The combinatorial entropy of mixing

Huyskens, P. L., and M. C. Haulait-Pirson. 1985. Anew expression for the combinatorial entropy of mixing in liquid mixtures.J. Mol. Liq. 31 135-151. 

It emanates from Eq. (5) that as mentioned before the combinatorial entropy of mixing stabilizes the mixture and that X < 0 favors miscibility of the components, especially, in the case of high-molar-mass polymers (each r large) when the combinatorial entropy of mixing tends to very small values. 

Fig. 2. Variation of the parameter X (curve 4) and its constituents - interaction (/), free-volume (2) and size-effect (2) - as a function of reduced temperature according to Eq. (14). The parameters used are XJb = — 1 x 1CT4, T2 = 6 x 10-4, p2 = 3 x 10 5. The combinatorial entropy of mixing at = 0.5 and r = 1000 is given by the horizontal dashed straight line...

This is often called the combinatorial entropy of mixing. There are other contributions to the entropy that this simple model does not deal with. Free volume, for example, which in the lattice model approach can be handled by allowing for holes (empty sites) on the lattice. This is outside the scope of our discussion, however, but we will come back and qualitatively examine the effect of some of the factors we have neglected later, when we consider phase behavior. 

There is one more aspect of the entropy of mixing (Equation 11-13) that we need to mention. Let s say we were mixing 25 blue molecules with 75 red ones (because we re talking about the number of molecules, rather than moles, we will use k instead of R in the equation). The combinatorial entropy of mixing would be (Equation 11-16) ... 

The combinatorial entropy of mixing is usually taken in the form of the classical Flory-Huggins theory as... 

If the polymers are not of very high molecular weight and the combinatorial entropy of mixing is not negligible. 

Lichtenthaler et al. (55) determined interaction parameters for 22 solutes in poly(dimethyl siloxane) to test several expressions of the combinatorial entropy of mixing [Eq. (7)]. The magnitude of the interaction parameter is indeed directly dependent on the evaluation of the combinatorial contribution. The combinatorial contribution was computed following both the Flory-Huggins approximation and the multiple-connected-site model recently developed by Lichtenthaler, Abrams and Prausnitz (56). This model, which retains the Flory-Huggins term, also corrects for the bulkiness of the components of the mixture. Interaction parameters were computed through both approximations, showing the sensitivity of the results to the model chosen. 

Thg necessary conditions for miscibility are that G < 0 a that d G/d iji < 0, where mole fraction of the i component. In equation (1), the combinatorial entropy of mixing depends on the number of molecules present according to... 

Therefore, as the molar mass gets large the number of molecules becomes small, and the combinatorial entropy of mixing becomes negligibly small. 

The genesis of the UCST curve for polymer-solvent systems is usually ascribed to enthalpic interactions between the mixture components, which are relatively insensitive to pressure for these constant density systems. The UCST curve can be modeled well with a liquid solution model that adequately accounts for specific interactions between the segments of polymer and the solvent. Examples of interactions are hydrogen bonding and polar interactions. The model also needs to account for the combinatorial entropy of mixing solvent molecules with the many segments that make up a single polymer chain (Prausnitz, 1969). 

In Eq 2.39, ([)j is the volume fraction and Vj is the molar volume of the specimen i . The first two logarithmic terms give the combinatorial entropy of mixing, while the third term the enthalpy. For polymer blends Vj is large, thus the combinatorial entropy is vanishingly small — the miscibility or immiscibility of the system mainly depends on the value of the last term,... 

Free volume approach to the combinatorial entropy The combinatorial entropy of mixing can be more readily derived by a free volume approach which renders the assumptions inherent in the Flory-Huggins theory more transparently obvious. Anticipating what is to be presented in Section 3.3, vis-d-vis the equation-of-state theory, we present a brief account of this alternative derivation. 

Cooling an entropically stabilized dispersion decreases the contribution of the combinatorial entropy of mixing to the overall free energy of interpenetration. Whilst the free volume contribution also decreases significantly in absolute magnitude, the contact dissimilarity term is relatively unaffected by the drop in temperature, as discussed in the preceding section. It is therefore not... 

It might be expected that just below the UCFT, the enthalpies associated with the contact and free volume dissimilarities should impart enthalpic stabilization. Conversely, just above the LCFT (if accessible), the combinatorial entropy of mixing should give rise to entropic stabilization. Flocculation on cooling appears to result from the free volume contribution. This may explain why such flocculation is not always readily achieved in aqueous systems of this type. 

The formalism of the lattice was used for convenience to calculate the combinatorial entropy of mixing according to the method outlined in Section 8.2 for small molecules, including the same starting assumptions and restrictions. 

Equation 8.22 is the expression for the combinatorial entropy of mixing of an athermal polymer solution, and comparison with Equation 8.7 shows that they are similar in form except for the fact that now the volume fraction is foimd to be the most convenient way of expressing the entropy change rather than the mole fraction used for small molecules. This change arises from the differences in size between the components, which would normally mean mole fractions close to unity for the solvent, especially when dilute solutions are being studied. 

The combinatorial entropy of mixing of a polymer molecule and a solvent can also be obtained from a consideration of the statistical distribution of both molecules over a two-dimensional lattice. For an ideal solution this is given by ... 

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