Bragg—Williams theory

Nix and Shockley [6] gave a detailed review of the status of order-disorder theory and experiment up to 1938, with emphasis on analytic improvements to the original Bragg-Williams theory, some of which will be... 

AET atomic environment type BW Bragg-Williams (theory of ordoing)... 

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. 

Fig. 3. Schematic diagram for the variation with concentration of the partial molar heat of solution of the liquid noble metals into liquid tin, taken from reference 51. The numbers are the experimental and calculated AHt for the solutes, in cal/g atom, at the two concentrations of 0 and 0.02 mole fraction. Q-C labels the curves calculated by the quasichemical theory in first order B-W labels the curves calculated by the Bragg-Williams, or zeroth-order approximation, which assumes a random...

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. 

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]... 

The discussion of co-operative phenomena given here is based on the simple Bragg-Williams model. The modern theories of order-disorder changes have undergone rapid development recently. The situation in 1938 is admirably reviewed by Nix and Shockley 1 more recent summaries of both theoretical and experimental developments will be found in papers by Lipson and Wannier. See also Guggenheim,Rush-brooke, and footnote p. 305. 

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. 

Again, for simplicity we shall use the common notation F rather than < F >. Some theories enable one to define an approximate PD that depends on a set of parameters m and thus leads to an approximate free energy F(m). Such a parameter, for example, is the long-range order in the Bragg-Williams approxi-... 

However, as near-neighbour correlations are neglected completely in the Bragg-Williams approximation, the theory is unable to account for the specific heat and other anomalies in the isotropic phase. An attempt has been made to extend the theory by using the quasi-chemical or first... 

At the outset we emphasize, however, that F c) is not a well-defined quantity thermodynamic potentials are well defined for thermal equilibrium states only states with (d F (c)ldc )T < 0 violate the basis laws of statistical thermodynamics. For Ccoex < c < CcoeJ- the only well-defined free energy is the free energy which corresponds to the lever rule, i.e., F(c) = F(cil, T))X + F(c ,(7))(l - X), i.e., a linear function of the concentration. The double-well shape of F(c) obtained from mean-field theories, such as the Bragg-Williams approximation of binary mixtures, is an artifact of an uncontrolled approximation. So the singular behavior resulting at the spinodal should not be taken seriously. 

Many treatments have been developed in order to improve the BET theory [16, 17]. Among them we mention the model by Hill [16], essentially based on the same assumptions of BET theory, but accounting for lateral interactions among adatoms in the same layer (within the Bragg - Williams approximation). The obtained improved isotherm is considerably more complicated without leading to better agreement with experimental results [18) for these reasons it is not frequently utilized in practice - that occurs for other seemingly improved isotherms. 

The Plory-Huggins theory is a generalization of the BRAGG-WILLIAMS" approximation in the lattice model of binary solutions. The polymer is considered to consist of X segments equal in size to a solvent molecule. Hence x is the ratio of molar volumes of the polymer and solvent. N2 polymer molecules and Nj solvent molecules are placed randomly on a lattice of coordination number z. The volume fractions of solvent and polymer are then... 

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