Theory of departure from local equilibrium at the interface of a two-phase diffusion couple

Abstract

A simple model is developed to calculate the departure from local equilibrium at a moving two-phase interface in a binary diffusion couple. The model is based on the solution of a generalized diffusion equation derived by assuming that the diffusion flux of a species is proportional to the gradient of the functional derivative, with respect to concentration, of the total Helmholtz free energy. This total free energy is composed of the integral of two terms, a coarse-grained free energy per unit volume that reflects the bulk thermodynamic properties of the material and an energy density proportional to the square of the concentration gradient. Final results are conveniently represented by replacing the actual concentration profile with another having a jump discontinuity at a plane whose position is judiciously chosen within the region of high concentration gradient; values of composition and its derivative at the discontinuity are chosen such that the bulk solutions, when extrapolated to these values, are the same as for the complete profile. The composition deviations from thermodynamic equilibrium in each phase at the interface are found to be proportional to an assumed large interfacial resistivity; they contain a term proportional to the interface velocity and a second independent term proportional to the flux of one atomic species through the interface. Fluxes of A and B atoms, measured in the interface frame, are found to be related linearly to differences in chemical potential across the interface by a symmetric matrix which guarantees a decrease of net free energy but allows a given species to move contrary to its chemical potential gradient, thus giving rise to trapping in some circumstances.