Topics in the Theory of Elastic Networks and Soft-Matter Physics

技术概述：该奖项支持理论凝聚态物理广泛领域的理论研究和相关教育活动，特别关注机械刚性的性质和起源。PI将研究弹性响应和声子结构在一系列软盘或近软盘模型网络。这种网络出现在各种物理系统中，从网状玻璃到刚性软球的填料，到控制活细胞刚性的半柔性聚合物网络，再到具有不寻常特性的伸缩系统，当拉伸时，它们会在垂直于拉伸方向的方向上膨胀。提高对控制这些系统的力学和弹性响应的基础科学的理解有可能导致新材料和可能的新疾病治疗方法。大约在150年前，詹姆斯·克拉克·麦克斯韦（James Clerk Maxwell）对控制电、磁和光学现象的方程进行了最后的修饰，他提出了框架，以确定一个由连接的支柱组成的框架（例如可能在桥上发现的框架）是否机械稳定。这个框架今天仍然在建筑和微生物等不同的领域得到广泛应用。他特别指出了框架处于等静力状态的条件，也就是说，在这种情况下，框架有足够的支柱来支撑外部负载。该项目的重点是在麦克斯韦中心力刚度阈值附近的弹簧控制模型周期网络。PI将研究并尝试扩展一类新的二维最大形变晶格的成员，这些晶格的体模量消失，泊松比等于- 1，特别强调存在于表面瑞利波而不是体声子中的有限晶格中的零模式的性质和起源。他将研究这些体系的性质在加入次近邻力和弯曲力以及稀释后是如何变化的。拟周期和随机而非周期均衡格也将被研究，包括二维的随机菱形和五边形对称彭罗斯格及其在三维的二十面体推广。这些晶格可以通过各种方式进行操作，这应该允许对声子波函数的性质进行控制研究，特别是低能量的声子波函数，接近均衡极限，包括它们如何在空间中定位以及它们在多大程度上是表面波或体波。PI将研究二维和三维半柔性聚合物网络的一系列模型，包括二维kagome晶格的稀释和扭曲版本，其中增加了弯曲力以保持稳定，以及新构建的三维晶格，该晶格由无限长的直线和交联组成，这些交联与不超过四个其他交联连接在一起，就像实际系统中的情况一样。最后一个要研究的主题是弯曲力如何影响刚性渗透，在这种情况下，刚性框架在拆除支柱后变得松软。该奖项将有助于在一个高度跨学科的环境中培养年轻科学家，并使理论家和实验家以及不同学科的科学家之间广泛接触。非技术概述：该奖项支持凝聚态物理广泛领域的理论研究和相关教育活动，该领域的研究范围涉及各种各样的物质——从液体到结晶固体，从泡沫到钢梁，从橡胶到活细胞，从绝缘体到超导体。该研究项目的重点是研究刚性的性质和来源，从玻璃珠或沙粒的颗粒包装到赋予活细胞形式的细胞骨架，这是它们运动装置的重要组成部分。想象一下，用冰棍棍和末端的无摩擦销钉连接成一个框架。如果两根木棍用一根大头针连接在一起，它们就可以绕着大头针自由旋转：它们形成了一种软盘结构，可以在没有能量消耗或机械力的情况下扭曲。但是，连接在一个等边三角形中的三根木棍是刚性的：它们可以刚性地平移和旋转，但如果不弯曲或拉伸木棍，就不能改变三角形的形状。这个简单的观察可以推广到更复杂的结构，比如桥梁和建筑物：一个梁框架是软的，除非有足够数量的连接梁，否则它在机械上是不稳定的。1864年，詹姆斯·克拉克·麦克斯韦（James Clerk Maxwell）提出了控制电磁和光的方程，他提出了一套简单的数学规则，用于确定一般坐标系的稳定性。值得注意的是，在可见长度尺度的建筑结构中出现的结构图案也出现在微观长度尺度的凝聚态物质系统中，麦克斯韦的思想可以直接应用于它们。该奖项的研究将探索弹性刚度如何在微观网络中发展，从而对抗剪切和压缩力的稳定性。它将特别探索软盘扭曲的本质-例如它们是否延伸到整个样品，定位在特定点附近，或仅限于表面-在描述随机结晶固体和网络玻璃的模型中，准晶体在旋转五分之一圈时呈现相同的模式-普通晶体中不允许的对称性，生物聚合物网络是长链状分子的网络，彼此交叉，和其他材料。我们将要研究的一组特别有趣的材料有一个不寻常的特性，当它们沿着一个方向拉伸时，它们会膨胀而不是沿着垂直方向收缩。该奖项将有助于在一个高度跨学科的环境中培养年轻的科学家，在这个环境中，理论家和实验家以及不同学科的科学家之间有广泛的联系。PI是暑期学校、会议和研讨会的组织者和顾问委员会成员。

TECHNICAL SUMMARY:This award supports theoretical research and associated educational activities in the broad field of theoretical condensed matter physics with a particular focus on the nature and origin of mechanical rigidity. The PI will study elastic response and phonon structure in a range of floppy or nearly floppy model networks. Such networks occur in a variety of physical systems from network glasses, to packings of rigid soft spheres, to networks of semi-flexible polymers that control the rigidity of living cells, to auxetic systems that have the unusual property that when stretched, they expand in the direction perpendicular to the direction of stretch. Improved understanding of the fundamental science controlling the mechanics and elastic response of these systems has the potential to lead to new materials and possible new treatments for disease.Almost 150 years ago James Clerk Maxwell, who put the final touches on the equations that govern electric, magnetic, and optical phenomena, developed the framework, still very much in use today in fields as disparate as architecture and micro-biology, to determine whether a frame composed of linked struts, such as might be found in a bridge, is mechanically stable. He identified in particular the conditions under which a frame is isostatic, that is, under which there are just enough struts for the frame to support external loads. The project is focused on controlled model periodic networks of springs near the Maxwell central-force rigidity threshold. The PI will study, and attempt to expand the members of, a new class of two-dimensional maximally auxetic lattices, whose bulk modulus vanishes and whose Poisson ratio equals minus one, with a particular emphasis on the nature and origin of zero modes in finite lattices that reside in the surface Rayleigh waves rather than in bulk phonons. He will investigate how the properties of these systems change upon addition of next-nearest-neighbor and bending forces and upon dilution. Quasiperiodic and random rather than periodic isostatic lattices will also be studied, including random rhombus tilings and pentagonally symmetric Penrose tilings in two-dimensions and their icosahedral generalizations to three dimensions. These lattices can be manipulated in various ways that should allow for controlled studies of the nature of phonon wavefunctions, particularly low-energy ones, near the isostatic limit, including how they are localized in space and to what extent they are surface waves or bulk waves. The PI will study a range of models for networks of semi-flexible polymers both in two and three dimensions, including diluted and twisted versions of the kagome lattice in two-dimensions with bending force added for stabilization and a newly constructed three-dimensional lattice consisting of infinitely long straight lines with crosslinks that are connected to no more than four other crosslinks as is the case in real systems. The last topic to be investigated is how bending forces affect the rigidity percolation at which a rigid frame becomes floppy upon the removal of struts. This award will contribute to the training of young scientists in a highly interdisciplinary environment with broad contacts between theorists and experimentalists and among scientist of different disciplines. NONTECHNICAL SUMMARY:This award supports theoretical research and associated educational activities in the broad field of condensed matter physics, a field whose purview is the vast varieties of matter - from liquids to crystalline solids, from foams to steel girders, from rubber to living cells, from insulators to superconductors. This research project is focused on the nature and origin of rigidity in classes of materials ranging from granular packing of glass beads or sand grains to the cytoskeleton that gives form to living cells and is an essential part of their locomotion apparatus. Imagine a frame constructed by joining popsicle sticks with frictionless cotter pins through holes at their ends. If two sticks are joined at with a single pin, they will be free to rotate about that pin: They form a floppy structure that can be distorted without energy cost or mechanical force. But three sticks joined together in an equilateral triangle are rigid: They can be rigidly translated and rotated, but the triangular shape cannot be altered without bending or stretching the sticks. This simple observation generalizes to more complicated structures like bridges and buildings: a frame of beams is floppy and is not mechanically stable unless it has a sufficient number of links joining the beams. In 1864, James Clerk Maxwell, who gave us the equations governing electromagnetism and light, developed a simple set of mathematical rules for determining the stability of general frames.Remarkably, the structural motifs that occur in architectural structures at visible length scales also occur in condensed matter systems at microscopic length scales, and the ideas of Maxwell can be applied directly to them. Research in this award will explore how elastic rigidity, and thus stability against shear and compressional forces, develops in microscopic networks. It will explore in particular the nature of floppy distortions - for example whether they extend throughout the sample, are localized near particular points, or are restricted to the surface - in models that describe random crystalline solids and network glasses, quasicrystals which assume the same pattern when rotated one-fifth of a turn - a symmetry not allowed in ordinary crystals, biopolymer networks which are networks of long chainlike molecules that crisscross each other, and other materials. A particularly interesting set of materials that will be studied have the unusual property that when stretched along one direction they expand rather than contract along the perpendicular direction. This award will contribute to the training of young scientist in a highly interdisciplinary environment with broad contacts between theorists and experimentalists and among scientist of different disciplines. The PI is serves as an organizer of and on advisory board of summer schools, conferences and workshops.