Advances in Contact Angle, Wettability and

Adhesion

Scrivener Publishing100 Cummings Center, Suite 541J

Beverly, MA 01915-6106

Adhesion and Adhesives: Fundamental and Applied Aspects

The topics to be covered include, but not limited to, basic and theoretical aspects of adhesion; modeling of adhesion phenomena; mechanisms of adhesion; surface and interfacial analysis and characterization; unraveling of events at interfaces; characterization of interphases; adhesion of thin fi lms and coatings; adhesion aspects in reinforced composites; formation, characterization and durability of adhesive joints; surface preparation methods; polymer surface modifi cation; biological adhesion; particle adhesion; adhesion of metallized plastics; adhesion of diamond-like fi lms; adhesion promoters; contact angle, wettability· and adhesion; superhydrophobicity and superhydrophilicity. With regards to adhesives, the Series will include, but not limited to, green adhesives; novel and high-performance adhesives; and medical adhesive applications.

Series Editor: Dr. K.L. Mittal 1983 Route 52,

P.O.1280, Hopewell Junction, NY 12533, USAEmail: usharmittal@optimum.net

Publishers at ScrivenerMartin Scrivener (martin@scrivenerpublishing.com)

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Advances in Contact Angle, Wettability and

Adhesion

Edited by

K.L. Mittal

Volume 1

Copyright © 2013 by Scrivener Publishing LLC. All rights reserved.

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v

Contents

Preface xviiAcknowledgements xxi

Part 1: Fundamental Aspects 1

1 Correlation between Contact Line Pinning and Contact Angle Hysteresis on Heterogeneous Surfaces: A Review and Discussion 3

Mohammad Amin Sarshar, Wei Xu, and Chang-Hwan Choi 1.1 Introduction 31.2 Contact Line Pinning on Chemically Heterogeneous

Flat Surfaces 41.3 Contact Line Pinning on Hydrophobic Structured

Surfaces 71.4 Summary and Conclusion 14References 15

2 Computational and Experimental Study of Contact Angle Hysteresis in Multiphase Systems 19

Vahid Mortazavi, Vahid Hejazi, Roshan M D’Souza, and Michael Nosonovsky 2.1 Introduction 192.2 Origins of the CA Hysteresis 242.3 Modeling Wetting/Dewetting in Multiphase Systems 27

2.3.1 CA in Multiphase Systems 272.3.2 CA Hysteresis in Multiphase Systems 28

2.4 Experimental Observations 302.5 Numerical Modeling of CA Hysteresis 35

2.5.1 Background 352.5.2 The Cellular Potts Model 36

vi Contents

2.5.3 The Cellular Potts Modeling of Wetting 382.5.4 Results 40

2.6 Conclusions 44Acknowledgement 45References 45

3 Heterogeneous Nucleation on a Completely Wettable Substrate 49

Masao Iwamatsu 3.1 Introduction 493.2 Interface-Displacement Model 513.3 Nucleation on a Completely-Wettable Flat Substrate 54

3.3.1 d = 2-dimensional Nucleus 543.3.2 d = 3-dimensional Nucleus 62

3.4 Nucleation on a Completely-Wettable Spherical Substrate 65

3.5 Conclusion 69Acknowledgments 70References 70

4 Local Wetting at Contact Line on Textured Hydrophobic Surfaces 73

Ri Li and Yanguang Shan 4.1 Introduction 734.2 Static Contact Angle 76

4.2.1 Global Approach – Thermodynamic Equilibrium 774.2.2 Local Approach – Force Balance 79

4.3 Wetting of Single Texture Element 804.4 Summary 85References 85

5 Fundamental Understanding of Drops Wettability Behavior Theoretically and Experimentally 87

Hartmann E. N’guessan, Robert White, Aisha Leh, Arnab Baksi, and Rafael Tadmor

5.1 Introduction 875.2 Discussion 905.3 Conclusion 93References 94

Contents vii

6 Hierarchical Structures Obtained by Breath Figures Self-Assembly and Chemical Etching and their Wetting Properties 97

Edward Bormashenko, Sagi Balter, Roman Grynyov, and Doron Aurbach

6.1 Introduction 976.2 Materials and Methods 98

6.2.1 Fabricating Hierarchical Polymer Surfaces 986.2.2 Characterization of the Wetting Properties

of Polymer Surfaces 996.2.3 Plasma Treatment of the Surfaces 996.2.4 B.E.T Characterization of the Surfaces 100

6.3 Results and Discussion 1006.3.1 Morphology and Wetting Properties of the

Multi-scaled PC Surfaces 1006.3.2 Modifi cation of Wetting Properties of the

Multi-scaled Surfaces with Cold Radiofrequency Plasma Treatment 103

6.3.3 B.E.T Study of the Surfaces 1046.4 Conclusions 105Acknowledgements 105References 105

7 Computational Aspects of Self-Cleaning Surface Mechanisms 109

Muhammad Osman, Raheel Rasool, and Roger A. Sauer 7.1 Introduction 1097.2 Droplet Membrane 111

7.2.1 Governing Equations in Strong Form 1117.2.1.1 Surface Contact 1127.2.1.2 Line Contact 1137.2.1.3 Surface Roughness 113

7.2.2 Weak Formulation of the Governing Equations 1147.2.2.1 Finite Element Implementation 116

7.2.3 Model Verifi cation 1177.2.3.1 Force Equilibrium 117

7.2.4 Particle-Droplet Interaction 119

viii Contents

7.3 Flow Model 1217.3.1 Governing Equations 1217.3.2 Finite Element Implementation 1227.3.3 Normal and Tangential Velocities at the

Boundary 1257.4 Results 126

7.4.1 Multiscale View of Contact 1267.4.2 Computational Membrane Model 1277.4.3 Liquid Flow Model 1277.4.4 Particle-Droplet Interaction 128

7.5 Summary 129Acknowledgement 129References 129

8 Study of Material–Water Interactions Using the Wilhelmy Plate Method 131

Eric Tomasetti, Sylvie Derclaye, Mary-Hélène Delvaux, and Paul G. Rouxhet 8.1 Introduction 1328.2 Upgrading Wetting Curves 1338.3 Study of Surface-Oxidized Polyethylene 136

8.3.1 Introduction 1368.3.2 Experimental 1378.3.3 Results and Discussion 138

8.3.3.1 Surface Morphology and Composition 1388.3.3.2 Water Retention upon Emersion 1388.3.3.3 Surface Reorganization According

to the Environment 1418.3.4 Conclusion 143

8.4 Study of Amphiphilic UV-Cured Coatings 1438.4.1 Introduction 1438.4.2 Experimental 1448.4.3 Results and Discussion 144

8.4.3.1 Surface Morphology and Composition 1448.4.3.2 Wetting Measurements 148

8.4.4 Conclusion 151

Contents ix

8.5 Conclusion 151Acknowledgements 152References 152

9 On the Utility of Imaginary Contact Angles in the Characterization of Wettability of Rough Medicinal Hydrophilic Titanium 155

S. Lüers, C. Seitz, M. Laub, and H.P. Jennissen

9.1 Introduction 1569.2 Theoretical Considerations 156

9.2.1 Mathematical Basis 1569.2.2 Physical Basis 157

9.3 Materials and Methods 1589.3.1 Titanium Miniplates 1589.3.2 Profi lometric Surface Roughness 1599.3.3 Contact Angle Measurements 1599.3.4 Baseline Correction 1599.3.5 Calculation of Contact Angles 1609.3.6 Determination of Wetting Times 1609.3.7 SEM 1609.3.8 Nomenclature 160

9.4 Results and Discussion 1619.4.1 SEM of the uSLA Surface 1619.4.2 Characterization of Wetting of uSLA Surfaces 162

9.4.2.1 Ti-uSLA Surface Immediately after Etching 162

9.4.2.2 Ti-uSLA Surface after 24 Hours in 100 mM KH2PO4, pH 7.6 (Wet Storage) 164

9.4.2.3 Ti-uSLA Surface Stabilization by an Exsiccation Layer of Salt for 24 Hours (Dry Storage) 168

9.4.2.4 Hybrid Contact Angles on Ti-uSLA Surface 169

9.4.2.5 Wetting Times of Ti-uSLA Miniplates 169

x Contents

9.5 Conclusion 171Acknowledgement 171References 171

10 Determination of Surface Free Energy at the Nanoscale via Atomic Force Microscopy without Altering the Original Morphology 173

L. Mazzola and A. Galderisi 10.1 Introduction 17410.2 Materials and Methods 175

10.2.1 Nanoindentation Setup 175 10.2.2 Atomic Force Microscopy Setup 176 10.2.3 Focused Ion Beam Analysis 178 10.2.4 Profi lometric Analysis 179

10.3 Results and Discussion 180 10.3.1 Results 180 10.3.2 Discussion 181 10.3.3 Applications 186

10.4 Conclusion 188References 188

Part 2: Superhydrophobic Surfaces 191

11 Assessment Criteria for Superhydrophobic Surfaces with Stochastic Roughness 193

Angela Duparré and Luisa Coriand 11.1 Introduction 19311.2 Model and Experiments 194

11.2.1 Roughness Model and Data Analysis 194 11.2.2 Roughness and Contact Angle

Measurements 196 11.2.3 Nanorough Coatings 196

11.3 Results and Discussion 19711.4 Summary 200Acknowledgement 200References 201

Contents xi

12 Nanostructured Lubricated Silver Flake/Polymer Composites Exhibiting Robust Superhydrophobicity 203

Ilker S. Bayer, Luigi Martiradonna, and Athanassia Athanassiou 12.1 Introduction 204

12.1.1 Superhydrophobic Nanostructured Silver Synthesis from Precursors 204

12.1.2 Lubricated Silver Flakes in Polymer Composites 207

12.2 Experimental 210 12.2.1 Functionalization of Silver Flakes

with Stearic Acid (STA) 210 12.2.2 Preparation of Silver Flake/Polymer

Suspensions 211 12.2.3 Preparation of Superhydrophobic

Copolymer Blend/Silver Solutions 211 12.2.4 Preparation of Superhydrophobic

Laminates on Nonwovens 21312.3 Results and Discussion 214

12.3.1 Characterization of Nanostructured Silver Flakes 214

12.3.2 Surface Morphology and Wetting Properties of Superhydrophobic Coatings 215

12.3.3 Coating Deposition on Soft Hydrophobic Surfaces (Paraffi nic Films) 216

12.3.4 Wetting Characteristics of the Coatings Deposited on Paraffi nic Films Impregnated into Nonwoven Fabric Substrates 217

12.4 Conclusions 220References 220

13 Local Wetting Modifi cation on Carnauba Wax-Coated Hierarchical Surfaces by Infrared Laser Treatment 227

Athanasios Milionis, Roberta Ruffi lli, Ilker S. Bayer, Lorenzo Dominici, Despina Fragouli, and Athanassia Athanassiou 13.1 Introduction 22813.2 Experimental 229

xii Contents

13.2.1 Materials 229 13.2.2 Methods 229

13.2.2.1 Fabrication of SU-8 Micropillars 230 13.2.2.2 Preparation of Carnauba

Wax Solution 230 13.2.2.3 Spray Coating Technique 230 13.2.2.4 Heating by IR Laser Irradiation 231

13.2.3 Characterization of the Samples 23113.3 Results and Discussion 23113.4 Conclusions 238Acknowledgements 238References 239

Part 3: Wettability Modifi cation 243

14 Cold Radiofrequency Plasma Treatment Modifi es Wettability and Germination Rate of Plant Seeds 245

Edward Bormashenko, Roman Grynyov, Yelena Bormashenko, and Elyashiv Drori 14.1 Introduction 24514.2 Experimental 246

14.2.1 Materials and Methods 24614.3 Results and Discussion 24814.4 Conclusions 255Acknowledgements 255References 255

15 Controlling the Wettability of Acrylate Coatings with Photo-Induced Micro-Folding 259

Thomas Bahners, Lutz Prager, and Jochen S. Gutmann

15.1 Introduction 260 15.1.1 Surface Roughness and Wetting Behavior

in the Wenzel State 260 15.1.2 Wettability and Cleanability 262

15.2 The Process of Photo-induced Micro-folding 26415.3 Experimental 265

15.3.1 Acrylate Design 265 15.3.2 Application and Curing of Acrylate Lacquers 267 15.3.3 Characterization 267

Contents xiii

15.4 Review of Results 267 15.4.1 Morphology and Surface Properties of

Hydrophobic Surfaces 267 15.4.2 Morphology and Surface Properties of

Hydrophilic Surfaces 269 15.4.3 Post-Treatment of ‘Hydrophobic’ Acrylate

Layers 271 15.4.4 Dirt Take-up and Cleanability 272

15.5 Summary 274Acknowledgment 275References 275

16 Infl uence of Surface Densifi cation of Wood on its Dynamic Wettability and Surface Free Energy 279

M. Petric, A. Kutnar, L. Rautkari, K. Laine, and M. Hughes 16.1 Introduction 28016.2 Experimental 281

16.2.1 Materials 281 16.2.2 Surface Densifi cation Process 281 16.2.3 Heat/Steam Treatment 282 16.2.4 Oil Treatment 282 16.2.5 Contact Angle Measurements 283 16.2.6 Surface Free Energy 284

16.3 Results and Discussion 284 16.3.1 Compressed Thickness and Oven Dry

Density of Surface Densifi ed Wood 284 16.3.2 Dynamic Contact Angle 286 16.3.3 Surface Free Energy 290

16.4 Summary and Conclusions 294Acknowledgments 294References 295

17 Contact Angle on Two Canadian Woods: Infl uence of Moisture Content and Plane of Section 297

Fabio Tomczak and Bernard Riedl 17.1 Introduction 29717.2 Materials and Experimental Procedures 300

17.2.1 Materials 300 17.2.2 Methods 300

xiv Contents

17.3 Results and Discussion 30217.4 Conclusions 307Acknowledgement 307References 308

18 Plasma Deposition of ZnO Thin Film on Sugar Maple: The Effect on Contact Angle 311

Fabio Tomczak, Bernard Riedl, and Pierre Blanchet

18.1 Introduction 31218.2 Materials and Experimental Procedures 313

18.2.1 Materials 313 18.2.2 Procedures 314

18.3 Results and Discussion 31618.4 Conclusion 325Acknowledgements 326References 326

19 Effect of Relative Humidity on Contact Angle and its Hysteresis on Phospholipid DPPC Bilayer Deposited on Glass 329

Emil Chibowski, Konrad Terpilowski, and Lucyna Holysz19.1 Introduction 33019.2 Experimental 331

19.2.1 Materials 331 19.2.2 Deposition of DPPC Bilayers on Solid

Support by Solution Spreading 331 19.2.3 Methods 332

19.2.3.1 Contact Angle Measurements 332 19.2.3.2 Surface Images from Optical

Profi lometry 332 19.2.3.3 Surface Images from Scanning

Electron Microscope-Focused Ion Beam, SEM-FIB 332

19.3 Result and Discussion 333 19.3.1 Advancing and Receding Contact Angles 333 19.3.2 Apparent Surface Free Energy 334 19.3.3 Some Literature Data about DPPC

Layer Hydration 337

Contents xv

19.3.4 Optical Profi lometry Images of DPPC Bilayer 339

19.3.5 Surface Images of DPPC Bilayer from SEM-FIB 342

19.4 Conclusion 343Acknowledgments 344References 344

Part 4: Wettability and Surface Free Energy 347

20 Contact Angles and Surface Energy of Solids: Relevance and Limitations 349

Paul G. Rouxhet 20.1 Introduction 35020.2 Thermodynamic Background 35120.3 Determination of the Surface Energy of a

Solid from Contact Angles 354 20.3.1 The Owens-Wendt Approach 354 20.3.2 Overview of Other Methods 356 20.3.3 Comparison of Different Methods Applied

to Polymers and Surface-Modifi ed Polymers 35920.4 Wettability and Surface Composition of

Polypropylene Modifi ed by Oxidation 36420.5 Wettability and Surface Cleanliness of

Inorganic Materials 36820.6 Conclusion 371Acknowledgements 372References 373

21 Surface Free Energy and Wettability of Different Oil and Gas Reservoir Rocks 377

Andrei S. Zelenev and Nathan Lett 21.1 Introduction 37721.2 Experimental 379

21.2.1 Materials 379 21.2.2 Surface Free Energy Determination 380 21.2.3 Contact Angle and Surface Tension

Measurements 381

xvi Contents

21.3 Results and Discussion 38121.4 Conclusions 386References 387

22 Infl uence of Surface Free Energy and Wettability on Friction Coeffi cient between Tire and Road Surface in Wet Conditions 389

L. Mazzola, A. Galderisi, G. Fortunato, V. Ciaravola,

and M. Giustiniano

22.1 Introduction 39022.2 Theoretical Basis of the New Model 39122.3 Materials and Methods 398

22.3.1 Materials 398 22.3.2 Friction Tester 399 22.3.3 Hardness Measurements 399 22.3.4 Wettability, Surface Free Energy

and Surface Tension Determination 399 22.3.5 Profi lometric Analysis 402

22.4 Results and Discussion 40222.5 Summary and Conclusions 408Acknowledgement 409References 409

xvii

Preface

The history of contact angle and wetting can be traced back to the early 17th century. The putative seminal paper on this topic was published in 1805 by Thomas Young [An essay on the cohesion of fl uids, Phil. Trans. Roy. Soc., 95, 65-87(1805)]. In this paper he describes the balance of various forces (interfacial tensions) act-ing on a sessile liquid drop on a solid surface, which is popularly known today as the Young’s Equation. Apropos, there is no for-mal equation in this paper. Apparently, some brilliant individual transformed Young’s description into this equation. According to Prof. Robert J. Good [R.J. Good, Contact angle, wettability and adhesion, in: Contact Angle, Wettability and Adhesion, K.L. Mittal (Ed.) pp. 3-36, VSP, Utrecht 1993)] “Most surface and colloid chem-ists think of Thomas Young as the father of scientifi c research on contact angles and wetting. But probably the earliest direct recog-nition of wetting phenomena was given by Galileo [Galileo Galilei, Bodies that Stay Atop Water, or Move in it (1612)] who might be called the grandfather of the fi eld.”

Another momentous event is this fi eld occurred in 1997 when W. Barthlott and C. Neinhuis [W. Barthlott and C. Neinhuis, Purity of sacred lotus, or escape from contamination in biological surfaces, Planta, 202, 1-8(1997)] investigated the wetting properties of various plants and discovered extreme water-repellency (superhydropobic-ity) and self-cleaning mechanism of the sacred lotus (Nelumbo nucifera) and coined the term “Lotus Effect.” Since this discovery, there has been an explosive interest in the topic of superhydrophobicity and a legion of techniques have been described in the literature [see the book A. Carre’ and K.L. Mittal (Eds.) Superhydrophobic Surfaces, VSP/Brill, Leiden (2009)] to devise mechanically robust superhydrophobic sur-faces of a variety of materials. The antonymous fi eld of superhydrophi-licity has also attracted fervent interest from the research community. These days there is an ardent interest (both from fundamental and

xviii Preface

applied views) in modifying surfaces to alter their wetting behavior to render them superhydrophobic, superhydrophilic, oleophobic, oleo-philic, omniphobic, panphobic, amphiphobic. In other words, all kinds of “phobicities” and “philicities” are under intensive investigation.

Even a cursory look at the literature will evince that there is a brisk research activity regardingf contact angles and wetting/spreading from both fundamental and applied points of view. The wonderful world of wettability is very wide as it plays an extremely impor-tant role in many areas of human endeavor ranging from high-tech (microelectronics, micro-and nanofl uidics, MEMS and NEMS, bio-medical devices, for example) to the quite mundane (washing of clothes, spraying of insecticides/ pesticides on agricultural prod-ucts). Researchers have also studied the wettability behavior of skins of people (both males and females) from different origins and backgrounds. I wonder if wettability can be correlated to beauty! I should also add that all signals indicate that the interest in wetting phenomena will continue unabated.

Now coming to this volume, which is essentially based on the written accounts of papers presented at the Eighth International Symposium on Contact Angle, Wettability and Adhesion held in Quebec City, Quebec, Canada during June 13-15, 2012 under the aegis of MST Conferences. It should be recorded for posterity that all manuscripts were rigorously peer-reviewed, suitably revised (some twice or thrice) and properly edited before inclusion in this book. So this book is not a mere collection of unreviewed and unedited papers, rather it represents articles which have passed the rigorous scrutiny. Thus, these articles are of archival value and their standard is as high as any journal or even higher than many journals.

This book containing 22 articles is divided into four Parts as follows. Part 1: Fundamental Aspects; Part 2: Superhydrophobic Surfaces; Part 3: Wettability Modifi cation; and Part 4: Wettability and Surface Free Energy. The topics covered include: contact angle hysteresis on heterogeneous surfaces and in multiphase systems; fundamental understanding of drops wettability behavior; compu-tational aspects of self-cleaning surface mechanisms; utility of imag-inary contact angles in the characterization of wettability of rough surfaces; determination of surface free energy at the nanoscale via atomic force microscopy; superhydrophobicity and its assessment criteria; wettability modifi cation techniques for different materials; effects of cold RF plasma treatment on germination rate of plant

Preface xix

seeds; wettability of wood; wettability of DPPC bilayer; wettability, contact angles and surface free energy of solids; infl uence of surface free energy on friction coeffi cient between tire and road surface.

It is quite obvious from the above that this book comprising 22 articles written by world-renowned researchers covers many ramifi cations of contact angles and wettability. It represents a com-mentary on the contemporary research activity and refl ects the cumulative wisdom of a number of key researchers in this arena.

Yours truly sincerely hopes that anyone interested in staying abreast of the latest developments and perspectives in the domain of contact angle, wettability and adhesion will fi nd this compen-dium of great interest and value. Also I hope the information consolidated in this volume will serve as a fountainhead for new research ideas and applications.

xxi

Acknowledgements

Now comes the pleasant task of thanking those who were instru-mental in the birth of this book. First and foremost, I would like to express my most sincere thanks to the authors for their interest, enthusiasm, cooperation and contribution, without which this book could not be materialized. Second, my heart-felt thanks go to the unsung heroes(reviewers) for their time and effort in providing invaluable comments which most certainly enhanced the quality of these articles. The comments from the peers are sine qua non for maintaining the highest standard of a publication. Last, but not least, I am appreciative of the earnest interest and unwavering help of Martin Scrivener (publisher) in bringing this book to fruition.

Kash MittalP.O. Box 1280

Hopewell Jct., NY 12533E-mail:ushaRmittal@optimum.net

May 2, 2013

PART 1

FUNDAMENTAL ASPECTS

3

K.L. Mittal (ed.) Advances in Contact Angle, Wettability and Adhesion, (3–18) 2013 © Scrivener Publishing LLC

1

Correlation between Contact Line Pinning and Contact Angle Hysteresis

on Heterogeneous Surfaces: A Review and Discussion

Mohammad Amin Sarshar, Wei Xu, and Chang-Hwan Choi*

Department of Mechanical Engineering, Stevens Institute of Technology, Hoboken, New Jersey, USA

AbstractMicro- or nano-textured hydrophobic surfaces have attracted considerable interest due to their highly water-repellent property, and are called superhy-drophobic. Although such superhydrophobic surfaces typically exhibit high contact angles for water droplets, their adhesion and frictional properties such as contact angle hysteresis are signifi cantly affected by the dynamics of contact line pinning at the droplet boundary. However, a clear correlation between the contact line pinning and the contact angle hysteresis has not been revealed yet. In this paper, we review the literature reporting on their correlation, both for chemically and physically patterned heterogeneous surfaces, including our recent discovery on superhydrophobic surfaces. Then, we propose and discuss an appropriate new physical parameter that shows close and consistent correlation between the dynamics of contact line pinning and the contact angle hysteresis.

Keywords: Contact angle hysteresis, contact line pinning, heterogeneous surfaces, superhydrophobic

1.1 Introduction

When hydrophobic surfaces are roughened or patterned in proper length scale and morphology, air can be entrapped between the surface structures

*Corresponding author: Chang-Hwan.Choi@stevens.edu

4 Advances in Contact Angle, Wettability and Adhesion

(typically called as “Cassie state”) and the surfaces show highly non-wet-ting and slippery, so-called superhydrophobic property [1] that would be of great signifi cance in many applications such as in self-cleaning [2], hydro-dynamic friction reduction [3], anti-icing [4, 5], anti-corrosion [6], thermal/energy system [7], biotechnology [8], and micro- and nano-devices [9]. Known as “lotus effect” [10], such superhydrophobic surfaces generally result in high contact angle and low contact angle hysteresis for water drop-lets so that water droplets can easily roll off from the surfaces. However, also known as “petal effect” [11], if water droplets wet the surfaces either partially or uniformly with no air void retained (typically called as “Wenzel state”), the surfaces exhibit high contact angle hysteresis despite high apparent contact angle. In such a case, water droplets get strongly pinned on the surfaces and do not roll off even when the surfaces are tiled even greater than 90°. Such sticky surfaces for droplets are also of great impor-tance in many applications such as in spraying/coating [12], ink-jet printing [13], liquid transportation/analysis [14], and microfl uidics [15]. Recently it has also been shown that superhydrophobic surfaces, even in Cassie state, can cause more signifi cant contact line pinning and hence behave stickier than non-patterned planar hydrophobic surfaces, depending on the geom-etry and dimensions of surface patterns [16]. Such reports suggest that the pinning phenomena of droplets are affected in a complicate way by many surface parameters including physical morphology, chemical heterogeneity, and interfacial wetting states [17–21]. To date, a few different approaches have been applied to explain the direct correlation between the behav-iors of contact line pinning and the adhesion or frictional properties such as contact angle hysteresis for moving droplets. One of them is based on the effective contact area between the droplet and the solid surface [22–27], while the other one is based on the effective contact length [16, 28–32]. In this paper, we review the literature and discuss which physical parameters would be more relevant to correlate the dynamics of contact line pinning and the adhesion properties of heterogeneous surfaces such as contact angle hysteresis of superhydrophobic surfaces. Based on these, we also propose a non-dimensional surface parameter that can be universally applied to determine their correlation. Despite being simple, the new physical param-eter revealed in this paper should serve as a quick and effi cient criterion for the design and engineering of heterogeneous or superhydrophobic surfaces with tailored adhesion properties.

1.2 Contact Line Pinning on Chemically Heterogeneous Flat Surfaces

The physical insights into the dynamics of three-phase contact line on het-erogeneous or superhydrophobic surfaces can fi rst be obtained from the

Correlation between Contact Line Pinning 5

previous studies on the chemically heterogeneous surfaces with a fi nite number of fl at spots of higher hydrophobicity, or defects. For example, Joanny and de Gennes [33] analytically studied the contact angle hysteresis of solid surfaces with a single chemical defect at lower wettability state. As shown in Figure 1.1, the chemical defect with greater hydrophobicity causes pinning and distortion of contact line as the droplet crosses the defect and moves forward. In this case, the localized pinning force (F) exerted on the defect due to the deformation of the contact line can be estimated by Hooke’s law (F = kx) considering the spring of stiffness (k) defi ned as:

20; log

LF kx k

dpgq ⎛ ⎞= = ⎜ ⎟⎝ ⎠

(1.1)

where g is the surface tension of liquid, q0 is the quasi-equilibrium contact angle for an ideal surface with no defect, d is the diameter of a circular defect, and L is a cut-off length scale which can be either the droplet diam-eter or the capillary length. As a result, the pinning force exerted on the defect by the deformed contact line would be F = kxmwhere xm is the maxi-mum amplitude of the distortion of the contact line. This force vanishes to zero away from the defect. In this way, if the distortion of the contact line (or the overall shape of the contact line) is precisely known, the amount of the pinning force can be estimated precisely. Under the conditions of small equilibrium contact angle on the substrate with the small distortion of the contact line [33, 34], this approach gives reasonably good estimation of the pinning force. Equation 1.1 also implies what variables would be important in the dynamics of contact line movement. This approach can also be applied to the case of a small number density of the defects where the defects are not densely populated so that they do not act collectively

Depinningangle

Defect

Moving droplet

Three-phase contact linex

Direction ofmovement

t = t3t = t2

t = t1

Figure 1.1 Progress of three-phase contact line as it passes over a single defect. The contact line moves to upward direction with time (t) as a sequence of t1→ t2→ t3. While the dotted lines represent the contact lines before pinning (at t=t1) and after depinning (at t=t3), respectively, the solid line (at t=t2) shows the pinning and deformation of the contact line at the defect.

6 Advances in Contact Angle, Wettability and Adhesion

to cause the deformation of the interface. In the case the number density of defects increases and the defects are populated close to each other, the defects would behave collectively to deform the contact line. In such a case, the deformation of the contact line at each defect is comparatively smaller so that it results in a faceted droplet shape and the approach explained above is not appropriate to apply.

When the contact line recedes or advances over the multiple defects with a small number density, the pinning force per unit length of the con-tact line can be reduced to [34]:

( )cos cosnWR R 0g q q= − (1.2)

( )cos cosnWA A0g q q= − (1.3)

where WR and WA represent the dissipation energies per unit area (or pin-ning force per unit length) due to the deformation of contact line at a sin-gle defect in the receding (qR: receding contact angle) and the advancing (qA: advancing contact angle) motions of the droplet, respectively, and n is the total number of defects engaged in the contact line movement. By combining Equations 1.2 and 1.3, it leads to obtain an equation for contact angle hysteresis (cosqR – cosqA), such as:

( ) ( )cos cosn W WR A R Ag q q+ = − (1.4)

Physically, WR and WA have the same meaning as the localized pinning force (F) described in Equation 1.1. Thus, if the localized pinning force (F) is known for each defect engaged in the contact line movement, the effect of key surface parameters determining the contact angle hysteresis can be understood, such as the diameter of the defect (d) shown in Equation 1.1.

Experimentally, Cubaud and Fermigier [35] also studied the pinning force of a contact line on chemically heterogeneous surfaces. In the case of a small number density of defects, they proposed that one of the use-ful parameters, which would be more relevant to defi ne the pinning force than probing the mechanical deformation of the contact line (x, Figure 1.1), would be the angle which the two tangents to the contact lines at each side of the defect make (called depinning angle, Figure 1.1). In order to correlate with the depinning angle, they also introduced a new physical parameter fs for the defects, defi ned as:

f h

sdsTg =

(1.5)

where sT is the difference in spreading coeffi cients between the substrate and the defect and h is the thickness (height) of the droplet. They regulated the value of fs by varying the diameter of the defect (from 100 mm to 1800 mm ) and examined how the depinning angle would change for a single defect. Based on their experimental observation, they concluded that there should