Diffusion of interacting particles in discrete geometries: Equilibrium and dynamical properties

We expand on a recent study of a lattice model of interacting particles [Phys. Rev. Lett. 111, 110601 (2013)]. The adsorption isotherm and equilibrium fluctuations in particle number are discussed as a function of the interaction. Their behavior is similar to that of interacting particles in porous materials. Different expressions for the particle jump rates are derived from transition-state theory. Which expression should be used depends on the strength of the interparticle interactions. Analytical expressions for the self- and transport diffusion are derived when correlations, caused by memory effects in the environment, are neglected. The diffusive behavior is studied numerically with kinetic Monte Carlo (kMC) simulations, which reproduces the diffusion including correlations. The effect of correlations is studied by comparing the analytical expressions with the kMC simulations. It is found that the Maxwell-Stefan diffusion can exceed the self-diffusion. To our knowledge, this is the first time this is observed. The diffusive behavior in one-dimensional and higher-dimensional systems is qualitatively the same, with the effect of correlations decreasing for increasing dimension. The length dependence of both the self- and transport diffusion is studied for one-dimensional systems. For long lengths the self-diffusion shows a $1/L$ dependence. Finally, we discuss when agreement with experiments and simulations can be expected. The assumption that particles in different cavities do not interact is expected to hold quantitatively at low and medium particle concentrations if the particles are not strongly interacting.

Figure 1

The system, shown here between dashed lines, consists of an array of cavities connected by narrow passages. On the boundaries it is connected to uncorrelated cavities with the equilibrium distribution.

Figure 2

Measuring the self-diffusion: A concentration gradient of labeled particles (open green circles) is introduced under overall equilibrium conditions.

Figure 3

The probability distribution of cavity occupation $p_n^{\mathrm{eq}}$, Eq. (1), at different loadings and with $n_{\max }=13$, for $\left(\mathrm{a}\right)$ $f\left(n\right)=0$ $\left(\mathrm{b}\right)$ $f\left(n\right)=0.2n^2$ $\left(\mathrm{c}\right)$ $f\left(n\right)=-0.2n^2$. $\left(\mathrm{d}\right)$ $\Omega \left(n\right)$ for different interactions, at $\langle n\rangle =5$. The lines are a guide to the eye.

Figure 4

The inverse thermodynamic factor ${\Gamma }^{-1}=(\langle n^2\rangle -{\langle n\rangle }^2)/\langle n\rangle$ for $n_{\max }=13$ and different interactions.

Figure 5

[(a)–(c)] $\Omega \left(n\right)$ for different interactions and loadings, with $n_{\max }=13$. For $f\left(n\right)=0$ and $f\left(n\right)=0.2n^2$ the curves are shifted vertically so $\Omega (\langle n\rangle )=0$. For $f\left(n\right)=-0.2n^2$ the curves are shifted vertically so $\Omega \left(0\right)=0$. The lines are a guide to the eye. (d) Adsorption isotherms $\langle n\rangle \left(\mu \right)$. The curves are shifted horizontally so $\langle n\rangle \left(0\right)=0.5n_{\max }$.

Figure 6

Adsorption isotherms $\langle n\rangle \left(\mu \right)/n_{\max }$ for different values of $n_{\max }$ for the interaction $f\left(n\right)=-0.2n^2$. The curves are shifted horizontally to have $\langle n\rangle \left(0\right)=0.5n_{\max }$.

Figure 7

(a) $\Omega \left(n\right)$ and (c) $p_n^{\mathrm{eq}}$ for $n_{\max }=40,\langle n\rangle =20$, and $f\left(n\right)=-0.2n^2$. (b) $\Omega \left(n\right)$ and (d) $p_n^{\mathrm{eq}}$ for $n_{\max }=5,\langle n\rangle =2.5$, and $f\left(n\right)=-0.2n^2$.

Figure 8

The inverse thermodynamic factor ${\Gamma }^{-1}=(\langle n^2\rangle -{\langle n\rangle }^2)/\langle n\rangle$ for different $n_{\max }$ and $f\left(n\right)=-0.2n^2$.

Figure 9

Two cavities, A and B, divided by a transition-state surface TS. A particle is in the transition state.

Figure 10

One-dimensional system with $f\left(n\right)=-0.2n^2,n_{\max }=13,\langle n\rangle /n_{\max }=0.8$, and the rates of Eq. (29). $\left(\mathrm{a}\right)$ Self-diffusion as a function of the length $L$ of the system. The inset shows the same data, plotted as a function of $1/L$. The analytical fit is obtained using mathematica and was done for all lengths except $L=1$. $\left(\mathrm{b}\right)$ Transport diffusion.

Figure 11

$D_s,D_t$, and ${\Gamma }^{-1}$. (a) $f\left(n\right)=0$ and $n_{\max }=2$; (b) $f\left(n\right)=0$ and $n_{\max }=13$; (c) $f\left(n\right)=0.2n^2,n_{\max }=13$, and the rates of Eq. (29); (d) $f\left(n\right)=0.2n^2,n_{\max }=13$, and the rates of Eq. (28); (e) $f\left(n\right)=-0.2n^2,n_{\max }=13$, and the rates of Eq. (29); (f) $f\left(n\right)=0.000642n^2-0.0083n^3,n_{\max }=13$, and the rates of Eq. (28).

Figure 12

Single-particle memory $\tilde{p}\left(m\right)$ for different interactions and loadings and the rates of Eq. (29).

Figure 13

$p_n^{\mathrm{eq}}$ and $p\left(n\right|\widehat{n})$ for different $\widehat{n}$ at loading $\langle n\rangle =6,n_{\max }=13$, and the rates of Eq. (29) for (a) $f\left(n\right)=0.2n^2$ and (b) $f\left(n\right)=-0.2n^2$.

Figure 14

$f\left(n\right)$ of Table 1, $n_{\max }=13$, and the rates of Eq. (29). (a) $D_s,D_t$, and ${\Gamma }^{-1}$. (b) Close-up of diffusion curves for $5\le \langle n\rangle \le 12$. (c) $\Omega \left(n\right)$ and $p_n^{\mathrm{eq}}$ at $\langle n\rangle =6$ and $\langle n\rangle =9$.

Figure 15

$D_s$ and $D_t$ for $f\left(n\right)=-0.2n^2,n_{\max }=13$, and the rates of Eq. (29) for dimensions 1, 2, and 3.

Figure 16

Self-diffusion for $f\left(n\right)=0.2n^2,n_{\max }=13$, and the rates of Eq. (29) for dimensions 1, 2, and 3.