Miscibility gap and critical point behavior for a cubic margules solution model

Abstract

It is pointed out that a four parameter, binary solution model, in which the temperature independent enthalpy and excess entropy of mixing are cubic functions of the atom fraction, has provided quantitative fits to the thermodynamic properties of many metallic solution phases. It is therefore desirable to understand this semi-empirical model, in particular its miscibility gap and critical point behavior. It is shown that the latter is governed by two parameters, ϵ and η, that depend upon the temperature and the values of the four, temperature and composition independent, solution parameters. ϵ is the excess Gibbs free energy of mixing at x = 1 2 divided by RT/4. The parameter η is the difference in the excess chemical potentials at x = 1 2 divided by RT/4. The solution parameters define half of an infinite straight line in the ϵ-η plane called the constant-parameter trajectory, whereas the classical critical points fall along a parabola-like curve through ϵ = 2, η = 0. The appearance or disappearance of a miscibility gap is associated with the intersection of the constantparameter trajectory and the critical locus. The critical composition can be anywhere. Besides the normal miscibility gap, an inverted gap that first appears at high temperatures can occur and both types can occur in the same system. Equations for the gap width and asymmetry that are asymptotically correct near the critical temperature are developed. These show how the magnitude of the gap-width depends upon the solution parameters for a given temperature and critical point. The results are illustrated by application to some III–V systems and in obtaining a quantitative fit to the Pb-Ga system.