Static and Dynamic Configurations in Liquid Crystals

本项目主要研究液晶材料研究中出现的数学问题，特别是对静态构型流动模式的理解。研究的一个领域是在不同（强锚定和弱锚定）边界条件和外场条件下液晶能量最小解的缺陷和分子构型。这些能量包括对向列型液晶的osee - frank能量，对近晶液晶的Landau-de Gennes能量等。获奖者将应用偏微分方程解的正则性理论来解决这些问题。液晶材料在不同条件下也会经历复杂的相变。系统地研究了不同状态下构型的稳定性和分岔现象。最后，研究了不同类型液晶的动态特性。最近的实验和理论结果表明，分子方向场与流速场之间存在着有趣的耦合关系。因此，同时考虑这些矢量场并研究它们之间的相互作用是很重要的。提议者将研究这一耦合问题的理论和计算方面。重点是说明流对不同配置的影响。材料的各向异性特性将导致流动表现出非牛顿特征，这反过来对应于缺陷和不同的分子模式。作为物质的一种特殊形态，液晶在一个多世纪前就被发现了。液晶显示器（LCD）在工业上得到了广泛的应用。这类材料的基础研究非常重要。例如，缺陷的存在不仅影响器件的制造过程，而且提供了一种新的显示方式，其中缺陷有助于提高材料在外场下的精度和记忆性。本研究将集中于液晶的数学模型，特别是可能的缺陷配置及其随时间的行为。对静态和动态缺陷结构的研究将有助于改进此类器件的建模和设计。此外，提高对液晶的理解对更广泛的重要材料类别有好处，如聚合物和生物材料。

This project is focused on mathematical problems arising from thestudy of liquid crystal materials, especially on understandingstatic configurations flow patterns. One area of the research is to study the defect and molecule configurationsof minimizing solutions of the liquid crystal energies underdifferent (both strong and weak anchoring) boundary conditions andexternal fields. Such energies include the Oseen-Frank energy for nematicliquid crystals, the Landau-de Gennes energy for smectic ones, etc.. Theawardeee will apply the regularity theory for solutions of partial differential equations to these problems. Liquid crystal materials also experience complicatedphase transitions under different conditions. A systematic effort will bemade to study the stability of the configurations and bifurcationphenomena in different regimes. Finally this research addresses the dynamic properties of different types of liquid crystals. Recentexperimental and theoretical results indicate that there is an interestingcoupling between the molecule director field and the flow velocity field.Therefore it is important to consider these vector fields simultaneouslyand study the interaction between them. The proposer will investigateboth the theoretical and computational aspects of this coupling problem.The focus is to illustrate the flow effects on the different configurations. The anisotropic characteristics of the material will cause the flow to exhibit non-Newtonian features which in turn corresponds to defects anddifferent molecule patterns.As a special phase of matter, liquid crystals were discoveredmore than a century ago. Liquid crystal displays (LCD)have been widely used in industries. The fundamental researchof such materials is very important. For example, the presence of defectsnot only affects the manufacturing procedure of the devices, but alsoprovides a new kind of display where the defects contributes to higheraccuracy and memory of the material under external fields. This research willfocus on mathematical models for liquid crystals, in particular on possible defectconfigurations and their behavior over time. This study of static and dynamic defect structures will help to improve the modeling and designof such devices. Moreover, an improved understanding of liquid crystals has benefits for broader classes of materials that are important, such aspolymeric and biological materials.