Friction at the Atomic Level

Friction at the Atomic Level

Atomistic Approaches in Tribology

Motohisa Hirano

Author

Prof. Dr. Motohisa HiranoHosei UniversityFaculty of Science and EngineeringDepartment of Mechanical Engineering3-7-2, Kajino, KoganeiTokyoJapan

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vii

Contents

Preface xiii

1 Classical Theory and Atomistics 11.1 Law of Friction 11.2 The Origin of Friction 41.3 Atomistics in Tribology 6

2 Atomistic Models 92.1 Friction Models 92.2 Physical Essence of Mechanical Adiabaticity in Friction 11

3 Atomistic Locking and Friction 153.1 Theoretical Preliminaries 153.1.1 Model 153.1.2 Expression for Adiabatic Potential 173.2 Topological Description of Friction 193.2.1 Adiabatic Potential 193.2.2 Atomic Configurations of Surfaces 193.2.2.1 Variant P

𝛾(𝜌) Case 21

3.2.2.2 Invariant P𝛾(𝜌) Case 22

3.2.2.3 Restricted Invariant P𝛾(𝜌) Case 22

3.3 A More Realistic Case: A Relaxed Upper Body 223.4 Quasi-static Friction of α-Iron 243.4.1 Case (a) 243.4.2 Case (b) 25

4 Atomistic Origin of Friction 294.1 Friction Model 294.2 Static Friction 314.3 Energy Dissipation in Dynamic Friction 324.4 Criterion for Friction Transition 35

viii Contents

5 Superlubricity 435.1 A State of Vanishing Friction 435.2 How Does Friction Become Zero? 445.3 Nonadiabatic Motion of Atoms 455.4 Importance of High Dimensionality 46

6 Atomistic Simulation of Friction 496.1 Computer Simulation 496.2 Atomic Structure and Electronic States 516.2.1 Properties of Atoms 516.2.2 Electronic States 536.3 Cohesion of Solids 556.3.1 Cohesive Forces Between Molecules 556.3.2 Cohesive Forces in Solids 586.4 Crystal Binding 586.4.1 Ionic Crystals 596.4.2 Covalent Crystals 606.4.3 Metallic Crystals 616.4.4 Molecular Crystals 626.4.5 Hydrogen-Bonded Crystals 646.5 Interatomic Force and Interatomic Potential 666.6 Molecular Dynamics Method 686.6.1 Equations of Motion of Atoms 686.6.2 Numerical Integral 686.7 Simple Atomistic Model 696.7.1 Friction Model 696.7.2 Equation of Motion in Dimensionless Form 706.7.3 Friction Diagram 726.8 Energy Recurrence in Superlubricity 756.8.1 Energy Dissipation 756.8.2 Two-Dimensional Model Simulation 766.9 Realistic Systems 796.9.1 Friction Transition 796.9.2 Many-Body Interatomic Potentials 806.9.3 Stability of Superlubricity 82

7 Experimental Approach for Atomic Level Friction 857.1 Atomic Force Microscopy Techniques 857.2 Verification of Atomistic Theory 877.2.1 Static Friction Forces 877.2.2 Commensurability in Sliding Surfaces 88

8 Summary 998.1 Origin of Friction 998.2 Controlling Friction 100

Contents ix

A Physical Preliminaries 103A.1 Analytical Mechanics 103A.1.1 Coordinates and Transformation of a Coordinate System 103A.1.1.1 Cartesian Coordinate System 104A.1.1.2 Expression of Velocity and Acceleration in Polar Coordinates 104A.1.1.3 Three-Dimensional Polar Coordinate System 108A.1.1.4 Cartesian Curvilinear Coordinates 111A.1.1.5 Generalized Coordinates 113A.1.1.6 Generalized Momentum and Canonical Conjugate Variable 116A.1.1.7 Generalized Force 116A.1.2 Lagrange Equation of Motion and Variational Principle 118A.1.2.1 Lagrange Equation of Motion 118A.1.2.2 Application of Lagrange’s Equation of Motion 120A.1.2.3 Variational Principle and Euler–Lagrange Equation 123A.1.2.4 Principle of Virtual Work 126A.1.3 Hamilton’s Canonical Equation 129A.1.3.1 Hamiltonian 129A.1.3.2 Hamilton’s Canonical Equation 132A.1.3.3 Phase Space and Trajectory of Motion 132A.2 Fundamentals of Statistical Mechanics 134A.2.1 Kinetic Theory of Gases 134A.2.2 Principle of Equal a priori Probability and Ergodic Hypothesis 138A.2.3 Microscopic State 139A.2.4 Number of States and Density of States 142A.2.5 Entropy 144A.2.6 Thermal Equilibrium of a Coupled System 145A.2.7 Constant Temperature System: Canonical Ensemble 148A.2.8 Classical System at a Given Temperature 152A.3 Classical Mechanics with Vector Analysis 154A.3.1 Law of Motion 154A.3.2 Motion of Mass Point Expressed with a Vector 155A.3.3 Moment of Force Acting on Mass Point 157A.3.4 Angular Velocity Vector 157A.3.5 Outer Product and Rotation 158A.4 Vibration and Wave 159A.4.1 What is a Wave? 159A.4.2 Fundamental Relation 161A.4.3 Harmonic Oscillation 162A.4.4 Wave Function 164A.4.5 Wave Equation 167A.4.6 Traveling Wave 169A.4.7 Phase Velocity and Dispersion 170A.4.8 Group Velocity 172A.4.9 Three-Dimensional Wave: Plane Wave 175A.5 Lattice Vibration 179

x Contents

A.5.1 Lattice Vibration and Thermal Properties of Crystals 179A.5.2 Lattice Vibration of a One-Dimensional Crystal 184A.5.2.1 Model of a One-Dimensional Crystal 184A.5.2.2 Continuum Approximation 185A.5.2.3 Natural Vibration and Natural Frequency 187A.5.2.4 Dispersion Relation 189A.5.2.5 First Brillouin Zone 189A.5.3 Acoustical Mode and Optical Mode 191A.5.4 Lattice Vibration in a Three-Dimensional Crystal 196A.5.5 Phonon 197

B Mathematical Supplement 199B.1 Trigonometry 199B.1.1 Definition 199B.1.2 Addition Formula 200B.1.3 Basic Properties 202B.2 Taylor Expansion 204B.3 Complex Exponential Function 206B.4 Vectors and Geometry 208B.4.1 Equations of Line and Plane 208B.4.1.1 Equations of Line 208B.4.1.2 Equation of a Plane 209B.4.1.3 Equation of a Sphere and a Spherical Tangent Plane 214B.4.1.4 Application to Geometry 215B.5 Linear Algebra 216B.5.1 Determinant and Inverse Matrix 216B.5.1.1 Permutation 216B.5.1.2 Definition of a Determinant 217B.5.1.3 Characteristics of a Determinant 217B.5.1.4 Inverse Matrix 218B.5.1.5 Application of a Determinant 219B.5.2 Linear Equations: Cramer’s Formula 219B.5.3 Eigenvalue and Eigenvector 221B.5.3.1 Eigenvalue and Eigenvector of a Square Matrix 221B.5.3.2 Diagonalization of a Matrix 223B.5.3.3 Normal Form of a Quadratic Form Polynomial 225

C Data Analysis 227C.1 Fundamentals of Description of Physical Data 227C.1.1 Classification of Deterministic Data 228C.1.1.1 Sinusoidal Periodic Data 228C.1.1.2 Compound Periodic Data 229C.1.1.3 Almost Periodic Data 232C.1.2 Classification of Random Data 233C.1.2.1 Stationary Irregular Process 233C.1.2.2 Ergodic Irregular Process 234

Contents xi

C.1.3 Fundamental Properties of Random Data 235C.1.3.1 Squared Average: Average and Variance 235C.1.3.2 Probability Density Function 235C.1.3.3 Autocorrelation Function 237C.1.3.4 Power Spectral Density Function 237C.2 Signal Processing 239C.2.1 Analog Signal and Digital Signal 239C.2.2 Fourier Analysis 240C.2.2.1 Fourier Series 240C.2.2.2 Fourier Transform 242C.2.2.3 Discrete Fourier Transform 243C.2.3 Applications of Fourier Transform 246C.2.3.1 Impulse Response 246C.2.3.2 Analysis of a Linear System 250C.2.3.3 Equation of Motion 252

D Crystal Structure 255D.1 Periodicity of Crystals 255D.2 Crystal Structure 256D.2.1 Simple Cubic Structure 256D.2.2 Body-Centered Cubic Structure 256D.2.3 Face-Centered Cubic Structure 257D.2.4 Hexagonal Closed-Packed Structure 258D.2.5 Sodium Chloride Structure and Cesium Chloride Structure 259D.2.6 Diamond Structure 260D.3 X-ray Diffraction 261D.3.1 Diffraction Condition 261D.3.2 Reciprocal Vector 263D.3.3 Bragg’s Condition 264D.4 Various Crystal Data 264

E The SI (mks) Unit System 267E.1 Three Basic Units 267E.1.1 Unit of Length: Meter 267E.1.2 Unit of Time: Second 268E.1.3 Unit of Mass: Kilogram 268E.1.3.1 Atomic Mass Unit 268E.2 The SI (mks) Unit System 269E.3 The cgs System 273

F Practice for Verlet Algorithm 275

G Program Example of Molecular Dynamics for AtomisticModel 279

G.1 Annealing Program 279G.2 Sliding Program 281

xii Contents

H Table of Values 285

I Table of Relative Atomic Weights 287

References 289

Afterword 295

About the Author 297

Index 299

xiii

Preface

How much do we know about friction? We actually know how to utilize frictionsurprisingly well. Although we are not consciously aware of it in our daily lives,we are very familiar with the ways to deal with it. Whenever we turn the pagesof a book, or slide a heavy corrugated carton across the floor, we are employingnaturally learned tricks to cope with friction. Even the people in ancient Egyptin 2000 B.C. knew how to use rollers and oil to transport large rocks when con-structing pyramids. We also know that friction comes in various degrees; we arevery careful when climbing up or down wet steps while going out in the rain;and we learn the hard way when we slip and fall in leather shoes on a suddensnowy day. Friction is extremely sensitive to small changes: One drop of lubricantcan dramatically improve the performance of a machine. In professional sports,controlling friction can make the crucial difference between winning and losing.Athletes, who have put their heart and soul into winning, must exercise extremecare in waxing their ski boards, putting for win in golf, and judging the crucialmoment to exchange suitable tires during Formula One automobile races.

Friction has always been thought to exist constantly and eternally, but can fric-tion disappear? Recent studies are beginning to explore a world outside our com-mon sense. A theory in which superlubricity with zero friction appears duringcertain types of contact between surfaces has been proposed, bringing a new vistain research on friction. Experiments are being conducted worldwide, and interna-tional workshops on superlubricity are being held in various venues. Experimentsto find how losses from friction can be minimized are now being conducted fromthe viewpoints of atomistic theory.

The author began his research shortly after his graduation when he joined ateam that was developing an artificial satellite. Outer space is an extremely severeenvironment for machines. To complete the mission for the artificial satellite, itwas essential that friction be minimized. At that time in 1985, the approach ofcomet Halley after around 76 years had been a popular topic in the mass media(Next approaching date will be July 29 in 2061). After the end of the astronomi-cal show, a simple and naive question came to mind: “Why does friction occur?”Research into this question has a long history, dating back at least to experimentsby Leonardo da Vinci in the fifteenth century Italian Renaissance. The Frenchphysicist, Coulomb, and the German, Hertz are best known for their work inelectromagnetics and for SI units (LeSystè me International d ’Unitès), but theyand other famous physicists also studied friction. However, they found no clear

xiv Preface

answer to its fundamental cause. In olden times, people thought that frictionoccurred as the result of rough asperities on the surfaces mechanically lockingwith each other, but this was when the existence of atoms was not yet known. Atpresent, it is thought that friction is caused by the interactions between atomsthat become prominent when two smooth surfaces are brought together. Largerfriction is observed when the surfaces are smoother, which may seem opposite toour normal perception. The source of this misunderstanding can be found in ele-mentary textbooks on Newtonian mechanics. In problems in high-school-levelmechanics, a surface without friction is described as a smooth surface and a sur-face with friction, a rough surface. On the other hand, smooth surfaces can causeserious problems in industrial products because they have large friction. Underthe vacuum environment of outer space, smooth surfaces can become adheredspontaneously when they come together. It also seems that the question of thestatic friction coefficient being larger than the dynamic friction coefficient hasnot been solved. We must search on the atomic level for a clear interpretation ofthe friction phenomenon.

This book has been written on the basis of atomistics, which proves that allsubstances are composed of atoms. Atomistics was established after a long his-tory of controversy surrounding the atomic hypothesis, that is, the question ofwhether the structure of substances is continuous or discontinuous; this contro-versy began in the Greek era, around 500 B.C. The purpose of this book is toprovide the necessary knowledge for young researchers to understand the the-ory of friction on the atomic level, that is, the atomistics of friction, and to fur-ther advance the theory of friction. Just as in most fields of study, an enormousamount of effort is required in order to set up a new research theme in any newfield of science for which the intrinsic understanding is still developing, and toachieve a deep understanding of the problem. The resolve to learn and studyfrom an interdisciplinary viewpoint including both basic sciences such as physics,chemistry and biophysics, and applied sciences such as mechanics, electronics,and instrumentation is essential. I hope the young generations will challengeunknown fields. I realize this anew. Basics and motivation are most importantin anything. They are essential for achieving one’s goals. Whether the field isresearch or art, attaining the basic skills through continued basic training andhaving a strong motivation become the focus of one’s activities. Athletes focus onthe importance of having motivation more and more emphatically in interviewsafter the games. I believe that motivation is important, and that repetition ofmotivation, execution, and earning a sense of achievement will lead to progress.

1

1

Classical Theory and Atomistics

Many research workers have pursued the friction law. Behind the fruitfulachievements, we found enormous amounts of efforts by workers in every kindof research field. Friction research has crossed more than 500 years from itsbeginning to establish the law of friction, and the long story of the scientifichistory of friction research is introduced here.

1.1 Law of Friction

Coulomb’s friction law1 was established at the end of the eighteenth century [1].Before that, from the end of the seventeenth century to the middle of the eigh-teenth century, the basis or groundwork for research had already been done byGuillaume Amontons2 [2]. The very first results in the science of friction werefound in the notes and experimental sketches of Leonardo da Vinci.3 In his exper-imental notes in 1508 [3], da Vinci evaluated the effects of surface roughness onthe friction force for stone and wood, and, for the first time, presented the conceptof a coefficient of friction.

Coulomb’s friction law is simple and sensible, and we can readily obtain itthrough modern experimentation. This law is easily verified with current exper-imental techniques, but during the Renaissance era in Italy, it was not easy tocarry out experiments with sufficient accuracy to clearly demonstrate the uni-versality of the friction law. For that reason, 300 years of history passed after thebeginning of the Italian Renaissance in the fifteenth century before the frictionlaw was established as Coulomb’s law.

The progress of industrialization in England between 1750 and 1850, whichwas later called the Industrial Revolution, brought about a major change in theproduction activities of human beings in Western society and later on a globalscale. Research and development of machines necessary for various manufac-turing industries was promoted. Improvement in the performance of lubricationtechnology was required together with machine design technology and machineprocessing technology.

1 Charles Augustin de Coulomb, 1736–1806, France.2 Guillaume Amontons, 1633–1705, France.3 Leonardo da Vinci, 1452–1519, Italy.

Friction at the Atomic Level: Atomistic Approaches in Tribology, First Edition. Motohisa Hirano.© 2018 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2018 by Wiley-VCH Verlag GmbH & Co. KGaA.

2 1 Classical Theory and Atomistics

The laws of friction can be described as the following empirical laws.1. The friction force is proportional to the force acting in the direction perpen-

dicular to the surface of friction regardless of the apparent area of contact.2. The dynamic friction force is independent of the speed of sliding motion.3. The static friction force is greater than the dynamic friction force.

We can see friction at work in the various mechanical phenomena that sur-round us, and Coulomb’s law can explain most of the nature of the dry frictionof solid objects. For mechanical technology that supports industry, it goes with-out saying that friction is an important problem to be overcome. In the studyof mechanical engineering, mechanical design that takes friction and contactphenomena into account ensures the efficiency of machinery. That fact madea detailed understanding of the nature of friction essential and motivated theresearch for the laws of friction.

Leonardo da Vinci conceived of friction experiments out of his own interest inscience and interest in the shipbuilding technology of his day. His experimentalrecords pointed to the material of the objects and surface roughness as factorsthat affect friction. Those experimental results founded the conjecture that fric-tion is caused by mechanical locking of roughness on the surfaces of the objects.da Vinci also discovered that the friction force of dry solids is proportional to theweight of the object, which is perpendicular to friction force, and is independentof the area of contact far before the establishment of Coulomb’s law. That pro-portionality of friction force and weight is linked to coming up with the conceptof a coefficient of friction [4]. da Vinci also considered the difference betweensliding friction and rolling friction. He thus revealed facts and laws that wereentirely unknown before his research. After his work, the research on the originof the appearance of friction had to wait for the appearance of an understandingbased on atomistic theory and nanotechnology [5] for experimenting at theatomic level. Thus, for the next 200 years, the study of friction did not take thecenter stage in scientific research. The history of tribology and its related topicsare shown in Figure 1.1.

The friction laws were established in the seventeenth and eighteenth centuriesin France. At that time, shipbuilding, flower milling, and other industries thrived,and advances in mechanical design made the study of friction and mechanicalcomponents such as gears and bearings essential. On the foundation of advancedexperimental techniques, the study of friction moved forward from the workof Amontons, Coulomb, and others, resulting in a deeper understanding of thenature of friction and the laws that describe it.

Amontons explained the lawful behavior of friction and the friction lawssuggested by da Vinci through meticulous experimentation in 1699, proceedingwith research to clarify why the friction laws hold by determining the causes [2].Among the issues that Amontons tackled was the difficult problem of clarifyingwhether friction force is proportional to contact area. The common sense of thetime was that friction force is proportional to the area of contact. In fact, therewere experimental results that the friction force is proportional to the contactsurface area when the surface is coated with a film of oil or other lubricant.

1.1 Law of Friction 3

AD

1400

1500

1600

1700

1800

1900

2000

Leonardo da Vinci, 1442 – 1519 Science of friction

Boyle’s law

Experiment of friction

Intermolecular interaction

Dynamic friction theory

Establishment of friction law

Boyle–Charles’s law

Atomistics

Avogadro’s law

Brownian motion

Contact mechanics

Molecular theory

Quantum theory

Experiment of smooth surface

Mechanical adiabaticity

Matter wave

Leonhard Euler, 1707 – 1783

Robert Boyle, 1627 – 1692

Guillaume Amontons, 1633 – 1705

John Theophilus Desaguliers, 1683 – 1744

John Dalton, 1766 – 1844

Robert Brown, 1773 – 1853

Heinrich Rudolf Hertz, 1857 – 1894

J. A. Ewing, 1855 – 1935

Max Planck, 1858 – 1947

W. B. Hardy, 1864 – 1934

G. A. Tomlinson, 1855 – 1935

Louis de Broglie, 1892 – 1987

Amedeo Carlo Avogadro, 1776 – 1856

Charles Augustin de Coulomb, 1736 – 1806

Jacques Alexandere César Charles, 1746 – 1823

Figure 1.1 History of tribology.

Philippe de la Hire,4 who lived in the same generation as Amontons, approachedthat problem with precise experimentation and showed that the friction forceis proportional only to weight and is unrelated to the contact surface area in1706 [6].

As the mechanics of Isaac Newton5 was being systematized in the seventeenthand eighteenth centuries [7], there were attempts to incorporate friction forceinto the dynamics. At that time, friction force was a new force that was not dealtwith in dynamics. Antonie Parent6 solved the problem of an object taking fric-tion force into account as a static equilibrium problem and published a paper in1704 describing the concepts of the friction angle and friction cone [4]. UsingNewton’s mechanics as the foundation, Leonhard Euler7 solved the problem ofthe sliding motion of an object with friction and provided the first theoreticalbasis in dynamics for the static friction coefficient being larger than the dynamicfriction coefficient. The fact that the friction during sliding is often smaller thanstatic friction could be explained by assuming that the asperities on one surfacecould jump part of the way over the gap between asperities on the other [8]. Eulersolved the problem of belts and ropes wrapped around a cylinder as a dynamics

4 Philippe de la Hire, 1640–1718, France.5 Isaac Newton, 1642–1727, United Kingdom.6 Antonie Parent, 1666–1716, France.7 Leonhard Euler, 1707–1783, Switzerland.

4 1 Classical Theory and Atomistics

problem, showing that very large force is necessary for slippage of wrapped beltsor ropes [4].

Charles Augustin de Coulomb was born in Angouleme, France in 1736. Hemade contributions of particular note in the fields of electromagnetism andmechanics [1]. In electromagnetics, he is well known for deriving the law ofstatic electrical force. In the fields of physics and mechanical engineering, too, heis known for his great achievement in establishing the Coulomb’s law of friction.The eighteenth century in France was an era in which culture, economics, andindustry reached full maturity. There were strong gains in machine performanceand durability, and overcoming friction was a major obstacle for those achieve-ments. Before Coulomb, there were limits to the conditions that could be set inlaboratory experiments, but advancement in the rapidly developing mechanicaltechnology made it possible to obtain highly reliable practical data from actualmachines. The French Academy of Sciences offered an award for excellent, highlypractical research on friction. To meet the expectations, Coulomb submittedexcellent research results for various types of friction, including flat surfacefriction, rope friction, pivot bearing friction, and rolling friction. Coulombaccurately solved the problem of flat surface friction and compiled dry frictionexperiments and theory to demonstrate the principles behind the friction law.

1.2 The Origin of Friction

The Japanese scientist Norimune Sota8 wrote an interesting article on the scien-tific history of friction research [4]. The science of friction started in Italy dur-ing the Renaissance period in the fifteenth century. Leonardo da Vinci carefullyobserved and experimented on stones and wood found in daily life and intro-duced the concept of the friction coefficient. More than 200 years passed withoutany progress in friction research, until much discussion of the laws regarding fric-tion and the origin of friction started to happen in the seventeenth to eighteenthcenturies. The results of research were applied to engineering in the form of lubri-cation technology during the Industrial Revolution in the eighteenth century, andresearch by Coulomb and others were summarized as laws of friction.

The principles of how friction happens at contacting surfaces were discussedfrom the end of the seventeenth century to around the middle of the eighteenthcentury as mentioned, and Coulomb completed his surface-roughness model.Although surface roughness still sometimes could be an explanation of frictionalbehavior, the surface-roughness model basically fails to explain energy dissipationbecause of the gravitational force being the conservative force, as pointed out byJohn Leslie9 in 1804 [9].

In contrast, John Theophilus Desaguliers10 was aware of the importance ofintermolecular force [10]. His idea, which is the root of the molecular theory,is the complete opposite of the popular roughness theory, around the middle

8 Norimune Sota, 1911–1995, Japan.9 John Leslie, 1766–1832, United Kingdom.10 John Theophilus Desaguliers, 1683–1744, United Kingdom.

1.2 The Origin of Friction 5

of the eighteenth century. After Desaguliers, during the 100 years until thenineteenth century, only one British physicist Samuel Vince11 committed toDesaguliers’ idea. The molecular theory considers the atomistic origin of frictionto be the interaction of molecular forces at the surfaces where friction appears,as pointed out by James Alfred Ewing12 in 1877 [11]. Accordingly, this theoryclaims that a smoother surface means that the friction surfaces come together,increasing the interference between surface forces. Desaguliers extracted a fewmillimeter-sized pieces from a lead sphere, and found in 1725 that stronglypressing such pieces against each other resulted in strong bonding between thepieces [10]. Further observation of the remains after separation showed thatonly a fraction of the pressed surface had actually been in contact. This findingin 1725 gave rise to the prediction that “friction ultimately increases if surfacesare fully polished to very flat.” This prediction was proved by William BateHardy13 in 1919 with improvements in surface processing technologies [12]. Heis also well known as the first person to use the term boundary lubrication. Hecarried out experiments on the friction of glass surfaces and showed that glasssurfaces that are polished very carefully such as those in lenses have greaterfriction than glass with rough surfaces. He also found that tracks of wear causedby friction are initially about 1 μm wide, and as friction gradually increases wear,the width increases to about 50 μm. This experiment refuted the roughnesstheory and proved that friction is not only a problem of energy loss from theinteraction of molecular forces but also is a phenomenon in materials sciencethat accompanies fracture of the surface. The experiment done by Ragnar Holm14

in 1936 demonstrated that the friction between clean surfaces is high underhigh vacuum and that minute amounts of gas molecule adsorption significantlydecrease friction [13]. The modern ultrahigh-vacuum experiment of clean metalsurfaces by Buckley showed a correlation with electronic properties such as thenumber of d-electrons [14]. Strang’s experiment [15] done in 1949 proved thatmeasured up-and-down motion of a solid in sliding was very small, and thecorresponding work for the up-and-down motion was only 3–7% of the totalwork consumed by friction. These results showed the work for up-and-downmotion stemming from the surface roughness was negligible. Thus, moleculartheory gained evidence and became the foundation of the atomistics of friction.

On the other hand, regarding the friction model of actual surfaces, the con-tact model was refined through the concept of real area of contact proposedby Holm and Mises’s material yield theory in plastic deformation [16]. Relationsbetween friction forces and materials properties such as plasticity were investi-gated in detail in terms of adhesion theory based on shear models at the trulycontacting and adhesive element [17–20]. A pair of contact asperities can beapproximated as two spheres making elastic contact, that is, Hertzian contactby Hertz by15 [21]. The findings resulted in today’s lubrication technologies forhead-disk interfaces in contact start-stop-type magnetic information storage disk

11 Samuel Vince, 1749–1821, United Kingdom.12 James Alfred Ewing, 1855–1935, United Kingdom.13 William Bate Hardy, 1864–1934, United Kingdom.14 Ragnar Holm, 1879–1970, Germany.15 Heinrich Rudolf Hertz, 1857–1894, Germany.

6 1 Classical Theory and Atomistics

devices, and lubrication technologies on the small scale [22] will become evenmore important in miniature precision devices in the future.

1.3 Atomistics in Tribology

The work done by friction has a very different nature from the work done by grav-ity [4]. Work by gravity happens when objects are moved against gravity, which isalways acting on objects. In contrast, friction is the force required to slide objectsperpendicular to the direction of gravity. Once sliding motion starts, frictionappears as resistance against the sliding motion and results in work by friction.Therefore, friction has the interesting property that it appears when objects startsliding and disappears when objects stop. Even in interatomic forces, no work byfriction is generated as long as the combined interatomic force is perpendicular tothe sliding direction. Leslie did not agree with Desaguliers’ atomistic idea. Ewingstated in 1877, as mentioned, that friction force stems from molecular interac-tion at contacting surfaces. The British physicist Tomlinson [23] was the first toexplain the energy dissipation stemming from molecular interaction at the startof the twentieth century, in 1929. He should have been inspired by the modernatomistics established by the British chemist John Dalton.16

Modern atomistics was established after physics reached the level of atomsin the nineteenth century. Physics started to consider atoms around the mid-nineteenth century, although the original concept of atomistics itself, which isthat matter consists of atoms, is thought to have emerged in ancient Greece asparticle philosophy. The British physicist–chemist Robert Boyle17 tried to useparticle philosophy as the foundation of chemistry, and his attempt to buildchemistry upon particle philosophy materialized in the early nineteenth centuryas Dalton’s atomistics. Dalton postulated that objects with sizes that are toucheddaily, regardless of whether the objects are in gas, liquid, or solid state, consistof a vast number of very minute particles or atoms bound together by force.He thought that there is attraction and repulsion between atoms and that thebalance between these opposing forces results in the three states of gas, liquid,and solid. The attraction and repulsion between atoms was later explained onthe basis of the concept of electron energy levels and electron states in quantummechanics. Dalton’s atomistics was improved through corrections by AmedeoCarlo Avogadro18 and others. Although there were opponents to atomistics, itexplained many experimental findings about the materials properties of gases,Boyle’s law, diffusion and viscosity of gases, laws on heat conductivity, and thelaw of increasing entropy. Atomistics later provided an important foundationfor problems regarding the nature of heat. Physicists such as Hermann vonHelmholtz19 came to believe that atoms govern thermal motion. Tomlinson’spaper states early on that “friction is generally recognized to happen becauseof interactions between molecules that are very close to each other” [23]. Hetheoretically investigated the forces that appear in the relative motion of atoms

16 John Dalton, 1766–1844.17 Robert Boyle, 1627–1691, United Kingdom.18 Amedeo Carlo Avogadro, 1776–1856, Italy.19 Hermann von Helmholtz, 1821–1894, Germany.

1.3 Atomistics in Tribology 7

Figure 1.2 Tomlinson’s single-pairatom model for explaining energydissipation in friction [23]. Tomlinson1929. Reproduced with permission ofTaylor and Francis.

eF

EE′

F′

C B D

in the field of interatomic interactions at the contact surface, and succeededin rationally explaining the problem of how friction arises from interatomicinteractions at the contact surface, or how mechanical energy dissipates into heatenergy due to friction, by introducing the concept of mechanical adiabaticity,thereby opening the door to the atomic theory of friction. Figure 1.2 showsthe original model in the paper. It has been considered that two solid bodiesin contact and with relative sliding motion, and, for simplicity, a single atomD forming part of a body which is moving in the direction of EF past anotherbody, of which B and C form two atoms in the state of equilibrium characteristicof a solid. Let us suppose that the atom D in moving past B along the line EFapproaches B to within a distance of the attraction field but outside the range ofthe repulsion. The passage of D causes a slight disturbance in the position of B,which moves away from C, supposing C to be fixed. The atom D in proceedingfurther along EF then withdraws from B, which returns to its original position. Itis conceivable that B arrives back to its original position with some appreciablevelocity and therefore with some added energy, the aggregate of which mightcorrespond to the loss of energy in friction. How does a loss of energy occur infriction? The energy dissipation mechanisms are described in Chapters 2 and 4.

However, very little research on the atomistics of friction followed becauseof the difficulty in handling the complexity of actual non-well-defined surfacesbased on the theory. Friction research has been innovated with recent advancesin nanotechnology [5]. Friction research in ideal systems where many factorsof friction are identified has been difficult for experimental technology rea-sons; however, recent measurement technologies, including scanning probemicroscopy (SPM) [24–27] and technologies to control very clean well-definedsurfaces under ultrahigh vacuum, have enabled direct comparison betweentheoretical models and experiments [28, 29]. Theory can investigate in detailthe fundamental properties of interatomic interactions and the mechanismfor the appearance of friction generation using computational experiments onatomistic models [30]. Therefore, ideal friction experiments, where the originof friction are accurately identified, can be combined with atomic-scale frictionsimulations, and thus the adequacy of atomic-scale friction theory can now bedirectly verified. For example, atomic force microscopy (AFM) can accuratelymeasure the friction between the surface of a very sharp tip attached to the endof a cantilever and the surface of a sample using the optical lever method [31],which is a displacement measurement method. The latest experimental deviceshave enabled the first observations of friction without wear or damage [26].The adhesion theory cannot be used to investigate such friction without wear,and therefore it was necessary to clarify the origins of friction in terms ofatomistics [32].

9

2

Atomistic Models

Several models have been proposed to explain the origin of friction force. Somerelate to the mechanical locking of surface asperities and others to the atomisticorigin, that is, the molecular interactions between the constituent atoms of solids.A solution to the problems in understanding friction mechanisms in real sys-tems is achieved from the viewpoint of phenomenology by a priori assuming thatfrictional force exists. The experimental data usually measured in real systemscontains many unknown factors: surface roughness and poisoning by variouscontaminants. It is difficult, therefore, to study the origin of friction force fromthe experimental data available at present. Recent experimental studies, on theother hand, try to exclude many of the unknown factors by preparing well-definedsurfaces. The purity and completion of such surfaces can be detected by currentsurface analysis techniques such as scanning tunneling microscopy (STM). Thischapter considers the atomistic origin of friction force on clean surfaces by dis-cussing the atomistic models.

2.1 Friction Models

Uncovering the principles of energy dissipation in friction has been recognizedas an important problem for a long time. For friction phenomena caused by adhe-sion at the true contact area, which has been observed the most, friction energyhas been considered to dissipate by plastic deformation at the true contact area[13, 18, 19]. This is the basic concept of adhesion theory, which postulates thatbumps on the surface dig into the other surfaces and cause wear debris becauseof plastic deformation and the subsequent fracture, and the accumulation of suchbehavior results in energy dissipation. The principle is the same as the idea thatthe energy necessary for the deformation of bulk materials at the macroscopicscale is due to the dissipation by motion of dislocations and propagation of cracksin the material. However, friction experiments at the atomic scale mentioned inthe previous section revealed new friction phenomena that do not accompanyplastic deformation or wear, that is, wear-free friction [26], and thus the problemof energy dissipation in friction regained attention in relation to atomistics.

McClelland [32] built an atomistic model in which infinite planes slide againsteach other to investigate the problem of energy dissipation in a wear-free

Friction at the Atomic Level: Atomistic Approaches in Tribology, First Edition. Motohisa Hirano.© 2018 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2018 by Wiley-VCH Verlag GmbH & Co. KGaA.

10 2 Atomistic Models

A

A

B

B

(a)

(b)

(c)

z

x

A

B

Figure 2.1 Atomistic friction models. (a) Solid Asliding across solid B. (b) Independent oscillatormodel (Tomlinson’s model). (c) Frenkel–Kontorovamodel. Black part shows rigid body sliding towardthe right [32]. McClelland 1989. Reproduced withpermission of Springer.

friction model (Figure 2.1a). Atoms of an upper body do not interact with eachother in the independent oscillator model in Figure 2.1b, and so the model isfundamentally the same as Tomlinson’s model (Figure 1.2). Figure 2.1c is theFrenkel–Kontorova model, [33] described later, which will be shown in Figure6.11. This model assumes that strong forces such as metallic or covalent bondsact on atoms in solids, and relatively weak forces such as van der Waals forcesor hydrogen bonds act between atoms at the surfaces above and below. Theproperties of interatomic forces are described in Section 6.4. The idea behindthis model is that in the charge density wave (CDW) model, which is a physicalsystem that describes sliding motion similar to friction. It is well known thatCDWs [34] receiving from interactions between ionic crystals during slidingresult in unstable sliding when the interactions become even slightly strong,as mentioned later. In friction models with such properties, when the surfacesin contact are incommensurate, or when the ratio of periodicity along slidingsurfaces above and below is an irrational number, the two surfaces are found tobe able to slide without energy dissipation. Such sliding phenomenon withoutenergy dissipation may be unfamiliar in the field of tribology, but is well known toappear ubiquitously in some physical systems with two interacting periodicities[35]. Examples of such physical systems include CDW, ionic conductivity,epitaxial crystal growth, and adhesive atom layers. The Frenkel–Kontorovamodel is a theoretical model frequently used to describe such physical systems.Sokoloff showed that the Frenkel–Kontorova model for CDW can reproducephenomena such as stick-slip in friction, thus highlighting its usefulness as amodel for friction of solids [36].

The idea of commensurability in solid surfaces in contact is leading to newresearch fields in up-to-date theoretical and experimental research in nano-tribology, that is, atomic-level friction. A sliding friction system where aone-dimensional atomic chain interacts with a periodic potential was inves-tigated as a model of friction between ideal crystal surfaces such as theFrenkel–Kontorova model, including the kinetic energy term [37, 38]. In suchmodels of ideal crystals, the energy gain and loss of interatomic interactionenergies at the sliding surface cancel each other out and the total energy of the

2.2 Physical Essence of Mechanical Adiabaticity in Friction 11

sliding surface becomes invariant with sliding distance as long as the atomicstructure of the incommensurate surface in contact is the same after atomicrelaxation; therefore, the friction of an infinite system becomes zero at the limitof zero sliding speed [37, 38]. In contrast, if the interaction between surfacesbecomes larger than the interactions inside solids and exceeds a threshold,a structural phase transition, which is called Aubry transition [39] happenswhere locally commensurate structures appear at the incommensurate surfacein contact. In this case, atoms are locally pinned, and even when the solids areadiabatically and slowly slid, the pinned atoms rapidly break bonds becauseof sliding, causing nonadiabatic or noncontinuous motion, resulting in thedissipation of accumulated elastic energy. This is the principle of friction gen-eration postulated by Tomlinson [23]. But how do structural phase transitionsat incommensurate contact planes behave in various models? The occurrence ofphase transitions is determined by the competition of interatomic interactionsinside solids and between surfaces. Aubry transitions are likely to appear in theone-dimensional Frenkel–Kontorova model; hence, a state of vanishing friction,which is called superlubricity, is thought to occur only when intersurfaceinteractions are weak [32, 36]. In contrast, we will see in Chapters 5 and 6 thatthe high degree of freedom of atomic movement in models of high dimensionswas pointed out to be essentially important in the occurrence of superlubricity,and superlubricity without structural phase transitions was found to occur inrealistic three-dimensional systems with strong interactions such as metallicbonding [40].

Thus, the concept of superlubricity, or the phenomenon of zero friction,emerged from atomistics-based research on atomic scale friction [37, 38, 40, 41].

2.2 Physical Essence of Mechanical Adiabaticityin Friction

Tomlinson proposed an atomistic picture for the origin of the frictional forces.Let us describe an essence of his idea. Suppose a friction system consisting offour atoms numbered by 1, and 1′, 2′, and 3′ as seen in Figure 2.2. All atoms areassumed to interact with each other. The atom 1 forms a part of the upper body,which interacts with the other atoms of the upper body, and the atoms 1′, 2′, and3′ form the lower body.

We shall concentrate on the behavior of the atom 1 when the upper body slowlyslides against the lower. When the atom 1 is on the atom 2′, the atom 1 feels theattraction from the atom 2′, as seen in Figure 2.2a. During sliding, the atom 1moves toward the right direction. When the sliding displacement is small, thisis a process of storing the elastic energy, as seen in Figure 2.2b. When the atom1 goes beyond the certain distance, the attraction from the atom 3′ overwhelmsthat from the atom 2′. The atom 1 prefers the position on the atom 3′. It hasbeen, then, assumed that the atom 1 nonadiabatically or abruptly changes itsposition. The nonadiabaticity leads to transforming the elastic energy into thevibrational or kinetic energy of the atom 1, as seen in Figure 2.2c. The vibra-tional energy of the atom 1 may be considered to dissipate into the vibrational

12 2 Atomistic Models

1′

1

(a) (b) (c)

1 1

2′ 3′ 1′ 2′ 3′ 1′ 2′ 3′

Figure 2.2 The friction system consisting of four atoms numbered by 1, and 1′, 2′, and 3′. Allatoms are assumed to interact with each other. The atom 1, nonadiabatically (abruptly)changes its position during sliding. The nonadiabaticity leads to transforming the elasticenergy into the vibrational or kinetic energy of the atom 1. The vibrational energy of the atom1 may be considered to dissipate into the vibrational energies of the other atoms, that is, intothe thermal energy. This picture involves the irreversible physical process, that is, the energydissipation in its natural form.

energies of other atoms, that is, into the thermal energy. This picture involvesthe irreversible physical process, the energy dissipation in its natural form. If theatom 1 is assumed only to change its position slowly, the atom 1 may not takean excess kinetic energy, which is concluded from the adiabatic theorem [42].Here, we list some timescales relevant to the frictional systems. The sliding veloc-ity may be 10−3 to 101 m/s. The frequency of the atomic motion is about 1014 timesper second. The upper body may slide about 10−17 to 10−13 m per a frequency ofatom, which is very small compared with the atomic interdistance of an order of10−10 m. The change of the potential which the atom feels during the frequencytime of the atomic motion is very small; the parameter characterizing its change,(ΔT∕𝑣) × d𝑣∕dT , becomes 10−7 to 10−3. This consideration implies that the atom1 can adiabatically follow the change of the potentials yielded by sliding if theatom 1 does not change its position abruptly. As pointed out by Tomlinson [23],the assumption of the slow movement of the atom 1 fails to explain the energydissipation in the dynamic process of friction.

To clarify his idea, we describe this process using a simplified model [32, 36].The atom 1 interacts with the other atoms of the upper body whose coordinateis symbolically expressed by Q. The atom 1 also interacts with the atoms of thelower body, which is assumed to be rigid. We shall concentrate on the equilibriumposition of the atom 1 during sliding. The equilibrium position of the atom 1 canbe determined by minimizing the interaction potential energy given by

𝑣(Q, r) = 𝑣1(Q − r) + 𝑣2(r), (2.1)

where r is the position of the atom 1, 𝑣1(Q − r) describes the interaction betweenthe coordinate Q and the atom 1, and 𝑣2(r) the interaction between the atom andthe lower body. Q stands for the displacement coordinate of the sliding upperbody against the lower one. The equilibrium position of the atom is determined

2.2 Physical Essence of Mechanical Adiabaticity in Friction 13

(a) (b)

1

2′3′

3′

2′3′

2′

ΔE3′

(c)

(e)

(d)

Figure 2.3 The explanation of Tomlinson’s mechanism using the potential surface. The shapeof the potential surface 𝑣(Q, r) depends on Q. The equilibrium position of the atom 2 isindicated by a circle, and another possible equilibrium position by the dotted circles. The leftand the right local minima correspond to the equilibrium position of the atom 2′ and that onthe atom 3′, respectively. In the processes from (a) to (d), r(Q) continuously varies with Q. Atthe process in Figure 2.3(d), r(Q) sites on the saddle point of the potential surface. When oneproceeds further, r(Q) changes discontinuously from the left minimum to the right one, asshown in (d) and (e). Then, the potential energy difference ΔE between two local minimatransforms into the kinetic energy of the atom through the nonadiabatic change of theposition of the atom 2. The kinetic energy may be consumed into exciting the vibrations of thesurrounding atoms, that is, into the thermal energy. The ingredient of this process is anappearance of the discontinuity in the equilibrium positions of atoms.

as a function of Q, and hereafter we express this fact by introducing notationssuch as r(Q), ri(Q), and r(Q).

Tomlinson’s picture may be described using the potential surface as follows.The shape of the potential surface 𝑣(Q, r) depends on Q. Under the appropriateconditions, the potential surface takes the various shapes as Q varies, as shownin Figure 2.3. The equilibrium position of the atom 1 is indicated by a circle,and another possible equilibrium position by the dotted circle. The left and theright local minima correspond to the equilibrium positions of the atom 2′ and 3′,respectively. The behaviors of the frictional system corresponding to these figuresare shown in Figure 2.4. In the processes from Figure 2.3a–d, r(Q) continuouslyvaries with Q. At the process in Figure 2.3d, r(Q) sites on the saddle point ofthe potential surface. When one proceeds further, r(Q) changes discontinuouslyfrom the left minimum to the right one, as shown in Figure 2.2d and e. Then,the potential energy difference ΔE between two local minima transforms intothe kinetic energy of the atom by nonadiabatically changing the position of theatom 1. The kinetic energy further may be consumed into exciting the vibra-tions of the surrounding atoms, that is, into the thermal energy. The ingredient of

14 2 Atomistic Models

(a)

Q

Q Q

Q

Q

(b)

(c) (d)

(e)

Figure 2.4 The behaviors of thefrictional system corresponding tothe figures shown in Figure 2.3.

this process is an appearance of the discontinuity in the equilibrium positions ofatoms.

Tomlinson proposed the possible mechanism of the origin of the friction forces,but did not enquire whether or not his mechanism occurs in the realistic fric-tional systems. In Chapter 3, we discuss the criterion for the occurrence of hismechanism, and will conclude that Tomlinson’s mechanism is unlikely to occurin realistic systems [40].