Statistical approach to gelation

Flory [1,2] and also Stockmayer [3,4] used a statistical approach to derive an expression for predicting the extent of reaction at the time where gelation will occur by calculating when Xw approaches infinite size. This statistical approach in its simplest form assumes that the reactivity of all functional groups of the same type is the same and independent of molecular size and shape. It is further assumed that there are no intramolecular reactions between functional groups on the same molecule such as cyclization reactions. 

This critical value is, however, only an approximation leading often to an overestimation of the critical conversion value. The main reason to this failure is that gelation actually occurs, at least for some molecules, at a finite degree of polymerisation. Equations based on a statistical approach can provide better results in the estimation of gel point. 

The two approaches to the problem of predicting the extent of reaction at the onset of gelation differ appreciably in their predictions of pc for the same system of reactants. The Carothers equation predicts the extent of reaction at which the number-average degree of polymerization becomes infinite. This must obviously yield a value of pc that is too large because polymer molecules larger than Xn are present and will reach the gel point earlier than those of size Xn. The statistical treatment theoretically overcomes this error, since it predicts the extent of reaction at which the polymer size distribution curve first extends into the region of infinite size. 

Can the system be reacted to complete conversion without gelation If not, what is the extent of conversion of the acid functionality at the gel point calculated from (a) the Carothers equation and from the statistical approaches of (b) Flory-Stockmayer and (c) Macosko-Miller ... 

In experimental studies, where the loss of fluidity is taken as marking the gel point, the conversion at the observed gel point is almost always found to be higher than that (calculated) at the theoretical gel point. This can be explained by the model proposed by Bobalek et al (1964) for the gelation process, as shown in Fig. 5.8. According to this model, at the theoretical gel point, a number of macroscopic three-dimensional networks (gel particles) form and undergo phase separation. The gel particles so formed remain suspended in the medium and increase in number as reaction continues. At the experimentally observed gel point, the concentration of gel particles reaches a critical value and causes phase inversion as well as a steep rise in viscosity. The lower value of pc predicted by the statistical approach is also attributed to the occurrence of some wasteful intramolecular cy-clization reactions not taken into account in the derivation and also in some cases to the limited applicability of the assumption of equal reactivity of all functional groups of the same type, irrespective of molecular size. 

Although the assumptions of the simple statistical approach of Floiy (23) and Stockmayer (24) do not hold for PF resin, it was used to model the gelation behavior of these model systems because the real pyrolysis oil systems are so complex that the more elaborate modeling approaches could not be applied to these systems. 

The statistical structural model for the gelation behavior of cyanate-epoxide polyreactions was described by Bauer [69] using the cascade formalism, according to the approach by Dusek [70], which had been previously applied for the polycyclotri-merization of acetylene derivatives. 

Family [9] considered the conformations of statistical branched fractals (which simulate branched polymers) formed in equilibrium processes in terms of the Flory theory. Using this approach, he found only three different states of statistical fractals, which were called uncoiled, compensated, and collapsed states. In particular, it was found that in thermally induced phase transitions, clusters occur in the compensated state and have nearly equal fractal dimensions ( 2.5). Recall that the value df = 2.5 in polymers corresponds to the gelation point this allows gelation to be classified as a critical phenomenon. 

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