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 NN-sided polyhedral grains in a space-filling network by the corresponding “average NN-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.