Bending, Twisting and Packing: Geometry and Soft Materials

这个项目将探索软凝聚物质理论中的问题，重点放在那些提出并用几何方法解决的问题上。主要有两个主题。第一个主题是探讨层状系统的非线性弹性和缺陷的能量学。PI在近晶理论方面的最新进展表明，层间距和曲率之间存在微妙的相互作用。这可以用来得到非线性弹性力学的精确解，并且可以用来构造当分子的本征曲率有利时的变分解。这项工作与扭曲晶界相和由弯曲核介元组成的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。第二个是对大分子和纳米晶体自组装理论的新补充。这一理论的关键要素是纯粹的排斥势与面积最小、空间填充的结构或蜂窝之间的联系。这种相互作用与表明紧密堆积晶格受到青睐的熵论点并列。PI将发展这些想法，利用泡沫塑料中理想化多面体的新结果，形成晶格填充的平均场理论。他将用蒙特卡罗模拟来补充这项工作，以检验这些想法。智力价值建议的第一部分集中在近晶学的非线性理论上。虽然近晶学的弹性理论与超导体的朗道理论密切相关，但唯象理论却截然不同。从反常弹性到螺杆缺陷之间的幂定律相互作用，近晶中间相潜在的旋转不变性导致了微妙和令人惊讶的行为。这里提出的工作将集中在理论的内在几何公式上，该理论可用于研究缺陷组态，纯粹通过边界条件。这种几何方法允许PI将叶和孤子的数学结合在一起来研究这些系统，并提供了一种新的方法来研究这个系统。拟议的工作将受益于当前实验工作的数据以及与这些小组的互动。该提案的第二个主旨通过开发一种平均场方法来解决这些问题，进一步加强了干泡沫和坚硬球体之间的物理联系。总之，这些问题将导致材料几何这一新兴领域的进步。广泛的影响和外延这项研究计划横跨许多领域，包括化学、物理和数学。在过去的几年里，PI研究综合了这些领域的想法，特别是几何在材料中的作用。国际和平研究所根据为博尔德学院编写的软凝聚物质物理学课堂讲稿，撰写了一篇教学评论文章。这是这项工作的一个持续主题。除了泡沫塑料、填充和近晶理论的进步外，PI还将把当前几何方面的想法和结果带给材料界，并将使数学界面临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，有理由相信这样的研究努力将把所研究的问题、解决方法和进一步研究的方向交织在一起。

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.***