Step-Growth Polymerized Systems of type “A3 + A1”: A Method to Calculate the Bivariate (Molecular size) × (Square Radius of Gyration) Number Distribution

Step-growth polymerized systems of type “A3 + A1” are considered. The monomers bear, respectively, 3 or 1 reactive A group. During the reaction, an A group on one monomeric unit might react with an A group on another such unit, thus chemically coupling the two units involved. Complexly structured polymeric molecules are formed. The A3's act as branching points; the A1's as end cappers. At the end of the reaction, the population of molecules present in the reactor vessel varies in size and branching structure. A method is presented to calculate the bivariate (molecular size) × (square radius of gyration) number distribution. It is shown that within the class of molecules of the same size, their square radius of gyration follows a shifted gamma distribution. Two new molecular parameters are introduced: the D index and the G index. The method uses bivariate generating functions.