The Mathematics of Liquid Crystals - Analysis, Computation and Applications

液晶（LC）是物质的中间相或相，其物理性质介于常规固体和常规液体的物理性质之间。液晶在现代生活中无处不在，在多媒体技术、光学成像和生物医学等科学和工业领域有着广泛的应用。最大的液晶应用领域是显示技术，液晶显示器（LCD）占据了目前平板显示器市场的近90%。当优先考虑紧凑性、便携性和低功耗时，LCD是首选。LCD的性能受多种因素的复杂组合控制-外部影响、光学特性、对电场和磁场的响应、弹性效应和缺陷。至关重要的是通常知之甚少的潜在微观结构。事实上，在微观和宏观尺度之间传递信息的系统机制被认为是现代LC科学的主要挑战。数学和LC科学之间的相互作用是双重的。一方面，数学可以为液晶现象提供基本的见解，而这反过来又对控制，预测甚至工程LC特性至关重要。另一方面，液晶和液晶显示器的数学建模导致新的前沿问题，在不同的数学分支，如偏微分方程理论，拓扑学，代数几何，多尺度理论和逆问题。我的研究计划旨在（a）解决LC科学基础方面的关键数学问题，并辅以新颖的数值算法，（B）开发LC科学的跨学科方法，（c）将理论与工业LC应用相结合。这些问题是基本的科学兴趣，并有直接的关系，一类有前途的高分辨率低功耗显示器称为LCD。双稳态液晶显示器的独特之处在于，它们只需要功率来在光学对比状态之间切换，而不需要单独支持这些状态，例如Zenithally Bistable Nematic Device和Post Aligned Bistable Nematic Device。液晶显示器的数学理论有一个层次，从最详细的原子理论到最不详细的宏观（连续）理论。该领域的大多数数学工作都集中在宏观理论方法上，但仍存在一些悬而未决的问题。在我的研究计划中，我将首先在宏观理论框架中开发一个数学工具库。感兴趣的问题包括：（i）一些关键问题的几何和材料特性的双稳态和光学性能的影响和（ii）一个严格的数学理论，在LC中的缺陷。缺陷是材料中的局部缺陷的区域，并且液晶样品通常由这样的缺陷填充。缺陷在物理现象中起着至关重要的作用，但人们对它们的了解却很少。第二步是开发新的多尺度方法，将微观和宏观模型结合在一起。所提出的多尺度理论将易于分析，计算效率高，并将捕捉宏观行为的微观起源。这种方法也将应用于聚合物模拟，膜建模和肽和蛋白质建模。这些理论和数值工具将构成一个良好的理论基础，为LCD的。工业研究人员感兴趣的是理解几何形状和材料性质对（a）物理可观察状态的结构和光学性质以及（B）MEMS器件的开关特性的影响。这些问题将通过与工业界的积极合作来回答，以期优化现代液晶显示器并设计出适合特定应用的新设备。

Liquid crystals (LC) are mesophases or phases of matter with physical properties intermediate between those of conventional solids and conventional liquids. LCs are ubiquitous in modern life and have widespread applications in science and industry e.g. multimedia technology, optical imaging and bio-medicine. The largest LC application area is display technology, with liquid crystal displays (LCDs) occupying almost 90% of the current flat-panel display market. LCDs are preferred whenever compactness, portability and low power consumption are a priority. The performance of a LCD is controlled by an intricate combination of a variety of factors - external influences, optical properties, response to electric and magnetic fields, elastic effects and defects. Of key importance is the underlying microscopic structure that is often poorly understood. In fact, systematic mechanisms for transferring information between microscopic and macroscopic scales is recognized to be a major challenge for modern LC science.The interaction between mathematics and LC science is twofold. On the one hand, mathematics can give fundamental insight into liquid crystal phenomena which, in turn, is crucial for controlling, predicting and even engineering LC properties. On the other hand, the mathematical modelling of LCs and LCDs leads to novel cutting-edge problems in diverse branches of mathematics e.g. theory of partial differential equations, topology, algebraic geometry, multiscale theory and inverse problems. My research programme aims to (a) to address key mathematical questions in the foundational aspects of LC science complimented by novel numerical algorithms, (b) to develop a cross-disciplinary approach to LC science and (c) integrate theory with industrial LC applications. These problems are of fundamental scientific interest and have immediate relevance to a promising class of high-resolution low power consumption displays known as bistable LCDs. Bistable LCDs are distinctive in the sense that they require power only to switch between optically contrasting states but not to support these states individually e.g. Zenithally Bistable Nematic Device and Post Aligned Bistable Nematic Device.There are a hierarchy of mathematical theories for LCs, ranging from the most detailed atomistic theories to the least detailed macroscopic (continuum) theories. Most of the mathematical work in the field has focused on macroscopic theoretical approaches but a number of open questions remain. In my research programme, I will first develop an arsenal of mathematical tools in the macroscopic theoretical framework. The problems of interest include (i) some key questions related to the effect of geometry and material characteristics on bistability and optical properties and (ii) a rigorous mathematical theory for defects in LCs. Defects are regions of local imperfections in a material and liquid crystal samples are typically populated by such defects. Defects play a crucial role in physical phenomena and yet, they are poorly understood. The second step will be to develop new multiscale methodologies that can couple microscopic and macroscopic models together. The proposed multiscale theories will be analytically tractable, computationally efficient and will capture the microscopic origins of macroscopic behaviour. Such methodologies will also have applications to polymer simulations, membrane modelling and modelling of peptides and proteins. These theoretical and numerical tools will constitute a sound theoretical foundation for bistable LCDs. Industrial researchers are interested in understanding the effect of geometry and material properties on (a) the structure and optical properties of physically observable states and (b) the switching characteristics of the bistable devices. These questions will be answered in active collaboration with industry, with a view to optimize modern LCDs and design new devices tailored to specific applications.