Random Branching of Polymer Chains with Schulz-Zimm Distribution. 1. Bivariate Distribution and Related Formulae

Analytical solution for the bivariate distribution W(r,k) of the degree of polymerization r and the number of branch points k is obtained for the random branching of polymer chains that follow the Schulz-Zimm distribution, or in mathematical term, the gamma distribution. It is found that when the bivariate distribution is normalized to make the total area unity, the normalized distribution follows another gamma distribution. The bivariate distribution function enables one to determine various useful properties of the branched polymer system, including full weight fraction distribution W(r), the number- and weight-average degree of polymerization having k branch points, the expected branching density rho for a given degree of polymerization r or a given number of branch points k. The branching density at large r limit, rho(r -> infinity) is an important property to represent the branched polymer system, rather than the average branching density of the whole system, and the value of rho(r -> infinity) can be determined for the present random branched system. The effect of the distribution breadth of the primary chains can be investigated by using the formulae developed in this article.