Diffusion of Interacting Particles in Discrete Geometries

We evaluate the self-diffusion and transport diffusion of interacting particles in a discrete geometry consisting of a linear chain of cavities, with interactions within a cavity described by a free-energy function. Exact analytical expressions are obtained in the absence of correlations, showing that the self-diffusion can exceed the transport diffusion if the free-energy function is concave. The effect of correlations is elucidated by comparison with numerical results. Quantitative agreement is obtained with recent experimental data for diffusion in a nanoporous zeolitic imidazolate framework material, ZIF-8.

Figure 1

The model: particles enter cavities via particle reservoirs at certain chemical potential. Particles jump between different cavities through narrow passages. (a) Transport diffusion: a concentration gradient shows a current. (b) Self-diffusion: a concentration gradient of labeled particles is introduced under overall equilibrium conditions.

Figure 2

$D_s/D_0$, $D_t/D_0$, and ${\Gamma }^{-1}$ as a function of loading $\theta =⟨n⟩/n_{\max }$, $n_{\max }=13$; for (a) $\beta f(n)=0$, (b) $\beta f(n)=0.2n^2$, (c) $\beta f(n)=-0.2n^2$, and (d) $\beta f(n)$ that is subsequently concave, convex, and again concave. The red dashed lines (analytical solution) and squares (simulations) show the transport diffusion, and the blue dotted lines (analytical solution) and full circles (simulations) the self-diffusion (values on lhs axis). The analytical ${\Gamma }^{-1}$ (black full lines) are compared with the ratio of $D_s$ and $D_t$ (black stars) from the simulations (values on rhs axis).

Figure 3

Comparison of experimental data of methanol in ZIF-8 [39] with simulations from our model. The experimental self-diffusion and transport diffusion are represented by the blue full circles and red squares, respectively, with the corresponding results from simulations given by the blue dotted line and the red dashed line (values on lhs axis). The experimental (black open circles) and analytical (full black line) ${\Gamma }^{-1}$ are compared with the ratio of self- and transport diffusion (black stars) taken from the simulations (values on rhs axis). The inset shows $\beta f(n)$.