Grain boundary networks: Scaling laws, preferred cluster structure, and their implications for grain boundary engineering

Within the context of grain boundary engineering, where grain boundaries are classified as special vs. general, grain boundary networks are known to have non-random topologies and percolation thresholds which differ from randomly assembled networks. This non-random structure is due to crystallographically imposed local correlations among boundaries. In the present work, we simulate crystallographically consistent grain boundary networks and measure four network properties: the cluster mass distribution, the average radius of gyration, the connectivity length and the strength of the percolating cluster. We find that for very large lattices, behavior of the crystallographically consistent networks is well described by the scaling laws of standard percolation theory. However, at shorter length scales, the cluster mass distributions and radii of gyration are significantly non-random for both special and general boundaries as a result of the local correlations. In this regime, we observe strong preferences for some magic cluster structures, and the scaling laws of percolation theory fail. The critical length scale separating these two classes of behavior is on the order of three grain diameters; this represents a new critical length scale for the statistical description of microstructures, and may figure into many microstructure-property relationships.