Chapter 1 A Brief Introduction to Soft Matter 1
Chapter 2 Discovery of SoftMatter Quasicrystals and Their
Properties 6
2.1 SoftMatter Quasicrystals with 12 and 18fold Symmetries 6
2.2 Characters of SoftMatter Quasicrystals 10
2.3 Some Concepts Concerning Possible Hydrodynamics of SoftMatter
Quasicrystals 11
2.4 First and Second Kinds of TwoDimensional Quasicrystals 11
2.5 Motivation of Our Discussion in the Book 13
Chapter 3 Review in Brief of Elasticity and Hydrodynamics of Solid
Quasicrystals 16
3.1 Physical Basis of Elasticity of Quasicrystals, Phonons and Phasons 16
3.2 Deformation Tensors 19
3.3 Stress Tensors and Equations of Motion 21
3.4 Free Energy Density and Elastic Constants 23
3.5 Generalized Hookes Law 25
3.6 Boundary Conditions and Initial Conditions 26
3.7 Solutions of Elasticity 27
3.8 Generalized Hydrodynamics of Solid Quasicrystals 27
3.9 Solution of Generalized Hydrodynamics of Solid Quasicrystals 30
3.10 Conclusion and Discussion 31
Chapter 4 Equation of State of Some Structured Fluids 34
4.1 Overview on Equation of State in some Fluids 35
4.2 Possible Equations of State 36
4.3 Applications to Hydrodynamics of SoftMatter Quasicrystals 37
Chapter 5 Poisson Brackets and Derivation of Equations of Motion of
SoftMatter Quasicrystals 39
5.1 Brown Motion and Langevin Equation 39
5.2 Extended Version of Langevin Equation 40
5.3 Multivariable Langevin Equation, Coarse Graining 40
5.4 Poisson Bracket Method in Condensed Matter Physics 41
5.5 Application to Quasicrystals 43
5.6 Equations of Motion of SoftMatter Quasicrystals 43
5.7 Poisson Brackets Based on Lie Algebra 48
Chapter 6 Oseen Flow and Generalized Oseen Flow 54
6.1 NavierStokes Equations 54
6.2 Stokes Approximation 55
6.3 Stokes Paradox 55
6.4 Oseen Modification 56
6.5 Oseen Steady Solution of Flow of Incompressible Fluid Past Cylinder 56
6.6 Generalized Oseen Flow of Compressible Viscous Fluid Past a
Circular Cylinder 63
Chapter 7 Dynamics of SoftMatter Quasicrystals with 12Fold
Symmetry 71
7.1 TwoDimensional Governing Equations of SoftMatter Quasicrystals of
12Fold Symmetry 71
7.2 Simplification of Governing Equations 76
7.3 Dislocation and Solution 77
7.4 Generalized Oseen Approximation Under Condition of Lower Reynolds
Number 79
7.5 Steady Dynamic Equations Under Oseen Modification in Polar
Coordinate System 80
7.6 Flow Past a Circular Cylinder 82
7.7 ThreeDimensional Equations of Generalized Dynamics of SoftMatter
Quasicrystals with 12fold Symmetry 91
7.8 Possible Crack Problem and Analysis 93
7.9 Conclusion and Discussion 96
Chapter 8 Dynamics of Possible 5 and 10Fold Symmetrical
SoftMatter Quasicrystals 100
8.1 Statement on Possible SoftMatter Quasicrystals of 5 and 10Fold
Symmetries 100
8.2 TwoDimensional Basic Equations of SoftMatter Quasicrystals of
Point Groups and 100
8.3 Dislocations and Solutions 104
8.4 Probe on Modification of Dislocation Solution by Considering Fluid
Effect 105
8.5 Transient Dynamic Analysis 107
8.6 ThreeDimensional Equations of Point Group 10 mm SoftMatter
Quasicrystals 113
8.7 Conclusion and Discussion 116
Chapter 9 Dynamics of Possible SoftMatter Quasicrystals of
8Fold Symmetry 119
9.1 Basic Equations of Possible SoftMatter 8Fold Symmetrical
Quasicrystals 119
9.2 Dislocation in Quasicrystals with 8Fold Symmetry 121
9.3 Transient Dynamics Analysis 123
9.4 Flow Past a Circular Cylinder 131
9.5 ThreeDimensional SoftMatter Quasicrystals with 8Fold Symmetry of
Point Groups 8mm 134
9.6 Conclusion and Discussion 137
Chapter 10 Dynamics of SoftMatter Quasicrystals with 18Fold
Symmetry 139
10.1 SixDimensional Embedded Space 139
10.2 Elasticity of Possible Solid Quasicrystals with 18Fold Symmetry 141
10.3 Dynamics of Quasicrystals of 18fold Symmetry with Point Group
18mm 143
10.4 The Steady Dynamic and Static Case of First and Second Phason
Fields 147
10.5 Dislocations and Solutions 149
10.6 Discussion on Transient Dynamics Analysis 152
10.7 Other Solutions 153
Chapter 11 The Possible 7, 9 and 14Fold Symmetry Quasicrystals
in Soft Matter 155
11.1 The Possible 7Fold Symmetry Quasicrystals with Point Group 7m
of Soft Matter and the Dynamic Theory 155
11.2 The Possible 9Fold Symmetrical Quasicrystals with Point Group 9m
of Soft Matter and Their Dynamics 159
11.3 Dislocation Solutions of the Possible 9Fold Symmetrical Quasicrystals of
Soft Matter 162
11.4 The Possible 14Fold Symmetrical Quasicrystals with Point Group
14mm of Soft Matter and Their Dynamics 167
11.5 The Solutions and Possible Solutions of Statics and Dynamics of 7
and 14Fold Symmetrical Quasicrystals or soft Matter 170
11.6 Conclusion and Discussion 170
Chapter 12 An Application of Analytic Methods to Smectic A Liquid
Crystals, Dislocation and Crack 172
12.1 Basic Equations 172
12.2 The Kleman-Pershan Solution of Screw Dislocation 174
12.3 Common Fundamentals of Discussion 175
12.4 The Simplest and Most Direct Solving Way and Additional Boundary
Condition 176
12.5 Mathematical Mistakes of the Classical Solution 178
12.6 The Physical Mistakes of the Classical Solution 179
12.7 Meaning of the Present Solution 180
12.8 Solution of Plastic Crack 180
Chapter 13 Conclusion Remarks 186
內容試閱:
As wellknown quasicrystals with 12fold symmetry observed since 2004 in liquid crystals, colloids, polymers and nanoparticles have received a great deal of attention. In particular, 18fold symmetry quasicrystals in colloids were discovered in 2011. More recently the quasicrystals with 12fold symmetry were also found in giant surfactants. The formation mechanisms of these kinds of quasicrystals are connected closely with selfassemble of spherical building blocks by supramolecules, compounds, block copolymers and so on and are quite different from that of the metallic alloy quasicrystals. They can be identified as softmatter quasicrystals exhibiting natures of quasicrystals with softmatter characters. Soft matter lies in the behaviour of intermediate phase between solid and simple fluid, while the nature of quasicrystals exhibits importance of symmetry as they are highly ordered phase. These features are very complex yet extremely interesting and attractive. Hence, they have raised a great deal of attention of researchers in physics, chemistry and materials science.
All the observed softmatter quasicrystals so far are twodimensional quasicrystals. It is wellknown that twodimensional quasicrystals consist of only two distinct types, one being 5, 8, 10 and 12fold symmetries, the other being 7, 9, 14 and 18fold according to the symmetry theory. Therefore, two terminological phrases can be defined such as the first and second kinds of twodimensional quasicrystals respectively. The twodimensional solid quasicrystals observed so far belong to the first kind only, while softmatter quasicrystals discovered up to now can be in both kinds. This may imply that many new types of softmatter quasicrystals in addition to those with 12 and 18fold symmetries may be observed in the near future. Hence, the interdisciplinary studies on softmatter quasicrystals present great potential and hopeful research topics.
However, some difficulties exist in studying those new phases due to the complexity of their structures and lack of fundamental experimental data including the material constants to date. Furthermore, the theoretical studies are also difficult. For example, the symmetry groups of softmatter quasicrystals observed or possibly to be observed have not yet been well investigated although there are some work being done the details are not to be included in the book. In conjunction with this issue, the study on constitutive laws for phonons and phononphason coupling is still difficult.
In spite of these problems, there are potential efforts to undertake the study of these topics. For example, the softmatter quasicrystals as a new ordered phase are connected with broken symmetry or symmetry breaking, like those discussed in solid quasicrystals. Thus, the elementary excitations such as phonons and phasons are important issues in the study of quasicrystals based on the Landau phenomenological theory. For softmatter quasicrytals, furthermore, another elementary excitation, i.e., the fluid phonon must be considered besides phonons and phasons. According to the Landau school, liquid acoustic wave is fluid phonon refer to Lifshitz EM and Pitaevskii LP, Statistical Physics, Part 2, Oxford: ButterworthHeinemann, 1980. This is suitable for describing the liquid effect of softmatter quasicrystals, which can be seen as complex liquids or structured liquids. The elementary excitationsphonons, phasons and fluid phononand their interactions constitute the main feature of the new phase. They will be discussed as a major issue in the book. The concept of the fluid phonon is introduced in the study of quasicrystals for the first time. Related to this, the equation of state should also be introduced. With these two key points and in reference to the hydrodynamics of solid quasicrystals, the dynamics of softmatter quasicrystals can be established, but with an important distinction compared with that of solid quasicrystals. The present hydrodynamics cannot be linearized due to the nonlinearity of equation of state. To overcome the difficulty arising from other aspects in theory, we can draw from study of solid quasicrystals For example, Lubensky TC, Symmetry, elasticity and hydrodynamics in quasiperiodic structures, in Introduction to Quasicrystals, ed by Jaric MV, Boston: Academic Press, 199-289, 1988; Hu CZ et al, Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals, Rep. Prog. Phys., 631, 1-39, 2000; Fan TY, Mathematical Theory of Elasticity of Quasicrystals and Its Applications, Beijing: Science PressHeidelberg: SpringerVerlag, 1st edition, 2010, 2nd edition, 2016. This shows that the theory of solid quasicrystals is a basis for the present discussion, which provides an initial glimpse from the viewpoint of quantitative analysis into the rich phenomena of softmatter quasicrystals.
Some applications are given by describing the distribution, deformation and motion of softmatter quasicrystals. The mathematical principle and its applications require the assistance of other areas of knowledge, a part of which is briefly listed in the first six chapters of the book for more details, refer to Chaikin J and Lubensky TC, Principles of Condensed Matter Physics, New York: Cambridge University Press, 1995, and the others are introduced in the due computation. The computational results are preliminary and very limited so far, but verify partially the mathematical model, and explored in certain degree to distinguish the dynamic behaviour between softmatter and solid quasicrystals. In addition, the specimens and flow modes adopted in the computation might be intuitive, observable and verified easily. However, it does not mean that they belong to the most important samples.
The author would like to thank the National Natural Science Foundation of China and Alexander von Humboldt Foundation of Germany for their support over the years and Profs. Messerschmidt U in MaxPlanck Institut fuer Mikrostrukturphysik in Halle, Trebin HR in Stuttgart Universitaet in Germany, Lubensky TC in University of Pennsylvania, Cheng, Stephen ZD in University of Akron in USA, Wensink HH in Utrecht University and in Holland, Li XianFang in Central South University and Chen WeiQiu in Zhejiang University in China for their cordial encouragements and helpful discussions.