Solution models quasi-chemical

The modeling of solid solutions, when the solvent and the solute exhibit major behavioral differences, is not easy. The required models quickly become complex. The intervention of structure elements enables us to model such solutions using quasi-chemical equilibria. 

UNIFAC was built on the framework of a contemporary model for correlating the properties of solutions in terms of pure-component molecular properties and fitting parameters, viz. UNIQUAC (the universal quasi-chemical) model... 

By a statistical model of a solution we mean a model which does not attempt to describe explicitly the nature of the interaction between solvent and solute species, but simply assumes some general characteristic for the interaction, and presents expressions for the thermodynamic functions of the solution in terms of an assumed interaction parameter. The quasi-chemical theory is of this type, and we have noted that a serious deficiency is its failure to consider the vibrational effects in the solution. It is of interest, therefore, to consider briefly the average-potential model which does include the effect of vibrations. 

It is difficult to point to the basic reason why the average-potential model is not better applicable to metallic solutions. Shimoji60 believes that a Lennard-Jones 6-12 potential is not adequate for metals and that a Morse potential would give better results when incorporated in the same kind of model. On the other hand, it is possible that the main trouble is that metal solutions do not obey a theorem of corresponding states. More specifically, the interaction eAB(r) may not be expressible by the same function as for the pure components because the solute is so strongly modified by the solvent. This point of view is supported by considerations of the electronic models of metal solutions.46 The idea that the solvent strongly modifies the solute metal is reached also through a consideration of the quasi-chemical theory applied to dilute solutions. This is the topic that we consider next. 

Whereas the quasi-chemical theory has been eminently successful in describing the broad outlines, and even some of the details, of the order-disorder phenomenon in metallic solid solutions, several of its assumptions have been shown to be invalid. The manner of its failure, as well as the failure of the average-potential model to describe metallic solutions, indicates that metal atom interactions change radically in going from the pure state to the solution state. It is clear that little further progress may be expected in the formulation of statistical models for metallic solutions until the electronic interactions between solute and solvent species are better understood. In the area of solvent-solute interactions, the elastic model is unfruitful. Better understanding also is needed of the vibrational characteristics of metallic solutions, with respect to the changes in harmonic force constants and those in the anharmonicity of the vibrations. 

The final step in our quasi-chemical development is merely to recognize that a stoichiometric model for chemical association provides a correct description of %o. We imagine following a specific solute molecule of interest through chemical conversions defined by changes in the inner shell populations,... 

Pratt, L. R., and Rempe, S. B. (1999). Quasi-chemical theory and implicit solvent models for simulations. In Simulation and Theory of Electrostatic Interactions in Solution. Computational Chemistry, Biophysics, and Aqueous Solutions (L. R. Pratt and G. Hummer, eds.), vol. 492 of AIP Conference Proceedings, pp. 172-201. American Institute of Physics, Melville, NY... 

If the second term in the configurational entropy of mixing, eq. (9.42), is zero, the quasi-chemical model reduces to the regular solution approximation. Here, Aab is given by (eq. (9.21). If in addition yAB =0the ideal solution model results. 

Temkin was the first to derive the ideal solution model for an ionic solution consisting of more than one sub-lattice [13]. An ionic solution, molten or solid, is considered as completely ionized and to consist of charged atoms anions and cations. These anions and cations are distributed on separate sub-lattices. There are strong Coulombic interactions between the ions, and in the solid state the positively charged cations are surrounded by negatively charged anions and vice versa. In the Temkin model, the local chemical order present in the solid state is assumed to be present also in the molten state, and an ionic liquid is considered using a quasi-lattice approach. If the different anions and the different cations have similar physical properties, it is assumed that the cations mix randomly at the cation sub-lattice and the anions randomly at the anion sub-lattice. 

A more accurate type of quasi-chemical solution model was introduced by Guggenheim [E. A. Guggenheim, Mixtures (Oxford University Press, New York, 1952)] to account for specific A + B AB association corrections in both Hfs (x) and st(x). The quasi-chemical approach employs an explicit pair partition function for the equilibrium population of A B complexes in solution. More general associated solution models were also developed to incorporate AB complexes of other than 1 1 stoichiometry [A. D. Pelton and M. Blander. Metall. Trans. 17B, 805-15 (1986)]. Although such quasichemical and solution models can be more accurate, they are also more difficult to implement. 

The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

Modern theoretical developments in the molecular thermodynamics of liquid-solution behavior are based on the concept of local composition. Within a liquid solution, local compositions, different from the overall mixture composition, are presumed to account for the short-range order and nonrandom molecular orientations that result from differences in molecular size and intermolecular forces. The concept was introduced by G. M. Wilson in 1964 with the publication of a model of solution behavior since known as the Wilson equation. The success of this equation in the correlation of VLE data prompted the development of alternative local-composition models, most notably the NRTL (Non-Random-Two Liquid) equation of Renon and Prausnitz and the UNIQUAC (UNIversal QUAsi-Chemical) equation of Abrams and Prausnitz. A further significant development, based on the UNIQUAC equation, is the UNIFAC method,tt in which activity coefficients are calculated from contributions of the various groups making up the molecules of a solution. 

The present results address contributions essential to quasi-chemical descriptions of solvation in more realistic cases. An interesting issue is how these packing questions are affected by multiphasic behavior of the solution. In such cases, the self-consistent molecular field should reflect those multiphase possibilities just as it can in pedagogical treatments of nonmolecular models of phase transitions (Ma, 1985). 

Working out the historical quasi-chemical approximation in the present language for the two-dimensional Ising model of a binary solution will give perspective on the developments of this chapter. The model system is depicted in Fig. 7.15. Each site of the lattice possesses a binary occupancy variable, 5-, = —1,1 for the /th site. This will be interpreted so that Sj = 1 indicates occupancy of the /th site by one species, e.g. W (water), and = — 1 then indicates occupancy of that site by the other species, say O (oil). We write... 

There seems to be a well-developed folklore that judicious explicit inclusion of a small number of solvent (water) molecules can dramatically improve the accuracy of imphcit hydration models (Gilson et al, 1997). An important physical observation is that an appropriate inclusion of an inner shell only can capture most of the effects of the solvent on the solute of interest (Beglov and Roux, 1994 1995 Bizzarri and Cannistraro, 2002). The quasi-chemical approach is the theory for inclusions of that sort. 

There are many other equations, which have been proposed, that do not result from Wohl s method. Two of the most popular equations are the Wilson and the universal quasi-chemical theory (UNIQUAC) by Abrams and Prausnitz.These equations are based on the concept of local composition models, which was proposed by Wilson in his paper. It is presumed in a solution that there are local compositions that differ... 

This is referred to as the quasi-chemical model. More detailed solid solution models can be found in textbooks on metallurgical thermodynamics, for example, Chemical Thennodynamics of Materials, by C. H. P. Lupis (Elsevier Science Publishers, Amsterdam, 1983). 

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