  A Monte Carlo (MC) simulation method based on the Gillespie algorithm is
developed for the polymerization of branched polyetherimides from two back-bone
monomers [4,4'-bisphenol A dianhydride (BPADA) and m-phenylenediamine (MPD)], a
chain terminator [phthalic anhydride (PA)], and a branching agent
[tris[4-(4-aminophenoxy)phenyl] ethane (TAPE)]. This polymerization involves 4
reactions that all can be reduced to a condensation reaction between an amine
group and a carboxylic anhydride group. By comparing the MC simulation results
to the predictions of the Flory-Stockmayer theory on the molecular weight
distribution, we show that the rates of the 4 reactions in the MC simulations
should be set based on the concentrations of the functional groups on the
monomers involved in each reaction. Using the MC simulations, we show that the
Flory-Stockmayer theory predicts the molecular weight distribution well for
systems below the gel point that is set by the functionality of the branching
agent but starts to fail for systems around or above the gel point. However,
for all the systems the MC simulations can be used to reliably predict the
molecular weight distribution. Even for a macroscopic system, a converging
distribution can be quickly obtained through the MC simulations on a system of
only a few hundred to a few thousand monomers but with the same molar ratios of
monomers as in the macroscopic system.
