AN ANALYTIC MODEL FOR GRAIN-BOUNDARY EXPANSIONS AND CLEAVAGE ENERGIES

Using a pair-potential model for atomic interactions, the expansions and cleavage energies of grain boundaries are analysed. By decomposing the energies of grain boundaries into layer interaction energies it is readily seen that the significance of periodicity in the boundary plane is to provide additional terms in the boundary energy that are absent when the boundary is incommensurate. Cleavage energies of incommensurate grain boundaries are particularly easy to formulate and compute using layer interaction energies. They are calculated using a model for a relaxed incommensurate boundary that consists of rigid crystals on either side of the boundary plane that are allowed to float to attain the equilibrium expansion. Clear dependences of the expansions and cleavage energies on the interplanar spacings of incommensurate boundaries are derived analytically and confirmed by full atomistic relaxation for long-period commensurate boundaries. The factors giving rise to these dependences are elucidated, and the limitations of the model are assessed.