Generalized Dynamics of Soft-Matter Quasicrystals: Mathematical Models, Solutions and Applications

Preface to the Second Edition
Preface to the First Edition
Contents
Notations
1 Introduction to Soft Matter
	References
2 Discovery of Soft-Matter Quasicrystals and Their Properties
	2.1 Experimental Observation of Quasicrystalline Phases in Soft Matter
	2.2 Characters of Soft-Matter Quasicrystals
	2.3 Some Concepts Concerning Possible Generalized Dynamics on Soft-Matter Quasicrystals
	2.4 First and Second Kinds of Two-Dimensional Quasicrystals
	2.5 Motivation of Our Discussion in the Book
	References
3 Introduction on Elasticity and Hydrodynamics of Solid Quasicrystals
	3.1 Physical Basis of Elasticity of Quasicrystals, Phonons, and Phasons
	3.2 Deformation Tensors
	3.3 Stress Tensors and Equations of Motion
	3.4 Free Energy Density and Elastic Constants
	3.5 Generalized Hooke’s Law
	3.6 Boundary Conditions and Initial Conditions
	3.7 Solutions of Elasticity
	3.8 Hydrodynamics of Solid Quasicrystals
		3.8.1 Viscosity of Solid
		3.8.2 Hydrodynamics of Solid Quasicrystals
	3.9 Solution of the Hydrodynamics of Solid Quasicrystals
	3.10 Conclusion and Discussion
	References
4 Case Study of Equation of State in Several Structured Fluids
	4.1 Introduction of Equation of State in Some Fluids
	4.2 Possible Equations of State
	4.3 Applications to Dynamics of Soft-Matter Quasicrystals
	4.4 The Incompressible Model of Soft Matter
	References
5 Poisson Brackets and Derivation of Equations of Motion in Soft-Matter Quasicrystals
	5.1 Brownian Motion and Langevin Equation
	5.2 Extended Version of Langevin Equation
	5.3 Multivariable Langevin Equation, Coarse-Graining
	5.4 Poisson Bracket Method in Condensed Matter Physics
	5.5 Application of Poisson Bracket to Quasicrystals
	5.6 Equations of Motion of Soft-Matter Quasicrystals
		5.6.1 Generalized Langevin Equation
		5.6.2 Derivation of Generalized Dynamic Equations of Soft-Matter Quasicrystals
	5.7 Poisson Brackets Based on Lie Algebra
	5.8 On Solving Governing Equations
	References
6 Oseen Theory and Oseen Solution
	6.1 Navier–Stokes Equations
	6.2 Stokes Approximation
	6.3 Stokes Paradox
	6.4 Oseen Modification
	6.5 Oseen Steady Solution of the Flow of Incompressible Fluid Past Cylinder
	6.6 The Reference Meaning of Oseen Theory and Oseen Solution to the Study in Soft Matter
	References
7 Dynamics of Soft-Matter Quasicrystals with 12-Fold Symmetry
	7.1 Two-Dimensional Governing Equations of Soft-Matter Quasicrystals of 12-Fold Symmetry
	7.2 Simplification of Governing Equations
		7.2.1 Steady Dynamic Problem of Soft-Matter Quasicrystals with 12-Fold Symmetry
		7.2.2 Pure Fluid Dynamics
	7.3 Dislocation and Solution
		7.3.1 Limitation of Zero-Order Solution of Dislocation, Possible Modification Considering the Fluid Effect
	7.4 Generalized Oseen Approximation Under the Condition of Lower Reynolds Number
	7.5 Steady Dynamic Equations Under Oseen Modification in Polar Coordinate System
	7.6 Flow Past a Circular Cylinder
		7.6.1 Two-Dimensional Flow Past Obstacle
		7.6.2 Quasi-Steady Analysis—Numerical Solution by Finite Difference Method
		7.6.3 Numerical Results and Analysis
	7.7 Three-Dimensional Equations of Generalized Dynamics of Soft-Matter Quasicrystals with 12-Fold Symmetry
	7.8 Governing Equations of Generalized Dynamics of Incompressible Soft-Matter Quasicrystals of 12-Fold Symmetry
	7.9 Conclusion and Discussion
	References
8 Dynamics of 10-Fold Symmetrical Soft-Matter Quasicrystals
	8.1 Statement on Soft-Matter Quasicrystals of 10-Fold Symmetries
	8.2 Two-Dimensional Basic Equations of Soft-Matter Quasicrystals of Point Groups 10,  overline10
	8.3 Dislocations and Solutions
	8.4 Probe on Modification of Dislocation Solution by Considering the Fluid Effect
	8.5 Transient Dynamic Analysis
		8.5.1 Specimen and Initial-Boundary Conditions
		8.5.2 Numerical Analysis and Results
	8.6 Three-Dimensional Equations of Point Group 10mm Soft-Matter Quasicrystals
	8.7 Incompressible Complex Fluid Model of Soft-Matter Quasicrystals with 10-Fold Symmetry
	8.8 Conclusion and Discussion
	References
9 Dynamics of 8-Fold Symmetric Soft-Matter Quasicrystal Models
	9.1 Basic Equations of 8-Fold Symmetric Soft-Matter Quasicrystal Models
	9.2 Dislocation in 8-Fold Symmetric Soft-Matter Quasicrystals
		9.2.1 Elastic Static Solution
		9.2.2 Modification with Consideration of the Fluid Effect
	9.3 Transient Dynamics Analysis
		9.3.1 Specimen
	9.4 Flow Past a Circular Cylinder
	9.5 Three-Dimensional Systems with 8-Fold Symmetric Soft-Matter Quasicrystals
	9.6 Incompressible Model of the 8-Fold Symmetric Soft-Matter Quasicrystals
	9.7 Solution Example of an Incompressible Model
	9.8 Conclusion and Discussion
	References
10 Dynamics of 18-Fold Symmetric Soft-Matter Quasicrystals
	10.1 Six-Dimensional Embedded Space
	10.2 Elasticity of the Possible 18-Fold Symmetric Solid Quasicrystals
	10.3 Dynamics of 18-Fold Symmetric Quasicrystals with  18 mm Point Group
	10.4 The Steady Dynamic and the Static Case of the First and the Second Phason Fields
	10.5 Dislocations and Solutions
		10.5.1 The Zero-Order Approximate Solution for Dislocations in 18-Fold Symmetric Soft-Matter Quasicrystals
		10.5.2 Modification to the Solution (10.5.3) to (10.5.6) Considering the Fluid Effect
	10.6 Discussion on Transient Dynamics Analysis
	10.7 Three-Dimensional Equations of Generalized Dynamics of 18-Fold Symmetric Soft-Matter Quasicrystals
		10.7.1 Introduction
		10.7.2 Some Basic Relations
		10.7.3 Three-Dimensional Equations of Generalized Dynamics of Point Group 18 mm Soft-Matter Quasicrystals
	10.8 Incompressible Generalized Dynamics of 18-Fold Symmetric Soft-Matter Quasicrystals
	10.9 Other Solutions and Applications
	References
11 The Possible 7-, 9-, and 14-fold Symmetry Quasicrystals in Soft Matter
	11.1 The Possible 7-fold Symmetry Quasicrystals with Point Group 7m of Soft Matter and the Dynamic Theory
	11.2 The Possible 9-fold Symmetrical Quasicrystals with Point Group 9m of Soft Matter and Their Dynamics
	11.3 Dislocation Solutions of the Possible 9-fold Symmetrical Quasicrystals of Soft Matter
	11.4 The Possible 14-fold Symmetrical Quasicrystals with Point Group 14mm of Soft Matter and Their Dynamics
	11.5 The Numerical Solution of Dynamics of 14-fold Symmetrical Quasicrystals of Soft Matter
	11.6 Incompressible Complex Fluid Model
	11.7 Conclusion and Discussion
	References
12 Re-Discussion on Symmetry Breaking and Elementary Excitations
	References
13 An Application to the Thermodynamic Stability of Soft-Matter Quasicrystals
	13.1 Introduction
	13.2 Extended Free Energy of the Quasicrystal System in Soft Matter
	13.3 The Positive Definite Nature of the Rigidity Matrix and the Stability of the Soft-Matter Quasicrystals with 12-Fold Symmetry
	13.4 Comparison and Examination of Results of Soft-Matter Quasicrystals with 12-Fold Symmetry
	13.5 The Stability of 8-Fold Symmetry Soft-Matter Quasicrystals
	13.6 The Stability of 10-Fold Symmetry Soft-Matter Quasicrystals
	13.7 The Stability of the 18-Fold Symmetry Soft-Matter Quasicrystals
		13.7.1 A Brief Review on Some Fundamental Relations from the Dynamics of the Second Kind of Soft-Matter Quasicrystals
		13.7.2 Extended Free Energy of the Quasicrystals System of Second Kind
		13.7.3 The Positive Definite Nature of the Rigidity Matrix and the Stability of the Soft-Matter Quasicrystals with 18-Fold Symmetry
		13.7.4 Comparison and Examination
		13.7.5 Some Discussions
	13.8 Conclusion
	References
14 Applications to Device Physics—Photon Band Gap of Holographic Photonic Quasicrystals
	14.1 Introduction
	14.2 Design and Formation of Holographic PQCs
	14.3 Band Gap of 8-fold PQCs
	14.4 Band Gap of Multi-fold Complex PQCs
	14.5 Fabrication of 10-Fold Holographic PQCs
		14.5.1 Material and Writing System
		14.5.2 Experimental Results
	14.6 Band Gap of Cholesteric Liquid Crystal
	14.7 Conclusions
	References
15 Possible Applications to General Soft Matter
	15.1 A Basis of Dynamics of Two-Dimensional Soft Matter
	15.2 The Outline on Governing Equations of Dynamics of Soft Matter
	15.3 The Modification and Supplement to Eq. (15.2.1)
	15.4 Solution of the Dynamics of Soft Matter
	15.5 Conclusion and Discussion
	References
16 An Application to Smectic A Liquid Crystals, Dislocation, and Crack
	16.1 Basic Equations
	16.2 The Kleman-Pershan Solution of Screw Dislocation
	16.3 Common Fundamentals of Discussion
	16.4 The Simplest and Most Direct Solution and the Additional Boundary Condition
	16.5 Mathematical Mistakes of the Classical Solution
	16.6 The Physical Mistakes of the Classical Solution
	16.7 Meaning of the Present Solution
	16.8 Solution of Plastic Crack
	References
17 Conclusion Remarks