Average degrees of polymerization and molar masses of branched and hyperbranched condensation polymers: Recursive probability approach

This article extends the "in-out" recursive probability method of Macosko and Miller to the polycondensation of mixtures of polyfunctional monomers bearing A- and/or B-groups. Explicit relationships of the theoretical average degrees of polymerization and average molar masses are established in the most general case and can be applied to any linear, hyperbranched, or network polymerizations, leading in the latter case to the value of the conversion at the gel point. These relationships reduce to those of Stockmayer in the case of reactions involving monomers bearing either A- or B-groups. The effect of the elimination of a condensation byproduct is also examined. An approximate explicit expression of the mass-average molar mass is proposed, allowing calculations with an error in the 1-3% range when water is the byproduct. Some examples are treated to illustrate the method, including AA + BB + AB, AB + B-g, AB(f), AfB(g), AB(f) + B-g, and more complex polymerizations. The effect of the functionality of the core molecule on the polydispersity of hyperbranched polymers is examined for various systems.