Postgel regime

Li et al. [25] validated the use of the PKRCM for the case of MMA/EGDMA (methyl methacrylate) with gelation. Tobita and Hamielec used this method to model gelling systems, at both the pre- and postgelation regimes... 

Because both averages are divergent at a = a = 1 /(/ — 1), this is identified as the gel point. Since the gel point thus found is the point where the reactivity of A groups is 1 (complete reaction), networks are formed in the limiting state where all A groups are reacted. In other words, there is no postgel regime in this system. 

Therefore, in the postgel regime where the average reactivity is larger than the critical gel value a, we see that clusters of finite sizes are connected to the gel in the way such that the reactivity of the sol part stays at a constant value a = a. The relation (3.23) then gives... 

Stockmayer, and is called Stockmayer s treatment of the postgel regime (the solid line... 

This theoretical treatment is, however, not a unique way of describing the reaction in the postgel regime. Flory postulated that the reactivity of the sol part should be found... 

The molecular weight distribution function of the sol part is therefore given by replacing a by a in (3.19). The weight fraction of the sol in the postgel regime is then calculated to be... 

In the classical free statistics, the number of the functional groups on the surface of a tree-like cluster is of the same order of that of the groups inside the cluster, so that a simple thermodynamic limit without surface term is impossible to take. The equilibrium statistical mechanics for the polycondensation was refined by Yan [14] to treat surface correction in such finite systems. He found the same result as Ziff and Stell. Thus the treatment of the postgel regime is not unique. The rigorous treatment of the problem requires at least one additional parameter defining relative probability of occurrence of infra- and intermolecular reactions in the gel. 

In the postgel regime, we take the similar limit in the relation... 

In the postgel regime, we have infinite clusters. Let be the number of chains of species i in such macroscopic clusters. The total volume is... 

In the postgel regime, a chain participating in the gel must be in chemical equilibrium with an unassociated chain of the same species. This imposes the additional conditions... 

In the last term, is the number of polymer chains that belong to the network. It becomes a macroscopic variable after the gel point is passed. This additional term appears only in the postgel regime. The free energy 5(0) for a chain to be bound to the network depends on the concentration 0. It is negative, and its absolute value increases with the concentration because the network structure becomes tighter and denser with the concentration. 

In the postgel regime where 4>>4> (a > a ), we have an additional condition (7.6). The activity z of the solute molecule is related to the binding free energy of the gel. 

Within Stockmayer s treatment of the postgel regime, after a complex calculation, it turns out that there is no singularity up to the second derivatives, but the third derivative contains a discontinuity... 

In the postgel regime, the last term for the gel part is necessary [9,26],... 

If the higher-concentration phase lies in the postgel regime, the postgel form of v must be employed in the chemical potentials. These equilibrium conditions determine the total volume fractions and in each phase as well as the molecular distributions pj and in them. 

We generalize the treatments of the postgel regime by Pearson-Graessley [11-13] in classical gelation theory to multiple junctions [ 14]. Let (/c = 2,3,4,...) be the junction... 

Because only one vacancy separates the train into two parts, all lattice sites must be occupied for the system to percolate. In other words, there is no postgel regime in one dimension. 

Let us next find the fraction Poo of the percolated cluster in the postgel regime (p> pc). Let u be the probability for a functional group to be connected to the finite clusters only... 

Because the hydrophobic segments on polymer chains are partly exposed to water in the postgel regime, the solution tends to separate into two macroscopic phases by hydrophobic association. Thus gelation and phase separation compete as the temperature goes up. The solution reveals an interesting multi-critical phase behavior [33]. 

In the pregel regime (pp agrees with the total volume fraction of polymers, but it can be smaller than (p in the postgel regime. 

In the postgel regime we foUow Stockmayer s treatment, and assume that the degree of reaction a in the sol part remains at constant a. The corresponding value of z is given by z = f- 2y /(f — The polymer volume fraction of the sol part... 

In the postgel regime where hydrated gel networks exist, one needs consider the number N i) of polymer chains of type I involved in the network. Their number density is given by (i) = (i) / S2, and their volume fraction by (i) = n (i) (i). 

Finally, an isolated /-molecule should also be in equilibrium with an /-molecule attached to the gel network in the postgel regime. Hence we have... 

In the postgel regime, the variable z in these equations must be replaced by the smaller root z of Flory s condition xx(z) — xa,(z )- The parameter z refers to the conversion of the entire system, while z refers to that of the sol part only. 

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