Entropy combinatory

Obviously, B can never be negative in these cases to the extent that equation 5 is vaUd, miscibility can only be driven by combinatorial entropy, and this possibihty is maximized by matching the values of 5 and 5g as nearly as possible. In general, high molecular weight, nonpolar polymers are rarely miscible with each other. 

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. 

As in a previous case, the genera number of adsorbed initial molecules is 1 mole. If intermolecular bonds are great enough, molecules in the volume cannot transit, and there is no motion at the surface. In this case, we can omit terms connected with combinatorial entropy of molecule disposition in the volume, and, even more so, their difference ... 

Terms Gch-s and Gch-s are defined by combinatorial entropy, depending on the transposition of intermolecular bonds at the molecule, taking into account the geometry of the surface and the chain (some intermolecular bonds, for example, the ones at neighboring chain atoms cannot be performed as a consequence of the structure... 

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. 

This is identical to the projected entropy spr, Eq. (9), except for the last term. But by construction, the combinatorial entropy assumes that p0—the overall density—is among the moment densities retained in the moment free energy. The difference scomb — spt = —p0 In p is then linear in this density, and the combinatorial and projection methods therefore predict exactly the same phase behavior. 

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...

Lichtenthaler, R. N. Liu, D. D. Prausnitz, J. M., "Thermodynamics of Poly(dimethylsiloxane) Solutions. Combinatorial Entropy and Molecular Interactions," Ber. Bunsenges. Phys. Chem., 78, 470 (1974b). 

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) ... 

It ignores "free volume" and only accounts for combinatorial entropy,... 

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. 

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