Random Branching and Cross-linking of Polymer Chains, Analytical Functions for the Bivariate Molecular Weight Distributions

Cross-linking and branching of primary polymer molecules are investigated using the Galton–Watson (GW) process. Starting with the probability generating function (pgf) of the primary molecular weight distribution (MWD), analytical expressions are derived for the bivariate pgfs g(n_br, s) of branched polymers which depend also on the number of branch points n_br. The bivariate MWDs n(n_br, i) (i: number of molecular units) are then derived as Taylor expansions in s. All three cases of random branching: X-shaped (cross-linking), T-shaped (only one end takes part in the branching process), and H-shaped (both ends can take part in the branching process) are treated. An extension of the formalism does not require the construction of the pgf and allows the direct use of the MWD of the primary chains. However, using pgfs allows to go past the gel point and to determine the MWD and content of the sol. Explicit expressions are given for special distributions: the mono modal, the most probable, the Schulz-Zimm, the Poisson, and the Catalan distribution for the cases of X-shaped and T-shaped branching.