Bragg-William approximations

We now turn to a mean-field description of these models, which in the language of the binary alloy is the Bragg-Williams approximation and is equivalent to the Ciirie-Weiss approxunation for the Ising model. Botli these approximations are closely related to the van der Waals description of a one-component fluid, and lead to the same classical critical exponents a = 0, (3 = 1/2, 8 = 3 and y = 1. 

We have employed the Bragg-Williams approximation (BWA) to obtain rough estimates of the ordering/segregation critical temperatures. It is well known that the BWA usually overestimates critical temperatures (approximately by 20 %) in comparison with the exact value obtained from Monte Carlo simulations, or by other highly accurate methods of statistical mechanics. This order of accuracy Is nevertheless sufficient for our present purposes. 

Because much experimental work has been stimulated by the quasi-chemical theory, it is important to gain proper perspective by first describing the features of this theory.12 The term, quasichemical will be used to include the Bragg-Williams approximation as the zeroth-order theory, the Bethe or Guggenheim pair-distribution approximations as the first-order theory, and the subsequent elaborations by Yang,69 Li,28 or McGlashan31 as theories of higher order. 

It should be clear that the Mayer method provides a convenient and economic framework within which to correct the major omissions inherent in the use of the crudest order-disorder results, e.g. the quasi-chemical and Bragg-Williams approximations, for crystals with short-range forces. However, detailed calculations by this method do not appear to have been attempted so far. 

If the Bragg-Williams approximation (69) is employed to describe the attractive interactions in the adsorbed layer, and the transition state is taken to be free of such interactions, the rate can be described by... 

Applying the Bragg-William approximation (assuming that the H-H interaction is given by the mean field of the H-atoms) and with the concentration Ch = Nh/ N, the free energy is... 

The nc value for palladium-hydrogen is 0.25 from magnetic susceptibility measurements (52) with Tc = 564°K (52), and from P-C-T data, de Ribaupierre and Manchester (26) estimate nc = 0.29 and Tc = 566°K. The Bragg-Williams approximation gives a reasonable WfiH value by using an average value of nc, the critical temperature, and an analytical expression for ie(n) determined... 

The homogeneous dynamics of the system could be reproduced qualitatively describing the time evolution of the camphor coverage, 0, in the Bragg Williams approximation [33], following earlier suggestions by Frumkin [162] ... 

Using the lattice-gas theory of monolayer adsorption in the mean field (or Bragg-Williams) approximation gives the C,T dependence of n, i.e. the vacancy isotherm, in the form [402,403]... 

Further elaboration requires a model. We shall consider the Bragg-Williams approximation (sec. I.3.8d) in which only the enthalpic part of G is accounted for, the entropy is assumed to remain ideal. For gas adsorbates this leads to the FFG isotherm II.3.8.17] and A1.5a] and in solutions it gives rise to the Regular Solution model, both models being fairly widely applicable. For this approximation, for a binary solution we derived I.3.8.25]... 

These matters may be attended to quantitatively in the so-called Bragg-Williams approximation. On temporarily ignoring the variation of G with x in... 

Variation of the Gibbs Free Energy with Temperature in the Bragg-Williams Approximation... 

The FFG and quasi-chemical equations of state are both based on a lattice model, with the inclusion of a lateral interaction parameter w. For attraction w < 0, for repulsion w > 0. Equation 13.4.37] is in the Bragg-Williams approximation, where it is assumed that lateral interaction has no consequences for the configurational entropy. The quasi-chemiccd approximation is better in this respect, see sec. 1.3.8a, as is inferred from the fact that it can better account for phase equilibria. 

Eustathopoulos has also summarized some theoretical calculations of a for binary alloys, which are based on lattice models in which the interface layers are treated in the Bragg-Williams approximation, assuming complete atomic disorder in the interface and in the crystal. The calculated surface free energies are related to the surface free energies of the pure components and to the activities in the bulk phases. 

The results shown in Figures 2 and 4 are intuitively obvious, and reflect the well known fact that the critical temperature in the system depends primarily on the strength of molecular interactions. In particular, in the lattice gas models the maximum of Tc is reached for the system of particles characterized by a" corresponding to the highest interaction between adsorbed particles. This can be readily demonstrated by considering the prediction of a very simple mean-field theory in the Bragg-Williams approximation. 

The L-type, would follow the Langmuir model, which is site adsorption without any lateral interaction between the adsorbate molecules. The concavity of the curve, in normal scale, is always directed toward the concentration axis. The S-type would follow a more complex model in which lateral interactions between molecules are to be taken into account, using, e.g., the Bragg-Williams approximation [15]. A concavity of the adsorption isotherm directed toward the y-axis is a very strong indication of lateral interactions between molecules. If one looks at the lUPAC classification of gas adsorption isotherms [1], the same remark holds this type of concavity is related with phenomena involving interactions between adsorbate molecules capillary condensation, multilayer formation, 2-D phase changes, etc. 

In these approximations, as well as in higher ones, one finds that when w < 0 (attraction) there exists a critical temperature below which a first order phase change will be observed—a sudden condensation, as the equilibrium gas pressure is increased, from a dilute localized monolayer to a relatively condensed localized monolayer. For a plane square surface lattice of sites, the Bragg-Williams approximation gives — w/kTc = 1 and the quasi-chemical approximation — w/kTc = 1.386. 

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