Analysis of Partially Ordered Materials

项目目标是分析描述液晶和超导材料的非线性偏微分方程所描述的数学模型，重点研究液晶薄膜、铁电液晶材料、弯芯光纤等极调制液晶材料和高温超导材料。研究人员确定了描述这些现象的数学模型的解决方案的定性特征，包括材料的不同阶段的模式形成以及缺陷的位置、性质和数量。模型是高度非线性的，用非凸二阶能量表示。开发偏微分方程的方法来分析这些特征是该项目的一部分。该项目与材料应用有关。例如，物理学家提出用一层液晶材料薄膜覆盖胶体粒子，使其缺陷充当粒子之间的天然键合位点（即化学连接位置），从而构建胶体粒子的晶格。这种自组装允许创建类似于有机化学分子特征的功能化微米级物体，可用于基于颗粒的生物分析和催化，以及光子带隙材料。铁电液晶用于制造光学开关、纳米器件和显示器。液晶弯芯纤维已被物理学家提出用于模拟人造肌肉和其他生物应用。高温超导体被用于小型传感器，如鱿鱼（超导量子干涉探测器）和制造强力磁铁。该项目对这些应用的模型进行数学研究。

BaumanDMS-1109459 The goal of the project is the analysis of mathematical models described by nonlinear partial differential equations describing liquid crystal and superconducting materials, focusing on thin liquid crystal films, ferro-electric liquid crystal materials, polar-modulated liquid crystal materials such as bent-core fibers, and high-temperature superconducting materials. The investigators identify qualitative features of solutions to the mathematical models describing these phenomena, including pattern formation of various phases of the materials and the location, nature, and number of defects. The models are highly nonlinear and expressed in terms of nonconvex, second-order energies. Developing methods in partial differential equations to analyze these features is part of the project. The project is related to applications in materials. For example, physicists have proposed to build lattices of colloidal particles by coating each with a thin film of liquid crystal material in such a way that the film's defects act as natural bonding sites (i.e., chemical linking locations) between the particles. This self-assembly allows the creation of functionalized micron-sized objects similar to the molecules characteristic of organic chemistry, which can be used for particle-based bioassays and catalysis, and for photonic band-gap materials. Ferro-electric liquid crystals are used to make optical switches, nano-devices, and displays. Liquid crystal bent-core fibers have been proposed by physicists to model artificial muscles and other biological applications. High-temperature superconductors are used for small-scale sensors such as squids (superconducting quantum interference detectors) and to make powerful magnets. The project takes up the mathematical investigation of models for these applications.