Further refinements in the energy of grain boundaries

Earlier surface dislocation analysis of a grain boundary recognized the tendency of the grain-boundary surfaces to coalesce in order to reduce surface energy. The coalescence process is described by a distribution of surface dislocations on the grain-boundary surfaces. In the present paper, previous analysis is further refined. In particular, the sum of the Burgers vectors of the surface array of grain-boundary dislocations is not equal to the Burgers vector of the grain-boundary lattice dislocation. Instead, the Burgers vector of the surface array is determined as a function of the coalescence of the grain-boundary surfaces. The conservation of Burgers vectors of dislocations is used to predict the presence of a screening array of dislocations. The screening array of dislocations is determined by minimization of the total energy of the configuration. The distortion around the boundary is relaxed by the screening array. In general, the distribution of the screening array is two dimensional. This result has been proved by the presence of a minimum energy configuration for two sets of screening arrays of dislocations situated at different distances from the boundary.