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

作者：Partha P. Mitra, Pabitra N. Sen, Lawrence M. Schwartz · 发表于：Physical review. B, Condensed matter · 年份：1993 · DOI：10.1103/physrevb.47.8565 · 被引用次数：582 · 研究领域：NMR spectroscopy and applications、Advanced Neuroimaging Techniques and Applications、Soil and Unsaturated Flow

We investigate the time-dependent diffusion coefficient, D(t)=〈${\mathit{r}}^{2}$(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)/${\mathit{D}}_{0}$ =1-${\mathit{A}}_{0}$(${\mathit{D}}_{0}$t${)}^{1/2}$+${\mathit{B}}_{0}$${\mathit{D}}_{0}$t+O[(${\mathit{D}}_{0}$t${)}^{3/2}$], where ${\mathit{A}}_{0}$=4S/(9 \ensuremath{\surd}\ensuremath{\pi} ${\mathit{V}}_{\mathit{P}}$) and ${\mathit{B}}_{0}$=-HS/(12${\mathit{V}}_{\mathit{P}}$)-${\mathit{tsum}}_{\mathit{i}}$(${\mathit{L}}_{\mathit{i}}$/${\mathit{V}}_{\mathit{P}}$)f(${\mathrm{\ensuremath{\varphi}}}_{\mathit{i}}$). Here ${\mathit{D}}_{0}$ is the diffusion constant of the bulk fluid, S/${\mathit{V}}_{\mathit{P}}$ is the surface area to pore volume ratio, H is the mean curvature of the smooth portions of the surface, ${\mathit{L}}_{\mathit{i}}$ is the length of a wedge of angle ${\mathrm{\ensuremath{\varphi}}}_{\mathit{i}}$, and the function f(\ensuremath{\varphi}) is defined below. More generally, we consider partially absorbing boundary conditions, where the absorption strength is controlled by a surface-relaxivity parameter \ensuremath{\rho}. Here, the density of walkers (i.e., the net magnetization) decays as M(t)=1-\ensuremath{\rho}St/${\mathit{V}}_{\mathit{P}}$+..., and D(t) is defined as 〈${\mathi...