Connectivity and percolation behaviour of grain boundary networks in three dimensions

Grain boundary networks are subject to crystallographic constraints at both triple junctions (first-order constraints) and quadruple nodes (second-order constraints). First-order constraints are known to influence the connectivity and percolation behaviour in two-dimensional grain boundary networks, and here we extend these considerations to fully three-dimensional microstructures. Defining a quadruple node distribution (QND) to quantify both the composition and isomerism of quadruple nodes, we explore how the QNDs for crystallographically consistent networks differ from that expected in a randomly assembled network. Configurational entropy is used to quantify the relative strength of each type of constraint (i.e., first- and second-order), with first-order triple junction constraints accounting for at least 75% of the non-random correlations in the network. As the dominant effects of constraint are captured by considering the triple junctions alone, a new analytical model is presented which allows the 3-D network connectivity to be estimated from data on 2-D microstructural sections. Finally, we show that the percolation thresholds for 3-D crystallographically consistent networks differ by as much as ±0.07 from those of standard percolation theory.