Flory swelling theory

Various modifications of the Flory theory [4] are usually applied to describe the uncharged gels. Their crosslinking density can be simply calculated from the swelling degree using Eqs. (3.1) and (3.2) or analytical expressions for the Mc value (see, for example, Ref. [124]). 

The upper part of the phase diagram (Fig. 5.1) corresponds to good solvents. At low concentrations, polymer coils are far from each other and behave as isolated real chains (see Section 3.3.1.1). At temperatures for which the excluded volume interaction within each chain exceeds the thermal energy kT, they begin to swell. The Flory theory prediction for the size of swollen real chains with excluded volume v > /y/N is ihs, same as the result for a self-avoiding walk of thermal blobs [Eq. (3.77)] ... 

Hint Use Flory theory to estimate the swelling of a randomly branched... 

In this case, eqn (16.5.7) predicts an appreciable but continuous shrinking of the chain when temperature decreases, i.e. when z diminishes (see Table 16.7 and Fig. 16.11). Moreover, the same equation shows that the swelling is just equal to unity when z = zF = - (16/3)y = — 0.015. Comparing the calculated values of X with the measured values of jEg, we observe that the Flory theory is not realistic. 

It is of interest to note that experimental values of the parameter d characterizing the strain dependence of equilibrium swelling ratio for elastomers subjected to uniform biaxial extension closely approximate the theoretical values. It will be recalled that this parameter is specified by Eq. [6.1.21]. Moreover, the Flory theory defines it as d = 2/9 = 0.22(2),... 

The maximum degree of swelling of a network characterized by the volume fraction 2m of polymer and the number density v of chains are related in Ae Flory theory -22) by the Equation... 

The Flory theory is not rigorous, but it captures the two most essential features of the coil size variation in good solvents (1) that R oc N with v 0.6 for large N and (2) that the swelling coefficient a depends on the parameter z. More sophisticated theories and computer simulations point to v 0.588 which is very close to the Hoty exponent. 

The degree of swelling of superabsorbent polymers can be expressed by the following equation using Flory s swelling theory of ionic networks [5] ... 

The Ronca-Allegra theory (177), and Flory-Erman theory (3,178,182) are both based on the idea that effects of constraints are local and decrease with increasing strain and swelling. The basic difference between the two theories is that in the Ronca-Allegra theory the fluctuations of junctions become exactly affine as the undeformed state is approached, whereas in the Flory theory they are close to but below those of the affine state. 

In good solvents, the mean force is of the repulsive type when the two polymer segments come to a close distance and the excluded volume is positive this tends to swell the polymer coil which deviates from the ideal chain behavior described previously by Eq. (1). Once the excluded volume effect is introduced into the model of a real polymer chain, an exact calculation becomes impossible and various schemes of simplification have been proposed. The excluded volume effect, first discussed by Kuhn [25], was calculated by Flory [24] and further refined by many different authors over the years [27]. The rigorous treatment, however, was only recently achieved, with the application of renormalization group theory. The renormalization group techniques have been developed to solve many-body problems in physics and chemistry. De Gennes was the first to point out that the same approach could be used to calculate the MW dependence of global properties... 

The swelling of a network is described by the Flory-Rehner theory (38) which gives... 

According to Flory-Huggins theory, in the limit of x the critical x parameter is 0.5.(H) Below this value the polymer and solvent will be miscible in all proportions. Above this value, the solvent will not dissolve the polymer, but will act only as a swelling solvent. Thus, the pure solvent may not dissolve the polymer even though it is not crosslinked. If x is not , the critical value of x is larger, but the same qualitative arguments regarding mutual solubility of the solvent and polymer hold. Thus, the application of Equation 1 does not require that the pure solvent be able to completely dissolve the polymer, only that the solvent dissolve into the polymer by an amount that can be measured. 

Three common methods of measuring crosslinking (swelling, elastic modulus, and gel point measurements) have recently been critically appraised by Dole (14). A fourth method using a plot of sol + sol against the reciprocal dose has also been used extensively. However, Lyons (23) has pointed out that this relation, even for polyethylenes of closely random distribution, does not have the rectilinear form required by the statistical theory of crosslinking. Flory (19) pointed out many years ago that the extensibility of a crosslinked elastomer should vary as the square root of the distance between crosslinks. More recently Case (4, 5) has calculated that the extensibility of an elastomer is given by ... 

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