Mean-field Bragg-Williams theory

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]... 

Those theories of ordering that assume that the thermodynamics of ordering can be explained by Q alone are termed mean-field theories. Here two such theories are explored and compared that of Bragg and Williams (1934) and that of Landau (1937). While some attempt has been made to consider the pressure-dependence of order-disorder phenomena in minerals (Hazen and Navrotsky 1996), here I shall limit the discussion of these phenomena to their temperature-dependence alone. 

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. 

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. 

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