Chapter 11, Alternative discussion of local composition models

Chapter 11, Alternative discussion of local composition models

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Section 11.6 Local Composition Theory 381 Nevertheless, there are some obvious limitations to the assumption of a constant packing frac- tion. A little calculation would make it clear that the λ for liquid propane at T r = 0.99 is significantly larger that λ for toluene at T r = 0.619. Thus, a mixture of propane and toluene at 366 K would not be very accurately represented by the Flory-Huggins theory. Note that deviations of λ from each other are related to differences in the compressibilities of the components. Thus, it is common to refer to Flory-Huggins theory as an “incompressible” theory and to develop alternative theories to represent “compressible” polymer mixtures. Not surprisingly, these alternative theories closely resemble the van der Waals’ equation (with a slightly modified temperature dependence of the a parameter). This observation lends added significance to Gibbs’ quote: “The whole is simpler than the sum of its parts” and to Rayleigh’s quote: “I am more than ever an admirer of van der Waals.” 11.6 LOCAL COMPOSITION THEORY One of the major assumptions of regular solution theory was that the mixture interactions were independent of each other such that quadratic mixing rules would provide reasonable approxima- tions as shown in Section 10.1 on page 322. But in some cases, like radically different strengths of attraction, the mixture interaction can be strongly coupled to the mixture composition. That is, for instance, the cross parameter could be a function of composition. a 12 = a 12 ( x ). One way of treating this prospect is to recognize the possibility that the “local compositions” in the mixture might devi- ate strongly from the bulk compositions. As an example, consider a lattice consisting primarily of type A atoms but with two B atoms right beside each other. Suppose all these atoms were the same size and that the coordination number was 10. Then the local compositions around a B atom are x AB = 9/10 and x BB = 1/10 (notation of subscripts is AB A around B” ). Specific interactions such as hydrogen bonding and polarity might lead to such effects, and thus, the basis of the hypothesis is that energetic differences lead to the nonrandomness that causes the quadratic mixing rules to break down. Excess Gibbs models based on this hypothesis are termed local composition theories, and were first introduced by Wilson in 1964. 1 To develop the theory, we first introduce nomencla- ture to identify the local compositions summarized in Table 11.2 We assume that the local compositions are given by some weighting factor, ij , relative to the over- all compositions. 11.64 1. Wilson, G.M., J. Am. Chem. Soc . 86:127 (1964). Table 11.2 Nomenclature for local composition variables. Composition around a “1” molecule Composition around a “2” molecule x 21 - mole fraction of “2’s” around “1” x 12 - mole fraction of “1’s” around “2” x 11 - mole fraction of “1’s” around “1” x 22 - mole fraction of “2’s” around “2” local mole balance, x 11 + x 21 = 1 local mole balance, x 22 + x 12 = 1 x 21 x 11 ------- x 2
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Chapter 11, Alternative discussion of local composition models

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