Bending, Twisting and Packing: Geometry and Soft Materials

本专题将探讨软凝聚态理论中的问题，重点是那些提出并以几何方式解决的问题。主要有两个方面：第一个是探索层状系统的非线性弹性和缺陷的能量学。PI最近在近晶理论上的进展表明，层间距和曲率之间存在微妙的相互作用。这可以用来得到精确解的非线性弹性，并可用于构建变分的解决方案时，内禀曲率是有利的分子。本文的工作与扭晶界相和层状相的新结果相联系。PI将探索三周期表面的分解作为变分计算的起点。第二个是大分子和纳米晶体自组装理论的新补充。这个理论的关键要素是纯粹的排斥势和面积最小化、空间填充结构或蜂窝之间的联系。这种相互作用与熵参数并列，表明密排晶格是有利的。PI将发展这些想法，制定一个平均场理论的格子包装，采用新的结果理想化多面体泡沫。他将补充这项工作与蒙特卡洛模拟，以测试这些想法。智力MeritThe第一部分的建议侧重于非线性理论的smectics。尽管近场学的弹性理论与超导体的朗道理论密切相关，但其唯象却截然不同。从反常弹性到螺旋缺陷之间的幂律相互作用，近晶中间相的潜在旋转不变性导致了微妙而令人惊讶的行为。这里提出的工作将集中在一个固有的几何公式的理论，可用于研究缺陷的配置，纯粹通过边界条件。这种几何方法使PI能够将叶理和孤子的数学结合起来研究这些系统，并提出了一种新的方法来研究这个系统。拟议的工作将受益于目前的实验努力和与这些群体的互动的数据。该提案的第二个重点是通过发展一种平均场方法来解决这些问题，进一步加强了干泡沫和硬球物理学之间的联系。这些问题将共同推动材料几何学这一新兴领域的发展。更广泛的影响和影响这项研究计划涉及化学、物理和数学等多个领域。在过去的几年里，PI的研究已经综合了这些领域的想法，特别是几何在材料中的作用。PI写了一篇教学评论文章，基于为Boulder学校开发的软凝聚态物理学的讲义。这是这项工作的一个持续主题。除了在泡沫，包装和smectics理论的进展，PI将带来当前的想法和几何结果的材料社区，并将数学社区暴露在软物质中出现的一些挑战。幸运的是，这两个领域都非常活跃，有理由相信，像这样的研究工作将使所研究的问题、解决方法和进一步研究的方向相互交织。

This project will explore problems in soft-condensed matter theory with an emphasis on those that are posed and solved geometrically. There are two main thrusts.The first explores the nonlinear elasticity of layered systems and the energetics of defects. Recent progress by the PI on the theory of smectics has shown a subtle interplay between layer spacing and curvature. This can be exploited to get exact solutions to the nonlinear elasticity and can be used to construct variational solutions when intrinsic curvature is favored by the molecules. This work makes contact with new results on twist-grain-boundary phase and layered phases composed of bent-core mesogens. The PI will explore decompositions of triply-periodic surfaces as a starting point for variational calculations.The second is a new addition to the theory of self-assembly of macromolecular- and nano-crystals. The key element of this theory is a connection between purely repulsive potentials and area-minimizing, space-filling structures or honeycombs. This interaction is juxtaposed with entropic arguments that show that close-packed lattices are favored. The PI will develop these ideas to formulate a mean-field theory of lattice packings, employing new results on idealized polyhedra in foams. He will supplement this work with Monte Carlo simulations in order to test these ideas.Intellectual MeritThe first part of the proposal focuses on the nonlinear theory of smectics. Though the elasticity theory of smectics is closely related to the Landau theory of superconductors, the phenomenology is strikingly different. From the anomalous elasticity to the power-law interactions between screw defects, the underlying rotational invariance of the smectic mesophase leads to subtle and surprising behavior. The work proposed here will focus on an inherently geometric formulation of the theory that can be used to study defect configurations, purely via the boundary conditions. This geometric approach allows the PI to bring together the mathematics of foliations and solitons to study these systems and presents a fresh approach to this system. The proposed work will benefit from the data of current experimental efforts and interactions with those groups. The second thrust of the proposal furthers the connection between the physics of dry foams and hard spheres by developing a mean-field approach to these problems. Together, these problems will lead to progress in the emerging area of materials geometry.Broader Impact and OutreachThis research proposal spans many fields, including chemistry, physics, and mathematics. Over the past few years the PIs research has synthesized ideas from these fields, particularly the role of geometry in materials. The PI has written a pedagogical review article, based on lecture notes that were developed for the Boulder School on the Physics of Soft Condensed Matter. This is a continuing theme of this work. In addition to progress in the theory of foams, packing, and smectics, the PI will bring current ideas and results in geometry to the materials community and will expose the mathematics community to some of the challenges that arise in soft matter. Fortunately, both fields are very active and there is reason to believe that research efforts like this will intertwine and co-mingle the problems studied, their method of solution, and the direction of further research.***