Thermodynamics of a compressible lattice gas crystal: Generalized Gibbs-Duhem equation and adsorption

TL;DR

This paper extends thermodynamic principles to compressible lattice gas crystals, deriving a generalized Gibbs-Duhem equation and adsorption relation to analyze composition-strain coupling and vacancy formation.

Contribution

It introduces a generalized Gibbs-Duhem equation for compressible lattice gases by treating lattice sites as a thermodynamic variable, linking elastic energy and composition effects.

Findings

Reinstated Gibbs-Duhem relation as an adsorption equation.

Quantified vacancy creation tendency under isothermal, isobaric conditions.

Compared response functions for different thermodynamic ensembles.

Abstract

Compressible lattice gas models are used in material science to understand the coupling between composition and strain in alloys. The seminal work in this field is the 1973 Larch\'{e}-Cahn paper (Acta Metall. 21, 1051-1063). Single-phase crystals in Larch\'{e}-Cahn theory are stable under open constant pressure, constant temperature conditions. The Gibbs free energy does not have to match the product $\mu N$ of the number of particles $N$ and their chemical potential $\mu$. Similarly, the grand potential and the product $pV$ of pressure and volume $V$ may not add up to zero. Discrepancies already arise under hydrostatic stress. The elastic energy is not proportional to volume and the Gibbs-Duhem relation valid for liquids is violated. Extensivity is recovered by treating the number of lattice sites $M$ as an additional thermodynamic variable. The difference $ G-\mu N $ can be identified with $\nu M$ where $\nu$ is the thermodynamic force conjugate to $M$. The reinstated Gibbs-Duhem equation can be cast in the form of an adsorption equation and applied to quantify the tendency to vacancy creation under isothermal isobaric conditions. We have worked this out for a uniform one-component compressible lattice gas crystal. Shear stress is omitted. The coupling between composition and strain is implemented by decomposing pressure in a mechanical component depending on deformed density $N/V$ and an elastic term linear in the volume strain as determined by $V/M$. Various $\left( \mu, p, T \right) $ response functions are compared to the $\left( \mu, V, T \right) $ counterparts.