Simulating joint chain length and composition fractions from semi-batch ethylene copolymerization experiments

Linear low density polyethylenes produced using cy-olefin comonomers and heterogeneous Ziegler-Natta catalysts have broad chain length (CL) and copolymer composition (CC) distributions. Multiple types of active sites on the catalyst are the cause of these broad and often multimodal distributions that influence both end-use and processing behaviour of the copolymer. Measurement of the joint CL and CC distribution can be accomplished by cross-fractionating polyethylene by Temperature Rising Elution Fractionation (TREF) and Size Exclusion Chromatography (SEC). Using these techniques, copolymer can be separated into bins corresponding to specific CC and CL ranges. In this article, Stockmayer's (1945) bivariate distribution is used to develop a methodology for modelling the quantity of accumulated copolymer that corresponds to each specific bin of the joint CL and CC distribution. First, the instantaneous joint CL and CC distribution for polymer produced at each site type is integrated over specified finite ranges of composition and chain length. The end points of these ranges correspond to TREF fraction composition limits and selected chain lengths from SEC analyses. The instantaneous rate of polymer production from all site types in each of the bins is then calculated. The polymer accumulated in each bin over the course of a polymerization experiment is determined by numerically solving a set of ordinary differential-equations. This methodology can be used to predict experimental TREF and SEC cross-fractionation results for copolymer produced dynamically in semi-batch laboratory reactors. The resulting predictions can then be used with experimental results to estimate parameters in kinetic models describing ethylene copolymerization with multiple active site catalysts.