Bragg-Williams approach

Formula for the chemical potentials have been derived in terms of the formation energy of the four point defects. In the process the conceptual basis for calculating point defect energies in ordered alloys and the dependence of point defect concentrations on them has been clarified. The statistical physics of point defects in ordered alloys has been well described before [13], but the present work represents a generalisation in the sense that it is not dependent on any particular model, such as the Bragg-Williams approach with nearest neighbour bond energies. It is hoped that the results will be of use to theoreticians as well as... 

The most essential step in a mean-field theory is the reduction of the many-body problem to a scheme that treats just a small number of molecules in an external field. The external field is chosen such that it mimics the effect of the other molecules in the system as accurately as possible. In this review we will discuss the Bragg Williams approach. Here the problem is reduced to behaviour of a single chain (molecule) in an external field. Higher order models (e.g. Quasi-chemical or Bethe approximations) are possible but we do not know applications of this for bilayer membranes. 

As shown above, in the point approximation to the cluster variation method (essentially the Bragg-Williams approach), we carry out the counting of configurations by finding all fhe ways of distributing A and B atoms that guarantee an overall concentration x of A atoms and 1 — x of B atoms. The number of such configurations is... 

In order to construct the surface phase diagram, the just described stabihty determination of the different phases was repeated as function of x (no other stable phases appeared than those already mentioned). Additionally, the chemical potential scale had to be transferred to the temperature scale. For this purpose, the interaction between antisites was neglected, so that they can be treated as independent particles. The Bragg-Williams approach [120] yields A/u. = kT]n 2x— 1) and, eventually, the phase diagram of the surface, shown in Figure 11.18. Again,... 

There are serious difficulties in the derivation of the multi-component adsorption (which is typical of the reforming scenario), taking into account the role of adsorbate interactions. The only mathematically tractatable way to do it is, by following the Bragg-Williams lattice -gas Mean field field Approach (MFA). In this approach, the pair and higher-order correlations between the adsorbed molecules are ignored. With these assumptions, the adsorption isotherms within the MFA format can be easily written(8-l 1,17)... 

The greatest barrier in the application of the Multicomponent Fowler-Guggenheim or Bragg-Williams Lattice gas model to, a practical situation like Pet-reforming, is the absence of experimental interaction parameters. In the simulations of the earlier sections, representative values were used. In general, for an n component system, we need to fix n(n+l) / 2 interaction parameters of the symmetric W matrix (91 for a 13 component Model ). Mobil has used successfully a 13 lump KINPTR model(5), which essentially uses a Hougen-Watson Langmuir-Hinshelwood approach. This results in a psuedo-monomolecular set of reactions, which is amenable to matrix analysis. 

Now we switch to a thermodynamic model to look at the same process. The advantage is that the thermodynamic model is more general and not dependent on lattices or the Bragg-Williams approximation. The disadvantage is that it gives less physical insight into microscopic interactions. In this approach, experimentally measurable quantities are used in place of a lattice model. 

The individual variants of the lattice model differ fi om each other in the way the spatial distribution of the molecules of the individual components is taken into account. The simplest solution is the Bragg-Williams (B-W) approach which assumes a random distribution of molecules within the bulk phase. The thermodynamical meaning of this assumption is that the mixture is regular. In the adsorption layer, however, it is only in two dimensions (i.e., within the individual sublayers that a statistical distribution of molecules is assumed). Pioneering work in this field was published by Ono [92-94] and Ono and Kondo [95,96]. The method was later applied to the description of L/G interfaces by Lane and Johnson [97] and later taken up by Altenberger and Stecki [98]. Analytic isotherm equations have also been derived from the above... 

LRO parameter r approaches an equilibrium curve in accordance with the Bragg-Williams model (dashed line). 

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