Modeling, Analysis and Applications of Coupled Elasticity and Liquid Crystal Effects

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CaldererDMS-0909165 This project deals with modeling and analysis offerroelectric liquid crystals and hydrogels, with the goal ofstudying switching and hysteresis of devices made of suchmaterials. The project on liquid crystals focuses on newlydiscovered bent core ferroelectric phases that are capable ofsustaining polarization fields of just one order of magnitudebelow that of traditional solid ferroelectric compounds. Thegoal of the work on hydrogels is to model a cyclic membraneappropriate for application to the design of pulsating drugdelivery devices. Such types of elastic membranes are also veryrelevant to the study of fuel cells and filtrating devices. Bothproblems share distinctive phenomenology and mathematical issues,such as presenting a first order phase transition behaviorbetween two distinguished states. In ferroelectric liquidcrystals, they correspond to the oppositely polarized states withdistinct optical properties; a main goal is to optimize theswitching speed between them. Understanding and controllinghysteresis may help achieve optimal switching. This alsorequires a good understanding of the rheology of bent core liquidcrystal flow. Mathematical issues involve non-convexity,metastability, and coupling of Maxwell's equations with fluidflow and elasticity. Overall, ferroelectricity is an area ofliquid crystals rich in phenomenology, modeling, and mathematicalchallenges that remain largely unexplored. It is often the case that seemingly disparate problems ofscience and technology have common mathematical underpinnings. In this project, the investigator addresses two of theseproblems, with applications in pharmacology and fuel cells aswell as in optical devices, such as video and high speed Internetswitching. The underlying idea is the modeling and mathematicalstudy of "switching," with the goal of designing faster and moreefficient devices. In pharmaceutical applications, a switchconnects the two relevant states: release and non-release of thedrug to be administered. A goal of the investigator is to modela cyclic drug delivery membrane. Such periodically releasingdevices are believed to be especially relevant in hormonetherapies, where matching the natural body hormone cycle isperhaps as important as the replacement hormone itself. Inproton exchange membrane fuel cells, positive hydrogen ionsproduced at the anode travel through the membrane, separatingfrom the fuel and yielding electrons. A ferroelectric switchconnects optically distinct states. In these examples, energyloss is a common feature of switching dynamics, together withcycle lengthening in periodic devices. This phenomenon, known ashysteresis, occurs in many mechanical and magnetic systems. Theinvestigator harnesses the mathematical knowledge of hysteresisin other fields as an approach to understanding pharmacologicaland liquid crystal devices. Mentoring of graduate andundergraduate students as well as other educational activities isintertwined and integrated with the research aspects of the work.