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

Abstract

A mathematical model of RAFT polymerization processes is presented capable of predicting the full molecular weight distribution (MWD) through the use of probability generating functions (pgf). The bivariate distribution of the intermediate RAFT species is calculated. The model is able to work with the three kinetic mechanisms currently under discussion for explaining the observed behavior of this type of polymerization. For comparison purposes, the population balances are also solved by direct integration of the resulting equations. The results show that the pgf technique allows obtaining accurate solutions with very small computational times for systems of any average molecular weight. Spurious oscillations observed in the high molecular weight tail of the MWD can be easily disregarded. A sensitivity analysis over several of the kinetic constants is also performed, showing the effects of changing their values over several orders of magnitude. This analysis aims to showcase the enormous potential of the pgf technique for modeling and optimization of complex polymerization kinetics.

An Update on the Pivotal Role of Kinetic Modeling for the Mechanistic Understanding and Design of Bulk and Solution RAFT Polymerization

“…Hence, it is not surprising that deterministic methods for the full CLD calculation have been developed to a limited extent only. [47,52,[72][73][74][75][76][77] The most popular deterministic full CLD method is the discretized Galerkin h-p-method (dGhp), as explored by Wulkow et al [76] and applicable via the commercial package PREDICI. [25,33,51,55,[78][79][80][81][82][83] For RAFT polymerization, one of the earliest PREDICI contributions [25] paid attention to the influence of RAFT specific kinetic parameters on the temporal evolution of the CLD and monomer conversion.…”

“…Also more complex polymer architectures have been modeled in a later stage. [29,47,61,68,[84][85][86] For example, Junkers and Barner-Kowollik [29] used the PREDICI software package to find optimal reaction conditions for RAFT polymerization of acrylates, which is characterized by branch formation. Moreover, Chaffey-Millar et al [85] constructed the CLD of star polymers made via the so-called R-group approach, considering PREDICI output for the explicit simulation of the arm lengths.…”

“…Alternatively, the RAFT polymerization CLD can be obtained deterministically via solving probability generating functions by performing discrete Laplace transformations. [47,86] To reduce the computational cost in deterministic modeling techniques, the potential use of the pseudo steady state assumption (PSSA) has been intensively studied. [75,77,[87][88][89][90] By PSSA the number of ODEs can be reduced and a set of differential algebraic equations (DAEs) that is less stiff needs to be solved so that the computational cost is inherently reduced.…”

“…On the left side of the figure, RAFT cross-termination is considered, whereas on the right side this termination is neglected, as is often done in kinetic modeling studies to represent the SF theory or RAFT polymerization systems in which no retardation is observed. [32,47,72,75,111,132] For simplicity the same RAFT fragmentation rate coefficient for release of the R 0 and R i * species from the R 0 X*R i intermediate species (k frag(a,b) in Figure 1d) is considered.…”

“…Several research groups have focused on this important research task, both experimentally and via computer simulations. [25,[44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61] However, the number of kinetic studies considering the simultaneous impact and interaction of all relevant process parameters is limited.…”