Analysis of Nonlinear Systems Modeling Partially Ordered Materials

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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.