Statistics of Randomly Branched Polycondensates

Abstract

The combinatorial theory of so‐called “trail‐weighting functions,” variously averaged over all trails in distributions of linear or branched molecules which are random (or at least lack long‐range correlations) has been described and exploited for calculation of physical parameters in previous papers. A trail‐weighting function is some arbitrary function φ(n) of the number of distinct trails of length n in the graph of a given molecule. By extending the use of univariate to bivariate generating functions, we here derive the moments of the distribution, the Stokes radii, and second virial coefficients for osmotic pressure and for light scattering of random f‐functional polycondensates. It emerges that the z‐average radius of gyration, the z‐average reciprocal Stokes radius, and the second virial coefficient of osmotic pressure are related to the weight‐average degree of polymerization, and the weight‐average radius of gyration and the weight‐average reciprocal Stokes radius to the number‐average degree of polymerization.