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

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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.