A gaussian-based formalism for the representation of free energy as a function of composition of binary metallic solutio
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I.
INTRODUCTION
THEearliest thermodynamic description of isothermal real binary solution behavior comprised the laws of Raoult (ai = Ni) and Henry (a~ = 3'~ which applied only to the terminal solvent and solute regions, respectively, so that a usually large intermediate compositional region existed in real systems in which there was no known relation between activity and composition. It has been found that the use of more sophisticated solution models such as the one and two coefficient Margules equations (here termed regular and subregular, respectively, for reasons discussed later) enables the mathematical representation of the terminal solution behavior to be considerably extended, but for most real systems, the problem of the representation of the activity with respect to composition in the intermediate range remains unsolved. The first indication of a possible solution to this problem was suggested by Darken 1 during his studies of a quadratic formalism in relation to the behavior of metallic systems. Darken found by plotting In 3'2 against ~ for real systems that two extended and near linear terminal regions existed together with an intermediate undefined compositional region. Darken further extended this study by defining an excess stability function for binary systems, which may be written as follows: excess stability =
d 2 A F xs
d 2 A F xs
dN2
= d ( 1 - N2)2 =
2RTd In 3'2 d(1 - N2)2
[1] The excess stability is thus clearly constant in the terminal regions, and by plotting excess stability vs N2 for the data of a large number of binary systems, Darken found that these systems yielded peaks which faded to join the horizontal portions of the curve in the terminal regions. Darken suggested that the following Gaussian-based term may be used to represent these peaks: dLe-d2(N2-N2*)2
[2]
where the constants dl, dE, and N2* control the height, width, and location, respectively, of each individual peak.
J.D. ESDAILE is Senior Principal Research Scientist, C. S. I. R. O. Division of Mineral Engineering, P.O. Box 312, Clayton, Victoria, 3168, Australia. Manuscript submitted April 2, 1981. METALLURGICALTRANSACTIONS B
The present work is an extension of the concept of Darken. It employs a Gaussian term at the A F xs vs N2 level and the underlying formalism employs in place of the quadratic terms of Darken, either the Krupkowski3 or subregular solution models. It is shown later that these models, in conjunction with different Gaussian terms, yield A F xs values which are very close, despite wide variations of the exponent m in the Krupkowski equations from the regular solution value of two. For the integral terms, the Gaussian function, symbolized as A F xs, takes the form: A F xs = C ( e -h2[0-(1/2)]2 -
e -h2/4)
[3]
where C and h 2 a r e independent of composition; the term O is related to the atomic fraction N2 by the following equation: O = 0N2/(0N2 + N1)
[41
where the constant 0 is used to introduce asymmetry into the right-hand side of Eq. [3], i . e . , A F xs is symmetrical only if 0 is equal to
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