A study of morphologies in block copolymers and Langmuir films

复杂材料中出现的形态模式与其物理性质密切相关。例子包括嵌段共聚物中的层状、圆柱形和球形图案，中性和带电朗缪尔单层，以及近晶液晶薄膜。研究者和他的合作者基于统计物理第一原理衍生的数学模型来分析和预测形态模式。他们发展了处理非局部本构关系、奇异摄动和临界特征值的技术。这些数学现象被证明是自组织和模式形成的原因。研究者找到了有效的简化方法，将原来的无限维问题精确地简化为可管理的有限维问题。研究者还通过寻找和表征自由能欧拉-拉格朗日方程的一些不稳定解，研究了材料中图案成核的动态现象。这个项目扩展了我们对奇摄动变分问题的认识。它增强了经典理论，比如伽马收敛。复杂的材料，如嵌段共聚物，无处不在。用于室内装潢和床上用品的聚氨酯泡沫是由称为热塑性弹性体的多嵌段共聚物组成的，它结合了高温回弹性和低温柔韧性。普通盒式胶带使用三嵌段共聚物来实现压敏粘合。在道路建设中，嵌段共聚物与沥青混合，以减少极端低温和高温下的路面开裂和车辙。这个项目旨在加深我们对这些系统背后的数学理论的理解。它产生了表征和预测这些材料的机械、光学、电学、离子、势垒和其他物理性质的有效方法。它们帮助当今的合成化学技术生产结构精美的材料，以满足民用基础设施和制造业不断增长的需求。

Morphology patterns that appear in complex materials areintimately related to their physical properties. Examples includelamellar, cylindrical, and spherical patterns in block copolymers,neutral and charged Langmuir monolayers, and smectic liquidcrystal films. The investigator and his collaborators analyze andpredict morphology patterns based on mathematical models derivedfrom first principles of statistical physics. They developtechniques to deal with nonlocal constitutive relations, singularperturbations, and critical eigenvalues. These mathematicalphenomena are proved to be the reasons behind self-organizationand pattern formation. The investigator finds effective reductionmethods that accurately simplify the original infinite-dimensionalproblems to manageable finite-dimensional ones. The investigatoralso studies the dynamic phenomenon of pattern nucleation inmaterials by finding and characterizing some unstable solutions tothe Euler-Lagrange equations of the free energy. The projectexpands our knowledge of singularly perturbed variationalproblems. It enhances classical theories such as Gammaconvergence. Complex materials, such as block copolymers, are usedeverywhere. The polyurethane foams used in upholstery and beddingare composed of multi-block copolymers known as thermoplasticelastomers that combine high temperature resilience and lowtemperature flexibility. Common box tapes use triblock copolymersto achieve pressure-sensitive adhesion. Block copolymers areblended with asphalt in road construction to reduce pavementcracking and rutting at low and high temperature extremes. Thisproject seeks to deepen our understanding of the mathematicaltheories underlying these systems. It produces effective methodsthat characterize and predict the mechanical, optical, electrical,ionic, barrier, and other physical properties of these materials. They help today's synthetic chemistry technologies to produceexquisitely structured materials to meet an ever rising demandfrom civil infrastructure and manufacturing.