Generalized Dynamics of Soft-Matter Quasicrystals

Table of Contents

1. Frontmatter

2. Chapter 1. Introduction to Soft Matter

   Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

   The chapter delves into the intriguing world of soft matter, focusing on quasicrystals found in various systems such as liquid crystals, colloids, polymers, and surfactants. It begins by defining soft matter and its unique position between isotropic fluids and ideal solids. The chapter then explores the thermal energy per unit volume concept to explain the softness of soft matter and compares it with ideal solids. The hydrodynamics and generalized dynamics of soft-matter quasicrystals are discussed, highlighting the complexities in modeling their behavior. The chapter also introduces key material constants and their significance in understanding the mechanical properties of soft-matter quasicrystals. Additionally, it touches upon the challenges in determining wave speeds and the use of the Stokes and Oseen approximations in soft matter dynamics. The chapter serves as a foundation for further exploration into the fascinating properties and behaviors of soft matter.

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3. Chapter 2. Discovery of Soft-Matter Quasicrystals and Their Properties

   Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

   The chapter delves into the fascinating world of soft-matter quasicrystals, which possess long-range orientational order without translational symmetry. It begins with an introduction to the experimental observation of quasicrystalline phases in soft matter, highlighting the unique properties of one-, two-, and three-dimensional quasicrystals. The chapter then explores the theoretical underpinnings of quasicrystals, including their description using higher-dimensional space and the famous Penrose tiling. A significant portion of the chapter is dedicated to the discovery and properties of dodecagonal quasicrystals (DDQC) in diverse soft-matter systems such as dendrimers, block copolymers, and giant molecules. It discusses the formation mechanisms, structural features, and the role of entropy in random tilings. The chapter also touches on the metastable nature of DDQC phases and their potential for designing materials with tailored properties. Throughout, the chapter offers insights into the complex interplay between thermodynamic forces and molecular shapes that drive the formation of these intriguing structures.

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4. Chapter 3. Introduction on Elasticity and Hydrodynamics of Solid Quasicrystals

   Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

   The chapter delves into the fundamental principles of elasticity and hydrodynamics in solid quasicrystals, essential for understanding their dynamics. It begins with an overview of the elasticity theory, extending the crystalline description into higher dimensions. The core of this theory is the Landau phenomenological approach, which describes the structural transition through an order parameter. Phonons and phasons, the two elementary excitations, are discussed in detail, along with their respective deformation and stress tensors. The chapter also introduces the hydrodynamics of solid quasicrystals, incorporating viscosity and dissipation, and presents the governing equations of hydrodynamics. Despite the complexity, the chapter offers insights into the unique properties of quasicrystals, making it a valuable resource for researchers in the field.

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5. Chapter 4. Case Study of Equation of State in Several Structured Fluids

   Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

   The chapter delves into the fundamental properties of the equation of state in structured fluids, emphasizing its importance in condensed matter physics and general relativity. It presents case studies in superfluid liquid helium, liquid crystals, and explosion physics, highlighting the challenges and limitations of existing models. The chapter then introduces possible equations of state, drawing from thermodynamics theories and classical approximations. It explores the applications of these equations to the dynamics of soft-matter quasicrystals, discussing both compressible and incompressible models. The chapter concludes by noting the significance of the coefficient of isothermal compression in the dynamics of soft-matter quasicrystals and the need for experimental verification of computational results.

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6. Chapter 5. Poisson Brackets and Derivation of Equations of Motion in Soft-Matter Quasicrystals

   Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

   The chapter begins by introducing the Poisson bracket method for deriving the equations of motion of soft-matter quasicrystals, building on the foundational concepts of soft-matter quasicrystals established in previous chapters. The Poisson bracket method, originating from Landau and further developed by researchers such as Martin et al. and Lubensky et al., is applied to derive the equations of motion for both phonons and phasons, highlighting the unique properties of each. The chapter also explores the application of the generalized Langevin equation and the mathematical significance of Lie algebra in the derivation process. The resulting equations of motion provide a robust framework for understanding the dynamic properties of soft-matter quasicrystals, setting the stage for further research and applications in materials science and condensed matter physics.

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7. Chapter 6. Oseen Theory and Oseen Solution

   Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

   This chapter introduces the Oseen theory and its solution, focusing on liquid dynamics and the limitations of the Stokes equations. It discusses the Stokes paradox and the Oseen modification, which provides a successful solution for flow past a cylinder in viscous, incompressible fluids. The Oseen solution is significant for the study of soft-matter quasicrystals, offering a framework for understanding their dynamics despite their compressibility. The chapter concludes with a detailed derivation of the velocity field and stresses around a cylinder, highlighting the approximate nature of the Oseen solution and its value as a testing stage for soft-matter solutions.

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8. Chapter 7. Dynamics of Soft-Matter Quasicrystals with 12-Fold Symmetry

   Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

   This chapter delves into the dynamic properties and governing equations of soft-matter quasicrystals with 12-fold symmetry, building upon previous discussions. It focuses on the quantitative analysis of the dynamic equation system, incorporating hydrodynamics and thermodynamics. The chapter presents solutions to initial-boundary value problems for two-dimensional and three-dimensional fields, highlighting the importance of the equation of state and the complexity of the equations. It also discusses applications such as dislocation problems and flow past obstacles, showcasing the unique behavior of soft-matter quasicrystals compared to conventional liquids. The chapter concludes with a discussion on the validity and applicability of the generalized dynamics of soft-matter quasicrystals, setting the stage for further research in the field.

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9. Chapter 8. Dynamics of 10-Fold Symmetrical Soft-Matter Quasicrystals

   Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

   This chapter delves into the intricate dynamics of 10-fold symmetrical soft-matter quasicrystals, a topic that has garnered significant interest due to their unique properties. Unlike their 12-fold symmetry counterparts, these quasicrystals exhibit a strong coupling between phonons and phasons, which is crucial for understanding their behavior. The chapter also explores the transient dynamics of these quasicrystals, providing insights into wave propagation and the response of matter to waves. Additionally, the chapter discusses dislocation solutions and the effect of fluid on these solutions, offering a thorough analysis of the complex interactions within these materials. The inclusion of numerical analysis and results further enhances the chapter's value, making it a must-read for those interested in the dynamics of soft-matter quasicrystals.

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10. Chapter 9. Dynamics of 8-Fold Symmetric Soft-Matter Quasicrystal Models

    Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

    The chapter delves into the dynamics of 8-fold symmetric soft-matter quasicrystals, a lesser-explored area compared to 12-, 18-, and 10-fold symmetric quasicrystals. It begins by establishing basic equations and constitutive laws for phonons and phasons, using symmetry groups and the Poisson bracket method. The chapter then discusses elastic static solutions for dislocations, highlighting the complexity of these solutions compared to lower symmetries. It introduces transient dynamics analysis and numerical solutions for specimens under impact tension and flow past obstacles, showcasing the strong phason-phonon coupling unique to 8-fold symmetric quasicrystals. The chapter also extends to three-dimensional systems, providing governing equations and discussing the nature of wave propagation and diffusion. Additionally, it introduces an incompressible model for simplified mathematical treatment, though acknowledging the loss of some physical insights. The chapter concludes by emphasizing the importance of the strong phonon-phason coupling and the need for further study into three-dimensional problems and thermodynamic stability.

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11. Chapter 10. Dynamics of 18-Fold Symmetric Soft-Matter Quasicrystals

    Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

    The chapter delves into the groundbreaking discovery of 18-fold symmetric soft-matter quasicrystals, which differ significantly from previously reported quasicrystals with lower symmetries. The study discusses the unique properties of these quasicrystals, including their elasticity and dynamics, and how they can be described using a six-dimensional embedding space. The authors also explore the decoupling of phonon and phason fields, providing insights into the behavior of these quasicrystals under various conditions. Additionally, the chapter presents analytical solutions for dislocations and discusses the implications of these findings for the understanding and application of quasicrystals in materials science.

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12. Chapter 11. The Possible 7-, 9-, and 14-fold Symmetry Quasicrystals in Soft Matter

    Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

    The chapter delves into the fascinating world of quasicrystals with 7-, 9-, and 14-fold symmetries in soft matter, focusing on their mathematical dynamics and constitutive laws. It introduces generalized dynamics models suggested by Fan, highlighting the unique couplings between phonons and phasons. The discussion includes the point group descriptions and phonon-phason constitutive laws, leading to a comprehensive set of governing equations. Notably, the chapter provides numerical solutions for the dynamics of 14-fold symmetry quasicrystals under dynamic loading, offering unprecedented insights into their behavior. Additionally, it suggests an incompressible complex fluid model for these quasicrystals, paving the way for further research and understanding of these intriguing materials.

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13. Chapter 12. Re-Discussion on Symmetry Breaking and Elementary Excitations

    Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

    The chapter delves into the fundamental concepts of symmetry breaking and elementary excitations, with a particular focus on the Landau theory. It introduces and discusses phonons, phasons, and fluid phonons, which are essential excitations in quasicrystals. The author explores the interactions and implications of these excitations in both solid and soft-matter quasicrystals, highlighting the crucial role they play in understanding the dynamics and behavior of these complex structures. By revisiting the pioneering work of scientists like Planck, Einstein, Debye, and Landau, the chapter provides a historical context and builds on their theories to offer a contemporary perspective on quasicrystals. The discussion includes the classification of two-dimensional quasicrystals and the coupling or decoupling of phonons and phasons, emphasizing their significant impact on the motion of matter. Additionally, the chapter underscores the importance of fluid phonons in soft-matter quasicrystals, demonstrating their agreement with classical fluid dynamics. This comprehensive analysis not only validates the theoretical framework but also opens avenues for further research in the field of quasicrystals.

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14. Chapter 13. An Application to the Thermodynamic Stability of Soft-Matter Quasicrystals

    Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

    The chapter explores the thermodynamic stability of soft-matter quasicrystals, which have been observed in various materials such as liquid crystals, polymers, and colloids. Unlike solid quasicrystals formed under rapid cooling, soft-matter quasicrystals are created through self-assembly processes involving supramolecules and other building blocks. The stability of these quasicrystals has been a topic of debate since their discovery, with previous studies focusing on effective free energy approaches. This chapter introduces an alternative method using generalized dynamics and extended free energy, which simplifies the analysis and provides quantitative results based on material constants that can be experimentally measured. The stability is found to depend on the fluid effect, phonons, phasons, and their interactions. The chapter also discusses the stability of different types of soft-matter quasicrystals, including those with 12-, 8-, 10-, and 18-fold symmetries, highlighting the importance of the fluid effect and the coupling between various excitations. The approach presented offers a systematic, direct, and simple method for determining stability, with potential applications in materials science and related fields.

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15. Chapter 14. Applications to Device Physics—Photon Band Gap of Holographic Photonic Quasicrystals

    Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

    This chapter delves into the applications of device physics, specifically focusing on the photon band gap of holographic photonic quasicrystals (PQCs). It begins by highlighting the superiority of soft-matter quasicrystals over solid quasicrystals in terms of photonic band gap (PBG) properties. The construction of two-dimensional PQCs with different rotational symmetries using multi-beam holographic interference is discussed, followed by an investigation of their PBG properties using the finite element method (FEM). The results demonstrate that 10-fold and 12-fold PQCs are more effective in forming PBGs. The chapter also includes practical fabrication methods, such as single-prism holographic lithography, to create 10-fold PQCs. Additionally, the chapter introduces the concept of cholesteric liquid crystals (CLCs) and their unique optical properties, providing a broader context for understanding photonic structures. This chapter is essential for researchers and engineers aiming to design and fabricate advanced photonic integrated devices.

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16. Chapter 15. Possible Applications to General Soft Matter

    Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

    This chapter delves into the dynamics of general soft matter, building upon the unified perspective introduced in previous chapters on soft-matter quasicrystals. By excluding the complexities of quasiperiodic symmetry, the authors simplify the governing equations to focus on the fluid and elastic fields and their interactions. The chapter outlines the mass and momentum conservation equations, symmetry breaking rules, and elastic energies, providing a basis for understanding the dynamics of two-dimensional soft matter. It also discusses the modification and supplement of these equations for specific soft matter types, such as liquid crystals and polymers, and offers insights into solving the dynamics of soft matter. The chapter concludes by highlighting the importance of the equation of state in soft matter dynamics and the need for further research into material constants. It serves as a foundation for studying complex fluid dynamics in general soft matter, with practical applications in various industries.

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17. Chapter 16. An Application to Smectic A Liquid Crystals, Dislocation, and Crack

    Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

    This chapter delves into the intricate dynamics of smectic A liquid crystals, focusing on the complexities of dislocation and crack problems. It begins by reviewing the well-known Kleman-Pershan solution for screw dislocations, identifying critical mathematical and physical errors in this classical approach. The authors then present a novel solution that corrects these mistakes, offering valuable insights into the energy and stress fields associated with dislocations. This new solution not only resolves the paradoxes surrounding the classical approach but also provides a foundation for understanding the plastic crack problem in soft matter. The chapter concludes by suggesting a rupture criterion for determining the stability of liquid crystals, emphasizing the importance of experimental validation. Throughout, the chapter highlights the need for further research into the mathematical solvability of initial-boundary value problems in soft matter hydrodynamics.

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18. Chapter 17. Conclusion Remarks

    Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

    This chapter delves into the dynamics of soft-matter quasicrystals, extending the work on solid quasicrystals by Lubensky and symmetry group theory by Hu. It presents initial solutions to the governing equations of soft-matter quasicrystals, validating the theory and showcasing applications in quantum electronics and integrated photonics, particularly through the photonic band-gap. The chapter also discusses thermodynamic stability and general soft matter applications, highlighting the need for further experimental verification. Notably, it touches on the correlation between Frank-Kasper phase and soft-matter quasicrystals, and mentions the observation of tenfold symmetry soft-matter quasicrystals, pending open experimental reports.

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19. 18. Correction to: Introduction on Elasticity and Hydrodynamics of Solid Quasicrystals

    Tian-You Fan, Wenge Yang, Hui Cheng, Xiao-Hong Sun

    The chapter addresses a critical correction in the understanding of elasticity and hydrodynamics of solid quasicrystals, which is fundamental for the accurate modeling and experimental validation of their behavior. By providing the correct equation, it ensures that researchers and professionals can rely on precise data for their studies and applications. This correction is vital for advancing the field of soft-matter quasicrystals, enabling more reliable predictions and innovations in materials science.

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