Table of Contents
Preface xiii
1 Classical Theory and Atomistics 1
1.1 Law of Friction 1
1.2 The Origin of Friction 4
1.3 Atomistics in Tribology 6
2 AtomisticModels 9
2.1 Friction Models 9
2.2 Physical Essence of Mechanical Adiabaticity in Friction 11
3 Atomistic Locking and Friction 15
3.1 Theoretical Preliminaries 15
3.1.1 Model 15
3.1.2 Expression for Adiabatic Potential 17
3.2 Topological Description of Friction 19
3.2.1 Adiabatic Potential 19
3.2.2 Atomic Configurations of Surfaces 19
3.2.2.1 Variant P;; (;;) Case 21
3.2.2.2 Invariant P;; (;;) Case 22
3.2.2.3 Restricted Invariant P;; (;;) Case 22
3.3 A More Realistic Case: A Relaxed Upper Body 22
3.4 Quasi-static Friction of α-Iron 24
3.4.1 Case (a) 24
3.4.2 Case (b) 25
4 Atomistic Origin of Friction 29
4.1 Friction Model 29
4.2 Static Friction 31
4.3 Energy Dissipation in Dynamic Friction 32
4.4 Criterion for Friction Transition 35
5 Superlubricity 43
5.1 A State of Vanishing Friction 43
5.2 How Does Friction Become Zero? 44
5.3 NonadiabaticMotion of Atoms 45
5.4 Importance of High Dimensionality 46
6 Atomistic Simulation of Friction 49
6.1 Computer Simulation 49
6.2 Atomic Structure and Electronic States 51
6.2.1 Properties of Atoms 51
6.2.2 Electronic States 53
6.3 Cohesion of Solids 55
6.3.1 Cohesive Forces Between Molecules 55
6.3.2 Cohesive Forces in Solids 58
6.4 Crystal Binding 58
6.4.1 Ionic Crystals 59
6.4.2 Covalent Crystals 60
6.4.3 Metallic Crystals 61
6.4.4 Molecular Crystals 62
6.4.5 Hydrogen-Bonded Crystals 64
6.5 Interatomic Force and Interatomic Potential 66
6.6 Molecular Dynamics Method 68
6.6.1 Equations of Motion of Atoms 68
6.6.2 Numerical Integral 68
6.7 Simple Atomistic Model 69
6.7.1 Friction Model 69
6.7.2 Equation of Motion in Dimensionless Form 70
6.7.3 Friction Diagram 72
6.8 Energy Recurrence in Superlubricity 75
6.8.1 Energy Dissipation 75
6.8.2 Two-DimensionalModel Simulation 76
6.9 Realistic Systems 79
6.9.1 Friction Transition 79
6.9.2 Many-Body Interatomic Potentials 80
6.9.3 Stability of Superlubricity 82
7 Experimental Approach for Atomic Level Friction 85
7.1 Atomic Force Microscopy Techniques 85
7.2 Verification of AtomisticTheory 87
7.2.1 Static Friction Forces 87
7.2.2 Commensurability in Sliding Surfaces 88
8 Summary 99
8.1 Origin of Friction 99
8.2 Controlling Friction 100
A Physical Preliminaries 103
A.1 AnalyticalMechanics 103
A.1.1 Coordinates and Transformation of a Coordinate System 103
A.1.1.1 Cartesian Coordinate System 104
A.1.1.2 Expression of Velocity and Acceleration in Polar Coordinates 104
A.1.1.3 Three-Dimensional Polar Coordinate System 108
A.1.1.4 Cartesian Curvilinear Coordinates 111
A.1.1.5 Generalized Coordinates 113
A.1.1.6 Generalized Momentum and Canonical Conjugate Variable 116
A.1.1.7 Generalized Force 116
A.1.2 Lagrange Equation of Motion and Variational Principle 118
A.1.2.1 Lagrange Equation of Motion 118
A.1.2.2 Application of Lagrange’s Equation of Motion 120
A.1.2.3 Variational Principle and Euler–Lagrange Equation 123
A.1.2.4 Principle of VirtualWork 126
A.1.3 Hamilton’s Canonical Equation 129
A.1.3.1 Hamiltonian 129
A.1.3.2 Hamilton’s Canonical Equation 132
A.1.3.3 Phase Space and Trajectory of Motion 132
A.2 Fundamentals of StatisticalMechanics 134
A.2.1 Kinetic Theory of Gases 134
A.2.2 Principle of Equal a priori Probability and Ergodic Hypothesis 138
A.2.3 Microscopic State 139
A.2.4 Number of States and Density of States 142
A.2.5 Entropy 144
A.2.6 Thermal Equilibrium of a Coupled System 145
A.2.7 Constant Temperature System: Canonical Ensemble 148
A.2.8 Classical System at a Given Temperature 152
A.3 Classical Mechanics with Vector Analysis 154
A.3.1 Law of Motion 154
A.3.2 Motion of Mass Point Expressed with a Vector 155
A.3.3 Moment of Force Acting on Mass Point 157
A.3.4 Angular Velocity Vector 157
A.3.5 Outer Product and Rotation 158
A.4 Vibration andWave 159
A.4.1 What is a Wave? 159
A.4.2 Fundamental Relation 161
A.4.3 Harmonic Oscillation 162
A.4.4 Wave Function 164
A.4.5 Wave Equation 167
A.4.6 TravelingWave 169
A.4.7 Phase Velocity and Dispersion 170
A.4.8 Group Velocity 172
A.4.9 Three-DimensionalWave: PlaneWave 175
A.5 Lattice Vibration 179
A.5.1 Lattice Vibration and Thermal Properties of Crystals 179
A.5.2 Lattice Vibration of a One-Dimensional Crystal 184
A.5.2.1 Model of a One-Dimensional Crystal 184
A.5.2.2 Continuum Approximation 185
A.5.2.3 Natural Vibration and Natural Frequency 187
A.5.2.4 Dispersion Relation 189
A.5.2.5 First Brillouin Zone 189
A.5.3 Acoustical Mode and Optical Mode 191
A.5.4 Lattice Vibration in a Three-Dimensional Crystal 196
A.5.5 Phonon 197
B Mathematical Supplement 199
B.1 Trigonometry 199
B.1.1 Definition 199
B.1.2 Addition Formula 200
B.1.3 Basic Properties 202
B.2 Taylor Expansion 204
B.3 Complex Exponential Function 206
B.4 Vectors and Geometry 208
B.4.1 Equations of Line and Plane 208
B.4.1.1 Equations of Line 208
B.4.1.2 Equation of a Plane 209
B.4.1.3 Equation of a Sphere and a Spherical Tangent Plane 214
B.4.1.4 Application to Geometry 215
B.5 Linear Algebra 216
B.5.1 Determinant and Inverse Matrix 216
B.5.1.1 Permutation 216
B.5.1.2 Definition of a Determinant 217
B.5.1.3 Characteristics of a Determinant 217
B.5.1.4 Inverse Matrix 218
B.5.1.5 Application of a Determinant 219
B.5.2 Linear Equations: Cramer’s Formula 219
B.5.3 Eigenvalue and Eigenvector 221
B.5.3.1 Eigenvalue and Eigenvector of a Square Matrix 221
B.5.3.2 Diagonalization of a Matrix 223
B.5.3.3 Normal Form of a Quadratic Form Polynomial 225
C Data Analysis 227
C.1 Fundamentals of Description of Physical Data 227
C.1.1 Classification of Deterministic Data 228
C.1.1.1 Sinusoidal Periodic Data 228
C.1.1.2 Compound Periodic Data 229
C.1.1.3 Almost Periodic Data 232
C.1.2 Classification of Random Data 233
C.1.2.1 Stationary Irregular Process 233
C.1.2.2 Ergodic Irregular Process 234
C.1.3 Fundamental Properties of Random Data 235
C.1.3.1 Squared Average: Average and Variance 235
C.1.3.2 Probability Density Function 235
C.1.3.3 Autocorrelation Function 237
C.1.3.4 Power Spectral Density Function 237
C.2 Signal Processing 239
C.2.1 Analog Signal and Digital Signal 239
C.2.2 Fourier Analysis 240
C.2.2.1 Fourier Series 240
C.2.2.2 Fourier Transform 242
C.2.2.3 Discrete Fourier Transform 243
C.2.3 Applications of Fourier Transform 246
C.2.3.1 Impulse Response 246
C.2.3.2 Analysis of a Linear System 250
C.2.3.3 Equation of Motion 252
D Crystal Structure 255
D.1 Periodicity of Crystals 255
D.2 Crystal Structure 256
D.2.1 Simple Cubic Structure 256
D.2.2 Body-Centered Cubic Structure 256
D.2.3 Face-Centered Cubic Structure 257
D.2.4 Hexagonal Closed-Packed Structure 258
D.2.5 Sodium Chloride Structure and Cesium Chloride Structure 259
D.2.6 Diamond Structure 260
D.3 X-ray Diffraction 261
D.3.1 Diffraction Condition 261
D.3.2 Reciprocal Vector 263
D.3.3 Bragg’s Condition 264
D.4 Various Crystal Data 264
E The SI (mks) Unit System 267
E.1 Three Basic Units 267
E.1.1 Unit of Length: Meter 267
E.1.2 Unit of Time: Second 268
E.1.3 Unit of Mass: Kilogram 268
E.1.3.1 Atomic Mass Unit 268
E.2 The SI (mks) Unit System 269
E.3 The cgs System 273
F Practice for Verlet Algorithm 275
G Program Example of Molecular Dynamics for Atomistic Model 279
G.1 Annealing Program 279
G.2 Sliding Program 281
H Table of Values 285
I Table of Relative AtomicWeights 287
References 289
Afterword 295
About the Author 297
Index 299
