Pushing forward kinetic modeling tools for polymer circularity design and recycling

Abstract

Polymer circularity has received increasing attention due to ecological benefits, by which plastic waste should be reused or converted into high-value products in an economic framework balanced with virgin polymer production. From a chemical engineering point of view, the understanding of reaction kinetics and chemical modifications plays a crucial role in improving the process towards polymer circularity. This reaction kinetics is connected to molecular variations for which (micro)kinetic models are essential. In this perspective, the main kinetic simulation methods are summarized, focusing on their respective characteristics and challenges, besides differentiating between deterministic and stochastic methods. The application of kinetic simulations in polymer circularity processes is clarified in the form of three case studies, including (i) mechanical recycling with deliberate chemical modification by reactive extrusion, (ii) chemical recycling aiming at monomer recovery, and (iii) recycling-by-design aiming at vitrimer molecular design. Attention is also paid to the relevance of benchmarking the methods applied.

Graphical abstract

1. Introduction

Plastic products have been widely used in industry and daily life, due to their diverse and excellent physical and chemical properties [1]. Their global annual production capacity has risen to 8.3 billion tons in 2015 and still increases every year [2]. Such increasing production capacity creates nonnegligible environmental issues caused by plastics. However, only 9% of the generated plastic waste is recycled [3], with nearly 50% of the plastic waste going to landfills, ca. 20% incinerated, and the remaining part mainly ending as unmanaged dumps or leaking into the environment, which causes serious environmental problems. Hence, polymer circularity design and research are needed to deal with the increasing polymer production capacity and the resulting pollution.

In industry, mechanical recycling, chemical recycling, and recycling-by-design are noteworthy pathways to realize polymer circularity, as shown in Fig. 1. By mechanical recycling, plastic waste streams are separated, cleaned, (re-)melted, and pelletized to realize manufacturing again as final polymer products, which is often economically beneficial at least if the product quality can be guaranteed [4,5]. Upgrading via, e.g., reactive extrusion (REX) can result in high-performance products, compensating for more severe mechanical and recycling property losses [6,7].

Fig. 1.

Three polymer circularity (process) design paths.

By chemical recycling, plastic waste can be converted after a sorting step to oligomers or monomers for further synthesis in the absence of the property loss caused by conventional mechanical recycling. This is a green and efficient way of polymer circularity but more energy-intensive [[8], [9], [10], [11]]. In addition, recycling-by-design is an emerging method to realize polymer circularity [12]. Here one aims at the development of inherently sustainable functional materials, providing a new perspective for realizing polymer recycling. For example, one can target linear polymers degradable upon disposal and crosslinked polymers with dynamic exchange properties [13,14].

To fully exploit the three design paths included in Fig. 1, simulations that are based on chemicophysical variations are an important tool, as they can help to develop new processes, guide the operation of reactors and separation equipment, and automate the control of polymer circular processes [[15], [16], [17]]. The scale of such chemical simulation methods ranges from microscopic to macroscopic, including molecular simulation [18], (micro)kinetic simulation [19], flow field and reactor simulation [20], and process flow simulation [21]. Particularly, (micro)kinetic simulation can accurately characterize the evolution of the polymer molecular and microstructure in the reaction process as well as reflect the reaction rate(s). A multi-scale approach is eventually the most suited one, covering alongside the traditional micro-scale features of the meso‑scale, e.g., interphase phenomena, as well as the macro-scale dealing with concentration and temperature gradients as encountered during scale-up because of inevitable non-idealities.

An accurate representation of the core micro-scale is non-trivial as distributed macromolecular features are easily encountered. A variety of micro-scale structures exist such as blocks, grafts, and crosslinks, with high-performance polymers having a specific microstructure or morphology and thus specific properties [22,23]. For example, hyperbranched polymers have a low viscosity and high solubility [24], and rubbery (e.g., polybutadiene) chains grafted with vinyl monomers (e.g., styrene) have high impact resistance compared with conventional (linear) vinyl polymer [25].

It is thus paramount to control the effect of process/synthesis conditions on the microstructural variations, with a key role for kinetic simulations. Particularly, by using kinetic simulation methods a specific emphasis can be put on molar mass monitoring with a preference to track distributed molecular variations besides average properties. More in general real-time online monitoring of the reaction process by kinetic simulation facilitates the design and development of polymer recycling processes, as well as the automated operation and control of equipment.

The main purpose of this contribution is to summarize the (micro-scale) kinetic simulation methods relevant to polymer circularity and design. The kinetic simulation methods are first classified according to their features and application scope. Three case studies are then discussed to illustrate specific applications of kinetic simulation in polymer circularity, including mechanical recycling, chemical recycling, and recycling-by-design. This perspective highlights thus the mainstream kinetic simulation methods and provides valuable guidance for the design, operation, and control of polymer recycling by means of simulations.

2. Kinetic simulation methods

In this section, kinetic simulation methods applied to polymer circularity are categorized and organized, focusing on the characteristics and application of each method. The two main categories are deterministic and stochastic methods. The main characteristics and limitations (challenges) of both are also summarized in Table 1, including a further division according to submethods.

Table 1.

Characteristics and challenges of kinetic simulation methods.

Kinetic simulation method		Summary	Characteristics and challenges	Ref.
Deterministic methods	Direct methods: analytic methods	Solving the PBE analytically and obtaining an explicit solution about species concentration.	√ Obtaining explicit expressions for species; √ Suitable for exploring mechanisms; ! Only a few (intrinsic) systems have analytic solutions, and some of the solutions are complex.	[[26], [27], [28], [29], [30]]
	Direct methods: numerical analysis	Solving the PBE numerically. This method is based on the numerical analysis methods for ODEs such as the Runge-Kutta method. Commercial software can be used such as MATLAB®.	√ The solution is precise and intuitive. √ Particularly suitable for sufficiently accurate solution of systems with not too many species, or as an exact solution to verify the accuracy of a model. ! High computational cost. ! Unsolvable when the system is complex because the PBE equations are large in number and rigidity (mathematical stiffness).	[31]
	Direct methods: finite element method	Based on the variational original path in generalized theory. The approximation error process is minimized by function fitting to solve the system of ODE. Finite element methods often yield numerical solutions.	-The same as numerical analysis method.	[31]
	Method of moments	Statistical moments of polymer species are used to transform the PBE into ODE about moments, realizing the reduction in the number of equations	√ Low computational cost. √ Particularly suitable for the simulation of linear homopolymers. ! Low level of molecular detail; generally only average properties can be calculated unless a posteriori corrections are made; not that easy for more complex reaction schemes with the need for closure terms. ! Reaction rate coefficients are usually constants	[32,33]
	Steady-state approximation based	Treating the concentration of some species (especially reactive intermediate species, such as free radicals) in the polymer-based system as quantities which conserved quantities over time.	√ Simple derivation and low computational cost. ! Distribution of active species cannot be estimated. ! Inevitable error.	[34,35]
	Partial differential method	Assuming the chain length is a continuous variable, the system of ODE about time can be written as PDE about time and chain length, which can be solved using an appropriate method such as the Galerkin method in particular cases.	√ Continuous equations are often easier to be solved analytically and numerically. ! Serious errors when the chain length is small (e.g., less than 20). ! Less trivial for more complex reaction schemes.	[[36], [37], [38]]
	Fixed pivot method	Approximating the process based on continuous PBEs using a small number of discrete ODEs, which effectively reduces the number of equations compared to the discrete scheme. The fixed pivot method is essentially a discretization scheme for a continuous PBM.	√ Retaining the ability to compute distributional properties with the speed of calculation increased. ! The absence of conservation (for more complex cases) makes the fixed pivot method systematically inaccurate	[39,40]
Stochastic methods	Event-driven (conventional Gillespie-based) kinetic Monte Carlo	Gillespie established a random sampling algorithm for reaction kinetic simulations based on Markov chains. Most of the kMC methods are based on this algorithm.	√ High accuracy and easy to implement as no need for complex rate equation derivations √ Ability to track the reaction history √ Can be combined with convergence analysis, e.g. signal-to-noise ratio. ! High computational costs unless focus on averages only	[41,42]
	Conditional kinetic Monte Carlo	Distributed properties are assessed a posteriori based on given probability functions that depend on previous events in this method, particularly suitable for systems with missing initial conditions.	√ Advanced molecular characteristics can be assessed ! The prediction of the advanced molecular characteristics is prone to assumptions that are chosen conditionality, thus not accounting for the time history of kinetics.	[43]
	Tree-based (event-driven) kinetic Monte Carlo	Based on conventional kMC, a (binary) tree data structure is used to store information and improve the efficiency of random sampling and information access.	√ Reasonable level of molecular detail; √ Particularly suitable for linear copolymers; ! Slow if many reactive species and branched structures.	[[44], [45], [46]]
	Matrix-based (event-driven) kinetic Monte Carlo; with connected matrices coupled matrix-based Monte Carlo (CMMC).	Based on conventional kMC, a matrix is used to store information and improve the efficiency of random sampling and information access.	√ Particularly suitable for the storage of complex microstructures √ Very high level of molecular detail; ! Slow if many reactive species and groups are to be tracked chain by chain; recent developments result in faster simulations.	[47]

2.1. Deterministic methods

Deterministic (kinetic) modeling assumes that the reaction kinetic behavior is predictable for a reaction system with a continuous time variation. The core mathematical part is defined by population balance equations (PBEs), which are a set of ordinary differential equations (ODEs) for species concentrations with respect to (reaction) time (assuming homogeneity or a given location).

Depending on the form of the solution that needs to be obtained, a so-called direct method can be subdivided into analytical and numerical methods. The early literature usually employed analytical methods based on the mathematical principles of ODE to obtain an explicit functional expression for the species concentration evolution with time or conversion, because computers were not yet developed enough to support large numerical operations [27,48]. There are still a few reports on analytical methods in recent years [49]. However, this type of solution tends to apply only to simple cases with ideal assumptions. The derived analytical solutions may also be difficult to apply because of possible complex forms so analytical methods for solving the PBE are not commonly used.

Correspondingly, the numerical method is widely applied compared to the analytic method. PBEs are then numerically solved according to the principles of numerical analysis. Well-established and mature commercial solvers such as MATLAB® and FEniCS® can for instance be used. However, (de)polymerization reaction systems are complex and involve many species leading to a large number of equations, complicating the numerical integration. The coexistence of fast and slow reactions in polymer reaction engineering and recycling inevitably leads to high numerical rigidity. Hence, the direct method is not that widely used and is often considered as a reference to verify the accuracy of other models. The direct method is thus employed as a comparison to verify the accuracy of other mathematical models under simplified conditions [31,50].

To reduce the computational cost of the deterministic realistic kinetic simulation while guaranteeing the computational accuracy, simplifications are made on the PBE level. For example, a common way for a simplified intermediate species concentration calculation is the steady state approximation or assumption [34,51]. A chain length grid refinement according to for instance the fixed pivot method is a second example. The method of moments (MoM), which only deals with the calculation of averages, is a third example [32,52].

The steady-state approximation is particularly suitable for simulations with limited reaction scheme complexities, due to its rather basic derivation and small computational cost, although it inevitably introduces errors for a certain set of conditions [53]. The fixed pivot method in turn allows to adjust computational cost and accuracy according to the needs of the system and it can be used to calculate molar mass distributions (MMDs) in histogram format. However, the lack of generically applicable conservation equations leads to unavoidable and systematic errors. Partial differential methods transform PBE into a partial differential equation (PDE), which in some cases greatly reduces the computational cost [37,54]. Note that continuous PBE can be associated with errors compared to the more accurate results from discrete PBE, especially in case the chain length or mass is small. Hence, the continuous PBE model needs to be carefully chosen especially for the depolymerization simulation [54].

If only the average nature of the polymer-based process is needed and not the entire MMD, MoM is the appropriate choice. The core of MoM is to transform PBE into ODE for the statistical moments, in order to ensure the accuracy of the calculation under the condition of small rigidity compared with the PBE calculation, at low computational cost.

2.2. Stochastic methods

The stochastic methods are performed discretely, assuming that the time evolution of an arbitrary reaction system can be rigorously formulated by the chemical master equation (CME), using random variables to characterize instantaneous changes for each species in the system. The main purpose of stochastic methods is to solve this CME [41].

This became only feasible for chemical systems according to the event-driven kinetic Monte Carlo (kMC) method. Specifically, in 1977, a stochastic simulation algorithm (SSA) was proposed by Gillespie based on the Markov chain principle that enables the stochastic results to converge to the CME solution in case the sample size (control volume) is large enough, which is known as kinetic Monte Carlo (kMC) for reaction kinetic simulation [55]. As events are tracked one by one for all (macro-)species present the MMD can be obtained accounting for the reaction event history. Note that conditional kMC also exists, which allows advanced molecular macrostructures to be generated not explicitly depending on the kinetics, hence, less strict [43]. Distributed properties are then evaluated through a posteriori function for a given (fixed) probability function.

Based on the event-driven framework proposed by Gillespie, kMC methods have experienced many developments and spawned several categories in the last decades [56,57]. To reduce computational costs, a variety of acceleration methods have e.g., been introduced such as the τ-leaping acceleration method and the scaling acceleration method [58,59]. Improvements have also been made for the data storage structure, which can help to shorten the simulation time and widen the applicability for more dynamic simulations.

Specifically, Chaffey-Millar et al. [44] introduced binary tree data structures into kMC for information storage and extraction, and in general the method is named tree-based kMC, with e.g., higher order trees being introduced in Trigilio et al. [58] Tree-based kMC presents a simple and intuitive model for data storage and management whose logic is particularly suited for the kinetic simulation of linear polymer systems and MMD prediction. However, the method is not suitable for the simulation of complex systems because multiple sub-binary trees are needed in this situation, increasing simulation time and memory usage.

More interestingly, complex reaction systems or systems demanding more molecular detail are matrix-based Monte Carlo methods. In the most advanced format, interconnected matrices are exploited, resulting in coupled matrix-based Monte Carlo (CMMC), with high accuracy even for (highly) cross-linked systems [47]. A detailed benchmark study has been included in Wu et al., highlighting the consistency of CMMC with the deterministic MoM combined with numerical fragmentation on the distribution level (D-MoM-NF) [60]. As shown in Fig. 2, dealing with single phase grafting, both methods give very similar results at three selected styrene conversions. The D-MoM-NF method is although practically only applicable for the intermediate molar mass region, highlighting the added value of CMMC in view of a full molecular representation.

Fig. 2.

Comparison of molar mass distribution (MMD) or molecular weight distribution (MWD) as calculated by D-MoM-NF (Distribution - Method of Moments - Numerical Fractionation) and CMMC (coupled matrix-based kinetic Monte Carlo). (a) Topological structure of polystyrene grafted polymers and the graphical legend to be simulated. (b)-(d) Benchmarking (red) of CMMC and a deterministic method d-MoM-NF aiming at distributions as included in Wu et al. [60] for single phase grafting at different St conversion, with p/q the number of butadiene/styrene segments. The two methods align for the 29 topologies covered in the deterministic model, with CMMC also applicable up to high chain lengths and crosslinking amounts (blue).

The various mathematical methods have their characteristics and limitations. Overall, one needs to focus on a balance between computational speed/cost and simulation detail, to select suitable kinetic simulation methods according to the characteristics of the reaction system and desired output. In the next section, we describe how to select appropriate kinetic simulation methods for three case studies. This is done following the outline in Fig. 1, including mechanical recycling with upgrading (REX), chemical recycling through depolymerization and chemical design for crosslinked systems with molecular exchange features.

3. Case studies for kinetic simulation in polymer circularity

In this section, three case studies are put forward to illustrate the application of kinetic simulation in polymer circularity and design. These case studies deal with mechanical recycling, chemical recycling, and recycling-by-design.

3.1. Case study 1: mechanical recycling for polymer upgrading by reactive extrusion

Mechanical recycling is a preferred recycling route from an energy point of view [6,61]. It is however often accompanied by a loss of mechanical strength. If this loss becomes pronounced one needs to apply chemical recycling as covered in the next case study. Intermediately one can apply reactive extrusion (REX) or in general reactive or chemical modification, as considered in the present case study, to limit the loss of mechanical strength and even achieve high-performance polymer production.

REX combines multiple manufacturing processes in a single reactive unit such as melting, mixing, transport, pressure generation, homogenization, and product shaping [62]. Extruders can display desired mixing and heat transfer capabilities for continuous production of highly viscous systems. Notably, REX modeling requires tracking of the chemical reactions occurring in the extruder as these profoundly affect the microstructure of the product and thus the final properties [63]. For example, grafted polymers made by REX can enable the compatibility of separated phases when blending with other polymer products [60,64,65], which needs industrial-scale design although more complex modeling tools also account for macro-scale effects [[66], [67], [68]].

A practical downside of REX can be a less optimized process design with a too-long average residence time causing too many degradation reactions, also acknowledging temperature and self-heating phenomena. Moreover, the residence time distribution (RTD) affects the reaction kinetics and final product properties. Hence, an accurate RTD prediction is essential. In what follows, the recent RTD-based research at Ghent University is highlighted, which in its current phase deals with a more basic macro-scale description for a given temperature profile but uniquely described the micro-scale layer.

As shown in Fig. 3, De Smit et al. investigated the kinetics and RTD for REX using CMMC [69]. Initially focusing on polyolefin grafted maleic anhydride (subplot a-e), they considered the extruder at first sight as a series of multiple ideal reactors. Exchange streams (Fig. 3a) are additionally included, which realizes the coupling of microscale reaction kinetics with macroscale reactor simulation with non-idealities in bulk temperature and concentration. The RTD can be calculated per compartment, as shown in Fig. 3b, and different mass transfer situations can be simulated with different numbers of compartments and physically driven exchange parameters. For example, a more plug flow reactor is obtained with more compartments and forward exchange, as highlighted in Fig. 3c. Per compartment, the microscale kinetic simulation is based on the method already developed by the same group [70,71]. As shown in Fig. 3e, molecules staying longer in the extruder have a different chemical modification history with those that exit earlier.

Fig. 3.

Simulation of residence time distribution RTD, and chemical modification of individual molecules, for reactive extrusion (REX), aiming at grafting. (a) Multi-compartment modeling approach; (b) RTDs in different compartments, ultimately giving the observed one upon exit; (c) effect of number of compartments on RTD to highlight the plug flow limit; (d) grafting and side reactions for polyolefin modification (covered in a-e); (e) effect of RTD on the microstructure of graft polymer, selecting a few chains per residence time slice; (f) phase volume variations for grafting of rubbery phase: two-phase model application [70].

Globally, several simulation schemes can be applied for the microscale kinetic modeling per compartment, including MoM, the Monte Carlo method, a homogeneous phase model, and a two-phase model (Fig. 3f with grafting of polybutadiene) [70].

3.2. Case study 2: chemical recycling aiming at increased monomer yield by depolymerization design

A key challenge for the chemical recycling of polymers is to obtain a high yield of monomers (or oligomers) in a fast manner [72]. Opposed to mechanical recycling, degradation reactions are now desired and again their kinetics can be covered by both deterministic and stochastic methods. At first sight, one could focus on average-based solvers, as the main goal is the tendency to form smaller molecules. However, distribution effects can matter as certain degradation reactions are monomer unit-dependent and can go faster if certain functional groups are present. An inclusion of such effects is less trivial, increasing the computational cost. In what follows, the potential of MoM for depolymerization is highlighted, benefiting from recent work performed at Shanghai Jiao Tong University.

MoM has advantages such as a low computational cost and suitability for the real-time control of polymer depolymerization processes and the regulation of operations [32]. It can be coupled with other simulation techniques (e.g., flow field simulation via computational fluid dynamics; CFD) [73,74]. Traditionally MoM allows us to calculate the evolution of the average properties. Meanwhile, a variety of upgrading strategies have been applied to extend the applicability of moment-based methods, such as direct moment methods [75] and partial moment methods [50]. Two of such upgrading strategies for chemical recycling MoM are demonstrated in Fig. 4 [31,50], considering two main degradation reactions for linear polymers, namely random scission and chain-end scission whose mechanisms have been shown in Fig. 4a and d.

Fig. 4.

Examples of two upgrading strategies of Method of Moments (MoM) benchmarking with the analytic solution (focus on chain length distribution at different yields) [31,50]. k_r, k_e, k_-e represent the rate coefficients of random scission, chain-end scission and reverse chain-end scission; n_c is a parameter of the PBE-MoM model so that species with a chain length bigger than n_c can be ignored. A higher n_c facilitates the benchmarking.

Specifically, A new form of analytical solution for chain-end scission process is proposed based on which MoM is improved [31]. Compared with the conventional MoM, the upgraded MoM can accurately calculate the moments (Fig. 4b) and chain length distribution (Fig. 4c) during chain-end scission process. Meanwhile, a PBE-MoM model has been proposed based on a selection strategy to simulate depolymerization with simultaneous random and chain-end scission [50]. As shown in Fig. 4e, the model includes the critical chain length n_c as a model parameter. The kinetics of the system is described using the PBE for chain lengths less than n_c while the moment equations are used overall to describe the conservation relationships. As shown in Fig. 4f, with increasing n_c the simulation results converge to the actual solution, which means that choosing a sufficiently large n_c allows the upgraded MoM to be used for the simultaneous random and chain-end scission system.

3.3. Case study 3: recycling-by-design for more homogeneous covalent adaptable networks

Designing high-performance recyclable polymers through chemical methods is an emerging way to achieve polymer circularity, which is called recycling-by-design. In the present case study, we deal with network polymer recycling-by-design.

Conventional highly crosslinked networks are connected by irreversible covalent bonds, which makes crosslinking polymers mechanically strong but non-recyclable [76]. Interestingly, one can replace irreversible cross-linked covalent bonds with reversible covalent bonds in the original synthesis. The replaced dynamic bonds can be reversibly broken to achieve flow or stress relaxation under certain conditions (e.g., light, pH, and temperature), allowing the crosslinked polymer to be recycled under specific conditions.

Specifically, covalent adaptable networks (CANs) have attracted much attention in recent years [[77], [78], [79], [80]]. CANs have both the mechanical strength of traditional thermosets and the processability of thermoplastics, making them an ideal material for polymer recycling [81,82]. CAN synthesis follows a complex reaction system involving both crosslinking and exchange reactions, as shown in Fig. 5a focusing on acetoacetate groups with amine-based exchange potential. The many reactions and viscosity effects lead to the generation and evolution of various topological structures so that the kinetics of (dynamically) crosslinked polymers can thus be difficult to simulate accurately.

Fig. 5.

kMC simulation for covalent adaptable networks (CANs) (a) Crosslinking and exchange reactions in the formation of CANs. (b) Reactions (E: exchange and C: crosslinking) in the kinetic Monte Carlo (kMC) model are selected based on the reaction probabilities that depend on chemical and diffusion parameters. (c) These (input) parameters are based on model validation to experimental data, as explained in Liu et al. [72] to simulate various outputs. (d-g) The corresponding simulation results, with (d) comparison of simulated (line) and experimental (point) values for the evolution of the A (acetoacetate) functional group conversion with time. (e)-(g) Branching density, mass-averaged degree of polymerization (RDP), and mass fraction of the sol (g) at different temperatures.

In the early literature, only gel points could be predicted by deterministic methods in a polymer cross-linking system and elaborate microstructures could not be obtained [83]. Deterministic methods are unable to obtain detailed microstructures of CAN, and the kMC method is recommended for precise and detailed kinetic information. More in detail, the Olsen research group applied MC simulations to capture defects in the network topology during the crosslinking process to quantify the gel points [84]. They also used the same model to study the topology and elasticity of crosslinking systems composed of monomers with different degrees of functionality [85]. Furthermore, the formation process of a conventional polyurethane crosslinked network has been simulated by kMC to visualize and predict the characteristic properties of the process [86], and De Smit et al. [62] mapped individual exchange capabilities by CMMC, starting from the kMC network simulations of the De Keer et al. [19]

In addition, Bowman et al. used the Semenov-Rubinstein theory of linear dynamics of reversible networks to predict the behavior of Diels-Alder networks in the vicinity of the gel point [87]. A deterministic model describing the structure and kinetics of CAN-conjugated intermediates was developed, in which the relationship between the reaction equilibrium coefficient and the platform modulus was obtained.

Both Shanghai Jiao Tong University and Ghent University recently joined forces to study by stochastic means the synthesis of CAN (vitrimers) with vinylogous urethane dynamic bonds (Fig. 5a) [88]. A detailed kinetic model was developed using matrix-based kMC, considering experimentally measured data for model validation. The type of reactions in the kMC model was selected based on the reaction probabilities (Fig. 5b). As shown in Fig. 5c, they used Fourier transmission infrared spectroscopy (FT-IR) and rheological data to tune input intrinsic and diffusion parameters, obtaining kinetic plots and molecular properties such as branching density, crosslinking density, and mass fraction of the sol, as shown in Fig. 5d-g. It is shown that a higher temperature facilitates exchange and the formation of a more homogeneous network structure thanks to the later establishment of the gel point. A similar effect can be obtained by increasing the ratio of amine to acetoacetate groups.

4. Conclusion

Polymer circularity enables recycling and conversion of plastic wastes into high-performance products, in which kinetic simulation can help the identification of reaction schemes as well as support reactor operation and control.

In this perspective, kinetic simulation models of (de)polymerization processes are summarized and their application to polymer circularity is presented in the form of three case studies, which deal with upgrading by reactive extrusion (mechanical recycling), depolymerization (chemical recycling), and dynamic networks (recycling-by-design).

Generally, the direct method for solving PBE (basic reaction schemes) and the kMC method (basic and detail reaction schemes) are highly accurate, so these methods can always be used as a guideline for judging the accuracy of other kinetic methods. Adapted PBEs such as MoM enable a fast computation opening the door to control-based systems, whereas the current kMC methods are more recommended in case the molecular detail and reaction history reflection are the most important to capture.

It is clear that mathematical modeling has greatly assisted in our understanding and designing of the kinetics of polymer circularity processes. Each mathematical model has different strengths and challenges, with a balance between computational cost and simulation details being essential. It is necessary to choose the appropriate kinetic simulation method according to the technical details of a given system and the actual needs.

With developments in computing power and innovations in simulation methods, it can be expected that the accuracy and computational cost of kinetic simulations will be further optimized which will allow the simulation of more complex and detailed structures in a faster manner. In addition, a shift is to be expected to a better connection of all length scales in the modeling field, with a desire to treat the micro-scale in sufficient detail and facilitate the calculation of distributed properties in case the user or application demands this.

In parallel, the development of experimental techniques provides new ideas for the establishment and validation of kinetic models. In addition, multi-scale modeling will become an important direction because it can take into account more realistic application environments.

Declaration of competing interest

The authors declare that they have no conflicts of interest in this work.

Biographies

Jiang Wang received his master's degree from Shanghai Jiao Tong University, China, in 2023. He is currently a doctoral student with Professor Yin-Ning Zhou as his supervisor at Shanghai Jiao Tong University. His current interests are (de)polymerization kinetic modeling and polymer recycling.

Dagmar R. D'hooge received his Ph.D. in chemical engineering from Ghent University, Belgium, in 2010. He was a postdoctoral researcher in the Matyjaszewski Macromolecular Engineering Group at Carnegie Mellon University in 2011 and in the Macromolecular Architectures Research Team at Karlsruhe Institute of Technology in 2013. He has been appointed as an associate professor at Ghent University since 2016. In 2022, he is the elected head of the Department of Materials, Textiles and Chemical Engineering. His current interests are the design of polymerization, polymer processing and polymer recycling.

Zheng-Hong Luo (BRID: 09055.00.72970) received his Ph.D. from Zhejiang University, China, in 2003. He then joined Xiamen University, China, in 2003 and was promoted to Full Professor in 2012. In the same year, he moved his research group to Shanghai Jiao Tong University (SJTU). In 2016, he was awarded the National Science Fund for Distinguished Young Scholars of China and as a Distinguished Professor at SJTU. In 2019, he was awarded as a 2018 Yangtze River Scholar Professor by the Ministry of Education of China. His research interests are polymer reaction engineering and chemical reactor engineering.

Yin-Ning Zhou (BRID: 05530.00.13825) received his Ph.D. from Shanghai Jiao Tong University (SJTU), China, in 2016. From 2016 to 2018, he worked as a postdoctoral fellow at McMaster University, Canada. Since October 2018, he has been appointed as a tenure-check associate professor at SJTU and was promoted to Professor in August 2023. His research focuses on kinetic modeling of polymerization and sustainable polymer reaction engineering.