At long times the effective solute diffusivity can be described by the (modified) Hart-Mortlock and Maxwel-Garnett equations for diffusion parallel and perpendicular to the grain boundary respectively.In this paper we analyze for the first time the time dependence of the effective solute diffusivity for these conditions.We assume that there are local regions (delineated by the diffusion length) in the grains adjacent to the grain boundary where the solute is equilibrated with the grain boundary.We write equations for the effective solute diffusivity with this assumption.Comparison with Monte Carlo simulations shows that this is quite a reasonable approximation for solute diffusion paralel to the grain boundary.For diffusion perpendicular ot the grain boundary it is only a fair approximation unless the segreation is weak.
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