Bending, Twisting and Packing: Geometry and Soft Materials

弯曲、扭转和包装：几何形状和软材料

• 批准号: 0547230
• 负责人: Randall Kamien
• 金额: $ 55万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-03-01 至 2013-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0547230&HistoricalAwards=false
• 关键词: Bending; Twisting; Packing; Geometry; Soft

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.***

本项目将探讨软凝聚物质理论中的问题，重点是那些以几何方式提出和解决的问题。主要有两点。第一部分探讨了层状系统的非线性弹性和缺陷的能量学。PI在微晶学理论方面的最新进展表明，层间距和曲率之间存在着微妙的相互作用。这可以用来得到非线性弹性的精确解，也可以用来构造当分子倾向于固有曲率时的变分解。本工作与有关扭晶界相和由弯曲核介质组成的层状相的新结果有联系。PI将探索三周期曲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。该理论的关键要素是纯排斥电位与面积最小化、空间填充结构或蜂巢之间的联系。这种相互作用与表明密排晶格更受青睐的熵论证并列。PI将利用泡沫中理想多面体的新结果，发展这些思想来形成晶格填料的平均场理论。他将用蒙特卡罗模拟来补充这项工作，以测试这些想法。本文的第一部分主要讨论了拟模学的非线性理论。虽然仿电磁学的弹性理论与超导体朗道理论密切相关，但其现象学却截然不同。从异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和令人惊讶的行为。这里提出的工作将集中在一个内在的几何公式的理论，可用于研究缺陷的配置，纯粹通过边界条件。这种几何方法允许PI将叶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当前实验努力的数据和与这些团体的互动。该建议的第二个重点是通过发展平均场方法来解决这些问题，进一步建立干泡沫和硬球体物理之间的联系。总之，这些问题将导致材料几何这一新兴领域的进步。更广泛的影响和推广这项研究计划涉及许多领域，包括化学、物理和数学。在过去的几年里，pi的研究综合了这些领域的思想，特别是几何在材料中的作用。PI根据为博尔德学院开发的关于软凝聚态物理的课堂笔记撰写了一篇教学评论文章。这是这项工作的一个持续主题。除了在泡沫理论、填料理论和密实学方面取得进展外，PI还将为材料界带来几何领域的最新思想和成果，并将使数学界面临软物质领域出现的一些挑战。幸运的是，这两个领域都非常活跃，我们有理由相信，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