Continuous Displacement Formulation of Alloys

Abstract

The conventional permutational statistics of cooperative phenomena is extended to include displacement of atoms from their reference lattice points. The formulation is done in the cluster variation method (CVM) framework, by changing summations in energy and entropy to integrals. It is demonstrated for the pair approximation of CVM on phase-separating fcc binary alloys. The treatment can take into account local lattice distortion due to atomic size difference, the elastic effects, and the pressure effects. To compare stability of states under pressure, the grand potential Ω(T, V, {μ _i}) is extended to Z(T, p, {μ _i}) by a Legendre transform. Although the new function Z(T, p, {μ _i}) vanishes in equilibrium, and is called the ZERO-potential in the paper, it remains nonzero and is used to determine the coexisting phases when the chemical potentials are modified. Numerical calculations are done using the natural iteration technique on model inter-atomic potentials of the Lennard-Jones type. The numerical results of using Ω(T, V, {μ _i}) and Z(T, p, {μ _i}) potentials for phase-separating diagrams, for composition and pressure dependence of the lattice constant, and for the bulk modulus are reported and discussed.