A simple bisection technique for the calculation of a two-solid or two-liquid miscibility gap in binary metallic systems
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I.
INTRODUCTION
II.
A prerequisite for calculating the equilibrium composition of the miscibility gap boundaries of a liquid or solid phase of a binary system lies in the elucidation of the two compositional regions in which these boundaries must fall and in which converging calculations can be made. This is done by using the instability requirements of Gibbs and equations relating activities to composition. It is shown that these compositional regions are bounded on the inside by the spinodal, inside of which a single liquid is unstable against the smallest compositional fluctuation, and on the outside by regions in which only one liquid is stable, as shown in Figure 1. It is then shown that a simple bisection technique can be used in these intermediate zones to obtain the required convergence to the miscibility gap compositions. The convenience and accuracy of the method is indicated by the facts that it requires a desk calculator ( e . g . , a Hewlett Packard 9810) and only a rough estimate of the consolute point composition as a starting point for the calculations, and calculates with precision the miscibility gap compositions in concentrated and dilute solutions. The method of determination of the spinodal points (at dZAFM/dN 2 = 0) forms an important part of the calculation, and this permits the temperature and composition of the consolute point to be calculated with precision. The method is rigorously tested using the elementary regular solution equations whose symmetrical nature permits other means to be used for calculating the miscibility gap boundary composition. The example illustrated in Figure 1 is based on the three coefficient Margules equation of Gaye and Lupis 2 for the liquid phase in the Pb-Zn system at a temperature of 1000 K. Finally, the method is used to calculate the boundary compositions of the two liquid region of the Cu-Pb system following the correlation of the liquidus boundary and thermodynamic data of this system by the Gaussian plus Krupkowski formalism of Esdaile.
CALCULATION OF T H E MISCIBILITY GAP BOUNDARIES
A. Introduction
As a guide to illustrate the nature of this method, reference is made to Figures l(a) and (b) which show, respectively, the AF M vs Nzn and the apb and az. vs Nzo curves of the system Pb-Zn according to the following threecoefficient Margules equation of Gaye and Lupis at 1000 K. A F M ( J / g atom) = NpbNz.(13973Npb + 24224Nzn
+ 8638.6NpUVzn) + 1000R(Npb In Npb ~- gzn In Nz.) [1] The partial free energy equations were obtained from Eq. [1] by means of the Eqs. [2] and [3] AFz. = A F M + (1 - Nzn) dAF M
AFpb -- A F M + (1 - Upb)dAF--~M
METALLURGICAL TRANSACTIONS A
[3]
dNpb
The common tangent to the A F M vs Nzn curve of Eq. [1] as required by the present miscibility gap boundary solution is shown in Figure l(a). Figure l(b) illustrates some features of the activity vs composition curves which are utilized in the present method and includes the rectangle IJLK whose vertical sides represent the compositions of the miscibility gap boundaries and whos
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