Sublattice model

As for the thermodynamic modelling of Laves phases, this has been successfully performed for several alloy systems. Especially using the sublattice model (see 2.4.2.2) a number of phase diagrams containing Laves phases have been calculated taking into account homogeneity ranges and polytypism. 

Table 3.9. A sublattice model suggested for the main Laves phase types.

The random substitutional and sublattice models are intimately related, as a phase with random occupation of atoms on all sites can technically be considered as a phase containing a single sublattice and mathematically the same general equations used in both cases. However, it is useful to separate the two types as the sublattice models implicitly define some internal, spatial substructure and give rise to site occupations which define stoichiometric compounds. Also, and very importantly, and are governed by site occupation of the components in the various sublattices rather than the global concentration of the components themselves. 

The generalised multiple sublattice model (Sundman and Agren 1981)... 

Definition of site fractions. The multiple sublattice model is an extension of earlier treatments of the two-sublattice models of Hillert and Steffansson (1970), Harvig (1971) and Hillert and Waldenstrom (1977). It allows for the use of many sublattices and concentration dependent interaction terms on these sublattices. To woiic with sublattice models it is first necessary to define what are known as site fractions, y. These are basically the fiactional site occupation of each of the components on the various sublattices where... 

Essentially, sublattice models originate from the concepts of Temkin (1945) who proposed that two separate sublattices exist in a solid-state crystal for cations and anions. The configurational entropy is then governed by the site occupation of the various cations and anions on their respective sublattices. When the valence of the cations and anions on the sublattices are equal, and electroneutrality is maintained, the model parameters can be represented as described in Section 5.4.2. However, when the valence of the cations and anions varies, the situation becomes more complex and some additional restrictions need to be made. These can be expressed by considering equivalent fractions (/) which, for a sublattice phase with the formula (/, . .. )(M"", . ..), are given by... 

To overcome this problem an extension of the sublattice model was proposed by Hillert et al. (1985) which is now known as the ionic two-sublattice model for liquids. As in the previous case it uses constituent fractions as composition variables, but it also considers that vacancies, with a charge corresponding to the charge of the cations, can be introduced on the anion sublattice so that the composition can move away from the ideal stoichiometry and approach an element with an electropositive character. The necessary neutral species of an electronegative element are added to the anion sublattice in order to allow the composition to approach a pure element. The sublattice formula for the model can then be written as... 

To complete this section it is interesting to show the equivalence between the ionic two-sublattice model and the associate model as demonstrated by Hillert et al. (1985). Equation (5.62) can be simplified for a system (A - )p(B+ , J3°)q... 

Condition Eq. 18 is discussed based on two approaches the two-sublattice model and the FE soft mode dressed anharmonically, giving the inequaUty ... 

The conducting ion sublattice in FICs is generally considered molten . The molten sublattice model for fast ion conduction was first proposed by Strock (1936) on the basis of structural and thermodynamic data for Agl. In most FICs, the entropy of the phase transition to the FIC state is larger than the entropy of melting. For example, in Agl the entropy of the transition at 420 K from the -form to the a-form (FIC state) is 14.7 J deg mol , whereas the entropy of melting at 861 K is only 11 J deg mol . ... 

There are further models which do not introduce new ideas but use the above frame for more complicated types of interactions. Bari and SivardiSre215) discussed a two sublattice model in analogy to the molecular field theory of antiferromagnetism. In this case there are two different interaction constants, viz. the intrasublattice and the intersublattice interaction. They also expand the one-sublattice model by an Heisenberg type magnetic interaction term between the HS states. Such an interaction may only become important for degenerate spin states. 

Transverse components of the magnetization and restrictions of the two-sublattice model [9,10]... 

The magnetic data of CoO(I) and CoO(II) have been correlated with a two sublattice model and exchange parameters valuated. 

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