Analysis of Mathematical Models of Materials

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The proposed project is aimed at analyzing mathematical models for liquid crystalline and superconducting materials focusing on the electro-magnetic,electromechanical, and electro-optical interactions that take place withinthem. Mathematical models for superconducting materials to be analyzedare based on the Ginzburg-Landau energy, and the project will include aninvestigation of the qualitative behavior of mathematical solutions which are important for applications such as high-temperature, anisotropic superconducting materials, in the presence of applied electric and magnetic fields. Mathematical models for liquid crystals to be analyzed are based on De Gennes' models incorporating the Oseen-Frank energy terms as well asterms accounting for smectic layering (present in chiral materials),elasticity, and the influence of applied electric andmagnetic fields. A hierarchy of problems will be studied incorporatingmore and more complex nonlinear interactions. Special emphasis will be given to the analysis of scaling up from meso-scale models where thephysics is described to macro-scale models used in applications. Techniquesfrom mathematical modeling, partial differential equations, and finiteelasticity will be employed. The project will result in a better understanding of the responses ofhigh-temperature superconducting materials and liquid crystallinematerials to applied magnetic and electric fields based on their microscopic structure. Applications will include the design of appropriate superconductorsfor power transmission and the creation of powerful magnetic fields. Applications of the liquid crystal research include the design of smaller, faster, and more accurate optical switches, nanoscale electro-mechanicaldevices, and liquid crystal displays.