Morphology of Condensed Matter: Physics and Geometry of Spatially Complex Systems (Lecture Notes in Physics, 600)

✍ Scribed by Klaus R. Mecke (editor), Dietrich Stoyan (editor)

• Publisher: Springer
• Year: 2002
• Tongue: English
• Leaves: 428
• Edition: 2002
• Category: Library

✦ Synopsis

The morphology of spatially stuctured materials is a rapidly growing field of research at the interface of statistical physics, applied mathematics and materials science. A wide spectrum of applications encompasses the flow through porous and composite materials as well as microemulsions and foams. Written as a set of lectures and tutorial reviews leading up to the forefront of research, this book will be both a compendium for the experienced researcher as well as a high level introductory text for postgraduate students and nonspecialist researchers working in related areas.

✦ Table of Contents

Chapter 1
1 Introduction
2 Probabilistic Models of Structures
2.1 The Choquet Capacity T(K)
2.2 Calculation and Estimation of the Functional T(K)
2.3 Morphological Content of the Choquet Capacity
2.4 Minkowski Functionals of Dilated Sets
3 Change of Scale in Random Media
3.1 Introduction
3.2 Third Order Bounds of the Dielectric Permittivity
3.3 Third Order Bounds for Elasticity
3.4 Estimation of Overall Properties from Numerical Simulations
4 Fracture Statistics Models
4.1 Weakest Link Model and Boolean Random Functions
4.2 Critical Defect Fraction
4.3 Griffith Crack Arrest Criterion
4.4 Models of Random Damage
4.5 Fracture Statistics Models and Simulations
5 Conclusions
References
Chapter 2
1 Spatial Structures: Experimental Data and Stochastic Models
1.1 Generation of Model Media
1.2 Experimental Images from Microtomography
2 Characterization of Complex Morphologies via Minkowski Functionals
2.1 Integral Geometric Measures
2.2 Morphology of Model Systems
2.3 Characteristics of a Sandstone Sample
3 Reconstruction of Complex Morphologies via Minkowski Functionals
3.1 Reconstruction of Boolean Models
3.2 Reconstruction of Sandstone Images
4 Characterisation via Minkowski Functionals of Parallel Surfaces
4.1 Erosion-Dilation Operations
4.2 Discrimination of Model Composites
5 Conclusions
Acknowledgements
References
Chapter 3
1 Introduction
2 Pore Scale Topology
3 Pore Network Modeling
4 Topology of Continuous Fields
5 Numerical Simulations
6 Conclusions
References
Chapter 4
1 Introduction
2 Status of the Field
3 Principle of Nanotomography
4 Proof of Concept: Block Copolymer Microdomain Structure
5 Potential Applications
5.1 Polymer Films and Coatings
5.2 Metals
5.3 Ceramics
5.4 Composites and Filled Polymers
5.5 Semiconductor Devices
5.6 Possible Impact to Other Fields
6 Perspectives: Seeing, Feeling, and Understanding Materials
7 Summary
Acknowledgments
References
Chapter 5
1 Surfaces in Self-assembling Amphiphilic Systems
2 Free Energy Functionals
2.1 Interface Models
2.2 Ginzburg-Landau Models
3 Triply Periodic Minimal Surfaces (TPMS)
4 Parallel Surfaces
5 Surfaces of Constant Mean Curvaturve (CMC)
6 Random Surfaces
6.1 Microemulsion and Sponge Phases
6.2 Gaussian Random Fields
6.3 Phase Behavior of Random Surfaces
6.4 Monte Carlo Simulations of Triangulated Surfaces
6.5 Comparison with Experiments
7 Summary and Outlook
8 WWW Resources
References
Chapter 6
1 Introduction
2 Experimental
3 Generic Results on Mesoscopic Monolayer Textures
4 Free Energy of Monolayer Textures
5 Quantification of Experimentally Determined Domain Shapes
(A) Spatially Isolated Domains
(B) Crowded Domain Ensembles: Minkowski Function Assessment of Similarity of Textures [3, 34]
(C) Determination of Molecular Quantities from Simulated Domain Shapes
6 Conclusions
Acknowledgements
References
Chapter 7
1 Introduction
2 Model Systems
2.1 Ellipsoids
2.2 Gay-Berne Particles
2.3 Other Models
3 Properties of the Nematic Phase
4 Interfacial Properties
5 Conclusions
Acknowledgements
References
Chapter 8
1 Introduction
2 A Wide Range of Properties and Applications
2.1 Structure
2.2 Mechanics
2.3 Chemistry
2.4 Model Systems
2.5 Other
3 A Generic Model for Fluid Foams
3.1 Common Characteristics
3.2 The Simplest Foam
4 Position of the Problem
4.1 Energy
4.2 Local Rules: Plateau
4.3 What Is the Problem?
5 Electrostatic Analogy
5.1 Pressure, Geometry, Topology
5.2 Continuous Limit
5.3 Examples
6 The Energy of the Foam
6.1 Perimeter Increase due to a Defect
6.2 Several Topological Charges
6.3 Relation Between Pressure and Energy
7 The Pressure Field within the Foam
7.1 Measurements
7.2 Laplacian of the Pressure
8 The Minimum Perimeter Problem
8.1 Different Equilibrium States
8.2 Global Energy Minimum (Minimal Perimeter at Free Topology)
8.3 Ground State Configuration (Perimeter-Minimizing Pattern)
9 Conclusion
Acknowledgments
References
Chapter 9
1 Introduction
2 Basic Principles of Mathematical Morphology
3 Granulometry
3.1 Principle
3.2 Texture Classification Using Global Granulometries
3.3 Texture Segmentation Using Local Granulometries
3.4 Discrete Line Segments and Disks
4 Orientation of Directional Structures
4.1 Global Orientation Information
4.2 Local Orientation Information (Orientation Field)
5 Some Other Techniques
5.1 Morphological Covariance
5.2 Multiscale Surface Area Measurements
5.3 Extrema Analysis and Watershed Segmentation
6 Concluding Remarks and Summary Table
References
Chapter 10
1 Introduction
2 Going Beyond Scalar Minkowski Functionals – A Physical Motivation
3 The Hierarchy of Minkowski Valuations – Extending the Framework
4 Exploring Higher-Rank Minkowski Valuations
4.1 Some Mathematical Results
4.2 Examples
5 Physical Applications of Higher-Rank Minkowski Valuations
5.1 The Inner Structure of Spiral Galaxies
5.2 The Morphology of Galaxy Clusters
5.3 The Geometry of the Electric Charge Distribution in Molecules
6 Density Functional Theory Based on Minkowski Valuations
7 Conclusions
Acknowledgement
References
Chapter 11
1 Introduction
2 Homology Groups andBetti Numbers
3 A Theoretical Framework for Computational Homology
4 Computer Implementation
4.1 ConnectedComponents
4.2 Alpha Shapes andHomology
5 Example Data
5.1 Binomial Point Process
5.2 Sierpinski Triangle Relatives
Acknowledgements
References
Chapter 12
1 Introduction
2 The Euler Number of a Set and of Its Complement
2.2 Consistency Relation
3 Approximation of the Euler Number of a Set
3.1 General
3.2 The Integral Geometric Approach
3.3 Adjacencies, Tessellations and Graphs
3.4 The Euler Number with Respect to Adjacency
3.5 Pairs of Complementary Adjacencies
3.6 Sufficient Conditions for Correct Approximation
4 Applications and a Simulation Study
Acknowledgements
References
Chapter 13
1 Introduction
2 A Survey of Methods in Particle Statistics
2.1 Shape Ratios
2.2 Shape Functions
2.3 ModelingGr owth, Fracture and Abrasion
3 Gibbs Pixel-Particles
3.1 Energies and Shapes
3.2 Pixel-Particles and their Parameters
3.3 Energy Functions for Gibbs Pixel-Particles
3.4 The Number of Polyonominoes
3.5 A MCMC Scheme for SimulatingGib bs Pixel-Particles
3.6 A Gallery of Simulated Gibbs Pixel-Particles
References
Chapter 14
1 Introduction
2 B-Distances and Related Notions
2.1 B-Distances
2.2 Contact Vectors
3 Spatial Random Structures
3.1 Point Processes
3.2 Particle Processes
3.3 Particle Processes as Marked Point Processes
3.4 Random Sets and Grain Models
3.5 Grain Models with Convex Grains
3.6 Integrability Conditions
4 Contact Distribution Functions
4.1 The Capacity Functional
4.2 Contact Distributions
4.3 Estimators of Contact Distribution Functions
4.4 Regularity Properties
4.5 Generalizations of Contact Distributions
5 Poisson Processes and Boolean Models
5.1 The Stationary Poisson Process
5.2 The Stationary Boolean Model
5.3 The Spherical Contact Distribution of the Boolean Model
5.4 General Structuring Elements
5.5 The Linear Contact Distribution of the Boolean Model
5.6 The Mean Normal Distribution
6 Poisson Cluster Processes
6.1 Definition of a Poisson Cluster Process
6.2 Contact Distributions of a Poisson Cluster Process
6.3 Gauss–Poisson Processes
6.4 Neyman–Scott Processes
6.5 Asymptotic Behaviour of the Empty Space Hazard
7 Local Geometric Concepts
7.1 Support Measures
7.2 Smooth Convex Bodies
7.3 Additive Extensions of Support Measures
8 Poisson Cluster Models
8.1 Definition of a Poisson Cluster Model
8.2 The Spherical Contact Distribution of a Poisson Cluster Model
8.3 Examples
8.4 General Gauge Bodies
9 General Stationary Random Sets
9.1 The General Form of Direction Dependent Contact Distributions
9.2 First Derivatives and Surface Intensities
9.3 Mean Normal Measure and Dilation Volumes of Stationary Boolean Models
9.4 Second Derivatives
10 The Instationary Case
10.1 Introduction
10.2 The Instationary Poisson Process
10.3 General Poisson Processes and Boolean Models
10.4 Instationary Cluster Models
10.5 Direction Dependent Contact Distributions of the General Boolean Model with Convex Grains
10.6 A Boolean Model with Spherical Grains
11 Smooth Boolean Models and Second Order Information
11.1 Smooth Boolean Models
11.2 Contact Distributions with Local Information
11.3 Independent Marking with Respect to Tangent Points
11.4 The Case of Integrable Intensity Functions
11.5 Homothetic Particles
12 Final Remarks
12.1 General Sampling Schemes
12.2 Intrinsic Volumes of Parallel Sets
References
Chapter 15
1 Marked Point Sets
1.1 The Framework
1.2 Two Notions of Independence
1.3 Investigating the Independence of Sub-point Processes
1.4 Investigating Mark Segregation
2 Describing Empirical Data: Some Applications
2.1 Segregation Effects in the Distribution of Galaxies
2.2 Orientations of Dark Matter Halos
2.3 Martian Craters
2.4 Pores in Sandstone
3 Models for Marked Point Processes
3.1 The Boolean Depletion Model
3.2 The Random Field Model
3.3 The Cox Random Field Model
4 Conclusions
Acknowledgments
Appendix: Completeness of Mark Correlation Functions
References
Chapter 16
1 Introduction
2 Simple Jump Processes
3 Spatial Birth-and-Death Processes
3.1 The Simple Case
3.2 Notation for Spatial Jump Processes
3.3 Description of Spatial Birth-and-Death Processes
3.4 Detailed Balance and Local Stability Conditions
3.5 Coupling Construction
4 Perfect Simulation
4.1 Dominated Coupling from the Past
4.2 Upper and Lower Processes
4.3 Clan of Ancestors
5 Spatial Birth-and-Catastrophe Process
5.1 Description and Construction
5.2 Perfect Simulation
6 The General Case
Acknowledgement
Appendix: Notation Index
References
Chapter 17
1 Introduction
2 Fundamentals of the Theory of Marked Point Processes
2.1 Definition
2.2 Stationary Marked Gibbs Point Process
2.3 The Nguyen-Zessin-Georgii Equation for Stationary Marked Gibbs Point Processes
2.4 Marked Hard-Core Gibbs Point Processes
3 Estimation of Gibbs Process Parameters by the Takacs-Fiksel Method
4 Water Droplets on Naphtalin-Brom Surface
References
Chapter 18