## 5 Calphad Modeling of FK Phases
### Simple Models  
Note that a review on this subject is given in [51]. Since, it is commonly accepted that the mechanism for non-stoichiometry is substitutional in FK phases, it is easy to derive sublattice models for the most simple FK phases which contain only two crystallographic sites. The sublattice models for phases such as \(A15\) or \(C15\) may be described as follows (\(A\),\(B\))\({}_{3}\)(\(A\),\(B\)) and (\(A\),\(B\))(\(A\),\(B\))\({}_{2}\).  
The case of Laves phase \(C14\) is more complex since three sites are present. It has been suggested that the two sites occupied preferentially by \(B\) could be joined in order to get the same model as for the \(C15\) phase [52]. Other authors suggest, on the contrary, based on some experimental measurements, that the site occupancy may be significantly different between these two sites and that three sublattices should be used [53].  
On the other hand, phases like the \(\delta\) phase are so complex (14 sites) that they have to be simplified. A common model for this phase is \(A_{12}\)(\(A\),\(B\))\({}_{20}\)\(B_{24}\)[54, 55].  
Between these extreme examples, phases with a moderately high number of crystal sites (four to five) may be discussed.  
The \(\mu\) phase has been commonly described by grouping the two sites of CN12 in a single sublattice [56, 57] in models such as (\(A\),\(B\))\({}_{21}\)\(A_{18}\) or (\(A\),\(B\))\({}_{21}\)(\(A\),\(B\))\({}_{18}\). However, the recent detailed crystallographic study presented in Section 3.2 showed that, contrary to the expectation, the two sitesof CN12 have strongly different site occupancies and should be dissociated. Resulting model can be (_A_)\({}_{12}\)(_A_,_B_)\({}_{6}\)(_A_,_B_)\({}_{3}\)(_A_,_B_)\({}_{18}\). This proposed new model was applied to Nb-Ni system in [58].  
Over the years, many different models have been used for the \(\sigma\) phase (for a review see [22]). Some of them use grouping of the sites based on convenience and should be avoided (e.g., _A_\({}_{4}\)(_A_,_B_)\({}_{18}\)_B_\({}_{8}\)). More recent ones are based, as in the case of the \(\mu\) phase, on the expectation that Kasper rules are valid (e.g., (_A_\({}_{4}\)(_A_,_B_)\({}_{16}\)(_A_,_B_)\({}_{10}\)). Again, it has been shown that it was not the case and that, contrary to the expectation, sites of CN15 and CN14 have no particular reasons to be dissociated [22]. This yields to the proposal of a new model simpler that the previously used one: (_A_,_B_)\({}_{2}\)(_A_,_B_). This recent model has been used to describe the systems Mo-Re [37] and Cr-Fe [59].  
For the \(\chi\) phase, since the observed site occupancies are well accounted for by the Kasper rules, the issued proposal was (_A_,_B_)\({}_{5}\)(_A_,_B_)\({}_{12}\)(_A_,_B_)\({}_{12}\). [23]. A discussion was made in this article on the need to allow or not all the elements on the different sublattices in order to reduce the number of end-members generated by the model.  
A summary of these recommendations can be found in Table 1. These examples emphasize the need for accurate experimental crystallographic data for each phase to reveal how deviations from stoichiometry are accommodated in order to help define the model. In addition, it has been proved that the use of measured experimental site occupancies was quite useful in the Calphad approach to help define the relative stability of the different end-members generated by the model.