Nonlinear PDEs for Soft Matter Systems

PhillipsDMS-0604839 这位研究者和他的同事分析了由非线性偏微分方程描述的软物质系统（即液晶和超导材料）的物理模型。 他们专注于电磁，光学和机械的相互作用，以及热mechanicalexchanges内。 他们试图识别这些模型的解决方案中的定性特征，包括相变，缺陷的性质及其发展。采用了数学建模、偏微分方程、变分法和有限弹性力学等技术。 该模型是高度非线性的，并表示为非凸的，二阶能量。 开发分析和pdemethods来解决这些问题是该项目的一部分。 对于液晶，主要目标是对近晶C * 材料中的电驱动光开关事件进行理论分析。第二个目标是对液晶中由外加应力和相变引起的缺陷结构进行分析研究。 在弹性体中，电致和热致机械变形是研究干燥和溶胀弹性体的。 在超导性中，他们研究了以其中涡旋丝的形成和演化为特征的电流模式。 对于低温材料，他们用含时的Ginzburg-朗道方程研究了三维物体中涡丝的静态和动态特性。 对于高温超导性，他们研究了d波模型的静态解中的涡旋结构，以及层状劳伦斯-多尼亚赫模型的解中的电流模式. 理解软物质系统中的物理相互作用是设计过程的核心，在设计过程中，人们努力制造更小，更快，更精确的设备。 液晶被用来制造光学开关、纳米器件和显示器。液晶弹性体已经被物理学家提出来用于人造肌肉的模型和其他生物学应用。超导体用于小型传感器，如squids（超导量子干涉探测器）和制造强大的磁铁。 研究者所采取的方法是分析用非线性偏微分方程描述软物质系统的数学模型。 这项工作导致预测的定性功能，实际材料应该拥有和见解，应该是有用的设计过程，以及为实现numericalsimulations。

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.