Geometry of coincident-site lattice boundaries

Recently large-scale numerical computations, including the first principles calculations, regarding the interfacial energy between the two crystallines, which capture microscopic nature of materials concerned have been made. It has been believed that the details of the bonding at an interface are crucial in understanding the dependence of the interfacial energy on geometrical location of the interface. Nevertheless the similarity of the overall features of the curves of the interfacial energy obtained against variation of the parameters which specify the geometrical location of an interface for several materials suggests a common underlying physics which should not depend strongly on the material chosen. The present investigation is therefore to discuss the commonly observed characteristics in the energy curves from purely geometrical point of view. To this end I first clarify, taking the twist boundaries as an example to which the method is applied, the hierarchy in the geometrical construction of the coincident-site lattice by invoking the continued-fraction expansion algorithm for the transformation operator between the two lattice structures which are separated by the interface in question. Interfacial energies are then calculated against misorientation based on a two-dimensional version of the model put forth by du Plessis and van der Merwe, which turn out to reveal a fractal energy surface with cusps at every Sigma value. It is emphasized that these singularities are not attributable to the nature of the bonding but are of purely geometrical origin.