B Cascade theory of gelation

Cascade theory starts with the weight fraction generating function, or the probability generating function (p.g.f.) [17,18] 

The weight average, z-average, and higher averages are calculated by 

Starting with the 0-th generation, the p.g.f. for the total number of molecules in the first, second,..r-th generations, can be seen to be 

19 EHagram for the p.g.f. of the cascade tree. Each junction is accompanied by a factor u x). The 0-th generation has / branches, while the generations higher than the first have / — 1 branches. 

In the case of more complex mixtures of polydisperse functional primary molecules with the distribution function pf, F functions are 

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