Combinatorial entropy, mixing

Owing to low values of the combinatorial entropy mixing, miscibility in polymer-polymer systems requires the existence of strong specific interactions between the components, such as hydrogen bonding [Olabisi et al., 1979 Sole, 1982 Walsh and Rostami, 1985 Utracki, 1989]. The thermodynamic characterization of the interactions in miscible polymer blends has been the subject of extensive studies [Deshpande et al., 1974 Olabisi, 1975 Mandal et al., 1989 Lezeano et al., 1992, 1995, 1996 Farooque and Deshpande, 1992 Juana et al., 1994]. 

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. 

Most of the assumptions are based on idealized models, indicating the limitations of the mathematical methods employed and the quantity and type of experimental data available. For example, the details of the combinatorial entropy of a binary mixture may be well understood, but modeling requires, in large measure, uniformity so the statistical relationships can be determined. This uniformity is manifested in mixing rules and a minimum number of adjustable parameters so as to avoid problems related to the mathematics, eg, local minima and multiple solutions. 

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. 

It should be pointed out that in both these cases the degree of chlorination differs from PVC by around 10%. By any estimate, the heat of mixing in these cases should be quite unfavourable. For example, an estimate based on solubility parameters and using group contribution gives for PVC (6 = 19.28 J cm ) and chlorinated polyethylene (45wt.-%Cl) (5 = 18.77 J cm" ), hence for a 50/50 mixture AH is -1-0.065 J per cm of mixture. Together with unfavourable equation-of-state terms and a small combinatorial entropy contribution these mixtures would not be expected to be miscible. 

It is possible to simulate the spinodal curves of the phase diagram of polymer pairs using the Equation-of-state theory developed by Flory and co-workers. It is only, however, possible to do this using the adjustable non-combinatorial entropy parameter, Qjj. Another problem arises in the choice of a value for the interaction parameter Xjj. This is introduced into the theory as a temperature independent constant whereas we know that in many cases the heat of mixing, and hence is strongly temperature dependent. The problem arises because Xj was never intended to describe the interaction between two polymers which are dominated by a temperature dependent specific interaction. 

The link between GC quantities and the interacticm parameters of solution theories is readily established (39). In statistical theories of K>lution thermodynamics, the )lute activity is expressed as the sum of two terms, a combinatorial entropy and a noncombinatorial free energy of mixing. In the Flory-Hi ins approximation one has. 

It should be noted that Eq. (13) does not apply to the interaction parameter based on volume fractions (x), due to the inclusion of a temperature dependent combinatorial entropy. In computing partial molar heats of mixing at infinite dflution, it is essential that the correction for gas pha% nonideality ( n) be included, owing to the magnitude of (< 100—300 cal/mol). 

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... 

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