Mathematical modeling of the bivariate molecular weight—Long chain branching distribution of highly branched polymers.: A population balance approach

In the present study a population balance approach is described to follow the time evolution of molecular polymer properties in free-radical polymerizations. The model formulation is based on the fixed pivot technique (FPT) which was properly adapted in two dimensions in order to calculate the bivariate molecular weight–long chain branching distribution (MW–LCBD) for highly branched polymers like poly(vinyl acetate) (PVAc) and low-density poly(ethylene) (LDPE) produced in chemically initiated free-radical polymerization systems. The method assumes that the overall polymer population can be assigned to selected discrete points which are also called ‘grid points’. Accordingly, dynamic bivariate population balance equations (PBEs) are derived for the ‘live’ and ‘dead’ polymer chains, which are solved at the specified grid points. Thus, the infinite size of differential molar balance equations is reduced to a smaller and feasible system of differential equations. The validity of the numerical calculations is examined via a direct comparison of simulation results obtained by the 2-D FPT with available experimental data of the molecular weight distribution (MWD), number and weight average molecular weights (Mn,Mw)(Mn,Mw) and number average degree of branching (Bn)(Bn) of highly branched polymerization systems (i.e., Vac and LDPE). In general, the 2-D FPT can provide very accurate predictions of the molecular and branching characteristics of highly branched polymers in a relatively short time. It is important to point out that, to our knowledge, this is the first time that the joint MW–LCBD for branched polymers has been calculated by a sectional grid method via the direct solution of the governing PBEs for both ‘live’ and ‘dead’ polymer chains.