Solution models Bragg-Williams

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]... 

This shows that the Bragg-Williams model predicts that the transition temperature is a direct function of the exchange energy for the order-disorder process. Furthermore, the equilibrium condition dddQ = 0 gives the solution... 

Figure 11. r dependence of Q for members of the ilmenite-hematite solid solution, determined from neutron powder diffraction (solid symbol Harrison et al. 2000a) and quench magnetization (open symbols Brown et al. 1993). Sohd hues are fits using a modified Bragg-Williams model. 

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... 

Figure 26.1 7 (a) Theoretical predictions for the heat capacity C near the critical temperature, for two-dimensional systems. The Bragg-Williams mean-field lattice model of Chapter 25 leads to a triangular function, while the exact solution of the two-dimensional Ising model shows a sharp peak. Source R Kubo, in cooperation with H Ichimura, T Usui and N Hashitsome, Statistical Mechanics, Elsevier Pub. Co., New York (1 965). (b) Experimental data for helium on graphite closely resembles the Ising model prediction. Source RE Ecke and JC Dash, Phys Rev B 28, 3738 (1983). 

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... 

Note.- We have seen that relation [2.70] of the strictly-iegular solution model was called the Bragg and Williams zero-order approximation. Similarly, in view of relation [2.89], the quasichemical solution is called an approximation of order 1. The order at hand is, in fact, the power to which the exponential appearing in relation [2.89] is raised power zero for the Bragg and Williams model, and power 1 for the quasi-chemical model. 

In order to evaluate the functions g(5) and E s), we need to know the distribution of the atoms on the lattice for the given value of s. Two models have been developed the Gorsky, Bragg and Williams model and the quasi-chemical model. The hypotheses upon which these models are based are similar, respectively, to those used for the model of a strictly-regular solution (see section 2.3.3) and those used for Fowler and Guggenheim s quasi-chemical solution model (see section 2.3.5). 

This maximum value suggests a definition of a critical temperature for this system. However, an another critical temperature is usually introduced to describe order-disorder transitions [24]. Its determination is strongly connected to the Bragg-Williams model. In this model, the solution of... 

---