Williams model

Figure 9.10 Order parameter a for the Bragg-Williams model as a function of reduced temperature.

R Williams. Model building in mathematical programming. Wiley, Chichester, 1988. 

The original Williams model (1967) was modified according to the suggestion made by Fredrickson (1976). 

H.P. Williams, Model Building in Mathematical Programming, third ed., Wiley, Chichester, 1997. 

Detailed calculations on the condensed phases of biphenyl have been carried out by the variable shape isothermal-isobaric ensemble Monte Carlo method. The study employs the Williams and the Kitaigorodskii intermolecular potentials with several intramolecular potentials available from the literature. Thermodynamic and structural properties including the dihedral angle distributions for the solid phase at 300 K and 110 K are reported, in addition to those in the liquid phase. In order to get the correct structure it is necessary to carry out calculations in the isothermal-isobaric ensemble. Overall, the Williams model for the intermolecular potential and Williams and Haigh model for the intramolecular potential yield the most satisfactory results. In contrast to the results reported recently by Baranyai and Welberry, the dihedral angle distribution in the solid state is monomodal or weakly bimodal. There are interesting correlations between the molecular planarity, the density and the intermolecular interaction. 

Williams modelled this behavior using the Dugdale model and derived a growth law of the form ... 

In the Bragg-Williams model studied in 6, not only G (given by 19.54) but also 8 and F change continuously on passage through the Curie point. On the other hand the second derivatives... 

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. 

Figure 4. The dependence of the free energy on the degree of order, Q, given by the Bragg-Williams model, at three temperatures (above E, below E and at E)-...

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 9. The results of the Monte Carlo simulations of ordering of A1 and Si in the cordierite framework (from Thayaparam et al. 1995) and feldspar (Meyers et al. 1998), compared wi the equilibrium order parameter predicted by the Bragg-Williams model and the pair CVM model, shown on a rescaled temperature axis.

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 ordering behaviour in ilmSO, ilm90 and ilmlOO appears to be fully reversible, but the data close to Tc can only be fitted with a critical exponent for the order parameter, /3, which is of the order of 0.1, which does not correspond to any classical mean field Landau-type model. Instead, a modified Bragg-Williams model is required, that describes the free energy phenomenologically in terms of a configurational entropy alongside an enthalpy that contains terms up to... 

I.T. Show, M.B. Show, and L.R. Williams, Modeling the Contributions of Cross-Reacting Chemicals in Immunoassays in Aquatic Toxicology and Hazard Assessment 12th vol., ASTMSTP 1026, ed. by U.M. Cowgill and L.R. Williams, American Society for Testing and Materials, Philadelphia, 1989, pp. 21-33. 

Fig. (29). Williams model study on the spirosuccinimide ring system of asperparaline A (123) [94]...

Study no dynamical properties have been calculated and hence it is not possible to comment on the performance of the Williams model in predicting dynamical properties accurately. However, it is worthwhile to note that an earlier study of solid benzene using the Williams intermolecular potential found that it leads to a rather tightly packed structure [31]. 

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