Kinetic Modeling of Non‐Linear Polymerization

Abstract

Summary: Recent developments of a method based upon population balances of generating functions of polymer chain length distributions (CLD) are presented. The calculation of the CLD and how to take into account chain length dependent reactivity are discussed. Prediction of polymer properties is also possible but only easily done for the average molecular radius of gyration; some results are presented for a radical polymerization including transfer to polymer and propagation with terminal double bonds.

Kinetic Modeling of Hyperbranched Polymer Synthesis through Atom‐Transfer and Nitroxide‐Mediated Radical Polymerization of Vinyl/Divinyl Monomers

“…In the context of CRP systems, it is also often considered that dormant polymer molecules contain only one capping unit (monofunctional assumption). These and other approximations are avoided with the present approach, as shown in [16][17][18][19][20][21]. It should be noted that this theory is also applicable after gelation but this analysis is beyond the scope of the present paper as stressed above.…”

“…It should be mentioned that, although in principle, changes of reactivity with chain length (the most notorious example being the change of termination rate constant) can be computationally dealt with [21], the increased computing time (2 or 3 powers of ten) is often not always worth the effort. Multiple intramolecular reactions within non-linear molecules is a more serious issue, since no exact way of dealing with all reaction possibilities is feasible, and good approximations are still to be found.…”

“…Significant modeling efforts [10] were also performed using the pseudo-kinetic method approach and the post gelation period is analyzed by means of the Flory-Stockmayer theory of gelation. An extensive comparison between the present general kinetic approach and alternative methods used to describe non-linear polymerization have been presented in [16][17][18][19][20][21]. These comparisons show the often important deleterious effect of approximations such as the pseudo-steady state for radical concentrations and neglecting the existence of polymer molecules with more than one radical (monoradical assumption), neglecting the existence of polymer molecules with more than one terminal/pendant double bond.…”

Modeling of Acrylamide/N,N′-Methylenebisacrylamide Solution Copolymerization

“…25,26 Moreover, intramolecular cyclizations are known to be dominant in this system, reducing the amount of PDB available for multiradical formation (which in the present case is mediated by crosslinking only). Also assuming the same reactivity of sol and gel is a typical choice when it comes to crosslinking free-radical polymerization, 5,23 whereas assumptions (4)- (6) are typically employed in the present model. 23,27 As for the employment of pseudokinetic constants (assumption (6)), some literature works report the necessity to employing a chain-length dependent termination in the frame of polymer modification.…”

“…Kinetic models usually rely on the solution of population balance equations (PBEs), which describe among other properties, the chain length distribution (CLD) of the polymer and the cross-linking density. In cross-linking copolymerization, multidimensional PBE are typically needed, because at least the numbers of units and pendant double bonds must be employed as internal coordinates. − Despite the great deal of information these models provide, their numerical solution, especially when the entire chain length distribution is needed, requires a significant computational effort.…”