Particle Packing Problems

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This award will support work on fundamental aspects of particle packing problems. There are many open questions that will be pursued. Can sphere packings with a diversity of density and disorder be identified in d-dimensional Euclidean space? What are the densest packings of spheres in dimension greater than three? What are the densest packings of nonspherical objects in two and three dimensions? Can random packings ever fill space more densely than ordered packings (implying disordered or ``glassy" ground states)? For non-tiling nonspherical particles, can an upper bound on the maximal density be derived that is always strictly less than unity? Specifically, the following seven general areas will be explored: (1) jammed sphere packings with anomalously low densities; (2) identification of growing length scales in the approach to the glass transition; (3) dense spheres packings in three-dimensional space with a size distribution; (4) jammed disordered polyhedron packings; (5) maximally dense packings of nonspherical particles; (6) maximally dense sphere packings in high dimensions; and (7) studies of jamming on the unit sphere in various space dimensions. Among other activities, this award will be used to support the research of graduate students seeking their Ph.D. degrees.Packing problems, such as how densely solid objects fill space, have fascinated people since the dawn of civilization, and continue to intrigue scientists because of their connection to a host of problems that arise in the physical sciences, mathematics, engineering and biology. While optimal packing problems are intimately related to solid states of condensed matter, disordered sphere packings have been employed to model the glassy state of matter and granular media. Dense packings of nonspherical particles in low dimensions are relevant to problems in materials science and biology. Sphere packings in high dimensions is of importance in communications theory. Pure mathematicians have a longstanding interest in packing problems. This is a multidisciplinary project that links the applied mathematics, statistical and condensed-matter physics, materials science, engineering, biology, communications, and pure mathematics communities. Important scientific advances and practical outcomes that could potentially emerge from the proposed research include a deeper understanding of the nature of low-temperature states of matter (e.g., crystal ground states, the glass transition and the mysterious occurrence of disordered ground states), granular media, mechanically stable low-weight network solids, identification of new alloy crystal structures and new high-pressure phases of matter, and insights concerning the manner in which biological cells or organelles pack.