Phase behavior of binary lattice mixtures

We begin with the simplest case of a confined binary mixture, which is a symmetric mixture confined between diemically homogeneous, nonsclective planar substrates (slit-pore). Tlie grand-potential density governing the equilibrium properties of such a mixture is given by Eq. (D.29) for the special case Xb = X = 1 a = s- These equilibrium states are obtained tn 

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Similarly to the description of real phase behavior of mixtures of low-molar-mass components, mixture models based on activity coefficients can be formulated. Whereas in the case of low-molar-mass components the models describe the deviations from an ideal mixture, the models for polymer solutions account for the deviations from an ideal-athermic mixture. As a starting point for the development of a model, all segments are placed on a lattice (Figure 10.3). Polymer chains will be arranged on lattice sites of equal size, where the number of occupied lattice sites depends on the segment number r. For a quasi-binary polymer solution, all other places are occupied with solvent segments. 

The gas-lattice model considers liquids to be a mixture of randomly distributed occupied and vacant sites. P and T can change the concentration of holes, but not their size. A molecule may occupy m sites. Binary liquid mixtures are treated as ternary systems of two liquids (subscripts 1 and 2 ) with holes (subscript 0 ). The derived equations were used to describe file vapor-Uquid equilibrium of n-alkanes they also predicted well the phase behavior of -alkanes/PE systems. The gas-lattice model gives the non-combinatorial Helmotz free energy of mixing expressed in terms of composition and binary interaction parameters, quantified through interaction energies per unit contact area (Kleintjens 1983 Nies et aL 1983) ... 

This chapter demonstrates how to calculate phase diagrams and solubility isotherms for binary and ternary supercritical mixtures. As Johnston has pointed out (Wong, Pearlman, and Johnston, 1985 Johnston, Peck, and Kim, 1989), no single model will work for all situations. As the equations describing molecular interactions in dense fluids become more accurate, we can expect our abilities to model complex phase behavior to improve. At present, using a cubic equation of state or a lattice-gas equation appears to offer the best compromise between accuracy and ease of application. 

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