Pattern Formation at Interfaces (eBook)

Preface 6
Table of Contents 8
Interfacial patterns and waves in liquid layers and thin films 9
Contents 9
1 Introduction 11
2 Phenomenology of Bénard instabilities 14
2.1 Physical mechanisms of patterns and waves 14
2.2 Application-oriented aspects 17
2.3 Dimensionless numbers and time scales 19
2.4 Other instability mechanisms in very thin liquid films 21
3 Basic equations and boundary conditions 23
3.1 Non-negligible gas thermal conductivity 26
3.2 Generalized one-sided modeling of evaporation 27
3.3 Reference states 30
3.4 Linear stability analysis 32
Monotonic modes 34
Oscillatory modes 36
3.5 Direct numerical simulations 43
4 Simplified models for non-equilibrium patterns 45
4.1 The Swift-Hohenberg equation and its variants 45
4.2 Basic symmetries of Bénard set-ups 47
4.3 Symmetries and amplitude equations 49
Bifurcation of rolls 50
Bifurcation of hexagonal patterns 51
4.4 Long-wave order-parameter equations for patterns 55
5 Acknowledgments 61
Bibliography 62
Nonlinear dynamics of fronts 65
Contents 65
1 Introduction 67
2 Reaction-diffusion systems 69
Calcium waves in eggs of fish and amphibiae 72
Chemotaxis 72
Animals’ disease 72
3 The Fisher-Kolmogorov equation 73
3.1 Stationary solutions 73
3.2 Front solutions 75
3.3 Motion of the front edge 76
3.4 Non-generic fronts 78
4 Multistability 79
4.1 Fronts between locally stable phases 79
4.2 Lyapunov functional 81
4.3 Allen-Cahn equation 83
4.4 Interaction between kinks 85
4.5 Phase transition in an external field 88
4.6 Domain wall pinned by an inhomogeneity 90
4.7 Curved fronts of the phase transition 93
5 Combustion fronts 96
5.1 Formulation of the problem 96
5.2 Plane stationary front 98
5.3 Dynamics of curved fronts 101
5.4 Linear stability theory of the planar front 105
Monotonic instability 106
Oscillatory instability 107
5.5 Nonlinear development of front instabilities 108
Monotonic longwave instability 108
Oscillatory shortwave instability 110
Bibliography 111
Three Dimensional Film Dynamics 112
Contents 112
1 Introduction 114
1.1 Basic equations 114
1.2 Instabilities 114
1.3 Pattern formation – Examples 118
1.4 Types of instabilities 120
2 Thick films with undeformable surface – Pure fluids 124
2.1 The basic equations 124
Scaling to non-dimensional quantities 124
Decomposition of the velocity field, basic equations 124
Boundary conditions 126
A. Bottom Plate 126
B. Surface of the fluid 126
2.2 Linear stability analysis 127
2.3 Numerics 128
The method 129
Stability 130
Accuracy 132
2.4 Results 132
Closed upper surface 132
Free upper surface 133
3 Thick films with undeformable surface – binary mixtures 133
3.1 The basic equations and boundary conditions 134
3.2 The linear problem – codimension-two point 137
3.3 Nonlinear solutions 137
4 Reduced description – Order parameter equations 140
4.1 Order parameters 140
4.2 The Ginzburg-Landau equation 141
4.3 The Swift-Hohenberg equation 144
Gradient expansion 145
Stripes, hexagons, and squares 146
5 Thin films with a deformable surface 151
5.1 Reduced two-dimensional description – perfect fluids 152
The shallow water equations 153
Numerical solutions 155
5.2 Reduced two-dimensional description – viscous fluids 156
The lubrication approximation 156
Laplace pressure and gravity 158
The disjoining pressure and ultra-thin films 161
6 Spinodal dewetting 163
6.1 Ultra-thin isothermal films 163
Normal form 163
Numerical solutions 164
Metastable region and nucleation 166
Physical values 166
6.2 Externally heated thin films 168
Thin film equation and parameters 168
The disjoining pressure 169
Fluid parameters 170
Holes or drops? 171
6.3 Time dependent numerical solutions 173
Normal Form 173
Results: the horizontal layer 174
6.4 The inclined layer 177
Bibliography 178
Thin Film and Droplet Patterns Shaped by Surface Forces 180
Contents 180
1 Basic Equations 182
1.1 Evolution Equation 182
1.2 Disjoining Pressure 182
1.3 Effective Mobility 183
1.4 Contact Angle 184
2 Quasistationary Motion 186
2.1 Perturbation Expansion 186
2.2 Translational Solvability Condition 187
2.3 Motion due to Asymmetry of Contact Angles 189
3 Interactions Mediated by the Precursor 190
3.1 Moving Droplet on a Precursor Film 190
3.2 Mass Transport through the Precursor 192
3.3 Coarsening 193
3.4 Migration of Interacting Droplets 195
4 Chemical Self-Propulsion 196
4.1 Substrate Modification 196
4.2 Traveling Bifurcation 200
4.3 Non-diffusive Limit 201
4.4 Relaxation to a Stationary Pattern 204
4.5 Scattering 204
5 Thickness Fronts 208
5.1 Static Thickness Fronts 208
5.2 Evaporation and Condensation 210
5.3 Fluxes and Mobility of the Front 212
5.4 Solvability Condition 213
6 Evaporative Patterns 215
6.1 Straight-line front 215
6.2 “Pancake” and “hole” 217
6.3 Solution in a comoving frame 219
6.4 Zigzag instability 221
Acknowledgement 224
Bibliography 224
Interfacial Phenomena in Materials Science 225
Contents 225
1 Equilibrium crystal shape 227
2 Growth of a spherical crystal nucleus 233
3 Mullins-Sekerka instability 238
4 Dendrites 240
5 Surface diffusion and surface-diffusion-controlled interface shape 245
6 Elastic instability of solid epitaxial films and self-assembly of quantum dots 251
Bibliography 258
The Physics and Analyses of Interfacial Instabilities that Arise from Phase Change 260
Contents 260
1 The Physics and Analysis of Instability During the Solidification of a Pure Material 262
1.1 The Physics of the Instability 262
1.2 The Model 267
1.3 The Base Solution 270
1.4 The Perturbation Equations 271
2 Evaporative Instability - Linear Theory 277
2.1 Introduction 277
2.2 The physical model 277
2.3 Physics of the phase-change problem without convection 278
2.4 Physics of the phase-change problem with convection 279
2.5 The mathematical model 280
2.6 The base state solution and the perturbed equations 283
Results of calculations and explanation 286
2.7 Stabilizing effect of the vapor flow 292
3 Evaporative instability in bilayer systems - Nonlinear theory 295
3.1 Introduction 295
3.2 The physical model 296
3.3 Mathematical model 296
3.4 Nonlinear analysis 299
3.5 Perturbation expansions 299
3.6 Overview 300
3.7 Perturbed equations 302
3.8 Discussion 308