4SL Model for FCC, HCP, and BCC
Ordered FCC, HCP and BCC solution phases handled by the 4SL model requires four sublattices for substitutional ordering and can additionally have an interstitial sublattice. A 4SL model is especially useful for modelling different kinds of ordered phases that are based on the same disordered phase, such as L10 and L12 based on FCC_A1, and B2, D03, L21 and B32 based on BCC_A2 etc.
For ordered FCC or HCP phases, these four substitutional sublattices represent four corners of the regular tetrahedron on these lattices, all of which are the nearest neighbours. These corners are equivalent lattice points, thus all G parameters for each end-member with the same elements, but distributed on different sites, must be identical. It should be emphasized that the end-member energy here represents the ordering energy rather than the formation energy of the compound.
The constraints on the parameters in the 4SL model can be derived based on the symmetry of the lattice:
GA:B:B:B=GB:A:B:B=GB:B:A:B=GB:B:B:A
GA:A:B:B=GA:B:A:B=GA:B:B:A=GB:A:A:B=GB:A:B:A=GB:B:A:A
GB:A:A:A=GA:B:A:A=GA:A:B:A=GA:A:A:B
LA,B:*:*:*=L*:A,B:*:*=L*:*:A,B:*=L*:*:*:A,B
The asterisk * means that the interaction parameter is independent of the occupation of that sublattice. For the disordered phase to be completely disordered, i.e. that all site fractions are equal on all four sublattices, all constraints must be correct.
For ordered BCC phases, the situation is a bit more complicated, as the four sublattice ordering phase represents an irregular tetrahedron with two pairs of sites that are next nearest neighbors. Thus, for an A-B binary solution phase, with A located on two sublattice sites and B on the other two, the end-member described by GA:A:B:B has four nearest neighbour bonds between A and B atoms, whereas the end-member described by GA:B:A:B has two nearest neighbour bonds between A and B atoms and two next nearest neighbour bonds. Many parameters thus have a relation:
GA:B:B:B=GB:A:B:B=GB:B:A:B=GB:B:B:A
GB:A:A:A=GA:B:A:A=GA:A:B:A=GA:A:A:B
GA:A:B:B=GB:B:A:A
GA:B:A:B=GA:B:B:A=GB:A:A:B=GB:A:B:A
LA,B:*:*:*=L*:A,B:*:*=L*:*:A,B:*=L*:*:*:A,B
The GA:B:B:B term represents D03, GA:A:B:B B2 and the GA:B:A:B B32 ordering. There are also two kinds of reciprocal interaction parameters:
LA,B:A,B:*:*;0…9
LA,B:*:A,B:*;0…9
Kusoffsky et al. (2001) have shown the influence of different parameters available in the 4SL model applied to fcc ordering. The possibility to use the 4SL model to BCC ordering has been studied by Sundman et al. (2009) in the Al-Fe system.