# Towards a Kinetic Modeling of the Changes in the Electrical Properties of Cable Insulation during Radio-Thermal Ageing in Nuclear Power Plants. Application to Silane-Crosslinked Polyethylene

^{1}

^{2}

^{*}

## Abstract

**:**

^{−1}) at different temperatures (21, 47, and 86 °C). The changes in the physico-chemical and electrical properties of Si-XLPE throughout its exposure were determined using Fourier transform infrared spectroscopy coupled with chemical gas derivatization, hydrostatic weighing, differential scanning calorimetry, dielectric spectroscopy and current measurements under an applied electric field. From a careful analysis of the oxidation products, it was confirmed that ketones are the main oxidation products in Si-XLPE. The analytical kinetic model for radio-thermal oxidation was thus completed with relatively simple structure–property relationships in order to additionally predict the increase in density induced by oxidation, and the adverse changes in two electrical properties of Si-XLPE: the dielectric constant ${\mathsf{\epsilon}}^{\prime}$ and volume resistivity R. After having shown the reliability of these new kinetic developments, the lifetime of Si-XLPE was determined using a dielectric end-of-life criterion deduced from a literature compilation on the changes in R with ${\mathsf{\epsilon}}^{\prime}$ for common polymers. The corresponding lifetime was found to be at least two times longer than the lifetime previously determined with the conventional end-of-life criterion, i.e., the mechanical type, thus confirming the previous literature studies that had shown that fracture properties degrade faster than electrical properties.

## 1. Introduction

^{−2}and 10

^{−1}Gy·h

^{−1}) and temperature (between 30 and 50 °C), polymer insulation is expected to perish because of oxidation both initiated by the polymer radiolysis and the thermal decomposition of POOH [6,7,8]. The pioneering researchers in this field were Gillen and Clough [9], who proposed a kinetic model for predicting the degradation of a PVC cable jacket that was exposed to radiation at a low ageing temperature in order to understand its surprisingly rapid ageing over 12 years under 0.25 Gy·h

^{−1}at 43 °C. However, this model fails to predict the predominance of thermal ageing at very low dose rates.

_{F}) of the hydroperoxide concentration, corresponding to the onset of the rapid auto-acceleration of the oxidation reaction when triggering the thermal initiation, typically: [POOH] = [POOH]

_{F}≈ 1.6 × 10

^{−1}mol·L

^{−1}[8].

^{5}Hz at room temperature), where the polymer initially displays an almost ideal insulation behavior [20,24,25]. These bands are the manifestation of a physical phenomenon called “orientation polarization” or “Debye polarization”, occurring when groups of atoms have a permanent dipole moment [24]. This is particularly the case for polar oxygenated groups, such as carbonyls and hydroxyls, which are easily oriented in the direction of the applied electrical field. That is the reason why, in initially nonpolar polymers, such as low-density (LDPE) and crosslinked low-density polyethylene (XLPE), oxidation leads to a significant increase in the dielectric constant ${\mathsf{\epsilon}}^{\prime}$ from typically $2.3\pm 0.3$ up to values higher than 4.5 [14,26,27,28,29,30,31], whereas the volume resistivity R dramatically drops from typically 10

^{16±2}Ω·cm to an asymptotic value of around 10

^{12±1}Ω·cm [31,32,33,34,35,36].

- According to Lorentz [37] and Lorenz [38], ${\mathsf{\epsilon}}^{\prime}$ would be related to the molar polarization P of a given dielectric material as follows:$$\mathrm{P}=\frac{{\mathsf{\epsilon}}^{\prime}-1}{{\mathsf{\epsilon}}^{\prime}+2}\mathrm{V}$$$${\mathsf{\epsilon}}^{\prime}=\frac{1+2\left(\mathrm{P}/\mathrm{V}\right)}{1-\left(\mathrm{P}/\mathrm{V}\right)}$$$$\mathrm{P}={{\displaystyle \sum}}^{}{\mathrm{P}}_{\mathrm{i}}\hspace{1em}\mathrm{and}\hspace{1em}\mathrm{V}={{\displaystyle \sum}}^{}{\mathrm{V}}_{\mathrm{i}}$$

- According to Darby [41], as electrical forces caused by polarizability and polar moment also determine the cohesive energy, a relationship should be expected between the dielectric constant ${\mathsf{\epsilon}}^{\prime}$ and the solubility parameter ${\mathsf{\delta}}_{\mathrm{sol}}$. Based on a literature compilation of the ${\mathsf{\epsilon}}^{\prime}$ and ${\mathsf{\delta}}_{\mathrm{sol}}$ values reported for common polymers, Darby proposed the following empirical proportionality:$${\mathsf{\delta}}_{\mathrm{sol}}\approx 7{\mathsf{\epsilon}}^{\prime}$$i.e.,$${\mathsf{\epsilon}}^{\prime}\approx 1.4\times {10}^{-1}{\mathsf{\delta}}_{\mathrm{sol}}$$It should be recalled that ${\mathsf{\delta}}_{\mathrm{sol}}$ is related to the molar attraction constant F as follows:$${\mathsf{\delta}}_{\mathrm{sol}}=\frac{\mathrm{F}}{\mathrm{V}}$$$$\mathrm{F}={{\displaystyle \sum}}^{}{\mathrm{F}}_{\mathrm{i}}\hspace{1em}\mathrm{and}\hspace{1em}\mathrm{V}={{\displaystyle \sum}}^{}{\mathrm{V}}_{\mathrm{i}}$$

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Radio-Thermal Ageing Conditions

^{60}Co γ-ray source at different temperatures. All the exposure conditions are summarized in Table 2. It should be noted that the ageing experiments numbered 1, 3 and 4 were performed at three distinct dose rates (8.5, 77.8 and 400 Gy·h

^{−1}, respectively) at low temperature close to ambient in order to investigate the effect of dose rate on the oxidation kinetics. In contrast, the ageing experiments numbered 1 and 2 were performed at almost the same dose rate (8.5 and 6.0 Gy·h

^{−1}, respectively) but at two different temperatures (47 and 86 °C) in order to investigate the effect of temperature on the oxidation kinetics.

#### 2.3. Experimental Characterizations

#### 2.3.1. Physico-Chemical Analyses

^{−1}with a Perkin Elmer FTIR Frontier spectrometer (Perkin Elmer, Villebon-sur-Yvette, France), after averaging the 16 scans that were taken at a resolution of 4 cm

^{−1}. For each film, at least three FTIR measurements were performed. However, because a large variety of hydroxyl and carbonyl products are formed during the radio-thermal ageing of Si-XLPE and their main IR absorption bands are often overlapped [46,47,48,49,50,51], FTIR spectroscopy was coupled with chemical gas derivatization, with ammonia (NH

_{3}) acting as the gaseous reagent. Indeed, NH

_{3}is well known for transforming carboxylic acids into carboxylates, and esters and anhydrides into primary amides, thus inducing a significant shift of the IR absorption bands of these two carbonyl products along the wavenumber axis [49]. This chemical deconvolution method has been extensively detailed in the literature for linear PE, for instance in references [49,51].

^{−1}) of the FTIR spectrum of Si-XLPE during its radio-thermal ageing in air under 77.8 Gy·h

^{−1}at 47 °C (a) before and (b) after NH

_{3}treatment. The subtraction of these two spectra, i.e., $\left(\mathrm{c}\right)=\left(\mathrm{b}\right)-\left(\mathrm{a}\right)$, facilitates the calculation of the carbonyl products consumed, including anhydrides (centered at about 1778 cm

^{−1}), linear esters (1740 cm

^{−1}) and carboxylic acids (1714 cm

^{−1}), and also the products formed during the NH

_{3}treatment, including amides (1670 cm

^{−1}) and carboxylates (1555 cm

^{−1}). In addition, the FTIR spectrum after NH

_{3}treatment (b) shows the carbonyl products that have not reacted with NH

_{3}, including cyclic esters (i.e., γ-lactones, centered at about 1773 cm

^{−1}), aldehydes (1736 cm

^{−1}) and ketones (1720 cm

^{−1}). The concentration of these different carbonyl products [P=O] was determined by applying the classical Beer–Lambert’s law:

^{−1}), OD is the optical density of its IR absorption band (dimensionless), $\mathsf{\epsilon}$ is its molar extinction coefficient (L·mol

^{−1}·cm

^{−1}), and ep is the film thickness (cm).

^{−1}(see Figure 2). The corresponding molar extinction coefficient is also reported in Table 3. In fact, the concentration of alcohols was deduced from the total concentration of hydroxyls by subtracting the concentrations of hydroperoxides and carboxylic acids.

^{−1}under a nitrogen flow of 50 mL·min

^{−1}.

^{−1}at 47 °C. As expected for a crosslinked polymer, the melting of the Si-XLPE occurred in a relatively wide temperature domain, typically ranged between 30 °C and 125 °C, with the maximum value of the main endothermic peak being located at around 114 °C. During the radio-thermal exposure, a gradual increase in the area under the main endothermic peak can be observed, thus indicating that an efficient chemi-crystallization process has occurred.

^{−1}[60].

#### 2.3.2. Electrical Measurements

^{−2}–10

^{6}Hz.

^{2}) were deposited on specimens using a plasma cold sputtering system. An electric field equal to 5 kV·mm

^{−1}was applied through a Keithley 2290E-5 DC power supply (Keithley Instruments, Cleveland, Ohio, USA). The current was recorded through a Keysight B2980A (Keyseight Technologies, Santa Rosa, California, USA) and the volume resistivity R (expressed in Ω·cm was finally obtained through the following equation:

^{−1}, E is the applied electric field in V·cm

^{−1}and J is the current density in A·cm

^{−2}.

## 3. Foundations of the Kinetic Model

^{−7}< I < 5.0 × 10

^{−1}Gy·s

^{−1}) at a low temperature close to ambient has been detailed in previous publications [6,7,8]. As a reminder, the main feature of this mechanistic scheme is that oxidation is initiated by both the polymer radiolysis (1R) and the thermal decomposition of POOH in bimolecular mode (1T):

- Initiation:
- (1R) PH + hν → P
^{•}+ ½H_{2}(${\mathrm{r}}_{\mathrm{i}}={10}^{-7}{\mathrm{G}}_{\mathrm{i}}\mathrm{I}$) - (1T) 2POOH → P
^{•}+ PO_{2}^{•}(k_{1})

- Propagation:
- (2) P
^{•}+ O_{2}→ PO_{2}^{•}(k_{2}) - (3) PO
_{2}^{•}+ PH → POOH + P^{•}(k_{3})

- Termination:
- (4) P
^{•}+ P^{•}→ Inactive products (k_{4}) - (5) P
^{•}+ PO_{2}^{•}→ Inactive products (k_{5}) - (6) PO
_{2}^{•}+ PO_{2}^{•}→ Inactive products + O_{2}(k_{6})

^{•}and PO

_{2}

^{•}designate an oxidation site, an hydroperoxide, alkyl and peroxy radicals, respectively. δ, λ, and μ are stoichiometric coefficients. r

_{i}, G

_{i}and k

_{j}(with j = 1, …, 6) are the radiochemical initiation rate, the radical yield and the rate constants, respectively.

- 1
- Oxidation is mainly initiated by the polymer radiolysis that occurs throughout the exposure (i.e., ${\mathrm{r}}_{\mathrm{i}}\gg 2{\mathrm{k}}_{1}{\left[\mathrm{POOH}\right]}^{2}$), with the thermal decomposition of POOH being an additional (but secondary) source of radicals for the long term;
- 2
- The radical species reach a steady-state regime from the early periods of the radio-thermal exposure (i.e., $\mathrm{d}\left[\mathrm{Rad}\right]/\mathrm{dt}=0$).

- The concentration of POOH:$$\left[\mathrm{POOH}\right]={\left[\mathrm{POOH}\right]}_{\infty}\frac{1-\mathrm{b}\mathrm{Exp}\left(-\mathrm{Kt}\right)}{1+\mathrm{b}\mathrm{Exp}\left(-\mathrm{Kt}\right)}$$$${\left[\mathrm{POOH}\right]}_{\infty}={\left(\frac{{\mathrm{k}}_{3}\left[\mathrm{PH}\right]}{2{\mathrm{k}}_{1\mathrm{b}}}{\left(\frac{{\mathrm{r}}_{\mathrm{i}}}{2{\mathrm{k}}_{6}}\right)}^{1/2}\frac{\mathsf{\beta}\mathrm{C}}{1+\mathsf{\beta}\mathrm{C}}\right)}^{1/2}$$$$\mathrm{K}=2{\left(2{\mathrm{k}}_{3}\left[\mathrm{PH}\right]{\mathrm{k}}_{1\mathrm{b}}{\left(\frac{{\mathrm{r}}_{\mathrm{i}}}{2{\mathrm{k}}_{6}}\right)}^{1/2}\frac{\mathsf{\beta}\mathrm{C}}{1+\mathsf{\beta}\mathrm{C}}\right)}^{1/2}$$$$\mathrm{b}=\frac{{\left[\mathrm{POOH}\right]}_{\infty}-{\left[\mathrm{POOH}\right]}_{\mathrm{ini}}}{{\left[\mathrm{POOH}\right]}_{\infty}+{\left[\mathrm{POOH}\right]}_{\mathrm{ini}}}$$
_{ini}and [POOH]_{∞}are the initial and steady concentrations of hydroperoxides, respectively. As for the weakly pre-oxidized samples, it is usually observed that: ${\left[\mathrm{POOH}\right]}_{\infty}\gg {\left[\mathrm{POOH}\right]}_{\mathrm{ini}}$ [51,63]. It can thus be considered that: b ≈ 1.

- The concentration of carbonyls:$$\begin{array}{c}\left[\mathrm{P}=\mathrm{O}\right]=\left[{\mathsf{\gamma}}_{1\mathrm{CO}}\frac{{\mathrm{k}}_{3}\left[\mathrm{PH}\right]}{2}{\left(\frac{{\mathrm{r}}_{\mathrm{i}}}{2{\mathrm{k}}_{6}}\right)}^{1/2}\frac{\mathsf{\beta}\mathrm{C}}{1+\mathsf{\beta}\mathrm{C}}+{\mathsf{\gamma}}_{6\mathrm{CO}}\frac{{\mathrm{r}}_{\mathrm{i}}}{2}{\left(\frac{\mathsf{\beta}\mathrm{C}}{1+\mathsf{\beta}\mathrm{C}}\right)}^{2}\right]\mathrm{t}\\ +2{\mathsf{\gamma}}_{1\mathrm{CO}}\frac{{\mathrm{k}}_{3}\left[\mathrm{PH}\right]}{\mathrm{K}}{\left(\frac{{\mathrm{r}}_{\mathrm{i}}}{2{\mathrm{k}}_{6}}\right)}^{1/2}\frac{\mathsf{\beta}\mathrm{C}}{1+\mathsf{\beta}\mathrm{C}}\left(\frac{1}{1+\mathrm{b}\mathrm{Exp}\left(-\mathrm{Kt}\right)}-\frac{1}{1+\mathrm{b}}\right)\end{array}$$

- The oxygen consumption:

_{O2}in the exposure environment according to the classical Henry’s law:

^{−8}mol·L

^{−1}·Pa

^{−1}regardless of the temperature [64]. As an example, in the case of an ageing in air under atmospheric pressure for which ${\mathrm{P}}_{\mathrm{O}2}=0.21\times {10}^{5}\mathrm{Pa}$, Equation (24) finally leads to: $\mathrm{C}=3.8\times {10}^{-4}\mathrm{mol}\xb7{\mathrm{L}}^{-1}$.

^{−1}corresponds to the critical value of the oxygen concentration ${\mathrm{C}}_{\mathrm{C}}$ above which oxygen excess is reached:

^{−1}) at low temperatures close to ambient (i.e., 47 and 21 °C). However, a poorer agreement was obtained under the lowest dose rate (i.e., 6.0 Gy·h

^{−1}) at the highest temperature (86 °C) because, in these critical radio-thermal exposure conditions, thermal initiation becomes of the same order of magnitude as (if not greater than) the radiochemical initiation. For context, the values of the different kinetic parameters used for these simulations have been recalled in Table 4.

## 4. Main Oxidation Products

^{−3}L·mol

^{−1}[51].

## 5. Calculation of the Changes in Density

- As recalled in Section 2.3, the incorporation of “heavy” atoms, such as oxygen, into a polymer structure initially containing many “light” atoms (i.e., carbon and hydrogen) induces an increase in its density [32,54,55,56,57,58]. Since crystals are impermeable to oxygen, oxidation only occurs in the amorphous phase where it thus induces an increase in ${\mathsf{\rho}}_{\mathrm{a}}$.
- In Si-XLPE, oxidation leads to a predominance of chain scissions over crosslinking [8]. Chain scissions progressively destroy the macromolecular network from which short linear fragments are extracted, which can easily migrate towards crystalline lamellae when the amorphous phase is in a rubbery state. The integration of these short fragments with crystalline lamellae induces a chemi-crystallization, i.e., a thickening of crystalline lamellae and an increase in the crystallinity ratios (i.e., ${\mathrm{X}}_{\mathrm{C}}$ and ${\mathrm{V}}_{\mathrm{C}}$).

_{CRU}and M

_{CRU}are the total number of atoms and the molar mass of the CRU, respectively. As an example, for unoxidized Si-XLPE: ${\mathrm{M}}_{\mathrm{CRU}\mathrm{ini}}=28\mathrm{g}\xb7{\mathrm{mol}}^{-1}$ and ${\mathrm{N}}_{\mathrm{CRU}\mathrm{ini}}=6$, so that: ${\mathrm{M}}_{\mathrm{a}\mathrm{ini}}=4.67\mathrm{g}\xb7{\mathrm{mol}}^{-1}$.

_{2}molecules chemically consumed per carbon atom (${\mathrm{n}}_{\mathrm{O}2}$). From these three quantities, two key ratios can be deduced in turn: ${\mathsf{\Delta}\mathrm{M}}_{\mathrm{a}}/{\mathsf{\Delta}\mathrm{n}}_{\mathrm{O}2}$ then ${\mathsf{\Delta}\mathrm{M}}_{\mathrm{a}}/{\mathrm{Q}}_{\mathrm{O}2}$. The calculation of these different properties has been detailed in Appendix A for when hydroperoxides are the main oxidation products (i.e., for hydroperoxidized PE). This calculation can easily be generalized to all other oxidation products. The corresponding results are reported in Table 6.

## 6. Prediction of Electrical Properties

_{O2}is expressed in mol·L

^{−1}:

^{−1}) at low temperatures close to ambient (i.e., 47 and 21 °C). However, a poorer agreement is obtained under the lowest dose rate (i.e., 6.0 Gy·h

^{−1}) at the highest temperature (86 °C) because, in these critical radio-thermal exposure conditions, thermal initiation becomes of the same order of magnitude as (if not greater than) the radiochemical initiation.

## 7. Proposal of an End-of-Life Criterion

_{2}, the following values of ${\mathsf{\epsilon}}^{\prime}$ and R were reported in the literature: ${\mathsf{\epsilon}}^{\prime}\approx 3.9$ [71] and $\mathrm{R}\approx {10}^{12}\mathsf{\Omega}\xb7\mathrm{cm}$ [72]. These are indeed of the same order of magnitude as the previous asymptotic values.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Evaluation of the Effect of Oxygen Consumption on the PE Density When Hydroperoxides Are the Main Oxidation Products

_{2}molecules per carbon atom is:

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**Figure 1.**Changes in the carbonyl and carboxylate region of the FTIR spectrum of Si-XLPE during its radio-thermal ageing in air under 77.8 Gy·h

^{−1}at 47 °C: (

**a**) before NH

_{3}treatment; (

**b**) after NH

_{3}treatment; (

**c**) subtraction of the two previous spectra to evidence the consumption of carbonyls and the formation of carboxylates during the NH

_{3}treatment.

**Figure 2.**Changes in the hydroxyl region of the FTIR spectrum of Si-XLPE during its radio-thermal ageing in air under 77.8 Gy·h

^{−1}at 47 °C.

**Figure 3.**Changes in the melting peak of Si-XLPE during its radio-thermal ageing in air under 77.8 Gy·h

^{−1}at 47 °C.

**Figure 4.**Density of amorphous phase versus oxygen consumption for Si-XLPE aged in the various radio-thermal environments under study.

**Figure 5.**Volume fraction of crystals versus oxygen consumption for Si-XLPE aged in the various radio-thermal environments under study.

**Figure 6.**Density versus oxygen consumption for Si-XLPE aged in the various radio-thermal environments under study.

**Figure 7.**Changes in the dielectric constant (normalized by its initial value ${\mathsf{\epsilon}}_{\mathrm{ini}}^{\prime}$) of Si-XLPE in the various radio-thermal environments under study. Comparison between simulation with Equation (50) (solid lines) and experimental data (symbols).

**Figure 8.**Dielectric absorption versus dielectric constant for Si-XLPE aged in the various radio-thermal environments under study.

**Figure 9.**Volume resistivity versus dielectric constant for Si-XLPE aged in the various radio-thermal environments under study.

Group | P_{i} (cm^{3}·mol^{−1}) | F_{i} (J^{1/2}·cm^{3/2}·mol^{−1}) | V_{i} (cm^{3}·mol^{−1}) |
---|---|---|---|

4.7 | 275 | 16.1 | |

10 | 560 | 10.8 | |

15 | 590 | 18 | |

15.8 | 825 | 28.5 | |

9.6 | 725 | 9 | |

14.8 | 925 | 12.8 |

Ageing No. | Dose Rate (Gy·h ^{−1}) | Dose Rate (Gy·s ^{−1}) | Temperature (°C) | Maximum Duration (h) | Maximum Dose (kGy) |
---|---|---|---|---|---|

1 | 8.5 | 2.36 × 10^{−3} | 47 | 12,800 | 109 |

2 | 6.0 | 1.67 × 10^{−3} | 86 | 16,267 | 98 |

3 | 77.8 | 2.16 × 10^{−2} | 47 | 3830 | 298 |

4 | 400 | 1.11 × 10^{−1} | 21 | 668 | 269 |

Oxidation Products | ν (cm^{−1}) | ε (L·mol^{−1}·cm^{−1}) | Reference for ε |
---|---|---|---|

Hydroxyls | 3420 | 70 | [51] |

Anhydrides | 1778 | 730 | [53] |

γ-Lactones | 1773 | 720 | [52] |

Esters | 1740 | 590 | [51] |

Aldehydes | 1736 | 270 | [51] |

Ketones | 1720 | 300 | [51] |

**Table 4.**Values of the kinetic parameters used for modeling the oxidation kinetics of Si-XLPE in the various radio-thermal environments under study [8].

T (°C) | 21 | 47 | 47 | 86 |

I (Gy·h^{−1}) | 400 | 77.8 | 8.5 | 6.0 |

G_{i} | 8 | 8 | 8 | 8 |

k_{1b} (L·mol^{−1}·s^{−1}) | 5.0 × 10^{−9} | 2.5 × 10^{−7} | 2.4 × 10^{−7} | 1.0 × 10^{−5} |

k_{2} (L·mol^{−1}·s^{−1}) | 10^{8} | 10^{8} | 10^{8} | 10^{8} |

k_{3} (L·mol^{−1}·s^{−1}) | 1.6 × 10^{−3} | 1.9 × 10^{−2} | 1.9 × 10^{−2} | 3.6 × 10^{−1} |

k_{4} (L·mol^{−1}·s^{−1}) | 8.0 × 10^{11} | 8.0 × 10^{11} | 8.0 × 10^{11} | 8.0 × 10^{11} |

k_{5} (L·mol^{−1}·s^{−1}) | 1.2 × 10^{10} | 7.0 × 10^{10} | 9.0 × 10^{10} | 2.4 × 10^{11} |

k_{6} (L·mol^{−1}·s^{−1}) | 5.0 × 10^{4} | 1.0 × 10^{6} | 2.0 × 10^{6} | 6.0 × 10^{7} |

γ_{1CO} (%) | 90 | 70 | 75 | 100 |

γ_{6CO} (%) | 90 | 70 | 75 | 100 |

**Table 5.**Relative proportions (expressed in mol%) of the different oxidation products in the Si-XLPE aged in the various radio-thermal environments under study.

T (°C) | 21 | 47 | 47 | 86 |

I (Gy·h^{−1}) | 400 | 77.8 | 8.5 | 6.0 |

[POOH] (mol%) | 16.2 | 19.9 | 15.9 | – |

[Alcohols] (mol%) | 7.3 | 4.0 | 6.6 | 2.7 |

[Anhydrides] (mol%) | 1.5 | 1.9 | 0.9 | 3.7 |

[γ-Lactones] (mol%) | 1.0 | 1.5 | 2.1 | 3.0 |

[Linear esters] (mol%) | 4.5 | 3.7 | 3.1 | 16.7 |

[Aldehydes] (mol%) | 11.5 | 14.4 | 15.4 | 25.1 |

[Ketones] (mol%) | 35.3 | 30.2 | 37.6 | 31.0 |

[Carboxylic acids] (mol%) | 22.7 | 24.3 | 18.4 | 17.8 |

**Table 6.**Constitutive repeating unit (CRU) for various oxidation products formed in PE. Corresponding values for the molar mass (${\mathrm{M}}_{\mathrm{CRU}})$ and the total number of atoms of the CRU (${\mathrm{N}}_{\mathrm{CRU}}$ ), the number of O

_{2}molecules chemically consumed per carbon atom (${\mathrm{n}}_{\mathrm{O}2}$ ) and three key ratios: ${\mathsf{\Delta}\mathrm{M}}_{\mathrm{a}}/{\mathsf{\Delta}\mathrm{n}}_{\mathrm{O}2}$, ${\mathsf{\Delta}\mathrm{M}}_{\mathrm{a}}/{\mathsf{\Delta}\mathrm{Q}}_{\mathrm{O}2}$ and ${\mathsf{\Delta}\mathsf{\rho}}_{\mathrm{a}}/{\mathsf{\Delta}\mathrm{Q}}_{\mathrm{O}2}$.

CRU | ${M}_{\mathrm{CRU}}(g\xb7{\mathrm{mol}}^{-1}\mathbf{)}$ | ${N}_{\mathrm{CRU}}$ | ${n}_{O2}$ | $\Delta {M}_{a}/\Delta {n}_{O2}(g\xb7{\mathrm{mol}}^{-1})$ | $\Delta {M}_{a}/\Delta {Q}_{O2}(g\xb7{\mathrm{cm}}^{3}\xb7{\mathrm{mol}}^{-2})$ | $\Delta {\mathit{\rho}}_{\mathit{a}}/\Delta {Q}_{O2}(g\xb7{\mathrm{mol}}^{-1})$ |
---|---|---|---|---|---|---|

14p + 32 | 3p + 2 | 1/p | 7.56 | 124.44 | 37.33 | |

14p + 16 | 3p + 1 | 1/2p | 7.56 | 124.44 | 37.33 | |

14p + 14 | 3p − 1 | 1/2p | 12.44 | 234.24 | 70.27 | |

14p + 15 | 3p | 1/2p | 10.00 | 175.69 | 52.71 | |

14p + 31 | 3p + 1 | 1/p | 8.78 | 149.23 | 44.77 | |

14p + 30 | 3p | 1/p | 10.00 | 175.69 | 52.71 |

**Table 7.**Comparison between the lifetimes determined using a dielectric or a mechanical end-of-life criterion for Si-XLPE in the various radio-thermal environments under study.

I (Gy·h^{−1}) | 400 | 77.8 | 8.5 | 6.0 |

T (°C) | 21 | 47 | 47 | 86 |

${t}_{F}\left({\mathsf{\epsilon}}^{\prime}\right)$ (days) | 67 | 184 | 629 | 289 |

${t}_{F}\left({\mathsf{\epsilon}}_{R}\right)$ (days) | 32 | 43 | 246 | - |

$\mathbf{Ratio}{t}_{F}\left({\mathsf{\epsilon}}^{\prime}\right)/{t}_{F}\left({\mathsf{\epsilon}}_{R}\right)$ | 2.1 | 4.3 | 2.6 | - |

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**MDPI and ACS Style**

Hettal, S.; Suraci, S.V.; Roland, S.; Fabiani, D.; Colin, X. Towards a Kinetic Modeling of the Changes in the Electrical Properties of Cable Insulation during Radio-Thermal Ageing in Nuclear Power Plants. Application to Silane-Crosslinked Polyethylene. *Polymers* **2021**, *13*, 4427.
https://doi.org/10.3390/polym13244427

**AMA Style**

Hettal S, Suraci SV, Roland S, Fabiani D, Colin X. Towards a Kinetic Modeling of the Changes in the Electrical Properties of Cable Insulation during Radio-Thermal Ageing in Nuclear Power Plants. Application to Silane-Crosslinked Polyethylene. *Polymers*. 2021; 13(24):4427.
https://doi.org/10.3390/polym13244427

**Chicago/Turabian Style**

Hettal, Sarah, Simone Vincenzo Suraci, Sébastien Roland, Davide Fabiani, and Xavier Colin. 2021. "Towards a Kinetic Modeling of the Changes in the Electrical Properties of Cable Insulation during Radio-Thermal Ageing in Nuclear Power Plants. Application to Silane-Crosslinked Polyethylene" *Polymers* 13, no. 24: 4427.
https://doi.org/10.3390/polym13244427