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。 实例包括嵌段共聚物中的层状、圆柱形和球形图案，中性和带电的朗缪尔单分子膜，以及近晶液晶膜。研究者和他的合作者基于统计物理学第一原理推导出的数学模型来分析和预测形态模式。 他们开发技术来处理非局部本构关系，奇异摄动，和关键特征值。 这些现象被证明是自组织和模式形成的原因。 研究人员发现有效的reductionmethods，准确地简化原来的无限维问题，以管理有限维的。 通过对自由能Euler-Lagrange方程的不稳定解的寻找和表征，本文还研究了材料中图形形核的动力学现象。该投影扩展了我们对奇摄动变分问题的认识。 它增强了经典理论，如伽玛收敛。 复杂的材料，如嵌段共聚物，到处都在使用。用于室内装潢和床上用品的聚氨酯泡沫由称为热塑性弹性体的多嵌段共聚物组成，该热塑性弹性体结合了联合收割机高温弹性和低温柔性。普通的盒式胶带使用三嵌段共聚物来实现压敏粘合。嵌段共聚物在道路建设中与沥青混合，以减少路面裂缝和车辙在低温和高温极端。该项目旨在加深我们对这些系统背后的数学理论的理解。 它产生了有效的方法来表征和预测这些材料的机械，光学，电学，离子，屏障和其他物理特性。它们帮助当今的合成化学技术生产精美的结构材料，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.