Earlier author of present report has proposed as alternative of method of Newton (or Newton-Raphson method ) for solving of system of equations of equilibria for two phases, which requires the task of initial approximation, the original algorithm of calculation (as U-algorithm ) of PD of binary systems was realized in [1]. The U-algorithm is generalization of Maxwell method ("equality of areas for P-V coordinates for phase transformation gas->liquid of pure substances"), include the equality of areas in coordinates ¶G/¶x - x, where G-molar Gibbs energy, x-molar fraction of 2-nd component for closed binary systems.
In present work covariant form of the system of equations for two-phase equilibria for closed nCS is received, including special case for general-spherical coordination axis. In the general case set of equilibrium equations is written in hyperspherical system of coordinates, which represent the generalization of the spherical system coordinates. This set is divided into two sebsets of equations (see, e.g.[2]).The first subset involves as the equality for radial part of the gradient of the Gibbs molar energies for two phases with respect to the vector of phase compositions calculated at the points of composition vectors for the equilibrium phases for as scalar equation in the form of scalar multiplication of gradinent of the Gibbs molar energy for out of any phase on vector of tie-line, which equal of difference between the Gibbs molar energy for two phases at tie-lines ends [2]. The second subset involves the equalities of the angular components of the gradients of the Gibbs molar energies for two phases calculated at the points of composition vectors, which connect the pole of hyperspherical system of coordinates with ends of tie-line. For three–component system (3CS) the hyperspherical system of coordinates transformed in polar system of coordinates and second subset transformed in only one equation of angular components of the gradients of the Gibbs molar energies for two phases.
Numerical procedure of solution set of equilibrium equations for 3CS divided into three stages in searching for two-phase tie-line in isothermal-isobaric section of phase diagram. At first, calculated the curve of the intersection of the surfaces Gibbs molar energies as functions of compostion for two phases and selection along curve the pole point for polar system coordinates. Thereafter all the space of independent variables (ra,rb,ja) is divided into two subspaces: the first is a two-dimensional subspace ER2 formed by the radial components of composition vectors for both phases (ra, rb); the second is the one-dimensional Ej1 for 3CS (or Eqin-2 – (n-2)-dimension space for nCS), formed by the angular variable j for 3CS (or angular variables q1, q2,…, qn-2 for nCS). For every fixed pole of the spherical system coordinates in space ER2 and for every fixed angle j for 3CS (or q1, q2,…, qn-2 for nCS) we solved the subset from two equations using generalized Maxwell equal-area rule in coordinates ¶G/¶r – r and seek the tentative tie-line [2]. Finally, at the third stage we calculate of root of equation for angular components of gradients of the Gibbs molar energies for two-phases by using “angle dichotomy” (search of root of equation by changing angle variable into one-dimensional space) for three-component system.
As result we find a stable two-phase tie-line among the family of tentative tie-lines for fixed pole in 3CS. This algorith we have used for development of autonomic computer program for calculation as tie-line for 3CS for as thermodynamic properties as function temperature at two-phase three-component alloys [3].
The work is part of a research projects under the terms RFBR N 02-03-32621, the Netherlands-Russian Federation cooperation program (NWO-RFBR grant 047.014.008) and grant B0056 of the Russian Federation Target Program “Integration”.
Reference
1. A.L.Udovsky, V.N.Karpushkin, E.A.Kozodaeva. CALPHAD,1995, v.17,№ 3.
2. A.L.Udovsky.Doklady Physics.2001, v. 46, N 4, p.247-250.
3. M.V.Kupavtsev, A.L.Udovsky, M.Jacobs, H.A.J.Oonk . Abstract at CALPHAD XXXIII Conference.
E-mail udovsky@ultra.imet.ac.ru