Beauty of Explicit Dispersity (Đ) Equations in Controlled Polymerizations

Dispersity (Đ) as a critical parameter indicates the level of uniformity of the polymer molar mass or chain length. In the past several decades, the development of explicit equations for calculating Đ experiences a continual revolution. This viewpoint tracks the historical evolution of the explicit equations from living to reversible-deactivation polymerization systems. Emphasis is laid on displaying the charm of explicit Đ equations in batch reversible-deactivation radical polymerization (RDRP), with highlights of the relevant elegant mathematical manipulations. Some representative emerging applications enabled by the existing explicit equations are shown, involving nitroxide-mediated polymerization (NMP), atom transfer radical polymerization (ATRP), and reversible addition-fragmentation chain transfer (RAFT) polymerization systems. Stemming from the several outlined challenges and outlooks, sustained concerns about the explicit Đ equations are still highly deserved. It is expected that these equations will continue to play an important role not only in traditional polymerization kinetic simulation and design of experiments but also in modern intelligent manufacturing of precision polymers and classroom education.

Molecular Pathways for Polymer Degradation during Conventional Processing, Additive Manufacturing, and Mechanical Recycling

The assessment of the extent of degradation of polymer molecules during processing via conventional (e.g., extrusion and injection molding) and emerging (e.g., additive manufacturing; AM) techniques is important for both the final polymer material performance with respect to technical specifications and the material circularity. In this contribution, the most relevant (thermal, thermo-mechanical, thermal-oxidative, hydrolysis) degradation mechanisms of polymer materials during processing are discussed, addressing conventional extrusion-based manufacturing, including mechanical recycling, and AM. An overview is given of the most important experimental characterization techniques, and it is explained how these can be connected with modeling tools. Case studies are incorporated, dealing with polyesters, styrene-based materials, and polyolefins, as well as the typical AM polymers. Guidelines are formulated in view of a better molecular scale driven degradation control.

Evolution of Molar Mass Distributions Using a Method of Partial Moments: Initiation of RAFT Polymerization

We describe a method of partial moments devised for accurate simulation of the time/conversion evolution of polymer composition and molar mass. Expressions were derived that enable rigorous evaluation of the complete molar mass and composition distribution for shorter chain lengths (e.g., degree of polymerization, Xn = N < 200 units) while longer chains (Xn ≥ 200 units) are not neglected, rather they are explicitly considered in terms of partial moments of the molar mass distribution, μxN(P)=∑n=N+1∞nx[Pn] (where P is a polymeric species and n is its’ chain length). The methodology provides the exact molar mass distribution for chains Xn < N, allows accurate calculation of the overall molar mass averages, the molar mass dispersity and standard deviations of the distributions, provides closure to what would otherwise be an infinite series of differential equations, and reduces the stiffness of the system. The method also allows for the inclusion of the chain length dependence of the rate coefficients associated with the various reaction steps (in particular, termination and propagation) and the various side reactions that may complicate initiation or initialization. The method is particularly suited for the detailed analysis of the low molar mass portion of molar mass distributions of polymers formed by radical polymerization with reversible addition-fragmentation chain transfer (RAFT) and is relevant to designing the RAFT-synthesis of sequence-defined polymers. In this paper, we successfully apply the method to compare the behavior of thermally initiated (with an added dialkyldiazene initiator) and photo-initiated (with a RAFT agent as a direct photo-iniferter) RAFT-single-unit monomer insertion (RAFT-SUMI) and oligomerization of N,N-dimethylacrylamide (DMAm).

Model Predictive Control with Integrated Model Reduction for a Continuous Lactide Ring-Opening Polymerization Process

Poly(lactic acid) production has received increasing attention, mainly due to its inherent biodegradable thermoplastic properties and to its renewable-resource-based composition. This process is affected by changes in the operating conditions and by raw material impurities which influence the reaction rate and degrade the polymer properties. As the system model is multivariable with coupled dynamics and constraints, linear model predictive control (LMPC) is employed here. A model reduction technique is proposed to obtain an approximate linear representation of the nonlinear system around the operating point to minimize the calculation cost of the controller. The proposed LMPC approach is validated by simulation and is compared to a proportional-integral controller and a nonlinear model predictive control. It is found that LMPC has a superior performance in terms of off-spec time when a disturbance occurs in the feed, and it can restore the target conditions better and faster.

A general model for the ideal chain length distributions of polymers made with reversible deactivation

A general model is developed for the distribution of polymers made with reversible deactivation. The model is applied to a range of experimental systems including RAFT, cationic and ATRP.

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

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 x_n), 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 x_n 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.

Jacket temperature regulation allowing well-defined non-adiabatic lab-scale solution free radical polymerization of acrylates

Conventional batch solution free radical polymerization of n-butyl acrylate with thermal initiators such as AIBN is known to be strongly exothermic and influenced by highly activated side reactions such as backbiting and β-scission.

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

An innovative coupled matrix-based Monte Carlo (CMMC) concept has been applied to successfully assess the detailed description of the molecular build-up of linear and non-linear chains in the free-radical induced grafting of linear precursors chains.

Spatiotemporally Controlled Photoresponsive Hydrogels: Design and Predictive Modeling from Processing through Application

Photoresponsive hydrogels (PRHs) are soft materials whose mechanical and chemical properties can be tuned spatially and temporally with relative ease. Both photo-crosslinkable and photodegradable hydrogels find utility in a range of biomedical applications that require tissue-like properties or programmable responses. Progress in engineering with PRHs is facilitated by the development of theoretical tools that enable optimization of their photochemistry, polymer matrices, nanofillers, and architecture. This review brings together models and design principles that enable key applications of PRHs in tissue engineering, drug delivery, and soft robotics, and highlights ongoing challenges in both modeling and application.

Progress in Reaction Mechanisms and Reactor Technologies for Thermochemical Recycling of Poly(methyl methacrylate)

Chemical or feedstock recycling of poly(methyl methacrylate) (PMMA) by thermal degradation is an important societal challenge to enable polymer circularity. The annual PMMA world production capacity is over 2.4 × 10⁶ tons, but currently only 3.0 × 10⁴ tons are collected and recycled in Europe each year. Despite the rather simple chemical structure of MMA, a debate still exists on the possible PMMA degradation mechanisms and only basic batch and continuous reactor technologies have been developed, without significant knowledge of the decomposition chemistry or the multiphase nature of the reaction mixture. It is demonstrated in this review that it is essential to link PMMA thermochemical recycling with the PMMA synthesis as certain structural defects from the synthesis step are affecting the nature and relevance of the subsequent degradation reaction mechanisms. Here, random fission plays a key role, specifically for PMMA made by anionic polymerization. It is further highlighted that kinetic modeling tools are useful to further unravel the dominant PMMA degradation mechanisms. A novel distinction is made between global conversion or average chain length models, on the one hand, and elementary reaction step-based models on the other hand. It is put forward that only by the dedicated development of the latter models, the temporal evolution of degradation product spectra under specific chemical recycling conditions will become possible, making reactor design no longer an art but a science.

The Competition of Termination and Shielding to Evaluate the Success of Surface-Initiated Reversible Deactivation Radical Polymerization

One of the challenges for brush synthesis for advanced bioinspired applications using surface-initiated reversible deactivation radical polymerization (SI-RDRP) is the understanding of the relevance of confinement on the reaction probabilities and specifically the role of termination reactions. The present work puts forward a new matrix-based kinetic Monte Carlo platform with an implicit reaction scheme capable of evaluating the growth pattern of individual free and tethered chains in three-dimensional format during SI-RDRP. For illustration purposes, emphasis is on normal SI-atom transfer radical polymerization, introducing concepts such as the apparent livingness and the molecular height distribution (MHD). The former is determined based on the combination of the disturbing impact of termination (related to conventional livingness) and shielding of deactivated species (additional correction due to hindrance), and the latter allows structure-property relationships to be identified, starting at the molecular level in view of future brush characterization. It is shown that under well-defined SI-RDRP conditions the contribution of (shorter) hindered dormant chains is relevant and more pronounced for higher average initiator coverages, despite the fraction of dead chains being less. A dominance of surface-solution termination is also put forward, considering two extreme diffusion modes, i.e., translational and segmental. With the translational mode termination is largely suppressed and the living limit is mimicked, whereas with the segmental mode termination occurs more and the termination front moves upward alongside the polymer layer growth. In any case, bimodalities are established for the tethered chains both on the level of the chain length distribution and the MHD.

Role of External Field in Polymerization: Mechanism and Kinetics

The past decades have witnessed an increasing interest in developing advanced polymerization techniques subjected to external fields. Various physical modulations, such as temperature, light, electricity, magnetic field, ultrasound, and microwave irradiation, are noninvasive means, having superb but distinct abilities to regulate polymerizations in terms of process intensification and spatial and temporal controls. Gas as an emerging regulator plays a distinctive role in controlling polymerization and resembles a physical regulator in some cases. This review provides a systematic overview of seven types of external-field-regulated polymerizations, ranging from chain-growth to step-growth polymerization. A detailed account of the relevant mechanism and kinetics is provided to better understand the role of each external field in polymerization. In addition, given the crucial role of modeling and simulation in mechanisms and kinetics investigation, an overview of model construction and typical numerical methods used in this field as well as highlights of the interaction between experiment and simulation toward kinetics in the existing systems are given. At the end, limitations and future perspectives for this field are critically discussed. This state-of-the-art research progress not only provides the fundamental principles underlying external-field-regulated polymerizations but also stimulates new development of advanced polymerization methods.

Method of Moments Applied to Most-Likely High-Temperature Free-Radical Polymerization Reactions

Many widely-used polymers are made via free-radical polymerization. Mathematical models of polymerization reactors have many applications such as reactor design, operation, and intensification. The method of moments has been utilized extensively for many decades to derive rate equations needed to predict polymer bulk properties. In this article, for a comprehensive list consisting of more than 40 different reactions that are most likely to occur in high-temperature free-radical homopolymerization, moment rate equations are derived methodically. Three types of radicals—secondary radicals, tertiary radicals formed through backbiting reactions, and tertiary radicals produced by intermolecular chain transfer to polymer reactions—are accounted for. The former tertiary radicals generate short-chain branches, while the latter ones produce long-chain branches. In addition, two types of dead polymer chains, saturated and unsaturated, are considered. Using a step-by-step approach based on the method of moments, this article guides the reader to determine the contributions of each reaction to the production or consumption of each species as well as to the zeroth, first and second moments of chain-length distributions of live and dead polymer chains, in order to derive the overall rate equation for each species, and to derive the rate equations for the leading moments of different chain-length distributions. The closure problems that arise are addressed by assuming chain-length distribution models. As a case study, β-scission and backbiting rate coefficients of methyl acrylate are estimated using the model, and the model is then applied to batch spontaneous thermal polymerization to predict polymer average molecular weights and monomer conversion. These predictions are compared with experimental measurements.

Synthesis of block cationic polyacrylamide precursors using an aqueous RAFT dispersion polymerization

Synthesis of cationic polyacrylamides (CPAMs) by introducing cationic polymer precursors followed by chain extension of acrylamide (AM) homopolymer blocks via RAFT polymerization is a promising approach for engineering high-performance CPAMs. However, the aqueous solution polymerization of AM usually leads to high viscosity, thus limiting the solid content in the polymerization system. Herein a novel approach is introduced that uses a random copolymer of AM and methacryloxyethyltrimethyl ammonium chloride (DMC) as a macro RAFT chain transfer agent (mCTA) and stabilizer for aqueous RAFT dispersion polymerization of AM. The AM/DMC random copolymers synthesized by RAFT solution polymerization, having narrow dispersities (Đ _s) at different molecular weights and cationic degrees (C _s), could serve as the mCTA, which was confirmed by mCTA chain extension in aqueous solution polymerization of AM under different C _s, solid contents, AM addition contents, extended PAM block lengths, and mCTA chain lengths. The block CPAMs had a Đ value of less than 1.2. A model was developed using the method of moments with consideration of the diffusion control effect, for further understanding the chain extension kinetics. Predicted polymerization kinetics provided an accurate fit of the experimental data. The AM/DMC random copolymers were further used for aqueous RAFT dispersion polymerization of AM under different polymerization temperatures, C _s, and mCTA chain lengths. The resulting products had a milky appearance, and the block copolymers had Đ _s of less than 1.3. Higher C _s and longer chain lengths on mCTAs were beneficial for stabilizing the polymerization systems and produced smaller particle sizes and less particle aggregation. The products remained stable at room temperature storage for more than a month. The results indicate that aqueous RAFT dispersion polymerization using random copolymers of AM and DMC at moderate cationic degrees as a stabilizer and mCTA is a suitable approach for synthesizing CPAM block precursors at an elevated solid content.

Modeling of Miniemulsion Polymerization of Styrene with Macro-RAFT Agents to Theoretically Compare Slow Fragmentation, Ideal Exchange and Cross-Termination Cases

A 5-dimensional Smith-Ewart based model is developed to understand differences for reversible addition-fragmentation chain transfer (RAFT) miniemulsion polymerization with theoretical agents mimicking cases of slow fragmentation, cross-termination, and ideal exchange while accounting for chain length and monomer conversion dependencies due to diffusional limitations. The focus is on styrene as a monomer, a water soluble initiator, and a macro-RAFT agent to avoid exit/entry of the RAFT leaving group radical. It is shown that with a too low RAFT fragmentation rate coefficient it is generally not afforded to consider zero-one kinetics (for the related intermediate radical type) and that with significant RAFT cross-termination the dead polymer product is dominantly originating from the RAFT intermediate radical. To allow the identification of the nature of the RAFT retardation it is recommended to experimentally investigate in the future the impact of the average particle size (dp) on both the monomer conversion profile and the average polymer properties for a sufficiently broad dp range, ideally including the bulk limit. With decreasing particle size both a slow RAFT fragmentation and a fast RAFT cross-termination result in a stronger segregation and thus rate acceleration. The particle size dependency is different, allowing further differentiation based on the variation of the dispersity and end-group functionality. Significant RAFT cross-termination is specifically associated with a strong dispersity increase at higher average particle sizes. Only with an ideal exchange it is afforded in the modeling to avoid the explicit calculation of the RAFT intermediate concentration evolution.

From n-butyl acrylate Arrhenius parameters for backbiting and tertiary propagation to β-scission via stepwise pulsed laser polymerization

A stepwise method to estimate the Arrhenius parameters for backbiting, tertiary propagation, and β-scission in acrylate radical polymerization.

Coupled stochastic simulation of the chain length and particle size distribution in miniemulsion radical copolymerization of styrene and N-vinylcaprolactam

Kinetic Monte Carlo modeling is applied for the coupled simulation of the chain length and particle size distribution in isothermal batch miniemulsion copolymerization of styrene and N-vinylcaprolactam.

Deterministic Approaches for Simulation of Nitroxide-Mediated Radical Polymerization

Since its development in the last decades, controlled radical polymerization (CRP) has become a very promising option for the synthesis of polymers with controlled structure. The design and production of tailor-made materials can be significantly improved by developing models capable of predicting the polymer properties from the operating conditions. Nitroxide-mediated polymerization (NMP) was the first of the three main variants of CRP to be discovered. Although it has lost preference over the years against other CRP alternatives, NMP is still an attractive synthesis method because of its simple experimental implementation and environmental friendliness. This review focuses on deterministic methods employed in mathematical models of NMP. It presents an overview of the different techniques that have been reported for modelling NMP processes in homogeneous and heterogeneous media, covering from the prediction of average properties to the latest techniques for modelling univariate and multivariate distributions of polymer properties. Finally, an outlook of model-based design studies of NMP processes is given.