.60)

where t0 is the duration of the initial stage of spreading, when the capillary regimeof spreading is not applicable; and ω is an effective lubrication parameter, whichhas been discussed and estimated in Section 4.1. It is important to emphasizethat the effective lubrication parameter, ω, is independent of the drop volume anddepends solely on the porous layer properties. According to Equation 4.60, thecharacteristic time scale of the first stage of spreading is

, (4.61)

where .

λ η= <<t

t*

max

1

L tV

t t( )

.

.=

+( )10 43

0 1

0

0 1γωη π

tL L

Vηη

γωπ

* =

0 0

3

0

3

10 4

L L t0 0= ( )



Combination of Equation 4.59 and Equation 4.60 gives:

Substitution of this expression into the first Equation 4.58 gives the followingexpression for the velocity of the drop base expansion, v+:

. (4.62)

Substitution of Equation 4.48 into the second Equation 4.58 gives the fol-lowing expression for velocity of the drop base shrinkage, v:

. (4.63)

Substitution of the two latter equations into Equation 4.57 results in:

(4.64)

The only unknown function now is the radius of the wetted region inside theporous layer, l(t), which is determined in the following text.

Inside the Porous Layer outside the Drop (–∆∆∆∆ < z < 0, L < r < l)

The liquid flow inside the porous layer obeys the Darcy equation

.

The solution of the preceding equations is

(4.65)

θπ

ηγω

=

+( )−410

0 1 0 3

0

0 3Vt t

. ..

.

vV

t t+ =

+( )

0 14 10 1

0 3 0 1

0

0 9.

. .

.πγωη

vm l

V m l

dl

dt− =−( )

23

42 3

02

2

1 3

π

π θ

/

/

∆

∆

dL

dt

V

t t=

+( )

0 14 10 1

0 3 0 1

0

0.

. .

.πγωη 99

2 3

02

2

1 3

23

4−−( )

π

π θ

/

/

m l

V m l

dl

dt

∆

∆

10

r rr

p

rv

K p

rp∂

∂∂∂

= = − ∂∂

,η

p A K r B

vAr

p= − +

=

( ) lnη/



where A and B are integration constants that are determined using the boundaryconditions for the pressure at the drop edge, r = L(t), and at the circular edge ofthe wetted region inside the porous layer, r = l(t). The latter boundary condition is:

, (4.66)

where

is the capillary pressure inside the pores of the porous layer, and r* is a charac-teristic scale of the pore radii inside the porous layer.

The boundary condition at the drop edge is

, (4.67)

where pd is an unknown pressure. It is further shown that . However,we keep this small value for a future estimation.

Taking into account the two preceding boundary conditions, both the inte-gration constants, A and B, can be determined, which gives the following expres-sion for the radial velocity according to Equation 4.65:

. (4.68)

The velocity at the circular edge of the wetted region inside the porous layeris:

.

Combination of the two preceding equations gives the evolution equation forl(t):

. (4.69)

p p p r l ta c= − =, ( )

prc ≈ 2γ*

p p p r L ta d= + =, ( )

p pd c<<

vK p p

rl

L

p c d=+( )

η ln

dl

dtv

r l=

=

dl

dt

K p p

ll

L

p c d=+( )

η ln



An estimation of the time scale tmax can be made according to Equation 4.69and taking into account Equation 4.49 as follows

,

or

. (4.70)

Comparison of the estimated values of tη* according to Equation 4.61 and oftmax according to Equation 4.70 shows that under all our experimental conditions(see the following text), the following inequality tη* << tmax is satisfied.

Omitting the small term, pd, and substitution of Equation 4.69 into Equation4.64 gives the following system of differential equations for the evolution of boththe radius of the drop base, L(t), and that of the wetted region inside the porouslayer, l(t):

,

(4.71)

. (4.72)

Let us make the system of differential equations (Equation 4.71 and Equation4.72) dimensionless using new scales whereLmax is the maximum value of the drop base, which is reached at the time instanttm (that has to be determined). The same symbols are used for the dimensionlessvariable as for corresponding dimensional variables (marked with an overbar).The system of Equations (Equation 4.71 and Equation 4.72) transforms as:

, (4.73)

l

t

K p

ll

L

p cmax

maxmax

max

*

/

ln≈

η

tl

lL

K p

Vl

Lm K pp c p c

max

maxmax

*

max

*

ln

/

ln≈ =

20

η π ∆ /ηη

dL

dt

V m l=

−( )

0 14 100

20 3

0

.

.

π

πγωη

∆ ..

.

/

ln

1

0

0 9

02

1 23t t

m K p L

V m ll

L

p c

+( )−

−( )

π η

π

∆

∆

dldt

K p

llL

p c=/η

ln

L L L l l l t t t= = =/ / /max max max, , ,

dL

dt

t

l l

lm

m m

=+( )

−( ) +( )−( )2

3 1 1

10 9

21 3

20

τ

χ

.

.

.

ln

33

0 92

23

1 1t

L

ll

L+( )

−−( ) +

τ χ.

ln



, (4.74)

where

.

Thus, the latter system includes only two dimensionless parameters, thefirst one is very small, and the second one changed insignificantly under ourexperimental conditions because of a weak logarithmic dependence on lmax/Lmax.

Accordingly, the two dimensionless dependencies should fall ontwo almost universal curves, which is in very good agreement with our experi-mental observations (see Results and Discussion).

According to Equation 4.73, the dimensionless velocities of the expansion ofthe drop base, , and the shrinkage, , are as follows:

. (4.75)

Figure 4.5 shows dimensionless velocity and calculated according toEquation 4.75. It appears that:

(1) The first stage is very short. Here, the capillary spreading prevails overthe drop base shrinkage caused by the liquid imbibition into the poroussubstrate;

(2) The spreading of the drop almost stops after the first stage of spreading,and the shrinkage of the drop base is determined by the suction of theliquid from the drop into the porous substrate.

Let us consider the asymptotic behavior of system (4.71 and 4.72) over thesecond stage of the spreading. According to Figure 4.5, over the second stage ofthe spreading, velocity of the expansion of the drop, v+, decreases. To understandthe asymptotic behavior, the term corresponding to v+ on the left-hand site ofEquation 4.71 is omitted. This gives:

(4.76)

dl

dtl

l

L

=+

1

1 χ ln

τ χ= << =t tlL0 1 1/ max

max

max

, / ln

τ χ, ;

L t l t( ), ( )

v+ v−

vt

l l

l

t

m

m m

+ =+( )

−( ) +( )−( )2

3 1 1

10 9

21 3

20 3

τ

χ

.

.

.

ln ++( )=

−( ) +

−τ χ

0 92

23

1 1.

,

ln

vL

ll

L

v+ v−

dLdt

m K p L

V m llL

p c= −−( )

23

02

π η

π

∆

∆

/

ln

,



whereas the second Equation 4.72 is left unchanged. The system of differentialequations (Equation 4.72 and Equation 4.76) can be solved analytically. For thispurpose, Equation 4.76 is divided by Equation 4.72, which gives

.

If is used instead of l, the latter equation takes the following form:

,

which can be easily integrated and the solution is

, (4.77)

where C is an integration constant. Let us rewrite Equation 4.54 using the samedimensionless variables:

FIGURE 4.5 Dimensionless velocity of spreading (v+,– solid line) and velocity of drop

shrinkage (v–,– dotted line) on dimensionless time, calculated according to Equation 4.75.Intersection of these two dependences determines the value of the dimensionless time

–tmax

≈ 0.08, when the radius of the drop base reaches its maximum value,–Lmax = 1 (in dimen-

sionless units).

1.00.50.00

5

10

15

V+

V–

–

–

V–

t

dL

dl

m L l

V m l= −

−( )2

3 02

ππ∆

∆

V V m l= −02π ∆

dL

dV

L

V=

3

V C L= 3



. (4.78)

Comparison of Equation 4.77 and Equation 4.78 shows that the dynamiccontact angle asymptotically remains constant over the duration of the secondstage. This constant value is marked in the following text as θf. Let us introduce

,

which is the value of the dynamic contact angle at the time instant when themaximum value of the drop base is reached. Then Equation 4.78 can be rewrittenas:

(4.79)

and the latter relationship should be a universal function of the dimensionlesstime,

–t. This conclusion agrees well with our experimental observations (see

Results and Discussion). It is necessary to emphasize that in the case underconsideration, the constancy of the contact angle has nothing to do with thecontact angle hysteresis; there is no hysteresis in our system here. θf is not areceding contact angle but forms as a result of a self-regulation of the flow inthe drop–porous layer system.

The system of equations (Equation 4.71 and Equation 4.72) includes sevenparameters, five of which can be measured directly (V0, γ, η, m, and ∆ are theinitial volume of the drop, the liquid–air interfacial tension, the liquid viscosity,the porosity of the porous layer, and the thickness, respectively), and two addi-tional parameters, ω and Kppc, which should be determined independently. It isnoteworthy that the porous layer permeability and the capillary pressure alwaysenter as a product, that is, this product can be considered as a single parameter. Aprocedure for the independent determination of an effective lubrication coeffi-cient, ω, has been discussed in Section 4.1.

Experimental Part

Silicone oils SO20 (viscosity 0.218 P), SO50 (viscosity 0.554 P), SO100 (vis-cosity 1.18 P), and SO500 (viscosity 5.582 P) purchased from Prolabo, Productospara Laboratorios Quimicos, a Spanish company, were used in the spreadingexperiments. The viscosity of oils were measured using the capillary EnglerViscometer VPG-3 at 20 ± 0.5˚C. Cellulose nitrate membrane filters purchasedfrom Sartorius (type 113), with pore size 0.2 and 3 µm (marked by the supplier),

VL

VL= 4 3

0

3max

πθ

θ πm m

V

Ll= −0

32

41

max

( )

θθm

ml l

L= − −( ) ( )1 12 2

3

/



were used as porous layers. These membranes are referred to as the membrane,0.2 µm and the membrane, 3 µm, respectively. Pore size distribution and perme-ability of membranes were tested using the Coulter Porometer II.

The pore size of the membrane, 0.2 µm, falls in the range 0.2–0.38 µm, withthe average pore size 0.34 µm. The permeability of the same membrane is equalto 12 l/min*cm2 (air flux at the transmembrane pressure 5 bar). The permeabilityof the membrane, 3 µm, is 2.5 l/(min*cm2) at the transmembrane pressure 0.1 bar.All membrane samples used are plane parallel circles of radius 25 mm andthickness in the range from 0.0130 to 0.0138 cm. The porosity of the membranesranges between 0.65 and 0.87. The porosity was measured using the differencein the weight of dry membranes and membranes saturated with oil. Membraneswere dried for 3–5 hours at 95˚C and then stored in a dry atmosphere prior tospreading experiments.

The same experimental device as described in Section 4.1 (Figure 4.2) wasused for monitoring the spreading of drops over initially dry porous layers. Thetime evolution of the radius of the drop base, L(t), the dynamic contact angle,θ(t), and the radius of the wetted region inside the porous layer, l(t), weremonitored. The porous wafer 1 (Figure 4.2) was placed in a thermostated andhermetically closed chamber 2, where zero humidity and fixed temperature (20 ±0.5˚C) were maintained. The distance from the wafer to the tip of the dosatorranged from 0.5 to 1 cm in different experiments. The volume of drops was setby the diameter of the separable replaceable capillary of the dosator in the range1–15 µl.

Experiments were carried out in the following order:

• The membrane was placed in the chamber and left in a dry atmospherefor 15–30 minutes.

• A light pulse produced by a flash gun was used to synchronize thetime instant when the drop started to spread and recorded using bothvideo-tape recorders (a side view and a view from above).

• A droplet of silicone oil was placed onto the membrane.

Each run was carried out until complete imbibition of the drop into themembrane took place.

Independent Determination of Kppc

As mentioned previously, the permeability of the porous layer and the capillarypressure always enter as a product, i.e., as a single coefficient. Additional exper-iments were carried out to determine this coefficient. For this purpose, the hori-zontal imbibition of the liquid under investigation into the dry porous sheet wasundertaken. Rectangular sheets, 1.5cm∗3cm, were used. These porous sheets werecut from the same membranes used in the spreading experiments. Each sheet wasimmersed to a length of 0.3–0.5 cm into a liquid container, and the position ofthe imbibition front was monitored over time. In the case under investigation, a



unidirectional flow of liquid inside the porous substrate took place. Using Darcy’slaw, we can conclude that

(4.80)

where d(t) is the position of the imbibition front inside the porous layer. It wasfound that all runs d2(t)/2 proceed along a straight line, whose slope gives usKp pc value. According to Reference 3, Kppc should be independent of the testedliquid viscosity.

The measured values of Kp pc are presented in Table 4.2, which shows thatthe coefficient, Kp pc, for each type of membrane is independent of the testedliquid within the experimental error. Average values of Kp pc for each membranewere used in the calculations.

Results and Discussion

According to our observations, the whole spreading process can be subdividedinto two stages (see Figure 4.6 as an example): fast spreading over the first severalseconds until the maximum radius, Lmax, of the drop base is reached. During thefirst stage, an imbibition front inside the membrane expands slightly ahead of thespreading drop. After that the drop base starts to shrink slowly, and the imbibitionfront expands until the drop completely disappears. An example of the timeevolution of the radius of the drop base and the radius of the wetted region insidethe porous layer is provided in Figure 4.7.

In all our spreading experiments, the drops remain spherical over the durationof both the first and the second stages of the spreading process. This was crosschecked by reconstructing of the drop profiles at different time instants of spread-ing, fitting those profiles by a spherical cap:

,

TABLE 4.2The Measured Values of Kp pc

Membrane Pore Size, µµµµm Liquid Kppc, dyn

3 SO20 (1.2 ± 0.4)*10–4

3 SO100 (1.77 ± 0.03)*10–4

3 SO500 (1.6 ± 0.2)*10–4

0.2 SO5 (3.4 ± 0.3)*10–5

0.2 SO100 (3.1 ± 0.3)*10–5

d t K p tp c2 2( ) ,= /η

h z R r rcenter center= + − −( )2 2



where (rcenter, zcenter) is the position of the center of the sphere, R is the radius ofthe sphere. rcenter, zcenter, and R are used as fitting parameters. The fitting is basedon the Levenberg–Marquardt algorithm. In all cases, the reduced Chi-square valueis found less than 10–4. The fitted parameter R gives the radius of curvature ofthe spreading drops at different times.

The edge of the wetted region inside the porous layers was always circular.Drops remained in the center of this circle over the entire duration of the spreadingprocess. No deviations from cylindrical symmetry or instabilities were detected.

The spherical form of the spreading drop allows the measurement of theevolution of the dynamic contact angle of the drop. In all cases, the dynamiccontact angle decreases very fast over the first stage of spreading until a constantvalue is reached, which is referred to as θf. The dynamic contact angle θf, remainsconstant over the main part of the second stage.

FIGURE 4.6 Time sequence of spreading of SO500, volume 8.7 µl over the membranewith pore size 3 µm (side view): (a) t = 0.5 sec (after deposition); (b) 3 sec; (c) 12 sec;(d) 22 sec; (e) 36 sec.

(a)

(b)

(c)

(d)

(e)



An example of the evolution with time of the dynamic contact angle is

4.8 shows that the dynamic contact angle decreases fast over the duration of the

FIGURE 4.7 Development over time of the drop base, L, and the radius of the wettedregion inside the porous layer, l. The same drop as in Figure 4.6: SO500 drop, placed ontothe membrane with pore size 3 µm. Solid lines calculated according to Equation 4.71 andEquation 4.72.

FIGURE 4.8 Development (over time) of the dynamic contact angle. The same drop asin Figure 4.4 and Figure 4.5: SO500 drop, placed onto the membrane with pore size 3 µm.Solid line calculated according to Equation 4.96 and Equation 4.97.

1000 100 10

0.1

0.2

0.3

0.4

0.5

L, �

, cm

t, s

60

80

40

20

0 100 200 300 400 500 600 700

t, s

0

θ, d

egre

e


presented in Figure 4.8 for the same drop as in Figure 4.6 and Figure 4.7. Figure


first stage of the spreading when the radius of the drop base expands to itsmaximum value, Lmax. The dynamic contact angle remains almost unchanged overthe duration of the second stage of spreading. Note that this behavior has nothingto do with the hysteresis of the contact angle because it does not occur in oursystem, that is, silicone oils on nitrate cellulose membranes. Solid lines in Figure4.7 and Figure 4.8 represent the results of numerical integration of the system ofequations (Equation 4.71 and Equation 4.72). The short final period (just beforethe drop disappears) is not covered by our calculations. Close to this final stagethe calculation errors increased, which was caused by a division by a very smallquantity on the second term in the right-hand side of Equation 4.71.

Table 4.3 shows that the final value of the dynamic contact angle, θt (lastcolumn in Table 4.3), depends on the volume of the drop, as well as on theviscosity of the liquid, and hence, θt is determined solely by hydrodynamics.

Figure 4.9a presents experimentally measured dependencies of the radius ofthe drop base and the wetted region inside the porous layer on time for differentsilicone oils, porous layers, and drop volumes. All relevant values are summarizedin Table 4.3. The main result appears in Figure 4.9b, which shows that allexperimental data (the same as in Figure 4.9a) fall on two universal curves ifdimensionless coordinates are selected as follows:

where Lmax is the maximum value of the drop base, which is reached at the timeinstant tm. The same symbols (with and without overbar) are used for dimension-less values as for dimensional ones. The scale lmax is determined by Equation 4.49,and the time scale tmax is given by the Equation 4.70.

The measured values of Lmax, lmax, and tmax for all experimental runs are givenin Table 4.3. Figure 4.9b shows that the dimensionless time is about 0.08 ≈λ << 1 as it was stated above. The dimensionless time 1 corresponds to the timeinstant when the drop is completely sucked by the porous substrate. Solid curvesin Figure 4.9b represent the solution of the system of differential equations(Equation 4.73 and Equation 4.74). If the parametersτ andχ change, then boththe theoretical curves remain inside the array of experimental points. In this sensethey represent universal relationships.

The twelfth column in Table 4.3 gives the experimental values of the dynamiccontact angle, θm, which the drop has when the drop base reaches its maximumvalue, Lmax. These values were used for plotting the time evolution of the dynamiccontact angle, θ/θm. Figure 4.10 shows that all experimental points fall on a singleuniversal curve, as predicted by Equation 4.79.

The solid line in Figure 4.10 is a result of calculations according to Equation4.79, where dimensionless dependencies are taken from the previousFigure 4.9b.

L L L l l l t t t= = =/ / /max max max, , ,

tm

L t l t( ), ( )


Spread

ing o

ver Poro

us Su

bstrates

349

TABLE 4.3Notations Used and Calculated/Predicted Values

LiquidNotation

on Figures

Membrane Pore Size,

µµµµmV0,

ml*103

∆∆∆∆,mm m-Porosity

tmax

Slmax

cmlmax

TheoryLmax cm

Lmax Theory

θθθθm

Degreeθθθθt

Degree

SO20 � 0.2 3.1 0.114 0.85 102 0.318 0.317 0.179 0.184 20.0 12SO20 0.2 9.0 0.116 0.72 440 0.585 0.584 0.314 0.31 12.6 11.4SO20 � 0.2 15.6 0.116 0.73 814 0.77 0.766 0.387 0.39 14.2 12.1SO20 3 3.8 0.136 0.87 10.9 0.345 0.319 0.196 0.198 25.9 22.3SO20 � 3 5.5 0.134 0.83 17.1 0.428 0.398 0.223 0.234 20.3 18.6SO100 × 3 8.6 0.137 0.82 186 0.493 0.494 0.257 0.274 18.5 11

SO100 � 3 14.5 0.138 0.77 354 0.659 0.659 0.332 0.34 15.5 17.2SO500 ▫ 3 3.6 0.138 0.89 296 0.306 0.306 0.174 0.179 22.6 19.4SO500 � 3 8.7 0.138 0.88 851 0.477 0.478 0.264 0.286 19.1 17.3SO500 � 3 14.5 0.136 0.78 1660 0.66 0.660 0.339 0.34 15.8 16.5



FIGURE 4.9 (a) Measured dependences of radii of the drop base (L, mm) and radii of thewetted region inside the porous layer (l, mm) on time (t, s). All relevant values are sum-marized in Table 4.3. (b) The same as in Figure 4.13a but using dimensionless coordinates:L/Lmax, l/lmax, and t/tmax, where Lmax is the maximum value of the drop base, which is reachedat the moment tm. The same symbols (with overbar) are used for dimensionless values asfor dimensional ones. The scale lmax is determined by Equation 4.49 and the time scale tmax

is given by Equation 4.70. Solid lines according to Equation 4.73 and Equation 4.74.

1500

t, s

1000 500 0 0

1

2

3

4

5

6

7

8

L, �

mm

SO 20 V0 = 3.1 mm3, r = 0.2 μ

SO 100 V0 = 14.5 mm3, r = 3 μ

SO 500 V0 = 14.5 mm3, r = 3 μ

SO 500 V0 = 8.7 mm3, r = 3 μ

SO 500 V0 = 3.6 mm3, r = 3 μ

SO 100 V0 = 8.6 mm3, r = 3 μ

SO 20 V0 = 5.5 mm3, r = 3 μ

SO 20 V0 = 3.8 mm3, r = 3 μ

SO 20 V0 = 15.6 mm3, r = 0.2 μ

SO 20 V0 = 9.0 mm3, r = 0.2 μ

SO 20 V0 = 3.1 mm3, r = 0.2 μ

SO 100 V0 = 14.5 mm3, r = 3 μ

SO 500 V0 = 14.5 mm3, r = 3 μ

SO 500 V0 = 8.7 mm3, r = 3 μ

SO 500 V0 = 3.6 mm3, r = 3 μ

SO 100 V0 = 8.6 mm3, r = 3 μ

SO 20 V0 = 5.5 mm3, r = 3 μ

SO 20 V0 = 3.8 mm3, r = 3 μ

SO 20 V0 = 15.6 mm3, r = 0.2 μ

SO 20 V0 = 9.0 mm3, r = 0.2 μ

Theory

0.0 0.5 1.0

0.0

0.5

1.0

0.0

0.5

1.0

L/L m

ax

�/� m

ax

t/tmax



APPENDIX 1

The slip boundary condition is used in the following text for the sake of simplicity.The slippage coefficient is taken according to the Section 4.1. The similar resultscan be deduced using Brinkman’s equations for the description of the liquid flowinside porous layers.

The slippage condition at the drop–porous layer interface is

(A1.1)

where

is the velocity inside the porous substrate; ηp is an effective viscosity inside theporous layer (see Reference 3), and pa – pp is the pressure inside the porous layer,which may be different from the pressure in the spreading drop. The porous layerthickness, ∆, is assumed to be much bigger than Brinkman’s length, δ = .Hence, both velocities v0 and u0 change stepwise at the drop–porous layer inter-face: the jump of the first velocity is given by Equation A1.1, whereas the jumpof the second velocity is

FIGURE 4.10 Dynamic contact angle on the dimensionless time. Solid line according toEquation 4.79.

θ/θ m

5

4

3

2

1

00.0 0.2 0.4 0.6 0.8 1.0

t/tmax

Theory

SO 20 V0 = 3.1 mm3, r = 0.2 µ

SO 100 V0 = 14.5 mm3, r = 3 µ

SO 500 V0 = 14.5 mm3, r = 3 µ

SO 500 V0 = 8.7 mm3, r = 3 µ

SO 500 V0 = 3.6 mm3, r = 3 µ

S100 V0 = 8.6 mm3, r = 3 µ

SO 20 V0 = 5.5 mm3, r = 3 µ

SO 20 V0 = 3.8 mm3, r = 3 µ

SO 20 V0 = 15.6 mm3, r = 0.2 µ

SO 20 V0 = 9.0 mm3, r = 0.2 µ

η ηδ

∂∂

=−

=+

vz

v v

z

pp

0

0

vK p

rpp

p

p=∂∂η

K pηp



. (A1.2)

This means that the vertical velocity changes stepwise from the value givenby Equation A1.2 on the drop–porous layer interface to zero inside the porouslayer. Equation A1.1 becomes

. (A1.3)

Substitution of Equation A1.2 and Equation A1.3 into Equation 4.45 gives:

(A1.4)

where

according to Reference 3. The latter equation describes the evolution of thespreading profile of the drop, both in space and time. The only unknown depen-dence left is the pressure inside the porous layer, pp.

The conservation law inside the porous layer

is used to determine this pressure. The latter equation is integrated over z from∆ to 0 inside the porous layer, using condition (A1.2) and Darcy’s law to expressthe velocity components, using the pressure gradient. After certain transforma-tions, the final equation becomes

,

(A1.5)

uK

r rr

hr

pp

pp

0 = − ∂∂

∂∂

+

η δγ

vK p

rh

r r rr

hr

p

p

p

p

0 1=∂∂

+ ∂∂

∂∂

∂∂

ηδ γη

∂∂

= ∂∂

∂∂

−

− ∂

∂ht

K

r rr

hr

pK

rp

pp

p

pη δγ

η rrrh

p

r

r rr h h

r r

p∂∂

− ∂∂

+( ) ∂∂

∂γη

αδ3

13

13 2

∂∂∂∂

rr

hr

,

α ηη

= <p

1

10

r

rv

r

u

z

∂( )∂

+ ∂∂

=

1 1r r

rp

r

p

r rr

h

r r rr

hp p∂∂

∂∂

− = − ∂

∂∂∂

∂∂

∂∆δ

γδ ∂∂

− ∂

∂r r rr

1∆δ

γ ∂∂∂

h

r



which describes the radial distribution of the pressure inside the porous layer.Equation A1.5 shows that

(1) Under the spherical part of the spreading drop, the pressure inside theporous layer remains constant and equal to the pressure inside the drop,that is

(A1.6)

and hence, the liquid does not flow inside the porous layer under thespherical part of the drop.

(2) The pressure inside the porous layer changes from the constant value(A1.6) to the pressure outside the drop close to r = L in a narrow regionwith a scale

(A1.7)

as stated earlier (see Equation 4.51 through Equation 4.53).

Let us introduce a new local dimensionless variable, x, as follows:

(A1.8)

After omitting certain small terms, Equation A1.5 becomes

(A1.9)

where f = h/δ is the dimensionless profile of the drop in this narrow transitionregion. The latter scale is selected in the same way as in Section 4.1.

Equation A1.9 can be directly integrated using the following boundaryconditions:

. (A1.10)

. (A1.11)

The boundary condition (A1.10) follows from Equation A1.6, whereas con-dition (A1.11) still includes the unknown pressure, pd, which is determined below.

pLp

∞ = 2γθ

ξ δ= <<∆ L

xr L

x= − −∞ < <ξ

, 0

′′ − = − ′′′( )′ + ′′

p p f f fp p

γδξ2

p pp p( )−∞ = ∞

p pp d( )0 =



The solution of Equation A1.9, which satisfies both boundary conditions(A1.10) and (A1.11) is

(A1.12)

Let us assume that

. (A1.13)

This assumption means that the pressure inside the porous layer in the narrowregion under consideration is much higher than the capillary pressure in the drop.This is a reasonable pressure to expect because the pressure just inside the porouslayer determines the drop suction by the porous layer. In this case Equation A1.12reduces to

. (A1.14)

In order to determine the unknown value, pd, the solutions (4.65) and (4.68)for the pressure distribution inside the porous layer outside the drop are used.The solutions (4.65) and (A1.14) should give the same radial velocity from bothsides at r = L. Thus we have:

. (A1.15)

The latter equation justifies the neglect of pd, relative to pc, in the above-mentioned main text.

4.3 SPREADING OF LIQUID DROPS OVER THICK POROUS SUBSTRATES: COMPLETE WETTING CASE

Let us extend the study of Section 4.2 to the case of spreading of small siliconeoil drops (capillary spreading regime) over porous solid substrates thicker thanthe drop size. The spreading of small silicone oil drops (capillary regime ofspreading) over various dry thick porous substrates (permeable in both normaland tangential directions) was experimentally investigated in this section. Thetime evolution of the radii of both the drop base and the wetted region on thesurface of the porous substrate were monitored. It was observed that the total

p p e f f f x y dp dx

x

= + ′′′( )′ + ′′

−( )∫γδξ3

0

sinh yy.

p pd p>> ∞

p p ep dx=

ppL l

L

pL

l

L

pdc

c c=+

≈ <<1

ξ

ξ

ln ln



duration of the spreading process can be divided into two stages: a first stage,when the drop base expands until its maximum value is reached, and a subsequentsecond stage, when the drop base shrinks. It was found that the dynamic contactangle remains constant during the second stage of spreading. This fact has nothingto do with the contact angle hysteresis, as there is no hysteresis in the system.Appropriate scales are used, with a dimensionless time, to plot the dimensionlessradii of the drop base and of the wetted circle on the surface of the poroussubstrate, the relative dynamic contact angle, and the effective contact angle insidethe porous substrate. All these experimental data fall onto universal curves, whenthe spreading of different silicone oils is done on porous substrates of similarpore size and porosity [14].

The spreading of small liquid drops over thin porous layers saturated withthe same liquid (Section 4.1) or a dry porous layer (Section 4.2) has beenconsidered by appropriately matching flows in both the spreading drop and theporous substrate.

In Section 4.2, the spreading of silicone oil drops over various dry microfil-tration membranes (permeable in both normal and tangential directions) wasdiscussed. Plotting the dimensionless radii of the drop base and of the wettedregion inside the porous layer using a dimensionless time scale, all the experi-mental data fell on two universal curves. According to the theory presented inSection 4.2: (1) the dynamic contact angle dependence on the same dimensionlesstime should be a universal function, (2) the dynamic contact angle should changerapidly over an initial short spreading stage and remain constant over the remain-ing duration of the spreading process. This fact has nothing to do with the contactangle hysteresis, as there was no hysteresis in the system under consideration inSection 4.2. These conclusions were in good agreement with experimental obser-vations (Section 4.2).

In the present section, we extend our study in Section 4.2 to the spreadingof small silicone oil drops (capillary spreading regime) over different poroussubstrates, whose thickness is much bigger than the drop size. A number ofsimilarities with the case of the spreading over thin porous substrates (Section4.2) is found.

THEORY

As already mentioned in the introduction to Chapter 3, at small capillary numbers,

,

the drop profile remains spherical in shape over the main part of the spreadingdrop. We also concluded in Section 4.1 that at the spontaneous spreading, thecapillary number is always small, except for a short initial stage. Based on thiswe concluded that during the spreading over porous substrate, the liquid drops

CaU= <<ηγ

1



remain spherical in shape (see Figure 4.11). Hence, the drop profile over thespherical part (assuming the low slope profile) is:

(4.81)

where V(t) is the drop volume, h(t, r) is the drop profile, L is the radius of thedrop basis, and r is the radial coordinate.

Equation 4.81 gives the following value of the dynamic contact angle, θ,(tanθ ≈ θ):

, (4.82)

hence,

. (4.83)

The spreading process is a superposition of two motions: (1) the spreadingof the drop over the already saturated part of the surface of the porous substrate,which results in an expansion of the drop base, and (2) the shrinkage of the dropbase caused by the imbibition into the porous substrate. Hence, we can write thefollowing balance equation:

FIGURE 4.11 Spreading of liquid drops over dry porous substrates: (1) spherical drop;(2) wetted region inside the porous substrate (modeled by a spherical cap); (3) dry partof the porous substrate; L(t) — radius of the drop base; l(t) — radius of the wetted circleon the surface of the porous substrate; θ(t) — dynamic contact angle of the spreadingdrop, and ψ(t) — effective contact angle inside the porous substrate.

–– ––

–––

–θ ψ1

0–– –

L

Vd

Vpl

3

2

h t rV

LL r r L t( , ) ( ), ( )= − <2

42 2

π

θπ

= 43

V

L

LV=

41 3

πθ

/



, (4.84)

where and denote unknown velocities of the expansion and the shrinkageof the drop base, respectively.

Using Equation 4.83 we get:

. (4.85)

Over the whole duration of the spreading over the porous layer, both thecontact angle and the drop volume can only decrease with time. Accordingly, thefirst term on the right-hand side of Equation 4.85 is positive, and the second oneis negative.

Comparison of the Equation 4.84 and Equation 4.85 yields

(4.86)

Let l(t) be the radius of the wetted region on the surface of the poroussubstrate. This unknown quantity cannot be determined without the numericalintegration of Brinkman–Darcy equations inside the porous substrate and cou-pling with the flow in the drop. However, we can draw some conclusion basedon the analysis given in Section 4.2. According to this analysis, the wholespreading process can be subdivided into two stages: during the first stage, v+

prevails, and v– dominates during the second stage of spreading. Let us consider the second stage of spreading. During this stage, the first

term on the right-hand side of Equation 4.85 can be neglected, and this equationreduces to:

. (4.87)

The latter equation after inserting the expression of θ (4.82) takes the form:

,

dL

dtv v= −+ −

v+ v−

dL

dt

V d

dt V= −

+

13

4 13

44

1 3

2

1 3

πθθ

π θ

/ /ddV

dt

vV d

dt

vV

+

−

≡ −

>

≡ −

13

40

13

4

4

1 3

2

πθθ

π θ

/

11 3

0

/

.dVdt

>

dL

dt V

dV

dt=

13

42

1 3

π θ

/

dL

dV

L

V=

3



which can be easily integrated. Its solution is

, (4.88)

where C is an integration constant. Comparison with Equation 4.82 yields

(4.89)

over the duration of the second stage of the spreading. This conclusion agreeswell with experimental observations [14] (see Results and Discussion), as wellas with observations in Section 4.2. Note that, in the case under consideration,as in Section 4.2 the constancy of the contact angle has nothing to do with thecontact angle hysteresis; there is no hysteresis in the system under consideration.θf is not a receding contact angle but forms as a result of a self-regulation of thedrop–porous layer system.

Inside the Porous Substrate

The analysis of experimental data shows that the radius of the wetted regioninside the porous layer is proportional to l3. If the drop base (Figure 4.11) isassumed to be a point source of liquid, then the shape of the wetted area insidethe porous substrate is a hemisphere with an increasing radius and a constanteffective contact angle ψ(t). Obviously, “a point source” assumption is too simplean approximation, as both radii, l(t) and L(t), are of similar size. Yet, it wouldhelp understanding the essence of the process. Hence, let us assume that thewetted region inside the porous substrate is of a spherical cap form with changingeffective contact angle ψ(t) (Figure 4.11).

Let Vp = (V0 – V)/m, where Vp is the volume of the liquid inside the poroussubstrate at time t; V0 is the initial volume of the drop, and m is the porosity.Under the above assumption, the liquid volume in the porous substrate, Vp, canbe expressed as (Figure 4.11)

, (4.90)

where l is the radius of the wetted circle on the outer surface of the poroussubstrate. Equation 4.90 enables us to calculate the time evolution of ψ using theexperimental data.

Experimental Part

Silicone oils S5 (viscosity 0.05 P), S100 (viscosity 1.0 P), and S500 (viscosity5.0 P) purchased from Brookfield Engineering Laboratories Inc. (Middleboro,

V C L= 3

θ θ( )t constf= =

V lp =−( ) +π ψ ψ

ψ3

1 23

2

3

cos ( cos )

sin



MA, USA) were used in the spreading experiments. Glass filters, J. Bibby ScienceProducts, Ltd, and metal filters Sintered Products, Ltd, both purchased fromClaremont Ltd (Broseley, UK) were used as porous substrates. The diameters ofglass filters were 5.0 cm, 2.9 cm, and 2.9 cm, and their thicknesses were 2.5 mm,1.9, and 2.2 mm, respectively. The diameter of metal filters was 5.6 cm, and theirthickness was 1.9 mm. Pore size distribution and permeability of membraneswere tested using the Coulter Porometer II (Coulter Electronic Ltd, Luton, UK).

Glass filters with three different pore size distributions were used. The averagepore sizes were 3.7 µm, 4.7 µm, and 26.8 µm, respectively; their porosities were0.56, 0.53, and 0.31, respectively. The permeability of the same membraneswere 1.8 l/min/cm2; 1.9 l/min/cm2, and 11.5 l/min/cm2 (air flux, transmembranepressure 0.1 bar). Metal filter (Cupro-Nickel): average pore size 26.1 µm, andporosity 0.32. The porosity of filters was measured using the difference in theweight of the filters saturated with oil and the dry filters. Filters were dried for3–5 h at 95˚C and then stored in a dry atmosphere prior to the spreading exper-iments. All relevant information and values are summarized in the Table 4.4.

TABLE 4.4Characteristics of Porous Substrates and Drops Used

Material, Figure, Symbol Porosity

Average Pore Size,

µµµµm ηηηη, PV0,µµµµl

tmax, sec

Lmax, mm

Lmax, mm

θθθθm, grad

ψψψψmax, grad

Glass Figure 4.12

∆

0.53 4.7 0.05 5.0 0.64 2.42 3.10 25.8 38

Glass Figure 4.12

▫

0.53 4.7 1 5.9 12.0 2.30 3.20 24.4 39

Glass Figure 4.12

�

0.53 4.7 5 8.2 60.0 2.58 3.50 23.6 42

Cupro-NickelFigure 4.13

▫

0.32 26.1 5 8.2 15.7 2.38 3.20 35.0 52

GlassFigure 4.13

�

0.31 26.8 5 6.8 18.52 2.40 3.20 25.5 44

GlassFigure 4.14

▫

0.56 3.7 5 8.0 36.0 2.53 3.20 21.5 30

GlassFigure 4.14

�

0.31 26.8 5 6.8 18.52 2.40 3.20 25.5 44



Figure 4.2 shows a schematic description of the set-up. The time evolutionof the radius of the drop base, L(t), the dynamic contact angle, θ(t), and the radiusof the wetted circle on the surface of the porous substrate, l(t), were monitored.All porous substrates, before the start of the experiment, were placed for half anhour in a KOH aqueous solution inside an ultrasonic bath; later, they were rinsedout by plenty of Milli-Q water (Milli-Q water system, made by MILLIPORES.A., France) in an ultrasonic bath. After that, they were dried for 2 hours in a110˚C dry atmosphere. Porous substrates were stored in a dry atmosphere beforestarting the experiments.


• The dry porous substrate was placed in a dry atmospheric chamberand left there for 15–30 minutes,

• A light pulse produced by a flash gun was used to synchronize thetime instant in both video recorders when the drop started to spread,

• A droplet of silicone oil was placed onto the porous substrate.

Each run was carried out until the complete imbibition of the drop into theporous substrate took place.


According to the observations, the whole spreading process can be subdivided

spreads until the maximum radius, Lmax, of the drop base is reached, which isfollowed by the second stage, when the drop radius decreases. Over the durationof the first stage, the imbibition front inside the membrane grows slightly aheadof the spreading drop front. Subsequently, the drop base starts to shrink until thedrop completely disappears, and the imbibition front grows until the end of theprocess. Examples of the time evolution of the radius of the drop base and theradius of the wetted circle on the surface of the porous substrates are presentedin Figure 4.12a, Figure 4.12b, Figure 4.13a, and Figure 4.14a.

In all experiments, the drops remained spherical over the whole spreadingprocess. This was cross checked by reconstructing the drop profiles at differenttime instants of spreading, fitting those drop profiles by a spherical cap:

,

where (rcenter, zcenter) is the position of the center of the sphere, and R is the radiusof the sphere. rcenter, zcenter, and R are used as fitting parameters. The Leven-berg–Marquardt algorithm was used for fitting. In all cases the reduced Chi-squarevalue was found smaller than 10–4. The fitted parameter R gives the radius ofcurvature of the spreading drops at different times.

h z R r rcenter center= + − −( )2 2


into two stages (Figure 4.12a and Figure 4.12b): the first stage is when the drop


FIGURE 4.12 (a) Spreading of different silicone oils over identical dry porous glass filters(in dimensional form). Glass porous filter: porosity 0.56, average pore size 4.7 µm,permeability 1.9 l/min · cm2; (1) silicone oil η = 0.05 P; (2) silicone oil η = 1 P; (3) siliconeoil η = 5 P (see insets). The same symbols show the evolution of l(t) and L(t) with time.Upper parts: L(t), radius of the base of the spreading drop; lower parts: l(t), radius of thewetted circle on the surface of the porous glass filter. (b) Spreading of different siliconeoils over identical dry porous glass filters. (The same data as in Figure 4.12a, usingappropriate dimensionless coordinates.) Glass porous filter: porosity 0.53, average poresize 4.7 µm, permeability 1.9 l/min/cm2; (1) silicone oil η = 0.05 P; (2) silicone oil η =1 P; (3) silicone oil η = 5 P (see insets); Lmax — maximum value of the radius of the dropbase; lmax — maximum value of the radius of the wetted circle on the surface of the glassfilter; tmax — total duration of the process (see Table 4.4).

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0.01 0.1 1 10

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

L, m

m

�, m

m

–1 –2 –3

t, s

1.0

0.8

0.6

0.4

0.2

0.0

0.0 0.2 0.4 0.6 0.8 1.0 0.0

0.2

0.4

0.6

0.8

1.0

t/tmax

L/L m

ax

�/� m

ax

–1 –2 –3



The wetted area on the surface of the porous substrate was always circular.Drops remained in the center of this circle over the whole duration of the spreadingprocess. No deviations from cylindrical symmetry or instabilities were detected.

The spherical form of the spreading drop allows measuring the evolution ofthe dynamic contact angle of the spreading drops. In all cases, the dynamic contactangle decreased very fast during the first spreading stage and remained constant

FIGURE 4.12 (continued) (c) Spreading of different silicone oils over identical dryporous glass filters. Dynamic contact angle vs. time in dimensionless units. Glass porousfilter: porosity 0.53, average pore size 4.7 µm, permeability 1.9 l/min/cm2; (1) silicone oilη = 0.05 P; (2) silicone oil η = 1 P; (3) silicone oil η = 5 P (see insets); θm — dynamiccontact angle value at the time when the maximum value of the radius, Lmax, of the dropbase is reached (see Table 4.4). (d) Spreading of different silicone oils over the same dryporous glass filter. Evolution of the effective contact angle inside the porous glass filterswith the dimensionless time. Glass porous filter: porosity 0.53, average pore size 4.7 µm,permeability 1.9 l/min/cm2; (1) silicone oil η = 0.05 P; (2) silicone oil η = 1 P; (3) siliconeoil η = 5 P (see insets); ψmax — maximum value of the effective contact angle (see Table 4.4).

θ/θ m

5

4

3

2

1

00.0 0.2 0.4 0.6 0.8 1.0

t/tmax

–2–1

–3

1.0

0.8

0.6

0.4

0.2

0.0 0.2 0.4 0.6 0.8 1.00.0

ψ/ψ

max

t/tmax

–3–2–1



during the second spreading stage. This constant value of the contact angle isdenoted below as θf.

Examples of the of the dynamic contact angle with respect to the evolutionwith time are presented in Figure 4.12c, Figure 4.13b, and Figure 4.14b. Thesefigures show that the dynamic contact angle decreases during the first spreadingstage when the radius of the drop base grows up to its maximum value, Lmax.Within experimental error, the dynamic contact angle remains unchanged duringthe second spreading stage.

Spreading of Silicone Oil Drops of Different Viscosity over Identical Glass Filters (Figure 4.12a to Figure 4.12d)

Figure 4.12a to Figure 4.12d present experimental results on the spreading ofsilicone oil drops of different viscosity over identical glass filters. Silicone oilsS5, S100, and S500 were used in these spreading experiments. The diameter ofthe glass filter was 2.9 cm, its thickness 1.9 mm, its porosity 0.53, and its averagepore size 4.7 µm.

From Figure 4.12a to Figure 4.12d, experimental data marked by symbol ∆∆∆∆correspond to the spreading of the silicone oil S5 (drop volume V0 = 5.0 µl; themaximum radius of the drop base Lmax = 2.42 mm; the maximum (final) radiusof the wetted circle on the outer surface of the glass filter lmax = 3.10 mm; andthe total duration of the spreading tmax = 0.64 sec); experimental data marked bysymbol �� correspond to the spreading of the silicone oil S100 (drop volume V0 =5.9 µl; the maximum radius of the drop base Lmax = 2.30 mm; the maximum(final) radius of the wetted circle on the outer surface of the glass filter lmax =3.20 mm; and the total duration of the spreading tmax = 12.0 sec), and experimentaldata marked by symbol Ο correspond to the spreading of the silicone oil S500(drop volume V0 = 8.2 µl; the maximum radius of the drop base Lmax = 2.58 mm;the maximum (final) radius of the wetted circle on the outer surface of the glassfilter lmax = 3.50 mm; and the total duration of the spreading tmax = 60.0 sec).

In Figure 4.12a, the time evolution of both the radius of the base of thespreading drops and the radius of the wetted circle on the outer surface of theglass filter for silicone oils of different viscosity are presented using experimentaldata in dimensional form. Figure 4.12a shows that the kinetics of the spreadingand imbibition varies for drops of different size and different viscosity. Conse-quently, the total duration of the spreading process, the maximum radius of thedrop base, and the radius of the wetted circle on the outer surface of the glassfilter vary considerably. However, if, as in Section 4.2, we rescale quantities asL/Lmax, l/lmax, and t/tmax, then all experimental data fall into two universal curvesas shown by Figure 4.12b.

According to Section 4.2 the evolution of reduced dynamic contact angle,θ/θm, with the dimensionless time, t/tmax, should be universal. Here, θm is the valueof the dynamic contact angle that is reached when the radius of the drop basereaches its maximum value (the end of the first stage of spreading). The sameprocedure is used in the case of spreading over thick porous substrates. In



Figure 4.12c, the reduced dynamic contact angle, θ/θm, is plotted versus thedimensionless time t/tmax.

This plot shows that: (1) all three experimental curves fall into a singleuniversal curve, and (2) the dynamic contact angle, as the theory predicts, remainsconstant during the second spreading stage.

In Figure 4.12d, the evolution of the relative effective dynamic contact angle,ψ/ψmax, inside the porous glass filter with the dimensionless time is presented. Itis noteworthy that in all three cases, all the data follow a single universal curve.

These three experimental runs show that the spreading behavior of drops ofdifferent viscosities and volumes on the same thick porous substrate is identical,if appropriate dimensionless coordinates are used.

Spreading of Silicone Oil Drops over Filters with Similar Properties but Made of Different Materials (Figure 4.13a–Figure 4.13c)

In this series of experiments, the spreading of drops of the same silicone oil S500,over different substrates was studied. We wanted to check if the universal behaviorfound in the previous case remains valid even when different porous substratesmade of different materials, glass, and metal filters, are used.

In Figure 4.13a–Figure 4.13c: �� = the spreading of the silicone oil S500 (dropvolume V0 = 8.2 µl; the maximum radius of the drop base Lmax = 2.38 mm; themaximum radius of the wetted circle on the outer surface of the porous substratelmax = 3.20 mm; the total duration of the spreading tmax = 15.7 sec over the metal

FIGURE 4.13 (a) Spreading of silicone oil (η = 5 P) over dry porous glass and metalfilters. Radii in reduced coordinates. (1) metal filter: porosity 0.32, average pore size26.1 µm (see insets); (2) glass filter: porosity 0.31, average pore size 26.8 µm (see insets),Lmax — maximum value of the radius of the drop base; lmax — maximum value of the radiusof the wetted circle on the surface of the glass filter; tmax — total duration of the process(see Table 4.4).

1.0

0.8

0.6

0.4

0.2

0.0

1.0

0.8

0.6

0.4

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0

L/L m

ax

t/tmax

�/� m

ax

–1

–2



filter (Cupro-Nickel with diameter 5.6 cm, thickness 1.9 mm, average pore size26.1 µm, and porosity 0.32); Ο – the spreading of the silicone oil S500 (drop volumeV0 = 6.8 µl; maximum radius of the drop base Lmax = 2.40 mm; maximum radiusof the wetted circle on the outer surface of the porous substrate lmax = 3.20 mm;and the total duration of the spreading tmax = 18.52 sec over the glass filter withdiameter 2.9 cm, thickness 2.2 mm, porosity 0.31, and average pore size 26.8 µm).

FIGURE 4.13 (continued) (b) Spreading of silicone oil (η = 5 P) over dry porous glassand metal filters. Dynamic contact angle vs. time in dimensionless units. (1) metal filter:porosity 0.32, average pore size 26.1 µm (see insets); (2) glass filter: porosity 0.31, averagepore size 26.8 µm (see insets); θm — dynamic contact angle value at the time when themaximum value of the radius of the drop base, Lmax, is reached (see Table 4.4). (c) Spreadingof silicone oil (η = 5 P) over dry porous glass and metal filters. Evolution of the effectivecontact angle inside the porous filters with relative time. (1) metal filter: porosity 0.32,average pore size 26.1 µm (see insets); (2) glass filter: porosity 0.31, average pore size 26.8µm (see insets); ψmax — maximum value of the effective contact angle (see Table 4.4).

0.20.00

1

2

3

4

θ/θ m

t/tmax

0.4 0.6 0.8 1.0

–1–2

1.0

0.8

0.6

0.4

0.2

0.0 0.2 0.4 0.6 0.8 1.00.0

ψ/ψ

max

t/tmax

–2–1



Figure 4.13a presents the dependence of the dimensionless radius of the dropbase (left ordinate) and that of the dimensionless radius of the wetted circle onthe outer surface of the porous substrate (right ordinate) on the dimensionlesstime. Figure 4.13b presents the dependence of the relative dynamic contact angleon the dimensionless time. Figure 4.13c presents the dependence of the effectivedynamic contact angle inside the porous substrate on the same dimensionlesstime.

The curves show that the spreading of drops of different size on poroussubstrates made of different materials with, however, similar porosity and averagepore size, fall on universal curves if, as in the previous case, the same dimen-sionless coordinates are used. Thus, the universal spreading behavior over poroussubstrates does not depend on the material of the substrate.

Spreading of Silicone Oil Drops with the Same Viscosity (ηηηη ==== 5P) over Glass Filters with Different Porosity and Average Pore Size (Figure 4.14a to Figure 4.14c)

In this section the spreading of silicone oil drops with the same viscosity overglass filters with different porosity and average pore size is investigated to checkif the universal behavior found in the two previous sections is still applicable.

From Figure 4.14a to Figure 4.14c: �� – the spreading of silicone oil S500(drop volume V0 = 8.0 µl; maximum radius of the drop base Lmax = 2.53 mm;maximum radius of the wetted circle on the surface lmax = 3.20 mm; and the total

FIGURE 4.14 (a) Spreading of silicone oil (η = 5 P) over different dry glass filters. Radiiof spreading in reduced coordinates. (1) glass filter: porosity 0.56; pore size 3.7 µm (seeinsets); (2) glass filter: porosity 0.31; pore size 26.8 µm (see insets). Lmax — maximumvalue of the radius of the drop base; lmax — maximum value of the radius of the wettedcircle on the surface of the glass filter; tmax — total duration of the process (see Table 4.4).

1.0

0.8

0.6

0.4

0.2

0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

t/tmax

L/L m

ax

I/Im

ax

–1

–2



duration of the spreading tmax = 36.0 sec over the glass filter with diameter 5.0 cmthickness 2.5 mm, porosity 0.56, and average pore sizes 3.7 µm); Ο = thespreading of the silicone oil S500 (drop volume V0 = 6.8 µl; maximum radius ofthe drop base Lmax = 2.40 mm; maximum radius of the wetted circle on the outer

FIGURE 4.14 (continued) (b) Spreading of silicone oil (η = 5 P) over different dry glassfilters. Dynamic contact angle vs. time in dimensionless units. (1) glass filter: porosity0.56; pore size 3.7 µm (see insets); (2) glass filter: porosity 0.31; pore size 26.8 µm (seeinsets). θm — the dynamic contact angle value at the time when the maximum value ofthe radius of the drop base, Lmax, is reached (see Table 4.4). (c) Spreading of silicone oil(h=5 P) over different dry glass filters. Evolution of the effective contact angle inside theporous filters with time in dimensionless units. (1) glass filter: porosity 0.56; pore size 3.7µm (see insets), (2) glass filter: porosity 0.31; pore size 26.8 µm (see insets). ψmax — themaximum value of the effective contact angle (see Table 4.4).

5

4

3

2

1

0.0 0.2 0.4 0.6 0.8 1.00

θ/θ m

t/tmax

–2

–1

1.0

0.8

0.6

0.4

0.2

0.0 0.2 0.4 0.6 0.8 1.00.0

ψ/ψ

max

t/tmax

–2

–1



surface of the glass filter lmax = 3.20 mm; and the total duration of the spreadingtmax = 18.52 sec over the glass filter with diameter 2.9 cm, thickness 2.2 mm,porosity 0.31, and average pore sizes 26.8 µm).

Figure 4.14b and Figure 4.14c show that the relationships between the relativedynamic contact angle, θ/θm, and the effective dynamic contact angle inside theporous substrate, ψ/ψmax, on the dimensionless time, t/tmax, exhibit universalbehavior again. However, the dependence of the dimensionless radius of the dropbase, L/Lmax, and that of the dimensionless radius of the wetted circle on thesurface of the porous substrate, l/lmax, on the dimensionless time, t/tmax, deviatefrom universal behavior during the first spreading stage. The duration of the firststage, when the drop base increases with time and reaches its maximum value,is shorter in the case of the spreading of silicone oil drops over the glass filterwith a smaller average pore size than over the glass filter with a larger averagepore size (Figure 4.14a).

In conclusion, we can safely say that the spreading behavior over poroussubstrates is mostly, if not entirely, determined by the porosity and the averagepore size of the porous substrate and differs if these two characteristics aredifferent.

Conclusions

Experiments were carried out on the spreading of small silicone oil drops (cap-illary regime of spreading) over various dry thick porous substrates (permeablein both normal and tangential directions). The time evolution of the radii of boththe drop base and the wetted region on the surface of the porous substrate weremonitored. It has been shown that the overall duration of the spreading processcan be divided into two stages: a first stage, when the drop base grows until amaximum value is reached, and a second stage, when the drop base shrinks. Ithas been observed that the dynamic contact angle remained constant during thesecond spreading stage. This fact is supplied by a heuristic argument and hasnothing to do with hysteresis of contact angle, as there is no hysteresis in thesystem. Using appropriate scales, the dimensionless radius of the drop base, theradius of the wetted circle on the surface of the porous substrate, the dynamiccontact angle, and the effective contact angle inside the porous substrate havebeen plotted using a dimensionless time.

Experimental data shows that the spreading of silicone oil drops over drythick porous substrates exhibits a universal behavior if: (1) porous substratesmade of different materials with, however, similar porosity and average pore sizeare used and (2) if appropriate dimensionless coordinates are introduced to depictthe data. However, if porous substrates with different porosity and average poresize are used the dynamic of both the radius of the drop base, L/Lmax, and the radiusof the wetted circle on the outer surface of the porous filter, l/lmax, behave differ-ently during the spreading process. Yet, both the relative dynamic contact angle,θ/θm, and the effective contact angle inside the porous substrate, ψ/ψmax, showuniversal behavior.



4.4 SPREADING OF LIQUID DROPS FROM A LIQUID SOURCE

In this section, we shall consider a liquid drop being created and then spread overa solid substrate with a liquid source. We shall look at both cases, complete andpartial wetting, and for small and large drops. Then, we expect to observe spread-ing and forced flow caused by the liquid source in the drop center. Both capillaryand gravitational regimes of spreading shall be considered [15].

For conditions of complete wetting, the spreading is an overlapping of twoprocesses: a spontaneous spreading and a forced flow caused by the liquid sourcein the center. Both capillary and gravitational regimes of spreading are considered,and power laws are deduced. In both cases of small and large droplets, theexponent is a sum of two terms: the first term corresponds to the spontaneousspreading, and the second term is determined by the intensity of the liquid source.In the case of a constant flow rate from the source, the radius of spreading isgiven by the following law, R(t) ~ t0.4, in the case of the capillary spreading, andR(t) ~ t0.5 in the case of gravitational spreading. In the case of partial wetting,droplets spread with a constant advancing contact angle (at small capillary num-bers). This yields R(t) ~ t 1/3. Experimental data are in good agreement with thetheoretical predictions.

The spreading of liquid drops over solid nonporous substrates has beeninvestigated in Section 3.1, Chapter 3; for conditions of complete wetting, thespreading of small droplets is governed by the capillary law of spreading (Equa-tion 3.25), which is

,

where R is the radius of the drop base, t is time, and λc, is a preexponential factor,which is determined by the disjoining pressure isotherm in Section 3.2. Similarto the case of complete wetting, the spreading of bigger drops is governed bygravity according to Equation 3.36 in Section 3.1:

.

For small drops, the capillary law of spreading is in excellent agreement withexperimental data, as was shown in Section 3.2. Over time, small droplets spreadout, and the radius of the drop base increases with time and, hence, should be atransition from a capillary regime of spreading to the gravitational regime. Thistransition has been experimentally confirmed [16] in the case of spontaneousspreading. Note that, in the case of complete wetting, the dynamic contact angle

R tV

t tc in( ) . ( )

.

.=

+0 65

30 1

0 1λ γη

R tgV

t tc( ) . ( )

/

/=

+0 78

31 8

1 8ρη



tends to zero over time, and droplets spread out completely (see Section 3.1 andSection 3.2 for details).

Spreading of liquid drops in the case of partial wetting has been less inves-tigated, and it is less understood. The main problem in this case is the presenceof contact angle hysteresis, which was considered in Section 1.3 and Section 3.10.The hysteresis phenomenon is usually associated with nonhomogeneity/rough-ness of the solid substrates. However, it has been shown in Section 3.10 that anS-shaped disjoining pressure isotherm leads to the presence of the contact anglehysteresis even on smooth homogeneous substrates. In the case of partial wetting,the droplet spreads out until a static advancing contact angle is reached. Afterthat the droplet does not spread out on a macroscale but still spreads out on amicroscale (see Section 3.10). If the droplet is “gently” pushed from inside bypumping liquid from the orifice at its center, then the droplet spreads out with aconstant advancing contact angle, which is equal to the static advancing contactangle. The term gently means that the capillary number remains small during thespreading process.

The spreading of liquid drops for both cases, complete and partial wetting,when liquid is injected into the droplets from a small orifice at their center isconsidered in the following text.

THEORY

Let us consider the spreading of a small liquid droplet over a solid substrate inthe presence of liquid source in the drop center (Figure 4.15).

It is assumed that the shape of the drop remains axisymmetrical, and hence,a cylindrical coordinate system, (r, z), is used in the following text, where r isthe radial distance from the center, and z is the vertical coordinate. Because ofsymmetry, the angular component of the velocity vanishes, and all other unknowns

FIGURE 4.15 Schematic presentation of the spreading in the presence of the liquid sourcein the drop center. R(t) — radius of the drop base; θ — contact angle; (1) liquid drop;(2) solid substrate with a small orifice in the center; (3) liquid source (syringe).

3

1

2

θR(t)



are independent of the angle. It is assumed in the following text that the Reynoldsnumber is Re << 1 and that the drop profile has a low slope; that is,

,

where h(t, r) is the drop profile. Using these assumptions, we arrive at the equation(Equation 3.11, Section 3.1), which describes the profile of the spreading liquid

, (4.91)

which is referred to in the following text as the equation of spreading. The liquid is assumed nonvolatile and injected from the center according to

a prescribed time rate V ′(t); hence, the drop volume obeys the following conser-vation law:

, (4.92)

where now V(t) is fixed by the pumping rate. Two unknown functions are to be determined: the liquid profile, h(t,r) and

the radius of the spreading drop base, R(t), which is referred to in the followingtext as the radius of spreading. The pressure inside the spreading drop, p(t,r),can be determined via the liquid profile, h(t,r). This expression includes severalcomponents (see Section 3.1. and Section 3.2), where we now keep only twocomponents: capillary and gravitational parts:

p = pa – γK + ρgh, (4.93)

where pa is the pressure in the ambient air; γ is the liquid–air interfacial tension;ρ and g are the liquid density and gravity acceleration, respectively; K is thecurvature of the liquid–air interface. In the low slope approximation (ε << 1), thecurvature is

,

hence, Equation 4.93 becomes:

. (4.94)

ε = ∂∂

<<hr

2

1

∂∂

= ∂∂

∂∂

ht r r

r hpr

13

3

η

20

π r h dr V t

R t( )

( )∫ =

Kr r

rhr

= ∂∂

∂∂

1

p pr r

rhr

gha= − ∂∂

∂∂

+γ ρ1



Let us introduce scales in radial and vertical directions, r*, h*, respectively.Then the following dimensionless quantities are introduced

,

where dimensionless values are marked by an overbar. In dimensionless form,Equation 4.94 can be rewritten as

.

In this expression we have two parameters:

.

The first dimensionless parameter estimates the intensity of capillary forces andthe second one that of the gravity force.

If capillary forces prevail, then we have the capillary regime of spreading;that is if

.

If the gravity dominates, then the gravitational regime of spreading takesplace, therefore

if .

The length

,

which was already introduced in Section 3.1, is the capillary length. Hence, thecapillary regime (R(t) < a) is the initial stage of spreading of small drops, whereasthe gravitational regime is the final stage of spreading of small drops (R(t) > a)

rrr

hhh

or r rr h hh= = = =* *

* *, ,

p ph

r r rr

hr

gh ha= − ∂∂

∂∂

+γ ρ*

**2

1

γ ρh

rgh*

**2

and

γ ρ γρ

h

rgh r

g*

** *2

>> <<or

γ ρ γρ

h

rgh r

g*

** *2

<< >>or

ag

= γρ



or the overall regime of spreading of big drops. In the following text, we considerthe spreading of small drops with a transition from the capillary to the gravita-tional regime of spreading over time.

Note that, in Equation 4.93 and Equation 4.94, we ignored the role playedby the disjoining pressure and hence, we cannot consider the drop profile in thevicinity of the three-phase contact line. However, here we are not going tocalculate the drop profile. Instead we try similarity solutions of the equation ofspreading (4.91) in the case of both the capillary and gravitational regime ofspreading (that is, initial and final stages of spreading of small drops). This methodallows us to calculate time evolution of the radius of spreading, but the preexpo-nential factor includes an unknown dimensionless integration constant. Based onthe consideration presented in Section 3.2 this constant can be calculated; how-ever, in this section, it is extracted from experimental data.

If the initial drop size is small enough, then the effect of gravity can beignored. Accordingly, the drop radius, R(t), has to be smaller than the capillarylength,

The liquid is injected through the orifice in the drop center with the flow rateV ′(t), where V(t) is an imposed function of time. In the case of spontaneousspreading, V(t) = V0 = const, where V0 is the constant volume of the spreadingdrop. In the case of constant liquid flow rate from the liquid source, V(t) = I t,where I is the intensity of the liquid source.

Mass conservation of liquid (Equation 4.92) demands that V(0) = 0 if therewas not a drop at the initial time.

In Appendix 3, we deduce a condition when the drop spreads according tothe power law, and the general possible form of V(t) compatible with the powerlaw of spreading is deduced. In the same Appendix 3 we deduce dependences ofthe radius of spreading with time in the case of complete wetting (both capillaryand gravitational regime of spreading) and in partial wetting (small capillarynumbers, Ca = Uη/γ).

In the case of the constant source of liquid I in the drop centre, the followingdependencies for the radius of the drop base are deduced in Appendix 3.

During the initial (capillary) stage of the spreading of small drops, the radiusof the drop base should follow

(4.95)

whereas at the final (gravitational) stage of the spreading of small drops, theradius of the drop base follows the law

R tg

( ) ≤ γρ

.

R tI

tc( ) ,

.

.=

α γ

η

30 1

0 4



(4.96)

where αc, αg are the dimensionless constants to be determined from experimentaldata.

In the case of partial wetting, the static hysteresis of the contact angle deter-mines the spreading behavior at low capillary numbers, Ca << 1, whereis the rate of spreading. This condition was always satisfied under our experi-mental conditions.

Because of the contact angle hysteresis, the drop does not move if the contactangle, θ, is in the range θr < θ < θa, where and are the static advancing andstatic receding contact angles, respectively (see Section 3.10). In our experimentalprocedure (Figure 4.15), we are interested in the static advancing contact angle only.

If the capillary number, Ca, is very small, which is the case in our experi-ments, then the advancing contact angle does not vary significantly. It is assumedthat the contact angle, θ, does not vary over duration of the spreading experimentand remains equal to its static value θa.

In this case, the radius of the drop base should follow

(4.97)

In spite of the similarity between expressions for the radius of spreading(Equation 4.95 and Equation 4.96 in the case of complete wetting and Equation4.97 in the case of partial wetting), there is one significant difference betweenthese two spreading processes: if the liquid source is closed, then in the case ofcomplete wetting, the drop will continue to spread out according to the law R(t) ~t 0.1 (in the case of capillary regime), or R(t) ~ t1/8 (in the case of gravitationalregime). However, in the case of partial wetting, the drop will stop spreading assoon as the liquid source is closed.

EXPERIMENTAL SET-UP AND RESULTS

Materials and Methods

The spreading of silicone oil and aqueous droplets over glass substrates wereinvestigated. Silicone oil was purchased from Brookfield Engineering of Middle-boro, MA. Its viscosity was measured using the rheometer AR1000 (TA Instru-ments) at 25˚C. Density was measured by the weight method and for measuringsurface tension, the Tensiometer (White Electrical Instrument Co. of Worcester-shire, U.K.) was used. The following values were found: dynamic viscosity η =91.0 cP, density ρ = 0.96 g/cm3, surface tension γ = 22.5 dyn/cm.

Microscope glass slides (76 x 26 mm; Menzel-Glaser GmbH of Braunsch-weig, Germany) were used for spreading experiments. Circular orifices of

R tgI

tg( ) ,

/

.=

α ρ

η

31 8

0 5

U R t= �( )

θa θr

R tI

ft

a

( )( )

./

/=

θ

1 3

1 3



diameters 0.5mm were drilled in their centers. A liquid was injected throughthose orifices in the glass substrate with a constant flow rate using a HarvardApparatus syringe pump. This produced a liquid drop over the solid substrate.Constant flow rate resulted in a linear increase of the drop volume with time.The time evolution of the radius of the base of the spreading drops was monitored.

The glass slides were cleaned by immersing them in a chromic acid solutionfor 2 h followed by rinsing 10 times with distilled water and twice with ultrapure water, then drying in an oven at 70˚C for 30 min. Each cleaned and driedslide was used only once. At least three runs were conducted for each experimentalcondition, and average values are reported in the following text.

The diagram of the experimental set-up is shown in Figure 4.16. All exper-iments were carried out at 25 ± 0.5˚C. The solid substrate, 1 (Figure 4.16), wasfixed in the ring; a syringe (4) was positioned in the center of the substrate andconnected to the Harvard Apparatus syringe pump (18). The droplets of siliconeoil or water (3), were formed due to the injection. The following flow rates wereused: 0.005 ml/min; 0.01 ml/min, and 0.02 ml/min.

The spreading process was recorded using a CCD camera (5, 10) (Figure 4.16)and a VHS recorder (6) (11). The camera has been equipped with filters havinga wavelength of 640 nm. Such an arrangement suppresses illumination of theCCD camera by the scattered light from the substrate and hence, results in ahigher precision of the measurements. The source of light (13) was used duringexperiments. The camera and a VHS recorder were connected to a computer (8).Images were analyzed using Drop Shape Analysis FTA 32.

FIGURE 4.16 Schematic presentation of the experimental set-up for monitoring theadvancing and receding contact angles on a smooth substrate. (1) porous substrate; (2)hermetically closed, thermostated chamber; (3) liquid drop; (4) glued in syringe needle,positioned in the center of the solid substrate 1 (connected to the Harvard Apparatussyringe pump 18); (5, 10) CCD cameras; (6, 11) VCRs; (7) mirror; (8) PC; (9,14) telephotoobjectives; (12) collimating lens; (13) light source; (15) flash gun; (16) optical windows;(17) upper syringe; (18) Harvard Apparatus syringe pump.

17

7

162

1312 16 14 10 8

13

4

2CCD

1CCD

15

9 5 6

1116

18




The kinetics of spreading of silicone oil (complete wetting) and aqueous droplets(partial wetting) over glass substrates was investigated. The spreading processwas caused by both the spontaneous spreading and the injection of liquid throughthe liquid source in the center of drops.

Complete wetting. Two stages of spreading of small silicone oil drops havebeen observed: the first initial stage — a capillary regime, and the second finalstage — a gravitational regime. The experimental time dependences of radius ofspreading of silicone oil drops over glass surface are presented in Figure 4.17.Three experiments with different injection velocities (0.005 ml/min, 0.01 ml/min,and 0.02 ml/min) were conducted. In each experiment, the two above-mentionedstages of spreading are observed in Figure 4.17.

The data obtained from the initial stage of spreading correspond to thecapillary stage (R(t) < a). Equation 4.95 can be rewritten,

(4.98)

which is a linear function of time in lg–lg coordinates.

FIGURE 4.17 Radius of spreading vs. time for the spreading of silicone oil drops onglass surface (log–log plot), diameter of the orifice 0.5 mm: � capillary stage, � gravita-tional stage, I = 0.005 ml/min, experiment 1; � capillary stage, ▫ gravitational stage, I =0.01 ml/min, experiment 2; ♦ capillary stage, ◊ gravitational stage, I = 0.02 ml/min,experiment 3; dashed line fitted according to Equation 4.103; solid line fitted accordingto Equation 4.104.

lg ( ) lg . lg ,

.

R tI

tc=

+ ⋅α γη

30 1

0 4

15

12

8

6

4

3

2

1

2 4 6 10 20 40 60 100 300

Time (sec)

R (

mm

)



According to Equation 4.98, the experimental data were fitted according to

lg R(t) = X + n lg t, (4.99)

where intercept, X, and slope, n, are the fitting parameters. The fitted exponents,n, (Table 4.5) show good agreement with the theoretically predicted exponent 0.4.

The fitted value of X was compared with Equation 4.98, and the unknowndimensionless constant, αc, was determined as follows:

(4.100)

Using this equation, the constant αc was calculated for each set of data (Table4.5). The average value ± S.D. obtained was αc = 2.57 ± 0.18. The modelpredictions are shown in Figure 4.17 by dashed lines.

In the case of the gravitational regime of spreading (R(t) > a), experimentaldata were compared with the theoretical predictions according to Equation 4.96.This equation can be rewritten as:

(4.101)

According to Equation 4.101, experimental data were fitted according to:

lg R(t) = Y + n lg t, (4.102)

where Y and n are the fitting parameters. The theoretical predictions are shownin Figure 4.17 by solid lines. The fitting of the data resulted in an average value

TABLE 4.5Spreading of Silicone Oil Drops over Glass Surface with the Orifice in the Center

Injection Flow Rate Capillary Stage Gravitational Stage

Imm3/sec ααααc n ααααg n

0.0833 2.7589 0.4007 2.6402 0.48930.1667 2.5426 0.4005 2.1811 0.50000.3333 2.4040 0.4005 1.8833 0.4991

α ηγcI

X=

3

0 1.

exp( )

lg ( ) lg . lg .

/

R tgI

tg=

+ ⋅α ρη

31 8

0 5



of the exponent, n, in Equation 4.102 equal to 0.4961, which shows good agree-ment with the theoretically predicted exponent 0.5. Using the fitted value Y andEquation 4.101, the unknown dimensionless constant, αg, was calculated as:

. (4.103)

The average value of the constant ± S.D. was αg = 2.64 ± 0.38. The averagedrop height is 3.6 ± 0.3 mm, which is in good agreement with the calculatedvalue 3.9 mm.

Partial wetting. The partial wetting case was investigated using the spreadingof water droplets over the same glass surfaces. Three experiments with differentinjection velocities 0.005 ml/min, 0.01 ml/min, and 0.02 ml/min were presentedin Figure 4.18. The spreading behavior was compared with the theoretical pre-diction according to Equation 4.97, which can be written as:

. (4.104)

During the spreading of water droplets over glass surfaces, the advancingcontact angle does not vary significantly. The average value of the contact anglewas calculated for each experimental run, and the average advancing contact

FIGURE 4.18 Radius of spreading vs. time for the spreading of water droplets on glasssurface (log–log plot), diameter of the orifice 0.5 mm: � I = 0.005 ml/min, experiment 1;� I = 0.01 ml/min, experiment 2;♦ I = 0.02 ml/min, experiment 3; solid line drawnaccording to Equation 4.105.

α ηρg

gIY=

3

1 8/

exp( )

lg ( ) lg( )

lg/

R tI

ft

a

=

+θ

1 313

R (

mm

)

40.5

0.6

1.0

1.4

2.0

3.0

4.0

6 10 20 40 60 100

Time (sec)



angle was determined from three experimental runs. The average advancingcontact angle θa ± S.D. is 54 ± 2˚. This value was used in Equation A2.34 forthe calculation of the function f(θa), which was substituted in Equation 4.104.The dashed line in Figure 4.18 was plotted according to Equation 4.104. Notethat Equation 4.104 does not include any fitting parameters. Figure 4.18 showsthat our experimental data are in good agreement with the theoretically predictedlaw (4.104).

Conclusions

The spreading of liquid over solid substrates when there is liquid injection throughan orifice is investigated from both theoretical and experimental points of view.Two cases of spreading over a glass substrate with a diameter of the orifice 0.5 mmwere studied: spreading of silicone oil droplets (complete wetting) and spreadingof water droplets (partial wetting). In the case of silicone oil spreading, tworegimes of spreading were observed: capillary regime and gravitational regime.A theory has been developed for the cases of complete wetting and partial wettingat low capillary numbers. In all three cases, power laws of spreading werededuced: capillary regime of spreading according to Equation 4.95, gravitationalregime of spreading according to Equation 4.96, and partial regime of spreadingaccording to Equation 4.97. Experimental data validated our theoretical depen-dences of the radius of spreading on both time and injection velocity in bothcases of complete and partial wetting.

APPENDIX 2

Let us introduce the following similarity coordinate and functions,

, (A2.1)

where ϕ(ξ), H(t) are two new unknown functions. Note that we are interestedonly in the time evolution of the radius of spreading, R(t).

Substitution of (A2.1) into the conservation law, Equation 4.92, results in

. (A2.2)

Let us select the unknown function H(t) as

. (A2.3)

ξ ϕ ξ= =r

R t

h t r

H t( ), ( )

( , )( )

2 2

0

1

π ξϕ ξ ξR t H t d F t( ) ( ) ( ) ( )=∫

H tV t

R t( )

( )

( )=

2 2π



Then, Equation A2.2 can be reduced to

. (A2.4)

To move further it is necessary to specify the equation of spreading (4.5),which determines the drop profile h(t, r) or ϕ(ξ). Two different cases are underconsideration: complete and partial wetting cases.

Capillary Regime, Complete Wetting

In this case, according to Equation 4.94, the pressure inside the spreading dropcan be written as

.

This expression should be substituted into Equation 4.91, which yields thetime evolution of the drop profile, h(t, r):

. (A2.5)

Note that the omission of the action of surface forces results in the well-known singularity on the moving three-phase contact line (see Section 3.1).

Substitution of the similarity coordinate and function using Equation A2.1and Equation A2.3 into Equation A2.5 results in

where an overdot denotes differentiation with time, whereas ′ means differenti-ation with respect to the similarity variable, ξ. This equation can be rewritten as

(A2.6)

ξϕ ξ ξ( )d =∫ 10

1

p pr r

rhra= − ∂

∂∂∂

γ 1

∂∂

= − ∂∂

∂∂

∂∂

∂∂

h

t r rrh

r r rr

h

r

13

1 3

ηγ

��

H tH t R t

R tH t

R t( ) ( )

( ) ( )( )

( )( )

(ϕ ξ ξϕ ξ γ

η− ′ = −

3

4

4 ))( ) ( ) ,

1 13

ξξϕ ξ

ξξϕ ξ′( )′

′

′

3 34

4

3

3

ηγ

ϕ ξ ηγ

ξ� �H t R t

H t

R t R t

H t

( ) ( )

( )( )

( ) ( )

( )− ′ϕϕ ξ

ξξϕ ξ

ξξϕ ξ( ) ( ) ( ) .= − ′( )′

′

′1 13



Equation A2.6 should depend on the similarity coordinate only, and it shouldnot include any time dependence. This is possible only if, simultaneously, thefollowing two relations are satisfied

, (A2.7)

where B1 and B2 are unknown constants. Both constants should be positivebecause H(t) and R(t) are both increasing functions of time.

Let α = B1/B2 and divide the first equation in (A2.7) by the second equation.It results in:

,

that upon integration yields:

, (A2.8)

where C is an integration constant, and α is still an unknown exponent. Substitution of Equation A2.8 into Equation A2.7 results in the following

time evolution of the radius of spreading, R(t),

, (A2.9)

which shows that 4-3α should be positive, that is, α < 4/3.Equation A2.9 and Equation A2.3 allow determination of the unknown func-

tion H(t):

. (A2.10)

Using Equation A2.10, Equation A2.9, and Equation A2.3, we can concludethat the following relation should be satisfied:

. (A2.11)

3 34

4 1

3

3 2

ηγ

ηγ

� �H t R t

H tB

R t R t

H tB

( ) ( )

( ),

( ) ( )

( )= =

� �H

H

R

R= α

H t CR t( ) ( )= α

R tB C

t( )( )

/( )

/( )= −

−

−4 33

23

1 4 3

1 4 3α γη

α

α

H t CB C

t( )( )

/( )

/( )= −

−

−4 33

23

4 3

4 3α γη

α α

α α

V t CB C

t( )( )= −

+− +

−24 3

32

32

4 3 2

4 3π α γη

αα α

α



This relation shows that the similarity mechanism considered in the precedingtext is possible only if the dependency V(t) is defined by the power law (A2.11).

Let us assume now that

, (A2.12)

where a′ and b are constants imposed by the source. The exponents in EquationA2.11 and Equation A2.12 should be equal, and hence

.

Substitution of this expression into Equation A2.9 gives the following spreadinglaw:

. (A2.13)

Accordingly, the exponent is the sum of two terms: the first term, 0.1, stemsfrom the spontaneous spreading (see Chapter 3), and the second term, 0.3b, isdetermined by the liquid source.

In the case of constant flow rate of the liquid,

(A2.14)

where I is the intensity of the liquid source. The comparison of exponents inEquation A2.14 and Equation A2.9 yields

or α = 1/2. On the other hand, α = B1/B2 or B2 = 2B1 though B1 is the onlyunknown constant. The comparison of the preexponential factors in EquationA2.14 and Equation A2.9 gives

. (A2.15)

The spreading law according to Equation A2.13 now takes the following form:

. (A2.16)

V t a tb( ) = ′

α = −+

4 21 3b

b

R t C t b( ) . .= ⋅ +0 1 0 3

v I t= ·

124 3

= +−

αα

CI

B=

310 1

1 4ηπγ

/

R tB I

t( )

.

.=

5

241

3

3

0 1

0 4γηπ



The only unknown constant in Equation A2.16 is the dimensionless constant B1.

The equation of spreading (A2.6) can be rewritten now as

(A2.17)

Remember, this equation describes the behavior of the drop profile, ϕ(ξ), nottoo close to the moving three-phase contact line (valid only away from the rangeof the action of surface forces). Let us try to include the disjoining pressure actioninto Equation A2.17. In the case of complete wetting, the disjoining pressureisotherm is

,

where A is the Hamaker constant. Now Equation 4.8 should be rewritten as

,

and this expression should be substituted in the equation of spreading (4.5). Usingthe same similarity coordinate, ξ, and the drop profile, ϕ, according to EquationA2.1, we arrive at the following equations for the determination of the unknownfunction, ϕ:

(A2.18)

where

is a dimensionless constant, which characterizes the intensity of the action ofsurface forces. Equation A2.18 shows that the same similarity property is validfor the full Equation A2.18, when the action of surface forces is taken into account,as for Equation A2.17.

B B1 132

1 1ϕ ξ ξϕ ξξ

ξϕ ξξ

ξϕ ξ( ) ( ) ( ) ( )− ′ = − ′( )′

′′

′

.

Π( )hA

h=

3

p pa r r

rh

rgh

A

h= − ∂

∂∂∂

+ −γ ρ13

B BB

1 13 1

32

1 1ϕ ξ ξϕ ξξ

ξϕ ξξ

ξϕ ξ χϕ

( ) ( ) ( ) ( )− ′ = − ′( )′ +(( )

,ξ

′

′

χ πη

= 103

A

I



Equation A2.18 must satisfy the following boundary conditions:

, (A2.19)

which are the symmetry conditions in the drop center. It should also satisfy

, (A2.20)

where infinity means the drop edge. It has been shown in Section 3.2 that thedrop profile tends asymptotically to zero inside “the inner region,” which is theboundary condition (A2.20). The conservation law (A2.10) should also be takeninto account.

The unknown parameter, B1, should be small according to the previous con-sideration in Chapter 3. In this case, the whole drop can be subdivided into tworegions: the outer region, which has a spherical shape, and the inner region, wherethe inner coordinate should be introduced. Matching of these two regions allowsthe determination of the unknown parameter, B1, via the dimensionless Hamakerconstant, χ. It has been shown in Section 3.2 that the dependence of B1 on theparameter χ is a weak one. It has been shown also in Section 3.2 that the lackof proper asymptotic behavior of Equation A2.18 does not allow precise deter-mination of the unknown constant B1.

That is why we use Equation A2.16 in the following form

, (A2.21)

where the dimensionless parameter αc is determined in the following text usingexperimental data.

Gravitational Regime, Complete Wetting

In this case, Equation 4.94 becomes , and substitution of the latterequation into the equation of spreading (Equation 4.91) results in

. (A2.22)

Using the same similarity coordinate and function (A2.1), we conclude thatrelations (A2.2–A2.4) are still valid. Using the same procedure as the one previ-ously mentioned, we can transform Equation A2.22 to

′ = ′′′ =ϕ ϕ( ) ( )0 0 0

ϕ ξ ξ( ) ,→ → ∞0

R tI

tB

c c( ) ,

.

.

.

=

=

α γη

απ

30 1

0 4 13

05

24

11

p p gha= + ρ

∂∂

= ∂∂

∂∂

ht

gr r

r hhr

ρη3

3

��

H tH t R t

R tg H t

R t( ) ( )

( ) ( )( )

( )( )

(ϕ ξ ξϕ ξ ρ

η− ′ =

3

4

2 ))( ) ( ) .

1 3

ξξϕ ξ ϕ ξ′

′



This equation can be rewritten as

(A2.23)

Equation A2.23 should depend on the similarity coordinate only; that is, itshould not include any time dependence. This is possible only if the followingrelations are satisfied simultaneously:

, (A2.24)

where D1 and D2 are unknown dimensionless constants. Both constants shouldbe positive (or zero) because H(t) and R(t) are both increasing functions of time.

Let β = D1/D2 and divide the first equation in (A2.24) by the second equation.That results in

,

which upon integration yields

, (A2.25)

where G is an integration constant, and β is still the unknown exponent. Substitution of Equation A2.25 into Equation A2.24 results in the following

time evolution of the radius of spreading, R,

, (A2.26)

which shows that 2-3β should be positive, that is, β < 2/3.Equation A2.26 and Equation A2.25 allow the determination of the unknown

function H(t):

. (A2.27)

Using Equation A2.26, Equation A2.27, and Equation A2.3, we can conclude

. (A2.28)

3 32

4 3

ηρ

ϕ ξ ηρ

ξ� �H t R t

gH t

R t R t

gH t

( ) ( )

( )( )

( ) ( )

( )− ′′ = ′

′ϕ ξξ

ξϕ ξ ϕ ξ( ) ( ) ( ) .1 3

3 32

4 1 3

ηρ

ηρ

� �H t R t

gH tD

R t R t

gH tD

( ) ( )

( ),

( ) ( )

( )= = 22

� �H

H

R

R= β

H t GR t( ) ( )= β

R tgD G

t( )( )

/( )

/( )= −

−

−2 33

23

1 2 3

1 2 3β ρη

β

β

H t GgD G

t( )( )

/( )

/( )= −

−

−2 33

23

2 3

2 3β ρη

β β

β β

V t GgD G

t( )( )= −

+− +

−22 3

32

32

2 3 2

2 3π β ρη

ββ β

β



Thus, we see that the similarity mechanism considered in the preceding textis possible only if the dependency V(t) is defined by the power law (A2.28).

Let us assume now that the liquid source produced the liquid in the sameway as in the case of the capillary regime; that is, according to Equation A2.12.

The exponents in Equation A2.28 and Equation A2.12 should be equal, andhence in

.

Substitution of this expression into Equation A2.26 gives the following spreadinglaw:

, (A2.29)

that is, the exponent is the sum of two terms: the first term, 1/8, stems from thespontaneous gravitational spreading (see Section 3.1), and the second term, 3b/8,is determined by the liquid source.

In the case of constant flow rate of the liquid,

(A2.30)

and the comparison of exponents in Equation A2.30 and Equation A2.12 results in

or β = 0. On the other hand β = D1/D2, hence, D1 = 0. Then, D2 is the onlyunknown constant. According to the first Equation A2.24 and Equation A2.25,D1 = β = 0 means that, in the case of gravitational spreading, the maximumheight of the spreading drop remains constant when the source of liquid followsEquation A2.30. The comparison of the preexponential factors in Equation A2.30and Equation A2.28 gives

.

The spreading law according to Equation A2.26 takes the following form now:

. (A2.31)

β = −+

2 21 3b

b

R t const t b( ) / /= ⋅ +1 8 3 8

V t I t( ) =

122 3

= +−

ββ

GI

gD=

34 2

1 4η

πρ

/

R tgD I

t( ) /

/

.=

2

31 6 2

3

3

1 8

0 5ρηπ



The only unknown quantity in Equation A2.31 is the dimensionless constantD2, which is left unknown and was determined experimentally. Note that theconstant D2 can be, in general, determined in the following way: (1) a narrowzone close to the drop edge, where capillary forces become important should beconsidered, and (2) matching of asymptotic solutions (capillary zone as an innerzone and the gravitational zone as an outer zone) should be made, which allowsthe determination of the numerical constant D2.

In order to determine the unknown constant, we rewrite Equation A2.31 as

, (A2.32)

and the constant αg is obtained using experimental data.The latter expression allows the determination of the constant thickness of

the spreading drop during the gravitational regime of spreading. CombiningEquation A2.3 and Equation A2.32 would result in

, (A2.33)

which is independent of time as predicted in the preceding text. It means thatduring the gravitational regime of spreading from a liquid source with constantflow rate intensity, the drop spreads like a “pancake.”

Partial Wetting

The drop volume can be expressed in terms of the spreading radius and the contactangle as follows:

, (A2.34)

where θa is the static advancing contact angle. We assume that the capillarynumber is very small and hence, the contact angle does not change duringspreading. Hence, f(θa) also remains constant. Combination of Equation A2.10and Equation A2.34 results in

.

R tgI

tD

g g( ) ,

/

. /=

=

α ρη

απ

31 8

0 5 1 6 23

23

1 8/

HI

gg

=

12

1 4

αη

ρ

/

V t R t f fa aa a( ) ( ) ( ), ( ) tan tan= = +

3 2

6 23

2θ θ π θ θ

R tV t

f a

( )( )

( )

/

=

θ

1 3



In the case of the constant flow rate from the liquid source (Figure 4.15),according to Equation A2.30, the preceding equation gives

, (A2.35)

which is compared with our experimental observations in the case of partialwetting.

REFERENCES

1. Brinkman, H., J. Chem. Phys., 20, 571, 1952.2. Whitaker, S., The Method of Volume Averaging, Kluwer Academic Publishers,

Dortrecht/Boston/London, 1999, p. 221.3. Starov, V.M. and Zhdanov, V.G., Colloids Surf. A: Physicochem. Eng. Aspects,

192, 363, 2001. 4. Kalinin, V.V. and Starov, V.M., Colloid J. (USSR Academy of Sciences, English

Translation), 51(5), 860, 1989. 5. Starov, V.M., Kosvintsev, S.R., Sobolev, V.D., Velarde, M.G., and Zhdanov, S.A.,

Spreading of liquid drops over dry porous layers: complete wetting case, J. ColloidInterface Sci., 252, 397–408, 2002.

6. Kornev, K.G. and Neimark, A.V., J. Colloid Interface Sci, 235, 101, 2001.7. Neogi, P. and Miller, C.A., J. Colloid Interface Sci., 92(2), 338, 1983.8. Starov, V.M., Kalinin, V.V., and Chen, J.-D., Adv. Colloid Interface Sci., 50, 187,

1994. 9. Kalinin, V. and Starov, V., Colloid J. (USSR Academy of Sciences, English

Translation), 48(5), 767, 1986.10. Marmur, A., Adv. Colloid Interface Sci., 19, 75, 1983.11. Joanny, J.-F., J. Phys., 46, 807, 1985.12. Starov, V.M., Kosvintsev, S.R., Sobolev, V.D., Velarde, M.G., and Zhdanov, S.A.,

J. Colloid Interface Sci., 252, 397, 202.13. Nayfeh, A.H., Perturbation Methods, Wiley-Interscience, New York, 1973.14. Starov, V.M., Zhdanov, S.A., and Velarde, M.G., Langmuir, 18, 9744, 2002.15. Holdich, R., Starov, V., Prokopovich, P., Hjobuenwu, D., Rubio, R., Zhdanov, S.,

and Velarde, M., Colloids Surf., A: Physicochem. Eng. Aspects, 2005.16. Cazabat, A.M. and Cohen-Stuart, M.A., J. Phys. Chem., 90, 5849, 1986.17. Brinkman, H., Appl. Sci. Res., A1, 27, 1947.

R tI

ft

a

( )( )

/

/=

θ

1 3

1 3


389

5 Dynamics of Wetting or Spreading in the Presence of Surfactants

INTRODUCTION

In this chapter we consider the kinetics of spreading of surfactant solutions overhydrophobic and porous substrates. In spite of the wide use of these processes,currently we are not in the position to answer even basic questions in this area,such as how surfactant molecules are transferred in a vicinity of the three-phasecontact line.

In the case of aqueous surfactant solution, our knowledge of the behavior ofthe transition zone from the meniscus to thin films in front is very limited.Disjoining pressure isotherms in the presence of surfactants are investigated inthe case of free liquid films [30], and much less is known in the case of liquidfilms on solid support. At the moment we cannot present a clear physical pictureof the equilibrium of liquids or menisci ion the presence of surfactants.

The most important problem of surfactant transfer in a vicinity of the three-phase contact line is to be investigated here.

In Chapter 1 and Chapter 2 we showed that the Young’s equation does nothave any theoretical basis and should be replaced by the Deriaguin–Frumkinequation for equilibrium contact angle. In this chapter, in view of our limitedknowledge of surfactant behavior in the vicinity of the moving three-phase contactline, we decided to use the Young’s equation for the quasi-equilibrium contactangle for the description of slowly developing time-spreading processes overhydrophobic surfaces. We realize that any conclusion based on this semiempiricalrelation should be understood accordingly as semiempirical. However, ourhypothesis on adsorption of surfactants on a bare hydrophobic substrate in frontof the moving meniscus allows us to develop some theoretical predictions, whichare in reasonable agreement with known experimental data (Section 5.2 to Section5.4; Section 5.7). However, the main question how this transfer goes on is leftunanswered.

The situation is even less investigated in the case of simultaneous spreadingand imbibition into porous substrate (Section 5.1). We present some theoreticaland experimental investigations of the process (Section 5.1), which should beconsidered as a first step in this direction.



In Section 5.6 we consider the much more theoretically understood processof flow caused by surface tension gradient (Maramgoni flow). We show that theflow caused by the point source of surfactant on the surface of thin aqueous filmis caused by the surface tension gradient only. All other forces can be disregarded.This process recently became a powerful tool for investigation of the newlyrecognized phenomenon called superspreading.

5.1 SPREADING OF AQUEOUS SURFACTANT SOLUTIONS OVER POROUS LAYERS

In this section we shall follow the track of Chapter 4 for the case of surfactantaqueous solutions like sodium dodecyl sulfate (SDS, an anionic surfactant). Weshall start with the spreading problem of big drops, albeit small enough to allowneglecting gravity, over porous solid substrates [1]. The porous substrate shouldbe thin in the sense used in Chapter 4. We shall be considering various SDSconcentrations: zero (pure water) and concentrations below, near, and above thecritical micelle concentration (CMC). The overall spreading process would bedivided into three stages: in the first stage, the drop base expands until itsmaximum value is attained; the contact angle decreases very fast. In a secondstage, the radius of the drop base remains constant, whereas the contact angledecreases linearly with time. Finally, in the third stage, the drop base shrinks butthe contact angle remains constant, while the wetted area inside the porous solidsubstrate expands all the time [1].

Appropriate scales were used with a plot of the dimensionless radii of thedrop base, of the wetted area inside the porous substrate, and the dynamic contactangle on the dimensionless time.

The experimental data show that the overall time of the spreading of dropsof SDS solution over dry thin porous substrates decreases with the increase ofsurfactant concentration; the difference between advancing and hydrodynamicreceding contact angles decreases with the surfactant concentration increase; andthe constancy of the contact angle during the third stage of spreading has nothingto do with the hysteresis of contact angle but is determined by hydrodynamicreasons. It is shown using independent spreading experiments of the same dropson nonporous nitrocellulose substrate that the static receding contact angle isequal to zero, which supports our conclusion on the hydrodynamic nature of thereceding contact angle on porous substrates.

In Chapter 4 the spreading of liquid drops over thin porous layers saturatedwith the same liquid (Section 4.1) or dry (Section 4.1) was investigated in thecase of complete wetting. Brinkman’s equations were used for the description ofthe liquid flow inside the porous substrate.

In this section we take up the same problem in the case where a drop spreadsover a dry porous layer as in partial wetting. Spreading of a big drop (but stillsmall enough to neglect the gravity action) of aqueous SDS solutions over “thinporous layers” (nitrocellulose membrane) is considered in the following text.


Dynamics of Wetting or Spreading in the Presence of Surfactants 391

EXPERIMENTAL METHODS AND MATERIALS [1]

Spreading on Porous Substrates (Figure 4.4)

Aqueous solutions of SDS were used in the spreading experiments in zero concen-tration (pure water) and concentrations below CMC, near CMC, and above CMC.

SDS was purchased from Fisher Scientific UK Ltd. of Leicestershire, U.K.and used as obtained, without further purification. Nitrocellulose membranes,purchased from Millipore of Billerica, MA, with average pore size a = 0.22, a= 0.45, and a = 3.0 µm (marked by the supplier) were used as a model of thinporous layers.

The same experimental chamber as in Chapter 4 (Figure 4.2) was used formonitoring of the spreading of liquid drops over initially dry porous substrates.The time evolution of the radius of the drop base, L(t), the dynamic contact angle,θ(t), and the radius of the wetted area inside the porous substrate, l(t), wasmonitored (Figure 4.4). The porous substrate 1 (Figure 4.2) was placed in athermostated and hermetically closed chamber 2, where 100% relative humidityand fixed temperature were maintained. The initial volume of the droplets rangedbetween 1.47µl and 10.45 µl.

Care was taken so that all interfaces in the syringe and the attached replaceablecapillary had been saturated with surfactants before the actual solutions wereused. The drop was applied on the surface by pumping manually the piston ofthe syringe. The distance between the forming drop and the surface was keptminimal to avoid collateral inertia effects.


• The dry membrane (initially stored in 0% humidity atmosphere) wasplaced in the chamber with a 100% humidity atmosphere and left for15 to 30 min.

• A light pulse produced by a flash gun was used to synchronize bothvideotape recorders (a side view and a view from above).

• A droplet of liquid was placed onto the membrane.

Each run was carried out until the complete imbibition of the drop into theporous substrate took place.

Measurement of Static Advancing and Receding Contact Angles on Nonporous Substrates

A modified experimental setup (as compared with the previous case) was usedfor measurements of static advancing and receding contact angles on a nonporousnitrocellulose substrate. Figure 4.19 shows the schematic presentation of thesample chamber for monitoring of the advancing and receding contact anglesduring droplet spreading over a smooth nonporous nitrocellulose substrate. Thetime evolution of the radius of the drop base, L(t,) and the dynamic contact angle,θ(t), was monitored.



The nonporous substrate under investigation 1 (Figure 4.19) was attached bydouble-sided tape to a solid substrate and was placed into a thermostated andhermetically closed chamber 2, where controlled fixed humidity and fixed tem-perature were maintained. A glued-in syringe needle 4 was positioned in thecenter of the solid substrate and connected to the Harvard Apparatus syringepump 18 (Figure 4.19).

The nonporous nitrocellulose substrate was prepared as follows:

• The nitrocellulose substrate was attached to the solid substrate 1(Figure 4.19) using double-sided sticky tape, ensuring that no air bub-bles could be trapped.

• The nitrocellulose substrate attached to the solid substrate was placedinto an acetone atmosphere. Acetone vapor was used to seal any poresthat may have been open, and the nitrocellulose substrate was left inthe acetone atmosphere for approximately 20 min until the surfacebecame transparent.

• After that, the nitrocellulose substrate attached to the solid substrate 1was placed into a sealed container filled with air, allowing the surfaceto reach equilibrium after the acetone treatment.

• A hole was made in the center of the nitrocellulose substrate, allowingpenetration of the liquid through a glued-in syringe needle 4 connectedto the Harvard Apparatus syringe pump 18 (Figure 4.19). This alloweddrawing the fluid in or out of the surface and monitoring the dropletvolume according to a prescribed rate.

The experimental runs were carried out in the following order:

• The speed at which the fluid would be drawn out of the drop was fixedin the Harvard Apparatus syringe pump 18 (Figure 4.19). The refillrate was varied between 0.01 and 0.1 µl per minute.

• The nitrocellulose substrate 1 was placed into the experimental cham-ber 2, and the needle was connected to the pump 18 (Figure 4.19).

• The experimental chamber was closed, and the fan was switched onto equilibrate the atmosphere inside the chamber.

• After approximately 20 min the fan was switched off and video record-ers were switched on.

• The droplet of the liquid was deposited onto the surface under inves-tigation, using the upper syringe 17 into the center of syringe needle 4(Figure 4.19).

• After 1 min of recording, the pump was turned on to draw off the liquidfrom the drop.

The volume of the spreading drops should remain constant in the case ofspreading over nonporous nitrocellulose surface. The constancy of the volume ofthe spreading drops was monitored during these experiments, which confirmedthat the substrate used was nonporous.




In all our spreading experiments, the drops remained as spherical shapes over thewhole spreading process. This was cross-checked by reconstructing of the dropprofiles at different time instants of spreading, fitting those drop profiles by aspherical cap (see Chapter 4, Section 4.1 and Section 4.2). In all cases, the reducedchi-square value was found to be less than 104.

The wetted area inside the porous layer was mostly circular. In some cases itwas of an ellipsoidal form, and it was taken into account for the calculation of thedrops volume. Drops remained in the center of this spot over the duration of thespreading process. In all cases, saturation of the membrane in the vertical directionwas much faster than the whole duration of the spreading–imbibition process; thatis, the membrane was assumed always saturated in the vertical direction (Section4.2). A schematic presentation of the process is presented in Figure 4.4.

According to our observations, the whole spreading process can be subdividedinto three stages (see Figure 5.1). During the first stage, the contact angle, θ,rapidly decreases, whereas the radius of the drop base, L, increases until itsmaximum value, Lmax, is reached. The first stage was followed by the secondstage, when the radius of the drop base, L = Lmax, was constant, and the contactangle, θ, decreased linearly with time. During the third stage, the radius of thedrop base decreased and the contact angles remained constant. At the final thirdstage, the drop base shrank until the drop completely disappeared, and the imbi-bition front expanded until the end of the process. The spherical form of thespreading drop allows measuring the evolution of the contact angle of the spread-ing drops. The contact angle, θ, during the first stage, decreased very fast; duringthe second stage, the contact angle, θ, decreased much slower and linearly, andthe contact angle remained a constant value over the duration of the third stage.This constant value of the contact angle is referred to in the following text as θm.

All relevant experimental data are summarized in the Table 5.1.In this section we present a brief theoretical explanation why the dependency

of the contact angle during the second stage of the spreading is a linear functionof time.

During the second stage,

(5.1)

where tm corresponds to the beginning of the second stage. This means thataccording to Equation 4.72 from Chapter 4, Section 4.2,

(5.2)

where l is the radius of the circular edge of the wetted region inside the porouslayer, Kp is the permeability of the porous layer in the tangential direction, pc isthe capillary pressure on the wetting front inside the porous layer, lm is the radius

L L t tm= >max , ,

dldt

K p

l lL

l t lp cm m= =

/η

ln, ( ) ,

max



of the circular edge of the wetted region inside the porous layer at the momenttm, and η is the liquid dynamic viscosity. This equation can be easily integrated,which gives

,

or

. (5.3)

FIGURE 5.1 Spreading of pure aqueous drops over nitrocellulose membrane, a =0.22 µm. L/Lmax — dimensionless radius of the drop base; l/lmax — radius of the wettedarea; θ/θm — dynamic contact angle; t/tmax — dimensionless time; (1) first stage: θ rapidlydecreases while L increases until the maximum value Lmax; (2) second stage: L is constant,L = Lmax, and θ decreases linearly with time; (3) third stage: L decreases and θ remainsconstant.

tmax = 20.72 sec V0 = 2.9 mm3

tmax = 35.28 sec V0 = 5.3 mm3

1.0

0.8

0.6

0.4

0.2

0.0

1.0 0.8 0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0 0.0

0.2

0.4

0.6

0.8

1.0 θ/θ m

Nitrocellulose membranes, a = 0.22 μm. SDS = 0%

θr

t/tmax

L/L m

ax

�/� m

ax

θa

3

3

2 1

2

1

ll

eLl

l

eL

K pt tm

m p cm

2 2 2ln ln ( )

max max

− = −η

lK p

l

eL

t t

ll

eLl

eL

p cm

mm

2

2

2= − +

η ln( )

ln

lnmax

max

maxx



The term

TABLE 5.1Experimental Data Used in Figure 5.1 through Figure 5.5

Pore Size,Figure, Symbols

AveragedPore

Size, a µµµµm

SDS Concentration,

%

V0, µµµµl Initial

Volume of Drop

tmax, sec

Lmax, mm

lmax, mm

θθθθm, grad

tm,

sec

Figure 5.1 squares

0.22 0 2.9 20.72 1.49 3.38 51 0.36

Figure 5.1 circles

0.22 0 5.3 35.28 1.85 4.13 50 0.36

Figure 5.2 squares

3.0 0.1 2.1 2.96 1.47 2.30 31 0.36

Figure 5.2 triangles

3.0 0.1 3.7 4.92 2.01 3.40 24 0.64

Figure 5.2 circles

3.0 0.1 9.7 13.12 2.59 5.28 35 0.48

Figure 5.3 diamonds

0.22 0 2.9 20.72 1.49 3.38 51 0.36

Figure 5.3 circles

0.22 0.1 10.5 75.2 2.78 5.79 32 0.4

Figure 5.3 squares

0.22 0.2 7.4 33.84 2.45 4.67 27 0.44


0.22 0.5 2.6 8.2 2.03 2.90 20 0.12

Figure 5.4 diamonds

0.45 0 6.1 8.44 2.83 4.35 44 0.52

Figure 5.4 circles

0.45 0.1 9.5 10.28 2.46 4.92 38 0.35

Figure 5.4 squares

0.45 0.2 7.2 8.44 2.83 4.35 21 0.36


0.45 0.5 7.1 6.84 2.75 4.80 23 0.08

Figure 5.5 diamonds

3.0 0 5.9 108.04 1.89 4.21 46 21

Figure 5.5 circles

3.0 0.1 9.7 13.12 2.59 5.28 35 0.48

Figure 5.5 squares

3.0 0.2 6.8 3.96 2.65 4.43 21 0.28

Figure 5.5 circles

3.0 0.5 5.1 2.12 2.39 3.97 22 0.12

lnmax

l

eL



in the denominator of the equation is a slow changing function of l; that is, theright-hand side of the equation is almost indistinguishable from the linear functionof time. Equation 5.3 can be rewritten as

. (5.4)

According to Equation 4.48 and Equation 4.54 from Chapter 4, Section 4.3,

and

, (5.5)

where ∆ is the thickness of the porous layer. Combining Equation 5.4 andEquation 5.5 results in

. (5.6)

This equation shows that the contact angle decreases linearly with time duringthe second stage of spreading.

Everywhere in the following text the time evolution is presented for both theradius of the base of the spreading drops and the radius of the wetted area usingdimensionless coordinates. The drops were of different volumes and differentSDS concentrations. The total duration of the spreading process, tmax, the maxi-mum radius of the drop base, Lmax, and the final radius of the wetted area, lmax,varies considerably depending on the drop volume, SDS concentration, the aver-aged pore size, and porosity of nitrocellulose membranes. It has been suggestedin Section 4.2 that the following dimensionless values, L/Lmax, l/lmax, and t/tmax,be used. The same dimensionless values are used in the following text. Accordingto Section 4.3, the reduced contact angle dependency, θ/θm, against dimensionlesstime, t/tmax, is used in the following text, where θm is the value of the dynamiccontact angle, which is reached at the moment t = tm (the end of the first stageof the spreading process).

Figure 5.1 shows that spreading behavior of drops of different volumes overthe same porous substrate has a universal character in dimensionless coordinates.

l At B AK p

l

eL

B

ll

eLl

e

p cm

m

2

2

2= + = =,

ln,

ln

lnmax

max

ηLL

tK p

l

eL

mp c

max max

ln−

2

η

V t V m l t( ) ( )= −02π ∆

θπ

ππ π

ππ

= = − = −43

02

303 3

2V

L

V m l t

L

V

L

m

Ll t

m m m m

∆ ∆( )( ))

θπ

ππ

ππ π

= − + = − +V

L

m

LAt B

m A

Lt

V03 3 3

0

max max max

( )∆ ∆

LLB

m

Lmax max3 3

−

ππ

∆



However, not all of the experimental data have shown such universal behavior.Some experimental runs differ during the first stage of spreading in dimensionlesscoordinates (Figure 5.2). This difference becomes bigger if the overall time ofspreading is shorter (droplets of different volumes).

In Figure 5.3 to Figure 5.5 (0.22 µm, 0.45 µm, and 3.0 µm nitrocellulosemembranes, respectively) the time evolution is presented of the radius of the dropbase, the radius of the wetted area inside the porous substrates, and the contactangle at different SDS concentrations.

Figure 5.3 to Figure 5.5 show that the second stage of spreading becomesshorter in dimensionless coordinates with the increase in SDS concentrations.Contact angles show the universal constant behavior during the third stage ofspreading for each of the concentrations.

FIGURE 5.2 Spreading of droplets of 0.1% SDS solution over nitrocellulose membrane,a = 3.0 µm; L/Lmax — dimensionless radius of the drop base; l/lmax — radius of the wettedarea; θ/θm — dynamic contact angle; t/tmax — dimensionless time.

1.0

0.8

0.6

0.4

0.2

0.0

1.0 0.8 0.6 0.4 0.2 0.0 0.0

1

2

3 0.0

0.2

0.4

0.6

0.8

1.0 θ/θ m

Nitrocellulose membranes, a = 3.0 μm. SDS = 0.1%

t/tmax

L/L m

ax

�/� m

ax

tmax = 2.96 sec V0 = 2.1 mm3

tmax = 4.92 sec V0 = 3.7 mm3

tmax = 13.12 sec V0 = 9.7 mm3



Advancing and Hydrodynamic Receding Contact Angles on Porous Nitrocellulose Membranes

Using presented experimental data on the spreading of drops of aqueous SDSsolutions over dry porous substrates, the values of advancing, θa, and hydrody-namic receding, θrh, contact angles were extracted as a function of the SDSconcentration. We are using in the following text the term hydrodynamic recedingcontact angle and the symbol θrh to distinguish it from the static receding contactangle, which is found equal to zero (see the following text on static hysteresis ofthe contact angle of SDS solution drops on smooth nonporous nitrocellulosesubstrate). The advancing contact angle, θa, was defined at the end of the firststage when the drop stopped spreading (the radius of the drop base reached itsmaximum value). The hydrodynamic receding contact angle, θrh, was defined atthe moment when the drop base started to shrink.

FIGURE 5.3 Spreading of droplets of SDS solutions over nitrocellulose membrane, a =0.22 µm; L/Lmax — dimensionless radius of the drop base; l/lmax — radius of the wettedarea; θ — dynamic contact angle; t/tmax — dimensionless time.

1.0

0.8

0.6

0.4

0.2

0.0

1.0 0.8 0.6 0.4 0.2 0.0 0

30

60

θ

90 0.0

0.2

0.4

0.6

0.8

1.0

t/tmax

L/L m

ax

�/� m

ax

Nitratecellulose membranes, a = 0.22 μmSDS = 0.0%; V0 = 2.9 mm3; tmax = 20.72 sec SDS = 0.1%; V0 = 10.5 mm3; tmax = 75.2 sec SDS = 0.2%; V0 = 7.4 mm3; tmax = 33.84 sec SDS = 0.5%; V0 = 2.6 mm3; tmax = 8.2 sec



In Figure 5.6, experimental data on the apparent contact angle hysteresis aresummarized. This figure shows that the advancing contact angle, θa, decreaseswith SDS concentration; the hydrodynamic receding contact angle, θrh, on thecontrary, slightly increases with SDS concentration.

These experimental runs show that (1) the difference between advancing andreceding contact angles becomes smaller with the increase in the SDS concen-tration (Figure 5.6), and (2) the dimensionless time interval when the drop basedoes not move also decreases with the increase in the SDS concentration.


1.0

0.8

0.6

0.4

0.2

0.0 90

60

θ

30

0 0.0 0.2 0.4 0.6

t/tmax

0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

�/� m

ax

L/L m

ax

Nitratecellulose membranes, a = 0.45 μmSDS = 0.0%; V0 = 6.1 mm3; tmax = 8.44 s SDS = 0.1%; V0 = 9.5 mm3; tmax = 10.28 s SDS = 0.2%; V0 = 7.2 mm3; tmax = 8.44 s SDS = 0.5%; V0 = 7.1 mm3; tmax = 6.84 s



Static Hysteresis of the Contact Angle of SDS Solution Drops on Smooth Nonporous Nitrocellulose Substrate

In the previous parts of this section the spreading of drops at different SDSconcentrations on nitrocellulose membranes of various pore sizes was considered.In all cases during the third stage of spreading, the radius of the drop base, L,shrank, and the hydrodynamic receding contact angle, θrh, remained constant.The duration of the third stage of spreading increases with the SDS concentrationincrease. It is necessary to note that the behavior of drops of aqueous SDSsolutions during the third stage of spreading (partial wetting) is remarkably similarto the behavior during the second stage of spreading in the case of completewetting (Chapter 4, Section 4.2).


1.0

0.8

0.6

0.4

0.2

0.0

1.0 0.8 0.6 0.4 0.2 0.0 0

30

60

90 0.0

0.2

0.4

0.6

0.8

1.0 θ

t/tmax

L/L m

ax

�/� m

ax

Nitratecellulose membranes, a = 3.0 μmSDS = 0.0%; V0 = 5.9 mm3; tmax = 108.04 s SDS = 0.1%; V0 = 9.7 mm3; tmax = 13.12 s SDS = 0.2%; V0 = 6.8 mm3; tmax = 3.96 s SDS = 0.5%; V0 = 5.1 mm3; tmax = 2.12 s



It was found previously in this section that the advancing and hydrodynamicreceding contact angles are strongly dependent on the SDS concentration onporous nitrocellulose membranes.

Next, results are presented on measurements of static advancing and staticreceding contact angles on smooth nonporous nitrocellulose substrate for differentSDS concentrations. The idea is to compare hysteresis of contact angles on thesmooth nonporous nitrocellulose substrate with the hysteresis contact anglesobtained earlier on porous nitrocellulose substrates at corresponding SDSconcentrations.

Static advancing or receding contact angle values were obtained using theexperimental procedure described in Chapter 4, Section 4.5 (see Figure 4.19).The droplet was pumped using the syringe, and dynamics of the droplet spreadingwere monitored. The final value of the contact angle under this experimentalcondition was equal to the static value of the advancing contact angle.

The static advancing contact angle of pure water on nonporous nitrocellulosesubstrate was found approximately to be equal to 70˚. The static advancing contactangle decreases with the increase of SDS concentration (Figure 5.7). This trendcontinues until the CMC is reached. At concentrations above the CMC, theadvancing contact angle remains constant and approximately equals 35˚.

The receding contact angle values were obtained using the same experimentalsetup (Figure 4.19). In this case the contact angle dynamics were investigatedusing linearly decreasing droplet volume.

The nonzero value of the static receding contact angle was found only in thecase of pure water droplets. In all other cases (even at the smallest SDS concen-trations used 0.025%), the static receding contact angle was found equal to zero

FIGURE 5.6 Porous nitrocellulose substrates. Apparent contact angle hysteresis variationwith SDS concentration. Nitrocellulose membranes of different average pore sizes. Opensymbols correspond to the advancing contact angle, θa. The same filled symbols correspondto the hydrodynamic receding contact angle, θrh.

SDS concentration, %

Nitrocellulose membranes

30

20

10

00.0 0.1 0.2 0.3

a = 0.22 µma = 0.45 µma = 3.0 µm

0.4 0.5

40

50

60

θ a, θ

rh (d

egre

e)



in all the concentration ranges used: from 0.025% (ten times smaller than CMC)to 1% (five times higher than CMC).

Both static receding and static advancing contact angles on smooth nonporousnitrocellulose substrate against SDS concentration are presented in Figure 5.7.

The results highlight a linear decline of the static advancing contact anglefrom 70o at 0% SDS (pure water) to approximately 35˚ at the value of the CMC(2.4%); after that, the static advancing contact angle reaches a steady value thatremains constant irrespective of further increase in the SDS concentration. Incontrast to this, the static receding contact angle is approximately equal to 45˚for the pure water and is equal to zero in the presence of SDS even at concen-trations as low as 0.025%.

Comparison of Figure 5.6 and Figure 5.7 shows:

• The advancing contact angle dependence on SDS concentration onporous nitrocellulose substrates is significantly different from the staticadvancing contact angle dependence on nonporous nitrocellulose sub-strates. This means that, in the case of porous substrates, the influenceof both the hydrodynamic flow caused by the imbibition into the poroussubstrate and the substrate roughness change significantly the advanc-ing contact angle.

• The hydrodynamic receding contact angle in the case of the poroussubstrates has nothing to do with the hysteresis of the contact angleand is completely determined by the hydrodynamic interactions in away similar to the complete wetting case in Section 4.3.

FIGURE 5.7 Nonporous nitrocellulose substrate. Advancing and receding contact anglesvariation with SDS concentration. Open symbols correspond to the static advancing contactangle, θa. Filled symbols correspond to the static receding contact angle, θr.

70

60

50

40

30

20

10

00.0 0.2 0.4

SDS concentration, %0.6 0.8 1.0

Cont

act a

ngle,

deg

ree

θa Advancing contact angleθr Receding contact angle



Conclusions

Experimental investigation were carried out on the spreading of small drops ofaqueous SDS solutions (capillary regime of spreading) over dry nitrocellulosemembranes (permeable in both normal and tangential directions) in the case ofpartial wetting. Nitrocellulose membranes were chosen because of their partialhydrophilicity. The time evolution was monitored for the radii of both the dropbase and the wetted area inside the porous substrate.

The total duration of the spreading process was subdivided into three stages:

• The first stage, when the drop base expands until the maximum valueof the drop base is reached; the contact angle rapidly decreases duringthis stage.

• The second stage, when the radius of the drop base remains constantand the contact angle decreases linearly with time.

• The third stage, when the drop base shrinks and the contact angleremains constant.

The wetted area inside the porous substrate expands during the whole spread-ing process.

Appropriate scales were used with a plot of the dimensionless radii of thedrop base, the wetted area inside the porous substrate, and the contact angle onthe dimensionless time.

Experimental data presented in this section show:

• The overall time of the spreading of drops of SDS solution over dry, thin,porous substrates decreases with the increase of surfactant concentration.

• The difference between advancing and hydrodynamic receding contactangles decreases with the surfactant concentration increases.

• The constancy of the contact angle during the third stage of spreadinghas nothing to do with the hysteresis of contact angle but is determinedby hydrodynamic reasons, and the spreading behavior becomes similarto the case of the complete wetting.

5.2 SPONTANEOUS CAPILLARY IMBIBITION OF SURFACTANT SOLUTIONS INTO HYDROPHOBIC CAPILLARIES

In this section, a theory is developed to describe a spontaneous imbibition ofsurfactant solutions into hydrophobic capillaries, which takes into account themicelle disintegration and solution concentration reduction close to the movingmeniscus as a result of adsorption, as well as the surface diffusion of surfactantmolecules. The theory predictions are in good agreement with the experimentalinvestigations on the spontaneous imbibition of the nonionic aqueous surfactant



solution, Syntamide-5, into hydrophobized quartz capillaries [2,3]. Also, a theoryof the spontaneous capillary rise of surfactant solutions in hydrophobic capillariesis presented, which connects the experimental observations with an adsorptionof surfactant molecules in front of the moving meniscus on the bare hydrophobicinterface [4].

Pure water does not penetrate spontaneously into hydrophobized quartz cap-illaries; however, surfactant solutions penetrate spontaneously, and the penetrationrate depends on the concentration of surfactant. Both the air–liquid interfacialtension, γ, and the contact angle of the moving meniscus, θa, are concentrationdependant, where a subscript a indicates the advancing contact angle.

It is obvious that adsorption of surfactant molecules behind the movingmeniscus results in a decrease of the bulk surfactant concentration from the capillaryinlet in the direction of the moving meniscus. However, as we show in this section,the major process, which determines penetration of surfactant solutions into hydro-phobic capillaries or spreading of surfactant solutions over hydrophobic sub-strates, is the adsorption of surfactant molecules onto a bare hydrophobic substratein front of the moving three-phase contact line. This process results in a partialhydrophilization of the hydrophobic surface in front of the meniscus or drop,which, in its turn, determines spontaneous imbibition or spreading.

It is easy to understand why the adsorption in front of the moving meniscuson a hydrophobic substrate is so vital in the case of the hydrophobic substrate.Let us consider the very beginning of the imbibition process, when a meniscusof a surfactant solution touches, for the first time, an inlet of the hydrophobiccapillary. The contact angle, θa, at this moment, is bigger than π/2, and the liquidcannot penetrate into the hydrophobic capillary. Solid–liquid and liquid–air inter-facial tensions, γsl and γ, respectively, do not vary with time on the initial stagebecause the adsorption of surfactant molecules onto these surfaces is a fast processas compared with the rate of imbibition. The only interfacial tension that canvary is the solid–air interfacial tension, γsv. If the adsorption on the solid–airinterface does not occur, then the spontaneous imbibition into a hydrophobiccapillary cannot take place spontaneously because the advancing contact angleremains above π/2. However, if the adsorption of surfactant molecules on thebare hydrophobic surface in a vicinity of the three-phase contact line takes place,then the solid–air interfacial tension, γsv, grows with time. After some criticalsurface adsorption, Γsvcr , is reached, the advancing contact angle reaches π/2.Only after that can the spontaneous imbibition process start. This considerationshows that there is a critical bulk concentration, C*, below which Γsv remainsbelow its critical value, Γsvcr, and the spontaneous imbibition process does nottake place.

Let us consider expression (1.2) from Section 1.1 in Chapter 1 for the excessfree energy, Φ, of the droplet on a solid substrate:




where S is the area of the liquid–air interface; P = Pa – Pl is the excess pressureinside the liquid, Pa is the pressure in the ambient air, and Pl is the pressure insidethe liquid; the last term on the right-hand side gives the difference between theenergy of the part of the bare surface covered by the liquid drop as comparedwith the energy of the same solid surface without the droplet.

This expression shows that the excess free energy decreases if (1) the liquid–vapor interfacial tension, γ, decreases, (2) if the liquid–solid interfacial tension,γsl, decreases, and (3) the solid–vapor interfacial tension, γsv, increases.

Let us assume that, in the absence of the surfactant, the drop forms anequilibrium contact angle above π/2. If the water contains surfactants, then threetransfer processes take place from the liquid onto all three interfaces: surfactantadsorption at both (1) the inner liquid–solid interface, which results in a decreaseof the interfacial tension, γsl, (2) the liquid–vapor interface, which results in adecrease of the interfacial tension, γ, and (3) transfer from the drop onto thesolid–vapor interface just in front of the drop. As we already noticed in theforegoing text, all three processes result in decrease of the excess free energy ofthe system. However, adsorption processes (1) and (2) result in a decrease ofcorresponding interfacial tensions, γsv and γ ; but the transfer of surfactant mole-cules onto the solid–vapor interface in front of the drop results in an increase oflocal free energy; however, the total free energy of the system decreases. Thatis, surfactant molecule transfer (3) goes via a relatively high potential barrier and,hence, goes considerably slower than adsorption processes (1) and (2). Hence,they are fast processes as compared with the third process (3).

In the case of partial wetting, the capillary imbibition in the horizontaldirection proceeds according to the following dependency:

(5.7)

where l is the length of the part of the capillary filled with the liquid, R is theradius of the capillary, η is the liquid viscosity, the advancing contact angle, θa,is below π/2: 0 < θa < π/2, and t is time.

Pure water does not penetrate spontaneously into hydrophobic capillaries,and shows the advancing contact angle, θa > π/2. This means that the liquid canonly be forced into the capillary. However, the advancing contact angle is adecreasing function of the surfactant concentration, and at some critical concen-tration, C*, is equal to π/2. This means that above C*, the surfactant solutionpenetrates spontaneously into hydrophobic capillaries.

In the case of the imbibition of surfactant solutions into hydrophobic capil-laries, the penetration is controlled by both the surfactant molecules transfer andthe liquid viscosity, according to Equation 5.7. In this case the aim is to revealthe mechanism of the penetration and to determine the concentration of surfactantmolecules near the meniscus Cm < C0, where C0 is the surfactant concentration

� = Rtaγ θ

ηcos

,2



at the capillary inlet. Figure 5.8 shows that, in the case of Syntamid-5, the advanc-ing contact angle, θa, exceeds 90o at concentration C*, which is slightly under 0.05%.

In the following text we show that both the bulk diffusion and the surfacediffusion of surfactant molecules (including that on the unwetted portion of thecapillary in front of the moving meniscus) play an important role, and a theoryis presented for this case.

THEORY

As we already mentioned in the introduction to this chapter, we know surprisinglylittle about the behavior of surfactant solutions in the vicinity of the three-phasecontact line. That is why in this chapter we are using Young’s equation fortheoretical treatment. We showed in Chapter 2 that the equation does not have afirm theoretical basis.

Let us consider a dependency of Ψ(Cm) = γ(Cm) cosθa(Cm) on the concentra-tion of surfactant, Cm, on the moving meniscus. According to the triangle rule, itcan be calculated as

(5.8)

where γsv and γsl are solid–vapor and solid–liquid interfacial tensions. Accordingto Antonov’s rule, the dependency of two interfacial tensions on the concentrationcan be presented as

(5.9)

FIGURE 5.8 The influence of concentration of aqueous surfactant solutions (Syntamide-5,molecular weight 420) on the advancing contact angle, θa (curve 1) and γ cos θa (curve 2)measured on a flat hydrophobized quartz surface. C* marks the critical surfactant concen-tration, below which the surfactant solution does not spread. Broken line (3) according toEquation 5.12.

0 0.1 0.2C, %

0

1

2

3

γ co

s θa

C∗

Adva

ncin

g con

tact

angl

e, θ a

degr

ee

90

891

3 2

88

87

Ψ( ) ( ) ( ),C C Cm sv m sl m= −γ γ

γ γ γ γ γsv m sv sv sl mC C( ) ; ( )= −

+ =+∞

∞ +∞

0 1ΓΓ

ΓΓ ssl sl

0 1−

+−∞

∞ −∞

ΓΓ

ΓΓ

γ ,



where superscripts 0 and ∞ mark zero and the complete coverage of hydrophobicadsorption sites, respectively; subscripts + and – mark adsorption just behind andjust in front of the moving meniscus, respectively; and Γ is the surface adsorptionof surfactant molecules.

Note that adsorption of surfactant molecules results in a decreasing ofsolid–liquid interfacial tension, that is, However, adsorption of sur-factant molecules on the bare hydrophobic interface in front of the movingmeniscus results in a local increase of the solid–vapor interfacial tension, that is,

The initial contact angle on the bare hydrophobic interface isassumed to be bigger than π/2, that is,

It is assumed in the following text that both adsorption isotherms are linearfunctions of the surfactant concentration below CMC (which is the only caseconsidered below) and remain constant above CMC. This means that

(5.10)

at concentrations below CMC.Both a spontaneous imbibition and a spontaneous capillary rise into hydro-

phobic capillaries are sufficiently slow processes, that is, we assume in thefollowing text a condition of local equilibrium on the moving three-phase contactline. According to this assumption, the equality of chemical potentials of adsorbedsurfactant molecules should be satisfied across the three-phase contact line, thatis, where Γ- and Γ+ are jumps of adsorption across themeniscus surface (Figure 5.9), and ΦSL, ΦSV (in kT units) are corresponding valuesof the energy of surfactant molecules at solid–water and solid–air interfaces,respectively. From the latter equation we conclude that

It is obvious that Φsv is higher than Φsl, and hence, Γ+ < Γ–. The relation can berewritten using Equation 5.10 ast

(5.11)

Substitution of Equation 5.9 through Equation 5.11 into Equation 5.8, andhaving in mind the latter inequalities, we can conclude, after rearrangements, that

(5.12)

γ γsl sl0 0− >∞ .

γ γsv sv0 0− <∞ .

γ γsv sl0 0 0− < .

Γ− = G Csl m ,

ln ln ,Γ Φ Γ Φ− ++ = +sl sv

Γ ΓΦ Φ+

−=−( )exp

.sv sl

ΓΦ Φ+ = =

−( )G C GG

sv m svsl

sv sl

,exp

.

Ψ( ) ,*C C Cm m= −( )α



where

According to Equation 5.12 Ψ(Cm), dependency should be a linear functionof concentration at Cm > C*, which is in good agreement with experimentalobservations (line 3 in Figure 5.8 at concentrations C > C*) in a range of surfactantconcentration under consideration in this section.

We now try to solve theoretically the problem of a spontaneous imbibitionof surfactant solutions into hydrophobic cylindrical capillaries, taking intoaccount the transfer and the surface diffusion of surfactant molecules as well asadsorption on the bare hydrophobic surface in front of the moving meniscus. Thelocation of the moving meniscus in the capillary is l(t) (Figure 5.9).

The transfer of surfactant molecules in the filled portion of the capillary isdescribed by the convective diffusion equation

where C(t,x,r) is the local concentration of surfactant; D is the diffusion coeffi-cient; t, x, and r are time, axial, and radial coordinates, respectively; and v(r) isthe axial velocity distribution.

Integration of this equation over radius from 0 to R, where R is the capillaryradius, results in

FIGURE 5.9 Distribution of surfactant concentrations along the capillary length duringa spontaneous imbibition process: volume concentration, C, and surface concentration,a = 2Γ/R.

�

�(t)x

a+ a–

a(x)

c(x) cm

c0

a0

2�

0

α γ γ γ γ= −( ) + −( )

>∞

∞∞

∞G Gsvsv sv

slsl slΓ Γ

0 0 0,, , .*C sl sv= = − >αα α γ γ

00 0 0 0

∂∂

= ∂∂

+ ∂∂

∂C t x rt

DC t x r

xD

r rr

C t x r( , , ) ( , . ) ( , , )2

2

1∂∂

− ∂∂ ( )

r xv r C t x r( ) ( , , ,



The second term on the right-hand side of the equation is equal to

,

where Dsl is the surface diffusion coefficient over the filled portion of the capillary.Hence, the two equations result in

A characteristic time scale of the equilibration of the surfactant concentrationin a cross section of the capillary, τ ~ R2/D ≈ 0.1 sec, if we use for estimationsR ~ 10 µm and D ~ 10–5 cm2/sec. A characteristic time scale of the spontaneouscapillary imbibition or rise into hydrophobic capillaries is much bigger than0.1 sec (see Figure 5.10 and Figure 5.11). This means that the surfactant concen-tration is constant in any cross section of the capillary and depends only on theposition, x (Figure 5.9), that is, C = C(t, x). We also assume that the adsorptionequilibrium in any cross section is also reached. Taking this into account, theequation can be rewritten after both sides are divided by R2/2 as:

, 0 < x < l(t), (5.13)

where D and Dsl are diffusion coefficients of surfactant molecules in the volumeand over the wetted capillary surface; v = dl/dt is the meniscus velocity; and

(5.14)

where Fsl = (2/R) Gsl .

∂∂

= ∂∂∫t

rC t x r dr Dx

rC t x r dr

R

( , , ) ( , , )0

2

2

00

R

r R

R DC t x r

r

xr

∫

+ ∂∂

− ∂∂

=

( , , )

vv r C t x r dr

R

( ) ( , , )0∫

− ∂∂

= ∂∂

− ∂∂=

DC t x r

r tD

xr R

sl

( , , ) Γ Γ2

2

∂∂

= ∂∂∫t

rC t x r dr Dx

rC t x r dr

R

( , , ) ( , , )0

2

2

00

2

2

R

slRt

Dx

xr

∫

− ∂∂

− ∂∂

− ∂∂

Γ Γ

vv r C t x r dr

R

( ) ( , , )0∫

∂ +∂

= ∂∂

+ ∂∂

− ∂∂

( )a Ct

DC

xD

sla

xv

Cx

2

2

2

2

a x tR

G C x t F C x tsl sl( , ) ( , ) ( , ),= =2



Concentration below CMC

Let C(x, t) and a(x, t) be the local surfactant concentrations in the bulk solutionand in the adsorbed state on the capillary surface. A constant surfactant concen-tration C(0, t) = C0 < CMC is kept at the capillary inlet. In this case the surfactanttransport in the filled portion of the capillary obeys Equation 5.13.

FIGURE 5.10 The time evolution of the imbibition length l (mm) with time, t (min) foraqueous solutions of Syntamide-5 in a horizontal hydrophobized quartz capillary, R = 16µm. (1) C0 = 0.05%; (2) C0 = 0.1%; (3) C0 = 0.4%; (4) C0 = 0.5%; (5) C0 = 1%.

FIGURE 5.11 Spontaneous capillary rise in a vertical hydrophobized quartz capillary(R = 11 µm), Syntamid-5 surfactant solution (C0 = 0.1 %) [3]. Time evolution of theimbibition length, l (mm), on time, t(min).

10987654321

0 1 2 3

12

1

2

3

4

55

4

3

�, cm

�, m

m

t min1/2

0 15 30 45 60 75 90t, min1/2

25

20

15

10

5

�, m

m



Substitution of expression (5.14) into Equation 5.13 results in

0 < x < l(t), (5.15)

where Def = (D + Fsl Dsl) is the effective diffusion coefficient. On the nonwettedportion of the capillary surface, l(t) < x, only surface diffusion takes place, that is:

(5.16)

where Dsv is the diffusion coefficient of surfactant molecules on the nonwettedhydrophobic capillary surface.

Equation 5.15 and Equation 5.16 are solved in the following text using thefollowing boundary and initial conditions:

C(0,t) = C0; a(∞,t) = 0, (5.17)

l(0) = 0; a(x, 0) = 0. (5.18)

The following condition of the mass balance on the moving meniscus surfaceshould be satisfied:

(5.19)

where l– and l+ represent two points located on the opposite sides of the meniscus:on the liquid phase side, l–, and on the unwetted side in front of the movingmeniscus, l+ (Figure 5.9). Condition (5.19) expresses the conservation of massat the moving meniscus and the moving three-phase contact line.

As we already mentioned, the imbibition process is the slow one, and theduration is shown in Figure 5.10 and Figure 5.11. It allows us to assume in thefollowing text a condition of local equilibrium on the moving three-phase contactline. Hence, we conclude in the same way as in Equation 5.11:

.

The energy of the surfactant molecule on a bare hydrophobic substrate in frontof the moving meniscus, Φ+, is higher than the corresponding energy in theaqueous solution, Φ–, and, hence, a+ < a–.

( ) ,12

2+ ∂

∂= ∂

∂− ∂

∂F

Ct

DC

x

dldt

Cxsl ef

∂∂

= ∂∂

at

Da

xsv

2

2,

Dax

DCx

Dax

a asv

l

sv

l

∂∂

− ∂∂

+ ∂∂

= −+ −

−( ++ ) ,dldt

aa

+−

+ −

=−( )exp Φ Φ



Equation 5.15 and Equation 5.16 with boundary (5.19) and initial conditions(5.17) to (5.18) have a solution only if the concentration on the moving meniscusremains constant; that is, Cm = const. This corresponds to the experimentallyobserved law of the spontaneous imbibition, l = K , where the constant K is tobe determined.

The solution of Equation 5.15 and Equation 5.16 is tried in the following textin the following form C = C(ξ), a = a(ξ), where

.

Transformations of Equation 5.15 and Equation 5.16 yield:

(5.20)

(5.21)

Solution of these equations has the following form:

(5.22)

(5.23)

Integration constants A1, A2, B1, and B2 should be determined using initialconditions (5.19). This results in

(5.24)

(5.25)

t

ξ = x

t

− + = −ξξ ξ ξ2

12

2

2

dCd

F Dd C

d

K dCdsl ef( ) ,

− =ξξ ξ2

2

2

dad

Dd a

dsv .

C AFD

KD

dsl

ef ef

( ) exp( )ξ ξ ξ

ξ

= − + +

∫1

2

0

14 2

ξξ + B1.

a AD

d Bsv

K

( ) exp .ξ ξ ξξ

= −

+∫2

2

24

AC C

FD

KD

m

sl

ef ef

K10

2

0

14 2

= − −

− + +

exp( )ξ ξ∫∫

=

d

B C

ξ

; ,1 0

AF C

Dd

B F Csv m

svK

sv m22

2

4

= −

−

=∞

∫ exp

; .ξ ξ



Substitution into boundary condition Equation 5.19 expressions from Equa-tion 5.24 and Equation 5.25 yields the following transcendental equation:

(5.26)

K value governs the imbibition rate according to Equation 5.7 and can be rewrittenusing Equation 5.12 as

. (5.27)

Substitution of this equation into Equation 5.26 gives the required equationfor the determining of the unknown concentration on the moving meniscus, Cm.

Equation 5.26 and Equation 5.27 give the solution of the problem underconsideration. To solve Equation 5.26, the following coefficients should beknown: diffusion coefficients, D, Dsl, and Dsv; adsorption constants, Gsl and Gsv;capillary radius, R; solution viscosity, η; concentration at the capillary inlet, C0;and the coefficient α in Equation 5.12 or Equation 5.27.

Numerical analysis of the final equation for K shows that its values increasesas R, C0, D, and Dsl increase, or Dsv decreases. This is due to the fact that thehigh surface mobility of the surfactant on the unwetted portion of the capillaryreduces the concentration on the meniscus, Cm, thereby inhibiting the imbibition.

Concentration above CMC

In the case of the surfactant concentration at the capillary inlet, C0, above CMC,diffusion and adsorption are accompanied by the destruction of the micelles. Ifthe surfactant concentration at the capillary inlet is higher than CMC, then thetotal surfactant concentration, C, can be presented as C = Cmol + CM, where Cmol

and CM are concentrations of free surfactant molecules and molecules insidemicelles, respectively. Concentration of free molecules, Cmol, remains approxi-mately constant above CMC and equal to CMC (Cc below). This means that (1)decrease in the surfactant concentration goes through disintegration of micelles,

KF F

CC

DF KD

sl svm

efsl

ef

2

11

40

2

( )

exp( )

− =−

−

− + +

exp( )1

4 2

2

0

FD

KD

sl

ef ef

Kξ ξ∫∫

−

−−

−

d

F DKD

D

sv svsv

sv

ξ

ξ

exp

exp

2

2

4

4

∞

∫K

dξ

.

Kr C Cm= −α

η( )*

2



whereas the concentration of the free surfactant molecules remain constant, and(2) the wetting depends on the concentration of the free molecules, and hence,remains independent of the concentration unless the concentration is below CMC:γ(Cc)cosθ(Cc) = ϕc = const. This means that the K value, which determines theimbibition rate, is constant and is equal to

(5.28)

until the concentration at the meniscus, Cm, is higher than CMC.The values of Kc for C0 > CMC~1% obtained in experiments with Syntamide-

5 [2,3] were of the order of 10–1, which corresponds to the contact angle (a =89˚, that is, only slightly different from 90˚).

Unlike the case when the concentration at the capillary inlet is below CMC,the meniscus moves with velocity l = Kc , where Kc is given by Equation 5.28and does not vary with time. In this case, the surfactant adsorption on the capillarysurface is accompanied by a continuous decrease of concentration, Cm, near themeniscus from Cm = C0 > CMC at the beginning of the imbibition process at t =0 to Cm = CMC or Cm = Cc at the end of this first fast stage.

After Cm reaches CMC, the condition Kc = const is no longer valid. The Kvalue decreases below Kc, and the imbibition rate slows down. After Cm = CMCis reached on the meniscus, a further reduction of concentration, Cm, causesseparation of the micelle front (where C = CMC) from the meniscus surface. Themicelle front movement is governed, as shown in the following text, by the samelaw lM = KM , where KM < K. The further stage of impregnation occurringwhen Cm < Cc is described in a similar way as in the previous section: theconcentration no longer varies but remains const and smaller than CMC.

The variations of the impregnation process associated with the above-mentioned phenomenon is illustrated in Figure 5.10. A sharp change in the rateof impregnation at some distance l = lc is indeed observed for Syntamide-5solutions at C0 > CMC.

Values of lc, which correspond to the first fast stage of the imbibition process,are estimated in the following text. As mentioned previously, the first fast stageof the imbibition is determined by the dissociation of micelles close to the movingmeniscus. Let us assume an adsorption of micelles on the meniscus according toReference 8, which is adopted according to the linear law Γ = GM (Cm – Cc),where Cm – Cc is the micelles concentration at C0 > Cc, and GM is the correspond-ing adsorption constant. Diffusion of micelles is neglected in the following dis-cussion because of the short duration of the first stage. The mass balance on themoving meniscus during the first fast stage of the imbibition is

(5.29)

K r C C constc c= −

=α η

*,2

t

t

( )( )

, ( )F F Cdldt

Gd C C

dtC C C Csl sv c M

m cm c− = − − − = −0 0 cc ,



where (Fsl – Fsv)Cc = const is the limiting adsorption on solid surface (micellesdo not adsorb); the difference Cm – Cc is equal to the micelle concentration. Theleft-hand side of Equation 5.29 characterizes the surfactant adsorption rate on thenewly wetted surface, and the right-hand side the micelle disintegration rate fromthe adsorbed layer of micelles.

The solution of Equation 5.29 under the initial condition t = 0, Cm – Cc =C0 – Cc has the following form:

(5.30)

The left-hand side of the equation vanishes as all the micelles near themeniscus disintegrate, i.e., the concentration Cm reduces to CMC. Equation 5.30allows determining the instant t = tc when Cm = Cc, which is the end of the firstfast stage of the imbibition. Using this condition and Equation 5.30, the lengthof the fast imbibition can be determined as

(5.31)

For R ~ 2*103 cm, C0 = 2%, Cc = 0.1%, lc = 0.5 cm (in agreement withexperimental observations), we get

~ 25

in an agreement with Reference 8.The first fast stage of the imbibition is followed by the second slower stage.

Now, concentration on the moving meniscus, Cm, is below CMC. Two regionscan now be identified inside the capillary: the first region, from the capillary inletto some position that we mark as lM(t), where concentration is above CMC andthe solution includes both micelles and individual surfactant molecules; the sec-ond region, from lM(t) to l(t), where concentration is below CMC and onlyindividual molecules are transferred. Concentration is equal to CMC at x = lM (t).Consideration in the second region, lM (t) < x < l(t), is similar to that at concen-tration below CMC. That is why only the transport in the first region is consideredin the following text.

Inside the first region, 0 < x < lM(t), concentration of free surfactant moleculesis constant and equal to CMC [9]. Hence, the transfer is determined by thediffusion of micelles and convection of all molecules. As mentioned in thepreceding text, total concentration, and Cmol remain constant andequal to CMC; hence,

C t C C CC G G K t

R Gm c cc sl sv c

M

( ) .− = −( ) −−( )

0

2

l K tCC

rG

G Gc c c

c

M

sl sv

= = −

−( )

0 12

.

G

G GM

sl sv−( )

C C Cmol M= + ,



(5.32)

where DM is the diffusion coefficient of micelles, and C = Cc +CM is the totalconcentration. Adsorption on membrane pores is determined by the concentrationof the free molecules, which is constant in the first region and so is the adsorption.It is the reason why the diffusion of adsorbed molecule in the first region isomitted in Equation 5.32. Transfer of surfactant molecules in the second region(micelles-free region) is described by Equation 5.15.

Boundary conditions on the moving boundary between the first and secondregions, lM(t), are as follows:

(5.33)

As before, we assume that

(5.34)

where K is given by Equation 5.27, that is, expressed via unknown concentrationon the moving meniscus, Cm, and KM is a new unknown constant.

Let a similarity variable be introduced now in the same way as in the caseof concentration below CMC, that is, ξ = x/ in Equation 5.32, Equation 5.15,and Equation 5.16. Using boundary conditions (5.33) and (5.19), the followingsystem of two nonlinear algebraic equations can be deduced:

(5.35)

.

Two unknown values in this system of equations are concentrations on themoving meniscus, Cm, and unknown constant, KM. It is necessary to rememberthat K is expressed via Cm according to Equation 5.27.

∂∂

= ∂∂

− ∂∂

Ct

DC

x

dldt

CxM

2

2,

DCx

DCx

C l tM

x lM

ef

x lM

M

∂∂

= ∂∂

= − = +

, ( , )) .= Cc

l t K t l t K tM M( ) , ( ) ,= =

t

KF F

CC

DF KD

sl sv

c

mef

sl

ef

2

11

4

2

( )

exp( )

− =−

−

− + +

exp( )1

4 2

2FD

KD

sl

ef efKM

ξ ξKK

sv svsv

sv

d

F DKD

D∫−

−

−

ξ ξ

exp

exp

2

2

4

4

∞

∫K

dξ

.

D

C CK

D

DK

M

cM

M

MM

( ) exp

exp

0

2

2

4

12 2

−

− −

ξ ξ

=− −

∫0

214

K ef

c msl

e

M

d

D

C CF K

D

ξ

( ) exp( )

ff

sl

ef efK

F

D

K

Dd

− + +

exp( )1

4 2

2ξ ξ ξMM

K

∫



It is possible to show, using Equation 5.35, that KM < K; that is, the borderbetween the first and the second regions moves slower than the meniscus. Thediffusion coefficient of micelles is much smaller than that of individual molecules.Thus, actually, the system of Equation 5.35 includes a small parameter Dm /Def <<1, which means that the second equation can be omitted in the zero approximation,and the constant, K, is only slightly different from the same value in the case ofconcentration below CMC. That is, during the second stage, the rate of sponta-neous imbibition is only slightly higher than in the case when concentration isbelow CMC. Experimental data in Figure 5.10 confirm this conclusion.

Now we can conclude that the two-stage process of the imbibition (as pre-sented in Figure 5.10) takes place only if the concentration on the capillary inletis above CMC. It is necessary to emphasize that the duration of the fast stage isthe shorter the thinner the capillary is. K values during the second stage of theimbibition are only slightly higher than in the case where concentration is belowCMC.

Spontaneous Capillary Rise in Hydrophobic Capillaries

A further confirmation of the adsorption of surfactant molecules in front of themoving meniscus on the bare hydrophobic substrate is a phenomenon of thespontaneous capillary rise of surfactant solutions in hydrophobic capillaries.

Let us consider a spontaneous capillary rise of surfactant solutions in hydro-phobic capillaries.

Figure 5.11 shows the results of one of the capillary rise experiments withSyntamide-5 solution in a vertical hydrophobized quartz capillary [10]. Theobserved time evolution of the imbibition length, l(t), follows l(t) = K depen-dency at the initial stage of the process. The value of K determined from theslopes of the l/ dependencies correspond to the advancing contact angle value,θa, being only a few seconds less than 90˚. At such θa values the capillary risewould be expected to stop as soon as the liquid reached a height of lmax = 10–3 cm.However, it does not stop at this height but goes up to a height of 3–4 cm withalmost constant K. The only explanation of this phenomenon is that the meniscusrises in the capillary following the surface diffusion front of surfactant molecules,which hydrophilizes the bare hydrophobic capillary surface in front of the movingmeniscus. At each position, l, the meniscus curvature must satisfy the followingequilibrium condition:

(5.36)

where Ψ(Cm) = γ(Cm) cosθa(Cm), ρ is the density of the solution, and g is thegravity acceleration. As θa is very close to π/2 according to Equation 5.1, we canuse a linear dependency (5.12):

t

t

2Ψ( )( ),

CR

gl tm = ρ

Ψ( ) ( ).*C C Cm m= −α



Using Equation 5.12 we can rewrite Equation 5.36 as

(5.37)

During the initial stage of the capillary rise, l(t) = K (see Figure 5.11).This is possible only if

Cm = C* + B , and (5.38)

where the constant K is to be determined.Equation 5.38 shows that the case under consideration is governed by a

completely different mechanism as compared with the case of the horizontalimbibition (where Cm remains constant over time). In the case of the spontaneouscapillary rise in hydrophobic capillaries, Cm(t) does not remain constant but mustincrease as the capillary rise progresses. The comparison of Figure 5.11 andFigure 5.10 shows that the time scale of the spontaneous capillary rise is around100 times bigger than the corresponding time scale in the case of the capillaryimbibition into horizontal capillaries.

The maximum height of the capillary rise, lmax, is reached after the concen-tration on the meniscus, Cm, becomes equal to the concentration at the capillaryentrance, C0. After that the capillary rise stops. Using Equation 5.37, lmax isdetermined as

(5.39)

Thus, the experimental observation presented in Figure 5.11 corresponds tol(t) << lmax, that is, the initial stage of the capillary rise.

In the following text the problem of the spontaneous capillary rise of surfac-tant solutions in hydrophobic capillaries is considered in the case when concen-tration at the capillary inlet is below CMC. In this case, the transport of surfactantmolecules is described by Equation 5.15 and Equation 5.16, and boundary con-ditions (5.17) through (5.19). The substantial difference from the spontaneouscapillary imbibition is that now the relation between l(t) and the concentrationon the moving meniscus, Cm, is given by relation (5.37), which shows that Cm isan unknown function of time. Using these equations and boundary conditions,we show in the following text that l(t) dependency on time can be calculated,and it is proportional to the square root of time at the initial stage of the capillaryrise (see Appendix 1 for details).

The solution in Appendix 1 shows that, at the initial stage of capillary rise,l(t) develops as

l tgR

C Cm( ) ( ).*= −2αρ

t

t B KgR= ρα2

,

lgR

C Cmax *( ).= −20

αρ



(5.40)

where κ and ω are defined in Appendix 1. This dependency agrees with experi-mental observations in Reference 3 (see Figure 5.11). At the final stage of thecapillary rise, l(t) levels off as

According to Equation 5.39 and Figure 5.11, in the experimentpresented in Figure 5.11. The experimental value of Kexp in Equation 5.40, cal-culated according to Figure 5.11, is Kexp ~ 5·10–3 cm/sec1/2. In experimentspresented in Figure 5.11, ω ~ 1. Using these estimations we conclude κ ~ 10–5

sec–1, which coincides with the value calculated in Appendix 1 according toEquation A1.11, if we assume that Fsv << Fsl, D ~ 10–5 cm2/sec, using estimationGSL ~ 10–5 cm [8], and the value of α is taken directly from Figure 5.8.

APPENDIX 1

To simplify a mathematical treatment of the problem, we disregard in the fol-lowing discussion the diffusion of surfactant molecules in front of the movingmeniscus. That means only the adsorption in front of the moving meniscus istaken into account. Under this simplification, the process of the capillary rise ina hydrophobic capillary is governed by Equation 5.15 with boundary condition(5.19).

The concentration of surfactant changes considerably only in close proximityto the moving meniscus. That means the surfactant concentration differs from theconcentration at the capillary inlet, C0, in a narrow region between l(t)–δ(t) andl(t), where δ(t) is to be determined. At the boundary l(t)–δ(t), the concentrationof surfactant is equal to the concentration at the capillary inlet, C0, and thederivative of the concentration at this point is zero (a smooth transition to theconstant concentration).

Let us introduce for the sake of convenience a new unknown function Z =C0 – C, which satisfies the following equation and boundary conditions:

(A1.1)

From Equation 5.37, , where

.

l t K t K l( ) , ,max= = 2κω

l t lt

( ) ,max= − +

11ω

κ

lmaxexp ~ 35 mm

( ) , ( ).1 02

2+ ∂

∂= ∂

∂− ∂

∂< <F

Zt

DZ

x

dldt

Zx

x l tsl ef

C C Al tm = +* ( )

AgR= ρα2



Boundary condition (5.19) takes the following form using the preceding equations:

(A1.2)

Other boundary conditions are:

(A1.3)

(A1.4)

Integration of Equation A1.1 after simple manipulations using boundaryconditions (A1.3) and (A1.4) results in

(A1.5)

In the following text we find a solution that satisfies the integral balanceEquation A1.5 and boundary conditions (A1.2) through (A1.4). The simplestsolution, which satisfied boundary conditions (A1.3) through (A1.4), is as follows:

(A1.6)

where both l(t) and δ(t) dependencies are to be determined. Substitution of thisexpression into boundary condition (A1.2) and Equation A1.5 gives, after somerearrangements, a system of two differential equations for the determination oftwo unknown dependencies l(t) and δ(t):

where The following initial conditions should be satisfied:

(A1.8)

DZx

F F C Aldldtef

x l

sl sv

∂∂

= −( ) +( )=

* .

Z l Z l C C Al( ) , ( ) ,*− = = − −δ 0 0

∂∂

== −

Zx

x l δ

0.

ddt

ZdxD

FZx

FF

Cl

l

ef

sl x l

sl

sl− =∫ =

+∂∂

++

δ1 1 00 − −( )C Al

dldt* .

Z C C All x= − −( ) − −

0

2

1* ,δ

ddt

C C AlD

F

C C Al Fef

sl

s0

06

13− −( ) =

+− −( )

+**δ

δll

sl

ef

FC C Al

dldt

dldt

D

F

C C Al

1

2

0

0

+− −( )

=− −( )

*

*

∆ δ CC Al*

,

+( )

(A1.7)

∆F F Fsl sv= − > 0.

l( ) ( ) .0 0 0= =δ



If the first equation in (A1.7) is divided by the second equation, then anequation for δ(l) dependence can be obtained. This equation can be solved andthe solution substituted into the second equation in system (A1.7), which givesthe following equation for l(t) determination:

(A1.9)

where

(A1.10)

These definitions show that χ >> 1; hence, Equation A1.9 can be rewritten as

(A1.11)

This equation can be easily solved, and the solution is as follows:

(A1.12)

If u << 1 (initial stage of the process), then from Equation A1.12 we concludethat

(A1.13)

At the final stage of the process, 1 – u << 1, Equation A1.12 gives

(A1.14)

5.3 CAPILLARY IMBIBITION OF SURFACTANT SOLUTIONS IN POROUS MEDIA AND THIN CAPILLARIES: PARTIAL WETTING CASE

Let us now consider the imbibition of surfactant solutions into porous solidsubstrates, which are partially wetted by water. We shall see that this case is

dudt

uu u u

u= −+ + −

=λω χ

( )( )( )

; ( ) ,1

20 0

2

u l l lgR

C CC

C C

Def

= = −( ) =−

=

/ max max **

*

, , ,2

4

00

αρ

ω

λ11

3

22

0

0

+( )=

−( )F

FF l

f C

F C Csl

sv sv∆∆

max *

, .χ

dudt

uu u

u= −+

=+

=κω

κ λχ

( )( )

; , ( ) .1

20 0

2

uu

uu t+ +

−+ + − =( )

( ) ln( ) .ω ω κ11

2 1

u t= 2κω

.

ut

= − +1

1ωκ

.



considerably different from that of hydrophobic porous media. To be concrete,we shall continue studying the imbibition in a cylindrical capillary whose wallsare partially wetted by water, recalling that such a capillary can be used to modela porous medium. At variance with the case of hydrophobic capillaries, here,water can penetrate into the capillary even in the absence of surfactants on themoving meniscus. The presence of surfactant molecules on the moving meniscuslowers the contact angle, and hence a higher capillary pressure builds behind themeniscus. Consequently, the imbibition rate increases with the increase of con-centration of surfactant molecules on the moving meniscus.

The moving meniscus covers fresh parts of the capillary walls where surfac-tant molecules have not adsorbed yet. Thus, with the imbibition process, there isthe simultaneous adsorption of surfactant molecules onto fresh parts of the cap-illary walls in a vicinity of the moving meniscus. The amount adsorbed isinversely proportional to the radius of the capillary, i.e., the thinner the capillary,the higher the adsorption is. On the other hand, the imbibition rate is lower inthinner capillaries (higher friction). This gives more time to diffusion to bring newsurfactant molecules to cover the fresh part of the capillary walls. Thus, we havetwo competing, opposite, trends. Indeed, if the capillary radius is smaller thansome critical value, then adsorption goes faster than the imbibition process, andall surfactant molecules are adsorbed on the capillary walls, leaving nothing forthe meniscus where the concentration vanishes. In such circumstance, the imbi-bition rate of a surfactant solution becomes independent of the surfactant con-centration in the feed solution with a value equal to that of pure water [11]. Thesetheoretical conclusions are in agreement with experimental observations [11].

The kinetics of the capillary imbibition of aqueous surfactant solutions intohydrophobic capillaries has been investigated earlier in Section 5.2. It has beenshown that the rate of imbibition is controlled by the adsorption of the surfactantmolecules in front of the moving meniscus on the bare hydrophobic surface ofthe capillary. This process results in a partial hydrophilization of the surface ofthe capillary in front of the moving meniscus and provides the possibility for theaqueous surfactant solution to penetrate into the initially hydrophobic capillary.Therefore, no surfactant molecules on the meniscus, no imbibition. In the followingtext, the imbibition of surfactant solutions into the porous substrates, which arepartially wetted by water, is considered. It is shown that the situation in this caseis considerably different from the case of hydrophobic porous media.

THEORY

The imbibition of aqueous surfactant solutions into a single cylindrical capillarywith walls partially wetted by water is considered in this section. A single capillaryis used as a model of a porous medium.

The situation in this case is different from the case of hydrophobic capillaries(Section 5.2); water can penetrate into the capillary even in the absence ofsurfactant molecules on the moving meniscus. However, the presence of surfactant



molecules on the moving meniscus results in a lower contact angle (as comparedwith pure water), and hence, in a higher capillary pressure behind the meniscus.As a result, the imbibition rate increases with the concentration of surfactantmolecules on the moving meniscus.

The moving meniscus covers fresh parts of the capillary walls where surfac-tant molecules have not adsorbed yet. This means that the imbibition process isaccompanied by the simultaneous adsorption of surfactant molecules onto freshparts of the capillary walls in a vicinity of the moving meniscus. The adsorptionis inversely proportional to the radius of the capillary; that is, the thinner thecapillary, the higher the adsorption. On the other hand, the rate of imbibition islower in thinner capillaries (higher friction). This gives more time for diffusionto bring in new surfactant molecules and cover the fresh part of the capillarywalls. Therefore, there are two competing, opposite, trends. This means that ifthe capillary radius is smaller than some critical value, then adsorption goes fasterthan the imbibition process, and all surfactant molecules are adsorbed on thecapillary walls; therefore, there is nothing left for the meniscus, and the concen-tration on the moving meniscus remains zero. This qualitative considerationshows that if the capillary radius is smaller than some critical value, then the rateof the imbibition of surfactant solutions remains independent of the surfactantconcentration in the feed solution and equals that of pure water.

This qualitative conclusion is justified in the following text using the theo-retical consideration of the capillary imbibition of aqueous surfactant solutionsinto cylindrical capillaries whose walls are partially wetted by water.

Let us consider an imbibition of surfactant solution from a reservoir with afixed surfactant concentration, C0 (feed solution), into a thin capillary with radiusR << L (Figure 5.12), where L is the capillary length. The capillary walls arepartially wetted by pure water (at zero concentration of surfactant), that is,

(5.41)

FIGURE 5.12 Imbibition of a surfactant solution in thin capillary; C0 — concentrationat the capillary entrance; Cm — concentration on the moving meniscus; �(t) — positionof the moving meniscus.

x

x L

cm

Cm

C0

C0

0

R

�(t)

γ θ γ γ( ) cos ( ) ( ) ( ) ,0 0 0 0 0a sv sl= − >



where are the liquid–air, the solid substrate–vapor, the solid substrate–liquid interfacial tensions, and the advancing contact angle, respectively. All thesevalues are concentration dependent.

Let Cm be the bulk concentration of surfactant behind the moving meniscus;then,

(5.42)

It is assumed that the imbibition process goes sufficiently fast, and transferof the surfactant molecules on the bare surface in front of the moving meniscuscan be neglected because this process goes much slower (see Section 5.2). Hence,the solid–vapor interfacial tension, γsv, does not depend on the surfactant concen-tration and remains equal to its value at zero concentration. It is also taken intoaccount in Equation 5.42 that is a decreasing function of the surfactantconcentration. Equation 5.42 shows that is an increasingfunction of the concentration, with the maximal value, reachedat CMC and the minimal value, reached at zero surfactantconcentration.

Concentration below CMC

Let the surfactant concentration at the capillary entrance be below CMC, C0 <CCMC.

The transfer of surfactant molecules in the filled portion of the capillary isdescribed by the convective diffusion equation as in Section 5.2,

,

where C(t,x,r) is the local concentration of surfactant; D is the diffusion coeffi-cient; t, x, and r are time, axial, and radial coordinates, respectively; and v(r) isthe axial velocity distribution.

Integration of the preceding equation over radius from 0 to R, where R is thecapillary radius, results in

γ γ γ θ, , ,sv sl a

γ θ γ γ γ γ( ) cos ( ) ( ) ( ) ( ) ( )C C Cm a m sv sl m sv sl= − > −0 0 0

== γ θ( ) cos ( ).0 0a

γ sl mC( )Ψ( ) ( ) cos ( )C C Cm m a m= γ θ

Ψ Ψmax ( ),= CCMC

Ψmin ( ) ( ),= −γ γsv sl0 0

∂∂

= ∂∂

+ ∂∂

∂C t x r

tD

C t x r

xD

r rr

C t x r( , , ) ( , . ) ( , , )2

2

1∂∂

− ∂∂ ( )

r xv r C t x r( ) ( , ,

∂∂

= ∂∂∫t

rC t x r dr Dx

rC t x r dr

R

( , , ) ( , , )0

2

2

00

R

r R

R DC t x r

r

xrv

∫

+ ∂∂

− ∂∂

=

( , , )

(( ) ( , , ) .r C t x r dr

R

0∫



The second term on the right-hand side of the equation is equal to

,

where Dsl is the surface diffusion coefficient over the filled portion of the capillary,and Γ is the surface concentration. Hence, the preceding two equations result in

A characteristic time scale of the equilibration of the surfactant concentrationin a cross section of the capillary is τ ~ R2/D ≈ 10–3 sec, if we use for estimationsR~1 µm and D ~ 10–5 cm2/sec. A characteristic time scale of the spontaneouscapillary rise into partially hydrophilic capillaries is around 10 sec (see Figure 5.15),which is much bigger than 10–3 sec. Hence, the surfactant concentration is con-stant in any cross section of the capillary and depends only on the position, x(Figure 5.12), that is, C = C(t, x).

Taking this into account, the preceding equation can be rewritten, after bothsides are divided by R2/2, as

0 < x < l(t), (5.43)

where

is the averaged velocity (the same symbol is used for the averaged velocity asfor the local one). The averaged velocity, v, is equal to the meniscus velocity, thatis,

, (5.44)

where is the position of the moving meniscus. Substitution of Equation 5.44into Equation 5.43 and neglecting the surface diffusion results in the following

− ∂∂

= ∂∂

− ∂∂=

DC t x r

r tD

xr R

sl

( , , ) Γ Γ2

2

∂∂

= ∂∂∫t

rC t x r dr Dx

rC t x r dr

R

( , , ) ( , , )0

2

2

00

2

2

R

SLRt

Dx

xrv

∫

− ∂∂

− ∂∂

− ∂∂

Γ Γ

(rr C t x r dr

R

) ( , , ) .0∫

2 22

2

2

2R tCt

DC

xD

R xv

CxSL

∂∂

+ ∂∂

= ∂∂

+ ∂∂

− ∂∂

Γ Γ,

vR

rv r dr

R

= ∫22

0

( )

vd

dt= �

�( )t



equation, which describes the concentration profile inside the filled portion ofthe capillary:

0 < x < l(t). (5.45)

As a result of adsorption on the capillary wall, the concentration of thesurfactant molecules on the moving meniscus, Cm, is lower than that at thecapillary entrance and should be determined in a self-consistent way.

Solution of Equation 5.45 should be subjected to the following boundaryconditions:

, (5.46)

, (5.47)

and the last boundary condition on the moving front is

(5.48)

The preceding condition expresses the conservation of mass at the movingmeniscus and the three-phase contact line. In order to deduce this boundarycondition, the following procedure should be undertaken: (1) a local coordinateshould be introduced in a narrow region close to the moving meniscus; (2) localtransport equations should be integrated over this narrow region; and (3) limitsfrom the filled portion of the capillary should be calculated, which gives condition(5.48).

Because R << L, the liquid flow inside the capillary is the simple Poiseuilleflow, which means that

where η is the dynamic viscosity, assumed to be independent of surfactantconcentration. It is also assumed in the following discussion that the surfactantconcentration remains constant at the moving meniscus (similar to the case ofhydrophobic capillaries in Section 5.2). Solution of the preceding equation usinginitial condition l(0) = 0 gives

(5.49)

2 2

2R tCt

DC

x

dldt

Cx

∂∂

+ ∂∂

= ∂∂

− ∂∂

Γ,

C t C( , )0 0=

C t l t Cm( , ( )) =

2Γ( ).

( )

CR

ddt

DCx

m

x t

�

�

= − ∂∂ =

dldt

R C CR l

R C Cm a m m a m= =2

82 1

ηγ θ γ θ( )cos ( ) ( )cos ( )

44ηl,

l t K t KR C Cm a m( ) ,

( )cos ( ).= = γ θ

η2



Substitution of Equation 5.49 into Equation 5.45 results in

(5.50)

with boundary conditions (5.46) and (5.47), and

(5.51)

which is the consequence of (5.48).Let us introduce the following similarity coordinate

and solution of Equation 5.50 is assumed to depend on the similarity coordinateonly. In this case, Equation 5.50 and boundary conditions (5.46), (5.47), and (5.51)take the following form:

(5.52)

(5.53)

(5.54)

(5.55)

where

is a dimensionless parameter. In the case of aqueous surfactant solutions D ~10–5 cm2 sec, η ~ 10–2 P, γ ~ 102 dyn/cm; therefore, the last parameter can beestimated as

.

2

2

2

2r tCt

DC

x

K

t

Cx

∂∂

+ ∂∂

= ∂∂

− ∂∂

Γ,

Γ( ),

C K

K tD

Cx

m

x K t

= − ∂∂ =

ξ = x

K t,

λ ξ2 12

1′′ = ′ − ′ +

C C

RCΓ ( ) ,

C C( ) ,0 0=

C Cm( ) ,1 =

′ = −CC C

Dm m( )

( ) ( ),1

2Γ Ψ

η

λ η22

2 4= =D

K

DR CmΨ( )

λ2910

~− cmR



In the following text we consider only capillaries with R > 0.1 µm = 10–5 cm.In this case, .

The first consequence of Equation 5.52 to Equation 5.55 is that the problemunder consideration is really a similarity one.

To further simplify the mathematical treatment of the problem under consid-eration, the simplest adsorption isotherm is adopted:

(5.56)

In this case, Equation 5.52 and Equation 5.55 can be rewritten as follows ifthe concentration on the moving meniscus is above zero (the case of zero con-centration is considered separately in the following discussion):

(5.57)

(5.58)

with boundary conditions (5.53) to (5.54). Note that according to Equation 5.42,Ψ, in the case under consideration (according to Equation 5.56), is independentof the concentration and equal to its maximal value, Ψmax.

In the following discussion the smallness of the parameter λ2 is utilized. Thissmall parameter is a multiplier at the highest derivative in Equation 5.57. Thismeans that matching of asymptotic solutions can be used. Let us introduce thefollowing local variable, z:

. (5.59)

Using the new variable, the inner solution of Equation 5.57 satisfies thefollowing system:

(5.60)

and the boundary condition

(5.61)

λ2 410 1< <<−

ΓΓ

( ),

,.C

C

C=

>

=

∞ 0

0 0

λ ξ ξ2 1 0 1′′ = ′ −( ) < <C C , ,

′ = − ∞CD

( ) ,max12

Γ Ψη

z = −1 ξλ

′′ = − ′ < < ∞

′ =

=

∞

C zC z

CD

C Cm

,

( )

( )

max

0

02

0

Γ Ψ λη

C C( ) .∞ = 0



Solution of the problem (5.60) is

(5.62)

The following equation for the determination of the unknown concentrationon the moving meniscus, Cm, yields, using boundary condition (5.21) and solution(5.62):

or

(5.63)

The concentration on the moving meniscus should be positive, Cm > 0;therefore, the following requirement should be satisfied:

(5.64)

or

(5.65)

Let us introduce the following notation:

(5.66)

Two cases are considered here: (1) R < Rcr and (2) R > Rcr.In the first case, (1), condition (5.64) is violated at any concentrations in the

feed solution between zero and CMC. This means that concentration of surfactantmolecules on the moving meniscus is equal to zero at any concentration fromthis range. Hence, there are two regions behind the moving meniscus. The first

C z CD

z dzm

z

( ) exp .max= + −( )∞ ∫Γ Ψ λη2

22

0

/

C CD

m0 3 22= + ∞Γ Ψmax

/,

λ πη

C CD Rm = − ∞0 2

Γ Ψπηmax .

CD R0 2

> ∞Γ Ψπηmax ,

RD C

> ∞Γ Ψ2

022

πη

max .

RD C

cr = ∞Γ Ψ2

22

πη

max .CMC



region is close to the capillary entrance, where the concentration is changing fromC0 in the feed solution to zero on the moving border between two regions. Thefirst region is followed by the second region where concentration remains zeroover the duration of the whole process. Let the moving border between these tworegions be

(5.67)

where β < 1 is a value to be determined.In this case, the concentration on the meniscus remains zero, and the meniscus

moves slowly, according to Equation 5.49:

(5.68)

The concentration profile in the first region is a solution of the followingproblem (using the same similarity variable as before):

(5.69)

(5.70)

(5.71)

(5.72)

Condition λ << 1 is assumed to be satisfied; this means that

(5.73)

where χ is a new unknown value. Solution of the problem (5.69) to (5.73) givesthe following equation for the determination of an unknown value, χ:

or

(5.74)

�1( ) ,t K t= β

l tR

t( ) .min= Ψ2η

λ ξ ξ β2 1 0′′ = ′ −( ) < <C C , ,

′ = − ∞CD

( ) ,minβ βη

Γ Ψ2

C C( ) ,0 0=

C( ) .β = 0

β λχ= −1 ,

CD R0

2

2= −( )∞Γ Ψπ

ηχmin exp ,

χ πη

=

∞ln .min

/

Γ ΨC D R0

1 2

2



The main conclusion from this consideration is that the adsorption processin sufficiently thin capillaries consumes all surfactant, and the imbibition is notinfluenced by the presence of surfactants in the feed solution at any concentration.

Let us consider the second case when the capillary radius is bigger than thecritical value determined by Equation 5.66, that is, R > Rcr.

If the concentration in the feed solution is low enough, condition (5.65) isviolated when the concentration on the moving meniscus, Cm, is equal to zeroand the meniscus moves slowly according to Equation 5.68. It is worth notingthat the capillary radius is assumed to be bigger than that in the previous case.

If, however, the concentration in the feed solution, C0, is high enough, con-dition (5.65) is satisfied, and concentration of the surfactant molecules is differentfrom zero on the moving meniscus; the imbibition process goes faster according to

(5.75)

Hence, if the capillary radius is bigger than the critical value, then the wholeconcentration range in the feed solution can be subdivided into two parts: thelow concentration range

when the adsorption consumes all surfactant molecules, the concentration on themoving meniscus is equal to zero, and the meniscus moves slowly according to(5.68); and the high concentration range

when the adsorption does not consume all surfactant molecules, the concentrationon the moving meniscus is different from zero, and the meniscus moves fasteraccording to (5.75).

In Figure 5.13, against the concentration in the feed solution, C0, isschematically plotted according to the simplified theoretical model discussed.

If a more realistic case of adsorption isotherm (approximated by a Langmuirtype isotherm)

l tR

t( ) .max= Ψ2η

C CD Rcr0 2

< = ∞Γ Ψπηmax ,

C CD Rcr0 2

> = ∞Γ Ψπηmax ,

�2 /t

ΓΓ

( ),

*,

*

cc c c

c=

<

=

∞

ω

ω



is adopted, then the dependency presented in Figure 5.13 changes continuouslyfrom lowest to the highest value instead of the stepwise change.

Concentration above CMC

If the concentration in the feed solution, C0, is above CMC, then, after some shortinitial period inside the capillary, two zones form (similar to Section 5.2): in thefirst region, close to the capillary entrance, the concentration inside the capillaryis higher than CMC. This region is followed by the second region where con-centration is below CMC. The concentration is equal to CMC at the borderbetween these two regions. The consideration similar to that in Section 5.2 andpresented in the preceding discussion shows that the main conclusion remainsunchanged in this case: there is a critical radius of the capillary, below which theconcentration on the moving meniscus remains zero at any concentration in thefeed solution.

The concentration on the moving meniscus, Cm, is below CMC. Two regionscan be identified inside the capillary: the first region, from the capillary inlet tosome position, lM(t), where concentration is above CMC, and the surfactantsolution includes both micelles and individual surfactant molecules: and thesecond region, from lM(t) to l(t), where concentration is below CMC and onlyindividual surfactant molecules are transferred. The concentration is equal toCMC at x = lM(t). Consideration in the second region, lM(t) < x < l(t), is similarto that at concentration below CMC. That is why only the transport in the firstregion is considered in the following discussion.

Inside the first region, 0 < x < lM(t), concentration of free surfactant moleculesis constant and equal to CMC (see Section 5.2). Hence, the transfer is determinedby the diffusion of micelles and convection of all molecules. The total concen-tration, and Cmol remain constant and equal to CMC; hence,

FIGURE 5.13 Dependency of permeability on surfactant concentration: theoretical pre-dictions. (1) thin capillary, the radius is below critical value; (2) thick capillary, the radiusis above critical value.

2

1

Ccr C0

�2 /t

C C Cmol M= + ,



(5.76)

where DM is the diffusion coefficient of micelles, and C = Cc + CM is the totalconcentration of surfactant molecules. Adsorption on membrane pores is deter-mined by the concentration of the free molecules, which is constant in the firstregion, and so is the adsorption. This is why the diffusion of adsorbed moleculesin the first region is omitted in Equation 5.76. Transfer of surfactant moleculesin the second region (micelles-free region) is described by Equation 5.45.

Boundary conditions on the moving boundary between the first and secondregions, lM(t), are as follows:

(5.77)

As before, we assume that

(5.78)

where K is given by Equation 5.49; that is, it is expressed via unknown concen-tration on the moving meniscus, Cm, and KM is an unknown value to be determined.

Let a similarity variable be introduced in the same way as in the case of aconcentration below CMC; that is, ξ = x/K in Equation 5.45 and Equation5.76. Using boundary conditions (5.77), expressions (5.78), and boundary con-ditions (5.54) to (5.55), the system of two nonlinear algebraic equations can bededuced for determination of two unknown values: concentration on the movingmeniscus, Cm, and the position of the boundary, KM. This system includes a smallparameter

,

which is the ration of diffusion coefficients of micelles and free surfactant mol-ecules. Using this new small parameter, it is possible to show that solution of thementioned system only slightly deviates from the solution in the previous casewhen concentration is below CMC.

This means that the constant K and the expression for the critical radius (5.66)are only slightly different from the same values in the case of concentration belowCMC. Hence, the previous conclusion concerning the existence of the criticalradius remains valid even at concentrations above CMC, which is confirmed inthe following text by our experimental data.

∂∂

= ∂∂

− ∂∂

Ct

DC

x

dldt

CxM

2

2,

DCx

DCx

C l tM

x lM x lM

M

∂∂

= ∂∂

== − = +

, ( , ) CCc.

l t K t l t K tM M( ) , ( ) ,= =

t

ε = <<D

DM 1



Experimental Part

Figure 5.14 shows the sample chamber for monitoring the permeability of theinitially dry porous layers. The time evolution of the permeability front wasmonitored. The membrane (1) was fastened on a lifting up/down device (2) andplaced in a thermostated and hermetically closed chamber (3), where 100%humidity (to prevent evaporation from the wetted part of the membrane) and fixedtemperature (20 ± 0.5˚C) were maintained. To prevent temperature fluctuations,the chamber was made from brass and, in the chamber walls, several channelswere drilled that were used for the pumping of a thermostating liquid. Thechamber was equipped with a fan. The temperature was monitored by a thermo-couple. A box with water was used to keep absolute humidity inside the chamber.On the bottom of the chamber, a small Petri dish (4) with different water solutionsof SDS was placed.

The chamber was equipped with optical glass windows (5) for observationof the imbibition front of the surfactant solution. A CCD camera (6) and VCR(7) were used for storing the sequences of the imbibition. Automatic processingof images was carried out on a PC (8) using an image processor, Scion Image.The duration of each experimental run was in the range from 2.5 to 30 sec. Thediscretization of time in the processing ranged from 0.04 to 2 sec in differentexperimental runs; the size of pixel in an image was 0.01 mm.


• The membrane was placed in the chamber and left in atmosphere of100% humidity for several minutes.

• The membrane was immersed vertically (0.1–0.2 cm) into a containerwith SDS solution. After that, the position of the imbibition front wasmonitored over time.

• Several runs for each membrane type and each concentration of SDSsolution were carried out.

FIGURE 5.14 Schematic presentation of the experimental setup. (1) membrane; (2) liftingup and down device; (3) thermostated chamber; (4) Petri dish with an SDS solution; (5)optical glass windows 5; (6) CCD camera; (7) video tape recorder; (8) PC; (9) light source.

8

CCD

9 5

2

6 75

1

34



A rectangular membrane sample 1.5 cm∗3 cm was used. Porous sampleswere cut from the cellulose nitrate membranes purchased from Millipore ofBillerica, MA. Three different membranes with averaged pore size were used:0.22 µm, 0.45 µm, and 3.0 µm. Each membrane sample was immersed 0.1–0.2cm into a liquid container, and the position of the imbibition front was monitoredover time.

Results and Discussions

It was confirmed that in all our experimental runs, gravity action could bedisregarded. A unidirectional flow of liquid inside the porous substrate took place.Using Darcy’s law, we can conclude that

(5.79)

where now is the position of the imbibition front inside the porous layer, Kp

is the permeability of the porous membrane, and pc is an effective capillarypressure inside the porous sample. The permeability of the porous layer and thecapillary pressure enter as a product in Equation 5.79; that is, as a single coeffi-cient. Experiments were carried out to determine this coefficient, and its depen-dency on the surfactant concentration if any. It was found that in all runs, l2(t)proceeds along a straight line whose slope gives us the Kp pc value.

According to our previous notations, Kppc = K2η.Figure 5.15 is the example of the imbibition of 0.1% SDS solution into

0.22 µm nitrocellulose membrane. This dependency is in good agreement withEquation 5.79.

FIGURE 5.15 Example of the time evolution of the imbibition front. SDS concentration0.1%, nitrocellulose membrane with averaged pore size 0.22 µm.

�2( ) ,t K p tp c= /η

�( )t

5

4

�, m

m

3

2

1

0

Nitrocellulose membrane Pore size 0.22 μm

SDS-0.1%

t, s302520151050



Figure 5.16 presents Kppc dependency on the concentration of the SDS in thefeed solution for three different membranes. Kppc in the case of membranes with0.22-µm and 0.45-µm averaged pore size is independent of concentration. How-ever, in the case of membranes with 3.0-µm averaged pore size, Kppc increaseswith SDS concentration. This means that the critical radius, Rcr, is somewherein between 0.45 and 3.0 µm. Figure 5.16 confirms our conclusion concerning theexistence of the critical pore radius below which permeability is independent ofsurfactant concentration.

5.4 SPREADING OF SURFACTANT SOLUTIONS OVER HYDROPHOBIC SUBSTRATES

We shall now study the spreading of aqueous surfactant solutions over hydro-phobic surfaces. The spreading of surfactant solutions over hydrophobic surfacesis considered in the following text from both theoretical and experimental pointsof view. Water droplets do not wet a virgin solid hydrophobic substrate. It is shownin this section that the transfer of surfactant molecules from the water droplet ontothe hydrophobic surface changes the wetting characteristics in front of the dropon the three-phase contact line. The surfactant molecules increase the solid–vaporinterfacial tension and hydrophilize the initially hydrophobic solid substrate justin front of the spreading drop. This process causes water drops to spread overtime. The time of evolution of the spreading of aqueous surfactant solution dropletsis predicted and compared with experimental observations. The assumption that

FIGURE 5.16 Dependency of kpc on concentration of SDS solutions for nitrocellulosemembranes with different averaged pore sizes. Remains constant in the case of membraneswith averaged pore size both 0.22 µm (line 1) and 0.45 µm (line 2). Increases withsurfactant concentration in the case of membrane with averaged pore size 3 µm (curve 3is drawn simply to guide the eyes).

SDS-concentrations, %

kpc

0.04

0.03

0.02

0.01

0.000.0 0.2 0.4 0.6 0.8 1.0

1

2

3

0.22 µm‘Millipore’

0.45 µm3.0 µm



surfactant transfer from the drop surface onto the solid hydrophobic substratecontrols the rate of spreading is confirmed by our experimental observations.

Surfactant adsorption on solid–liquid and liquid–vapor interfaces changes thecorresponding interfacial tensions. Liquid motion caused by surface tension gra-dients on liquid–vapor interfaces (Marangoni effect) is the most investigatedprocess (see Section 5.6). The phenomena produced by the presence of surfactantmolecules on a solid–vapor interface have been studied less. In Section 5.2, theimbibition of surfactant solutions into thin quartz capillaries was investigated.Spreading and imbibition of surfactant solutions into both hydrophobic andhydrophilic surfaces (Section 5.2, Section 5.3, the current section, and Section5.7) revealed various new and intriguing phenomena. In this section we addressthe problem of aqueous surfactant solutions spreading over hydrophobic surfacesfrom both the theoretical and experimental points of view [14].

THEORY

Let a small water drop be placed on a hydrophobic surface. If the drop is smallenough, then the effect of gravity can be ignored. Accordingly, the radius of thedrop base, R(t), has to be smaller than the capillary length, a, and hence,

where ρ and γ are the liquid density and liquid–vapor interfacial tension, respec-tively; g is the gravity acceleration.

First of all, let us consider expression (1.2) from Chapter 1, Section 1.1 forthe excess free energy, Φ, of the droplet on a solid substrate:

where S is the area of the liquid–air interface; is the excess pressureinside the liquid, Pa is the pressure in the ambient air, and Pl is the pressure insidethe liquid; the last term on the right-hand side gives the difference between theenergy of the part of the bare surface covered by the liquid drop as comparedwith the energy of the same solid surface without the droplet (Figure 5.17).

The foregoing expression shows that the excess free energy decreases if (1) theliquid–vapor interfacial tension, γ, decreases, (2) if the liquid–solid interfacialtension, γsl, decreases, and (3) the solid–vapor interfacial tension, γsv, increases.

Let us assume that, in the absence of surfactant, the drop forms an equilibriumcontact angle above π/2. If the water contains surfactants, then three transferprocesses take place from the liquid onto all three interfaces: surfactant adsorptionat both (1) the inner liquid–solid interface, which results in a decrease of thesolid–liquid interfacial tension, γse, (2) the liquid–vapor interface, which results

R t ag

( ) ≤ = γρ

,


P P Pa l= −



in a decrease of the liquid–vapor interfacial tension, γ, and (3) transfer from thedrop onto the solid–vapor interface just in front of the drop. As we already noticedpreviously, all three processes result in a decrease of the excess free energy ofthe system. However, adsorption processes (1) and (2) result in a decrease ofcorresponding interfacial tensions, γsv and γ ; but the transfer of surfactant mole-cules onto the solid–vapor interface in front of the drop results in an increase ofa local free energy. However, the total free energy of the system decreases. Thatis, surfactant molecule transfer (3) moves via a relatively high potential barrier,and hence, considerably slower than adsorption processes (1) and (2). Therefore,they are fast processes as compared with the third process (3).

The transfer of surfactant molecules onto the unwetted (hydrophobic)solid–vapor interface in front of the liquid has been shown in Section 5.2 to playan important role in the wetting of hydrophobic surfaces.

All three surfactant transfer processes are favorable to spreading, as theyresult in both an increase of the spreading power, γsv – γ – γsl, and hence, adecrease in the contact angle. As mentioned previously, the transfer of surfactantmolecules from the drop onto the solid–vapor interface in front of the drop resultsin an increase of local surface tension, γsv. Hence, it is the slowest process thatwill be the rate-determining step. Let us define the initial contact angle by

(5.80)

with the initial values of solid–vapor, solid–liquid, and liquid–vaporinterfaces, respectively. The term initial means that, although the adsorptionprocess on the liquid–vapor and solid–liquid interfaces has been completed (theyare fast processes), the solid–vapor interface still has its initial condition as abare hydrophobic interface without any surfactant adsorption. At this initialinstant of time, a process of slow transfer of surfactant molecules starts from thedrop onto the solid–vapor interface. Let Γs(t) be the instantaneous value ofsurfactant adsorption onto the solid surface in front of the liquid drop on thethree-phase contact line, and Γe be the equilibrium surface density of adsorbed

FIGURE 5.17 Sketch of the geometry of a drop placed on a solid substrate.

γsv(t)

γ0

γ0sl

θ Γs(t)

R(t)

cos ,θ γ γγ

π00 0

0 2= − ≥sv sl

γ γ γsv sl0 0 0, ,



surfactant molecules that will eventually be reached. The driving force of theprocess is proportional to the difference Γs(t) – Γe. Hence, the surfactant adsorp-tion behavior with time is described by

(5.81)

with the initial condition that

at t = 0, (5.82)

and τs = 1/α is the time scale of surfactant transfer from the drop onto thesolid–vapor interface at the three-phase contact line. Let us assume that

(5.83)

where the prefactor αT is determined by thermal fluctuations only; ∆E is an energybarrier for surfactant transfer from the liquid drop onto the solid–vapor interface;k and T are Boltzmann’s constant and absolute temperature, respectively; Ξ is afraction of the drop liquid–vapor interface covered with surfactant molecules. Weassume that the position of surfactant molecules on a hydrophobic solid interfaceis the hydrophobic “tails down.”

We have assumed that transfer of surfactant molecules onto the hydrophobicsolid interface takes place only from the liquid–vapor interface. It is difficult toassess the contribution of surfactant molecule transfer along the solid surfacefrom beneath the liquid. However, experimental data presented in the followingtext in this section support our assumption (although they do not prove it deci-sively). The drop surface coverage, Ξ, is an increasing function of the bulksurfactant concentration inside the drop, whose maximum is reached close to theCMC. It follows from Equation 5.83 that at low surfactant concentrations insidethe drop, the characteristic time scale of the surfactant molecules transfer, τs,decreases with increased concentration, whereas above the CMC, τs levels offand reaches its lowest value. Both of these effects are observed in experimentalresults in the following discussion (compare Figure 5.20).

As the drop adopts a position according to the triangle rule, the contact angle,θ(t), is determined by the relationship

(5.84)

d t

dtt

se s

ΓΓ Γ

( )= − ( ) α ,

Γs ( )0 0=

α α= −

T

EkT

Ξ ∆exp ,

cos ,θγ γ

γt

tsv sl( ) =( ) − 0

0



where γsv(t) is the instantaneous solid–vapor interfacial tension at the three-phasecontact line. This dependency is determined by Γs(t). According to Antonov’s rule,

(5.85)

where γsv∞ is the solid–vapor interfacial tension of the surface completely covered

by surfactants, and Γ∞ is the total number of sites available for adsorption. Hence,the final value of the contact angle can be determined from Equation 5.84 as

. (5.86)

According to Equation 5.85, the solid–vapor interface in front of the spreadingdrop changes its wettability with time: from highly hydrophobic at the initialstage to partially hydrophilic at the final stage.

Using Equation 5.85 in Equation 5.84 yields the instantaneous contact angle

(5.87)

where cosθ0 is given by Equation 5.80, and the positive value of λ is

Equation 5.81 with initial condition (5.82) yields the solution

(5.88)

Using (5.88) in Equation 5.87 gives the final expression for the instantaneouscontact angle

(5.89)

A simple geometrical consideration (Figure 5.17) shows that the radius ofthe wetted spot, R(t), occupied by the drop can be expressed as

γ γ γsv svs

svs

tt t( ) =

( )+ −

( )

∞

∞ ∞

ΓΓ

ΓΓ

0 1 ,

cos θ γ γγ

∞∞

= −sv sl0

0

cos cos ,θ θ λtts( ) = +

( )∞

0ΓΓ

λ γ γγ

= −∞sv sv

0

0.

Γ Γs et t( ) = − −( )( )1 exp .α

cos cos exp .θ θ λ αt te( ) = + − −( )( )∞0 1

ΓΓ



(5.90)

where V is the drop volume, which is supposed to remain constant, and the contactangle, θ, is given by Equation 5.89.

Equation 5.89 and Equation 5.90 include two parameters: the dimensionlessparameter β = λΓe /Γ∞ and the parameter α with dimension of inverse of time. Itfollows from Equation 5.89 that β = cosθ∞ – cosθ0 > 0, where θ ∞ is the contactangle after the spreading process is completed. If both values of the contact angle,θ0 and θ∞, have been measured, β can be determined. Hence, only α is used inthe following discussion to fit the experimental data.

Let us introduce a dimensionless wetted area, S(t), as

where X = cosθ, (cosθ0 ≤ X ≤ cosθ∞), which, using Equation 5.89, becomes X =cosθ∞ − βe-αt.

It follows that both dS(t)/dt and dS(X)/dX are always positive, and the secondtime derivative is

(5.91)

Two different situations are possible: (A) if the second derivative (5.91)changes sign, then the spreading rate can go via an inflection point, whereas (B)if the second derivative (5.91) is always negative, the spreading rate dS(t)/dtdecreases with time. Case A corresponds to high surfactant activity,

whereas case B corresponds to “low surfactant activity,”

R tV( ) =

+

6 1

23

2

1 3

2

1 3π θ θ

/

/

tan tan

,

S tR t

V( )

( )

tan tan/

/

=

=

+

2

2 3

2 3 26

1

23

2πθ θ

= +

−( ) +( )2 3 1 3 2 3

1

1 4 2/ / /

,X

X X

d S t

dt X XX

dS X

dXX

2

2

22

1 22

( )( )( )

cos( )=

− +−( ) − +∞α θ 33 2X cos ( cos )θ θ∞ ∞− −

cos . ,θ∞ ≥ −( ) ≈49

10 1 0 961

cos . .θ∞ < −( ) ≈49

10 1 0 961



Using Equation 5.86, the preceding two conditions can be rewritten as

at a high surfactant activity, and

at low surfactant activity.Under experimental conditions described in the following text, only case B

was observed and, hence, low surface activity surfactants were used, whereas inReference 15, high surface activity surfactants (superspreaders) were apparentlyused.

Experiment: Materials

Two types of substrate were used: a PTFE film and a polyethylene (PE) wafer.The latter substrate was prepared by crushing granules of polyethylene compo-sition (softening point is 100°C) between two clean glass plates under an appliedpressure 1 kg/cm2 at temperature 110°C. Transparent wafers of circular sectionwith radius 1.5 cm and thickness 0.01 cm were used [14].

The cleaning procedure of PTFE and PE wafers was as follows: the surfaceswere rinsed with alcohol and water, then the substrates were soaked in a sulfo-chromic acid from 30 to 60 min at temperature 50°C. The surfaces were thenwashed with distilled water and dried with a strong jet of nitrogen. The equilib-rium macroscopic contact angles obtained were 105° and 90° for PTFE and PEsubstrates, respectively (for pure water droplets).

Aqueous solutions of sodium dodecyl sulfate (SDS) from Merck with weightconcentration from 0.005 up to 1% (the CMC of the SDS is 0.2%) were used inspreading experiments.

Monitoring Method

The time evolution of the contact line was monitored by following VCR imagesof drops. The images were stored using a CCD camera and a recorder at 25 framesper second. The automatic processing of images was carried out using the imageprocessor Optimas. In the case of spreading over PE, the initial contact angle ofthe drop was less than 90°, and the drop was observed from above. The observedwetting area of the drop was monitored, and the wetting radius was calculated.For the PTFE substrate, we used a side view of the drop, and hence, the wettingradius was determined directly.

Simple mass balance estimations show that time variation of surfactant con-centration inside the spreading drops can be neglected in our experiments (thoughit may become important in experiments of longer duration).

γ γ γsv sl∞ > +0 961 0 0.

γ γ γsv sl∞ < +0 961 0 0.



Water adsorption in front of the spreading drops was neglected because ofthe hydrophobic nature of substrates used (in contrast to spreading of evaporatingdrops [16]). The Peclet number in all experiments was so small that surfactantdiffusion in front of the drop was neglected.


According to observations in Reference 14, all drops were of spherical shape,and no disturbances or instabilities were detected. Immediately after deposition,the drops had a contact angle, which differed slightly from the equilibrium angleof pure water on the same substrate. After a very short initial time, the dropsreached a position referred to in the following discussion as the initial position.After that, for 1–15 sec, depending on the SDS concentration, the drops remainedat the initial position. Then the drops started to spread until a final value of thecontact angle was reached, and the spreading process was completed.

In Figure 5.18, the evolution of the spreading radius of a drop over PTFEfilm at 0.05% SDS concentration is plotted. In Figure 5.19, a similar plot is givenfor 0.1% SDS concentration. In both figures, the solid lines correspond to thefitting of the experimental data by Equation 5.89 and Equation 5.90, with τs =1/α used as a fitting parameter.

Figure 5.20 shows that, qualitatively, the τs dependency agrees with thetheoretical prediction and tends to support our assumption concerning the mech-anism of surfactant molecule transfer onto the hydrophobic surface in front ofthe drop.

Similar results were obtained for the spreading over the polyethylene substratefor concentrations below CMC. However, in this case, the spreading behavior of

FIGURE 5.18 Time evolution of the spreading of a water drop (aqueous solution C =0.05% SDS; 2.5 ± 0.2 µl volume) over PTFE wafer. Error bars correspond to the errorlimits of video evaluation of images (pixel size).

R, m

m

1.10

1.08

1.06

1.040.1 1 10 100

t, s



drops at concentrations above CMC is drastically different with increasing SDSconcentration (Figure 5.21). The rate of spreading is increased so much that at1% concentration, the power law with the exponent 0.1 (solid line) fits experi-mental data reasonably well. This clearly shows a transition to a different mech-anism of spreading, which can be understood in the following way. In our previousconsiderations, the influence of the viscous flow inside the drop was completelyignored. This means that it was assumed that τs >> τvis, where τvis is a time scale

FIGURE 5.19 Time evolution of the spreading of a water drop (aqueous solution C = 0.1%SDS; 2.5 ± 0.2 µl volume ) over PTFE wafer. Error bars are the same as in Figure 5.18.

FIGURE 5.20 Fitted dependency of τs on surfactant concentration inside the drop (spread-ing over PTFE wafer). Error bars correspond to the experimental points scattering indifferent runs; squares are average values.

R, m

m

1,600

1,575

1,550

0.1 1 10 100t, s

5

4

3

τ, s

2

1

0.1 1c, %



of viscous relaxation. In this case, τs decreases so considerably that the mentionedinequality becomes invalid, and now, τs ~ τvis becomes valid.

5.5 SPREADING OF NON-NEWTONIAN LIQUIDS OVER SOLID SUBSTRATES

In this section the spreading of drops of a non-Newtonian liquid (Ostwald–de Waele liquid) over horizontal solid substrates is theoretically investigated inthe case of complete wetting and small dynamic contact angles. Both gravitationaland capillary regimes of spreading are considered. The evolution equationdeduced for the shape of the spreading drops has self-similar solutions, whichallows obtaining spreading laws for both gravitational and capillary regimes ofspreading. In the gravitational regime case of spreading, the profile of the spread-ing drop is provided [17].

The spreading of liquids over solid surfaces has been studied from boththeoretical and experimental points of view in Chapter 3 (Section 3.1 and Section3.2), where investigations have dealt with the kinetics of spreading of Newtonianliquids. Both gravitational and capillary spreading regimes have been considered,and the spreading laws have been established. It has been shown in Section 3.1and Section 3.2 that the singularity at the three-phase contact line is removed bythe action of the surface forces (disjoining pressure).

The theoretically predicted spreading laws for gravitational and capillaryregimes have been deduced as and , respectively, where R(t)is the radius of the base of the spreading drop, and t is the time. Comparison ofthe predicted spreading laws with experimental data in Chapter 3 has shownexcellent agreement.

FIGURE 5.21 Spreading of SDS solution over polyethylene substrate, concentrationabove CMC. Dependency of spreading radius on time R(t) = A ·tn, where A and n are fittedparameters. The case n = 0.1 is shown by a solid line.

1.4

1.2

1.0

0.8

0.610 20 30 40

t, s

-C = 0.3%-C = 0.5%-C = 0.7%-C = 1.0%

n = 0.053

n = 0.09n = 0.091n = 0.1

R(t) = A . tn

R(t)/

A

R t t( ) ~ /1 8 R t t( ) ~ /1 10



However, a number of liquids (polymer liquids and suspensions [18,19]) showa non-Newtonian behavior. The aim of this section is to extend the similaritysolution method used in Chapter 3 to the case of spreading of non-Newtonianliquids (Ostwald–de Waele liquids) over solid surfaces and to deduce the corre-sponding spreading laws for both gravitational and capillary regimes of spreading.

GOVERNING EQUATION FOR THE EVOLUTION OF THE PROFILE OF THE SPREADING DROP

The problem is solved under the following assumptions:

1. Complete wetting case.2. The dynamic contact angle is low.3. Reynolds number is low, Re << 1.4. The rheological properties of the liquid are determined by the viscosity

dependency, η(S), on the shear deformation rate, S [18].

The first assumption allows us not to consider the flow in the vicinity of thethree-phase contact line, where the influence of the surface forces become impor-tant. These forces influence only a preexponential factor in the spreading lawaccording to Section 3.1 and Section 3.2. In the case of Newtonian liquids, thepreexponential factor has been found to be almost insensitive to the details of thesurface forces (Section 3.2). This provides a justification of the adopted procedurediscussed in the following text.

The second assumption means that R* >> H*, where R*, H* are characteristicscales in the radial and axial directions, respectively. In the case of the completewetting assumptions, (2) and (3) are always satisfied at the final stage of spreading.

Let us consider a drop of an incompressible non-Newtonian liquid withdensity ρ and surface tension γ, which spreads over a horizontal solid substrate.The density, viscosity, and the pressure gradient in the surrounding air areneglected. The solid substrate is assumed to be rigid and nondeformable.

Both the axisymmetric and cylindrical problems of spreading (Figure 5.22)are considered in the following discussion.

FIGURE 5.22 Cross section of the spreading drop.

r

z

H(t)

R(t)

h(r, t)



The liquid flow inside the spreading drop obeys the following equations:

Incompressibility condition

, (5.92)

and Navier–Stokes equations, which are considerably simplified using assump-tions (1)–(4):

, (5.93)

(5.94)

where (r,ϕ, z) is the cylindrical coordinate system, the z-axis coincides with theaxis of symmetry, and z = 0 corresponds to the solid substrate; all functions areindependent of the angle, ϕ, because of symmetry; g is the gravity acceleration;p is the pressure; and vr and vz are radial and axial components of the velocityvector, respectively.

Let

be components of the deformation rate tensor. The parameter S is expressed interms of the components of the deformation rate tensor as

.

Under assumption (2) this expression becomes

, (5.95)

No-slip conditions are adopted on the solid substrate:

(5.96)

10

r rrv

v

zrz∂

∂ ( ) +∂∂

=

− ∂∂

− =p

zgρ 0

− ∂∂

+ ∂∂ ( ) ∂

∂

=p

r zS

vzrη 0,

Dvrrrr= ∂

∂, D

vrr

ϕϕ = , D Dvz

v

rrz zrr z= = ∂

∂+ ∂

∂

12

, Dv

zzzz=

∂∂

S D D D Drr zz rz= + + +( )2 22 2 2 2ϕϕ

Sv

zr= ∂

∂

2

v zz = =0 0, ;



(5.97)

Let the profile of the spreading drop be z = h(r,t), which is to be determined.Boundary conditions on the free liquid–air interface include the kinematic con-dition

(5.98)

and the conditions for the normal and tangential components of the stress tensor

(5.99)

(5.100)

where pa is the pressure in the surrounding air and small terms are omitted basedon assumption R* >> H*. This assumption means ; that is, the low slopeapproximation is valid.

Using the continuity Equation 5.92 and the kinematic condition (5.98), anequation describing the evolution of the drop profile becomes

. (5.101)

Integration of Equation 5.93 over with boundary condition (5.99) results inthe following expression for the pressure distribution:

(5.102)

This equation shows that ∂p/∂r is independent of z. Integration of Equation5.94 over with boundary condition (5.100) gives

. (5.103)

v zr = =0 0, .

∂∂

= − ∂∂

= ( )ht

v vhr

z h r tz r , , ;

p pr r

rhr

z h= − ∂∂

∂∂

=a γ 1, ;

η Svz

z hr( ) ∂∂

= =0, ,

′ <<h 2 1

∂∂

+ ∂∂

=∫h

t r rr v zr

h1

00

d

z

p p g h zr r

rhr

= + −( ) − ∂∂

∂∂

a ρ γ 1

.

z

η ∂∂

∂∂

= − ∂∂

−( )v

z

v

z

p

rh zr r

2



Integration over of this equation results in the following expression for theradial component of the velocity:

, (5.104)

where the function G(x) is determined as

, . (5.105)

Substitution of the expression for the radial velocity (5.104) into Equation 5.101gives an equation that describes the profile of the spreading drop:

, (5.106)

where an effective viscosity, ηef (y), is determined as

. (5.107)

In the case of the spreading of a cylindrical drop (plane symmetry, two-dimensional drop), a similar consideration using a Cartesian coordinate systemyields the following equation:

, (5.108)

where

. (5.109)

z

v Gp

rh z G

p

rhr = ∂

∂−( )

− ∂∂

∂pp

r∂

G x F y y

x

( ) = ( )∫ d0

η F x F x x2 ( )( ) ( ) ≡

∂∂

= ∂∂

∂∂

∂∂

h

t r rr

hp

r

hp

r

1

3

3

ηef

13

1 2 3

0

ηef

y

y y G y y G z dz− − −= − ∫( ) ( ) ( )

∂∂

= ∂∂

∂∂

∂∂

h

t r

hp

r

hp

r

3

3ηef

p p g h zh

r= + −( ) − ∂

∂a ρ γ2

2



In Equation 5.108 and Equation 5.109, r is the Cartesian coordinate perpen-dicular to the z-axis. The effective viscosity, ηef (y), is determined by the samerelations (5.105) and (5.107) as in the axisymmetric case.

Equation 5.106 to Equation 5.108 and Equation 5.102 to Equation 5.109 canbe rewritten in the following form:

, (5.110)

, (5.111)

where m = 0 and m = 1 correspond to the case of the cylindrical and axisymmetricdrops, respectively.

Substitution of Equation 5.111 into Equation 5.110 results in the followingnonlinear differential equation, which describes the evolution of the profile of thespreading drop:

, (5.112)

Conservation of the drop volume reads:

, , (5.113)

where V is the drop volume in the axisymmetric case and the cross section areain the case of cylindrical drops; R(t) is the location of the three-phase contact line.

In the case of Ostwald–de Waele liquid [18]:

η(S) = k S (n-1)/2, n > 0, (5.114)

where n < 1 corresponds to pseudoplastic fluids (can be as low as 0.1 for somenatural rubbers [18]), and n > 1 corresponds to dilatant fluids.

∂∂

= ∂∂

∂∂

∂∂

h

t r rr

hp

r

hp

r

mm1

3

3

ηef

p p g h zr r

rh

rmm= + −( ) − ∂

∂∂∂

a ρ γ 1

∂∂

= ∂∂

∂∂

− ∂∂

∂∂

∂∂

h

t r rr

h gh

r r r rr

h

rm

mm

m

1

13 ρ γ

∂∂

− ∂∂

∂∂

∂3

1η ρ γef h gh

r r r rr

mm hh

r∂

V h r t r rm m

R t

= ( )( )

∫20

π , d h R t t( )( ) =, 0



In this case, ηef (y) is

(5.115)

and Equation 5.112 transforms into

The position of the three-phase contact line, R(t), is the time-dependentcharacteristic horizontal scale. Let us introduce the time-dependent characteristicthickness of the drop, H(t), using conservation law (5.113) as V = 2πm H(t) Rm+1(t),or

. (5.117)

Self-similar solutions of Equation 5.116 are tried in the following text in thefollowing form:

, . (5.118)

According to Equation 5.113 and Equation 5.117, the dimensionless dropprofile, ˆζ(r), satisfies the following conditions:

(5.119)

In the following two sections, two limiting cases of spreading are considered,when either capillary or gravitational forces can be neglected. In the case ofgravitational regime of spreading, the capillary forces can be neglected if

.

ηefn

n

nyn

nk y( ) = + −2 1

3

1 1

,

∂∂

=+

∂∂

∂∂

− ∂∂

−

+

ht

nn

kr r

r h sign ghr r

nm

mn

n

2 111

2 1

ρ γ 11 1

r rr

hr

ghr rm

m∂∂

∂∂

∂∂

− ∂∂

ρ γrr r

rhrm

mn∂

∂∂∂

1

.

(5.116)

H tV

R tm m( ) = ( )+2 1π

h r t H t r, ˆ( ) = ( ) ( )ζ r R t r= ( ) ˆ

ζ ζˆ ˆ ˆ , .r r rm( ) = ( ) =∫ d 1 1 00

1

R t ag

( ) >> = γρ



In the capillary regime of spreading, the capillary forces dominate, that is,

.

GRAVITATIONAL REGIME OF SPREADING

In this case, the spreading Equation 5.116 transforms into

.

Assuming the solution in the form (5.118), this equation yields

(5.120)

The preceding equation shows that the radius of spreading, R(t), should satisfythe following equation:

, (5.121)

where is a dimensionless constant. If R(t) is selected according to Equation5.121, then Equation 5.120, which describes the dimensionless drop profile, ˆζ(r),becomes

(5.122)

Equation 5.122 should be solved with the following boundary conditions:

, (5.123)

which is the symmetry condition in the drop center, and

R t ag

( ) << = γρ

∂∂

=+

∂∂

∂∂

+h

t

n

n

g

k r rr h sign

hn

mm

n

n

2 11

1 2 1ρ/

rr

h

r

n

∂∂

1/

− + + ′ =

+

+V R

Rm r

nn

gk

m m21

2 1

2πζ ζ

ρ

�( ) ˆ

∂∂

+

+( ) +( )

1 2 2

2 3 12

1 1n

m

n

n

m n

n

m

V

Rr rπ ˆ ˆ

r̂r mn

n nζ ζ ζ2 1 1+

′( ) ′

sign .

�Rn

n

g

k

VR

n

m

n nn=

+

+( )− +λ ρ

π2 1 2

1 22

/ /(( ) +( )+ m n1 1 /

λ

λ ζ ζ ζ ζ ζm rr

rm

m n n+( ) + ′ = − ′ ′+( )11 2 1 1

ˆˆ

ˆ sign/ //.

n( )′

′ =ζ ( )0 0



. (5.124)

Conservation law (5.119) gives the third equation for the determination ofthe unknown parameter λ.

The solution of Equation 5.121 with initial condition R(0) = R0 is

, (5.125)

where the spreading exponent is

. (5.126)

In the case of the Newtonian liquid (n = 1), this exponent is 1/(3m + 5).Thus, at n < 1: α < 1/(3m + 5), and the drop spreads slower than the Newtonianliquid; at n > 1: α > 1/(3m + 5), and the drop spreads faster than the Newtonianliquid. The dependence of the spreading exponent α on n, according to Equation5.126, in the case of axisymmetric spreading (m = 1), is shown in Figure 5.23.

Multiplying Equation 5.122 by , and after integration, taking into accountthe symmetry condition (5.123), results in

,

FIGURE 5.23 Axisymmetric (m = 1) gravitational regime of spreading. Spreading expo-nent α (Equation 5.126) vs. n; n = 1 — Newtonian fluid.

ζ( )1 0=

R t Rn

n

g

k

Vn

m

n

( ) = ++

+( )0

1 2

12 1 2

λα

ρπ

/ //

/

n

Rt

1

01 α

α

α

α =+( ) + −

n

m n2 2 1( )

r̂ m

λ ζ ζ ζ ζˆ ˆ sign/ /r rm m n n n+ +( )= − ′ ′1 2 1 1

1/8

0.010.0

0.1

0.2

0.3

0.4

1/3

α

0.1 1 10 100n



The solution of this equation with boundary condition (5.124) is

. (5.127)

Substitution of solution (5.127) into conservation law (5.119) gives an equa-tion for the determination of the unknown constant λ:

,

or, in terms of the gamma function (5.112),

.

The preceding equation has the following solution:

. (5.128)

It is possible to check that If n → 0, then at m = 0:

;

at m = 1:

.

Dependence of λ on n at m = 1 is shown in Figure 5.24.Substitution of the expression (5.128) into the solution (5.127) gives the

dimensionless drop profile in the case of the gravitational regime of spreading:

ζ λ λ= ++

−( )

+( ) ++( )

sign ˆ/

/n n n

nn

nr

2 1

1 221

1

sign/

/

/λ λ

n nn

n n

n

n

+( )+( )

+( ) +( )+( )

+( )−

21 2

3 2

2

11 xx x x

n m n n( ) =+( ) −( ) +( )∫ 1 2 1

0

1

1/ / d

sign/

/

( )/λ λ

n nn

n n

n

n

n

+( )+( )

+ +( )+( )

+( )2

1 2

3 2

2

1

Γ +++

++

+( ) +( ) + +

32

11

3 1 1

n

m

n

n n m

Γ

Γ( )(nn

n n

+

+( ) +( )

=2

1 2

1)

λ = ++

+( ) +( ) + + ++( )

( )( )

n

n

n n m n

n n12

3 1 1 2

31 Γ

nn n

n

n

m

n

+( ) +( )

++

++

1 2

32

11

Γ Γ

+n

n

2

λ → + → ∞m n1, .if

λ ~ ./

1 012198998

1

→ ∞n

λ ~ ./

1 763584622532

1

⋅

→ ∞n



. (5.129)

In the case of axisymmetric spreading (m = 1) of a Newtonian liquid (n = 1),this equation gives

, (5.130)

which coincides with the solution obtained in Chapter 3, Section 3.1. The profilesof axisymmetric spreading drops at different n according to Equation 5.129 areshown in Figure 5.25.

CAPILLARY REGIME OF SPREADING

In this case, Equation 5.116 becomes

(5.131)

FIGURE 5.24 Axisymmetric (m = 1) gravitational regime of spreading. Dimensionlessconstant λ (Equation 5.128) vs. n; n = 1 — Newtonian fluid. Solid line = according toEquation 5.128; broken line = asymptotic dependence λ ~ 1.7635846·(225/32)1/n at n << 1.

00 10 20 30 40 50

2

4

6

8

10

λ

n

–1–2–3

ζ ˆ ( )

( )( )

r n

n n m n

n n( ) = +

+( ) +( ) + + +

+( ) +( )1

3 1 1 2

1 2Γ

++

++

−( +

Γ Γn

n

m

n

r n

32

11

1 1ˆ )) +( )1 2/ n

ζ = −( )83

1 21 3

ˆ/

r

∂∂

= −+

∂∂

∂∂

+

ht

nn k r

rr h sign

r

n

m

mn

n

2 11

1

2 1

γ/

11 1

r rr

hr r r r

rm

mm

m∂∂

∂∂

∂∂

∂∂

∂hhr

n

∂

1/

.



Assuming the solution of Equation 5.131 in the form (5.117) and (5.118) yields

This equation shows that a self-similar solution exists if the radius of spread-ing, R(t), satisfies the following equation:

. (5.133)

In this case, the dimensionless drop profile is determined by the followingordinary differential equation:

,

(5.134)

FIGURE 5.25 Axisymmetric (m = 1) gravitational regime of spreading. Drop profiles(Equation 5.129) at different values of n: - - - - -, n = 1; – - – - –, n = 0.5; ----------, n = 2.

1

0 1n = 1n = 0.5n = 2

2

4 ζ

r

∧

∧

− + + ′ = −+

+

V R

Rm r

nn km m2

12 12

1

πζ ζ γ�

( ) ˆ/nn

m

n

n

m n n m m

V

R r

r

2

1 12 2

3 2 5 2π

× ∂∂

+

+ + +( / / ) ˆ

ˆˆ̂ sign

ˆˆ

ˆˆr

rr

rrm

mm

mm1 1′( )′

′

′( )′

ζ ζ

′

+( )1

2 1 5 132

/

/ . ( . )

n

n nζ

�Rn

n k

VR

n

m

n n n

=+

+( )−

+ +

λ γπ2 1 2

1 2 5/ / mmn m

n

+2

λ ζ ζ ζm rr

rr

rm

mm

m+( ) + ′ = ′( )′

1

1 1ˆˆ

ˆ signˆ

ˆ

′

′( )′

′

+( )1

1

2 1

ˆˆ

/

/

rr

mm

n

n nζ ζ

′



where is a dimensionless constant, which is different from the previous case(gravitational regime of spreading).

The solution of Equation 5.133, with the initial condition R(0) = R0, has thefollowing form:

, (5.135)

where the spreading exponent is

, (5.136)

which gives 0.1 in the case of the axisymmetric spreading of a Newtonian liquid(m = n = 1) and 1/7 in the case of the cylindrical drop spreading (n = 1, m =0). The dependence (5.136) in the case of axisymmetric spreading, m = 1, isshown in Figure 5.26.

Comparison of Equation 5.126 and Equation 5.136 shows that capillary andgravitational regimes of spreading give the same dependence ifn >> 1.

Multiplying Equation 5.134 by and integrating yields

,

FIGURE 5.26 Axisymmetric (m = 1) capillary regime of spreading. Spreading exponentα (Equation 5.136) vs. n.

1001010.10.010.0

0.1

0.2

0.3

0.4

1/3

n

α

λ

R t Rn

n k

Vn n n

mn( ) = +

+

( )

+( )+0

1 2

21

2 1 2

λα

γ

π

/ /

(( )

//

nR

t

01 α

α

α

α =+ + +

n

mn m n2 5 2

R t t m( ) ~ /( )1 2+

r̂ m

λ ζ ζ ζˆ ˆ signˆ

ˆˆ

ˆr rr

rr

rm mm

mm

m+ = ′( )′

′

′1 1 1 (( )′

′

++( )1

2 1

/

/

n

n n Cζ



where C is an integration constant. Taking into account that the functions andshould be finite and the symmetry condition in the drop center, ζ′(0) =

this equation becomes

(5.137)

According to Section 3.2, an alternative way of solving Equation 5.131 is asfollows. The whole drop is subdivided into two parts: the main spherical part(outer solution) and the narrow region close to the three-phase contact line (innersolution). The volume of the liquid in the narrow inner zone can be neglected ascompared with the main spherical part. In order to determine the liquid flow inthe inner region, the surface forces action should be introduced in this narrowregion. However, the solution in the inner region gives only a preexponentialfactor in the spreading law. Its dependence on the details of the flow has beenfound insignificant in the case of Newtonian liquids and compete wetting. In thecase of non-Newtonian liquids and the complete wetting case, surface forces inthe vicinity of the three-phase contact line can be of a very complex nature. Weassume that the influence of these complex and unknown surface forces givesonly a correction of a preexponential factor as in the case of Newtonian liquids.That is why the flow in the inner region is not considered in the followingdiscussion.

Accordingly, the right-hand side of Equation 5.137 is small everywhere exceptfor a narrow vicinity of the three-phase contact line, where approaches zero,and hence, the volume of the liquid in this small region can be neglected. Thus,the central part of the spreading drop is

, (5.138)

that is, the parabolic cap. The dynamic contact angle, θ, is determined by thefollowing relationship (tanθ ≈ θ):

. (5.139)

From Equation 5.133,

,

ζ( ˆ)r′′ζ ( ˆ)r′′′ =ζ ( ) ,0 0

1 1

ˆˆ ˆ .

rr r

mm n n n′( )′

′

= − −ζ λ λ ζsign

ζ

ζ =+( ) +( )

−( )m mr

1 3

21 2ˆ

θπ

= + ++

( )( )m m V

Rm m

1 3

2 2

U

nn k

Vn

m

n n

λ γπ2 1 2

1 2

+

+( )/ /

=

++ + +

− +

n m

n mn mmR

( )

( )

25 2

2



where . Substitution of this equation into Equation 5.139 results in thefollowing dependence of the dynamic contact angle of the spreading drop on therate of spreading, U:

,

(5.140)

or

. (5.141)

For the case of an axisymmetric spreading of a drop of Newtonian liquid(n = m = 1), Equation 5.141 gives the Tanner’s law (see Chapter 3, Section 3.1and Section 3.2). It is interesting to note that spreading law (5.141) is independentof the drop volume only in the case of Newtonian liquids. If n ≠ 1 then the right-hand side of Equation 5.141 depends on the drop volume.

DISCUSSION

The spreading of drops of non-Newtonian liquids (Ostwald–de Waele liquids)over horizontal solid substrates was theoretically investigated. An equation wasdeduced that describes the liquid profiles of axisymmetric and cylindrical spread-ing drops. The problem was solved under the following assumptions: (1) completewetting, (2) low dynamic contact angle approximation, (3) low Reynolds number,Re << 1, and (4) the rheological properties of the liquid are determined by theviscosity dependency, η(S), on the shear deformation rate, S. In the case ofcomplete wetting, the second and third assumptions are always valid at the finalstage of spreading and allow a considerable simplification of the description ofthe spreading.

Both gravitational and capillary spreading regimes of the spreading wereconsidered. In the gravitational regime case, the spreading law and the profile ofthe spreading drop have been completely determined and given by Equation 5.125and Equation 5.129, respectively. In the case of the capillary regime, the spreadinglaw and the apparent contact angle of the spreading drop have been calculatedand given by Equation 5.135 and Equation 5.140, respectively.

In the case of a Newtonian liquid (n = 1), the spreading laws for gravitationaland capillary regimes, Equation 5.125 and Equation 5.135, coincide with thosefound earlier in Chapter 3. If n < 1, an axisymmetric drop spreads slower than adrop of a corresponding Newtonian liquid with the same volume. If n > 1, thespreading exponents for gravitational and capillary regimes are greater than thosein the case of Newtonian liquids.

U R t= �( )

θλ π

= + + +

++ + +

( )( )

( )

m mn

n

Vn m

n mn m

m1 3

2 1

2

2

5 2

−+ + +

++ + +

1

5 2

2

5 2

n

n mn m

m

n mn m n mkU

γ

( +++ + +

2

5 2

)

n mn m

Un

n k V m

n mn

n m

=+

+

−+λ γ π θ

2 12

1

1 1

2( )

( )(( )

( )

m

n mn m

n m

+

+ + ++

3

5 2

2



It is interesting to note that both capillary and gravitational axisymmetricregimes of spreading give the same power law if n >> 1.

5.6 SPREADING OF AN INSOLUBLE SURFACTANT OVER THIN VISCOSE LIQUID LAYERS

In this section we present the results of the theoretical and experimental studyof the spreading of an insoluble surfactant over a thin liquid layer. Initial concen-trations of surfactant above and below CMC have been considered. If the con-centration is above CMC, two distinct stages of spreading are found: (1) the firststage is the fast one, and it is connected with micelles dissolution, and (2) thesecond stage is the slower one, when the surfactant concentration becomes belowCMC. In the second stage, the formation of a dry spott in the center of the filmis observed. A similarity solution of the corresponding equations for spreadinggives good agreement with the experimental observations [13].

When a drop of a surfactant solution is deposited on a clean liquid–airinterface, tangential stresses on the liquid surface develop. They are caused bythe nonuniform distribution of surfactant concentration, Γ, over a part of theliquid surface covered by the surfactant molecules, leading to surface stressesand flow (Marangoni effect) [12]:

, (5.142)

where η and u are the liquid dynamic shear viscosity and tangential velocity onthe liquid surface located at height h, respectively; (r,z) are radial and verticalcoordinates; and γ(Γ) is the liquid–air interfacial tension whose linear dependencyon surfactant surface concentration we assume in the following discussion. Thesurface tension gradient-driven flow induced by the Marangoni effect moves thesurfactant along the surface, and a dramatic spreading process takes place. Thenthe liquid–air interface deviates from an initially flat position to accommodatethe normal stress also occurring in the course of motion.

We restrict considerations in the following discussion to insoluble surfactants.Note that though a surfactant may be soluble, there are cases such that nonsolu-bility conditions can be used during a certain initial period in the spreadingprocess. Let us consider two characteristic time scales associated with surfactanttransfer: (1) τd accounts for the transfer from the liquid–air interface to the bulk,and (2) τa accounts for the transfer from the bulk back to the interface. In bothcases these characteristic time scales depend on an energy difference betweencorresponding states. Let us consider an aqueous surfactant solution: the latterhas both a hydrophilic head and a hydrophobic tail. Let Ehl, Eta, Etl be the energies(in kT units) of head–water, tail–air, and tail–water interactions. Using thesenotations, the energy of a molecule in an adsorbed state at the interface is Ead =Eta + Ehl, whereas, for the same molecule in the bulk, it is Eb = Etl + Ehl. Then,

R t t( ) ~ /1 3

η∂

∂γ ∂

∂u r h

z

d

d r

,( )=

ΓΓ



τd = τ⋅exp(Eb – Ead) = τ⋅exp{Etl – Eta}, and τa = τ⋅ exp(Ead – Eb) = τ⋅exp{–(Etl –Eta)}, where τ is determined by thermal fluctuations. Generally, the tail–waterinteraction energy is considerably higher than that of the tail–air interaction( where is an interaction energy per hydrophobic unit, and n is anumber of those units in each tail). Consequently, τd /τa = exp{2(Etl – Eta)} >> 1,and transfer from the interface to bulk is a much slower process than the reverseone. If the duration of a spreading experiment is shorter than τd, then, during thatexperiment, the surfactant can be considered as insoluble. Otherwise, if t > τd,the solubility of the surfactant in the liquid must be taken into account. In thelatter case, surfactant transfer to the bulk liquid tends to make concentrationuniform both in the bulk and at the interface, and the result is a substantialdecrease of the surfactant influence of that type (Marangoni flow).

Usually, surface diffusion can be neglected as compared to convective transfer(5.163). Indeed, from Equation 5.142, we have as a characteristic scale of surfacevelocity:

,

where are characteristic scales of interfacial tension, initial film thick-ness, and length in a tangential direction, respectively. The diffusion process overthe liquid surface scales is

,

where are the surface diffusion coefficient and a characteristic scale ofsurfactant concentration on the surface, respectively. Then, the ratio of diffusion toconvective flux can be estimated as 1/Pe = ∼ 10–8 << 1, for Ds ∼ 10–5 cm2/s(5.164) and γ* ≈ 102 dyn/cm. Here, Pe is the mean Peclet number. This estimationshows that surface diffusion can be neglected everywhere except for a smalldiffusion layer, which is disregarded in the following discussion.

We now consider two different cases: (1) when concentration in a droplet ofsurfactant solution, which is placed in the center of a liquid film, is above CMC,and (2) when such concentration is below CMC. In the first case, the spreadingprocess involves two stages: (1) the faster one when the surfactant concentrationis determined by the dissolution of micelles. This stage yields the maximumattainable surfactant concentration in the film center, and it is independent oftime. The duration of that stage is fixed by the initial amount of micelles in thedrop. (2) The second slower stage takes place when the surfactant concentrationchanges in the film center, but the total mass of surfactant remains constant. Inboth cases, a similarity solution provides a power law predicting the position ofthe moving front as time proceeds.

E nEtl tl≈ 1, Etl1

uh

L** *

*

≈ γη

γ * * *, ,h L

DL

s

Γ*

*2

Ds , *Γ

D Hsη γ/ * *

r tf ( )



THEORY AND RELATION TO EXPERIMENT

The motion of a thin liquid layer with initial thickness is considered underthe action of an insoluble surfactant on its open surface. For simplicity we assumethat surface tension varies linearly with the surface concentration of surfactant,

(5.143)

where is the interfacial tension of the pure water–air interface, and corre-sponds to the maximum attainable surface concentration (in equilibrium with amicelle solution).

We use, in the following discussion, dimensionless parameters and variables,and we use the same symbols as for dimensional quantities. The subscript isused to mark initial or characteristic values.

We further assume that ε = H*/L* << 1; hence, neglect of the nonlinear partof the interface curvature. A dimensionless Bond number, accountsfor the ratio of the gravitational force to the Marangoni forcing, where ρ is theliquid density, and g is the gravity acceleration. In our experiments, a water filmwith H* ≈ 0.1 mm is used, the Bond number is about 5*10–3 << 1 and hence, thegravity action can be safely neglected.

Under these conditions, the evolution equations for mass balance for waterand surfactant concentration on the surface are as follows:

(5.144)

(5.145)

where h(t,r) is the film thickness at time t; r is the radial coordinate, and Γ(t,r)the surfactant concentration on the surface. Equation 5.144 and Equation 5.145,after performing the integration, become (see Appendix 2 for details):

(5.146)

(5.147)

and are to be solved subject to the following boundary conditions:

H*

γ γ α( ) , ,*Γ Γ Γ Γ= − < <at 0 m

γ * Γm

*

ρ αgH* *,2 / Γ

∂∂

∂∂

ht r r

ru dz

h

+ =∫10

0

,

∂∂

∂∂

Γ Γt r r

ru t h+ ( )( ) =10, ,

∂∂

∂∂

γη

∂∂

γ ∂∂

∂∂

ht r r

rh

r r rr

hr

= −

−1

3

3 ααη

∂∂

hr

2

2Γ

,

∂∂

∂∂

γη

∂∂

γ ∂∂

∂∂

Γ Γt r r

rh

r r rr

hr

= −

1

2

2

−−

αη

∂∂

hrΓ

,



(5.148)

(5.149)

(5.150)

(5.151)

Let us introduce the following dimensionless variables and values:

The time scale deserves a comment. The capillary number for the spreadingprocess under consideration is very small:

∼ = .

On the other hand,

;

hence, a new time scale can be introduced as

,

where are, respectively, the time scale that actually governs the spreadingprocess and the characteristic velocity scale. Using these estimations we conclude:

for ,

∂∂ ∂

∂hr r

rh= = =3

30 0, ,at

h r→ → ∞1, ,at

∂∂Γr

r= =0 0, ,at

Γ → → ∞0, .at r

hh

Hr

rL

ttt

tL

H

→ → → →

= =

* * * *

**

* *

*

, , , ,

,

Γ ΓΓ

Γηα

β ε γ22

ααΓ*

.<< 1

t*

CaU= η

γ *

10 10 102 1 2− − / 10 15− <<

CaU L= =ηγ

ηγ τ

*

* *

*

*

τ ηγ*

*

*

= L

Ca

τ* *,U

t Ca*

*

*

*τγα ε

δ= = ≈ <<−

Γ10 13 ε ≈ −10 2



as it is the case in the experiments discussed in the following text. If we nowintroduce the dimensionless time

,

then in Equation 5.144 and Equation 5.145 we have

.

We show in the following text that the time evolution of the position of the movingfilm front is

∼ .

If we take into account the initial value, �, of rf (t), this dependence takes the form

,

where � ∼ 1 represents the contribution of the initial condition. According to ourchoice τ ∼ 1; hence,

,

and thus,

Multiplying Equation 5.146 by r, we conclude, after integration, that

(5.152)

ττ

= t

*

δ ∂∂τ

∂∂

=t

r tf ( ) t0 5 0 25

0 5 0 25

. .

. .

÷÷

=

τδ

rf ( ). .

τ τδ

≈ +

÷

�0 5 0 25

τδ

>> �

r t tf ( ) ~ .. .

. .τδ

=÷

÷0 5 0 25

0 5 0 25

r h dr−( ) =∞

∫ 1 00

,



which reflects the conservation law for the liquid.In our experiments, a droplet of surfactant solution with concentration above

CMC was placed in the center of a water film. Experimental observations showthat two distinct stages of spreading take place: (1) a first faster stage, and (2) asecond, slower stage. During the first stage, there is dissociation of micelles;hence, surface concentration of single molecules is kept constant during thatstage, and . Choosing as a characteristic scale for surfactantconcentration, we have, in dimensionless form, the following boundary conditionduring the first stage:

(5.153)

The first stage lasts until all micelles are dissolved. The duration of thatstage is considered in the following discussion. Past , a second stage starts.During the second stage, the total mass of surfactant remains constant; hence,the following boundary conditions apply:

(5.154)

and

(5.155)

where the characteristic scale for Γ is now selected as

,

with being the total amount of surfactant molecules in the droplet.

The First Spreading Stage

The spreading process in this case is described by Equation 5.146 and Equation5.147 with boundary conditions (5.148–5.151, and 5.153). According to Appendix 3,the influence of capillary forces can be neglected for t >> β, and Equation 5.146and Equation 5.147 become

(5.156)

Γ Γ( , )t m0 = Γ Γ* = m

Γ( , ) .t 0 1=

t1*t1*

∂∂Γr

r= =0 0, ,at

r drΓ =∞

∫ 10

,

Q

L*

*2 2π

Q *

∂∂

∂∂

∂∂

ht r r

rh

r=

12

2 Γ,



(5.157)

Equation 5.156 and Equation 5.157 cannot satisfy the boundary conditionsat r → ∞, and a shock-like spreading front forms (for the derivation of theseconditions see Appendix 3).

In our case, h+ = 1, Γ+ = 0; hence,

.

Then, using conditions (A3.3 and A3.4), we get

(5.158)

(5.159)

Equation 5.159 actually implies two conditions:

(5.160)

(5.161)

The matching of asymptotic expansions at the moving shock front (seeAppendix 5) shows that both conditions (5.160, 5.161) must be satisfied.

Let us now introduce a new variable

Then we have

(5.162)

where 0 < r < ; the constant ν is determined in the following discus-sion. In condition (5.152), the upper limit of integration must be replaced by ν;hence,

∂∂

∂∂

∂∂

Γ Γ Γt r r

r hr

=

1

.

∂∂Γ+ =r

0

�r h hrf ( ) ,− −−− = −1

12

2 ∂∂Γ

Γ Γ− −

−+

=�r hrf

∂∂

0.

Γ− = 0,

�r hrf = − −−∂

∂Γ

.

ξ = r

t1 2/.

h t r f t r( , ) ( ), ( , ) ( ),= =ξ ϕ ξΓ

r t tf ( ) = ν



(5.163)

which is compatible with definitions (5.162). Equation 5.156 and Equation 5.157,after using Equation 5.162, become:

(5.164)

(5.165)

with the corresponding boundary conditions that follow from Equation 5.158,Equation 5.160, and Equation 5.161. We have

These boundary conditions, after simple transformations, become

(5.166)

If we again change the variables using

, (5.167)

the new functions Ψ and G satisfy the same equations (5.164, 5.165) with thevariable µ, where 0 < µ < 1, and the following boundary conditions at µ = 1:

Ψ(1) = 2, G(1) = 0, and G′(1) = π. (5.168)

Then the problem does not depend on the unknown parameter ν.

ξ ξ ξ νν

f d( ) ,=∫2

02

− ′ = ′( )′ξ ξξ

ξ ξ ϕ ξf f( ) ( ) ( ) ,1 2

− ′ = ′( )′12

1ξϕ ξξ

ξ ξ ϕ ξ ϕ ξ( ) ( ) ( ) ( ) ,f

ν ν ν ϕ ν

ϕ ν

ν ν ϕ ν

f f

f

( ) ( ) ( ),

( ) ,

( ) (

− = − ′

=

= − ′

1

0

2

2

)).

ϕ ν

ν

ϕ ν ν

( ) ,

( ) ,

( ) .

=

=

′ = −

0

2

4

f

µ ξν

µ ϕ ν µ= = =, ( ), ( )f GΨ 2



Calculated dependencies of dimensionless film thickness Ψ(µ) and surfaceconcentration G(µ) are presented in Figure 5.27, which shows that a substantialdepression is formed in the film center. During the second stage, the depressionbecomes a dry spot right in the middle of the film.

In order to determine the unknown constant, ν, let us consider the total massof surfactant, Q(t), during the first spreading stage. We have

or

(5.169)

The left-hand side of Equation 5.169 does not depend on time t; hence, thesame is true for the right-hand side. Let us denote by q a constant value to beexperimentally determined from the duration of the first stage. Then, Equation5.169 takes the form

(5.170)

FIGURE 5.27 Theoretical predictions for dimensionless profile Ψ(µ) and surfactant sur-face concentration G(µ) during the first spreading stage (calculated according to equationsand boundary conditions (5.164, 5.165, 5.168)).

8

6

4

2

0.80.60.40.2µ

ψ

G

Q t r t r dr L t d( ) ( , ) ( ) ,* *= =∞

∫ ∫2 20

2

0

π π ξϕ ξ ξν

Γ Γ

ξϕ ξ ξπ

ν

( )( )

.* *

dQ t

L t=∫ 2 2

0Γ

ξϕ ξ ξν

( ) .d q=∫0



On the other hand, according to definition (5.167),

that may fix the ν value. Unfortunately, the integral

diverges due to the singularity of the dependence G(µ) at µ = 0. Hence, we canonly write that ν4 ∼ q, or

(5.171)

Let be a dimensional time scale of the duration of the first spreading stage;t1*/t* is the corresponding dimensionless time. If we choose t* = t1*, then, usingthe definition of the time scale, we get the corresponding value of the tangentiallength scale


(5.172)

where is the total amount of surfactant that is initially placed on the filmsurface, which is supposed to be known, are the drop-let volume and surfactant concentration in the droplet, respectively.

Unfortunately, the derived similarity solution cannot satisfy boundary condi-tion (5.153) at the origin as the concentration dependence on radial coordinatediverges in a vicinity of the origin. Thus, it is more convenient to redefine acharacteristic scale of surfactant surface concentration from Equation 5.172 usingthe condition q = 1 in Equation 5.172 (this choice gives the same characteristicscale during both stages of the spreading process). The ν value is still undeter-mined, but we shall show later on how to determine it. At time , the secondstage of the spreading process starts.

ν η η η4

0

1

G d q( ) ,∫ =

η η ηG d( )0

1

∫

ν ≈ q1 4/ .

t1*

Lt H

** * *

/

.=

1

1 2α

ηΓ

qQ

L= *

* *

,2 2π Γ

Q*

Q V C V C* * * * *= , where and

t1*



The Second Spreading Stage

In this stage, the film profile and the surfactant concentration obey the samesystem of equations (5.146) and (5.147) with boundary conditions (5.148 to 5.151)and (5.154 and 5.155).

Let us introduce

then the solutions of Equation 5.146 and Equation 5.147 are

where two unknown functions obey the following system ofequations:

(5.173)

(5.174)

Equation 5.174 can be integrated using condition (5.151), which gives ϕ(ξ) →0, at ξ → ∞, that is,

Thus, either

(5.175)

or

(5.176)

ξ = r

t1 4/;

h t r f t rt

( , ) , ( , )( )

,/

= ( ) =ξ ϕ ξΓ1 2

f ( ) ( )ξ ϕ ξand

ξ ξ ξ β ξξ

ξ ξ ξ23

243

12′ = ′( )′

′− ′f

ff f( )

( )( ) ( )ϕϕ ξ( ) ,

′

( ( )) ( ) ( ) '( )ξ ϕ ξ ξϕ ξ β ξξ

ξ ξ2 221

4′ = ( )′

′−f f f (( ) ( ) .ξ ϕ ξ′

′

ξϕ ξ ξ β ξξ

ξ ξ ξ ϕ ξ( ) ( ) ( ) ( ) ( )− ′( )′

′− ′2

142f f f

= 0.

ϕ ξ( ) ,= 0

′ = − + ′( )′

′

ϕ ξ ξξ

β ξξ

ξ ξ( )( )

( ) ( ) .4 2

1f

f f



In the first case, from Equation 5.173 we conclude that

(5.177)

which is valid at the periphery of the spreading part of the film.Equation 5.177 describes decaying capillary waves on the film surface.

Indeed, if we introduce a new local variable ζ = (ξ – λ)/χ with

near the moving edge λ (to be defined in the following discussion), we get fromEquation 5.177

The asymptotic behavior of this equation, according to Chapter 3 (Section3.5), yields

which describes decaying capillary waves ahead the advancing front.In the opposite case, when Equation 5.176 is valid, we obtain from Equation

5.173 using Equation 5.176 that

(5.178)

The value of λ is determined as a point where ϕ(λ) = 0, or from Equation5.176 and condition (5.155):

(5.179)

ξ ξ β ξ ξξ

ξ ξ2 343

1′ = ′( )′

′

f f f( ) ( ) ( )

′

,

χ βλ

=

43

1 3/

′′′ = −f

f

f( )

( )

( ).ς ς

ς1

3

f e A A( ) ( cos sin ), ,ς ς ς ςς

≈ + + → ∞−

132

32

21 2 at

ξ ξ ξ β ξξ

ξ ξ ξ ξ23

31

2′ = ′( )′

′

+

f

ff

f( )

( )( )

( )

′

.

24 2

1

0

=

− ′( )′

′

∫ ξ

ξβ ξ

ξξ ξ

λ

ff f

( )( ) ( ) ddξ.



The solution of Equation 5.176 is

where the integration constant is vanishing in accordance with Equation 5.179.It is easy to find a solution of the problem under consideration in the zeroth

approximation by setting β = 0 in Equation 5.176, Equation 5.177, and Equation5.179 (see Appendix 5). It is also possible to find an exact expression for λ usinga zeroth order solution from Appendix 6 and Equation 5.176, Equation 5.177,and Equation 5.179, but it is not our aim.

In conclusion of the theoretical consideration, let us summarize the resultsobtained. For the dimensional radius of the moving axisymmetric front, we havefor the first stage of spreading

(5.180)

with ν still undetermined. For the second stage, we find

(5.181)

where (sec) is the duration of the first stage of spreading,

As the front position must be the same at time according to both Equation5.180 and Equation 5.181, then ν = λ, and Equation 5.180 becomes

(5.182)

Our theory predicts that the layer thickness decreases in the center with avanishing value at the origin, which is the dry spot. When comparing withexperiments, we must take into account both the finite precision in the measure-ments of the film thickness and the possibility of evaporation during the experi-ment that may lead to discrepancy between our theory predictions and experi-mental results. Indeed, under experimental conditions, we can measure the film

ϕ ξ ξξ

β ξξ

ξ ξ( )( )

( ) ( )= − ′( )′

′

4 2

1f

f f∫ dξ

ξ

λ

,

r Lt

tt tf =

≤ν **

/

*, ,1

1 2

1(cm)

r Lt

tt tf =

≈ ≥λ λ**

/

/*, , ,

1

1 4

5 412(cm)

t1*

LH t Q

L*

* * *

/

**

*

, .=

=αµ πΓ Γ1

1 2

22

t1*

r Lt

tt tf =

≤λ **

/

*, .1

1 2

1(cm)



thickness only for values higher than a certain thickness, h*; we can consider thatthicknesses below h* constitute the dry spot. Using a zeroth order solution ofEquation A6.3, we obtain that

(5.183)

where rd (t) is the dry spot radius, whose motion according to the precedingequation obeys the same power law as the front of the film.

Moreover, evaporation has more pronounced influence at smaller thicknesses(h < h*) and that influence progressively increases with time. Hence, evaporationin the liquid film near the center of the layer helps further film thinning duringthe second stage.

Experimental Results

Observations of the spreading of a surfactant on a thin layer of liquid have beenperformed using aqueous solutions of sodium dodecyl sulfate at a concentrationc = 20 g/l above CMS (the critical micellar concentration of the SDS is 4 g/l).The thin layer of liquid is prepared by coating the bottom of a borosilicate glassPetri dish of diameter 20 cm with 10 ml of distilled water. The resulting thicknessis then H = 0.32 ± 0.01 mm. If carefully washed, this layer does not dewet duringthe time of the experiment. With a syringe, a drop of the surfactant solution,volume 3 µl, is put on the surface of this water layer. When touching the watersurface, the surfactant spreads on it, and this motion is followed using a smallamount of talc powder as a marker and a 25-Hz video camera to record it.

The spreading of surfactants makes the water flow away from the initiallocation of the drop, thus creating a depression where only a thin film of liquidsubsists. The periphery of this depression, i.e., the liquid front, has a sharpincrease in thickness. If the layer is horizontal and the drop is carefully placed,there is no preferred direction, and the edge is circular but some modulation mayappear in the experiment after a few seconds. Note that the surfactant occupiesmore surface area than the depressed zone as the talc powder is pushed ahead ofit. The dependence on time of the radius of the surfactant patch is given in Figure5.28a using log–log plot. The two successive stages described earlier in our theoryare clearly seen in Figure 5.28a. First, the short period when the surfactant spreadsfollowing the power law ∼ t 0.60 ± 0.15. The exponent is not determined with ahigh precision because this first stage is too fast (about 0.1 sec), which is onlythree times the time resolution (0.04 sec). The value 0.60 ± 0.15 agrees well withthe earlier given theoretical prediction 0.5, Equation 5.182.

At the end of the first stage, the motion abruptly slows down, and the movingfront follows a different power law ∼ t 0.17 ± 0.02. The new exponent is smallerthan the theoretical prediction of 0.25 given by Equation 5.181. In Figure 5.28b,the radius of the shallow region, i.e., the dry spot radius, is plotted for the same

221 4

2 1

Hr

th r t

hH

dd* / *

*

*

/

, ( )λ

λ

= =

or22

1 4t / ,

r tf ( )

r tf ( )



six experimental runs. We observe the power law dependence ∼ t0.25 ± 0.05,which is the value predicted by the theory, Equation 5.183. According to ourmeasurements, the ratio

is about 1/3, although it slowly changes with time during the second stage. Thus,from Equation 5.183 we conclude that h* ≈ 0.07 cm, the lowest thickness thatcan be detected by our experimental method.

FIGURE 5.28 (a) Radius of the spreading front vs. time. Points correspond to six differentexperimental runs. (b) Radius of the dry spot vs. time. Points correspond to six differentexperimental runs.

100

10 0.01 0.1 1 10

log r

f (m

m)

log t (s)

100

10 0.01 0.1 1 10

log r

f (m

m)

log t (s)

1/4

r td ( )

r t

r td

f

( )( )



In conclusion, the radius of the dry spot moves with the speed predicted bythe theory, whereas the front edge of the moving part of the layer proceeds slowerthan theoretically predicted during the second spreading stage. The discrepancymay be due to one or both of the following reasons:

1. Gravity action, which is much more pronounced at the front higheredge of the layer than at the lower edge (dry spot). Although the Bondnumber is very low in our experiments, flow reversal onset cannot beruled out during the second stage.

2. Our assumption that the surfactant used is insoluble during the wholeduration of the experiment may not be fully correct. Unfortunately, wehave been unable to estimate the desorption time, τd. If that time isreached during our experiments, dissolution of surfactant in the bulkmay be significant at the higher front edge.

APPENDIX 2

Derivation of Governing Equations for Time Evolution of Both Film Thickness and Surfactant Surface Concentration

At ε << 1, as we already noticed in Chapter 3, the Navier–Stokes equations reducesto

(A2.1)

p = p(r), (A2.2)

(A2.3)

where p(r), v(r,z), z are pressure, axial velocity, and axial coordinate, respectively.The following boundary condition must be satisfied:

u(r,0) = v(r,0) = 0, (A2.4)

(A2.5)

(A2.6)

where pa is the pressure in the ambient air. Solution of Equation A2.1 withboundary conditions (A2.4), (A2.5) gives

dpdr

u

z= η ∂

∂

2

2,

10

r rru

vz

∂∂

∂∂( ) + = ,

η γ γ α∂ ( )

∂= ∂

∂= ∂

∂= − ∂

∂u r h

z rdd r r

,,

ΓΓ Γ

p pr r

rhra= − ∂

∂∂∂

γ,



. (A2.7)

In order to derive Equation 5.144, we make use of a condition at the freeliquid–air interface:

. (A2.8)

After integration of Equation A2.3 over z, from 0 to h, and using the result ofintegration into Equation A2.8, we get Equation 5.144. Substitution of EquationA2.7 into Equation 5.144 and Equation 5.145 yield Equation 5.146 and Equation5.147.

APPENDIX 3

Influence of Capillary Forces during Initial Stage of Spreading

Let us estimate the influence of capillary forces during a short initial stage ofspreading when it is significant. Later on it can be neglected everywhere exceptfor thin boundary layers that we consider negligible. A solution of governingEquation 5.146 and Equation 5.147 is assumed in the following form:

, (A3.1)

where f and ϕ are two new unknown functions; and the constant ω is to bedetermined. Substitution of expressions (A3.1) into Equation 5.146 and Equation5.147 results in

(A3.2)

(A3.3)

where ξ = r/tω. There are two ways to choose ω:

ur r r

rh

r

zzh= −

−

−1

2

2

η∂∂

γ ∂∂

∂∂

ααη

∂∂Γr

z

∂∂

∂∂

h

tu r h

h

rv r h+ =( , ) ( , )

h fr

t

r

t=

=

ω ωϕ, Γ

ω ξ ξξ

ξ β ξ γξ

ξ ξωtf

f

tf

f′ = ′( )( )′

′−( )

( )1

3

3

4

2(( ),

ξ ϕ ξω2 2t′( )

′

ω ξϕ ξξ

ξϕ ξ β ξ γξ

ξ ξωtf

tf′ = ′( )( )′

( ) ( )( )1

2

2

4

′′− ′( )

′f

t

( ),

ξ ϕ ξω2



(i) If we require t = t4ω or ω = 1/4, then Equation A3.1 and Equation A3.2become

(A3.4)

(A3.5)

Equation A3.4 and Equation A3.5 show that the influence of the sur-factant grows with time.

(ii) If we require t = t2ω or ω = 1/2, then Equation A3.2 and Equation A3.3become

(A3.6)

(A3.7)

Equation A3.6 and Equation A3.7 show that the influence of capillary forcesdecay with time. According to Equation A3.1, the spreading law is rf (t) ≈ t1/4

during the time period when the capillary force influence is dominant, and rf (t) ≈t1/2 during the time period when the influence of the surfactants is dominant. Here,rf (t) marks the location of the moving boundary of the spreading front. It followsfrom Equation A3.4 and Equation A3.5 that capillary force influence is dominantif β >> t1/2, or,

. (A3.8)

In the same way, from Equation A3.6 and Equation A3.7 we find that thecapillary force influence is negligible, and the influence of the surfactant isdominant if

. (A3.9)

14

13

31 2ξ ξ

ξξ β ξ γ

ξξ ξ′ = ′( )( )′

′−f

ff t

f( )

( ) /22

2( )

,ξ ϕ ξ′( )

′

14

12

2

ξϕ ξξ

ξϕ ξ β ξ γξ

ξ ξ′ = ′( )( )′

′−( ) ( )

( )ff t11 2/ ( ) .f ξ ϕ ξ′( )

′

12

13

3 2

ξ ξξ

ξ β ξ γξ

ξ ξ ξ′ = ′( )( )′

′−f

ft

ff

( )( ) ( ))

,2

′( )

′

ϕ ξ

12

12

2

ξϕ ξξ

ξϕ ξ β ξ γξ

ξ ξ′ = ′( )( )′

′−( ) ( )

( )ft

f ff ( ) .ξ ϕ ξ′( )

′

t << β2

t >> β



Thus, the capillary force influence is significant during a very short timeinterval only. As β << 1, we can safely consider just the asymptotic behaviorwhen the surfactant influence is dominant and condition (A3.9) is satisfied. Notethat by omitting the capillary force action we neglect the highest derivatives inEquation 5.146 and Equation 5.147; hence, thin layers arise where the capillaryforce action is of the same order magnitude as the surfactant action.

APPENDIX 4

Derivation of Boundary Condition at the Moving Shock Front

Multiplication of Equation 5.156 and Equation 5.157 by r and integration overr from r1 to r2, where r1 < rf (t) < r2, where r1, r2 are some constant values, yields

, (A4.1)

. (A4.2)

Where we use the following abbreviation: fi = f (ri). The left-hand side of EquationA4.1 and Equation A4.2 can be transformed in the following way:

where f± = f (t, rf ±). If we now consider limits r1 tends to rf from below (↑), andr2 tends to rf from above (↓) when both integrals on the left-hand side of theequation vanish. Then, from Equation A4.1 and Equation A4.2, using the samelimits , we conclude that

(A4.3)

(A4.4)

which are the required boundary conditions at the shock front.

d

dtrh t r dr

r h

r

r h

rr

r

( , ) = −∫ 2 22

2 1 12

1

2 21

2

∂∂

∂∂

Γ Γ

d

dtr t r dr r h

rr h

rr

r

Γ Γ Γ Γ Γ( , ) = −∫ 2 2 2

21 1 1

1

1

2

∂∂

∂∂

ddt

rf t r drddt

rf t r dr rf t r drr

r

rf

( , ) ( , ) ( , )(

= +∫1

2

tt

r

r

r t

f f

r

f

r r f rf t r

t

)

( )

( , )

2

1

1

∫∫

=

+−� ∂∂

rr t

f f

r t

rf

f

dr r r f rf t r

tdr

( )

( )

( , ),∫ ∫− ++� ∂

∂

2

r r r rf f1 2↑ ↓,

�r h h hr

hrf − + +

+−

−−( ) = ∂∂

− ∂∂

12

2 2Γ Γ,

�r hr

hrf Γ Γ Γ Γ Γ Γ

− + + ++

− −−−( ) = ∂

∂− ∂

∂,



APPENDIX 5

Matching of Asymptotic Solutions at the Moving Shock Front

Let us introduce a new local variable

where χ(t) << 1 is a new unknown length scale to be determined. Neglecting thesecond curvature in Equation 5.146 and Equation 5.147, and introducing unknownfunctions in the following form Γ = χ(t)Φ(ξ), h = F(ς), we get

. (A5.1)

It follows from this equation that χ(t) → 0, at t → ∞. Unknown functionsΦ(ς) and F(ς) obey the following equations:

(A5.2)

(A5.3)

After integration of Equation A5.2 and Equation A5.3 with boundary condi-tions F → 1, Φ → 0, at ς → ∞, we get

(A5.4)

(A5.5)

Thus, either

Φ = 0, (A5.6)

and

(A5.7)

ς ξ νχ

= −( )

,t

χ β( )

/

tt

=

1 3

ν ′ = ′′′ − ′

′

FF F

F2

3

32Φ ,

ν ′ = ′′′ − ′( )′Φ Φ ΦΦF F F2 2 .

ν F F F F−( ) = ′′′ − ′123

3 2Φ ,

Φ Φν − ′′′ + ′( ) =F F F2 2 0.

′′′ = −F

F

F

32

13

ν,



or

(A5.8)

and

(A5.9)

From Equation A5.8 and Equation A5.9 we conclude that

(A5.10)

Equation A5.10 shows that boundary conditions (5.166) at the shock frontare the only possible conditions that can be matched with the inner solution. Fromthe above derivation we conclude that in the boundary layer

hence, vanishes from the point of view of the outer solution. Thus, both conditions(5.160) and (5.161) must be satisfied at the shock front.

APPENDIX 6

Solution of the Governing Equations for the Second Stage of Spreading

Putting β = 0 in Equation 5.176 and Equation 5.177, we get

, (A6.1)

with boundary conditions that follow from Appendix 4, Equation A4.3, andEquation A4.4,

. (A6.2)

Then, the solution of Equation A6.1 is

(A6.3)

′′′ =−( )

FF

F

3 23

ν,

′ =−( )

Φν 3

2

F

F.

F → ′ → − → −∞24

, , .Φ ν ςat

Γ Φ=

( )β ς β

t

1 3

1 3

/

/~ ;

ff

f'( )

( ), '( )

( )ξ ξ

ξϕ ξ ξ

ξ= = −2

4

f ( ) , ( ) , '( )λ ϕ λ ϕ λ λ= = = −2 08

f ξ ξλ

ϕ ξ λ ξλ( ) =

( ) = −2

8

2 2

, ln .



Substitution of f(ξ) in Equation 5.178 gives

. (A6.4)

Note that the solution (A6.3) satisfies Equation 5.176 to Equation 5.178 atarbitrary β.

5.7 SPREADING OF AQUEOUS DROPLETS INDUCED BY OVERTURNING OF AMPHIPHILIC MOLECULES OR THEIR FRAGMENTS IN THE SURFACE LAYER OF AN INITIALLY HYDROPHOBIC SUBSTRATE

Let us study in this final section the spontaneous spreading of a drop of a polarliquid over a solid when the amphiphilic molecules (or their amphiphilic frag-ments) of the substrate surface layer may overturn, creating hydrophilic parts onthe surface. Such a situation may occur, for example, during the contact of anaqueous drop with the surface of a polymer whose macromolecules have hydro-philic lateral groups capable of rotating around the backbone, or during thewetting of polymers containing surface-active additives or Langmuir–Blodgettfilms composed of amphiphilic molecules. It is shown in the following discussionthat drop spreading is possible only if there is lateral-side interaction betweenneighboring amphiphilic molecules (or groups). This interaction leads to tangen-tial transfer of the “overturned state” to some distance ahead of the advancingthree-phase contact line, making it partially hydrophilic. This kind of “self-organization” of its surface layer lowers the interfacial free energy due to theemergence (or adsorption) of polar groups at the surface.

The quantitative theory describing the kinetics of droplet spreading is devel-oped in the following text with allowance for this mechanism of self-organizationof the surface layer of a substrate in contact with a droplet [20].

A number of researches have been published (see, for example, Reference 21to Reference 24) demonstrating that interaction of a polymer with a polar liquid(first of all, with water and aqueous solutions) may result in the spontaneousrearrangement (self-organization) of its surface layer, providing the minimizationof the interfacial free energy due to the emergence (or adsorption) of polar groupsat the surface. Depending on the structure of the polymer macromolecules, thephysical state of the polymer, and other factors, such a rearrangement may involvevarious forms of molecular motion, from the reorientation of individualamphiphilic side groups (when their rotation around the backbone is allowed) tothe diffusion of macromolecules as a whole. Each of these processes may occuron different time scales. If the characteristic time scale of self-organization iscomparable with the time of measurement, this process may be observed whilestudying the contact angle (and size) of an aqueous droplet on time of contactwith a polymer [25]. Such a prolonged spreading of aqueous droplets occurs alsoduring the study of the wettability of model systems such as apolar polymers

λ = 25 4/



containing low-molecular-weight amphiphilic additives (surfactants) capable ofadsorption and reorientation at the polymer–liquid interface [25].

Langmuir was the first to mention the possible reorientation (overturning) ofamphiphilic molecules in contact with a polar liquid [26]. Later, experimentaldata were obtained that demonstrated the occurrence of such a process in monolayers and Langmuir–Blodgett films composed of long-chain fatty acids [27] andat the surface of mixtures of such acids with paraffin [28] during contact with anaqueous droplet, resulting in gradual droplet spreading. To start, the rate ofspreading was analyzed in terms of formal chemical kinetics [27,28], and themechanism of the process (the dynamic situation in the vicinity of the three-phasecontact line) was not considered at all. Recently, such a situation has also beenobserved for the spreading of a droplet over the surfaces of polymers, when theamphiphilic groups of their macromolecules are capable of reorientation byrotating around the macromolecule backbone.

The aim of this section is to analyze the mechanism of the spontaneousspreading of a droplet of a polar liquid induced by the overturning of amphiphilicmolecules (or their fragments) in the surface layer of a solid substrate and todevelop a quantitative theory describing this process.

THEORY AND DERIVATION OF BASIC EQUATIONS

Let us consider the spreading of an aqueous droplet over a solid substrate, ignoringthe evaporation of the liquid, i.e., assuming that the droplet volume remainsconstant during spreading. It is assumed also that the substrate is smooth, hori-zontal, and (which is important for further discussion) contains in the surfacelayer rotationally mobile amphiphilic molecules (or amphiphilic fragments ofmolecules; for brevity, we hereafter mention only molecules) capable of overturningin the plane perpendicular to the surface and incapable of lateral motions in theplane of the substrate. Hence, each of the amphiphilic molecules may be in oneof two states: (1) nonoverturned (normal), i.e., when the hydrophilic head groupof the molecule is oriented downward into the substrate whereas the hydrophobictail is directed upward into the second phase (air or water) in contact with thesubstrate; and (2) overturned state, i.e., in the opposite (as compared to theprevious case) orientation of hydrophilic and hydrophobic moieties of the mol-ecule (Figure 5.29). Let and be the total number of amphiphilic moleculesper surface area of the substrate capable of overturning and the number ofoverturned molecules, respectively; then p = N/N∞ is the probability to find anoverturned molecule. Let us consider the process of transition of a system to theequilibrium state during the contact of the substrate surface with the air, providedthat initially the system was somehow disturbed from an equilibrium state.

Let us consider for simplicity a linear chain of amphiphilic molecules, becausea switch to the two-dimensional case is self evident. Let pi (t) be the probabilityof the occurrence of molecule at the instant t in the overturned state; then, thesame probability at the time will be equal to

N∞ N

it t+ ∆



(5.184)

where P with subscripts are the probabilities of direct and reverse overturningsper unit time: Pv is the overturning probability caused by the interaction with theair of the amphiphilic molecule that has not been overturned previously; Psv isthe overturning probability caused by the interaction with underlying moleculesthe molecule that was overturned previously; and Pd and Pb are the probabilitiescorresponding to the direct and the reverse overturnings of a molecule due to itsinteraction with neighboring molecules. Only the interactions with the nearestneighbors are accounted for, i.e., with the molecules having numbers i + 1 andi – 1.

All the probabilities in Equation 5.184 may be written in the following form:

, (5.185)

, (5.186)

, (5.187)

, (5.188)

where constants αv, αsv, and β entering into the definitions of probabilities inEquation 5.185 and Equation 5.188 have the dimensionality of reciprocal timeand are determined using the energy of molecular interaction with surroundingphases in overturned and nonoverturned states:

FIGURE 5.29 Polymeric substrate containing rotationally mobile amphiphilic chains incontact with air or water; v — air, w — water, s — polymeric substrate. 1 — amphiphilicchains in the normal state, 2 — amphiphilic chains in the overturned state.

Water or air 2-overturned state

1 normal state Substrate

S

W V

p t t p t P t P t P t P ti i SV V d b+( ) = ( ) + − + −∆ ∆ ∆ ∆ ∆ ,

P pv v i= −( )α 1

P psv sv i= α

P p p p pd i i i i= −( ) + −( ) − +β 1 11 1

P p p p pb i i i i= −( ) + −( ) − +β 1 11 1



, (5.189)

, (5.190)

, (5.191)

where

;

subscripts t, h, v, and s correspond to the hydrophobic tail of a molecule, to itshydrophilic head group, to the air, and to the underlying molecules of the sub-strate, correspondingly; denotes the corresponding values in the absence of anyinteraction, i.e., determined only by thermal fluctuations; and z is the number ofneighboring molecules, i.e., z = 2 for a one dimension, and z = 4 for the two-dimensional case. It follows from definitions (5.189) and (5.190) that αv << αsv.

Let us consider Equation 5.191 in more detail. If expressions (5.189) and(5.190) involve both overturnings due to the thermal fluctuations and those causedby the interaction with the surrounding media, then, in contrast to these expres-sions, Equation 5.191 involves only the overturnings related to the interactionsbetween the neighbors; hence, random overturnings caused by thermal fluctua-tions should not be taken into account because they do not result in the transferof the overturned state. This is why the unity is subtracted in Equation 5.191.

Substituting expressions (5.185) and (5.188) into Equation 5.184 yields

Taking the limit in the last expression at ∆t → 0 results in

(5.192)

Let a be the mean distance between molecules capable of overturning. Then,Equation 5.192 may be rewritten in the following form:

α αv v= − exp Φ

α αsv v= exp Φ

β α χ= ( ) − exp 1

χ = − = + − −z

U U

RT

U U U U

RTth tt

vts hv tv hs2

, Φ

α

p t t p t

tp t p t

i iv i sv i

+( ) − ( )= − ( ) − ( )∆

∆α α1

++ ( ) + ( ) − ( ) − − ( )− + −β p t p t p t p ti i i i1 1 11 2 −−

( ){ }+ ( )p p ti t i1

dp t

dtp t p t p t p

iv i sv i i i

( )= − ( ) − ( ) + ( ) ++α α β1 1 −− ( ) − ( ) 1 2t p ti .



. (5.193)

The latter part of Equation 5.193 is a discrete analogy of the second-orderspatial derivative. Hence, using the continuous coordinate x ≈ ia, we obtain thefollowing second-order partial differential equation instead of Equation 5.193:

, (5.194)

where p(t, x) is the probability of the occurrence of an overturned molecule attime t at point x, and D = β a2 is the effective diffusion coefficient of the molecule-overturned state along the surface. The extension of Equation 5.194 to the plane(two-dimensional) case of our further interest is, as was stated before, a straight-forward procedure: it is reduced to the simple substitution of the partial second-orderderivative with respect to one coordinate x for the sum of the partial second-orderderivatives with respect to x and y, because, in the two dimensional case, p =p(t, x, y).

According to our previous consideration, the transfer of the overturned stateis determined only by the interactions between adjacent molecules and shouldvanish in two cases: (1) in the absence of interactions between adjacentamphiphilic molecules, i.e., at χ = 0 (in this case, β = 0 and hence D = 0); and(2) upon unlimited increase in the distance a between adjacent molecules, i.e.,when the surface concentration of molecules capable of overturning tends to zero;in this case, interactions between adjacent molecules also vanish. Indeed, let usassume that Uth, Utt, and are determined by dispersion interactions only. Inthis case, χ(a) ~ B/a6, where B is a constant expressed as usual via the polariz-abilities of hydrophilic head groups and hydrophobic tails: B = (2Ath – Att – Ahh)z/3,where Ath, Att, and Ahh are the corresponding Hamaker constants. Consequently,D(a) ~ β[expB/a6 – 1]a2 ~ βB/a4 → 0, with an increase in distance a betweenadjacent amphiphilic molecules.

In the equilibrium state, the probability p does not depend either on time orcoordinate; this equilibrium state is further denoted by pv, which is readilydetermined from Equation 5.194:

(5.195)

From definitions (5.189) and (5.190), we obtain that

,

i.e., pv is a small value.

dp t

dtp t p t a

p tiv i sv i

i( )= − ( ) − ( ) +

( ) ++α α β1 2 1 pp t p t

a

i i− ( ) − ( ) 1

2

2

∂∂

= −( ) − +∂ ( )

∂p

tp p D

p t x

xv svα α1

2

2

,

Uhh

pvv

v sv sv v

=+

=+

αα α α α

11 /

.

αα

sv

vv= ( ) >>exp 2 1Φ



Let us discuss now the events occurring underneath the aqueous droplet atthe solid–water interface. In this case, instead of equation 5.184, we arrive at

, (5.196)

where the subscript w refers to water. Expressions for the probabilities per unittime and are similar to those occurring previously, with correspondingconstants and which can be obtained from relationships (5.185) and(5.186), and (5.189) and (5.190), respectively, by substituting the subscript winstead of v.

Considerations similar to those used for deriving Equation 5.194 results, inthis case, in the following equation:

. (5.197)

The equilibrium value of the probability of the occurrence of an amphiphilicmolecule in the overturned state,

, (5.198)

is determined, as in the case of contact with the air, from equation 5.197. Unlikethe case of contact with the air, the probability is not a small value; on thecontrary, it is close to one. It is this difference in probabilities that provides forthe possibility of the aqueous droplet spreading over the initially hydrophobicsurface.

Note that, in the absence of lateral interactions between adjacent amphiphilicmolecules, the aqueous droplet may not spread over the surface under consider-ation despite the effect of overturning of molecules with the hydrophilic portionsupward, underneath the water. Indeed, in the absence of interactions betweenadjacent molecules, let the necessary quantity of molecules be overturned withtheir hydrophilic portions upward (Figure 5.30a), and the substrate surface under-neath the aqueous droplet become sufficiently hydrophilic so that the droplet edgecan move into the new position presented in Figure 5.30b. However, in the absenceof lateral transfer of the overturned state of the amphiphilic molecules in thesubstrate, the surface both in front of the droplet edge in Figure 5.30b and outsidethe edge are still in the initial hydrophobic state, thus forcing the droplet edge toreturn immediately to the initial position (Figure 5.30a). Thus, in the absence ofthe lateral transfer of the overturned state described by the diffusion term inEquation 5.194 and Equation 5.197, the spreading of the aqueous droplet overthe surface becomes impossible. However, if the adjacent amphiphilic molecules

p t t p t P t P t P t P ti i w sw d b+( ) = ( ) + − + −∆ ∆ ∆ ∆ ∆

Pw Psw

αw αsw ,

∂∂

= −( ) − +∂ ( )

∂p

tp p D

p t x

xw swα α1

2

2

,

pww

w sw

=+α

α α

pw



interact with each other and the lateral transfer of the overturned state due tothese interactions is possible, the droplet edge moves over the surface, and thismotion is determined exactly by the rate of the lateral transfer of the overturnedstate of the substrate molecules.

BOUNDARY CONDITIONS

According to the theory described above, the propagation of the overturned stateof amphiphilic molecules along the surface under the droplet is described byEquation 5.197, and the propagation beyond the droplet by Equation 5.194. It isnow required to formulate the boundary conditions for the probability, p(t, r), atthe boundary of the spreading axisymmetric droplet, i.e., at r = r0 (t) (Figure 5.31).

In view of the assumption of the absence of evaporation, the droplet volume,V, remains constant during the spreading, and it is assumed also that the dropletis small enough, that is, the gravity action may be neglected. The time scale ofthe spreading process is so big (see the following discussion) that the deformationsof the drop profile caused by the spreading or flow can be neglected: this meansthat the capillary number is enormously low, Ca << 1. According to the Intro-duction to Chapter 3, the droplet retains the shape of a spherical segment duringthe spreading

(5.199)

where h and r0 are the maximum height and radius of the base of the spreadingdroplet, respectively (Figure 5.31). The droplet height, h, is determined by rela-tionship where is the current value of the contact angle of thespreading droplet. Substituting this relationship into equation 5.199, we express

FIGURE 5.30 Impossibility of spreading of a water droplet on a hydrophobic substratewithout lateral interaction between neighboring chains (explanation in the text).

(a)

(b)

V h r h const= +( ) =π6

3 02 2 ,

h r= 0 2tan( ),θ/ θ( )t



the radius of the base of the spreading droplet via the current value of the contactangle as

. (5.200)

Let us denote surface (interfacial) tensions for the water–air (constant value),substrate–air, and substrate–water interfaces directly at the boundary of thespreading droplet by and respectively. Note that the values of

and near the droplet edge differ from the constant values of surfacetensions at the polymer–air and polymer–water interfaces far from the edge ofthe spreading droplet and in the depth of the droplet, respectively. It is assumedthat Young’s equation is satisfied at any moment at the boundary of the spreadingdroplet

, (5.201)

where

.

FIGURE 5.31 Spreading of a spherical droplet: r0 (t) — radius of the base of a drop,θ(t) — dynamic contact angle, h — height at center of a drop.

γsw

h

γsv

γ

θ

pinpv

xr0(t) doutd

pout

ppw

r tV

0

1 3

2

6 1

23

2

( ) =

+

π θ θ

/

tan tan

1 3/

γ γ, ( ),sv t γ sw t( ),γ sv t( ) γ sw t( )

cos( ) ( )

cos ( )θ γ γγ

θ= − = +sv swt tt0 ∆

cos , ( )( ) ( )θ γ γ

γγ γ

γγ γ0

0 0 0

= − = − − −sv sw sv sv swtt t∆

00sw

γ



In this equation, the superscript 0 marks corresponding initial values ofinterfacial tensions. Note that the ∆(t) value is positive and increases with timebecause the γsv(t) value rises in time due to possible appearance of hydrophilichead groups of the amphiphilic molecules at the substrate surface; in contrast,the γsw(t) value decreases with time due to overturning of molecules under thedroplet.

Let us emphasize once more that

1. In view of the lateral interaction between adjacent amphiphilic mole-cules of the substrate, the overturned state may be extended beyondthe boundary of the spreading droplet, resulting in an increase insurface tension of the substrate, γsv(t), in front of the moving droplet(Figure 5.31).

2. Interfacial tensions γsv(t) and γsw(t) do not remain constant near thedroplet boundary but vary depending on the coordinate in a closevicinity of the boundary of the moving droplet (see the followingdiscussion). Hence, interfacial tensions in the close vicinity of the edgeof the moving droplet, or, in a more formal manner, the limiting valuesof these tensions at r → r0(t) from the inner and outer droplet sides,enter Young’s equation 5.201.

The corresponding limits of the degree of overturning inside and outside thedroplet are denoted by pin and pout, respectively (Figure 5.31). Evidently, the ∆(t)value is not an explicit function of time but depends on time in an implicit mannervia the values of pin, pout, i.e., ∆ = ∆ (pin, pout). As shown, in view of the equalityof chemical potentials of amphiphilic molecules between the inner and outerboundaries of a droplet, the pout value is expressed as a function of pin. Hence, inview of the latter dependence, actually ∆ = ∆ (pin).

Let us use Antonov’s rule to determine unknown dependence ∆ (pin, pout),which means the additivity of the formation of the interfacial tensions.

Let γ ∞sw and γ 0sw be the interfacial tensions under the droplet in the case whenall molecules are overturned (all hydrophilic head groups are oriented upward)and when neither of these molecules is overturned (all hydrophilic head groupsare oriented downward), respectively. Similar surface tensions outside the dropletare denoted by γ∞

sv and γ 0sv , respectively. According to the assumption of the

additivity, the interfacial tensions in the closest vicinity of a droplet acquire thefollowing form:

(5.202)

Substitution of relationships (5.202) into Young’s equation (5.201) yields thefollowing expression for the dependence ∆(pin, pout) under consideration:

γ γ γ

γ γ γ

sv sv out sv out

sw sw in

p p

p

= − +

= − +

∞0

0

1

1

( ) ,

( ) ssw inp∞ .



. (5.203)

The obtained dependence ∆(pin, pout) is a linear function with respect to bothvariables.

As mentioned previously, the spreading process under consideration is veryslow (the time scale is hours). This allows employing the principle of localequilibrium accepted in nonequilibrium thermodynamics. In accordance with thisprinciple, chemical potentials of overturned and nonoverturned amphiphilic mol-ecules remain equal from both sides of the droplet boundary, i.e.,

. (5.204)

The µ(p,χ) dependence may be given, for example, in accordance with theFlory–Huggins theory [29] in the following form: µ(p,χ) = ln p + χ (1 – p)2. Inthe following discussion we consider only two limiting cases of weak and stronglateral interactions between amphiphilic molecules. At this stage, it is enough totake into account that, according to the equality of chemical potentials (5.204),pout and pin are interrelated by the known dependence pout = ϕ(pin). In view of allthat has been said previously, the value of ∆ is dependent on only one variable,i.e., ∆ = ∆(pin).

In view of the equality

from Equation 5.200, we obtain

(5.205)

where

(5.206)

∆( , )p p p pin outsv sv

outsw sw

in= − + −∞ ∞γ γγ

γ γγ

0 0

µ χ µ χ( , ) ( , ) ,p pU U

RTin outhv hw= + ℑ ℑ = −

tancoscos

/θ θ

θ211

1 2

= −+

r pV

G pin in0

1 36

( ) · ( ),/

=

π

G pAA

AA

A

in( ) ,

(

/ /=

−+

+ −+

1

11

311

1 6 1 3

pp pin in) cos ( ).= +θ0 ∆



Equation 5.205 allows determining the final droplet radius, r∞, at the end ofthe spreading process using the known dependence G(pin). To this end, it isnecessary to substitute the expression for the equilibrium fraction of overturnedmolecules under droplet pw from relationship (5.198) into Equation 5.205, whichyields

(5.207)

It is possible to verify that G(p) is an increasing function of p; i.e., r0(t)increases, approaching its final value determined by Equation 5.207.

Let us now consider the formulation of boundary conditions on a movingdroplet edge. Two boundary conditions are required because the edge itself moves,and the law of this movement should be determined.

The first boundary condition expresses the balance of the number of over-turned molecules at the boundary of the moving droplet, and it has the followingform:

(5.208)

and it is necessary to set the second boundary condition relating the pin and pout

values. The other boundary conditions are straightforward: the symmetry in thedroplet center

, (5.209)

and the tendency of the fraction of overturned molecules far from the droplet tothe equilibrium value at the substrate–air interface determined from Equation5.195 as

(5.210)

We use the condition of equality of chemical potentials of overturned mole-cules to the right- and left-hand sides of the moving boundary of a droplet, r0(t),which determines the dependence

(5.211)

rV

G pw∞ =

61 3

π

/

· ( ).

− + = −= − = +

Dpr

Dpr

p pd r

r r t r r t

in out

∂∂

∂∂

0 0

0

( ) ( )

( )(tt

d t)

,

∂∂

p

rr=

=0

0

p p rv→ →∞, .

∆ ∆in in out outp pµ µ( ) ( ).=



To analyze the dependences of chemical potentials, we employ the expres-sions resulting from the Flory–Huggins theory [29] modified to take into accountthe interactions of amphiphilic molecules with the environment and the under-lying molecules of a substrate, namely, interactions of their hydrophilic headgroups with water and their hydrophobic tails with the substrate molecules (underthe droplet), and interactions of hydrophilic head groups with the air and hydro-phobic tails with underlying substrate molecules (outside the droplet). This leadsto the following expression:

, (5.212)

where χ is the known parameter of interaction of amphiphilic molecules witheach other according to expression (5.191); in the case under consideration, thevalue of z is equal to 4.

In the case of weak interactions between adjacent molecules, i.e., at ,from expression (5.212) we conclude that . As a result, wearrive at the Boltzmann distribution

. (5.213)

In the case of strong interaction between the neighboring molecules, i.e.,, we obtain

. (5.214)

Hence, it results in equality of overturned fractions: pout = pin.The value of pout is always smaller than or equal to pin, thus enabling us to

solve Equation 5.212 for the arbitrary case and to express pout as a function ofpin, which was stated before by Equation 5.211.

Thus, the problem of droplet spreading acquires the following form: thedependence p(t, r) under the droplet at 0 < r < r0(t) is described by the equation

, (5.215)

and the dependence p (t, r) outside the droplet, r > r0 (t), is described by theequation

, (5.216)

ln ( ) ln ( )p p p pin in out out+ − = + − + ℑχ χ1 12 2

χ << ℑln lnp pin out= + ℑ

p pout in= −ℑexp( )

χ >> ℑ

ln ( ) ln ( )p p p pin in out out+ − = + −χ χ1 12

∂∂

α α ∂∂

∂∂

p

tp p D

r rr

p

rw sw= − − +( )11

∂∂

α α ∂∂

∂∂

p

tp p D

r rr

p

rv sv= − − +( )11



with the following boundary conditions: condition of symmetry in the dropletcenter (5.209), condition (5.210) far from the droplet, condition (5.205) express-ing the radius of the spreading droplet via pin, condition (5.208) expressing theequality of fluxes at the moving boundary of the droplet r = r0(t), and relationship(5.211) expressing the equality of chemical potentials of overturned moleculesnear the moving edge of the droplet.

Solution of the Problem

We perform the solution of the preceding problem introducing a number ofsimplifying assumptions whose validity is checked in the following text.

It is obvious that the value of p under the main part of the spreading dropletis independent of the coordinate but changes only with time due to the interactionof the amphiphilic molecules with the aqueous phase. Let us denote this coordi-nate-independent value by pd(t), which, according to Equation 5.215, satisfies thefollowing equation:

with the initial condition . The solution of the problem is

(5.217)

It follows from Equation 5.217 that the characteristic time scale of moleculeoverturning ttr* is equal to ttr* = 1/(αw + αsw).

In a narrow region with width δ near the droplet edge (from the inner side),the diffusion term in Equation 5.215 becomes of the same order of magnitude asthe term describing the overturning of molecules due to their interaction withwater. Let us introduce dimensionless values y = (r0(t)r)/δ, λ = αs/αw; τ = t/t*;ξ(t) = r(t)/r*, where r* = (6V/π)1/3, and new unknown function g (t, y) = p (t, r) –pd (t). The time scale t* is selected below.

Rewriting Equation 5.215 in dimensionless form using the introduced vari-ables, we obtain

. (5.215)

Accounting for the smallness of δ/r*, we conclude that

.

d pd t

p pdw d sw d= − −α α( ) ,1

p pd v( )0 =

p t p p p td w v w w sw( ) ( )exp( ( ) ).= + − − +α α

∂∂

= − +( ) + ∂∂

g

tg

D g

yw swα α

δ2

2

2

∂∂

∂∂ τ

∂∂ δ

ξ ∂∂ δ

ξg

t t

g g

y

r

t

g

y

r

t= + ≈1

*

*

*

*

*

� �



In this case, Equation 5.215 acquires the following form:

(5.218)

All terms in Equation 5.218 should be of the same order of magnitude; hence,the characteristic values are interrelated by the following relationships

Let us compare ttr* and t*: i.e., t*

is much larger than the characteristic time of overturning of amphiphilic mole-cules under the main portion of the droplet. This implies that under the droplet,p = pw, and changes occur only in the narrow region with the width δ and aredescribed by the equation

, (5.219)

with boundary conditions

, (5.220)

and

g(0) = pin – pw. (5.221)

As the desired function g(τ, y) depends on τ as a parameter, the solution ofEquation 5.219 satisfying conditions (5.220) and (5.221) may be readily obtained,and the expression for p acquires the form

. (5.222)

Let us perform similar transformations in the narrow region from the outerside of the droplet front

rt

g y

yg y

D g yw sw

∗

∗

∂ ( )∂

= − +( ) ( ) +∂ (

δτ

ξ α α τδ

τ,,

,�2

2 ))∂y2

.

δ α α δ= +( ) =∗ ∗D t r Dw sw , ./

t t t r D rtr w sw w sw∗ ∗ ∗ ∗∗ = + = + = >>/ / /( ) ( ) ;α α α α δ 1

∂ ( )∂

= − ( ) +∂ ( )

∂

g y

yg y

g y

y

τξ τ

τ,,

,�2

2

g y→ → ∞0,

p p p p yw in w= + −( ) − +

exp� �ξ ξ2 4

2

δ α α δout v sv out outD y r r t q p t r= +( ) = −( ) = −, ( ) , ( , )0 ppv.



In this case, the dimensionless equation describing the probability p in theδout region outside the droplet has the following form:

(5.223)

with boundary conditions

, (5.224)

and

(5.225)

Assuming that αw ∼ αsv >> αv ∼ αsw, we can conclude that δ = δout, and fromEquation 5.223 we obtain

(5.226)

Let us rewrite boundary condition (5.208) to dimensionless form:

(5.208′)

Using expressions (5.222) and (5.226), we arrive at an equation describingthe motion of the droplet boundary:

(5.227)

Equation 5.227 has the following solution:

. (5.228)

Using condition (5.205), we obtain

. (5.229)

− ++

∂ ( )∂

= ( ) +∂ ( )α α

α ατ

ξ ττ

w sw

v sv out

q y

yq y

q y,,

,�2

∂∂yout2

,

q yout→ → +∞0,

q p pout v( ) .0 = −

p p p p yv out v out= + −( ) + +

exp� �ξ ξ2 4

2

D py

D py

rt

p py y

in outδ δξ∂

∂+ ∂

∂= −( )

= =− +0 0

*

*

.�

( ) ( ) (p p p p p pin w out v in−− +

− −+ +

= −� � � �ξ ξ ξ ξ2 24

2

4

2 oout ) .�ξ

�ξ = − − +− −

p p p p

p p p pw in out v

w out in v( ) ( )

�ξ ξτ τ τ τ

= = = =dd r

d rd

dd

G pd Gd p

d pdin

in

in1 0

*

( )



Let us rewrite Equation 5.228 taking into account representation (5.229):

(5.230)

Assuming that pout ∼ pv << pin, and pw ∼ 1, we have the final equation for pin

(5.230′)

Taking into account the smallness of pout as compared to one, we obtain fromEquation 5.202 that the value of γsv changes slightly and remains close to theinitial value γ 0

sv. In this case, expression (5.203) may be rewritten in the followingform:

, (5.231)

and the function A(pin) acquires the form

. (5.232)

Thus, according to the proposed theory, the droplet spreads in a completelydifferent manner from what was suggested in Reference 27; i.e., the equilibriumconcentration of overturned amphiphilic molecules (or their fragments) is estab-lished rapidly (as compared to the characteristic time of spreading) under themain portion of the drop and retains its value over the course of the entirespreading process. All changes occur only within the narrow region in the vicinityof the perimeter of the spreading droplet.

COMPARISON BETWEEN THEORY AND EXPERIMENTAL DATA

Equation 5.230 contains one unknown parameter, t*, which is the characteristictime of the propagation of the overturned state, i.e., the characteristic time scaleof droplet spreading. Parameter t* was used as an adjustment parameter for thecomparison of the theoretical predictions according to the described theory andexperimental data reported elsewhere [27,28].

In Figure 5.32, the time dependences of the contact angle of an aqueous dropletat the surface of paraffin containing stearic acid at various concentrations, C, arepresented. Lines show the solutions of Equation 5.230, and the symbols denote

d pd

p p p p

G p p p p p

in w in out v

in w out in vτ

= − − +

′ −( ) −( )(( ) .

d pd t

p t

G p pin in

in in

=−( )

′( )1 *

.

∆ = − = − − −

∞ ∞γ γγ

γ γγ

γ γγ

sw swin

sv sw sv swp0 0 0 0

pp pin in= −∞(cos cos )θ θ0

A p pin in( ) cos (cos cos )= + −∞θ θ θ0 0



experimental data [28] for three different concentrations, C. The values of t* werefound as follows:

t* = 50.5 h at C = 0.6 wt %,t* = 27.5 h at C = 2.0 wt %,t* = 24.0 h at C = 9.0 wt %.

These values show that, as the concentration of stearic acid increases, thecharacteristic time of propagation of the overturned state decreases. This is due toan increase in diffusion coefficient, D, because of the decreasing average distancebetween acid molecules capable of overturning.

In Figure 5.33, the time dependences of the contact angle of aqueous dropsat a Langmuir–Blodgett film composed of stearic acid at various temperaturesare presented. Here, symbols represent experimental data from Reference 27, andlines drawn correspond to the solutions of Equation 5.230′. The deduced depen-dence of parameter t* on temperature is shown in Figure 5.34. Characteristic timet* of the propagation of the overturned state decreases with temperature, whichmay be explained by an increase in the rotational mobility of molecules capableof overturning, and simultaneously, the diffusion coefficient, D, of the overturnedstate of stearic acid molecules increases.

FIGURE 5.32 Time dependences of contact angle, θ, of water droplets at the surface ofparaffin containing stearic acid of different concentrations in wt% (experimental data fromReference 27): (1) C = 0.6; (2) C = 2.0; (3) C = 9.0. Solid lines according to Equation 5.230.

Cont

act a

ngle,

deg

120

110

100

90

80

70

600 5 10 15 20

Time, h

1

2

3



FIGURE 5.33 Time dependences of the contact angle, θ, of water droplets at the surfaceof a Langmuir–Blodget film formed by a stearic acid at various temperatures in °C.Experimental data from Reference 28. (1) t* = 13.5 h; (2) t* = 15.5 h; (3) t* = 23.0 h; (4)t* = 25.0 h; (5) t* = 28.5 h. Solid lines according to Equation 5.230′.

FIGURE 5.34 Dependence of characteristic time of a spreading of drops, t*, on temperature.

Cont

act a

ngle,

deg

100

110

90

80

70

0 10 20 30 40 50 60Time, h

1

2

3

4

5

200

150

100

50

012 28242016

t ∗ , h

Temperature, °C



REFERENCES

1. Zhdanov, S., Starov, V., Sobolev, V., and Velarde, M., J. Colloid Interface Sci.,264, 481–489, 2003.

2. Zolotarev, P.P., Starov, V.M., and Churaev, N.V., Colloid J. (Russian Academy ofSciences, English Translation), 38, 895, 1976.

3. Churaev, N.V., Martynov, G.A., Starov, V.M., and Zorin, Z.M., Colloid Polym.Sci., 259, 747, 1981.

4. Starov, V., J. Colloid Interface Sci., 270, 180, 2003.5. Zorin, Z.M., Iskandaryan, G.A., and Churaev, N.V., Colloid J. (Russian Academy

of Sciences, English Translation), 40, 671, 1978.6. Berezkin,V.V., Deryagin, B.V., Zorin, Z.M., Frolova, N.V., and Churaev, N.V.,

Dokl. AN SSSR, 225,109, 1975.7. Berezkin, V.V., Zorin, Z.M., Iskandaryan, G.A., and Churaev, N.V., Trans. VIIth

Int. Congr. Surfactants, B2, 329, Moscow 1978.8. Berezkin, V.V, Zorin, Z.M., Frolova, N.V., and Churaev, N.V., Colloid J. (Russian

Academy of Sciences, English Translation), 37, 1040, 1975.9. Shinoda, K., Nakagawa, T., Tamamushi, B., and Isemura, T., Colloidal Surfac-

tants: Some Physico-Chemical Properties, Academic Press, New York, 1963.10. Starov, V., Spontaneous rise of surfactant solutions into vertical hydrophobic

capillaries, J. Colloid Interface Sci., 270, 180–186, 2003.11. Starov, V.M., Zhdanov, S.A., and Velarde, M.G., Capillary imbibition of surfactant

solutions in porous media and thin capillaries: partial wetting case, J. ColloidInterface Sci., 273(2), 589–595, 2004.

12. Levich, V.G., Physicochemical Hydrodynamics, Prentice Hall, Englewood Cliffs,NJ, 1962.

13. Starov, V.M., de Ryck, A., and Velarde, M.G., J. Colloid Interface Sci., 190, 104,1997.

14. Starov, V.M., Kosvintsev, S.R., and Velarde, M.G., Spreading of surfactant solu-tions over hydrophobic substrates, J. Colloid Interface Sci., 227, 185–190, 2000.

15. Stoebe, T., Lin, Z., Hill, R.M., Ward, M.D., and Davis, H.T., Langmuir, 12, 337,1996; Langmuir, 13, 7270, 7276, 1997.

16. Reyes, R. and Wayner, P., J. Heat Transfer, 118, 822–830, 1996.17. Starov, V.M., Velarde, M.G., Tjatjushkin, A.N., and Zhdanov, S.A., On the spread-

ing of generalized newtonian liquids over solid substrates, J. Colloid InterfaceSci., 257, 284–290, 2003.

18. Pearson, J.R.A., Mechanics of Polymer Processing, Elsevier Applied SciencePublishers, London and New York, 1985.

19. Carre, A. and Eustache, F., Langmuir, 16, 2936, 2000.20. Starov, V.M., Rudoy, V.M., and Ivanov, V.I., Spreading of a droplet of polar liquid

induced by the overturning of amphiphilic molecules or their fragments in thesurface layer of a substrate, Colloid J. (Russian Academy of Sciences, EnglishTranslation), 61(3), 374–382, 1999.

21. Andrade, J.D. and Chen, W.-Y., Surf. Interface Anal., 8(6), 253, 1986.22. Andrade, J.D., Ed., Polymer Surface Dynamics, Plenum Press, New York, 1988.23. Lewis, B.K. and Ratner, B.D., J. Colloid Interface Sci., 159(1), 77, 1993.24. Miyama, M., Yang, Y., Yasuda, T., Okuno, T., and Yasuda, H., Langmuir, 13(20),

5494, 1997.



25. Rudoy, V.M., Stuchebryukov, S.D., and Ogarev, V.A., Colloid J. (English Trans-lation), 50(1), 199, 1988.

26. Langmuir, I., Science, 87, 1938, p. 493.27. Yiannos, P.N., J. Colloid Sci., 17(4), 334, 1962.28. Rideal, E. and Tadayon, J., Proc. R. Soc. (London) A, 225(1162), 346, 1954.29. Flory, P.J., Principles of Polymer Chemistry, Cornell University Press, Ithaca, New

York, 1990.30. Exerowa, D. and Kruglyakov, P., Foam and Foam Films: Theory, Experiment,

Application, Vol. 5, Studies in Interface Science, Elsevier, New York, 1988.


501

ConclusionsIn the first three chapters of this book, we have shown that all equilibrium andkinetics properties of liquids in contact with solid substrates can be describedusing a unifying approach by considering the simultaneous action of disjoiningand capillary pressure (save cases where gravity or other external fields apply).Using this approach the overall equilibrium liquid profile can be subdivided intothree parts: (i) the bulk of the liquid (meniscus or drop), where only capillary(and, eventually gravity) forces act, (ii) thin equilibrium films in front, where thesurface forces dominate (in the form of disjoining pressure action), and (iii) atransition region in-between, where both capillary and surface forces are equallyimportant. The main conclusion of Chapter 1 through Chapter 3 is that trueprogress in the area demands the consideration of disjoining pressure action ina vicinity of the apparent three-phase contact line. Though roughness and heter-ogeneity (chemical or otherwise) of the solid substrate affects wetting conditions,further progress in understanding of wetting and spreading must be based on theinclusion of surface forces action into consideration.

The major dominant quantity used to describe the liquid-solid substrate inter-action is the disjoining pressure isotherm and its dependence on the thickness ofthe layer. The latter dependence has been experimentally investigated only for alimited range of liquid film thicknesses and only for flat liquid films or layers.Further experimental work needed to get such dependence of the disjoining pres-sure isotherm on the thickness in the whole range of thickness (including over-saturation and the region of unstable flat films). Theory needed to really understandthe structural component of the disjoining pressure isotherm, and to understandhow the disjoining pressure is expressed in the case of non-flat liquid layers.

We have shown that in the case of complete wetting there is a reasonableagreement between the theory predictions and the progress is related to the factthat in such a case the disjoining pressure isotherm is well understood. Thesituation is drastically different in the case of partial wetting. We believe that thelack of progress in this case is related to the lack of understanding of theimportance of disjoining pressure in those circumstances. Note, in colloid andinterface science substantial progress was achieved only after the importance ofsurface forces action was understood. Hence, consideration of wetting and spread-ing processes on real, rough, and heterogeneous (chemical or otherwise) surfacestaking into account surface forces action in the case of partial wetting appearsas the most challenging problem. In Chapter 2 some problems in this area wereconsidered.

In Chapter 4 we have provided results about the kinetics of spreading andimbibition when liquids are in contact with porous solid materials. An importantconclusion is that the behavior of liquids in contact with porous materials is



substantially different from the corresponding processes with non-porous mate-rials. We also deduced the novel universal laws in this chapter. However, the latteruniversal dependencies have been deduced only in the case of spreading over thinporous layers and kinetics of wetting and imbibition in the case of thick porouslayers is still a challenge. The latter is especially true if surfactants are involved.

In Chapter 5 we have considered how the spreading of aqueous surfactantsolutions is affected by the presence of surfactants. It is difficult to imagine ourpresent-day life without surfactants (soaps, shampoos, detergents, washing liq-uids, etc.). Although some understanding has been accumulated in the case ofsurfactants acting on liquid-air interfaces where the transport processes are deter-mined by the Marangoni effect, much less is known about the behavior ofsurfactants in the vicinity of the three-phase contact line. Here we are very farfrom the level of understanding offered in Chapter 1 through Chapter 3. It is thereason why we used a semi-empirical model to understand the role of surfactantsin a vicinity of the three-phase contact line. In some cases such an approach leadsto predictions and explanations of available experimental data. However, under-standing of the real mechanism of surfactants transfer in the vicinity of the three-phase contact line demands further research efforts. Recently new surfactants liketrisiloxanes have attracted the attention of scientists and industrialists due to theirunusual properties that have led to call them “superspreaders.” We did not touchthe subject in this book, however, as understanding the nature of superspreadingbehavior is still a challenging open problem.

FREQUENTLY USED EQUATIONS

NAVIER–STOKES EQUATIONS

Navier–Stokes equations in cylindrical coordinate system (r, ϕ, z) are

ρϕ

ϕ ϕvvr

v

rv

vvz

v

rprr

r rz

r∂∂

+ ∂∂

+ ∂∂

−

= − ∂

∂

2

+ ∂∂

+ ∂∂

+ ∂∂

+ ∂∂

−ηϕ

2

2 2

2

2

2

2

1 1 2v

r r

v v

z rvr r

r r r r22 2

∂∂

−

v v

rrϕ

ϕ,

ρϕ

ϕ ϕ ϕ ϕ ϕvv

r

v

r

vv

v

z

v v

r rp

r zr∂

∂+

∂∂

+∂∂

+

= − ∂1

∂∂

+∂∂

+∂∂

+∂∂

+∂∂

ϕ

ηϕ

ϕ ϕ ϕ ϕ2

2 2

2

2

2

2

1 1v

r r

v v

z r

v

r++ ∂

∂−

22 2r

v v

rr

ϕϕ ,


Conclusions 503

If the Reynolds number is small, then the latter equations become the Stokesequations:

In any case, we have four equations for four unknown functions,

The viscose stress tensor is:

ρϕ

η

ϕvv

r

v

r

vv

v

zpzr

z zz

z∂∂

+ ∂∂

+ ∂∂

= − ∂

∂

+ ∂∂∂

+ ∂∂

+ ∂∂

+ ∂∂

2

2 2

2

2

2

2

1 1v

r r

v v

z r

v

rz z z z

ϕ..

∂∂

+∂∂

+ ∂∂

+ =vr r

v v

zvr

r z r10ϕ

ϕ.

01 12

2 2

2

2

2

2= − ∂

∂+ ∂

∂+ ∂

∂+ ∂

∂+ ∂

∂pr

v

r r

v v

z rvr

r r r rηϕ

−−∂∂

−

22 2r

v v

rrϕ

ϕ,

01 1 12

2 2

2

2

2

2= − ∂

∂+

∂∂

+∂∂

+∂∂

+∂

rp v

r r

v v

z r

v

ϕη

ϕϕ ϕ ϕ ϕ

∂∂+ ∂

∂−

r r

v v

rr2

2 2ϕϕ ,

01 12

2 2

2

2

2

2= − ∂

∂+ ∂

∂+ ∂

∂+ ∂

∂+ ∂

∂pz

v

r r

v v

z r

v

rz z z zη

ϕ

,

∂∂

+∂∂

+ ∂∂

+ =vr r

v v

zvr

r z r10ϕ

ϕ.

v r z v r z v r z p r zr z( , , ), ( , , ), ( , , ), ( , , ).ϕ ϕ ϕ ϕϕ

σ η σ ηϕϕ

ϕ ϕrr

rr

rpvr r

v v

r

v

r= − + ∂

∂= ∂

∂+

∂∂

−

2

1,

σσ ηϕ

σ ηϕϕϕ

ϕϕ= − +

∂∂

+

=

∂∂

+ ∂p

r

v vr

v

z r

vrz2

1 1, zz

∂

ϕ



where σrϕ is the tangential stress in the case of pure rotational flow, σzr is thetangential stress in the case of pure axial flow, and σzz in the normal stress to theflat surface.

NAVIER-STOKES EQUATIONS IN THE CASE OF TWO-DIMENSIONAL FLOW

In the case of a low Reynolds number, the latter equations become the Stokesequations:

In any case, these are three equations for three unknown functions, vx, vy, p. The viscose stress tensor appears as

The last component is the tangential stress in the case of a plane parallel flow.

σ η σ ηzzz

zrz rp

v

z

v

rvz

= − + ∂∂

= ∂∂

+ ∂∂

2 ,

ρ ηvvx

vy

vy

px

v

x

vx

x x x x∂∂

+ ∂∂

= − ∂∂

+ ∂∂

+ ∂∂

2

2

2

yy

vv

xv

v

ypy

vx

yy

y y

2

2

∂∂

+∂∂

= − ∂

∂+

∂ρ η

∂∂+

∂∂

∂∂

+∂∂

=

x

v

y

vx

v

y

y

x y

2

2

2

0

0

0

2

2

2

2

2

= − ∂∂

+ ∂∂

+ ∂∂

= − ∂∂

+∂

px

v

x

v

y

py

v

x x

y

η

η∂∂

+∂∂

∂∂

+∂∂

=

x

v

y

vx

v

y

y

x y

2

2

2

0

σ η σ η σ σ ηxxx

yyy

xy yxxp

v

xp

v

y

v= − + ∂∂

= − +∂∂

= = ∂∂

2 2, ,yy

v

xy+

∂∂


Conclusions 505

CAPILLARY PRESSURE

Capillary pressure is determined as γK, where γ is the interfacial tension (liq-uid–air or liquid–liquid), and K is the mean curvature of the interface.

Let h be the equation that describes the interface. Following are expressionsfor the mean curvature:

The general case in the Cartesian coordinate system, the interface profile, ish(x,y):

.

The axisymmetric case on the inner or outer surface of the cylindrical capillaryof the radius a, the interface profile, is h = h(x), where x is the axial coordinate

,

Here, the upper sign corresponds to the outer surface, and the lower sign corre-sponds to the inner surface; ′ means the differentiation with x.

The axisymmetric droplet on the plane substrate, the interface profile, is h(r),where r is the radial coordinate:

′ means the differentiation with r.

LIST OF MAIN SYMBOLS USED

GREEK

γ Interfacial tensionθ Contact angleΦ Excess free energyη Dynamic viscosityΠ Disjoining pressure

K

h

x

h

x

h

y

=

∂∂

+ ∂∂

+ ∂∂

2

2

2 23

1

/22

2

2

2 23

1

+

∂∂

+ ∂∂

+ ∂∂

h

y

h

x

h

y

//2

Kh

h

h

a h h= ′′

+ ′( )′

± + ′( )1 123 2

21 2/ /

( )∓

Kr

d

dr

rh

h= ′

+ ′( )1

1 21 2/



LATIN

g Gravity accelerationP Excess pressure p PressureS SurfaceT Absolute temperature in °KV Volume

SUBSCRIPTS

v Vapors Solidl Liquid* Characteristic scale or initial valuea Ambient aire Equilibriums Surface forces or saturatedc Capillary


Documents

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