Prediction of molecular weight distributions in polymers using probability generating functions

Abstract

We review the work of our group in the area of mathematical modelling of polymerisation reactors and processes, with emphasis in the prediction of molecular weight distributions using probability generating functions (pgfs). We apply the method to a polymerisation example for which the analytic solution is known, and show that its predictions are very accurate. We also compare the performance of the method to that of the straightforward integration of the mass balances of the reacting system, showing that the pgf method results in very fast calculations. We end by discussing ongoing and future applications of the method. © 2011 Canadian Society for Chemical Engineering

Development of Molecular Weight Distribution in ATRP with Radical Termination

“…[18] There are many different modeling approaches taken in the investigation of CRP systems, from the early use of kinetic modeling and method of moments, [19][20][21][22][23][24][25][26][27][28][29][30][31][32] to the more recent modeling studies, which utilized Monte Carlo method, PREDICI software, and other approaches. [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48] Out of all these studies, only some discussed the modeling of CLD, even though the knowledge of CLD is essential in determining polymer properties and in understanding reaction mechanism and kinetics. Several methods that have been employed in the literature to predict the CLD in CRP system include, but not limited to, Monte Carlo method, probability generating functions, and direct integration method.…”

“…Several methods that have been employed in the literature to predict the CLD in CRP system include, but not limited to, Monte Carlo method, probability generating functions, and direct integration method. [39][40][41][42][43][44][45] Our group has recently developed an analytical expression for the full CLD of ATRP with termination by analogously relating the activation/deactivation cycles in ATRP to a series of continuous stirred tank reactors (CSTRs). [45] By using this analogy, the well-known concepts for CSTRs are applicable to ATRP system, allowing the derivation of full CLD, while accounting for the effect of termination, as shown in Equation (1).…”

Mathematical Modeling of Molecular Weight Distributions in Vinyl Chloride Suspension Polymerizations Performed with a Bifunctional Initiator through Probability Generating Functions

“…A good value of Np is the one that yields the lowest value of the performance variable SSQ2. Initial guesses for Np should lie between 5 and 12 . MWD values presented here were always obtained for the optimum Np values.…”

Modeling of RAFT Polymerization using Probability Generating Functions. Detailed Prediction of Full Molecular Weight Distributions and Sensitivity Analysis

“…This technique consists in transforming the mass balances of polymeric species, characterized by the number of monomer units, to the pgf domain, producing as a result balances for the pgf transform of the MWD. Discrete points of the full MWD are recovered from the pgf domain by applying an appropriate numerical inversion technique . It is important to point out that this method does not require any prior knowledge of the shape of the MWD, and can deal with complex kinetic mechanisms.…”

“…A combination of univariate and bivariate pgfs are applied to the population balances to model the MWD of the different polymeric species.…”