Nonlinear theory of defects in nematic liquid crystals; Phase transition and flow phenomena

This is a summary of some recent mathematical advances in the theory of defects in nematic liquid crystals. It also includes some new results concerning disclinations (line defects) which have not been published elsewhere.     

Studies of layering and chirality of smectic A* liquid crystals

We discuss mathematical and physical aspects of the phase transition from nematic to smectic A liquid crystals. The first approach deals with analyzing a model obtained from the Maier-Saupe theory of nematic by taking into account that elongated liquid crystal molecules present distinguishable ends. Moreover, we represent long range microscopic interactions by means of nonlocal free energy functionals. The smectic configurations emerge as solutions of the extended nematic theory, through a modulation process. The second part of the article deals with energy minimization of the de Gennes free energy for smectic A* liquid crystals, and with the study of uniform twist grain boundary (TGB) structures. The goal is to mathematically justify parameter regions of the phase diagram of the transition between nematic and smectic A liquid crystals. Both approaches complement each other from the point of view that, while the first one deals with mechanisms causing layer arrangements, the second approach focuses on how chirality and layer effects interact, in a system with preassumed periodicity. The A* notation refers to chiral liquid crystals.     

Virial inversion and density functionals

We prove a novel inversion theorem for functionals given as power series in infinite-dimensional spaces. This provides a rigorous framework to prove convergence of density functionals for inhomogeneous systems with applications in classical density function theory, liquid crystals, molecules with various shapes or other internal degrees of freedom. The key technical tool is the representation of the inverse via a fixed point equation and a combinatorial identity for trees, which allows us to obtain convergence estimates in situations where Banach inversion fails.     

Continuous theory of dislocations and disclinations in a two-dimensional medium

A system of equations describing mobile defects in a two-dimensional Cosser at continuum, i.e. in a medium whose motion is determined by the displacement field and rotation field independent of it, is obtained.

The basic equations of the static theory /1–5/ and dynamic continuous theory /6–12/ of defects (dislocations and disclinations) are known for a three-dimensional medium, obtained by a variety of methods. A dislocation model of the misalignment surfaces used in describing the Martensitic transformations /2, 13/ is proposed. The dislocation representations were used in /14–16/ to describe the grain boundaries, and the difference dislocations within the boundaries of separation were studied in /17, 18/. The dislocation structure of internal boundaries of separation was described in /19, 20/ using the differential geometry characteristics (torsion and curvature tensors, non-holonomic object) of three-dimensional media. Surface dislocations and disclinations of the separate Volterra distortions-type were studied in /21/, with liquid crystals and various biological objects indicated as the suitable areas of application of these concepts.     

Skyrmion States in Chiral Liquid Crystals

We analyze static configurations for chiral liquid crystals in the framework of the Oseen–Frank theory. In particular, we find numerical solutions for localized axisymmetric states in confined chiral liquid crystals with weak homeotropic anchoring at the boundaries. These solutions describe the distortions of twodimensional skyrmions, known as either spherulites or cholesteric bubbles, which have been observed experimentally in these systems. We outline relations to nonlinear integrable equations and use the relations to study the asymptotic behavior of the solutions. Using analytic methods, we build approximate solutions of the equilibrium equations and analyze the generation and stabilization of these states in relation to the material parameters, external fields, and anchoring boundary conditions.     

基于液晶引流效应的上板游离态液晶盒数值计算研究

以小分子液晶Leslie-Ericksen理论为基础,建立液晶盒上基片游离状态的计算模型,在给出边界初始条件的基础上,运用二阶Runge-Kutta(龙格-库塔)法与中心差分法对方程组进行时空离散,通过MATLAB编写计算程序,调整计算参数得出液晶盒厚度、液晶盒两端所施加电场参数对液晶引流的影响.结果表明:液晶指向矢的大小随液晶盒上下基板所施加电场的交变而交变,并引起液晶引流速度大小交变;随着液晶盒厚度的增加,液晶盒上基片在一个周期内移动的位移也在增加;液晶盒两端所施加电场的占空比基本上不影响液晶盒上基片速度的大小,对液晶盒上基片速度最大值出现的时间点影响很大;通过与实验数据相对比,液晶盒上基片位移数值大小在同一个数量级,运动轨迹比较吻合.     

Possible splitting of energy levels of atoms or ions in octahedral crystals due to sets of equivalent neighbouring defects

Divalent ions when doped in alkali halide single crystals will go in alkali sites and form associate pairs with neighbouring cation vacancies or other impurity defects. As there will be sets of equivalent positions for such associated pairs, their energy levels will be degenerate. The reducible representations of these associated pairs involving first to sixth nearest neighbours have been reduced to the corresponding irreducible representations of the point group O _h , for cases having rock-salt structure as well as for cases having cesium chloride structure. It is pointed out that if there is a rapid hopping of the vacancies or the defects in the crystal one may expect because of the possible interactions in the system, a splitting of the energy levels of the pairs as indicated by the different irreducible representations. It is stated that this kind of splitting may be expected also in systems in which associated magnetic pairs are formed by two neighbouring spins.     

应用图论分析与最优化理论来数据挖掘大规模水牛普里昂蛋白结构数据

图论、最优化理论显然在蛋白质结构的研究中大有用场. 首先, 调查/回顾了研究蛋白质结构的所有图论模型. 其后, 建立了一个图论模型: 让蛋白质的侧链来作为图的顶点, 应用图论的诸如团、 $k$-团、 社群、 枢纽、聚类等概念来建立图的边. 然后, 应用数学最优化的现代摩登数据挖掘算法/方法来分析水牛普里昂蛋白结构的大数据. 成功与令人耳目一新的数值结果将展示给朋友们.     

On nematic liquid crystals with variable degree of orientation

Energy minimizing static configurations of nematic liquid crystals with variable degree of orientation can be described by a map which minimizes certain degenerate variational integrals. This mathematical model was recently posed by J. L. Erickson. In this paper, we shall prove the existence, local uniqueness, and global regularity of minimizers. We shall also estimate the size of possible defects.     

进化博弈决策机制设计综述

进化博弈论是一门交叉性强的综合性理论,在国内外已得到广泛研究和应用.本文系统论述了进化博弈的决策机制及其特点,指出了进化博弈研究的趋势,进化博弈论可作为中国科技工作者学习、研究和应用的有力工具.     

Blow-up of the Smooth Solution to the Compressible Nematic Liquid Crystal System

In this paper, we investigate the blow-up of the smooth solutions to a simplified Ericksen-Lesile system for compressible flows of nematic liquid crystals in different dimensional case. We prove that whether the smooth solution of the Cauchy problem or the initial-boundary problem to the nematic liquid crystal system will blow up in finite time. The main technique is the construction of the functional differential inequality.     

固体平板上超薄水膜去湿润过程的分子动力学模拟

运用分子动力学模拟研究固体表面超薄水膜的失稳和破裂过程．结果表明薄膜中小扰动将失稳,并在初始阶段线性增长．但固体和液体的相互作用对扰动的初期增长影响较小．最小厚度的下降导致薄膜发生破裂．此后破裂边缘以一定的动态接触角后退．与宏观理论预测一致,边缘半径随时间的变化与时间平方根成正比．若固液相互作用较强,将引起破裂时间延迟,动态接触角减小,固体表面附近的液体密度增加．     

LOW MACH NUMBER LIMIT OF A COMPRESSIBLE NON-ISOTHERMAL NEMATIC LIQUID CRYSTALS MODEL

In this paper, we study the low Mach number limit of a compressible nonisothermal model for nematic liquid crystals in a bounded domain. We establish the uniform estimates with respect to the Mach number, and thus prove the convergence to the solution of the incompressible model for nematic liquid crystals.     

Global energy conservative solutions to a system of variational wave equations

This paper is concerned with a system of variational wave equations which is the Euler–Lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts. The global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy is established by using the method of energy-dependent coordinates and the Young measure theory.     

Effects of CZSi furnace modification on density of grown-in defects

During large diameter Czochralski silicon growth, heat zone and argon flow influence the formation of defects in silicon crystal by changing the distribution of temperature. Different silicon crystals with various density of grown-in defects were grown by replacing the popular heater with the composite heater and changing the popular argon flow into a controlled flow. The experimental results have been explained well by the numeric simulation of argon flow.     

Stability of the vortex defect in the Landau–de Gennes theory for nematic liquid crystals

We analyze the radially symmetric solution corresponding to the vortex defect (the so-called melting hedgehog) in the Landau–de Gennes theory for nematic liquid crystals. We prove the existence, uniqueness and stability results of the melting hedgehog.     

Time‐periodic solution to the compressible nematic liquid crystal flows in periodic domain

In this paper, we consider the time‐periodic solution to a simplified version of Ericksen‐Leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals with a time‐periodic external force in a periodic domain in . By using an approach of parabolic regularization and combining with the topology degree theory, we establish the existence of the time‐periodic solution to the model under some smallness and symmetry assumptions on the external force. Then, we give the uniqueness of the periodic solution of this model.     

Weak complicial sets I. Basic homotopy theory

This paper develops the foundations of a simplicial theory of weak ω-categories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets provides a common generalisation of the theories of (strict) ω-categories, Kan complexes and Joyal's quasi-categories. We generalise a number of results due to the current author with regard to complicial sets and strict ω-categories to provide an armoury of well behaved technical devices, such as joins and Gray tensor products, which will be used to study the weak ω-category theory of these structures in a series of companion papers. In particular, we establish their basic homotopy theory by constructing a Quillen model structure on the category of stratified simplicial sets whose fibrant objects are the weak complicial sets. As a simple corollary of this work we provide an independent construction of Joyal's model structure on simplicial sets for which the fibrant objects are the quasi-categories.     

A New Notion of Defect for Holomorphic Curves and Their Sequences

In this paper, we study transcendental curves in ⁿ and also infinite sequences of rational curves using methods of the value distribution theory of holomorphic mappings and, in particular, the notions of Nevanlinna and Valiron defects. In contrast to the classical case, we study the defects of points in ⁿ rather than of divisors. It is shown how the main notions of the theory should be changed to make them meaningful in this context. Analogs of the main theorems are proved. It is established that the sets of defective points are sparse in the sense of an adequately introduced Hausdorff h-measure.     

Computability of String Functions Over Algebraic Structures Armin Hemmerling

We present a model of computation for string functions over single-sorted, total algebraic structures and study some basic features of a general theory of computability within this framework. Our concept generalizes the Blum-Shub-Smale setting of computability over the reals and other rings. By dealing with strings of arbitrary length instead of tuples of fixed length, some suppositions of deeper results within former approaches to generalized recursion theory become superfluous. Moreover, this gives the basis for introducing computational complexity in a BSS-like manner. Relationships both to classical computability and to Friedman's concept of eds computability are established. Two kinds of nondeterminism as well as several variants of recognizability are investigated with respect to interdependencies on each other and on properties of the underlying structures. For structures of finite signatures, there are universal programs with the usual characteristics. For the general case of not necessarily finite signature, this subject will be studied in a separate, forthcoming paper.