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

Calderer

DMS-0909165

This project deals with modeling and analysis of

ferroelectric liquid crystals and hydrogels, with the goal of

studying switching and hysteresis of devices made of such

materials. The project on liquid crystals focuses on newly

discovered bent core ferroelectric phases that are capable of

sustaining polarization fields of just one order of magnitude

below that of traditional solid ferroelectric compounds. The

goal of the work on hydrogels is to model a cyclic membrane

appropriate for application to the design of pulsating drug

delivery devices. Such types of elastic membranes are also very

relevant to the study of fuel cells and filtrating devices. Both

problems share distinctive phenomenology and mathematical issues,

such as presenting a first order phase transition behavior

between two distinguished states. In ferroelectric liquid

crystals, they correspond to the oppositely polarized states with

distinct optical properties; a main goal is to optimize the

switching speed between them. Understanding and controlling

hysteresis may help achieve optimal switching. This also

requires a good understanding of the rheology of bent core liquid

crystal flow. Mathematical issues involve non-convexity,

metastability, and coupling of Maxwell's equations with fluid

flow and elasticity. Overall, ferroelectricity is an area of

liquid crystals rich in phenomenology, modeling, and mathematical

challenges that remain largely unexplored.

It is often the case that seemingly disparate problems of

science and technology have common mathematical underpinnings.

In this project, the investigator addresses two of these

problems, with applications in pharmacology and fuel cells as

well as in optical devices, such as video and high speed Internet

switching. The underlying idea is the modeling and mathematical

study of 'switching,' with the goal of designing faster and more

efficient devices. In pharmaceutical applications, a switch

connects the two relevant states: release and non-release of the

drug to be administered. A goal of the investigator is to model

a cyclic drug delivery membrane. Such periodically releasing

devices are believed to be especially relevant in hormone

therapies, where matching the natural body hormone cycle is

perhaps as important as the replacement hormone itself. In

proton exchange membrane fuel cells, positive hydrogen ions

produced at the anode travel through the membrane, separating

from the fuel and yielding electrons. A ferroelectric switch

connects optically distinct states. In these examples, energy

loss is a common feature of switching dynamics, together with

cycle lengthening in periodic devices. This phenomenon, known as

hysteresis, occurs in many mechanical and magnetic systems. The

investigator harnesses the mathematical knowledge of hysteresis

in other fields as an approach to understanding pharmacological

and liquid crystal devices. Mentoring of graduate and

undergraduate students as well as other educational activities is

intertwined and integrated with the research aspects of the work.