Dresden 2003 – scientific programme

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M: Metallphysik

M 1: Hauptvortrag Glicksman

M 1.1: Invited Talk

Monday, March 24, 2003, 09:30–10:00, IFW A

Topological Considerations for 3-D Network Systems — •Martin E. Glicksman — Materials ScienceEngineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA — Alexander von Humboldt Preissträger, Institut für Metallkunde und Metallphysik, RWTH-Aachen, Germany

The problem of space filling in dynamical network structures, such as liquids, glasses, polycrystals, foams, and certain biological tissues is both basic and of long standing. A geometric constraint is developed for networks that specifies allowed pathways, or orbits, through H-K phase space, where H is the mean curvature, and K is the Gaussian curvature of the polyhedral forms comprising the network. Algebraic topology imposes additional requirements that the integral Gaussian curvature for each network forms is a conserved quantity for compact objects of zero geometrical genus. These objects may represent atomic coordination shells in liquids, cells in biological tissues, grains in polycrystals, and bubbles in a foam. The theory developed is based on representing network elements, such as shells, cells, grains and bubbles, as uniform trihedral polytopes, then adding curvatures to satisfy local thermodynamic equilibrium on relevant length scales. Our analysis yields several discrete dispersion laws that predict the volumetric and areal growth rates for polyhedral elements comprising a network micro(nano)structure as a function of their topological class. The new results extend to three dimensions the now half-century old von Neumann’s law that provides basic topological requirements for space filling in two dimensions. The new three-dimensional dispersion laws should prove useful for constructing more accurate models of liquid and glass structures, grain growth, and foam coarsening kinetics. They also help to clarify several long-standing issues, such as Kelvin’s conjecture on space-filling criteria required for three-dimensional network structures. Algebraically derived topological relations can provide analytical benchmarks to test numerical simulations, to guide further quantitative experiments on network dynamics in three-dimensional structures, and to assist in deriving important statistical measures for glasses, liquids, polycrystalline materials, foams, and some biological tissues.