Correlated grain-boundary distributions in two-dimensional networks

In polycrystals, there are spatial correlations in grain-boundary species, even in the absence of correlations in the grain orientations, due to the need for crystallographic consistency among misorientations. Although this consistency requirement substantially influences the connectivity of grain-boundary networks, the nature of the resulting correlations are generally only appreciated in an empirical sense. Here a rigorous treatment of this problem is presented for a model two-dimensional polycrystal with uncorrelated grain orientations or, equivalently, a cross section through a three-dimensional polycrystal in which each grain shares a common crystallographic direction normal to the plane of the network. The distribution of misorientations theta, boundary inclinations phi and the joint distribution of misorientations about a triple junction are derived for arbitrary crystal symmetry and orientation distribution functions of the grains. From these, general analytical solutions for the fraction of low-angle boundaries and the triple-junction distributions within the same subset of systems are found. The results agree with existing analysis of a few specific cases in the literature but present a significant generalization.