This book highlights the mathematical models and solutions of the generalized dynamics of soft-matter quasicrystals (SMQ) and introduces possible applications of the theory and methods. Based on the theory of quasiperiodic symmetry and symmetry breaking, the book treats the dynamics of individual quasicrystal systems by reducing them to nonlinear partial differential equations and then provides methods for solving the initial-boundary value problems in these equations. The solutions obtained demonstrate the distribution, deformation and motion of SMQ and determine the stress, velocity and displacement fields. The interactions between phonons, phasons and fluid phonons are discussed in some fundamental materials samples. The reader benefits from a detailed comparison of the mathematical solutions for both solid and soft-matter quasicrystals, gaining a deeper understanding of the universal properties of SMQ.
The second edition covers the latest research progress on quasicrystals in topics such as thermodynamic stability, three-dimensional problems and solutions, rupture theory, and the photonic band-gap and its applications. These novel chapters make the book an even more useful and comprehensive reference guide for researchers in condensed matter physics, chemistry and materials sciences.
Author(s): Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun
Series: Springer Series in Materials Science, 260
Edition: 2
Publisher: Springer
Year: 2022
Language: English
Pages: 253
City: Singapore
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
