Short-time behavior of the diffusion coefficient as a geometrical probe of porous media

We investigate the time-dependent diffusion coefficient, D (t) =〈r^2 (t) 〉/ (6t), of random walkers in porous media with piecewise-smooth pore-grain interfaces. D (t) is measured in pulsed-field-gradient spin-echo (PFGSE) experiments on fluid-saturated porous media. For reflecting boundary conditions at the interface we show that for short times D (t) /D₀ =1-A₀ (D₀t) ^1/2+B₀D₀t+O (D₀t) ^3/2, where A₀=4S/ (9 V₏) and B₀=-HS/ (12V₏) -tsum₈ (L₈/V₏) f (₈). Here D₀ is the diffusion constant of the bulk fluid, S/V₏ is the surface area to pore volume ratio, H is the mean curvature of the smooth portions of the surface, L₈ is the length of a wedge of angle ₈, and the function f () is defined below. More generally, we consider partially absorbing boundary conditions, where the absorption strength is controlled by a surface-relaxivity parameter. Here, the density of walkers (i. e. , the net magnetization) decays as M (t) =1-/V₏+. . . , and D (t) is defined as 〈r^2 (t) 〉ₒ/ (6t), where 〈r^2 (t) 〉ₒ is the mean-square displacement of surviving walkers. When 0 we find that the coefficient A₀ of the D₀t term in the above equation is unchanged, while the coefficient of the linear term changes to B₀+/ (6V₏). Thus, data on D (t) and M (t) at short times may be used simultaneously to determine S/V₏ and. The limiting behavior of D (t) as is also discussed.