Lambert diffusion in porous media in the Knudsen regime: Equivalence of self-diffusion and transport diffusion

We study molecular diffusion in nanopores with different types of roughness under the exclusion of mutual molecular collisions, i.e., in the so-called Knudsen regime. We show that the diffusion problem can be mapped onto Levy walks and discuss the roughness dependence of the diffusion coefficients $D_s$ and $D_t$ of self- and transport diffusion, respectively. While diffusion is normal in $d=3$, diffusion is anomalous in $d=2$ with $D_s\sim \ln t$ and $D_t\sim \ln L$, where $t$ and $L$ are time and system size, respectively. Both diffusion coefficients decrease significantly when the roughness is enhanced, in remarkable disagreement with earlier findings.

Figure 1

Geometry of the pores considered in this paper: (a), (c) two realizations of the generalized random Koch curve generator in $d=2$ and 3, respectively. (b) $2d$ pore of length $L=h$ and roughness $\nu =2$. (d) $3d$ pore of length $L=2h$ and roughness $\nu =2$.

Figure 2

The distribution $P\left(\mid x\mid \right)$ is plotted vs $\mid x\mid$ for the $2d$ pores (open symbols) and for the $3d$ pores (filled symbols) of $\nu =0$ (circles), $\nu =1$ (squares), and $\nu =3$ (triangles). The data of $d=3$ have been shifted down by a factor of 100. The lines of slopes $-3$ and $-4$ are guides to the eye. The average was taken over ${10}^5$ trajectories.

Figure 3

(a) The scaled mean square displacement $⟨x^2\left(t\right)⟩∕\left(t\ln t\right)$ (averaged over ${10}^5$ trajectories) is plotted vs $t$ in $d=2$. (b) The self-diffusion coefficient $D_s=⟨x^2\left(t\right)⟩∕\left(2t\right)$ is plotted vs $t$ in $d=3$. The different symbols indicate different roughness of $\nu =0$ (circles), 1 (squares), and 3 (triangles).

Figure 4

$D_t∕⟨u_x⟩$ calculated from $\vec{j}$ (large open symbols) and from $f_t$ for $x_{\min }=0$ (small open symbols) and $x_{\min }=0.4h$ (black symbols) for the $2d$ systems of $\nu =0$ (circles) and $\nu =3$ (triangles). The data from $\vec{j}$ and $f_t$ match perfectly when $x_{\min }=0$, whereas major deviations occur for $x_{\min }>0$.

Figure 5

(a) The scaled probability $f_{t}L{\left(\ln L\right)}^{-1}$ (averaged over ${10}^6$ trajectories) is plotted vs the system length $L$ for the $2d$ pores. (b) The transport diffusion coefficient $D_t=f_{t}L⟨u_x⟩$ is plotted vs the system length $L$ for the $3d$ pores. (The same symbols as in Fig. 3.)