Flory-Huggins theory mixing entropy

We concluded the last section with the observation that a polymer solution is expected to be nonideal on the grounds of entropy considerations alone. A nonzero value for AH would exacerbate the situation even further. We therefore begin our discussion of this problem by assuming a polymer-solvent system which shows athermal mixing. In the next section we shall extend the theory to include systems for which AH 9 0. The theory we shall examine in the next few sections was developed independently by Flory and Huggins and is known as the Flory-Huggins theory. 

Remember that the hump which causes the instability with respect to phase separation arises from an unfavorable AH considerations of configurational entropy alone favor mixing. Since AS is multiplied by T in the evaluation of AGj, we anticipate that as the temperature increases, curves like that shown in Fig. 8.2b will gradually smooth out and eventually pass over to the form shown in Fig. 8.2a. The temperature at which the wiggles in the curve finally vanish will be a critical temperature for this particular phase separation. We shall presently turn to the Flory-Huggins theory for some mathematical descriptions of this critical point. The following example reminds us of a similar problem encountered elsewhere in physical chemistry. 

Aspler and Gray (65.69) used gas chromatography and static methods at 25 C to measure the activity of water vapor over concentrated solutions of HPC. Their results indicated that the entropy of mixing in dilute solutions is mven by the Flory-Huggins theory and by Flory s lattice theory for roddike molecules at very nigh concentrations. 

Numerical parameter employed in the Flory-Huggins theory, to account for the contribution of the noncombinatorial entropy of mixing and the enthalpy of mixing to the Gibbs energy of mixing. 

Flory-Huggins Theory. The simplest quantitative model for AGmx that includes the most essential elements needed for polymer blends is the Flory-Huggins theory, originally developed for polymer solutions (3,4). It assumes the only contribution to the entropy of mixing is combinatorial in origin and is given by equation 3, for a unit volume of a mixture of polymers A. and B. Here, pt and... 

To apply the procedure outlined above to a polymer, it is necessary to use the Flory-Huggins theory of polymer solution, which takes into account the entropy of mixing of solutes in polymers caused by the large difference in molecular size... 

The entropy of mixing can be evaluated approximately from statistical mechanics, by applying the Flory-Huggins theory (27,28)... 

Using the lattice model, the approximate value of W in the Boltzmann equation can be estimated. Two separate approaches to this appeared in 1942, one by P. J. Flory, the other by M. L. Huggins, and though they differed in detail, the approaches are usually combined and known as the Flory-Huggins theory. This gives the result for entropy of mixing of follows ... 

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

According to the Flory-Huggins theory the mixing entropy S - S is given by... 

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 Flory-Huggins equation is interesting on several scores. First, it contains no lattice parameters. Second, the dominant term is In v, which physically means that the mixing of the polymer and solvent occurs because of the additional space available to the solvent when the domains of the polymer molecules become accessible to the solvent. The detailed structure of the polymer is irrelevant, rods being just as effective in providing space for the solvent molecules as polymer coils. This point is stressed because it is sometimes claimed that polymer molecules dissolve as a result of their increase in configurational entropy in solution. This is contrary to the precepts of the Flory-Huggins theory. 

Flory-Huggins theory This theory calculates the free energy of mixing of pure amorphous polymers with pure solvent. The entropy and the enthalpy of mixing can be calculated separately, and the following relationship applies ... 

Most of the shortcomings of the Flory-Huggins theory have been overcome by the use of the free volume theory (1). The entropic contribution to xi was attributed to the difference in free volume between the solvent and the polymer. The increase in the value of Xi with concentration was explained on the basis of the ordering of the solvent molecules on increasing the segment concentration. The phase separation near the critical temperature of the solvent was attributed to the decrease in entropy on mixing the solvent and polymer under such conditions. 

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