Computing Radius of Gyration Distributions for Reactor Populations of Highly Random‐Branched Polymers

This paper describes a method for finding the radius of gyration distributions from the architectures of branched polymers, using a statistical mechanical theory that employs a graph‐theoretical representation of the connectivity in the molecules. The method requires finding the smallest eigenvalues of the Kirchhoff matrix, which is of size n × n for a molecule with n monomer units. Since this is difficult for large n , we propose a coarse graining method reducing the matrix size to N × N at minimum for a molecule with N branches. The error is found to be acceptable for a still reasonable computational effort. In previous work we have presented a model for the synthesis of branched molecular architectures from kinetic data under the conditions of a continuously stirred tank reactor (CSTR) using Monte Carlo (MC) sampling. This MC model also utilises a graph–theoretical representation of molecules, which is readily used in the radius of gyration calculation. It is shown how changing parameters in the MC model, such as the length distribution of the primary polymers and the residence time distribution (RTD) in the reactor, influences the radius of gyration. Exact agreement is found to the Zimm and Stockmeyer theory when assuming an absence of RTD, but effectively smaller radii are obtained with RTD. The architectures' averaged ratio of radii for branched and linear molecules of the same weight, as well as the gyration radius distribution shape, turn out to be a function of N only. Finally, the model is applied in simulating the separation of a complete reactor population of molecules using size exclusion chromatography (SEC). It is shown that we can thus predict the molecular weight distribution at a certain hydrodynamic volume, and hence of one SEC fraction.