The grand partition function of dilute biregular solutions
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I.
INTRODUCTION
TIlE classical modes of regular solutions (regularity) is based on the random distribution of equal-sized molecules (atoms) in a substitutional lattice, where the heat of mixing is given by the sum of nearest-neighbor pairwise interactions, t~l Despite such a simple premise, the regularity explains well most of the key features of real solutions, and it enjoys wide practical application. An attempt was made to count both first and second nearest-neighbor interactions strictly within the premise of the conventional regularity, and it resulted in the biregularity3 21 All the thermodynamic properties of a biregular solution were expressed by a single equation of grand partition function (GPF), which gives birth to triplet equations for a ternary solution. The biregularity triplets show a third-order dependence on solution composition, i.e., one order higher than the triplets of Darken's quadratic formalism, t3XThe present article explores various thermodynamic characteristics of the GPF for dilute biregular solutions, as compared with the other dilute solution models. II.
GRAND PARTITION FUNCTION OF B I R E G U L A R SOLUTIONS
Multicomponent biregular solutions can be expressed by the combination of several ternary systems, as shown elsewhere, t2] Thus, considering a ternary solution enables us to discuss the biregularity at large. Consider a ternary biregular solution consisting of N~ atoms of species 1, N2 atoms of species 2, and N3 atoms of species 3. Its GPF (,=) can be expressed by Eq. [1.1], as shown in Table I. While the biregularity constant el23 (Eq. [1.5]) does not become zero under any circumstances, it reduces to Eq. [1.7] when the interaction between a central atom and its second nearest-neighbors is MEGURU NAGAMORI, formerly Visiting Professor, Tohoku University, Japan, is with the Centre de Recherches Minrrales, Quebec Government, Ste-Foy, PQ, Canada GIP 3W8. KIMIHISA ITO, Associate Professor, is with the Department of Materials Science and Engineering, Waseda University, Tokyo 169, Japan. MOTONORI TOKUDA, Professor, is with the Institute for Advanced Materials Processing (IAMP), Tohoku University, Sendal 980, Japan. Manuscript submitted July 16, 1993. METALLURGICALAND MATERIALSTRANSACTIONS B
negligibly small. Under these conditions, the permeation constant fj of an intervening nearest-neighbor j approaches zero. Biregularity then converges to the classical regularity (Eq. [1,21). Ideality is defined as a limiting case of the regularity (or Eq. [1.3]), where the internal energy of mixing approaches zero, or wu = w0 = 0. The equilibrium condition of a ternary biregular solution is given m by 01n=-/aNi=0
(i= 1,2,3)
[1]
Application of Eq. [1] to Eq. [1.1] directly yields three Raoultian activity coefficients (Yt, Y2, Ya), as shown in Table II. Thus, the GPF is now replaced by the triplets (Eqs. [2.1] through [2.3]), which in turn express all the thermodynamic properties of a ternary biregular solution. Equations [2.1] through [2.3] are valid for all ternary and binary compositions,
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