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 within them. Mathematical models for superconducting materials to be analyzed are based on the Ginzburg-Landau energy, and the project will include an investigation 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 as terms accounting for smectic layering (present in chiral materials), elasticity, and the influence of applied electric and magnetic fields. A hierarchy of problems will be studied incorporating more and more complex nonlinear interactions. Special emphasis will be given to the analysis of scaling up from meso-scale models where the physics is described to macro-scale models used in applications. Techniques from mathematical modeling, partial differential equations, and finite elasticity will be employed.
The project will result in a better understanding of the responses of high-temperature superconducting materials and liquid crystalline materials to applied magnetic and electric fields based on their microscopic structure. Applications will include the design of appropriate superconductors for 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-mechanical devices, and liquid crystal displays.
