Applied Mathematics, Modeling, and Experimental and Computational Analysis of Liquid Crystals

Calderer 9704714该项目由多学科活动办公室和数学与物理科学局数学科学司以及工程局土木工程与机械系统司共同资助。拟议的研究在工业应用的背景下，解决并结合了“液晶”和“铁性材料”(固体和流体)研究中的数学和计算问题。这项提议的一个基本主题是发展对在存在外场的液晶和铁性材料中观察到的静态和流动模式的理解。(例如，在处理铁性固体时，一个目标是达到较低的矫顽场临界值，以抑制损耗；在铁电液晶中，人们还寻求提高器件开关速度。)虽然“织构”和“缺陷”的出现往往会阻碍结构制造工艺的结果，但它也可能在光学应用中带来期望的特征，如提高器件的精度和显示记忆。在这个框架内，人们可以就固体和液晶系统中的织构和缺陷的控制制定一个相关的共同问题，目的要么是加强这种结构，要么是消除它们。这个研究团队的目标是通过探索广泛的方法和技术来帮助理解这些问题：作为一个整体的液晶和铁性系统的建模、数学、计算和实验。计划修改并统一一些可用的描述，以便以更普遍的方式理解铁电性质。这样的研究最终可能会提出一种设计新的、更好的铁电材料(固体和/或液晶)的方法，具有“大的极化”和“小的强迫磁场”。该建议还涉及(非牛顿)液晶流的各个方面，例如各种区域的缺陷和不稳定性。课程将特别关注对薄膜加工流程的分析以及对自由边界问题的研究。奇点和分叉理论的结果，以及对对称性所起作用的评价，已经被整合到有限元技术中，以建立一些经典牛顿流动的基本不稳定机制。这些数值技术现在已经足够成熟，可以应用于具有更复杂物理和多参数的更具技术重要性的流动的研究。拟议的研究涉及“液晶”和“铁性材料”(固体和液体)的数学和计算建模，在其工业应用的背景下。在处理液晶时，我们打算构建和分析“智能显示设备”的数学模型，包括铁电系统和复合材料。其主要目标是在增大显示器尺寸的同时保持良好的光学分辨率，并实现较低的电场或磁场开关值。作为我们研究计划的一部分，我们打算从铁性固体领域中提取已经存在的数学和物理信息，以帮助我们研究铁电液晶。反过来，液体分子的有序性和更大的自由度可以为改进铁电“传感器”和“换能器”的建模和设计提供良好的反馈，特别是为了这类器件的小型化。我们研究的主要问题之一是研究如何使用聚合物液晶材料来设计典型的固体器件。(前一种材料的制造和激活成本往往较低)。从不同的角度来看，铁电液体流动建模中的数学问题出现在此类材料的制造过程中，并带来了一类全新的数学问题，这些问题与传统上在牛顿流体流动研究中出现的问题有关。总体而言，我们打算将独立的研究领域聚集在一起，并在利用物理和现象学类比的基础上，在它们之间进行数学和计算方法的转移。

Calderer 9704714 This project being jointly funded by the Office of Multidisciplinary Activities and by the Division of Mathematical Sciences of the Directorate for Mathematical & Physical Sciences and also by the Division of Civil and Mechanical Systems of the Directorate for Engineering. The proposed research addresses and combines mathematical and computational issues in the study of "liquid crystals" and "ferroic materials", both solids and fluids, within the context of industrial applications. One underlying theme of this proposal is to develop an understanding of static and flow patterns observed in liquid crystals and ferroic materials in the presence of external fields. (For instance, in dealing with ferroic solids, a goal is to achieve lower critical values of coercive fields in order to curb dissipation; in ferroelectric liquid crystals one also seeks to increase device switching speed.) While the occurrence of "texture" and "defects" tends to hinder the outcome of structural manufacturing processes, it may also bring out desirable features in optic applications, such as improvement of the accuracy and display memory of the devices. Within this framework, one can formulate a relevant common problem in terms of the control of texture and defects in solid and liquid crystal systems, with either the intent to enhance such structures or to eliminate them. It is the goal of this research team to contribute towards the understanding of such questions by exploring a broad spectrum of methods and techniques: modeling, mathematical, computational and experimental in liquid crystal and ferroic systems as a whole. It is planned to revise and, perhaps, unify some of the available descriptions so that the ferroelectric nature can be understood in a more universal manner. Such a study could ultimately suggest a way of designing new and better ferroelectric materials (solids and/or liquid crystals) with "large polarization" and "small coerci ve fields". This proposal also addresses aspects of (non-Newtonian) liquid crystal flow such as defects and instabilities of various regimes. Special attention will be devoted to the analysis of processing flows of thin films as well as to the study of free-boundary problems. Results from singularity and bifurcation theory, and an appreciation of the role played by symmetry, have been integrated within the finite-element technique to establish the fundamental instability mechanisms of a number of classical Newtonian flows. These numerical techniques are now mature enough to apply to the study of more technologically important flows with more complex physics and multiple parameters. The proposed research deals with mathematical and computational modeling of "liquid crystals" and "ferroic materials" (both, solids and liquids), within the context of their industrial applications. In dealing with liquid crystals we intend to construct and analyze mathematical models of "smart display devices", involving ferroelectric systems as well as composites. The main goal is to preserve good optical resolution with increasing display sizes, and achieve low switching values of electricor magnetic fields. As part of our research plan, we intend to draw already existing mathematical and physical information from the field of ferroic solids, in order to help us in the studies of ferroelectric liquid crystals. In turn, the ordering properties and larger degrees of freedom of liquid molecules may provide good feedback for improving the modeling and design of ferroelectric "sensors" and "transducers", with the special aim towards miniaturization of such devices. One of the main issues of our research is to study how to use polymeric liquid crystal materials to design typically solid devices. (The former materials are often cheaper to manufacture and to activate). From a different point of view, mathematical problems in flow modeling of ferroelectric liquids arise in the manufacturing processes of such materials, and bring a whole new class of mathematical questions that relate to those that traditionally arise in studies of Newtonian fluid flow. Overall, we intend to bring together independent fields of research and carry out a transfer of mathematical and computational methods among them based on exploiting physical and phenomenological analogies.