On the Use of Oxidation Induction Time as a Kinetic Parameter for Condition Monitoring and Lifetime Evaluation under Ionizing Radiation Environments

The durability of polymeric materials is closely linked to their degradation under specific operating conditions when different stressors—general or specific, such as high temperature, sunlight or ionizing radiation, solvents, or mechanical stresses—act simultaneously, causing degradation. In the case of electrical cables, the durability of the electrically insulating materials used in their construction is an important parameter to ensure their operational security. In this work, we studied the degradation state of various types of electrical insulating materials from cables used in particle acceleration systems under European Organization for Nuclear Research (CERN) conditions (e.g., Super Proton Synchrotron, SPS) as a function of time and irradiation dose. A simple kinetic model was proposed based on the exponential decrease in the antioxidant amount in polymeric insulations. The onset oxidation time (OIT) values, used as an indicator of antioxidant concentration, were obtained from isothermal differential scanning calorimetry (DSC) and chemiluminescence (CL) measurements. Fourier transform infrared (FTIR) measurements were used to assess the degradation state and identify polymeric materials. The practical applicability of such a model in diagnosing degradation and in the subsequent evaluation of the remaining service life is of interest, as it can be adapted to a broad range of operating conditions and materials.


Introduction
Condition monitoring and the lifetime evaluation of different products that incorporate polymer materials are important research directions for applications in electrical engineering and related fields. Because of their organic structure, most polymer materials are quasi-unstable under environment-specific exploitation conditions, e.g., elevated temperatures, solar radiation, electric fields, ionizing radiations, mechanical stress, etc. Most real cases include the presence of atmospheric oxygen and combinations of the factors mentioned above in varying proportions, which can complicate predictions. Therefore, significant differences would be frequently observed between the laboratory and real cases because of differences between the simulated laboratory conditions and the real world, e.g., different temperature ranges or cycles, different dose rate values, or combined action of different stress factors during specific periods [1][2][3].
The lifetime of polymeric materials would not be so intensely studied if the condition of the polymeric materials did not decisively influence the functionality of the more complex aggregates that incorporate these materials, such as electrical cables, pipes carrying material necessary for such periodic mechanical tests. Moreover, the amount of available material sample from a given aged item is frequently very low, and the space for whiteness materials is also drastically limited. Hence, monitoring and evaluating a cable's residual lifetime in situ becomes essential in the development of safety strategies [26].
Outstanding pioneering work in lifetime estimation of materials for nuclear power plants was done by Clogh and Gillen, who proposed the use of accelerating factors for aging under multiple stress conditions (temperature and radiation) within the time-temperaturedose rate superposition methodology (see for example [27]), as well as by Schonbacher and Tavlet at CERN, who proposed initial qualification of different materials (based on mechanical tests) for use in radiation environments [28,29].
More elaborated kinetic models are based on kinetic constants related to the specific degradation mechanisms [13,30]. As such models include rate constants of the processes involved in degradation, they are very accurate in lifetime prediction. However, owing to the many approximations necessary to simplify the calculations, such as neglecting or considering simplified equations for stabilizer evaporation or for the temperature dependence of evaporation rate from a polymer matrix, the concordance with real aging cases is sometimes unsatisfactory. The influence of the polymer molecular structure and morphology on either the degradation mechanism or additive compatibility could be another cause of the inconsistency of the models relative to real cases. Other limiting factors in the development of both reliable and simple kinetic models include input data, such as the values of some kinetic constants, which, if material-dependent (determined previously in time-consuming laboratory experiments), result in a material-dependent model because of the specific values of the coefficients and constants [31].
Besides the abovementioned problems, methods based on extrapolation could present further disadvantages, such as: (i) the oxidation mechanism is not constant over a wide temperature range [24]; (ii) the results of extrapolations performed over physical transition points (such as melting of crystallinity), as well as accelerated aging experiments at temperatures in the range of these transitions, may be affected by significant errors [24], as the oxidation mechanism changes as crystalline domains are formed and grow with decreased temperature; (iii) the occurrence of limited oxygen diffusion processes, which can result in significant deviations from the Arrhenius-type dependence [25].
This paper is devoted to the analysis of experimental data of isothermal DSC and chemiluminescence (CL) collected from samples of various materials aged under different stress conditions, mainly in radiation fields and air. The aim is to show the applicability of some simple kinetic considerations and a degradation model for ethylene-based polymer materials. The proposed kinetic model can evaluate the maximum irradiation dose and the residual lifetime of polymer materials for a cable in use. Furthermore, this kinetic model can be applied to determine the lifetime of polymer materials exposed to other environmental factors, such as elevated temperatures, UV exposure, natural weathering, etc.

Materials
The polyethylene-based materials used in this study, namely low-density polyethylene (LDPE), linear low-density polyethylene (LLDPE), high-density polyethylene (HDPE), crosslinked polyethylene (XLPE), ethylene-propylene-diene copolymer (EPR), and ethylene vinyl acetate copolymer (EVA), were obtained from two main sources: laboratory-available polymer samples and samples taken from various types of commercially available electric cables (instrumentation and control, high voltage and medium voltage). Most of the studied cables were used in CERN's particle acceleration systems (e.g., Super Proton Synchrotron (SPS) and Large Hadron Collider (LHC)). For the structure of the cables studied in this work, as well as for the nature of the incorporated materials, please see the Supplementary Material. The commercial names of the polymers used in cable fabrication, as well as their producers, cannot be disclosed for commercial reasons. However, all the cables and the polymers used in their construction were produced in Europe.

Irradiation
The materials discussed here were aged under different conditions, and their oxidation kinetics before and after exposure were compared. For example, the aging effect of radiation fields was studied on either material samples previously separated from the cables or on materials sampled from irradiated cables. Gamma 137 Cs irradiation was applied at ICPE-CA (dose rate: 0.3 kGy/h in either air or nitrogen as inert atmosphere), whereas 60 Co irradiation was applied at Ionisos (Dagneaux, France; dose rate: 1.5 kGy). Cable witnesses were exposed in pre-established positions along CERN SPS and LHC accelerators, and periodically, the irradiated witnesses were sampled for condition monitoring of the real cables installed in the respective area.

Characterization Methods
The oxidation induction time (OIT) was determined from DSC and chemiluminescence measurements under similar temperature conditions (isothermal mode). DSC measurements were performed with Setaram DSC 131 (Setaram, Caluire, France) at CERN and Setaram DSC 131 EVO (Setaram, Caluire, France) at CERN and INCDIE ICPE-CA for comparison studies. OIT from DSC was obtained according to [32]. Similar curves of signal vs. temperature were obtained with a Lumipol 3 (Polymer Institute of Slovak Academy of Sciences, Bratislava, Slovakia) chemiluminescence instrument with kinetic parameters of significance similar to those of DSC's ( Figure 1). cially available electric cables (instrumentation and control, high voltage and medium voltage). Most of the studied cables were used in CERN's particle acceleration systems (e.g., Super Proton Synchrotron (SPS) and Large Hadron Collider (LHC)). For the structure of the cables studied in this work, as well as for the nature of the incorporated materials, please see the Supplementary Material. The commercial names of the polymers used in cable fabrication, as well as their producers, cannot be disclosed for commercial reasons. However, all the cables and the polymers used in their construction were produced in Europe.

Irradiation
The materials discussed here were aged under different conditions, and their oxidation kinetics before and after exposure were compared. For example, the aging effect of radiation fields was studied on either material samples previously separated from the cables or on materials sampled from irradiated cables. Gamma 137 Cs irradiation was applied at ICPE-CA (dose rate: 0.3 kGy/h in either air or nitrogen as inert atmosphere), whereas 60 Co irradiation was applied at Ionisos (Dagneaux, France; dose rate: 1.5 kGy). Cable witnesses were exposed in pre-established positions along CERN SPS and LHC accelerators, and periodically, the irradiated witnesses were sampled for condition monitoring of the real cables installed in the respective area.

Characterization Methods
The oxidation induction time (OIT) was determined from DSC and chemiluminescence measurements under similar temperature conditions (isothermal mode). DSC measurements were performed with Setaram DSC 131 (Setaram, Caluire, France) at CERN and Setaram DSC 131 EVO (Setaram, Caluire, France) at CERN and INCDIE ICPE-CA for comparison studies. OIT from DSC was obtained according to [32]. Similar curves of signal vs. temperature were obtained with a Lumipol 3 (Polymer Institute of Slovak Academy of Sciences, Bratislava, Slovakia) chemiluminescence instrument with kinetic parameters of significance similar to those of DSC's ( Figure 1).

Figure 1.
Typical isotherm curves of oxidation in DSC and CL measurements indicating the kinetic parameters that can be calculated from these curves: OIT, tmax (time to maximum of oxidation peak in either DSC and CL), and t1/2 (time to reach half of the oxidation peak) are the main parameters used in this work; other parameters include the oxidation rate (vox = the slope of the tangent at the inflexion point of the ascendent peak edge), offset of the oxidation peak, and the thermal effect of oxidation (Hox). The inset shows a typical thermogram of the non-isothermal heating (10 °C/min) of a polyethylene sample in an inert (nitrogen) atmosphere in DSC; the melting endotherm can be Figure 1. Typical isotherm curves of oxidation in DSC and CL measurements indicating the kinetic parameters that can be calculated from these curves: OIT, t max (time to maximum of oxidation peak in either DSC and CL), and t 1/2 (time to reach half of the oxidation peak) are the main parameters used in this work; other parameters include the oxidation rate (v ox = the slope of the tangent at the inflexion point of the ascendent peak edge), offset of the oxidation peak, and the thermal effect of oxidation (∆H ox ). The inset shows a typical thermogram of the non-isothermal heating (10 • C/min) of a polyethylene sample in an inert (nitrogen) atmosphere in DSC; the melting endotherm can be observed; the parameters used for characterization of the melting process are the melting temperature (T m or T m ) and the melting endotherm (∆H m ).
For DSC measurements, the samples (either as thin chops or slices with 0.2 mm thickness cut with a Jasco HS-1 vertical slicer (Jasco Corp., Tokyo, Japan)) were warmed from room temperature to the measurement temperature under nitrogen flow (50 mL/min) with a typical heating rate of 10 • C/min (see the heating diagram in Figure 1, inset). This procedure avoided sample degradation under warming and enabled the measurement of the typical transition temperatures (crystallinity melting).

Kinetic Model: Methodology and Premises
Degradation of polymers is a complex process that involves changing their physical and chemical structure under the action of various stressors, either physical (e.g., heat, light, ionizing radiation, mechanical stress, ultrasound, solvents, etc.) or chemical (e.g., oxygen, ozone, metal ions, various chemical compounds, etc.) [36,37]. As a result, the useful properties of polymers can degrade [38,39].
It is widely accepted that the most important and frequent form of degradation of polymeric materials is oxidation. The organic nature of polymers makes them susceptible to oxidative degradation under ambient conditions. The oxidative degradation processes of different polymers are governed by specific mechanisms determined by the chemical peculiarities of the main chain and the adjacent groups. [40,41]. In general, oxidative degradation of polymers occurs in their amorphous zone, and for many usual polymers, such as polyethylene, polypropylene, ethylene-based copolymers, and natural and synthetic rubbers, the degradation mechanism involves a chained process based on free radicals and hydroperoxide reactions as precursors of oxidation degradation products [42]. Note that the polymer examples mentioned above contain only carbon atoms within their main backbone (also called hydrocarbon polymers). Their oxidation mechanisms are similar and can be described by the so-called Bolland-Gee scheme [34] shown in Scheme 1.
For DSC measurements, the samples (either as thin chops or slices with 0.2 mm thickness cut with a Jasco HS-1 vertical slicer (Jasco Corp., Tokyo, Japan)) were warmed from room temperature to the measurement temperature under nitrogen flow (50 mL/min.) with a typical heating rate of 10 °C/min (see the heating diagram in Figure 1, inset). This procedure avoided sample degradation under warming and enabled the measurement of the typical transition temperatures (crystallinity melting).

Kinetic Model: Methodology and Premises
Degradation of polymers is a complex process that involves changing their physical and chemical structure under the action of various stressors, either physical (e.g., heat, light, ionizing radiation, mechanical stress, ultrasound, solvents, etc.) or chemical (e.g., oxygen, ozone, metal ions, various chemical compounds, etc.) [36,37]. As a result, the useful properties of polymers can degrade [38,39].
It is widely accepted that the most important and frequent form of degradation of polymeric materials is oxidation. The organic nature of polymers makes them susceptible to oxidative degradation under ambient conditions. The oxidative degradation processes of different polymers are governed by specific mechanisms determined by the chemical peculiarities of the main chain and the adjacent groups. [40,41]. In general, oxidative degradation of polymers occurs in their amorphous zone, and for many usual polymers, such as polyethylene, polypropylene, ethylene-based copolymers, and natural and synthetic rubbers, the degradation mechanism involves a chained process based on free radicals and hydroperoxide reactions as precursors of oxidation degradation products [42]. Note that the polymer examples mentioned above contain only carbon atoms within their main backbone (also called hydrocarbon polymers). Their oxidation mechanisms are similar and can be described by the so-called Bolland-Gee scheme 34 shown in Scheme 1.
In order to suppress the environmental degradation of the mentioned polymers and to extend their service life, stabilizing additives are used [44,45]. Antioxidants are largely used with many polymers because of their capacities to either annihilate free radicals (radical scavenger-type antioxidants) or decompose hydroperoxides in a way that is not harmful to the polymer. Hence, the chain-breaking antioxidants (e.g., phenols, amines, quinones, etc.) compete the reaction of free alkyl radicals (R • ) with oxygen; other free radicals, such as peroxyl (ROO • ) and alkoxyl (RO • ), can also be annihilated.
As a consequence of antioxidant interference in the oxidation mechanism, an induction period of oxidative degradation is observed with stabilized polymers as compared to their antioxidant-free counterparts. This induction period is clearly observable in ther-mal analysis of isothermal measurements, such as DSC and CL, performed in oxidative atmospheres (e.g., air or oxygen). The duration of the induction period is proportional to the concentration of the antioxidant and decreases with increasing the temperature. When measured in an isothermal mode during thermal analysis, the oxidation induction period is called oxidation induction time (OIT).
Given the above considerations, the proposed methodology for the kinetic model is based on the following premises: (a) The effect of the stress factors, irrespective of their nature, consists of molecular scissions, leading to free radicals. These species react rapidly with the surrounding oxygen in air and trigger oxidation chain reactions (Scheme 1), leading to material deterioration and implicitly shortening their lifetime. In the case of ethylene polymers, lifetime diminution is caused by the progressive consumption of antioxidants. The initiation of degradation (produced by breaking of labile chemical bonds of the polymer) occurs by the action of different stress factors (high temperatures, light, radiation, electric field, etc.). The formed free radicals are subsequently trapped by oxygen, and the degradation propagates as chain reactions of free radicals thus formed. On average, each free radical formed subsequently involves the reaction of five oxygen molecules [46]. The second component of propagation consists of reactions of hydroperoxides. Even in the absence of oxygen, stress factors can induce free radicals, producing polymer degradation by scission and eventual crosslinking.
The subsequent air exposure of such material results in its oxidation due to the reactions of the trapped free radicals [47].  [2,35,45]. Therefore, OIT measurement can be used to "titrate" the antioxidant content of the sample. Thus, decreasing kinetics of OIT can be used to describe the kinetics of antioxidant depletion. Given the direct connection between OIT and material lifetime, decreasing OIT kinetics can be used to evaluate the lifetime at moderate temperatures specific to service conditions. As previously mentioned, the main problems for extrapolations are (i) the existence of a transition point (e.g., melting point of crystallinity), which can cause changes in the degradation mechanism; and (ii) the influence of temperature on the rates of different reactions. Hence, the degradation mechanism is expected to be different at higher temperatures as compared to moderate temperatures (due to the activation or deactivation of specific reactions). (d) We hypothesize that under similar (quasi-constant) exposure conditions of a material, the resulting OIT reduction is proportional to the time spent in mild conditions specific to aging in real conditions. Thus, OIT can be used as a revealing agent of the degradation effect produced under mild conditions and not as an accelerated aging test. In this case, the lifetime of the material under real conditions is equal to the exposure time resulting in a zero OIT value. The above hypothesis was confirmed by experimental data (see Figure 2), enabling a "bypass" of the problem of extrapolation, with the OIT used as a "marker" of the remaining lifetime of the material. Figure 2a shows that the experimental data present a progressive decrease in the OIT with an increase in the irradiation time for three different types of materials. The experimental points can be fitted well to an exponential decay. This behavior appears attractive for predictions because it corresponds to a first-order kinetics. Hence, the lifetime, which corresponds to the exposure time (under service conditions) until the OIT reaches a zero value (under the laboratory test conditions), can be easily estimated. This behavior has Polymers 2022, 14, 2357 7 of 22 been experimentally observed for polymeric materials based on single ethylene monomers or their mixtures (copolymers), which follows a Bolland-Gee-type oxidation scheme (this scheme does not apply to polystyrene, which preponderantly decomposes, even under oxidative conditions at elevated temperatures).
remaining lifetime of the material. Figure 2a shows that the experimental data present a progressive decrease in the OIT with an increase in the irradiation time for three different types of materials. The experimental points can be fitted well to an exponential decay. This behavior appears attractive for predictions because it corresponds to a first-order kinetics. Hence, the lifetime, which corresponds to the exposure time (under service conditions) until the OIT reaches a zero value (under the laboratory test conditions), can be easily estimated. This behavior has been experimentally observed for polymeric materials based on single ethylene monomers or their mixtures (copolymers), which follows a Bolland-Gee-type oxidation scheme (this scheme does not apply to polystyrene, which preponderantly decomposes, even under oxidative conditions at elevated temperatures).  Figure S1); 2-C4 cable (irradiated as bulk; Figure S2); 3-C3 cable (irradiated as strap; Figure S3); 4-C9 cable, white insulation (irradiated as bulk-encapsulated; Figure S4), 5-C9 cable, white insulation (irradiated as bulk-free ends, Figure S4); * laboratory codes.
A similar decrease in OIT with irradiation dose was observed for other insulation materials under controlled-dose irradiation conditions (Figure 2b), irradiated either as straps or as a bulk cable from where the samples for DSC measurements were subsequently taken.
In many practical cases of long-term exposures, the intensity of the factors inducing degradation can be considered quasi-constant (as an average), although their momentary cyclical variations may be significant; hence, the above kinetic considerations could be extended to such cases as well.

Kinetic Model for Evaluating the Lifetime of Polymeric Materials
If we consider the process of antioxidant depletion (1), where A is the antioxidant: The rate of antioxidant consumption is described by the rate of OIT decrease and is given by Equation (2):  Figure S1); 2-C4 cable (irradiated as bulk; Figure S2); 3-C3 cable (irradiated as strap; Figure S3); 4-C9 cable, white insulation (irradiated as bulk-encapsulated; Figure S4), 5-C9 cable, white insulation (irradiated as bulk-free ends, Figure S4); * laboratory codes.
A similar decrease in OIT with irradiation dose was observed for other insulation materials under controlled-dose irradiation conditions (Figure 2b), irradiated either as straps or as a bulk cable from where the samples for DSC measurements were subsequently taken.
In many practical cases of long-term exposures, the intensity of the factors inducing degradation can be considered quasi-constant (as an average), although their momentary cyclical variations may be significant; hence, the above kinetic considerations could be extended to such cases as well.

Kinetic Model for Evaluating the Lifetime of Polymeric Materials
If we consider the process of antioxidant depletion (1), where A is the antioxidant: The rate of antioxidant consumption is described by the rate of OIT decrease and is given by Equation (2): where [OIT] is the momentary value of OIT (which is proportional to the antioxidant concentration at a given time, t), and k is the rate constant. Integration of Equation (2) leads to Equation (3): where C is the integration constant, the chemical significance of which is obtained if t = 0 (initial reaction time); and C is the natural logarithm of the initial OIT value: If C is substituted in Equation (3), the relation between the antioxidant concentration and the time (of aging) becomes: In the case of ionizing radiation aging, the parameter t can be replaced by the integrated dose, D: where k is the oxidation rate constant depending on the irradiation dose (k = k/D r , where D r is the dose rate).
Equations (5) and (6) describe the aging process of ethylene-based materials in ionizing radiation fields using the measured OIT values and the kinetic model presented above.
Several cases of application of this simple kinetic model are discussed below.

Evaluation of the Remaining Service Life (t x ) When the Elapsed Aging/Service Time Is Known
As previously mentioned, the lifetime, t x , of a polymeric material can be assumed to be the time required for total consumption of the incorporated antioxidant, i.e., the total aging duration required to obtain a zero value for OIT under specific aging conditions. Equation (5) can be used for t x evaluation if the k value is known. It can be calculated from Equation (5) if two OIT values are measured during two different aging periods, e.g., OIT 0 for initial (unaged) material and OIT t at aging time t.
By imposing a limit condition for OIT t (OIT t →0) in Equation (7), meaning the end of life of the polymeric material, the service life (t x ) can be calculated. However, using an exactly zero value makes Equation (7) mathematically impossible to solve, so a low OIT value must be imposed as a limit, e.g., 10 s (0.167 min).
Using different values, i.e., 100 s, will not result in significant differences in the evaluated lifetime. However, imposing very low OIT values, such as 10 −10 s, results in considerably longer lifetime values, although it is reasonable to maintain near-practically measurable values for OIT limit. Consequently, the estimated lifetime will be slightly lower than the accepted value from mechanical tests (e.g., time to reach 50% elongation), enabling a safety reserve. A more extensive discussion on the correlation of the OIT values with the mechanical properties of polymeric materials will presented in a forthcoming paper.

Evaluation of the Service Life (t x ) of Polymeric Insulation When the OIT 0 Value Is Not Known
An essential advantage of this simple kinetic model is that it enables the evaluation of the residual lifetime of material, even if the initial value of oxidation induction time (OIT 0 ) is not available. In such a case (which is more frequent in practice), it is enough to measure two OIT values at two different moments during the service t 1 and t 2 (t 1 can be measured, even at the moment of service start). Equations (9)-(11) are applicable in this case for the expression of the kinetic model, rate constant, and remaining lifetime calculations, respectively. Another advantage of the application of Equations (9)-(11) is that the OIT 0 value would be not representative of materials stored for long periods before use, such as cables in storage.

Evaluation of the Maximum Supported Dose (D x ) for a Cable in Use
For the practical cases wherein the dose is carefully monitored, such as nuclear power plants or particle accelerators, a similar first-order kinetic model can be used, whereby the time is replaced by the integral dose, D. The value k can be calculated (Equation (12)) if the OIT values at two different irradiation doses are known: for the cable in the initial state (D 0 ) and the same material aged at dose D, respectively.
By imposing a limit condition of OIT D close to zero, the maximum supported dose can be calculated if the OIT 0 and OIT D 1 or two OIT values at two different doses, D 1 and D 2 , are known, using Equations (13) or (16), respectively.
Similar to the above, another approach to the simple kinetic model allows for the evaluation of the maximum supported dose without the need to know the OIT value for the cable in its initial state but only the OIT D 1 and OIT D 2 values obtained on two similar cable samples irradiated under the same conditions at two different doses (D 1 and D 2 ; D 2 > D 1 ).

LDPE/XLPE Insulation Materials
An example of the use of DSC as an analytic tool, based on crystallinity melting, for easy discrimination between different ethylene/substituted ethylene polymers is shown in Figure 3. The melting peaks are located at temperatures characteristic of the respective types of polymers. Note that FTIR spectra already confirmed the polyethylene nature of these polymers. Hence, DSC and FTIR spectroscopy can be used as a rapid quality test. The corroboration of the information inferred by both techniques enables accurate establishment of the polymer type, the fluctuation of properties from one batch to another, and an eventual substitution of the suitable polymer during the fabrication process.
in Figure 3. The melting peaks are located at temperatures characteristic of the respective types of polymers. Note that FTIR spectra already confirmed the polyethylene nature of these polymers. Hence, DSC and FTIR spectroscopy can be used as a rapid quality test. The corroboration of the information inferred by both techniques enables accurate establishment of the polymer type, the fluctuation of properties from one batch to another, and an eventual substitution of the suitable polymer during the fabrication process.   lishment of the polymer type, the fluctuation of properties from one batch to another, and an eventual substitution of the suitable polymer during the fabrication process.    Table 1 shows an example of a kinetic model application in the case of the same material exposed to irradiation (γ 137 Cs) in different sample forms, namely (i) thin strap (thin strap previously sampled from cable insulation by a turning machine), (ii) bulk-free (sampled from an irradiated cable fragment with free ends), and (iii) bulk-encapsulated (sampled from an irradiated cable fragment with wax-capped ends). The rate constant (k ) is lower for the sample irradiated as a strap at a lower dose rate as compared to bulk samples due to the free access of oxygen over the whole strap surface and the dose-rate effect in the presence of oxygen [48]. There is also a certain difference between the degradation rates of the samples irradiated as bulk with or without limited contact with air; the prevention of air access resulted in a lower degradation rate and, consequently, in a longer lifetime. The latter case is probably closer to the inner parts (far from the ends) of a real cable, whereas the former (bulk air or uncapped) would be close to the situation of real cable ends (at the connectors). As a result, except for the surface layer, which is in direct contact with the air, irradiation occurs mainly in the absence of oxygen. Hence, crosslinking is dominant. After irradiation under these conditions, a relatively large amount of free radicals may remain trapped in the polymer, leading to slow post-irradiation oxidation as they combine with the oxygen diffused slowly into the bulk [10,48]. The presence of trapped species in irradiated polymers can be easily evidenced in chemiluminescence by the initial signal increase assigned to the sample's history [35]. The effects of the dose rate are illustrated in the case of polymeric materials from witness cables aged under real service conditions (Table 2) in the SPS CERN accelerator.  Figure S5) irradiated in the CERN-SPS accelerator (isothermal DSC measurements, T = 200 • C, air). This behavior can be understood in terms of antioxidant consumption through two competing processes. One is the reaction with free peroxy radicals, for which the rate depends on available oxygen (diffusion-limited), and the other is antioxidant consumption through direct reaction with the radiation-induced alkyl free radicals of the polymer. Increased dose rates can enhance the reaction share in the latter process, resulting in accelerated antioxidant consumption, lower OIT values, and lower lifetime values. Table 2 also shows that the different insulation materials result in significantly different OIT 0 values; the highest OIT 0 is observed for black insulation, whereas lower values are observed for green/yellow insulation. These different values could be due to different levels of incorporated antioxidants, as well as possible influences of the dye additives on thermal oxidation kinetics [2]. Furthermore, the specific influence of the additive systems can also explain the slight differences observed in the radiation behavior of the four insulation materials.

Irradiation
The zero OIT values measured for the samples irradiated at 105 kGy and 144 kGy ( Table 2) are in agreement with the observation that these samples are seriously damaged by irradiation, presenting cracks ( Figure 5) and friability. This confirms the viability of the proposed kinetic model. This behavior can be understood in terms of antioxidant consumption through two competing processes. One is the reaction with free peroxy radicals, for which the rate depends on available oxygen (diffusion-limited), and the other is antioxidant consumption through direct reaction with the radiation-induced alkyl free radicals of the polymer. Increased dose rates can enhance the reaction share in the latter process, resulting in accelerated antioxidant consumption, lower OIT values, and lower lifetime values. Table 2 also shows that the different insulation materials result in significantly different OIT0 values; the highest OIT0 is observed for black insulation, whereas lower values are observed for green/yellow insulation. These different values could be due to different levels of incorporated antioxidants, as well as possible influences of the dye additives on thermal oxidation kinetics [2]. Furthermore, the specific influence of the additive systems can also explain the slight differences observed in the radiation behavior of the four insulation materials.
The zero OIT values measured for the samples irradiated at 105 kGy and 144 kGy ( Table 2) are in agreement with the observation that these samples are seriously damaged by irradiation, presenting cracks ( Figure 5) and friability. This confirms the viability of the proposed kinetic model.

Figure 5. Appearance of cracks in XLPE insulations (IC1 cable) after three years of irradiation in SPS.
A frequently used parameter for characterization of the radiation strength of a material is the radiation index (RI), which is defined in IEC 60544-4 [49] as the log10 of the absorbed dose in gray (rounded to two significant digits) at which the elongation at break is reduced to 50% of its initial value under specified conditions of irradiation and tests. As we assumed that the mechanical properties of a material should fail after the OIT reaches a zero value, it is reasonable to assume that RI can be evaluated in this manner from OIT measurements. Hence, considering the Dx values calculated from OIT for several polyethylene insulation materials subjected to gamma irradiation under controlled conditions (Figure 6), the RI values calculated with the formula RI = log10Dx are similar to those previously reported for PE materials by Tavlet and Schönbacher based on mechanical tests [28,29] using similar dose and dose-rate conditions. The practical importance of these results is that using thermal analysis methods instead of mechanical tests for degradation diagnosis and lifetime prognosis makes it possible to reduce both the amounts of required A frequently used parameter for characterization of the radiation strength of a material is the radiation index (RI), which is defined in IEC 60544-4 [49] as the log 10 of the absorbed dose in gray (rounded to two significant digits) at which the elongation at break is reduced to 50% of its initial value under specified conditions of irradiation and tests. As we assumed that the mechanical properties of a material should fail after the OIT reaches a zero value, it is reasonable to assume that RI can be evaluated in this manner from OIT measurements. Hence, considering the D x values calculated from OIT for several polyethylene insulation materials subjected to gamma irradiation under controlled conditions (Figure 6), the RI values calculated with the formula RI = log 10 D x are similar to those previously reported for PE materials by Tavlet and Schönbacher based on mechanical tests [28,29] using similar dose and dose-rate conditions. The practical importance of these results is that using thermal analysis methods instead of mechanical tests for degradation diagnosis and lifetime prognosis makes it possible to reduce both the amounts of required samples and laboratory time and to increase the amount of information inferred by the measurements. Hence, improved condition monitoring of in-use cables can be achieved.

EPR Copolymer Insulation Materials
Ethylene-propylene copolymers are a class of synthetic rubbers obtained by copolymerizing ethylene and polypropylene, possibly in combination with other chemical compounds (diene, in a proportion of a few percent) to form ethylene-propylene-diene monomer rubber (EPDM). Both EPR and EPDM polymers are of interest for the production of cable insulation due to their low production costs and their interesting properties, such as mechanical flexibility, thermal stability, good dielectric properties in humid conditions, Figure 6. RI values of irradiated LDPE cable materials (strap, γ 137 Cs, D r = 0.4 kGy/h); C9 was irradiated as bulk (free ends).

EPR Copolymer Insulation Materials
Ethylene-propylene copolymers are a class of synthetic rubbers obtained by copolymerizing ethylene and polypropylene, possibly in combination with other chemical compounds (diene, in a proportion of a few percent) to form ethylene-propylene-diene monomer rubber (EPDM). Both EPR and EPDM polymers are of interest for the production of cable insulation due to their low production costs and their interesting properties, such as mechanical flexibility, thermal stability, good dielectric properties in humid conditions, high resistance to tree formation and corona effect, low hydrophobicity index, abrasion resistance, and high thermal range of use (−60 • C and + 160 • ) [5,6,19,[50][51][52]. Due to its rubber-like properties, EPR is used in the manufacture of many distribution cables up to 35 kV [53] or even 150 kV in direct current (DC) applications [54], cables for nuclear power plants [55,56], etc.
The cables used at CERN, known as "EPR cables", contain EPR in both the insulation and the sheath, with differing chemical natures. The sheath has a high PP content and a higher level of crystallinity. The PP/ethylene ratio is high in semicrystalline sheath materials. ATR-FTIR and non-isothermal DSC methods allow for differentiation between these two types of EPR-based cable components. DSC non-isothermal measurements show a softening peak at 153 • C (Figure 7a), and ATR/FTIR spectroscopy shows a high absorption band at 1380 cm −1 (due to bending vibration of the C-H bonds in CH 3 (Figure 7b). As in the case of LDPE, the OIT values of EPR-based insulation decrease exponentially with the irradiation dose (Table 3), suggesting the possibility of applying the proposed kinetic model to evaluate the lifetime of these materials. For the validation of the model, samples of HV cable with EPR insulation were subjected to irradiation at different irradiation doses and dose rates with both free and wax-encapsulated ends.  As in the case of LDPE, the OIT values of EPR-based insulation decrease exponentially with the irradiation dose (Table 3), suggesting the possibility of applying the proposed kinetic model to evaluate the lifetime of these materials. For the validation of the model, samples of HV cable with EPR insulation were subjected to irradiation at different irradiation doses and dose rates with both free and wax-encapsulated ends. Table 3. Degradation model parameters and lifetime evaluation of EPR insulation of C2 cable (isothermal DSC measurements, T = 200 • C, air; Figure S6). The methodology for assessing the maximum dose and lifetime for EPR materials is similar to that applied to LDPE-based materials. A similar cable with EPR insulation, with free access to oxygen (free ends), was irradiated under use conditions in SPS for one year. As the total dose received by the cable during this period was unknown, the life of this cable could be estimated based on OIT values under specific use conditions.

EVA Copolymer Insulation Materials
Due to their elastic properties, poly(ethylene-co-vinyl) acetate (EVA) copolymers, which are widely used in the production of wires and cables, incorporate a high concentration of stabilizers and mineral fillers for both high stability and fire resistance without halogen [57]. Figure 8 shows a comparison of the DSC curves for XLPE and EVA materials. Such curves enable easy discrimination between crystalline polymers, such as PE, and amorphous copolymers, such as EVA in our case. The melting endotherm is weak in the case of the copolymer (with Tm at~93 • C) because of its typically low crystallinity. The vinylacetate (VA) content was graphically estimated at 16% based on Tm values using the relationship proposed by Son and Choi (2019) [58]. The methodology for assessing the maximum dose and lifetime for EPR materials is similar to that applied to LDPE-based materials. A similar cable with EPR insulation, with free access to oxygen (free ends), was irradiated under use conditions in SPS for one year. As the total dose received by the cable during this period was unknown, the life of this cable could be estimated based on OIT values under specific use conditions.

EVA Copolymer Insulation Materials
Due to their elastic properties, poly(ethylene-co-vinyl) acetate (EVA) copolymers, which are widely used in the production of wires and cables, incorporate a high concentration of stabilizers and mineral fillers for both high stability and fire resistance without halogen [57]. Figure 8 shows a comparison of the DSC curves for XLPE and EVA materials. Such curves enable easy discrimination between crystalline polymers, such as PE, and amorphous copolymers, such as EVA in our case. The melting endotherm is weak in the case of the copolymer (with Tm at ~93 °C) because of its typically low crystallinity. The vinylacetate (VA) content was graphically estimated at 16% based on Tm values using the relationship proposed by Son and Choi (2019) [58]. ATR-FTIR spectroscopy (Figure 9) confirms the DSC data. The most specific band of EVA copolymer is at 1738 cm −1 , assigned to the carbonyl group in esters. This structural band makes the ATR/FTIR method challenging to apply when monitoring the degradation of these materials. ATR-FTIR spectroscopy (Figure 9) confirms the DSC data. The most specific band of EVA copolymer is at 1738 cm −1 , assigned to the carbonyl group in esters. This structural band makes the ATR/FTIR method challenging to apply when monitoring the degradation of these materials.
Polymers 2022, 14, x FOR PEER REVIEW was evaluated from the ratio of the peak areas at 730 cm −1 and 720 cm −1 [59], wh close to the value determined from DSC measurements.  (Figure 10b). This behavior suggests that the propo model can be applied to these materials for evaluation of lifetime and maxi ported irradiation dose (Table 4). Oxidation induced by stress factors results in weaker peaks located at 1680-1720 cm −1 . The group of four peaks located between 3300 and 3600 cm −1 and the intense band between 500 and 700 cm −1 are due to the mineral filler. As the vinyl acetate comonomer is free of OH groups, the spectrum of the copolymer should be free of absorption bands in the region of 3200-3500 cm −1 ; however a small, straight band is observed at around 3430 cm −1 [59], especially for copolymers with high VA content. These peaks are the result of intramolecular hydrogen bonds within the VA moiety (see the inset in Figure 9 for hydrogen bonds). However, even if such a peak exists, it is probably masked by more intense bands of the mineral filler. Furthermore, based on the IR spectra, the VA content of 17% was evaluated from the ratio of the peak areas at 730 cm −1 and 720 cm −1 [59], which is quite close to the value determined from DSC measurements.
Similar to the polymeric materials presented above (i.e., LDPE, XLPE, and EPR), EVA exhibits the same behavior in terms of OIT values decreasing over time (Figure 10a), irrespective of irradiation dose (Figure 10b). This behavior suggests that the proposed kinetic model can be applied to these materials for evaluation of lifetime and maximum supported irradiation dose (Table 4).
hydrogen bond in vinyl-acetate groups.
Similar to the polymeric materials presented above (i.e., LDPE, XLPE, and EPR), EVA exhibits the same behavior in terms of OIT values decreasing over time (Figure 10a), irrespective of irradiation dose (Figure 10b). This behavior suggests that the proposed kinetic model can be applied to these materials for evaluation of lifetime and maximum supported irradiation dose (Table 4).

Radiation-Induced Oxidation Profiles in Cable Depth by OIT Measurements
In the case of irradiated thick insulation samples, the radiation-induced oxidation is typically heterogeneous due to the influence of atmospheric oxygen. Figure 11a shows that OIT data vs. sampling depth fit logarithm functions. Furthermore, the OIT values from the wax-encapsulated samples (C3 cable bulk-encapsulated) are systematically higher than those of the non-encapsulated sample. Similar data from another HV cable (C4 cable) are presented in Figure 11b.
The effect of the irradiation dose in general diminution of OIT values is clearly observed. Representative OIT values illustrate the heterogeneous degradation of the insulation material in sample depth. The 5 mm sheath in oil-impregnated paper and a PET layer adherent to the PE insulator explain the reduced oxidation in the inner region compared to the outer region, where the OIT value was zero.

Radiation-Induced Oxidation Profiles in Cable Depth by OIT Measurements
In the case of irradiated thick insulation samples, the radiation-induced oxidation is typically heterogeneous due to the influence of atmospheric oxygen. Figure 11a shows that OIT data vs. sampling depth fit logarithm functions. Furthermore, the OIT values from the wax-encapsulated samples (C3 cable bulk-encapsulated) are systematically higher than those of the non-encapsulated sample. Similar data from another HV cable (C4 cable) are presented in Figure 11b. The effect of the irradiation dose in general diminution of OIT values is clearly observed. Representative OIT values illustrate the heterogeneous degradation of the insulation material in sample depth. The 5 mm sheath in oil-impregnated paper and a PET layer adherent to the PE insulator explain the reduced oxidation in the inner region compared to the outer region, where the OIT value was zero.
The tightly packed structure of C1 cable appears to provide good protection against the ingress of air around the insulator, with the black sheaths covering both sides of the insulator, probably as oxygen barriers. As a result, the OIT values found for both encapsulated and non-encapsulated cables are similar, with significant OIT values measured on both sides of the insulator (Figure 12). This behavior is different from that observed for cables C3 and C4, which presented zero values of OIT on the outer side of the insulator, even at a relatively long distance (2-3 cm) from the cable end. The tightly packed structure of C1 cable appears to provide good protection against the ingress of air around the insulator, with the black sheaths covering both sides of the insulator, probably as oxygen barriers. As a result, the OIT values found for both encapsulated and non-encapsulated cables are similar, with significant OIT values measured on both sides of the insulator (Figure 12). This behavior is different from that observed for cables C3 and C4, which presented zero values of OIT on the outer side of the insulator, even at a relatively long distance (2-3 cm) from the cable end.
CL measurements ( Figure 13) lead to similar conclusions, showing that the oxidation degree decreases from the surface in contact with air towards the inner material zones, following the direction of oxygen diffusion in the insulating material.
The results prove the capability of the DSC method to explore the oxidation profile of a bulk sample, evidencing the inhomogeneous oxidation in cable depth. In the case of cable insulation, the sheaths, the gas-tight connectors, and the jackets contribute to inhomogeneous oxidation by hindering air access. Considering the possible presence of inhomogeneous oxidation, the correct selection of the relevant samples from an aged cable is mandatory. For large-diameter insulators, such as in HV cables, it is necessary to analyze samples from both the external and internal surfaces of the insulator, as well as from the middle zone of the insulator in order to obtain the transversal oxidation profile following the electric field direction, which is most important for the insulator aging state. adherent to the PE insulator explain the reduced oxidation in the inner region compared to the outer region, where the OIT value was zero.
The tightly packed structure of C1 cable appears to provide good protection against the ingress of air around the insulator, with the black sheaths covering both sides of the insulator, probably as oxygen barriers. As a result, the OIT values found for both encapsulated and non-encapsulated cables are similar, with significant OIT values measured on both sides of the insulator (Figure 12). This behavior is different from that observed for cables C3 and C4, which presented zero values of OIT on the outer side of the insulator, even at a relatively long distance (2-3 cm) from the cable end. CL measurements ( Figure 13) lead to similar conclusions, showing that the oxidation degree decreases from the surface in contact with air towards the inner material zones, following the direction of oxygen diffusion in the insulating material. The results prove the capability of the DSC method to explore the oxidation profile of a bulk sample, evidencing the inhomogeneous oxidation in cable depth. In the case of cable insulation, the sheaths, the gas-tight connectors, and the jackets contribute to inhomogeneous oxidation by hindering air access. Considering the possible presence of inhomogeneous oxidation, the correct selection of the relevant samples from an aged cable is mandatory. For large-diameter insulators, such as in HV cables, it is necessary to analyze samples from both the external and internal surfaces of the insulator, as well as from the middle zone of the insulator in order to obtain the transversal oxidation profile following the electric field direction, which is most important for the insulator aging state.

DSC-CL Correlation
Chemiluminescence is a technique characterized by very high sensitivity in detection of the oxidation processes (CL: luminescence produced due to a chemical reaction). The emission of chemiluminescence is closely related to oxidation, a sufficiently exothermic process to produce excited molecular species with enough energy for deactivation, leading to light emission in the visible range [60]. The sampling is similar to DSC method, but

DSC-CL Correlation
Chemiluminescence is a technique characterized by very high sensitivity in detection of the oxidation processes (CL: luminescence produced due to a chemical reaction). The emission of chemiluminescence is closely related to oxidation, a sufficiently exothermic process to produce excited molecular species with enough energy for deactivation, leading to light emission in the visible range [60]. The sampling is similar to DSC method, but the sample's surface is explored with a photomultiplier tube coupled with electronic signal amplification systems, making these systems exceptionally sensitive [61].
Eliminating the contribution of other endothermic (molecular decomposition) or exothermic (crystallization) processes considerably simplifies the CL curve and contributes to increased accuracy of measurements [61]. However, the most important limitation of the method is the self-absorption of the CL signal in dark and/or high-mineral-charge materials.
The experimental results obtained by DSC-CL comparative analysis performed on insulating cable materials (based on LDPE, XLPE, EPR, and EVA), showed an excellent correlation (r 2 > 0.99) of data, especially on LDPE and XLPE materials ( Figure 14). In the case of EPR and EVA samples, lower linear correlation (r 2 : 0.92-0.94) between the two types of measurements was observed. This can be explained either by the fact that the samples are not melted and the thermal contact with the heated element is poor or because the samples are opaque and, therefore, the amount of signal that reaches the CL photomultiplier tube is less than that for PE. Both techniques are suitable for aging characterization in an ionizing radiation environment for the polymeric materials used to produce the electric cable. samples are not melted and the thermal contact with the heated element is poor or because the samples are opaque and, therefore, the amount of signal that reaches the CL photomultiplier tube is less than that for PE. Both techniques are suitable for aging characterization in an ionizing radiation environment for the polymeric materials used to produce the electric cable. The CL method can provide clearer and richer information related to both the oxidation mechanism and the effect of stabilizers [35]. However, in the case of cable materials, its effectiveness is limited by their opacity or color. Thus, for current practical applications, the use of the DSC technique appears most beneficial because it also offers the opportunity to highlight feature transitions (e.g., melting temperature, glass transition, etc.), which can be used to identify the nature of the polymeric material.

Conclusions
We propose a kinetic model based on the exponential decrease in oxidation induction time (OIT) to evaluate the lifetime of electrical insulating polymeric materials in electrical cables in ionizing radiation environments. The model also allows for determination of the maximum radiation dose that the polymeric material can support.
This model has been tested on dozens of cable types with polymeric materials of different chemical nature (LDPE, HDPE, XLPE, PE foam, EPR, and EVA) in use in CERN's particle acceleration systems. However, it can be extended to evaluate the lifetime of polymeric materials exposed to other stress factors, such as temperature, UV, natural aging, etc.
The results of the evaluation of the lifetime of polymeric materials from cables exposed to radiation environments using the proposed kinetic model correlate very well (>95%) with lifetime evaluation data resulting from mechanical testing (50% reduction in elongation at break). These results will be the subject of a future paper.
Supplementary Materials: The following supporting information can be downloaded at: www.mdpi.com/xxx/s1, Figure S1: Structure of C1 cable (General purpose, High Voltage -HV-coaxial cable); Figure S2: Structure of C4 cable (General purpose, HV/HF coaxial cable); Figure S3: Structure of C3 cable (General purpose, HV/HF-High Frequencies-coaxial cable); Figure S4: Structure of C9 cable (General purpose, I&C-Instrumentation and Control-cable); Figure S5: Structure of IC1 cable (General purpose, I&C-Instrumentation and Control-cable); Figure S6: Structure of C2 cable (General purpose, HV, coaxial cable); Figure S7: Structure of IC2 cable (General purpose, I&C-Instrumentation and Control-cable). The CL method can provide clearer and richer information related to both the oxidation mechanism and the effect of stabilizers [35]. However, in the case of cable materials, its effectiveness is limited by their opacity or color. Thus, for current practical applications, the use of the DSC technique appears most beneficial because it also offers the opportunity to highlight feature transitions (e.g., melting temperature, glass transition, etc.), which can be used to identify the nature of the polymeric material.

Conclusions
We propose a kinetic model based on the exponential decrease in oxidation induction time (OIT) to evaluate the lifetime of electrical insulating polymeric materials in electrical cables in ionizing radiation environments. The model also allows for determination of the maximum radiation dose that the polymeric material can support.
This model has been tested on dozens of cable types with polymeric materials of different chemical nature (LDPE, HDPE, XLPE, PE foam, EPR, and EVA) in use in CERN's particle acceleration systems. However, it can be extended to evaluate the lifetime of polymeric materials exposed to other stress factors, such as temperature, UV, natural aging, etc.
The results of the evaluation of the lifetime of polymeric materials from cables exposed to radiation environments using the proposed kinetic model correlate very well (>95%) with lifetime evaluation data resulting from mechanical testing (50% reduction in elongation at break). These results will be the subject of a future paper.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.