Flory’s treatment

For more than two decades researchers have attempted to overcome the inadequacies of Flory s treatment in order to establish a model that will provide accurate predictions. Most of these research efforts can be grouped into two categories, i.e., attempts at corrections to the enthalpic or noncombinatorial part, and modifications to the entropic or combinatorial part of the Flory-Huggins theory. The more complex relationships derived by Huggins, Guggenheim, Stavermans, and others [53] required so many additional and poorly determined parameters that these approaches lack practical applications. A review of the more serious deficiencies... 

In his original calculations, Flory found a value of v = 3/5 which seems to be quite close to the value given by Eq. (9). This accord, however, was proved to be fortuitous as a result of the mutual cancellation of two terms neglected in Flory s treatment [30]. 

In Flory s treatment the term (ry /M is rearranged into where (rl) refers to the ideal, unperturbed state. The expansion factor a (not to be confused with the same symbol used in Section III, A) is defined as which reduces to unity in an ideal solvent or, in Flory s... 

It is noted that the left side of Eq. (32) is exactly identical in form with the -function for rigid particles (Section III, D). In Flory s treatment the numerical coefficient on the right side of the equation should be a universal constant, designated by him as This constancy has been confirmed... 

The random walk approach, which is based on Flory s treatment of the polymer chain in solution in this case the surface was considered as a reflecting barrier. 

However, the conformation statistics in Flory s treatment gives the conformational free energy, rather than the conformational entropy adapted in the Gibbs-DiMarzio theory. In addition, W was calculated with respect to the fully ordered state therefore. In W = 0 simply implies the return to the fully ordered state, rather than frozen in a disordered state. Furthermore, reflects the static semi-flexibility, while the glass transition should be related with the d3mamic semi-flexibility of polymer chains. Therefore, fundamental assumptions of the Gibbs-DiMarzio thermodynamic theory are misleading. 

Examining the details involved in solvent interactions reveals a story which is particularly complicated by the interplay between enthalpic and entropic components. This is apparent even in Flory s treatment of traditional polymers, where the solutions often apply generally, but not specifically. While Moores study of solvents showed a general trend toward better folding of his hydrophobic mPE foldamers in more polar solvents [49] treatment of the data required rejection of chlorinated solvents, and did not include aromatics, apart from the anecdotal evidence that they did not unfold the aromatic backbone. This is a strongly interacting system which will require much more work to understand at a fundamental level. 

Fig. 3.9 Weight fraction of the gel part plotted against the reactivity. (Stockmayer s treatment (S), Flory s treatment (F), and Ziff-Stell s treatment (ZS).)...

This takes a value larger than o = 2//. Therefore, in Flory s treatment, cycle formation is allowed in the gel network. The number of independent cycles, or the cycle... 

The difference in the two approaches was later clarifled from a kinetic point of view by Ziff and Stell [13]. It was shown that Stockmayer s treatment allows reaction neither between sol and gel, nor between gel and gel. The increase of the gel fraction is only by the cascade growth of the sol clusters to infinity, while in Flory s treatment sol and gel interact, and reaction within the gel is also allowed. Ziff and Stell proposed a new treatment in which intramolecular reaction of the gel is not allowed but reaction between sol and gel is allowed. Their result on the gel fraction is shown in Figure 3.9 by the broken line (ZS). 

Similar calculation on the basis of Flory s treatment gives the factor —4B instead of B. The sign changes, but the discontinuity remains. 

Flory s treatment gives [-4/(/ - 1)]C instead of C. Again, the sign changes, but the discontinuity remains. Collecting all results, we come to the conclusion that the ideal model of thermoreversible gelation treated here shows a third-order phase transition that is analogous to the Bose-Einstein condensation. 

Following Flory s treatment of semiflexible polymer solutions (Flory 1956), the entropy part of partition function for polymer solutions is given by... 

Following Flory s treatment for semiflexible polymer solutions [35], the total partition function of polymer solutions can be described as... 

A notable aspect of Flory s treatment of the chain expansion in a good solvent is the asymptotic behavior for long chains... 

If we adopt Flory s assumptions for the coordinates S of the normal modes of the network instead of the coordinates R in Flory s treatment we arrive at the same expression for the free energy of deformation as Flory, however with one important difference the number of degrees of freedom contributing to this term in the elastic energy is 31 whereas in Flory s theory it is 3 Np. For networks with f = 4 Np = whereas in general... 

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