Mutual influence of molecular diffusion in gas and surface phases

We develop molecular transport simulation methods that simultaneously deal with gas- and surface-phase diffusions to determine the effect of surface diffusion on the overall diffusion coefficients. The phenomenon of surface diffusion is incorporated into the test particle method and the mean square displacement method, which are typically employed only for gas-phase transport. It is found that for a simple cylindrical pore, the diffusion coefficients in the presence of surface diffusion calculated by these two methods show good agreement. We also confirm that both methods reproduce the analytical solution. Then, the diffusion coefficients for ink-bottle-shaped pores are calculated using the developed method. Our results show that surface diffusion assists molecular transport in the gas phase. Moreover, the surface tortuosity factor, which is known to be uniquely determined by physical structure, is influenced by the presence of gas-phase diffusion. This mutual influence of gas-phase diffusion and surface diffusion indicates that their simultaneous calculation is necessary for an accurate evaluation of the diffusion coefficients.

Figure 1

Schematic of concurrent gas and surface diffusion in a cylindrical pore. Rather than undergoing desorption immediately after molecule-surface collision, molecules move along the surface over a certain distance according to a given surface diffusion coefficient and time.

Figure 2

Dependence of diffusion coefficient on ${\left(D_{\mathrm{s}}{t}_{\mathrm{s}}\right)}^{0.5}/L_{\mathrm{d}}$ in a cylindrical pore, as calculated by the test particle method. The calculated results are compared with the analytical solutions given by Eq. (4).

Figure 3

Comparison of diffusion coefficients for a cylindrical pore with isotropic or directional surface diffusion. The Knudsen number is set to infinity. The isotropic curve is calculated from Eq. (4), while the directional diffusion data are calculated by the test particle method.

Figure 4

Diffusion coefficients calculated by the test particle and MSD methods for ${\left(D_{\mathrm{s}}{t}_{\mathrm{s}}\right)}^{0.5}/L_{\mathrm{d}}=1$. The modified MSD results are obtained on the basis of Eq. (6), i.e., $D_{\mathrm{MSD}}{\left[\left(t_{\mathrm{g}}+t_{\mathrm{s}}\right)/t_{\mathrm{g}}\right]}^{-1}$.

Figure 5

MSDs under the conditions $D_{\mathrm{s}}/\left(v{L}_{\mathrm{d}}\right)=1000$ and $t_{\mathrm{s}}/\left(L_{\mathrm{d}}/v\right)=0.001$. The contributions of gas- and surface-phase diffusions to the overall MSD are shown separately, with dashed and dotted lines, respectively.

Figure 6

Ink-bottle pore structure: (a) three-dimensional image, and (b) cross-sectional drawing.

Figure 7

Diffusion coefficients for ink-bottle pores, calculated by the MSD method. The results are fitted to Eq. (8). The Knudsen number is set to infinity.

Figure 8

Surface tortuosity factors of ink-bottle pores, calculated from surface-phase MSDs and Eq. (8).

Figure 9

Diffusion coefficient contributions from gas-phase diffusion in ink-bottle pores.