The investigator and his colleague analyze mathematical models, described by nonlinear partial differential equations, of soft matter systems, namely liquid crystalline and superconducting materials. They focus on electro-magnetic, optical, and mechanical interactions as well as thermo-mechanical exchanges within them. They seek to identify qualitative features in the solutions for these models including phase transitions, the nature of defects, and their development. Techniques from mathematical modeling, partial differential equations, the calculus of variations, and finite elasticity are employed. The models are highly nonlinear and expressed in terms of nonconvex, second order energies. Developing analysis and pde methods to address these features is part of the project. For liquid crystals a major goal is to carry out a mathematical analysis for the event of electrically driven optical switching between bistable ferroelectric states in a smectic C* material. A second goal is to carry out an analytic study of the defect structure that appears in liquid crystals brought on by applied stresses and phase transitions. In elastomers the investigators study electrically and thermally driven mechanical deformations in dry and swollen elastomers. In superconductivity they examine current patterns characterized by the formation and evolution of vortex filaments within them. For low temperature materials they investigate both stationary and dynamic features of vortex filaments in three dimensional bodies, using the time-dependent Ginzburg-Landau equations. For high temperature superconductivity they study the vortex structure in stationary solutions to a d-wave model as well as current patterns in solutions for the layered Lawrence-Doniach model.
Understanding physical interactions in soft matter systems is central to the design process where one strives to make smaller, faster, and more accurate devices. Liquid crystals are used to make optical switches, nano devices, and displays. Liquid crystal elastomers have been proposed by physicists to model artificial muscles and other biological applications. Superconductors are used for small scale sensors such as squids (superconducting quantum interference detectors) and to make powerful magnets. The approach the investigators undertake is to analyze mathematical models that describe soft matter systems in terms of nonlinear partial differential equations. The work leads to predictions of qualitative features that the actual materials should possess and insights that should be useful for the design process as well as for the implementation of numerical simulations.
