Random Vulcanization of Preformed Chains

The principles of the calculation by means of the cascade theory is sketched in Fig. 58 and compared with the random polycondensation. Instead of selecting a single monomeric unit as root of a tree, a whole primary chain is placed on the zero-th generation, and the same is done for all the other primary chains from the cross-linked polymer. 

The formalism of construction a path-weight generating function appears at first sight the same as for the random polycondensates, with the essential difference, however, that s and U in the equations for a polycondensate with functionality y for the monomers 

The generating function for enumerating paths of length n is actually the scattering function of isolated primary chains of DP = y. 

The overall scattering function of the cross-linked system is obtained, as usual, by differentiation of U0 at s = 1, which with Eq. (E.19) yields 

We close this section with a few remarks. First 

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