Flory-Huggins athermal solution

The contribution due to size differences of the molecules is given by the Flory-Huggins athermal solution equation. The constituent atoms (other than hydrogen) m, are used as a measure of molecular size ... 

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. 

Since the 0 s are fractions, the logarithms in Eq. (8.38) are less than unity and AGj is negative for all concentrations. In the case of athermal mixtures entropy considerations alone are sufficient to account for polymer-solvent miscibility at all concentrations. Exactly the same is true for ideal solutions. As a matter of fact, it is possible to regard the expressions for AS and AGj for ideal solutions as special cases of Eqs. (8.37) and (8.38) for the situation where n happens to equal unity. The following example compares values for ASj for ideal and Flory-Huggins solutions to examine quantitatively the effect of variations in n on the entropy of mixing. 

Activity Coefficient at Infinite Dilution. A procedure similar to that employed by Wilson will be used here to obtain an expression for the excess Gibbs energy. Wilson started from the Flory and Huggins expression" 2 for the excess free energy of athermal solutions, but expressed the volume fractions in terms of local molar fractions. We selected Wilson s approach from a number of approaches, because it provided a better description of phase equilibria and because the interactions that count the most are the local one, but started from the more... 

The comparison of the experimental solubilities [4,5] of Ar, CH4, C2H6 and CsHg in the binary aqueous mixtures of PPG-400, PEG-200 and PEG-400 with the calculated ones is presented in Figs. 1-3 and Table 2. They show that Eq. (4) coupled with the Flory-Huggins equation, in which the interaction parameter x is used as an adjustable parameter, is very accurate. The Krichevsky equation (1) does not provide accurate predictions. While less accurate than Eq. (4), the simple Eq. (2) provides very satisfactory results without involving any adjustable parameters. It should be noted that Eq. (4) coupled with the Flory-Huggins equation with X (athermal solutions) does not involve any adjustable parameters and provides results comparable to those of Eq. (2). 

Eq. (4) combined with the Flory-Huggins equation with adjustable parameter x (the value of the parameter x is given in parenthesis). ° Eq. (4) combined with the Flory-Huggins equation with parameter = 0 (athermal solution). 

This is the Flory-Huggins (FH) equation for athermal solutions [5, 6]. The equation formally appears symmetric to all components of a mixture, but in fact it describes the great asymmetry between the polymer and the small solvent in solution. Consider the solution of a small solvent and a polymer that is r times the volume of the solvent. The solvent may well be the monomer, here designated as component 1, and the polymer may be an r-mer as component 2. Setting V2/V1 in Equation (4.367) gives... 

When molecular size differences, as reflected by liquid molal volumes, are appreciable, the following Flory-Huggins size correction for athermal solutions can be added to the regular solution free energy contribution... 

Since polymer solutions in principle do not fulfill the rules of the ideal mixture but show strong negative deviations from Raoult s law due to the difference in molecular size, the athermal Flory-Huggins mixture is usually applied as the reference mixture within polymer solution thermodynamics. Starting from Equation [4.4.11] or from... 

Most solutions of rodlike polymers are not athermal. Heat-of-mixing terms can be added to the chemical potentials, just as in the ordinary Flory-Huggins theory of solutions. When the interaction parameter % is large enough, the solution will separate into a dilute phase and a concentrated phase, just as for any polymer solution. There will still be a miscibility gap between dilute isotropic solutions and ordered concentrated solutions. A typical phase diagram for a solution of rods with x = 100 is shown in Figure 9.3. The concentrated phase is now very concentrated and ordered. 

The well-known mean-field incompressible Flory-Huggins theory of polymer mixtures assumes random mixing of polymer repeat units. However, it has been demonstrated that the radial distribution functions gay(r) of polymer melts are sensitive to the details of the polymer architecture on short length scales. Hence, one expects that in polymer mixtures the radial distribution functions will likewise depend on the intramolecular structure of the components, and that the packing will not be random. Since by definition the heat of mixing is zero for an athermal blend, Flory-Huggins theory predicts athermal mixtures are ideal solutions that exhibit complete miscibility. 

To apply the theoretical result of Flory and Huggins to real polymer solutions, i.e., to solutions that are not athermal, it has become common practice to add to the configurational part of the entropy, a semiemperical part for the residual contribution. In other words, we add a term that, if there is no difference in free volumes, is given by the enthalpy of mixing which we recast it in the form of... 

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