Sub-regular solution

The regular solution model (eq. 3.68) is symmetrical about xA = xB =0.5. In cases where the deviation from ideality is not symmetrical, the regular solution model is unable to reproduce the properties of the solutions and it is then necessary to introduce models with more than one free parameter. The most convenient polynomial expression with two parameters is termed the sub-regular solution model. 

Still, the strain enthalpy is of particular importance. An elastic continuum model for this size mismatch enthalpy shows that, within the limitations of the model, this enthalpy contribution correlates with the square of the volume difference [41,42], The model furthermore predicts what is often observed experimentally for a given size difference it is easier to put a smaller atom in a larger host than vice versa. Both the excess enthalpy of mixing and the solubility limits are often asymmetric with regard to composition. This elastic contribution to the enthalpy of mixing scales with the two-parameter sub-regular solution model described in Chapter 3 (see eq. 3.74) ... 

Kaufman (1968b) also made it clear that the use of more realistic descriptions, such as sub-regular solution models, would necessitate the determination of many more parameters and thought that "Until such time as our knowledge of solution theory and the physical factors which control the properties of solutions might permit these parameters to be determined, it is better to continue with a simpler model. This conclusion was of course also conditioned by the limited computer memory available at the time, which prevented the use of more complex models with the subsequent increase in number of parameters which they entailed. 

Both the Muggianu and Kohler equations can be considered symmetrical as they treat the components in the same way and do not differentiate between them. This is true for another method suggested by Colinet (1967) which is derived differently than either the Kohler or Muggianu equations but which can be reduced to the Muggianu equation (Hillert 1980). Toop s equation (1965) is essentially different in that it considers one of the binary systems does not behave in the same way as the others and the extrapolation is based essentially on two of the binaries having identical mole-fraction products with the mole-fiaction products of the third being different. For a sub-regular solution Toop s equation breaks down to (Hillert 1980)... 

The regular model for an ionic solution is similarly analogous to the regular solution derived in Section 9.1. Recall that the energy of the regular solution model was calculated as a sum of pairwise interactions. With two sub-lattices, pair interactions between species in one sub-lattice with species in the other sub-lattice (nearest neighbour interactions) and pair interactions within each sub-lattice (next nearest neighbour interactions), must be accounted for. 

Let us first derive the regular solution model for the system AC-BC considered above. The coordination numbers for the nearest and next nearest neighbours are both assumed to be equal to z for simplicity. The number of sites in the anion and cation sub-lattice is N, and there are jzN nearest and next nearest neighbour interactions. The former are cation-anion interactions, the latter cation-cation and anion-anion interactions. A random distribution of cations and anions on each of... 

NA i [ l is easily derived when the cations are assumed to be randomly distributed on the cation sub-lattice. The probability of finding an AB (or BA) pair is 2Xa+Xb+ in analogy with the derivation of the regular solution in Section 9.1. iVA+B+ is then the product of the total number of cation-cation pairs multiplied by this probability... 

The equations for the regular solution model for a binary mixture with two sublattices are quite similar to the equations derived for a regular solution with a single lattice only. The main difference is that the mole fractions have been replaced by ionic fractions, and that while the pair interaction is between nearest neighbours in the single lattice case, it is between next nearest neighbours in the case of a two sub-lattice solution. 

With Amix//m = 0 the ideal Temkin model for ionic solutions [13] is obtained. If deviations from ideality are observed, a regular solution expression for this mixture that contains two species on each of the two sub-lattices can be derived using the general procedures already discussed. The internal energy is again calculated... 

Despite the above problems, mixing parameters estimated from miscibility gap information will still be an improvement over the assumption of an ideal solid-solution model, ao parameters estimated from data in Palache et al. (20) and Busenberg and Plummer (21) are presented in table I for a few low-temperature mineral groups. Because of the large uncertainties in the data and in the estimation procedure, a sub-regular model is usually unwarranted. As a result, these estimated ao values presented should be used only for solid-solution compositions on a single side of the miscibility gap, i.e. only up to the given miscibility fraction. 

Saxena (1973). A general account of the mixing properties of crystalline solutions, with detailed discussions of the van Laar, regular (one coefficient, symmetric), "sub-regular" (two-coefficient, asymmetric) and quasi-chemical models. Margules and other equations for i presented for binary... 

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