Fractals, scaling and growth far from equilibrium

Transcription

1 Fractals, scaling and growth far from equilibrium Paul Meakin Department of Physics, University of Oslo CAMBRIDGE UNIVERSITY PRESS

2 Preface xiii Chapter i Pattern Formation Far From Equilibrium 1.1 Power Laws and Scaling The Logistic Map The Variety of Patterns in Nature Euclidean Patterns Cellular Patterns 2j Spiral and Helix Patterns Labyrinthine Patterns 32 ' Fluid Convection Patterns Moving-Boundary Processes Solidification Growth from Solution Solidification of Impure Materials Viscous Fingering Pattern Selection ' Anisotropy and Growth Velocity Laplacian Growth Instabilities Characteristic Lengths Beyond Linear-Stability Analysis 5/ 1.5 Solution of Interface Equations of Motion vn

3 Vlll Contents i i Numerical Solution of the Non-Local Equations Local Models 53 Complex and Disorderly Aggregates 59 Polymers 60 Scaling Symmetry 61 Notation 62 Monte Carlo Methods Additional Information Patterns Chapter 2 Fractals and Scaling Self-Similar Fractals Statistical Self-Similarity Lacunarity JO Determination of the Fractal Dimensionality J The Devil's Staircase Simple Rules Finite-Size Effects and Crossovers Power Law Distributions Scaling Corrections to Scaling in Multiscaling Fractal Trees and Inhomogeneous Fractals Self-Affine Fractals Generation of Self-Affine Surfaces The Geometry and Growth of Rough Surfaces Characterization of Self-Affine Rough Surfaces Finite-Size Effects and Crossovers Status Long Range Persistence Multifractals Universality Additional Information 766 Chapter 3 Growth Models Cluster Growth and Cluster Surfaces Lattice Animals Random Walks 773

4 Contents ix Self-Avoiding Random Walks Indefinitely Growing Walks Ij The Diffusion-Limited Growth Walk IJJ Random Walks on Random Substrates Active Random Walk Models Cluster Growth Models 7 S The Eden Model Ballistic Aggregation The Diffusion-Limited Aggregation Model The Dielectric Breakdown Model The Scaling Structure of DLA Other Aspects of DLA Diffusion-Limited Annihilation Percolation and Invasion Percolation Growth Models for Percolation Invasion Percolation Diffusion Fronts and the Effect of Gradients Directed Percolation The Screened Growth Model Faceted Growth Models Packing Models Growth Models Related to DLA Homogeneous Perturbations Inhomogeneous Perturbations 25, Attractive Interaction Model Growth on Fibers and Surfaces Simplified DLA Models Noise Reduction and Deterministic Models Lattice Structure Effects Models with Quenched Disorder Growth in High-Dimensionality Spaces Theoretical Methods Mean Field Theories Wedge Growth Theories Real-Space Renormalization Theories Other Approaches Additional Information 325

5 Contents Chapter 4 Experimental Studies DLA Processes Electrochemical Deposition Fluid-Fluid Displacement Experiments Thin Films and Interfaces Dissolution, Melting and Erosion of Porous Media Solidification and Crystallization Dielectric Breakdown Growth Probability Distributions Dense Branching Morphology Electrochemical Deposition Thin Films Fluid-Fluid Displacement Spherulites Percolation Invasion Percolation Displacement in Complex Fluids Polymer Solutions Colloidal Systems Foams Fractal Systems Other 2-Dimensional Patterns Additional Information 400 Chapter 5 The Growth of Surfaces and Interfaces The Structure and Growth of Rough Surfaces Basic Surface Growth Equations Surface Diffusion Universality Classes Exponent Scaling Relationships The Kuramoto-Sivashinsky Equation Simple Models Eden Growth Models Ballistic Deposition Models Solid-on-Solid Models The Polynuclear Growth Model Directed Polymers Langevin Dynamics Simulations 432

6 Contents XI Directed Percolation Theoretically Motivated Models Surface Growth with Weak Non-linearity Correlated Noise Non-Gaussian Noise Growth on Rough Substrates Models with Quenched Disorder Models and Simulation Results Universality Classes Exponent Scaling Relationships Experiments Fluid-Fluid Displacement Experiments The Growth of Cell Colonies Phase Boundaries and Grain Boundaries Deposition Experiments Erosion Experiments Electrochemical Deposition Corrosion and Oxidation Some General Comments 5.6 Thin Film Growth Models The Effects of Surface Diffusion Step Edge Dynamics Anomalous Scaling Porous and Amorphous Films Anisotropic Surfaces The Huygens Principle Model Oblique Incidence and Shadowing Models Oblique Incidence Ballistic Deposition Models Ballistic Fans Shadowing Models Cluster Shapes and Faceted Growth Additional Information 573 Appendix A Instabilities 574 A. 1 The Mullins-Sekerka Instability 574 A.2 The Saffman-Taylor Problem 580

7 Xll Contents Appendix B Multifractals 5S5 B.i Generation of Simple Multifractal Sets 586 B.2 Characterization of Multifractal Sets 597 B.3 Applications to Non-Equilibrium Growth 597 B.3.1 Quenched and Annealed Averages 605 B.3.2 Mass Multifractals 606 References 608 Index 663 Color plates are between pp. 242 and 243.