Branching in Chain Reactions

In Chap. C we have discussed in some detail the application of the cascade theory to polycondensates in their unperturbed state. In Chap. D some experimental results were already given for cross-linked or vulcanisated linear chains. In this chapter we shall now outline in brief how cross-linking chain reactions or the vulcanization of preformed chains of an arbitrary length distribution can be treated by cascade theory. Second, we shall discuss how heterogeneities in branching or a rigidity of a certain domain can be taken into account. 

Radical chain reactions are characterized by the coupled three stage reaction 

Evidently, a monomeric unit chosen as the root of a tree has two statistically unlike functional groups, but of the same reactivity. Hence, the generation function for the zero-th generation is the product of the two functional group generating functions F j and Ff 

The path-weight generating function is obtained in the same way as in the case of the ABf polycondensates. 

For the average of this path-weight generating function, one arrives after differentiation at 

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