Study of Diffusion in a One-Dimensional Lattice-Gas Model of Zeolites: The Analytical Approach and Kinetic Monte Carlo Simulations

Abstract

The diffusion of molecules adsorbed in a one-dimensional channel with side pockets is investigated in the framework of a one-dimensional lattice-gas model. The model can describe the molecules migration in some type of zeolites. We obtained the exact expression for the free energy of this model. Using the local equilibrium approximation we derived the analytical expressions for the diffusion coefficients. The concentration dependencies of the center-of-mass and Fickian diffusion coefficients are calculated for some representative values of the lateral interactions between molecules. The theoretical dependencies are compared with the numerical data obtained by the kinetic Monte Carlo simulations. The data obtained by the two completely different methods coincide amazingly well in the whole concentration and wide interaction regions.

Appendix

We derive here the analytical expression for the diffusion coefficient for molecules performing the long jump successions. In the low concentration region $\left( {\theta < p/(p + 1)} \right)$ all molecules occupy deep pocket sites. We need the probability of a molecule jump succession from some initial $i$th to the final $f$th $d$ site. The molecule performs slow jump from the $i$th site, $n$ fast jumps and occupies the $f$th site at the distance $ma$ from the $i$th site. The probability of this succession is a product of the probabilities of the elementary migration acts: the probability of the first slow jump from the $i$th to the channel $s$ site $\nu {\text{exp}}( - \varepsilon_{p} ),$ the probability of the sequence $n$ fast jumps $W(n,m)$ and the probability to occupy a $d$ site by the final jump $w_{d}$. As we supposed that the probability for a fast jump is equal for all NN sites, then the probability to jump in the NN $s$ site is $w = [p(1 - \theta_{p} ) + 2]^{ - 1} .$ The probability to occupy a $d$ site is $w_{p} = p(1 - \theta_{p} )[p(1 - \theta_{p} ) + 2]^{ - 1} .$ The probability of the fast jump succession is simply the product of the fast jump rates $w^{n} .$ The number of the different jump successions which transfer the molecule to a distance $ma$ after $n$ jumps is equal to

$$ \frac{n!}{{\left[ {\frac{1}{2}(n + m)} \right]!\left[ {\frac{1}{2}(n - m)} \right]!}} \equiv C_{(n + m)/2}^{{{\kern 1pt} n}} ,\quad \left| m \right| \le n, $$

where ${\text{C}}_{k}^{{{\kern 1pt} n}}$ is the binomial coefficient.

The square of the length of this jump sequence $L^{2} (n,m)$ is obviously equal to $(ma)^{2} .$ To obtain the effective jump length $L$ one should average over all sequences

$$ L^{2} = w_{p} a^{2} \sum\limits_{n = 0}^{\infty } w^{n} \sum\limits_{m = - n}^{n} m^{2} C_{(n + m)/2}^{{{\kern 1pt} n}} $$

(26)

The inner sum is equal $n\;2^{{{\kern 1pt} n}}$ [13]. The infinite geometric series is easily calculated. The final result is the following simple expression

$$ L^{2} = w_{p} a^{2} \sum\limits_{n = 0}^{\infty } n(2w)^{n} = a^{2} /p(1 - \theta_{p} ) $$

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