A1-L$1_0$ phase boundaries and anisotropy via multiple-order-parameter theory for an fcc alloy

Abstract

The dependence of thermodynamic properties of planar interphase boundaries (IPBs) and antiphase boundaries (APBs) in a binary alloy on an FCC lattice is studied as a function of their orientation. Using a recently-developed diffuse interface model based on three non-conserved order parameters and the concentration, and a free energy density that gives a realistic phase diagram with one disordered phase (A1) and two ordered phases (L$1_2$ and L$1_0$) such as occurs in the Cu-Au system,we are able to find IPBs and APBs between any pair of phases and domains, and for all orientations. The model includes bulk and gradient terms in a free energy functional, and assumes that there is no mismatch in the lattice parameters for the disordered and ordered phases. We catalog the appropriate boundary conditions for all IPBs and APBs. We then focus on the IPB between the disordered A1 phase and the L1${}_0$ ordered phase. For this IPB we compute the numerical solution of the boundary value problem to find its interfacial energy, $\gamma$, as a function of orientation, temperature, and chemical potential (or composition). We determine the equilibrium shape for a precipitate of one phase within the other using the Cahn–Hoffman “$\xi$-vector” formalism. We find that the profile of the interface is determined only by one conserved and one non-conserved order parameter, which leads to a surface energy which, as a function of orientation, is “transversely isotropic” with respect to the tetragonal axis of the L1${}_0$ phase. We verify the model's consistency with the Gibbs adsorption equation.