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

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 Larché-Cahn paper [F. C. Larché and J. W. Cahn, Acta Metall. 21, 1051-1063 (1973)]. Single-phase crystals in Larché-Cahn theory are stable under open constant pressure, constant temperature conditions. The Gibbs free energy does not have to match the product μN of the number of particles N and their chemical potential μ. 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 - μN can be identified with νM where ν 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 μ,p,T response functions are compared to the μ,V,T counterparts.