Boundary motion in polyhedral space-filling networks

摘要

Kinetics, topology, and geometrical combinatorics are combined to impose spacefilling requirements on network structures comprised of polycrystalline grains, foam bubbles, or biological cells. The theory developed here centers on representing network cells as uniform N-hedra, with face curvatures that satisfy Young-Laplace thermodynamic equilibria at contact faces and triple lines. The analysis yields analytic kinetic relations that predict the volumetric growth rates for irregular polyhedral cells comprising a 3-dimensional network microstructure. These results extend to three dimensions the von Neumann-Mullins law, which provides the well-known kinetic relation that is valid for tessellations in two dimensions. The 3-d kinetic laws derived here may prove useful for constructing more accurate models of grain growth and foam coarsening, and for clarifying several longstanding issues on space-filling criteria required for three-dimensional networks. The availability of algebraically-derived topological relations might also provide benchmarks to test numerical simulations and to guide further quantitative experiments on network dynamics in three-dimensional microstructures.