Time-dependent diffusion in tubes with periodic partitions

The presence of obstacles leads to a slowdown of diffusion. We study the slowdown when diffusion occurs in a tube, and obstacles are periodically spaced identical partitions with circular apertures of arbitrary radius in their centers. The mean squared displacement of a particle diffusing in such a system at large times is given by <Delta x(2)(t)>=2D(eff)t, t -> infinity, where D-eff is the effective diffusion coefficient, which is smaller than the particle diffusion coefficient in the tube with no partitions, D-0. The latter characterizes the short-time behavior of the mean squared displacement, <Delta x(2)(t)>=2D(0)t, t -> 0. Thus, the particle diffusion coefficient decreases from D-0 to D-eff as time goes from zero to infinity. We derive analytical solutions for the Laplace transforms of the time-dependent diffusion coefficient and the mean squared displacement that show how these functions depend on the geometric parameters of the tube. To obtain these solutions we replace nonuniform partitions with apertures by effective partitions that are uniformly permeable for diffusing particles. Our choice of the partition permeability is based on the recent result for the corresponding effective trapping rate obtained by means of boundary homogenization. To establish the range of applicability of our approximate theory we compare its predictions with the results found in Brownian dynamics simulations. Comparison shows excellent agreement between the two at arbitrary value of the aperture radius when the tube radius does not exceed the interpartition distance. (C) 2009 American Institute of Physics. [doi: 10.1063/1.3224954]