Atomistic models of a grain boundary in a square lattice

The classical theory of grain boundaries is based on Bollmannʹs O-lattice analysis. Special significance is given to those atomic sites in the lattice of grain 2 which would also be lattice sites in grain 1 if that grain was extended into the domain of grain 2. Little attention is given to the orientation of the grain boundary with respect to the lattices of the two grains. Several authors have shown that in some systems the preferred interface between two phases is governed by the requirement that lattice planes in the two phases should meet along a line lying in the interface. This is a different problem from that of determining the preferred orientations of the boundaries between two grains of the same phase. Using the simple model of a square two-dimensional lattice and an appropriate mathematical algorithm, we attempt an atomistic analysis of this latter problem.