Time-dependent diffusion coefficients in periodic porous materials

We derive an approximate solution for the Laplace transform of the time-dependent diffusion coefficient, D(t), of a molecule diffusing in a periodic porous material. In our model, the material is represented by a simple cubic lattice of identical cubic cavities filled with a solvent and connected by small circular apertures in otherwise reflecting cavity walls, the thickness of which can be neglected. The solution describes the decrease of D(t) from its initial value, D(0) = D, where D is the diffusion constant in the free solvent, to its asymptotic value, D(infinity) = D-eff, which is much smaller than D. A simple heuristic formula for the mean-squared displacement of the diffusing molecule is suggested. The theoretically predicted results are in good agreement with the data obtained from Brownian dynamics simulations.