Combinatorial entropy of mixing

The entropy of mixing for mixtures of dissimilar components is an important contribution to the ability to achieve miscibility. The determination of the entropy of mixing begins with the Boltzmann relationship  

For polymers, the assumption is made that the lattice is comprised of N cells with a volume of V. Each polymer molecule occupies volumes V] and V2, respectively, with each mer unit occupying a volume, Vmer- The molecular volume, V), is equal to the product of Vmer and the number of mer units. For solvents, the number of mer units is 1. The volume fractions (pi and (p2 are represented by the equations  

With the assumptions noted above for placement of polymers in the lattice, the substitution of the assumptions into Eq. 2.11 and Eq. 2.10 leads to  

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The separation of Hquid crystals as the concentration of ceUulose increases above a critical value (30%) is mosdy because of the higher combinatorial entropy of mixing of the conformationaHy extended ceUulosic chains in the ordered phase. The critical concentration depends on solvent and temperature, and has been estimated from the polymer chain conformation using lattice and virial theories of nematic ordering (102—107). The side-chain substituents govern solubiHty, and if sufficiently bulky and flexible can yield a thermotropic mesophase in an accessible temperature range. AcetoxypropylceUulose [96420-45-8], prepared by acetylating HPC, was the first reported thermotropic ceUulosic (108), and numerous other heavily substituted esters and ethers of hydroxyalkyl ceUuloses also form equUibrium chiral nematic phases, even at ambient temperatures. 

Statistical thermodynamic mean-field theory of polymer solutions, first formulated independently by Flory, Huggins, and Staverman, in which the thermodynamic quantities of the solution are derived from a simple concept of combinatorial entropy of mixing and a reduced Gibbs-energy parameter, the X interaction parameter. 

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. 

Donohue, M.D. Prausnitz, J.M. Combinatorial entropy of mixing molecules that differ in size and shape simple approximation for binary and multicomponent mixture. Can. J. Chem. 1975, 53, 1586. 

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) ... 

Polymers were thought to be usually immiscible due to their low combinatorial entropy of mixing. Any small unfavourable heat of mixing, positive AH, would thus preclude miscibility. However, many pairs of polymers are now known to show specific interactions such as hydrogen bonds which result in a favourable heat of mixing. It is part of the intentions of this review to stress the importance of these specific interactions and to show that a consideration of these interactions is essential to an understanding of the phenomena and theory of polymer miscibility. 

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. 

This combinatorial entropy of mixing for low-molar-mass species is given by... 

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. 

The preceding derivation makes clear, in the context of the free volume approach, what assumptions are necessary to derive the Flory-Huggins combinatorial entropy of mixing. Specifically, these are that the solvent and polymer possess identical free volume fractions and that the total free volume is conserved on mixing. 

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. 

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