Graphlike state of matter. V. Theory of Rayleigh scattering from randomly cross-linked chains of arbitrary primary distributions

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Abstract
The concepts and vocabulary of graph theory provide a simpler and more systematic approach to the combinatorics of branched polymer systems than the usual ad hoc methods based on probability theory. Formulas of remarkable simplicity are derived for the experimentally accessible z ‐average particle scattering factor Pz (θ) of polymer distributions arising from (a) the random condensation of distributions of star‐shaped monomers of variable functionality and (b) randomly cross‐linked systems of arbitrary primary chain distributions (vulcanizates). As usual, the weight average degree of polymerization and the mean square radius (and higher moments) are simply obtained from Pz (θ) when the subchains obey Gaussian statistics.

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