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 Merit The 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 Outreach This 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.
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Tying a knot in a piece of string is something we learn in kindergarten. What does it mean, however, to tie a knot in a field? In our work, we have explored this question. A whirlpool in the ocean is not just a line where there is no water – it is also a place where the water swirls around. More importantly, we could measure the whirlpool, its "vorticity," by measuring the flow of the water at a distance, avoiding the fate of mythical sailors by getting too close. The knotted field is the whole fluid flow around the vortex or set of vortices. Even without seeing the vortices, possibly tied into knots, the whole ocean has the knot imprinted upon it. The physics and mathematics of this phenomena is one focus of our work. On a more applied side, porous materials are of great industrial use because of their high surface areas. They are essential to filtering of water in ceramic filters and scrubbing of car exhaust through catalytic converters. Their use in chemical synthesis, particularly biochemicals is increasingly important. The experimental group at the University of Colorado Boulder discovered a new porous material that assembles spontaneously. We have developed the requisite tools to characterize, model, and control these systems through electric, magnetic, and optical fields. We are now working on questions of stability and robustness of these complex molecular orderings.
