The Flory argument

Minimization of the sum of the elastic energy R /N) and the repulsive energy [ Npj (R)] yields the famous Flory result. Fisher is usually given credit for generalizing the Flory argument to d-dimensions. ... 

The classical model for a polymer chain with self-excluded volume is the SAW. Let be the appropriate self-consistent field (SCF) for a SAW of N steps but of any size R, By analogy with the Flory argument, we are tempted to assume that for a SAW is proportional to the mean density of occupied sites,, where the volume in p (i ) has been replaced by its mean value ATva because we are considering the total ensemble of iV-step SAWs and not just the subset of size R, The SCF for a SAW must satisfy certain properties. As will be shown below, this choice for is not self-consistent for < 4. 

One can apply Flory arguments to calculate the end-to-end distance, R, as a function of the number of monomers for the case of a completely isolated charged chain (i.e., no counterions or salt ions). In the Flory argument the free energy is... 

Despite the fact that the Flory argument is a simple mean-field theory, it works very well for a number of cases, often producing estimates for df which differ from the exact result by only a few percent or less for d < d c- Here d c = 4d/ 2 - d) is the upper critical dimension above which self-avoidance is irrelevant. For a hnear polymer eq. (9.44) works extremely well for d three dimensions, the theory predicts u = fdf= i/5, which is very close to the best renormaUzation groups estimates of 0.588. Percolation clusters at the percolation threshold are another example where eq. (9.45) works well. Since d 4/3 in all dimensions,eq. (9.45) predicts that 2([Pg.553]

The above argument shows that complete overlap of coil domains is improbable for large n and hence gives plausibility to the excluded volume concept as applied to random coils. More importantly, however, it introduces the notion that coil interpenetration must be discussed in terms of probability. For hard spheres the probability of interpenetration is zero, but for random coils the boundaries of the domain are softer and the probability for interpenetration must be analyzed in more detail. One method for doing this will be discussed in the next section. Before turning to this, however, we note that the Flory-Huggins theory can also be used to yield a value for the second virial coefficient. 

We see from the above argument that, within the Flory theory of gels, the concentration dependence of x is the driving force for the transition in neutral gels. Hence, to understand the mechanism of the phase transition of gels on a molecular level, we must identify the microscopic interaction which makes x depend on the concentration. For this purpose, we must specify not only the... 

A somewhat similar equation has been given by Ptitsyn (219) on the basis of related arguments, and the deviations from the Flory relation are in the same direction. Thus there seems to be general agreement that some modification in the Flory expression for the long-range interaction effects on chain dimensions is needed. 

The Oishi-Prausnitz model cannot be defined strictly as a lattice model. The combinatorial and residual terms in the original UNIFAC and UNIQUAC models can be derived from lattice statistics arguments similar to those used in deriving the other models discussed in this section. On the other hand, the free volume contribution to the Oishi-Prausnitz model is derived from the Flory equation of state discussed in the next section. Thus, the Oishi-Prausnitz model is a hybrid of the lattice-fluid and free volume approaches. 

The excluded volume problem of polymer chains was taken up early in 1943 by Flory [6]. His arguments based on the chemical thermodynamics brought the conclusions (i) the existence of the Flory point ( point) where two body interactions apparently vanish, and (ii) that in non-solvent state chains behave ideally-... 

Flory has shown from empirical data that for a homologous series the velocity constant measured under comparable conditions approaches an asymptotic limit as the chain length increases. He therefore proposed the equal reactivity principle, which states in essence that the intrinsic reactivity of all functional groups is constant, independent of the molecular size. This principle is in apparent contradiction to the theoretical prediction of low chemical reactivity for macromolecules discussed above. Flory used the following arguments to explain the equality of reactivity between macromolecules and their low-molecular-weight analogs. 

When our coil is exactly at the Flory point, we say that it has a quasiideal behavior. The prefix quasi is used to recall that some interactions are still present because the three-body term has some residual effects (apart from the renormalization of v). Thus, some subtle correlations remain at T=Q. Since they are probably too small to be observed, we shall not insist very much on their properties. Mathematically, however, they are associated with unusual logarithmic factors. The origin of such factors can be understood from the following crude argument. 

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