Molecular Adhesion and Its Applications: The Sticky Universe

Molecular Adhesion and Its Applications The Sticky Universe

Molecular Adhesion and Its Applications The Sticky Universe Kevin Kendall The University of Birmingham Birmingham, England

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I wish I could derive all phenomena of nature, by some kind of reasoning, from mechanical principles; for I have many reasons to suspect that they all depend on certain forces by which the particles of bodies are either mutually attracted and cohere in regular figures or are repelled and recede from each other I. NEWTON, 1687

FOREWORD

At the beginning of the twentieth century, engineers and technologists would have recognized the importance of adhesion in two main aspects: First, in the display of friction between surfaces — at the time a topic of growing importance to engineers; the second in crafts requiring the joining of materials — principally wood—to form engineering structures. While physical scientists would have admitted the adhesive properties of glues, gels, and certain pastes, they regarded them as materials of uncertain formulation, too impure to be amenable to precise experiment. Biological scientists were aware also of adhesive phenomena, but the science was supported by documentation rather than understanding. By the end of the century, adhesion and adhesives were playing a crucial and deliberate role in the formulation of materials, in the design and manufacture of engineering structures without weakening rivets or pins, and in the use of thin sections and intricate shapes. Miniaturization down to the micro- and now to the nano-level of mechanical, electrical, electronic, and optical devices relied heavily on the understanding and the technology of adhesion. For most of the century, physical scientists were aware that the states of matter, whether gas, liquid, or solid, were determined by the competition between thermal energy and intermolecular binding forces. Then the solid state had to be differentiated into crystals, amorphous glasses, metals, etc., so the importance of the molecular attractions in determining stiffness and strength became clearer. Cross-linked rubbers and composites designed at the macro- and micro-level were developed to extend the range of materials available for engineering purposes. Adhesion at the molecular scale, at surfaces and interfaces, was recognized to be a vital factor determining performance. Biological sciences were not excluded from this explosion of knowledge. The study of cell structure and cell behavior, including material transport across membranes, cell division, and cell adhesion, raised aspects of adhesion already familiar in physical colloid systems. Then the rise of molecular biology in the last

vii

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Foreword

30 years has brought adhesion into prominence at all levels of organization in biological systems. Certainly there is a vast literature, and especially a voluminous research canon, associated with the science of adhesion. However, the literature is fragmented and diffuse because adhesion is involved in all areas of endeavor. The engineering literature is somewhat more ordered because of the need to agree good practice and safety protocol. It is nevertheless compartmentalized. Even so, it is not easy to align scientific knowledge with engineering practice in many fields of application. One possible exception is computer modeling, which is at the cutting edge of advances both in science and engineering though the emphasis is rather different. No doubt, in the future, we shall see adhesion modeled at the molecular level and tracked through to engineering practice with the aid of computers. Remarkably, there is no scientific monograph covering the state and current knowledge of adhesion. Nor is there an engineering treatise to take the reader onto a representative range of applications. This is not because we have lacked leading scientists or engineers or gifted teachers in the twentieth century. Presumably, they have been too busy in a field of rapid progress. Now the challenge of promoting a unified account of molecular adhesion, extending it to basic laws and technical practice and onto applications has been taken up by Kevin Kendall. His enthusiasm for the subject and his experience in academe and industry shines through this comprehensive treatise. It is a book that can be read from cover to cover, or a laboratory and design manual to be dipped into as work demands. It benefits enormously from the distillation of a vast subject through a single mind. Sir Geoffrey Allen FRS

PREFACE

Molecular adhesion is one of the most fundamental concepts in science. Molecules tend to be stuck together to form crystals, liquids, composite materials, assembled structures, colloids, rocks, pastes, living cellular creatures, and so forth. Our universe may be expanding against the force of gravity, but each local bit of the universe is firmly stuck together by molecular adhesion. Explaining this across the interdisciplinary boundaries of chemistry, physics, engineering, and bioscience is the objective of this book. The argument is at undergraduate teaching level, but the specific examples and references are geared for research specialists. The laws we remember from school are the laws of motion. Movement is interesting whereas stasis is boring. Newton made the gravitational law of adhesion exciting by using it to explain the movement of planets and satellites. Yet our Earth is largely static; stuck together by molecular adhesion. Our bodies lie in the tenuous skin of mobile material at the Earth’s surface, which explains our fascination with movement, leading to Newton’s Laws of Motion. To suggest laws of adhesion is almost a joke, rather like one of those Andy Warhol movies where nothing happens. But molecular adhesion is interesting precisely because it limits the movement we want; the movement of a car on a road, the movement of cornflakes onto our plates. Laws of adhesion must exist and should be revealed. Four centuries ago, Galileo famously said “It moves”; this century we are saying “It sticks”. Previously, we could only detect adhesion by this limit of movement. The single way to test for adhesion was by breaking the bond. Now nondestructive tests are becoming possible using the new technique of atomic force microscopy at the molecular level. Thus adhesion can be distinguished from, then related to, fracture. We have to understand both making the joint and breaking it to obtain a rational picture of adhesion as a whole. A second major advance is in computer modeling which enables us to describe the interactions of the many thousands of ix

x

Preface

atoms which participate in adhesion events. Adhesion is cooperative; the adhesion of 1000 atoms is different from the adhesion of 1 atom. Roughly 6000 articles are written each year on adhesion but these are in widely varying disciplines which may not be immediately accessible. This book cannot quote all these papers, nor can it present a comprehensive critique of the documents, but it can provide a skeleton of logic and a common agreed language for describing adhesion phenomena in those different areas, together with an assessment of the pivotal contributions in the literature. Individual researchers should find, in the framework provided here, a place to fit their own observations. Many books on surface chemistry contain a short chapter on adhesion. But such accounts are seldom satisfactory. Clearly, adhesion stems from the strong attractive forces between molecules. However, the connections between molecular forces and phenomena seen in soiling, cements, adhesives, corrosion, catalysis, or slime mold reproduction are not normally made explicit. Similarly, there are several texts on adhesion for engineers, though most engineers, following Coulomb and Hertz, have ignored adhesion. In a typical book on Contact Mechanics, only 1% deals with adhesion. Engineering books tend to be dominated by mathematical derivations and hardly acknowledge that molecules exist. But without molecular force, there is no adhesion. In this book I have emphasized the observations of phenomena based on adhesion, keeping the mathematical description to a minimum, concentrating on useful results rather than analytical manipulations, trying to show the connection between molecules and mechanics. The book is in three parts. The first introduces the background and lays the fundamental tenets of the subject which really go back to Isaac Newton. He experimented on the contact of glass lenses, trying to interpret the results in terms of molecular adhesion long before the idea of molecules existed. The second part of the book seeks to establish the laws and mechanisms of adhesion, and the third to explain the applications and benefits of molecular adhesion in the practical world. In the first part, the aim is to unravel the many ideas and theories which have been proposed to account for adhesion phenomena, to pin down the key observations which have led to our current state of thinking, and to establish three “laws of adhesion” which account for the phenomenology. The second part then goes on to establish the three laws on a more quantitative and theoretical level which can be tested by new theories of computer modeling and by new measurements such as Atomic Force Microscopy. Finally, in the third part, this theory of molecular adhesion is applied to eight important areas of technology, where the effects of intermolecular forces are dominant. These areas will be familiar in most industries. They include adhesion of particles, colloids, pastes, gels and cells, the adhesion of nanomaterials, of films and coatings, the fracture

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of adhesive joints, and composite materials. A concluding chapter points to the future of molecular adhesion science. My hope is that the adhesive gulf between chemists, engineers, and biologists can be joined, while simultaneously helping those materials scientists, dentists, powder technologists, cancer specialists, etc., who are fascinated by adhesion effects. If so, thanks are due to my wife for her constant support, to Professor Mai for allowing me to work in his department on a sabbatical in 1997, to Professor Tabor who gave me the stimulus to think about the issues in this book, and to many colleagues who have debated, theorized and experimented on this subject with me over the past 30 years. If not, please email me on [email protected], fax me with your comments on +44 (0) 121 414 5377, or write me at the Department of Chemical Engineering, University of Birmingham, Birmingham, UK.

CONTENTS

Part I 1.

BACKGROUND AND FUNDAMENTALS

Introduction to Molecular Adhesion and Fracture: the Adhesion Paradox

1.1. 1.2. 1.3. 1.4. 1.5. 1.6.

The Adhesion Paradox Adhesion at the Molecular Level Theory of Adhesion Adhesive Technologies Adhesion in Nature Interdisciplinary Nature of Adhesion; Purpose of this Book 1.7. Review of Adhesion Literature 1.8. References

2.

Phenomenology of Adhesion Effects: Fracture Stranger than Friction

2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9.

Adhesion Phenomena Friction Gravitational Attraction Electrostatic and Magnetic Attractions Adhesion between Nuclear Particles Demonstration of the Molecular Adhesion Force Probing Molecular Adhesion: the Range of Molecular Attractions Definition of Molecular Adhesion References

3 6 8 9 12 15

20 22 22

25 26 28 29 30 32 34 37 39 40 xiii

CONTENTS

xiv

3.

Theories and Laws of Molecular Adhesion: All Molecules Adhere

3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9. 3.10.

PART II 4.

5.

Three Adhesion Fallacies Critical Observations of Adhesion The Laws of Molecular Adhesion From Molecules, through Mechanisms, to Mechanics Jumping into Contact Cracking Molecules Apart Adhesion Is Really Three Things: Making, Equilibrium, and Breaking Adhesion in the Scanning Probe Microscope The Atomic Force Microscope References

41 42 44 46 48 50 52 54 56 57 60

LAWS AND MECHANISMS

Evidence for the First Law of Adhesion: Surfaces Leap into Contact

63

4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9. 4.10.

64 65 68 70 72 73 75 77 79 80

The Problem of Obtaining Reliable Adhesion Obreimoff ’s Experiment Tabor and Winterton’s Experiment Extension by Israelachvili and Tabor Finer Means Smoother Bradley’s Adhesion Rule The Significance of Bradley’s Rule The New Science of Atomic Force Microscopy (AFM) How Smooth Is an Atom References

Intermolecular Forces: the New Geometry of Computer Modeling

5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10.

The Two Parameter Model of Atomic Forces Experimental Evidence for Models of Molecular Forces Direct Measurement of Molecular Forces Intermolecular Forces from Bulk Properties The New Geometry of Computer Modeling Structuring of Hard Spheres Experiments on Spherical Polymer Particles Computer Model of the Crystallization Process Effect of Adhesion on the Structuring Process References

83 84 86 88 89 91 93 95 96 100 101

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CONTENTS

6.

Evidence for the Second Law of Adhesion: Contamination Reduces Adhesion

6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. 6.10. 6.11.

7.

Experiments to Show that Adhesion Is Reduced by Contamination Polymer Adhesion Affected by Contaminants Studies of Surface Species by the Adhesion Method Surface Forces Measured in Liquids Jumping in Steps as Molecules are Squeezed Out Adhesion with Water Present at Surfaces Adhesion of Wet Surfaces in the Atomic Force Microscope Influence of Polymers on Wet Adhesion Restructuring of Surfaces and Interfaces The Nanoscale Fountain Pen References

Influence of the Adhesion and Fracture Mechanism: The Third Law

7.9.

Problem of the Wide Range of Adhesion Energy Values Hierarchy of Mechanisms Controlling Adhesion The Simplest Failure of Adhesion Chemical Breakage of Adhesive Joints Brownian Adhesion The Cracking Mechanism Fracture Mechanics: Thermodynamic Theory of Cracking Experimental Proof that Stress does not Cause Cracking of Adhesive Joints Consistency of the Brownian Mechanism with Fracture

7.10. 7.11. 7.12.

Elasticity in the Adhesion Mechanism Roughness as a Strong Mechanism References

7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7. 7.8.

Mechanics

8.

More Intricate Mechanisms: Raising and Lowering Adhesion

8.1. 8.2. 8.3.

Roughness and Contamination as Hysteresis Mechanisms Dwell-time Effect Adhesive Drag

103 104 106 108 112 114 116 118 122 126 128 130

133 134 135 136 138 139 141 143

145 147 149 151

153 155 156 158 160

CONTENTS

xvi

8.4. 8.5. 8.6. 8.7. 8.8. 8.9. 8.10. 8.11.

PART III 9.

Adhesive Drag and Hysteresisr Measurements Crack Stopping Rolling, Tack and Adhesive Hysteresis Adhesive Dislocations Stringing or Crazing Aggregation Mechanisms Charge Separation and Electrical Effects References

163 165 167 170 171 173 176 177

APPLICATIONS AND BENEFITS

Adhesion of Particles: Deformation, Friction, and Sintering

181

9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9.7. 9.8. 9.9. 9.10. 9.11.

182 184 187 188 191 193 197 200 203 205 209

Contact of Spheres The JKR Contribution The Nature of Adhesive Contact Roughening the Surfaces Effect of Roughness on Particle Adhesion Friction of Fine Particles Elastic Sintering of Fine Particles Hysteresis and Drag in the Contact of Spheres Plastic Contact of Particles Sintering of Particles by Diffusion Mechanisms References

10. Adhesion of Colloids: Dispersion, Aggregation, and Flocculation

10.1. 10.2. 10.3. 10.4. 10.5. 10.6. 10.7. 10.8. 10.9. 10.10. 10.11.

Ubiquity of Colloids Colloids as Adhesion Sensors Electrical Stabilization of Particle Dispersions Point of Zero Charge; Adhesion Dominates Secondary Minimum and Further Complex Interactions Effect of Dissolved Polymer on Colloid Adhesion Particles with Strongly Bonded Polymer Growing Crystals Comminution of Colloids Growing Uniform Colloidal Particles References

213 214 215 219

223 226 229 231 232 235 238 242

CONTENTS

11.

12.

xvii

Pastes and Gels: Effects of Adhesion on Structure and Behavior

245

Importance of Pastes and Gels 11.1. Different Structures of Sols, Pastes, and Gels 11.2. Structure and Properties of a Gel 11.3. Elastic Modulus of Silica Gels 11.4. Shrinkage of Gels 11.5. Ultimate Structure of a Gel Product 11.6. Origin of Defects in Pastes 11.7. Fracture of Gels, Especially Cements 11.8. Paste Structure and Rheology 11.9. 11.10. Controlling the Sol Gel Transition 11.11. References

246 248 249

Adhesion of Biological Cells: the Nature of Slime

275

Introduction and Importance Models of Cell Contact: The Polymer Coating Cell Membrane and Cytoplasm: Effects on Contact Spot Roughness and Cell Surfaces 12.4. Cell Adhesion by Probe Methods 12.5. Cell Adhesion by Flow Methods 12.6. Cell Counting Methods 12.7. New Approach to Cell Adhesion 12.8. Experimental Results 12.9. 12.10. Application to Practice 12.11. Problematic Theories of Cell Adhesion 12.12. References

275 277

12.1. 12.2. 12.3.

13. Nano-adhesion: Joining Materials for Electronic Applications

13.1. 13.2. 13.3. 13.4. 13.5. 13.6. 13.7.

The Size Effect in Adhesion: Small Is Beautiful Adhesion of Plastic Contacts Adhesion of Single Atoms Stretching Single Molecules in the Atomic Force Microscope Adhesion Strength of Small Features Cleaning Particles from Wafers Adhesion in Electrophotography

252 254 257

260 263 266 268

272

279 281 284 287 290 291 293 296

299 301

305 306 309 310 311 313 315 317

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CONTENTS

13.8. 13.9. 13.10. 13.11. 14.

Polymer Synthesis for Coating Silicon Wafers Molecular Control of Nano-adhesion Visualizing Adhesion Using Micro-focus X-rays References

Films and Layers: Adhesion of Coatings

327

Complexity of Films and Coatings Ideal Experimental Arrangement Testing Methods for Adhesion of Films Wedging of Films: Direct Linkage with Zero Friction Elastic Linkage during Wedging Change in Elastic Linkage as the Crack Progresses Elastic Linkages Easing Failure of Film Adhesion Ultimate Adhesion: Pull-off and Indentation Deforming the Substrate: Pull-off, Stretching, and Indentation; Elastic, Plastic 14.10. Amplifying Mechanisms: Roughness, Elastic Arrest, Deflection, Losses 14.11. References

328 330 331 333 334 336 338 340

Fracture and Toughness of Engineering Adhesive Joints

353

Importance of Bonded Structures A Model of Bridge Collapse Definition of Joint Toughness History of the Failure of Lap Joint Theory The Correct Theory of Lap Joint Strength Consequences of this Theory of Lap Joint Failure Strengthening of a Lap Joint by Prestressing More Complex Overlapping Joints Various Adhesive Joint Geometries Summary of Engineering Adhesive Joints References

354 355 356 358 361 362 364 366 369 372 373

14.1. 14.2. 14.3. 14.4. 14.5. 14.6. 14.7. 14.8. 14.9.

15.

319 321 323 325

15.1. 15.2. 15.3. 15.4. 15.5. 15.6. 15.7. 15.8. 15.9. 15.10. 15.11.

16. Composite Materials: Held Together by Adhesion at Interfaces

16.1. Particulate Composites 16.2. Effect of Interfaces 16.3. A Crack Meeting an Interface 16.4. Delamination at the Interface

343 347 350

375 375 377 380 384

CONTENTS

16.5. 16.6. 16.7. 16.8. 16.9. 16.10. 16.11.

xix

Tough Laminates Healing the Interface Crack: Interfacial Dislocations The Overall Picture: Crack Stopping, Deflection and Healing The Problem with Composites: Bending and Compression Adhesion of Fibers in Composites Adjusting the Interface Adhesion in Fiber Composites References

17. The Future of Molecular Adhesion and the Sticky Universe

389 392 395 397 400 403 405

409

Adhesion Problems Solved Interesting Mechanisms Does Adhesive Strength Exist ? Adhesion at the Nanometer Level and Molecular Scale 17.5. Improved Theory of Adhesion by Computer Calculation 17.6. New Adhesion Applications 17.7. References

410 412 415

Index

423

17.1. 17.2. 17.3. 17.4.

416 419 420 421

1 INTRODUCTION TO MOLECULAR ADHESION AND FRACTURE: THE ADHESION PARADOX

For we must learn from the Phaenomena of Nature what Bodies attract one another and what are the Laws and Properties of the Attraction ISAAC NEWTON, Opticks,1 p. 376

Isaac Newton was obsessed with the attractions between the objects which we see about us. His work is perhaps the finest illustration of the curious paradox which is addressed in this book; that the bodies around us are strangely schizophrenic. Sometimes things stick together with remarkable strength, for example, a jumbo jet hangs together in mid-air despite the fact that it is made of 2 million separate parts. At other times, things fall apart very easily, for example, the drug particles in an asthmatic’s inhaler flow readily into the mouth and lungs to prevent dangerous attacks. How is it that things sometimes stick very well and at other times hardly at all? Newton could see this problem but could not understand it. He first compiled his famous Laws of Motion, which completely ignored attraction or adhesion between bodies. His outlandish postulate, originally developed by Galileo, was that bodies should travel in straight lines at constant velocity. This represented a force of inertia. In other words Newton assumed for the purpose of his theory of motion that there was no attraction or adhesion between objects. This is a stunning theoretical idea, which goes against all our everyday experience. On Earth, adhesion is a most common thing. However, once space travel became possible, and rockets could escape from the Earth’s pull, it became 3

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clear that spaceships as shown in Fig. 1.1(a) would obey Newton’s laws, and go straight, if they could get far enough away from the planets. To explain the fact that ordinary earthbound objects do not travel in straight lines at constant speed, like the cannon ball in Fig. 1.1(b), Newton had to produce two further ideas; gravity, which turned out to be extremely convincing; and friction or viscosity, which we have been fudging and striving to explain ever since. Gravity was the first adhesive force to be understood, mainly because of Newton’s pioneering insight. He proposed that a unifying attractive force acts between all objects in the universe. This all-pervading gravitational force has two interesting properties; first it is proportional to the masses of the objects; secondly it seems to act from the center of the objects and decreases with the square of separation. Thus, in the simple case of a small sphere like the Earth attracted to a large sphere like the Sun, the motion of the Earth can be predicted by combining Newton’s two laws; the law of inertia which assumes zero attraction plus the gravitational law of universal attraction. Both these laws conserve energy, so the planets can be viewed as a clockwork mechanism in which energy is converted without loss between the motion (i.e. inertia) and the attraction (i.e. gravitation), as shown in Fig. 1.2.

INTRODUCTION TO MOLECULAR ADHESION AND FRACTURE

5

This theory of gravitational adhesion was instantly successful and allowed the motions of planetary bodies to be predicted accurately. It was immediately clear that cannonballs on Earth should travel in parabolic paths and that planets like the Earth should move in elliptical curves around the Sun, as shown in Fig, 1.2, because this followed from a combination of the two forces, one the inertial force from the law of motion, and the other the gravitational adhesion force acting towards the center of the Sun. Of course, Newton recognized that the clockwork universe must be gradually running down. Energy is lost in tidal movements, and also as the planets plough through cosmic dust. This is a process which must eventually result in the planets sticking together, as observed2 in the finding of strange shaped asteroids (Fig. 1.1(c)). Of the 100 million asteroids larger than 1 km, there should be many thousands sticking together in a peanut shape, like asteroid Toutatis which was imaged by radar in 1992, and asteroid Castalia observed in 1989. This energy loss happens more obviously on Earth, where you can see tea leaves in a stirred cup of tea gradually slowing down and sticking together. Newton invented the idea of viscosity to explain this. Viscosity is the resistance which a body experiences when it falls through a molecular medium. For example, the Titanic, when it sank, did not fall to the ocean floor very quickly. The water resisted the downward movement of the ship, since the water molecules had to be swept out of the way, giving a viscous resistive force. In space, there are very few molecules to sweep out of the way, so Newton’s theory of inertia and gravitation works very well, and the planets do not slow down perceptibly. On Earth, molecules are abundant and their viscous resistance cannot be ignored. Thus, on Earth, we have to consider the combination of three forces; inertia, gravity and viscosity, as shown in Fig. 1.3. To use a political analogy, the forces are of two types: the first type is represented by inertia and gravity, which are conservative because they exchange energy between classes without loss; viscosity is more of a socialist force, because it stirs things up to spread the energy equally between all bodies.

6

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1.1. THE ADHESION PARADOX This book is about a paradox, the adhesion paradox, which is just as challenging as Newton’s ideas about motion and gravity. In fact, the adhesion paradox can be visualized in a similar way. Common experience tells us that ordinary objects do not stick together easily. A cup does not stick to its saucer; your foot does not adhere to the floor. Cathedrals were designed on the basis that the stones do not stick together but are simply held in place by gravity.3 Engineers have therefore been able to predict the behavior of structures, bearings, and cars by presuming that adhesion is zero. Yet it is evident that the small component parts of the cup, i.e. the grains and ultimately the atoms and molecules, stick together extremely well. Otherwise the cup would crumble to dust or would evaporate into the gas phase. Similarly your car would fall apart very easily, as my cars often do. Therefore we have to decide what is the most fundamental nature of matter. Does it stick or not? We can short-cut to the answer given in Chapter 3. The first Law of Adhesion is that “All atoms adhere with considerable force.” This is as strange and surprising as the law of motion enunciated by Newton. How could our world operate if all the atoms were stuck fast and immobilized? Cars would not work because the tires would stick to the road. Indeed, the car engine parts would all seize together and stop moving. The paradox is that the car is made up of thousands of parts, most of which do stick together rather well. But some do not. This book sets out to explain this paradox. First imagine what would happen to our planet if this first law of adhesion were taken at face value. The sands and soils which we grow our food on would stick together to form strong solid rocks. Sand grains would adhere so strongly that the beach would resemble sandstone. Soil would transform into mudstone, so tough that plants would not be able to push their roots through it and would perish. Any particles blown out of volcanoes would float around in the air until they touched a surface. Then they would stick immediately to that surface, making windows opaque and clogging machinery. The Earth’s atmosphere would be as clear and transparent as that on Mars. We know from the fossil record that

INTRODUCTION TO MOLECULAR ADHESION AND FRACTURE

7

sand beds can transform into sandstone and mud into mudstone, but this only happens over long times under extremes of pressure, temperature, or chemical modification. Clearly, something is retarding this adhesion under normal earthly conditions. A clue to this problem comes from experiences of space travel.4 When the first lunar modules landed, dust did leap onto the windows, making them opaque. Furthermore, the solar panels became obscured by detritus which stuck firmly to the surfaces. Also, the hinges on the spacecraft doors did tend to seize up. This proved the idea that clean surfaces in the vacuum of space were much more adhering than the dirty, wet, oxygen-covered surfaces here on Earth. Thus we conclude that contamination of the surfaces is very important in reducing the atomic adhesion to low values, well below what we should expect from the first law of adhesion. The second clue came from the study of electrical contacts and friction of bearings in the 1930s, as illustrated in Fig. 1.4. Holm5 and Bowden and Tabor6 realized that bodies apparently in contact were not truly touching over the geometric contact area. This was confirmed later by attempts to pass ultrasonic waves through contacts.7 The waves were inhibited at the contact, suggesting that only a small area of the bodies was really in molecular contact. It became clear that a design for an electrical contact could not be made on the basis of apparent

8

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geometry. Perhaps one square centimeter of metal appeared to be in contact, whereas measurements revealed that only one square millimeter was actually passing electrical current. In the same way, the frictional force required to shear the contact was much smaller than expected. The ultrasonic transmission was also reduced. Thus the paradox of adhesion being unpredictably large or small can be explained by these two complexities of contact at the molecular level; contamination and roughness.

1.2. ADHESION AT THE MOLECULAR LEVEL Consider the adhesion of two bodies on the Earth’s surface, as shown in Fig. 1.5. However much we clean or polish the surfaces to make them smooth and free from contamination, we realize that the contact between the bodies cannot be made easily at the molecular level. Magnifying a small portion of the contact region, by a factor of one hundred million, to observe the atomic structure of the material, it is evident that the atoms of the material do not make contact at all. In the first place, the surfaces are covered with layers of foreign atoms like oxygen and also by contaminating molecules such as water, which prevent true atomic contact between the solid bodies. Secondly, the surfaces are not smooth at the atomic scale, but are rough and jagged, like mountain ranges placed on top of

INTRODUCTION TO MOLECULAR ADHESION AND FRACTURE

9

each other,6 so that large gaps exist to reduce the surface interaction down to an even lower value. Thus, the picture of adhesion developed in Fig. 1.5 very much resembles the ideas generated by Newton about the laws of motion, (remember Fig. 1.3). The first law of adhesion, that all atoms adhere, is true in space when solid surfaces are smooth at the 0.1 nm dimension. However, in the Earth’s environment, we have to account for contaminant molecules, which reduce adhesion considerably. This is the second law of adhesion, described in more detail in Chapter 3. Both these effects are conservative, in that no energy need be lost. But there is another problem, that of surface roughness which reduces true contact enormously and which therefore diminishes adhesion even more. Roughness is an example of a mechanism, rather like viscosity in Newton’s theory, which is complex, variable, and capable of energy losses. Thus it is cannot be explained easily and must be debated at length. In addition to roughness, many other lossy mechanisms such as hysteresis, restructuring, etc., can be found, leading to a rich variety of adhesion phenomena. These are described in Chapter 7 onwards. Grouping all these mechanisms together, we conclude that there is a third law of adhesion: “the mechanism has a large influence on the macroscopic adhesion, even though the molecular adhesion remains constant.” Summing up these arguments, it is evident that molecular adhesion has several layers of complexity leading to the adhesion paradox. Our macroscopic world of ordinary experience is seen to be false at the molecular level. All atoms and molecules adhere. The moral we draw from this is clear: we must distrust all artificial ideas which are glibly produced to explain adhesion. Ideas such as keying, adhesives, suction, and so forth are utterly unnecessary. Atoms stick together naturally. It is the most fundamental property of atoms to adhere. The odd and unexpected conclusion is that our ordinary experience of adhesion is close to zero, even though we theoretically expect the adhesion of atoms to be large. This brings us to ask the question, “How large should adhesion be in theory?”

1.3. THEORY OF ADHESION Because of the variable and unpredictable nature of adhesion, it has been a common practice through the centuries to adopt a safe approach and design structures with the worst possible scenario in mind. In other words, engineers have traditionally adopted a theory of zero adhesion. This was certainly true of medieval bridge and cathedral builders who desired that only compression forces existed in their constructions, ensuring there were no tensions at the mortar joints to open up cracks. Figure 1.6(a) shows a brick lintel which gives a tension, causing a crack in the mortar adhesive. By devising first the corbelled arch and

10

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then the curved arch, these tensions could be overcome through the compressions acting from the ends, as shown in Fig. 1.6(b), relieving any cracking stresses in the mortar.3 Similarly, the dentist does not rely on adhesion when he fills a tooth, but prefers to undercut the cavity, so that the filling is held by mechanical locking of the mercury amalgam or dental cement, as shown in Fig. 1.7. The tooth would thus have to break to release the filling material, which can become detached from the tooth by shrinkage, cracking, or chemical attack. By the same argument, the designer of a retaining wall to hold back a soil mass does not presume that the soil has any internal adhesion between the particles, but instead designs the structure on the basis that the soil mass can slide freely on the plane shown by the dotted line. He knows by experience that the soil can have a considerable strength when dry. Dried clay and soil bricks are still

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11

widely used for building houses in developing countries. However, he also knows that, if the soil is wetted or changed in acidity, then the soil can be enormously weakened and can fail disastrously. All these practices illustrate the lack of theoretical knowledge about adhesion over the centuries. Newton’s gravitational theory had suggested that universal forces existed holding masses together. Unfortunately, although gravitation can hold a rock onto the Earth’s surface, it is much too small a force to account for the observed values of adhesion between two cemented rocks. The next important step in understanding molecular adhesion forces arose from the kinetic theory of matter, which was developed towards the end of the 19th century. Strangely, like Newton’s laws of motion, this theory was developed on the crazy assumption that no adhesion existed between the atoms or molecules of a gas. The need for a kinetic theory of matter arose from microscopic study of particles immersed in liquids.8 In 1827, a botanist called Brown had been observing, through his microscope, tiny pollen grains swimming in water. The water prevented the pollen grains from sticking together, giving a close approximation to zero adhesion. Brown could see that each grain of pollen was dancing in the watery suspension as though bombarded by invisible impacts, randomly hitting the particle from all directions. He concluded that the liquid was composed of very small atomic or molecular particles, too tiny to be visible in his microscope, in constant motion. The collisions of the invisible particles on the pollen grains were causing the dancing movement, as shown in Fig. 1.8.

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This was an amazing breakthrough in understanding the nature of molecules. It has still not been fully absorbed into our consciousness after two centuries. Most of us still believe that material is almost continuous and nearly static, whereas in reality it is always molecular and continually moving. This is the model of matter which led eventually to the theory of intermolecular adhesion. But the immediate effect of Brown’s observation was to stimulate theoretical argument about the properties of gases which, to a first approximation, behaved as though there was no adhesion between their constituent atoms. On this assumption, Clausius, Maxwell, Boltzmann, and their co-workers generated the mathematical theory describing the behavior of perfect gases.9 The speed of the moving molecules was then recognized to be the measure of temperature. Van der Waals in 1873 was the first to consider that molecules must have attraction for each other, and that such attraction can be seen in a deviation from the perfect gas equations.9 This idea, that all bodies attract each other with a considerable adhesive force, was the beginning of a logical theory of adhesion, providing the first law, an idea which will be studied in much more detail in Chapters 3 and 5. Van der Waals’ model explained how a gas would condense into a liquid or solid as a result of the attractive forces, once the temperature was reduced to slow the Brownian motion. This cooling of a material to form an adhering solid is one of the most powerful adhesive processes, that of thermoplastic adhesion. The idea of a “van der Waals” adhesive force accounting for the effects of adhesive processes has been a potent one, allowing the well-known empirical adhesive technologies to be explained to some extent.

1.4. ADHESIVE TECHNOLOGIES Although theories of adhesion have only been developed over this past century, the technological arts of adhesion processes have existed since prehistoric times. Man’s inventiveness in finding ways of sticking objects together to fashion beautiful structures is perhaps the finest illustration of human intelligence. There is a wide range of products and processes, shown in Table 1.1, which utilize adhesion. Most of these processes and products have grown empirically from primitive beginnings which are now almost forgotten. Perhaps the earliest and most striking example is the cave paintings discovered at Altamira in Spain and Lascaux in France in the nineteenth century.10 These paintings of bison and horses date from the last ice age, some 12,000–17,000 years ago, and illustrate the strong adhesion of fine pigments to the cave wall (see Fig. 1.9). Presumably, the ancient artists had ground up the

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colored minerals and plants to make a dispersion of fine particles in water to produce a paint which adhered successfully to the surface. Another powerful invention was that of Indian Ink, a suspension of carbon black which, after drying became waterproof. Such inks came originally from China some 5000 years ago.11 Lampblack was prepared by burning pine wood and collecting the soot in a furnace container. The fine black powder from the top of the furnace was mixed with glue made by boiling animal skins, together with other additives like crushed pearl, egg white and musk, pounded 30,000 times to break up aggregates, then strained through cloth to produce a fine ink. This was a superb example of the interaction of the collagen polymer solution with the carbon particle surface, to give entirely new properties. Of similar vintage was the discovery of the sintering process to make clay bricks which were weather-resistant. Mud and clay have been used as building materials for millenia. Clay in particular was important because of its mould-

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ability when wet; it could be plastically formed into elegant and large shapes. After drying, the clay was very strong but suffered from the disadvantage that it became weak again on rewetting. This problem was cured by heating the clay articles to high temperatures such that the particles adhered more strongly together. Temperatures of 700° C were sufficient to prevent rapid degradation by water. But above 1100°C, the material became totally resistant to moisture.11 Such technology has been extended over the past century to produce ceramic materials of all kinds, from electronic packaging, to magnets for electric motors, to nuclear fuel pellets. A somewhat later technology, the manufacture of paper, was created about 2000 years ago by the Egyptians, who found that beating papyrus reed stems together, then drying them, gave a matted fibrous surface which was ideal for writing.12 The cellulose fibers in the reed stems made good contact which hardened and strengthened as water was removed (see Fig. 1.10). Again, the problem was to waterproof the paper, and this can now be done by using synthetic polymer to stick the fibers more permanently. Later, mineral fibers like asbestos were used, but these proved to be a health risk when ingested. Now, there is a huge fibers industry, making polymer, glass and carbon textiles which can be adhered to give beautiful structures, such as fishing rods, cars, yachts, and the Millenium Dome. Other ancient examples of the use of natural earths and minerals include the application of bitumen as a hot melt adhesive to stick rocks together, known in

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biblical times, and that of volcanic ash as a cement which would harden and remain strong in the presence of water. The Romans wished to build durable seawalls and found that a certain ash from the region of Pozzuoli, so-called pozzolanic material, would harden after mixing with water and would not be weakened by further immersion in sea water. Some of the sea defences built with this material are surviving to this day. Interestingly, the pozzolanic cement has lasted better than the rocks in the original walls. It was not until 1824 that Joseph Aspdin managed to make such a cement synthetically by heating clay and limestone together in a furnace, then grinding the product to a fine powder which he called Portland cement. This has now been developed to be the largest synthetic material on the planet.13 A more recent adhesive technology which has only come to fruition in the 20th century is that of synthetic polymer latex. When Columbus travelled for the first time to the New World five hundred years ago, he found that the natives played games with a rubber ball, which they had made by gathering the natural latex from certain trees. This milky fluid exuding from the tree-bark could be dried and used in several interesting ways: as a glue to stick things together; as a waterproof coating for fabrics; or as an elastic material for ball games. The natural latex was somewhat unstable and putrescible, but once the trick of adding ammonia as a stabiliser was discovered, the latex could be stored, transported, and used for many applications. The secret of artificially making such milky rubber dispersions was found by Hofmann 14 and co-workers at Bayer in 1913. There was a need at that time to find substitutes for natural rubber for the manufacture of tires. By taking a synthetic rubber precursor, for example butadiene, which is an organic liquid, adding it to water in the presence of a dispersing agent such as blood serum, then shaking, a milky fluid like the natural polymer latex could be produced, as shown in Fig. 1.11. This has been enormously successful for producing the polymer materials which lie at the heart of our modern civilization. Typical applications are adhesives, paints, condoms, tires, window frames, clothes and shoes. Essentially, we have found a synthetic replacement for the natural sticky latex adhesive material we see in nature.

1.5. ADHESION IN NATURE Long before people made their investigations into adhesion technologies, natural phenomena were occurring on the Earth’s surface, illustrating the processes of adhesion which we now understand to some degree. For example, volcanic ash was deposited over many regions. This could cement together when wet to produce the porous rock which encapsulates the ruins of Pompeii, for example. Similarly, solid particles were deposited in seas and river estuaries. These adhered together under the influence of chemistry,

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temperature, and pressure to form sandstones, mudstones and ultimately metamorphic rocks. Sandstone-like products can be made synthetically by heating compacted sand at alkaline pH around 13 under 100 atmosphere pressure of steam. A peculiar type of silica rock which has obviously been formed by adhesion of small particles, is opal. Its structure was only seen in detail when it was inspected by electron microscopy in the 1960s by Saunders and Murray,15 as shown in Fig. 1.12. It became clear that the structure was regular at the level, near the wavelength of light, because opals gave colorful light scattering patterns. Microscopy proved that this coloring effect was caused by the opaline structure of uniform spherical silica particles arranged in almost perfect order. It seems likely that the perfect spheres were precipitated from silicic acid solution in some ancient sea. Over a long time, the spheres would sediment together, causing the regular structure to appear. Then the water dried out to leave the spheres in contact, which would then give substantial adhesion by the effects of pressure, temperature, and water vapor. Synthetic opal can be made by this route. Another example of the variation of adhesion in natural rocks is that of fossils. When rocks are formed by the adhesion of the fine particles of a mineral,

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any embedded creatures will influence the structure and give variations in the adhesive strength. Thus, when such a rock is fractured, the cleavage tends to follow the weak interface between the rock and the fossil, thus revealing the mark of ancient organisms in the Earth’s crust, as shown in Fig. 1.13. However, living things provide the most interesting examples of adhering structures. The earliest fossils indicate that primitive living cells were sticking to rock surfaces to form clumps. These fossils are the stromatolites,16 some 2000

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million years old, which were found in the siliceous chert rocks near Lake Superior. The algal cells could also secrete limestone, calcium carbonate, which could cement to form larger structures on which the living cells could proliferate. The huge coral reefs which abound in tropical seas show clearly the change in living matter which became possible when cells evolved beyond the slimy, filmforming stage to build substantial rocky structures from excreted minerals. In certain organisms, such as the diatoms, this adhesive growth has a fascinating beauty, as shown in Fig. 1.14. Extraordinary architectures are organized by the cells, somehow by templating the adhesion of the silica nanoparticles. About 500 million years ago, the most successful adhesive skeleton structure evolved in the mollusc, with superb properties: mother of pearl (Fig. 1.15). There are about 60 000 species of mollusc living today, each one depositing a shell material for protection from predators. This shell is excreted from the organism, but builds up in an organized lamellar form, with each crystal of the shell separated from its neighbor by a thin film of protein. The structure of such kinds of mother of pearl has been linked with the remarkable toughness of the shells. They are a hundred times more crack-resistant than the inorganic limestone material itself.17 Some of these molluscs, like the limpet, also secrete protein to glue themselves to the rocks below. Many investigations have been carried out to determine the nature of this wonderful, water-resistant adhesive. This secretion of adhesive glue reached its perfection in the spider, which evolved some 300 million years ago. Although the structure of spiders’ webs may

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look obvious, there are still questions about how the silk interacts with the watery adhesive droplets condensed on them (see Fig. 1.16). For example, in certain webs the silk is pulled into the droplets and this provides the silk with more give.18 Obviously, one of the most compelling reasons for studying adhesion is the fact that our bodies are grown and held together by these forces. From the moment that the sperm cell attaches itself to the egg, and the egg adheres to the wall of the womb, we live by the attractive forces of adhesion. Of course, we would end up as mere blobs of jelly without the strong adherence of the calcium

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phosphate bone particles to build our skeleton. Our skin would be weak and soggy if platelets of polymer had not adhered sufficiently to provide elasticity. Our brains could not work if the nerve cells had not made adhering communicating contacts. On the other hand, it is a blessing that our red blood cells do not stick together because capillary flow would then be impossible. Unfortunately, some cancer cells are not sticky enough and spread because the malignant cell adhesion is too low to prevent migration. The conclusion is that we wish to understand and control both high and low adhesion within our bodies.

1.6. INTERDISCIPLINARY NATURE OF ADHESION; PURPOSE OF THIS BOOK It is evident from this brief overview of adhesion technology, nature, and science that many different specialists have an interest in the definition of what adhesion is, in the observing and measurement of the adhesive forces, in the chemical modification of surfaces to control adhesiveness, in improving adhesion so that aircraft do not fall apart, and in reducing adhesion so that cars do not seize up. In other words, adhesion is an interdisciplinary subject. Indeed, it connects many different subjects because of its common function and value. We set out to address these issues, attempting to build a bridge of adhesion between the different disciplines. In the next chapter, the old debates about the phenomena and theories of adhesion are rehearsed. This allows us to focus on acceptable definitions of molecular adhesion. Then the logical theory of adhesion is expounded by defining the ideas, the terms, and symbols more carefully in order to work out the connections between them in Chapter 3. Thus, the laws of adhesion begin to be recognizable in a general form. Two new sciences are developing now to enlarge our ideas about the laws of adhesion: Atomic Force Microscopy (AFM), which for the first time has allowed us to measure molecular forces; and computer modeling (CM), which has pushed the theory of molecular adhesion onto a new level. These new arguments are described in Chapters 3 and 5 to give an overview of the three laws of adhesion. Having defined our framework of adhesion theory and measurement, it is then possible to cover the different application areas of adhesion science in Chapters 9–16. Starting with the adhesion of particles, which is fundamental to a molecular argument, we move on to colloids, pastes, and cells. Then the industrial areas are covered in electronics, films, adhesive joints, and composite materials (see Fig. 1.17). Finally, there is a discussion of the future of adhesion and how the description of adhesion phenomena may develop in the years to come. There is a problem that in some areas, such as fracture of joints, the mathematical analysis has become so detailed that it is incomprehensible to the

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chemist. Yet this mathematics often does not take into account the fact that molecules exist, whereas the chemist starts his theoretical argument from the molecular premise. This gulf of assumptions has to be crossed. Chemists are fascinated by adhesion because they know that different molecules will provide different levels of tackiness; physicists are challenged by the fact that adhesion is very difficult to measure; material scientists see that all substances contain a multiplicity of interfaces which dominate the properties; geologists recognize that the soils and sands of the Earth’s crust are controlled by adhesion; biologists see adhesion as the connection between cells and living structures. And engineers of all kinds, from dentists, to cement constructors, through aircraft designers, computer scientists, chemical engineers, powder technologists, and pharmacists, see adhesion as a common thread in their subjects. Unfortunately, there is a lack of common language and methodology across the boundaries of these disciplines. Would an aircraft designer understand what the parasitologist is talking about in a discussion of adhesion molecules like integrin? At present, the word “adhesion” itself means different things in separate

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subject areas. Some authors have gone so far as to say that molecular adhesion does not exist, and that the phenomena of sticky objects can be described by other mechanisms.19 This book is needed to rationalize the science and technology of molecular adhesion across the whole range of specialisms. It is important that we all agree what the word “adhesion” means, with a definition acceptable to both dentists and rocket engineers. A blood cell adheres one million times less than a sticky label, and that label sticks one million times less than an aircraft wing. The several scales at which adhesion operates are illustrated in Fig. 1.17 which shows molecules adhering to a cell surface, cells adhering to paint, paint on a wing and the wing on an aircraft. We have to solve the problems of measuring over such widely ranging scales and of finding suitable units and standards.

1.7.

REVIEW OF ADHESION LITERATURE

A large number of books and journals deal with molecular adhesion. Hirschfelder20 edited a fine work on Intermolecular forces, and Israelachvili21 considered the application of such forces to interfaces, and to colloidal and biological systems. Surface chemistry books have been written by Bikerman,19 Gregg,22 and Adamson.23 A number of excellent books on colloids exist, including those by Myers,24 Everett,25 Shaw26 and Hunter.27 A good book on surface forces in relation to catalysis was produced by Somorjai.28 Journals on molecular adhesion are rare. There is the Journal of Adhesion which has been edited by Lou Sharpe for more than 30 years. And there is the Journal of Adhesion and Adhesives, published by Butterworth. Colloid journals are more common because of their great industrial significance. These include the Journal of Colloid and Interface Science, Advances in Colloid and Interface Science, Colloids and Surfaces, Surface and Colloid Science, Colloid Polymer Science, and Progress in Surface and Membrane Science. Books on the applications of adhesion are much more common. Kinloch’s book on adhesion and adhesives29 is very interesting. Newer books have emerged recently on structural adhesives30, on particle adhesion,31 and on the energetics of adhesion mechanics.32 This is the first book on adhesion to attempt to give an overall picture.

1.8. REFERENCES 1. Newton, I., Opticks, Smith and Walford, London, 1704 (reprinted Dover, New York, 1952, p 376). 2. Kowal, C.T., Asteroids: Their Nature and Utilisation (2nd edn.) Wiley, London, 1996, pp 100–3. 3. Gordon, J.E., Structures, Penguin, London, 1978.

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4. Lee, L-H., In Fundamentals of Adhesion and Interfaces, eds. Rimai, D.S., DeMejo, L.P. and Mittal, K.L., VSP, Utrecht, 1995, pp 73–94. 5. Holm, R., Electrical Contacts, Springer Verlag, Berlin, 1967. 6. Bowden, F.P. and Tabor, D.T., Friction and Lubrication of Solids, Clarendon Press, Oxford, Part I 1950, Part II 1964. 7. Kendall, K. and Tabor, D., Proc. R. Soc A 323, 321–40 (1971). 8. Perrin, J.B., Atoms, Constable, London, 1923. 9. Maxwell, J.H., Theory of Heat, Longman Green, London, 1871. Brush, S.G., Statistical Physics and the Atomic Theory of Matter, Princeton University Press, Princeton, NJ, 1983. van der Waals, J.H. PhD Thesis, University of Leiden, 1873. 10. Gombrich, E.H., The Story of Art, Phaidon, London, 1998, p 40. 11. Needham, J., Science and Civilisation in China, Cambridge University Press, Cambridge, 1985, pp 239–47. 12. Allen, K., Int. J. Adhesion Adhesives 16, 47–51 (1996). 13. Bye, G.C., Portland Cement, 2nd edn, Thomas Telford, London, 1999, ch. 1. 14. Hofmann, German Patent 250690 to Bayer Farbenfabriken, 1913. 15. Sanders, J.V. and Murray, M.J., Nature, 275, 201–3 (1978). 16. Attenborough, D., Life on Earth, Collins, London, 1980, pp 18–23. 17. Currey, J.D., Proc. R. Soc. B 196, 443–63 (1977). 18. Dawkins, R., Climbing Mount Improbable, Penguin, London, 1997, p 34. 19. Bikerman, J.J., Physical Surfaces, Academic Press, New York, p. 119. 20. Hirschfelder, J.O. (ed.) Intermolecular Forces, Wiley Interscience, New York, 1967. 21. Israelachvili, J.N., Intermolecular and Surface Forces, Academic Press, New York, 1985. 22. Gregg, S.J., Surface Chemistry of Solids, Reinhold, New York, 1961. 23. Adamson, A.W., Physical Chemistry of Surfaces, 5th edn. Wiley Interscience, New York, 1990. 24. Myers, D., Surfaces, Interfaces and Colloids, VCH, New York, 1990. 25. Everett, D.H., Basic Principles of Colloid Science, Royal Society of Chemistry, London, 1988. 26. Shaw, D.J., Introduction to Colloid and Surface Chemistry, Butterworth, London, 1970. 27. Hunter, R.J., Foundations of Colloid Science, Clarendon Press, Oxford, 1987. 28. Somorjai, G.A., Introduction to Surface Chemistry and Catalysis, Wiley Interscience, New York, 1994. 29. Kinloch, A.J., Adhesion and Adhesives: Science and Technology, Chapman & Hall, London, 1987. 30. Adams, R.D., Comyn, J. and Wake, W.C., Structural Adhesive Joints in Engineering, Chapman & Hall, London, 1997. 31. Podczeck, F., Particle-Particle Adhesion in Pharmaceutical Powder Handling, Imperial College Press, London, 1998. 32. Maugis, D., Contact, Adhesion and Rupture of Elastic Solids, Springer, Berlin, 1999.

2 PHENOMENOLOGY OF ADHESION: FRACTURE STRANGER THAN FRICTION

Two polish’d marbles, . . . by immediate contact stick together ISAAC NEWTON, Opticks1

Newton was convinced that bodies brought into close contact should adhere strongly. However, the experimental demonstration of this phenomenon was not so readily achieved. Newton reasoned that the imperfection of surfaces was the most important factor inhibiting contact, as described in Chapter 1. Therefore he developed better methods for polishing glass lenses to upgrade their quality. This was the work that brought him to famous recognition in 1671 when he first displayed at the Royal Society his marvelous new reflecting telescope lenses for improved observation of the stars and planets. However, glass and hard substances like metals are the worst substances for revealing adhesion phenomena. That is why such materials are excellent in ball bearings, which last for years and rarely stick or seize up. So Newton only observed adhesion sporadically. He could not get reliable adhesion between the glass lenses. It turns out that Newton would have been much more successful in proving his ideas on adhesion if he had used the rubbery material which Columbus had brought back from the New World. Such soft material sticks far better than glass. Also, it has become evident that large bodies, like Newton’s glass lenses, are less likely to show adhesion than small ones. If Newton had done his experiments on fine glass fibers, which are easily drawn down from the melt to in thickness, five times finer than a human hair, then he would have quickly seen the reliable adhesion he was expecting from his theoretical arguments. The 25

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conclusion is that molecular adhesion is observed best on small, smooth, soft objects. Let us review some of these well-known adhesion phenomena.

2.1. ADHESION PHENOMENA Many phenomena lead us to believe that molecular adhesion exists. We recognize that, just as a small solid particle sticks to a surface, so a droplet of water adheres to glass, and we realize, as Young and Laplace did in the early 1800s, that this can be explained by attractions between the water molecules and the glass. It is evident that this is an intensely local and short-range attraction, because changing the size of the droplet to make it very small does not influence the adhesion, as shown in Fig. 2.1. Using liquids in this way, it is also very easy to show that the two major resisting effects mentioned in Chapter 1, surface contamination and roughness, are readily observable. Figure 2.1(b) shows that, when a thin, single layer of wax molecules is laid on the glass surface, then the water drop will not spread easily. Similarly, when the surface is roughened, the droplet is even more resistant to spreading. But it is evident that the roughness effect is different from the wax, which weakens both adhesion and detachment; the roughness weakens adhesion but strengthens detachment of the water from the glass. In other words it resists both processes of adhesion and detaching: it is hysteretic, in other words a process rather like viscosity in Newton’s theory of forces.

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27

The problem with adhesion is that we often use the same word to describe quite different situations. For example, when we split a log with an axe, as in Fig. 2.2(a), we can see the cellulose fibers separating by fracture within the wood, and we conclude that the wood is made up of cellulose fibres strongly adhering together. This is a valid description. However, when one describes the adhesion of a tire to a road surface2 or of a railway wheel to a steel rail, as in Fig. 2.2(b), this means something entirely different. In fact, there is no significant adhesion force in this case as one tries to lift the wheel from the track. Instead, the writer is referring to the force required to make the wheel skid along the surface. This phenomenon should be called friction and not adhesion. If the railway was built on the surface of the Moon, however, there could well be significant adhesion then between wheel and rail because the surfaces are much cleaner, giving improved contact. Friction would then also increase. In another example, Velcro fabric or a zip fastener is said to produce adhesion between two pieces of material as shown in Fig. 2.2(c). But such devices operate on a hook and eye principle, in which the hooks of one piece becomes entangled with the loops of another. The separation of such Velcro joints requires friction to slide the fibers across each other, plus deformation of the curved hooks to obtain release. No adhesion between the materials is necessarily needed. Similarly, a book on the “Adhesion of piles in stiff clays”3 is more about frictional sliding of the pile as it is pulled out of the clay, rather like the fiber pullout test described in Chapter 16. However, once the pile is removed, clay may then be seen adhering to the steel surface. Clearly, both friction and adhesion can be observed simultaneously in this case, as shown in Fig. 2.2(d). The conclusion from this discussion is that word “adhesion” should be used only to describe those phenomena in which a normal force is needed to separate

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materials, e.g. by fracture, and not to describe frictional effects where sliding or shear is the force.

2.2. FRICTION Friction is therefore different from adhesion. It is worth considering the differences in more detail because the friction problem is as old as that of adhesion,4 becoming known as tribology after the Greek word “tribos” meaning rubbing, during the celebrated studies of Bowden and Tabor.5 Friction had obviously been important in producing fire during ancient times, and preventing friction was valuable in construction and in chariot building, as shown by Egyptian wall-paintings which showed lubricant being poured in front of a moving stone slab about four thousand years ago. Friction was known to be similar to adhesion because it was affected by surface contamination and also by time of contact. Leonardo da Vinci had demonstrated the two laws of friction, which were rediscovered later by Amontons and Coulomb, and wrote in his notebooks Codex Atlanticus and Codex Arundel that “friction produces double the amount of effort if the weight be doubled” and “friction... will be... equal... although the contact may be of different breadth and length.” This last is extremely surprising at first glance and is illustrated by Leonardo’s diagram shown in Fig. 2.3. An engineer would surely expect the force to rise if the area of contact was larger. But this was not so for most cases. The explanation of these old laws was put forward by Bowden and Tabor following Holm’s studies6 of electrical conduction through metal contacts. It was demonstrated by elegant theoretical and practical arguments that the molecular contact between two metal blocks was very much smaller than expected from the simple geometrical appearance. But as the load increased, the molecular contact increased in proportion to load, as shown in Fig. 2.3(b), because the pressure was so high at the small points of contact with the material flowed plastically. Thus the friction also rose proportionally to the normal load. The difficulty with this theory was that no adhesion was usually observed after the normal load was removed. If the materials had been in molecular contact, causing cold-welding at the high pressure points, then surely some residual adhesion would be noticed when it came to separating the objects. The ingenious explanation put forward was that the elastic recovery of the materials would be sufficient to break the adhesive joints as the load was lifted. In fact, with clean metals, adhesion was generally observed, and transfer of metal fragments from one surface to the other could be detected, thus proving that strong adhesion had occurred in small localized contact regions.

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2.3. GRAVITATIONAL ATTRACTION In addition to friction, which is often confused with adhesion, there are several other phenomena which give attractions between bodies. These phenomena can therefore mix up the issues of molecular adhesion. As Newton wrote: “the attractions of Gravity, Magnetism and Electricity reach to very sensible distances, and so have been observed by vulgar Eyes, and there may be others which reach to so small distances as hitherto escape observation.”1 Gravity is an ever-present attractive force between bodies which must contribute to the adhesion between objects. This gravitational adhesion can be calculated from Newton’s famous law. Consider two spheres, 10 mm in diameter, mass 1 g each, in contact as shown in Fig. 2.4. The force of attraction is readily calculated from Newton’s gravitation law to be a very small force, so small that it can normally be neglected when compared with molecular adhesion. One chemical bond is typically 1 nN in force, a thousand times larger than the gravity force between two 10 mm diameter spheres. The conclusion is that we can usually neglect Newton’s gravitation in comparison to molecular forces. Although it appears from this calculation that we can ignore the gravitational force, there is a subtle influence of gravity which has a large adhesive effect,

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which can also be confused with molecular adhesion. This effect arises from the gravity of the Earth which pulls the atmospheric gases (nitrogen, oxygen, argon) etc. down onto the planet’s surface. This creates an atmospheric pressure which can act to give suction pad adhesion. Imagine a rubber suction pad sitting on a surface, as shown in Fig. 2.5(a). This seals around the rim so that gas cannot get in. When a force is applied to pull the suction pad off the surface, as in Fig. 2.5(b), a vacuum is created and the pressure of the Earth’s atmosphere resists this pull to give an apparent adhesion. Since the atmospheric pressure is l00 kPa, i.e. l00 kN per square meter, this is a surprisingly large adhesive effect which arises solely from gravitation. A pad only 40 mm in diameter can support a 10kg weight. It is obvious that this suction effect is not true molecular adhesion. Suction pads work better as you dive beneath the ocean where the gravitational pressure increases. But they get worse as you climb a mountain, eventually becoming useless in space. Molecular adhesion should remain the same, at the bottom of the sea or out in space, independent of gravitational effects.

2.4. ELECTROSTATIC AND MAGNETIC ATTRACTIONS Yet another confusing adhesive influence is that of electricity and magnetism. Adhesion of bodies as a result of electrical charging was known to the Greeks. Rubbing glass or amber with cloth would make the material attract small

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pieces of dust or paper. A particularly eerie sensation is when your hair stands up as a result of electrical fields. The same effect is obtained by rubbing a balloon. The rubber material picks up an electrical charge, allowing the balloon to stick to a window. It is evident that such adhesion is different from true molecular adhesion because moisture, or nuclear radiation, allows the charge to leak away and the balloon drops off. Priestley7 described such phenomena in 1767 in his book History and Present State of Electricity and suggested that the law of electrical attraction should be the same as Newton’s gravitational principle. This allowed Cavendish to provide an experimental proof four years later, but he did not publish his work and it was not found again for more than a century.8 Meanwhile, Coulomb in 1785 had constructed a mechanical balance for measuring the repulsive forces between two equally charged pith balls, showing that Newton’s law of inverse squares was indeed operating. The electrical force seemed to act from the center of the ball, and decayed with the square of the distance from the center. Thus such forces act over considerable distances and are easily distinguished from molecular adhesion forces, which act only a few nanometers from a surface. Coulomb then showed by means of his mechanical balance that the inverse square law operated also for magnets. It later became clear that magnetism resulted from flowing electrical charges and that the inverse square law applied also to electrical currents, i.e. electrons flowing in wires. Thus, the field of electromagnetism became systematically understood in terms of Coulombic interactions, caused by very small charged particles, i.e. electrons and their flow. Electrons were shown to be ubiquitous in all known atoms and also to be responsible for chemical reactions.9 For example, the intense and fiery reaction of sodium with chlorine was shown to be the transfer of a single electron from one sodium atom to its neighboring chlorine atom, as shown schematically in Fig. 2.6. The main problem with electrons is that they are much smaller than atoms, so they cannot be treated by our ordinary concepts of particles. Instead, they have properties of both waves and particles and so must be described by quantum theory.9 The other problem is that electrons are negatively charged and therefore

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repel each other. If electrons repel, and atoms are bunches of electrons, how then can atoms be attracted towards each other? The answer to this question is complex and is rooted in the quantum theory proposed by London10 in 1937, and summarized in several books on intermolecular forces. 11,12 The principle can be understood by thinking of the outer electron in the sodium atom rotating rapidly around its nucleus, as shown in Fig. 2.7. Because the electron is negative and the nucleus is positive, at any instant there is a separation of negative and positive charge, even though the average charge separation is zero over a period of time. This is an instantaneous dipole, like a tiny magnet, which can line up a similar sodium atom at a distance. This gives an instantaneous dipole–dipole interaction which is always attractive. Such a “London” force is the cause of the general attractive adhesion between all atoms. London showed that this force was very different from simple electrostatic interactions because it dropped off with distance more rapidly, with separation to the power of 7. This explains why molecular forces are short range, and only operate when bodies are in close contact. The conclusion from this brief discussion is that molecular adhesion is caused by electromagnetic forces, but not the simple forces that operate in electric motors or between magnets. Those Coulombic forces can be either attractions or repulsions and obey Newton’s inverse square law. By contrast, molecular adhesion is caused by London forces which are always attractive and which fall off extremely rapidly with separation. In other words, molecules are made up of both electrons and protons which balance their charges exactly to give zero Coulombic adhesion. But the slight separation of charges in the molecules gives overall attractive London adhesion. The sub-atomic particles themselves adhere in quite a different way as described below.

2.5. ADHESION BETWEEN NUCLEAR PARTICLES Electrons (very small sub-atomic particles) are therefore the fundamental cause of molecular attractions. These were the first elementary particles to be

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discovered, identified, and measured, for example in the classic experiments of J. J. Thomson more than a century ago. It was evident from those studies that an electron is much smaller than an atom, almost 2000 times less massive than a hydrogen atom. We can therefore view molecular adhesion as the bringing together of electron clouds. The other fundamental particles (the proton and neutron) were found later, as the model of the atomic nucleus was developed. They stick together in an entirely different way. It was apparent that in the nuclei of all the elements there exists a bunch of protons and neutrons sticking together. This adhesion of the nuclear particles has been theoretically defined in terms of the strong interaction, one of the three distinct forces in nature. The strong nuclear force hold the protons and neutrons together. This strong nuclear adhesive force plays no part in molecular adhesion, which we know to be dominated by the second force, electromagnetic interaction. Finally, the third force of nature is gravitation, which is too weak to account for molecular adhesion. The simplest way to view the three forces of nature is in terms of density. Newton’s weak gravitational force dominates our Solar System where the average density is around On Earth, where the density is about electromagnetic forces are dominant and these are the main topic of this book. However, in the densest stellar objects, neutron stars, the strong force is dominant because the density is that of nuclear particles, around The adhesion between such particles must be immense. Although this strong adhesion between nuclear particles is not evident on Earth, except within atomic nuclei, it controls neutron stars, which were originally proposed as a theoretical idea in 1934.13 This was not supported by experimental observations until the pulsars were discovered in 1967.14 These seemed to be rotating neutron stars, a concept which is accepted today. A neutron star has a mass around kg and a radius near 10 km, leading to the huge density mentioned above. The star is compressed by the large gravitational

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attraction of the mass and is supported by the outward thrust of the neutrons, to give the structure shown in Fig. 2.8. Because of the variety of these forces acting in the universe, it is rather important to define molecular adhesion in a careful and simple way so that there is no confusion between the three known forces: strong nuclear, electromagnetic, and gravitational.

2.6. DEMONSTRATION OF THE MOLECULAR ADHESION FORCE Consider now an experiment which demonstrates molecular adhesion simply and unequivocally. Take a sample of very fine, pure solid powder, say 4 g of diameter grains of titanium dioxide. This powder can be poured into a hard steel pelleting die, as shown in Fig. 2.9(a). Coulombic forces can be prevented by the presence of a radioactive source to ionise the air and leak away any stray electrons. After pressing the powder with a force, say l00 kN, the pellet can be ejected from the die in one piece. The powder particles then stick together with considerable strength. Of course, the tablet is porous because the titanium dioxide grains are so hard that they cannot squash together plastically. But where the powder particles touch each other, the molecules of titanium oxide are in close proximity as a result of the large force which has urged them into molecular contact. Then, the short-range adhesion forces discussed above can act to pull the particles strongly together, thereby resisting external stresses. By testing the pellet in bending or tension, it is easy to find that the compacted pellet has the properties of a porous titania ceramic. It is strong, elastic, and brittle, but not so strong as dense transparent titania which has no pores to weaken it. Typically, the pellet contains 50% by volume of air cavities,

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which make the pellet crack readily. Further evidence of molecular adhesion is obtained by heating the pellet in a furnace. At temperatures around 1200°C, the particles begin to diffuse, and the molecular adhesion within the pellet is so powerful that the pellet gradually shrinks to exclude the pores and become dense. The molecular adhesion forces are sufficient to pull the grains into complete and perfect contact. This process is the sintering phenomenon. William Hyde Wollaston15 first described this type of adhesion experiment in 1829. He was interested in making dense and strong wires from platinum and other rare metals, such as palladium and osmium, which he had just discovered. Platinum is so hard and refractory that it is extremely difficult to work by ordinary melting and casting techniques. Wollaston prepared the platinum in fine particle form by precipitating the metal from an acid solution which had been used to remove impurities. This produced a muddy mixture of water and particles which were cleaned by washing, then dispersed by milling in a wooden mortar and pestle. Wood was used to limit contamination since it would burn out later. The difficulty was converting the fine metal powder into a dense block. To achieve this, Wollaston needed to get the particles sticking together in a dense packing. First, he constructed a mechanical press, shown in Fig. 2.10, consisting of a brass cylinder into which fitted an iron piston around 25 mm diameter. The barrel of the cylinder was tapered so that the pellet could be ejected after compaction. After placing the platinum mud in the barrel, then covering it with blotting paper to allow the water to soak out, and greasing the piston with lard, Wollaston pressed on the ram to apply a force of about 30 tonnes weight to the powder mass. This force was sufficient to increase the packing of the fine particles under the ram from a loose state of 20% packing to a porous pellet near 50% dense. This pellet was ejected in one adherent piece and was “hard and firm” suggesting that each platinum particle was adhering to its neighbors by molecular forces.

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Heating of the pellet with burning charcoal was sufficient to remove moisture and organic lubricant. Then the pellet was raised to white heat in a Staffordshire coke furnace. This caused the pellet to contract as the particles sintered together. Pounding the hot pellet with a hammer produced a material which was 99% dense and which produced platinum wire of the “highest tenacity.” Of course, platinum is not as hard as titanium oxide and so the deformation of the particles during the compaction is more plastic. The adhesion between plastic particles during compaction is even more striking than that between elastic oxide grains. This is readily demonstrated by compressing potassium bromide powder in a steel die, as routinely done for infrared analysis. During the compression of the grains, the pressure at the contact points becomes larger than the yield pressure and consequently the contact spots enlarge until all the porosity has been excluded, as indicated schematically in Fig. 2.11. On removal from the die, the compacted pellet is seen to be fully transparent and completely dense. The pellet is also very strong, elastic, and brittle, comparable to a piece of solid potassium bromide made by other methods. Thus the conclusion is that the compaction force has brought the grains into molecular contact, generating adhesion. Further force has sheared the material close to the contacts, allowing more intimate molecular contact until all the particle surfaces are adhering strongly. The lessons we learn from such demonstrations are fourfold: all bodies can be made to adhere together; finer particles stick more easily; force is usually required to push the bodies into molecular contact; and deformation, especially plastic or diffusive flow, allows more extensive contact to give maximal adhesion.

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Skeptics may say that the adhesion developed in the above experiment can be explained by other well-known ideas. These sorts of argument leveled against molecular adhesion phenomena can be listed. There is first the suction argument, which says that the particles are acting as rubber suction pads, and merely sealing around the edges; this is easily shown to be false because the pellets are just as strong in vacuum, where suction pads fall apart. Secondly, there is the mechanical keying argument, which suggests that the particles behave like Velcro, with little hooks and eyes to cause adhesion. This is easily answered because when you look at the surfaces by electron microscopy, the particles are often extremely smooth and shiny. There are no hooks and eyes. Moreover, the smoother particles show stronger adhesion than rough particles. A third argument is that the particles are electrically charged, to give electrostatic attractions. This is readily disproved by doing the experiment in the presence of ionizing radiation to leak away the electrons. The adhesion is not affected. The fourth argument is moisture; a critic will say that the particle surfaces are damp and this is like the adhesion of wet sand, which sticks more than dry sand. Again this can be disproved by evacuation; in fact, the dryer the particles, the better they adhere. Finally, a desperate argument is Newton’s gravitation. But a rapid calculation shows that this is much too small to cause the adhesion observed. So the conclusion must be that molecular adhesion is the most rational explanation of the experiments described above. There is obviously a force of molecular adhesion which draws bodies together; as a consequence there must therefore be an energy of adhesion, because energy is the summation of force acting over a distance. The experiments above define the force; to understand the energy we must now look at the range of action of the adhesion.

2.7. PROBING MOLECULAR ADHESION: THE RANGE OF MOLECULAR ATTRACTIONS A simple way to distinguish molecular adhesion from electrostatic, gravitational, or liquid bridge adhesion is to observe the range of action of the force. Newton knew of the long-range action of those forces and was tantalized by the much shorter range of molecular adhesion. The two most easily distinguishable forces of attraction are suction and liquid bridge adhesion. The suction pad, as explained in Fig. 2.5, relies on atmospheric pressure to hold the bodies together. This pressure remains constant as the pad is pulled away from its substrate. The same is true of a liquid droplet acting to glue two balls together; the force of adhesion is constant as the balls separate, as shown schematically in Fig. 2.12. Thus these two types of adhesion

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are tough; the force stays high with distance and so a great deal of energy is required to pull the bodies apart. The energy is the area under the line. Electrostatic adhesion is not so tough because now the force falls off with the square of distance from the centers. But the force is still of long range and can be measured with distance on a large scale. You can measure the distance with a meter ruler, for example. Newton’s gravitation force also follows this curve but the force is very small. However, molecular adhesion is very different. This falls off in a very short distance of separation. As a consequence, these molecular adhesion forces cannot be measured with a meter ruler, but need a nanometer scale. The adhesion force may be high when the molecules are touching, but even a separation of one nanometer causes the force to drop almost to nothing. Thus the surfaces snap apart in a brittle fashion, totally different from the other types of adhesion force. The area under the curve is very small. In other words, the energy of molecular adhesion may be negligible. Taking this energy for one square metre of joint, we define the work of adhesion W in Joules per square metre. The other feature of molecular forces which is evident from this comparison is seen when the balls are brought back together. The suction pad and the liquid drop make contact again very easily. Also the electrostatic bond is readily renewed as the balls touch again, to give the same strength as before. But the molecular adhesion is not easily regained. The smallest speck of dust, or contamination by a single layer of foreign molecules, can prevent the molecular

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bonding. In other words, the surfaces cannot be replaced in exactly the same position to reinstate the original bond. That is the fundamental cause of the adhesion paradox discussed in Chapter 1. Molecular adhesion is not reliable or easily repeatable, because molecules cannot easily be put back in exactly the same position. As Newton recognized, this short range of action of molecular forces has made the force measurement extremely difficult. Only in the past few years or so has it been possible to make the necessary measurements at the nanometer level to prove these ideas. Two problems have been overcome. The first one was the measurement technique. Optical methods, such as multiple beam interferometry and laser optical levers, have been developed to measure atomic distances. Piezoelectric actuators were invented to control nanomovements. The second issue was surface smoothness. Smooth surfaces of oxides such as mica, silica or alumina have been found. Also tiny smooth probes have been made on a large scale by electronic wafer fabrication routes. These new techniques have allowed a new and proper definition of molecular adhesion.

2.8. DEFINITION OF MOLECULAR ADHESION Consider a definition of molecular adhesion which allows it to be distinguished from all the other known forms of attractions between bodies: molecular adhesion is the force experienced when bodies make contact at the molecular level, with gaps near molecular dimensions. This definition raises a number of questions which will be addressed in the following chapters. The obvious question relates to the origins and laws of molecular adhesion. How can one measure and interpret such phenomena? Clearly, molecular adhesion forces have the same origins as the forces of cohesion which hold solids and liquids together. These can be understood in terms of the heats of melting or evaporation, the elastic stiffnesses, or the chemical reactivities of materials, as described in Chapter 5. A second question concerns the tools which we can use to observe molecular adhesion more directly. New methods, such as Atomic Force Microscopy and Computer Simulation, have emerged over the past 30 years to play a large part in our understanding of the effects, as covered in Chapters 5 and 6. Finally there is the frontier of adhesion. The further our study of adhesion penetrates, the richer the mechanisms and the applications, as described from Chapter 7 onwards. Eventually we must ask in the final chapter “What is the future of molecular adhesion?”

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2.9. REFERENCES 1. Newton, I., Opticks, Smith and Walford, London, 1704 (reprinted Dover, New York, 1952, p 376). 2. Walters, M.H. and Smith, J.G., In: Adhesion II ed K.W. Allen, Applied Science, London, 1978, pp 145-66. 3. Tomlinson, M.J., Adhesion of Piles in Stiff Clays, London Construction Industry Research and Information Association, London, 1970. 4. Dowson, D., History of Tribology, Longman, New York, 1979. 5. Bowden, F.P. and Tabor, D., Friction and Lubrication, Methuen, London, 1967. 6. Holm, R., Electrical Contacts, Springer Verlag, Berlin, 1967. 7. Priestley, J., History and Present State of Electricity, J. Dodsley, London, 1767. 8. Singer, C., A Short History of Scientific Ideas, Clarendon Press, Oxford, 1959, p 354. 9. Pauling, L., The Nature of the Chemical Bond, 3rd edn., Cornell University Press, Ithaca, NY, 1960. 10. London, F., Trans. Faraday Soc. 33, 8–26 (1937). 11. Mahanty, J. and Ninham, B.W., Dispersion Forces, Academic Press, New York, 1976. 12. Margenau, H. and Kestner, N.R., Theory of Intermolecular Forces, Pergamon, Oxford, 1971. 13. Baade, W. and Zwicky, F., Phys. Rev. 45, 138 (1934). 14. Shapiro, S.L. and Teukolsky, S.A., Black Holes, White Dwarfs and Neutron Stars, Wiley Interscience, New York, 1983, pp 241–66. 15. Wollaston, W.H., Phil. Trans. R. Soc. 119, 1–8 (1829).

3 THEORIES AND LAWS OF MOLECULAR ADHESION: ALL MOLECULES ADHERE

I found the place in which they touched to become absolutely transparent, as if they had there been one continued piece of glass ISAAC NEWTON,1 Opticks, p. 194

The ideas that Newton developed about the sticking together of bodies were quite advanced compared with those of Galileo who had died just a few years earlier. Although Galileo,2 from his observations of the planets, had come to the conclusion that “all parts of the Earth do congregate… ever striving… at union”, and thus had a vision of adhesion as a general property of matter, he had only produced a vague and shadowy theory of this notion. He speculated that there were two causes: one was the abhorrence that nature had for the vacuum created as bodies were separated; the other was the viscous resistance needed to deform the materials. Galileo was wrong on both these matters. In contrast, Newton pressed glass lenses together and considered that the fundamental particles of the glass should become intimately bonded at the transparent contact area (see Fig. 3.1). He found adhesion in some cases. From the observations, he could address the common fallacies of the time, including the idea that vacuum was necessary, and the other common notions that the bodies require glue or have to key together to produce an adhesive effect. Let us consider these fallacies first before going on to state the laws that follow from Newton’s logic. 41

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3.1. THREE ADHESION FALLACIES The first fallacy, that vacuum causes adhesion, was understandable because of Galileo’s influential statement above and also because vacuum was invented by Newton’s contemporary, Galileo’s pupil and secretary Torricelli, who built the original mercury barometer around the time of Newton’s birth, generating the first artificial vacuum in the closed tube above the mercury column. The great force of vacuum on the Earth’s surface was later demonstrated by von Guericke when Newton was 13 years old. It was a tremendous demonstration, where two teams of 8 horses failed to separate the “Magdeburg hemispheres” after the air was withdrawn by von Guericke’s air pump. We can dismiss the vacuum theory of adhesion easily today, as in Chapters 1 and 2, because astronauts have found that, in space, suction does not give adhesion, whereas the molecular adhesion of particles in space is better than on Earth. We know therefore that the adhesion due to vacuum is only an indirect form of gravitational adhesion, varying from planet to planet. The suction effect cannot explain molecular adhesion. The second fallacy is more difficult because it arises from the adhesion paradox; that if we break a cup, the parts do not stick together merely by putting the pieces back in contact. Usually, we have to insert some adhesive or glue to join and stick the fragments of pottery back. So we presume that materials do not naturally adhere but require an adhesive layer in between. Similarly, when building a house, we find that the bricks will only stick together if we put mortar in between to make a strong joint, as shown in Fig. 3.2. The same sort of argument is put forward to explain why you cannot make sandcastles from dry sand. You need to use damp sand because the water acts as a glue to hold the sand grains together. This was the question that Newton puzzled over. If you could get the bricks or the sand grains into molecular contact, he mused, then they should stick. We now know that this is true, because when you take small particles with smooth surfaces, then bring them into close proximity, typically within 10 nm of each other, then the particles leap into contact. All known particles, whether rubber,

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gold, mica, or gelatin, do this. It seems that molecular adhesion is a very general and universal phenomenon. Thus the adhesion paradox is really caused by the problem of getting the bodies close enough together in the first place. The molecular adhesion forces are of such short range that they cannot act even over micrometer gaps. What is more, it has been shown quite clearly that putting water or adhesive on the particles reduces the molecular adhesion.3 Molecular adhesion works best in the vacuum of space. Already on Earth, the atmosphere of oxygen, moisture and organic contamination is known to reduce molecular adhesion by orders of magnitude. The conclusion is inescapable; adhesives reduce the adhesion between clean bodies. The third fallacy is insidious because it stems again from our common-sense knowledge of making furniture and building cars. It is universally obvious that wooden chairs are best assembled using mortise and tenon joints, as shown in Fig. 3.3(a). A male key is made from one piece of wood and this fits snugly into the female slot in the other piece. If the joint is finely made, then glue is

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unnecessary. The artisan inserts glue anyway for a belt and braces solution. A cruder example is the nailed wood, illustrating the old adage of adhesive technology: “When all else fails, use whopping great nails.” Finally, the jigsaw convinces us as infants that mechanical keying is the source of adhesion. This keying idea has been applied to particles; for example, Evans4 stated, “strength of (powder) compacts results mainly from the interlocking of irregularities on the particle surfaces.” This concept is totally false. The same idea has also been applied to the binding of antigens to antibodies.5 It is a macroscopic idea which we have presumed is also true at the nanometer level. However, it is an idea which cannot be correct. There are no Velcro hooks and eyes at the molecular level. Newton knew this and dismissed the notion of “hook’d atoms.” He found that the smoother he polished his glass marbles, the better they adhered. Atomically smooth surfaces stick best of all. These conceptual problems have arisen because we assume that macroscopic behavior applies to molecules. Some critical observations show this to be a false premise.

3.2. CRITICAL OBSERVATIONS OF ADHESION The problem of extrapolating down to molecular dimensions from macroscopic experience can be seen in two simple cases.6 First, consider the macroscopic task of sieving pebbles in the garden, as shown in Fig. 3.4(a). If the mesh size is 20mm, then the 10mm pebbles can easily be shaken through the holes. Each pebble shows no adhesion to the mesh, nor to its neighboring pebbles. But imagine trying the same experiment with l00 nm diameter pigment particles. First of all, the pigment particles tend to stick together to form clumps, as shown in Fig. 3.4(b). Secondly, the small particles stick to the sieve and clog up the mesh, even when the holes are l000 nm, far larger than the particle size. Sieving such tiny particles is impossible unless the adhesion is reduced, say by adding surface contaminant molecules.

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In the same way, consider the problem of getting a molecular size key into its receptor lock (Fig. 3.5(a)). The key sticks prematurely to the lock before it can enter, so the mechanism cannot work, as shown in Fig. 3.5(b), unless the molecular adhesion is reduced, by coating the receptor with contaminant atoms. A second example of the difference between macroscopic and molecular behavior is seen in the designing of machines. We know how to design machines like cars. To make a clutch mechanism between two rotating shafts, we have to press the two shafts together to get them both to rotate synchronously, as illustrated in Fig. 3.6(a). Alternatively, we could glue the shafts together to make a solid joint between the two shafts. The case of two rotating shafts in a molecular, nanoscale machine is quite the opposite. The shafts adhere spontaneously, leaping into contact, and provide

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the clutch mechanism naturally. The problem here is getting the nanoscale shafts apart again. Either a force must be applied to separate the shafts or debonding molecules must be inserted at the junction to separate them. These examples demonstrate the fallacy of extrapolating our macroscopic experiences down to the molecular level. The commonsense macromodels of suction, of adhesives, and of keying are not applicable at the nanoscale and lead us to error when interpreting molecular adhesion. It is useful to plot a logarithmic scale showing the boundary between the macro and nano worlds (Fig. 3.7). At the macroscopic engineering scale to the right of the diagram, above adhesion is not normal, and we need adhesives, keys and friction to get things to stick together. By contrast, to the left of the diagram, below l00 nm, adhesion is dominant and it is difficult to keep things apart. Everything is adhesive at this molecular scale unless we insert contaminant molecules to reduce adhesion. In between, from l00 nm to we have the transition zone, where adhesion is variable. Sometimes things adhere and sometimes not, depending critically on the mechanism. It is interesting that many technological materials like latex, butter, cosmetics, paper, ink, cements, etc., together with living things like diatoms, viruses, bacteria, algae, and so forth, inhabit this transition zone. Here, adhesion is highly sensitive to chemical influences and to physical conditions. So sticking may be finely controlled, may vary enormously, and may also evolve over time. Having defined the main adhesion fallacies, and determined where adhesion can always be found and predicted at the nanometer level, it is now possible to summarize the laws of adhesion which we believe to be universal.

3.3. THE LAWS OF MOLECULAR ADHESION The laws of molecular adhesion are:

1. All atoms and molecules adhere with considerable force. More simply, if two solid bodies approach to nanometer separations, then they will jump

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into contact as a result of molecular adhesion. This behavior differs from our ordinary engineering experience. 2. The effect of contaminant “wetting” molecules is to reduce adhesion, or even to make the bodies repel each other. In other words, adhesives reduce molecular adhesion. 3. Molecular adhesion forces are of such short range that various mechanisms can have large effects. Examples of such mechanisms are surface roughness, Brownian motion, cracking, viscous deformation, etc. These mechanisms lead to a rich variety of adhesion phenomena which may cause macroscopic adhesion to vary, even though the molecular adhesion remains the same. These laws at first sight seem to go against our common experience. Indeed, they may seem to be incorrect when seen for the first time. Rather like Newton’s laws of motion, the above statements strip away the interference of other effects such as gravity, friction, viscosity and geometry which dominate our everyday experience. They reveal the chemical reality of natural electronic forces between atoms. The best analogy is with Brownian motion.7 Before Robert Brown in 1827 observed the incessant spontaneous movement of pollen grains in water, it was believed that bodies were static. The breakthrough that Brown, and later Perrin, made was to recognize that this static appearance is false at the nanometer level. Molecular motion affects every particle in the universe. Whereas engineers had believed (and, in many cases, still believe) that objects stay put, we now know that all bodies in the cosmos are moving with an energy 3kT/2 where k is Boltzmann’s constant and T is the absolute temperature. This idea of perpetual particle motion is quite foreign to most people, and engineers can usually ignore it without too much error, because the energy of movement is very small, around J under ordinary conditions, too small to influence a car, for example, which would require perhaps 100 J of energy to give reasonable movement, 1022 times larger than the Brownian energy. However, designers of nanoscale machines must take this movement into account because objects below 100 nm in size are moving significantly. Just as Perrin concluded that a fluid’s “apparent repose is merely an illusion” because the fluid molecules are in a state of eternal and spontaneous motion, so must we believe that all molecules adhere strongly, even though macroscopic objects appear nonsticky. The apparent lack of adhesion we see in engineering situations is really an illusion because adhesion is universal at the molecular level, according to the first law of adhesion above. However, there is a serious conundrum here because it seems impossible that particles can be in constant Brownian movement, where it is necessary for particles to collide and bounce off each other, yet also sticking together, which would cause agglomeration and

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ultimate static behavior. That is the problem addressed in Chapter 5. There has to be a balance between Brownian movement and sticking, as illustrated in Fig. 3.8. The conclusion is that, within a Brownian system of moving objects, there must always be some adhering particles. In other words, a Brownian system cannot be fully dispersed in reality.8 Here then is a mechanism, Brownian motion, which has an enormously strong influence on adhesion phenomena. There are many other mechanisms which must also be accounted for. For example, why do the pages of this book not stick together to form a solid cellulose mass. This is not a ridiculous question. Goodyear, the inventor of rubber crosslinking, was producing waterproof mail bags 180 years ago by coating them with rubber. Unfortunately, the sheets did stick together completely and Goodyear went bankrupt. By chemically crosslinking the rubber to make it more elastic, removing the tacky contact mechanism of the polymer, he was able to solve this adhesion problem. The laws of adhesion have stripped away all the interfering mechanisms to get at the underlying reality. Now we must add the mechanisms back in one by one to understand the ultimate complex mechanics of adhesion phenomena.

3.4. FROM MOLECULES, THROUGH MECHANISMS, TO MECHANICS We normally observe adhesion at the macroscopic level, say by peeling a film from a surface as in Fig. 3.9. In order to understand the force which is necessary to pull the film off, we have to connect this mechanical picture with the molecular adhesion forces which we know to be universal at the nanometer level. The macroscopic mechanism looks simple, because we grip the film and pull it off by applying a force (Fig. 3.9(a)). However, it is obvious that the applied force is not acting directly at the contact. The film has to bend and apply leverage to the adhesive region. So there is a complex mechanical system operating. To understand this we need mechanics. In particular, we apply the continuum mechanics theory in Chapter 14.

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If one looks closer, using a microscope, at the point where the polymer film is detaching from the oxide surface, at the micrometer level (Fig. 3.9(b)), then it is evident that there is a crack traveling along the interface between the polymer and the glass. This crack is the mechanism by which the polymer material detaches from the surface. All brittle adhesive joints fail by cracking. This is a mechanism which involves elastic deformations and creation of new surfaces. It can be analyzed by the energy balance theory described in Chapter 7. Looking even closer at the precise point of detachment of the polymer, we begin to resolve the oxide molecules in the substrate, vibrating but not bodily moving from their crystalline positions. By contrast, the polymer chains are moving bodily in a liquid-like motion, and the ends of the chains are continually and rapidly making contact with the oxide, then breaking away. So the molecular picture is one of Brownian movement, where the molecules only stick onto the surface periodically. Thus, an equilibrium of molecular adhesion can be established, such that the point of detachment is where the rate of making contact is equal to the rate of breaking. This is the kinetic theory of molecular adhesion which will be discussed in Chapter 5. The conclusion from these arguments is that adhesion needs to be seen at three distinct levels. Adhesion is a molecular phenomenon, so we must first examine closely the molecular behavior. But it is not possible generally to apply forces directly to the molecules. The forces are usually acting through a mechanism, often a cracking mechanism. Cracks can easily be observed in adhesive joints and the detailed nature of the mechanisms have been established.9 Finally, the forces are applied to the crack through a particular system of mechanics, in this case a peeling system. The way in which the peeling is carried out can have a large influence on the adhesion forces observed, even though the molecular picture remains the same. Thus we need to study adhesion on three levels: from molecules, through mechanisms, to mechanics. This overview explains why there have been enormous problems of communication between adhesion scientists in different applications. Engineers

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tend to be concerned with the gross macroscopic mechanics and produce lengthy mathematical explanations which do not normally mention molecules. At the other extreme there are chemists who synthesize molecules which alter the adhesion at the atomic level. But these behave differently in the different macroscopic tests. In between are the material scientists who study the microscopic cracking behavior, desperately trying to reconcile the test protocols with the molecular reality. In this book, the objective is to study these three views of adhesion together, starting with molecules, going on to the mechanisms and finally understanding the mechanics.6 First, the molecular behavior becomes evident because molecules jump into contact.

3.5. JUMPING INTO CONTACT The skeptical engineer is wary of the idea that molecules leap into contact with their surrounding surfaces. After all, car parts on an assembly line do not suddenly jump together and adhere strongly to fashion the finished vehicle. However, at the molecular level it is quite obvious that such events occur naturally, and a nanoscale car could be self-assembled by these normal adhesive forces. The problem is that the nano-engine would not work because all its parts would seize together. Proof of this is found in three simple experiments. The first experiment is the one interpreted by Thomas Young10 in 1805 and shown in Fig. 3.10. A droplet of water is lowered towards a plastic surface. As the drop approaches the polymer, very little happens until the gap between water and solid is extremely thin, around 1000 nm. This shows that the molecular adhesion forces do not extend far out from the surfaces. But at a certain separation, the water suddenly jumps onto the plastic and spreads rapidly across it. The final angle of contact is a measure of the molecular attractions, small angle indicating strong adhesion, large angle showing small adhesion. In this example, it is clear that the water molecules are in constant motion, because small particles of pollen within the droplet can be seen dancing with the

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Brownian movement. Thus, the equilibrium of the droplet can be viewed as the rapid wetting and dewetting of the polymer surface by each individual molecule of water. If a force is applied to the drop, by tilting the plate, then the water can detach from the surface quite easily, while making new contact on the other side as the droplet rolls down the surface. Molecular adhesion at equilibrium is clearly only small in this instance, because the water does not strongly adhere to the polymer. The second experiment showing molecular adhesion onto surfaces was described by Daniell in 1839.11 He took a piece of hot charcoal, which is composed of very fine strands of carbon with a large surface area, and showed how, on cooling, the charcoal would absorb large quantities of gas. He wrote of “a force of heterogeneous adhesion” which was causing the gas molecules to stick to the solid charcoal surface. The same effect can be observed in the famous BET12 apparatus in which a sample of porous material like charcoal is cleaned by heating to 200°C in a nitrogen gas stream, then cooled to liquid nitrogen temperatures (see Fig. 3.11). The nitrogen absorbs onto the charcoal, as can be seen from the reduction in pressure of the gas in the closed vessel. This is such a well-understood effect that it is used to measure the molecular surface area of powders by the number of nitrogen molecules adhering to the surface. In this case, the adhesion is again reversible and the gas molecules can be removed by heating the sample. A third experiment which shows unequivocally that molecules leap into adhesive contact was performed by Johnson, Kendall and Roberts in 1970.3 This experiment was similar to Newton’s original test on glass telescope lenses (Fig. 3.1) but used rubber surfaces because they adhere much more reliably than glass. Roberts had developed a way of moulding rubber in concave glass lenses to produce remarkably smooth elastomeric spherical surfaces as shown in Fig. 3.12. The rubber composition was mixed and then pressed hot into the glass lens. After

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cooling, the rubber lenses could be peeled out of the moulds and then brought into contact (Fig. 3.12(b)). As the two smooth spherical surfaces approached each other, within a few micrometers of contact, the familiar Newton’s ring pattern could be seen in the narrow gap between the smooth surfaces. Then, as the rubber lenses were moved still nearer, a sudden jump of the rubber was observed and the black contact spot grew rapidly to a large size as the rubber deformed and spread under the influence of molecular adhesion (Fig. 3.12(c)). The appearance of this was very similar to the liquid drop spreading over a glassy polymer surface. To get the rubber lenses apart, a tensile force had to be applied to overcome this molecular adhesion. It is this cracking apart of adhering surfaces which we consider next.

3.6. CRACKING MOLECULES APART The adhesion of the rubber spheres is revealed unequivocally by their leaping into contact. However, the precise nature of this contact needs to be understood if the adhesion of the spheres is to be well defined. Consider the two rubber surfaces magnified so the contact region in the equilibrium contact shows up more clearly, as in Fig. 3.13(a). The rubber, after jumping into contact, has spread to form a flat circular contact spot about one millimeter in diameter. This flattening is a general mechanism of adhesion, similar to the spreading of the liquid droplet described previously in Fig. 3.10. However, the rubber cannot spread far because it is elastic and not liquid like the droplet. The flattening and spreading of the rubber causes an elastic resistance to build up in the rubber, storing energy, much as an elastic band does when stretched. This stored energy can be seen in the change in shape near the contact, from spherical to neck-like. Eventually, this elastic resistance

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stops the spreading and the equilibrium contact is attained. The size of the contact spot at this stage is a measure of the molecular adhesion. A large spot is a measure of large adhesion. This spreading and equilibrium is an example of the third law of adhesion. The molecules jump into contact with a particular energy, but that energy has to be absorbed into the system by a mechanism which can be quantified. This mechanism is hugely important because it affects the forces of adhesion enormously. For example, if the rubber is not perfectly elastic when flattened, then the contact spot size is different. The same molecules can be in contact, yet a slight change in the mechanism can raise or lower macroscopic adhesion by orders of magnitude, as we will see later in Chapters 7 and 8. In Fig. 3.13, the mechanism for separation of the particles is seen to be elastic cracking. The neck-like shape is precisely what would be expected of a crack. This can be shown by applying a tensile force as in Fig. 3.13(b). As the force is applied, the neck extends, the crack opens, and the contact spot shrinks as the crack penetrates the interface. If the tension force is small, then the contact will not break completely, but reaches a new equilibrium position, at a smaller contact size. This is an equilibrium crack of the kind first described by Griffith in 1920.13 He showed that such cracks can be described by an energy balance theory, in which the surface energy of the crack is held apart by the elastic energy in the rubber, plus the potential energy of the tensile force. This theory is a concept of pure mechanics which does not require any molecular interpretation, but is expressed in mathematical terms as a continuum theory. So we conclude that the molecules jump into contact, but they are impeded by the crack mechanism, which can be explained by fracture mechanics. Thus we follow the argument from molecules, through mechanism, to mechanics.

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As a larger and larger tensile force is applied to pull the rubber spheres apart, the contact gets smaller and smaller in diameter. However, the contact spot does not shrink gradually down to zero. Instead, a point of instability is reached where the crack speeds up suddenly and the spheres rapidly come apart. This is shown diagrammatically in Fig. 3.14. The crack can be in equilibrium over a certain range of forces, but above a particular tension, i.e. the pull-off force, the spheres catastrophically jump apart as the crack moves suddenly through the contact spot. This pull-off force is another measure of the molecular adhesion between the spheres. A large force indicates large adhesion.

3.7. ADHESION IS REALLY THREE THINGS: MAKING, EQUILIBRIUM, AND BREAKING After examining the way in which a particle sticks to a surface, as above, it becomes apparent that adhesion is not a single process, but one which we can separate into three different but related actions: jumping into contact, achieving a certain black spot size, then cracking apart as a tensile force is applied. The first adhesion phenomenon is the most convincing: all particles leap spontaneously onto surfaces, showing that the molecular attractions do not

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depend on any extraneous influences, such as adhesive molecules or surface keying. Thus adhesion can be measured by looking at the distance covered by the leap. A long jump means strong adhesion. This ingenious method of quantifying adhesion was first applied systematically by Tabor and Winterton in the late 1960s and will be described in detail in Chapter 4. The second phenomenon, the achievement of a black spot resulting from adhesion forces, was the idea mentioned by Newton as a measure of true molecular contact, but this was not measured sensibly until 1971.3 This black spot should really be the equilibrium size which balances the molecular adhesion forces trying to enlarge the black spot, in competition with the elastic forces in the rubber trying to force the particles apart. This balance defines the size of adhesion. A large spot means large adhesion. Clearly, this is a dynamic equilibrium at the molecular level, even though the black spot seems to be static when viewed macroscopically with a magnifying glass. The third phenomenon, that of detachment of the particles from each other by applying a pull-off force, is the test of adhesion which is most familiar to us. We increase the tension force applied to the particles until they just come apart, and define that force as the adhesion force. A large force means large adhesion. However, this is a very difficult experiment to carry out because the final detachment is an instability which is hard to reproduce exactly each time. Thus, many different values of adhesion can be found for the same samples in such tests, depending on the rate of loading, the precise moment of detachment, etc., leading to considerable unreliability in such measurements. These three adhesion measures of molecular adhesion, in the contact makeand-break process, also allow us to test the second law of adhesion, that contaminant molecules always decrease the attraction between bodies. Consider immersing the rubber spheres in water and repeating the adhesion experiment. The results show that all three indicators of molecular adhesion: (the jumping into contact, the size of the contact spot, and the pull-off force) are diminished by the presence of the water molecules. Adhesives reduce adhesion! When the rubber spheres are brought together under water, they can approach much closer before the jumping occurs. This suggests that the attractive force pulling the spheres together is reduced. Once the spheres have leaped into contact, the contact spot can be seen expanding to its equilibrium value. But now in wet conditions, the contact spot size is much smaller, indicating that the adhesion is less. Similarly, when the spheres are pulled apart under water, the force required is about ten times less than in dry conditions, showing a much reduced adhesion (see Fig. 3.15). Thus it is evident that the presence of the water molecules on the rubber surfaces has diminished the molecular adhesion, in accordance with the second law. These reductions in adhesion produced by contaminants become obvious when studied in the atomic force microscope.

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3.8. ADHESION IN THE SCANNING PROBE MICROSCOPE The macroscopic demonstrations described above, repeating Newton’s original tests on the adhesion of glass lenses, give an insight into the laws of adhesion. Newton had seen two of the three indicators of molecular adhesion: the black, flattened contact spot, and the pull-off force. But he had not seen the spontaneous jumping into contact. The reason was that, with large hard spheres, the adhesion could not be established reproducibly because surface roughness and dust particles prevented true molecular contact. Rubber spheres get over that problem to a certain extent because they are softer. The other approach is to make the spheres smaller and smaller. For small particles, roughness is less of a problem and adhesion is then always observable, even between the hardest surfaces. As particles are made very small, they begin to approach the size of single molecules, or ultimately single atoms, which always adhere. It would be a superb experiment if two molecules could be gripped, brought together, and then pulled apart to measure their molecular adhesion. Unfortunately no-one has yet found a way to do this (see Chapter 13). And the very act of gripping the molecules would change their character by preventing Brownian motion, and also distorting their electronic attractive fields of influence. A much more practical experiment became possible with the invention of the atomic force microscope by Binnig, Quate and Gerber14 in 1986. This device

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followed from Binnig and Rohrer’s work at IBM on the contact between a fine metal probe and a very flat surface, such as a silicon wafer used in electronic chips. Binnig, Rohrer and colleagues15 were working in the tradition of Ragnar Holm on the electrical conduction through metal contacts. Holm had seen that only a small part of an electrical contact actually conducts electrons, and concluded that the true molecular contact was small. Binnig and co-workers used finer and finer metal probes to show that the ultimate contact between a wire and a surface was a single atom. When the sample was moved beneath the tip, using a piezoelectric scanner, it received electrons from each atom in turn and produced an atomic picture of the surface, as shown in Fig. 3.16. This was the first time that the strange diamond patterned arrangement of adatoms on the surface of a silicon wafer had been seen directly, although it had been suspected from previous spectroscopic measurements of silicon surfaces. For this remarkable instrument, now called the scanning tunneling microscope, Binnig and Rohrer received the Nobel Prize in 1986.

3.9. THE ATOMIC FORCE MICROSCOPE (AFM) Around that time, there was a sudden realization that many different kinds of probe could give atomic resolution at surfaces. Electrical conduction, heat conduction, or acoustic transmission could all be used in principle. The scanning tunneling microscope, based on electron conduction, was immediately successful because it was so simple. A probe could be made by taking a metal wire and cutting it with scissors to give a pointed end. An atomically smooth surface could be prepared by pulling graphite flakes apart with sticky tape. Then a picture of the

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carbon atoms on the surface could be obtained using the piezoelectric scanner. But electron conduction was only one way in which atoms could be detected. That could only be used on electrical conductors, thus preventing measurements on polymers, oxides, biological materials, etc. Therefore, Binnig, Quate and Gerber14 devised the atomic force microscope (AFM) shown schematically in Fig. 3.17. The probe was now much smaller and lighter so as to detect the molecular adhesion forces. These forces caused a slight movement of the probe which could be observed by several different sensor methods. However, laser detection, as shown in Fig. 3.17, turned out to be most convenient. Small silicon cantilever probes were made by etching a silicon wafer. Laser light was reflected from the top of the silicon cantilever probe, and entered a detector where any deflection could be registered. A typical adhesion experiment using the AFM was performed by raising the sample towards the probe while observing the cantilever deflection. When the probe was distant from the sample, nothing happened, but as the probe came within range of the adhesion forces, the probe was attracted and pulled down, as shown by the result of Fig. 3.18. As the sample is pushed nearer to the probe, the attractive force increases rapidly and the probe then jumps into contact with the sample. This appears as a sudden deflection of the cantilever. Then, once the probe is stuck to the sample surface, the movement of sample and cantilever are equal, as shown by the linear contact deflection in Fig. 3.18. Moving the sample in the opposite direction can then be used to pull the probe off the surface. This result is shown in Fig. 3.19. The deflection of the cantilever is reversed along the same linear path as the probe sticks to the sample at first. Then, at a certain deflection of the cantilever, the probe jumps off the

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surface. This separation of the probe usually occurs at a different point from the attractive jumping into contact. Thus, there is a hysteresis in the adhesion process; the making of contact is not reversible during the breaking stage. It is evident from these results that two of the adhesion processes (making contact and breaking contact) can be observed in the atomic force microscope. The size of the contact spot can also be detected if some additional technique for sensing contact spot size is used, for example, electrical contact or thermal contact. Thus the three stages of adhesion can be found in the AFM. It has become clear from such measurements that all materials adhere in this test, verifying the first law of adhesion. These results will be considered in more detail in Chapter 4. Another great benefit of the AFM is that the sample and probe can be immersed in liquid, so that adhesion can be tested in a wide range of environments. The immediate conclusion from such tests is that the adhesion is reduced by such immersion, thus verifying the second law of adhesion. Finally, the curious phenomena of time dependence, of irreversibility, and of hysteresis have

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been observed generally in these nanoscale experiments. Consequently, it is clear that the mechanism of adhesion is complex, as stated in the third law. It is now necessary in the following chapters to deal with these laws in greater depth.

3.10. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Newton, I., Opticks, Smith and Walford, London, 1704 (reprinted Dover, New York, 1952, p 194). Galileo, Dialogue on the Two Chief Systems of the World, Elsevier, Amsterdam, 1632. Johnson, K.L., Kendall, K. and Roberts, A.D., Proc. R. Soc. A 324, 301–13 (1971). Evans, C., Metals Handbook, 9th edn. vol.7, Powder Metallurgy, Am. Soc. Metals, OH, 1984, p 288. Bettelheim, F. A. and March, J. General, Organic and Biochemistry, Harcourt Brace, London, 1998, p 785. Kendall, K., Science, 263, 1720–5 (1994). Perrin, J.B., Atoms (translated by D.L. Hammick), Constable, London, 1923. Kendall, K., Liang, W. and Stainton, C., Proc. R. Soc. Lond. A454, 2529-33 (1998). Kendall, K., J. Adhesion Sci. Technol. 8, 1271–84 (1994). Young, T., Miscellaneous Works, vol 1, Murray, London, 1805, p 418. Daniell, J.F., An Introduction to Chemical Philosophy, J.W. Parker, London, 1839. Brunauer, S., Emmett, P.H. and Teller, E. J. Am. Chem. Soc. 60, 309 (1938). Griffith, A.A., Phil. Trans. R. Soc. A 221, 163–98 (1920). Binnig, G., Quate, C.F. and Gerber, C. Phys. Rev. Lett. 56, 930 (1986). Binnig, G., Rohrer, H., Gerber, D. C. and Weibel, E. Phys. Rev. Lett. 50, 120–3 (1983).

4 EVIDENCE FOR THE FIRST LAW OF ADHESION: SURFACE LEAP INTO CONTACT

For by pressing such Glasses together their parts easily yield inwards, and the Rings thereby become sensibly broader ISAAC NEWTON, Opticks,1 p. 201

The quest for molecular adhesion started with Newton’s two fundamental ideas; that polishing glasses smoother would allow more intimate contact, and that the small movements of the surfaces towards each other could be measured by the interference patterns formed by light waves in the narrow gaps between the bodies. Newton had seen the black spot and the colored rings at the contact of two telescope lenses, though he deferred to Robert Hooke in his earlier observations of “ye apparition of a black spot at ye contact of two convex glasses,”2 when he made his famous statement about seeing further “by standing on ye shoulders of Giants.” Of course, Newton did not know about molecules, nor about molecular forces and their influence on elastic deformation. So he could not find the simple connection between the black spot size, the applied force, and the molecular adhesion. Two hundred years were to elapse before these relations could emerge. However, Newton’s idea of seeking out smooth surfaces and of measuring their very small movements in contact through interference methods persisted and ultimately led to the successes described in this chapter. The first experimenters, for example Tomlinson and Bradley, followed Newton by working on glass and this work reached its peak in the measurements of Derjaguin and Sparnaay in the early 1950s. However, glass was too rough and was replaced by smooth mica which had been first studied in 1930 by Obreimoff. Using mica allowed 63

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molecular contact to be studied in depth by Tabor, Winterton and Israelachvili. Polymer surfaces were also smooth enough to give reliable adhesion, as defined initially by Rivlin, leading eventually to the currently accepted theory of molecular adhesion.

4.1. THE PROBLEM OF OBTAINING RELIABLE ADHESION The first law of adhesion states that all atoms adhere. Therefore we would expect that blocks of material placed in good contact should stick together. This idea had been studied by Tyndall in 1875 and investigated further by Budgett in 1911.3 The objective was to show that adhesion between flat plates could be larger than that expected from atmospheric pressure. Steel blocks used as gauges in the metal working industries were highly polished to produce flatness much less than the wavelength of light, i.e. less than 100 nm. Tyndall placed these in contact by “wringing.” In other words he pressed and twisted them together in a sliding motion (see Fig. 4.1). The adhesion between the plates was then considerable and not affected much by placing the plates in a vacuum, demonstrating that atmospheric pressure was not the cause of the adhesion. However, it was clear to Budgett, when he repeated Tyndall’s observations, that the surface condition of smoothness and contamination was also vital (see Fig. 4.2). He showed that, when the surfaces were contaminated with dust, then the adhesion was reduced. More importantly, Budgett demonstrated that, when the surfaces were cleaned with solvent to remove grease and water, the blocks fell apart easily. Thus, the adhesion force was due mainly to the pulling apart of the contaminant film. McBain and Lee showed later that thin films of many materials could give a large adhesion, with strengths up to 40 MPa, when used as glue between smooth metal plates.4

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It was clear that measuring adhesion of “flat, clean” surfaces was doomed to failure. Budgett showed especially that the steel plates were reactive and that the contaminant moisture could generate rust-like reaction products which would also stick the plates together with large adhesion. This effect will be described in Chapter 11. Hardy and his colleagues also showed that long chain acids could give stronger adhesion, especially if time was allowed for ordering to occur.5 But the main advance came by changing the material from steel or glass to mica.

4.2. OBREIMOFF'S EXPERIMENT Obreimoff,6 working in the Physics Institute of Leningrad, where large sheets of a perfect type of Muscovite mica were available from the White Sea area near Chupa, knew of the problems of putting polished flat surfaces together, and recognized that optical contact could produce adhesion, but not satisfactorily. “Perfect” optical contact still contained gaps up to 100 atoms wide. However, he had observed that freshly split mica foils could be put back together to adhere with considerable force and set out to investigate this unique effect. His paper was most significant because it identified for the first time the three processes involved in adhesion; the jumping into contact, the equilibriation of the joint, and the pulling apart of the mica sheets. In addition, Obreimoff saw that evacuating the apparatus improved the adhesion, and also found electrical discharges which proved that adhesion was essentially an electromagnetic phenomenon. The mica was cut using a razor blade into 2 × 20 × 50 mm blocks, and a glass wedge with a smooth and rounded end was used to split a foil 0.1 mm thick from the top surface. The experiment was placed in an evacuated tube and a glass hammer containing an iron mass was moved with a magnet to press the wedge into the mica to promote splitting (Fig 4.3). The region at which separation occurred was viewed under a microscope and Newton’s interference fringes were seen in the narrow gap between the split surfaces. By measuring the positions of

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the interference fringes, Obreimoff was able to determine the shape of the mica strip which was being wedged from the block, as shown in Fig. 4.4. The point of separation of the mica foil from the block, that is the crack line, could not be seen directly because this kind of interference experiment only detects gaps down to about 50 nm. But the shape of the bent mica foil could be accurately measured and was shown to be parabolic. In other words, the strip was behaving as a simple leaf spring and its shape was not affected by the molecular adhesion forces. This was an important observation because it proved that the molecular forces were only acting across the small gap near the line of separation. Thus the molecular forces could be neglected in terms of the large scale behavior of the system. Engineers therefore safely ignore such forces when calculating the shapes of machinery. The mathematics of these calculations is given in Chapter 14, essentially employing the energy conservation method of Obreimoff.

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By carrying out a series of splitting experiments on different mica samples, Obreimoff was able to show by balancing energies that, in air at ordinary pressure, the energy required to fracture the material was After splitting the mica, Obreimoff found that the surfaces would spontaneously jump back together when he removed the glass wedge. He measured the energy of this spontaneous adhesion and it was about substantially less than the original adhesion energy. Then he pushed the wedge back in to measure the adhesion formed between the foil and the block by the jumping process and found the adhesion energy was around so it was taking more energy to split the adhering mica than was recovered on the jumping together. Thus he found some energy loss or adhesive hysteresis in this process. He also realized that it took some time for the splitting to reach equilibrium; the fringes moved for quite a time after the wedge was fixed, around 15 s. This was the first observation of the rate effect on adhesion. Perhaps of most significance was the result obtained when the air was evacuated from the vessel around the mica. Adhesion was increased as the air was pumped out. This observation supports the second law of adhesion; that adhesion is reduced by air molecules which contaminate the mica surfaces (see Chapter 6). As these contaminant molecules were removed by evacuation, the energy of adhesion was then increased to and impressive electrical discharges were seen around the mica samples at 1 nb pressure. This proved that the adhesion was connected with electromagnetic forces between the atoms in the mica crystal. However, it did not seem possible for the bonds themselves to have such a high energy, so the conclusion was that huge energy dissipation was occurring during this high vacuum adhesion test. Further development of this cleavage technique was made by Gilman while he was at the General Electric Research Laboratory in Schenectady, New York. He attempted to measure the surface energy of a wide range of crystals by the Obreimoff method.7 He made two main changes to the experiment: first, he worked in liquid nitrogen to avoid any plasticity effects; second he found the wedge method to be too frictional and used direct tension applied to the split ends of the crystal to drive the crack. The rig is shown in Fig. 4.5. Gilman concentrated on separation of the crystal planes by the crack; he did not make any observations of the crack healing to produce adhesive joining of the cleaved surfaces. So he did not fully demonstrate reversibility. However, he restricted plasticity to low levels and obtained surface energies comparable with those estimated from elastic modulus, as shown in Table 4.1. The next improvement came from the strict control of the separation of the surfaces, to show the jumping into contact in the most dramatic way, as demonstrated by Tabor and Winterton.

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4.3. TABOR AND WINTERTON'S EXPERIMENT Tabor and Winterton8 followed Obreimoff’s idea but made several advances in the method, which allowed the jumping to contact to be quantified more centimetres in size. This smoothness allowed repeatable and reliable molecular because this was atomically smooth over large areas, up to several square perfectly. First they recognized that it was essential to use Muscovite mica contact. Previous experiments by Deryagin and Abrikosova,9 later followed by Kitchener and Prosser,10 then Black et al, 11 had all used polished silica or silicate glass, but these were too rough to obtain true molecular contact, although the jumping due to the attractive forces had been seen. However, Tabor and Winterton decided to locate the contact of the mica more precisely by gluing thin foils of mica onto glass cylinder formers, such that the crossed cylinder contact gave a circular contact spot, reminiscent of the Newton black spot. The arrangement is shown in Fig. 4.6. Second, the movement of the foils towards each other to forge the contact was controlled precisely using a piezoelectric actuator. This was simply a strip of piezoceramic which moved when an electric field was applied to electrodes on each face. By this means the movement could be dialed in electrically with subnanometer precision while retaining high stiffness. This allowed the compliance of the system to be adjusted by suspending the upper foil on a spring steel strip. The stiffer this spring steel was, the nearer the surfaces could approach before jumping was observed. The schematic arrangement is drawn in Fig. 4.7.

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The third innovation was to use multiple-beam interferometry rather than Newton’s rings. This technique had been much developed by Tolansky12 in the 1940s to be ten times more sensitive than Newton’s fringes, and was therefore capable of resolving single-atom steps on a mica surface. To obtain the sharper fringes, it was merely necessary to evaporate a thin metal layer on the back surface of the mica foils before gluing them in position. The multiple reflections of white light between these half-silvered surfaces gave fine fringes which could resolve very small movements of about 0.3 nm. For example, in the laboratory conditions, air of 50–80% relative humidity was used. This caused a thin layer of moisture to be adsorbed on the mica. By measuring the fine multiple beam fringes before and after moisture adsorption, it was deduced that the total thickness of water between the two foils was 0.7 nm.13 In order to perform an adhesion experiment using this apparatus, two precautions had to be taken: the apparatus had to be protected from vibrations so the work was done at night on vibration isolators; also, the surfaces had to be

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exposed to ions and moisture to relieve any free charges clinging to the mica. The bottom foil could then be raised to meet the top foil, while noting the position by means of the interference fringes. As the surfaces got nearer, deflection of the steel spring was noticed and at a critical separation, the mica leaped into contact. This jump distance depended on the spring stiffness as expected, stiffer springs being necessary to allow close approach. The results are shown in Fig. 4.8. For jump distances between 5–15 nm, the results fitted the Hamaker theory which states that the attractive van der Waals force is proportional to cylinder diameter and inversely proportional to the square of the gap. The Hamaker constant was found to be These experiments were the first to reveal the true van der Waals forces at close approach. Previous tests had only shown the weaker, retarded forces which apply for gaps larger than 20 nm. These theories will be considered in Chapter 5.

4.4. EXTENSION BY ISRAELACHVILI AND TABOR After this stunning piece of work, Winterton joined the Nuclear Laboratories of the Electricity Generating Board at Berkeley, eventually becoming a lecturer at Birmingham University, while Tabor continued with a new PhD student, Jacob Israelachvili, who continued to develop the mica “surface force apparatus” as it was soon to become known. His immediate contribution was to push the jump distance from 5 nm down to 1.5 nm and then to measure the effect of stearic acid monolayers on the mica surfaces.14,15

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This incremental advance was achieved with better designed equipment, especially in relation to the spring support and the piezotransducer. The main problem with the earlier kit was that the spring had to be removed in each experiment. By making the spring a double-leaf and by adjusting its stiffness with a movable clamp, a whole series of tests could be carried out without dismantling the apparatus. Essentially the same results were obtained as before, but to closer approaches, and with less scatter. Stearic acid also did not affect the jump distances much, showing that the mica and acid have roughly similar Hamaker constants. So Israelachvili’s contribution at this point was to verify and extend Winterton’s measurements. However, the redesign paved the way for work on adhesion of mica when immersed in liquids, which will be described in detail in Chapter 6. Although the stearic acid coating did not affect the jump distance, it did change the attractive forces at separations less than 2 nm. Stearic acid-coated mica surfaces obeyed the Hamaker equation down almost to molecular contact, essentially giving a work of adhesion around Mica surfaces under damp atmospheric conditions are known to have a work of adhesion nearer , and it is evident therefore that the attractive forces increase greatly at separations below 2 nm for normal mica.16 The surface force apparatus cannot measure such forces. Moreover, the surfaces begin to deform substantially under these large forces and the theory is no longer valid. A plot of the results is given

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in Fig. 4.9 to show the development of these large forces compared to the results above. This graph is plotted with the attractive force as negative by convention. The mica apparatus thus only covers a small part of the force range.

4.5. FINER MEANS SMOOTHER Mica is unusual in its remarkable smoothness over large areas. However, many crystals and glasses are smooth over micrometer sized regions, so another approach to smoothness is by making finer test samples. This was the idea adopted by Tomlinson17 in 1928, to contribute to the lively debate at that time about the rate of change of molecular attractions with separation, which will be considered in detail in Chapter 5. Tomlinson heated and drew fresh fibers of fused silica to perform adhesion experiments, bringing the crossed fibers together to observe the contact point, which he estimated to have a black spot size less than and measuring the force of adhesion by elastic deflection of the fiber, as shown in Fig. 4.10. He also formed spherical blobs on the ends of the fibers and tested these in the same way. Tomlinson was careful to release electric charges by ionising the air, and also made sure that the silica was dry to avoid questions about surface moisture. The best adhesion was observed immediately on cooling from red heat. Also marked damage was seen after adhesion of the surfaces, proving that the forces of adhesion were large enough around the black spot to crack the glass material. A typical fiber was in diameter, about the same as a human hair, and the adhesion force measured was This force was readily determined by the 5 mm deflection of the fibre just before detachment. The interesting feature of the experiment reported by Tomlinson was the sudden attachment of the fibers when contact was approached. The surfaces leaped into contact! He took trouble to apply no force pushing the surfaces together, and concluded that “the molecular

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attractions acting at the instant of geometrical contact are sufficient to draw the fibres together.” The adhesion force seemed to be in proportion to the diameter of the bodies. This was the strange result which was not at all expected, and which was explained a few years later by Bradley.

4.6. BRADLEY'S ADHESION RULE Bradley18 had read Tomlinson’s paper and developed an improved method of measuring the adhesion, together with a better theory based on London’s wave mechanics theory of the forces between molecules. By adding up the London forces for all the molecules in two rigid spheres, Bradley came to the conclusion that the adhesive force required to separate them should be proportional to the sphere diameter, as shown in Fig. 4.11. He also showed that the force should be proportional to the work of adhesion W of the spheres, that is the energy required to separate one square meter of interface reversibly. Thus he produced his famous equation for adhesion of spheres shown in Fig. 4.11. Bradley then constructed an apparatus for measuring the force required to separate two silica spheres from adhesive contact (see Fig. 4.12). The rig could be evacuated to remove moisture and other contamination. Heaters were used to bake the glass and a radioactive source ionized the gas to leak away any stray charges. The silica balls were heated to incandescence immediately before the measurements. A deflection was applied to the bottom sphere until it detached from the upper sphere, giving the results shown in Fig. 4.13. Bradley found that the adhesion remained constant as gas was evacuated, so water was obviously not the cause of the adhesive force. However, the spheres were probably too large to give the molecular contact which Tomlinson had observed on his ten times smaller fibers. The force was proportional to diameter which fitted the theory, but the force was several times smaller than expected, and smaller than Tomlinson observed. This problem became evident when Bradley

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attempted to use sodium borate spheres instead of fused silica. Although adhesion was seen, it was variable and Bradley suggested the surfaces were rough as a result of reaction with water. The conclusion was that these spheres were still too large and insufficiently smooth to obtain reliable molecular adhesion. Another significant issue was the deformation and flattening of the spheres at the point of contact. Newton had stated that this occurred as the spheres were

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pulled together by adhesion forces. Tomlinson worked out from the well-known Hertz equations of elasticity, described in Chapter 9, that the contact spot diameter should be around for the contact of his silica fibers. Bradley took no account of this idea and his theory was based on the assumption of rigid spheres. This cannot be correct for elastic particles. Derjaguin19 attempted a solution of this question in 1934 by combining the Hertz and Bradley ideas, but his answer was not quite right. The final solution was obtained in 1971, showing that Bradley’s equation requires only slight numerical modification, by about a factor of 2, to when elastic deformation is taken into account (see 20 Chapter 9). So elastic stiffness hardly affects adhesion, and therefore it should not matter whether we stick stiff diamond or compliant rubber spheres together, we should get the same adhesion force for the same work of adhesion. This is something of a shock. The most surprising conclusion from these arguments is that the adhesion of particles is in proportion to their diameters. This result is of outstanding importance because it means that adhesion dominates all other forces at dimensions below particle size.

4.7. THE SIGNIFICANCE OF BRADLEY' S RULE The discovery that solid bodies jump into adhesive contact under the influence of the molecular attractions was enormously significant. This experiment was certainly known to Tomlinson17 in 1928 and was studied both by Obreimoff6 and Derjaguin and Abrikossova9 some time later. Once you see this phenomenon, you become convinced that molecular adhesion exists. When you observe it in different situations (on mica, on glass, on metals, on polymers) then you realise it is a universal observation that applies to all bodies when there are no contaminant molecules to stop the adhesive interaction of the particles. In a sense, although it came 100 years after Brown, this observation stands with Brownian motion as a critical breakthrough. Before Brownian motion was seen, engineers thought that matter was continuous and static. Suddenly they were aware that it was molecular and moving, though this molecularity and movement could often be ignored at the macroscopic level because the atoms and energies were so small. In a similar vein, engineers normally treat objects as nonadhering. Wheels roll and particles flow macroscopically without sticking. But it turns out that this is an illusion brought about by the small values of the work of adhesion, and by Bradley’s rule. Bodies should always stick but this depends greatly on size. Size is important. Consider the wheel of a truck, which is one meter in diameter. This does not adhere significantly to the road surface, and can be lifted up without significant sticking. However, applying Bradley’s rule that the adhesion force should be the

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product of the diameter and the work of adhesion, which we know to be about then the adhesion force should be 0.1 N. This is the theoretical prediction shown in Fig. 4.14 on the Bradley’s rule line. It is a small force, as can be seen by the logarithmic scales in the diagram. Compared to the gravity force acting on the truck tire, this theoretical adhesion force is a million times smaller. But we also know that the tire is so rough that it does not make perfect molecular contact with the road surface, so the measured adhesion is another million times less, as shown on the dotted curve. Thus it is clear that the adhesion at the engineer’s level is very much smaller than the weight, and can normally be neglected. Racing car tires are an exception to this because they are made much smoother to obtain better grip. In this case the adhesion is small but measurable. The interesting thing about Fig. 4.14 is the way the forces change as the size of the body becomes smaller. The force of gravity falls with the cube of diameter and so drops rapidly for smaller bodies, whereas the adhesion falls more slowly. Thus for smooth particles, there is a transition around 1 mm diameter, where gravity and adhesion are equal. Thus 1 mm dust particles should cling to the ceiling if they are smooth enough to obey Bradley’s rule. Fortunately, the surfaces are rough and so the bottom curve applies and the transition is near Thus, when we are dusting, we tend to find particles which are just about visible. However, geckoes and lounge lizards can use Bradley’s rule, and can stick to ceilings by improving their smoothness.

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Consider now a bacterium which is in diameter. This now behaves in a completely different way to the truck tire, as seen in Fig. 4.14. Because of the rapid decline in gravitational force with diameter, the weight of a bacterium is now extremely small, less than 1 pN, or less than a single weak chemical bond force. But adhesion, according to Bradley’s rule, has not declined so fast and is around for a smooth bacterium or 1 nN for a rough surface contact. Thus a bacterium in dry conditions will always stick to a surface and cannot behave like a truck tire which exhibits zero adhesion. A key problem with dry bacteria is releasing them from surfaces. When wet, the adhesion is decreased substantially, as described in Chapter 6, but bacteria can still show strong adhesive behavior even under that condition (see Chapter 12). The conclusion from this argument is that Bradley’s rule can explain the transition from the macroscopic engineering world, where nothing sticks, to the nanoworld where everything sticks.21 The transition for smooth spheres is 1 mm. Larger than 1 mm, ball bearings roll around and behave as we expect from our common experience. Below this they should stick. However, roughness introduces another dimension into this argument because true molecular contact is not achieved. This shifts the transition down to smaller particles, around in size. Thus there is a gray area of transition where particles can behave in a schizoid way, depending on surface roughness, sometimes sticking, sometimes not, between one micrometer and one millimeter in particle size Therefore we must consider more carefully what we mean by surface roughness as we move down from the engineering to the molecular scale. The best way to do this experimentally is by atomic force microscopy.

4.8. THE NEW SCIENCE OF ATOMIC FORCE MICROSCOPY (AFM) In the atomic force microscope, a fine pointed probe, typically l00nm diameter, is brought into close proximity with a smooth sample, which is controlled in position by a piezotube. The overall scheme was described in Chapter 3. Here the objective is to describe adhesion experiments as the probe is brought into adhesive contact with dry surfaces. Figure 4.15 shows the probe as it nears the sample. On the right is an atomic model of the near-contact region showing how the tip is from 10–100 nm diameter under normal circumstances, though the shape varies from tip to tip.22 One of the key problems in the AFM is the fact that the movement of the piezotube and tip must be added together. Clearly it is desirable to have as stiff a cantilever as possible so that most of the displacement is governed by the piezotube. Then the jumping of the tip onto the surface of the sample can be studied. It is important to remove all moisture in such experiments; otherwise, condensed capillary films obscure the results.

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One of the issues is to provide a realistic theory for the summing of London forces over the geometry of a probe tip. This has been carried out by various groups, and found to differ from the standard Hamaker expressions for sphere approaching flat, especially when the probe was very fine, e.g. less than l0nm diameter at the tip. The results obtained for the force on the tip as a function of separation could then be explained, and the Hamaker constant determined. Alternatively, a small sphere could be glued to the cantilever, in which case the

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standard equation applied. The results shown in Fig. 4.16 were obtained by Larson et al.23 All experiments in the AFM gave attractive jumping behavior in vacuum or air. In water and other liquids, jumping behavior could be observed as in Fig. 4.16, but could be stopped by stiffening the cantilever. The other question to be asked relates to the effect of surface roughness on such tests. Ultimately, the AFM probe can sense the different force curves across a surface, which then appears bumpy in force terms, even though it is perfectly flat in molecular terms.

4.9. HOW SMOOTH IS AN ATOM? This begs the question of what an atom is. Especially we have to ask whether the atom itself can be rough. The atomic hypothesis, that all matter could be divided into its ultimate constituents, whose shape and movement dictated the properties of the material, can be traced back to the Greek philosophers Leucippus and Democritus.24 Epicurus thought that the atoms could stick together because of hooks, sticks and holes which provided a mechanical bond, but that idea has been rejected through the centuries, largely because it is illogical to think of the ultimate particles of matter having extra, smaller bits attached. In particular, Newton thought the idea of “hook’d atoms” to be ridiculous and instead postulated the idea of forces acting from atom to atom, though generally he called the atoms “particula” or “corpuscula.” Of course it was known that atoms packed to form certain geometrical shapes of crystals, and this had stimulated Newton’s contemporary, Robert Hooke, to suggest that the atoms were spheres that were packing in regular order, rather as Plato had earlier conceived that atoms filled space as regular geometrical bodies, e.g. cubes, tetrahedra, octahedra, and icosahedra, as shown in Fig. 4.17. When Dalton invented the idea of atoms as participants in chemical reactions, he had the problem of how to stick atomic models together.25 He instructed his friend, Peter Ewart, to prepare a set of wooden balls, drill holes into the balls and insert wooden pegs to hold the model in place. These models appeared around 1810, about the same time that Wollaston was postulating the structuring of atoms into regular shapes to form molecules,26,27 the word first

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used by Avogadro in 1811. This idea, that atoms adhere to fill space, has been a very successful one and has led us to the idea of an atomically smooth surface, that is the idea of a plane filled with atoms. We then easily conceive of a rough surface because it has steps, vacancies, protruberances and so forth. So roughness is readily defined at this level, as shown in Fig. 4.18. The problem comes when we try to see roughness in the atoms themselves. With the scanning probe microscope, we can now distinguish features on the atomically smooth surface. But these are essentially variations in the force fields emanating from the atoms. The force field stretches some distance from the surface; it has a range of action which is quite large. The atom is smooth, and the plane of atoms is smooth, but the forces which act from the atoms vary significantly, and it is these forces which determine adhesion. So we must distinguish carefully what is the smooth plane of the atoms and what is the varying extent of the forces exerted by the atoms. Quantum effects must also be taken into account ultimately, but can be ignored to a first approximation.28 Once we have overcome this problem of definition, it is then possible to specify in detail what the forces are and how they act with Brownian motion to give adhesion. That is the purpose of the next chapter.

4.10. REFERENCES l. Newton, I., Opticks, Smith and Walford, London, 1704 (reprinted Dover, New York, 1952, p 201). 2. Newton, I. In: Turnbull, H.W. The Correspondence of Isaac Newton, vol I, Cambridge, Cambridge University Press 1959. 3. Budgett, H.M., Proc. R. Soc. A 86, 25–35 (1912). 4. McBain, J.W. and Lee, W.B., Proc. R. Soc. A 113, 606–620 (1927). 5. Hardy, W.B. and Nottage, M.E., Proc. R. Soc. A 112, 62 (1926). 6. Obreimoff, J.W., Proc. R. Soc. A 127, 290–297 (1930). 7. Gilman, J.J., J. Appl. Phys. 31, 2208–2218 (1960).

EVIDENCE FOR THE FIRST LAW OF ADHESION 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

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Tabor, D. and Winterton, R.H.S., Proc. R. Soc. A 312, 435–450 (1969). Deryagin, B.V and Abrikosova, I.I., Disc. Faraday Soc. 18, 33 (1954). Kitchener, J.A. and Prosser, A.P., Proc. R. Soc. A 242, 403 (1957). Black, W., de Jongh, J.G.V, Overbeek, J.T.G. and Sparnaay, M.J., Trans. Faraday Soc. 56, 1597 (1960). Tolansky, S., Multiple Beam Interferometry of Surfaces and Films, Oxford, Oxford University Press, 1948. Bowden, F.P. and Tabor, D., The Friction and Lubrication of Solids, part II Oxford, Oxford University Press, 1964, p 413. Israelachvili, J.N. and Tabor, D., Nature 236, 106 (1972). Israelachvili, J.N. and Tabor, D., Proc. R. Soc. A 331, 19–38 (1972). Bailey, A.I. and Kay, S.M., Proc. R. Soc. A 301, 47–56 (1967). Tomlinson, G.A., Phil. Mag. 6, 695–712 (1928). Bradley, R.S., Phil. Mag. 13, 853–62 (1932). Derjaguin, B.V., Kolloid Zeits 69, 155–64 (1934). Johnson, K.L., Kendall, K. and Roberts, A.D., Proc. R. Soc. A 324, 301–13 (1971). Kendall, K., Science, 263, 1720–25 (1994). Drummond, C.J. and Senden, T.J., Colloids Surfaces A 87, 217–34 (1994). Larson,I., Drummond, C.J., Chan, D.Y.C. and Grieser, F., J. Am. Chem. Soc. 115, 11885 (1993). Raos, N., Chem. Britain, February, 31-3 (1997). Rouvray, D.H., Chem. Indust., 4, 587–90 (1997). Wollaston, W.H., Phil. Trans. R. Soc. 98, 96 (1808). Wollaston, W.H., Phil. Trans. R. Soc. 103, 62 (1813). Von Baeyer, H.C., Taming the Atom: The Emergence of the Visible Microworld, Random House, New York 1992, p 980.

5 INTERMOLECULAR FORCES: THE NEW GEOMETRY OF COMPUTER MODELING

Where Attraction ceases, there a repulsive virtue ought to succeed ISAAC NEWTON, Opticks,1 p. 395

Isaac Newton almost certainly had in his mind the idea that adhesive attractions exist between the smallest parts of bodies, that is between the atoms and molecules. He had no way of measuring the size of these atoms nor of defining the way in which the forces vary with distance, although he predicted that “microscopes may at length be improved to the discovery of the Particles of Bodies”2, a kind of prophesy of the invention of electron and atomic force microscopes 300 years later. However, Newton was equally certain that the attraction could not be continuous, but must eventually be opposed by a repulsion as the atoms made contact, as indicated by the quote at the start of this chapter. The equilibrium between atoms must then be at the balance point where the force of attraction equals the force of repulsion, as suggested in Fig. 5.1. The purpose of this chapter is to consider the theory of these two forces between atoms: the attractive force which gets stronger as the atoms approach and the repulsion which acts when the atoms make contact. This leads us to a model of adhesion which has two parameters in the simplest case. Having established the forces, a computer model can then be defined to take into account the incessant motion of the atoms. This argument leads to the concept of a new geometry: the natural molecular adhesive geometry of the universe. Adhesion is shown to have a remarkably sensitive action on this geometry. 83

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5.1. THE TWO-PARAMETER MODEL OF ATOMIC FORCES The idea that there were two terms in the expression for the force between atoms and molecules was recognized by Newton, but really began to take shape this century when a number of theorists, like Mie for example, started to formulate mathematical equations to describe both an attractive term and a repulsive term,3 as shown in Fig. 5.2. In this diagram, the potential energy between two atoms is plotted on the vertical axis as the atoms are separated a certain distance, shown horizontally. Conventionally, the attraction is viewed as a negative potential given by the first term of the theory, and the repulsion is seen as positive, i.e. the second term. The two terms added together gave the total picture of the energy as shown in Fig. 5.2. From this energy, the force F between the atoms can readily be calculated by taking the gradient of the curve at any point, as shown on the top curve. Thus the force is zero at the minimum of the energy curve. The minimum of the force curve is where the bond breaks as the atoms are pulled apart. This is the maximum tension which can be supported by the attraction. The top curve shows that there is a minimum in the energy at a separation This is the equilibrium point, that is the point at which the repulsion and attraction balance. It is an “energy well” by analogy with a water well. Thus the atoms sit at the bottom of the energy well. Lennard-Jones4 applied this two-parameter model in the 1920s and early 1930s when there was much activity in trying to understand the quantum nature of the atom. Lennard-Jones, who was 30 years old and only called Jones when he suggested his memorable potential while at Trinity College Cambridge, foresaw the future, becoming Professor of Physics at Bristol before moving to be

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Professor of Chemistry at Cambridge. At the atomic bond level, there is no difference between physics and chemistry. Lennard-Jones, with his colleagues Taylor and Dent,5,6 realized that all known intermolecular forces are electromagnetic in nature. They range from the Coulombic forces between ions, to dipole forces between polar molecules, to the weak van der Waals forces which act between all atoms and which are responsible for adhesion. These adhesive forces, sometimes called London, London–van der Waals or dispersion forces, are always attractive because they result from instantaneous dipoles in one atom and their induced dipoles in a neighboring atom. The Lennard-Jones equation can be written

giving the energy w of the interaction as a function of the two parameters, the bond energy and the bond length. By inserting values of these two parameters into the equation, Lennard-Jones could calculate the properties of the atomic

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bond, especially confirming the measured viscosity of the gas, its equation of state, the spacing of atoms in a crystal, and the compressibility of the solid. The power of the repulsive term was between 9 and 20 and was typically 12. When this power was high, e.g. 20, then it approximated to a “hard-sphere” potential, such that the repulsion was very much like a hard wall, as we will see under Section 5.5 in the computer modeling. Later it was found that a whole range of two-parameter models could be used to describe the experimental information. For example, Morse7 fitted an exponential curve to both the attractive and the repulsive potentials. Buckingham had a power law attraction and an exponential repulsion. However, the power law expression of Lennard-Jones was most memorable because the attractive term corresponded very neatly with the London derivation of the van der Waals, force between two nonpolar molecules. London8 was working on a way to calculate the interaction between two atoms by solving Schrödinger’s equation of quantum mechanics,9 starting from the known atomic structure and the fundamental constants. In practice this approach is too complex, so approximations and compromises are necessary. Generally there are three types of approximation to solving the quantum theory: ab initio calculations semiempirical calculations, and model calculations. Ab initio calculations are available for only the simplest systems e.g. H–H, the bonding between two hydrogen atoms. Semiempirical calculations are used most because they combine quantum theory with some experimental results to give a good approximation to the truth. Model calculations replace the atoms or molecules with a more tractable model. For example, London replaced the molecule with a harmonic oscillator, allowing much simpler mathematical analysis to give his attractive behavior which fitted van der Waals experiments. It is this fit to experiment which we consider next.

5.2. EXPERIMENTAL EVIDENCE FOR MODELS OF MOLECULAR FORCES The data required to support and verify the above theories is remarkably sparse and largely specific to certain systems, such as the noble gases He–He and Ar–Ar, helium and argon molecules. Essentially there are four methods for determining the parameters in the atomic and molecular models. The first and most primitive one is the geometrical approximation which describes atoms as hard spheres with a certain radius. For example, you can determine the dimensions of the argon atom by measuring the spacing of the crystal lattice of solid argon, for example by X-ray diffraction. The second method is based on spectroscopic measurements to find the vibrational frequencies of the bonds. The third and most direct method is the scattering of molecular beams. Finally, there are the methods based on the bulk properties of the materials, for example

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gas law deviations such as viscosity used by Lennard-Jones, heats of sublimation, elastic properties, etc. The ordinary chemist’s view of atoms filling space like rigid balls is a reasonable approximation in many cases, despite the fact that quantum theory does not lead to a precise dimension of an atom. For example, the molecular model of the double helix of DNA was built by Crick and Watson assuming rigid geometric shapes for the atoms. Electrons around an atom do not, in fact, have a definite envelope, but are fuzzy according to quantum theory, with their effect dropping off exponentially as you move away from the atom. But the values of the range parameter in the Lennard-Jones equation are reasonably predictable and have been tabulated,9 as shown in Table 5.1, assuming hexagonal close packing of the spherical entities. This follows our expectations that atoms get larger as we go down a group, for example from Li to Cs. The atoms are accumulating more electrons and so should occupy more space. It is also easy to understand why the atomic radius should decrease as you move across a period, say from Li to F. The extra electrons are going into the same shell, but the added protons in the nucleus pull the shell in to give a more compact atom. Of course, ionization makes a large difference to these atomic radii. A cation such as is only half the size of the original sodium atom, having lost a bulky electron shell. Anions like are twice the size of their precursor fluorine atoms, having gained an electron. Molecules can also be constructed from these geometrical ideas.10 A molecule is a bunch of atoms bonded together, such that it behaves as a single unit, only connecting weakly to its neighboring molecules. Thus, methane can be considered to be a spherical molecule with a diameter of 400 pm. Decane is nothing like spherical and so can be considered as a cylindrical molecule 1.54 nm long and 400 pm diameter, but with possibilities of bending and rotating at each carbon linkage as illustrated in Fig. 5.3.

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The question then arises, “how do these molecules interact in terms of the forces between their constituent atoms?”

5.3. DIRECT MEASUREMENT OF MOLECULAR FORCES The most direct information on intermolecular forces has been obtained from scattering experiments on molecular beams. In particular, the repulsive forces have been elucidated by the neutral beam measurements of Amdur11 and his co-workers. The depth of the energy well and its bond length have been measured for many atoms and molecules using the “rainbow scattering effect” and interference phenomena below the “rainbow angle.” Typical results are shown in Fig. 5.4 for the potential well between potassium and the noble gases argon, krypton, and xenon.12

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The information gleaned by these techniques for a wide range of atomic and molecular interactions is contained in computer modeling packages, such as MSI13 which are now widely used in drug design and in many other applications. Much information can be gathered on web sites such as that at the Royal Institution in London.14 This information can then be used to put together molecular models.15,16

5.4. INTERMOLECULAR FORCES FROM BULK PROPERTIES Intermolecular forces are important because they dictate many of the macroscopic properties of the bulk material, for example the behavior of the gas and liquid phases, the crystalline structure, the elastic modulus, the ultimate strength, the heat of sublimation, and the chemical reactivity. Of course, the intermolecular forces are also responsible ultimately for the adhesion between bodies. Table 5.2 illustrates some of the properties which have been predicted from a knowledge of intermolecular forces. But the properties of gases are easiest to understand. In 1873, van der Waals was the first person to show how intermolecular forces must be introduced to explain the properties of gases. Gases exist because each atom has so much kinetic energy that the adhesion between the atoms is relatively insufficient to cause the atoms to stick together. Thus the compression of a volume V of atoms by a pressure P is dominated by the kinetic energy of the atoms as shown in Fig. 5.5. The kinetic energy of each atom is 3kT/2 where k is Boltzmann’s constant and T is the absolute temperature. As the gas is compressed at constant temperature, the energy of the atoms remains constant and so the perfect gas law, PV= constant, is obeyed for a perfect gas.

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Van der Waals could see two things wrong with this argument. The first objection was that the atoms have a certain volume which restricts the total compression. Once the piston is pressed to this atomic volume then the law must fail. Secondly, the atoms have a certain attraction to each other, resulting from the atomic adhesion, and if this becomes comparable to the kinetic energy of the atoms, then the perfect gas law must be inadequate. These two effects are illustrated in Fig. 5.6. At low packing, the gas law is extremely accurate, but as the atoms are pressed closer together, the PV rises because the volume of the atoms is no longer negligible. When the adhesion between the atoms increases towards kT then there is an extra deviation from the perfect gas behavior because the adhesion between

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the atoms also increases the pressure. Eventually, when the atomic attraction gets above about 10kT, the gas condenses into a compact phase because of the strong adhesion between the atoms. This is the sort of behavior we must now consider in terms of computer modeling techniques.

5.5. THE NEW GEOMETRY OF COMPUTER MODELING Originally, Greek philosophers thought that the universe was continuous and that the world could be described by lines, areas, and volumes, according to the geometry (literally “earth-measurement”), set down by Euclid, for example, around 300 BC. It became evident a few centuries ago that shapes are not continuous but are composed of similar but smaller shapes as they become more magnified. Thus a tree looks more complex the more it is studied on a finer scale, as shown in Fig. 5.7. This concept is the basis of fractal geometry which has been described by Mandelbrot.17 However, there must be a limit to this process as we approach atomic dimensions. At the atomic level, matter begins to look absolutely uniform, so that a crystal of argon is composed of exactly equal spheres arranged in a regular manner. However, this is not a valid picture according to the ideas we have been developing earlier in this book. Following van der Waals, we must conclude that the true geometry of the universe is not just affected by shape and size. It is also greatly affected by the random motion of the atoms and also by the adhesive forces of interaction between them. Unfortunately van der Waals only had the crudest mathematical tools to describe these additional features, and so could not formulate a realistic model. But now, with the emergence of computers, it has become possible to put the three elements of a realistic atomic model together: the spherical atoms, the random motions, and the attractive forces, to give a new vision of the world.

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The pioneering studies of this type of model, now known as molecular dynamics, were carried out by Alder and Wainwright in 1957 and the following decade.18 Wood and Jacobson, at the same time, obtained similar results by a slightly different approach.19 Berni Alder was working at the Lawrence Radiation Laboratories in California trying to bounce theoretical, nonadhering spheres around inside a computer. Since computers were rather limited in power at that time, a million times less potent than today’s machines, he could only model small numbers of spheres from 4 to 32 up to a maximum of 500. He found that, when the spheres had a lot of space to move around, then the motion was disordered like a gas, as in Fig. 5.8(a). But at high packing, as shown in Fig. 5.8(b), the spheres jostled together to produce an ordered structure which was crystalline in nature, although the spheres were still essentially fluid in their ability to move freely without any adhesion. This structuring was eventually proved to be a “first-order phase transition” and happens to be one of the few discoveries made by computer. Carl Stainton20 has repeated those early computer calculations of Alder and Wainwright using a modern desktop machine to show the detailed nature of the phase transition. Like Alder, he used periodic boundaries so that the spheres did not encounter obstacles in the model (Fig. 5.9(a)). As a particle left the cell during the calculation, it was assumed to re-enter from the other side. The particles obeyed Newton’s laws of motion in three dimensions and bounced perfectly off one another with no energy loss. When the particles were reduced in diameter to lower the packing density in the cell, the phase transition shown in Fig. 5.9(b) was found. The phase transition appeared suddenly, as the close-packed sphere structure was being diluted by shrinking each sphere diameter, when the spheres occupied 49% of the volume, i.e. at 0.49 packing fraction. The pressure was calculated by working out the impacts of the spheres on one square meter of the cell in one second. As the volume of the spheres was reduced during the computation, the

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pressure was seen to drop steadily in the structured phase. However, at 0.49 packing fraction, the pressure rose and the spheres were then seen to be in random, rather than ordered, positions. It is important now to consider this phase behavior in more detail.

5.6. STRUCTURING OF HARD SPHERES The computer study of this phase transition in the 1960s and 1970s led rapidly to a keen understanding of how hard, nonadhering spheres behaved as they were compacted towards the dense state. There were two outstanding problems: what was the range of the transition between random and fully structured packings and what was the ultimate stable form of the structured material? The first question was soon answered by Hoover and Ree,21,22 who looked at the upper transition at 0.55 packing fraction, above which the structure should be completely structured, called the melting transition because that is where melting (i.e. randomizing) first started to occur as the structured material was diluted. In the early computations, this upper transition did not appear because there were too few spheres to obtain random and structured phases together in the sample. Only when computing power had increased was it possible to see both transitions easily. These are shown in Fig. 5.10. The lower transition at 0.495 volume fraction became known as the freezing transition because that was where the random fluid first began to structure during compaction.

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The second problem, that of the preferred geometry in the structured state, was much more difficult because there were three possible candidate structures, and all of them were rather similar in energy. The crux of the problem was finding the lowest energy structure because this should be the one most stable and eventually dominant. Alder and his colleagues23 had considered this problem for a long time, but had not come up with the answer. The nature of the different possible structures can be seen by packing tennis balls in a box, as shown in Fig. 5.11. The bottom layer, marked A, goes down as a hexagonal close-packed bed, as shown by the solid circles. Putting the next layer on top gives the structure with the sphere centers in positions B, shown by the broken circle, marked B. The next ball to go on top of this layer, shown as a dotted circle, is then very important because it determines the structure. If it goes in the position marked C then the structure is face-centred cubic i.e. fcc, whereas if it goes back to A, then the structure is hexagonal close-packed, i.e. a hcp structure. This structure goes ABABAB, in contrast with fcc which goes ABC ABC ABC. If the layers are random, for example ABCACBCA..., then the packing is random hexagonal close-packed i.e. rhcp. All the three hexagonal structures go to 74% packing at the maximum limit. However, the energies of the packings are different at the close-packing levels, leading to a preference for the lower energy state. Alder had considered this question and predicted fcc would win. Woodcock demonstrated this result by computation in 1997 and the conclusion has been verified by several groups since then. 24–27 The energy difference was extremely small, about kT/1000, but this was still sufficient to force the fcc structure to dominate the hcp. These ideas could be checked in practice by comparing the computer results with experiments on polymer latex particles.

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5.7. EXPERIMENTS ON SPHERICAL POLYMER PARTICLES Of course, it is impossible to find particles with zero adhesion to confirm the theoretical argument that spheres should form a face-centered cubic structure above 0.49 volume fraction. However, certain types of polymer latex can be made to approximate to the theoretical nonadhesion conditions above. Polymethyl methacrylate particles were polymerized in a dispersion and polymer molecules were grafted to the surface to provide a steep repulsive force on close approach. The solvent was chosen to reduce the van der Waals force to a low level, by matching the refractive index to the spheres, obtaining an almost transparent suspension from the milky preparation. Pusey and van Megen28 prepared dispersions over a range of volume fractions and looked at light scattering from structures forming in the suspensions. Sedimentation occurred as the heavy polymer particles sank slowly to the bottom of the sample tubes. They observed the phase transition predicted from the computer theory. Crystalline regions could be seen as opalescent patches above 0.5 packing fraction. Above 0.58, the suspension seemed to be too viscous to form the crystals and this was interpreted as a glassy regime. Pusey and his colleagues29 then investigated the detailed structure of the crystalline regions by examining the diffraction patterns of the scattered light to distinguish fcc from hcp and rhcp structures. Under dilute conditions of slow

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sedimentation, they observed fcc structuring, but at higher concentrations found a mix of fcc and hcp. This suggests that equilibrium was not fully established, or that gravity was having a large influence on the results. The theory of hard spheres ignored gravity. Since that time, a large number of experimental tests have been carried out on polymer spheres,30–32 and also on silica spheres which simulate the opalescent patterns in natural opal.33,34 Generally, the broad predictions of the hard-sphere theory were verified but the fcc structure was not totally dominant. In order to distinguish the small forces acting on the suspensions, a number of experiments were carried on the Space Shuttle Columbia.35 There in the microgravity of space, it should be possible to compare the experiments with the computer predictions more realistically, since the computations assumed zero gravity. The spheres were 508 nm diameter polymer particles with a grafted surface layer. Packing fractions from 0.5–0.6 were examined over several days. The experiment consisted of stirring the suspension to randomize it, then observing the light scattering pattern to look for structural peaks. The diffraction patterns are so different, as shown in Fig. 5.12, that the result is unequivocal; fcc occurs on Earth but rhcp occurs in space. It was a surprise to find that a sample of packing 0.537 did not give fcc packing in space, but rhcp instead, whereas it gave fcc on earth. Also, samples crystallized quicker in space than on earth. For example, a glassy packing of 0.619 volume fraction remained random for a year on Earth whereas it crystallized in space in 4 d. Some of these odd effects require further investigation. One possible cause of these problems is the time required for the crystallization process, which is considered next.

5.8. COMPUTER MODEL OF THE CRYSTALLIZATION PROCESS As spheres bounce off each other without any adhesion or loss of energy, they exist most naturally in a random state, like a gas. But as the spheres are

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compacted together above a packing fraction of 0.55 they become constrained to form hexagonally packed regions. If the number of spheres is small, say 32 in Berni Alder’s early simulations, then the switch from random to structured is rapid and occurs equally for every single sphere. All the spheres cooperate in forming a single crystal structure, as shown in Fig. 5.13(a). This cannot happen for a large number of particles. With very large numbers, it is more likely that the structuring starts at several nucleation points, which leads to a polycrystalline structure in the final state, as shown in Fig. 5.13(b). This is what we see in opal, where there are many different regions of color, signifying crystallites usually a few millimetres in size, at different orientations to their neighbors, spread throughout the large gemstone. This problem has not been studied by computer modeling previously. The task we set ourselves was to increase the number of spheres and to observe these different crystalline regions forming in the zero adhesion model, while observing the structure of each crystal. Carl Stainton set up 12000 spheres in his desktop computer, started them buzzing around in a three-dimensional box with periodic boundaries, and then compacted them to 0.57 packing fraction, where structuring was then observed. The structures were picked out on the computer by their particular signatures, and then followed as they grew with time. Obviously, the growth of structure is random and is different in each simulation, depending on the position of all the spheres at the starting point. In other words, structuring is a random process. However, the general pattern of the structural growth can be observed to form some overall average conclusions about the ultimate geometry. Although the face-centered cubic crystal is the one that is most stable, having the lowest energy, it was surprising to find that this phase was not the first to be nucleated. In fact, the body-centered cubic, i.e. bcc lattice, was the one that

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tended to grow first in the simulation, closely followed by fcc and hcp. This suggests that Ostwald’s step rule is being followed. Ostwald found that the stable crystal does not emerge first in crystallization experiments. The phase closest in energy to the random phase appears first and then mutates with time into the stable crystal structure. This happens because the random phase more quickly adapts to the bcc rather than fcc even though the bcc is less stable. The bodycentred cubic structure is less well packed than fcc, having 0.68 packing fraction rather than 0.74, as is shown in Fig. 5.14. As the model was observed, small regions of bcc structure were seen appearing. These crystals were unstable and readily transformed into a faulted mixture of fcc and hcp structures by a slip of the lattice. Thus fcc regions closely followed the appearance of bcc material, with hcp lagging behind. The nuclei were not rounded or symmetrical but percolated through the surrounding random lattice. Thus, the conventional picture of spherical nuclei shown in many textbooks was proved false in these simulations. Slow conversion and growth of the nuclei occurred, but with an unpredictable path, giving differing amounts of fcc and hcp. There seemed to be no one single pathway through to the final stable fcc structure. As the crystal regions began to dominate, with the random phase disappearing, the development of fcc structure became spasmodic as sudden conversions of material took place. These jumps are seen in the simulation of 64,000 particles shown in Fig. 5.15. The final structure of the particles after a long period of computing is shown in Fig. 5.16. It was evident that the structure was dominated by fee shown as dark spheres, while most of the random material shown as light had disappeared. But there were still significant regions of bcc and hcp. The fcc was not all in the same orientation but had typically formed 3 crystallites in this size of packing. This suggests that 20 000 particles are required for each crystallite when nucleation is spontaneous and not dependent on any outside stimulation. Obviously, artificial

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nuclei could be added to promote fcc crystallization. This should lead to fewer polycrystalline regions in the final body.

5.9. EFFECT OF ADHESION ON THE STRUCTURING PROCESS The computer model described above is significant because it shows that the crystalline face-centered cubic geometry of our planet can arise from two simple rules: random movement of spheres together with compaction pressure to pack the spheres together past the 0.55 packing fraction. Thus a gas of nonadhering moving spherical molecules must structure in this way as it is compressed to a high density. Structure arises from randomness, not from solidity, and can arise in the gas phase. However, if the molecules adhere to each other, then this adhesion must pull the spheres together in addition to the compaction pressure, perhaps changing the structure. This is the question now addressed: how does the attractive potential of the atoms influence the structuring of the spheres? Alder had attempted to answer this question in the late 1950s but computing power was insufficient and he could not resolve it. But by following Alder’s lead, adopting a two-parameter model of adhesion, we have found it possible to show that adhesion has an enormous effect on the structuring process. This idea was tested by Carl Stainton in the model of 64,000 spheres. He applied the simplest possible potential model, the square well shown in Fig. 5.17, for comparison with the hard sphere case considered before. The two cases were run on the computer one after the other, with identical starting configurations, but with adhesion introduced in the second run.36 The adhesion was first inserted at a low level, less than kT, and was then gradually increased to show its effect. The results showed that a small amount of adhesion strongly influenced the crystal growth phenomenon. In the first place the nuclei became more rounded, whereas in the hard-sphere case they had been percolating. Secondly, the crystals grew at different rates in the presence of adhesion.

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A typical picture showing these two effects is given in Fig. 5.18. Adhesion of less than 1 kT had little effect on the model, but adhesion of 2 kT slowed down the crystallization, leaving a lot of random material, which is shown as light colored spheres in Fig. 5.18. It was noticable too that the nuclei were more rounded in this case, as expected, because there was now an interfacial energy between random and structured phases, arising from the adhesion force. The conclusion is that adhesion between atoms is enormously important to structure. An incredibly small energy of attraction between spheres can cause them to structure in a different way, starting from the same initial conditions. The energy of a typical chemical bond is around per atom. We are seeing here that an adhesion 100 times less than this is sufficient to cause structure change. Thus the slightest adhesion can be of great significance, much greater than previously realized.

5.10. 1. 2. 3. 4. 5. 6. 7. 8. 9.

REFERENCES

Newton, I., Opticks, Smith and Walford, London, 1704 (reprinted Dover, New York, 1952, p 395). Newton, I., Opticks, Smith and Walford, London, 1704 (reprinted Dover, New York, 1952, p 261). Israelachvili, J.N., Intermolecular and Surface Forces, Academic Press, London, 1985, p 8. Lennard-Jones, J.E., Proc. R. Soc. A 106, 441–77, 709–18 (1924). Lennard-Jones, J.E. and Taylor, P.A., Proc. R. Soc., A 109, 476–508 (1925). Lennard-Jones, J.E. and Dent, B.M., Proc. R. Soc., A 112, 230–4 (1926). Morse, P.M., Phys. Rev. 34, 57 (1929). London, F., Trans. Faraday Soc. 33, 8–26 (1937). Shriver, D.F. and Atkins, P.W., Inorganic Chemistry 3rd Edition, Oxford, Oxford University Press, 1999, p 24. 10. Israelachvili, J.N., Intermolecular and Surface Forces, Academic Press, New York, 1985, chapter 7. 11. Amdur, I. and Jordan, J.E., In: Molecular Beams (Advances in Chemical Physics Vol 10) ed. J Ross, Interscience, New York 1966, chapter 2.

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12. Bernstein, R.B. and Muckerman, J.T., In: Intermolecular Forces ed. J.O. Hirschfelder, Interscience, New York, 1967, pp 389–486. 13. MSI, Molecular Simulations Inc, USA. 14. www.Ri.ac.uk/potentials/, S Woodley, Royal Institution, London. 15. Leach, A.R., Molecular Modeling: Principles and Applications, Longmans, Harlow, 1996. 16. Gans, W., Fundamental Principles of Molecular Modeling, Plenum, New York, 1996. 17. Mandelbrot, B.B. and Evertsz, C.J.G., Nature 348, 143 (1990). 18. Alder, B.J. and Wainwright, T.E., J. Chem. Phys. 27, 1208–9 (1957). 19. Wood, W.W. and Jacobson, J . D . , J Chem. Phys. 27, 1207–8 (1957). 20. Stainton, C., PhD Thesis, University of Keele, 2000. 21. Hoover, W.G. and Ree, F.H., J. Chem. Phys. 47, 4873–8 (1967). 22. Hoover, W.G. and Ree, F.H., J. Chem. Phys. 49, 3609–17 (1968). 23. Alder, B.J., Carter, B.P. and Young, D.A., Phys. Rev. 183, 831–3 (1970). 24. Woodcock, L.V., Nature 385, 141–3 (1997). 25. Bolhuis, P.G., Frenkel, D.. Mau, S.C. and Huse, D.A., Nature 388, 235–6 (1997). 26. Bruce, A.D., Wilding. N.B. and Ackland, G.J., Phys. Rev. Lett. 79, 3002–5 (1997). 27. Speedy, R.J., J. Phys. Condens. Mater. 10, 4387–91 (1998). 28. Pusey, P.N. and van Megen, W., Nature 320, 340–2 (1986). 29. Pusey, P.N., van Megen, W., Bartlett, P., Ackerson, P.J., Rarity, A.G. and Underwood, S.M., Phys. Rev. Lett. 63,2753–6(1989). 30. Rutgers, M.A., Dunsmuir, J.H., Xue, J.Z., Russel, W.B. and Chaikin, P.M., Phys. Rev. B 93, 5043–6 (1996). 31. Elliot, M.S., Bristol, B.T.F. and Poon, W.C.K., Physica A 235, 216–23 (1997). 32. Dux, C. and Versmold, H., Phys. Rev. Lett. 78, 1811–4 (1997). 33. Miguez, H., Meseguer, F., Lopez, C., Mifsud, A. Moya, J.S. and Vazquez, L., Langmuir 13, 6009– 11 (1997). 34. Van Blaaderen, A., Science 282, 887–8 (1998). 35. Zhu, J., Min, L., Rogers, R., Meyer, W, Ottewill, R., STS-73 Space Shuttle Crew, Russel, W.B. and Chaikin, P.M., Nature 287, 883–5 (1997). 36. Stainton, C., van Swol, F., Kendall, K. and Woodcock, L.V., submitted (2001).

6 EVIDENCE FOR THE SECOND LAW OF ADHESION: CONTAMINATION REDUCES ADHESION

The Attraction of the Glass is almost balanced and rendered ineffectual by the contrary Attraction of the liquor ISAAC NEWTON, Opticks,1 p. 371

Newton found that dry glass lenses stuck together best when he “took two Object glasses,... pressed them slowly together . . . and then slowly lifted the upper Glass from the lower”.2 As soon as water was introduced, the glass lenses, which would stick together strongly in dry conditions, were less adhesive. We are very familiar with these effects; paper starts to come apart when wet, paint begins to fall off surfaces after immersion in water, and fibreglass yachts are weaker as water penetrates along the interface between the fibres and the polymer matrix. A typical test of adhesive reliability is to boil a joint in water for a couple of hours to see if it fails. It usually does. Since ancient times we have cleaned our clothing by washing in water. This releases the dirt particles more readily from the cloth fibers. Although it is possible to suck dirt particles out of cloth by vacuuming or shaking, washing is much more effective. In addition, when we add detergent, the dirt comes off even more easily. So water, and additives such as polymers which make water wetter, are very important in reducing adhesion. The problem, shown in Fig. 6.1, is that liquids also seem to act as adhesives under certain conditions. For example, wet sand sticks together better than dry sand. If we wish to agglomerate powder, we add water and stir the powder to compact it. Similarly, the best way to make an adhesive joint is to wet the 103

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adherends with a liquid adhesive, then to solidify the adhesive; the familiar thermoplastic glue method. Also, when sticking a rubber suction pad to a surface, it is best to wet it first. How can we explain these seemingly opposite effects of contamination on adhesion?

6.1. EXPERIMENTS TO SHOW THAT ADHESION IS REDUCED BY CONTAMINATION The problem of showing that adhesion is reduced by contamination is largely one of finding clean surfaces in the first place. On Earth, all surfaces rapidly become contaminated by layers of foreign molecules such as oxygen, water or oil. Thus, clean surfaces do not normally exist in our terrestrial environment. In space, on the other hand, there are fewer contaminants and so things naturally stick together better. Thus early experimenters like Bradley evacuated their apparatus to simulate space conditions. Obreimoff3 in 1930 first did this by cleaving mica under vacuum. He showed that the energy required was much greater than in air, as described in Section 4.2, obtaining values up to The cleavage energy dropped by a factor of 13 when the experiment was carried out in air. He also observed that the time needed for debonding was different in different atmospheres. For example, some days were needed at low pressure whereas only a few seconds were required in the laboratory conditions. This suggested that time was necessary for the molecules to move onto the surfaces to cause contamination, and reduce adhesion. About the same time, there was a controversy in England about the effect of liquids on adhesion of glass. Stone4 reported to the Philosophical Magazine that

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glass beads would only stick together when damp. As the beads dried, they fell apart, much as dry sandcastles weaken. This observation was not accepted by Tomlinson5 of the National Physical Laboratory who suggested that the beads were not smooth, thus confusing the results. He carried out further experiments on the quartz fibers he had used previously to demonstrate molecular adhesion in 1928,6 as described in Section 4.5. In the first experiment, Tomlinson sealed fused silica fibers in a brass cylinder, heated them to about 1000°C, and observed their adhesion as he manipulated them inside the chamber. The adhesion was substantial. To prove that dry air had no effect on the adhesion, he circulated air through sulfuric acid, calcium chloride towers, phosphorus pentoxides and roasted pumice to remove all moisture. He then found that the adhesion remained high for six weeks. But when the fibers were exposed to damp air, the adhesion was reduced to almost nothing in a few hours. His conclusion was that smooth silica adheres when dry but not much when wet. The reason for Stone,s misleading observations, that the surfaces of glass beads are rough and become filled with water, will be explained later in Chapter 7, as illustrated in Fig. 6.2. Bailey and Kay7 followed up these phenomena by showing that the work of cleavage of mica was in dry air, dropping to in hexane, at 50% relative humidity, and to in water. In other words, the presence of foreign molecules, either gas or liquid, was sufficient to reduce adhesion considerably. These experiments have been further pursued in great depth by Israelachvili and his co-workers, by taking atomically smooth flakes of mica and bringing them together in controlled conditions, as described later in Sections 6.4, 6.5 and 6.6. More recently, many adhesion experiments have been carried out in the atomic force microscope, showing most convincingly that adhesion falls when contamination is present. These are described in Section 6.7. Oxide surfaces such as mica are high energy surfaces and hence are very sensitive to contaminants. By contrast, polymer surfaces are lower in energy and show more constrained and reproducible effects. For example, the energy of adhesion of hydrocarbon polymers to surfaces is generally around too low to give the huge changes in adhesion seen with mica. However, contamination

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still reduces adhesion significantly. This effect is used in practice when molding thermoplastic objects. Mold release agents, typically stearic acid or similar molecules, are added to polymers before injection moulding to allow them to release easily from the metal tools, as shown in Fig. 6.3. The release agent adheres strongly to the metal tool, thus reducing the adhesion of the polymer, which can then be demolded with ease.8,9 The acid molecule is drawn as having two parts, the long tail hydrocarbon chain, and the adhesive head, which reacts and sticks strongly to the metal surface, to remain after the polymer is peeled off. Rubber adhesion experiments, performed in 1971,10 were the first quantitative demonstration of adhesion reduction by contaminants wetting the surfaces. These experiments are very readily reproducible and are described in detail next.

6.2. POLYMER ADHESION AFFECTED BY CONTAMINANTS A simple test of polymer adhesion is to take a table tennis bat and cover it with a smooth layer of rubber, as shown in Fig. 6.4. A ping-pong ball then adheres to the smooth surface sufficiently to support its own weight. The rubber surface is so smooth that it looks shiny and liquid-like, seeming to wet the ball as it makes molecular contact, but in reality there is no fluid present. When the rubber is immersed in water, alcohol or other liquid, the ball drops off.11 An apparatus for studying this effect in more detail was described by Roberts and Tabor12 in 1971, as shown in Fig. 6.5. They had been particularly interested in the effect of lubricant films under car tires or beneath windshield wipers. Previously, Blok had tried to study these thick contaminant layers by placing a smooth plastic sheet on the tire rubber surface.13 But Roberts showed that this was unnecessary. If the rubber was molded against an optical glass surface, then the rubber itself acted as a mirror and enabled Newton’s rings to be

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seen. The movement of the rings allowed the shape of the deforming rubber to be calculated. The rubber and glass surfaces were brought into close proximity through the water and the interference fringes were seen. Then the rubber jumped into contact with the glass, at first trapping some water islands within the contact region. But these water globules gradually disappeared as the water diffused out to the edge of the contact, suggesting that a very thin layer of water was remaining between

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rubber and glass. In particular, after equilibrium was reached, the size of the black contact spot could be measured accurately. This was seen to be much less than the size of the contact under dry conditions, proving that the surface energy of the glass and rubber in contact was reduced by the presence of water, approximately by a factor of ten. Chapter 9 shows the proportionality between work of adhesion and the cube of black spot diameter. In an experiment on equal rubber surfaces under dry conditions,7 the black spot diameter was 1.37 mm, equivalent to a work of adhesion of which is twice the surface energy of each individual surface. Thus the rubber surface energy was But when the surfaces were immersed in water, the black spot fell to 0.63 mm in diameter, suggesting a work of adhesion reduced to corresponding to a wet surface energy of Thus the presence of water at the adhesive contact diminished adhesion by a factor of nearly ten. This agreed satisfactorily with the wetting contact angle of 66° measured for water on the rubber surface, the first time that the famous Young equation for wetting of a solid by a liquid had been verified directly. However, this work was soon repeated in the USA.

6.3. STUDIES OF SURFACE SPECIES BY THE ADHESION METHOD Chaudhury and his colleagues at Dow Corning Corp and Lehigh University in Pennsylvania followed up these early results, especially looking at the surface treatment of the rubber to give a variety of surface species.14,15 The polymer selected was polydimethyl siloxane elastomer (PDMS or silicone rubber) which was made by crosslinking a liquid polymer to give a very elastic rubbery network. This is similar to the polymer once used in breast enhancement surgery. Because this polymer does not go glassy until its temperature is reduced to – 120°C, it is exceptionally resilient at room temperature and displays almost complete elastic behavior, making it ideal for measurement of polymer adhesion when contamination is present. In the first experiments, the adhesion of the PDMS rubber to itself was measured to give a work of adhesion of significantly smaller than the ordinary hydrocarbon rubbers used previously. Then a number of liquids were used to wet the rubber/rubber contact and the black spot size was measured under contaminated conditions. The results depended very much on how the contaminant liquid wetted the rubber surface, as shown in Fig. 6.6. The wetting liquid caused the black spot to decrease whereas the nonwetting liquid caused the black spot to increase in size. For example water, which does not wet PDMS rubber but has a contact angle of 102° as shown in Fig. 6.5(b), gave a work of adhesion of In this case the liquid makes the

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rubber adhere more strongly. But methanol, which wets the rubber, as in Fig. 6.5(a), gave a work of adhesion of thus reducing adhesion. We can begin to understand these phenomena in terms of work of adhesion by considering the following thought experiment,16 shown diagrammatically in Fig. 6.7. Imagine two cubes of solid which are pulled apart in a liquid. This requires a work of adhesion where the subscripts SL correspond to the solid/liquid interface. However, the process can be done another way, by first separating the blocks to give a vapor bubble, which requires a work corresponding to solid/vapor (SV) interfaces, then allowing the liquid meniscus to wet the surfaces to give fully immersed bodies. This wetting gives back energy where is the surface tension of the liquid/vapor (LV) interface. Since these two processes reach the same end-point, the two energy changes must be equal, so we can add energies, leading to the relation shown in the figure. In conclusion, the wettability is vital to the adhesion.

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When the contact angle is 90° and the liquid droplet forms a little hemisphere on the surface, then no work is done by the wetting, and the work of adhesion in liquid is the same as in vapor. But if the liquid wets the surface then adhesion must be reduced. This is equivalent to Young’s original theory of 1805, which he did not express in symbols but only in words.16 All the results above were consistent with this equation. Thus the second law of adhesion is true for wetting liquids. To check this law out more rigorously, Chaudhury and Whitesides15 made 11 different mixtures of methanol and water, ranging from 5% methanol to 100%, and used them to verify Young’s equation. First they measured the surface tension of each mixture, then they measured the contact angle of each mixture on a flat surface of the PDMS rubber. This enabled them to calculate the term cos in the equation of Fig. 6.7. Plotting this as the vertical axis in Fig. 6.8 allowed a comparison with the work of adhesion of the rubber measured in the 11 methanol/water mixtures. This graph shows the straight line plot expected from Young’s equation, together with the experimental points which fall squarely on the theoretical line. This experiment was a convincing proof that the equilibrium work of adhesion could be measured over a wide range of chemical conditions. The solid rubber was found to have a surface energy very similar to the high molecular weight liquid, suggesting that the methyl groups are sticking outwards at the surface. Then Chaudhury and Whitesides found a way of modifying their silicone rubber surface to change its chemical character. The PDMS polymer was exposed to an oxygen plasma for a short period, as shown in Fig. 6.9, creating a thin layer of silica on the surface, about 3 nm thick. By treating this silica layer with molecules of siloxane, single molecular layers, i.e. monolayers of particular structures, could be formed at the rubber surface.

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The method of applying the organic monolayer to the surface was very easy. A solution of the silane molecule was made in paraffin oil, at a concentration of between 1–2%. This was evacuated in a desiccator containing the silica-surfacedrubber samples. After a short time, the surfaces were found to be covered with a monolayer. Five long chain silane compounds were studied, as shown in Table 6.1. In each case, the chlorine atoms reacted with the hydroxyl groups on the silica to bond the long chains to the surface, as indicated in Fig. 6.9(b). These values show the remarkably sensitive changes in adhesion levels brought about by the changing chemical groups at the surface. The work of adhesion of the fluorinated surface was only half that of a typical hydrocarbon rubber. Moreover, the hysteresis exhibited by these surfaces was very small, indicating that the systems were near equilibrium, with little frictional energy loss. In addition, the results were consistent with contact angle measurements.

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The overall conclusion was that adhesion energetics in liquids could be readily understood by this technique. However, the adhesion forces varying with separation were not easy to measure by this method, because of the large deformations across the contact. Force measurement required the surface force apparatus of Israelachvili, to maintain the surfaces in rigid cylindrical form and to keep a known gap between the surfaces.

6.4. SURFACE FORCES MEASURED IN LIQUIDS The attractive forces between smooth mica surfaces immersed in liquids have been much measured over the past 30 years by Israelachvili and his colleagues. Israelachvili joined the staff at the Australian National University in Canberra during 1973 and started to do the unthinkable by carrying out experiments in the Mathematics Department. He began to work with Adams in modifying the surface force apparatus previously built in Cambridge.17 The idea was to fill it full of water and other liquids. A schematic of the equipment is shown in Fig. 6.10. Two flakes of mica, about thick, were cleaved and glued onto curved glass formers to give the crossed cylinder geometry described in Section 4.3. These mica surfaces could be brought together by a three-stage mechanism: an upper coarse screw which positioned the lower mica surface; a bottom screw which adjusted the mica to 1 nm distance, and the piezotube which controlled the movement to 0.1 nm. Light was passed through the mica via a microscope

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objective to give the multiple beam interference fringes for measuring the gap between the mica surfaces. The lower mica flake was suspended on a cantilever spring of stiffness so that the forces between the surfaces could be determined by the deflection. Special precautions had to be taken to purify the water to stop particles getting between the mica, inhibiting molecular contact. The early experiments showed that the attraction between the mica was much reduced by the presence of water. Results displayed in Fig. 6.11 show that adhesion dropped by an order of magnitude compared to previous measurements in air.18 The benefit of this smaller force was that measurements could be made at smaller gaps. The main difference between air and liquid is that other forces start to intrude, especially double-layer forces at long range and solvation forces at ranges less than a nanometer. The double-layer forces can be seen quite easily in the surface force apparatus. At pH around 6 in potassium nitrate solutions of various concentrations, the force was a repulsion which increased exponentially as the gap closed. For higher concentrations of salt, the slope increased as expected from DLVO theory (see Chapter 10). Results are shown in Fig. 6.12 for several concentrations of ranging from to The results were found to fit the DLVO theory over a wide range of separations above l0 nm, when the Hamaker constant was taken as At closer approach the repulsion increased faster. There was no clear-cut evidence of the primary minimum of attraction which should be

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expected below 5 nm when van der Waals forces should dominate. However, it is known that, with siliceous surfaces, the repulsion is continuous up to close contact, as a result of hydration layers.

6.5. JUMPING IN STEPS AS MOLECULES ARE SQUEEZED OUT The key question about adhesion of surfaces in the presence of contaminating molecules is what happens as the last few molecules are squeezed out from between the surfaces. It has been known since Langmuir’s time19 that certain contaminant molecules can stick very strongly to a surface in a monolayer and that further multilayers can build up with progressively weaker bonds as shown in Fig. 6.13. Bowden studied this for 30 years especially to find the conditions for removing the final tenacious bonded layer.20 In one of the last papers before he died, Bowden21 described experiments to clean-up surfaces of diamond, sapphire, quartz and other oxides. This was important for space exploration because the harsh clean conditions of space could cause bearings to seize up. Bowden and his

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co-workers showed that heating the surfaces in a vacuum of bar would remove much contamination. Alternatively, rubbing the surfaces together in a vacuum about 1000 times was sufficient to dislodge the molecules. Then strong adhesion and seizure could be observed. Horn and Israelachvili22 addressed this problem using smooth mica surfaces immersed in an inert liquid whose molecules were approximately spherical and of diameter 0.9 nm, octamethylcyclotetra siloxane or OMCTS. The idea was that this would not bond strongly to the mica surfaces (which anyway were contaminated with water and gas molecules), but would gradually be squeezed out of the way by the surfaces as they adhered together on close approach. The results shown in Fig. 6.14 indicate that the force of adhesion fluctuates significantly as each molecular layer is removed, revealing how the ordinary van der Waals forces are modulated by the molecular nature of the contamination. In conclusion, the surfaces cannot now jump together in one leap down the van der Waals curve; instead, the surfaces jump together in a number of steps which depend on the size of the contaminant molecules. In this case, the ultimate adhesion of the mica surfaces was weak, with an adhesion energy of This was beneficial because no damage was then seen on the mica surfaces and the experiment could be repeated time and time again. However, this low adhesion most likely was due to contaminant water molecules strongly bonded to the mica in the first mololayer. But the most significant result of this experiment, and of subsequent work by Horn and his colleagues23,24 on other solvent molecules, was the demonstration that the system could now sit stably at several different stages of adhesion, shown by the zero force points in Fig. 6.14. In other words, there is not just one single

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adhesive state between two surfaces. Several different states of adhesion can exist depending on how many layers of molecules have been squeezed out from the gap. This was a suggestion first put forward in 1973.25 Obviously, the primary and strongest adhesion will exist when the surfaces are in intimate molecular contact. One layer of foreign contaminant molecules will diminish adhesion by an order of magnitude. Further contaminant layers will diminish adhesion even more. It is clear from Fig. 6.14 that the size of the contaminant molecule is the most important factor in this argument, since the periodic jump distance was roughly equal to molecular diameter, larger molecules producing weaker adhesion. These results were confirmed on a number of other solvent systems, including benzene, cyclohexane and carbon tetrachloride, which all behave as fairly rigid spheres.26 Oscillations of attraction and repulsion were observed for up to ten molecular layers. With more flexible molecules, such as n-octane and 2,2,4-trimethyl pentane, the oscillations died faster, after about four molecular layers. Polar molecules, e.g. propylene carbonate or methanol, gave oscillations plus double-layer repulsions. But water was the most interesting solvent, which must be considered in more detail.

6.6. ADHESION WITH WATER PRESENT AT SURFACES Pashley and Israelachvili27,28 carried out a detailed study of mica surfaces approaching each other through water and dilute electrolyte solutions in an attempt to find “hydrate crystal layers.” These layers had been inferred from the structure of damp clay, which is known to swell in water and to have a distinct lubricious surface, quite different from normal oxides. In the 1930s such clay had been investigated by the X-ray diffraction method,29 which showed that the clay plates moved apart in water to distances of 0.25 and 0.55 nm, about the diameter of one or two layers of water molecules, as shown schematically in Fig. 6.15.

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Thus water appeared to be “structured” near the surface of the oxide plates. However, this structuring of water was not easy to study and became a particularly controversial subject, especially when “polywater” was suggested around 1967 by Derjaguin and his followers,29 who found that water in silica capillaries became solidified, supposedly by long-range hydration structure forces. This spurious idea, which was supported by a number of research groups,30 was ultimately shown to be related to sodium and silica impurities dissolving through solution near the glass surfaces to give a gelling phenomenon in the water. Pashley found that the structural forces only extended about 1 nm from the surface and not the micrometer distances suggested by Derjaguin. Pashley at first could not see any structuring forces in his 1981 paper using 1 M KCl solutions. However, this was because of the behavior of his spring system in bringing the mica surfaces together. Once he investigated the gaps below 1 nm more closely, especially with dilute KCl, e.g. then he found the stepwise jumping of the surfaces corresponding to removal of water molecular layers, as shown in Fig. 6.16. Similar experiments on pure silica surfaces showed the same sort of repulsion near contact but there was no sign of the stepwise jumping effect.31 The measurements of force could only be conducted in the regions shown by the black lines in Fig. 6.16. These measurements showed steep repulsions. If too great a force of compression was applied, then the mica jumped closer by removing one layer of water molecules, onto the next black line, which was again steeply repulsive. Eventually, the mica surfaces made contact to give a work of adhesion of In addition to this molecular contact state, there were two other stable points of adhesion corresponding to one and two molecular layers of water between the mica, respectively. In Fig. 6.16, these stable states are pointed

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out by arrows. These were located at gaps which agreed reasonably with the hydrated layers found in clays, around 0.25 and 0.55 nm. When the mica plates were well separated, i.e. more than 2 nm apart, the results fitted the DLVO theory of double-layer repulsions corresponding to a potential of – 78 mV, which amounts to 40% coverage of the mica with positively charged potassium ions. These long-range forces were those previously measured by Adams and Israelachvili.17,18 Thus there appeared to be three types of force acting between the mica surfaces: the long-range DLVO forces acting from 5 nm out, to be considered in more detail in Chapter 10; intermediate range repulsion acting from 2–5 nm; and finally the oscillatory jumping behavior operating from contact out to 1.7 nm gaps. These close-in oscillatory forces in water have only been detected with extended mica surfaces. They do not seem to have been observed in the atomic force microscope studies which we consider next. 6.7. ADHESION OF WET SURFACES IN THE ATOMIC FORCE MICROSCOPE The presence of water at the contact between probe and substrate in the atomic force microscope (AFM) makes life easier by reducing adhesive force and preventing the vicious jumping together of the surfaces. This allows simple measurement of adhesion, together with the prospect of atomic resolution of the surfaces. One particular reason for studying surfaces under water is to observe the growth and dissolution of the crystal defects and growth spirals on the surface. This has been much addressed by Hansma and his associates at Santa Barbara who looked especially at calcite surfaces.32,33 They attached a fluid flow cell to the AFM as shown in Fig. 6.17 and then placed freshly cleaved “Iceland Spar,” pure calcium carbonate, under the tip.

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The nature of the surface was observed as liquid was circulated continuously through the cell. The flow velocity had no effect on crystal growth, confirming that the growth was not diffusion limited at the flow rate employed. The crystal plane observed was the (1014) face which does not cross any of the strong covalent bonds, i.e. the C–O bonds, in the material. The images revealed the flat terraces of the crystal with spiral step patterns standing out from the surface, as shown below in Fig. 6.18. The terraces were flat at the molecular scale and growth occurred by movement of the steps outwards across the surface as described originally in the Burton, Cabrera and Frank model.34 The start of the spiral was seen to be a small pit in the surface, which was generally a screw dislocation. The spirals rotated as the steps grew and this rotation rate governed the growth rate. Measurements could readily be made of the step movement with time, Fig. 6.19(a) and of the effect of the supersaturation of the solution, Fig. 6.19(b). Typically the steps were monomolecular, i.e. 0.3 nm, and grew linearly with time at a few nm per second. The speed was independent of step spacing, showing that the adsorption on the terraces was not very influential, with a diffusion length < 2.5 nm. As the supersaturation increased, the step velocity rose rapidly, in proportion to the concentration. Dissolution of the spirals was also seen as the solution concentration was lowered. If the steps of the spiral were curved, they straightened as dissolution took place. This showed that the step velocity depended on orientation, with the slowest speed dominating. Any monomolecular islands on the surface disappeared very quickly, suggesting that growth on the terraces was not favorable without steps.

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In this wet calcite system, it was also possible to obtain atomic resolution using the AFM.33 This observation was followed up in detail by Ohnesorge and Binnig in 1993.35 They showed that the force on the probe tip was important if such atomic resolution was to be achieved. If the force exceeded 0.1 nN, then the tip would wipe away the steps and leave a perfectly flat and ordered crystal. To show the effect of the force on the atomic imaging, they moved the tip very sensitively towards the surface and produced scans showing the surface atoms as the tip position was changed above the surface. The first force could be detected at a separation of 1.2 nm between tip and crystal surface, when the cantilever force constant was A sharp tip of diameter 20 nm was used, giving a typical force curve as shown in Fig. 6.20. The zero point was chosen as the switchover from attraction to repulsion. Approach was seen to be slightly different from retraction, as shown by the

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broken line, but there was no jumping instability and little hysteretic behavior. The force sensitivity of the cantilever was around 1 pN. This meant that the tip could be positioned to about 0.3 nm, where the maximum van der Waals attraction was 45 pN as expected between two single atoms at equilibrium separation. Therefore, only one or two atoms could have contributed to this force curve. This allowed the atomic resolution which was seen on scanning the tip across the sample. When the gap z was initially 1 nm, the scanning showed dark spots corresponding to the attractions of oxygen atoms protruding slightly from the crystal plane. These spots corresponded to the known positions of oxygen atoms in the crystal structure of calcium carbonate, shown in Fig. 6.21. As the tip was scanned nearer to the surface, gradually closing the gap, the oxygen atoms appeared as larger rounded blobs, which eventually lost contrast as the tip began to pick up forces from the other atoms. The initial contrast was as good as that obtained by scanning tunneling microscopy. The conclusion was that adhesion of wet calcite was small and well controlled, such that reversible adhesion curves could be obtained with a sharp AFM probe tip, giving atomic resolution of the surface oxygen atoms.

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6.8. INFLUENCE OF POLYMERS ON WET ADHESION Polymers in solution have an enormous effect on the adhesion between surfaces. Such polymers are used as lubricants, as thickeners, as colloid stabilisers, as binders, glues, and inks. Also they are ubiquitous in biological systems. Their practical significance is large but understanding their effects remains to be explored to a great extent. This section describes some experimental observations of the adhesion forces and draws a schematic theoretical picture of the effects. There are three kinds of polymer to be found dispersed in liquids; the first is the insoluble polymer colloid found in a latex dispersion, for example polyvinyl acetate used in latex glues, Fig. 6.22(a). This is essentially a suspension of plastic beads of very small size. If the polymer is slightly soluble, then the solvent penetrates into the polymer beads and swells them up, as shown in Fig. 6.22(b), so that the polymer chains begin to interact slightly with neighbouring chains. For very soluble polymers, the swelling is complete, as in Fig. 6.22(c), such that the chains intermingle fully and the solution thickens or gels. Polymers can be converted from highly soluble to slightly soluble by adding salt, by increasing polymer concentration, or by changing the temperature such that the solution “clouds.” This means that the polymer chains are balling up and expelling solvent. This occurs at the theta temperature. Polymers are therefore molecules whose size can change enormously as this swelling or shrinking occurs. The insoluble polymer bead may be l0 nm in diameter, but on swelling it can grow to 30 nm, while on full dissolution it can extend to 100 nm. This means that when the polymer molecules sit on a surface, the adhesive force can also vary substantially. An example of this effect was described by Klein36 who, in 1980, used the Israelachvili apparatus to measure polystyrene molecules at mica surfaces. At a temperature of 24° C where the polymer was not very soluble, the results shown in Fig. 6.23 were seen.

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The high molecular weight polystyrene started to show an attraction between the surfaces just above one molecular diameter. As the molecules were compressed together a repulsion was then observed at a gap of about 20 nm. The lower molecular weight material behaved in somewhat similar fashion but with a repulsion at 8 nm gap. When the temperature was raised to 35°C, to give improved solvation of the polystyrene, the attraction was reduced by half. The results can be understood in terms of the forces between rubbery swollen polystyrene. As the layers approach, there is a van der Waals attraction, but the adsorbed particles then press into each other to give the repulsion (Fig. 6.24). The polymer is too strongly adherent for the adsorbed layer to be squeezed out by the compressive force. A key feature of long-chain polymer adsorption is the length of time needed to reach equilibrium. This was shown by Luckham and colleagues37 by studying the force of adhesion with time as the mica surfaces sat in a high molecular weight (1,120,000) polymer solution, polyethylene oxide in water. After 2–3 h in the solution, the mica showed a clear attraction, starting at a gap of l00 nm and

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reaching a maximum adhesion at 50 nm gap, as shown in Fig. 6.25. Smaller gaps gave a strong repulsion. After a longer period in the polymer solution, 6–8 h, the attraction disappeared and only repulsion was seen. After 48 h, the repulsion was strong and started at a gap of 200 nm. It was evident that equilibriation in the presence of a long-chain polymer was a lengthy business. A more practical way to study the interactions between oxide surfaces in polymer solutions is to use the Atomic Force Microscope (AFM). Milling, with his colleagues, has performed a series of experiments using a diameter silica sphere glued to the silicon nitride AFM cantilever, gradually brought towards a flat silica surface in solutions of polyelectrolyte, for example sodium polystyrene sulfonate or sodium polyacrylate. Well defined polymer samples were used to make sure all the molecules were of similar length, and the effects of polymer concentration and salt content were investigated.38–40 There was little hysteresis in these measurements under dilute conditions. A Nanoscope AFM from Digital Instruments was used because of its convenient wet cell arrangement. The spring constant of the cantilever, was measured by the resonance method as different known masses were added. The silica was cleaned by brief boiling in ammoniacal hydrogen peroxide solution, followed by washing in ultrapure water. Several molecular weights of sodium polyacrylate were used from 33 000 to 99 000. As expected, the higher molecular weight gave similar behaviour but at somewhat longer length scales. At a low polymer concentration of 39 ppm, there was an attraction with a minimum around separation, together with a repulsion at smaller gaps. However, at high concentrations, the minimum collapsed to a gap near with a much stronger attraction, followed by oscillatory behavior at larger separations (See Fig. 6.26). This oscillating behavior was very reminiscent of that observed with structured hydration layers described in Sections 6.5 and 6.6. It seemed like the polymer molecules were becoming compressed as the

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concentration was increased (Fig. 6.27), and the oscillations represented the force required to squeeze each layer of molecules out from the gap. Putting electrolyte into the polymer solution had the same effect of collapsing the polymer molecules, again showing the oscillatory behavior. The theory of these effects has been considered by a number of authors. Early work41,42 suggested that rigid rod-like molecules could be excluded from between charged surfaces. More recently, a number of theorists have suggested structuring mechanisms in polymer solutions to explain the oscillatory forces.43,44 None of these seems entirely adequate to describe polyelectrolytes though some agreement has been reached on uncharged, nonadsorbing polymer solutions.45

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In conclusion, it is evident that the interface can easily be restructured by polymer molecules adhering to it in a rich variety of ways. This restructuring of surfaces is a general phenomenon which must be discussed next.

6.9. RESTRUCTURING OF SURFACES AND INTERFACES We are rather familiar with the effects that small amounts of additives can have on surfaces. For example, one part per million of carbon monoxide in a gas stream can poison a platinum catalyst surface and inhibit oxidation reactions. Similarly, five parts per million of potassium ferricyanide can stop the crystallization of sodium chloride from brine solution. The example given in the previous section is another illustration of such surface effects; a few parts per million of sodium polyacrylate can produce a repulsion between oxide surfaces, thus inhibiting adhesion. In adhesion studies, the first demonstration of this effect was by Lord Rayleigh, who showed in 1899 that a single layer of fatty acid at a surface could reduce surface tension, adhesion, and friction.46 Since that time there has been enormous effort to study the structure of surfaces and interfaces to understand these substantial phenomena. It has become clear over the past 50 years that clean solid surfaces have a different structure from the bulk, and that this difference in structure must depend on any impurities or adhering bodies brought down onto the surface. Several examples are described in Somorjai’s book.47 Consider the surface of a solid shown schematically in Fig. 6.28. There are two simple reconstructions that can take place: the first is a simple shrinking of the clean surface atoms into the bulk, as they are pulled in by the unbalanced atomic forces, rather like the surface tension effect in a liquid; the second is a reordering that can occur if the surface atoms can move from side to side, as in the (100) crystal face of silicon. Here the surface atoms are loosely packed and move both down and sideways in the two top layers, as shown in Fig. 6.28(b). So, clean surfaces tend to restructure to satisfy the unbalanced atomic forces. When foreign atoms adsorb onto such surfaces, further reconstruction is possible. For example, the chemisorption of contaminant atoms can destroy the clean surface reconstructions described above. Alternatively, new structures may form, as when carbon is chemisorbed on nickel (100) surfaces.47 If such carbon-coated surfaces were brought together in an adhesion experiment, the carbon would have to diffuse out before full Ni–Ni adhesion could be attained. Such diffusion and restructuring effects could explain the observed changes of adhesion with time. Also, hysteresis in adhesion values could then be accounted for. The importance of these interface structuring phenomena stems from the large number of possible adhesion states that could exist as different molecules are arranged at the surfaces. Little is known about the possible triggering or

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switching mechanisms that could arise from these structures. Two simple examples are shown in Fig. 6.29. In the first example, surfactant molecules are placed on a surface but at low concentration. The surfactant is van der Waals bonded and so sits flat on the surface. However, when the concentration of the molecules is increased, the molecules are forced to stand upright and so reduce the adhesion significantly by separating the surfaces by the extra chain length. In the second example, reported by Joyce and colleagues,48 molecules were adsorbed onto a gold surface. As the

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molecular layer was probed with the AFM, there was a gradual change in the apparent length of the molecule. This was interpreted as a restructuring of the surfactant at the surface. Such effects may allow us to control adhesion at interfaces artificially in future, by writing controlled structures.

6.10. THE NANOSCALE FOUNTAIN PEN Writing or painting is one of the oldest adhesive technologies known to man, dating back 4000 years.49 It involves dipping a stick or brush in an ink, then transferring the ink to paper by capillary forces. Until recently, the finest writing tips and nibs have been hundreds of micrometers in diameter. But now a method for writing with nano-tips has been developed, enabling lines to be drawn at the 30 nanometre scale.50,51 Thus it becomes possible to restructure real surfaces artificially at the molecular level. The principle is shown in Fig. 6.30, illustrating how the invention arose from the perennial problem of capillary condensation in the fine gap between tip and surface. An atomic force microscope (AFM) tip is positioned close to a flat gold substrate. If the air is damp, then moisture appears in the very fine capillary gap and this strongly influences the probe, causing imaging problems.52 But this water can also be useful in allowing organic molecules to move from the tip onto the substrate. In a typical experiment,53 a silicon nitride tip was dipped into a solution of 1-octadecanethiol in acetonitrile for 1 min, followed by blow drying with difluoroethane. The tip was then brought down onto the substrate at 45% relative humidity, where a meniscus is known to form, and left in contact for 2 min. The substrate was made by evaporating 30 nm of polycrystalline gold onto a cleaved mica sheet. After removing the tip, the image could be scanned by lateral force microscopy, detecting the change in frictional conditions on the substrate where

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the organic molecules were deposited. It was evident that some of the octadecanethiol had migrated onto the gold, forming a disc about 400 nm in diameter, as shown in Fig. 6.3l (a). At longer times, the discs got bigger as more molecules transferred. By moving the tip along the surface, a line 100 nm wide was formed, allowing a grid to be traced out, Fig. 6.31(b). To obtain good shapes it was important for the substrate to be as smooth as possible. Therefore, the gold film was annealed for 3 h at 300°C to give (111) planes of gold on the mica substrate. The molecules of octadecanethiol were then found to adhere strongly to this substrate in an ordered monolayer with spacing of 0.5 nm, in good agreement with previous studies.53 The ink was then stable on the surface and was not damaged as the lateral force probe moved across it to measure the pattern. The finest line width achieved was 15 nm from a l0 nm probe tip. In a further development of this novel writing process, Mirkin and his colleagues went on to produce a “two color” writing process at the nanometer level.54 This process depended on the fact that the layer of molecules adhered strongly to the gold and was not washed off by a second pass of the probe with its water meniscus. 16-mercaptohexadecanoic acid was chosen as the first ink. This was drawn into polygon shapes on a gold surface, and could be seen as a high friction track by lateral force AFM. The probe was then changed to an octadecanethiol coated tip and this was used to overwrite the first pattern in a square raster. Finally the surface was scanned to show that the two patterns had not interacted during the writing, as indicated in Fig. 6.32. The mercaptohexadecanoic acid gave a white contrast due to high friction, showing the polygon shapes, whereas the octadecanethiol gave a low friction shown as a dark square background. Gold itself gave intermediate friction. The conclusion was that nanostructures could be produced by AFM probes writing

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with molecular inks, to form ordered monolayer patterns. These could be used to control adhesion very finely.

6.11. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

Newton, I., Opticks, Smith and Walford, London 1704 (reprinted Dover, New York, 1952, p 371). Newton, I., Opticks, Smith and Walford, London 1704 (reprinted Dover, New York, 1952, p 197). Obreimoff, J.W., Proc. R. Soc. A 127, 290–97 (1930). Stone, W., Phil. Mag. 9, 610–14 (1930). Tomlinson, G.A., Phil. Mag. 10, 541–4 (1930). Tomlinson, G.A., Phil. Mag. 6, 695–8 (1928). Bailey, A.I. and Kay, S.M., Proc. R. Soc. A 301, 47–56 (1967). Owen, M.J. and Jones, D., In: Polymer Materials Encyclopaedia, ed. J.C. Salamone, CRC Press, Boca Raton, FL 1996, p 7688. Release agents In: Encyclopaedia of Polymer Science and Engineering, 2nd Edn, Wiley Interscience, New York, 1988, vol 14, p 4 1 1 . Johnson, K.L., Kendall, K. and Roberts, A.D., Proc. R. Soc. A 324, 301–313 (1971). Kendall, K., Contemp. Phys. 21, 277–97 (1980). Roberts, A.D. and Tabor, D., Proc. R. Soc. A 325, 323–45 (1971). Blok, H. and Koens, H.J., Proc. Inst. Mech. Eng. 180, 221 (1965). Chaudhury, M.K. and Whitesides, G.M., Science 255, 1230–2 (1993). Chaudhury, M.K., and Whitesides, G.M., Langmuir 7, 1013–25 (1991). Kendall, K., Sci. Prog. Oxf. 72, 155–71 (1988). See also Young, T., Phil. Trans. R. Soc 95, 65–87 (1805). Israelachvili, J.N. and Adams, G.E., Nature 262, 774–6 (1976). Israelachvili, J.N. and Adams, G.E., J. Chem. Soc. Faraday Trans. 74, 975–1001 (1978). Langmuir, I., Science 88, 430–2 (1938). Bowden, F.P. and Hughes, T.P., Proc. R. Soc. A 172, 263–9 (1939). Bowden, F.P. and Hanwell, A.E., Proc. R. Soc. A 295, 233–43 (1966). Horn, R.G. and Israelachvili, J.N., J. Chem. Phys. 75, 1400–11 (1981). Christenson, H.K. and Horn, R.G., Chem. Phys. Lett. 98, 45–8 (1983). Horn, R.G., Biochim. Biophys. Acta 778, 224–8 (1984). Kendall, K., J Adhesion 5, 179–202 (1973). Israelachvili, J.N., Intermolecular and Surface Forces, Academic Press, London 1985 pp 198–201. Pashley, R.M. and Israelachvili, J.N., J. Colloid Interface Sci. 101, 511–23 (1984).

CONTAMINATION REDUCES ADHESION 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.

48. 49. 50. 51. 52. 53. 54.

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Israelachvili, J.N. and Pashley, R.M., Nature 306, 249–50 (1983). Derjaguin, B.V and Churaev, N.V., Nature 244, 430–1 (1973). Lippincott, E.R., Stromberg, P.R., Grant, W.H. and Cessac, G.L., Science 164, 1482–7 (1969). Grabbe, A. and Horn, G.R., J. Colloid Interface Sci. 157, 375–83 (1993). Gratz, A.J., Hillner, P.E. and Hansma, P.K., Geochim. Cosmochim. Acta 57, 491–5 (1993). Hillner, P.E., Manne, S., Gratz, A.J. and Hansma, P.K., Ultramicroscopy 42–44, 1387–93 (1992). Burton, W.K., Cabrera, N. and Frank, F.C., Phil. Trans. R. Soc. A 243, 299–358 (1951). Ohnesorge, F. and Binnig, G., Science 260, 1451–6 (1993). Klein, J., Nature 288, 248–250 (1982). Luckham, P.F. and Klein, J., J. Chem. Soc. Faraday Trans. 86, 1363–8 (1990). Milling, A.J., J. Phys. Chem. 100, 8986–93 (1996). Milling, A.J. and Vincent, B., J. Chem. Soc. Faraday Trans. 93, 3179–83 (1997). Milling, A.J. and Kendall, K., in press (2000). Hoagland, D.A., Macromolecules 23, 2781 (1990). Scheujens, J.M.H.M. and Fleer, G.J., J. Phys. Chem. 84, 178 (1980). Chattellier, X. and Joanny, J-F., J. Physique II 6, 1990 (1996). Dahlgren, M.A.G. and Leermakers, F.A.M., Langmuir 11, 2996 (1995). Fleer, G.J., Scheutjens, J.M.H.M., Cohen-Stuart, M.A., Cosgrove, T. and Vincent, B., Polymers at Interfaces, Chapman and Hall, London (1993). Strutt, W. (Lord Rayleigh), Phil. Mag. 48, 331–5 (1899). Somorjai, G.A., Introduction to Surface Chemistry and Catalysis, John Wiley, New York, 1994, chapters 2 and 6. Joyce, S.A., Thomas, R.C., Houston, J.F., Michalske, T.A. and Crooks, R.M., Phys. Rev. Lett. 68, 2790 (1992). Ewing, A.C., The Fountain Pen: A Collectors Companion, Running Press, Philadelphia, PA 1997. Hong, S., Zhu, J. and Mirkin, C.A., Science 286, 523–5 (1999). Jackman, R.J., Wilbur, J.L. and Whitesides, G.M., Science 269, 664–5 (1995). Binggeli, M. and Mate, C.M., Appl. Phys. Lett. 65, 415 (1994). Piner, R.D., Zhu, J., Xu, F., Hong, S. and Mirkin, C.A., Science 283, 661–6 (1999). Alves, C.A., Smith, E.L. and Porter, M.D., J. Am. Chem. Soc. 114, 1222 (1992).

7 INFLUENCE OF THE ADHESION AND FRACTURE MECHANISM: THE THIRD LAW

The Attractions of Gravity, Magnetism and Electricity reach to very sensible distances and so have been observed by vulgar Eyes, and there may be others which reach to so small distances as hitherto escape Observation ISAAC NEWTON, Opticks,1 p. 376

We have seen in the previous chapters that adhesion phenomena can be explained generally by two fundamental laws: the first is that all objects jump together as a result of universal atomic and molecular attractions; the second is the effect of intervening contaminant atoms which reduce the molecular attraction, often causing ‘jumping in steps’ as each contaminating molecular layer is squeezed out. However, there is still a significant problem of understanding adhesion effects, because the adhesive forces are so short-range, as Newton recognized above, that the two basic laws of adhesion may be obscured by other mechanisms such as Brownian motion, cracking, elastic deformation, roughness, energy dissipation, etc. which can alter the adhesion by many orders of magnitude. For example, a surface roughness of only ten nanometers may be sufficient to stop bodies adhering. Such roughness has no effect on gravitational, magnetic, or electrostatic attraction; indeed, such surfaces look highly smooth and polished to the unaided eye. The purpose of this chapter is to describe some of these adhesion mechanisms, especially starting with the Brownian and cracking mechanisms, then leading on to a whole range of subsidiary mechanisms which make adhesion 133

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science so interesting. The conclusion is that the mechanism can alter the measured adhesion value by an enormous amount, even though the atomic or molecular adhesion remains the same. This is the third law of adhesion.

7.1. PROBLEM OF THE WIDE RANGE OF ADHESION ENERGY VALUES The fundamental problem of adhesion measurement is that the measured energies required to break interfaces apart are too wide-ranging when compared to theory. If adhesion energy R is defined as the experimental energy measured to break 1 of interface, then measurements show that this adhesion energy can range from negative values, where joints fail spontaneously when immersed in liquid, to very small values when colloidal particles remain separate and stable for long periods, to the very large values of needed in engineering adhesive joints which hold aircraft together. The values of the theoretical molecular bond energies, i.e. equilibrium work of adhesion W, occupy only a small range from and cannot possibly explain the full scale of measured adhesion. These values are plotted in Fig. 7.1, and we can see immediately that we need two logarithmic scales to describe the results, one for attractions and another for repulsion. How can this extraordinary range of values be explained?2,3 It is evident from this diagram that the adhesive world would be very boring if simple atomic or molecular bonding under clean conditions were the only thing that could happen. The fact is that simple, clean chemical bonding can only explain the range of adhesion energies from around Contamination of the surfaces by foreign molecules can then explain reductions in adhesion as shown in Chapter 6. Further contamination can also be the source of negative, in other words repulsive, adhesion which pushes joints apart spontaneously, again described in Chapter 6 and also later in Chapter 10. To explain the wide adhesion variations shown in Fig. 7.1 we have to introduce some other mechanisms, giving us the third law of adhesion. Some of these are very obvious and have been mentioned earlier, for example, roughness

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which creates gaps between atoms on the surface, thereby reducing adhesion immensely; cracking which allows joints to break easily; and elastic deformations which cause atoms to be pushed closer to each other as forces are applied during compaction, thus increasing atomic contact and adhesion. However, there is a severe problem explaining the high values of adhesion which are necessary for engineering applications. These high values are described in more detail in Chapters 13–16. We have to introduce some amplification mechanisms if we are to explain such values rationally. These amplification mechanisms can operate at the molecular scale or at the macroscopic level. It is the variety of these subtle mechanisms which makes the subject of adhesion so fascinating, both to chemists who wish to consider molecules, and to engineers who think in terms of continuum mechanics. Thus the third law of adhesion, that mechanisms can change the measured adhesion enormously, even though molecular adhesion remains constant, is a most significant and ill-understood law.

7.2. HIERARCHY OF MECHANISMS CONTROLLING ADHESION To obtain a convincing picture of adhesion phenomena, it is important to construct a hierarchy of mechanisms, with the most fundamental effects at the top, leading to the additional mechanisms below. This is shown in Fig. 7.2. In this diagram, it becomes easier to identify the fallacious arguments which are often produced to explain adhesion effects. For example, chemists are prone to explain joint strengths in terms of strong chemical bonding and are puzzled by the difficulties encountered in sticking strongly bonded substances like diamond together. The problem with strong covalent bonding is that the joint is brittle and fails readily by cracking, thus making it appear fragile. By contrast, weak bonding of chewing gum to a carpet can lead to persistent and tough adhesion because of

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the large plastic deformation needed to stretch the viscoelastic gum. So strong or weak bonding in themselves cannot explain adhesion phenomena satisfactorily; we need to invoke the hierarchy of mechanisms. At the top of the hierarchy is the most basic cause of adhesion, i.e., the energy required to break molecular bonds. This can be overcome most simply by heating the joint to melt or vaporize the bonds; or the energy can be provided by dissolving the bond chemically; or it can be done by applying a force to break the bond mechanically. In second place in the hierarchy is the effect of contamination in reducing adhesion energy. As shown in Chapter 6, contamination of the surfaces by wetting liquids and other adhering molecules must reduce adhesion. If wetting is complete, then repulsive forces may be observed. Third place down the hierarchy belongs to the well-known mechanisms of Brownian debonding and cracking failure. Lower still are elastic deformation and roughness effects. These are the mechanisms to be considered in this chapter. All these illustrate the third law, that the mechanism can strongly influence adhesion, while keeping the molecular situation constant. Finally we come to the more complex mechanisms which will be considered in Chapter 8, the inelastic effects, the nonequilibrium phenomena, and plasticity of surfaces leading to observations of stringing, aggregation, etc. Clearly, these mechanisms and sub-mechanisms can then be further combined to give even greater complexity and richness of effects. First, let us consider the basic mechanisms and work our way down this hierarchy.

7.3. THE SIMPLEST FAILURE OF ADHESION Consider the simple melting failure of an adhesive joint as shown in Fig. 7.3. The joint was made by taking a plate of smooth glass, placing a large drop of water upon it, then laying a strip of polyethylene polymer onto the surface of the

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water. The contact angle of water on the polymer surface was about 90° so that no adhesive capillary forces were observed. The joint was then reduced in temperature so that the water froze. The polythene adhered to the ice below the freezing point, giving a peeling force of about 100 mN for a strip width of 10 mm. However, above the melting point, the polymer was released from the water easily with little resistance, about 1 mN. This idea, that a liquid can be frozen onto a solid surface to produce a useful adhesive bonding, is one of the most powerful in adhesion technology; it is the thermoplastic adhesive method. It is worth considering the mechanism of this change in adhesion as the joint melts. The magnified view of the breaking point is shown in Fig. 7.4. When the water is liquid, the contact angle is almost 90° and as the plastic film is pulled, the water rolls back to leave a clean polymer surface. This process has very little viscous resistance and so the force required to pull the film from the liquid is all being used to create a new water surface. In other words the force is equivalent to the surface energy (i.e. surface tension) of water which is giving 0.72 mN for a 10 mm wide strip, a very small force. When the water is frozen, this process is not possible, and now the adhesion fails by propagation of a crack which can be seen moving to separate the ice and the plastic film. There is now quite a bit of resistance to this crack propagation and this is the cause of the increase in force. The energy changes which apply in this case are somewhat different and are treated later in Section 7.7. This process of freezing and melting the joint was quite reversible. It was obvious that mechanical forces did not play any part in destroying the adhesion between the pieces as the joint was melted. Clearly, therefore, mechanical stresses are not the most fundamental cause of adhesion failure in this instance. Adhesive joints fail when the molecular bonding is reduced by melting.

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7.4. CHEMICAL BREAKAGE OF ADHESIVE JOINTS In another simple test of adhesion shown in Fig. 7.4, a thin film of epoxy resin was polymerized onto a glass surface, and the adhesion was good. The film required considerable force to wedge it off the glass surface. However, when water was placed at the contact edge between epoxy and glass, the adhesion between epoxy and glass was soon reduced to a very low value, such that the epoxy film floated away from the glass surface. Observation in the optical microscope showed that a film of water intruded at the interface between epoxy and glass, moving inwards from the outside edge, and gradually lifting off the polymer. Essentially, the adhesive bond between epoxy and glass had been dissolved by the water without the application of any external force. Again, it is evident that mechanical force is not always the main cause of breakage for adhesive joints. In this case, the joint is separated by a chemical process. In terms of energies, this is quite easy to understand. The work of adhesion of the fluid epoxy to the glass under “dry” conditions is approximately This is large enough for the liquid epoxy to wet the glass when forming the joint to make molecular contact. Then the polymerization process solidifies the resin, thus forming a solid bond with the glass. Of course, the glass surface at the molecular level is not truly dry because it has reacted with any water vapor around in the atmosphere to form a thin surface layer of hydroxyl groups. However, when liquid water is allowed near the glass surface, the water molecules strongly wet the glass to form a thick layer of water, as described in Section 6.6, and this produces repulsive forces which push the epoxy off the glass. The work of adhesion between epoxy and glass is negative under these conditions and so the adhesive bond falls apart spontaneously. Expressed in terms of the equation given in Fig. 6.7,

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where we now distinguish the wetting angles of the two different surfaces of the joint. The work of adhesion of the dry joint, is now exceeded by the wetting energy of the water on the glass, and so the wet joint has a negative energy, and so must come apart. This shows that adhesive joints must be treated with great caution where contamination with wetting molecules is possible, because such contaminants can essentially dissolve the adhesive bond. In practice, treatments of the glass to make it less reactive to water are applied and these can then give good moisture resistance, as described in Section 16.10. From these arguments, we see that adhesion is essentially a chemical process where the bonds can be broken by heating or chemical attack. The bonds pull themselves apart by Brownian motion, without the need for mechanical force.

7.5. BROWNIAN ADHESION The idea that molecules are involved in the adhesion process tells us that Brownian movement must be important. All molecules are in random motion. In fluids, the molecules wander around at great speeds in all directions, whereas in solids, the motion is tethered but the localized vibrations are still important. This means that an adhesive contact must not be viewed as a statical concept but as a dynamic one. Unfortunately, most books on adhesion fail to mention Brownian movement.4 A useful exercise is to compare the situation in which a liquid drop sits on a polymer surface with that of a smooth rubber sphere adhering to a glass plate, as in Fig. 7.6. Figure 7.6(a) shows a liquid drop which, to the naked eye, looks static and in a stable wetting equilibrium. However, microscopic examination of the liquid

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near the contact, where we can insert some tiny latex spheres as markers, reveals a seething motion of the liquid molecules as they evaporate, diffuse across the surface, and leap back into the droplet, moving randomly in all directions. The situation in the rubber sphere, Fig. 7.6(b) is similar in that the rubber molecules are all waving around in continuous motion. However, the molecules in this rubber case are linked together to form an elastic body rather than a liquid one. But each rubber molecule, as depicted in Fig. 7.6(c), can continually make and break contact with the glass. So, although the contact appears to be constant in size, it is actually expanding and contracting very rapidly, attaining a thermal equilibrium in which the rate of expansion of the contact is equal to the rate of contraction. This is the equilibrium model of contact which we established in Sections 3.3 and 3.4. Of course, for this to be valid, the adhesion bond energies of the molecules have to be around kT, in other words rather weakly adhering. Brownian adhesion occurs at very small dimensions, from 0.1–1000 nm, where thermal diffusion is dominant. Because of the small sizes, Brownian adhesion is difficult to observe experimentally, but it can be seen with small dispersed particles such as colloids and blood cells, for example (see Chapters 10 and 12). One result from this theory is that fine particles will stick together to form doublets whose number depends on the interparticle adhesion.5,6 Consider a number of fine particles immersed in dispersing fluid and undergoing Brownian motion in a box as in Fig. 7.7(a). With zero adhesion, no doublets should form. However, adhesion causes doublets to develop, as in Fig. 7.7. The number of doublets divided by the total number of particles is related to the adhesion in a definite way, according to statistical mechanics.7 In a dilute system, the ratio of doublets to singlets can be calculated if simplifying

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assumptions are made about the two-parameter model of adhesion forces, as defined in Chapter 5. This theory gives a straight line dependence of doublets on packing fraction of particles, as in Fig. 7.7(b). As the adhesion increases, more doublets must be observed. This fits experimental results.5 One conclusion of this theory is that adhesive bonds between bodies are not permanent but are random, depending on parameters such as adhesion energy, range of forces, temperature, and concentration. You do not need to pull things apart to test their adhesion. In the Brownian example, adhesive bonds are being made continually, and at equilibrium they are being broken equally fast. By altering the conditions we merely push the equilibrium one way or the other. For example, more particles could be made to adhere by putting more particles into the box, thus increasing the concentration. Conversely, the doublets could also be fractured by taking particles out of the box. This dilution must break doublets apart, as shown in Fig. 7.7(b).

7.6. THE CRACKING MECHANISM When we first think of adhesion experiments, we imagine that we can carry out the sort of tests shown schematically in physical chemistry textbooks, as shown in Fig. 7.8(a), in order to measure the surface forces as the bodies approach and begin to adhere. This was the experiment of Tabor and Winterton, and of Israelachvili.8,9 In fact, they could only obtain gaps down to around 20 nm by this means. Remembering that 99% of adhesion energy is below 1 nm, we know that full molecular adhesion was not being measured in those tests. In practice, this idealized experiment is impossible because, when the surfaces get close together, the adhesion force increases very rapidly and an instability occurs such that the surfaces jump into contact. Essentially, the adhesion force is so strong that it overcomes the elastic resistance of the materials. So the situation shown in Fig. 7.8(b) cannot exist; instead, the system goes to the position shown in Fig. 7.8(c), however much we try to control the positions of the surfaces. This is the crack geometry. Molecular contact is made over part of the surface, and there is no contact over the rest of the bodies.

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This elastic cracking mechanism is very common in adhesion studies, except when adhesion is much weakened by contamination. It allows us to make two excellent advances: the first is the experimental one of seeing the adhesion experimentally by optical interference; the second is the theoretical approximation that adhesion is perfect on one side of the crack line and zero on the other. This is the fundamental premise of fracture mechanics explained in the next section, allowing the use of a one parameter model. Almost all our knowledge of adhesion, in the days before atomic force microscopes, has been derived from the optical study of cracks at adhesive interfaces. From Newton onwards, through Obreimoff, Tabor and his followers, the direct viewing of the contact spot and its surrounding crack has been the key to adhesion measurement. Figure 7.9 shows the basic measurement technique. The upper part shows the light rays being reflected from the crack surfaces as first explained by Newton, while the lower part shows the appearance looking down at the crack front. Cracks have been seen in this way from ancient times as beautiful colored regions inside glass, for example, by Vitruvius. 10 The reason for the visibility of cracks is seen in Fig. 7.9. A light source shining on an adhesive joint is not much reflected at the adhesive interface, because the difference in refractive index at the interface is not large. Therefore, in reflection the adhered interface looks dark because almost no light is reflected

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from it. However, where the crack has penetrated the interface, there is now a small air gap near the crack tip and this causes strong reflections from the crack faces, because the difference in refractive index between air and solid is large. Therefore the crack looks like a bright region intruding into the dark adhesive contact zone. Moreover, because of the narrow gap near the crack tip, interference fringes can be seen as the two reflected beams, one from the top surface, the other from the bottom surface of the crack, interfere. This gives a series of colored bands if white light is used, or bright fringes in the dark field if a monochromatic light source is employed. From the spacing of these bands, the gap between the cracked surfaces can be measured down to nanometer resolution, or subnanometer if the surfaces are silvered to give multiple reflections. By applying forces to the cracked body, the crack can be made to move and observations can be made using the reflected light. The most interesting situation occurs when the crack can be opened up and then closed repeatedly to demonstrate reversible cracking. This idea of reversibility of cracks was due to Griffith11 who conceived the idea that a crack could be in thermodynamic equilibrium. In other words, energy put into the crack by the mechanical cracking force should be converted completely into molecular free energy of the new surface. This was the start of fracture mechanics.

7.7. FRACTURE MECHANICS: THERMODYNAMIC THEORY OF CRACKING Griffith first applied the theoretical idea that the crack line separates the full contact area from the zero contact region. In other words, as the crack passes through a joint, the surfaces are separated instantly from molecular contact to full separation. The area swept out by the crack therefore demands a certain work of adhesion which is fully supplied by the energy fed in by the forces and elastic deformations. If the crack closes, the theory demands that the molecular adhesive forces can give all their energy back to the external system. Thus the Griffith theory is an equilibrium energy conservation, or thermodynamic, argument as shown in Fig. 7.10. Griffith, who was working on strong glass fibers at the Royal Aircraft Establishment in Farnborough, thought that he had verified his theory experimentally for fracture of glass by measuring the cracking forces and showing they were consistent with the surface energy of molten glass, extrapolated down to the temperature of the cracking experiment. Unfortunately, Griffith made two errors. One was a numerical error in his theory, later corrected in 1924.12 The other mistake was to use glass as his ideal brittle material. It is now known that glass does not display reversible fracture, and that substantial energy losses occur in

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dissipative processes around the crack tip, stopping crack closure and preventing equilibrium. His cracks were not truly in equilibrium. The first truly reversible adhesion cracking experiments were carried out by Obreimoff13 in 1930 on mica and by Johnson et al. 14 in 1971 using smooth elastic rubber spheres. The diameter of the black contact zone was measured in reflected light, and plotted against the applied force to compare with the thermodynamic cracking theory. The results were reasonably reversible and fitted the thermodynamic work of adhesion theory. These experiments are described more fully in Chapters 4 and 9. Griffith used a difficult geometry which required complicated mathematics. The simplest form of his theory was found by Rivlin15 who applied the Griffith argument of energy conservation during cracking to the peeling adhesion of elastic paint films in 1944. Imagine an elastic polymer film peeling under a force F from a rigid glass substrate as in Fig. 7.11. The crack can be observed moving at steady speed along the interface by looking through the glass with reflected light. After a while, the crack has moved a distance c. The area of interface broken by this crack movement is bc where b is the width of the peeling film. Therefore the energy expended to create new surfaces by breaking the molecular

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bonds is Wbc, where W is the thermodynamic work of adhesion (i.e. the reversible energy required to break of molecular bonds at the interface). The work done by the force is force times distance i.e. Fc, which is all presumed to go into surface energy. Therefore the peel equation is F = Wb, which is very similar to Bradley’s rule described in Section 4.6. Of course there is elastic deformation energy in the bent elastic film, from the time when the force was first hung on the film. But this remains constant during peeling and so does not supply any energy to the surfaces. It is merely a constant energy term which moves along with the crack. Consequently it does not change during the energy balance. We also assume that there are no stretching or dissipation energy terms as the film is detached. This theory presumes that the crack can also heal at the same force. In practice, the force has to be slightly reduced for healing to be seen. For the most perfect elastic system, there is a force which can be suspended on the film whereby the crack does not know whether to peel or heal. The crack is essentially in thermodynamic equilibrium in which a slight increase in force will cause separation, and a slight decrease will cause healing. This is the situation to which the peel equation F = Wb can be applied. The peel equation derived by Rivlin above is most interesting because it seems to have no connection with the strength of the interface, that is the stress required to pull the interface apart. The idea that solid materials require a stress or pressure to tear them asunder goes back to Galileo and his treatise on the two sciences of mechanics and strength of materials.16 This has been a remarkably persistent idea which has not been justified by work on brittle materials. For example, it has been known for many years that glass can fail at a whole range of different stresses, depending on the chemistry at the surface. Indeed, that was the whole point of Griffith’s work, to understand why the failure stress of glass can vary so much. The concept of a constant fracture stress may be justified for failure of plastic materials in which the notion of yield stress is satisfactory. However, stress has little place in the equilibrium theory of molecular bond breakage adopted in this chapter. The equation F = Wb shows quite clearly that the work of adhesion W is the main material property resisting cracks. This is a chemical parameter describing the energy needed to overcome molecular attractions. Only the width of the strip is then relevant to the adhesion force. There is no area term in the equation, so stress cannot be involved. This can be proved experimentally.

7.8. EXPERIMENTAL PROOF THAT STRESS DOES NOT CAUSE CRACKING OF ADHESIVE JOINTS It is clear from the peel equation F=Wb that stress in the elastic material can be altered while the crack criterion does not change. For example, as a film is

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made thicker, the force required for equilibrium peeling does not rise, even though the stress at equivalent points in the film has fallen, as shown in Fig. 7.12. Thus, peel cracks can propagate when the stress is low and also when the stress is high. In other words, this equation supports the general conclusion from the Griffith thermodynamic argument that stress does not determine cracking. On the contrary, cracking is a spontaneous chemical phenomenon like Brownian motion, whose equilibrium shifts like a chemical reaction according to the various energy terms in the system, including mechanical energy and other energy terms. A direct experimental verification of this idea can be obtained in the peel test geometry. Consider the two peel tests shown in Fig. 7.13. Imagine applying a force F to the rubber film in the geometry of Fig. 7.13(a). Increase the force until peeling is on the point of occurring. This force creates a certain complex stress distribution in the elastomer, which is sticking to the rigid substrate by rubber/rubber bonds.

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The stress distribution in Fig. 7.13(a) is not changed if the rigid glass substrate is replaced by an equal rubber film peeling in the upward direction, as in Fig. 7.13(b), neglecting Poisson contraction effects. If stress were dictating fracture, then peeling would not quite occur in Fig. 7.13(b), just as the crack did not move in Fig. 7.13(a). However, peeling does occur in the geometry of Fig. 7.13(b) when measured experimentally,17 and this is readily explained in terms of the energy theory because there are now two displacements and forces in Fig. 7.13(b), giving a mechanical energy input twice that of Fig. 7.13(a) for the same interface resistance, that is, a peeling force at equilibrium of F = Wb/2, showing that peeling should now occur even though the film stresses remain the same as when no peeling was observed. This experiment was performed using ethylene propylene rubber (EPR) crosslinked with dicumyl peroxide.17 The rigid substrate was polymethylmethacrylate (PMMA) covered by a thin film of EPR. A rubber strip was contacted with the rigid surface for 1 h before applying the peel force. The load F was set at 12 mN on a 10 mm wide strip, and the crack was observed through the transparent rubber. No peeling was seen over a period of 30 min in the geometry of Fig. 7.13(a). Then two equal EPR strips were held in contact for 1 hour and the same load of 12mN was applied in the geometry of Fig. 7.13(b). Peeling was then seen at a crack speed of as expected from the equilibrium energy analysis. This proves that stress is not the fundamental factor in cracking. Brownian motion is really the key to cracking as discussed next.

7.9. CONSISTENCY OF THE BROWNIAN MECHANISM WITH FRACTURE MECHANICS The Brownian mechanism is an atomistic model, describing molecular bonds which are in rapid thermal motion, with local fluctuations and statistics of adhering molecules which interact with intermolecular potentials. In contrast, the fracture mechanics is a global continuum model which satisfies the conservation of energy principle and the particular equation of state of the material at large scales. There is no way that we can model every atom or molecule in a block of material. It is far better to understand the overall macroscopic behavior of a material in terms of the continuum mechanics of elasticity theory, for example. However, it is equally evident that such continuum theories must fail as we approach molecular dimensions where adhesive failure is occurring. It is far too simplistic to suggest that a single variable in the equation of state (e.g. stress or pressure) should dictate adhesion. This might have worked for the Magdeburg hemispheres but does not work for molecular forces. Instead, consider the situation depicted in Fig. 7.14. This shows that, at large scales we should treat

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the materials by continuum mechanics, but at the molecular scale we must consider molecular models where adhesion is governed by a two-parameter interaction potential and by statistical mechanics. Clearly these two models are consistent with each other and merge when the crack is looked at over a wide range of scales as in Fig. 7.14. At macroscopic resolution, the crack seems to be in equilibrium at a particular loading (Fig 7.14(a)). There does not seem to be any motion at the crack tip. However, when viewed at the atomistic level (Fig 7.14(b)) the crack tip is seen to be in rapid thermal Brownian motion. The reacted molecules form the adhered region to the left of the crack tip, whereas the unreacted molecules lie to the right at the open crack surface. The crack tip is not a static point in this model. It is wandering kinetically from right to left as the molecules spontaneously break and then rebond. Cracking is thus viewed as a chemical reaction between molecules at the crack tip. The force applied to open or close the crack is not the cause of reaction, i.e. peeling or healing, at the crack tip. The reaction is happening spontaneously and equally in both directions, causing the crack to open and close spontaneously at the molecular scale. Applying the crack driving force merely shifts the chemical equilibrium in one particular direction, either opening or closing the crack. This model allows us to define the work of adhesion as the sum of all the molecular adhesion energies over of interface. Therefore, if n is the number of molecular bonds per square meter, then the work of adhesion W is given by In this model, it is assumed that the molecular bonds are independent such that the energies are simply additive. This assumption is nearly true for elastomers which are cross-linked networks of long chain molecules, largely moving independently. The individual molecules are above their glass transition temperature, and so can move like independent fluid chains at the rubber surface while restrained globally by the elastic network.

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Work of adhesion is a much better parameter for describing solid adhesion than the surface energy which is often used to describe liquid surface attractions. Surface energy (i.e. surface tension) as a concept is extremely useful for liquids because many liquid surfaces have no resistance to flow at low rates, and so reach an equilibrium shape which is dominated by surface energy. Thus, surface energy of liquids is readily measured by the deformation of liquid surfaces. Unfortunately, the surfaces of solids are elastic under ordinary conditions. These elastic forces are so large that they dominate the surface tension and so the measurement of solid surface energy has proved very difficult. Bikerman18 was particularly critical of the concept of solid surface energy for that reason. As he correctly pointed out, if surface tension of solids exists, why are solids not as smooth as liquids? However, surface energy has been invoked as an important factor in evaporation of solids, in crystal nucleation and growth, in grinding and agglomeration of powders,19 in fracture, in extension of thin wires, and so forth. Certainly, the adhesion of bodies depends on the work of adhesion W and if the bodies are identical and smooth, we can take where is the surface energy of the solid as shown by Johnson et al.14 However, surface energy of solids is seldom enough to overcome elasticity, which is the next major mechanism to be discussed. 7.10. ELASTICITY IN THE ADHESION MECHANISM Although in Section 7.7, the adhesion of the peeling rubber was not affected by the elasticity of the film, and although Bradley’s rule shows that elasticity of spheres has only a small influence on their adhesion (Section 4.6), these are exceptional examples. Normally adhesion is strongly influenced by elasticity. The reason is that elasticity allows movement, and this movement usually acts to enhance the cracking of the joint. In terms of the energy balance theory of Section

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7.7, as the material stretches, more energy (i.e. force times stretch distance) is pumped into the crack, which then converts the energy into new surface. Two examples of this elastic stretching mechanism are shown in Fig. 7.15. In the first case, of the fiber pull-out, the fiber is stretched along its length by the force and this causes a crack to run along the interface. In the second example, of the butt joint, in which an elastic adhesive layer is joining two relatively rigid plates, the adhesive is stretched all through its volume and this eventually causes a crack to run in from the edge. These examples are considered in much more detail in Chapters 15 and 16. Essentially, these systems behave according to Griffith’s argument, that the elastic energy put into stretching can be released as new surface energy as the crack overcomes the work of adhesion of the interface. By going through the mathematical analysis of the energy balance, it can easily be shown that the force required to cause cracking increases with work of adhesion, elastic modulus, and dimension of the elastic material. However, the stress required, that is the adhesion pressure, increases as the material gets thinner. The equation is shown in Fig. 7.16, which gives some results for failure of polymer butt joints, showing how the strength S depends on adhesion W, modulus K and thickness d. The result fits our perceptions of adhesion, in that we expect the force to rise as the work of adhesion rises and also as the elastic modulus increases, i.e. as we use stiffer adhesive. The surprise is that the joint also gets stronger as we use less adhesive by making the joint thinner. This is the origin of the well-known injunction on adhesive products: “use the minimum amount of this product.” This is the adhesive salesman’s nightmare which was first explained in 1971.20 In summary, it is evident that adhesion energy can remain constant, while the apparent strength of an adhesive joint can vary strongly, depending on the elasticity and thickness of the adhesive. That is why the concept of “adhesive joint strength,” much loved by engineers as a criterion of design, is wrong.

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7.11. ROUGHNESS AS A STRONG MECHANISM There is no doubt that the roughness of surfaces plays a large part in adhesion. There are two large effects: if we wish to glue surfaces together on the one hand, we roughen the surfaces; on the other hand, if two rubber surfaces are tacky and sticky, one applies powder to the surfaces to roughen them and prevent molecular contact. Thus roughness has two seemingly opposite effects, promoting strong gluing in one case, while preventing tackiness in the other, as shown in Fig. 7.17. These apparently paradoxical effects are explained quite simply: roughness is a form of hysteresis, always acting against you, whether making or breaking the joint. This adhesion hysteresis is reminiscent of contact angle hysteresis where roughness retards the formation of the wetting angle of liquids on surfaces. The nature of this hysteresis may be tested by roughening a polymer surface, then wetting it with silicone rubber, and allowing it to cross-link to a solid elastomeric state. The liquid rubber penetrates the roughness on the surfaces to make true molecular contact, as in Fig. 7.17(a). When a force is applied to peel the rubber away from the plastic, the force required is substantially more, typically three times more, than that from an optically smooth surface. This can be explained by suggesting that the work of adhesion of the materials is the same, but the roughened surface has three times the geometrical area as a result of roughening. By taking the same rough surface and placing it in contact with an optically smooth rubber block, it is evident that making contact is much impeded by the roughness, as in Fig 7.17(b). The adhesion is then substantially less than that experienced between two smooth surfaces. This last experiment was carried out systematically by Fuller and Tabor in 1975.21 They used silicone rubber which was moulded into smooth glass concave lenses to produce spherical bodies which could be contacted with an acrylic plastic flat of varying roughness, prepared by bead blasting, as shown in Fig. 7.18.

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The pull-off force was measured as the roughness was increased. The other variables investigated were the curvature of the rubber surfaces and their elastic modulus. Curvature was found to make little difference to the results, but the elasticity was found to be very important. When the rubber was made stiffer, by cross-linking it more strenuously to give a high elastic modulus, the adhesion decreased significantly. The results are shown in Fig. 7.19 for three different stiffnesses of rubber. On the left-hand axis, the adhesion force relative to that of smooth surfaces is plotted against center-line-average roughness on the bottom axis. This roughness was measured using a stylus profilometer instrument and ranged from in value. It was clear that adhesion fell off systematically with roughness, especially for the stiffer rubber which gave almost no adhesion at roughness. The random roughness of surfaces can be modeled by a statistical distribution, as first shown by Johnson22 and later much expanded by others.23 Using such a statistical theory, Fuller and Tabor defined an adhesion parameter which was the asperity height divided by the maximum extension an asperity could withstand before adhesive fracture. This adhesion parameter increased with roughness and elastic modulus but decreased with work of adhesion and asperity

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radius. Thus the increase in modulus was shown to be equivalent to an increase in roughness, thereby explaining why very stiff materials, like ceramics and metals, do not adhere much even when polished to very high smoothness. Soft materials stick best. Having shown in this chapter how the simplest mechanisms, such as cracking, elastic deformation, and roughness, can affect adhesion, in Chapter 8 it is now possible to discuss more complex mechanisms, especially those which lead to energy loss and hysteresis.

7.12. REFERENCES l. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

Newton, I., Opticks, Smith and Walford, London, 1704 (reprinted Dover, New York, 1952, p 376). Kendall, K., Science 263, 1720–5 (1994). Kendall, K., Sci. Prog. 72, 155–71 (1988). Adams, R.D., Comyn, J. and Wake, W.C., Structural Adhesive Joints in Engineering, Chapman and Hall, London 1997. Kendall, K., Liang, W. and Stainton, C., Proc. R. Soc. A 454, 2529–33 (1998). Attenborough, F. and Kendall, K., J Adhesion 2000 in press. Stainton, C., PhD Thesis University of Keele, 2000. Tabor, D. and Winterton, R.H.S., Proc. R. Soc. A 312, 435–50 (1969). Israelachvili, J.N. and Tabor, D., Nature 236, 106 (1972). Vitruvius The Ten Books of Architecture (translated by M.H. Morgan) Dover, New York, 1960, p 63. Griffith, A.A., Phil. Trans. R. Soc. A 221, 163–198 (1920). Griffith, A.A., Proc. Inter. Congr. Appl. Mech. Delft. 1924, 55–63. Obreimoff, J.W., Proc. R. Soc. A 127, 290–7 (1930). Johnson, K.L., Kendall, K. and Roberts, A.D., Proc. R. Soc. A 324, 301–13 (1971). Rivlin, R.S., Paint Technol. 9, 215–8 (1944). Galileo, Two Sciences (1638), translated by S. Drake, Wisconsin University Press. Kendall, K., J. Adhesion Sci. Technol. 8, 1271–84 (1994). Bikerman, J.J., Phys. Stat. Sol. 10, 3–26 (1965). Kendall, K., Alford, N.McN. and Birchall, J.D., Nature 325, 110–22 (1987). Kendall, K., J. Phys. D: Appl. Phys. 4, 1186–95 (1971). Fuller, K.N.G. and Tabor, D., Proc. R. Soc. A 345, 327–42 (1975). Johnson, K.L., Contact Mechanics, Cambridge University Press, Cambridge, 1985, chap 13. Hills, D.A., Nowell, D. and Sackfield, A., Mechanics of elastic contacts, Butterworth-Heinemann, Oxford, 1993, chap 14.

8 MORE INTRICATE MECHANISMS: RAISING AND LOWERING ADHESION

Those Particles receding from one another with the greatest Force, and being most difficultly brought together, which upon Contact cohere most strongly ISAAC NEWTON, Opticks,1 p. 396

As Newton suggested in the quote above, there often seems to be some barrier to adhesion which prevents the full molecular adhesion from being attained. Therefore, it is often necessary to press materials forcibly together or to treat the surfaces in some special way to obtain the best adhesion results. In the previous chapter, we have seen how surface roughness is one such barrier; asperities have to be squashed flat if true contact is to be attained. Or, alternatively, the surfaces can be softened to allow the roughnesses to be compressed more easily. The roughness mechanism interacts with the elastic mechanism to influence adhesion. The purpose of this chapter is to follow other interactive mechanisms, especially those which lead to adhesive hysteresis, where there is resistance both to making and breaking the joint, as shown in Fig. 8.1. As two bodies are brought together, Fig. 8.1 (a), the force of adhesion is experienced but this does not rise fully to the equilibrium value, shown in Fig. 8.1(b). Something seems to prevent full adhesion. Then, when the surfaces are separated in Fig. 8.1(c), the force required to break the joint is much higher than equilibrium. There seems to be a barrier to separation. Thus a hysteresis loop is traced out and it is evident that energy is not conserved in the adhesion process. Some energy has been wasted in overcoming the barriers and this is the adhesive hysteresis. Since we believe that adhesion forces are conservative, i.e. no energy 155

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should be lost in the ideal case where two smooth clean surfaces stick together by molecular attractions, then we have to look for specific mechanisms causing hysteresis. Sometimes, this hysteresis can be thousands of times larger than the equilibrium adhesion.

8.1. ROUGHNESS AND CONTAMINATION AS HYSTERESIS MECHANISMS Obviously, roughness is a barrier which inhibits molecular contact and strong adhesion between surfaces. But more significantly, the roughness does not usually deform elastically as the surfaces make contact,2 as shown in Fig. 8.2. Thus, when the surfaces are pushed together, plastic deformation of the asperities occurs and this dissipates energy, thus providing a hysteresis mechanism. In the ultimate development of this process, the surfaces are deliberately sheared together during the contacting step, thus causing large local heating and softening at the asperity contacts, which then become more squashed, to allow very strong adhesion. This is used in friction welding to join materials together.3 Another roughness-dependent mechanism for adhesion hysteresis can be seen in humid atmospheres. As the rough surfaces are brought together, the

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asperities make contact, leaving capillary gaps between the surfaces, as shown in Fig. 8.3. Water condenses in these gaps. Because the condensing water is in a thin film form, it has two substantial effects on the adhesion. First, it gives a capillary attraction resulting from the curvature of the meniscus and the surface tension of the liquid. For a wetting liquid in a narrow gap, this pressure can be very large, as shown in Fig. 8.3(b). Thus, with time of contact, the adhesion between the surfaces can increase due to this capillary condensation. In addition, water can react to give hydration products at many surfaces, and these colloidal products can glue the surfaces together. An excellent example is the rusting together of steel plates in contact under humid conditions, to give extremely strong adhesion. The second effect of the water in the surface gaps is the viscous resistance of the liquid inhibiting rapid separation. The more rapidly the surfaces are pulled apart, the more this viscous resistance is exhibited. This viscous resistance to separation was mentioned by Galileo and was quantified by Stefan 130 years ago.4 In summary, we can see that adhesive hysteresis can arise from roughness itself, and also from contamination condensing in the gaps caused by roughness. This latter effect has been one of the greatest controversial areas of adhesion over the years, because people have believed that moisture is the cause of adhesion between particles. This is not generally true. The problem is illustrated in Fig. 8.4. Two atomically smooth spheres adhere by molecular forces, described in detail by JKR theory,5 which corrected Bradley’s original rule (see Section 9.2). However, rough spheres hardly adhere at all because of the poor atomic contact, as in Fig. 8.4(b). But when water condenses in the roughness gaps, measurable adhesion is obtained, as shown in Fig. 8.4(c). The liquid bridge filling the gap between the solid particles has reduced the solid/solid adhesion at the asperities, but has increased the sphere attractive force through the capillary meniscus pressure. This can be confused with JKR theory or Bradley’s rule because it has roughly the same form, depending on particle diameter D and surface tension of the liquid. McFarlane and Tabor6 verified the liquid bridge equation for glass spheres in vapours of several liquids, including water, glycerol, decane, octane and alcohol.

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Later, Fisher and Israelachvili7 showed the same result for mica surfaces even down to vapor pressures of 0.1, where only the odd molecular layer is condensed. Much more condensation is needed for rougher surfaces where it is necessary to fill the gaps. However, once the particles are too wet, then the capillary force drops and adhesion diminishes. These results are highly significant for agglomerating and coating granules.8 It is clear from these examples that the presence of contamination at the interface is a key issue in adhesive hysteresis.

8.2. DWELL-TIME EFFECT A very general adhesive effect which stems from such mechanisms as those above is the “dwell-time” phenomenon. Two surfaces are brought into contact and left for a time. The adhesion is then found to have increased. Further time of contact leads to further increase as shown in Fig. 8.5. This effect could, for example, be a result of the capillary condensation described above. When the surfaces are first in contact, the adhesion is low because roughness inhibits the short-range attractions. But as condensation occurs in the gaps, the adhesion rises with time. Another possible cause of this effect is the creep of the interfacial contact caused by the gradual squashing of roughnesses. When two solids are placed in contact, the true atomic contact area tends to grow slowly with time because the material is not perfectly elastic. So, even when the atomic adhesion remains constant, hysteresis can occur from this junction growth. This has been measured particularly for polymers.9,10

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A still more common cause of this effect is the expulsion of contamination from the space between the surfaces, shown schematically in Fig. 8.5. In a particular experiment with elastomer,11 shown in Fig. 8.6, natural rubber was mixed with 2.5% sulfur and cross-linked by heating at 145°C in contact with a smooth glass lens to give a spherical rubber surface. Immediately after demolding, this rubber was very smooth and adherent to glass surfaces, Fig. 8.6(a). However, after the rubber sample had been in air for several days, the surfaces were no longer so smooth, and observation showed that tiny particles of sulfur had diffused out of the rubber to contaminate the surface, Fig. 8.6(b). These particles were around 50 nm in size and reduced adhesion by an order of magnitude when tested with glass, Fig. 8.6(c). After some hours in contact, the adhesion to the glass had increased back to its original high level, and no sulfur particles could be seen at the contact, Fig. 8.6(d). The sulfur particles had been pushed back into solution in the rubber by the presence of the glass surface. In conclusion, when contaminant molecules exist at a contact between two bodies, they will move towards their equilibrium positions by flow or diffusion.

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Adhesion will then increase as a function of time until a new equilibrium is attained. This is a major cause of the adhesive dwell-time effect. One of the best understood examples is that of mica surfaces, pressed into contact through water, as described in Section 6.6. The plot of force versus separation is not smooth in this case, but shows sharp oscillations as layers of water molecules are squeezed out of the gap between the surfaces.12 When plotted as an energy diagram, this shows that the surfaces can exist in several metastable adhesive states, depending on how many water layers have been removed, as shown in Fig. 8.7. This type of diagram could be used to describe the growth of adhesion with time as mica sheets lie together over a period of time. The water molecules would gradually diffuse out from the gap and the mica would gradually move into closer contact. To achieve this, the mica has to overcome the energy barriers which become higher as true contact is approached. Thus we would expect the adhesion to depend both on temperature, which speeds up diffusion, and on the force applied to push the material closer. This is considered next.

8.3. ADHESIVE DRAG Once we have recognized that the contaminant molecules introduce an oscillating interaction energy between surfaces, as illustrated in Fig. 8.7, then we see that more complex adhesion effects must follow. For example, time effects must be observed because the contaminant molecules cannot get into position

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instantly. They require time to diffuse into and out of the interface. Moreover, the contamination at the interface will depend on the force we apply to the joint and the Brownian energy kT of the molecules which drives the diffusion process as the contamination escapes. Consider as an example the situation shown in Fig. 8.8. The interface between the two surfaces can exist in the two metastable states with adhesion energies and Imagine first that the surfaces are in state and we wish to pull them apart into state To do this we apply a peeling force and this must be sufficient to overcome the energy barrier, with the help of the Brownian energy kT. This problem of molecules coming apart across energy barriers was first solved by Eyring in the early 1940s.13 There are two forces required according to this theory; one to provide the reversible work of adhesion and the second to overcome the energy barrier. Thus the total force can be expressed as the sum of two terms in the peeling equation given in Fig. 8.8. The second term is an energy loss term which appears as heat. Clearly this depends on the rate of peeling V, and also on the temperature variant constants A and B. The higher the rate of peeling, the greater the force required and hence the larger the energy dissipated. Also, the higher the temperature, the faster the peeling. Such behavior is well known for adhesive joints, such as those between silicone rubber and acrylic sheet, as shown in Fig. 8.9. In this example, at low speeds, the adhesion leveled off at a low value, corresponding to an apparent reversible work of adhesion of 0.3 J m at very low velocities of peeling.14 We will study the precise nature of this equilibrium value in the next section. But at high speeds, the adhesion increased very strongly. This is the adhesive drag effect. Very similar results were obtained by Russian experimentalists in the 1950s.15 However, the Russian schools seemed to devise explanations of this adhesion behavior based on charge separation or

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diffusion.16,17 While there is no doubt that both charge separation and diffusion occur, the problem is explaining their place in the hierarchy of mechanisms. The other significant aspect of adhesive drag is its relation to the surface contamination present on the surface. For example, an alkyd paint film was painted on a glass surface, cross-linked, and then peeled off. For comparison, the same experiment was carried out on a glass surface coated with dimethyldichloro silane, as shown in Fig. 8.10. These results gave similar behavior to that of silicone on acrylic, with an apparent equilibrium work of adhesion at low speeds, plus a velocity-dependent

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peeling force at higher speed. However, there were two substantial differences: first, the apparent work of adhesion was ten times too high at second, the presence of a silane coating had an enormous effect on the adhesive drag but not on the apparent equilibrium work of adhesion. This fall in adhesion due to one layer of molecules at the surface is akin to a catalytic effect: the monolayer is not changing the equilibrium, but is having a large effect on kinetics by reducing the energy barrier to peeling. Clearly adhesive drag is not the whole story. There must be other energy losses in addition.

8.4. ADHESIVE DRAG AND HYSTERESIS MEASUREMENTS The problem of measuring adhesion, in general, is that the curves for peeling have a similar shape, with an apparent work of adhesion plus a large kinetic adhesion drag, but we are not sure exactly where the equilibrium is. So it is important to devise experiments to study both making and breaking the joint in order to define the precise equilibrium point. Three typical experiments are shown in Fig. 8.11. More details are given in Sections 4.2, 9.8 and 13.2. Figure 8.11(a) shows a wedging experiment, rather like that used by Obreimoff on mica.18 The film is detached by wedging, then the wedge is withdrawn slightly to allow healing. Figure 8.11(b) shows a sphere contact experiment, for example the JKR experiment,5 in which a smooth sphere is allowed to make contact with a surface, then detached with a force. Figure 8.11(c) illustrates a peeling film experiment in which the peel force is raised to peel the film, then lowered to heal the strip back onto the smooth substrate.19 In each of these tests, the speed of movement of the crack front can be measured by observing the detachment line through the transparent materials, on both peeling and healing. The measured adhesion energy R, worked out from the force using the appropriate equation (e.g. R = F/b for peeling), is then plotted against the

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crack velocity on logarithmic scales, as shown in Fig. 8.12. Both peeling and healing curves can be shown on the same logarithmic plot. This curve defines the adhesive drag on peeling and healing, and shows that equilibrium is not fully attained, but lies between the two asymptotes. At very low speeds of crack propagation through the adhesive joint, to the left of Fig. 8.12, the peeling and healing curves should coincide. However, it was found experimentally that there was always a gap between the curves, which was small for silicone rubbers but larger for less elastic materials. This gap was defined as the adhesive hysteresis. The equilibrium work of adhesion was somewhere within this gap, around but could not be found exactly in this experiment. Only by removing all the energy losses in the experiment would it be possible to attain true adhesive equilibrium. Such energy losses, as we have seen in Sections 8.1 and 8.2, could be caused by a number of mechanisms including roughness, impurities, inelastic deformation, etc. However, one important energy loss which was explained was the effect of the viscoelastic behavior of the polymer. This was studied by varying the crosslink density of the rubber, to alter the loss of elastic energy as the material relaxed. As the viscoelastic loss increased, so did the adhesive hysteresis, as shown in Fig. 8.13. These results demonstrated that the viscoelastic relaxation in the rubber could stop the peeling well away from the equilibrium point. This idea of crack stopping as a result of energy loss in the system is an interesting nonlinear mechanism, which we look at next.

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8.5. CRACK STOPPING The adhesion behavior of elastic materials seems to be understandable in terms of the breaking of molecular bonds. Only two terms appear to be necessary to explain this: an equilibrium work of adhesion which amounts to breaking all the bonds reversibly; and a kinetic term, which relates to the force required to jump over the energy barrier between adhesive states. Some well-known results from Deryaguin’s group20 are shown in Fig. 8.14 for peeling of cellulose nitrate from glass, fitting this model. The question is, “how does hysteresis arise in this model?” One particular answer is crack-stopping which can occur if the peeling film loses energy by

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some flow process.21 This is the “chewing gum effect.” Although chewing gum does not have a strong adhesion to surfaces, it is extremely difficult to remove because it stretches extensively and the peeling energy does not get to the adhesive bonds but is dissipated in the stretching. That is why the best way to remove chewing gum from carpets is to freeze it in cold gas to make it glassy. Then the gum comes off easily in a brittle manner.22 A model for this chewing gum effect can be built by taking an elastic rubber film peeling from a surface, measuring the crack speed as the crack travels along the interface. Then compare this with the same elastic film which has been cut in the middle, then the ends joined together with chewing gum, as shown in Fig. 8.15. The peeling stops at the point where the chewing gum is holding the cut film together. This shows that the sudden viscoelastic relaxation of the gum is sufficient to extract the energy from the crack and enhance the adhesion of the rubber. Quite clearly, the molecular adhesion at the interface remains unchanged; only the relaxation of the chewing gum can be implicated in the apparent increase of adhesion. It may be shown that the relaxation produces two distinct effects on the adhesion. The first is an amplifying effect on the adhesion energy caused by the effective fall in elastic modulus of the gum from E to The second is an additional resistance to cracking caused by the rate of change of elastic modulus. Figure 8.16 shows these two effects graphically. Figure 8.16(a) gives the amplifying hysteresis factor which magnifies the peel force by an order of magnitude. Figure 8.16(b) shows how the crack speed falls as a result of the

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second term, slowing hyperbolically with time. The crack can suddenly stop after travelling a short distance. Both these terms can give the sort of behavior which is observed in real adhering systems, where the adhesion force is much larger than expected from equilibrium concepts.

8.6. ROLLING, TACK AND ADHESIVE HYSTERESIS In considering the mechanism of adhesive hysteresis, where energy is lost during a complete cycle of making and breaking an adhesive joint, it is worth considering the example of a cylinder sitting on a plane surface, as illustrated in Fig. 8.17. As the cylinder makes adhesive contact, the crack moves out to heal the surfaces together, Fig 8.17(a), then during breaking, Fig. 8.17(b), the crack moves

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in the opposite direction to separate the surfaces, giving an energy curve like that in Fig. 8.12. This cycle is carried out in “tack testers” to measure the stickiness of paints and polymers. Tack is essentially an empirical measure of adhesive drag and hysteresis. During rolling, both these processes are combined because one of the cracks is closing while the other is opening. Thus, we expect that rolling resistance for smooth elastic materials should be given by the adhesive hysteresis and drag experienced in direct adhesion tests. Tabor and his colleagues first described rolling friction in terms of a hysteresis effect, but they concentrated on the energy lost in viscoelastic deformation of the material, especially for lubricated rubber surfaces.23,24 Tabor showed very clearly that a rubber with high deformation loss gave higher rolling resistance than one with low loss, even when there was zero adhesion. High loss rubber was also more resistant to sliding on a wet gritty surface. This concept proved rather important in the skid resistance of car tires in wet conditions. Tabor patented the invention of using lossy rubber for skid resistance, and was involved in patent battles with tire companies. Of course, cars also need low rolling resistance and so the trick is to use low loss rubber in the tire walls and high hysteresis rubber on the treads. However, even when the rubber is highly elastic, for example silicone elastomer which has little deformation loss, there can be strong rolling friction as a result of adhesive hysteresis. Such rolling friction is strongly dependent on short-range molecular forces, as can be demonstrated by placing a small droplet of water or a few specks of dust on the surface. The rolling resistance is then much reduced. This type of test is embodied in the rolling tack instrument for defining “tackiness”.25 In order to prove that tackiness and adhesive hysteresis were strongly connected, the following experiments were carried out,26 as shown in Fig. 8.18. A cylindrical glass roller was placed on a smooth rubber strip and allowed to roll down under a controlled angle. The same optically smooth rubber was then peeled at various speed from the glass to measure the break energy as in Fig. 8.18(b). The dwell-time effect was evaluated by doing the peeling test after

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several periods of adhesive contact, from 10–100,000s. Finally, the smooth rubber was allowed to heal back onto the glass by inclining the glass at a suitable angle. In all cases the speed of the crack moving through the adhesive interface was measured optically. The results are shown in Fig. 8.19, indicating the significant dwell-time, drag and hysteresis effects. These were then compared with rolling friction in Fig. 8.20. The energy required to break the bond was the most significant term, modified substantially by the dwell-time effect. At a rolling speed of the rolling contact (3 mm wide) was bonded for 300 s which corresponded to the break energy curve of Fig. 8.19. Faster rolling speeds gave lower dwell time so the rolling resistance was predictably lower than the peeling break curve. It was predicted that, at high speed, the rolling friction would decrease because dwell-

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time was too low. Also it was expected that dwell-time would dominate at low speeds, so that rolling resistance would increase. Rolling speeds below 10ms gave a rapid slow down of the rolling as the dwell-time effect took over. Hyperbolic slowing of the roller was seen with time.26 It was noted that the rolling results could be predicted extremely well from the peel data.

8.7. ADHESIVE DISLOCATIONS A similar but inverse effect occurs when a bubble is formed at an adhesive interface. This concept was first described in 1975 and has since been evaluated as a mechanism for strengthening and plasticity in laminates together with toughness in composites26–28 (see Chapters 15 and 16). Whereas the rolling cylinder has a contact spot surrounded by a gap, this interface bubble consists of a small region of gap, fully surrounded by contact. You can think of the interface bubble like a ruck in a carpet. The difference in this case is that, unlike the carpet, attractive molecular forces are pulling the surfaces together to compress the elastic material, thus storing elastic energy within it. The bubble is essentially a dislocation, when viewed by comparison with a Taylor crystal dislocation shown in Fig. 8.21(c). Within the bubble, at equilibrium, there must be a balance between the attractive forces pulling the surfaces into contact and the elastic energy pushing them apart. If the bubble is compressed to make contact at the interface, then released, the bubble opens up until the equilibrium of adhesive energy and elastic energy is attained. Alternatively, if pressure is applied to the material, the dislocation moves along the interface, rather like the rolling of the cylinder on the rubber strip in Fig. 8.21(a). Again, the dislocation movement is dominated by the hysteresis. But in this case, the bubble gradually gets shorter as it moves and eventually disappears. Interface dislocations, like the one described above can be formed in several situations. Schallamach first saw such interface bubbles rippling across a

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frictional contact between smooth rubber and glass,29 as shown in Fig. 8.22(a). In that instance, there was no actual sliding of rubber on glass, merely the propagation of “waves of detachment” i.e. interface dislocations through the contact region. In a related experiment, on the pull-out of a smooth rigid fiber from a rubbery matrix, interface bubbles were observed forming at the interface between fiber and matrix.27 Further loading caused these to travel along the fiber, gradually getting smaller and disappearing. Additional dislocations then followed before gross failure occurred. Model dislocations were also seen in the low angle peeling of rubber films and in the failure of laminates.31,32 Adhesive joints could be strengthened by a factor of three by the introduction of interface dislocations. From these arguments, it is clear that, when elastic surfaces are sufficiently smooth to peel and heal, then cracking is not the only mechanism for relative movement between two bodies; the surfaces can first separate and then recombine in the form of interface dislocations. This peeling demands energy whereas healing recovers energy, the net energy expended being given by the adhesive drag and hysteresis described above.

8.8. STRINGING OR CRAZING The mechanisms described in the previous sections are fundamentally molecular, but are enhanced by mechanical losses in the bulk of the material. A particular mechanism which operates in adhesive joints is stringing. It was mentioned by Rivlin,33 who had observed the way in which sticky tape adhesive pulled out into fibers near the crack tip. In other words, the crack line did not remain straight but broke up into fibrous filaments which seemed to hold the crack faces together, as shown diagrammatically in Fig. 8.23. In fact, this type of separation process is bound to happen if a thin layer of elastic material, like sticky tape adhesive, is trapped between two more rigid layers which are pulled apart. The stress within the thin layer is a bulk stress

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because the material is constrained in all three dimensions. This bulk energy can be partly released by cavitation in the elastic body or by growth of adhesive cracks as shown in Fig. 8.24. The pressure then becomes two-dimensional because one of the constraints is released, giving up energy which can drive the debonding. These adhesive cavities can often be seen in bullet-proof glass laminates where polyvinyl butyral is used to glue thick plates of glass together. If the polymer is stressed, then fingery cracks penetrate along the interfaces to cause the characteristic crazed pattern. It has been suggested that this pattern is viscous in nature, the so-called Taylor instability. But it can happen with perfectly elastic rubbers, so viscous flow cannot be the cause. The thinner the rubber, the finer the craze cracks. Crazing mechanisms have been invoked to describe the attack of methanol on polystyrene,34 to describe how plastic flow in a metal can retard a crack,35 and to describe the effects of fibres in composites.36 It can readily be proved that the brittle Griffith cracking mechanism is softened by the crazing phenomenon, making the adhesion appear tougher as shown in Fig. 8.25. The key factor is how much stress is supported by the strings to pull the crack faces together. If this crack-tying stress is high then the adhesive joint can appear almost ductile and

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fails more gracefully and predictably. The energy balance analysis applied to the interface crack shown in Fig, 8.25 proves that the stress applied to break the joint is the Griffith stress plus the tying stress. Thus the crazing mechanism means that the strength of the joint stays higher as the debonding propagates.

8.9. AGGREGATION MECHANISMS We have seen above how the interaction between various mechanisms can lead to reduction or enhancement of adhesion. These are amplification effects which multiply the atomic and molecular bond energies by substantial factors. One other important question relates to structures built up by adhesive forces. Since the atomic forces have only a short range, around 1 nm, then it would seem logical that long-range structures seen at the micrometer or millimeter scale should not be caused by adhesive forces. However, it turns out that structure can be strongly influenced by adhesive forces. In other words, there is a structural amplification effect which can extend for long distances in an adhering system. One obvious example of this is opal, the gemstone formed by adhesion of silica particles over geological time. The colors seen by diffraction of white light from the particles suggest that the structuring of the particles extends for millimetres, a million times further than would be expected from the range of atomic forces. The same sorts of colors can be seen in polymer latex dispersions. It is important to inquire about the origins of such structures. The first factor to take into account is that, even with zero adhesion, structure must form as the number of particles is increased in a Brownian system. This was first demonstrated by computer modeling in the 1950s.37 In other words, adhesion is not necessary for structure. Indeed, if particles are made

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to adhere suddenly in a suspension, then disordered gels are obtained because each particle sticks in a random way to its neighbors, as shown in Fig. 8.26. Thus adhesion inhibits structure. Figure 8.26(a) shows a 3D computer simulation of spheres without adhesion.38 These have shuffled together with Brownian movement to give a predominantly face-centered-cubic packing, which is the most stable closepacked structure. Because there was no adhesion, the movement of the spheres was not restricted, so that each ball could find its best position after many collisions with its neighbors. By contrast, in Fig. 8.26(b), the spheres have adhered on their first collision with neighbors and so the structure ends up random and imperfect. This example shows quite clearly that short-range adhesion forces lead to disordered structures. Only when the adhesion is small can ordering occur by the Brownian collision process. Later, as when opal is formed by drying, the spheres adhere strongly in their ordered positions to give the gemstone beauty. The problem of how adhesion affects structuring is an old one, going back to the early theories of nucleation of a droplet from a vapor,39 a crystal from a gas,40 or a crystal from a melt or glass.41 The arguments about such phase separation have carried on over many years, as summarized by Zettlemoyer and others.42,43 A clump of particles, i.e. a nucleus, has to appear within the random gas by a nucleation process, and this clump can then grow as further particles stick onto it, as shown in Fig. 8.27. This picture of nucleation and growth of structure has now been changed by computer modeling of particles using molecular dynamics.44 Stainton has studied the emergence of structure in a random bunch of uniform spheres as they are compacted together to increase their volume fraction. He looked at the structure

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in the computer model by two techniques: order parameter and Voronoi polyhedra. If particles clicked together in a regular order, then he could pick out the type of order and the number of particles in that ordered clump. As expected, he found that ordering occurred above 50% packing of spheres and this happened without any adhesion. There was not just one type of nucleus formed in the assembly; three different kinds of nucleus were seen, body-centered cubic, hexagonal closepacked, and face-centered cubic. These grew at different rates, with face-centred cubic dominating eventually, as shown in Fig. 8,28(a). The other feature of this model was that the nucleus was not a simple spherical entity as envisaged in the early theories. Instead it was a complex random structure permeating through the sphere packing, as shown in Fig.

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8.28(b). Most interesting was the observation that strong adhesion between particles tended to prevent structure formation. Only 5 kT of adhesive energy was necessary to prevent crystallization, presumably by forming a disordered gel structure.

8.10. CHARGE SEPARATION AND ELECTRICAL EFFECTS Although we have recognized that adhesion stems from the electromagnetic interactions of atoms and molecules, we have not much mentioned the electrical mechanisms which are associated with adhesion. Yet it is clear, when we strip adhesive tape from a surface or take off our polyester sweater, that electrical forces are operating and that sparks can often be seen as the adhesive bond is ruptured. Again, when surfaces are immersed in water, it is evident that ions can sit on the surfaces and produce electrical potentials which cause attractions and repulsions. Similarly, dust can be collected by passing over electrodes at high voltages. The dust adheres strongly to the plates of such electrostatic precipitators. Similarly, flow from hoppers can be switched by applying electric fields across the exit valve, and explosions in powder containers may be caused by sparks stemming from build up of contact potentials. Thus, electrical discharge mechanisms in adhesion are strong and must be explained. The principle on which this book is based is that adhesion phenomena can all be explained in terms of interactions between electrons in atoms and molecules. The weak adhesive interactions, such as van der Waals forces, can be achieved with nonpolar molecules generating instantaneous induced dipoles, as explained by London, without any free charge movement. Such forces are universal and always attractive, with an energy around i.e. 0.01 eV By contrast, strong ionic, metallic, or covalent bonds require electron exchange between atoms. Free charges can then appear. The energy is then much higher, around 1 eV per molecule or Such electron exchange can lead to the build up of a double layer of electrical charges which result both in adhesive force and also in electrical discharges during separation. Much work has been performed in Russia on this topic,45 and there have been many basic studies through the 20th century,46 especially in relation to the Xerox process for photocopying.47 For metals and semiconductors, the mechanism of charging is well understood in terms of electron transfer across the surfaces in contact. However, for polymers, the charging mechanism is less well appreciated, although empirical lists of polymers can be drawn up as in Fig. 8.29. The results of such contact experiments are highly dependent on the presence of moisture and on the addition of ionic salts to the polymers. Especially interesting are those studies with polymers blended with ionomers. In a model of

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ionic transfer published recently,48 polymers containing different concentrations of ionomers were contacted and the charge measured with a Kelvin probe, which is a metal tip brought into contact with the surface, then removed. Figure 8.30 shows the schematic mechanism and the results. In conclusion, there are many mechanisms which can alter the level of molecular adhesion between solid bodies. Some of these may be elastic and reversible while others are time dependent, leading to drag and hysteresis. Thus, adhesion measurements cannot generally be explained in terms of a simple adhesion energy or range. The mechanisms of adhesion can magnify or reduce the adhesion force by several orders of magnitude. 8.11. REFERENCES 1. Newton, I., Opticks, Smith and Walford, London, 1704 (reprinted Dover, New York, 1952, p 396). 2. Bowden, F.P. and Tabor, D., Friction and Lubrication of Solids, Clarendon Press, Oxford, Part II,

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9. 10. 11. 12. 13. 14. 15. 16.

17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.

CHAPTER 8 Threadgill, P., TW1 reprint 513/2/97, 1–4 (1997) TW1, Cambridge UK. Stefan, N.N., Z. Akad. Wiss. Wien. 11, 713 (1874), see Taylor, P., Proc. R. Soc. A 108, 12 (1924). Johnson, K.R., Kendall, K. and Roberts, A.D., Proc. R. Soc. A 324, 301–13 (1971). McFarlane, J.S. and Tabor, D., Proc. R. Soc. A 202, 224–43 (1950). Fisher, L.R. and Israelachvili, J.N., Colloid Surf. 3, 303–19 (1981). Ormos, Z.D., In: Powder Technology and Pharmaceutical Processes, eds D. Chulia, M. Deleuil, and Y. Pourcelot, Elsevier, Amsterdam, 1994, pp 359–76. Persson, B.N.J., Sliding Friction, Springer, Berlin, 1998, pp 53–1. Kendall, K., J. Adhesion 7, 55–72 (1974). Kendall, K., Science 263, 1720–5 (1994). Israelachvili, J.N. and Pashley, R.M. Nature 306, 249–50 (1983). Glasstone, S., Laidler, K.J. and Eyring, H., Theory of Rate Processes, McGraw Hill, London, 1941, p 339. Kendall, K., J. Adhesion 5, 179–202 (1973). Krotova, N.A., Kirillova, Y.M. and Derjaguin, B.V., Z. Fiz. Chim. 30, 1921 (1956). Derjaguin, B.V., Krotova, N.A. and Smilga, V.P., Adhesion of Solids, Consultants Bureau, London, 1978, chap 2. Voyutskii, S.S., Autohesion and Adhesion of High Polymers, Wiley Interscience, New York, 1963, chap 1. Obreimoff, J.W. Proc. R. Soc. A 127, 290–97 (1930). Kendall, K., J. Phys. D: Appl. Phys. 11, 1519–27 (1978). Krotova, N.A., Kirillova, Y.M. and Deryaguin, B.V., Z. Fiz. Chim. 30, 1921 (1956). Kendall, K., Int. J. Fracture 11, 3–12 (1975). Gent, A.N. and Petrich, R.P., Proc. R. Soc. A 310, 433–9 (1969). Tabor, D., Proc. R. Soc. A 229, 198 (1955). Eldredge, K.R. and Tabor, D., Proc. R. Soc. A 229, 181 (1955). Voet, A. and Geffken, C.F., Ind. Eng. Chem. 43, 1614 (1955). Kendall, K., Wear 33, 351–8 (1975). Kendall, K., J. Mat. Sci. 10, 1011–14 (1975). Kendall, K., Nature 261, 35–6 (1976). Kendall, K., Phil. Mag. 43, 713–29 (1981). Schallamach, A., Wear 17, 301 (1971). Kendall, K., J. Phys. D: Appl. Phys. 11, 1519–27 (1978). Kendall, K., Phil. Mag. 36, 507–15 (1977). Rivlin, R.S., Paint Technol 9, 215–7 (1944). Kendall, K., Clegg, W.J. and Gregory, R.D., J. Mater. Sci. Lett. 10, 671–4 (1991). Dugdale, D.S., J. Mech. Phys. Sol. 8, 100–4 (1960). Bowling, J. and Groves, G.W., J. Mater. Sci. 14, 443–8 (1979). Alder, B.J. and Wainwright, T.E., J. Chem. Phys. 31, 459–66 (1959). Stainton, C., PhD Thesis, University of Keele, 2000. Farkas, L., Z. Phys. Chem. 125, 236 (1927). Kaischew, R. and Stranski, I.N., Z. Phys. Chem. A 170, 295 (1934). Volmer, M. and Weber, A., Z. Phys. Chem. 119, 227 (1926). Zettlemoyer, A.C. (ed.) Nucleation Phenomena, Adv. Colloid Interface Sci. 7 (1977). Schmelzer, J., Ropke, G. and Mahnke, R., Aggregation Phenomena in Complex Systems, WileyVCH, Chichester, 1999. Stainton, C. and Kendall, K., to be published, 2001. Derjaguin, B.V., Krotova, N.A. and Smilga, VP., Adhesion of Solids, Consultants Bureau, New York, 1978. Harper, W.R., Proc. R. Soc. A 205, 83–89 (1989). Dessauer, J.H. and Clark, A.E., Xerography and Related Processes, Focal Press, New York, 1965. Diaz, A.F., In: Fundamentals of Adhesion and Interfaces, eds. DeMejo, L.R., Rimai, D.S. and Sharpe L.H., Gordon and Breach, Amsterdam, 1999, pp 111–22.

9 ADHESION OF PARTICLES: DEFORMATION, FRICTION, AND SINTERING

Particles attract one another by some Force, which in immediate Contact is exceeding strong ... and reaches not far from the particles with any sensible effect ISAAC NEWTON,1 Opticks, p. 389

Newton’s statement above appears very definite and unequivocal. However, he recognized that there were problems in achieving sufficient contact between particles to develop the short-range attractive forces. He found that “dry Powders are difficultly made to touch one another so as to stick together.”1 A small dust grain or surface roughness at the contact could prevent the full development of the attractions. Indeed, the modern subject of soil mechanics generally assumes that there is zero adhesion between particles.2 If a 1 mm diameter glass ball is placed on a smooth glass plate, it usually rolls around and does not stick. However, when a smooth rubber sphere is placed on the glass, then it does stick and refuses to roll easily, as shown in Fig. 9.1. This goes against molecular expectations because one would predict both glass and rubber spheres to adhere similarly with a force around Evidently, it is important to study the detailed nature of the contact between particles in order to understand these differences. The purpose of this chapter is to define the contact between spheres, to understand how the adhesion is developed, and to use this knowledge to explain phenomena such as powder friction and sintering, which are vitally important to the smokes, dusts, soils, sands and particulate masses which occupy much of the planet. 181

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9.1. CONTACT OF SPHERES Although Newton had measured the black spot at the contact of glass spheres, and also seen the black spot expand and contract reversibly as the spheres were pressed together, he did not pursue the relationship between the spot size and load. Almost 200 years were to elapse before Hertz defined the connection published in 1882.3 Hertz was a 23-year-old assistant to Helmholtz in Berlin when he was stimulated by Newton’s rings and derived the elastic theory of sphere contact in his Christmas vacation in 1880.4 He found that the spot diameter increased with the cube root of load F, showed that the elastic modulus E, Poisson’s ratio v and sphere diameter D were also important, and verified his equation

which applies to equal spheres, by measuring contact spots for glass and metal spheres, as shown in Fig. 9.2. For spheres of different diameters and and materials of different elastic constants, the effective diameter and effective modulus could be substituted into Equation (9.1), where and The main contribution of Hertz was to understand that the spheres press into each other to give a hemispherical pressure distribution, with maximum pressure P in the middle, falling as to zero pressure at the edge of the contact spot, where z is the diameter of a circle within the contact spot of diameter d. He also knew that the centers of the two spheres approached each other because of such pressure by a distance given by

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All these expressions assumed that no adhesion or friction existed at the contact between the spheres. This was an excellent assumption for large loads such as those experienced in ball bearings, under train wheels, and where car tires meet a road surface. The experimental measurements then fitted the theory very well. However, when the load was zero, or even tensile, as in the experiments conducted by Bradley5,6 in 1932 on adhering spheres, it was clear that the spheres were still deforming, but now the pressure was generated by the molecular attractions and not by the external load. The simplest way to take this into account was to assume that the molecular attractions were acting like an external load and to use Equation (9.1) to describe the results, as Derjaguin7,8 did in 1934. His basic thermodynamic argument was correct because it equated the work done by the surface attractions against the work of deformation in the elastic spheres. However, the deformation Derjaguin used was not exactly correct because he did not take into account the effect of surface attractions on the pressure distribution at the contact. The geometry assumed by Derjaguin is shown in Fig. 9.3 with the incorrect Hertzian pressure distribution. The sphere surfaces were pulled together by the surface attractions and the movement of the spheres was equivalent to pressing with a load F, which could then be calculated by an energy balance. The surface energy was the contact area times the work of adhesion W i.e. The potential energy in the load F was which was from the Hertz Equation (9.2). Adding these two energy terms gave the total energy U

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By the principle of energy conservation

Therefore

Because this equation was so similar to Bradley’s equation, for the adhesion of rigid spheres, Derjaguin concluded from this analysis that the elastic contact deformation made little difference to the adhesion force.

9.2. THE JKR CONTRIBUTION The correct solution to the problem of contact between elastic spheres with surface adhesion was obtained by Johnson, Kendall and Roberts9 37 years later. This came about because Roberts and Kendall had both been supervised by Tabor while studying for doctorates in Cambridge, while Johnson had collaborated over many years with Tabor on the contact problems associated with friction and lubrication.10 Roberts,11 while observing the contact of rubber windscreen wiper blades on glass, had noticed that the contact spot was much larger than he expected from Hertz theory under dry conditions, yet approached the Hertz predictions rather precisely when wetted with soapy water. Kendall12 had been measuring the contact spot size between polymer, glass, and metal surfaces using optical and

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ultrasonic methods, and became convinced that adhesion made the Hertz equation incorrect at low loads because the contact spot was larger than expected, as illustrated in Fig. 9.4. The problem was to explain such increased values of contact size. Long before these experimental measurements, Johnson13 had attempted to do this by showing that the pressure distribution within an adhesive contact could be described by adding two simple stress distributions together. However, he was puzzled by the resulting infinite stresses at the edge of the contact which he therefore expected to fail under the high tension. Figure 9.5 shows the way the component stresses add within the contact region. The solution came on a cold February night in 1970 when Roberts had visited Kendall in Derby where he was working on wheel to rail adhesion at British Railways Technical Centre. After discussing the problem over dinner, it

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became clear that the answer lay in applying Derjaguin’s method of Equation (9.3) to Johnson’s stress distribution. Johnson did the mathematics an evening later to provide the correct equation, the so-called JKR equation, for the elastic contact spot diameter d of equal spheres, diameter D and elastic constants E and v, with short-range work of adhesion W:

Roberts and Kendall then did more experimental work on rubber/rubber and gelatin/poly(methyl methacrylate) contacts. These results fitted Equation (9.4) extremely well, as shown in Fig. 9.6, and allowed the work of adhesion to be scaled to the observations. It turned out later that a similar mathematical argument had been produced by Sperling in 1964, but he had found no experimental evidence to support his theory.13 From the contact spot size at zero load, where

the work of adhesion fitting the results for dry rubber contact was and that for gelatin on poly(methyl methacrylate) was When water was present at the rubber contact, the work of adhesion dropped to and this was consistent with Young’s equation for the contact angle of 66° measured for water droplets sitting on the smooth rubber. This was the first time that Young’s equation had been verified by direct measurement. When 0.01 M sodium dodecyl sulfate (SDS) solution was the immersion medium, the rubber contact size fitted the Hertz equation down to the lowest loads obtainable, showing that the work of adhesion was less than This was consistent with the wetting behavior of the soap solution on the rubber surface.

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9.3. THE NATURE OF ADHESIVE CONTACT This theory and its supporting experimental evidence changed the conception of adhesive contact considerably (Fig. 9.7). Not only was it evident that molecular adhesion could have a considerable effect for small particles and for small loads, where the particles were significantly attached by the adhesive forces, but also the elastic contact was acting as its own measuring device which sensed adhesion. In short, molecular adhesion between solids could be measured by observing the size of Newton’s black spot, while knowing the elasticity and geometry of the particles. The black contact spot was a molecular adhesion sensor. The assumption of very short-range molecular force, which acts only within the contact spot, is reasonable for contact sizes larger than a few nanometers though much discussion has raged on this.4 There is then an “infinite stress” at the contact edge, exactly the same kind of stress singularity found in cracking problems, as shown by Griffith. Obviously, such an “infinite stress” cannot exist in reality. Near the crack tip, the molecules must be fluctuating rapidly with Brownian movement and the crack will be making and breaking many times per second. The one parameter model fails at the molecular level and a two parameter model is required.13 When tensile loads were applied to pull the spheres apart, an equilibrium contact spot size could be obtained as the load was reduced, but below a certain contact size, equilibrium could no longer be found and the surfaces then came apart rather quickly at a load given by

where D was the diameter of the equal spheres. This result showed that Derjaguin was not quite right when he said that elastic deformation did not much change the

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adhesion result from that for rigid bodies.4 Elastic modulus does not enter Equation (9.8). However, the size of the force does change slightly with the assumption of elasticity because Equation (9.8) differs from Bradley’s equation for rigid spheres. Of course, the adhesion is changed much more by two other effects: the first is by roughening the surfaces which reduces adhesion enormously; the second is by adhesion kinetics and hysteresis, which slow down the approach to contact equilibrium and produce irreversibility in the process. Two obvious ways exist to observe these contact spot effects: direct optical observation and ultrasonic inspection.

9.4. ROUGHENING THE SURFACES Measuring the contact between rough surfaces is difficult because the black contact spot then becomes broken up, and contact is then only made at a few localised contact spots. This difference between smooth surface and rough surface contact is illustrated in Fig. 9.8. This shows the contact between glass lenses pressed together, and underneath gives the appearance of the Newton’s ring interference pattern observed in reflected light, looking through the lenses. The circular black contact spot is seen for the smooth lenses (Fig. 9.8(a,b)) and this expands as the lenses are pressed together further and contracts as the lenses are unloaded. To see what happened for rough surfaces, the smooth lenses were rubbed together to scratch the contact.

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The Newton’s ring pattern then changed substantially. A few smaller black spots could be seen where the roughened surfaces were making molecular contact, but the rings showed the surfaces were now being held apart by the debris at the contact. A single black spot could not now be produced however hard the lenses were pushed together. A simple way to quantify this effect is in terms of contact stiffness S, defined as the extra load dF required to produce a small movement of the lenses towards each other, i.e.

where F is the force on the lens and is the lens displacement. This contact stiffness can be worked out for smooth spheres without adhesion from the Hertz Equation (9.2) to give the expression for the Hertz contact stiffness

which shows that the stiffness should increase with the size d of the black spot, that is with the cube root of load. But the interesting thing about stiffness is that it can be measured by passing ultrasonic waves through the contact.12,14 Figure 9.9(a) shows the results of ultrasonic transmission measurements through a glass lens pressed into a glass flat. The more the glass was pressed together, the more ultrasonic signal penetrated the black spot, allowing the stiffness to be plotted, and demonstrating agreement with Equation (9.10) above. However, after the glass surfaces had been roughened by sliding the lens across the glass plate, the ultrasonic transmission fell, showing a lower stiffness which now varied with and not with the Hertz This would be expected from either plastic flow of material at the contact, or from elastic deformation of multiple contacts.10, 15

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Such behavior is important to the understanding of frictional phenomena. It has been known since Amontons10 in the 17th century that friction, the force required to slide one solid over another, increases in proportion to the load pressing the solids together. This is clearly inconsistent with the Hertz equation for contact area. In the Hertz case, the contact diameter increases with thus the contact area increases with and therefore the friction force should also increase with at odds with the Amontons observations. However, the behavior shown in Fig. 9.9(b), after sliding causes scratching of the surfaces, is consistent with Amonton’s Law. The contact size increases with so that the contact area, and therefore friction, is then proportional to load The effect of ultrasonic vibration at contacts has an interesting history. Originally, Fitzgerald16 had pressed an ultrasonic transducer onto a coin or ball bearing and heard a buzzing from the contact, which he associated with quantum effects. Hopkins17 soon showed that a much simpler explanation for the buzzing was the Hertz stiffness of the contact spot, which allowed the contact to resonate at a frequency much lower than the sphere resonant frequency. Then Kleesattel18,19 and his colleagues began to use the buzzing effect to measure the plastic contact deformation under an indenter, and thus invented the ultrasonic hardness tester. This was a breakthrough in hardness determinations for metal parts because rapid measurements could be made, and the damage to the surface was much smaller than the traditional Vickers diamond test. The ultrasonic method can also be used to measure molecular adhesion. According to Hertz, the contact stiffness is zero at zero load and so the ultrasonic transmission should be zero. But experiments show that ultrasonic waves are transmitted through contacts at zero load. This proves that adhesive molecular

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forces are pulling the surfaces together. From the JKR theory, the contact stiffness at zero load is

giving the transmission behavior for glass lenses shown in Fig. 9.10. This method is especially useful for contact between rough surfaces where the contact spot is complex.

9.5. EFFECT OF ROUGHNESS ON PARTICLE ADHESION It is obvious that a particle sitting on a surface will adhere differently when there is roughness. This is clear from Fig. 9.11 which shows three situations: smooth surfaces, wavy surfaces with large diameter roughness, and rough surfaces with small diameter asperities. The smooth sphere on a smooth surface shown in Fig. 9.11 (a) requires a removal force

where the minus sign indicates tension and the force is twice that for two equal spheres (Equation (9.8)) because However, when the particle is lying in a large diameter hollow, the force increases, and when the particle lies on the small diameter asperities of Fig. 9.11(c) the force decreases. Thus, the change in adhesion for simple surface roughnesses can be readily explained by the JKR theory. Because of the importance of particle adhesion in photocopiers,20 Mizes21 used an atomic force microscope to measure both surface roughness and adhesion simultaneously and showed that they were related by this argument. He demonstrated on four different surfaces that the adhesion fluctuations were directly related to the surface curvature fluctuations, as shown in Fig. 9.12.

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The problem gets worse when more complex surface roughnesses exist at the particle contact. A realistic description of contact under these circumstances was provided by Greenwood and Williamson,22 who showed that three parameters were needed to describe the contact adequately: the standard deviation of the height distribution, the radius of each asperity, and N the number of asperities per unit area. They then presented data for real surfaces which satisfied the relationship Fuller and Tabor23 used such a model to show that adhesion was reduced drastically by minor surface roughness. They made the simplifying assumption that all the asperities had the same curvature, but had heights following a Gaussian distribution. Experimental results on the adhesion of smooth rubber

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to roughened poly(methyl methacrylate) fitted the theoretical curve in Fig. 9.13 very satisfactorily. These results are relevant to the removal of particles from contaminated surfaces by air jets. Zimon and his followers24,25 had observed glass spheres sticking to surfaces and had measured the air velocity necessary to dislodge them. The results shown in Fig. 9.14 show that a higher velocity is needed for smaller particles, because the adhesion force is larger in accord with Equation (9.12). However, a much smaller velocity is required for rough surfaces. Understanding the mechanism of detachment is difficult since the flow is complex and both rolling and sliding of particles can occur. Rolling seems to explain the results in terms of the JKR model.26

9.6. FRICTION OF FINE PARTICLES It is evident from the discussion above that large particles behave entirely differently to small particles because of roughness affecting adhesion. Large particles will generally have roughnesses on their surfaces which prevent good adhesion, whereas small particles, of diameter less than the asperity diameter, will appear smooth and will therefore obey the JKR equations. We must therefore reevaluate the early theories of soil mechanics and friction which treat all particles as equal. Coulomb was the first person to put together a theory of soil behavior in order to design dams and retaining walls.27 Consider the experiment shown in Fig. 9.15 in which a powder bed containing damp silica sand loaded with a

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weight is pushed along a silica plate by a traction force Compare this with a solid block of wet silica loaded under the same conditions. Coulomb’s friction theory for particles was based on the idea that the traction force divided by the plate area was given by an adhesion pressure K plus a friction term where was the friction coefficient, i.e.

The problem with this equation is threefold. First, the friction coefficient calculated from Equation (9.14) in the experiment of Fig. 9.15(a) does not match that measured in Fig. 9.14(b). 0.5mm sand grains give whereas two large quartz blocks give Second, the friction of sand grains increased as the particles got smaller, rising to 0.6 for wet sand 0.01 mm in diameter. Finally, the friction of fine powders is not constant as the normal pressure is increased but falls substantially at high loads, in conflict with Equation (9.14). These discrepancies are explained by the JKR theory as first defined by Barquins, Maugis and colleagues.28,29 Although Equation (9.14) is the simplest expression which takes into account both the external load and the molecular adhesion, it cannot be correct for elastically deforming bodies because there is then an extra term arising from the interaction of the applied load and the adhesive forces. From the JKR expression for a particle on a flat surface, Fig. 9.16, the effective load pressing the particle into the surface is

so that the shear force is

which has its first two terms similar to Coulomb’s equation but now has the extra third term showing the interaction of the external and surface forces.

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The results on crossed polymer fibers obtained by Briscoe and Kremnitzer30 shown in Fig. 9.16 gave good agreement with Equation (9.16). The friction force increased more rapidly than Coulomb’s law predicted at low loads but approached Coulomb’s law at high loads. The value of the work of adhesion W from these friction experiments was near close to the value determined in direct adhesion experiments on the same fibers. The logical conclusion from these arguments is that the friction coefficient of powders must vary substantially with both particle diameter and applied pressure. This fits the experience of soil mechanics where it is known that finer grains give higher friction. Some results for wet sand are shown in Fig. 9.17. The macroscopic silica samples gave a friction coefficient of 0.33, whereas the finer particles gave an apparently higher value up to 0.66 for grains.31 Clearly, this apparent rise in Coulomb law friction follows from Equation (9.16) which leads to the idea that the apparent friction coefficient should rise for small grains at low pressures according to the equation

This fitted the experimental results very well when the work of adhesion was taken to be 2 J m, which is larger than expected for wet silica. Hysteresis in the measurements is the probable cause of the discrepancy.28 A similar frictional behavior has been found in atomic force microscope experiments.32

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A platinum tip, diameter 280 nm, was brought into contact with an atomically smooth mica surface in ultrahigh vacuum. Although contact area was not measured directly, the friction results showed a good fit to JKR theory, as illustrated in Fig. 9.18. Sliding friction could be measured even when a tensile force was applied to pull the probe from the substrate. These results were similar to those of Homola et al who measured friction and contact area between smooth

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mica, and also found agreement with JKR theory.33 Johnson has interpreted several of these results in terms of detailed contact mechanics.34

9.7. ELASTIC SINTERING OF FINE PARTICLES The question of what happens when a bunch of particles comes together was asked by Newton, who found the process difficult. Yet if all particles adhere strongly as described above, they should leap into contact, then deform and squash closer to each other as a result of molecular adhesion. It is a paradox that particles which adhere strongly do not do this. Instead they form weak, loose treelike structures because each particle sticks where it touches, as shown in Fig. 9.19(a). This instant adhesion therefore prevents good compaction, which can only be achieved if the particles do not adhere strongly but can wander around to find dense close-packing positions, Fig. 9.19(b). The process of particles coming into adhesive contact is agglomeration. This should be distinguished from the subsequent processes of deformation and shrinkage which happen during sintering. Sintering is the reduction in surface area of particles, driven by the urge to reduce the surface free energy of the material. It is clear that sintering of two elastic particles can occur by the JKR mechanism because the surface area of a sphere, originally is reduced by the contact spot area formed by elastic deformation, i.e. in Fig. 9.20, so that the fraction of elastic sintering is or from Equation (9.7)

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If the agglomeration process has been allowed to proceed to close-packing then there are twelve such contact spots on each sphere, Fig. 9.20(b), so that the sintered fraction is 12 times that of Equation (9.18). Putting in numbers shows that particles of 1 GPa modulus will be about 1% sintered when closepacked under a work of adhesion This sintering idea was first proposed to account for the coalescence of rubber latex particles by Bradford and his colleagues in 1951, 35–37 but they presumed that the particles were viscous liquids. However, there is no doubt that elastic latex particles can also coalesce to form strong, dense coatings. Generally, though, latex polymers used in paints and adhesives are near their glass transition temperature and are not fully elastic. Brown38–40 and followers thought that capillary forces were also important in coalescence as the water dried out, and Voyutskii41 believed that diffusion was also necessary to account for the time dependence and irreversibility of the coalescence phenomenon. Obviously, a key feature of coalescence in paints and adhesives is that water does not redisperse the particles, so the products are water resistant. 42 The JKR explanation of latex coalescence was proposed in 1982. Padget had observed the hexagonal structure of coalesced rubber latex (Fig. 9.21 (a)) and Kendall had measured the contact spot sizes between latex particles using electron microscopy (Fig. 9.21(b)). When the results were plotted in Fig. 9.21(c), they fitted the JKR equation and macroscopic observations, taking the elastic modulus to be 5.64MPa and the work of adhesion to be The theory of coalescence required two stages42: the first was an agglomeration step which was driven by drying of the latex film, allowing the particles to be pushed together into close-packed adhesive contact, but with small adhesion because of the presence of water; the second was a sintering step in which the work of adhesion increased as the last water was removed and elastic deformation occurred with shrinkage.

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The adhesion of particles by such mechanisms is vitally important in Pharmaceuticals,43 xerography,44 semiconductors,45 printing, and agriculture. Many articles are written on these topics each year. A particular contribution has been made by Rimai, Demejo and Bowen in understanding the adhesion of toner particles which must transfer from a photoconductor to a receiver.46 JKR behavior was observed for glass spheres on polyurethane, as shown in Fig. 9.22. Curious effects of large deformation,47 engulfment48 and hysteresis were seen. This hysteresis is to be considered next.

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9.8. HYSTERESIS AND DRAG IN THE CONTACT OF SPHERES When particles make elastic contact with a surface, equilibrium is not attained immediately. Time is required for the contact spot to enlarge, and more time is needed for the contact to separate when a tensile force is imposed. This is adhesive drag. Indeed, equilibrium may never be attained. On making the contact, the spot size has a certain diameter at a given load. When breaking the contact at an identical load, the contact spot is bigger. This is known as adhesive hysteresis, which was observed by Drutowski in 1969. These effects may be studied systematically with smooth elastomer spheres at zero load as shown in Fig. 9.23.49

To obtain the making contact curve, the loading screw was adjusted until Newton’s ring fringes appeared. Then, suddenly, on further approach, the surfaces leaped into contact and the growing contact spot was recorded on the TV camera. The diameter increased rapidly for about one hundred seconds and then remained constant for days. To measure the breaking contact curve, the glass plate was pressed momentarily into the rubber and then released. The large contact spot formed by the initial load then rapidly decreased in diameter for about a hundred seconds and then remained constant. The final contact diameter at zero load was almost twice that for making contact. This is adhesive hysteresis, which may be viewed as similar to contact angle hysteresis. A drop of liquid placed on a surface grows out to form a circle of contact, but when a large volume droplet is shrunk to form an equal size drop, the contact circle is larger, as shown in Fig. 9.24. These observations demonstrate that adhesive drag and hysteresis are not caused by the solid nature of the materials, but by intrinsic properties of the interface, moderated by lossy effects within the materials. The phenomena of adhesive drag and hysteresis are relevant to the tack of solids. Certain adhesive tapes need to be tacky so as to “grab” the attaching

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surface. Originally, this concept derived from the thumb test in which the immediate force of adhesion to a sticky tape was estimated by hand. Wetzel50 devised an instrumented version where a hemispherical probe was rested on a tacky surface for a short time and the force of attachment measured. This depended greatly on time, temperature and pull-off rate.51 Both Gent and Kaelble worked on these ideas in the 1960s,52,53 taking the view that the rheological behavior of the materials was most important. To obtain a more general explanation of these complex effects, Kendall54, 5 separated out the drag and hysteresis terms. The drag was viewed as an interface reaction which was kinetically controlled by a surface energy barrier, whereas the hysteresis was viewed as a crack stopping effect brought about by the lossy relaxation of the inelastic material. The influence of these two separate phenomena was best demonstrated by measuring the contact spot size at various times and temperatures, as shown in Fig. 9.25.

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The experiment was conducted as before, measuring the contact spot size d both making and breaking the contact, then calculating the adhesive energy and crack speed v from the TV record using the equation derived from equation 9.7

where D was the sphere diameter, E its Young’s modulus, and v its Poisson’s ratio. The measured adhesive drag at 20° C was fitted by a power law expression giving as a function of crack speed

and the temperature dependence was fitted to an Arrhenius type of curve. Then it was discovered that the crack stopping effect became more noticeable at lower temperatures as the viscoelastic loss in the rubber increased. The viscoelastic loss was quantified in terms of the relaxation constant C where

C was 0.01 for the rubber at room temperature, but increased at lower temperatures. The results were fitted to the following equation which was solved numerically:

where x is the distance travelled by the crack and the term is a crack stopping term which increases with the relaxation of the rubber, as described in Chapter 8. Essentially, this is a crack blunting expression. The theory gave a reasonable description of the results as shown in Fig. 9.25, explaining why the adhesive drag and hysteresis of rubber both increase substantially at low temperature. This theory also explained the high hysteresis of uncrosslinked (i.e. raw) natural rubber. Raw rubber is rather tacky, as Goodyear found originally when he became bankrupt as a result of his products sticking together. But crosslinking the rubber with sulfur has a marked effect on reducing tackiness. The contact hysteresis experiments showed the cause of this effect by comparing the adhesion energy versus crack speed curves for raw and crosslinked rubber spheres, as shown in Fig. 9.26. Both materials fitted the same adhesive drag curve at crack speeds higher than But the results deviated at lower speeds because the crack stopping occurred differently for the two materials. The raw rubber showed crack stopping rather quickly as the crack proceeded through the contact spot, whereas the crosslinked rubber allowed the crack to penetrate further before stopping. This could be explained by the higher relaxation constant for the raw rubber, 0.08 as opposed to 0.01. The results gave a reasonable fit to the theory of Equation (9.22). Thus the enhanced tack was interpreted as a larger contact spot

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caused by premature crack stopping as the crack blunted. This crack blunting becomes enormously important for plastically deforming materials.

9.9. PLASTIC CONTACT OF PARTICLES It was during the 1950s that experiments showed plastic deformation of particles caused by adhesive contact with surfaces. Krupp56 originally proposed that the adhesion forces could rise to such large values that they would exceed the elastic limit of the material and induce irreversible, plastic flow. He allowed small gold spheres to rest on a surface and then estimated adhesion by rotating the surface at high speed to dislodge the spheres by centrifugal force. The gold particles were soft and became permanently flattened by the adhesive contact deformation, as shown by electron microscope observations, Fig. 9.27(a). Only small particles, less than in diameter, would show this effect. Krupp explained this in terms of the equations for London–van der Waals attractive forces between rigid spheres, together with the Hertz equations of contact. Because the attraction is proportional to particle diameter, the force at the particle contact decreases with D. However, the elastic area of the contact spot decreases faster, from the Hertz Equation (9.1), with Thus, as the particle gets smaller, the contact pressure must rise to the point at which plastic deformation occurs. Such plastic effects were shown dramatically by Easterling and Tholen57,58 in electron micrographs of metal particles sticking together. They showed that

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dislocations and plastic twins could be observed emanating from the highly stressed contact region, as shown in Fig. 9.27(b). They also showed the strains in the surrounding elastic crystals by a special imaging technique. Scanning electron microscope studies were performed on polystyrene spheres sitting on polished silicon surfaces by Rimai, Demejo and Bowen.59 The bulk polymer had a Young’s modulus of 2.55 GPa and a yield stress of 10.8 MPa when measured on a testing machine. With such a low yield point it was estimated that the particles should be plastically deformed under the adhesion forces. Therefore they applied the plastic deformation theory of Maugis and Pollock60 to fit the results, as shown in Fig. 9.28. This gave the expression for contact diameter d in terms of sphere diameter D

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where W was the work of adhesion and Y the yield stress. The work of adhesion was from these results, but this value was sensitive to the yield point measurements and may have been overestimated. This mechanism of plastic coalescence can provide a route by which fine powders can sinter, just as elastic deformation gave Equation (9.18). For closepacked plastic particles the sintered fraction is

where H = 3 Y is the hardness of the material. This equation shows that even very hard l00nm diameter particles with H of 1 GPa will be 10% sintered if the work of adhesion is This concept of plastic sintering was first used by Wollaston to compact platinum powder together, and is now regularly used to make powder metal components, obtaining up to 90% density at room temperature by pressing micrometer size metal powders in dies (see Fig. 9.29).61 The process has also been used to sinter nanosize ceramic particles at modest temperatures62 well below the melting point. For example, 10 nm titania particles sintered to a dense product at 500° C, whereas micrometer scale powders required more than 1000°C. After plastic compaction, the resulting powder compact is then fully densified by high-temperature sintering, raising the temperature in an inert atmosphere to a point where diffusion can occur to transport material into the pores.

9.10. SINTERING OF PARTICLES BY DIFFUSION MECHANISMS The simplest example of diffusion-controlled sintering is the fusion of two spherical liquid droplets under the action of molecular attractions. This was considered by Frenkel63 in his famous theory published in 1945 and later discussed by Kuczynski.64

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Frenkel imagined two spherical liquid droplets meeting slowly so that the surfaces became drawn together by the molecular surface attractions, as in Fig. 9.30(a). The liquid surfaces should leap into contact as a result of such attractions, but the growth of the contact spot must then be limited by the viscous resistance to flow of the liquid spheres. At short times, the movement of the spheres towards each other should be proportional to time t and inversely related to the viscosity as indicated in Fig. 9.30(b), giving the contact spot diameter d in the equation

in which W is the work of adhesion. This eventually pulls the two droplets into a single larger sphere as illustrated in Fig. 9.30(c). Equation (9.25) is interesting because it is quite different from the elastic and plastic sintering Equations (9.7) and (9.23) given earlier. In all cases, the sintering is driven by the molecular work of adhesion W, but the resistance to coalescence may be elastic, plastic, or viscous. The elastic case is simplest because the work of adhesion pulls the spheres together into an equilibrium with the elastic resistance, storing the energy reversibly in the elastic strain field. The contact spot size is then constant. By contrast, for the viscous case, coalescence occurs progressively with time, and the contact spot increases irreversibly until the two droplets merge to form a single large drop. The plastic case is intermediate in that flow only occurs above the flow stress, and the flow is then so fast that the process is assumed to be independent of time. From Equation (9.25) it is evident that the droplets move towards each other as the viscous coalescence occurs. The linear shrinkage of a cubic packing of such droplets would therefore be given by showing that shrinkage proportional to time should be experienced, as found experimentally for the glass spheres in Fig. 9.31 studied by Cutler and his colleagues.65,66 As the temperature was increased, the sintering occurred faster, as

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expected from a decrease in glass viscosity. However, the relation between adhesion energy and viscosity is not easy to define by this method. There are two problems of sintering which are not explained by the equations such as (9.26) above. The first is that sintering should proceed in proportion to time, but instead stops prematurely; the second is that the expected shrinkage rate should be larger for smaller diameter particles but often is not. Both these problems are connected to the fact that powder compacts contain agglomerates, which get stronger and more numerous as the particles get finer. Agglomerates compacted together as in Fig. 9.32 sinter like larger particles of diameter D and so shrink more slowly than expected from the primary particle size. Additionally, agglomerates can act like foreign inclusions which inhibit

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sintering by not allowing uniform shrinkage. The molecular adhesion force is so feeble that it is readily retarded by such slight heterogeneities. Another key problem with diffusive sintering is that it can occur by several diffusion mechanisms as shown in Fig. 9.33. Viscous and plastic flow are two simple possibilities but another five are readily distinguished, including solidstate diffusion, grain boundary diffusion, surface diffusion, gas phase transport, and liquid layer transport. These inevitably form a neck at the particle contact, to reduce the sharp curvature of the contact region. Herring67 showed that the time t required for these diffusion processes scaled with particle size D to a power n which was 1 for viscous flow, 3 for bulk diffusion, 4 for grain boundary diffusion etc., i.e., Although these mechanisms transport material, they may not cause shrinkage and pore removal. The alternative possibility is grain and pore growth, in which the structure coarsens without shrinkage, as shown schematically in Fig. 9.34. These various diffusion models were considered by Kuczynski, Coble and followers in an effort to control the final structure of fired particulate materi-

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als.68–70 A particular success of this endeavor was the invention of the alumina envelope for sodium vapor lamps. To make these transparent products from sintered powder is difficult because pore volume has to be reduced well below 1% to prevent light scattering. This can only be achieved by adding magnesium oxide to inhibit coarsening and by firing in atmospheres such as hydrogen so that pores can diffuse easily out of the material. Since the discovery of this process in 1959, and its detailed study by Brook and colleagues, production has risen to 16 million per year.71–72

9.11. REFERENCES 1. 2. 3. 4.

5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

Newton, I., Opticks, Smith and Walford, London, 1704 (reprinted Dover, New York, 1952, p 397). Bolton, M., A Guide to Soil Mechanics, Macmillan, London, 1979, chap 4. Hertz, H., Miscellaneous Papers, ed. P. Lenard, Macmillan, London, 1896, p 146. Johnson, K.L., Contact Mechanics, Cambridge University Press, Cambridge, UK 1985; see also Johnson, K.L. and Greenwood, J.A., J. Colloid Int. Sci. 192, 326–33 (1997); see also Maugis, D., Contact, Adhesion and Rupture of Elastic Solids, Springer, Berlin (1999) 284–296. Bradley, R.S., Phil. Mag. 13, 853–62 (1932). Bradley, R.S., Trans. Faraday Soc. 32, 1088 (1936). Derjaguin, B.V., Kolloid Z. 69, 155 (1934). Derjaguin, B.V., Krotova, N.A. and Smilga, V.R., Adhesion of Solids (translated by R.K. Johnson), Consultants Bureau, London, 1978, pp 423–42. Johnson, K.L., Kendall, K. and Roberts, A.D., Proc. R. Soc. A 324, 301–13 (1971). Bowden, F.P. and Tabor, D., Friction and Lubrication of Solids, Part 1 (1950), Part 2 (1964), Clarendon Press, Oxford. Roberts, A.D., Engng Mater. Des. 11, 579 (1968). Kendall, K., PhD Thesis, University of Cambridge, 1969. Johnson, K.L., Br. J. Appl. Phys. 9, 199 (1958). See also Sperling, G., Doktor-Ingenieurs Dissertation, Technischen Hochschule Karlsruhe, 1964, p75. See also Leng, Y.S., Hu, Y.Z. and Zheng, L.Q., Proc. R. Soc. A 456, 185–204 (2000). Kendall, K. and Tabor, D., Proc. R. Soc. A 323, 321–40 (1971). Archard, J.F., Nature 172, 918 (1951). Fitzgerald, E.R., J. Acoust. Soc. Am. 36, 2086 (1964). Hopkins, I.L., J. Acoust. Soc. Am. 38, 145 (1965). Kleesattel, C. and Gladwell, G.M.L., Ultrasonics 6, 175 (1968). Kleesattel, C. and Gladwell, G.M.L., Ultrasonics 7, 57 (1969). Schein, L.B., Electrophotography and Development Physics, Springer Verlag, New York, 1988. Mizes, H.A., In: Advances in Particle Adhesion, ed. D.S. Rimai and L.H. Sharpe, OPA, Amsterdam, 1996, 155–65. Greenwood, J.A. and Williamson, J.B.P., Proc. R. Soc. A 295, 300–19 (1966). Fuller, K.N.G. and Tabor, D., Proc. R. Soc. A 345, 327–40 (1975). Zimon A.D., Adhesion of Dust and Powder, Consultants Bureau, New York, 1982. Taheri, M. and Bragg, M., J. Aerosol Sci. Technol. 16, 15–20 (1992). Soltani, M. and Ahmadi, G., In: Advances in Particle Adhesion, ed. D.S. Rimai and L.H. Sharpe, OPA, Amsterdam, 1996, pp 105–23.

210 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67.

CHAPTER 9 Coulomb, C.A., Mem. Acad. Sci. Savants Etrangers 7, 343–82 (1773). Kendall, K., Nature 319, 203–5 (1986). Barquins, M., Maugis, D. and Courtel, R., C. R. Acad. Sci. Paris 280B, 49–52 (1975). Briscoe, B.J. and Kremnitzer, S.L.,J. Phys. D: Appl. Phys. 12, 505–15 (1979). Rowe, P.W., Proc. R. Soc. A 269, 500–27 (1962). Carpick, R.W., Agrait, N., Ogletree, D.F. and Salmeron, M., Langmuir 12, 3334–40 (1996). Homola, A.M., Israelachvili, J.N., McGuiggan, P.M. and Hellgeth, J.W., Wear 136, 65–84 (1990). Johnson, K.L., Langmuir 12, 4510–13 (1996). See also Sridhar, I., Johnson, K.L. and Fleck, N.A., J. Phys. D: Appl. Phys. 30, 1710–9 (1997). Dillon, R.E., Matheson, L.A. and Bradford, E.B., J. Colloid Sci. 6, 108 (1951). Henson, W.A., Taber, D.E. and Bradford, E.B., Ind. Engng. Chem. 45, 735 (1953). Bradford, E.B. and Vanderhoff, J.W., J. Macromol. Chem. 1, 335 (1966). Brown, G.L., J. Polym. Sci. 22, 423 (1956). Sheetz, D.P., J. Appl. Polym. Sci. 9, 3759 (1965). Myers, R.R. and Knauss, C.J., J. Paint Technol. 40, 315 (1968). Voyutskii, S.S. and Starkh, B.W., Physical Chemistry of Film Formation from High Polymer Dispersions, Moscow 1954. Kendall, K. and Padget, J.C., Int. J. Adhesion Adhesives 2, 149–54 (1982). Lamm, K.K. and Newton, J.M., Powder Technol. 73, 267 (1992). Mastrangelo, Photo. Sci. Eng. 26, 194 (1982). Menon, V.B., Michaels, L.D., Donovan, R.P. and Ensor, D.S., In: Particles on Surfaces 2: Detection, Adhesion and Removal, ed. K.L. Mittal, Plenum, New York, 1989, pp 297–306. Rimai, D.S., Demejo, L.P. and Bowen, R.C., In: Fundamentals of Adhesion and Interfaces, eds. D.S. Rimai, L.P. Demejo and K.L. Mittal, VSP, Utrecht, 1995, pp 1–23. Lee, A.E., J. Colloid Interface Sci. 64, 577–8 (1978). Rimai, D.S., Demejo, L.P. and Bowen, R.C., In: Advances in Particle Adhesion, eds. D.S. Rimai and L.H. Sharpe eds., Gordon and Breach, Netherlands, 1996, pp 139–54. Kendall, K., J. Adhesion, 7, 55–72 (1974). Wetzel, R.H., ASTM Bull, TP72, 221 (1957). Counsell, P.J.C., Aspects of Adhesion vol 7, eds. D.T. Alner and K.W. Allen, Transcripta, London, 1973, p 202. Gent, A.N. and Petrich, R.P., Proc. R. Soc. A 310, 433 (1969). Kaelble, D.H., J. Colloid Sci. 19, 413 (1964). Kendall, K., J. Adhesion 5, 179-202 (1973). Kendall, K., J. Polym. Sci. Polym. Phys. 12, 295–301 (1973). Krupp, H., Adv. Colloid Interface Sci. 1, 1 1 1 (1967). Easterling, K.E. and Tholen, A.R., Ada Met. 20, 1001–8 (1972). Tholen, A.R., In: Microscopic Aspects of Adhesion and Lubrication, ed. J.M. Georges, Elsevier, Amsterdam, 1982, pp 263–77. Rimai, D.S., Demejo, L.P. and Bowen, R.C., In: Fundamentals of Adhesion and Interfaces eds. D.S. Rimai, L.P. Demejo, and K.L. Mittal, VSP, Utrecht, 1995, pp 1–23. Maugis, D. and Pollock, H.M., Acta Met. 32, 1323 (1984). Wollaston, W.H., Phil. Trans. R. Soc. 119, 1–5 (1828). Gleiter, H., In: Deformation of Polycrystals: Mechanisms and Microstructure, eds N. Hansen et al, Risoe National Laboratory, Roskilde, 1981, p 15. Frenkel, J., J. Phys. USSR 9, 385 (1945). Kuczynski, G.C., J. Appl. Phys. 20, 1160–3 (1949). Henrichsen, R.E. and Cutler, I.B., Proc. Br. Ceram. Soc. 12, 155 (1970). Cutler, I.B., J. Am. Ceram. Soc. 52, 11 (1969). Herring, C., J. Appl. Phys. 21, 301–3 (1950).

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68. Kuczynski, G.C., Trans. Am. Inst. Miner. Metall. Eng. 185, 169–78 (1949). 69. Coble, R.L., J. Appl. Phys 32, 793–9 (1961). 70. Coblenz, W.S., Dynys, J.M., Cannon, R.M. and Coble, R.L., In: Sintering Processes ed. G.C. Kuczynski, Materials Science Research, vol. 13, Plenum, New York, 1980, pp141–57. 71. Brook, R.J., In: Treatise on Materials Science and Technology, vol. 9, Academic Press, New York, 1976, pp 331–64. 72. Bennison, S.J. and Harmer, M.P., Ceramic Trans. vol 7, Am Ceram Soc, Westerville, OH, 1989, pp 13–19. 73. Rhodes, W.H. and Wei, G.C., In: Advanced Ceramic Materials, ed. R.J. Brook, Pergamon, Oxford, 199l, pp 273–6.

10 ADHESION OF COLLOIDS: DISPERSION, AGGREGATION, AND FLOCCULATION

The resisting Power of fluid Mediums . . . is very nearly as the D iameter and the Velocity of the spherical Body together I SAAC N EWTON , Opticks, 1 p. 365 (precursor of Stokes Law for viscous force on a particle, F = 3πηD V, used in equation 10.2).

Since, from the first law of adhesion, particles always adhere, the natural state of particulate materials is aggregated as in Fig. 10.1 (a), with each particle stuck fast to its neighbors. However, when the particles become contaminated with liquid, the second law of adhesion applies and the adhesion is reduced. In certain circumstances, this reduction in adhesion proceeds so far that the particles remain separate or dispersed in the contaminating medium, as in Fig. 10.1(b). Such bizarre behavior is a challenge to the theorist because it seems to go against the basic laws of adhesion. The purpose of this chapter is to show that several complex mechanisms cause the particles to behave strangely. The most important mechanism is Brownian movement, followed by electric charging, followed by adsorption of polymer molecules at the surface.

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10.1. UBIQUITY OF COLLOIDS The colloidal state of materials is extraordinary on several counts.1,2 First of all, it is so ubiquitous in nature and such a powerful influence on our lives. For example, the silvery colors of mountain rivers indicate the presence of minute clay particles which are carried downstream to form huge deltas as the particles are flocculated and adhered by salt in the ocean. In another example, clouds in the sky turn darker in color as the fine water droplets adhere together before a storm. Without clouds, formed from fine particles of water dispersed in the atmosphere, our earthly planet would be uninhabitable. Yet the detailed process by which clouds exist in dispersed form, then aggregate to precipitate as rain is not fully understood. As a final example, consider the blood in our bodies. The fine red corpuscles must flow easily to penetrate the smallest capillaries, yet must coagulate when the blood vessels are damaged to prevent fatal hemorrhage. Such fine chemical tuning of the aggregation process demands a sophisticated explanation. The second extraordinary feature of colloids is the problem that colloids should not really exist according to the first law of adhesion. Just as the clouds should not stay up in the atmosphere, because water is much denser than air, so the droplets should not remain separate but should coalesce under van der Waals attractions to form one huge water mass. The necessity of this coagulation is best explained by considering the enormous surface area of the cloud droplets. Each droplet is spherical because it strives to minimize its surface area. In the same way, the droplets should aggregate to reduce the overall surface and form a single large sphere, going from Fig. 10.2(a) to (b). The energies involved in this coalescence are known from experiments like the Wilhelmy plate shown in Fig. 10.2(c). A thin wetted plate, width L, dips into the water and is sucked in by

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the surface force. Thus a force F has to be applied to resist the engulfing surface energy where

so that can be measured as The factor of 2 arises because both faces are pulled down by the water meniscus. When the water droplets coalesce, to reduce the total surface energy, the surface energy is released as heat and it is evident that the temperature rise of the water is inversely proportional to the droplet diameter. Thus, Fig. 10.3 shows that 1 nm diameter water drops can be boiled by the coalescence phenomenon, illustrating the thermodynamic driving force for aggregation. So the driving force for coalescence of colloidal particles is very large. Why, then, do clouds remain so stable in the sky? This question brings us to the third amazing feature of colloidal particles, their sensitivity to small changes in environmental conditions. Thus, clouds can be seeded by small dust particles to cause rain, clay suspensions can coagulate when they reach the sea, and bleeding can be stopped by a styptic pencil of aluminum chloride. The conclusion is that colloidal dispersions are in a metastable state which can switch easily into the aggregated state. Since we can detect the change very sensitively, it follows that colloids are adhesion sensors, which we can use to study molecular forces.

10.2. COLLOIDS AS ADHESION SENSORS Colloidal particles offer us a direct bridge into the molecular adhesion world, allowing us to see molecular adhesive phenomena with the naked eye. This is

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because our eyes are tuned to light about in wavelength, so that particles near this size give us maximum color change as coagulation events occur. The original observation using optical microscopy by Robert Brown of the incessant random movement of pollen grains suspended in water is an example of how molecules reveal their motion by impacts on a particle suspended in a molecular fluid. Brown showed in 1827 that such small particles could act as molecular sensors. Thus a diameter particle in water at room temperature randomly moves about in one second under molecular impacts. The mathematics of the movement were described by Einstein3 and the molecular nature of the motion proved by Perrin4 who showed that Avogadro,s number could be obtained by observing the equilibrium concentration of colloidal particles at various heights in water. Essentially, Perrin demonstrated that the fine particles were behaving like large molecules. Thus, Brownian movement explains the dispersed nature of colloids. The particles obey the kinetic theory, each particle on average displaying a kinetic energy of 3kT/2, undergoing elastic collisions with its neighbors, so the particles do not fall easily under gravity. The use of color change in a colloid to measure adhesion was first shown by Faraday5 who had prepared gold dispersions by precipitation from solution and marveled at their brilliant red color and stability over time. Preparations he made are still standing at the Royal Institution in London without significant coagulation. By adding salt to the gold suspension, Faraday showed that the particles immediately adhered as shown by the change in color from red to blue. He also demonstrated that gelatin molecules in solution could protect the gold particles from adhering. Addition of salt failed to produce the color change in the presence of gelatin, showing the stabilizing effect of polymer. The optical method for observing the adhesion of colloidal particles is shown in Fig. 10.4. This was later much improved by Tyndall who studied adhesion of smoke particles.5 Of course, as the particles stick together, their appearance changes substantially. At the most fundamental level, the Brownian motion of fine particles can be

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monitored by laser dynamic light scattering (photon correlation spectroscopy) in which a laser beam scattered from the dispersion displays a speckle pattern which changes with time. The autocorrelation function of the scattered beam is a measure of the diffusion coefficient D which is related to particle diameter D by the Stokes-Einstein result

where is the viscosity of the fluid in which the particles are dispersed. Thus the coagulation of the particles gives aggregates of larger diameter D, which reduces the diffusion coefficient and changes the speckle pattern. Ordinary white light can also be used to see the adhesive agglomerates forming. As the particles agglomerate, their Brownian velocity drops. The time required for a certain Brownian displacement increases with D3, where D is the particle diameter. Thus fine colloids can crash out very quickly as they undergo adhesive collisions in a short time, whereas larger aggregates move less and therefore do not collide but sediment under gravity. Light scattered by the particles increases with D6 so that a fine dispersion which looks transparent suddenly turns white and opaque during coagulation, allowing the process to be monitored through its turbidity. Our eyes are also sensitive to the changes in structural and flow properties of the suspensions as the particles adhere. Thus a flocculated suspension appears curdled and stringy whereas a good dispersion looks creamy and uniform. Very slight changes in molecular adhesion can be detected by such appearances. The simplest example of this is the coagulation of blood when two samples of different blood groups are mixed. Bringing a foreign surface into contact with a colloidal dispersion can also cause adhesion to occur. Bongrand6 has shown that, as particles drift over a surface at low speed, they can be captured by the surface to form temporary adhesive bonds, so that the particle velocity seems to jump from zero to the flow velocity and back again, as shown in Fig. 10.5 (see Chapter 12.6). As a particle is observed flowing slowly over the surface at constant speed, the particle velocity is suddenly seen to drop to zero. The particle has been captured by the surface. A short time later, the particle begins to move again at the original steady speed, as shown in Fig. 10.5(b). Using these observations, Bongrand claimed he could measure extremely low adhesion, around 0.1 kT of bond energy. The use of colloidal particle movement to measure fundamental parameters has a long and distinguished history. One of the most stimulating experiments in this tradition was that of Millikan7 who used this technique to measure the electronic charge in 1915. He used a single droplet of oil or mercury which was formed in a cloud by spraying the liquid into a chamber, as shown in Fig. 10.6. As the drops fell slowly under gravity, some of them passed through a small hole in a

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metal plate. Underneath this metal plate was another parallel plate which could be charged electrically from a battery to produce a uniform electrical field, adjustable until the electrical uplift on the oil drop balanced the gravitational force. This drop could then be held stationary in the optical microscope while the other droplets fell and disappeared. One drop could thus be isolated for detailed observation. The air in the vicinity of the droplet was ionized by Xrays. As Millikan observed a single drop, he found sudden changes in the velocity which he attributed to the picking up of a single unit of charge e from the ionized air. By balancing the Stokes viscous resistance against the applied electrical force Ee, Millikan obtained the charge on the electron, In these expressions, is the viscosity of air, D the drop diameter, and E the applied electric field across the plates calculated from the voltage divided by the gap. This experiment is most significant because it gives a clue to the extra ordinary lack of adhesion between many colloidal particles. If the particles are charged, then they will experience electrical repulsions which could be greater than the molecular adhesion. Then the particles are held apart by electrical forces. Adhesion can consequently be altered by varying the electrical nature of the fluid,

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for example by dissolving ionic salts in water. The salts generally reduce the electrical repulsion and promote adhesion. Clearly, electrical repulsion mechanisms are of great significance in allowing stable particle dispersions to exist, as noted by Schultze and Hardy8 at the turn of the century, following experiments on flocculation due to salt addition.

10.3. ELECTRICAL STABILIZATION OF PARTICLE DISPERSIONS The detailed argument, DLVO theory, for the stability of colloidal dispersions was originally worked out by Derjaguin and Landau9 in the Soviet Union and by Vervey and Overbeek10 in the Netherlands. However, the first convincing direct tests of these theories were not carried out until the work of Israelachvili and Adams11 using molecularly smooth mica sheets immersed in various liquids with electrolytes present. Consider two spheres of oxide, for example mica, interacting in water containing ions such as magnesium or calcium with pH controlled to 5.8 by addition of hydrochloric acid solution (HCl) as considered by Pashley and Israelachvili.12 The model of the electrical charges around the spheres is as shown in Fig. 10.7(a) which indicates that the spheres have picked up negative charges from the solution, leaving a cloud of positive ions in the gap between the spheres and also in the solution surrounding the particles. This separation of the charges near the colloid surfaces, the so-called double layer, was postulated by , Gouy and Chapman.13 Using Maxwell s equations with several approximations, the repulsive force between the spheres can be calculated to be2

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where is the permittivity of the liquid, is the permittivity of free space, k is Boltzmann’s constant, T is temperature, z is the valence of the ion, is the inverse of the double layer thickness, is the surface potential, h is the gap, and D is the diameter. Pashley and Israelachvili measured the force as the gap was varied, Fig. 10.7(b), and compared the results with the sum of the van der Waals attractive force and the repulsion given by Equation (10.3). They obtained good agreement, but when the gap was reduced to about 5 nm, the surfaces jumped into contact, because the van der Waals force exceeded the electrical double layer repulsion. An even more sensitive test of these electrical repulsions was discovered by Prieve and his colleagues around 1990.14 They allowed a small sphere, diameter suspended in water, to fall towards a glass surface under gravity, as shown in Fig. 10.8(a). An argon laser beam prevented the particle from moving sideways. As the gap between the particle and glass reached about 200 nm, the particle began to scatter the evanescent light internally reflected from a helium neon laser shining laterally along the glass. Brownian movement of the particle up and down caused a flickering of this light which could be used to measure the particle position to within 1 nm. The probability p(h) of finding the particle at a particular height h is given by Boltzmann,s equation

where is the potential energy of the sphere at height h. Thus the potential energy can be worked out as a function of height and shown to result from two forces: first, the electrical repulsion pushing up due to the double layer; second, the gravitational force pressing down. The results fitted the theoretical curve, demonstrating that this was a method with high force resolution, because it used kT as a gauge to measure 0.001 pN,

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almost a million times better than the surface force apparatus and a thousand times better than the AFM. The resolution of distance is 1 nm. From these ideas, the overall picture of colloid stability can be derived, as shown in the potential energy diagram of Fig. 10.9(a). This gives the potential energy of a sphere as it approaches its neighbor from a large separation. The two forces (van der Waals which is always attractive and electrical double layer which is always repulsive) combine to give the DLVO resultant curve. At first the particle experiences a repulsion, but when the gap reduces to a few nm, the attraction dominates and the particle is accelerated into an adhesive energy well. Within this well, the particle makes molecular contact and then experiences the strong Born repulsion which prevents further movement. This detailed potential curve can be modeled more simply by the square well approximation of Fig. 10.9(b). Widely separated particles give no interaction. But at a certain range the particle experiences a repulsive barrier of energy and if it is sufficiently energetic, the particle can leap over the barrier to adhere in the square well of range and depth where it then meets the hard sphere repulsion as the particles make molecular contact. The stability of the colloid may be interpreted as the difficulty which the particles experience in getting over the energy barrier. Particles which have been prepared in the fully dispersed state, for example by precipitation, have a distribution of velocities with an average thermal energy 3kT/2. Particles in the distribution with energy greater than have sufficient impetus to cross the barrier, so then doublets will form in the adhesive potential well. Treating the formation of doublets as a reaction

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the rate of formation of doublets is proportional to the rate constant times the concentration of particles squared, The fraction of particles with sufficient energy to cross the barrier is given by

where is the total number of particles and k is Boltzmann’s constant. Therefore, the rate of appearance of doublets is proportional to i.e.

The rate of formation of doublets in this model is a measure of the colloidal instability. This has been plotted in Fig. 10.10 to show how the height of the energy barrier can have a large effect on the apparent stability of the suspension. It is evident from this relationship that a colloidal dispersion, which takes 1 min to reach its equilibrium level of doublets when there is no energy barrier, will take more than a century when the energy barrier is 18 kT. It is worth noting from the results of Fig. 10.7(b) that the energy barrier for oxide particles is of the order of whereas 18 kT is ten times lower than this, i.e. at room temperature. Thus oxide particles can remain kinetically stable in water for very long periods under normal conditions due to this energy barrier. Thus the final equilibrium state of the particles may not be attained in reasonable experimental times. However, the final equilibrium state can be predicted exactly for this model using the methods of statistical mechanics.15

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For a dilute system with low adhesion, the ratio of doublets to singlets is given by the equation

where

is the volume fraction of particles and k is Boltzman’s constant.

10.4. POINT OF ZERO CHARGE; ADHESION DOMINATES To adjust the height of the energy barrier in an aqueous suspension, it is simplest to alter pH by adding acid or alkali. The double layer of charge around each particle then changes systematically as shown in Fig. 10.11 for aluminum oxide particles, with no energy barrier at the point of zero charge. The idea of an electrical charge sitting on the surface of such particles goes back to nineteenth century observations on clay particles moving in an electric field, readily applied by dipping two electrodes into the suspension. In 1892, Picton and Linder16 found that sols of sulfur, gold, silver and platinum were negatively charged whereas iron oxide, chromium oxide and aluminum oxide, were positive. In the hands of Smoluchowski17 these experiments led to a model of colloidal particle velocity u in an electric field E, giving the equation which defined the zeta potential This deceptively simple equation in which was the permittivity of the liquid, the permittivity of free space, the potential on the sphere surface (i.e. the zeta potential) and the viscosity) was derived by equating the electrical

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force on a charged sphere, qE where charge with the Stokes viscous resistance to movement, The particle radius a in these equations then dropped out, showing that all sizes of particles should travel at the same speed in the electric field. In Fig. 10.11, the alumina particles are positively charged under acid conditions, at pH up to 7. Above pH 9, as ammonium hydroxide is added to make the suspension more alkaline, the particles are negatively charged. In between, around pH 8, the potential on the particles switches from positive to negative. This is the point of zero charge at which the energy barrier is zero, so that adhesion is freely expressed and aggregation occurs most rapidly. Under such conditions, the suspension tends to gel and to exhibit maximum viscosity. This point of zero charge may be detected very sensitively by applying an oscillating electrical field to the suspension using the acoustosizer instrument (Fig. 10.12). The acoustosizer comprises a tank with electrodes stuck to the glass walls. These electrodes are fed with a high frequency electrical pulse which causes the charged particles in the colloid to vibrate. The vibrations travel along acoustic delay lines and are picked up by the piezo-electric detectors. Obviously, at the point of zero charge, there are no oscillations in the colloid, so that zero signal is obtained from the instrument. However, when the particles become charged as the pH is altered, acoustic signals are picked up, and by measuring several frequencies, sufficient information can be obtained to calculate the particle size and the zeta potential.19 The great benefit of this device, making it much superior to optical methods for determining zeta potential, is that it can be used on concentrated suspensions of the kinds used in paints, foods, minerals processing, ceramic industries, etc. It also has much potential for miniaturization and online monitoring because it is so robust.

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At the point of zero charge, there is no repulsive electrical force on the particles and so the full adhesion between the grains is developed. If this adhesion is strong, then each Brownian collision between singlet particles will produce a doublet. This was the case considered by Smoluchowski20 in 1917 and extended by Fuchs21 in 1934. The theory was based on the idea that colloidal particles behave like molecules which can react to form a bimolecular compound. Thus the rate of appearance of doublets is proportional to the square of singlet concentration N per unit volume by the law of mass action, and is limited by the Brownian diffusion coefficient to give

where k is Boltzmann’s constant and is the viscosity of the liquid. This equation is only true for dilute suspensions at early times, since triplets and further multiplets are formed thereafter. Higashitani and Matsuno22 used electronic counting of polystyrene latex in diameter to confirm this model, and obtained the results shown in Fig. 10.13, describing the disappearance of singlets and the growth of doublets, triplets, etc. Doublets grew within a few minutes, but then triplets started to appear, followed by quadruplets, and so on. The doublets then began to diminish. The results could be described by a simple theory originally produced by Smolu-

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chowski who assumed that collisions between clusters of equal size dominate, The number of aggregates containing n particles was given at time by

where and t were constants. Shearing such dispersions can speed up the aggregation process by allowing more collisions to occur. This effect is scaled by the Peclet number

which relates the applied shear rate to the Brownian shear rate where is viscosity and D is particle diameter. Diffusion dominates at low Pe whereas flow dominates at high Pe. The effect of shearing is complicated by the fact that the collision rate is always enhanced at low Pe but not always at high Pe. In addition, high shear rates tend to tear weak aggregates apart, which is often the objective when dispersing pigments and other powders into liquids.23 This problem is illustrated by the plot of doublet formation rate as shear rate is increased in Fig. 10.14.24 The expected behaviour at low shear does not seem to match up with the high shear predictions.

10.5. SECONDARY MINIMUM AND FURTHER COMPLEX INTERACTIONS The problem of understanding the adhesion between colloidal particles is exacerbated by the differences in adhesion that can often be observed. Some aggregates appear to be very strongly bonded, whereas others seem weak and

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easily broken apart. Often, there have been attempts to define strong flocs as agglomerates and to call weak floes aggregates. Such distinctions are false because it is evident that a whole range of agglomerate strengths can be found. It is necessary to define the precise interaction energy diagram to clarify the nature of the adhesion. The most significant feature of the DLVO theory of colloids was that it demonstrated the possibility of complex particle interactions displaying two minima, a primary minimum to the left and a smaller secondary minimum to the right as shown in Fig. 10.15. These may be viewed schematically in terms of six parameters; three energies: and plus three ranges and In this case, the particles can exist in three possible states: gas-like particles undergoing Brownian motion at long range, soft adhesion in the secondary minimum; and hard adhesion in the primary minimum. There is also the energy barrier between these adhesive states, slowing the approach to the primary minimum. Imagine 1000 particles in a computer model, interacting with the above potential set at values shown in Fig. 10.16, starting from the fully dispersed state. At first, doublets are observed forming in the secondary minimum, and these reach an equilibrium number after a short time. Then, doublets are seen forming in the primary minimum, taking a longer time to equilibriate because of the energy barrier. Thus, two kinds of adhesive aggregates appear in the final equilibrium situation: hard doublets in the primary minimum and soft doublets in the secondary minimum. The numbers of these two types of doublets may be derived, at equilibrium for dilute suspensions with low adhesion, using statistical mechanics theory15 to calculate the ratio of doublets to singlets

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It may be seen from these equations that the number of hard doublets is as expected from previous calculations, for example Equation (10.7). However, the number of soft doublets depends on both and Of course, if the energy barrier is sufficiently high, the process of forming primary aggregates will be sluggish. Then, the doublets will be mainly soft aggregates in the secondary minimum. For more concentrated dispersions with higher adhesion, the equations above are inaccurate. Computer simulation is then the best way to obtain correct predictions of aggregates. New types of aggregates should be seen as simulations are carried out on particles with secondary minima. Such aggregates should be partly made up of hard bound particles and partly of soft bound particles. The conclusion is that mixed hard/soft aggregates could be formed in the case of the secondary minimum potential. When even more complex interactions exist between colloidal particles, it may be expected that further potential barriers and energy minima may exist between the spheres.25,26 These are shown in Fig. 10.17. Horn and Israelachvili measured the interaction between mica surfaces in octamethylcyclotetrasiloxane, an inert liquid with roughly spherical molecules of diameter 0.9 nm. They interpreted the oscillating force measured between the surfaces as contact was approached as a solvation force due to denser packing of the liquid molecules at separations of one, two, or three diameters, etc. This effect had been postulated by Abraham,27 Chan et al., and Snook et al.28,29

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Evidently, with such a complex interaction, the particles can form several types of aggregates with different and mixed adhesion values. The ratio of doublets to singlets in the nth minimum at equilibrium is given by

10.6. EFFECT OF DISSOLVED POLYMER ON COLLOID ADHESION Having considered the effects of Brownian movement and electrical forces on colloid adhesion, it is now necessary to address the third important factor: soluble polymer molecules. Ever since Faraday added gelatin to his gold sols, polymers have been known to provide extra stability to dispersions. However, polymers can also cause flocculation, as shown in 1939.30 In fact, certain polymers are used commercially both as stabilizers and as flocculants. For example, low molecular weight polyacrylic acid (Fig. 10.18(e)) is much used to reduce viscosity of clays and oxides in water by reducing particle adhesion, whereas high molecular weight polyacrylic acid is used as a flocculant for removing fine particles during water treatment. The purpose of this section is to explain this duality by considering the interactive potential between particles in the presence of dissolved polymer molecules. The way in which acrylic acid monomer is linked in long chains which form random coils in aqueous solution is shown in Fig. 10.18. Dissolved polymer molecules behave as particles (Fig. 10.18(e)) which exhibit internal Brownian movement. The difference between a polymer particle and a clay particle is that the polymer can change its size because it is swollen by

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good solvents. Consequently, each segment can pulsate within the molecule, moving apart from other segments, and opening up to fill a larger volume. Also the polymer molecule can interact in a variety of ways with a solid surface. For example it can be strongly bound by one of its ends to the surface, or it can be weakly bound at several points along its length. In the weakly bound state, shown in Fig. 10.19(a), most of the polymer molecules coil up in solution to form particles which are considerably smaller than the extended chain length. A fully extended chain made up of N segments each of length l would be Nl long, whereas a polymer chain is known to coil up in

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a random configuration to give a distance between the ends of the molecule.31 However, some of the molecules are adsorbed onto the particle surface, and these are in a dynamic equilibrium, constantly jumping on and off the particle. The force acting between silica surfaces in polyacrylic acid solution has been measured by atomic force microscopy,32 giving the interaction curve shown in Fig. 10.19(b). At large separations, the particles follow the van der Waals attraction curve as the silica surfaces come closer together. But then the polymer molecules cause a repulsion as the polymer molecules resist desorption from the surfaces. This repulsion increases to a certain point but then the particles jump into a secondary minimum. As the particles are pushed further together, the polymer is squeezed out of the gap and eventually the silica particles make intimate contact in the primary minimum. This curve is interesting because it shows a complex interaction with three minima. Typically, the secondary minimum occurred at a gap of l0nm and the tertiary minimum at a gap of 25 nm This tertiary minimum was quite weak, with an attraction energy of about 10kT, leading to soft adhesive behavior. However, the secondary minimum was very deep, around 30kT, giving strong adhesion compared to Brownian movement. Thus, polyacrylic acid can have both stabilizing and flocculating effects in this system. The repulsions prevent strong adhesion of the particles, but the secondary minimum can produce aggregates with substantial strength. This aggregation can be viewed as the result of depletion of polymer molecules in the narrow gap between the particles, a gap so small that polymer tends to be excluded. The osmotic pressure of polymer molecules is diminished in the gap and so the particles are sucked together.33,34 Such complex restructuring of the interfaces, with polymer molecules changing positions substantially as the particles approach, is quite different from the situation when polymer is strongly bonded to the particles.

10.7. PARTICLES WITH STRONGLY BONDED POLYMER Particles can be made with strongly bound soluble polymer molecules at the surface as shown schematically in Fig. 10.20. When two such particles approach through the solvent, there is a slight van der Waals attraction at large separations. But as the gap equates to two polymer molecule diameters, an increasing repulsion is observed. This eventually breaks down into the primary minimum if sufficient force is applied to debond the polymer molecules. Thus, there is a small secondary minimum in this situation, separated from the primary minimum by a large energy barrier. Such particles behave very much like hard spheres, as shown in Fig. 10.20(c), because there is

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little interaction under ordinary dispersion conditions, followed by strong repulsion when the particles collide. Latex particles, such as polystyrene or poly(methyl methacrylate), surfacegrafted with water soluble polymer, behave in this way. For example, Pusey35 and his colleagues have made latex dispersions with strongly bonded polymer chains at the surface, matching the refractive index of latex and solvent to reduce van der Waals forces to a minimum. These dispersions appeared to be fully stable and exhibited phase transitions when concentrated to particle volume fractions of 0.5, producing ordered “colloidal crystals” which displayed opalescent colors. The conclusion was that the particles were behaving as hard spheres with zero adhesion. Similar results were found by Klein and Luckham36 for strongly bound polymer on mica surfaces at full coverage in a good solvent. Figure 10.21 (a) shows the interaction energy for poly(oxyethylene) layers strongly adsorbed onto mica surfaces from toluene at 23°C. Again, the repulsion was similar to a hard sphere interaction. Of course, this interaction depends a great deal on the solvent and on the time allowed for equilibration. Restructuring of the polymer can occur with time to give repulsion at long times but attraction at short times as shown by the results of Fig. 10.21(b).

10.8. GROWING CRYSTALS The nature of polymer adsorption on surfaces has an enormous effect on crystals nucleated and growing from concentrated solutions. For example, it is

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known that small quantities of polyacrylic acid can inhibit the formation of calcium scale on pipes and boilers. In a particular example of this phenomenon, calcium sulfate hemihydrate, Plaster of Paris, was mixed with water containing 0.5% of polyacrylic acid. The mixture remained fluid and pasty over extended times as long as twelve months. Normally, calcium sulfate hemihydrate reacts with water within a few minutes to form the dihydrate which precipitates as needle-like crystals to form a hard white solid. This is the normal setting reaction for Plaster of Paris. The influence of the polymer molecules is illustrated in Fig. 10.22(b). Polymer chains

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sit on the surface of very small crystal nuclei of the dihydrate and prevent their growth. This effect may be understood in terms of the concept of critical nucleus size for growth of crystals. Consider a small particle of material of diameter D immersed in another medium as shown in Fig. 10.23(a). If the particle is a droplet of liquid sitting in another liquid, such that an interfacial energy exists between the two liquids, then the particle has an excess surface energy which causes a pressure excess inside the particle according to Laplace’s equation. This excess pressure causes the particle to dissolve more readily, such that the solubility S of small particles is greater than the ordinary solubility according to which is Kelvin’s equation,1 in which is the interface energy, the molar volume, D the particle diameter, R the gas constant and T the absolute temperature. Below a certain critical size, therefore, particles will dissolve, whereas above that size the particles will grow, depending on the surface energy. Thus, small nuclei can be kept in solution by addition of surface active polymers. For example, calcium sulfate dihydrate nuclei can be suppressed by polyacrylic acid molecules. The mechanism can be understood with reference to Fig. 10.23(b). The surface energy of the particle is increasing with the square of particle diameter, whereas the bulk internal energy grows negatively with the cube of diameter. Small particles increase in energy as they grow because of their large surface. The critical nucleus size occurs when the bulk energy takes over from the surface energy so that the total energy begins to fall with increasing size. A similar effect is observed when organic molecules are used as templates in the formation of zeolite structures formed by cooking aluminosilicate gels. The

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addition of particular molecules to the hydrothermal reactor mixture can lead to a variety of structures developing. Certain molecules seem to act as void fillers, e.g. methanol and ethylenediamine. In other situations, several molecules can produce particular structures and are seen as structure directing molecules rather than true templates. But in particular instances, only one molecule seems to be effective in producing a structure, e.g. triquaternary amine producing ZSM-18. In this case, true templating seems to be achieved. Molecules can be grown by computer modeling to fill the void space available in the desired structure, using the condition that the van der Waals interactions between the atoms of template and host are optimized. 10.9. COMMINUTION OF COLLOIDS A favorite method for producing colloidal suspensions is to comminute a powder in a ball or bead mill. Typically, water is used as the fluid, into which 10– 20% volume fraction of powder is suspended. Zirconia beads are generally used as the grinding medium because of their hardness and wear resistance. The beads are stirred vigorously and impact the powder particles to fracture them into fragments. This process of comminuting particles to make fine powders is an extremely inefficient process. Most of the energy input merely produces heat, with only a small fraction ending up as surface energy of fine particle product. It has been estimated that up to 10% of a country’s energy use is expended on grinding processes of this sort. A model of the particle cracking mechanism was proposed in 1978,37’38 as shown in Fig. 10.24. Two beads approach each other at speed, and impact a particle which is trapped between them. The particle contains a crack which is then propagated through the particle to break it into two fragments. Evidently, this crack is not a

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simple tensile crack as described by the Griffith theory of fracture. Instead, the crack is driven by the bending of the sample arms as shown in Fig. 10.24(b.) For an elastic particle, it was shown that the compressive force F required to drive a long crack was37

where b was the breadth of sample, d its width, E its elastic modulus, and R the fracture energy. This equation showed quite clearly that small particles should appear stronger because the mean compressive stress F/bd increased with as the particles were reduced in size. Such a “size effect” is an old idea, since Gauthey had observed in the eighteenth century that brittle substances like rocks seemed to get stronger when made smaller.39 Of course, if the stress required for cracking rises in this way, then the plastic mechanism of deformation will ultimately take over as the compressive stress F/bd equals the plastic yield stress Y. This is shown in Fig. 10.25 which illustrates the changeover from brittle to ductile failure as the particle size was reduced. These experiments were performed on model particles into which long cracks of controlled length were inserted before compressing. Large samples cracked in brittle fashion, but when the samples were made less than 4 mm in size, the samples squashed plastically and the cracks did not propagate. Essentially, for small samples, not enough elastic energy could be stored in the material to cause cracking, so that plastic deformation was preferred. Solving the equations led to the critical size for the brittle-ductile transition

This model explained the well-known comminution limit for ball-milling of fine particles. After prolonged milling, particles reach a certain size, and it is impossible to make them smaller by continued grinding.40 For example, Equation

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(10.17) predicts a crushing limit of around 1 µm for calcium carbonate, compared with an average measurement of 0.8 µm after prolonged milling.41 Such plasticity of compressed calcium carbonate particles had been observed earlier, 42 following Boddy’s original microscopic observations of plasticity in compressed fine grains of coal.43 The comminution limit is shown in Fig. 10.26 which gives bead milling results for calcium sulfate dihydrate crystals in water at 0.2 volume fraction. 44 The particle size starts at 20 µm, as measured by the Malvern light scattering instrument, and gradually decreases to 15 µm after 300 s of bead milling. This is the comminution limit in distilled water. However, when the nucleation inhibitor, polyacrylic acid, was added to the suspension, two effects were observed. First, the comminution was more rapid. Second, the comminution limit was reduced to around 4 µm, suggesting that the surface energy of the calcium sulfate crystals was reduced by a factor of 4 through the adsorption of the polymer. This is an example of the Rehbinder45 effect, an increase in grinding due to surfactant. One of the reasons for studying calcium sulphate was its relatively high solubility in water, Clearly, a soluble crystal will be strongly etched by the solvent and so the surface flaws will be removed. This should inhibit grinding, 46 just as Joffe found that salt crystals were less brittle after dissolving their surfaces in water. Less soluble crystals, such as calcium carbonate, solubility should grind faster because the surface flaws remain intact. This is shown by the results of Fig. 10.26(b). However, when such calcium carbonate crystals were etched in hydrochloric acid, filterwashed and comminuted, they ground more slowly, suggesting that the natural flaws had been removed. The

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comminution limit appeared to be the same. After a few days standing, the grinding rate reverted to its original value, showing that the flaws had recovered.

10.10. GROWING UNIFORM COLLOIDAL PARTICLES Because ball milling reaches a limit, it is impossible to make very fine particles by this method. Therefore, growing particles from vapor or solution phases has been the preferred approach to making fine dispersions. This growth method is also attractive because the particles can be controlled in shape, and can also be grown to the same size, which is useful in many applications. For example, sintering of particles is eased if the particles are nearly equal in diameter. Such fine, equal particles cannot be produced by comminution. In this section we discuss the several processes which have been developed for growing particles. In the simplest method for making a colloidal dispersion, a metal was evaporated in a low pressure environment, and allowed to contact a solvent in a rotating flask, as shown in Fig. 10.27(a). The metal vapor condensed to form solid particles in the large volume of the flask, then these particles were picked up in the solvent to form a suspension which could be drained off.45 Alternatively, metal particles such as titanium were evaporated into a vessel, collected on a cold finger, oxidized by admitting oxygen, then scraped off into a compaction cell where their sintering could be studied in ultra-high vacuum conditions. Very fine particles were formed by such vapor condensation, typically 4nm diameter for palladium and 12 nm for titanium dioxide, with rather a narrow size distribution.

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Of course, vapor phase methods have been used from ancient times to produce carbon black particles for mixing into inks. The carbon particles grow as a smoke in the reaction of a carbon-containing fuel at elevated temperature.46 These soot grains can vary in size substantially from around down to 10 nm, depending on the precise conditions of reaction. The ultimate small carbon particle formed by such vapor phase reaction is the Buckminsterfullerene molecule which is 1 nm in diameter, containing 60 carbon atoms in a spherical arrangement.47 The problem with vapor deposition of particles is that interparticle adhesion tends to stick the grains together into chain-like masses which are then rather intractable, forming either fluffy, low density powders or hard compacted agglomerates. It is preferable to process the particles in a solvent medium which reduces adhesion to low levels so that the grains can be handled more readily. Since the early preparations of gold sols by Faraday in 1857, made by boiling aqueous solutions of gold chloride, many monosized dispersions have been found, usually by trial and error methods.48 Many of these empirical techniques were developed for applications such as photography, in which the properties of the silver bromide particles were critical. Another application was for electronic inks which required very fine dispersed conducting particles.49 A more recent application is for magnetic oxide particles which can be used to bind to cells in medical treatments.50 The particles with their adhering cells can then be removed using a magnetic separator. Over the years there has been a gradual improvement of understanding in the growth of monosize particles from solution. For example, Zsigmondy51 and Turkevitch52 with their colleagues showed how to improve the gold particles studied by Faraday. A major application was gold inks used to produce the decorative patterns on pottery. Eventually it became clear that three steps shown in Fig. 10.28 had to be controlled if uniform particles were to be produced: nucleation was the first step in which many small growth centers around 10nm diameter were formed spontaneously in the solution; growth was the second step,

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by which these nuclei expanded to form large particles up to µm diameter; aggregation was the final step, by which the particles adhered to form agglom erates. If nucleation continued while growth was proceeding, or if aggregation occurred, then suspensions with a wide range of sizes and shapes would result. Therefore it was necessary to separate and control these three processes. 53 A wide range of monodisperse particle systems have been grown. G enerally the nucleation is produced by concentrating the solution above the critical supersaturation. Then the conditions are suddenly changed to prevent further nucleation but to allow growth to continue. For example, the seeds of nuclei may be transferred to another solution with the right conditions for growth. Alternatively, the mixture containing the nuclei may be diluted or heated or pH changed to lower the supersaturation to a suitable level for growth, but without further nucleation. By controlling the growth rate, the shape of the particles can then be adjusted, since different crystal faces will grow at different rates depending on conditions. Thus, silver bromide may be grown as cubes, or as 8sided or 14sided polygons merely by changing pH, as shown in F ig. 10.29. A particularly neat way to make very pure oxide dispersions was devised by 54 Stober, Fink and Bohn in 1968. Tetraethyl silicate was dissolved in alcohol and reacted with water in the presence of acid catalyst. Very fine nuclei of silica were formed, but these aggregated immediately to form uniform spherical clusters 55 159 nm in diameter which could then be grown further by controlled addition of tetraethyl silicate. The final particles were porous and of high surface area

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because of this aggregation mechanism, but could be condensed to full density by heating to 500°C. Another powerful way to control the nucleation and growth of particles is by the use of surfactants. This is the dominant industrial method used in growing fine polymer dispersions, for example of polyacrylics. A surfactant such as SDS is dissolved in water and the monomer is added together with initiator. Nucleation then occurs, producing fine polymer particles about 10nm in size. Further controlled addition of monomer allows the spherical particles to grow to the desired product size, typically 200 run diameter. Addition of surfactant to a nucleating system can have an enormous stabilizing effect. For example, during the precipitation of AgCl through mixing solutions of NaCl and the presence of surfactant increased the stability of the 6 nm nuclei by a factor of 6 over simple aqueous solutions.56 The nuclei showed no further growth and only small tendency to aggregate. Presumably, the surfactant molecules are interacting strongly at the surface of the nucleus, to alter the adhesion of the nuclei as shown in Fig. 10.30(a). Of course, the surfactant molecules can also phase separate to form micelles within the solvent as shown in Fig. 10.30, in addition to other liquid crystal phases which can strongly influence particle nucleation and growth. For example, the formation of a hexagonal liquid crystal phase for 55% Tween surfactant dissolved in 45% water caused crystal growth of copper sulfate to be much inhibited, producing monosize dispersions of crystals around in diameter.57

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10.11. REFERENCES l. Newton, I., Opticks, Smith and Walford, London, 1704 (reprinted Dover, New York, 1952, p 365). 2. Everett, D.H., Basic Principles of Colloid Science, Royal Society of Chemistry, Letchworth, 1988. Russel, W.B., Saville, D.A. and Schowalter, W.R., Colloidal Dispersions, Cambridge University Press, Cambridge, 1989. 3. Einstein, A., In: Investigations on the Theory of Brownian Movement, ed. R. Furth, Dover, New York, 1956 (translated A.D. Cowper) Ann. Phys. 19, 371–81 (1906). 4. Perrin, J., Brownian Motion and Molecular Reality, Taylor and Francis, London, 1910. 5. Faraday, M., Phil. Trans. R. Soc. 147, 184 (1857). 6. Pierres, A., Benoliel, A. and Bongrand, P., Faraday Disc. 111, paper 24 (1999). 7. Millikan, R.A., The Electron, University of Chicago Press, Chicago, 1917. 8. Hardy, W.B., Proc. R. Soc. 66, 110–25 (1900). See also Overbeek, J.T.G., Pure. Appl. Chem. 52, 1151 (1980). 9. Derjaguin, B.V and Landau, L.D., Acta Physicochim USSR 14, 633–62 (1941). 10. Vervey, E.J.W. and Overbeek, J.T.G., Theory of the Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948. 11. Israelachvili, J.N. and Adams, G.E., J. Chem. Soc. Faraday Trans. I, 74, 975–1001 (1978). 12. Pashley, R.M. and Israelachvili, J.N., J. Colloid Interface Sci. 97, 446-55 (1984). 13. Gouy, G., J. Phys. Rad. 9, 457–68 (1910). Chapman, D.L., Phil. Mag. 25, 475–81 (1913). 14. Prieve, D.C. and Walz, J.Y., Appl. Opt. 32, 1629 (1993). Prieve, D.C., Adv. Colloid Int. Sci. 82, 93–135 (1999). 15. Stainton, C., PhD Thesis, Keele University, 1998. 16. Picton, H. and Under, S.E., J. Chem. Soc. 61, 148 (1892). 17. Smoluchowski, M. von, Bull. Int. Acad. Sci. Cracovie 8, 182–200 (1903). 18. Hunter, R.J., The Zeta-potential in Colloid Science, Academic Press, London, 1981. 19. Hunter, R.J., Colloid Surf. A141, 37–65 (1998). 20. Smoluchowski, M. von, Z Phys. Chem. 92, 129–68 (1917). 21. Fuchs, N., Z. Phys. 89, 736–43 (1934). 22. Higashitani, K. and Matsuno, Y., J. Chem. Eng. Japan 12, 460–5 (1979). 23. Parfitt, G.D., (ed.) Dispersion of Powders in Liquids 3rd edn, Applied Science, London, 1981. 24. Russel, W.B., Saville, D.A. and Schowalter, W.R., Colloidal Dispersions, Cambridge University Press, Cambridge, 1989, p297. 25. Kendall, K., J. Adhesion 5, 179–202 (1973). 26. Horn, R.G. and Israelachvili, J.N., J. Chem. Phys. 75, 1400–411 (1981). 27. Abraham, F.F., J. Chem. Phys. 68, 3713–16 (1978). 28. Chan, D.Y.C., Mitchell, D.J., Ninham, B.W. and Pailthorpe, B.A., J. Chem. Soc. Faraday Trans. 76, 776–84 (1980). 29. Snook, I.K. and van Megen, W., J. Chem. Phys. 70, 3099–105 (1979). 30. Napper, D.H., Polymeric Stabilization of Colloidal Dispersions, Academic Press, New York, 1983. 31. Yamakawa, H., Modern Theory of Polymer Solutions, Harper and Row, London, 1971. 32. Milling, A.J. (ed) Surface Characterization Methods (Surfactant Science Series vol 87), Marcel Dekker, New York, 1999. 33. Asakura, S. and Oosawa, F., J. Chem. Phys. 22, 1255–6 (1954). 34. Asakura, S. and Oosawa, F., J. Polym. Sci. 33, 183–92 (1958). 35. Pusey, P.N., In: Liquids, Freezing and the Glass Transition, Les Houches Session LI, eds. D. Levesque, J.-P. Hansen and J. Zinn-Justin North Holland, Amsterdam, 1991, pp 763–942. 36. Klein, J. and Luckham, P.F., Macromolecules 17, 1041–8 (1984). 37. Kendall, K., Proc. R. Soc. A 361, 245–63 (1978).

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Kendall, K., Nature 272, 710–1 (1978). Timoshenko, S., History of Strength of Materials, McGraw Hill, London, 1953, p 58. Lowrison, G.C., Crushing and Grinding, Butterworth, London, 1974, chaps. 1–5. Gregg, S.J., Chem. Ind. 611 (1968). Steier, V.K. and Schonert, K., Dechema Monog 69 pt.1, Dechema, Frankfurt, 1972, p 167. Boddy, R.G.H.B., Proc. Conf. Ultrafine Structure Coal and Coke, BCURA, London, 1943, p 336. Whitfield, R. and Kendall, K., unpublished results. Granquist, C.G. and Buhrmann, R.A., J. App. Phys. 47, 2200–19 (1976). Siegel, R.W., Ramasamy, S., Hahn, H., Zhongquan, L, Ting, L. and Gronsky, R., J. Mat. Res. 3, 1367–72 (1988). Kratschmer, W, Lamb, L.D., Fostiropoulos, K. and Huffman, D.R., Nature 347, 354 (1990). Matijevic, E., (ed.) MRS Bull. 14, 18–46 (1989). Ferrier, G.G., Berzins, A.R. and Davey, N.M., Platinum Metals Rev. 29, 175 (1985). Matijevic, E., Langmuir 2, 12 (1986). Zsigmondy, R. and Thiessen, P.A., Das Kolloide Gold, Akademic Verlag, Leipzig, 1925. Turkevitch, J., Gold Bull. 18, 86 (1985). Sugimoto, T., Adv. Colloid Interface Sci. 28, 65 (1987). Stober, W., Fink, A. and Bohn, E., J. Colloid Interface Sci. 26, 62–6 (1968). Bogush, G.H. and Zukowski, C.F., J Colloid Interface Sci. 142, 19–34 (1991). Dvolaitzky, M., Ober, R., Taupin, C., Anthore, R., Auvray, X., Petipas, C. and Williams, C., J. Disp. Sci. Technol. 4, 29 (1983). Ward, J.I. and Friberg, S.E., MRS Bull. 14, 41–46 (1989).

11 PASTES AND GELS: EFFECTS OF ADHESION ON STRUCTURE AND BEHAVIOR

Is it not from the mutual Attraction of the Ingredients that they stick together for compounding these Minerals ISAAC NEWTON, Opticks,1 p. 385

As we saw in the last chapter, colloids often exist in the dispersed or sol state where adhesion between particles is unusually low, as a result of repulsive energy barriers. When the adhesion barrier is overcome, by heating, by adding salt, or by introducing polymer, then coagulation occurs and the composition turns into a paste or gel. We think of a paste as a semi-fluid material which can flow like a liquid when the shear stress is high enough. Latex paint is an example of such a dispersion shown in Fig. 11.1. In the sheared liquid paint, the adhesion forces are low, and the material is fluid. However, once the paint is on the surface, the shear stress is low and the adhesion between the particles can solidify the paint, making it gel. The gel structure holds the paint up against gravity, thereby preventing dripping down the wall. In the example above, the adhesion forces are small, so the paint gel is soft. Paint gel is much softer than a silica gel made by drying the particulate precipitate made from acidified sodium silicate solution. In this case the silica particles are extremely adherent and can resist large stresses. However, even such a hard, glassy silica gel can be moulded by the application of large pressures, of the order of 1000 atm (100 MPa). Thus, gels and pastes are distinguishable from sols in terms of the deformation behavior or rheology. A liquid sol is soft and fluid, flowing under 245

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very small forces. A gel is hard and elastic, resisting deformation, but plastic at high stresses. A paste is somewhere in between. The explanation is that pastes and gels differ largely in the adhesion forces acting between the particles in the suspension. A sol exhibits no interparticle adhesion. A gel displays large adhesion, so large that the final material can be used for construction, like cement. In contrast, the paste is a mixture of liquid and gel characteristics with intermediate adhesion. It is essential to understand how molecular adhesion influences such characteristics, because this determines how a liquid can turn into a gel, at the so-called liquid/gel transition point, where the paint stops flowing and becomes solid.

11.1. IMPORTANCE OF PASTES AND GELS Concentrated pastes, composed of powders mixed into liquids (Fig. 11.2), are enormously important in our lives.1,2 Natural dispersions such as mud, clay, and blood have been the traditional base of large industries since prehistoric times. Modeled on these are bread and concrete, made by milling solids into powder, then mixing the resultant fine flour into water to form a paste. Subsequently the paste is gelled into a solid material by further processing, for example by baking the bread, reacting the concrete, or drying the paint latex to adhere the particles together. These are the biggest man-made materials in terms

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of world production. More than a billion tonnes of cement are produced annually and 22 million tonnes of paint are mixed by 8000 manufacturers. Modern examples of pastes are lead acid battery materials which are formed by grinding lead and adding water, alumina ceramics manufactured from calcined bauxite, slurried and extruded, latex made by polymerizing a monomer in water, and tarmac formed from a mixture of gravel and molten bitumen. Highly technical products such as paints, electronic inks, and polymer composites are more complex types of pastes which have had major impacts over the past fifty years.4,5 There are also many intermediate processes in the minerals and chemicals industries where slurries are dried, filtered, centrifuged, and variously compacted. The easing of such paste processes is an important economic interest.6,7 The purpose of this chapter is to show how such improvements can be brought about by the understanding of the adhesive forces acting between particles in sols, pastes, and gels. The liquid/gel process varies for different products, ranging from cooling (bitumen, polymer composites), drying (mud, ink), reacting (cement, lead paste), baking (bread, ceramic), or treating with drugs (blood, cancer cells). Final properties may be fairly insensitive to the structure of the paste, or they may be critically dependent on small microstructural features. For example, the elastic modulus of sintered dense alumina does not vary much with the treatment of the original paste, whereas its mechanical strength and electrical breakdown resistance are highly sensitive to defects at the 1 to scale, defects which can arise during processing of the suspension before the sintering stage. These defects are very important if we wish to improve the paste, to produce better products as shown later in Section 11.8. Consider an example of a typical ceramic paste/gel process, that of making extruded zirconia tubes for a fuel cell application.8,9 This is a demanding service which requires thin ceramic with high strength, good electrochemical performance, and no leaks. The starting material is zirconium carbonate which is dissolved in sulfuric acid and purified by solvent extraction to remove impurities such as silicon. Hydrated zirconia is precipitated from the sulfate, then washed and calcined in an oven near 800°C to grow fine crystallites of zirconia as shown in the flow sheet of Fig. 11.3. These are bead-milled in water with dispersing agent and polymer extrusion aid to give a slurry which is filtered through a mesh to remove any agglomerates. After removing excess water, the paste thickens and can be extruded through a tube die to give wall thickness. The tubes are dried and fired to 1400°C to produce a fully dense product of high strength. The various stages in this process, especially the solidification of the fluid dispersion to the hard gel, i.e., the liquid/gel transition, dictate the structure and properties of the final product.

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11.2. DIFFERENT STRUCTURES OF SOLS, PASTES, AND GELS The structure of a sol or liquid suspension, as shown in Fig. 11.4(a), is the simplest colloid dispersion, described in detail in Chapter 10. Particles are independent from neighboring grains, swimming in the solvent with Brownian movement. Each particle can collide with its neighbor but the collision is not adhesive and so the particles remain fully dispersed. Such a sol can be modeled by the hard sphere approach. For spherical particles of equal size and zero adhesion, the structure is random at low concentrations. As more particles are added to the dispersion, the spheres begin to jostle each other into regular positions at a volume fraction of 0.494, the freezing transition at which the disordered structure starts to become ordered. While the concentration is increased, further structuring occurs until the dispersion is fully organized in the face-centered cubic arrangement at volume fraction 0.545. The viscosity of the sol increases as volume fraction is raised, shows a blip at the disorder/order transitions, then continues to increase, becoming infinite at volume fraction 0.74 which is the maximum packing attainable for uniform spheres.

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A gel is totally different from a liquid dispersion because adhesion between particles now dominates and the particles are fully connected (Fig. 11.4(b)). Every particle has at least one connection to a neighbor, but many particles have two connections to form chains, and some particles may have 12 connections which is the maximum for uniform spheres, forming densely packed structures. The structure of gelled particles is therefore static, and the fluid medium merely fills the pores in the structure. The liquid can be removed, if the gel is strong enough, and the gel can stand by itself in a vacuum. However, the gel may collapse if it is not too well connected. Clearly, the important property of a gel is its coordination, in other words the number of adhesive contacts per particle. While a coordination number of 2 gives chain-like, random, voluminous, weak gels, a coordination number of 12 gives close-packed, structured, dense, strong gels. The difference between a sol (liquid dispersion) and a gel is therefore adhesion. Ideal sol particles have zero adhesion, are in constant thermal motion, and are pushing outwards to escape from their containing fluid. By contrast, gel particles adhere strongly, are fixed by adhesive contacts, and are pressing inwards as a result of molecular attractions to collapse the gel and form more contact spots. A paste is intermediate between a sol and a gel because the adhesion between particles is of medium strength, weak enough for some particles to move in the suspension, but strong enough to cause gelled clumps to form. Thus the paste exhibits some features of a sol and some of a gel, as shown in Fig. 11.4(c). It can be fluid under some circumstances, yet solid under others. The paste exists near the sol–gel transition, where a sol is changed into a gel by certain processes. Imagine, for example, that the adhesion between particles in a sol is increased by adding salt to the dispersion to reduce the electrical repulsions, as described in Chapter 10. The particles begin to stick together and the dispersion becomes more viscous. At low forces, the suspension does not flow at all; it has become solid. But flow is possible when the force is increased. Thus, the structure of the paste is variable at different forces. This variation in structure leads to interesting ranges of behavior. To understand this behavior it is necessary to define the properties of a gel, to compare this with the sol discussed in Section 10.8 and to see what happens at the transition between sol and gel.

11.3. STRUCTURE AND PROPERTIES OF A GEL The impressive thing about gels is their solidity. It seems odd that a fluid dispersion like silica sol, composed of 20 nm diameter particles of silica, can dry to form a solid block of material which behaves like glass. The product is

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transparent, hard, strong, and elastic. However, it differs from glass in being very porous, capable of absorbing about half its own volume of water. In order to explain the properties of the gel, we assume that it is made up of spherical particles of equal size arranged in a cubic packing, as shown in Fig. 11.5. Each sphere makes six adhesive contacts with its neighbors. Evidently it is the solid nature of these molecular adhesive contacts which gives the gel its solidity. If all these contacts were broken, then the gel would revert to its fluid sol structure. The most significant feature of the gel structure is therefore adhesion which pulls the particles into molecular contact, as described in detail in Chapter 9. The spheres thus deform elastically to give a contact spot diameter d at zero applied load given b y 1 0 – 1 1

where W is the work of adhesion, D the particle diameter, E the Young’s modulus of the particles and v their Poisson ratio. This equation was derived in Chapter 9 using the energy balance model of adhesion.13 Two properties of the gel immediately become apparent from that analysis. The first is the force required to separate the particles against the adhesive attraction. This separation force F is

which, when divided by the area of the cubic structure unit, leads to a theoretical tensile strength for the gel of

This equation shows that the strength of a gel can be high enough for structural applications if the particle size is small enough. For example, a cement gel with particle size of 10nm and a work of adhesion of would have a theoretical tensile strength of 12MPa according to this equation. This is the value measured in tensile tests of Portland cement. However, the problem is more

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complex because of the composite structure of the cement product (Chapter 16). Also flaws and cracks in the gel reduce the strength, while nonequilibrium cracking tends to increase strength as described in Section 11.6. Strength is difficult because it is so dependent on defects in the structure. Two much more interesting properties of gels are elasticity and shrinkage. These properties do not depend on defects so much. By measuring the elastic modulus of a gel, it should be possible to determine the work of adhesion for the particles because the elastic deformation is dictated largely by the contact spots between the grains,11 with spot diameter d proportional to as in Equation (11.1). In the same way, a measurement of shrinkage should determine the increase in work of adhesion, because this causes the contact spots to expand and pull the particles together as the gel dries or is treated chemically.14 Behavior of gels was first calculated from the elastic properties of a single contact between two spheres in 1985 and presented in a lecture which was published a year later.12 The contact stiffness arises because the spheres move together as a compression load F is applied to the two spheres. Under zero load the distance between the sphere centres is D, as shown in Fig. 11.6(a), but when a force F is applied (Fig. 11.6(b)) the spheres move together an extra distance which depends on the adhesion between the spheres. The stiffness is defined as

and is directly proportional to the contact spot diameter d according to the equation

Originally this was derived by Kendall et al11 but there was an error of in the arithmetic, which was corrected by Thornton and Yin.15 The value of this equation is that it shows the stiffness to be a measure of the adhesion, because combining Equations (11.5) and (11.1) gives

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This enables the calculation of the Young’s modulus E* of the gel because this is the stress divided by the strain to give

In other words, the elastic modulus of the cubic structured gel, E*, should increase with adhesion W, particle elasticity E and Poisson’s ratio v, but decrease as the particle diameter D rises. These equations are interesting when we apply them to common gels such as silica, alumina, titania, or zirconia. For example, for diameter silica particles, E = 70 GPa, because of the hydrated surface and is 0.3. The contact spot diameter turns out to be 32 nm from Equation (11.1), only 3% of the particle diameter. It is salutary to note that such tenuous adhesive contact between the particles is responsible for the gel behavior. The Young’s modulus of the gel from Equation (11.7), assuming a cubic structure, is 0.75 GPa. This is close to the measured values for silica gel, which is about a hundred times more compliant than silica itself. Let us now consider the preparation and measurement of such gels.

11.4. ELASTIC MODULUS OF SILICA GELS Gels may be prepared from powders in various ways, including smoke deposition, dry compaction, paste drying or phase separation followed by leaching. A convenient method which can be used for a wide range of particle sizes is plastic processing in which the powder is mixed into a polymer solution, strongly sheared to break down the powder agglomerates, then extruded or pressed to form samples which can then be dried and heated to remove the water and the organic material.16 In a typical experiment, shown in Fig. 11.7, silica powder (OX50, Degussa) of diameter 50 nm was mixed in a 30% polyvinyl alcohol aqueous solution (KH17s, Nippon Gohsei) using a two-roll mill. After mixing for ten minutes, the composition appeared transparent, indicating that the large agglomerates had largely been broken down to the primary particles. The plastic sheet was pressed for several hours at 5 MPa to remove air bubbles, and the sheet was dried, cut into strips, and heated to 500°C

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to remove water and polymer, giving a strong gel sample. Because of the polymer content, the material did not crack during drying, despite the considerable shrinkage. Then the strips could be bend tested to measure the Young’s modulus E* of the gel. When the elastic modulus results were plotted, in Fig. 11.8(a), it was clear that the value of E* increased rapidly with the volume fraction of silica in the gel. The results fitted an expression of the form

This strong dependence on packing fraction could be explained in terms of the structure of the gel. The theory of Equation (11.7) had been worked out using a simple cubic model where there were 6 contacts for every particle, but only one contact dictated the Young’s modulus of the gel. If one carries out the same calculation for the three other regular packings (the cubical tetrahedral, the tetragonal sphenoidal, and the hexagonal close-packed structures) then the Young’s modulus increases almost as as shown in Fig. 11.8(b). The reason for this is that the coordination number per particle is increasing as the square of volume fraction while the stiffening effect of the particles themselves is also rising as leading to a fourth power dependence overall. Although this theory of the dependence of gel modulus on structure cannot be exact, because one can imagine structures which will not fit Equation (11.8), the expression is useful in describing the experimental results. More precise theories have been postulated, for example that of Batchelor et al.,17,18 who suggested that thermal conductivity K* of an isotropic gel is related to the particle conductivity K by

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where D is particle diameter and d the contact spot diameter, but this demands knowledge of the particle coordination number n which is not measurable by any method at the present time. However, the experimental results gave a particle diameter dependence of as shown in Fig. 11.9(a), fitting Equation (11.8). Equation (11.8) is beneficial because it requires only one structural parameter, the volume fraction which is easily measured by a density determination. Of course, the structure of many gels is not uniform, but contains density variations which act as defects. A high concentration of defects can influence shrinkage, whereas strength can be affected at much lower levels. It is known from studies of large particle beds that the general state of a compacted gel is nonuniform,19 since the load is not transmitted evenly through all the contacts, but is supported preferentially by a small number of contacts (Fig 11.9(b)).

11.5. SHRINKAGE OF GELS The problem of structure of gels is important when one considers the shrinkage of the particle assembly during drying or sintering. Gels display anomalous shrinkage behavior. A typical anomaly is the strange shrinkage on changing temperature or altering the chemical environment. Silica gel, for example, shrinks when it is heated under ordinary atmospheric conditions, whereas dense silica expands. The reason, of course, is that heating silica gel drives off the adsorbed water, thus changing the chemical environment and increasing the work of adhesion between the fine silica particles. This increase in adhesion, which pulls the grains together, is much more noticeable than the thermal expansion of the silica bonds as temperature rises.

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Bangham20 and his colleagues, after Meehan’s21 initial observations, had proposed that a gel was in a highly deformed compressive state, and had presumed that a change of work of adhesion in the gel would result in a proportional shrinkage strain From the measured strain of various gels, such as coal and charcoal, the work of adhesion could be calculated in the presence of different gases.22 However, that theory is not consistent with the elastic deformation of spherical gel particles,14 which shows that shrinkage strain is proportional to the two-thirds power of adhesion, i.e. A typical result for thermal shrinkage of silica gel is shown in Fig. 11.10(a). The solid silica particles expand as shown in the upper graph. In contrast, the gel shrinks up to 200°C, then expands until 450°C, then shrinks above 500°C. Subtracting the thermal expansion gives the resultant effect of the increased work of adhesion in the bottom line. Through removal of water from the gel surface, the work of adhesion must increase by a factor of 3 overall to give this shrinkage. Above 800°C, the gel shrinks irreversibly by diffusive movement of the silica, the sintering process driven by work of adhesion. Sintering shrinkage was originally considered by Frenkel21,22 who showed that two viscous spheres of diameter D would gradually fuse together at their point of contact by viscous flow, giving a shrinkage strain

where W is the work of adhesion, t is time and the viscosity. This equation, which presumes that the structure is cubic, was verified by measuring the sintering shrinkage of an assembly of glass spheres (Fig. 11.10(b)).25 Shrinkage strain increased linearly with time and fitted Equation (11.10), though the values

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of W and were not accurately known. However, it is clear that shrinkage must depend on the local coordination number and will go faster in a close-packed system than in a loosely packed gel. When the gel is not homogeneous, the shrinkage varies through the sample and the gel pulls itself apart. This mechanism is readily proved by taking a gel and artificially disturbing its structure by mixing foreign particles into it during the compaction stage. When heated, the uniform structured gel shrinks easily as sintering of the particle contacts occurs (Chapter 9). However, the disturbed structure gel does not shrink so easily, as shown in Fig. 11.11. Just a few percent of foreign structure can inhibit sintering substantially.26 Figure 11.11 shows three possible structures of a gel: the top is uniform cubic, assumed by Frenkel, the middle is random, and the bottom non-uniform. This problem was revealed by Rhodes and his colleagues27,28 who studied the sintering of 20 nm particle size gels of zirconia and titania. They found that when the gel was made by drying a very stable colloidal dispersion, to give a uniform structure, the sintering occurred at unusually low temperatures, 1100°C for zirconia and 800°C for titania. By contrast, when the gels were made from agglomerated powders of the same particle size, the sintering temperature rose to 1600°C for zirconia and 1200°C for titania, indicating that the structure was now degraded and nonuniform. The conclusion was that the method of making the gel had a strong influence on the gel performance, because the process controlled the uniformity of structure. This was proved by Clegg et al.16 who mixed fine silica by several methods and showed that plastic mixing gave the best sinterability. Silica powder was pressed into compacts, or stirred in water to make a paste, followed by drying. Both these gels failed to sinter until the temperature was raised to 1500°C, suggesting that large agglomerates were inhibiting shrinkage (Fig. 11,12).

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However, when the powder was plastically mixed on a two-roll mill with polymer solution, the sintering then occurred at 900° C, the temperature expected for 50 nm diameter silica particles. This result indicated that the gel structure attained by plastic processing was almost perfect.

11.6. ULTIMATE STRUCTURE OF A GEL PRODUCT A fascinating question about gel products relates to the ultimate gel structure and final properties which can be achieved from paste processing.29 Conventional products are clearly not perfect because they can be improved by small changes in the processing method. For example, it was found that the addition of a small amount of polymer to a cement paste mix, followed by intense shearing, improved the bending strength of the hardened product by an order of magnitude, while increasing the reliability by a factor of 5 as shown in Fig. 11.13.30,31 This diagram presents the results as a plot of failure probability on the vertical axis versus stress on the horizontal axis. Each experimental point represents a strength measurement on a cement strip. Probability is worked out by putting the strength values in order from the highest for sample 1 to the lowest at sample N. Probability for the nth sample is n/(N + 1). This plastic mixed material was so strong that it could be extruded into useful springs (Fig. 11.13(b)).

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It was demonstrated that the flaw size in the material had been reduced from a few millimeters to about by the polymer addition followed by shearing to break up agglomerates. Further improvements were difficult to obtain because the particle size of the large cement grains was suggesting that the size of the particles themselves gave the ultimate limit to properties. Quantitative interpreta-

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tion of the improvements was possible through the Griffith-type equation32 relating bend strength to surface flaw length f

where was the fracture toughness of the cement, that is its crack resistance, in this case Figure 11.14(a) shows a bend test arrangement for measuring the cracking of cement bars. Equation (11.11) is plotted in Fig. 11.14(b) to compare with bend test results on plastic mixed cement samples with edge notches of different lengths cut into the samples with a diamond saw to give various flaw lengths f. The ultimate strength of the cement paste was obtained when the flaw size f was the size of the large particles. Also shown are the results for an ordinary mix of cement, which was weaker by a factor of 20. Consider this theory in relation to the strength of alumina, one of the Earth’s most abundant oxides. Figure 11.15 compares the theoretical tensile strength of alumina with measured values for samples made by three different methods: whisker growth, melting/freezing, and powder extrusion followed by sintering. It is evident that the paste processed material at 0.5 GPa is much inferior to melt formed alumina, which gives a strength of 7 GPa.33 Also shown is the theoretical ultimate strength of 46 GPa for alumina,34 but this value cannot be attained by melt processing, which gives a strength of 7 GPa because of the mobile dislocations which allow yielding of the alumina. Powder processing gives strengths around 0.5 GPa, fourteen times less than melt processing, suggesting from Equation (11.11), inserting toughness of for alumina, that the samples contained edge defects some in length.35 It is astonishing that a material made by extruding particles in size should

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contain such large flaws. The strength of paste processed alumina would be ten times higher if the flaw size was equal to the particle size in this case. Therefore, it appears that flaws are being generated during processing of the concentrated paste. Consider the possible sources of such strength limiting defects.

11.7. ORIGIN OF DEFECTS IN PASTES The nature of defects limiting the strength of materials has been of continuing concern since Griffith32 showed that glass could be strengthened substantially by removing surface flaws. Since the largest defects give the greatest weakness, from Equation (11.11), the concept of a hierarchy of flaws has developed36,37 whereby a material may be improved by removing the largest flaws from the overall defect population. The largest flaws are gas bubbles, large grains, and contaminant inclusions. These can all be removed by evacuation, by controlled thermal treatment, and by working under clean room conditions. The problem is that, even when ceramic pastes are made under the cleanest conditions, the fired products still contain defects between 30 and in length,38 suggesting that there are other processes which can generate large flaws in pastes. For example, diameter titania particles were ball-milled in water with dispersing agent, sedimented to give a fluid concentrate containing no particles larger than cast, dried, and fired to give dense titania ceramic. The bend strength was 200 MPa and Weibull modulus 5, typical strength and reliability for ceramic (Fig. 11.16(a)). This corresponded to flaws some in size, taking toughness as Aggregates of this size could be observed when the dried titania slurry was redispersed, and when the slurry was slightly acidified to destabilize it. The conclusion was that aggregates were growing within the paste during the slurry concentration process. However, when the

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titania slurry was mixed with a polymer under high shear conditions, much higher strengths and reliability were obtained,38 as shown by the results in Fig. 11.16(b). Strength was increased to 750 MPa and Weibull modulus to 25. It is now necessary to consider the forces which may contribute to the growth and removal of defects in these paste processes. One obvious mechanism by which defect structures grow in pastes is analogous to the growth of crystal grains during metal casting (Fig. 11.17). As the particles become compacted during moulding, for example by drying,

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filtering, or sedimenting, they cannot remain in a random arrangement but must form ordered structures to increase their packing fraction for purely geometric reasons. Thus, ordered structures appear from perfect random dispersions during the concentration process,39,40 giving a phase separation between ordered and disordered material, leading to density variations, “phase boundaries,” and flaw development. The hard sphere phase diagram is illustrated in Fig. 11.17, demonstrating that flaws can form by this ordering process at all temperatures. However, it is known that this ordering may be prevented by making the particles slightly different in size or by freezing the random structure so quickly that the diffusive ordering process has no time to occur.41–45 A second, less obvious, mechanism for flaw generation arises from the molecular adhesive attractions which exist between particles in suspension. When these attractions are large, it is well known that the particles flocculate to give agglomerates which can clearly act as flaws (see Chapter 10). As the interparticle adhesion is reduced by adding ions, surfactants, and adsorbing polymers, it has been thought that perfectly dispersed particles can be produced. However, it is now known that aggregated structures form at low adhesion and at low particle concentrations.46–51 Such structures can be found as doublets at low concentrations. The interparticle adhesion is directly related to the ratio of doublets to singlets (Chapter 10). But as the particles are packed closer together, triplets and higher-order aggregates are formed, with a distinct peak at a particular size, typically 20 times larger in diameter than the original particles. Figure 11.18 shows a result for silica spheres in diameter. The large aggregates were

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called multiplets, to distinguish them from floes. Whereas floes increase with time, and are not usually reversible, the multiplets appeared to be stable with time and disappeared on dilution. As the dispersion was concentrated, the multiplets acted as nuclei on which the structured regions of ordered particles grew at the disorder/order transition near 0.5 volume fraction. This theory allows one to calculate the adhesion between silica particles from the structure of opal, knowing the connection between multiplet volume and the adhesion parameters (Fig. 11.19). Opal is formed by the gradual deposition in prehistoric times of a stable sol of silica particles about in diameter.42 The average size of the structured opalescent regions is 2 mm. If we assume that each ordered region starts from a multiplet which acts as a growth point for the ordered structure, then there must have been one multiplet in 0.008ml of suspension, i.e. a multiplet volume fraction of From the linear plot of multiplet volume fraction versus particle volume fraction in Fig. 11.19, an adhesion parameter, the gradient of the curve, can be calculated as a very small value indicating an adhesion a million times less than kT. Clearly, there were very weak adhesive forces acting in the silica dispersion from which the opals derived.

11.8. FRACTURE OF GELS, ESPECIALLY CEMENTS The problem of the strength of gels stretches back into antiquity, since mud, clay, cement and brick are all gels which have been used as construction materials for millennia. How does the gel structure determine strength? This was the question which had exercised Lucretius52 in the first century BC. He had observed water seeping through rock and had associated weakness with porosity:

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“The more vacuum a thing contains within it, the more readily it yields.” On the other hand, Vitruvius53 had observed that long pores were more damaging than short ones and he had suggested that: “Cracks make bricks weak.” In recent times these separate schools of thought have persisted, despite their conflicting nature. It is evident that a gel may be very porous with a large volume of cavities at the nanometer scale, softening the material, but there is no doubt that a crack at the millimeter scale, with almost zero volume, will have a very damaging effect on strength. Experimentally, many results have shown the fall in strength of a gel as porosity is increased. This is true of compressive strength, tensile, and bending strength. It is also true for a wide range of substances made by different methods: natural rock and firebrick,54 alumina and zirconia,55 iron,56 cement,57 plaster of paris,58 and ice,59 as shown in Fig. 11.20. A typical empirical expression used to describe the results is that due to Ryshkewitch55 and Duckworth,60

in which is the strength at zero porosity, is the strength at porosity p, and b is a constant found to lie between 1.3–9. Much effort has been directed towards explaining the value of b for different types of structure. Eudier’s,61 model which was originally proposed by Tabor62 to account for the loss in bonding over the pore area, suggests a value of b around 3, whereas Knudsen’s63 gives a value near 9. Many similar models have appeared over the years.64,65 Rumpf66,67 produced a theory for the strength of agglomerates which was based on adhesion between the particles, and showed that strength must be larger for better-packed finer particles, and found tensile strengths around 0.1 MPa for calcium carbonate compacts. However, none of the authors above considered the effects of cracks on the fracture process. When notches were cut into silica, alumina, and zirconia gels,

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the strength was directly related to crack length in a Griffith manner, and the fracture energies were extremely low, indicating that gels are the most brittle materials known to man.68,69 The crack length was the most important parameter, but pore volume had two influences on the Griffith equation; reducing elastic modulus E* and also reducing fracture energy R* of the gel. The equation below gave a good fit to results for silica gel and also for cement strength (Fig. 11.21)

in an edge notched bend test where and were the porosity-dependent gel modulus and fracture energy, respectively, and c was the notch length. If a spherical particle model is adopted, then E* is known from Equation (11.8) and R* can readily be calculated by the same approach to give

so that the final expression for gel tensile strength takes the form

showing that the packing fraction is most important, but work of adhesion is vital and both particle diameter D and crack length c play a significant role. This model also explains the powerful effects of binders added to gels. In agglomeration processes, small amounts of soluble material are often added to the mix to enhance strength. Just a few percent of binder drying out on the gel surfaces can glue the particles together and increase strength by an order of magnitude. Figure 11.22 shows the effect of using an alumina binder from a sol–

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gel precursor to increase the strength of a silicon carbide gel.70 The model, assuming that the binder material sits uniformly around the particles to increase the contact spot diameter, showed that strength rose with binder volume fraction to the power

11.9. PASTE STRUCTURE AND RHEOLOGY From the arguments above, it is clear that a sol differs from a gel mainly in the adhesion between the particles. The gel is solid because all the particles are in strong adhesive contact. By contrast, the sol is liquid because there is zero adhesion between the particles, thus allowing Brownian movement of the grains. Between these two extremes lies the paste, where adhesion is comparable with 3kT/2, the thermal energy of each particle in the suspension. Consequently, some regions of the paste at the microscopic scale are solid and some are fluid. These regions are constantly changing because of the movement of some particles, which can collide with other particles in the suspension, to make and break adhesive structures which may be thus be changed in geometry. For example, Fig. 11.23(a) shows a paste structure which is solid because a continuous network extends through the material. Figure 11.23(b), on the other hand, shows the situation after strong shearing, which breaks down some of the connections in the structure to form agglomerates which can flow. These two structures behave entirely differently when a shear force is applied. The connected structure is elastic and does not flow at low stress, whereas the disconnected structure can flow at the smallest stress.

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This structural change on shearing can explain the interesting rheology of pastes. At first the paste is stiff and solid, but as the shear is applied, the structure gradually loosens and the material becomes more fluid. In the same way, when the shearing stops, the solid structure can form again as adhesive bonds are rejoined. There is a stiffening with time after the force is removed, as the partial gel structure rebuilds itself. To understand these effects, the adhesive sphere computer model may be used.71 Several thousand spheres are arranged in the computer and allowed to interact according to an adhesive potential. Each sphere moves with constant velocity until it approaches a neighbor. Then, at a short distance before contact, it experiences an attractive potential followed by a hard collision. By varying the attractive energy and the range of the potential, the changes in structure can be observed, and the effect of applying shear forces modelled. Essentially, this model repeated the hard sphere calculations of Woodcock,72 but with adhesion. The shear resistance and viscosity dependence on adhesion are shown schematically in Fig. 11.24. It should be noted that these arguments only apply to particles smaller than about There are two reasons for this: the first is that larger particles will generally not adhere too strongly because they are not smooth enough; the second is that Brownian movement is not very significant above this dimension. Large particles like sands and gravel interact mainly through friction, not adhesion. Therefore they obey Coulomb’s law of soil mechanics73,74

where F is the force required to shear a bed of particles, A is the small adhesion force which can often be neglected, is the friction coefficient, and W the normal

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load on the bed. Such noncohesive powder assemblies have also been computer simulated by sphere models,75,76 which can also be used to describe the fracture of gels.77 Computer models offer the best hope of understanding the transition from sol to gel, a regime much more complex than the simple fluid or solid systems.

11.10. CONTROLLING THE SOL-GEL TRANSITION Many paste processes have a solid product as their end point. Precipitating, coagulating, sedimenting, cementing, filtering, drying, and compacting are all used to convert a fluid into a solid. This is the sol–gel transition at which the particles experience adhesion forces which bond them together from fluid to solid. The problem is understanding how the transition takes place. The easiest way to visualize the transition is to imagine a perfect sol with zero adhesion, then suddenly to switch on adhesive forces between the particles. The fluid suspension will immediately go solid. Later, the solid gel may be fluidized again by removing the adhesion force. In practice, this experiment is most readily carried out by adding a coagulating chemical to the slurry. The added species change the work of adhesion of the particles, which then stick together. An example is the gelling of silica sol by addition of acid. In this case, the long-range colloidal repulsion between the silica spheres is removed and the particles then stick by thermal collisions to form a tree structure of gel (Fig. 11.23(a)) which is readily modeled in the computer. This process has been observed in two dimensions with particles floating on the surface of a liquid.78

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More usually, the particles gel because they are pressed together by an external force. Two modern examples of this type of gelation are electrorheological and magneto-rheological fluids. Consider a dispersion of particles in a fluid, across which an electrical field is applied via two electrodes, as in Fig. 11.25(a). The particles are attracted to each other by the electric field and adhere to form chains at the electrodes. The fluid in this region goes solid because it is much more difficult to shear the adhering chains of particles. When the field is removed, the gel returns to its liquid state as the chains break up.79 Over the years this effect was used in electrostatic precipitators to collect airborne dust in smokestacks,80,81 where the dust was collected as a solid cake. Winslow in 194982 found a similar but reversible effect with silica particles mixed in oil. The slurry stiffened when the field of around was applied but became fluid again when the field was removed. In modern formulations, carbonaceous particles of diameter and of resistivity are mixed to around 40% volume fraction in a silicone fluid of resistivity Although the movement of the particles first takes place by dielectrophoresis,83 i.e. the force on an induced dipole due to an electric field, the clamping force between contacting particles is much enhanced by electrical conduction.84,85 A model for the electric clamping force was obtained by considering the electrical constriction at a contact spot between two particles.86,87 Large attractive forces were found in the narrow gap just outside the contact region, where a high electric field was developed. The deformations caused by the electrical attraction have not been solved for spheres, but can be calculated for flat ended cylinders.85 Two cylinders of diameter D make a small central contact spot of diameter d (Fig. 11.25(b)). When a force F is applied to break this adhesive contact, the

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deformation around the crack is known to give a gap h at radius r where (Fig. 11.25(c))

with an adhesive breaking force of

If an electrical current is now passed through the contact, as in Fig. 11.26(a), a strong electric field acts across the gap, giving a uniform electrical pressure acting to close the crack, since the potential difference increases at the same rate as the gap thickness away from the contact spot. This electrical pressure can dominate the molecular adhesion to give a breaking stress

where is in air, V is the voltage applied, E is the Young’s modulus of the particle, and is its Poisson’s ratio. This equation explains the dependence of breaking stress for semiconducting surfaces in contact under electric fields as shown by the results in Fig. 11.26(b).88 Electro-rheological fluids are interesting because they can be used in vibration isolators, dampers, valves, and clutches.89 Magneto-rheological or ferrofluids, consisting of magnetizable particles mixed in a liquid, give a similar gelation when a magnetic field is applied. These find applications in seals and bearings.87 There are many ways in which external forces can be applied to particles in a paste. The simplest method is to press the paste between rigid plates in a die compaction experiment, Fig. 11.27(a). Usually, a spray-dried granulate is used,

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with an agglomerate size around Such spherical aggregates flow like ball-bearings and fill the mould easily. If the experiment is done in air, the gas flows out readily as the mold closes, and the agglomerates touch, deform, and gradually coalesce as the pressure is increased. Eventually, a fully compacted pellet of strongly adhering grains can be ejected from the mold. Typically, the packing of the powder (volume of powder/volume of mold) is plotted against logarithm of pressure to determine the point at which granule coalescence starts (Fig. 11.27(b)). A better way to observe this is to pass an ultrasonic wave through the pellet during compaction, as shown in Fig. 11.28(a). The wave speed is seen to increase sharply when a gel network forms. On releasing the pressure, the wave speed falls

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to a level corresponding to the gel modulus. The shape of this curve, Fig. 11.28(b), shows the balance between the applied pressure forcing the particles together, and the adhesion force holding the particles in molecular contact. Thus, this ultrasonic experiment can be used as a measure of work of adhesion for the particles. For a wave speed increase by a factor 1.17, the work of adhesion was given by90,91

where D was the grain diameter, the nominal pressure supported by the compact, and the volume fraction of particles in the compact. A value of was obtained for monosize silica spheres of diameter A similar method based on electrical conduction through a carbon pellet was also useful.92 Die compaction of agglomerates has been simulated in a computer sphere model by Thornton et al.19 When the particles are in a liquid paste, the liquid is usually expelled through pores in the mold, and the particles form a filter cake. The mechanism by which this occurs has been modeled by Woodcock et al.93 The application of forces to powder beds, sometimes immersed in liquids, is the subject of soil mechanics.94 This considers particles to interact via Coulomb’s law, Equation (11.16), but also takes into account the hydrodynamic forces acting on the individual particles. Unfortunately, as we have seen, Coulomb’s law is not correct for particles which experience molecular adhesion, so the friction coefficients found in soil mechanics theories seem to vary. Friction seems to increase as the particles get smaller because smaller particles adhere more strongly.71 Soil mechanics is therefore a difficult science.

11.11. REFERENCES 1. Newton, I., Optick’s, Smith and Walford, London, 1704 (reprinted Dover, New York, 1952, p 385). 2. McKay, R.B. (editor) Technological Applications of Dispersions Surfactant Science Series vol 52, Marcel Dekker, New York, 1994. 3. Kendall, K., Powder Technol., 58, 151–61 (1989). 4. Parfitt, G.D. (ed.) Dispersion of Powders in Liquids, 3rd edn, Applied Science Publishers, London, 1981. 5. Harper, C.A. (ed), Handbook of Thick Film Hybrid Microelectronics, McGraw Hill, New York, 1974. 6. Wankat, PC., Rate Controlled Separations, Elsevier, London, 1990, pp 623–727. 7. Bridger, K., Tadros, M.E., Leu, W. and Tiller, P.M., Sep Sci Technol. 18, 1417 (1983). 8. Kendall, K. and Prica, M., 1st European Solid Oxide Fuel Cell Forum, ed. U Bossel, Luzern, 1994, pp 163–170. 9. Staniforth, J. and Kendall, K., J Power Sources 86, 401–3 (2000). 10. Kendall, K., Powders and Grains 93, ed. Thornton, C., Balkema, Rotterdam, 1993, pp 25–31. 11. Kendall, K., Alford, N.McN and Birchall, J.D., Proc. R. Soc. A 412, 269–83 (1987).

PASTES AND GELS 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57.

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Kendall, K., Alford, N.McN and Birchall, J.D., Special Ceramics 8, 255–65 (1986). Johnson, K.L., Kendall, K. and Roberts, A.D., Proc. R. Soc. A324, 301–13 (1971). Kendall, K., Materials Forum, 11, 61–70 (1988). Thornton, C. and Yin, K.K., Powder Technol 65, 153–156 (1991). Clegg, W.J., Alford, N.McN, Birchall, J.D. and Kendall, K., J. Am. Ceram. Soc. 72, 432–6 (1989). Batchelor, G.K. and O’Brien, R.W. Proc. R. Soc., A 355, 313 (1977). Thornton, C., Report No PM-0392/1, Aston University, 1992. Thornton, C. and Sun, G., Powders and Grains 93, ed. C. Thornton, Balkema, Rotterdam, 1993, pp 129–34. Bangham, D.H. and Fakhoury, R, Nature, 122, 681 (1928); Proc. R. Soc. A 130, 81 (1930). Meehan, F.T., Proc. R. Soc. A 115, 199 (1927). Yates, D.J.C., Trans. Br. Ceram. Soc. 54, 272 (1955). Frenkel, J., J. Phys. USSR. 9, 385 (1945). Kuczynski, J.C., J. Appl. Phys. 20, 1160 (1949). Henrichsen, R.E. and Cutler, I.B., Proc. Br. Ceram. Soc. 12, 155 (1970). Clegg, W.J., Alford, N.McN and Birchall, J.D., Br. Ceram. Proc. 39, 247 (1987). Rhodes, W.H., J. Am. Ceram. Soc. 64, 19 (1981). Yan, M.F. and Rhodes, W.H., Mater. Sci. Eng. 61, 59 (1981). Birchall, J.D., Alford, N.McN and Kendall, K., Mater. Sci. Technol. 2, 329–36 (1986). Birchall, J.D., Howard, A.J. and Kendall, K., Nature 289, 388–90 (1981). Kendall, K., Birchall, J.D. and Howard, A.J., Phil. Trans. R. Soc. A 310, 139–53 (1983). Griffith, A.A., Phil. Trans. R. Soc. A 221, 163–96 (1920). Mallinder, F.P. and Proctor, B.A., Phil. Mag. 13, 197 (1966). Kelly, A. and Macmillan, N.H., Strong Solids, Clarendon Press, Oxford, 1986, p 6. Alford, N.McN., Birchall, J.D. and Kendall, K., Nature 330, 51–3 (1987). Lange, F.F., J. Am. Ceram. Soc. 66, 396–8 (1983). Lange, F.F. and Metcalf, M., J. Am. Ceram. Soc. 66, 398–406 (1983). Kendall, K., Alford, N.McN., Clegg, W.J. and Birchall, J.D., Nature 339, 130–2 (1989), Pusey, P. N. and van Megen, W., In: Physics of Complex and Supramolecular Fluids eds. S.A. Safran and N.A. Clark, Wiley Interscience, New York, 1987, pp 673–98. Russel, W.B. and Sperry, PR. Prog. Organic Coatings 23, 305–24 (1994). Alder, B.J. and Wainwright, T.E., J. Chem. Phys. 31, 459 (1959). Sanders, J.V. and Murray, M.J., Nature 275, 201 (1978). Hoover, W.G. and Ree, F.H., J. Chem. Phys. 49, 3609 (1968). Pusey, P.N., J. Phys. (Paris) 48, 709 (1987). Poon, W., Pusey, P.N. and Lekkerkerker, H., Physics World 9, 27–32 (1996). Liang, W., Austin, J.C. and Kendall, K., J. Mater. Sci. Lett. 17, 951–2 (1998). Kendall, K. and Liang, W, Colloid Surf 131, 19–201 (1998). Kendall, K., Liang, W. and Stainton, C., J. Adhesion 67, 97–109 (1998). Kendall, K. and Liang, W., http://www.keele.ac.uk/depts/ch/inorganic/paper.html Kendall, K., Liang, W. and Stainton, C., ICF 9, Sydney April 1997, Pergamon, Amsterdam pp 2433–40. Kendall, K. and Liang, W., Br. Ceram Trans 96, 92–95 (1997). Lucretius, On the Nature of the Universe, translated by R. Lathom, Penguin, London, 1951, p 43. Vitruvius, The Ten Books of Architecture, translated by M.H. Morgan, Dover, New York, 1960, p 43. Griffith, J.H., Ceram. Abstr. 18, 35 (1939). Ryshkewitch, E., J. Am. Ceram. Soc. 36, 65 (1953). Goetzel, C.G., Iron Age 150, 82 (1942). Feret, R., Bull. Soc. Encourage. Indust. Nat. II, 1604 (1897).

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63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94.

CHAPTER 11 Lambe, C.M. and Offutt, J.S., Bull. Am. Ceram. Soc. 33, 272 (1954). Frankenstein, G., US Army Cold Regions Res.and Eng. Lab. Tech. Rep. No 172, 1–36 (1968). Duckworth, W., J. Am. Ceram. Soc. 36, 68 (1953). Eudier, M, Powder Metall, 9, 278 (1962). Tabor, D., In: Mechanical Properties of Non-Metallic Brittle Materials, ed. H. Walton, Butterworths, London, 1958, p 48. Knudsen, P.P., J. Am. Ceram. Soc. 42, 376 (1959). Schiller, K.K., In: Mechanical Properties of Non-Metallic Brittle Materials, ed. H. Walton, Butterworths, London 1958, p 35. Assur, A. Natl. Acad. Sci. Nat. Res. Council US Publ. 598, 106 (1958). Rumpf, H., Int. Symp. on Agglomeration, ed. W.A. Knepper, Interscience, London, 1962, pp 379– 418. Rumpf, H., Chem. Ing. Tech. 42, 538 (1970). Kendall, K., AIP Conf Proc. 107, pp 78–88 (1984). Kendall, K., Powder Metall. 31, 28–31 (1988). Birchall, J.D., Alford, N.McN and Kendall, K., J. Mater. Sci. Lett. 6, 1456 (1986). Stainton, C., PhD Thesis, Keele University, 2000. Hopkins, A.J. and Woodcock, L.V, J. Chem. Soc. Faraday Trans. 86, 2121–32 (1990). Kendall, K., Nature 319, 203–5 (1986). Coulomb, C.A., 1773 Mem. Acad. Sci. Savants Etrangers 7, 343–82. Cundall, P.A. and Strack, O.D.L., Geotechnique 29, 47–65 (1979). Thornton, C., Yin, K.K. and Adams, M.J., J. Phys. D: Appl. Phys. 29, 424–35 (1996). Van Baars, S., ICF9 2, 917–26, (1997). Abrahams, F.F., Phys. Rep. 80, 341 (1981). Bridgestone R&D Division, 3-1-1 Ogawa higashi-cho, Kodaira-shi, Tokyo. McLean, K.J., J. Air. Pollut. Control Assoc. 27, 1100 (1977). Lowe, H.J. and Lucas, D.H., J. Appl. Phys. (Suppl 2) S40 (1953). Winslow, W.M., J. Appl. Phys. 20, 1137 (1949). Jones, T.B., Electromechanics of Particles, Cambridge University Press, Cambridge, 1995, chap.3 See, H., and Saito, T. Rheol. Acta. 35, 233–11 (1996). Kendall, K., J. Phys. D: Appl. Phys. 24, 1072–5 (1991). Dietz, P.W., and Melcher, J.R. Ind. Eng. Chem. Fundam. 17, 28 (1979). Moslehi, G.B. and Self, S.A. IEEE Trans. Ind. Appl. 1A-20, 1598 (1984). Stuckes, A.D. Proc. Inst. Elec. Eng. Land. 103, 125 (1956). Rosensweig, R.E. Science 204, 57–60 (1979). Kendall, K., Br. Ceram. Trans. 89, 211–3 (1990). Kendall, K., Technol. Powder 66, 101–4 (1991). Kendall, K., J. Phys. D: Appl. Phys. 23, 1329–31 (1990). Warr, S. PhD Thesis, University of Bradford 1992. Horne, M.R. Proc. R. Soc. A 310, 21–34 (1969).

12 ADHESION OF BIOLOGICAL CELLS: THE NATURE OF SLIME

I wish I could derive all phenomena of nature‚ by some kind of reasoning‚ from mechanical principles ISAAC NEWTON‚ 16871

In Newton’s time‚ the concept of the biological cell did not properly exist. The establishment of the cell theory of life had to wait until the 1830s. Yet the measurements of blood circulation created by Harvey‚ who had been at Padua from 1601–2 while Galileo was perfecting his mechanistic views of motion and materials‚ had brought about a change in thinking during Newton’s life‚ enabling living matter to be interpreted mechanistically. And the burgeoning field of microscopy‚ which was discussed frequently in the Royal Society by Robert Hooke who published his famous book in 1665‚2 when Newton was 23 years old‚ had allowed tiny eggs and structures to be resolved and categorized for the first time‚ as shown in Fig. 12.1. Indeed‚ Hooke used the word “cellula” to describe plant cell structures. So much had these ideas taken hold that‚ while Newton was still a young man‚ Harvey’s notion that “ex ovo omnia”3 (all things come from eggs) was becoming widely accepted. Thus it gradually became clear that man is a peculiar aggregate of cells sticking together by adhesive processes. This chapter examines such adhesion in detail.

12.1. INTRODUCTION AND IMPORTANCE Cell adhesion is one of the most dominant variables in biology.4‚5 It defines the distinction between single-cell and multicellular organisms‚ as in Fig. 12.2(a). 275

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It dictates whether cells can survive and move over substrates (Fig. 12.2(b)). It is important to metastasis of cancer cells‚ to infection of cells by bacteria and viruses (Fig. 12.2(c))‚ to creation of nerve connections‚ to mating of cells and to other contact and communication phenomena. Yet the observation and measurement of cell adhesion are difficult because the cells have small diameters‚ between and the adhesion force is generally low‚ typically 0.1–100 nN. Usually‚ cell adhesion must be studied in a microscope. Attachment of cells may then be observed in relation to the chemical environment‚ and the cells may be pulled apart and sensed by various techniques including direct mechanical probing‚ shearing with a flow‚ or cytometry (cell counting) to determine the adhesion quantitatively. There are several statistical problems with such observations. The first is Brownian movement of cells‚ which are small enough to be pulsating with

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molecular impacts as shown in Section 12.8 later. Secondly‚ there is the natural variation between cells giving rise to differences in adhesion between populations. Consequently‚ cell adhesion values cannot be measured as precisely as other adhesive joints. These statistical variations cause lingering doubts about the precise mechanisms of cell adhesion. The interesting thing about biological cells is their intermediate size range between the nanometer sizes where everything sticks and the centimetre dimensions where nothing adheres‚ as explained in Section 4.7. Thus biological cells can vary in their adhesion very markedly depending on shape‚ elasticity‚ surface roughness‚ and surface chemistry‚ as explained in the mechanisms of Chapters 4–8. In particular‚ the cell can often control its own adhesion‚ as in the slime mold‚ using some mechanism to change from a single cell to a structure containing 100‚000 or more cells. Some of this variety of adhesion may be straightforward double layer effects (see Chapter 10) but there is no doubt that more complicated mechanisms may be involved and that particular polymer molecules‚ now identified as adhesins‚ integrins‚ lectins‚ etc.‚ can be influential. The purpose of this chapter is to outline the basic adhesion principles‚ starting from a definition of cell surface structure‚ moving on to measurement methods‚ then finally discussing the peculiar molecular influences.

12.2. MODELS OF CELL CONTACT: THE POLYMER COATING Our models of cell contact have developed from electron microscope observations and other nanoscale techniques for studying the detailed structure of cellular junctions‚ together with the deformation of cells as they stick. Essentially there are two situations: (a) where a cell meets another cell‚ e.g.‚ blood cells touch‚ and (b) when a cell meets a neighboring surface‚ e.g.‚ algae sit on a rock. In each case it is important to understand the local environment of the contact region‚ as shown in Fig. 12.3. This contact area is usually dominated by polymer layers‚ both on the outer surface of the cell and on the substrate. Such slimy polymeric films often form the bulk of the solid material in a cell colony‚ between 50–90%. The usual picture is dominated by the layer of polymer on each surface. On the cell this is called the sheath‚ capsule‚ or glycocalyx whereas on the glass it is called the biofilm. Such layers were not easily seen by optical microscopy whereas the cell membrane was more visible. The cell membrane is a back-toback layer of phospholipid surfactant molecules which is often ruffled at the 100 nm level‚ as shown schematically in Fig. 12.3‚ with various projections such as proteins or filaments projecting from it.

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Zobell6 noted the fuzzy polymer coating surrounding the membrane in 1943 and vigorous studies followed in the 1960s. Staining methods were found and described in detail.7 The polymer material was seen to comprise proteoglycan‚ a protein chain around 200 000 D long with around 100 glycosaminoglycan chains around 20 000 D hanging from it.8 These molecules could be several micrometers long‚ anchored on the surface at several points‚ forming the slimy film of thickness from 100–2000 nm. The cell sheath also contains other glycoproteins‚ the so-called cell adhesion molecules or CAMs‚ which have been implicated in the mediation of cell adhesion. A large number of such molecules have now been identified since the original work by Gerisch9 on slime mold (see Section 12.7). A list of these is given in Fig. 12.4‚ together with other molecules such as the nectins and

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immunoglobulins which have been isolated from cellular material and shown to change cell adhesion.10 Fibronectin has two chains of 225‚000 D appearing as two strands around l00nm long.11 Proteins are also deposited on any solid substrate which has come into contact with the suspension holding the cells. Many studies have shown that a film of polymeric material builds up on all surfaces‚ such as glass or polymer‚ which have held the cell dispersion. These slimy films may have an effect on the cell adhesion to that surface. For example‚ when fibronectin was coated onto polished metal surfaces of titanium and cobalt alloys‚ the cell adhesion increased by a factor of 2.6.12 In this manner‚ the adhesion molecules act like the silane coupling agents used to coat glass fibers in Section 16.10.

12.3. CELL MEMBRANE AND CYTOPLASM: EFFECTS ON CONTACT SPOT Intimate cell contact and subsequent deformation are largely defined by the cell membrane and its inner gelatinous contents‚ the cytoplasm‚ whose structure is shown schematically in Fig. 12.5. The principal structural element of the membrane is the lipid bilayer made up of surfactant molecules comprising phospholipids (for example‚ phosphatidyl choline)‚ cholesterol‚ and glycolipids‚ with the hydrophobic chains of the lipids

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sandwiched between the polar end groups. These lipids are in a highly mobile state and are thought to form phase separated domains like Langmuir–Blodgett films on water. Overall‚ the surface charge is negative but not so simple as a colloid particle. Attempts to explain cell adhesion in terms of DLVO theory (see Section 10.5) have met with only limited success. Within this membrane structure sit protein molecules‚ in about equal weight to the lipids‚ and these cannot be readily removed because they are anchored by ten or twenty hydrophobic amino acids.13 For example‚ a red blood cell membrane contains largely glycophorin molecules which have 131 amino acid residues and about 100 saccharide residues arranged as 16 short carbohydrate chains about 1 nm long.14 Sticking out from the membrane surface are larger molecules such as the adhesion molecule made of polypeptide chains about 5 nm long in total. The fact that these membrane structures are in constant motion has been known for many years‚ because fused cells with fluorescent labeled antibodies of different colors rapidly merge to give uniform color. The diffusion coefficient is between for lipid to for antibodies. Despite the complexity of internal cell structure‚ which comprises the containing membrane‚ gelatinous interior cytoplasm‚ internal granular bodies‚ fibrous skeleton‚ and nuclei‚ each cell may be treated approximately as a spherical viscoelastic shell containing a viscous fluid. The model of an outer shell with an inner fluid describes very well the deformation of red blood cells and also of artificial vesicles made by sonicating phospholipids in water.15 The shell dictates the equilibrium while the fluid contents dictate the rate of approach to equilibrium. A red cell needs three numbers to describe its response time of 0.1 s: an area compressibility a shear modulus and a viscosity Pa s.16 A red blood cell is atypical because its surface is smooth. More typically‚ a phagocyte cell has a ruffled surface with more than twice the surface area of an equivalent sphere. This cell has no resistance to deformation until the ruffles are pulled flat. After that the cell is stiff and ruptures when extended more‚ as in the experiment of Fig. 12.6. Deformation of the structure is important as contact between two cells becomes established in the sequence of events illustrated in Fig. 12.7. This scheme is derived from electron microscope images‚ originally described by

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Brightman and Palay‚17 to demonstrate that the geometry of the contact zone can change systematically as the membranes approach each other. The membrane is shown as a parallel set of lines‚ 7 nm apart‚ representing the phospholipid bilayer. In the first type of contact‚ Fig. 12.7(a)‚ the cell membranes remain some tens of nanometers apart‚ though the cells are evidently touching‚ the so-called zona adherens. The gap is presumably filled with the sheath of polymer coat which surrounds each cell‚ acting as a steric barrier to full membrane contact. Figure 12.7(b) shows the next stage of contact‚ the desmosome‚ in which the gap remains large but structures inside the cell build up over a period of 30 min. Then‚ Fig. 12.7(c)‚ some hours after initial contact‚ the cells can move into apparent molecular proximity‚ called “zona occludens.” If this region extends‚ then phagocytosis can occur causing the larger cell to engulf the smaller one‚ Fig. 12.7(d). Alternatively‚ the cells can gain more intimate contact and can interpenetrate to form a gap junction where direct communication is provided by apparent gaps in the membrane‚ Fig. 12.7(e).18 Ultimately‚ cells can fuse by total integration of cell membranes to form a larger cell‚ Fig. 12.7(f). This happens during mating‚ for example. These changes in cell contact are strongly related to the roughness and projections which can be seen on the cell surface.

12.4. ROUGHNESS AND CELL SURFACES The roughnesses on a cell surface are extremely interesting because they vary so much from species to species‚ and because they are known to be strongly

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involved in cell adhesion. Already we have mentioned the ruffled membrane surface‚ which can straighten out as it allows a cell to spread while maintaining a constant volume of cytoplasm‚ as shown in Fig. 12.8. Much more complex projections emanating from cell surfaces have also been observed by optical and electron microscopy. Whereas some cells are very smooth‚ others project pseudopodia and microvilli‚ as shown in Fig. 12.9. Pseudopodia are fluid excrescences‚ stringy feet which probe several micrometers into the fluid medium. Microvilli are much smaller‚ around 200 nm in length‚ surrounding bundles of filaments which form the internal cell skeleton. Such projections seem to help adhesion. Phagocytes do not produce pseudopodia and cannot attach to glass.19 On bacteria‚ the most obvious projections are flagellae which push the cell through the liquid. These are about 20 nm in diameter‚ helical in shape and made of a protein‚ flagellin‚ which grows out to several in length. Typically‚ E. coli has 5–10 flagellae which impart a random tumbling motion. They drive the cell towards surfaces where adhesion might occur.

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Fimbriae were identified as the filaments responsible for developing bacterial adhesion to other cells or surfaces.20 Various types were distinguished‚ produced by E. coli‚ salmonella and many other types of bacteria. Typically these projections were long and 7 nm in diameter‚ with up to 400 per cell. When the bacterial adhesion was measured by a blood cell aggregation test‚ or by a microscope counting of adhering cells‚ the fimbriae were inhibited by Dmannose addition. Pili are similar looking structures but are used to transfer genetic information.21 They are typically 9 nm diameter and several long with 1–3 pili per cell. The pili can be identified because they attract bacteriophages which are thought to transfer nucleic acid into the cell. Cells like to adhere to rough surfaces. A well-known case is that of artificial bone for implants‚ made from porous ceramic material such as foamed hydroyapatite; the cells like pores about diameter to grow in. In a particular recent example‚ cells stuck better to substrates with multiple grooves. Micronscale grooves were cut into a titanium surface and various cells were observed moving and growing on the grooves.22 Fibroblasts‚ the cells responsible for new tissue deposition during healing‚ moved along the grooves which ranged from in depth and wide. On the other hand‚ macrophages‚ white blood cells which remove foreign matter‚ became trapped in the grooves. The adhesion molecule actin was seen forming at the grooves by fluorescence after 5 min. Tubulin‚ vinculin and other adhesion molecules were observed later. Healing of wounds was speeded by presence of the grooved substrate‚ compared to a flat. Another important issue is the chemical structure of the surfaces onto which the cells attach. It is well known that cells will only thrive on certain surfaces where the adhesion is fully developed. For example‚ fibroblasts will not grow while moving freely in a culture suspension‚ but will proliferate while adhering on glass or plastic. This is the so-called “anchorage dependence” of cells. This concept was tested in an elegant experiment by Folkman and Moscona.23 They used bovine aorta endothelial cells which grow usually on plastic tissue culture dishes made from polystyrene. By coating the plastic dishes with increasing thicknesses of polyHEMA (poly(2-hydroxyethyl methacrylate)‚ the adhesiveness of the cells could be varied bit by bit. The experiment and its results are shown in Fig. 12.10. An alcoholic solution of polyHEMA was pipeted into a plastic culture dish and allowed to evaporate to leave a very thin coating of polyHEMA on the surface. Using higher concentrations of polyHEMA allowed thicker films to be built up‚ from 3.5 nm– which reduced the cell adhesion systematically. The cells would stick well to the thin layers but not to the thick ones. This effect was seen some minutes after plating the cells onto the surface. The cells adjusted their size to reach a stable shape‚ which was almost spherical

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for the thick polyHEMA films and strongly flattened for the thin films. Afterwards‚ these shapes did not change over a 7 d period‚ even though the film was swelling slightly in the water. The rate of DNA synthesis in the cells was measured by incubating with followed by autoradiography. The results showed that the DNA synthesis stopped in the spherical nonadhering cells but proceeded rapidly in the adherent cells. An interesting question was whether this inhibition of DNA synthesis was due to adhesion or to shape. By crowding the cells to make them push against each other‚ the cells became more spherical. The DNA synthesis was less in this case‚ proving that cell shape is the important variable. Now that the model of the cell structure is established‚ it is necessary to discuss methods for measuring adhesion of cells.

12.5. CELL ADHESION BY PROBE METHODS The most direct method for observing cell adhesion is microscopy. Cells may be observed sticking to each other or to another body‚ then spreading out along the surface. A qualitative measure of adhesion is the size of the contact. Red blood cells make only small contact whereas muscle cells make extended contact‚ as shown in Fig. 12.11.

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If the cells were of known deformation characteristics‚ like the polymer latex discussed in Section 9.7‚ then it would be possible to quantify the adhesion from the deformation of the cells. Unfortunately‚ this is not generally possible because cells are not spherical‚ not uniform‚ and not constant with time. Probing the cells mechanically does offer a way forward. As early as 1944‚ Coman used a glass fiber to detach epithelial cells from surrounding tissue.24 From the fiber bending he obtained a force of He then attempted to show that cancer cells were less adhesive. Lately‚ the probe method has been much developed by Evans and his colleagues.16 He used an inverted microscope and sucked the cells onto micropipettes. Cells could then be pressed together and pulled apart to determine the adhesion forces‚ as illustrated in Fig. 12.12.25‚26 By varying the suction pressure‚ the energy of adhesion could be obtained. In principle it is possible to obtain force separation curves but this has not been achieved‚ probably because the cells are so compliant. The same technique has been used to measure the adhesion of lecithin vesicles in salt solution. The contact and separation in this case were reversible and the work of adhesion was This compared with the work of adhesion of ordinary red cells in blood plasma of and for red cells bonded by lectin.

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Atomic force microscopy (AFM) has become very popular as a way of measuring cell adhesion. In an AFM experiment on living glial cells, Parpura and colleagues imaged the cultured cell as it sat on the surface, then increased the force on the probe until it removed the cell from the substrate. While the analysis of the force action was difficult, the conclusion was that adhesion of different cells could be compared using this technique.27 In similar experiments using agarose beads coated with specific molecules, the AFM was used to measure specific bonding between receptors and ligands. The specificity of reaction was confirmed by blocking the active site.28 The AFM is particulary useful for studying molecular adhesion interactions using a functionalized tip and a receptor attached to a smooth surface. A typical experiment was that by Luckham and Smith who investigated the adhesion of cholera toxin B subunit to its receptor The geometry of the contact is shown in Fig. 12.13. The molecule was diluted to spread it over the glass surface in methanol/chloroform. The layer was then deposited on silane coated glass using a Langmuir trough. The cholera toxin molecule was attached to the silicon probe tip using glutaraldehyde to crosslink it. Then the tip was lowered towards the surface in water to detect the bonding force, with the aim of detecting a single toxin/receptor interaction. Many force measurements showed that the adhesion of one bond was 0.09 nN, but two bonds, i.e., 0.18 nN, and three bonds, i.e. 0.27 nN could be observed. These forces are compared with other measurements of other molecular pairs in Table 12.1. The benefit of the atomic force microscope is that it can define the surface of cells at molecular resolution, showing the mechanisms in more detail. For example, red blood cells were collected in EDTA K2 and smeared across a steel disc.32 Fixing of the cells was achieved with glutaraldehyde. The cells were dried and scanned in the AFM. At high resolution, down to 5 nm, structures were

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observed on the cell membrane. In patients with spherocytosis‚ 80 nm “blebs” or protrusions were seen on the surface. Thus‚ the AFM can distinguish variation in adhesion across a cell surface. This is not possible with other techniques.

12.6. CELL ADHESION BY FLOW METHODS In practical cell culture applications‚ fluid flow methods of estimating cell adhesion have been most useful. In particular‚ the radial flow chamber shown in Fig. 12.14 was developed to study both deposition and removal of cells at surfaces.33 The apparatus has been used to measure adhesion of many cell types to different surfaces‚ including diatoms34 and pseudomonas fluorescens.35 The cell is shown in Fig. 12.14(a). The cell culture is grown to produce a fluid containing a concentration of cells. This fluid is pumped down a tube to impact onto the test plate‚ then to move radially along the plate. Shear stress is highest at the inlet in the middle of the plate‚ dropping off with radius according to the equation

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where Q is the flow rate‚ the viscosity‚ and h the gap. The results show that there is a critical radius for cell adhesion and another for cell detachment. Different test surfaces give different characteristic curves as shown in Fig. 12.14(b). The interpretation of such curves in terms of the cell adhesion force or energy is difficult because it depends greatly on the assumptions about the cell geometry and the flow regime‚ However‚ this method is of practical utility for measurement of biofouling which is of enormous interest to industry.36 Another flow method for studying cell adhesion was developed by Bongrand and his associates.37 The idea was to use a laminar flow along a plane surface‚ such that the force on each cell was less than the strength of a single ligand– receptor bond‚ but sufficient to make the free cells move with a velocity of a few micrometers per second. By watching the movement of cells‚ bonding events‚ and breaking events could be defined. Then the influence of adhesion molecules on these events could be observed. The basic scheme is shown in Fig. 12.15. A rectangular cavity 1 mm deep was cut in a plastic block and a glass coverslip glued to it with silicone rubber to form a microscope cell through which cell dispersions could be pumped with a motorized syringe. The shear rate G was around In this situation‚ the force experienced by a spherical cell bound to the substrate is about38

Where D was the cell diameter‚ G the shear rate‚ and was the viscosity. Because the viscosity of dilute aqueous solutions is 0.0007 Pa s and a typical cell diameter is the force is which is around 1 pN‚ much smaller than the force of a single bond. Of course‚ the cells some distance from the wall are moving and also rolling but these do not stick to the surface and are neglected. The experiment consists in observing the cells and only observing those close to the wall. These move at constant slow speed until there is an adhesion

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event; then they stop‚ but start moving again later once the bond breaks due to Brownian collisions. A well defined arrest and continuation is shown in Fig. 12.16(a). A large population of T-cell hybridoma B10BR cells coated with anti CD8 antibodies was observed with the video microscope and cell arrests were measured as the shear rate was increased in the range to give the results shown in Fig. 12.16(b). In the full line of this graph‚ the fraction of cells showing one stop‚ i.e. greater than 1 s‚ was plotted as the shear rate was increased. A significant fraction of cells showed binding events‚ which fell as the shear rate increased. Once the cells adhered‚ some could stick very strongly within a few seconds. In this case‚ the cells made molecular contact with the surface and required a large increase in flow rate‚ typically 1000 times larger‚ to detach them. Thus there were two adhesion steps‚ an initial single bond‚ followed by a multiple strong bond. The interesting feature of the experiments was the effect of antibodies on the glass surface. The glass was treated with ethanol‚ then with glutaraldehyde in phosphate buffer‚ followed by incubation with monoclonal antibodies. In the control experiments‚ random antibodies were used‚ but in the broken curve of Fig. 12.16 anti-CD8 antibodies were applied. This had a strong effect on increasing adhesion. Also‚ the adhesion events were more permanent. These experiments were supported by further tests on receptor-coated beads to show that similar phenomena occurred.39 Beads diameter were coated with streptavidin and observed flowing across mica surfaces‚ either control or coated with biotin. The adhesion molecules caused 5–13 times more arrests.

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12.7. CELL COUNTING METHODS The third method widely used for measuring cell adhesion is cytometry or cell counting‚ first employed by Orr and Roseman.40 They passed a cell dispersion through a Coulter counter and measured the loss of single cells as aggregates formed. The counter in this case did not detect the aggregates themselves but could‚ in principle‚ have measured the whole size distribution of aggregated cells. The Coulter counter was invented by Wallace H Coulter in the late 1940s and its mode of action has found widespread use in the blood and cell industries.41 It is shown schematically in Fig. 12.17. Cells are dispersed in an isotonic solution which is stirred and pumped through a fine orifice. An electrode is positioned on each side of the orifice‚ passing a small current through the conducting fluid as it flows through the hole. As a cell goes through the hole‚ it causes an electrical blockage proportional to its diameter. Each pulse is counted and sized by the computer to give a distribution of cell sizes. In more recent versions of this method‚ cells are labeled with fluorescent molecules‚ and specific adhesive bonds between cells can then be measured using a laser and detector.42 This has mainly been used to measure cell–cell interactions in the immune system‚ especially those involving toxic cells and target cells. The method can also be used to separate specific adhered cells. In a typical experiment‚42 mucosal epithelial cells were mixed with bacterial and candida organisms. By counting the aggregates in the electronic apparatus‚ the adhesion could be quantified. Alternatively‚ the engulfment of nanoparticles by human phagocytes could be quantified by labeling the particles using fluorescent molecules‚ then counting the fluorescing cells.42

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12.8. NEW APPROACH TO CELL ADHESION Recently, a new method for measuring and understanding cell adhesion has been devised.43,44 The objective was to remove the need for probes because these damage the cells and change the conditions. Also, the idea was to produce an absolute measure of cell adhesion which did not demand new definitions of binding. It seemed logical to define adhesion of cells in terms of the two parameter model of adhesion interactions described in Sections 5.1, 5.9 and 7.5. Consider a dilute dispersion of uniform spherical particles as shown in Fig. 12.18.45 These spheres experience Brownian motion and therefore diffuse in all directions, causing collisions between the particles. If there is adhesion between the particles, then a collision has a chance of creating a doublet, that is two particles adhering together at the point of contact. If the adhesive bond is weaker than kT, then thermal collisions can break this bond in a period of time. The spheres will then separate and move apart. Thus there is a dynamic equilibrium between joining and separation, giving a certain number of doublets in the suspension at equilibrium, after a suitable time has elapsed for diffusion to take place. High adhesion should give a larger number of doublets and lower adhesion a smaller number. Hence there is a definite connection between sphere adhesion and the equilibrium number of doublets observed in a dilute suspension. Of course, there are several assumptions in this argument. The main premise is that the spheres are all identical. This is not true of most cells which are known to have distributions of various molecular species on their surfaces. However, it is possible in principle to filter out any rogue doublets formed by unusually tacky cells. Equilibrium can then be re-established. Repeating this filtering and equilibration procedure several times should lead to a point where the remaining cells are more nearly equal. A second assumption is that the cells are spherical and equal in diameter. In fact, human red cells are dimpled and range in size

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between The errors in the argument caused by such problems are not yet known but are being investigated by computer modeling. The most interesting consequence of the above idea‚ that cell adhesion may be measured by observing the number of doublets at equilibrium in a dilute suspension‚ is that an exact mathematical solution can be found under certain circumstances‚ depending on the interaction between spheres when they collide. The simplest situation is that shown in Fig. 12.19 where a particle approaches its neighbor at constant speed until‚ at a certain separation‚ the particles are attracted to each other with an energy If this energy remains constant until the spheres touch rigidly at the point of contact‚ then the square well potential is revealed. The approaching sphere travels at constant speed‚ is accelerated into the potential well‚ reflects rigidly on contact‚ and then is decelerated as the particles move apart. This “hard sphere square well” which was first used by Alder and Wainwright46 in 1961 can be solved exactly to predict the number of doublets in a suspension. The mathematical result is that the ratio of doublets to singlets is proportional to the volume fraction of the cells and depends on the range and the energy of the well according to the equation below

The conclusion of this argument is that a plot of doublet to singlet ratio versus particle volume fraction should yield a straight line passing through the origin. The gradient of the line is a measure of the adhesion which depends on range and energy of the interactions. Thus a high gradient signifies high adhesion and a low gradient low adhesion as shown below in Fig. 12.20. Thus an adhesion number can be defined as the gradient of this plot‚ to give a measure of the bonding of the

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cells. The experimental objective was to define this nondimensional adhesion number for three different species of red cells: horse‚ rat‚ and human.

12.9. EXPERIMENTAL RESULTS The important conclusion from these arguments is that cell adhesion must depend on cell concentration. This is obvious in retrospect but has not been stated before. Cells will appear to stick more in proportion to their volume fraction. Of course‚ this is a general law which applies to all reversibly adhering Brownian particles. Red blood cells‚ erythrocytes‚ were used because of their low and reversible adhesion. Cells were prepared from three species‚ human blood from North Staffordshire Hospital‚ fresh horse blood in EDTA‚ and fresh rat blood from Central Animal Pathology Ltd. Each blood sample was washed six to seven times in phosphate buffered saline to remove the nonred-cell components‚ before suspending in physiological saline solution‚ then examined by both optical and Coulter tests. Each species of cell was treated in three ways to judge the effect of surface adhesion molecules: by adding glutaraldehyde‚ fibronectin‚ and papain. The optical apparatus is shown in Fig. 12.21. The cells were placed in an accurately defined space within a glass cell which was imaged using a video microscope at 40× magnification. Each cell could then be clearly seen moving around with Brownian movement‚ while not overheating as occurred at l00× magnification. Pictures of the cells were taken at random locations in the cell and the numbers of doublets and singlets were counted by image analysis software. Taking the ratio of doublets to singlets‚ the adhesion number was obtained. The collision and adhesion events could be observed in experiments as shown in Fig. 12.21(b) which shows one field of view. There were several doublets which could be counted.

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The second set of experiments to measure the doublet numbers used the Coulter counter‚ which was set up in standard mode to count the individual red cells‚ as shown by the results of Fig. 12.22(a). The strong peak showed a symmetrical distribution of single cells at a volume fraction near At higher concentration‚ a shoulder appeared at a 13% higher diameter‚ Fig. 12.22(b)‚ and this was interpreted as a doublet peak. At still higher concentration

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of the red cells‚ the shoulder increased in size (Fig. 12.22(c))‚ indicating that more doublets formed as the blood cells became more numerous. The number of doublets was measured and divided by the singlet peak to obtain the ratio This was then plotted as a function of cell volume fraction to give the curve shown in Fig. 12.23. The results showed the doublets increasing in proportion to concentration and allowing the adhesion number to be found by determining the gradient. For human cells this was 420. Horse and rat erythrocytes were then tested in the same way and shown to give significantly higher adhesion. Baskurt et al.‚47 have shown that the aggregation of such cells is increased over human cells‚ but volume fraction effects were not taken into account. Popel et al.‚48 recognized that horse cells stick better and this was attributed to the athletic nature of the animal. Table 12.2 quantifies the difference of adhesion in terms of the adhesion number These results show conclusively that rat cells are almost twice as sticky as human red cells‚ while horse erythrocytes are almost twice as adhesive as rat cells. Whether this can be explained in terms of the higher energy of the bonds‚ as defined by Equation (12.1)‚ or in larger range of bonds remains to be determined.

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Addition of surface active molecules to the cell suspension was also studied. The results for human cells are illustrated in Fig. 12.23 which shows that fibronectin increased the adhesion whereas glutaraldehyde reduced it. The effect of additives on horse erythrocytes is shown in Table 12.3. The control sample of horse cells in isoton showed somewhat weaker adhesion than the sample shown in Table 12.2. Such variation was found to be common in different samples of horse blood. Differences between animals in type‚ age‚ etc.‚ and also in blood cell conditioning had a distinct influence which will be described in separate papers. It is evident from the results that glutaraldehyde reduced the adhesion by about 25% whereas fibronectin increased the adhesion by 10% and papain by 20%‚ changes which were comparable with the effects seen on human red cells but disappointingly small compared with the effects anticipated.

12.10. APPLICATION TO PRACTICE Let us focus on three adhesion questions of wide interest: why cells move when they are adhered to a substrate‚ why cells gather in multicell organisms‚ and how to stop unwanted cells sticking‚ growing and fouling on substrates such as teeth. The question of cell movement was studied by Palecek et al.49 Using CHO B2 cells on silane coated glass‚ with fibrinogen or fibronectin on the surface to promote adhesion in Optimem 1 solution‚ they observed that the migration speed of the cells was related to cell adhesion in a particular way. At low adhesion and at high adhesion the cells moved slower‚ with an optimum adhesion of 20 nN for maximum speed‚ as shown in Fig. 12.24(a). Migration speed was measured by video camera with image analysis to give the position of the cell centroid over a period. Adhesion was measured in a fluid flow device‚ at 50% cell removal in 5 min‚ assuming a hemispherical model. The same cell speed relation to adhesion was observed when the cell integrin (fibronectin receptor) was modified by genetic engineering‚ or when the binding affinity of receptor and adhesion

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molecule were modified. This means that several possibilities may emerge for moderating cell movement‚ for example in cancer treatment.50 The forces causing cells to gather together into clumps‚ as shown in Fig. 12.24(b)‚ have been studied by a large number of workers‚ particularly with slime mold Dictyostelium discoideum.51 Of great significance is the change from single cells‚ living separately on an agar gel surface‚ to a single clump of 100‚000 cells which moves as a slug‚ by a concerted movement to a central point. The starting stimulus is starvation. Certain cells then begin to emit pulses of the attractant‚ cyclic adenosine monophosphate (cAMP)‚ every ten minutes. Surrounding cells respond to the cAMP signal and move towards the source a short distance of Then‚ 20 s after receiving the signal‚ each cell emits a further cAMP pulse‚ while remaining insensitive to cAMP for several minutes. This behavior causes a series of cAMP waves to diffuse outwards and each cell to move inwards as shown in Fig. 12.24(b). Eventually the cells reach the center and aggregate to form a slug which can then move as a single body‚ and which crawls away to fruit eventually. The slug is differentiated because the aggregation centre forms the front tip which ultimately produces the fruiting body. Recent experiments have studied the speed of mutant slime molds on different surfaces and investigated the effects of adhesion on the phenomenon. For example‚ the cell contact with the substrate was periodic and the cycle times were much larger on mica than on glass.52‚53 One of the most important problems of cell adhesion is dental plaque‚ which is formed when bacteria such as streptococcus mutans and streptococcus sanguis adhere to tooth surfaces‚ causing decay (caries) and gum disease (gingivitis).54‚55 Dental plaque can be removed by brushing or prevented by chemical additives‚ such as chlorhexidine‚ which kill the bacteria. Study of dental plaque is normally done on hydroxy apatite which is the main component of dental enamel.

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Polymethyl methacrylate is also used as a model for dentures and stainless steel as a model for orthodontic brackets.56 Cells are grown in cultures‚ then applied to hydroxyapatite beads coated with saliva‚ incubated for 90 min on a rotating wheel‚ and tested for adhesion. Normally the test consists in counting the adherent cells after loose material has been rinsed off. Counting can be by radioactive labeling during growth with The effect of additives such as chlorhexidine‚ zinc citrate‚ or trichlorohydro diphenylether can be assessed by this method.57 Adhesion is normally defined as the difference between simple rinsing of the surfaces and vigorous scraping‚ scrubbing‚ or sonication. The plaque is revealed by staining with a dye such as erythrosine or by scraping with a probe. After removal‚ the cells can be cultured and observed by microscopy.58 Anti-adhesive agents have been sought to prevent the sticking of the bacteria to teeth but generally these have not been successful‚ and bacteriocides are normally used. The problem is the mechanism of the bacterial adhesion‚ as shown in Fig. 12.25. It is thought that a film of polymer from saliva first attaches to the hydroxyapatite. Then bacteria attach to this coating. The bacteria produce a complex layer of cell sheath which glues the cells to each other. The composition of saliva is therefore significant. Roger et al.‚ compared salivas with and without lysozyme.59 Lysozyme gave a substantial reduction in S. mutans adhesion. Lactoperoxidase was even more beneficial in reducing adhesion‚ whereas a nonspecific protein‚ albumin‚ had no effect. Another possibility is that collagen within dentin is the key material of attachment once the enamel is penetrated.60 Strains of lactobacillus which did not bind collagen adhered much less. Infection could be reduced by providing alternative collagen binding sites in the mouth.

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12.11. PROBLEMATIC THEORIES OF CELL ADHESION Adhesion of cells is one of the most fascinating topics. It is directly relevant to our human, cellular condition. It is vital to disease, hygiene, cancer, growth, memory, and so forth. More technically, cell adhesion falls at the boundary of molecular and engineering adhesion, where Brownian motion is still important and the adhesion energy is around kT, such that the bonds form and break easily under ordinary conditions. It is also the most complex form of adhesion in this book; there are difficult geometries, complicated viscoelastic and structural behavior, a variety of chemical reactions, colloidal forces, and enormous ranges of polymer molecules present. For these reasons we should be cautious in developing simplistic models of cell adhesion. It is likely that cells adhere in several stages. The first approach is dominated by diffusion and colloid forces. Then there are polymer interactions, followed by surface restructuring, intimate contact, and deformation. It therefore seems likely that simple "adhesion molecule" models of cell adhesion are wrong. We have already seen in Section 3.1 that “lock and key” models of adhesion must be oversimplistic, though they proliferate in textbooks. The function of cell adhesion molecules is more likely to depend on kinetic mechanisms of adhesion, rather than simple bonding models. Drag, hysteresis, and restructuring may well dominate. In other words, the cell adhesion molecules remove adhesion barriers and act like catalysts. Only by understanding the true thermodynamic adhesion, then modifying it through complex mechanisms as shown in Chapters 7 and 8, will we attain a firm knowledge of cell adhesion. An example of the unexpected interaction between receptor and ligand proteins was found by Liebert and Prieve61 using the extremely sensitive method for measuring attractive and repulsive forces between a particle and a glass slide (see Section 10.3). An immunoglobulin G (IgG) protein molecule was covalently attached to a carboxylated polystyrene particle, while a protein A (SpA) molecule was covalently attached to a glass slide, as shown in Fig. 12.26. The polystyrene sphere was then allowed to fall towards the glass slide and the potential energy was measured from the Brownian motion. The results showed that the coated surfaces, Fig. 12.26(a), gave a stronger attractive force than that experienced between clean surfaces or between the blocked receptor surfaces. This suggests that there is an additional specific attraction between the surfaces coated with the IgG and SpA molecules. The surprising feature was that this occurred at long range compared to the protein molecule dimensions. Thus the specific adhesion is not related to specific molecular contact and to the lock and key mechanism but to the energy and range of the molecular forces. Another problem was discovered recently, when it was found that red cell adhesion energy was not much altered by adhesion molecules although the cells

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coagulated when the adhesion molecules were added.43‚45 In most biology textbooks‚62 there is a lock and key theory of how small additions of adhesion molecules can cause strong attachments between cells. The usual diagram is shown in Fig. 12.27 where a ligand or antigen molecule fits neatly into a receptor or antibody molecule. This model cannot be correct for the following reasons:

1. As Newton showed‚ the best interaction between surfaces is obtained with molecularly smooth bodies; roughness always interferes with adhesion.

2. The lock and key mechanism cannot work at the molecular level because the key would stick before it got in the lock. Imagine opening a door with a strongly magnetic key; it would stick prematurely. 3. Adhesion molecules do not increase the adhesion energy by much. For example‚ in Section 12.9 it was demonstrated that fibronectin had only a

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10% effect on absolute adhesion.43 How can such a small energy have such a large effect? The only possibility is that the adhesion molecules operate by a triggering mechanism which produces a large influence on the whole system. Two possibilities are shown in Fig. 12.27 (b) and (c). The first graph shows a phase separation trigger mechanism. The two phases in a cell suspension are divided by the curve‚ which is a phase separation line. Above the curve‚ the cells are free whereas below the curve the cells are stuck. Imagine the cells sitting at point A in the diagram‚ where they seem very stable to increases or decreases in volume fraction. However‚ when the adhesion is increased slightly by adding an adhesion molecule‚ the system moves below the phase boundary and adhesion occurs. Alternatively‚ consider the mechanism in Fig. 12.27(c). Here the cells are not in a stable adhesion energy well but are repelled by an energy barrier. The adhesion molecule function is to overcome this energy barrier at the surface‚ thus allowing adhesion for only a small energy change. This is a kinetic mechanism in which the adhesion molecule acts as a catalyst to lower the energy barrier.

12.12. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Christianson‚ G.E.‚ In the Presence of the Creator‚ Free Press‚ New York‚ 1984. Hooke‚ R.‚ Micrographia‚ London‚ 1665. Harvey‚ W.‚ On the Generation of Animals‚ London‚ 1651‚ frontispiece. Bongrand‚ P.‚ Physical Basis of Cell–Cell Adhesion‚ CRC Press‚ Boca Raton‚ FL‚ 1988‚ chaps 1–10. Curtis‚ A.S.G.‚ In: Adhesion 2‚ ed. K..W. Allen‚ Applied Science Publishers‚ London‚ 1978‚ pp 1–21. Zobell‚ C.E.‚ J. Bacteriol. 46‚ 39–56 (1943). Patrick‚ S. and Larkin‚ M.J.‚ In: Microbial Biofilms‚ eds. Denger‚ Gorman & Sussman‚ Blackwell Scientific‚ Oxford‚ 1993‚ pp 109–46. Roden‚ L.‚ In: The Biochemistry of Glycoproteins and Proteoglycans‚ ed. W.J. Lennarz‚ Plenum‚ New York‚ 1980‚ p 267. Gerisch‚ G.‚ Curr. Top. Develop. Biol. 14‚ 243 (1980). Matyas‚ G.R.‚ Evers‚ D.C.‚ Radinsky‚ R. and Morre‚ D.J.‚ Exp. Cell. Res. 162‚ 296 (1986). Engel‚ J.‚ Oedermatt‚ E. and Engel‚ A.‚ J. Molec. Biol. 150‚ 97 (1981). Kornu‚ R.‚ Kelly‚ M.A. and Smith‚ R.L. J. Orthopaed. Res. 14‚ 871–7 (1996). Davies‚ D.R. and Metzger‚ H.‚ Ann. Rev. Immunol. 1‚ 87 (1983). Gomperts‚ B.‚ The Plasma Membrane‚ Academic Press‚ London‚ 1977‚ p 1. Evans‚ E.A. and Parsegian‚ V.A.‚ NY Acad. Sci. 13‚ 416 (1983). Evans‚ E.A.‚ In: Physical Basis of Cell-Cell Adhesion ed. P. Bongrand‚ CRC Press‚ Boca Raton‚ FL‚ 1988‚ pp 91–123‚ 173–89. Brightman‚ M.W. and Palay‚ S.L.‚ J. Cell. Biol. 19‚ 415–39 (1963). Pitts‚ J.D. and Finbow‚ M.‚ J. Cell. Sci. 4‚ 239 (1986). Van Oss‚C.J.‚ Infect. Immun. 4‚ 54–9 (1971). Duguid‚ J.P.‚ Smith‚ I.W. Dempster‚ G and Edmunds‚ P.N.‚ J. Pathol. Bacteriol. 92‚ 107–38 (1955).

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43. 44. 45. 46. 47.

48. 49. 50. 51. 52. 53. 54. 55.

CHAPTER 12 Ottow‚ J.C.G.‚ Ann. Rev. Microbiol. 29‚ 79–108 (1975). Wilkinson‚ C. and Curtis‚ A.‚ Biomaterials 18‚ 1573 (1997). Folkman‚ J and Moscona‚ A.‚ Nature 273‚ 345–9 (1978). Coman‚ D.R.‚ Cancer Res. 4‚ 625 (1944). Evans‚ E. and Leung‚ A.‚ J. Cell. Biol. 98‚ 1201 (1984). Evans‚ E. and Metcalfe‚ M.‚ Biophys. J. 46‚ 423 (1984). Parpura‚ V.‚ Haydon‚ P.G.‚ Sakaguchi‚ D.S. and Henderson‚ E.‚ J. Vac. Sci. Technol. A 11‚ 773–5 (1993). Moy‚ V.T.‚ Florin‚ E.L. and Gaub‚ H.E.‚ Colloid Surf. A 93‚ 343–8 (1994). Luckham‚ P.F. and Smith‚ K.‚ Faraday Discuss. 111‚ paper 23 (1999). Moy‚ V.T.‚ Florin‚ E.L. and Gaub‚ H.E.‚ Science 266‚ 257 (1994). Boland‚ T. and Ratner‚ B.D.‚ Proc. Nat. Acad. Sci. USA 92‚ 5297 (1995). Zachee‚ P.‚ Snauwaert‚ J.‚ Vandenberghe‚ P.‚ Hellemans‚ L. and Boogaerts‚ M.‚ Br. J. Haematol. 95‚ 472–81 (1996). Fowler‚ H.W. and McKay‚ A.J.‚ In: Microbial Adhesion to Surfaces‚ eds. Berkeley. Lynch‚ Melling and Rutter‚ Academic Press‚ London‚ 1980‚ pp 143–61. Woods‚ D.C. and Fletcher‚ R.L.‚ Biofouling 3‚ 287–303 (1991). Duddridge‚ J.E.‚ Kent‚ C.A. and Laws‚ J.F.‚ Biotechnol. Bioeng. 24‚ 153–64 (1982). French‚ M.S. and Evans‚ L.V.‚ In: Algal Biofouling eds. L.V Evans and K.D. Hoagland‚ Elsevier‚ Amsterdam‚ 1986‚ pp 79–100. Pierres‚ A.‚ Tissot‚ O.‚ Malissen‚ B. and Bongrand‚ P.‚ J. Cell. Biol. 125‚ 945–53 (1994). Goldman‚ A.J.‚ Cox‚ R.G. and Brenner‚ H.‚ Chem. Eng. Sci. 22‚ 653–60 (1967). Pierres‚ A.‚ Benoliel‚ A. and Bongrand‚ P.‚ Faraday Discuss. 111‚ paper 24 (1999). Orr‚ C.W. and Roseman‚ S.‚ J. Membrane Biol. 1‚ 109 (1969). Coulter Electronics Inc‚ Luton‚ UK. Segal‚ D.M.‚ In: Physical Basis of Cell Adhesion‚ ed. P. Bongrand‚ CPC‚ Boca Raton‚ FL‚ 1988‚ pp 157–72. Gorman‚ S.P.‚ McCafferty‚ D.F.‚ and Anderson‚ L.‚ Lett. Appl. Microbiol. 2‚ 97–100 (1996). Rolland‚ A.‚ Merdrignac‚ G.‚ Gouranton‚ J.‚ Bourel‚ D.‚ LeVerge‚ R. and Genetet‚ B.‚ J. Immunol. Meth. 96‚ 185–93 (1987). Kendall‚ K.‚ Liang‚ W. and Stainton‚ C.‚ Proc. R. Soc. A 454‚ 2529–33 (1998). Kendall‚ K. and Liang‚ W.‚ Colloid Surf. 131‚ 193–201 (1998). Kendall‚ K. and Attenborough‚ F‚ J. Adhesion 2000‚ in press. Alder B.J. and Wainwright‚ T.E.‚ J. Chem. Phys. 31‚ 459–66 (1959). Baskurt‚ O.K.‚ Farley‚ R.A. and Meiselman‚M.J.‚ Am. J. Physiol. Heart Circul. Physiol. 42‚ H2604– 12(1997). Popel‚ A.S.‚ Johnson‚ P.C.‚ Kavenesa‚ M.V. and Wild‚ M.A.‚ J. Appl. Physiol. 77 1790–4 (1994). Palecek‚ S.P.‚ Loftus‚ J.C.‚ Ginsberg‚ M.H.‚ Lauffenberger‚ D.A. and Horwitz‚ A.F.‚ Nature 385‚ 537–40 (1997). Varner‚ J.A.‚ Cheresh‚ D.A.‚ Curr. Opin. Cell Biol. 8‚ 724–30 (1996). Gerisch‚ G.‚ Hulser‚ D.‚ Malchow‚ D. and Wick‚ U.‚ Phil. Trans. R. Soc. B 272‚ 181–92 (1975). Xu‚ X.X.S.‚ Kuspa‚ A.‚ Fuller‚ D.‚ Loomis‚ W.F. and Knecht‚ D.A.‚ Develop. Biol. 175‚ 218–26 (1996). Schindl‚ M.‚ Wallraff‚ E.‚ Deubzer‚ B.‚ Wittke‚ W‚ Gerisch‚ G. and Sackmann‚ E. Biophys. J. 68‚ 1177–90(1995). Nyvad‚ B. and Kilian‚ M.‚ Caries Res. 24‚ 267–72 (1990). Addy‚ M.‚ Slayne‚ M.A. and Wade‚ W.G.‚ In: Microbial Biofilms eds. Denger‚ Gorman & Sussman‚ Blackwell Scientific‚ Oxford‚ 1993‚ pp 167–85.

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56. Eliades‚ T.‚ Eliades‚ G. and Brantley‚ W.A.‚ Am. J. Orthodont. Dentofac. Orthoped. 108‚ 351–60 (11995). 57. Saxton‚ C.A.‚ J. Periodontol. 57‚ 555–61 (1986). 58. Heginbothom‚ M.‚ Fitzgerald‚ T.C. and Wade‚ W.G.‚ J. Clin. Pathol. 43‚ 253–6 (1990). 59. Roger‚ V.‚ Tenovno‚ J.‚ Lenander Lumikari‚ M.‚ Sonderling‚ E. and Vilja‚ P.‚ Caries Res. 28‚421–8 (1994). 60. McGrady‚ J.A.‚ Butcher‚ W.G.‚ Beighton‚ D. and Switalski‚ L.M.‚ J. Dental Res. 74‚649–57 (1995). 61. Liebert‚ R.B. and Prieve‚ D.C.‚ Biophys. J. 69‚ 66–73 (1995). 62. Bettelheim‚ F.A. and March‚ J.‚ General‚ Organic and Biochemistry‚ Harcourt Grace‚ Fort Worth‚ TX p 785.

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Microscopes may at length be improved to the discovery of the Particles of Bodies ISAAC NEWTON, Opticks,1 p. 261

Although Newton could not find the fundamental particles of matter, he had no doubt that they existed and could be located by some future microscope invention. This invention arrived in the early 1980s, 300 years after Newton’s prediction, when Binnig, Rohrer, Gerber and Weibel demonstrated atomic resolution2 using the scanning tunneling microscope (STM). Previously, transmission electron microscopy (TEM) had seen rows of molecules in the 1950s and resolved atomic lattices in the 1960s and 1970s. The great step forward with the STM was the ability to adhere individual atoms to surfaces; to remove them and to form patterns at the atomic level.3 This raised the possibility of writing all the books ever written on a postage stamp.4 The relevance of these ideas to electronics is embodied in Moore’s curve,5 shown in Fig. 13.1, which describes the annual increase in the number of logic circuits on a computer chip. The packing of circuits onto chips has risen exponentially with time since 1970. This means that conducting paths, junctions, and insulating films have been getting progressively smaller, heading towards atomic dimensions, where ultimately there will be 1018 circuits on each chip. This poses large problems of adhesion, ranging from the difficulty of bonding very fine wires to the issue of removing fine particle contamination. 305

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This chapter assesses the effect of miniaturizing on adhesion issues. First, it is demonstrated that adhesion stress improves as bodies get smaller. But there is a limit of elasticity which has a strong influence. The ultimate adhesion experiment is with a single atom. This experiment is on the point of being carried out. It has important implications for the future of adhesion science.

13.1. THE SIZE EFFECT IN ADHESION: SMALL IS BEAUTIFUL The good news is that adhesion appears to increase as objects get smaller, in sofar as the stress required to remove a wire from its contact is increased. However, the energy needed for removal decreases so the bad news is that adhesive joints become more brittle with miniaturization, giving the possibility of unexpected and catastrophic failure. These ideas were first generated by Galileo6 in his famous book Two Sciences. At that time, there was great argument about the bending strength of beams. We all know that a wooden cantilever, as depicted in Fig. 13.2, will

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eventually break as the weight is increased. The longer the cantilever, the more likely it is to break. Because there may be knot holes and cracks in a longer beam, the conclusion before Galileo came on the scene was that the longer beam should be weaker because of the higher probability of finding a defect. As Galileo reported, “they assign the cause to the imperfections of matter which is subject to many variations and defects”. Galileo demonstrated for the first time that this probability argument was false with the apparatus shown in Fig. 13.2. Judging by the vegetation on his test machine, technicians were in short supply even then. Galileo took the imaginative step of considering the beam to be pivoted around the axis shown in the diagram. By taking moments about this pivot, knowing that force times distance (i.e. energy) must balance,

proving by this first ever energy analysis that the stress in the beam rises in proportion to its length and inversely with the square of its thickness. Although Galileo’s stress equation was later shown to have a slight numerical error of because the outer tensile stress is the overall basis of deflection and strength of beams was established by the above argument, showing why long beams fail at low loads. This idea led Galileo to the general principle that “the larger the structure is, the weaker . . . it will be.” This is the same principle that operates as adhesion is used for smaller and smaller circuits. Adhesion appears to increase as the contact gets smaller. This idea was proved for the first time in 1971 when the energy analysis of adhesive joint equilibrium was applied to the geometry shown in Fig. 13.3, which represents a rigid wire making flat contact with an elastic substrate.6

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As the pull-off force is applied, the elastic substrate deforms into the shape shown in Fig. 13.3(b) and a crack starts at the edge of the contact and moves through the interface to cause rapid fracture. The energy balance analysis can be carried out for this geometry by considering the three energy terms involved in the cracking;

1. surface energy

2. potential energy; the deflection z of a rigid punch diameter d in contact with an elastic substrate under load F was given by Boussinesq in 18858,9

Therefore the potential energy is 3. the elastic energy is half the potential energy and of opposite sign, that is

Adding these three terms and applying the condition of energy conservation as the contact diameter d decreases

Therefore

The conclusion from this argument, which was verified by experiment, was that the adhesion force decreases as the wire diameter is reduced. But instead of the force going down with contact area, that is it goes with In other words, the stress required for adhesion failure increases for finer wires. The adhesion seems to get stronger with Some numbers are plotted in Fig. 13.4 for elastic platinum wires on a silicon substrate.

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The reason for this strengthening of the joint is the cracking mechanism. Although there does not seem to be a crack at the edge of the wire, there is a virtual crack because the rigid material can be replaced by an elastic half-space as shown in Fig. 13.4(b). The stress in the elastic material rises to infinity at the edge according to the Boussinesq analysis, because of the pressure distribution. This infinite stress is similar to that causing cracking in the original Griffith theory.10 However, as Maugis and Barquins11,12 have discussed, this infinite stress was a problem for 100 years until 19717 when it was realized that molecular adhesion could resist it. The upper limit of the curve in Fig. 13.4 is the stress to debond a single atom from the surface.

13.2. ADHESION OF PLASTIC CONTACTS Of course, the high stress at the edge of the contact must cause plastic deformation eventually as the force rises, as shown in Fig. 13.5. Ultimately, this plastic flow will dominate the deformation and the elastic solution of Equation (13.6) will no longer hold. The force will reach a limit given by13

where H is the hardness, related to the yield stress Y by H = 3 Y. Plastic flow is important to the cold welding of metal wires during pressure bonding of the material to the substrate. Oxide films must be disrupted and virgin material extruded through cracks in the oxide to give good adhesion. The more clean metal makes contact, the better the adhesion. Unless the contact is made in vacuum, a large surface deformation, between 10–90% for metals is needed to disperse the contamination.14

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13.3. ADHESION OF SINGLE ATOMS The possibility of adhering single atoms to surfaces in a controllable manner was first demonstrated using the scanning tunneling microscope (STM). First, in 1987, it was shown that an atom could be transferred from a tip to a germanium surface.15 Then, it became possible to pin a molecule to a graphite surface,16 using phthalate molecules, and increasing the tip potential to 3.5 V. Finally, atoms were moved around controllably on a nickel surface using the STM.17 Later it proved possible to test the vibrational spectroscopic properties of single acetylene molecules sitting on the surface.18 Xenon atoms sitting on a (110) nickel surface were chosen for one series of experiments.17 The nickel surface was completely flat and smooth at the atomic level, but there were corrugations in the interaction potential between nickel and xenon which allowed the xenon atom to adhere to the surface and not be disturbed by the scanning tip, as shown in Fig. 13.6(a). To achieve this, the STM was placed in an ultra-high-vacuum system cooled to 4K so that no measurable contamination occurred over several weeks. The stability was such that experiments on a single atom could be carried out for days at a time. The nickel sample was cleaned by ion sputtering and annealing, then observed after dosing with xenon atoms to the desired coverage as shown in Fig. 13.6(b). At a tip voltage of 0.01 V a current of A was drawn, and each xenon atom then appeared as a 0.16nm bump on the surface. The atoms were not shifted by this scanning method. To move the xenon atom across the surface, the tip was placed directly above the atom, then lowered towards the atom, changing the tunneling current to a higher value than before, up to A. With this interaction between the tip and the xenon atom, it was now possible to drag the atom across the surface to a new position. This was done at low speed, Then the tunneling current was reduced to A as the tip was raised to leave the xenon atom attached in its new position. Although the tip was made from tungsten wire, the true identity of

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the outermost atoms on the tip was not known exactly. For any given tip and voltage, there was a certain closeness of the tip to the xenon atom to allow it to be moved across the surface. This closeness varied with direction. Movement across the row of nickel atoms required more close contact than movement along the row. Thus, greater adhesion between tip and xenon atom was required to cross the nickel atom potential barrier. The sequence of events during movement of a xenon atom is shown in Fig. 13.7. Using this method, xenon atoms were positioned in uniform arrays, 0.5 nm apart on the nickel surface, occupying every other unit cell on the unreconstructed nickel surface atoms. It was evident that such arrays could be used as electronic devices. For example, a single cobalt atom sitting on a copper (111) surface was probed in the STM and found to exhibit a magnetic resonance.19 This could be used as an information storage or switch element.

13.4. STRETCHING SINGLE MOLECULES IN THE ATOMIC FORCE MICROSCOPE Recently, a new AFM technique has been used to stretch single polymer molecules adhering to surfaces.20,21 An AFM tip is brought into contact with adsorbed polymer chains on a smooth surface of glass or gold. Then the tip is pulled from the surface and a force/extension curve plotted out for the polymer molecule. The scheme of the experiment is shown in Fig. 13.8, together with a typical result. A polymer, typically polyacrylic acid of molecular weight 450 000, was dissolved in millipore water to a concentration of at a pH of 4. A small amount of this solution was deposited on a clean glass cover slip and dried in air to form a thin polymer film.22 The substrate was then rinsed with water to leave only the strongly adsorbed molecules. An AFM tip was then brought down onto the surface in a 1 mM solution. As the tip touched the surface, the

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thickness of the polymer film could be estimated from the shape of the deflection curve. Typically, the adsorbed polymer layer was 50–90 nm thick. Then the tip was retracted and the force/extension curve was measured as the polymer molecules were stretched out, as in Fig. 13.8(b). The force increased with extension until eventually fracture occurred. If the polymer chain was stretched and then relaxed before fracture, then the curve was reversible, suggesting that an equilibrium curve was obtained. Only about 30% of the tip contacts resulted in an extension curve as shown. Often, only a single curve was obtained, suggesting that one molecule was involved. Since the tip was 60 nm in diameter, compared to the radius of gyration of the polymer of 35 nm, this idea that one molecule can be addressed is reasonable. Sometimes several tension peaks were obtained, corresponding to more than one molecule attachment. These results were discarded. The model used to describe these results was the freely jointed chain model of the polymer molecule, originally due to Kuhn and Gruen,23 which treats the molecule as a chain of independent segments of length l. The force of retraction is then entropic and given by the Langevin function where L is the total chain length, n the number of segments and E is the elastic segment stiffness. This was found to fit the results at low forces but required an extra term20 at high loads. This term was due to the Hookean stretching of the elements in the chain This theory gave a good fit to the results, as shown in Fig. 13.9(a). The segment length turned out to be 0.64 nm and the segment elasticity was In one particular example, that of dextran, the results showed a kink in the

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force/extension curve Fig. 13.8(b) and this was attributed to a transformation of the glucose ring in the polymer backbone. In another example, where titin proteins were extended, the curves showed periodic, sawtooth peaks, which indicated that the protein was unfolding by way of individual immunoglobulin domains. Each force peak was around 200 pN and spaced 25 nm apart as expected from the length of the polymer unit. The unfolding was completely reversible after full relaxation for a period of seconds. For polyacrylic acid, the force could be as high as 1 nN and the extension up to It was suggested that this force was the adhesion force between polymer and tip. This would correspond to an adhesion strength of around 5 GPa if the chains were adhering in a close-packed formation. This indicates that single polymer chains can adhere with an extremely high strength, 50 times higher than the normal polymer adhesion strength in bulk samples. The theoretical strength of the polymer chain itself is around 100 GPa. Of course, one molecule can debond at the theoretical value, whereas larger assemblies break by cracking or flow and this diminishes the strength of larger polymer samples substantially, as shown in the next section. The yield strength of a bulk polymer may typically be around 70 MPa.

13.5. ADHESION STRENGTH OF SMALL FEATURES As features are made smaller on microelectronic substrates, with lines approaching 100 nm width, it is important to understand how strongly such features adhere. Consider, for example, a line on a silicon wafer which is scratched by a probe as shown in Fig. 13.10(a). How much shear stress is required to remove the feature from its adhesive bond with the surface?

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A model experiment of this kind has been carried out on nanocrystals of sodium chloride grown as cubes on a silicon substrate. An AFM tip was used to shear the cube from the substrate and measure the stress required.24 Crystals were grown on glass substrates by evaporating a dilute sodium chloride solution. The tip of the AFM was scanned across the surface to locate the crystals and to measure their size from the flat square top surface. The crystals were cubes up to 50 nm in size. Then, the contact force was raised to a high level and the tip was used to break the crystals from the surface. Humidity was found to be very important and was controlled to high precision in the microscope. Once the probe touched a crystal, the lateral force rose to a critical value, then fell as the cube sheared across the surface. The fracture stresses were calculated from the peak force and the crystal interface area, and plotted as a function of crystal size in Fig. 13.11(a). Two striking observations were drawn from this curve: in the first place, the stress was much larger than expected for a soft material like NaCl,

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which normally shears around 1 MPa pressure; secondly, the adhesion stress increased further as the crystals became smaller. This was the same sort of behavior expected from Equation (13.6) which shows that adhesion strength in general increases for smaller objects as where d is the size of the specimen. A shear stress of 65 MPa was required to remove a 20 nm crystal. Humidity also had a strong effect as shown in Fig. 13.8(b). This could be explained by an equation of the type

where was the fall in work of adhesion due to a single monolayer, k was a constant, p was the relative humidity, and was the relative humidity at half monolayer coverage.

13.6. CLEANING PARTICLES FROM WAFERS One of the main causes of failure in semiconductor devices is particles sticking to the surface, shorting out the conductor lines. Some of these contaminant particles arise from the wafer polishing operations needed to produce flat surfaces with roughness less than 1 nm. The polishing process is shown schematically in Fig. 13.12. A silicon wafer is held in a vacuum chuck which rotates to give a polishing action. Surface roughnesses are removed from the wafer by the polishing medium held in the pores of the polyurethane (PU) polishing membrane which is spun on the rotating table. This medium consists of 7% concentration of 200 nm silica particles suspended in water with ammonia and surfactant additives.25 After the polishing operation is complete and the wafer is flat to 0.3 nm, leaving about 100 silica particles per square centimeter of surface, the surface needs to be cleaned. A range of processes is used, from washing with wet brushes to megasonic

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vibration at 760 kHz and 640 W power. Alternatively, laser radiation can be used to lift the particles from the surface. One of the main problems is that the wafer cleaning is much more difficult after the polishing process. As the wafer becomes smoother, particles adhere more securely. There are several possible reasons for this. As the surface is made cleaner, the work of adhesion W increases so that the force F for removal, F = WD, increases. The presence of water or humidity keeps the adhesion down, but if the particles dry, then the adhesion can increase significantly. Figure 13.12b shows a particle on a wafer with a water meniscus condensed from the humid atmosphere. Silicic acid corrosion products form in the meniscus over a period of weeks, gluing the particle to the surface and giving strong adhesion, up to 26

Laser cleaning of organic contamination from wafers can also be achieved by irradiating the surface with an excimer laser as shown in Fig. 13.13(a). The laser can remove around 90% of the particles by instantaneously heating the surface, causing a thermal jolt which dislodges the particles. To prevent the particle falling back onto the wafer, the silicon is upside down and heated to 60°C to give a thermophoretic force which pushes the particles into the cooler zone away from the surface.27 This thermophoretic force arises because gas molecules on the hot side of the particles are impacting with more energy than those on the cold side, thereby pushing the particles towards the cold region. Epstein in 192928 showed that the force was proportional to diameter and to temperature gradient and dominates gravity for particles below as shown in Fig. 13.13(b). The force is given for a temperature gradient of It varies in the same way as the van der Waals force but is several orders of magnitude smaller under these conditions.

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13.7. ADHESION IN ELECTROPHOTOGRAPHY One of the most stimulating examples of adhesion applications during the past 50 years has been the development of electrophotography by Xerox and other companies. This process allows documents to be copied using a completely dry process, in which fine particles are electrostatically transferred from an electrical image to paper, then thermoplastically fused in heated rollers. A flow diagram of the process is shown in Fig. 13.14. The device is fascinating because there are so many points at which adhesion is controlled as the particles are transferred through the machine. The heart of the device is the photoconductor, a material which requires both a voltage and light for electron mobility. At the start, a photoconducting moving belt is charged electrostatically, then exposed to the optical image, such that the light discharges the potential in the bright areas, leaving charged regions where the black letters appeared in the document. These charged letters are then developed by brushing the belt with toner particles, which are polymer beads carrying fine black pigment. The toner particles adhere to the belt by both electrostatic and van der Waals forces, giving a powdery black image on the belt which can now be discharged by the eraser to give only van der Waals adhesion. By applying an electric field, the van der Waals force can be overcome to transfer the image to a paper sheet, where it forms a powdery image adhering by both electrostatic and van der Waals forces. The paper then passes through heated rollers to fuse the polymer particles, fixing the image on the paper thermoplastically. A flow diagram of the process is shown in Fig. 13.14.

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Any residual material on the belt can then be removed by erasing the electrical charge and sweeping it with a vacuum brush. Then the belt starts on its next copying cycle. To understand the adhesion science involved in this magnificent process, it is best to consider the experiment devised by Quesnel and Rimai, shown below in Fig. 13.15. In this experiment, two sizes of polymer beads were mixed with magnetic particles to tribocharge them Fig. 13.5(a). After a while, the charge on each particle became uniform and equal. Then the polymer particles could be deposited on the photoconducting sheet. Another plastic sheet with an electrode on one side was placed on top, giving a separation provided by the large size beads. Then a voltage was applied to the two electrodes Fig. 13.5(b). At a critical voltage, the small polymer beads jumped across to the other electrode in the cell. This occurred because the electrostatic force applied to each sphere overcame the van der Waals adhesion force. The theory shows that there are three forces acting on the particles in this experiment: first, there is the molecular adhesion force due to van der Waals force, where D is the diameter of the small spheres and W is the work of adhesion; second, there is the electrostatic charge on the particle which pulls it onto the surface of the plastic film, giving a force where is the charge density on the particles and is the permittivity of free space; and third, there is the applied electric field V which acts to make the particles jump across the gap of thickness These three forces are drawn schematically in Fig. 13.16 and balanced in the equation below

From the results as the voltage was gradually raised, Fig. 13.16(b), it was evident that the particles all seemed to jump at the same voltage, within about 20%. Different diameters of particles were then tested, Fig. 13.16(c), and it was clear that the molecular adhesion force dominated for small particles, with the electrostatic force becoming significant above The work of adhesion

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for the polystyrene was calculated to be lower than the known value, probably as a result of surface roughness. For the spheres, the electrostatic charge contributed only 10% to the total force, even though the charge on each particle was around Thus, electrostatic adhesion can be used to overcome molecular adhesion in a controllable way, providing the particle sizes are in the right range.

13.8. POLYMER SYNTHESIS FOR COATING SILICON WAFERS In order to vary the surface properties of electronic materials, coatings need to be applied such that the interaction with the surface is optimized to a very fine degree. There are two ways to achieve this: one using short chains which naturally form self-assembled monolayers (SAMs),29 the other building long chain molecules which bind to the surface at several points to form polymer brushes, as shown in Fig. 13.17.30 The SAMs, around 3 nm thick, can be deposited from a Langmuir trough by dipping and withdrawing a smooth wafer. Polymer brushes, on the other hand, are formed by drying dilute solutions after spin coating on the wafer to form surface films 20–30 nm thick.31 In a typical example of a SAM, a methoxytriethylene

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glycol (EG3-OMe) terminated alkane thiol molecule was synthesized and laid down on an evaporated gold film on a smooth substrate. This molecule was shown to produce helical and amorphous chains which resisted soiling. The same molecule on a silver coated wafer produced planar structures which did not resist soil adhesion. Such adhesion forces were investigated using atomic force microscopy (AFM) with a hydrophobic tip in deionized water, showing extremely long range repulsions out to 60 nm which collapsed to a 3 nm repulsion after addition of salt. A longer-chain polyethylene glycol formed a polymer brush which gave a repulsion at 6 nm, approximately the polymer layer thickness.32 The understanding of how such polymer molecules attach to and influence the properties of surfaces has been considered by de Gennes and his colleagues.33 Perhaps most important is the phase separation that can occur on the surfaces, where the molecules arrange with certain groups binding to the surface and other groups sitting on the outside. Figure 13.18 shows how such phase separation can occur. A triblock copolymer typically has a central hydophilic group and two hydrophobic chains dangling from it, as shown in Fig. 13.18(a). The molecules readily phase separate to form micelles in solvent, Fig. 13.18(b), or complex structured regions on a surface, Fig. 13.18(c). To make such polymer molecules with complex block structures which readily phase separate, living anionic polymerization can be used.34 The monomers are first prepared. In a triblock copolymer, these could be 4 vinyl phenoldiphenyl ethylene for the central hydrophilic block; polystyrene for one end chain and poly(methyl methacrylate) for the other end block. The molecular scheme is illustrated in Fig. 13.19. The OH group was protected by reaction with tert-butyldimethylsilyl chloride (TBDMS) in order to obtain the living anionic polymerisation. Diphenylethylene (DPE) was used to lower the living chain reactivity. The monomers were added in order; styrene, styrene-o-TBDMS, DPE, MMA to the solvent tetrahydrofuran (THF) at 78°C in nitrogen with stirring. The reaction was terminated with methanol and precipitated by hexane to give the product PS-bpoly(styrene-o-TBDMS)-b-PMMA which was dried in vacuum at 130°C then

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reacted with dilute HCl to remove the protective TBDMS group, revealing the OH groups. The final result was a polymer with a central hydrophilic block of 1 kDa molecular weight, with side chains each 10 kDa in length. When the polymer was tested by NMR and GPC, the molecule was as predicted from the weights of reactants added, with molecular weight distribution of 1.1–1.2. When spin coated onto a silicon substrate, the polymer phase separated into islands which could readily be seen at the level in the AFM, as indicated in Fig. 13.20. The lower region of the polymer film was 10 nm thick and the upper region was a further 10 nm thick. Thus, a complex film structure can form quite naturally at the wafer surface by the condensation of the block copolymer into its equilibrium state.

13.9. MOLECULAR CONTROL OF NANO ADHESION Phase separation of polymer molecules is clearly one mechanism by which complex surface structures can develop to alter the adhesion. Another method which has recently come to light is that of molecular interdigitation.35 According

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to this idea, illustrated in Fig. 13.21, contact between an AFM probe and a substrate can be mediated by the detailed structure of the adsorbed molecule. A simple dialkyl sulphide monolayer formed on the fold surfaces gave a uniform film leading to low adhesion, Fig. 13.21(a). Moreover, as the chain length was increased, the adhesion fell systematically, roughly with the square root of molecular length, as in Fig. 13.21(b). However, if the dialkyl molecule has one chain much longer than the other, then the adhesion can increase, because the longer chains can then interpenetrate the gaps on the opposite film. This is shown in Fig. 13.22(a). The original molecule had equal chains which could close-pack to form a dense monolayer on each surface. Asymmetrical molecules were then prepared etc, gradually increasing one are of the molecule to a maximum length These were adsorbed onto the gold coated probe and substrate by dipping in a 1-3 mM solution in THF for a period of at least 12 hours. The materials were removed, rinsed in THF and ethanol, then dried in pure nitrogen. The adhesion force was then found to increase with chain length, as shown in Fig. 13.22(b), rather than decreasing. The increase was considerable. It could be explained by supposing that the longer alkyl chains could penetrate into the opposing film, displacing the ethanol solvent. Removing the chains would therefore require energy to get the solvent back into contact with the film. The sliding friction force was also measured and found to be doubled for the interdifitating films. Friction coefficient was 0.12 for the equal chain lengths. By contrast, the friction coefficient rose to 0.25 for the chain length monolayers. The overall conclusion is that fine molecular structure can influence the adhesion force very significantly.

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13.10. VISUALIZING ADHESION USING MICRO-FOCUS X-RAYS Many adhesive systems are used in the microelectronic industry, and it is important for quality control to look at the interface to detect any defects, bubbles, debonds or cracks which could be a cause of failure in the service lifetime. A common instrument for studying interfaces in this way is the microfocus X-ray machine shown schematically in Fig. 13.23. An electron gun produces a beam of electrons which is focused in a fine spot onto a special cooled target. The resulting beam of X-rays has a spot size near and this can produce shadow images on a photomultiplier screen to detect bubbles or cracks at the level. The images are obtained in real time and can be analyzed immediately with image enhancement software to give quantitative information about microelectronic packaging. Typical applications are package voiding or delamination, wire bonding anomalies and lead frame distortion.36 The imaging of objects in the micro-focus X-ray device depends on the density difference between the objects, 2% being a typically resolvable contrast. For example, gold wires are readily resolved within a plastic package, whereas aluminum wires are less visible. The wire bonds can be checked easily from above to see wire droop, or from the side to detect ball lift or cracking of the wire in the necked-down region just above the ball-bond (see Fig. 13.24) The use of real-time micro-focus X-ray systems with versatile sample manipulation has emerged as an excellent inspection technique in package

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evaluation. A particularly good example is the detection and measurement of voids in the die attachment. Such detachment can have an important effect on heat dissipation from the device. Figure 13.25 shows a number of void regions in an attachment area viewed by microfocus X-ray. This brings us in the next chapter to the adhesion of films and coatings.

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13.11. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

33. 34. 35. 36.

Newton, I., Opticks, Smith and Walford, London, 1704 (reprinted Dover, New York, 1952, p 261). Binnig, G., Rohrer, H., Gerber, C. and Weibel, E., Phys. Rev. Lett. 49, 57 (1982). Eigler, D.M. and Schweizer, E.K., Nature 344, 524 (1990). Feynman, R.P., Eng. Sci. 23, 22 (1960). Moore, G.E., Proc. IEEE IEDM, 75CH1023–1, 11 (1975). Galileo, G., 1638, Two Sciences (translated S. Drake) Wisconsin University Press, pp 12–13. Kendall, K., J. Phys. D: Appl. Phys. 4, 1186–95 (1971). Boussinesq, J., Application des potentiels a l’etude de l’equilibre et du mouvement des solides elastiques, Gauthier-Villars, Paris 1885 (new edition, Blanchard, Paris, 1969). Barquins, M. and Maugis, D., J. Mec. Theor. Appl. 1, 331–57 (1982). Griffith, A.A., Phil. Trans. R. Soc. A 221, 163–98 (1920). Maugis, D. and Barquins, M., J. Phys. D:Appl. Phys. 16, 1843–74 (1983). Maugis, D. and Barquins, M., J. Phys. D:Appl. Phys. 11, 1989–2023 (1978), see Maugis, D., Contact, Adhesion and Rupture of Elastic Solids, Springer, Berlin 1999, ch 4. Johnson, K.L. Proc 14th IUTAM Congress on Theoretical and Applied Mechanics, Delft, Sept 1976, North-Holland, Amsterdam, 1977, pp 133–43. Tylecote, R.F., The Solid Phase Welding of Metals, E Arnold, London, 1968. Becker, R.S., Golovchenko, J.A. and Swartzentruber, B.S., Nature 325, 419–21 (1987). Foster, J.S., Frommer, J.E. and Arnett, P.C., Nature 331, 324–6 (1988). Eigler, D.M. and Schweitzer, E.K., Nature 344, 524–6 (1990). Stipe, B.C., Rezaei, M.A. and Ho, W., Science, 280, 1732–5 (1998). Manoharan, H.C., Lutz, C.P. and Eigler, D.M., Nature 403, 512–5 (2000). Rief, M., Oesterhelt, F., Heymann, B. and Gaub, H.E., Science 275, 1295–7 (1997). Rief, M., Gautel, M., Oesterhelt, F., Fernanderz, J.M. and Gaub, H.E., Science 276, 1109–12 (1997). Li, H., Liu, B., Zhang, X., Gao, C., Shen, J. and Zou, G., Langmuir 15, 2120–4 (1999). Kuhn, W and Gruen, E., Kolloid Z. 101, 248 (1942). Hariadi, R.F., Langford, S.C. and Dickinson, J.T., J. Appl. Phys. 86, 4885 (1999). Busnaina, A.A. and Dai, F., J. Adhesion. 67, 181–93 (1998). Feng, J.W., Moumen, N. and Busnaina, A.A., Proc. Adhesion. Soc. 23, 349–52 (2000). Wu, X., Sacher, E. and Meunier, M., Proc. Adhesion. Soc. 23, 360–2 (2000). Epstein,P., Z. Phys. 54, 537 (1929). Zhao, X.M., Wilbur, J.L. and Whitesides, G.M., Langmuir 12, 5504 (1996). Milner, S.T., Science 251, 905 (1991). Binder, K., Advances in Polymer Science vol 138, Springer Verlag, Berlin, 1999. Harder, P., Grunze, M., Dahint,R., Whitesides, G. and Laibinis, P.E.J., J. Phys. Chem., B 102, 426– 36 (1998). Raphael, E. and deGennes, P.G., J. Phys. Chem. 96, 4002 (1992). Wang, J., Kara, S., Kang, H., Long, T.E., Li, Q.and Ward, T.C., Proc. Adhesion. Soc. 23,337–9 (2000). Van der Vegte, E.W., Subbotin, A., Hadziioannou, G., Ashton, P.R., and Preece, J.A., Langmuir, 16, 3249–56 (2000). Lehmann, D., Proc. Int. Electro Packaging Conf. 1993, pp 828–35.

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The…vitrified Part of the Metal…by covering the Metals in form of a thin glassy Skin, causes these colours ISAAC NEWTON, Opticks,1 p. 221

Surface coatings comprise a major application of adhesion technology. Often, the adhesion must be sufficient to withstand horrendous scratching, bending, impact, and environmental abuse during service. Thus, a range of pragmatic tests has grown up to monitor the complex practical adhesion behavior, without much basic understanding of the underlying mechanisms. For example, in Fig. 14.1 the coated panel is indented, cut, scratched, bent, or impacted, often in a corrosive environment. Afterwards, the remanent adhesion is tested by attempting to strip the coating from the deformed region using standard adhesive tape. The dilemma is that, if the adhesion can be measured by detaching the coatings, then it is clearly insufficient for the application and should be improved by better surface cleaning or pre-treatments. Of course, as Newton realized, all solid surfaces are covered by a coating anyway, because the surface by its very nature differs from the bulk material. It must restructure, adsorb molecules from the atmosphere, or form an oxide film instantly after its formation. The purpose of this chapter is, first, to recognize this fact and then to define the importance of augmenting and enhancing surface films. Then the ideal film adhesion geometry is described and tested by peeling, wedging, shrinking, and so forth. The fundamental adhesion mechanisms are then defined and the various molecular treatments are discussed. 327

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14.1. COMPLEXITY OF FILMS AND COATINGS The production of films and coatings is a huge industry with a wide range of applications. From the earliest human art forms of cave painting, metal burnishing, and pottery glazing, the decorative and property enhancing features of adherent surface films have been appreciated and improved. The cave paintings at Altamira are believed to be 15 000 years old,2 yet most modern films and coatings have been developed during the past century, demonstrating a rapid rate of technological advance. An example of such development is paint, which is generally applied from solution or suspension to harden and adhere under ambient conditions. A wide range of reactive polymer systems is now being produced, based on vinyl acetate, alkyd, urethane and other chemistries. Total world paint production is around 22 Mtonnes per annum. Another example of a modern surface coating is the thermoplastic film used on sheet steel to protect it from corrosion. This coating must adhere so well that the steel can be pressed to shape and cut while retaining good bonding, as shown in Fig. 14.2. Such thermoplastic film materials are manufactured at the scale. Yet another typical example is poly(vinylidene fluoride) or PVDF film which is coextruded onto plastic food containers to slow gas diffusion, either oxygen ingress which spoils food or carbon dioxide egress which flattens beer and carbonated drinks.

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The sheer variety of processes used to apply coatings to surfaces is fascinating. Melt coatings, solvent based paints, latex formulated with water, reactive coatings such as polyurethanes, polyesters, and epoxies, electrochemical deposition, powder coating, spraying, dipping, and evaporation techniques have all been employed.3,4 Of course, even before painting, all metal surfaces are covered with a natural growth of adsorbed vapor plus an oxide layer which is formed by instant reaction of the clean metal with the surrounding atmosphere, as shown schematically in Fig. 14.3. The interphase region is complex because it is rough, deformed by grit blasting treatments, and contains several interfaces, especially if a phosphate or anodic oxide layer is deposited before painting. For example, anodizing in phosphoric acid grows structures on aluminum surfaces, improving the adhesion of polymer coatings by providing crack resistant interface geometries.5–7 The natural oxide coating alone amounts to 10 Mte of material formed each year on metal surfaces. Such layers are enhanced by heating, polishing, nitriding, carburizing, ion implantation, anodizing in an electrolyte, or coating. Corrosionresistant metal coatings have been traditionally made by dipping steel sheets in a molten zinc bath. Metal and specialized ceramic coatings may be applied by vapor deposition routes which include both chemical and physical processes. Ceramic coatings also include those clays and oxides used to coat paper surfaces and those glassy layers made to cover tiles and pottery. They include coatings

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used on magnetic tapes and discs. Some ceramic coatings such as the insulating layers in gas turbine combustion chambers or in metal hip joints are sprayed on in a flame or plasma, solidifying on contact with the cool surface. Diamond-like layers are grown from reactive gases. Advanced coatings are now widely used in the electronics industry to provide active or insulating layers. Obviously, it is difficult to define adhesion in such complex situations with roughness, multiple interfaces, and complicated chemical contaminants. In order to understand the complex action of these structures it is first necessary to study the ideal, smooth, plane interface. The purpose of this chapter is to define the ideal situation first, then show the effects of elasticity, which can act both to increase adhesion and to reduce it. Then the more complex mechanisms of roughness, chemical treatment, crack deflection, and energy dissipation are described. Such mechanisms are also shown to display opposing effects, sometimes increasing adhesion and sometimes reducing it.

14.2. IDEAL EXPERIMENTAL ARRANGEMENT An ideal adhesive film system can be made by the method originally used by Roberts,8 shown in Fig. 14.4(a). He moulded gum rubber against an optically smooth surface such as glass, crosslinked the polymer by heating with sulfur or peroxide, peeled the rubber from the surface to reveal an optically smooth polymer surface, then adhered this material to another smooth surface to form the joint. Alternatively, low temperature reacting materials such as silastomers or

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polyurethanes were used to give very smooth surfaces when reacted against glass.9–12 In the results described here, ethylene propylene rubber (Enjay 404) was twin-roll milled with 0.32% sulfur and 2.7% dicumyl peroxide, then pressed at 160°C against a glass plate for 1 h to crosslink the material around 1 mm thick before chilling to room temperature. The rubber sheet was cut into strips 10mm wide which could be peeled from the glass and re-adhered to a clean glass surface for adhesion testing. This gave a well-defined interface which could be detached by several means, while viewing the interface crack in reflected light.13 Several experimental precautions needed to be taken when measuring the fracture. The first was recognition of the dwell-time effect in which adhesion increased with time of contact between rubber and glass. The interface was therefore made a fixed time, e.g., one hour, before the cracking experiments. A second phenomenon was the relation of crack speed to applied force, shown in Fig. 14.4(b). As the force or the temperature was raised, so the peeling crack speed increased. Crack speed and temperature were therefore held constant when making the adhesion force measurements. The “break” value of adhesion under these conditions was higher than the equilibrium value W for the interface. Adhesion energy R was the name given to these measured values. The surfaces were healed together to see clearly the approach to equilibrium adhesion of the system during “make” of the joint. On healing, Fig. 14.4(b), the measured adhesion energy was always less than the equilibrium value W. Hysteresis, the difference between the adhesion energies during healing and peeling, was observed even after long times.

14.3. TESTING METHODS FOR ADHESION OF FILMS The adhesion of such adhering films may be tested in a number of other ways shown schematically in Fig. 14.5. In these tests, only the film is deformed

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by the applied force, so that the substrate can be presumed rigid. The mechanism of failure is cracking along the interface, driven by force applied in various ways. Adhesion is normally defined as the force required to detach the film. Unfortunately this force varies drastically from one test to another. For example, a film may detach at a low force in the peel test but the same film can require a much larger force of detachment in the lap test. Thus, it is necessary to disbelieve anyone who talks of the strength of film adhesion measured in one of these tests. The fact is that adhesive strength, as defined by the force of detachment divided by the contact area, can be any value you want, depending on the test method. Therefore, test data should be compared using the interface cracking theory presented below. This cracking theory shows that geometrical and elastic differences between the tests cause the different adhesion forces. In an ideal reversible adhesion experiment, this theory will give the work of adhesion W at the interface in terms of the applied force and the geometrical and elastic parameters. In most cases, however, the adhesion is not reversible and the theory gives the adhesion energy or work of fracture R, that is the energy required to fracture of interface. R is usually much larger than the thermodynamic value W. To interpret such large energies requires some insight into the detailed mechanisms of the cracking process. The most obvious feature of many adhesive interfaces is brittle failure. It is a rare interface which exhibits good resistance to cracks. Consequently, once a flaw

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has been initiated in a typical coated surface, a crack propagates readily along the interface, often accelerating catastrophically under a small driving force. Such behavior is similar to the fracture of glass or ceramic, showing the sudden explosive failure, low strength, poor reliability, sensitivity to defects introduced during the processing, and ease of damage by impact or by environmental agents. Adhesive films must be designed like glass or ceramic parts and this is a challenge to the average engineer. In the tests shown in Fig. 14.5, only the film is deformed. More rarely, the underlying substrate is also deformed, for example during pull-off, stretching or bending shown in Fig. 14.6, and this causes detachment of the film, but now the force is different from that found for rigid substrates. For example, in the pull-off test of Fig. 14.6(a), the substrate deformation dominates the failure process and decreases the adhesion force enormously Let us now deal with all the above tests in more detail.

14.4. WEDGING OF FILMS: DIRECT LINKAGE WITH ZERO FRICTION Perhaps the easiest way to detach a film from a surface is to scrape it with a sharp blade, driving a wedge along the interface. As Fig. 14.7 shows, this process opens a crack ahead of the wedge, and this crack progressively detaches the film as the wedge is pushed in further. Wedging was originally treated in 1930 by Obreimoff who produced a description of the crack ahead of the knife and verified his result experimentally using mica cleavage observed through optical interference.14 The simple connection between the wedge force and the adhesion in the case of zero friction15 is shown in Fig. 14.7(b). As the wedging force drives the knife a distance c, the crack also moves the same distance. The work done by the knife is the force times the distance, if there is no frictional resistance. At equilibrium, this work is converted completely into creating new surfaces which requires work of adhesion times the area of broken interface, Wbc, where b is the width of the film. Since the elastic deformation remains constant as

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the film detaches uniformly, the elastic energy in the system can be ignored. It does not change with crack length and so cannot drive the crack. Energy is conserved if there is zero friction, so the final equation for equilibrium fracture is

the same equation obtained for reversible peeling (Section 7.7), which is another direct linkage between adhesion force and molecular bonding, with no contribution from elastic or other effects. This equation demonstrates that, under reversible cracking conditions, a very small force is needed to wedge a film from a surface, even if the bonding is the strongest chemical bonding available, because wedging is a direct mechanism for converting mechanical energy into surface energy. As shown in Chapter 5, a strong bond would give a work of adhesion around leading to a wedging adhesion force of 10 N per meter of film width, a feeble resistance to failure. In practical applications, such as polymer coated steel sheets, the adhesion energy needed industrially is at least 100 times this value, or better yet 10,000 times, so it is important to consider mechanisms which can produce such amplification, as shown later in Section 14.10.

14.5. ELASTIC LINKAGE DURING WEDGING There are two other ways of finding the adhesion energy from the wedging test as shown in Fig. 14.8: measuring the crack extension ahead of the wedge, first achieved by Obreimoff, or measuring the force normal to the direction of cracking, in the cantilever beam or peeling test first analyzed by Rivlin. 16 If

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the film is transparent, then the crack extension is readily visible in reflected light and the separation can easily be measured as a function of crack length c using the optical interference fringes seen in the crack gap. At the tip of the wedge, the separation must be equal to the wedge thickness w, as shown in Fig. 14.8(a). The analysis now requires a theory for the elastic deformation of the film around the crack, because the mechanism involves an elastic linkage between the applied force and the crack, since the elastic energy changes as the crack extends. If it is assumed that the film bends like a thin beam fixed at the tip of the crack, then the deflection w at the end of the beam is from simple beam theory. Actually, the deflection will be slightly more than this because of extra deflection around the crack tip, which tends to increase the angle of the bent film at the tip from zero to around 5°. 17 The stored elastic energy in the beam is then given by Fw/2, that is This can also be expressed in terms of deflection w rather than force F as There are three energy terms which contribute to the cracking: the potential energy in the load F, the elastic energy stored in the bent film, and the surface energy in the cracked interface. Writing down these terms:

Thus the total energy in the system is

For equilibrium, dU/dc = 0 Therefore

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Alternatively the equilibrium can be expressed in terms of the vertical force (Fig. 14.8(b))

Equation (14.2) is useful because it shows that the work of adhesion can be measured by looking at the crack length ahead of the wedge. Obreimoff’s results are shown in Fig. 14.9 for various wedge and film thicknesses. They fit the theory remarkably well. The value of this method, as Obreimoff demonstrated, is that it can be used to follow environmental changes in the work of adhesion as the crack is exposed to water, acid, etc. He showed, for example, that the adhesion energy in vacuum was whereas in normal air conditions the adhesion energy dropped to about Equation (14.3) is interesting because it demonstrates that the force of adhesion varies grossly as the geometry changes. The force needed to lift off the film when the force is applied vertically is much different than when the force is applied horizontally as in Equation (14.1). Putting in some reasonable numbers for a low adhesion polymer surface coating, Pa, and the vertical force is 12 N per metre of film width whereas the horizontal force is only 0.1 N. This extra force is needed because the elastic linkage has to take energy from the elastic field to drive the crack, whereas the direct linkage converts mechanical energy directly into molecular crack energy.

14.6. CHANGE IN ELASTIC LINKAGE AS THE CRACK PROGRESSES The cracking of an adhesive film interface by a normal force is amusing because the elastic mechanism alters as the crack extends, as shown in Fig. 14.10.

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A very short crack exists at the start (Fig. 14.10(a)), where c is much less than h, and this is like a Griffith crack, requiring a high force given by

This is written in a somewhat unfamiliar form, but may be seen to depend on the parameter which appears in all the elastic linkage equations for cracking, but scaled by a h/c term, in this case to the power After the crack has extended such that the crack length c is comparable to h, then Equation (14.3) applies and the scaling is with h/c to the power 1. This obviously requires a lower force. When the crack extends still further, the elastic energy term becomes constant with crack length and then the force is independent of elasticity, giving a direct linkage between force and molecular adhesion, corresponding to the Rivlin peel equation F = Wb, which leads to a still smaller detachment force (see Section 7.7). Thus, the adhesion force for an elastic film can range over many different values depending on the elastic linkage, which is governed by the specific geometry and force application in the test method. But this wide range of adhesion forces can be explained by the mechanics of cracking in terms of the single molecular parameter W, the work of adhesion. Short cracks give a force dropping with intermediate cracks give a force which falls with c, and long cracks give a constant force. In the above examples the elastic linkage makes detachment more difficult in terms of adhesion force. Now let us consider two examples where peeling is made easier by elastic linkage.

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14.7. ELASTIC LINKAGES EASING FAILURE OF FILM ADHESION There are some situations in which the elastic energy helps to propagate the crack and thus decrease the adhesion force. The first, shown in Fig. 14.11(a), occurs when the peeling angle of the film is reduced from 90° to lower values. As Rivlin showed, the potential energy in the load is now changed to so that the force must be raised to continue peeling. When the force is raised, the elastic film begins to stretch significantly, storing elastic energy in the uniformly extended elastic material. The condition for equilibrium of the crack is then

The results for peeling of an elastomer from glass at various angles are shown in Fig. 14.11(b). As the angle was reduced, the peel force rose, but eventually levelled out at

which is the equation for lap failure of a flexible film in contact with a rigid surface, which applies also to shrinkage of films, to lap joints, and to testing of composite materials (see Equation (14.8) and Sections 15.3 and 16.3). This equation applies especially to the situation shown in Fig. 14.12(b), in which a stress is applied to compress a film which then detaches by cracking along the interface. Equation (14.6), originally postulated and proved in 1973, rather similar to the equation for fibre debonding of Gurney and Hunt18 and Outwater and Murphy, has been “rediscovered” regularly since that time.19,20 Shrinkage is another phenomenon which demonstrates how elastic energy can reduce peeling force, or even cause spontaneous detachment of an adhering coating. The shrunk film was formed by taking a strip of smooth elastomer, stretching it, and adhering it to the glass surface in the stretched condition as in

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Fig. 14.12(a). It was demonstrated21 that the elastic energy stored in the film for a crack length c was where was the residual strain in the film, b the width, E the Young’s modulus, and d the thickness. On cracking, this energy was converted into surface energy Wbc, so the criterion for cracking under a peel force was

and thus for spontaneous cracking under zero peel force

In this equation, essentially similar to the condition for cracking of a lap joint22 or removal of a film by applying a compression (Fig. 14.12(b)), W is the work of adhesion, E is the Young’s modulus of the film, h its thickness and the residual elastic strain (or the elastic stress) in the coating. Experimental results confirmed this theory for elastomers adhering to glass. For biaxial tension, is replaced by where v is Poisson ratio. This is an interesting result because it is evident that there is a strong size effect, with thinner films adhering better at the same strain (or stress) condition. The fracture mechanics analysis for elastic linkage must give a size effect since the crack is driven by stored energy. Thin films cannot store as much energy because of their low volume and so adhere better. Equation (14.8) has itself been “rediscovered” periodically.23–26 Brittle films can also crack during the failure process, especially if the coating is under tension due to shrinkage, as shown in Fig. 14.13. In this case the adhesion fracture energy is larger than the interface toughness, and the stored energy in the film causes cracking of the film rather than of the interface. Obviously, defects are needed within the coating to initiate the cracks which then propagate in a regular pattern.

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If there are sufficient defects to initiate cracks, eventually a pattern of cracks will form with a characteristic spacing s given by

which agreed with results on films grown by laser ablation on a

27,28

14.8. ULTIMATE ADHESION: PULL-OFF AND INDENTATION It is evident from the arguments above that an adhering film will only detach by cracking at the interface when sufficient energy can be pumped into the system. Since energy is force times distance, the most adherence will be obtained for the stiffest geometries, which resist elastic displacement and therefore cannot move to store much energy. Two examples of the stiffest geometries are shown in Fig. 14.14. Figure 14.14(a) shows a thin elastic film adhering to a rigid substrate. A rigid disc has been adhered to the top surface of the film and a force is being applied to pull off the coating. For simplicity, the film surrounding the disc has been cut away so that a crack can initiate around the edge of the film and propagate inwards to give an adhesion measurement. If the film is not cut, then there is a question of how to initiate the crack underneath the disc, a process which would depend on defects at the interface. Imagine that the crack has propagated to leave a diameter d of film in remaining contact. The elastic energy stored in the extended film is , where K is the bulk modulus of the elastic film and h its thickness. The potential energy in the applied load is twice this and negative, so the sum of elastic and

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potential energies is surfaces is

341

. Since the surface energy of the broken crack then the total energy U is

so that conservation of energy gives dU/dd = 0, i.e.

This means that the adhesion force can be very large if the film thickness is sufficiently small. Results confirming Equation (14.10) for thin gelatine films with are shown in Fig. 14.15.29 This theory explained the old results of Meissner and Merrill30 who found that poly(methyl methacrylate) films seemed to adhere stronger in thinner layers (Fig. 14.15(b)). In fact this enigmatic result had been known for many years in the

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field of metal solder films, and there was much experience in industry that the thinnest possible films gave the best adhesive performance.31 Tubes of rubber adhesive would carry the message “Use the minimum amount,” an admonition almost unknown on packs of soap powder or tubes of toothpaste. It is interesting to note that the high adhesion of thin films in pull-off tests had earlier been interpreted wrongly in terms of reduced statistics of flaw sizes at lower thicknesses.32–34 The essence of that argument was that interface cracking needed to be initiated at flaws within the adhesive film. As the volume of the film became smaller, for thinner films, the probability of finding a suitable flaw was reduced. Thus, a thinner adhesive gave better strength. Gent and others confirmed that the true explanation of the higher adhesion in thinner films was the poor energy storage capacity of a thin layer. The energy balance theory of adhesive cracking must therefore lead to the conclusion that thin films give a higher adhesion force at the same work of adhesion, as shown by Equation (14.10) above.35-37

The same reasoning applies to an elastic surface coating which is indented by a rigid punch, as shown in Fig. 14.16(a). In this case the crack is caused by the squeezing of the film from the gap between the punch and the substrate. This causes a ring crack to be initiated just outside the punch contact region, as shown by some early experiments in 1974.38 Results for indentation cracking of an adhesive interface are shown in Fig. 14.16(b). Films of silicone rubber adhering to poly(methyl methacrylate) were indented with a steel pointer and the diameter of the debonded circle was measured. The debonding increased with force in accord with a simple theory of elastic cracking. However, the forces required to crack the interface were much larger in this geometry than in peeling or wedging. These results were confirmed by Evans and Hutchinson and others who have used more complicated fracture mechanics analysis to measure adhesion energies for films.39–41

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14.9. DEFORMING THE SUBSTRATE: PULL-OFF, STRETCHING AND INDENTATION; ELASTIC, PLASTIC When the adhering film is attached to a deformable substrate, then more energy can be fed into the crack and adhesion is reduced in consequence. This was demonstrated in the pull-off geometry shown in Fig. 14.17(a), where a coating of poly(methyl methacrylate), Perspex from ICI, was adhering to a body of smooth compliant material, in this case solidified gelatine solution. This is the case of a rigid coating adhered to an elastic substrate. Applying a pull-off force to the Perspex film, which was in the form of a disc, deformed the surface of the substrate very noticeably, as shown in Fig. 14.16(a), and a crack then began to run along the interface from the outer edge to fracture the adhesive bond. This crack could be held at various diameters d by reducing the force F until healing began to occur. The adhesion force could then be plotted as a function of diameter to compare with the cracking theory (Fig. 14.17(b)), giving good agreement with the equation

In this case the adhering film is rigid compared to the substrate and so the coating thickness was not important. If the coating and substrate have similar deformations, then Equation (14.11) would have to be combined with Equation (14.10) to provide the correct solution. The film adhesion could then vary with film thickness and elastic constants in an interesting way. These considerations apply to the simple tensile test of film adhesion shown in Fig. 14.18(a), which illustrates a thin polymer film adhering to a thicker polymer sheet.42 The edge of the film is lifted up before the test to ensure that a long starter crack is in position. When the sheet was stretched elastically, the

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crack propagated steadily along the adhesive interface and could be observed through the transparent material. The energy balance theory of adhesion showed the equation for work of adhesion to be

where F was the stretching force, b the width, c the thickness of the adhering film and its Young’s modulus, h the thickness of the substrate, and its modulus. These values were consistent with peeling measurements of the film adhesion using Equation (14.1). Such experiments have been useful in determining the interface fracture energies of laminates.43,44 Often it is more convenient is such tests to apply the forces in elastic bending, as shown in Fig. 14.19(a). Now consider a film adhering to a polished block of aluminum which can bend plastically, as in Fig. 14.19(b). Before the test, one edge of the film is peeled off to give a starter crack. The aluminum sheet is then bent in a four-point test jig and the total strain comprising a small elastic component and a large plastic element, at the interface is measured. The elastic polymer film is seen to bend away from the metal surface in the debonded region and eventually the crack starts to propagate along the smooth interface. The condition that this crack

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should just move is given by Equation (14.8) in which the work of adhesion is given by where E is the modulus of the film, h its thickness, and its strain. The theories above presume that the coating remains elastic at all times, with the plastic deformation only altering the elastic energy stored within the film. However, in many cases, the surface coating is also cracked or plastically damaged by the applied force, and this damage can contribute substantially to the debonding process. Consider the two situations depicted in Fig. 14.20. In 14.20a the plastic damage due to an indentation is confined to the film and the substrate remains elastic. In the simplest model of plastic deformation of the film only, the material under the indenter is assumed to squeeze out to apply a stress to the surrounding film, storing elastic energy within it. Propagation of the crack along the interface then decreases the stored elastic energy according to the equation

which is similar to Equation (13.6) but taking into account the lack of Poisson contraction in the film plane.45,46 Experimentally, the indenter is loaded onto the film, then released. The debond circle can then be seen and its diameter measured. Of course, the initiation of the interface crack is a problem, unless the indenter also causes some ring cracking around the indenter contact circle. The formation of these ring cracks may trigger the delamination. Usually, in such indentation experiments, plastic flow also arises in the substrate under the film as shown in Fig. 14.20(b). This has been analyzed by Swain and his colleagues.47 As the indenter was pressed into the surface, ring cracks were observed just outside the indenter contact region. Sometimes radial cracks were also seen in the film.48 Subsequently, the substrate was pushed down

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under the film, which then delaminated to a diameter D. The theoretical analysis gave a delamination criterion

where is the maximum radial stress, h the film thickness, E its Young modulus, d is the indentor contact diameter, D is the delamination diameter, and K is a correction factor in which is the displacement of the coating and the actual displacement of the indenter contact circle. This equation was used to determine the adhesion energy of titanium nitride films to silicon giving and tantalum nitride to aluminium nitride, giving values probably quite close to the thermodynamic work of adhesion for these systems. Drory and Hutchinson have considered the case where the film is not much deformed by the indenter but the substrate undergoes large plastic flow (Fig. 14.21 (a)). For diamond-like films on titanium alloy they obtained good agreement between the diameter of delamination/diameter of indentation versus film thickness (Fig. 14.21(b)). In these experiments the diamond-like film was found to be under a biaxial compressive stress of 7 GPa and the adhesive energy was found to be very much higher than the equilibrium work of adhesion, which must be nearer It was claimed that plastic deformation of the substrate accounted for much of the discrepancy.50 The results suggested that the film would leap off the substrate spontaneously under the influence of the compressive energy if the film thickness grew to its critical value Scratching takes place when the indenter is dragged along the surface after indenting the coating. Heavens noted that the film would debond at a critical load as the indenter was pressed down onto the coating.51 This is a much more complex mode of film delamination which can give useful test data but is difficult

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to analyze. However, it is clear that there should be a transition to debonding as the load is increased, just as debonding should be preferred for thicker films.29,52,53

14.10. AMPLIFYING MECHANISMS: ROUGHNESS, ELASTIC ARREST, DEFLECTION, LOSSES We have seen how the adhesion of coatings and films varies remarkably with the elasticity, geometry and loading condition of the test method. However, it is also essential to understand how the delaminating crack can be inhibited by various mechanisms which amplify the adhesion to give adhesion energies up to far higher than the thermodynamic values of known to apply to smooth reversible interfaces. First, consider surface roughness. This causes a paint film to make more extensive contact with the substrate, once the liquid has wetted the surface fully. Once a crack runs along this interface, more energy has to be expended because the contact area is larger than that of smooth surfaces. This was demonstrated by taking an optically smooth glass sheet and grit-blasting half of it to increase the roughness, then depositing a coating such that the film started on the smooth surface and crossed into the roughened region, as in Fig. 14.22(a). When the coating was peeled from the smooth glass, the peel force was around as expected for van der Waals forces. But when the peel crack reached the rough region, the peel force increased by a factor near 10. The roughness of the interface had amplified the adhesion to give an effective adhesion energy

where A was the atomic area of the smooth surface and roughened surface.

the atomic area of the

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The only problem with this amplification was that the surfaces could not heal together subsequently because the rough asperities could not be perfectly interlocked to regain the molecular contact as shown in Fig. 14.22(b). The healing of rough surfaces was treated by Fuller and Tabor.54 Of course, if the roughness becomes very large, then the crack has difficulty following the interface. So the film partly debonds from the substrate and partly goes through the film material. If the film is composed of a tough polymeric material then this fracture process dissipates much energy and the adhesion energy can rise by a factor of 1000. The downside of this mechanism is that the paint may not be able to penetrate such tortuous roughnesses during the application of the liquid coating, in which case the adhesion may fall. Kinloch55 describes many experiments aimed at showing these effects, either through mechanically generated roughness or chemically formed asperities. It is evident that many chemical treatments do not increase the molecular adhesion but merely extend the surface of contact. Another amplifying effect may be described as elastic “roughening,” in which the interface may be smooth but it contains regions of higher elastic modulus56 (Fig. 14.23(b)). As the interfacial crack approaches the stiff region, it is arrested, then accelerated as it passes the obstacle. These regions may be reinforcing fibers contained in the paint for example. The adhesion energy increased to

where was the elastic modulus of the stiffened film and the ordinary film modulus. This can amplify adhesion by a factor up to 100. Another way of increasing film adhesion is to adjust the interface adhesion energy so that it is comparable with the fracture energy of the polymer coating. Under such conditions, the interface crack prefers to deflect into the bulk of the coating material, as illustrated in Fig. 14.24(a). Once this happens, the interface crack has to be started again and this requires much extra force. It is rather like

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peeling an orange with a fragile skin. The peel may not adhere well but it also breaks off and makes peeling much more difficult. Perhaps the most important amplifying effect is the viscoelastic or plastic nature of the film at high applied stresses near the interface crack. This may be modeled by the experiment shown in Fig. 14.24(b). An elastic film is peeling uniformly from a glass substrate at low peel force corresponding to van der Waals adhesion, with a line force F/b of around At a certain point, the film has been cut and fitted with a plastic hinge made of aluminum foil. When the crack reaches this point, the plastic hinge extracts the energy from the peeling load and the crack stops.57 A much higher force is required to restart the peeling as explained in Equation (14.4). The new adhesion energy is

giving an amplification factor which could be around 10,000. Introducing such plastic or viscoelastic relaxations into the coating material can be achieved by adding rubber particles or by altering the side chains on the polymer. Perhaps the most common way to amplify the adhesion of coatings is to treat the surface with a coupling agent or primer, which either improves the adhesion directly, or prevents attack by corrosive agents such as water. In the literature there are numerous examples of such surface treatments, for example by silanes, organotitanates and organozirconates.58,59 These react with the polymer film, with themselves and with the substrate. It is interesting that reaction within the bulk polymer itself can give a large increase in adhesion, especially when the reaction leads to cross-linking or grafting of the polymer chains to increase their length. The molecular weight of the polymer was found to have the largest effect on adhesion, depending on the type of side chain on the polymer.60,61 These results also show a paradoxical behavior of chemical modification to adhesion. Crosslinking the polymer improves the molecular weight effect at the

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interface and improves adhesion. However, crosslinking also makes the polymer more elastic and reduces the viscoelastic adhesion contribution. Thus, chemical treatment by this method can go both ways.

14.11. REFERENCES 1. Newton, I., Opticks, Smith and Walford, London, 1704 (reprinted Dover, New York, 1952, p 221). 2. Seymour, R.B. and Mark, H.F. (eds.) Organic Coatings: Their Origin and Development, Elsevier, New York, 1990, pp. 1–2. 3. Myers, R.R. and J.S. Long, J.S. (eds.) Treatise on Coatings, Marcel Dekker, New York, 1975. 4. Paul, S., Surface Coatings, Science and Technology, John Wiley, New York, 1985. 5. Snogren, R.C., Handbook of Surface Preparation, Palmerton, New York, 1975. 6. Packham, D., Dictionary of Adhesion, 1995. 7. Kinloch, A.J., Adhesion and Adhesives, Chapman & Hall, London, 1987, chap 4. 8. Roberts, A.D., Eng. Mater. Des. 11, 579–582 (1968). 9. Kendall, K.,J. Phys. D: Appl. Phys. 4, 1186–95 (1971). 10. Kendall, K., J. Adhesion. 5, 105–117 (1973). 11. Kendall, K., J. Adhesion. 5, 179–202 (1973). 12. Barquins, M. and Maugis, D., J. Adhesion. 13, 53–65 (1981). 13. Kendall, K., J. Adhesion. Sci. Technol. 8, 1271–84 (1994). 14. Obreimoff, J.W., Proc. R. Soc. A 127, 290–97 (1930). 15. Kendall, K. and Fuller, K.N.G., J. Phys. D: Appl. Phys. 20,1596–600 (1987). 16. Rivlin, R.S., Paint Technol. 9, 215–8 (1944). 17. Kendall, K., J. Adhesion. 5, 105–117 (1973). 18. Gurney, C. and Hunt, J., Proc. R. Soc. A 299, 508–524 (1967). 19. Williams, J.G., Strain Anal. Engng. Des. 28, 237 (1993). 20. Drory, M.D. and Hutchinson, J.W., Proc. R. Soc. A 452, 2319–41 (1996). 21. Kendall, K., J. Phys. D: Appl. Phys. 6, 1782–87 (1973). 22. Kendall, K., J. Phys. D: Appl Phys. 8, 1449–52 (1975). 23. Chiang, S.S., Marshall, D.B. and Evans, A.G., Surfaces and Interfaces in Ceramic and Ceramic Metal Systems, eds. J. Parks and A.G. Evans, Plenum, New York, 1981, pp 603–17. 24. Evans, A.G. and Hutchinson, J.W., Int. J. Sol. Struc. 20, 455–66 (1984). 25. Marshall, D.B. and Evans, A.G., J. Appl. Phys. 56, 2632–8 (1984). Also see Rossington, C. and Khuri-Takub, B.T., J. Appl. Phys. 56, 2639–44. 26. Williams, J.G., Strain Anal. Engng. Des. 28, 237 (1993). 27. Thouless, M.D.,Ann. Rev. Mater. Sci. 25, 69–96 (1995). 28. Thouless, M.D., Olssen,E. and Gupta, A., Acta Metall. Mater. 40, 1287–92 (1992). 29. Kendall, K., J. Phys. D: Appl. Phys. 4, 1186–95 (1971). 30. Meissner, H.P., and Merrill, E.W., ASTM Bull. 151, 80–3 (1948). 31. Chadwick, R., J. Inst. Metals 62, 277 (1938). 32. Meissner, H.P. and Baldauf, G.H., Trans. ASME 73, 697 (1951). 33. Bikerman, J.J., Science of Adhesive Joints, Academic Press, New York, 1968, p 273. 34. Dukes, W.A. and Bryant, R.W., J. Adhesion. 1, 48 (1969). 35. Gent, A.N., Rubber Chem. Technol 47, 202 (1974). 36. Gent, A.N., J. Polymer. Sci. A 2, 283 (1974). 37. Hilton, P.D. and Gupta, G.D., Design and Engng. Conf., ASME, Philadelphia, April 1973, paper 73-De-21, 1973.

FILMS AND LAYERS: ADHESION OF COATINGS 38. 39. 40. 41. 42. 43. 44. 45.

46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61.

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Kendall, K., Final Year Student Thesis, Monash University, 1974, unpublished results. Evans, A.G. and Hutchinson, J.W., Int. J. Sol. Struc. 20, 455–66 (1984). Dawson, D., Mechanics of Coatings, Elsevier, New York, 1989, pp 429–34. Rickerby, D.S., Surf. Coat. Tech. 36, 541–59 (1984). Kendall, K., Proc. R. Soc. A 344, 287–302 (1975). Clegg, W.J., Alford, N.McN., Kendall, K., Button, T.W. and Birchall, J.D., Nature 347, 455–7 (1990). Evans, A.G., Ruhle, M., Dalgleish, B.J. and Charalambides, PG., Mater. Sci. Eng A 126, 53–64 (1990). Boismier, D.A., Kriese, M.K. and Gerberich, W.W., eds. B.L., Karihaloo, Y-W., Mai, M.I., Ripley, and R.O., Ritchie, ICF9, Advances in Fracture Research, Pergamon, Amsterdam 6, 3065–73 (1997). King, R.B., Int. J. Sol. Struct. 23, 1657–64 (1987). Weppelmann, E.R., Hu, X.Z. and Swain, M.V., J. Adhesion Sci. 8, 611–24 (1994). Jensen, H.M., eds. BL., Karihaloo, Y-W., Mai, Y-W., M.I., Ripley, and R.O., Ritchie, ICF9, Advances in Fracture Research, Pergamon, Amsterdam 5, 2423–31 (1997). Drory, M.D. and Hutchinson, J.W., Proc. R. Soc. A 452, 2319–41 (1996). Tvergaard, V. and Hutchinson, J.W., J. Mech. Phys. Sol. 41, 1119–35 (1993). Heavens, O.S., J. Phys. Radium 11, 355–9 (1950). Mittal, K.L., J. Adhesion Sci. Technol. 1, 247 (1987). Rickerby, D.S., Surf. Coat. Tech. 36, 541–59 (1988). Fuller, K.N.G. and Tabor, D., Proc. R. Soc. A 345, 327 (1975). Kinloch, A.J., Adhesion and Adhesives, Chapman & Hall, London, 1987, chap 3. Kendall, K., Proc. R. Soc. A 341, 409–28 (1975). Kendall, K., Acta Metall. 27, 1065–73 (1979). Kinloch, A.J., Adhesion and Adhesives, Chapman & Hall, London, 1987, pp 152–9. Plueddemann, E.P., Silane Coupling Agents, 2nd edn, Plenum Press, New York, 1991. Kendall, K., Br. Polym. J. 10, 35–8 (1978). Kendall, K. and Sherliker, F.R., Br. Poly. J. 12, 85–8 (1980).

15 FRACTURE AND TOUGHNESS OF ENGINEERING ADHESIVE JOINTS

There are therefore Agents in Nature able to make the Particles of Bodies stick together by very strong Attractions. And it is the business of experimental Philosophy to find them out. ISAAC NEWTON‚ Opticks,1 p. 394

As Newton observed, fine particles and smooth surfaces stick together significantly. Indeed it is a general rule of nature, encapsulated in the first law of adhesion, that all bodies adhere as a result of molecular attractions. The problem is translating this into engineering practice where it is necessary to take large pieces of structures and join them together to make assemblies which will survive drastic attacks by forces, by impacts, and by chemical corrosion. Broadly speaking, a structure resists force by virtue of its strength, impact by its toughness, and chemicals by its durability. Understanding these terms may be relatively easy in a general sense but gets more difficult as the different detailed failure mechanisms are investigated. The purpose of this chapter is to define these properties of engineering adhesive joints more carefully. When doing so, it becomes clear that strength is a misleading idea when it comes to designing engineering structures held together by adhesive bonds. Such structures are often brittle and can fail catastrophically. Toughness is often a more critical criterion than strength for these joints. The conclusion is that we have to be extremely cautious when designing engineering structures at the limit of their adhesive performance. 353

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15.1. IMPORTANCE OF BONDED STRUCTURES Our demand for huge macho structures is insatiable. Nations vie with each other to have the tallest buildings‚ the longest bridges or tunnels‚ and the largest aircraft and submarines. Yet‚ all such edifices must be made by piecing together small bits of material‚ much as the caddis fly larva joins grains of sand to build its tubular home shown in Fig. 15.1. A jumbo jet contains 2 million parts‚ half of which are fasteners. This technological idea goes back to the Egyptian era‚ when paper was invented by beating papyrus stems together to form an overlapping bond‚2 or by binding papyrus stems together to make overlap joints in boatbuilding‚ as illustrated on the tomb walls of Ka’emwese in the Valley of the Kings (Fig. 15.1(b)). Can we understand how such constructions will survive? The old railway bridge at Conwy‚ which was one of the first containing tensioned overlapping joints‚ shown under their original test in Fig. 15.2‚ has lasted for 150 years but we have no way of knowing how long this will continue.3 Tales of disastrous collapse are legion. Despite having ‘brick for stone‚ and bitumen for mortar‚4 the walls of Jericho came tumbling down‚ presumably because of the poor quality of the adhesive bond. The myth of Daedalus and his flying machine was based on the premature failure of the hot-melt wax adhesive used to bond the structure. It is worth remembering that more than two dozen major bridges have collapsed during the 20th century.5

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15.2. A MODEL OF BRIDGE COLLAPSE The fundamental nature of an adhesive bridge failure can be understood by building a model using adhesive smooth rubber strips as shown in Fig. 15.3(a). Three tapes were used to bridge a gap between two supports‚ overlapping the sticky surfaces together.

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When a weight was placed on the model bridge‚ as shown in Fig. 15.3(b)‚ the joints resisted failure until a certain critical load was reached. At this critical load‚ that is the failure load‚ the joints peeled apart and the model bridge collapsed. With 10 mm wide strips the failure load was 70 g. However‚ when impact load was used‚ the weight required was less‚ around 40 g. Under wet conditions the load was also lower‚ around 20 g. These experiments illustrate several important features of engineering adhesive joints. The most important feature is the sudden nature of failure. Once the peeling starts‚ it tends to proceed rather rapidly. In other words‚ adhesive joints are usually brittle. This is confirmed by the impact test. Low energies of impact cause failure. Corrosive agents like water reduce the strength of the joint. Taken overall‚ the conclusion is that adhesive joints behave rather like glass: they are brittle‚ easily broken by impact‚ and prone to corrosive surface damage. This is the main reason why adhesive joints have been treated with suspicion by engineers. The overriding property needed by a designer is toughness‚ that is the resistance to sudden failure. Mechanical or welded joints‚ of higher toughness‚ are often preferred over adhesives for this reason. Of course‚ sticky tapes were devised with the opposite of toughness in mind‚ because it was important to be able to pull the adhesives off and re-stick them. Originally‚ in the mid-1960s‚ Spencer Silver at the 3M company was playing around with the polymerization of organic monomers when he found a polymer material which was easily removed from paper and then stuck back on again.7 This polymer seemed to have no useful applications‚ but the inventor diligently worked on making tacky bulletin boards and sticky tapes with it. After several frustrating years of failure in trying to convince his colleagues of its usefulness‚ Silver found technical support in Robert Oliveira and application ideas from two inventive marketeers‚ Nicholson and Fry. Fry was looking for tacky bookmarks to help his hymn singing‚ so paper strips coated with the new adhesive were ideal. Once these ‘post-it’ notes were distributed freely around offices‚ the demand for them became intense. This was the breakthrough required for successful sales.

15.3. DEFINITION OF JOINT TOUGHNESS The invention of sticky notes was a technological triumph which came about because scientists were developing many different types of polymers in the midpart of the 20th century. For example‚ adhesive tapes had been a brilliant early advance.8‚9 Companies such as ICI had imagined products such as wallpaper which could be pulled off the walls‚ cleaned in the washing machine and then placed back on the wall. Around the same time‚ there was a great scientific interest in such weak‚ reversible adhesion10–12 because it seemed to be fundamental to the principles of molecular bonding.

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The main conclusion of that work was that the toughness of the adhesive joint could be defined thermodynamically as the work of adhesion W. Of course‚ this definition was only applicable in certain special equilibrium circumstances‚ when the materials were elastic and when the fracture occurred very slowly. The toughness W was measurable in a T peel test shown in Fig. 15.4. This idea explained the force F required to peel two sheets apart‚ depending only on W and the width of the strips b (see Section 7.7). The interesting thing about this concept of toughness was that‚ although it had been used to understand the adhesion of particles‚8 and films‚9‚10 before 1975 it had never been applied to the failure of the most important engineering adhesive joint‚ the overlapping joint‚ or lap joint as it is most commonly called‚ famous for its use from Egyptian times. Then‚ the first glimmering of a solution appeared when it became clear that the T peel test of Fig. 15.4 was easily converted into a lap test by a simple change of grip. This can readily be demonstrated with smooth elastic rubber strips as shown in Fig. 15.5. When the joint is pulled in this different way‚ the crack still moves along the adhesive interface but a higher force is required and the joint becomes more stretched. It is evident from this model that the lap joint peels in essentially the same way as the T joint‚ except for the extra elastic stretching of the material caused by the higher force required. If we assume that the force F depends on the work of adhesion W in the same manner as the peeling joint‚ but also on the elastic modulus E of the strips and on the size L of the strips‚ then it is obvious from a simple dimensional analysis that the force must be given by an equation of the kind

where C is a constant. This is an interesting equation for two reasons: first‚ because it has the same form as Griffith’s original equation for the fracture of brittle materials like glass;13 second‚ because it is quite different from the

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traditional engineer’s view that the force required to break a lap joint depends on the overlap area A like

We must now consider how these two different and conflicting ideas about adhesive joint failure have come to be accepted. An engineer must be able to hold two contradictory ideas‚ such as Equations (15.1) and (15.2)‚ in his mind at the same time. Unfortunately both ideas cannot be true simultaneously.

15.4. HISTORY OF THE FAILURE OF LAP JOINT THEORY A theory of the rupture of lap joints became necessary early in the 19th century when large structures began to be built by riveting iron sheets together. An excellent example is the Conwy tubular railway bridge built by William Fairbairn for Robert Stevenson’s railway into Wales in the 1840s.6 Tough rolled sheets of wrought iron had become available‚ allowing pure tensile structures to be built to supercede the cumbersome compression structures previously built of brick and stone. Tensile structures are lighter and more elegant because they do not depend on their weight to stand up. Instead‚ tensile structures depend on the high strength and toughness of the materials‚14 allowing novel bridge designs like that of Conwy to be attempted. To convince Stevenson that the Conwy bridge would not fail‚ Fairbairn carried out a detailed experimental study of riveted lap joint fracture.15 He must

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have persuaded Stevenson‚ because he became President of the Institute of Mechanical Engineers in London soon after. He showed quite conclusively that the riveted joint became distorted by the force as shown in Fig. 15.6(a)‚ the material bending to drive a crack through the rivet and tear it apart. It is easy to build a rubber model of this structure‚ simulating an adhesive lap joint‚ to show that the deformation is identical‚ as shown in Fig. 15.6(b).16 However‚ Fairbairn did not quantify his empirical observations. He stated explicitly that he had little interest in the calculation of lap joint failure and sought only a practical working joint to support his engineering structures. His paper in the Philosophical Transactions of the Royal Society15 therefore stands out starkly in contrast beside those contemporary works on the mechanical equivalent of heat or researches in electricity which led to superb mathematical formulations of those problems. In comparison‚ the theory of lap joints remained primitive. One of the reasons for this was the confusion over the deformation mode in the adhesive. Two distinct types of lap joint exist‚ as shown in Fig. 15.7. If the adhesive layer is thick and flexible‚ then most of the deformation occurs within the adhesive layer and the adherends have little influence. Such a joint is best avoided because it tends to be compliant and weak. It is far better if the adhesive is made thinner and thinner until all the deformation arises in the adhering plates so that the joint is as stiff as possible. Then the strength of the joint will be

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maximized. The best joint is made with the thinnest possible adhesive. Better yet‚ the best joint has zero adhesive thickness. Nearly a century after Fairbairn‚ in 1938‚ two ideas emerged from the new engineering of airframes which were to focus on the paradoxical notion embodied in Equations (15.1) and (15.2). Volkersen17 derived a stress analysis for the deformation of a lap joint‚ showing that infinite stresses could arise at the ends of a lap joint‚ and Chadwick18 measured the peel strength of soldered joints‚ raising the conundrum that a joint is much weaker in peeling than it is when overlapped. How can strength be different when the same adhesive is employed? In order to understand the confusion‚ it has to be remembered that all designers and engineers over the past 400 years have been educated in Galileo’s principle that failure depends on the stress‚ i.e. the negative pressure‚ in the material. This stress criterion of failure states that rupture occurs when the stress reaches a critical value. In mathematical terms this is Equation (15.2) which gives ultimate force proportional to area. Such a principle would mean that the force needed to break a lap joint would increase in proportion to the length of overlap. But as de Bruyne demonstrated‚19 this is not true for the lap joint‚ as shown in Fig. 15.8. The force increases at first with overlap‚ but then levels off.

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Still more interesting was the observation that the lap joint strength depended on the thickness and elastic stiffness of the plates being joined together. Thick plates gave weaker joints. This ties in with the well-known tradition that lap joints work best for thin foils like paper and plastic. High modulus sheets give stronger joints. Also‚ a joint is stronger if you clamp it with reinforcing plates. Clearly‚ this evidence points to Equation (15.1) being the more correct description of lap joint failure.

15.5. THE CORRECT THEORY OF LAP JOINT STRENGTH The correct theory of lap joint peeling can be derived with reference to Fig. 15.9 which shows a long joint which has already cracked substantially. The equation for failure can be obtained by applying the energy balance theory of adhesion to the elements A and B in Fig. 15.9. Consider what happens to each element as the crack penetrates along the interface.20 Element A stretches as the crack goes through‚ whereas element B relaxes because it ends up with no force on it after the crack has passed. There is no energy change around the crack tip because this remains constant as the crack moves a short distance through a long joint. Thus the region around the crack tip can be ignored in the calculation because only changes in energy can drive the crack. Consider the elastic energy stored in the shaded regions A and B before the crack passes. Elastic energy is calculated from the work done in stretching the element as the force is applied and this is shown in Fig. 15.10. The original elastic energy is equal in both A and B is

giving a total elastic energy of The final energy in element B is zero while that in A is overall elastic energy increases during the peeling by

Thus the

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However‚ the force F also moves when it stretches the element A and the work done in this movement is

So the total work released as the crack moves through the lap joint is In equilibrium‚ because energy is conserved‚ this excess work must equal the surface energy created by revealing the new open surfaces i.e. bzW. The equation for joint failure is therefore

Hence

15.6. CONSEQUENCES OF THIS THEORY OF LAP JOINT FAILURE This equation derived above for lap joint failure is surprising in a number of ways. It is certainly consistent with Equation (15.1) on dimensional grounds‚ and is equivalent to Griffith’s brittle fracture theory for glass.13 Moreover‚ it fits the puzzling historic results for lap joint failure which showed that the overlap length is not important for long joints‚ and the strength increases with sheet thickness d and stiffness E. Additionally‚ it is now clear why chemical environment can weaken the joint‚ because the failure depends on work of adhesion W‚ which decreases markedly with surface contamination (see Chapter 6). The most intriguing questions raised by this argument are related to the wellestablished idea of lap shear strength which is much used by engineers in designing bridges and other load bearing structures. In the first place‚ it is obvious

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from Equation (15.5) that the idea of strength cannot easily be applied to this joint‚ because the failure force is not proportional to area. In other words‚ the attempts that engineers have made to divide fracture force by the overlap area to give strength F / bl are doomed to failure because F / bl is not a constant. F / bl can be made into any number you desire merely by adjusting the values of l‚ d and E. A lap joint can have any strength you want! A second paradox is the use of the word “shear” to describe the fracture. It is evident from the calculation used to obtain Equation (15.5) that shear is not mentioned. The joint peels but does not slide or shear (Fig. 15.11). Only tension forces and displacements are needed to explain the failure of the joint. In fact‚ it would be far more logical to describe this failure as a tension failure‚ just as the Griffith equation describes tension failure. Of course‚ shear stresses exist around the crack tip‚ as in every crack geometry known‚ but the energy associated with these stresses remains constant as the crack moves and therefore cannot drive the crack. The most interesting comparison is between the T peel test and the overlap test‚ as illustrated in Fig. 15.11. Both of these joints fail by a cracking mechanism. Both cracks travel at constant speed under steady load. This distinguishes such cracks from Griffith cracks which accelerate. But the overlap joint requires considerably more force because the energy is injected into the crack by elastic stretching and not by direct movement as in the peel joint. This was first demonstrated with rubber strips in 1975.20 The strips were smooth after casting on glass surfaces‚ and showed reversible adhesion which increased with peeling speed as shown by the full line in Fig. 15.12. The theoretical prediction of overlap failure force was calculated from Equation (15.5) and plotted as the broken line in Fig. 15.12. Experimental

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measurements of lap joint cracking over the same speed range confirmed good agreement between theory and practice.20

15.7. STRENGTHENING OF A LAP JOINT BY PRESTRESSING One very significant difference between the peel joint and the lap joint is the effect of prestressing; that is‚ applying a force to the sheets before joining and then relaxing the stress after the joint is hardened. This is the operation which is used in prestressed reinforced concrete to improve the tensile strength of beams. The steel reinforcing wires are stretched‚ the concrete is then poured around them and hardened‚ and the force is then relaxed‚ effectively putting the concrete into compression. Figure 15.13 shows the similar effect of elastically stretching one of the sheets before overlapping and adhering to a normal sheet. After relaxing the force‚ the joint is seen to be curved because one sheet has pulled the other into a circular shape as it shrinks. Prestressing lap joints was mentioned by Gazis in 1962‚21 but without a good theory. The prestressed joint may now be tested by two methods; peeling or lap stretching‚ as shown in Fig. 15.14. In T peeling‚ the joint is weaker than before as a result of the prestress. The elastic energy stored in the material helps the crack to propagate. An energy balance shows that the peel force is now given by

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However‚ the lap joint breaking force depends on which sheet is gripped. It can be stronger or weaker than the joint without prestress. In Fig. 15.14(a) the application of the force has to overcome the prestress before any cracking can occur and so the force required for failure is higher:

whereas the force required for case b is lower because the prestress energy now helps to drive the crack:

Obviously‚ it is better to grip the joint as in Fig. 15.14(a) if a high strength is required. But there is a limit to such strengthening. Once the prestress is larger than 3 times the failure stress for the ordinary joint‚ there is enough stored elastic

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energy in the joint to make it fail spontaneously when the prestressing load is removed after joint hardening.22 In other words‚ the bending shown in Fig. 15.13(c) is so great when the prestress is removed that the joint tears itself apart spontaneously.

15.8. MORE COMPLEX OVERLAPPING JOINTS The solutions given above correspond to long cracks penetrating very long overlaps as shown in Fig. 15.15(a). In this case, the tensile forces F are lined up almost exactly along the adhesive interface. However, when the overlap is short, then the joint bends very significantly and the forces are at an angle to the adhering interface, causing substantial peeling of the sheets, and weakening the joint. This case is shown in Fig. 15.15(b). In this instance, peeling theory can be applied to calculate the force in terms of the peel angle

And23

Therefore

which is 40.8% weaker than the long joint explained in Equation (15.5).

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This equation can explain the early observations on overlap joints made by the early aircraft engineers who were trying to glue Mosquito aircraft together in World War II.24–26 The joint strength is lower for short overlaps‚ but still is not proportional to overlap length. Experimental testing of this equation was carried out on model rubber joints‚24 as shown by the results in Fig. 15.16. Joints were made of various overlap lengths‚ and the line force F / b at failure was measured at a constant crack speed of The joints longer than 5 joint thicknesses behaved as long joints and failed at however much the overlap was increased. The shortest joints failed at increasing as the overlap increased. Aircraft built in the early part of the 20th century were glued together with natural adhesives made from casein which is a protein made from milk.27 These adhesives worked satisfactorily except when the joints became damp‚ so that they absorbed moisture. They then became weak and smelt of old camembert cheese. It is claimed that the maintenance engineers used this fact as a nondestructive test‚ routinely smelling the bonded joints on the aircraft to check their quality. When they detected the camembert smell‚ the joints were replaced. This problem was largely overcome by using synthetic polymer adhesives which did not rot. Urea formaldehyde resins were used to bond the Mosquito structure together but these tended to be brittle. De Bruyne and his colleagues modified the chemistry to produce tougher polyvinyl formal/phenolic resol polymers which were used on the first Comet jet liner in the 1950s.28 One of the most complex aspects of the overlap joint is the question of how the crack starts. Figure 15.17 illustrates the end of an overlap and shows how

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slight modification of the geometry can have an influence on the initial stress concentration at the corner where the crack is initiated. This geometry change can occur quite naturally as a bit of adhesive squeezes out from the joint. Alternatively‚ the joint can be chamfered or machined to smooth out the sharp corner. If the stress concentration is reduced‚ then the joint appears stronger at the crack initiation. However‚ the propagation of the crack which causes failure is still dominated by Equation (15.5)‚ so such strengthening can give a false sense of security. In fact‚ it is easy to show that the sharp-edged

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joint requires about 40% extra force to start the crack than to propagate it. Thus the propagation equation is the conservative criterion for failure of these structures. Figure 15.18 shows crack speed measurements on a model joint pulled with a steady load. The crack initiated at the corner of the joint as expected. It then speeded up steadily‚ until it penetrated about one joint thickness along the interface. Then it travelled at constant speed in accord with Equation (15.5). From these results‚ the initiation force was found to be 1.39 times higher than the propagation load. This result fitted an approximate theory based on the idea that the strain energy in the shaded region of Fig. 15.18 remained constant as the crack initiated. This gave the initiation force

Of course‚ the precise calculation of the initiation force depends on an accurate stress analysis of the region around the crack. Many stress calculations have been made since Volkersen’s original estimates in 1938.17 The solutions depend greatly on the detailed geometry near the corner of the joint‚ and on the presence of small defects like bubbles and cracks. Often it is convenient to use finite element analysis to compute the stress picture.29 Once the stress pattern is known‚ the crack propagation criterion can then be calculated from the energy condition for cracking.30‚31 Alternatively‚ a plastic flow criterion may be invoked to provide a means of predicting flow at the high stress region. However‚ fracture mechanics methods are usually most powerful in calculating the failure events.32

15.9. VARIOUS ADHESIVE JOINT GEOMETRIES It is a surprise to many engineers when they find that the best overlap joints fail by a tension mechanism‚ with the stretching of the lapped sheets dominating the failure. However‚ it is possible for weaker joints to fail by shear‚ if the adhesive layer is thick and compliant‚ such that it deforms much more than the surrounding material‚ as indicated in Fig. 15.19. In this case‚ all the stored energy is in the elastically deforming adhesive material. The adherends then remain rigid and no longer contribute to the failure. Pure shear may be obtained in two types of loading. Figure 15.19(a) shows the obvious shear deformation of the adhesive as the lapped sheets are pulled. However‚ the material also deforms in pure shear if the sheets are pulled in the other direction (Fig. 15.19(b)). This is odd because it looks like a tensile failure at first. However‚ the adhesive is in shear because it cannot pull in from the sides‚ and therefore it is the shear modulus which is important. In this case we assume the joint width b is small compared to thickness d.

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From the energy balance theory of cracking‚ the force required to break these joints is readily calculated. The stress in the adhesive is force/area‚ i.e. F / bl‚ and this energy is the only term driving the crack if the displacement remains constant. If the crack moves a distance c‚ then the elastic energy loss is stress × stress × volume/2G‚ i.e. i.e. This must equal the energy of the new surface revealed by the crack‚ i.e. Wbc. Therefore the cracking criterion is

In the same way‚ the criterion for failure of a butt joint can be derived. The butt joint is perhaps the oldest type of adhesive geometries and was used in biblical times to join stones with a layer of bitumen.4 If the bricks are rigid‚ then all the deformation is in the adhesive layer. This is highly constrained and so is in pure hydrostatic pressure as a result of the applied force‚ as shown in Fig. 15.20(a)‚ because the adhesive cannot deform laterally. Using the same logic as above‚ it follows that the bulk modulus K of the adhesive material is now controlling the elastic energy stored in the joint. Thus the failure criterion is given by

This follows inevitably from the idea that the stiffest joint stores least energy and therefore requires a larger force to attain the fracture energy. Of course‚ the

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toughness of the joint‚ i.e. W‚ remains constant‚ unless there is dissipation within the body of the glue which increases the energy absorption with glue thickness. Thus‚ the brittleness of the joint remains the same with thickness. Figure 15.20(b) shows experimental results for solidified gelatin solution between two Perspex plates. The force needed for slow equilibrium fracture was proportional to joint area‚ and inversely dependent on the square root of thickness as predicted by Equation (15.14).33 It is evident from these arguments that the strength of an adhesive joint depends very much on the manner in which it is loaded and constrained. This is readily seen when a silicone elastomer is used to bond two Perspex plates together‚ at several different rubber thicknesses‚ then tested in peel as shown in Fig. 15.21. For large thickness of adhesive‚ the crack can be seen as a straight front moving uniformly along the rubber/Perspex interface. However‚ as the elastomer is made thinner‚ the crack front appears to form a wave as it tries to remove the extra constraint. Finally‚ for very thin adhesive‚ the crack breaks into sinuous fingers which thread their way through the interface. Essentially there are

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three modes of energy storage in the joint: bulk energy‚ shear energy‚ and tension energy. The fingers change the mode from bulk to shear to tension deformation. Thus the crack mechanism can change as the crack propagates. The breakup of the crack front in this way is similar to the crazing which is observed in polymers like polystyrene. The deformation of the material is changed from isostatic to tensile by increasing the crack tortuosity.34 An excellent paper on this effect has now been presented to show how the wavelength of the elastic instability increases linearly with film thickness h of the elastic adhesive35 giving = 3.94 h. Such sinuous cracks can often be seen in laminated glass where the polymer laminating layer has debonded. A further range of various adhesive cracking geometries has been analysed recently by Maugis36 using the fracture mechanics method. He considers the double cantilever test‚ the double torsion test and the blister test in addition to peeling and lap. The equations for W at cracking equilibrium are given in Fig. 15.22. 15.10. SUMMARY OF ENGINEERING ADHESIVE JOINTS It is evident from this brief discussion that engineering adhesive joints are complex and ill-understood. The simplistic criterion of “adhesive strength” often

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put forward by engineers is wrong when looked at in detail. Most joints fail by a brittle mechanism in which a crack propagates along the interface or glue-line. Thus adhesive joints should be treated like glass: brittle‚ unreliable‚ and prone to impact and defects. The equations of failure then resemble Griffith’s famous criterion‚ featuring W the work of adhesion‚ E the elastic modulus‚ and d the dimension of the sample. The stress at failure often depends on the thickness of the adhesive or of the adherends. Thus‚ energy dominates stress as the condition of failure for most joints. Better joints are those which multiply the energy of failure‚ for example with tough adhesives. Only when energy can be dissipated during the fracture can the joint become tougher. Much needs to be learned about such toughening energy dissipation mechanisms. 15.11. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

Newton‚ I. Opticks‚ Smith and Walford‚ London‚ 1704 (reprinted Dover‚ New York‚ 1952‚ p 394). Allen‚ K.‚ Int. J. Adhesion Adhesives 16‚ 47–51 (1996). Timoshenko‚ S.‚ History of the Strength of Materials‚ McGraw Hill‚ New York 1953‚ p 157. Bible‚ Genesis 11‚ 3. Broek‚ D.‚ The Practical Use of Fracture Mechanics‚ Kluwer‚ Amsterdam 1989‚ p17. Fairbairn‚ W.‚ An Account of the Construction of the Britannia And Conwy Tubular Bridges‚ John Weale‚ London‚ 1849. Ranganath Nayak‚ P and Ketteringham‚ J.M.‚ Breakthroughs‚ Rawson Associates‚ New York‚ 1986‚ pp 29–6. Benedek‚ I.‚ Development and Manufacture of Pressure Sensitive Products‚ Marcel Decker‚ New York‚ 1998‚ chap 1. Benedek‚ I. and Heymans‚ L.J.‚ Pressure Sensitive Adhesives Technology‚ Marcel Decker‚ New York‚ 1996‚ chap 1. Johnson‚ K.L.‚ Kendall‚ K. and Roberts‚ A.D.‚ Proc. R. Soc. A 324 301–13 (1971) Kendall‚ K. J. Phys. D: Appl. Phys. 4‚ 1186–95 (1971). Maugis‚ D. and Barquins‚ M.‚ In: Adhesion and Adsorption of Polymers‚ Part A‚ ed L.-H. Lee‚ Plenum‚ New York‚ 1980‚ pp 203–77. Griffith‚ A.A.‚ Phil. Trans. R. Soc. A 221‚ 163–98 (1920). Gordon‚ J.E.‚ The New Science of Strong Materials‚ Penguin‚ London‚ 1968‚ pp. 106–8. Fairbairn‚ W.‚ Phil Trans. R. Soc. A 2‚ 677 (1850). Kendall‚ K.‚ Phil. Mag. 36‚ 507–15 (1977). Volkersen‚ O.‚ Luftfahrtforschung 15‚ 41 (1938). Chadwick‚ R.‚ J. Inst. Metal. 62‚ 277 (1939). De Bruyne‚ N.A.‚ In: Adhesion and Cohesion‚ ed P Weiss‚ Elsevier‚ New York‚ 1962‚ p 46. Kendall‚ K.‚ J. Phys. D: Appl. Phys. 8‚ 512–22 (1975). Gazis‚ D.C.‚ In: Adhesion and Cohesion‚ ed P Weiss‚ Elsevier‚ New York‚ p51. Kendall‚ K.‚ J. Phys. D: Appl. Phys. 8‚ 1449–52 (1975). Kendall‚ K.‚ J. Adhesion 5‚ 105–117 (1973). Kendall‚ K.‚ J. Adhesion 7‚ 137–40 (1975). De Bruyne‚ N.A.‚ Aircraft Engng. 16‚ 115 (1944). Sheridan‚ M.L. and Merriman‚ H.R.‚ Am. Soc. Testing Mater. Spec. Tech. Bull. No 201‚ 33 (1957). Kinloch‚ A.J.‚ Proc. Inst. Mech. Eng‚ Preprint 08‚ 1–32 (1996). De Bruyne‚ N.A.‚ Proc. Conf. on Bonded Aircraft Structures‚ Bonded Structures Ltd‚ Cambridge‚ UK‚ 1957‚ pp1–9.

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29. Adams‚ R.D.‚ Comyn‚ J. and Wake‚ W.C.‚ Structural Adhesive Joints in Engineering‚ Chapman and Hall‚ London‚ 1997. 30. Groth‚ H.L.‚ Int. J. Adhesion Adhesives 8‚ 107 (1988). 31. Dillard‚ D.‚ Hinkley‚ J.A.‚ Johnson‚ W.S. and St Clair‚ T.L.‚ J. Adhesion 44‚ 51–67 (1994). 32. Fleck‚ N.A.‚ Hutchinson‚ J.W. and Suo‚ Z.‚ Int. J. Sol. Struc. 27‚ 1683–703 (1991). 33. Kendall‚ K.‚ J. Phys. D:Appl. Phys. 4‚ 1186–95 (1971). 34. Kendall‚ K.‚ Clegg‚ W.J. and Gregory‚ R.D.‚ J. Mater. Sci. Lett. 10‚ 671–4 (1991). 35. Ghatak‚ A.‚ Chaudhury‚ M.K.‚ Shenoy‚ V and Sharma‚ A.‚ Phys. Rev. Lett. 85‚ 4329–32 (2000). 36. Maugis‚ D.‚ Contact‚ Adhesion and Rupture of Elastic Solids‚ Springer‚ Berlin (1999)‚ chap 5.

16 COMPOSITE MATERIALS: HELD TOGETHER BY ADHESION AT INTERFACES

The parts of all homogeneal hard Bodies which fully touch one another, stick together very strongly ISAAC NEWTON, Opticks,1 p. 388

Glass is a uniform material, because every part of its structure is the same down to the molecular scale. However, other materials in common use, such as metals, polymers, ceramics, and composites, are not normally homogeneous in structure but are composed of domains separated by interfaces. The domains are readily detected by viewing polished surfaces with a microscope, as seen in Fig. 16.1, which reveals the individual crystal grains plus the separating interfacial boundaries, at which the grains are stuck together through molecular adhesion. Such interfaces can have huge effects on the properties of the material as Newton observed. However, the most significant property improvements are found when the interface structure and adhesion are optimized to particular values. This is best illustrated by considering particulate fillers added to polymers.

16.1. PARTICULATE COMPOSITES A typical example is rubber filled with colloidal carbon particles which are mixed into the polymer during processing. Hancock,2 one of the first scientists to 375

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study mixing of particles into polymers, decried the use of fillers in rubber because he viewed it as adulteration. However, he found very large beneficial influences of fine particulate sulfur, zinc oxide, and carbon, which could be mixed into the rubber to improve strength and toughness markedly. The particles themselves were not tough. Therefore the conclusion was that the interfaces between the particles and the rubber were responsible for the property improvement in the composite. This conclusion was supported by experiments which varied the interface systematically. For example, as the amount of carbon black was increased, thus increasing the interfacial region, the toughness improved, reaching a maximum at around 20 vol% of the material (Fig. 16.2). Also, when the carbon surfaces were graphitized by heating to change the molecular structure of the carbon, the toughening effect diminished, and when the particles were increased in diameter to reduce the interfacial area, the toughening was reduced.3–5

Since those early experiments on mixing particles into natural rubber, an enormous industry has grown up to mix powders into polymers for a wide range of applications. Some of these are merely to utilize former waste products such as wood chips or slate dust. Mixing these powders with a polymerizable resin such as polyurethane, styrene, or methyl methacrylate produces mouldable sheets with excellent properties. More technical applications include resin concretes for repairing roads or buildings, kitchen work-tops, sinks, and sanitary ware. These are made by grinding silica, calcium carbonate, or alumina trihydrate with monomer, casting, then polymerizing in a mold. In the most technical applications, particular powders and resins are used to obtain specific effects, such as wear resistance of silica filled acrylics for dental repair, or hydroxyapatite filled polymers for bone implants.

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16.2. EFFECT OF INTERFACES A simple demonstration that interface adhesion is vital to the composite material strength can be achieved by doping the interfaces with molecules which reduce the grain boundary adhesion. For example, adding copper to molybdenum reduces the intergrain adhesion considerably and the resulting composite is extremely brittle, falling apart at the grain boundaries. Only one layer of molecules covering the interface is sufficient to give this effect, so it follows that composite materials can be enormously sensitive to small amounts of impurities of this kind. Fig. 16.3 illustrates the sensitivity effect for cubic grains in size. The volume of interface molecules relative to the total material volume is of the order 3d/L where d is the molecule diameter and L is the grain size. If the contaminant molecule is 0.1 nm in diameter then 3d/L is 3 parts per million. It is evident from this estimate that very small quantities of interface contamination could ruin the properties of a composite material. Perhaps the most dramatic effect of interfacial adhesion on material properties is found with glass fiber reinforced epoxy resins. Both the glass and the epoxy are brittle materials with a resistance to cracks less than . But when the two brittle materials are combined by mixing glass fibers into liquid epoxy, followed by polymerizing the resin to solidify it, then the crack resistance

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of the composite structure can rise to several orders of magnitude higher than each component separately. Such high toughness is sufficiently useful for many structural applications, for example aircraft, cars, and buildings. This principle of improving glass by interposing polymer layers had been invented by Benedictus who found that glass flasks which had been used to boil cellulose nitrate solutions were much more impact resistant than clean flasks. Glass laminates made by sticking glass sheets together with polyvinyl butyral are now widely used in armor and in car windshields.

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Building houses with glass is still thought to be stupid by respected scientists.6 But glass can readily provide adequate strength and toughness, providing suitable adhesive interfaces are inserted to make the glass tough. Many buildings are now made from glass fiber reinforced plastic. Thus it is apparent that no material should be dismissed because it seems weak or brittle. In principle, even the most brittle materials such as diamond could be made tough by the introduction of appropriate adhesive interfaces. By the argument of the previous paragraph, this could be achieved through adding a few parts per million of chemical additive. Of course the additive would have to be smart enough to arrange itself in the structure illustrated in Fig. 16.4(b), which shows how a crack is stopped at interfaces. Such interfaces are found in fibrous or laminated structures. If the additive arranged itself uniformly, as in Fig. 16.4(a), then the crack could propagate easily and the material would be brittle. But if the additive produced controlled interfaces perpendicular to the crack path, as in Fig. 16.4(b), then the crack could be stopped. This idea, that cracks can be much inhibited by interfaces, is a relatively recent one, stemming from the works of Cook and Gordon.7 Previously, it had been thought that the strength of the fibrous components was the most important parameter.8 A good model for the effect of the interface in a composite is given by the bimetallic strip.9 This consists of two strips of metal with different expansion coefficients, stuck together at the interface. Separately, each strip remains flat when heated. However, when joined together, the strips bend as the temperature changes (Fig. 16.5). This new property of the composite material, i.e., bending on heating, depends on adhesion at the interface between the component strips, since no bending would occur if the strips were not bonded together. In the same way, two strips of material may exhibit little resistance to cracking when tested separately. But when joined adhesively, the strips may exhibit high crack resistance. This new property of composite toughness clearly depends on interface adhesion. But the material property which is now most important is elastic modulus, and not expansion coefficient, as revealed when the passage of a crack through an interface is analyzed.

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16.3. A CRACK MEETING AN INTERFACE The problem of a crack meeting an interface in a composite as shown in Fig. 16.6(a) was posed by Cook and Gordon7 who wrongly concluded that the strength of the interface was important. Zak and Williams10 followed by Rice and Sih11 also studied this problem using elastic stress analysis, but the problem was complicated mathematically and led to strange oscillating stresses near the crack

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tip. There were also substantial experimental difficulties because cracks inside the composite material could not easily be observed, though it was known from fracture surface inspection that a crack slowed down as it approached an inclusion, and then accelerated as it emerged through it. The solution to this problem was found by adopting a simpler crack geometry, shown in Fig. 16.6(b), in which an adhesive peel crack was viewed with a TV camera through a glass plate.12 Rubber peeling from glass was the experimental model. The crack speed was measured accurately as the crack approached an interface at which the elastic modulus of the material increased. This stiffening was achieved by moulding stiff fibers into the rubber over half its length.

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In the first experiments, the crack behavior for uniform rubber material was measured (Fig. 16.7(a)). As the force F applied to the crack increased, so did the crack speed. Identical behavior was observed for different rubber thicknesses and also for the stiffened rubber, demonstrating the simplicity of the peel crack system in which the peel force F depends only on the adhesive fracture energy R at a given crack speed, as shown by the peel equation (Chapters 7 and 13).

Clearly, factors such as film thickness and elastic properties do not affect elastic peel cracks according to this equation. However, when the crack approached an interface where the elastic modulus changed, in this case because stiff fibers were embedded in the rubber, the crack speed was seen to change substantially, as shown in Fig. 16.7(b). At first the crack traveled at constant speed under the steady load, as expected from Equation (16.1). Then, at the interface with the stiffer material, the crack slowed down by a factor of 100. Subsequently, as the crack penetrated the stiff material, it speeded up to regain its original constant speed after 15mm of further travel. When the crack moved from the stiff material towards the compliant rubber, the opposite effect was observed (Fig. 16.7(c)). The crack traveled at the same constant speed as before at constant load. But at the interface where the material was slacker, the crack speeded up significantly, and then slowed back down to reach its original constant speed after 5 mm of travel. The theoretical explanation of these effects can be derived from the energy theory of fracture.12 Consider a crack as it just penetrates into a stiffer region of material, shown in Fig. 16.8. When the crack is at the interface, it exhibits the shape of a long bent beam under load. However, when the crack tries to penetrate into the stiffer material, there is more elastic resistance to bending deformation

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and so the shape changes (Fig. 16.8(b)). These shapes were measured microscopically on peeling samples. The theoretical shapes were calculated from elastic beam theory. Since the stiffer material is more resistant to bending, the peeling load cannot deflect so much and consequently does less work. But the interface fracture work remains constant. Therefore the peeling force must be raised to maintain the same fracture work and peeling speed. Putting this theory mathematically the beam deflection is

where F is the force, L the beam length, E the elastic modulus, b the width, and d the film thickness. To maintain the crack propagation at constant speed, this deflection must remain constant, i.e. F/E is constant. Therefore, at the interface, the condition for cracking changes to

and since

Thus, while it is clear that the elastic modulus of a material may not affect its crack resistance as indicated in Equation (16.1), a change in elastic modulus at an interface toughens the material by a factor of This theory was used to calculate the full lines for comparison with experiment in Fig. 16.7. The conclusion from these arguments is that composites gain their toughness primarily from changes in elastic modulus at the interfaces. Thus, epoxy resin may have a toughness of only but with glass present this can rise substantially. Because the elastic modulus of glass is 70 GPa compared with 3.5GPa for the resin, the ratio of moduli is thus 20 which multiplies the epoxy resin toughness from to This theory also explains why

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ceramics are so difficult to toughen because the ratio of elastic moduli for ceramics is low. For example, the elastic modulus of zirconia is 200 GPa and that for alumina is 400 GPa. The possible elastic toughening effect at an interface in a zirconia/alumina composite is therefore only a factor of two.13

16.4. DELAMINATION AT THE INTERFACE The crack stopping effect described above presumes that the interfacial adhesion is perfect. An alternative effect is observed if the adhesion at the interface is reduced. Then the crack can deflect along the interface, preventing catastrophic failure by the Cook–Gordon7 mechanism, illustrated in Fig. 16.9. Imagine a block of material which has been cut in half and rejoined at the interface. In this case there is no change in elastic modulus at the interface, so the crack is not impeded by an interface mismatch. If the adhesion at the interface is good, then the crack can go straight through (Fig. 16.9(b)), but if the interface has poor adhesion, then the crack deflects along the interface and the catastrophic failure is prevented. The principle is that of a rope; one strand of the rope can break without much reducing the strength of the rope as a whole. But if the rope strands become bonded together with sufficient adhesion, then eventually the crack will go straight through and the whole rope fails. The problem is to find the interface adhesion which just prevents crack penetration of the interface, because this is the condition for maximum performance in bending or compression, while retaining the toughness stemming from the low interface adhesion. Cook and Gordon7 originally modeled this situation by numerical stress analysis and came to the conclusion that the interface strength was 5 times less than the material strength at the critical deflection point. A more satisfactory theory was obtained by considering the crack propagation criteria in the two directions14 at an interface a short distance c inside the block of solid material, as illustrated in Fig. 16.10. The crack approaching the interface is governed by the Griffith equation which gives the stress required to drive the crack through the interface:

Here the fracture toughness is the cohesive toughness of the material, and the block is assumed to be thick so that the term for plane strain conditions is included. In contrast, the stress required to initiate the crack along the interface is given by the lap-joint equation (Chapter 15):

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where is the interface adhesion energy. Dividing these equations gives the condition for the crack to just deflect:

showing that the ratio of cohesive to adhesive fracture energy is the important parameter which must be maintained at a value around 10, depending on geometry and Poisson’s ratio. This equation was experimentally verified using ethylene propylene rubber as the model material. When cross-linked with dicumyl peroxide, this rubber is both brittle and transparent, allowing cracks to be driven easily through the material and observed by eye. The benefit of using such rubber is that the cracks can be driven quite slowly, so that the deflection phenomenon can be seen very easily without high speed cameras.14 Interfacial adhesion could also be varied simply in this material by bringing two smooth surfaces of rubber together, then changing the crosslinking reaction to give different values of the interface toughness Consider the formation of the interface shown in Fig. 16.11. The main block of rubber was moulded against polished steel to give a smooth surface, then crosslinked for 30 min. A thin sheet of rubber was also cross-linked in a press for 30 min, using a glass mould to give a smooth surface, then adhered to the main block. The thin sheet was brought into good molecular contact with the block, as seen by observing the removal of bubbles at the interface, and the final composite was crosslinked for another 30 min to give the full rubber reaction. This experiment gave medium adhesion at the interface. By varying the times of the initial and final crosslinking, while

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maintaining the total 60 min reaction time, the interface adhesion could be altered over a wide range, while preserving the same bulk properties of the rubber material. Then a razor slit was cut into the rubber surface, a tensile stress was applied, and the crack was observed travelling through the interface, as in Fig. 16.11(c). The behavior of the crack when it reached the controlled interface was noted as a function of the interfacial adhesion. The cohesive and adhesive toughnesses were measured in the fracture tests shown in Fig. 16.12. Two cohesive tests, the Griffith and T tear tests, were used.

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These gave consistent cracking results. Also, two adhesive fracture tests, the lap test and the peel test, were employed. Again these were consistent. The results shown in Fig. 16.13 give the variation in toughness for the rubber and the interface after the different treatment times. It can be seen that the toughness increased significantly with crack speed, as expected from the arguments of Chapter 8. When a crack was driven towards the interface at a speed of the crack delection was observed at low adhesion, but not at high adhesion (Fig. 16.13(b)). The ratio of at the changeover transition was found to be 9.1, in good agreement with Equation (16.6). Another way to obtain the transition was to use a low adhesion interface and to increase the crack speed. It can be seen in Fig. 16.13(b), that, for this rubber system, the interface toughness increased more rapidly with crack speed than the cohesive toughness of the rubber. Thus the crack crossed through the interface at high speed whereas it was deflected at lower speed. By this method the transition ratio was found to be 11, somewhat higher than the theoretical prediction of 9.1. The energy balance theory of fracture also predicted that crack deflection should be somewhat easier in the centre of the sample than on the outer surface,

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as shown in Fig. 16.14(b). The theoretical condition giving the ratio of cohesive to interface fracture energies for deflection at the centre was The value obtained experimentally was 5.6,13 showing reasonable agreement with theory. The effect of a modulus mismatch on the crack deflection was also tested. It was demonstrated that for a modulus mismatch of a factor 10, the deflection criterion became easier, with

16.5. TOUGH LAMINATES These arguments have been applied to the development of tough ceramic laminates. In general, ceramic materials such as silicon carbide, aluminum nitride, or zirconium oxide are brittle. They fail catastrophically when a single crack penetrates the material. Using the arguments above, a new type of ceramic laminate has been invented, to prove that the introduction of correct interfaces raises the toughness considerably.15–17 There are two major problems with the standard methods of producing ceramic composites, such as mixing carbon fibers into glass ceramic or sintering ceramic powders around ceramic fibers,13,18,19 the methods originally developed in the early 1970s. The first difficulty is finding suitable fibers, sufficiently strong and refractory to reinforce ceramics, while maintaining an appropriate interface with the matrix.20 The second is the shrinkage which usually occurs during the processing of ceramic powders. Such shrinkage cannot be accommodated by the rigid fibers, so sintering is inhibited and cracking often observed. To avoid this shrinkage, vapor deposition of ceramic may be used, but this is generally time consuming and expensive.21 In the new composite structure, the design was based on the structure of a ceramic capacitor, with layers of silicon carbide ceramic around thick joined by graphite films about in thickness. Just as ceramic capacitors are manufactured, shown schematically in Fig. 16.15, the silicon carbide was made by mixing the fine powdered ceramic with an aqueous polymer solution to form a

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viscous, plastic dough which could be high shear mixed to break down agglomerates, pressed into sheets, and then rolled flat into tapes of extended area. The interface films were made by milling graphite into an aqueous polymer solution to form a viscous ink which was printed in a thin film onto the surface of the silicon carbide tapes. The tapes were stacked and pressed to form a laminate which was heated under argon to pyrolyze the polymer binders, and then sintered at 2040°C under argon for 30min to give a 98% dense body. The laminated structure shrank substantially, but the layers of silicon carbide and graphite did not crack during the process. As a control sample, silicon carbide tapes were also compacted to form a ceramic body without the interface layers, so as to measure the improvement stemming from the interfaces. To compare the laminated sample with the monolithic control, three-point bend tests were carried out as shown in Fig. 16.16. The elastic properties of both samples were similar, with a flex modulus of 450 GPa. The bend strengths were also comparable, with 500 MPa for the monolithic and 633 MPa for the laminated sample. However, the resistance to cracking of the samples was entirely different. When a notched sample of the monolithic material was bent, the load/displacement curve was typical of a brittle ceramic material, with sudden cracking failure after an initial elastic deformation (Fig. 16.16(a)). The fracture toughness of the sample was calculated to be with a low fracture energy of calculated from the area under the curve. By contrast, the sample with the laminated interfaces failed in a much more complex way, as illustrated in Fig. 16.16(b). The initial notch grew until it reached the first graphite interlayer. Then the crack stopped and was deflected along the interface, providing some extra energy dissipation. The load on the sample then had to be increased substantially to start another crack in the laminate. This load corresponded to a fracture toughness of More rapid crack growth then occurred but with delaminations at every graphite

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interlayer providing significant energy loss. The total work of fracture in this experiment, calculated from the area under the curve, was slightly higher than wood. Thus, laminating the silicon carbide raised the work required to propagate the crack by a factor of over 100. By varying the interface systematically, various levels of adhesion have been obtained, and it has been possible to quantify the critical condition for crack deflection for the laminate with a thin interphase of reduced adhesion. This turned out to be different from the criterion explained in Section 16.4 for a simple plane interface. Clegg and his colleagues22–28 found that, as the thickness of the graphite was increased, eventually reaching 10,000 atomic layers, the crack deflection became easier, giving a deflection criterion of for undoped graphite and 1.75 for doped graphite, much less than the value of 9.1 predicted by the simple theory, but closer to that found for a large elastic mismatch. In more recent papers, Clegg and his colleagues have verified those observations, and shown that several types of interface can be used to give crack deflection, to overcome the problem that graphite is too easily oxidized

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above 800°C. In particular, porous interfaces were shown to be effective. Interfaces between alumina and zirconia were also capable of causing crack deflection.

16.6. HEALING THE INTERFACE CRACK: INTERFACIAL DISLOCATIONS

Once the interface crack has initiated inside a composite material, then traveled some distance along the interface, there may be a tendency for the crack to heal, if the crack surfaces come into close proximity, so that the molecular adhesion forces can pull the smooth crack faces together again. The simplest situation in which this interface crack healing arises is shown in Fig. 16.17.29 This schematic diagram illustrates the observations made on a model composite laminate made from four strips of smooth, transparent rubber. This model (Fig. 16.17(a)) was made by sandwiching the two short strips between the two long ones, ensuring that the smooth rubber surfaces came into molecular contact to exclude air bubbles. Where the two short strips met in the middle of the laminate, there was a slight gap which opened up when tensile forces were applied to stretch the composite, as in Fig. 16.17(b). Once this gap had opened somewhat, four cracks were seen moving along the interfaces within the composite (Fig. 16.17(c)). However, these cracks did not move far before healing was observed, starting from the original gap in the middle of the laminate. The healing followed the interface cracks, leading to the situation shown in Fig. 16.17(d), where four “bubble cracks” could be seen in the laminate. Each “bubble crack”, so-called because it resembled a trapped air bubble within the interface, consisted of a crack traveling along its interface, pursued by a healing crack which eventually caught up with and arrested the original crack. The final result, shown in Fig. 16.17(e), was a laminate which looked much like the original material, except that the gap in the middle was now larger, and the interfaces contained four dislocations which stored elastic energy within the strained rubber.30–34 The nature of interfacial dislocations has been defined in chapter 8. The mechanism of formation is best studied in the simple laminate configuration shown in Fig. 16.18. When a tensile force is applied to a laminate which contains a long crack, the crack propagates at constant speed as shown by the experimental results of Fig. 16.18(a). The faces of the crack do not close up, because the application of the force causes the crack faces to open significantly. However, when a constraining force is applied to press the crack faces together, as illustrated in Fig. 16.18(b), the crack faces can touch, and healing is then observed. The energy theory of fracture shows that the constraining force first slows the crack a little, then accelerates it slightly before healing occurs. But after the healing takes place, the crack begins to slow down rapidly,

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eventually stopping as the healing front catches up with the crack. The fit of the theory to the experimental results is given in Fig. 16.18(b). After removing the forces, the left-hand edge of the laminate is displaced and the laminate is bent under the stored strain energy in the dislocated material. This natural process of interfacial cracking followed by healing is essentially the same process shown to produce interfacial dislocations in Chapter 8; that of peeling, stretching, and healing. On releasing the forces, the joint bends because residual strain energy is stored within it. Of course, if too much strain energy was stored, then the joint would tear itself apart on releasing the load. The presence of residual stresses means that the lap joint is stronger in the presence of the dislocation. More force must be applied to start a new crack running. This is important in a composite material because the increased force can trigger off cracks at other places in the composite, thereby spreading damage. Consider how this mechanism of interface failure could apply to a natural laminated material like mother of pearl. This nacreous substance is formed by the deposition of thin calcium carbonate crystals by the mollusc. Such plate-like crystals are typically long and thick, separated by a tenuous layer of polymeric protein material which forms only a few percent of the composite volume. Yet, despite this low polymer content, nacre displays significant plastic deformation when stretched, with strains up to 0.018, accompanied by white deformation bands in the composite structure.34–36 This behavior may be modeled by the structure shown in Fig. 16.19, which is seen to be formed out of lap joint sub-units in the same way as the model used in Fig. 16.17. When such a structure is stretched, it is evident that the joints between the ends of the nacre crystals could open up and cause interfacial dislocations to propagate along the flat crystal interfaces. The weakest joint opens first, then heals to form the dislocation. It cannot open further because the

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dislocation makes the joint stronger. Thus the next weakest joint opens up and the process spreads through the nacre to cause plasticity over a significant volume. The criterion for interfacial dislocations is that cracks start to move along the interface, but the crack opening is constrained to cause healing. Thus the condition for dislocation formation is the same as that for interface cracking in the lap joint (Chapter 15). Dislocations will form at a stress

and strain

These equations allow the calculation of the optimum thickness of the calcium carbonate plates in the nacre. For maximum energy dissipation in the material, the plates should be as thin as possible to maximize the number of dislocations. However, as the plates are made thinner, then the stress needed to cause dislocations is increased according to Equation (16.7). If the stress gets too large, then the plates fracture instead and the laminate is embrittled. Thus there is an optimum plate thickness which gives maximum energy absorption without plate cracking. If the plate strength is then the thickness d from Equation (16.7) is taking E = 150 GPa and This thickness agrees well with the observed nacre dimension shown in Fig. 16.19.

16.7. THE OVERALL PICTURE: CRACK STOPPING, DEFLECTION, AND HEALING From the arguments above, it is clear that the interfaces within composite materials play an enormous part in the failure process. A crack is stopped when it approaches an interface with stiffer material, it is deflected when the interface adhesion energy is low, and the interface appears plastic as a result of interface healing in certain situations. The problem of understanding these several effects acting together then becomes acute, and it is necessary to find a way of representing these three interacting phenomena in a simple manner. Two difficulties prevent easy understanding of these effects. The first is the complicated mathematical problem of describing cracks at interfaces. The second is making experimental observations deep within a composite structure. The theoretical investigation of stresses around a crack in an interface has proceeded since the early study by Williams,37 and the mathematical principles have been summarised by Rice38 and others. The mathematical formulation rapidly becomes too complicated.39 Most progress has been made by Hutchinson and his colleagues40 who attempted to explain the previous observations of crack stopping and crack deflection. Starting from the Dundurs41 definition of the

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elastic mismatch, and noting the earlier theoretical analyses of cracks at interfaces,42–46 they conclude that crack behavior at interfaces can be pictured on a plot of two ratios: the ratio of fracture energies of interface to material and the Dundurs mismatch parameter where is the Young’s modulus divided by for material 1. This plot, illustrated in Fig. 16.20, showed that the crack goes straight through the interface for high adhesion but is deflected for low adhesion, as shown many years before.12 The benefit of this diagram was its depiction of the interaction between the elastic and fracture energy effects. As the elastic mismatch increased, the deflection became easier. This graph applied only to semi-infinite media; not to a crack just penetrating the outer surface of a body, crossing a laminated strip, or traversing an interphase. It would be a good idea to include strain mismatch at the interface, resulting from interface dislocations, on the same diagram. Such a plot, shown schematically in Fig. 16.21, indicates whether a crack will be deflected by an interface under a frail range of conditions. The crack is deflected at all values of above the shaded surface, and goes straight through the interface at all values below. At high mismatch of elastic modulus or interface strain, the crack delects more easily and the adhesion of the interface can then be improved significantly to deliver enhanced composite behavior. Of course, the deflection is also influenced greatly by the way in which the forces are applied to the composite body in order to stimulate fracture. Forces can

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be applied to open the crack, to slide the crack faces or to twist them, so-called mode I, mode II, or mode III.47 The failure of composites under different loadings is a major problem which we discuss next.

16.8. THE PROBLEM WITH COMPOSITES: BENDING AND COMPRESSION Composite structures are remarkably tough when tested in tension. However, when stressed in other ways the composite may be more brittle than expected. Rope is a good example. The tensile properties are excellent, but the rope is floppy when bent or compressed. Similarly, wood is difficult to chop across the grain, but cleaves easily along the grain. The problem is to define the composite properties under different loading circumstances. Consider, for example, a simple laminate which is loaded in four different ways, tension, bending, compression, and peeling, shown in Fig. 16.22. In each case a crack travels along the interface, but the force required turns out to be completely different. This was first analyzed sensibly by Outwater and Murphy48 who studied the mechanism by which cracks move through a body, and applied an energy conservation theory to the results. A very simple view of such interfacial debonding in laminates was achieved by measuring the delamination in the four geometries of Fig. 16.22 using an interface made by pressing Perspex sheets together at 110°C for 30 min under a load of 10 tonnes. This interface was

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planar, visible through the material so that cracking could be followed easily, and of fracture energy around depending on the crack speed. By applying forces in tension, bending, compression, and peel, the difference in composite failure could be assessed for the same interface.49–51 From the energy balance theory of fracture, four equations were produced to explain the failure force F in the different geometries given the same interfacial fracture energy R. The simplest equation was for peeling

For tension, the force for debonding was

Where E was the elastic modulus and d the sample thickness. Compression failure was slightly easier than tension

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whereas bending failure was easiest of all, especially as the crack length c increased

These theories were plotted to compare with the experimental results in Fig. 16.23. Good agreement was obtained. The conclusion was that composites must give different performance characteristics depending on the loading method. Tension is best, followed by compression, with bending and peeling leading to weak behavior. There is no way around this problem, though it is clear from the equations above that the problem of bending can be diminished by reducing crack length at the interface, in other words by reducing flaw size between the laminae. An additional conclusion is that the idea of interfacial shear strength so often used to interpret interfacial failure of composites is false. The key material parameters dictating failure are adhesive energy R and elastic modulus E, together with geometrical design. The anomalous idea of interface shear strength had been embedded in the early literature8 and standards52–53 which assumed that interfaces could shear at a particular stress level. This stress-dominated shearing does not happen in reality. Instead, the interface peels and heals. It is clear that strength does not appear in any of the equations above. However, once the interface strength idea became established, especially after Broutman’s work on fibers,55,57

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it became extraordinarily tenacious. It still appears in the latest works of respected authors.56 Another way around the bending and compression problem is to make the composite isotropic. One way of doing this would be to weave fibers equally in six directions. This is an interesting geometrical problem which is equivalent to weaving a 3D Penrose tiling with fibers. If the fibers are all equal then the volume packing is only 0.2485, but if the fibers have different sections then this can be more than doubled.83

16.9. ADHESION OF FIBERS IN COMPOSITES Broutman had observed the detachment of a single fiber from a matrix in three types of test shown schematically in Fig. 16.24. In the first test, a resin block was stretched, and shear stresses developed at the embedded fiber ends, causing failure. The second test used Poisson’s ratio tensions developed across the neck of the compression sample to pull the resin from the fiber by interface tension. The third test was essentially a debonding and friction test as a fiber was pulled through a block of resin. Such tests have an interesting pedigree because they have traditionally been used on well-known macro-composites such as reinforced concrete.58 For example, the pull-out test has a long history in the rubber tire industry where it was important to measure the bonding of the elastomer to the steel wires used to stiffen the tire structure. Tire cord adhesion was found to increase by an order of magnitude when the steel was plated with copper to react with the sulfur used in cross-linking the elastomer. Interfacial strength has largely been discredited as a parameter for describing such failures. The energy method of fracture mechanics is more satisfying because it shows that failure is really a function of two separate variables; crack length and fracture energy, with other elastic, geometric, and kinetic factors also needing consideration. The correct fiber debonding equation based on the energy method was postulated theoretically by Gurney and Hunt60 in 1967, A failure criterion dependent solely on a critical stress or “interface strength” is altogether too simplistic. In the same way, the idea that a fiber should retain its high strength in a composite matrix is bizarre. Obviously, a matrix crack can attack the fiber and cause it to weaken considerably. Reductions by a factor of 2 were predicted by fracture mechanics and observed in strength tests on fibers embedded in a cracked matrix.61 A reasonable description of the test shown in Fig. 16.24(a), now known as the single fiber fragmentation test, has been provided by Nairn and his colleagues.59 The single fiber embedded in a large amount of matrix is stretched, causing fragmentation of the fiber as well as interface debonding, resin fracture, and other complex effects. In a measured situation with T50 carbon fibers in an

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epoxy matrix, the crack density was plotted as the strain was increased. The debonding of the fibers was also observed. By assuming that the energy theory of fracture held, the theoretical plot fitted the data when the fiber fracture energy was and the interface fracture energy was Such models can now be verified by direct measurement at the interface using Raman spectroscopy.62 The debond growth can also be used to determine the interface bonding energy. For example, a value of was obtained by measuring debond growth between E glass fibers and polymer matrix.53

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More direct measures of interface debonding energy are provided by the pull-out or push-out tests shown in Fig. 16.26. When a tensile force is applied to a fiber to extract it from its composite matrix, an interface crack eventually starts to run along the fiber. It is obvious from a simple fracture mechanics argument65 that the stress on the fiber to propagate the crack, assuming a very compliant matrix, must be given by an expression of the form

where D is the fiber diameter, L the bonded length, G the shear modulus of the matrix, R the interface fracture energy and the volume fraction of fibers in the composite. Thus the stress on the fiber at debond seems to rise for finer fibers and for higher volume fractions, as expected from the size effect in the Griffith theory of fracture. By the same line of reasoning, the strength of a fiber attacked by a deep matrix crack must be of the form61

where E is the Young’s modulus of fiber and matrix (assumed equal), R is the fiber fracture energy, v is Poisson’s ratio and D is the fiber diameter. This equation shows that a smaller diameter fiber must seem to be stronger within a cracked composite material. For fine fibers with very smooth surfaces, healing can also occur after debond, so that interfacial dislocations are produced. For fat fibers with rough surfaces, molecular contact cannot be reformed easily. Consequently frictional pull-out is then observed, with damage at the interface.

16.10. ADJUSTING THE INTERFACE ADHESION IN FIBER COMPOSITES Very soon after glass fiber composites began to be widely used, in the late 1940s, it became evident that the chemistry of the interface between polymer and

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fiber was crucial. This was a particular problem for fiberglass boats because water could be seen penetrating quickly along the glass/polyester interface, severely weakening the composite material. Such effects could be simulated by boiling composite samples in water and then bend testing them. Typically the bend strength fell from 400 MPa to 200 MPa during a two hour boiling test.66,67 Clearly this was due to the wettability of the fibers which caused spontaneous debonding (Chapter 6). The problem was solved by finding silane molecules which could be coated onto the glass fiber surfaces immediately after cooling from the glass melt spinning process. The chemical idea was to have one end of the molecule which would react with the glass surface by the mechanism shown in Fig. 16.27, whereas the other end contained a group which could react with the resin matrix. This was the coupling agent philosophy. Although this mechanism proved to be simplistic, because the silane formed clumped regions on the surface and within the polymer, the effects were significant, raising the dry composite bend strength to 600 MPa and giving a 5% reduction after boiling in water. Certain silanes proved to be more successful than others, with methacryl and amino propyl becoming favorites (Fig. 16.27).67 Now that carbon fibers are widely used in polymer matrix composites, a large number of experiments have been carried out to vary the surface chemistry and molecular adhesion at the interface.68–73 A typical production process for carbon fibers involves electrochemical treatment, “anodizing”, which introduces reactive groups onto the fiber surface. An additional complication was that the surface topography of the fibers was also modified by the treatment, making it difficult to distinguish the effects of chemistry and geometry. A recent conclusion by Drzal et al.69 was that both chemistry and roughness combined synergistically to increase fiber/matrix adhesion. Optimizing the electrolyte used during anodic treatments, from alkalis, nitric and sulphuric acids, dichromates through permanganates, has led to the adoption of ammonium bicarbonate as a useful species, readily washed off after treatment to leave various oxygenated groups on the carbon. Such groups were readily picked up by XPS and SIMS techniques. Although the fiber surface area did not change much during the process, the number of active surface sites rose by as much as 30%. However, some of these may have been within surface porosity. Plasma treatments of fiber surfaces have also been used extensively to introduce active groups onto carbon surfaces. Oxygen, nitrogen, ammonia, and argon have all been used to plasma etch surfaces. Contaminants are desorbed and weak crystallites removed. Alternatively a polymer forming organic compound can be used in the plasma to produce a surface coating on the fiber. Almost all organic compounds give a polymerized surface film which adheres strongly to the surface. By varying the coating systematically using mixtures of hexane and acrylic acid in the plasma, the adhesion and properties of carbon fiber Tenax T-

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5000 and Epoxy Epikote 828 were evaluated. The interface failure was thus varied from weak debonding to strong matrix cracking. In conclusion, the chemical nature of the fiber/matrix interface was found to be extremely important to composite behavior. There is no doubt that this applies to the many different kinds of reinforcements and matrices in common use,74–83 including cements, metals, ceramics, and polymers.

16.11. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

Newton, I., Opticks, Smith and Walford, London, 1704 (reprinted Dover, New York, 1952, p 388). Hancock, T., Personal Narrative, 1857, p. 76, American Chem. Soc. Centennial Volume, (1939). Kendall, K., Br. Polym. J. 10, 35–38 (1978). Kendall, K. and Sherliker, F.R., Br. Polym. J. 12, 85–8, 111–3 (1980). Boonstra, B.B., In: Reinforcement of Elastomers, ed. G. Kraus, London, Interscience, 1965, p. 529. Marder, M. and Fineberg, J. Phys. Today, September, 1996, pp24–9. Cook, J. and Gordon, J.E., Proc. R. Soc. A 282, 508 (1964). Kelly, A., Strong Solids, 2nd edn, Clarendon, Oxford, 1973. Hull, D., An Introduction to Composite Materials, Cambridge University Press, Cambridge, 1981, p3. Zak, A.R. and Williams, M.L., Trans. ASME J. Appl. Mech. 34, 967–74 (1963). Rice, J.R. and G.C. Sih, Trans. ASME J. Appl. Mech. 87, 418–23 (1965). Kendall, K., Proc. R. Soc. A 341, 409–428 (1975). Sambell, R.A.J., Bowen, D.H. and Phillips, D.C., J. Mater. Sci. 7, 663–75 (1972). Kendall, K., Proc. R. Soc. A 344, 287–302 (1975). Clegg, W.J., Alford, N.McN., Button, T.W., Birchall, J.D. and Kendall, K., Nature 347, 455–7 (1990). Clegg, W.J. and Kendall, K., JFCC Int. Workshop on Fine Ceramics, Nagoya, March 1992, Elsevier, Amsterdam, pp 143–8. Clegg, W.J. and Kendall, K., European Patent No. GB9002986, 1990. Prewo, K.M. and Brennan, J.J., J. Mater. Sci. 15, 463–8 (1980). Briggs, A. and Davidge, R.W., Mater. Sci. Eng. A 109, 363–72 (1989). Yajima, S., Hayashi, J. and Okamura, K., Nature 261, 683–5 (1976). Lamicq, P.J., Bernhard, G.A., Dauchier, M.M. and Mace, J.G., Am. Ceram. Soc. Bull. 65, 336–8 (1986). Clegg, W.J. and Seddon, L.R., Proc. 4th Int. Symp. Ceram. Mater. for Engines, Goteborg, Sweden, 1991. Phillipps, A.J., Clegg, W.J. and Clyne, T.W., Composites 24, 166–76 (1993). Clegg, W.J., Acta Metall. 40, 3085–93 (1992). Phillipps, A.J., Clegg, W.J. and Clyne, T.W., Composites 25, 524–33 (1994). Clegg, W.J., Howard, S.J., Lee, W, Phillipps, A.J. and Stewart, R.A., Composites 2, 337–50 (1995). Lee, W. Howard, S.J. and Clegg, W.J., Acta Metall. 44, 3905–22 (1996). Nutbrown, E.A. and Clegg, W.J., Key Engineering Materials 132–6 [III], 2021–24 (1997). Kendall, K., Int. J. Adhesion Adhesives, 1, 301–4 (1981). Kendall, K., Nature 261, 35–6 (1976). Kendall, K., Phil. Mag. 36, 507–15 (1977). Kendall, K., J. Phys. D:Appl. Phys., 11, 1519–27 (1978). Kendall, K., Phil. Mag. 43, 713–29 (1981).

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Kendall, K., Mater. Res. Soc. Symp. Proc. 40, 167–76 (1985). Currey, J.D., Proc. R. Soc. B196, 443–463 (1977). Jackson, A.P., Vincent, J.F.V. and Turner, R.M., Proc. R. Soc., B234, 415–440 (1988) Williams, M.L., Bull. Seismol. Soc. Am., 49, 199–204 (1959). Rice, J.R., J. Appl. Mech., 55, 98–103 (1988). Lazarus, V. and Leblond, J.B. ICF9, Sydney 1997, Pergamon, Oxford 1997, pp 1811–22. He, M. and Hutchinson, J.W. Int. J. Sol. Struc., 25, 1053 (1989). Dundurs, J. J. Appl. Mech., 36, 650–2 (1969). Cook, T.S. and Erdogan, F., Int. J. Eng. Sci. 10, 677–97 (1972). Erdogan, F. and Biricikoglu, V. Int. J. Eng. Sci. 11, 745–66 (1973). Goree, J.G. and Venezia, W.A. Int. J. Eng. Sci. 15, 1–27 (1977). Lu, M. and Erdogan, F., Eng. Fracture. Mech., 18, 491–528 (1983). Hutchinson, J.W. and Suo, Z. Adv. Appl. Mech. 29, 63–191 (1992). Broek, D. The Practical Use of Fracture Mechanics, Kluwer, Dordrecht, 1989, chap 1. Outwater, J.O. and Murphy, M.C. Mod. Plastics 47, 160 (1970). Kendall, K., J. Mater. Sci., 11, 638–44 (1976). Kendall, K., J. Mater. Sci., 11, 1263–6 (1976). Kendall, K., J. Mater. Sci. 11, 1267–9 (1976). ASTM Part 26 Test D2344–72 (1973) pp 413–6. ASTM Part 26 Test D2733–70 (1973) pp 764–7. Broutman, L.J., Interfaces in Composites, ASTM Special Technical Publication 452, American Society for Testing and Materials, New York, 1969. Broutman, L.J., Proc 25th SP1/RP Annual Technology Conf., Soc Plastics Ind, New York, 1970, paper 13-D. Kim, J-K. and Mai, Y-W. Engineered Interfaces in Fiber Reinforced Composites, Elsevier, Amsterdam, 1998. Broutman, L.J. and McGarry, F.J. 17th Ann. Tech. Manag. Conf. Reinf. Plastics Div. Soc Plastics Ind, New York, 1962, Section 1E, pp 1–8. Shiriajeva, G.V. and Andreevskaya, G.D. Plast. Massy 4, 43 (1962). Nairn, J.A. and Liu, Y.C. Composite Interfaces 4, 241–67 (1997). Gurney, C. and Hunt, J., Proc. R. Soc. A 299, 508–524 (1967). Kendall, K., Alford, N.McN. and Birchall, J.D., MRS 78, 181–7 (1987). Wadsworth, N.J. and Spilling, I. Br. J. Appl. Phys. (J. Phys. D) 1, 1049–58 (1968). Huang, Y. and Young, R.J. Comp. Sci. Technol., 52, 505–17 (1994). Wagner, H.D., Nairn, J.A. and Detassis, D. Appl. Comp. Mater. 2, 107–17 (1995). Kendall, K., J. Mater. Sci. , 10, 1011–4 (1975). Plueddemann, E.P., Molecular Characterisation of Composite Interfaces eds. Ishida and Kumar, Plenum Press, New York, 1985, pp 13–23. Plueddemann, E.P., Silane Coupling Agents, Plenum Press, New York 1988. Kim, J.K. and Mai, Y.W. Comp. Sci. Technol. 41, 333–78 (1991). Drzal, L.T., Sugiura, N. and Hook, D. Composite Interfaces 4, 337–54 (1997). Alexander, M.R. and Jones, F.R. Carbon 33, 569–80 (1995). Rand, B. and Robinson, R. Carbon 15, 257–63 (1977). Denison, P., Jones, F.R. and Watts, J.F. Surf. Interface Anal, 12, 455–60 (1988). Jones, C. Surf. Interface Anal. 20, 357–67 (1993). Bascom, W.D. and Chen, W–J. J. Adhesion 34, 99–119 (1991). Kettle, A.P., Beck, A.J., O’Toole, L., Jones, F.R. and Short, R.D. Comp. Sci. Technol. 57, 1023–32 (1997). Maso, F J.C. Interfaces in Cementitious Composites, E&FN Spon, London 1993.

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77. Piggott, M.R., Load Bearing Fiber Composites, Pergamon Oxford, 1980. 78. Chawla, K.K., Ceramic Matrix Composites, Chapman & Hall, London 1993. 79. Taya, M. and Arsenault, R.J. Metal Matrix Composites-Thermomechanical Behaviour, Pergamon, Oxford, 1989. 80. Kinloch, A.J., Toughened Plastics 1: Science and Engineering, American Chemical Society, Washington DC, 1993. 81. Hancox, N.L., Fiber Composite Hybrid Materials, Applied Science Publishers, London, 1981. 82. Daniel, I.M. and Ishai, O., Engineering Mechanics of Composite Materials, Oxford University Press, Oxford, 1994. 83. Parkhouse, J.G. and Kelly, A., Proc. R. Soc., A 454, 1889–1909 (1998).

17 THE FUTURE OF MOLECULAR ADHESION

I can calculate the motions of the heavenly bodies‚ but not the madness of the market ISAAC NEWTON‚ apocryphal

Newton was supposed to have made this remark in 1720 when all of London was clamoring for shares in the South Sea Company and Newton had taken up his post as Master of the Mint.1 He had begun to invest in 1713 when he held £2500 of stock‚ purchased more in the boom and is likely to have made a loss of £20‚000 when the market crashed‚ though his net losses are not known precisely. Predicting the future is a baleful business‚ except in hindsight. By the same principle‚ it is difficult to predict the future of adhesion science. All we can do is look back on the knowledge gained so far‚ then attempt to extrapolate into the future. The first general theory of adhesion was Newton’s universal law of gravity‚ which was extremely successful in describing the motions of planets. No correction was needed to this law until Einstein’s theory of relativity emerged two centuries later. Newton also had noted adhesion due to electrostatic and magnetic forces‚ which are well described in the theory of electromagnetism. The problem‚ as Newton recognized‚ was that none of these well-understood forces could account for the adhesion observed between bodies in close contact. Molecular adhesion was first noted by van der Waals as a force which fouled up the perfect gas laws‚ but it was only when quantum theory was developed in the 1930s that the electronic nature of the molecular forces could be understood. The London force is perhaps the best understood of these molecular forces; an 409

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induced dipole–dipole attraction which exists between all molecules. We also know of other electronic forces between atoms‚ for example the ionic or covalent bonding forces. All these act over a short range and give adhesive bonds which are quite different from the familiar gravitational or electrostatic forces which act at long distances with inverse square law dependence. Remember‚ 99% of van der Waals force acts below 1 nm separation‚ and this force acts from the surface rather than the center of a body‚ making it more difficult than gravity mathematically. The purpose of this chapter is to look back over the last decades to find a guide to future breakthroughs in adhesion. Let us first consider the problems of adhesion which have been solved‚ and then look forward to the definition and explanation of the remaining mysteries of molecular forces.

17.1. ADHESION PROBLEMS SOLVED One of the most significant problems which has been solved during the 20th century is the “adhesion paradox.” In all previous times‚ people believed that bodies did not stick naturally to each other (Fig. 17.1). Adhesion was thought to require some artificial device such as “mortise and tenon keying” or “adhesive layers” to fix things in contact. In fact‚ the major laws we learned at school demanded zero adhesion. Newton’s laws of motion presumed that bodies had no adhesion‚ and Maxwell’s famous kinetic theory of gases assumed zero adhesion between atoms and molecules. Similarly‚ distinguished engineers like Boussinesq and Hertz calculated the contact deformations of materials with the premise of nonadhesion. Most soil mechanics used in earth sciences still assume zero adhesion. All these laws are essentially erroneous and cannot be true exactly‚ although they are often good approximations because the electronic forces between atoms fall off so quickly with separation. Now there is overwhelming evidence to show that all atoms and molecules adhere. This is the first law of adhesion. The fact that we find such adhesion difficult to observe at our macroscopic scale‚ as Newton himself found in the

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seventeenth century‚ can be explained by three factors: first‚ roughness larger than the range of the molecular forces keeps surfaces apart; second‚ the modest value of molecular adhesion energy cannot give high adhesion by itself; third‚ the ubiquitous presence of contamination reduces the molecular attraction between surfaces (Fig. 17.2). Roughness of surfaces is perhaps the largest effect because even the slightest imperfection‚ a few nanometers in size‚ can push the bodies outside the range of atomic forces‚ thus preventing adhesion. Highly polished glass and metal surfaces‚ which appear perfect and mirror-like to the naked eye‚ can hold asperities a hundred times larger than the 1 nm gap which destroys adhesion. Only when it was appreciated that certain cleavage surfaces‚ such as mica‚ or crystal growth planes‚ like alumina plates‚ were atomically smooth was it apparent that adhesion could be obtained in macroscopic experiments‚ as first demonstrated by Obreimoff in 1930.2 The other advance was the comprehension that compliant surfaces like soft rubber could squeeze imperfections out of the way to achieve perfect molecular contact.3 If only Newton had worked on rubber rather than glass‚ he would surely have solved the adhesion paradox in 1666‚ the annus mirabilis when he made many of his great discoveries. Finally‚ the atomic

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force microscope proves that adhesion is universal. There is always an attraction between an AFM tip and a surface in vacuum. The second issue is the small values of intermolecular adhesive energies‚ which range from Such small values mean that reversible adhesion is difficult to observe for particles greater than 1 mm diameter‚ because gravity is then larger. This brings us to the second paradox of adhesion;4 that jumbo jets are held together by such puny energies. A small vandal could easily kick off a jumbo jet wing if the contacts were at equilibrium (Fig. 17.3). Fortunately they are not and the energy to break those joints is 100 000 times larger than expected from the molecular bond energy. Finally‚ there is the wonderful effect of contamination at surfaces. Our everyday experience tells us that surface adhesion can be improved by inserting superglue. This is a false idea. Glue diminishes the work of adhesion‚ as proved by considering the wetting of the surface by the glue‚ as shown schematically in Fig. 17.4. In another example‚ soil mechanics experts tell us that particle adhesion is increased by wetting the powder. The opposite is true: water diminishes the adhesion between dry particles in molecular contact. These observations are encapsulated in the second law of adhesion‚ that “contaminant molecules reduce molecular adhesion.” The reason why glue is successful is that it fills the gaps between the molecules‚ though lowering the adhesive energy. Because of the short range of molecular forces‚ filling gaps can increase the adhesion force by a factor 1000‚ while the overall adhesion energy decreases only slightly by a factor 10.

17.2. INTERESTING MECHANISMS Solving the problem of the adhesion paradox during the last century has brought a new understanding to the field of adhesion. We now appreciate that

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adhesion is the universal action of molecular forces which are powerfully influenced by small gaps and by contaminant molecules. But there is another factor to consider; that is‚ the wide range of mechanisms which can play a part in the adhesion process. For example‚ a joint formed between two surfaces by solidifying a thermoplastic adhesive can readily be broken by heating the joint to re-melt the adhesive. This is the Brownian mechanism of joint failure. Increasing kT‚ the energy of the adhering molecules‚ will eventually break any bond. By the same argument‚ in any system of adherent particles‚ the number of doublets must depend on the adhesion energy‚ on the range of the molecular force‚ and on the concentration of particles. As shown in Fig. 17.5‚ a dilute system must appear less adherent‚ whereas a concentrated system will give more adhesive bonds‚ even though the adhesion energy remains constant. This concept is the third law of adhesion‚ that is “adhesion can vary widely with the mechanism‚ even though the molecular bond energy remains constant.” A similar mechanism of failure is chemical—contaminant molecules can penetrate the adhesive interface by Brownian movement and reduce the energy so much that the joint falls apart. In other words‚ keeping kT constant‚ the molecular energy of adhesion can be reduced chemically to change the adhesion. This is just another way of saying that is really the dominant factor in adhesion. Water is the most common example of this effect. Paper falls apart when wet; detergents allow dirt to drop off clothes in a washing machine. The conclusion from this argument is that adhesive bonds are not static‚ continuous entities: on the contrary‚ they are seething molecular systems which are essentially statistical as shown in Fig. 17.6. Another previously neglected‚ yet dominant‚ mechanism of adhesion is the elastic deformation and cracking mechanism. Although Derjaguin5 had recognized the idea of elastic deformation of adhering particles in the early 1930s‚ following Obreimoff ’s inspired analysis of mica bonding‚ it was not until the late 1960s and early 1970s that the logical progression of these ideas took root after Williams et al6 analyzed the blister test and Outwater and Murphy7 considered

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the pull-out of fibers from composites. Even now‚ engineers use erroneous criteria of adhesive failure when designing adhesive systems. In particular‚ the discredited stress criterion of adhesive joint failure is still being recommended8 when it should be replaced by the energy criterion.9 The two distinct criteria for the failure force in a lap joint are shown in Fig. 17.7. Using the strength criterion‚ that the joint will fail when the average stress reaches a critical value several difficulties emerge. First‚ there is the problem that the experimentally measured strength varies with overlap length z‚ the joint seeming to get weaker as it gets longer. Secondly‚ the joint strength increases with the thickness d of the joined strips. Finally‚ and most critically‚ it is evident that the stress through the joint is not uniform but becomes large at the ends of the adhesive where a crack easily starts. On the other hand‚ the energy criterion recognizes that there should be a mathematically infinite stress at the ends for a perfectly elastic material. The joint

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should therefore fail by a cracking mechanism equally from each end. Then it is clear from the energy balance (Chapter 15) that the overlap is not important for a long joint‚ the work of adhesion is the main resistance to cracking‚ but the elastic modulus and strip thickness must be taken into account.10 On the energy criterion you can make a joint with any strength you like by adjusting the elastic and geometric parameters. The important thing to remember when examining these confusing points of view is that engineers have consistently been diverted since Galileo’s time. Galileo used an energy criterion when he first produced his famous theory of the strength of wooden beams. Of course‚ he also introduced the idea of pressure causing failure‚ as demonstrated by his student Torricelli. Unfortunately‚ the energy and stress criteria of failure cannot be generally consistent. It can be demonstrated quite unequivocally that the energy criterion is true for elastic materials whereas the stress criterion is false (Chapter 7.8).

17.3. DOES ADHESIVE STENGTH EXIST? The above discussion illustrates perfectly the third law of adhesion; that the adhesive behavior of the same bonded materials can vary enormously with the mechanism involved. An important result has been the equilibrium analysis of several different joint geometries according to the energy criterion of failure.9 Experimentally‚ the same materials were chosen for all the joint geometries. poly(methyl methacrylate) as the transparent substrate and solidified gelatin solution as the elastic adherend. This choice was dictated by the need to obtain reversible adhesion; in other words‚ the adhesion on making the joints was identical to that on breaking. The different geometries are shown in Fig. 17.8. From the elastic energy balance theory‚ the several equations for the failure force of each adhesive joint were calculated as shown. These equations were found to describe the experimental results rather well‚ providing the crack speed was kept low and uniform during the tests. The conclusion from these arguments

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was that adhesive strength did not exist. Instead‚ cracking failure was observed in all cases‚ essentially fitting a “Griffith-like” energy balance principle.11 Of course‚ the above experiments were devised to give weak and reversible adhesion in order to verify the theory. In engineering design‚ it is usual to go for more resistance to load‚ especially in the butt joint where the joint progressively requires more force as the adhesive thickness is reduced. For very thin adhesive layers‚ it was clear that the mechanism of failure was changing‚ and that cavitation could arise within the adhesive.12 Breakdown was observed within the adhesive layer‚ and ultimately this limited the load capability of the joint at a thickness near Ultimately‚ it is obvious that the breakage of single bonds will limit the strongest joint. This would give the maximum possible theoretical strength of the joint for a single molecular fracture event (Section 13.5). This is shown as the ultimate strength in the Griffith plot given in Fig. 17.9. The conclusion therefore is that adhesive strength only exists under certain conditions‚ for example when a single bond is broken‚ or where there is general failure at uniform stress throughout the material.

17.4. ADHESION AT THE NANOMETER LEVEL AND MOLECULAR SCALE This discussion about the adhesion of single molecular bonds is rather important. Looking back over the past few hundred years‚ it seems that our progress in adhesion study has been to understand more and more about the molecular meaning of adhesion. This has happened in several ways: first‚ molecular contact has been achieved reliably with elastomers‚ with mica‚ with fine particles‚ and with atomic force probes; second‚ very small forces have been measured by AFM and by other techniques (Section 10.3); finally‚ the ultimate

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adhesion experiment on single molecular bonds has been carried out with polymer molecules (see Section 13.3) and will soon be carried out on smaller molecules and even on atoms in the near future. Let us consider these areas in more detail. The greatest breakthrough in the past century has been the movement to finer and finer probes of adhesion. Of course‚ this process started with Wollaston13 and his early press for compacting platinum powders to ease the processing of intractable hard materials. Bradley and Tomlinson also played their part in demonstrating strong adhesion of microscopic silica bodies. But the emergence of fine latex polymers and ever finer pigments and nanoparticles has allowed us to see the ubiquitous nature of adhesion at scales below There is no doubt that all nanoparticles adhere to a great extent. This adhesion can be so large that strong compacts of high coordination are formed‚ or voluminous gels are created as the particles meet in strong linkages of low coordination as shown in Fig. 17.10. The effect of the particle adhesion on the elasticity and strength of gels has been described (Chapter 11). From such measurements‚ values of the work of adhesion have been calculated.14 Adhesion of fine particles is essential to our largest bulk industry‚ the cement manufacturing sector. It is also vital to the photocopying industry where the balance of electrostatic and molecular adhesion must be closely controlled. However‚ adhesion also causes substantial problems in our newest industry‚ the semiconductor manufacturing sector‚ where it is necessary to clean very fine adhering particles from wafer surfaces. Therefore‚ the mechanisms of particle contamination‚ and the methods for shifting the particles from the surface are vitally important. Figure 17.11 shows how the adhesion of particles needs to be controlled in the three applications. The understanding of adhesion at this nanoparticle level has been much enhanced by the atomic force microscope (Sections 3.8 and 4.8). Originally‚ this instrument was thought of as an extension to the scanning tunneling microscope invented by Binnig and colleagues‚15 in which the very small movements due to

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surface forces could be detected by the tunneling current.16 However‚ it turned out that the surface forces around 1 nN due to molecular bonds could easily be detected and measured by a simple laser beam deflection. This meant that surface force measurements were rather simple. Consequently‚ the study of surfaces using the atomic force microscope has blossomed. The instrument is not limited to conducting systems like the STM and so can be used generally in all applications because of the ubiquity of molecular adhesion forces. Particular advances have been made in measuring the breaking force of individual polymer bonds (Section 13.4) and also in manipulating single atoms on surfaces (Section 13.3). Single bonds have been formed between functionalized tips and receptor coated surfaces (Section 12.5). The several types of experiment are illustrated in Fig. 17.12. In future‚ these measurements will be further extended to study chemistry at the single-bond level‚ with enormous implications for the new nanochemistry field. The importance of such studies in future cannot be overemphasized. Equally important will be the theoretical interpretation of the results. Computer simulation is set to provide much of this theoretical framework.

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17.5. IMPROVED THEORY OF ADHESION BY COMPUTER CALCULATION There is little doubt that the theory of adhesion will improve rapidly as computer calculations become more rapid and sophisticated. These improvements will take place in three areas as shown schematically in Fig. 17.13: first is the enhanced understanding of molecular adhesion forces at the atomic level; second is the modeling of the statistical behavior of Brownian adhesive systems; finally‚ the analysis of adhesion in continuum mechanics terms will increase as specific adhesion computer packages become available. Already‚ chemistry is turning into a computerized subject‚ where mathematical modeling of molecules is just as important as molecular synthesis or chemical analysis. The structure of molecules can be worked out from the interatomic potentials using computer packages which are continuously under development at the large pharmaceutical companies (Section 5.3).17 The configurations of molecules‚ their reactions‚ and products will ultimately be predictable through the computer. At a somewhat larger scale‚ for colloidal particles or polymer molecules undergoing Brownian movement‚ it is becoming possible to predict the structuring and phase behavior as the particles collide and adhere stochastically.18 To carry out these calculations‚ the adhesive interactions between the particles need to be known. Then an ensemble of particles‚ say one million in number‚ can be set moving in the computer to find the equilibrium configuration of the system. Obviously this depends on three main variables (Section 5.9): the concentration of particles‚ their interaction energy‚ and the range of the adhesion. Such calculations will lead to a new understanding of aggregation and structuring

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processes. Understanding and controlling the growth of complex long-range structures through the influence of short-range molecular forces is still the greatest challenge. Finally‚ at the macroscopic engineering level‚ new advances will emerge in the estimation of the mechanisms of sticking and fracture of joints. At a primitive level‚ such calculations are mere stress analysis. But‚ as we have seen in Section 17.2‚ stress is of relatively minor importance in the behavior of adhesive systems. The energy balance and the mechanisms of attachment and detachment are much more significant. The one parameter adhesion model‚ defined by the Work of Adhesion‚ has succeeded in describing the cracking of macroscopic adhesive joints‚ where the range of molecular forces is so low that it can be ignored. However‚ it is evident that‚ as the molecular scale is approached‚ two parameters are needed to describe adhesion‚ ie molecular bond energy and range. Additionally‚ when contaminant molecules produce energy barriers at the interface‚ extra parameters are required. It is expected that rapid strides will be made in computational theory of these mechanisms. It is clear from the above discussion that computer modeling forms the bridge between the molecules and the continuum mechanics of adhesion. So the question that we asked in Section 3.4‚ how to understand the connection between molecular bonding and macroscopic engineering behavior‚ is on the verge of being answered. But much work needs to be done to forge this connection absolutely. Both bigger computer models and the establishment of “molecularcontinuum” models are still required for ultimate success.

17.6. NEW ADHESION APPLICATIONS As more sophisticated adhesion models become available‚ it is obvious that they will greatly influence the applications of adhesion‚ from colloids‚ pastes and gels to aerospace‚ electronics and cells. It seems likely that‚ just as photocopying‚ food‚ and aerospace drove adhesion science forward in the last 50 years‚ electronics and bioadhesion will push back the boundaries in the new millenium. We will move increasingly towards molecular applications. In the new electronic structures under development‚ interfaces between the various active materials will multiply many-fold. Adhesion in these cases does not merely involve mechanical contact‚ but also demands electronic‚ ionic‚ and optical integrity. Thus the study of adhesion will become more sophisticated to encompass these new demands. New structured materials will emerge‚ often with complex self-assembled architectures. At the nanometer level‚ it should be possible for an electronic device to come together by itself‚ driven by smart adhesion forces.

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The final frontier is bioadhesion. Biological structures do self-assemble. What is the nature of the adhesive force driving this process? And why does this go wrong in the case of cancer cells‚ for example? Many models of bioadhesion are simplistic as shown in Section 12.11. The idea of keying at the molecular level was dismissed by Newton three centuries ago. Also‚ the idea that an “adhesion molecule” can bond cells together is facile. Improved experimental methods for defining bioadhesion will be developed‚ some with unprecedented sensitivity. New arguments about ligands and receptors will be produced. There is a new and revolutionary chapter opening up in the explanation of the sticky universe.

17.7. REFERENCES 1. Elliot‚ L.‚ Guardian‚ Feb 28‚ 12 (2000). Christianson‚ G.E.‚ In the Presence of the Creator‚ Free Press‚ New York‚ 1984‚ p 571. “He could not calculate the madness of the people”‚ Joseph Spence Anecdotes‚ observations and characters of books and men ed. Samuel Singer‚ London‚ 1820‚ p 368. 2. Obreimoff‚ J.W.‚ Proc. R. Soc. A 127‚ 290–7 (1930). 3. Johnson‚ K.L.‚ Kendall‚ K. and Roberts‚ A.D.‚ Proc. R. Soc. A 324‚ 301–13 (1971). 4. Kendall‚ K.‚ J. Adhesion 5‚ 77–9 (1973). 5. Derjaguin‚ B.V.‚ Kolloid Z. 69‚ 155 (1934). 6. Williams‚ M.L.‚ J. Appl. Polym. Sci. 13‚ 29–40 (1969). 7. Outwater‚ J.O. and Murphy‚ M.C.‚ Mod. Plastics 47‚ 160 (1970). 8. Adams‚ R.D.‚ Comyn‚ J. and Wake‚ W.C.‚ Structural Adhesive Joints in Engineering‚ Chapman & Hall‚ London‚ 1997‚ p 61. 9. Kendall‚ K.‚ J. Phys. D: Appl. Phys. 4‚ 1186–95 (1971). 10. Kendall‚ K. J. Phys. D: Appl. Phys. 8‚ 512–22 (1975). 11. Griffith‚ A.A.‚ Proc. R. Soc. A 221‚ 163–98 (1920). 12. Patterson‚ M.‚ Lakrout‚ H. and Singer‚ I.L.‚ Proc. Adhesion Soc. 23‚ 71–3 (2000). 13. Wollaston‚ W.H.‚ Phil. Trans. R. Soc. 119‚ 1–8 (1829). 14. Kendall‚ K.‚ Alford‚ N.McN. and Birchall‚ J.D.‚ Proc. R. Soc. A 412‚ 269–83 (1987). 15. Binnig‚ G.‚ Rohrer‚ H.‚ Gerber‚ C. and Weibel‚ E.‚ Phys. Rev. Lett. 50‚ 120–3 (1983). 16. Binnig‚ G‚ Quate‚ C.F. and Gerber‚ C.‚ Phys. Rev. Lett. 56‚ 930 (1986). 17. Leach‚ A.R.‚ Molecular Modeling: Principles and Applications‚ Longmans‚ Harlow‚ 1996. 18. Stainton‚ C.‚ PhD Thesis‚ University of Keele‚ 2000.

INDEX

Actin, 283 Adatoms, 57 Adhesion energy, 115, 134, 163, 331–341, 346, 412 fallacies, 42–44 force, 336, 337 hysteresis, 163–165, 200–203, 331 molecules, 278, 280, 289, 296 number, 293–296 paradox, 6, 410 parameter, 192 of polymers, 122–125 sensor, 215–217 in space, 7, 30, 96, 104, 114 states, 115, 117, 126 stress, 306, 360, 414 in thermal equilibrium, 48–50, 140 in vacuum, 67 Adhesive dislocations, 170 drag, 200–203, 331 energy well, 221 strength, 150 tape, 356 Adhesives, 43, 55, 410 Agglomeration, 103, 158, 197, 198, 207, ch 10, 265 Aggregation, 213, 214, 215, 290 Agriculture, 199 Aircraft, 378 Alkyd, 328 Alumina, 39, 209, 223, 247, 258, 264, 392 Alumina trihydrate, 376

Aluminum, 322, 329 Aminopropyl silane, 404 Amontons, 28, 190 Anodizing, 329 Antibodies, 44 Antigens, 44 Apparent contact, 8 Arch, 10 Argon, 91 Armor, 378 Asbestos, 14 Asteroids, 5 Astronauts, 42 Atomic force microscope, 19, 56–59, 77–79, 118–121, 124, 128, 191, 196, 286, 305, 311, 314, 321, 412, 416, 417 Attractive force, 72

Bacteria, 77, 282, 298 Bacteriophage, 283 Ballbearings, 46, 77, 183 Bearings, 270 Benzene, 116 Bimetallic strip, 379 Binders, 125, 265 Bioadhesion, ch 12, 420 Biofouling, 288 Bitumen, 14, 247 Black spot, 54–56, 63, 182, 183, 188 Blister test, 372, 413 Blood, 15, 19, 214, 217, 285–296 Body centered cubic packing, 97 423

424 Boltzmann, 220, 222, 223 Born repulsion, 221 Boussinesq, 308, 410 Bradley’s rule, 73–77, 104, 145, 149, 183, 184, 188 Bricks, 42 Brick lintel, 9 Brittle failure, 332 Brittle/ductile transition, 236 Brownian motion, 11, 12, 47, 48, 75, 80, 139– 141, 146–149, 160–162, 174, 216–218, 225–230, 248, 266, 276, 289, 291, 293, 299, 413 Buildings, 378 Bullet-proof glass, 172 Butadiene, 15 Butt joint, 341, 415 Butyl aerylate copolymer, 199 239 Caddis fly, 354 Calcite, calcium carbonate, 118–121, 237–238 Calcium sulfate, 233, 237 CAMP, 297 CAMS, 278 Cancer, 19, 285, 297, 299 Capillary force, 198 Capillary gap, 128, 157 Carbon black, 239, 271, 376 Carbon fibers, 400, 404 Carbon tetrachloride, 116 Cars, 50, 168, 183, 378 Casein adhesive, 367 Cathedral, 6 Catalytic debonding, 163 Cave painting, 12 Cells, 46, ch 12 Cell contact, 279–280 Cell sheath, 277, 278, 298 Cellulose nitrate, 165, 378 Cement, 14, 246, 247, 257–266, 417 Ceramic, 224 Ceramic coating, 330 Ceramic capacitor, 389 Charcoal, 51 Chewing gum effect, 166 Cholera toxin, 286 Cholesterol, 279 Clay, 10, 13,46, 116, 118, 214 Cleaning, 310, 315

INDEX Cleavage, 105, 105 Clouds, 214 Clutch, 45, 270 Coalescence of spheres, 198–200, 214, 215 Coagulation, 225 Coated steel, 334 Cobalt, 311 Collagen, 13 Collisions, 11 Colloidal carbon, 376 Colloidal crystals, 232 Colloids, ch 10 Columbus, 25 Comet jet liner, 367 Comminution, 235 Compaction, 34, 252 Composites, 170, 172 Computer model of particle adhesion, 91–101, 271, 221–229, 419 Concentrated suspensions, 224, 246 Contact, 8, 182 Contact angle, 26, 50, 109, 138, 186 Contact angle hysteresis, 200–201 Contaminant, 64, 104, 309, 316, 411 Copper, 311 Copper sulfate, 241 Corbelled arch, 10 Corrosion, 316 Coulomb, x, 31, 32, 193–195, 271 Coulter counter, 290–296 Coupling agent, 349, 403–404 Cracks, 10, 49, 52–55, 142–149, 171, 251, 309 Crack blunting, 202–203 Crack deflection, 348 Crack meeting interface, 380–391 Crack stopping, 164–167, 202 Crack tying stress, 172 Crazing, 171–173 Crosslinking, 350 Crystals, 96–99, 119, 232–235 Cyclohexane, 116, 123 Cytometry, 276, 290–296 Cytoplasm, 279 D mannose, 283 Daedalus, 354 Dampers, 270 Debonding, 104 Decane, 88 Defects, 251, 258, 260–265, 307, 339, 340, 369

INDEX Deformation of spheres, 75, ch 9 Delamination, 384 Dental cement, 10 Dental enamel, 297–298 Derjaguin, 183, 184, 186, 187, 219, 413 Dextran, 312 Diamond films, 330, 346 Diatoms, 17, 287 Diffusion, 36, 119, 159–161, 198, 205, 291 Diffusion coefficient, 217, 280 Dipole, forces, 32, 410 Dislocations, 119, 170–171, 259, 392–394 Dispersion, ch 10 DLVO theory, 113, 118, 219–227, 280 DNA, 284 Doublets, 222, 261, 291–296, 413 Double torsion test, 372 Drug particles, 3 Drying, 252 Dugdale model, 173 Dust, 38, 181, 269 Dwell-time effect, 158–160, 331 Dynamic equilibrium, 291–296 E coli, 282 Einstein, 216 Elastic deflection, 72, 144, 149–150, 182–184, 250, 307 Elastic limit, 203 Elastic linkage, 334–339 Elastic mismatch, 396 Elastomer, 415 Electrical adhesion, 176–177, 317 Electromagnetic force, 32–34, 409 Electron microscopy, 277, 280, 282, 305 Electrons, 31, 176, 218 Electrophotography, 317–319 Electrorheology, 268–269 Electrostatics, 30, 409 Electrostatic precipitator, 176, 269 Energy balance theory, 53, 144, 149–150, 250, 308, 335, 361, 398, 414 Energy barrier, 160–163 Engines, 46 Engineering adhesive joints, 134, ch 15 Engulfment, 199, 215, 290 Epoxy, 138, 378, 383,402 Equilibrium adhesion, 49, 53, 134, 160, 187, 331 Erythrocytes, 285, 293–296 Ethylene propylene rubber, 147

425 Face centered cubic packing, 94–96, 174 Faraday, 216, 229, 239 Ferrofluid, 270 Fiber, 72, 171, 195, 403 Fiberglass, 103, 403 Fiber pull-out, 413 Fibroblasts, 283 Fibronectin, 279, 293 Fillers, 376 Film adhesion, 49 Fimbriae, 283 Finger cracks, 172 Flagellae, 282 Flaws, 251, 258 Flaw size, 399 Flaw statistics, 307, 342 Flocculation, ch 10, 261 Flocs, 227 Food, 224 Fossils, 16 Fractals, 91 Fracture mechanics, 143–146, 149, 263–265, 342, 369 Frenkel, 205–207, 255 Friction, 4, 27, 28, 29, 111, 128, 171, 190, 193–196 Fuel cell, 247 Fused silica, 72 Galileo, ix, 41, 42, 145, 157, 275, 306–307, 360, 415 Gas laws, 11, 90 Geckoes, 76 Gel, 117, 122, 174, 224, ch 11, 417 Gelatin, 186, 216, 341, 343, 415 Gent, 201, 342 Germanium, 309 Glass, 14, 41, 63, 103, 104, 143, 162, 165, 171, 172, 181, 193, 199, 206, 278, 286, 311, 341, 411 Glass fibers, 25 Glass fiber composites, 378 Glass houses, 379 Glass transition temperature, 148, 198 Glial cells, 286 Glue, 122, 151, 412 Glutaraldehyde, 286, 293, 296 Glycocalyx, 277 Glycolipid, 279 Gold, 128, 203, 216, 311, 320

426 Goodyear, 48, 202 Graphite, 57, 390 Gravity, 3, 4, 29, 76, 409 Griffith, 53, 143, 146, 150, 172, 187, 236, 258, 260, 309, 337, 357, 362, 363, 386, 416 Grinding, 235–237 Growing particles, 238–241 Hamaker constant, 70, 78, 113 Hancock, 375 Hardness, 205, 309 Harvey, 275 Healing, 283, 392 Hertz, x, 182–186, 189, 190, 410 Hexagonal packing, 94, 198 Hierarchy of adhesion mechanisms, 135 Hooke, 63, 79, 275, 312 Human hair, 72 Hydroxyapatite, 283, 297–298, 377 Hysteresis, 26, 59, 67, 111, 121, 124, 126, 151, 155–173, 188, 195, 199 Ideal adhesion experiment, 330–341 Immunoglobulin, 278, 313 Implants, 283 Indentation, 341, 345, 346 Indian ink, 13 Ink, 122, 128, 239 Instability, 54, 141 Integrin, 277 Interface shear strength, 399 Interfacial dislocation, 392–394 Interference fringes, 65, 113, 143, 182, 188, 200 Intermolecular forces, ch 5, Israelachvili, 70–72, 105, 112, 122, 141, 158, 219, 228 Jigsaw, 43 JKR theory, 51–54, 144, 149, 157, 163, 184–202, 411 Jumbo jet, 3, 411, 412 Jump distance, 71 Jump into contact, 55, ch 4 Kelvin probe, 177 Key, 43, 45, 410, 420 Kinetic theory, 11, 47–50, 90–101, 139–141, 159–164, 410 Kitchen tops, 376

INDEX Laminates, 170–171, 344, 384–400 Lampblack, 13 Langmuir, 114, 280, 319 Lap joint, 338, 357–366, 414 Lap shear strength, 362–363, 414 Laplace, 26 Laser, 217, 220, 290 Laser ablation, 340 Laser cleaning, 316 Latex, 46, 122, 198, 232, 245, 247, 417 Laws of adhesion, 46–48, chs 3–8 Laws of motion, 3 Lead acid battery, 247 Lennard-Jones, 84–87 Leonardo da Vinci, 15 Light scattering, ch 10 Limestone, 15 Lipid, 280 Liquid bridge, 37, 38, 157 Lizards, 76 London, 32, 85, 176, 203, 410 Lord Rayleigh, 126 Lubricants, 122 Lucretius, 263 Lunar module, 7 Magdeburg hemispheres, 42, 147 Magnetic attraction, 30 Magnetorheology, 268 Make and break, 55, 139–141, 163–165, 331 Mechanisms, ch 7, ch 8, 413 Meniscus, 104–106, 315 Mercaptohexadecanoic acid, 129 Methacrylsilane, 404 Methane, 88 Methanol, 110, 116, 172 Methoxytriethylene glycol, 320 Mica, 39, 63, 65–71, 105, 112, 123, 129, 144, 197, 219, 232, 297, 336, 411, 413, 416 Micelles, 241 Mineral processing, 224 Mold release agent, 106 Molecular attraction, 72 Molecular adhesion energy e, 100, 148, 176, 413 Molecular adhesion force, 34, ch 5 Molecular adhesion sensor, 187, 215 Molecular weight effect, 349 Molecule, 12, 31, 88–89 Mollusc, 17, 394 Monolayer, 106, 110–112, 114, 130, 163, 315

INDEX Moores curve, 305, 306 Mortar, 10, 43 Mortise and tenon, 43, 410 Mosquito aircraft, 367 Mother of pearl, 17, 394 Mud, 13 Multilayers, 114 Multiplets, 262 Nails, 44 Nano-adhesion, ch 13, 321 Nano-cngine, 50 Nano-particle, 290, 417 Nano-scale, 45, 277 Nano-world, 46 Nectins, 278 Neutron star, 33 Newton, x. 3–5, 25, 32, 33, 37, 39, 41, 47, 63, 74, 79, 83, 92, 103, 107, 133, 142, 181, 182, 188, 197, 245, 275, 300, 326, 409, 410, 411, 420 Nickel, 309 Nuclear force, 33 Nucleation, 97, 174, 240–241 Octadecanethiol, 128 OMCTS, 115 Obreimoff, 63, 65–67, 104, 142, 144, 163, 333– 336,411, 413 Octane, 116, 157 One parameter model, 142, 187, 420 Opal, 16, 97, 173, 263 Organotitanate, 349 Organozirconate, 349 Osmium, 35 Ostwald, 98 Oxide particles, 222 Oxygen, 7–8, 104, 121 Paint, 12, 103, 198, 245, 247, 328 Palladium, 35, 238 Papain, 296 Paper, 14 Papyrus, 354 Particulate composites, 376 Pastes, ch 11 Peel and heal, 171, 331 Peel angle, 338 Peeling of lap joint, 356

427

Pellets, 35–36, 270, 417 Perrin, 22, 47, 2 1 6 Phagocyte, 280 Pharmaceuticals, 199 Phase separation, 261, 280 Phospholipid, 279, 281 Photoconductor, 317 Piezoelectric, 39, 57, 68, 77, 224 Piles in clay, 27 Pigments, 12, 44 Pili, 283 Plaster of pans, 233 Plastic contact, 29, 203–206 Plastic flow, 29, 36, 172, 206–208, 309 Platinum, 35, 196, 308 Pollen, 11 Polyacrylic acid, 229, 311, 312, 313 Poly HEMA, 283 Polypropylene carbonate, 116 Polymer, 49, 103, 122–125, 158, 229–232, 311 Polymer brushes, 319 Polymer latex, 15, 94–98, 173, 198–200 Polymethyl methacrylate, 95, 147, 151, 161, 186, 232, 241, 342, 415 Polyoxyethylene, 232 Polystyrene, 123, 172, 204, 232 Polyvinyl acetate, 122 Polyvinyl butyral, 172 Portland cement, 15, 263–265 Polyurethane, 199, 315, 328, 331 Polyvinylacetate, 122, 328 Polyvinylbutyral, 172, 378 Polyvinylidene fluoride, 328 Porous interfaces, 392 Potassium bromide, 36 Powder adhesion, 44 Pozzolan, 14 Powder metal, 205 Prestressing, 364 Pretreatment, 326 Printing, 199 Probe tip, 77–78 Protein, 279, 280, 313 Proteoglycan, 278 Pseudopodia, 282 Pull-off force, 53, 54, 152 Pull-off test, 52–57, 333 Pure shear, 369 Racing car, 76

428 Railway wheel, 27 Rate effect on adhesion, 67, 163–165 Red cell, 21, 286, 291–296 Reinforced concrete, 364, 400 Removal of particles, 193, 313–319 Repulsion, 84, 123 Resin concrete, 376 Restructuring of surface, 126, 231 Reversibility, 67, 356, 357, 415 Rheology, 245, 266, 267 Road repair, 376 Rolling tack, 167–170 Roughness, 26, 135, 151–153 Rubber, 48, 107–109, 168 Rust, 65 Rutile, 78 Salmonella, 283 SAMS, 319 Sand grains, 42, 103, 193–195 Sandstone, 7 Sanitary ware, 376 Scanning tunneling microscope, 56, 57, 305, 310 Schallamach waves, 171 Scratching, 346, 347 Seals, 270 Secondary minimum, 226–228 Sedimentation, 95 Seizure, 115 Self assembly, 420 Semiconductors, 199, 270, 315 Shear stress, 287, 313, 314, 363 Short lap joint, 366 Shrinkage, 10 Sieving, 44 Silane, 162, 286, 349, 404 Silica, 16, 17, 39, 105, 117, 162, 173, 193, 263, 264, 376, 417 Silica gel, 245, 249–257 Silica spheres, 73, 173, 240, 262, 263, 271, 315 Silicon, 58, 124, 204, 346 Silicon carbide, 265, 389 Silicon nitride, 124, 128 Silicon wafer, 57, 308, 313, 315 Silicone rubber, 108, 151, 161, 342, 371 Silk, 18 Silver, 320 Silver bromide, 239, 240 Silver chloride, 241 Sintering, 13, 35, 197–209

INDEX Size effect, 236, 307, 339 Slime mold, 277, 297 Slug, 297 Smoluchowski, 223, 225 Smooth rubber sphere, 52, 139, 144, 181–195 Sodium, 32 Sodium chloride, 314 Sodium polystyrene sulfonate, 124 Sodium polyacrylate, 124 Soil, 10, 181, 271, 410 Sols, 223, ch 11 Soldered joints, 360 Sol/gel transition, 246, 249, 267, 268 Sperm, 19 Spheres, 95–97 Spherocytosis, 287 Spider web, 17 Steel, 64, 327 Steric barrier, 232, 281 Sticky tape, 57, 171, 356 Strength of adhesion, 145, 150 Strep. Mutans, 297 Strep sanguis, 297 Stress analysis, 369 Stringing, 171 Stromatolites, 17 Strong nuclear force, 33 Suction pad, 30 Sulfur, 159 Surface energy, 67, 108, 137, 149, 197, 215 Surface force apparatus, 70 Surface tension, 137, 149, 157, 158 Stearic acid, 70, 71 Stokes, 217, 218, 224 Tabor, 68–70, 106, 141, 142, 151, 157, 168, 184, 192, 263, 348 Tack, 48, 151, 167–169, 200–202 Tantalum nitride, 346 Tarmac, 247 Technology of adhesion, 12 Tension failure, 363 Testing adhesion, 331 Tetraethylsilicate, 240 Theory of adhesion, 9 Thermophoresis, 316 Thermoplastic, 12, 137, 317, 328, 413 Thickener, 122 Thumb test, 201 Tires, 15, 76, 168, 183

INDEX Tire cord adhesion, 400 Titanium dioxide, 34 Titanium nitride, 346 Toluene, 232 Tomlinson, 72, 75 Toner, 199, 317–318 Tooth, 10 Torricelli, 42 Tough laminates, 389 Toughness, 356, ch 15, 383, 387 Triblock copolymer, 320–321 Tribocharge, 318 Tungsten, 310 Turbidity, 216–217 Two parameter model, ch 5, 141, 187 Ultrasonics, 7, 270, 271, 315 Urethane paint, 328 Van der Waals, 12, 70, 85, 86, 91, 95, 115, 121, 123, 127, 176, 203, 214, 220–240, 316, 317, 318, 409 Velcro, 27, 37, 44 Vesicle, 280 Vibration isolator, 270 Virus, 46 Viscoelastic, 280

429 Viscosity, 4,5, 157, 206–208, 280 Vitruvius, 263 Volcanic ash, 16

Washing, 103, 315 Water, 7–8, 26, 103, Wax, 26 Wedging, 163, 333–334 Wetting angle, 108, 109, 138–139, 186, 412 Wheel adhesion, 75, 183 Wilhelmy, 214 Wire, 305–307 Wire bonding, 324 Wollaston, 35, 79, 205 Wood, 27, 391, 397 Work of adhesion, 38, 108, 109, 134, 145, 148– 150, ch 9, 251–252, 255, 271, 285, 315, 316, 331, 332, 337, 357, 362, 411, 415, 417

X rays, 218, 323–324 Xenon, 309 Xerox process, 176, 191, 199, 317–319

Yield stress, 205, 236, 313 Young, 26, 50, 108–109, 186