The modified quasi-chemical model: Part III. Two sublattices
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INTRODUCTION
IN the first two articles in the present series, the modified quasi-chemical model for short-range ordering (SRO) in the pair approximation was described for solutions in which the species mix on only one lattice or sublattice. The present article extends the treatment to mixing on two sublattices. In solid solutions, the existence of two sublattices is a manifestation of long-range ordering. For example, in a solid ionic solution, one can distinguish anionic and cationic sublattices. In a liquid solution, on the other hand, there is no long-range ordering and, strictly speaking, it is incorrect to speak of sublattices. In molten NaCl, for example, the Na+ and Cl⫺ ions should be treated as residing on one sublattice, but with a very high degree of SRO, such that the nearest neighbors of Na+ ions are almost exclusively Cl⫺ ions, and vice versa. Solutions of molten salts could, thus, be treated with the single-sublattice model described previously.[1,2] However, in such solutions, in which the degree of SRO is very high, it is conceptually and mathematically simpler to treat the liquid solution as if it consisted of two distinct sublattices. This does not preclude the possibility of a small number of cation-cation or anion-anion nearest neighbors, since these can be treated within the two-sublattice model as substitutional defects (cations on anion sites and anions on cation sites). In a solid solution, the ratio of the numbers of sites on the two sublattices is necessarily constant. However, in a liquid, this ratio can vary with composition. For example, in molten NaCl-CaCl2 solutions, the ratio of cation to anion sites varies from 1/1 to 1/2 as the composition varies from pure NaCl to pure CaCl2. A two-sublattice model for multicomponent molten salt solutions was developed in an earlier article,[3] but only for the case of random mixing of species on their respective sublattices. This was an extension and generalization of earlier work by Blander, Yosim, and Saboungi.[4,5,6] Shortrange ordering of first-nearest neighbors (FNNs) was introduced into the model by Dessureault and Pelton.[7] That is, account was taken of the fact that certain FNN (“cationanion”) pairs predominate. However, only reciprocal ternary [1,2]
systems (with only two species on each sublattice) were considered, and only for the case of an equal and constant number of sites on the two sublattices. In the present article, this model is generalized. Simultaneous SRO of FNNs and SNNs is not treated by the present model, since this is not possible within a pair approximation. However, this will be the subject of the next article in the present series. II. THE MODEL A. Definitions and Coordination Numbers The solution consists of two sublattices, I and II. Let A, B, C, . . . and X, Y, Z, . . . be the species which reside on sublattices I and II, respectively. In a salt solution, for example, A, B, C, . . . are the cations and X, Y, Z, . . . are the anions. As another example, in a spinel solid solution, sublattices I and II would be asso
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