# Analytical and Numerical Modeling of Degradation and Pyrolysis of Polyethylene: Measuring Aging with Thermogravimetry

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Model

#### 2.1. Analytical Model for Cross-Linked Polymer

#### 2.2. Analytical Degradation Model

#### 2.3. Analytical Expression for Volatile Fraction

#### 2.4. Coarse-Grained Kinetics of Pyrolysis

## 3. Materials and Methods

#### 3.1. TGA Experiments with Aged XLPE

^{3}and 0.71, respectively. Thermo-radiative aging was performed in the Panoza facility at UJV Rez, Czech Republic, with a

^{60}Co γ-ray source. The average temperature was 47 °C, and the average dose rate was 77.8 Gy/h. TGA experiments were conducted at VTT under a nitrogen atmosphere using a heating rate of 10 K/min. Char yield was checked by switching to air atmosphere and heating the crucibles up to 1000 °C.

#### 3.2. Numerical Aging and Decomposition Model

_{x}monomer units and the edges are the chemical bonds between them. Here, only one vertex and edge type is considered, i.e., the effect of different local chemistries and bonds is disregarded. The structure of the network is stored in an adjacency list, which contains the bonded neighbors of each monomer. Random scission is performed by removing a random edge from the graph, and conversely, random cross-linking is performed by adding a new edge between two random vertices. The gel fraction is computed by identifying the largest connected component in the graph and counting the fraction of vertices contained by it. The cycle rank of the gel is calculated as the difference between the number of edges and number of vertices in the gel.

#### 3.3. Evaporation Limit

## 4. Results

#### 4.1. Model for Unaged XLPE

#### 4.2. Aging of XLPE

#### 4.3. Aging Characterization by TGA

_{2}atmosphere to higher temperatures due to antioxidants [34,35,36]. To reduce the number of fitting parameters, cross-link densities were inferred from known data and initial fitting results. For the unaged sample, a cross-link density of 0.195 mol/l was calculated using Equation (4) based on an initially fitted ${M}_{\mathrm{w},\mathrm{chains}}\approx 5640\mathrm{g}/\mathrm{mol}$ and the known gel fraction of 71% [32]. For the aged samples, we calculated the number of cross-links, assuming ${p}_{\mathrm{xl}}=0.2$. Judging from the elongation at break values in Hettal et al. [32], the gel is likely still present after 42 days of aging. Their kinetic model can be used to estimate the number of scissions at 42 days, giving 0.91 mol/l. From Figure 4, it can be seen that if there were no simultaneous cross-linking reactions (${p}_{\mathrm{xl}}=0$), the gel would be completely lost after 42 days. Assuming ${p}_{\mathrm{xl}}=0.2$ gives a 64% reduction in cycle count for the sample aged for 42 days and 82% reduction after 84 days, which seems roughly appropriate.

_{sc}at 450 °C for bond dissociation, cross-link density, and fraction of bonds scissed $S=\left({n}_{b}-{n}_{b,0}\right)/{n}_{b,0}$. Our kinetic parameters are of the same order as the theoretical estimates based on literature data presented in Section 2.4 and comfortably inside the rather wide range of reported literature values for pyrolysis of various polyethylene grades [37,38]. Finally, we note that the model curves were calculated considering 150 °C as the initial temperature, as the experimental data showed mass loss only after 150 °C, indicating the absence of species that would vaporize below 150 °C. As such species would be expected as a result of the aging reactions, we believe that the slightly elevated radiochemical aging temperature (47 °C) was sufficiently high to slowly evaporate anything that would vaporize below 150 °C in TGA.

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Celina, M. Review of polymer oxidation and its relationship with materials performance and lifetime prediction. Polym. Degrad. Stab.
**2013**, 98, 2419–2429. [Google Scholar] [CrossRef] - Celina, M.; Linde, E.; Brunson, D.; Quintana, A.; Giron, N. Overview of accelerated aging and polymer degradation kinetics for combined radiation-thermal environments. Polym. Degrad. Stab.
**2019**, 166, 353–378. [Google Scholar] [CrossRef] - Burnay, S. Degradation of Polymeric Components in Nuclear Power Applications. Energiforsk AB Report 2018:480. Available online: https://energiforsk.se/program/polymera-material-i-karnkraft/rapporter/degradation-of-polymeric-components-in-nuclear-power-applications-2018-480/ (accessed on 1 June 2022).
- Chabira, S.F.; Sebaa, M.; G’sell, C. Oxidation and Crosslinking Processes During Thermal Aging of Low-Density Polyethylene Films. J. Appl. Polym. Sci.
**2012**, 124, 5200–5208. [Google Scholar] [CrossRef] - Suraci, S.V.; Fabiani, D.; Roland, S.; Colin, X. Multi scale aging assessment of low-voltage cables subjected to radio-chemical aging: Towards an electrical diagnostic technique. Polym. Test.
**2021**, 103, 107352. [Google Scholar] [CrossRef] - Vahabi, H.; Sonnier, R.; Ferry, L. Effects of ageing on the fire behaviour of flame-retarded polymers: A review. Polym. Int.
**2015**, 64, 313–328. [Google Scholar] [CrossRef] - Matala, A.; Hostikka, S. Pyrolysis Modelling of PVC Cable Materials. Fire Saf. Sci.
**2011**, 10, 917–930. [Google Scholar] [CrossRef] - Wang, Z.; Wei, R.; Ouyang, D.; Wang, J. Investigation on thermal stability and flame spread behavior of new and aged fine electrical wires. J. Therm. Anal. Calorim.
**2020**, 140, 157–165. [Google Scholar] [CrossRef] - Weon, J.-I. Effects of thermal ageing on mechanical and thermal behaviors of linear low density polyethylene pipe. Polym. Degrad. Stab.
**2010**, 95, 14–20. [Google Scholar] [CrossRef] - Boersma, A. Predicting the efficiency of antioxidants in polymers. Polym. Degrad. Stab.
**2006**, 91, 472–478. [Google Scholar] [CrossRef] - Wei, X.-F.; Linde, E.; Hedenqvist, M.S. Plasticiser loss from plastic or rubber products through diffusion and evaporation. NPJ Mater. Degrad.
**2019**, 3, 18. [Google Scholar] [CrossRef][Green Version] - Xu, A.; Roland, S.; Colin, X. Thermal ageing of a silane-crosslinked polyethylene stabilised with a thiodipropionate antioxidant. Polym. Degrad. Stab.
**2020**, 181, 109276. [Google Scholar] [CrossRef] - Barbosa, A.P.C.; Fulco, A.P.P.; Guerra, E.S.S.; Arakaki, F.K.; Tosatto, M.; Costa, M.C.B.; Melo, J.D.D. Accelerated aging effects on carbon fiber/epoxy composites. Compos. B Eng.
**2017**, 110, 298–306. [Google Scholar] [CrossRef] - Pablos, J.L.; Abrusci, C.; Marín, I.; López-Marín, J.; Catalina, F.; Espí, E.; Corrales, T. Photodegradation of polyethylenes: Comparative effect of Fe and Ca-stearates as pro-oxidant additives. Polym. Degrad. Stab.
**2010**, 95, 2057–2064. [Google Scholar] [CrossRef] - Fayolle, B.; Colin, X.; Audouin, L.; Verdu, J. Mechanism of degradation induced embrittlement in polyethylene. Polym. Degrad. Stab.
**2007**, 92, 231–238. [Google Scholar] [CrossRef] - Nyden, M.R.; Forney, G.P.; Brown, J.E. Molecular Modeling of Polymer Flammability: Application to the Design of Flame-Resistant Polyethylene. Macromolecules
**1992**, 25, 1658–1666. [Google Scholar] [CrossRef] - Vyazovkin, S. Computational aspects of kinetic analysis. Part C. The ICTAC Kinetics Project—The light at the end of the tunnel? Thermochim. Acta
**2000**, 355, 155–163. [Google Scholar] [CrossRef] - Nemeth, A.; Blaszó, M.; Baranyai, P.; Vidóczy, T. Thermal degradation of polyethylene modeled on tetracontane. J. Anal. Appl. Pyrolysis
**2008**, 81, 237–242. [Google Scholar] [CrossRef] - Levine, S.E.; Broadbelt, L.J. Detailed mechanistic modeling of high-density polyethylene pyrolysis: Low molecular weight product evolution. Polym. Degrad. Stab.
**2009**, 94, 810–822. [Google Scholar] [CrossRef] - Gascoin, N.; Navarro-Rodriguez, A.; Fau, G.; Gillard, P. Kinetic modelling of High Density PolyEthylene pyrolysis: Part 2. Reduction of existing detailed mechanism. Polym. Degrad. Stab.
**2012**, 97, 1142–1150. [Google Scholar] [CrossRef][Green Version] - Voter, A. Introduction to the Kinetic Monte Carlo Method. In Radiation Effects in Solids; Springer: Dordrecht, The Netherlands, 2007; Volume 235, pp. 1–23. [Google Scholar]
- Gillespie, D.T. Exact Stochastic Simulation of Coupled Chemical Reactions. J. Phys. Chem.
**1977**, 81, 2340–2361. [Google Scholar] [CrossRef] - Galina, H.; Lechowicz, J.B. Monte-Carlo modeling of degradation of polymer networks. Polym. Gels Netw.
**1998**, 6, 103–111. [Google Scholar] [CrossRef] - Adema, K.N.S.; Makki, H.; Peters, E.A.J.F.; Laven, J.; Van der Ven, L.G.J.; Van Benthem, R.A.T.M.; De With, G. Kinetic Monte Carlo simulation of the photodegradation process of polyester-urethane coatings. Phys. Chem. Chem. Phys.
**2015**, 17, 19962–19976. [Google Scholar] [CrossRef] - Bystritskaya, E.V.; Karpukhin, O.N.; Kutsenova, A. Monte Carlo Simulation of Linear Polymer Thermal Depolymerization under Isothermal and Dynamic Modes. Int. J. Polym. Sci.
**2011**, 2011, 849370. [Google Scholar] [CrossRef][Green Version] - Vinu, R.; Levine, S.E.; Wang, L.; Broadbelt, L.J. Detailed mechanistic modeling of poly(styrene peroxide) pyrolysis using kinetic Monte Carlo simulation. Chem. Eng. Sci.
**2012**, 69, 456–471. [Google Scholar] [CrossRef] - Peterson, B.K.; Formolo, M.J.; Lawson, M. Molecular and detailed isotopic structures of petroleum: Kinetic Monte Carlo analysis of alkane cracking. Geochim. Cosmochim. Acta
**2018**, 243, 169–185. [Google Scholar] [CrossRef] - Younker, J.M.; Saito, T.; Hunt, M.A.; Naskar, A.K.; Beste, A. Pyrolysis Pathways of Sulfonated Polyethylene, an Alternative Carbon Fiber Precursor. J. Am. Chem. Soc.
**2013**, 135, 6130–6141. [Google Scholar] [CrossRef] - Charlesby, A. Gel formation and molecular weight distribution in long-chain polymers. Proc. R. Soc. A
**1954**, 222, 542–557. [Google Scholar] [CrossRef] - St. Cholakov, G.; Wakeham, W.A.; Stateva, R.P. Estimation of normal boiling points of hydrocarbons from descriptors of molecular structure. Fluid Phase Equilib.
**1999**, 163, 21–42. [Google Scholar] [CrossRef] - Da Cruz, M.; Van Schoors, L.; Benzarti, K.; Colin, X. Thermo-oxidative degradation of additive free polyethylene. Part I. Analysis of chemical modifications at molecular and macromolecular scales. J. Appl. Polym. Sci.
**2016**, 133, 43287. [Google Scholar] [CrossRef] - Hettal, S.; Roland, S.; Sipilä, K.; Joki, H.; Colin, X. A new analytical model for predicting the radio-thermal oxidation kinetics and the lifetime of electric cable insulation in nuclear power plants: Application to silane cross-linked polyethylene. Polym. Degrad. Stab.
**2021**, 185, 109492. [Google Scholar] [CrossRef] - Colin, X.; Monchy-Leroy, C.; Verdu, J. Effect of gamma irradiation on tensile properties of low molecular weight polyethylene samples. Radiat. Phys. Chem.
**2011**, 80, 895–901. [Google Scholar] [CrossRef][Green Version] - Peltzer, M.; Wagner, J.R.; Jiménez, A. Thermal characterization of UHWMPE stabilized with natural antioxidants. J. Therm. Anal. Calorim.
**2007**, 87, 493–497. [Google Scholar] [CrossRef] - Busolo, M.A.; Lagaron, J.M. Antioxidant polyethylene films based on a resveratrol containing Clay of Interest in Food Packaging Applications. Food Packag. Shelf Life
**2015**, 6, 30–41. [Google Scholar] [CrossRef] - Kabir, A.S.; Li, H.; Yuan, H.; Kuboki, T.; Xu, C. Effects of de-polymerized lignin content on thermo-oxidative and thermal stability of polyethylene. J. Anal. Appl. Pyrolysis
**2019**, 140, 413–422. [Google Scholar] [CrossRef] - Westerhout, R.W.J.; Waanders, J.; Kuipers, J.A.M.; van Swaaij, W.P.M. Kinetics of the Low-Temperature Pyrolysis of Polyethylene, Polypropene, and Polystyrene Modeling, Experimental Determination, and Comparison with Literature Models and Data. Ind. Eng. Chem. Res.
**1997**, 36, 1955–1964. [Google Scholar] [CrossRef][Green Version] - Snegirev, A.Y.; Talalov, V.A.; Stepanov, V.V.; Korobeinichev, O.P.; Gerasimov, I.E. Autocatalysis in thermal decomposition of polymers. Polym. Degrad. Stab.
**2017**, 137, 151–161. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Comparison of numerical and analytical TGA curves. Numerical curve is for a system of 10

^{5}monomers with 700 chains and 615 cross-links (corresponding to ${M}_{\mathrm{w}}=4000\mathrm{g}/\mathrm{mol}$, ${\rho}_{\mathrm{xl}}=0.4$ mol/l). Analytical curves are for the same system with and without crosslinks. (

**b**) Effect of dispersity on the numerical TGA curves. Results are shown for dispersity of 2 sampled from both log-normal and Flory–Schulz distributions with the same ${M}_{w}$, and dispersity of 4 sampled from log-normal distribution. Here, the scission rate ${r}_{\mathrm{sc}}$ is taken to follow an Arrhenius law with the constants $A=1.00\times {10}^{14}{\mathrm{s}}^{-1}$ and $E=248\mathrm{kJ}/\mathrm{mol}$. Numerical results are averaged over 100 runs.

**Figure 2.**(

**a**) Weight average molecular weight and dispersity of the system as a function of the number of random bond scissions applied to an initial single chain consisting of 10

^{6}monomers. The results are averaged over 100 runs. Shading represents standard deviation. (

**b**) Initial molecular weight distributions as averages over 100 runs. Solid lines are Flory–Schulz distributions for 0.05% (56,000 g/mol) and 0.5% (5600 g/mol) of initial bonds scissed.

**Figure 3.**Evolution of the gel fraction as a function of the number of cross-links per chain for initially linear PE systems. Numerical results are averaged over 100 runs. Shading represents standard deviation.

**Figure 4.**Evolution of (

**a**) gel fraction and (

**b**) cycle rank as a function of aging reactions with different values for ${p}_{\mathrm{xl}}$. Initial XLPE has ${M}_{\mathrm{w},\mathrm{chains}}=5600$ g/mol and the total number of monomers is 10

^{6}. Solid and dashed lines are results from numerical and analytical models, respectively. Numerical results are averaged over 100 runs. Shading represents standard deviation.

**Figure 5.**TGA curves. (

**a**) Solid lines show experimental TGA curves, while dashed lines show the fitted model curves. (

**b**) Differentiated experimental TGA curves.

Aging Time d | M_{w,chains}g/mol | A 1/s | E kJ/mol | r_{sc} @ 450 °Cmol/L/s | ρ_{xl}mol/L | S |
---|---|---|---|---|---|---|

0 | 5630 | 1.00 × 10^{16} | 276 | 1.09 × 10^{−4} | 0.195 | 0 |

42 | 2390 | 1.16 × 10^{15} | 262 | 1.45 × 10^{−4} | 0.283 | 0.0067 |

84 | 1500 | 1.16 × 10^{15} | 262 | 1.45 × 10^{−4} | 0.372 | 0.0137 |

126 | 1220 | 1.16 × 10^{15} | 262 | 1.45 × 10^{−4} | 0.460 | 0.0180 |

168 | 903 | 1.16 × 10^{15} | 262 | 1.45 × 10^{−4} | 0.548 | 0.0260 |

210 | 724 | 1.16 × 10^{15} | 262 | 1.45 × 10^{−4} | 0.637 | 0.0337 |

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**MDPI and ACS Style**

Verho, T.; Vaari, J.
Analytical and Numerical Modeling of Degradation and Pyrolysis of Polyethylene: Measuring Aging with Thermogravimetry. *Polymers* **2022**, *14*, 2709.
https://doi.org/10.3390/polym14132709

**AMA Style**

Verho T, Vaari J.
Analytical and Numerical Modeling of Degradation and Pyrolysis of Polyethylene: Measuring Aging with Thermogravimetry. *Polymers*. 2022; 14(13):2709.
https://doi.org/10.3390/polym14132709

**Chicago/Turabian Style**

Verho, Tuukka, and Jukka Vaari.
2022. "Analytical and Numerical Modeling of Degradation and Pyrolysis of Polyethylene: Measuring Aging with Thermogravimetry" *Polymers* 14, no. 13: 2709.
https://doi.org/10.3390/polym14132709