Analysis of Nonlinear Systems Modeling Partially Ordered Materials

偏序材料非线性系统建模分析

• 批准号: 0306516
• 负责人: Daniel Phillips
• 金额: --
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-15 至 2008-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306516&HistoricalAwards=false
• 关键词: Analysis; Nonlinear; Systems; Modeling; Partially

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.

本研究的目标是分析部分有序材料的数学模型。重点是电磁，电光和机电相互作用，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。理解这些相互作用是设计过程的核心，从而努力使器件更小，更快，更精确。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，将为每种材料考虑一个层次的物理问题。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。在超导中，PI将试图展示具有大的稳定电流（钉扎电流）的解决方案。对近晶液晶感兴趣的一个原因是相对于介电液晶的光学状态之间的切换时间的显著减少。这里的一个主要目标是估计这些开关时间的模型考虑。本课程将采用数学建模、偏微分方程和有限弹性力学等技术，重点研究液晶中的机电和电光相互作用的数学模型以及描述超导体电磁特性的模型。关于液晶，我们解决的应用是由光学和生物系统的动机，并侧重于光学开关和纳米器件，包括检测器，执行器和阀门的设计。这些应用广泛应用于各种产品，如人造肌肉和监视器的液晶显示器。这些材料的分析模型的开发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。我们正在解决的现象的一个例子是液晶显示器中的切换时间。最近的研究表明，使用近晶型液晶的光学状态之间的切换时间显着减少，相对于那些标准的（x-ray）液晶。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，目标是捕获正确的物理特性，并准确估计和比较所考虑的模型的切换时间。