A set of phase-equilibrium equations in covariant form and its use for developing a general method of calculation of two-phase conodes in multicomponent systems

Abstract

In the last 25–30 years, thermodynamic calculations of phase diagrams for both binary and multicomponent systems received wide application. The computer programs [1–7] developed in various research centers of the world are of wide use. The numerical methods involved in these programs can be divided into two classes. In studies of the first class [2], the Gibbsenergy minimum is sought for a heterogeneous system. In studies of the second class, the set of phase-equilibrium equations (establishing the equality of chemical potentials for the components of various phases) is solved [1, 3–7]. In this case, either the Newton–Raphson iterative method [1] or the Nelder–Mead modified simplex method is used to minimize the objective function representing the sum of residuals of the phaseequilibrium equations [7]. An essential constraint of the former methods is the impossibility of guaranteeing attainment of the global minimum of the Gibbs energy for a heterogeneous system. This means that the calculated phase diagram can be both stable and metastable. An essential disadvantage of the latter methods is the necessity of selecting a successful initial approximation for starting the iterative process, for which there is no assurance that the iterations will converge to the desired solution. Thus, the calculating methods of both the first and second class render it principally impossible to create autonomous computer programs for calculating phase diagrams and thermodynamic properties of multicomponent alloys of systems with three or more components.