Nonlinear PDEs for Soft Matter Systems

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PhillipsDMS-0604839 The investigator and his colleague analyze mathematicalmodels, described by nonlinear partial differential equations, ofsoft matter systems, namely liquid crystalline andsuperconducting materials. They focus on electro-magnetic,optical, and mechanical interactions as well as thermo-mechanicalexchanges within them. They seek to identify qualitativefeatures in the solutions for these models including phasetransitions, the nature of defects, and their development. Techniques from mathematical modeling, partial differentialequations, the calculus of variations, and finite elasticity areemployed. The models are highly nonlinear and expressed in termsof nonconvex, second order energies. Developing analysis and pdemethods to address these features is part of the project. Forliquid crystals a major goal is to carry out a mathematicalanalysis for the event of electrically driven optical switchingbetween bistable ferroelectric states in a smectic C* material. A second goal is to carry out an analytic study of the defectstructure that appears in liquid crystals brought on by appliedstresses and phase transitions. In elastomers the investigatorsstudy electrically and thermally driven mechanical deformationsin dry and swollen elastomers. In superconductivity they examinecurrent patterns characterized by the formation and evolution ofvortex filaments within them. For low temperature materials theyinvestigate both stationary and dynamic features of vortexfilaments in three dimensional bodies, using the time-dependentGinzburg-Landau equations. For high temperaturesuperconductivity they study the vortex structure in stationarysolutions to a d-wave model as well as current patterns insolutions for the layered Lawrence-Doniach model. Understanding physical interactions in soft matter systemsis central to the design process where one strives to makesmaller, faster, and more accurate devices. Liquid crystals areused to make optical switches, nano devices, and displays. Liquid crystal elastomers have been proposed by physicists tomodel artificial muscles and other biological applications. Superconductors are used for small scale sensors such as squids(superconducting quantum interference detectors) and to makepowerful magnets. The approach the investigators undertake is toanalyze mathematical models that describe soft matter systems interms of nonlinear partial differential equations. The workleads to predictions of qualitative features that the actualmaterials should possess and insights that should be useful forthe design process as well as for the implementation of numericalsimulations.