Mathematical Sciences: Mathematical and Numerical Problems in Material Sciences and Fluid Mechanics

9623137 E提议的工作是PI长期项目“复杂材料和复杂流体”的初始阶段。该计划是探索和发展非平衡热力学，先进的数值技术，渐近和原型问题的严格数学分析的思想，以达到对聚合物液体，高温超导体等动力学问题的坚实，连贯和实用的理解水平。PI的工作将与化学家和实验物理学家密切合作。在这个初始阶段，PI建议研究以下问题。（1）. 涉及聚合物的相分离动力学和形态学。重点是理解驱动相分离的各向同性分子相互作用与产生取向顺序的各向异性分子相互作用之间的相互作用，以及由聚合驱动的相分离。（2）. 近晶液晶的几何缺陷。(3)。复杂流动的有效而精确的数值方法。这包括直接模拟高雷诺数湍流，湍流模型和聚合物液体，如液晶。重点将放在可靠、实际和可并行使用的方法上，这些方法可立即用于加强其他方面的进展。许多技术领域都需要用聚合物溶液制造微孔泡沫和复合材料。下一代平板显示器件最有前途的候选者是在液晶预聚体溶液中通过化学驱动相分离工艺制成的。虽然已经积累了大量的实验结果，无论是柔性聚合物还是刚性聚合物，但这些过程的动力学及其对形态学的影响却知之甚少。这些问题的极端复杂性阻碍了理论的理解。目前的提议是应用数学家、化学家和物理学家共同努力解决这个问题。该建议的一个关键组成部分是使用最先进的数值方法和计算机对这些过程进行数值模拟。聚合物通常很大，尺寸不均匀，这使得这个问题与普通二元液体的相分离有着根本的不同。＊＊＊

9623137 E The proposed work is the initial phase of PI'slong term project on ``Complex Materials and Complex Fluids''. The plan is to explore and develop ideas from non-equilibrium thermodynamics, advanced numerical techniques, asymptotics and rigorous mathematical analysis for prototypical problems, to reach a solid, coherent and practical level of understanding for the dynamical issues in polymeric liquids, high temperature superconductors, etc. PI's work will be carried out in close collaboration with chemists and experimental physicists. In this initial phase, PI proposes to study the following problems. (1). Dynamics and morphology of phase separation involving polymers. The emphasis is on understanding the interplay between isotropic molecular interaction which drives the phase separation, and the anisotropic molecular interaction which gives rise to orientational order, as well as phase separation driven by polymerization. (2). Geometric defects of smectic liquid crystals. (3). Efficient and accurate numerical methods for complex flows.This includes direct simulation of high Reynolds number turbulent flows, turbulence models and polymeric liquids such as liquid crystals. The emphasis will be on reliable, practical and parallelizable methods which can be used immediately to enhance progress on other fronts. %%% Many areas of technology calls for the manufacturing of microcellular foams and composite materials out of polymer solutions. The most promising candidate for the next generation of flat panel display devices are made by chemically driven phase separation processes in a liquid crystal-prepolymer solution. While a vast amount of experimental results have been accumulated, both for flexible and rigid polymers, the dynamics of these processes and the implication to morphology is very poorly understood. Theoretical understanding is hindered by the extreme complexity of these problems. The current proposal is a coordinat ed effort between applied mathematicians, chemists and physists to tackle this problem. A key component of this proposal is the numerical simulation of these processes using state-of-the-art numerical methods and computers. The fact that polymers are often large and non-uniform in their sizes makes this problem fundamentally different from the phase separation of ordinary binary liquids. ***