Going Beyond the Carothers, Flory and Stockmayer Equation by Including Cyclization Reactions and Mobility Constraints

Abstract

A challenge in the field of polymer network synthesis by a step-growth mechanism is the quantification of the relative importance of inter- vs. intramolecular reactions. Here we use a matrix-based kinetic Monte Carlo (kMC) framework to demonstrate that the variation of the chain length distribution and its averages (e.g., number average chain length xn), are largely affected by intramolecular reactions, as mostly ignored in theoretical studies. We showcase that a conventional approach based on equations derived by Carothers, Flory and Stockmayer, assuming constant reactivities and ignoring intramolecular reactions, is very approximate, and the use of asymptotic limits is biased. Intramolecular reactions stretch the functional group (FG) conversion range and reduce the average chain lengths. In the likely case of restricted mobilities due to diffusional limitations because of a viscosity increase during polymerization, a complex xn profile with possible plateau formation may arise. The joint consideration of stoichiometric and non-stoichiometric conditions allows the validation of hypotheses for both the intrinsic and apparent reactivities of inter- and intramolecular reactions. The kMC framework is also utilized for reverse engineering purposes, aiming at the identification of advanced (pseudo-)analytical equations, dimensionless numbers and mechanistic insights. We highlight that assuming average molecules by equally distributing A and B FGs is unsuited, and the number of AB intramolecular combinations is affected by the number of monomer units in the molecules, specifically at high FG conversions. In the absence of mobility constraints, dimensionless numbers can be considered to map the time variation of the fraction of intramolecular reactions, but still, a complex solution results, making a kMC approach overall most elegant.

Are RAFT and ATRP Universally Interchangeable Polymerization Methods in Network Formation?

“…9−12 Recently, theoretical improvements have been developed beyond the Stockmayer/Flory approach. 13 However, these theories included two idealized assumptions. The first is the equal reactivity of monomers and crosslinkers.…”

“…Significant effort has been devoted to understanding gelation processes and the architecture at the macromolecular, nanomolecular, and macroscopic scales. − Indeed, the network structure is intimately tied to the material properties; − therefore, it is critical to understand the impact of synthetic methods on network architecture and final material properties . Eighty years ago, the first theory of gelation was proposed by Flory and Stockmayer. − Recently, theoretical improvements have been developed beyond the Stockmayer/Flory approach . However, these theories included two idealized assumptions.…”

Procedures and Guidelines for Inputting and Output Smoothening of Kinetic Monte Carlo Distributions

“…[53][54][55][56][57] Model outputs may be prone to noise either, due to i) the desire to obtain fast simulation results, caused by using low (Monte Carlo) control volumes, and ii) the inevitability of tail regions involving only one molecule or species. For example, De Keer et al [58][59][60] predicted experimental log-MMD's, so-called size exclusion chromatography (SEC) traces, for condensation polymer networks and found that an increase of the stochastic simulation volume merely extends the noisy tail towards higher molar masses on the abscissa. This increase never leads to a physically expected downward curvature of the tail, which is a consequence of the network chemistry in which very large molecules will react with each other and create new noise in the (moving) tail, at least assuming that dif-fusional limitations are not limiting the further reaching of high chain length at one point.…”

Connecting Gas-Phase Computational Chemistry to Condensed Phase Kinetic Modeling: The State-of-the-Art

“…In the former mechanism, one has in essence initiation, chain initiation, and many propagations, until termination creates “dead” polymer molecules on a very short time scale ((m)second scale) [ 12 ]. In the latter mechanism, functional groups are combined step by a step, and a gradual transition from monomer to dimer to n -mer takes place [ 13 ].…”