Stochastic modeling techniques have emerged as a powerful tool to study the time evolution of processes in many research fields including (bio)chemical engineering and biology. One of the most appl...

Gillespie-driven kinetic Monte Carlo algorithms to model events for bulk or solution (bio)chemical systems containing elemental and distributed species

One of the challenges for modelling of polymerization kinetics is the detailed description of the molecular build-up of both linear and non-linear chains, specifically those with many grafts and crosslinks. Such grafted/crosslinked (co)polymers are relevant as emulsifiers, surface-modifying agents, coating materials, and compatibilizers. In the present work, we put forward a coupled matrix-based Monte Carlo (CMMC) concept to be successful in this respect, avoiding computational stiffness. The CMMC concept is illustrated for single phase grafting of polybutadiene (PB) with styrene (St) at 343 K by limiting the St conversion to 30%. Considering literature kinetic parameters, a benchmark for average characteristics as obtainable by deterministic method of moments simulations is first presented to then gradually extend the level of modelling output addressing (i) conventional grafting performance indicators (e.g. the grafting yield and mass ratio); (ii) univariate chain length distributions for all macrospecies types (polystyrene, PB, PB with only T grafts, PB with at least one H graft, etc.); (iii) bivariate St–butadiene distributions showing a compositional drift, due to the kinetic tendency to attack longer chains if they are sufficiently present and the competition between grafting and crosslinking; (iv) the explicit molecular build-up of individual molecules predicting the positioning of the St–Bd and St–St connectivity points and the chain formation history. It is demonstrated that the CMMC tool allows the simulation of the contribution and structure of molecules that are hard to access purely experimentally, so that in the long run, novel structure–property relationships are within reach. It is also showcased that consideration of elementary reactions is highly recommended and that even at 343 K, thermal self-initiation with St and related chain transfer reactions matter for a full appreciation of molecular variations. The current work also opens the pathway to identifying pragmatic equations for the experimentalist and online control benefiting from a detailed CMMC solution under any desired conditions.

Coupled matrix kinetic Monte Carlo simulations applied for advanced understanding of polymer grafting kinetics

Bulk and solution radical polymerization is important in daily live. A challenge is still to maximize polymerization rate and control over molecular characteristics such as the molar mass distribution. The last decades have made clear that kinetic modeling is indispensable with originally most focus on deterministic implementations such as the method of moments (MoM) and only more recently promising results for event-driven kinetic Monte Carlo (kMC) simulations that belong to stochastic methods. Computationally, a critical reaction is termination for which one has both distinguishable and nondistinguishable distributed species, which requires a delicate treatment of a stoichiometric factor 2. Proper benchmarking of MoM andkMC simulations demands thus a careful translation of this factor in the Monte Carlo (MC) reaction probabilities. However, limited attention has been paid in thekMC field to the detailed description of such translations. Here, a rigorous derivation is presented on the level of individual termination rates. Emphasis is on termination by recombination and disproportionation. It is highlighted that six types of factor 2 exist that all need to be incorporated with care, including an IUPAC-based one. The consistency is demonstrated by a successful benchmark of essential modeling results for free radical polymerization of methyl methacrylate.

Benchmarking Stochastic and Deterministic Kinetic Modeling of Bulk and Solution Radical Polymerization Processes by Including Six Types of Factors Two

The molar mass distribution (MMD), which is also known as the molecular weight distribution (MWD), is a key molecular property for a polymerization process, as it determines polymer strength and polymeric material deformation. Several MMD and related chain length distribution (CLD) representations are however used interchangeably, often leading to biased conclusions. Herein it is highlighted how the most interesting simulated CLD/MMD representations, i.e., the number, mass, z-based, and logarithmic CLD/MMD, can be translated into each other and how they need to be corrected to facilitate comparison with experimental size exclusion chromatography traces. Their relevance is highlighted by including case studies with increasing complexity in polymer products, addressing first linear, then branched, and ultimately crosslinked systems. For illustration purposes, the case studies are restricted to chain-growth polymerization, including free radical polymerization, cationic ring-opening polymerization, and free radical-induced grafting. All model parameters are taken from literature and kinetic Monte Carlo tools are utilized. Guidelines are also provided for both the experimentalist and modeler to maximize the reliability of CLD/MMD reporting in future kinetic studies.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/mats.202170005

Translating Simulated Chain Length and Molar Mass Distributions in Chain-Growth Polymerization for Experimental Comparison and Mechanistic Insight

A tree-based kinetic Monte Carlo (kMC) model is presented that differentiates between 38 end-group pairs for isothermal degradation of poly(styrene peroxide) (PSP). The binary trees allow for fast and accurate calculation of reaction probabilities, with mass-weighted binary trees for the accurate sampling of peroxide bond fissions and hydrogen abstractions along chains. The kinetic parameters are tuned via artificial neural networks (ANNs) to successfully predict literature experimental data, among other lumped product yields. ANNs are also utilized for sensitivity analysis to unravel the effects of individual reactions on the time evolution of experimental responses and other simulation outputs, including the variations of the chain length distributions of the macrospecies. PSP degradation is characterized by three stages of degradation considering both instantaneous and time-averaged concentrations. The first stage features rapid unzipping and results in the fast production of major products benzaldehyde and formaldehyde, the second stage features the most significant level of hydrogen abstractions involving PSP and other macrospecies types, and the third stage exhibits the consumption of the remaining peroxide bonds toward oligomeric species in a wide time frame until the degradation process is finalized by the depletion of peroxide bonds. This proof-of-concept study based on unprecedentedly detailed analyses of the chemistry via kinetic Monte Carlo paves the way to further improve our understanding of chemical recycling of solid plastic waste.

Diego Alberto Clemente‐Montes

Papers

Modeling of Reversible Deactivation Radical Polymerization of Vinyl Monomers Promoted by Redox Initiation Using NHPI and Xanthone