Coupling of orthogonal diffusion modes in two-dimensional nonhomogeneous systems

Abstract

Collective diffusion coefficient in a two-dimensional lattice gas on a nonhomogeneous substrate is investigated using variational approach. In our model particles reside and jump randomly between adsorption sites modeled as potential wells with different depths. Site blocking is the only allowed particle-particle interaction mechanism. It is shown that the value of the diffusion coefficient in one lattice direction depends nontrivially on the rate and the character of the particle jumps in other directions. The collective diffusion coefficient increases, eventually approaching values predicted within the mean-field approximation when the jump rate increases in the direction perpendicular to that in which the diffusion coefficient is measured. Analytical predictions of our model are supported by the Monte Carlo simulation performed for selected systems.

Collective diffusion in a non-homogeneous interacting lattice gas

“…Note that the diffusion denominator is fully determined by the equilibrium properties of the system. The diffusion numerator M(k) is shown in [27] (cf equations (28) and 29there) to be in the ka 1 limit:…”

“…in a series of follow-up works [23]- [28]. Most of these works deal with homogeneous onedimensional systems with short range interactions but some progress has also been made in two dimensions [23,28], non-homogeneous substrates [27,28] and systems with long range particle-particle interactions [26,29]. In this approach the collective diffusion coefficient is related to the lowest eigenvalue of a rate matrix which describes the kinetics of microscopic states of the system.…”

“…Then, the best estimate of the eigenvalue, guaranteed not to lie below the true one, is obtained in the spirit of the Ritz variational approach analogous to that used in quantum mechanics but properly generalized to account for a non-Hermitian character of the rate matrix. In most of the applications so far, except in [27,28], the designation 'variational' was not fully justified because variational parameters minimizing the eigenvalue were not used-they could be easily guessed for these relatively simple applications. For a non-homogeneous substrate, even for a non-interacting adsorbate considered in [27] and in two dimensions in [28] the systematic variational approach with variational parameters adjusted to minimize the eigenvalue is a necessity.…”

Collective diffusion of dense adsorbate at surfaces of arbitrary geometry

“…which is the variational formula used by us to solve various diffusional problems [26][27][28][29][30][31][32][33][34] . M ( k) and N ( k) have been referred to as the expectation value numerator and the normalization denominator, respectively.…”

Diffusion of particles interacting via long-range and oscillating forces in one dimension

“…An important background is provided in the works of Reed and Ehrlich 17 , an early summary by Gomer 18 , and in reviews by Danani et al 19 and Ala-Nissila et al 20 . The variational approach to the collective diffusion problem, used here, was developed in a series of works 21,22,23,24,25,26,27,28 and was shown to be a very efficient tool to analyze collective diffusion problems for various types of inter-particle interactions for homogeneous or inhomogeneous substrates either in one or two dimensions.…”