Chemical pressure and vacancies in crystals under nonhydrostatic stress

Abstract

Inhomogeneous stress is a driving force for diffusion of vacancies in crystals. The other way around a non-uniform distribution of vacancies induces stress. The accepted theory of composition-stress coupling in crystals is Larché-Cahn (LC) theory. Composition is defined in terms of the population of lattice sites with a limit of one-particle per site. Coupling is modelled by adding a compositional strain term to the elastic strain in the constitutive relation for stress. An alternative mechanism, proposed here, is letting the site binding energy vary with spatial (deformed) density. This generates a chemical pressure adding to the elastic stress. The governing equations for this model are derived using non-equilibrium continuum thermodynamic methods. The gradient of the chemical pressure has a dual function acting both as the drift force for migration and as an effective internal one body force in the Cauchy equation for the elastic stress. The practical evaluation presented in the paper is restricted to equilibrium properties. We examine the effect of externally applied non-hydrostatic stress, either in the form of surface tractions or a one-body force density (gravitation). The model system is a one component crystal with a fixed number of lattice sites. The number of particles can be variable but is always smaller than the number of lattice sites. The linear elastic response is modelled by the standard Lamé stress tensor. The results for systems deformed by surface tractions are in qualitative agreement with LC theory allowing for differences in the expression for the elastic moduli. Deviations are more serious for the crystal deformed by gravitation.