A general thermodynamic approach for diffusion on a lattice

TL;DR

This paper introduces a thermodynamic framework for modeling particle diffusion on lattices, linking microscopic interactions to macroscopic transport coefficients, and validates it through classical and surface diffusion models.

Contribution

It develops a unified thermodynamic approach that relates diffusion coefficients to thermodynamic properties and interactions, extending existing models with analytical and numerical validation.

Findings

Reproduces the Darken equation for substitutional diffusion.

Derives the non-diagonal diffusion matrix for surface diffusion of Langmuir particles.

Validates analytical predictions with numerical simulations across different potentials.

Abstract

This work presents a general thermodynamic approach to describe particle diffusion on a lattice, a model used to study transport processes in solids and on surfaces. By treating each lattice site as an open thermodynamic system, the effects of microscopic particle interactions are represented through the chemical potential. A fundamental relationship between the Onsager matrix ($L$) and its ideal-system counterpart ($L_\text{id}$, where interactions are neglected) using the determinant of the covariance matrix is demonstrated. This framework allows for the calculation of transport coefficients using the combination of their ideal values and thermodynamic properties. The general result is successfully applied to reproduce the Darken equation for substitutional diffusion in solids and to derive the non-diagonal diffusion matrix of the Zhdanov model for surface diffusion of Langmuir particles. In the last case, analytical predictions are further validated through numerical simulations across various interaction potentials.