## 5 Calphad Modeling of FK Phases  
### Basics of Calphad Modeling  
The aim of the Calphad modeling is to describe the Gibbs energies of each phase of a given system. This yields the possibility of combining these phases and systems in a database allowing to make predictions of the phase equilibria in a multi-component system. In the Calphad approach, generally, phases are described not only in their stable ranges of stability, but also outside in metastable states. The most convenient way of describing the Gibbs energy of a non-stoichiometric ordered phase is to use the so-called compound energy formalism, also called sublattice model [50]. It consists in dividing the phase into different sub-lattices in which atom mixing may or may not occur. The Gibbs energy is then defined by applying the thermodynamics of solution on each sublattice on which disorder is present. The reference term for the Gibbs energy is given by a linear combination of the Gibbs energies of so-called end-members which are the ordered compounds defined by the model.  
Figure 6: Fe site fraction in \(\sigma\) Cr-Fe, obtained by CWM-CVM at 500K (**left**) [28], and obtained by BWA at 1,000 K (right) [38]. Experimental points from [45].  
Let's take the example of a phase of stoichiometric composition \(AB_{2}\) with \(A\) on one site and \(B\) on another site, and having a homogeneity range extending on both side of the stoichiometric composition. If the mechanism is an anti-site defect for both hypo- and hyper-stoichiometric compositions, then the two sublattices may be used and the sublattice model would be written (\(A\),\(B\))(\(A\),\(B\))\({}_{2}\) in which the bold element designates the major element on each sublattice. This model generates four end-members of compositions \(AA_{2}\), \(AB_{2}\), \(BA_{2}\) and \(BB_{2}\), among which only one corresponds to a stable composition. One of the tasks of Calphad assessors is to assign values to the Gibbs energies of these compounds.  
One should be able, with this method, to describe both the composition range of the phase (in the preceding example, one may note that the phase is defined from pure \(A\) to pure \(B\)) and the configurational entropy in a correct manner (which would not be the case if the phase was treated with a single sublattice (\(A\),\(B\))).  
However, this approach requires the knowledge of the mechanism for non-stoichiometry (number of sites on which disorder is present and type of disorder). Let's imagine that the non-stoichiometry on the \(B\) rich side of the preceding compound is due to \(A\) vacancies, then the model should be written (\(A\),vac)\(B_{2}\) which gives rise to different end-members \(AB_{2}\) and pure \(B\) with a totally vacant \(A\) site, a different range of described composition (from \(AB_{2}\) to pure \(B\)), and different configurational entropy for the same composition.  
On the other hand, for crystal structures containing more than two crystallographic sites on which disorder occurs, the number of end-members to be evaluated varies exponentially with the number of sublattices. This yields the necessity to simplify the description and reduce the number of sublattices by (\(i\)) combining different sites in single sublattices and (\(ii\)) not allowing all the elements on the different sublattices.