Quantifying synergy for mixed end-scission and random-scission catalysts in polymer upcycling

Abstract

The environmental consequences of plastic waste are driving research into many chemical and catalytic recycling strategies. The isomerizing ethenolysis strategy for polyethylene upcycling combines three catalysts to affect two different actions: non-processive scission at chain ends, and scission at random interior points. We show that population balance equations (PBEs) based on the local density approximation (LDA) accurately describe the end-scission chemistry. We further show that the model can be simplified to a first-order PBE when started from a realistic molecular weight distribution. The simplification enables formulation and solution of a model that includes both end-scission and random-scission modalities. The mixture of catalysts (in theory) can exhibit a quantitative synergy, e.g., with the total number of cuts for the catalyst mixture exceeding that for the sum of its separate component catalyst actions. We develop equations to predict and optimize the synergistic acceleration.

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Introduction

Over 300 million tons of plastics are produced every year, yet less than 10% is recycled.^1–3 Most is incinerated, landfilled, or discarded into the natural environment. Chemical recycling processes, especially with catalysts to guide selectivity, have the potential to create high value chemicals from plastic waste.^4–17 Different catalysts for polymer upcycling operate in different ways: end scission or random scission, heterogeneous or homogeneous, processive or non-processive, etc. Processes that use a mixture of catalysts may even exploit synergies between different catalyst modalities.^18–26

Current research in catalytic polymer upcycling is mainly empirical,^27–40 with only a small fraction of studies attempting theoretical analysis of the underlying reaction mechanisms and reaction progress.^41–49 Based on an extrapolation from other branches of catalysis,^50–62 theoretical models and calculations will likely have a major impact on polymer upcycling, from understanding the chemistry, to developing new catalysts, to industrial process design. However, polymer upcycling presents several unique challenges for theory and computation.⁴⁹ The polymer mixture typically contains chains with millions of different molecular weights having different degrees and types of functionalization.

Population balance equations (PBEs) are analogous to the more familiar species balance equations (often called rate equations), but PBEs describe a continuous distribution rather than a short discrete list of reagents.^63,64 Several recent studies have used PBEs to model the evolution of molecular weight distributions (MWDs) and generation of products during catalytic polymer upcycling processes.^45,49

This study is motivated by recent experiments that combine polyethylene with excess ethylene and three catalysts: a double-bond isomerization catalyst, a metathesis catalyst, and a dehydrogenation catalyst.^18,20 The first two catalysts cooperate to affect end-scission via a complex isomerizing ethenolysis reaction. In brief, this reaction uses ethylene to excise CH₂ units from the ends of vinyl terminated polyethylene chains, leaving a propylene product and a shortened chain. Prior to the experimental works, Guironnet and Peters described the mechanism and modeled the double-bond isomerization and metathesis kinetics in isomerizing ethenolysis.⁴⁵ In the subsequent experiments, the third dehydrogenation catalyst was added to create double bonds at random interior locations.^18,20 The double bonds allow the chain to also be cut at random locations upon metathesis with ethylene. The random-scission events should result in more chains with double bonds at both ends, and thereby accelerate the isomerizing ethenolysis reaction.

At present, the isomerizing ethenolysis and dehydrogenation catalysts do not work together as intended.^18,20 However, end scission and random scission are common catalysis motifs, and nature has already combined them. For example, fungi use a cocktail of cellulases to depolymerize cellulose, with random scission by the endocellulases creating new chain ends which are then attacked by exocellulases. In the cellulase literature, the synergy between these enzymes was anticipated in many studies^65–68 and several works modeled the enzymatic reactions with discrete rate equations and population balance equations.^69–72 One population balance study demonstrated the synergy between end-scission and random-scission enzymes, although no recipe for the synergy calculation was provided.⁷²

In this work, we revisit the second-order PBE developed by Guironnet and Peters.⁴⁵ Using numerical solutions to the complete set of double-bond isomerization and metathesis rate equations as a standard, we show that the full PBE, as well as a simplified first-order version of the PBE, both yield accurate solutions for realistic initial MWDs. Then we combine the first-order model for end scission with birth and death terms in the PBE for random scission. We provide a new analytic solution to the combined PBE for end scission and random scission and predict the synergistic acceleration due to the combination of random-scission and end-scission catalysts. Finally, we discuss how these results might be used to optimize the catalyst mixture to achieve the maximum acceleration.

Revisiting the isomerizing ethenolysis model

Guironnet and Peters presented a kinetic model for polymer upcycling with tandem double bond isomerization and olefin metathesis catalysts.⁴⁵ They started with rate equations for all olefin species, and used pseudo-steady state approximations (PSSAs) and local density approximations (LDAs), to convert the massive ODE system into one Fokker–Planck-type equation:withandHere κ is the ratio of the metathesis rate over the isomerization rate and k_D is the isomerization rate, i.e., the frequency at which a double bond changes position on the olefin chain. In eqn (2) and (3), ṅ represents the chain shortening rate, and D_n represents the rate at which dispersity grows in the polymer distribution. Eqn (1) is different from typical PBEs because it does not invoke the coefficients as adjustable phenomenological parameters. Rather, the coefficients are expressions derived from details of the underlying mechanism. Detailed derivations of (1)–(3) as well as the starting rate equations, can be found in Guironnet and Peters.

There are several methods for modeling end-scission depolymerizations, including numerical methods and analytic solutions⁴⁴ for discrete rate equations^73,74 and for PBEs.^75–77 Guironnet and Peters compared numerical solutions for the complete set of double-bond isomerization and ethene metathesis rate equations to analytic solutions from the simplified PBE in eqn (1)–(3).

Guironnet and Peters focused on a sharp monodisperse initial MWD. They found that the PBE yielded highly accurate solutions for κ > 1, but less accurate solutions for κ < 1. In Fig. 1 we similarly compare numerical solutions to (analytic) PBE solutions, but we start with a more realistic polydisperse initial MWD. Note that the calculated MWDs from eqn (1) do not include the propylene products.

Fig. 1 Comparison between exact solution from rate equations and solutions of eqn (1) for (a) κ = 0.1; (b) κ = 1 (c) κ = 10.

The accuracy of eqn (1) decreases when olefin metathesis is slower than double bond isomerization, i.e., when κ < 1. In the derivation of eqn (1), the LDA ignores gradient [∂ρ/∂n] terms when applying the PSSA to the distribution of double bond positions. If we include gradient terms when applying the PSSA equations, the resulting PBE becomes nonlinear and no longer admits analytical solutions.

An analysis of error contributions suggests that the average “drift velocity” term, i.e., that containing ṅ, contributes negligible errors. The inaccuracies for κ < 1 are entirely due to an exaggerated increase in polydispersity, i.e., from an overestimation of D_n.

We can estimate the importance of polydispersity errors as follows. The predicted variance in the MWD over time can be calculated by the mean square displacement: Δn²(t) = Δn²(0) + 2D_nt,^78,79 where Δn²(t) is the variance of the MWD at time t. The time required to consume chains of an average initial length M_n(0) is:

Therefore, we have Δn_max² = Δn²(0) + 2D_nM_n(0)/ṅ where both D_n and ṅ can be written in terms of κ using eqn (2) and (3). The result gives an estimate for the variance in the MWD as the reaction approaches completion:When the added variance from the second derivative term [((4 + κ)/κ)^1/2M_n(0)] is smaller than the initial variance [Δn²(0)], it makes little difference whether the second derivative term is included or neglected. That is, for the ratio [Δn_max²/Δn²(0)], when it is close to 1, the dispersion term does not affect the results, and when it is much larger than 1, then the dispersion term becomes very important to the PBE. Indeed, for realistic initial MWDs the second derivative term may be omitted entirely. We illustrate this in Table 1:

Table 1 Inaccuracies emerging from the second order (2°) term in the LDA-derived PDE. The 2°-term is important for small initial molecular weights, but inconsequential for realistic PE materials with broad MWDs

	Parameters	Δnmax^2/Δn^2(0)	Dispersion from 2°-term
Fig. 1a	M n (0) = 200	6.78	Very important. Small errors matter
	Δn(0) = 15
	κ = 0.1
Fig. 1c	M n (0) = 200	2.04	Moderately important. OK to make small errors
	Δn(0) = 15
	κ = 10
Conk et al.	M n (0) = 3276	1.00	Unimportant. Can entirely omit 2°-term
	Δn(0) = 2000
	κ = 0.1

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Note that the MWD can be scaled with the mass of CH₂ to recover the distribution of chain concentrations, and that all calculations in Table 1 are performed in the number of carbons instead of Daltons for simplicity. The analysis in Table 1 suggests that, for initial MWDs like those from commercial plastics, the second-order term in eqn (1) can be entirely neglected.

In Fig. 2 we let ρ₀(n) be the initial MWD from the experiments of Conk et al. on isomerizing ethenolysis of polyethylene.²⁰Fig. 2a solves the equation using the LDA-derived PBE as written in eqn (1). Fig. 2b presents the solution for the same initial conditions and same PBE, but with the second-order term omitted. In this case, the PBE becomes:

and the solution (e.g., from the method of characteristics) is:

ρ(n, t) = ρ₀(n − ṅt) (7)

Note that the mass is not conserved because the propylene products are not included in ρ(n, t).

Fig. 2 (a) Predicted MWD evolving with time according to eqn (1) with κ = 1. (b) Predicted MWD evolving with time from eqn (6) with κ = 1.

There is no apparent difference between Fig. 2a and b, which confirms our prediction that the first order term in the LDA-derived PBE is sufficient for predicting the MWD evolution with realistic initial MWDs. We emphasize that this conclusion should be generally applicable to any end-scission upcycling mechanism, and not specific to isomerizing ethenolysis.

Tandem end scission and random scission

In this section, we build a model for tandem end-scission and random-scission catalysts. End scission results in a new chain of length that is slightly smaller than that of the parent chain. Because the changes in molecular weight from end scission are small, the result is a gradual drift-like term in the PBE. In contrast, random scission results in new chains of completely different lengths from the parent chain. Thus, random scission must be modeled using birth and death terms.⁸⁰ The combined result of both end scission and random scission is the PBE:Here ṅ is the rate of chain shortening on the end as in eqn (2), and thus will be a negative value. And k_c is the rate of random scission per carbon–carbon bond. The birth and death terms on the right-hand side of eqn (8) are explained in the references.^49,80 The combined equation was also obtained in work on end-scission and random-scission by cellulases.⁷²

To enable a systematic analysis of synergy, we pursue a nondimensionalization that balances end-scission and random-scission contributions. First, we nondimensionalize chain length using the initial mean chain length of the polymer, that is:

We refer to N as the relative chain length throughout the remainder of the manuscript. Correspondingly, we have:Then, we define the nondimensional time as the real time multiplied by the sum of two contributions: the frequency of new chain creation by a random scission and the frequency of entire chain digestion via end scission:Now, we define a new term α:which can be interpreted as the fraction of chains at the initial average length that get cut at a random location before complete digestion from the end.

With these definitions, the nondimensional form of eqn (8) becomes:

We solved eqn (13) numerically over nondimensional time τ by first converting the equations into a series of ODEs. Then, using a normal distribution as the initial condition, we use Scipy v1.4.1 (ref. 81) to solve the initial value problem. The backward differentiation formula (BDF) is chosen as the ODE solver for stability and convergence. The solutions are shown in Fig. 3a and b for the limiting cases where there is only random scission (ṅ = 0 and α = 1) and only end scission (k_c = 0 and α = 0), respectively. Fig. 3c shows the solutions for a case where both catalysts work together at ratio α = 0.95. The effects of random cuts are most pronounced when nearly all chains get randomly cut before they are consumed by end-scission.

Fig. 3 Time evolution of the relative chain length distribution for three cases. (a) Only random-scission, i.e., α = 1; (b) only end-scission, i.e., α = 0; (c) both end- and random-scission with α = 0.95. All three calculations begin from a normal distribution in relative chain length N with a mean of 1.0 and standard deviation of 0.075.

We also developed an analytical solution for eqn (13) using the strategy from Ziff and McGrady,⁸²

whereTo check the derivation, we confirmed that the analytic solution matches the numerical solutions. See ESI† for detailed derivations and results.

The solutions in Fig. 3 resemble those from analysis of combined end-scission and random-scission of cellulose by cellulase enzymes. In the following section, we exploit the non-dimensionalization to provide an expression that quantifies synergy of the end-scission and random-scission catalysts.

Quantifying synergy with cutting rates

When both end-scission and random-scission rates are nonzero, the number of chain ends will increase with time due to random scission. If excess end-scission catalyst is available, the overall rate of conversion should increase with time. To quantify the number of cuts, we first decompose ṅ as:

ṅ = k_EΔn (16)

where k_E is the number of end-scission events per time per chain, and Δn is the segment size per end scission. For isomerizing ethenolysis, Δn = 1. The rate of end scission is:Similarly, the rate of random scission is:r_E and r_R can be normalized by k_cM_n(0) + ṅ/M_n(0) and then written in terms of the dimensionless α. With these simplifications, the total (normalized) scission rate, with both random- and end-scission events occurring at the same time, is:Now to quantify the synergistic acceleration of the mixed catalyst system, we compare the rate in eqn (17) to that for limiting cases with end scission or random scission occurring separately.

For a system with only end scission, at the same physical time t, define the nondimensional time τ₀ as:

Then we solveto predict the time evolution of the molecular weight distribution ρ⁰_N(N, τ₀) with only end scission. Note that we are not actually changing the rate parameters for end or random scission relative to those in eqn (17). Rather, we are solving for the molecular weight evolution and scission rates as though the two scission mechanisms are occurring separately.

For end scission alone, the normalized scission rate is:

For the system with only random scission, we define τ₁ as:

τ₁ = k_cM_n(0)t = ατ (23)

Then we solveto predict the time evolution of the molecular weight distribution ρ¹_N(N, τ₁) with only random scission. Now the normalized rate with random scission alone isTo compare the rate with both catalysts operating simultaneously (eqn (17)) to the rates of their separate actions, we add the rates in eqn (22) and (23). In other words, we can directly compare r_E + r_R to r⁰_E + r¹_R.

We define the synergistic acceleration as η(α, τ) = (r_E + r_R)/(r⁰_E + r¹_R). From eqn (19), (22), and (25), the synergistic acceleration is a function of α, τ, and M_n(0)/Δn.

When α = 0.0 or α = 1.0, the synergistic acceleration factor η → 1.0. For all intermediate values it initially climbs to a value larger than unity. At long times, as the reaction completes, the acceleration factor drops below unity because the mixed catalyst system more rapidly reaches a point where all chains have been cut to propylene and butadiene.

Note that derivations from eqn (16)–(26) assume cutting rates are only determined by the number of chains and C–C bonds with no limitation on the catalyst amount. Situations where the rate of end scission becomes limited by the availability of catalyst will require additional analysis.

Fig. 4 plots η as a function of dimensionless time and for three different values of the initial chain length parameter M_n(0)/Δn. The first initial chain length parameter, M_n(0)/Δn = 40, corresponds approximately to the lengths of chains at the onset of isomerizing ethenolysis in the work of Conk et al.²⁰ The chains are unusually short because they are randomly dehydrogenated before the isomerizing ethenolysis step. Therefore, the chains of Conk et al. undergo metathesis with ethane to give short chains very early in the isomerizing ethenolysis process.

Fig. 4 Synergistic acceleration η vs. dimensionless time at different α values with (a) M_n(0)/Δn = 40; (b) M_n(0)/Δn = 200; (c) M_n(0)/Δn = 1000.

Results for the other initial chain length parameters in Fig. 4, M_n(0)/Δn = 200 and M_n(0)/Δn = 1000, show that the synergistic acceleration becomes a relatively weak function of chain length for long chains.

We can see that all mixtures of end-scission and random-scission catalysts initially accelerate beyond the rate of the limiting cases (α = 0 and α = 1). η first increases due to the newly generated chain ends from random scissions, and all the new chains are longer than the minimal cutting length. Then, as these chains reach the minimum length, η starts to decrease, ultimately dropping to 0, which indicates the complete consumption of chains.

The results in Fig. 4 allow us to design the depolymerization process by varying α to maximize the cutting rate. Interestingly, the α value that maximizes the synergistic acceleration seems to be related to M_n(0)/Δn. For M_n(0)/Δn = 40, the maximum synergy occurs for α = 0.98. For M_n(0)/Δn = 200, the maximum synergy occurs for α = 0.995, as shown in the ESI.† For M_n(0)/Δn = 1000, the maximum synergy occurs for α = 0.999, as shown in the ESI.† In each case, the value of α that maximizes the synergy is approximately α = 1 − Δn/M_n(0). This design rule should be regarded as a conjecture, because as yet we cannot say whether it remains valid for all values of M_n(0)/Δn nor provide a justification for its validity.

Conclusion

For realistic initial molecular weight distributions (MWDs), this paper shows that a previous kinetic model for isomerizing ethenolysis by Guironnet and Peters can be dramatically simplified to the form of a first-order population balance equation (PBE). Motivated by recent experiments, which demonstrated the isomerizing ethenolysis chemistry and also explored the effects of an additional dehydrogenation catalyst, we extended the kinetic model to cases with mixed end-scission and random-scission catalysts.

A dimensionless parameter α that emerges from the model quantifies the relative rates of end scission and random scission. We provided an analytic solution to the combined end-scission and random-scission equation for any α. We compare results from the two limiting cases of random-scission and end-scission to intermediate cases where both catalysts are working together. We show that, given sufficient end-scission catalyst, random scission causes a proliferation of new chain ends and thereby accelerates the overall depolymerization progress.

We developed a mathematical expression for the synergistic acceleration factor (η), i.e., the rate relative to the combined rate from the two independently occurring processes. η depends on the initial average molecular weight, on the reaction time, and on the value of α, i.e. the fraction of chains at the initial average length that get cut at a random location before they are completely digested from the chain ends. As time advances, η begins at 1.0 (no acceleration), and gradually climbs to a maximum value that depends on α and the initial molecular weight. The results should be useful in understanding and optimizing the mixture of catalysts to achieve the maximum acceleration. Specifically, our analysis suggests that the value of α (and accordingly the catalyst mixture) that maximizes the synergy is approximately α = 1 − Δn/M_n(0).

Author contributions

ZC was responsible for writing – original draft and editing, methodology, computational work, and data analysis. EE developed the analytical solution of PBE. BP was responsible for conceptualization, funding acquisition, project administration, supervision, and writing – review and editing.

Conflicts of interest

The authors declare no conflict of interest.

Acknowledgements

We thank members of the iCOUP team, Charles Sing, and Damien Guironnet for helpful discussions. This work was supported by the Institute for Cooperative Upcycling of Plastics (iCOUP), an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, via subcontract from Ames National Laboratory. Ames National Laboratory is operated for the DOE by Iowa State University under Contract No. DE-AC02-07CH11358.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d3re00390f

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