Self-similar evolution of network structures

A new theoretical approach is developed for the derivation of self-similar grain size distributions for three-dimensional polycrystals. The method is based on representing each class of irregular N-sided polyhedral grains in a space-filling network by the corresponding average N-hedron - or topological proxy. This method allows a more rigorous statistical mechanical derivation of the theoretical grain size distribution for three-dimensional polycrystals, and provides a clearer understanding of the assumptions underlying it, as well as clarification of its predictive limitations. Our method also yields several new results, namely, the self-similar distributions of the normalized cube-root grain volumes and of the number of faces per grain. A statistical comparison of the theoretical predictions with simulation data based on Brakke's Evolver and kinetic Monte Carlo methods is also provided. (c) 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.