Analysis of Nonlinear Systems Modeling Partially Ordered Materials

提出的研究目标是部分有序材料的数学模型分析。重点是电磁、光电和机电相互作用，强调在近晶液晶和弹性体中的铁电和压电响应以及超导体中的磁通钉接。理解这些相互作用是设计过程的核心，在这个过程中，人们努力制造更小、更快、更精确的设备。液晶和弹性体用于制造光学开关、纳米器件和显示器。它们也被物理学家用来模拟人造肌肉。超导体用于电力传输和制造极强的磁铁。考虑到日益复杂的相互作用，将考虑每种材料的数学问题层次。我们将特别强调从描述物理相互作用的中尺度模型到应用中使用的宏观尺度模型的放大。PI试图找到模型解决方案的定性特征。在超导方面，PI将尝试展示具有大固定电流（固定电流）的解决方案。人们对近晶液晶感兴趣的一个原因是，相对于介电液晶，近晶液晶在光态之间的切换时间大大减少。这里的主要目标是估计所考虑的模型的这些切换时间。将采用数学建模、偏微分方程和有限弹性等技术。建议的研究重点是研究液晶中机电和电光相互作用的数学模型和描述超导体电磁特性的模型。至于液晶，我们的应用是由光学和生物系统驱动的，重点是光开关和纳米器件的设计，包括探测器、执行器和阀门。这些应用广泛存在于人造肌肉和显示器液晶显示器等产品中。这些材料的分析模型的发展和分析是材料科学家和物理学家进行实验研究的重要补充。我们正在处理的一个例子是液晶显示器的切换时间。最近的研究表明，与标准（向列）液晶相比，近晶液晶在光学状态之间的切换时间大大减少。我们研究了这些材料的数学问题层次，考虑到越来越复杂的相互作用，目标是捕获正确的物理，并准确地估计和比较所考虑的模型的切换时间。

The goal of the proposed research is the analysis of mathematical modelsfor partially ordered materials. The focus is on electro-magnetic, electro-optical, and electro-mechanical interactions, emphasizing ferroelectric and piezoelectric responses in smectic liquid crystals and elastomers and flux pinning in superconductors.Understanding these interactions is central to the design process whereone strives to make smaller, faster, and more accurate devices. Liquid crystals and elastomers are used to make optical switches, nano devices, and displays. They have also been used by physicists to model artificial muscles. Superconductors are usedfor power transmission and to make extremely strong magnets. A hierarchy of mathematicalproblems will be considered for each material taking into account increasingly complex interactions. Special emphasis will be given to scaling-up frommeso-scale models where the physical interactions are delineated to macro-scale models used in applications. The PI seeks to find qualitative features ofsolutions for the models. In superconductivity the PI will try to exhibit solutionswith large stationary currents (pinned currents). One reason for the interest in smectic liquid crystals is the dramatic decrease in switching times between optical states relative to those for dielectric liquid crystals . A majorgoal here is to estimate these switching times for the models considered. Techniques from mathematical modeling, partial differential equations, and finite elasticity will be employed.The proposed research centers on the study mathematical models for electro--mechanical and electro-optical interactions in liquid crystals and models describing the electro--magnetic properties of superconductors. With regard to liquid crystals, the applications that we address are motivated by optical and biological systems and focuses on the design of optical switches and nanodevices, including detectors, actuators, and valves. These applications are found in a wide range of products such as artificial muscles and liquid crystal displays for monitors. The development and analyses of analytic models for these materials serves as an important complement to experimental investigations carried out by materials scientists and physicists. An example of the phenomena we are addressing is switching times in liquid crystal displays. Recent studies using smectic liquid crystals show a dramatic decrease in switching times between optical states, relative to those for standard (nematic) liquid crystals. We study a hierarchy of mathematical problems for these materials, taking into account increasingly complex interactions with the goals of capturing the correct physics and to accurately estimate and compare the switching times for the models considered.