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

CaldererDMS-0909165这个项目研究铁电液晶和水凝胶的建模和分析，目的是研究由这种材料制成的器件的开关和滞后。液晶项目的重点是新发现的弯曲铁电相，这种相能够维持比传统固体铁电化合物低一个数量级的极化电场。水凝胶研究的目标是建立一种适用于脉动给药装置设计的环状膜模型。这种类型的弹性膜也与燃料电池和过滤设备的研究非常相关。这两个问题都有独特的现象学和数学问题，例如在两个不同的状态之间呈现一阶相变行为。在铁电液体晶体中，它们对应于具有不同光学性质的相反的偏振态；一个主要目标是优化它们之间的开关速度。了解和控制迟滞现象可能有助于实现最佳切换。这也需要对弯曲芯液晶流的流变性有一个很好的了解。数学问题涉及非凸性、亚稳性以及麦克斯韦方程与流体流动和弹性的耦合。总体而言，铁电性是液晶领域中一个富有现象学、建模和数学挑战的领域，这些挑战在很大程度上仍未被探索。通常的情况是，看似不同的科学和技术问题有共同的数学基础。在这个项目中，研究人员解决了其中的两个问题，在药理学和燃料电池以及光学设备中的应用，如视频和高速互联网交换。其基本思想是对“切换”进行建模和数学研究，目标是设计更快、更高效的设备。在制药应用中，开关连接两种相关状态：释放和不释放要给药的药物。研究人员的一个目标是模拟一种循环药物释放膜。这种周期性释放的装置被认为在激素疗法中特别重要，在激素疗法中，匹配自然的身体激素周期可能与替代激素本身一样重要。在质子交换膜燃料电池中，在阳极产生的正氢离子穿过膜，从燃料中分离出来并产生电子。铁电开关连接光学上截然不同的状态。在这些例子中，能量损失是开关动力学的共同特征，与周期性器件中的周期延长一起。这种现象称为磁滞，存在于许多机械和磁性系统中。研究人员利用其他领域的滞后数学知识作为理解药理学和液晶设备的一种方法。对研究生和本科生的辅导以及其他教育活动与工作的研究方面交织在一起并结合在一起。

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.