Regular solution modelling

The equilibrium conditions given by eqs. (4.15) and (4.16) can in general be expressed through the activity coefficients. Using a solid-liquid phase equilibrium as an example we obtain 

These expressions can be simplified since the activity coefficient in the particular case of a regular solution can be expressed by the regular solution constant 2 through eqs. (3.84) and (3.85)  

These two simultaneous equations can then be solved numerically to calculate the solidus and liquidus lines. 

Let us now return to our hypothetical system A-B where we also consider the liquid and where the solid and liquid solutions are both regular (following Pelton and 

The two branches intersect at the eutectic point and the phase diagram thus relies on a single interaction parameter, Qlki, only. 

---

The simplest model beyond the ideal solution model is the regular solution model, first introduced by Hildebrant [9]. Here A mix, S m is assumed to be ideal, while A inix m is not. The molar excess Gibbs energy of mixing, which contains only a single free parameter, is then... 

The regular solution model can be extended to multi-component systems, in which case the excess Gibbs energy of mixing is expressed as... 

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. 

In the two-state model [20,21] the two different species interact and the interaction can be expressed using the regular solution model. Thus the Gibbs energy of the liquid is... 

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) ... 

The regular solution model, originally introduced by Hildebrand [2] and further developed by Guggenheim [3], is the most used physical model beside the ideal... 

In the derivation of the regular solution model the vibrational contribution to the excess properties has been neglected. However, as a first approximation the vibrational contribution can be taken as independent of the interaction between the different atoms, and this contribution can be factored out of the exponential and taken into account explicitly. The partition function of the solution is then given as... 

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... 

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. 

T entropy term in the quasi-regular solution model... 

Ghiorso M. S. and Carmichael L S. E. (1980). A regular solution model for meta-aluminous silicate liquids Applications to geothermometry, immiscibility, and the source regions of basic magma. Contrib. Mineral Petrol, 71 323-342. 

Non-Ideal Solutions Regular and Non-Regular Solution Models... 

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. 

Throughout the editorial stages of the emerging review it was continually necessary to spell out the differences between (a) the use of an ideal solution model, (b) the use of a regular solution model with parameters derived solely from atomic properties and finally (c) the use of interaction parameters derived by feedback from experiment. A proper luiderstanding of the differences between these three approaches lay at the heart of any realistic assessment of the value of calculations in relation to experimentally determined diagrams. 

I.2 Non-ideal mixing. In reality there are mixing energies associated with attractive and repulsive interactions between A and B atoms and a further excess mixing energy, G, must be considered. The simplest way to consider these interactions is via die regular solution model. In this case... 

---