The problem of calculating the effective diffusivity in nanocrystalline materials was addressed. Two simple 2-dimensional grain patterns were constructed, and equations were derived from the Hart–Mortlock and Maxwell–Garnett formulations. Monte Carlo computer simulation was used to test the validity of the various equations. It was shown that one of the derived equations provided an excellent description of the effective diffusivities up to a 40% volume fraction of atoms in the grain boundaries.

Diffusion in Nanocrystalline Materials. I.V.Belova, G.E.Murch: Journal of Physics and Chemistry of Solids, 2003, 64[5], 873-8