In contrast to using associates for liquid solutions is a sublattice approach in which cations and anions are mixed on respective lattice sites. With anions and cations assigned to specific sublattices, it is possible
to capture interactions and short-range order with species occupying the sites and additional energetic terms. The components can essentially be allowed to independently mix on each sublattice within the energetic constraints and the system free energy minimized.46 The approach has been successfully used by Gueneau et a/.47 to model the liquid in the U-O and Pu-O systems where ionic metal species reside on one lattice and oxygen anions, neutral UO2 or PuO2, charged vacancies, and O species on the other.

An improvement to the simple sublattice approach is the quasichemical method introduced by Fowler and Guggenheim48 and later further devel­oped by Pelton and coworkers.49-52 It approaches short-range order in liquids through the formation of nearest-neighbor pairs on a quasilattice. It thus differs significantly from the modified associate species approach such that in the quasichemical method short-range order is accommodated by com­ponents pairing and the energetically described extent of like and unlike components pairing. The technique thus avoids the paradox where a high

degree of short-range order in the associate species approach causes minimal end-member species con­tent and therefore fails in the limit to be a Raoult’s law solution.52

In the modified quasichemical approach for a sim­ple binary A—B system, the components are treated as distributed on a quasilattice and that an energetic term governs exchange among the pairs.

(A — A) + (B — B)=2(A — B) [19]

A parameter, Z, represents the nearest-neighbor coordination number such that each component forms Z pairs. For one mole of solution,

ZXA = 2nAA + 2nAB [20]

ZXb = 2»bb + 2»ab [21]

where the moles of each pair are nAA, nBB, and nAB. The relative proportions of each pair is Xj where

Xij = n, j / (»AA + «BB + »ab) [22]

The configurational entropy contribution is captured from the random distribution of the pairs over the quasilattice positions. The result is a heat of mixing of

Hmix = (Xab/2)Eqm [23]

where Eqm is the quasichemical model energetic parameter, and a configurational entropy

Sconfig = — R(Xa ln Xa + Xb ln Xb)

— R(Z/2)(XAAln(XAA/xA ) + Хвв1п(Хвв/Хв2)

+ Хав1п(Хав/2ХаХв)) [24]

Utilizing Gibbs free energy functions for the compo­nents and expanding EQm as a polynomial in a scheme for minimizing the system free energy pro­vide a system for optimization of the liquid using known thermochemical values and phase equilibria. Issues such as the displacement of the composition of maximum short-range order from 50% composition are dealt with by assuming different coordination numbers for each component. Greater accuracy is obtained by inclusion of the Bragg-Williams model, thus incorporating lattice interactions beyond nearest neighbors. The modification to the quasi­chemical model yields

Hmix = (Eqm/2)Xab + EbwXaXb [25]

The extension of the modified quasichemical model to ternary systems is directly possible using only binary model parameters.

An issue for the modified quasichemical model is that it fails at high deviations from random ordering, although that is generally not a problem because immiscibility will occur before the deviations grow too large. The model can also predict a large amount
of ordering that can result in a negative configu­rational entropy, a physical impossibility.53