On the thermodynamics of crystalline solids

Abstract

A formalism for the analysis of thermodynamic equilibrium in an arbitrarily stressed multicomponent crystal is developed. We assume that the energy per unit cell is a function of the following per-unit-cell quantities: the entropy, the mole number of each chemical species, and six independent dot products of the three vectors that define the cell. Body forces specific to each species are included, but chemical reactions and capillarity are excluded. We use a Gibbsian variational method but refer all quantities to the unvaried state which may be one of large strain. We obtain the familiar equations of thermal and mechanical equilibrium and the uniformity of generalized potentials μi +φi for each species; the φi are associated with the body forces and the μi are identified as chemical potentials since we show them to be equal locally to the chemical potentials of components present in a fluid with which the crystal is in equilibrium. The equations of equilibrium are then rederived under the assumption that certain equations of constraint hold among the variables that specify the mole numbers per unit cell of chemical species. This allows the treatment of a system composed of both mobile and immobile species, considered by Li, Oriani, and Darken, as well as a famous example of Gibbs in which a single component solid may be in equilibrium with three fluids having different chemical potentials of that component. We then introduce an extended set of variables in order to give an explicit description of point defects and to allow comparison of our treatment with those of Herring and of Larché and Cahn. Equations are obtained that allow the determination of equilibrium defect populations; when defects are in equilibrium, our previous equations are recovered.