Pregel regime

In nongelling mixtures, or a pregel regime of gelling ones, the total volume fraction should be given by 

The weight-averages of the aggregation numbers / and m in the clusters are then given 

---

In the pregel regime, we have = 1, and hence we And (3.17). For the multiple reaction which leads to... 

In the pregel regime of pairwise reaction, this equation reduces to (3.20). For multiple reaction, this equation gives... 

To derive a specific form of the equilibrium constants bi, let us introduce a simple model for the internal structure of clusters. Clusters are assumed to take a tree structure with no internal loops (Cayley tree). Cycle formation within a cluster is neglected. This is a crude approximation on the basis of the classical theory of gelation presented in Section 3.2 [5,6,7,8], but in fact it is known to work very weU at least in the pregel regime. 

In the pregel regime (a < a ), the volume fraction occupied by the polymer chains belonging to the sol must always be equal to the total polymer volume fraction . Thus, from (7.26), the total polymer volume fraction f and the extent of association a satisfy the relation... 

For monodisperse primary chains, we have a strictly two-component system, and the thermodynamic stability limit (spinodal) is given by a cj), T) = 0, where cr is the factor (7.123). Further, for such strictly binary systems, the critical solution point, if it exists in the pregel regime, can be found by the additional condition d A o/dcp = 0. The condition is given explicitly by... 

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

Solving the coupled equations (10.47) and (10.49) with respect to y and z, we can express them in terms of the total volume fraction coupled equations in the pregel regime are... 

---