Analysis of Partially Ordered Materials

鲍曼DMS-1109459 该项目的目标是分析由描述液晶和超导材料的非线性偏微分方程描述的数学模型，重点是液晶薄膜，铁电液晶材料，极化调制液晶材料，如双芯光纤和高温超导材料。 研究人员确定描述这些现象的数学模型的解决方案的定性特征，包括材料的各个阶段的图案形成以及缺陷的位置，性质和数量。 该模型是高度非线性的，并表示在非凸，二阶能量。 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，物理学家已经提出通过用液晶材料的薄膜涂覆每个胶体颗粒来构建胶体颗粒的晶格，使得膜的缺陷充当天然结合位点（即，化学连接位置）。 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尺寸物体，其可用于基于粒子的生物测定和催化，以及用于光子带隙材料。 铁电液晶用于制造光学开关、纳米器件和显示器。 物理学家提出了液晶双芯纤维来模拟人造肌肉和其他生物应用。 高温超导体用于小型传感器，如squids（超导量子干涉探测器）和制造强大的磁体。 该项目承担了这些应用程序的模型的数学研究。

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.