Topology and Kinetics of Network Structures
Martin E. Glicksman
Materials Science and Engineering Department
University of Florida
The mathematical problem of space filling and growth in network structures is both basic and of practical interest to physicists, materials scientists, and biologists. Network cells may represent various physical entities, such as crystal grains in polycrystals, bubbles in foams, atomic coordination shells in liquids, or even biological cells in tissues. The theory to be described is based on representing network cells by regular polyhedral proxies, called average N-hedra (ANHs) that satisfy space filling (topology) and thermodynamics (local equilibrium) over relevant length scales. The topological analysis used yields kinetic laws that predict average growth rates for isotropic networks as a function of discrete topological parameters, such as the network’s number densities of neighbor contacts, triple points, quadra-junctions, or total triple-line length. New results found for area shrinkage rates of three-dimensional polyhedral networks extend the now half-century old von Neumann ‘n-6’ law that provides well- founded predictions for quasi-2-d grain growth in polycrystalline films or thin shells. Predictions based on topological theory agree with Evolver computations of large 3-d grain assemblies, published independently by Kraynik et al., Wakai, and by Cox. Analytic relations derived for ANHs can provide benchmarks for testing numerical simulations of network behavior, as well as guides for designing quantitative kinetic experiments, and, eventually, critical statistical measures for characterizing network microstructures.