Analysis of Nonlinear Systems Modeling Partially Ordered Materials

本研究的目标是分析偏序材料的数学模型。重点放在电磁、电光和机电相互作用上，强调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扎。了解这些相互作用是设计过程的核心，人们努力制造更小、更快和更精确的器件。液晶和弹性体被用来制造光学开关、纳米器件和显示器。它们还被物理学家用来模拟人造肌肉。超导体被用来传输电力和制造极强的磁铁。将考虑到日益复杂的相互作用，为每种材料考虑一系列数学问题。将特别强调从中观模式到宏观模式的放大，在这些模式中，物理相互作用被描绘为应用中使用的宏观模式。PI试图找到模型解的定性特征。在超导中，PI将试图展示具有大的静止电流(钉扎电流)的解。人们对近晶液晶感兴趣的一个原因是，与介电液晶相比，光学状态之间的转换时间显著减少。这里的一个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、偏微分方程组和有限弹性等技术。建议的研究重点是研究液晶中电-机械和电光相互作用的数学模型，以及描述超导体电磁性质的模型。关于液晶，我们讨论的应用是由光学和生物系统驱动的，重点是光学开关和纳米器件的设计，包括探测器、执行器和阀门。这些应用广泛应用于人造肌肉和显示器液晶显示器等广泛的产品中。对这些材料的分析模型的开发和分析是对材料科学家和物理学家进行的实验研究的重要补充。我们正在解决的现象的一个例子是液晶显示器中的开关时间。最近使用近晶液晶的研究表明，与标准(向列)液晶相比，光学状态之间的转换时间显著减少。我们研究了这些材料的数学问题的层次结构，考虑到越来越复杂的相互作用，目标是捕捉正确的物理，并准确地估计和比较所考虑的模型的开关时间。

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.