Sub-lattice model

In simple solutions such as binary alloys, the components are distributed on a single lattice. More complex solutions may consist of two or more sub-lattices, and in a solution of simple ionic salts like NaCl and NaBr there is one sub-lattice for cations and one for anions. In these cases the interactions considered in the models are between next neighbouring pairs of atoms rather than nearest neighbour atoms, as is the case with a single lattice. Two sub-lattice models can also be applied to... 

Chapter S examines various models used to describe solution and compmmd phases, including those based on random substitution, the sub-lattice model, stoichiometric and non-stoichiometric compounds and models applicable to ionic liquids and aqueous solutions. Tbermodynamic models are a central issue to CALPHAD, but it should be emphasised that their success depends on the input of suitable coefficients which are usually derived empirically. An important question is, therefore, how far it is possible to eliminate the empirical element of phase diagram calculations by substituting a treatment based on first principles, using only wave-mecbanics and atomic properties. This becomes especially important when there is an absence of experimental data, which is frequently the case for the metastable phases that have also to be considered within the framework of CALPHAD methods. 

They can also be made identical in the general case if the conditions t = —vB/vA and j — 1. The equivalences break down in ternary and higher-order systems as there is the introduction of more compositional variables in the associate model than for the two-sublattice case. This was considered (Hillert et al. 1985) to demonstrate the advantages of the sub-lattice model, but as mentioned previously it turns out that the number of excess terms to describe Fe-Mn-S is very similar. 

It will be interesting to see how such treatments based on a sub-lattice model can be made more general. So far it has only been used for quite simple systems with ordered structures such as LI2 and LIq. It may be necessary to include more than four sub-lattices for complex ordered phases which are superstructures of these types, and to consider more than 1st or 2nd neighbour energies. Furthermore, the choice of clusters for the sro part must relate back to the sub-lattice model itself and it is difficult to see how the more complex clusters routinely handled by CVM models can be reconciled with the sub-lattice models used so far. 

The sub-lattice model is now the predominant model used in most CALPHAD calculations, whether it be to model an interstitial solid solution, an intermetallic compound such as 7-TiAl or an ionic solution. Numerous early papers, often centred around Fe-X-C systems, showed how the Hillert-Staffansson sub-lattice formalism (Hillert and Staffansson 1970) could be applied (see for example Lundberg et al. (1977) on Fe-Cr-C (Fig. 10.8) and Chatfteld and Hillert (1977) on Fe-Mo-C (Fig. 10.9)). Later work on systems such as Cr-Fe (Andersson and Sundman 1987) (Fig. 10.10) showed how a more generalised sub-lattice treatment developed by Sundman and Agren (1981) could be applied to multi-sub-lattice phases such as a. 

Temkin was the first to derive the ideal solution model for an ionic solution consisting of more than one sub-lattice [13]. An ionic solution, molten or solid, is considered as completely ionized and to consist of charged atoms anions and cations. These anions and cations are distributed on separate sub-lattices. There are strong Coulombic interactions between the ions, and in the solid state the positively charged cations are surrounded by negatively charged anions and vice versa. In the Temkin model, the local chemical order present in the solid state is assumed to be present also in the molten state, and an ionic liquid is considered using a quasi-lattice approach. If the different anions and the different cations have similar physical properties, it is assumed that the cations mix randomly at the cation sub-lattice and the anions randomly at the anion sub-lattice. 

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. 

With Amix//m = 0 the ideal Temkin model for ionic solutions [13] is obtained. If deviations from ideality are observed, a regular solution expression for this mixture that contains two species on each of the two sub-lattices can be derived using the general procedures already discussed. The internal energy is again calculated... 

Non-stoichiometry in solid solutions may also be handled by the compound energy model see for example a recent review by Hillert [16]. In this approach the end-member corresponding to vacancies is an empty sub-lattice and it may be argued that the model loses its physical significance. Nevertheless, this model represents a mathematically efficient description that is often incorporated in thermodynamic representations of phase diagrams. 

System Lattice Model Nearest neighbours Sub-lattice/... 

Figure 7.16. Comparison of Ni-Al Phase diagrams obtained by (a) using a hybrid CVM-CALPHAD approach (Tso 1992, Cacciamani 1997) and (b) CALPHAD approach incorporating a sub-lattice ordering model (Ansaia et al. 1995).

Muntean, M., Rector, D., Herling, D., Khaleel, M., and Lessor, D. Lattice-Boltzmann diesel particulate filter sub-grid modelling- progress report. SAE Technical Paper No. 2003-01-0835 (2003). 

Pomonis and Vickermann used model solid solution catalysts with a-Al203, Ti02, and Sn02 as the host compounds.105 They conclude that vanadium ions have to be present in order to obtain catalysts which are both active and selective. Furthermore a relatively facile electron exchange between the active site cations is necessary in order to give an energetically more stable surface oxygen state than on a localized atomic site. This is important both for the release of 02 and for the reoxidation of the active sites. The host lattice has an important role in the electronic isolation of the active sub-lattice. 

Fig.6. The B2(110) surface average (solid lines) and sub-lattice (dotted lines) concentrations of the segregant in AB model alloy as a function of reduced temperature calculated in the FCEM approximation for different segregation/order factors r (indicated near the plots). The difference in sub-lattice concentrations corresponds to the surface LRO parameter that vanishes at the surface transition temperature Tg that coincides with the bulk transition temperature T independently of r.

The structural unit model has been used successfully to predict the structures of grain boundaries in perovskite structured SrTiOs bicrystals [11.31-11.34]. The structural units observed for symmetric SrTiOs [001] tilt boundaries are shown in Fig. 11.8. In a similar manner to the isolated dislocation cores in YBCO, the structural units also appear to contain atomic positions where the cations are too close together. Again, depending on the structural unit, the close separation of the atomic columns can occur for either of the sub-lattice sites,... 

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