Self-diffusion at grain boundaries with a disordered atomic structure

A change in the free energy of a grain boundary is analyzed in the case when lattice vacancies come to the boundary and are then delocalized in its disordered atomic structure. It is shown that the free energy of the boundary is minimized at some excess atomic volume Δvb=Δvb*, whose value depends on the energy of vacancy formation in the crystal lattice and the boundary energy. The formation of a metastable localized grain-boundary vacancy as a result of thermal fluctuations of the density in a group of n0=\gMv/Δvb atoms (\gMv is the vacancy volume), followed by the jump of an adjacent atom into this vacancy, is taken as an elementary event of grain-boundary diffusion. Expressions for the activation energy of diffusion and the diffusion coefficient are derived for equilibrium (Δvb=Δvb*) and nonequilibrium (Δvb>Δvb*) boundaries.