Diffusion in the presence of periodically spaced permeable membranes

The diffusion of molecules in biological tissues and some other microheterogeneous systems is affected by the presence of permeable barriers. This leads to the slowdown of diffusion at long times as compared to barrier-free diffusion. At short times the effect of barriers is weak. In consequence, the diffusion coefficient D(t) decreases as a function of time. We derive an exact solution for the Laplace transform of D(t) for diffusion in a space separated into layers by equally spaced, parallel identical planes of arbitrary permeability. Additionally, we give an approximation to D(t) which is reasonably accurate over the whole range of the partition permeability from zero (the case of isolated layers) to infinity (the case of no barriers).