Manuela Da Cruz

Laetitia Van Schoors

Karim Benzarti

Xavier Colin
email: xavier.colin@ensam.eu

Thermo-oxidative degradation of additive free polyethylene. Part I. Analysis of chemical modifications at molecular and macromolecular scales

Keywords: ageing, degradation, kinetics, polyolefins

The effects of thermal oxidation on the molecular and macromolecular structures of additive free PE were investigated between 100 8Cand1408C in air in order to tentatively establish non-empirical structure/property relationships. In the first part, the changes in POOH concentration were assessed by three different analytical methods: iodometry, modulated differential scanning calorimetry, and Fourier transform infrared (FTIR) spectrophotometry after SO 2 treatment. All these methods provided very similar results until the end of the induction period, after which iodometry overestimated strongly POOH concentration because it titrates also other chemical species formed at high conversion ratios, namely double bonds. In parallel, the changes in carbonyl group concentration were determined by FTIR spectrophotometry after NH 3 treatment. As the accumulation kinetics of ketones, aldehydes, and carboxylic acids were closely interrelated, the question of their actual formation mechanisms in the current thermal oxidation mechanistic scheme was raised. An alternative reaction pathway was proposed for the bimolecular decomposition of POOH. In the second part, the corresponding changes in weight and number average molecular masses were monitored by high temperature gel permeation chromatography equipped with a triple detection technology. As both quantities decreased dramatically from the beginning of exposure and their ratio M w /M n tends toward the asymptotic value of 2 and it was concluded that a "pure" chain scission process operated. Finally, as the number of chain scissions perfectly correlates, the concentration sum of aldehydes and their oxidation products (i.e., carboxylic acids), it was also concluded that these carbonyl groups result exclusively from the b scission of alkoxy radicals.

INTRODUCTION

Polyethylene (PE) is commonly used in civil engineering and building applications due to its low cost, easy processability, good barrier properties, and chemical resistance. However, environmental factors such as ultraviolet (UV) radiation and seasonal temperature variations can cause its chemical degradation and, ultimately, its embrittlement at long term. [START_REF] Verdu | In Vieillissement des Plastiques; Afnor Technique[END_REF][START_REF] Singh | [END_REF] The incorporation of antioxidants during melt processing is, thus, necessary to delay the onset of oxidation and increase the lifetime. 3 However, in service conditions, antioxidants can be physically lost by water extraction and/or evaporation, but also consumed by chemical reactions. 4 When the polymer matrix is no longer sufficiently protected, the oxidation rate increases suddenly [START_REF] Mkacher | Vieillissement Thermique des gaines PE et PVC de Câbles Electriques[END_REF] and thenceforth, induces macromolecular and morphological changes leading to a catastrophic decay in fracture properties. [START_REF] Fayolle | [END_REF] Therefore, the study of PE durability is a multidisciplinary problem which requires addressing scientific challenges at different structural scales. At the lowest (i.e., molecular) scale, stabilization and oxidation mechanisms must be better understood in order to establish the most realistic mechanistic scheme as possible and deduce a non-empirical mathematical expression for the oxidation rate.

At the highest (i.e., macroscopic) scale, the changes in mechanical properties versus exposure time must be analyzed in detail in order to identify what quantities are the most sensitive to oxidative degradation and to define, on this basis, the most relevant end-of-life criterion for the application under consideration. In the case of a semi-crystalline polymer having its amorphous phase in rubbery state, such as polyolefins, it is usual to select the ultimate elongation for describing the sharp ductile/brittle transition because this property varies monotonically with exposure time (contrarily to tensile strength). 7 Thus, the conventional end-of-life criterion is a loss of 50% of the initial value of the ultimate elongation. [START_REF] Risch | Proceedings of the 59th International Wire & Cable Symposium[END_REF] At intermediary (i.e., macromolecular and morphological) scales, embrittlement processes must be clearly evidenced and elucidated. In most cases of PE oxidation, random chain scissions largely predominate over crosslinking [START_REF] Khelidj | [END_REF][10][11] and induce chemi-crystallization. 12 The conversion ratio of both processes is, thus, closely related to the changes in the molecular mass distribution. 12 From this rapid literature overview, it appears that the ideal approach for lifetime prediction of PE could be summarized by the following causal chain: At this stage of investigations, important efforts are still needed for determining all the structural variables playing a key role in this causal chain, but also for establishing non-empirical connections between these variables, that is, structure/property relationships.

The present article is focused on the thermal oxidation between 100 8C and 140 8C in air of additive free PE at both molecular and macromolecular scales. It must be recalled that thermal oxidation is a radical chain reaction initiated by the decomposition of its main propagation product: the hydroperoxide group (POOH), which is also the most instable product. [START_REF] Verdu | In Vieillissement des Plastiques; Afnor Technique[END_REF][START_REF] Khelidj | Vieillissement d'Isolants de Câbles en Poly ethylene en Ambiance Nucl eaire[END_REF] Indeed, the dissociation energy of its OAO bond is about 150 kJ/mol, against more than 325 kJ/mol for all other chemical bonds composing the polymer chain (including structural defects such as allylic CAH bonds). [START_REF] Colin | In Handbook of Multiphase Polymer Systems[END_REF] In its simplest form (i.e., in oxygen excess and in the absence of thermal and photochemical stabilizers), this reaction can be written such as: [START_REF] Khelidj | Vieillissement d'Isolants de Câbles en Poly ethylene en Ambiance Nucl eaire[END_REF] Initiation:

Iu ð Þ POOH ! 2P 1 c 1 P H @O 1 ð1 -c 1 ÞPAOH 1 c S S 1 H 2 O k 1u ð Þ Ib ð Þ2POOH ! P 1 PO 2 1 c 1 P H @O 1 1 -c 1 ð ÞPAOH 1 c S S 1 H 2 O k 1b ð Þ Propagation: II ð ÞP 1 O 2 ! PO 2 k 2 ð Þ III ð ÞPO 2 1 PH ! POOH 1 P k 3 ð Þ Bimolecular recombination: VIa ð ÞPO 2 1 PO 2 ! PO OP ½ cage 1 O 2 k 6a ð Þ VIb ð Þ½PO OP cage ! POOP 1 X k 6b ð Þ VIc ð Þ½PO OP cage ! P@O 1 PAOH k 6c ð Þ VId ð Þ½PO OP cage ! 2P 1 2c 1 P H @O 1 2ð1 -c 1 ÞPAOH 1 2c S S k 6d ð Þ
where PH, P8, PO 2 8, PO8, POOH, POOP, P H @O, P@O, PAOH, S, and X accounts, respectively, for secondary CAH groups (i.e., methylene groups), alkyl, peroxy and alkoxy radicals, hydroperoxides, dialkyl peroxides, aldehydes, ketones, alcohols, chain scissions, and crosslink nodes. k i are elementary rate constants of the chemical events under consideration. c S is the yield of bscission (leading to aldehydes), irrespectively of the molecularity of the initiation reaction, in competition with hydrogen abstraction (leading to alcohols).

From a practical point of view, it is more convenient to consider an apparent yield c 1 for the formation of aldehydes owing to the high difficulty to titrate them accurately by common spectrochemical methods, in particular by Fourier transform infrared (FTIR) analysis. Indeed, their titration requires extracting their elementary contribution from the composite absorption band relative to all types of carbonyl groups. However, deconvolution methods are in general based on too simple mathematical functions (frequently of Gaussian's type) of which the actual representativeness is highly questionable. [START_REF] Verdu | In Vieillissement des Plastiques; Afnor Technique[END_REF] Moreover, the molar extinction coefficients of carbonyl groups are in general poorly known. As an example, they would be ranged between 155 and 270 L/mol/ cm for aldehydes. [START_REF] Carlsson | [END_REF]16 The most relevant molecular variables, from a kinetic modeling point of view, are oxygen uptake and hydroperoxides concentration because their respective changes versus exposure time can be accurately simulated without the use of additional parameters than the previous rate constants. [17][18][19] Oxygen uptake can be easily measured by manometric methods. [20][21][22][23] These latter have recently reached a remarkable degree of sensitivity. 24 In contrast, the titration of hydroperoxides is more complicated and still controversial. Indeed, these chemical species are not only thermolabile, but are also formed in very low concentrations in polyolefin matrices (typically 1 mol/L). Nevertheless, three main titration methods of hydroperoxides are reported in the literature:

The spectrochemical approach consists in the determination of the concentration of reaction products by FTIR spectrophotometry after SO 2 (SO 2 -FTIR), [25][START_REF] Richaud | Durabilit e des G eotextiles en Polypropyle[END_REF][START_REF] Tireau | Propri et es a Long Terme des Gaines de Poly ethyle `ne Haute Densit e Utilis ees pour les Ponts a Haubans[END_REF][START_REF] Carlsson | [END_REF] NO (NO-FTIR), [29][30][31][32] or Fe 21 treatment (Fe 21 -FTIR). 33 The chemical approach consists in the titration of hydroperoxides by iodometry. [33][34][35] And the thermodynamical approach 33,36,37 is based on the measurement of the thermal decomposition enthalpy of hydroperoxides by differential scanning calorimetry (DSC) under nitrogen flow.

For the two first methods, significant discrepancies between the experimental results are reported in the literature. In the case of SO 2 -FTIR, these discrepancies are presumably due to different ways to process experimental data. As an example, Tireau et al. [START_REF] Tireau | Propri et es a Long Terme des Gaines de Poly ethyle `ne Haute Densit e Utilis ees pour les Ponts a Haubans[END_REF] used the absorption band characteristic to the reaction products between POOH and SO 2 (i.e., hydrosulfates) at 1042 cm 21 and a molar extinction coefficient of 470 L/mol/cm to estimate the POOH concentration. It can be noticed that other characteristic bands of hydrosulfates could have been chosen for making this quantification, for instance at 920 and 1195 cm 21 . 25,[START_REF] Carlsson | [END_REF] In contrast, Gardette et al. 35 In the case of iodometry, these discrepancies are presumably due to changes in the operating conditions of experimental protocols, especially in the reflux times of samples in sodium iodide solution, but also to different ways to filter the solution after refluxing. Some authors maintained the sample in solution for a sufficiently long time in order to dissolve it completely and thus, avoid diffusion problems into the sample thickness. 33 As an example, according to Richaud et al., 34 in the case of thermo-oxidized polypropylene (PP) films of 70 mm thickness, the reaction time needed for converting all hydroperoxides into alcohols is 30 min. In contrast, according to Carlsson et al., [START_REF] Carlsson | [END_REF] in the case of photo-and thermo-oxidized PE and PP films of 25 mm thickness, a reflux time of 5 min is enough for converting more than 95 mol % of POOH groups. However, for thicker samples having undergone an oxidation under c irradiation (which may then be partially cross-linked), this time should be increased to 60 min. In the case of radio-oxidized PE films of 120 mm thickness, the same authors found a good correlation between the POOH concentrations determined by iodometry and FTIR measurements at 3400 cm 21 . They also reported that some samples (especially photo-oxidized ones) had generated colloidal suspensions after refluxing, presumably due to an extraction of the small macromolecular segments formed by chain scissions. To avoid such a complication, they suggested filtering the solution under a nitrogen atmosphere through a filter of 5 mm pore size, just before performing spectrophotometric measurements.

In contrast, in the case of DSC, experimental results are scarce and poorly scattered. The use of modulated (MDSC) instead of conventional DSC is justified when the melting endotherm of the crystalline lamellae is very close to or is overlapped with the decomposition exotherm of hydroperoxides. This is the case for instance for PP. [START_REF] Richaud | Durabilit e des G eotextiles en Polypropyle[END_REF]37 Each one of these three titration methods has advantages and drawbacks. According to Carlsson et al., 32 those involving a chemical reaction in gas phase are the most reliable for identifying the functional groups, since they avoid the structural modifications (extraction of molecular additives and short macromolecular fragments, morphological restructuration of polymer matrix, etc.) that are usually observed in liquid phase. Thus, these authors 32 showed that SO 2 -FTIR is a reliable method in the case of PE oxidation, but can lead to misleading results in the case of PP oxidation because of the formation of complex POOH compounds. In contrast, according to Jacobson, 25 SO 2 -FTIR would be a poorly reliable method due to the thermal instability of hydrosulfates. Thus, it may only provide semi-quantitative information on the spatial distribution of hydroperoxides in the polymer matrix by coloring the areas rich in hydroperoxides. Finally Gardette et al. 35 indicated that iodometry can only be used if hydroperoxides are thermally stable throughout the experiment duration, typically for temperatures lower than 80 8C.

On the contrary, according to Ouldmetidji, 36 MDSC would present many advantages compared with SO 2 -FTIR and iodometry. Easier and faster to use, it would be also more sensitive.

Mall egol et al. 37 showed that the thermal decomposition enthalpy of hydroperoxides (DH POOH ) is proportional to their concentration. These authors also mentioned that the POOH decomposition is followed by the bimolecular recombination of macro-radicals and their addition onto unsaturations, both chemical reactions inducing the crosslinking of the polymer matrix in the DSC cavity. Thereafter, Richaud et al. 34 used the classical concepts of thermochemistry for expressing DH POOH as a linear combination of the dissociation energies of the different chemical bonds broken or formed during the POOH decomposition and the subsequent chemical events. In the case of thermo-oxidized PP, the experimental determination of DH POOH in the non-reversible MDSC signal was complicated by the overlapping of a narrow component of the melting endotherm, centered around 150 8C, with the wide decomposition exotherm of hydroperoxides ranged between 90 8C and 200 8C.

After having shown that this component is too small to change the quasi-symmetric shape of the decomposition exotherm and thus, can be reasonably neglected, Richaud et al. found a good correlation between MDSC and iodometry. 34 At last, Mallegol et al. 37 used MDSC to determine the molecularity of the POOH decomposition.

A large number of secondary molecular variables can be also used to check the validity of the proposed oxidation mechanistic scheme and the corresponding kinetic model, provided that assumptions are made on formation yields. Most authors mainly focused on carbonyl groups because these latter can be easily detected by FTIR spectrophotometry between 1680 and 1770 cm 21 . Some authors used chemical derivatization methods for demonstrating the presence of various types of carbonyl groups and identifying them. These methods induce a significant shift along the wavenumber axis of the absorption bands of carbonyl groups after reaction with a gaseous reagent, usually ammoniac (NH 3 ) or sulfur tetrafluoride (SF 4 ). As an example, NH 3 transforms carboxylic acids into carboxylates having a characteristic band around 1550 cm 21 . This gas also transforms esters into primary amides absorbing within the 1620-1680 cm 21 range. In contrast, SF 4 reacts with carboxylic acids to give acid fluorides absorbing within the 1810-1840 cm 21 range. As a result, these gaseous treatments allow also better monitoring the changes versus exposure time in the absorption bands of unreactive carbonyl groups such as: ketones, aldehydes, lactones, etc. Based on these results, Tireau et al. proposed to use the absorption bands of carbonyl groups before and after the total conversion of carboxylic acids and esters by NH 3 , but also the resulting absorption bands of carboxylates and primary amides, for estimating the concentration of all types of carbonyl groups formed during the photo-and thermo-oxidation of PE. [START_REF] Tireau | Propri et es a Long Terme des Gaines de Poly ethyle `ne Haute Densit e Utilis ees pour les Ponts a Haubans[END_REF] Pilichowski et al. 31 recognized that these gaseous treatments are of undeniable interest from an analytical point of view, but also argued that they are only effective for thin polymer films of about 100 mm thickness. According to Ollier-Dureault et al., 38 Delor et al., 39 and Maillot et al., 40 SF 4 and NH 3 treatments would lead to similar results, regardless of the operating conditions of experimental protocol or the way to exploit data.

Only a few authors examined the consequences of PE oxidation at the macromolecular scale. Two main processes were detected: random chain scissions and crosslinking. Chain scissions lead to a decrease in the weight and number average molecular masses (i.e., M w and M n ) and, when polydispersity index (PDI 5 M w / M n ) is initially higher than 2, they also induce a narrowing of the molecular mass distribution. Indeed, in the case of a "pure" chain scission process, PDI tends progressively toward 2. In contrast, crosslinking leads to an increase in the average molecular masses and a widening of the molecular mass distribution. All these macromolecular modifications induce a significant change in the rheological properties, in particular in the melt viscosity. [START_REF] Verdu | In Vieillissement des Plastiques; Afnor Technique[END_REF] In the case of PE thermal oxidation, the competition between both macromolecular processes is controversial. Indeed, according to Iring et al., 41 Fayolle et al., 42 and Tireau et al., [START_REF] Tireau | Propri et es a Long Terme des Gaines de Poly ethyle `ne Haute Densit e Utilis ees pour les Ponts a Haubans[END_REF] a "pure" chain scission process would operate since M w and PDI decrease sharply from the beginning of exposure. In contrast, according to Srivastava et al., 43 crosslinking would equilibrate chain scissions in the early periods of exposure since the melt viscosity remains almost constant, while the ultimate elongation and gel fraction increase. Finally, according to Weon 44 and Khabbaz et al., 45 chain scissions would be largely predominant over crosslinking throughout the thermal exposure duration. The existence of a secondary crosslinking process would be attested by the increase in the thermal decomposition temperature or PDI.

The objectives of the present article are twofold. First of all, the reliability of the main titration methods of hydroperoxides, that is, SO 2 -FTIR, iodometry, and MDSC, will be investigated. Secondly, the changes in average molecular masses will be determined by high temperature gel permeation chromatography equipped with a triple detection technology (HT-GPC). The most interesting results will be presented and interpreted in the light of the concentration changes of POOH, carbonyl species, and chain scissions. A peculiar attention will be paid on the possible correlations between the structural modifications taking place at both molecular and macromolecular scales.

EXPERIMENTAL

Materials

A medium density PE classified as PE80 and recommended for applications of pressure pipe and fittings was chosen for this study.

The unstabilized and unfilled PE powder was supplied by Borealis Company. Films of about 100 mm thickness were elaborated by compression molding with a Gibrite laboratory press under a pressure of 24 MPa for 90 s at 190 8C. This low value of thickness ensures a homogeneous oxidation in the whole sample thickness during thermal exposures in air up to temperatures of 200 8C.

After processing, PE films were characterized by conventional laboratory techniques. Their main characteristics are: melting point T m0 5 127.8 8C 6 1.0 8C, crystallinity ratio X C0 5 53% 6 3%, density q 0 5 0.935 g/cm, weight and number average molecular masses M w0 5 120 6 15 kg/mol, and M n0 5 8 6 2 kg/mol.

Oxidative Ageing Conditions PE films were thermally aged below and above their initial melting point at the temperatures of 80 8C, 90 8C, 100 8C, 110 8C, 120 8C, and 140 8C in air-ventilated ovens regulated at 61 8C.

Experimental Methods PE films were regularly removed from ovens and cooled at room temperature in a desiccator containing silica-gel. After complete cooling, they were characterized by a series of complementary analytical methods in order to determine the resulting changes at both molecular and macromolecular scales.

FTIR Analysis. The course of oxidation was monitored by FTIR spectrophotometry in a transmittance mode. The IR spectra of virgin and thermo-oxidized PE films were determined with a Perkin-Elmer Frontier device within the 4000-400 cm 21 range by averaging the 32 scans obtained with a minimal resolution of 4 cm 21 .

As previously observed by Tireau et al., [START_REF] Tireau | Propri et es a Long Terme des Gaines de Poly ethyle `ne Haute Densit e Utilis ees pour les Ponts a Haubans[END_REF] oxidation leads to two main modifications in IR spectra (Figure 1):

Between 3100 and 3700 cm 21 , the sudden appearance and rapid increase of a broad absorption band centered around 3400 cm 21 and assigned to the stretching vibration of the OAH bond of hydroperoxides, alcohols, and carboxylic acids.

Between 1690 and 1760 cm 21 , the sudden appearance and rapid increase of a narrower absorption band having a maximum at 1718 cm 21 and assigned to the stretching vibration of the C@O bond of many potential carbonyl compounds in particular: aldehydes, ketones, carboxylic acids, esters, and lactones.

The identification of these different oxidation products and the determination of their elementary contribution to both composite absorption bands have required to couple FTIR spectrophotometry with chemical derivatization techniques, in particular with SO 2 and NH 3 treatments (see next section).

In IR spectra, oxidation also results in a slow growth of the small absorption bands located at 965, 910, and 888 cm 21 and, respectively, characteristic to vinylene, vinyl, and vinylidene unsaturations (see inset of Figure 1). These latter are initially present in low concentration as structural defects in the PE matrix. Their concentration changes versus exposure time was determined using the Beer-Lambert's law and taking values of molar extinction coefficient of 160, 120, and 100 L/mol/cm, respectively. [START_REF] Mkacher | Vieillissement Thermique des gaines PE et PVC de Câbles Electriques[END_REF][START_REF] Tireau | Propri et es a Long Terme des Gaines de Poly ethyle `ne Haute Densit e Utilis ees pour les Ponts a Haubans[END_REF]46,47 POOH Titration by SO 2 -FTIR. Virgin and thermo-oxidized PE films were exposed in a closed glass vessel at room temperature in an atmosphere fully saturated in sulfur dioxide for converting all hydroperoxides into hydrosulfates according to the following reaction 25 :

POOH 1 SO 2 ! PSO 4 H (1)
Gaseous SO 2 was generated from a highly concentrated solution of Na 2 SO 3 into HCl (typically 2500 g/L) deposited at the bottom of the vessel. After a period of one week, when reaction (1) was completed, the films were analyzed by FTIR spectrophotometry.

Examples of IR spectra of thermo-oxidized PE films after and before SO 2 treatment are presented in Figure 2. As expected, SO 2 treatment leads to a significant reduction of the absorbance of the hydroxyl region (typically between 3100 and 3700 cm 21 ) and the formation of four new absorption bands characteristic to hydrosulfates at 1177 cm 21 (O@S@O), 1050 cm 21 (S@O), 910 cm 21 (SAOAC) and 593 cm 21 (CAS). In agreement with the literature, 25,[START_REF] Carlsson | [END_REF] the absorbance at 1050 cm 21 was chosen for calculating the POOH concentration with the Beer-Lambert's law taking a value of molar extinction coefficient of 155 L/mol/ cm.

POOH Titration by Iodometry. Hydroperoxide concentration was also determined by iodometry which is, to date, the most commonly used analytical method for PP in the literature because of its assumed high reliability. [START_REF] Tireau | Propri et es a Long Terme des Gaines de Poly ethyle `ne Haute Densit e Utilis ees pour les Ponts a Haubans[END_REF]31 This method is based on the reduction of the hydroperoxides by sodium iodide in an acidic medium according to the following reaction [START_REF] Richaud | Durabilit e des G eotextiles en Polypropyle[END_REF]33,37 :

POOH 1 3I 2 1 2H 1 ! POH 1 I 3 2 1 H 2 O (2)
Samples of about 10 mg were immerged in a 7 mL solution consisting in a mixture of isopropanol and acetic acid solvents (10:1) in a two neck glass flask equipped with a bulb condenser. When refluxing, a 4 mL solution composed of sodium iodide dissolved into isopropanol (at a concentration of 200 g/L) was added with a syringe throughout the side neck. After a period of 15 min, when reaction (2) was completed, the mixture was quenched up to room temperature with 25 mL of distilled water.

I 2 3 ions were titrated by UV spectrophotometry at 355 nm with a Perkin-Elmer Lambda 35 device. Their concentration was calculated with the Beer-Lambert's law taking a value of molar extinction coefficient of 25,000 L/mol/cm. The POOH concentration was then deduced by the following equation:

½POOH5 I 2 3 3V solution m PE 3q PE ( 3
)
where V solution is the volume of the solution, m PE is the mass of the PE film, and q PE is the PE density.

Examples of UV spectra of virgin and thermo-oxidized PE films after sodium iodide treatment are presented in Figure 3. One observes clearly an increase in the absorption band at 355 nm with the exposure time at 100 8C in air.

POOH Determination by Modulated DSC. Finally, the POOH concentration was determined by MDSC. This method is based on the measurement of the decomposition enthalpy of hydroperoxides in a neutral atmosphere, typically between 130 8C and 200 8C. In this temperature range, POOH decomposition is mainly bimolecular. 17,18 The resulting alkoxy and peroxy radicals rearrange into alkyl radicals, which then recombine by coupling or disproportionation. Thus, the simplified mechanistic scheme of POOH decomposition can be written such as:

Ia ð Þ Decompostion : 2POOH ! PO 1 PO 2 1 H 2 O DH 1 ð Þ ðIIaÞ b scission : PO ! P H @O 1 P DH 2a ð Þ IIb ð Þ H abstraction : PO 1 PH ! PAOH 1 P DH 2b ð Þ III ð Þ H abstraction : PO 2 1 PH ! POOH 1 P DH 3 ð Þ IVa ð Þ Coupling : P 1 P ! PAP DH 4a ð Þ IVb ð Þ Disproportionation : P 1 P ! PH 1 F DH 4b ð Þ
where F accounts for a vinylidene double bond and DH i are the elementary enthalpies of the chemical events under consideration.

The corresponding total decomposition enthalpy (expressed in kJ/ mol of POOH) is the algebraic sum of all elementary enthalpies:

DH theory 5DH 1 1a 2 DH 2a 1ð12a 2 ÞDH 2b 1DH 3 1a 4 DH 4a 1ð12a 4 ÞDH 4b ( 4
)
where a 2 and (1 2 a 2 ) are the respective partition coefficients between reaction paths (IIa) and (IIb), and a 4 and (1 2 a 4 ) are those between reaction paths (IVa) and (IVb). In the case of PE, but also of copolymers and terpolymers based on the ethylene monomer (e.g., EPR and EPDM), typical values for these coefficients are: a 2 5 0.7 and a 4 5 0.5. 10,48 Values of DH i were calculated from the classical concepts of thermochemistry, that is, as a Examples of non-reversible MDSC thermograms of virgin and thermo-oxidized PE films are presented in Figure 4. The top of the decomposition exotherm is indicated by an arrow. One observes clearly an increase then a decrease in the area of this peak with the exposure time in air at 100 8C.

The POOH concentration was then deduced with the following equation:

½POOH5 DH POOH DH theory (5)
Carbonyl Titration by FTIR2NH 3 . Virgin and thermooxidized PE films were exposed in a closed glass vessel at room temperature in an atmosphere fully saturated in ammonia for converting all carboxylic acids and esters, respectively, into ammonium carboxylates and primary amides according to the following reactions: 49,50

P 5O AOH 1 NH 3 ! P 5O AO 2 1 NH 4 1 (6)
P 5O AOAP 1 NH 3 ! P 5O ANH 2 1 PAOH (7)
Gaseous NH 3 was generated from a pure NH 3 solution deposited at the bottom of the vessel maintained under vacuum (i.e., 10 23 bars) for 10 min. After a period of 1 week, when reactions ( 6) and ( 7) were completed, [START_REF] Achimsky | Etude Cin etique de la Thermooxydation du Polypropylene[END_REF] the films were analyzed by FTIR spectrophotometry.

Examples of IR spectra of thermo-oxidized PE films after and before NH 3 treatment are presented in Figure 5. As expected, NH 3 treatment leads to a significant reduction of the absorbance of the carbonyl region (typically between 1690 and 1760 cm 21 ) and the formation of two new absorption bands characteristic to ammonium carboxylates and primary amides, respectively, centered around 1555 and 1680 cm 21 . The concentrations of carboxylic acids and esters were calculated with the Beer-Lambert's law from the absorbances at 1555 and 1680 cm 21 and taking values of molar extinction coefficient of 330 and 405 L/mol/cm, respectively. [START_REF] Richaud | Durabilit e des G eotextiles en Polypropyle[END_REF]50,[START_REF] Flett | [END_REF] An example of mathematical deconvolution (with Origin 8 commercial software) of the composite absorption band relative to carbonyl groups not having reacted with NH 3 is shown in the inset of Figure 5. It appears clearly that this remaining band is composed of three elementary contributions attributed to unsaturated and saturated ketones (at 1705 and 1719 cm 21 , respectively) and aldehydes (at 1736 cm 21 ). The total concentrations of ketones and aldehydes were calculated with the Beer-Lambert's law from the absorbances at 1705, 1719, and 1680 cm 21 and taking values of molar extinction coefficient of 300, 300, and 155 L/mol/cm, respectively. [START_REF] Tireau | Propri et es a Long Terme des Gaines de Poly ethyle `ne Haute Densit e Utilis ees pour les Ponts a Haubans[END_REF] An example of subtraction in the carbonyl region of the IR spectra obtained before and after NH 3 treatment is reported in Figure 6. The mathematical deconvolution (with Origin 8 commercial software) of the resulting composite absorption band allowed highlighting the elementary contributions of the two types of carbonyl groups having reacted with NH 3 , that is, carboxylic acids (at 1700 and 1713 cm 21 ) and esters (at 1738 cm 21 ). These contributions allowed confirming the concentration values of carboxylic acids and esters determined previously from the absorption bands of their reaction products with NH 3 . The total concentrations of carboxylic acids and esters were calculated with the Beer-Lambert's law from the absorbances at 1700, 1713, and 1738 cm 21 and by taking values of molar extinction coefficient of 680, 680, and 590 L/mol/cm, respectively. [START_REF] Tireau | Propri et es a Long Terme des Gaines de Poly ethyle `ne Haute Densit e Utilis ees pour les Ponts a Haubans[END_REF]53 To summarize, FTIR-NH 3 allowed detecting and titrating four main types of carbonyl groups during the course of PE thermal oxidation: ketones, carboxylic acids, aldehydes, and esters. To demonstrate that it is a quite reliable analytical method, the summation of the elementary concentrations of these carbonyl groups was compared with their total concentration determined with the Beer-Lambert's law from the absorbance at 1718 cm 21 before NH 3 treatment. To satisfy this equality, it was necessary to take a value of 335 L/mol/cm for the composite molar extinction coefficient which, according to literature, 5 is quite realistic.

RESULTS AND DISCUSSION

First of all, the course of the PE thermal oxidation was monitored at molecular scale.

Molecular Scale

Changes in POOH Concentration. Figure 7 displays the changes in POOH concentration versus exposure time in air between 100 8C and 140 8C determined by the three analytical methods under study (i.e., SO 2 -FTIR, iodometry, and MDSC).

One observes that POOH accumulate in PE films up to a maximum concentration ([POOH] S ) characterizing a steady-state, that is, an equilibrium state between their formation in propagation and their decomposition in initiation (see thermal oxidation scheme presented in introduction). The time to reach

[POOH] S is called "oxidation induction time" (t i ). It appears clearly that [POOH] S is an increasing function of temperature, whereas t i is a decreasing function of temperature.

However, as the substrate concentration depletes during the thermal exposure, the steady-state cannot be indefinitely maintained and thus, the POOH concentration ends up decreasing.

It is interesting to notice that the initial value of [POOH] is ranged between 1 3 10 22 and 3 3 10 22 mol/L which, according to literature, 17 corresponds to a very low pre-oxidation level of PE films. Such an order of magnitude was commonly used for simulating the thermal oxidation kinetics of unstabilized and unfilled PE matrices in the literature. 10,11,18 It appears clearly that SO 2 -FTIR and MDSC lead to very similar results. As previously said in introduction, MDSC is considered as the most relevant method for POOH titration by many authors. 33,36,37 To demonstrate that it titrates only POOH, complementary MDSC experiments were performed on thermo-oxidized PE films after SO 2 treatment, that is, after total conversion of POOH into hydrosulfates. As expected, the resulting POOH concentrations are very low: they are typically ranged between 3 3 10 23 and 3 3 10 22 mol/L (Figure 8). This upper boundary is of the order of magnitude of the initial POOH concentration of PE films (i.e., 10 22 mol/L), which roughly corresponds to the detection threshold of the two spectrochemical techniques under study (i.e., iodometry and SO 2 -FTIR). Because of its greater sensitivity, MDSC was chosen as the reference titration method in this study.

In contrast, the fact that SO 2 -FTIR gives access to the same POOH concentrations than MDSC is quite surprising given the numerous potential sources of error reported for this method in the literature. 25,[START_REF] Carlsson | [END_REF][START_REF] Henman | In Developments in Polymer Degradation[END_REF] It can be concluded that: (i) The experimental protocol developed in this study, but also the way to exploit results, are rigorous; (ii) Hydrosulfates are relatively stable at ambient temperature contrarily to the previous findings of Jacobson. 25 Finally, a kinetic analysis was performed to confirm definitively the reliability of both titration methods. In particular, it was tried to demonstrate the correctness of the values of [POOH] S and (t i ). Since POOH decomposition is mainly bimolecular in the temperature range under investigation, it was possible to derive the analytical expressions of both quantities from the thermal oxidation mechanistic scheme presented in introduction 17,18 :

POOH ½ S 5 k 3 ½PH 0 2 ffiffiffiffiffiffiffiffiffiffi k 1b k 6 p and t i 5 12ln Y 0 k 3 ½PH 0 ffiffiffiffi ffi k 1b k6 q (8)
where k 1b , k 3 , and k 6 are the respective rate constants of initiation, propagation, and apparent termination steps. Their respective Arrhenius laws have been determined in a previous publication. 10 As an information, they are recalled in Table I.

[PH] 0 is the initial concentration of methylene groups in the amorphous phase (i.e., 60 mol/L) and Y 0 is the reduced initial concentration of POOH:

Y 0 5 ½POOH 0 ½POOH S (9)
In a first approach, the detection threshold of the spectrochemical techniques (i.e., 3 3 10 22 mol/L) was assigned to [POOH] 0 .

Theoretical values of [POOH] S and t i were calculated with eq. ( 8). They are compared with experimental results in the Arrhenius diagrams of Figure 9. One observes a satisfying agreement between theory and experiment, which demonstrates the reliability of the SO 2 -FTIR and MDSC methods. It is also interesting to notice that the slopes of these straight-lines give access to the activation energies of [POOH] S and t i . They are of 20 and 118 kJ/mol, respectively.

Iodometry leads to the same results than SO 2 -FTIR and MDSC in melt state [Figure 7(d)]. In contrast, it overestimates strongly the POOH concentration after the end of the induction period in solid state [Figure 7(a-c)]. This concentration gap is an increasing function of temperature. It can be concluded that iodometry does not only titrate POOH, but also other types of chemical species which accumulate at high conversion ratios of oxidation process. However, these species disappear rapidly above the melting point, that is, presumably for a significant increase in molecular mobility. Among all the degradation products formed during the thermal oxidation of the PE matrix, only double bonds are capable of reacting with iodine. [START_REF] Guyader Coquillat | Vieillissement des Propergols a Matrice Polybutadie `ne: Mod elisation Cin etique de l'Oxydation[END_REF][START_REF] Tutorskii | [END_REF] According to Tutorskii et al., [START_REF] Tutorskii | [END_REF] this reaction would be written such as: The validity of this assumption was checked by FTIR spectrophotometry. Figure 10 This assumption can be also supported by kinetic considerations. Indeed, it is well known that changes in molecular mobility affect preferentially the rate of chemical reactions involving reactive species in very low concentration in the polymer matrix. In the case of thermal oxidation, these species are PO 2 8

radicals. In saturated hydrocarbon polymers, these radicals propagate oxidation by hydrogen abstraction. In contrast, in polydiene elastomers, an alternative path is the fast addition of PO 2 8 radicals onto double bonds. 57,58 For sake of simplification, this second propagation is usually neglected for PE (see thermal oxidation mechanistic scheme presented in "Introduction" section) because double bonds are formed in a too low concentration to really play a key kinetic role. Many authors considered that double bonds are mainly formed in termination steps by disproportionation of radical species. [START_REF] Khelidj | [END_REF] In the present case, it is suspected that their consumption by PO 2 8 peroxy radicals is facilitated by a sudden increase in molecular mobility, typically when the polymer reaches its liquid state. Changes in Carbonyl Concentration. Figure 11 displays the changes in the total concentration of carbonyl products versus exposure time in air between 80 8C and 140 8C determined by FTIR analysis of the composite absorption band centered at 1718 cm 21 . As already reported in previous studies [START_REF] Verdu | In Vieillissement des Plastiques; Afnor Technique[END_REF]10 and explained in introduction, one observes a sudden auto-acceleration of the oxidation reaction from the end of the induction period. Then, oxidation reaches a maximal rate characterizing a steady-state when t t i . This rate is an increasing function of temperature.

It is important to remember that four main types of carbonyl groups have been identified by NH 3 -FTIR in the experimental section: unsaturated and saturated ketones (at 1705 cm 21 and 1719 cm 21 , respectively), carboxylic acids (at 1700 cm 21 and 1713 cm 21 ), aldehydes (at 1736 cm 21 ), and esters (at 1738 cm 21 ). Figure 12 displays their concentration changes versus exposure time in air between 100 8C and 140 8C. As previously reported by Salvalaggio et al., 49 it appears clearly that ketones are majority before all other carbonyl products. They represent about 65 6 10 mol % of the totality of carbonyl groups whatever the exposure time and temperature, against about 15 6 5 mol % for aldehydes and carboxylic acids, and 5 6 2 mol % for esters. It can be thus concluded that the accumulation curves of the different carbonyl products are homothetic, that is, they result from the same oxidation kinetics. In particular, they exhibit the same induction period.

As an illustration, the concentration of ketones, determined for all the exposure times and temperatures under study, have been plotted versus the sum of the concentrations of aldehydes and carboxylic acids in Figure 13. One observes that all the points are put around a single master straight-line passing through the origin and having for slope:

½ketones ½aldehydes1½carboxylic acids 51:4 (10)
It is important to recall that carbonyl products can be only formed in initiation and termination steps. However, as the rate constants of both reactions exhibit very different values of activation energy (see Table I), there is only one way to get the kind of correlation of Figure 13: each reaction must lead to the formation of all types of carbonyl groups (at least, those which predominate, that is, ketones, aldehydes, and carboxylic acids).

The case of termination was already elucidated in a previous publication, which has led to the proposal of the thermal oxidation mechanistic scheme seen in introduction. [START_REF] Khelidj | Vieillissement d'Isolants de Câbles en Poly ethylene en Ambiance Nucl eaire[END_REF]59 According to this scheme, aldehydes result exclusively from the b scission of PO8 radicals (step VId). However, as the dissociation energy of their CAH bond is significantly lower (368 kJ/mol) than in methylene groups (393 kJ/ mol), [START_REF] Colin | In Handbook of Multiphase Polymer Systems[END_REF] aldehydes are highly sensitive to radical attacks. Therefore, their rapid oxidation leads to the formation of carboxylic acids, as proposed by many authors in the literature. 41,60 In first approach, one could simply reveal these latter into the thermal oxidation mechanistic scheme by replacing half of the aldehydes by carboxylic acids as already done, for instance, in the case of poly(ethylene terephthalate) (PET). 61 At last, ketones are formed by disproportionation of PO8 radicals (event VIc).

The case of initiation is more complicated because, in the temperature range under study, the main source of radicals source is the bimolecular decomposition of POOH (Ib). If the formation of aldehydes and carboxylic acids can be explained with the same chemical events evoked in termination, in contrast, there is no suggestion of formation mechanism of ketones starting from two POOH in the literature. To tentatively overcome such a deficiency, the bimolecular decomposition of POOH has been modified as follows:

Ia ð Þ2POOH ! ½PO 2 OP cage 1 H 2 O k 1a ð Þ Ib ð Þ PO 2 OP ½ cage ! PO 2 1 P 1 c 1 P H @O 1 1 2 c 1 ð ÞPAOH 1 c S S k 1b ð Þ Ic ð Þ½PO 2 OP cage ! POOH 1 P@O ðk 1c Þ
The next section will be dedicated to the consequences of thermal oxidation at macromolecular scale.

Macromolecular Scale. Figure 14 displays the changes in weight average molecular mass (M w ) and PDI versus exposure time in air between 100 8C and 140 8C determined by HT-GPC. As already reported in previous studies, [START_REF] Fayolle | [END_REF][START_REF] Tireau | Propri et es a Long Terme des Gaines de Poly ethyle `ne Haute Densit e Utilis ees pour les Ponts a Haubans[END_REF]42 one observes that both quantities decrease catastrophically from the beginning of exposure, which suggests that an intensive chain scission process operates during the thermal oxidation of PE. As PDI tends eventually toward the asymptotic value of 2, it can be concluded that this is a "pure" chain scission process (i.e., with absence of crosslinking). The rate of chain scissions is clearly an increasing function of temperature.

The absence of induction period confirms that HT-GPC is an analytical method more sensitive than common spectrochemical methods, in particular FTIR spectrophotometry. That is the reason why embrittlement of polyolefins is generally observed before the build-up onset of oxidation products. [START_REF] Fayolle | [END_REF] However, this scenario is less clear if one refers to the number of chain scissions.

Since one chain scission creates a new macromolecular chain and two new chain-ends, the number of chain scissions can be simply written such as:

1 M n 2 1 M n0 q 0 (11) Or S5 PDI M w 2 PDI 0 M w0 q 0 ( S5
) 12
where q 0 is the initial polymer density (expressed in kg/L if M n and M w are given in kg/mol).

The changes in S in air between 100 8C and 140 8C were calculated from eq. ( 12). They are compared with the corresponding changes in [POOH] in Figure 15. It appears clearly that chain scissions are not numerous enough to be detected by GPC for about the first half of the induction period (i.e., for t i /2). This is due to the fact that the decrease in M w is counterbalanced by the decrease in PDI in the early periods of exposure, thus maintaining M n close to its initial value M n0 . However, when PDI approaches its asymptotic value, M n ends up becoming lower than M n0 . Thenceforth, the rate of chain scissions increases suddenly up to reach a maximal value (r S ) when [POOH]

[POOH] S . All these observations allow classifying the different analytical methods under study according to their sensitivity order:

MDSCտSO 2 2FTIR Iodometry > GPC > standard FTIR
In the case of a bimolecular decomposition of POOH, an analytical expression for r S is:

r S 5c 0 S k 1b ½POOH 2 S 12c 00 S k 6 ½PO 2 2 S ( 13
)
where c 0 S and c 00 S are the respective apparent yields of chain scissions in initiation and termination.

The maximal concentration of PO 2 8 radicals has been reported elsewhere 17 :

½PO 2 S 5 2k 1b ½POOH 2 S k 3 ½PH 5 k 3 ½PH 2k 6 ( 14
)
Introducing eq. ( 14) into eq. ( 13) gives finally: Theoretical values of r S were calculated with eq. ( 15) taking c 0 S 5 c 00 S 5 0.6 throughout the temperature range under investigation. They are compared with experimental results in the Arrhenius diagrams of Figure 16. One observes a satisfying agreement between theory and experiment, which confirms the reliability of the HT-GPC method. The activation energy of r S is 146 kJ/mol.

Finally, the number of chain scissions, determined for all the exposure times and temperatures under study, have been plotted versus concentration sum of aldehydes and carboxylic acids in Figure 17. One observes that all the points are gathered around a single master curve passing through the origin. This correlation allows demonstrating the validity of the assumption (made in section "Changes in carbonyl concentration") that aldehydes and their oxidation products (i.e., carboxylic acids) result exclusively from the b scission of PO8 radicals.

The fact that the sum of their concentration remains lower than the number of chain scissions suggests that a non-negligible part of these carbonyl products is lost as volatile compounds during the thermal exposure. As an example, it is well known that acyl and carboxyl radicals, involved in the oxidation mechanism of aldehydes into carboxylic acids, can rearrange by b scission to give carbon monoxide and dioxide, respectively. 41 Such respective decarbonylation and decarboxylation reactions could be taken into account by lowering the values of the formation yields of carboxylic acids in initiation and termination steps.

To summarize, all these results highlight a strong correlation between the chemical processes induced by oxidation at molecular scale and their consequences at macromolecular scale.

CONCLUSIONS

The thermal oxidation of additive free PE was carefully investigated in air between 100 8C and 140 8C at both molecular and macromolecular scales with several complementary analytical methods. On one hand, the POOH concentration was determined by iodometry, MDSC and SO 2 -FTIR, whereas carbonyls products were clearly identified and titrated by NH 3 -FTIR. It was shown that iodometry does not only detect POOH, but also other chemical species formed at high conversion ratios of the oxidation process, namely double bonds. On the other hand, the weight average molecular mass and polydispersity index were determined by HT-GPC for deducing the corresponding number of chain scissions. The different sensitivity levels of these techniques have been identified.

A complementary kinetic analysis was performed on key ageing criteria such as: oxidation induction time, maximal concentration of POOH, and maximal rate of chain scissions, in order to show the efficiency of the kinetic modeling tool developed for predicting the thermal oxidation kinetics of PE in previous studies at the laboratory, but also for checking the correctness of experimental results.

Despite its apparent success, it appeared that this tool is unable to predict some important characteristics of the oxidation process such as: (i) The formation of a large variety of carbonyl groups (mainly ketones, aldehydes, and carboxylic acids); (ii) The existence of a correlation between the concentrations of these different carbonyl groups; (iii) The existence of a correlation between the number of chain scissions and the concentration sum of aldehydes and their oxidation products (i.e., carboxylic acids).

To tentatively overcome these deficiencies, it was proposed to improve the thermal oxidation mechanistic scheme from which the kinetic model is derived. These improvements will consist in adding: (i) An alternative reaction pathway for the bimolecular decomposition of POOH; (ii) The rapid oxidation of aldehydes into carboxylic acids; (iii) The loss of aldehydes and carboxylic acids as volatile compounds. The validity of these different assumptions will be checked in future studies.

APPENDIX

Let us consider the following simplified mechanistic scheme for the bimolecular decomposition of hydroperoxides in a neutral atmosphere:

I ð Þ Decomposition : 2POOH ! PO 1 PO 2 1 H 2 O ðDH 1 Þ IIa ð Þ b scission : PO ! P H @O 1 P ðDH 2a Þ IIb ð Þ H abstraction : PO 1 PH ! PAOH 1 P ðDH 2b Þ III ð Þ H abstraction : PO 2 1 PH ! POOH 1 P ðDH 3 Þ IVa ð Þ Coupling : P 1 P ! P2P ðDH 4a Þ IVb ð Þ Disproportionation : P 1 P ! PH 1 F ðDH 4b Þ
A rapid review of available data from organic chemistry allowed us compiling the dissociation energies of the different chemical bonds involved in this mechanistic scheme (Table A1).

The elementary enthalpies DH i of each chemical event were calculated from the classical concepts of thermochemistry as shown below. First of all, the decomposition of one POOH group (I) leads to the breaking of one OAO bond and one OOAH bond, and the formation of one HOAH bond, so that: At last, the coupling of two alkyl radicals (IVa) leads to the formation of one CHACH bond, so that:

DH 4a 5 2DH >CHACH< 5 2328kJ=mol (A5)
In contrast, the disproportionation of two alkyl radicals (IVb) leads to the formation of one CH5CH bond, so that:

DH 4b 5 DH ACH@CHA 5 2303kJ=mol (A6)
It is important to recall now that the total decomposition enthalpy of hydroperoxides (expressed in kJ/mol of POOH) writes:

DH theory 5 DH 1 1 a 2 DH 2a 1 1 -a 2 ð ÞDH 2b 1 DH 3 1 a 4 DH 4a 1 1 -a 4 ð ÞDH 4b (A7)
with a 2 5 0.7 and a 4 5 0.5.

The numerical application leads finally to:

DH theory 52291:2 kJ=mol:

Figure 1 .Figure 2 .

 12 Figure 1. IR spectra of PE films before (dash lines) and after thermal exposure at 120 8C in air (solid lines). The inset shows an enlargement of the absorption regions characteristic to unsaturations. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

  function of the dissociation energies of the different chemical bonds broken or formed in each chemical event. Their calculation is detailed in Appendix A. It comes finally: DH theory 5 2291 kJ/mol.Samples of about 5 mg were heated under a nitrogen flow of 50 mL/min from 20 8C to 220 8C with a heating rate of 1 8C/ min and a superimposed temperature modulation (of amplitude 5 0.159 8C and period 5 60 s) in the cavity of a TA Instruments Q1000 calorimeter. The decomposition enthalpy of hydroperoxides DH POOH was graphically determined in the non-reversible MDSC signal as the area under the exotherm located in the 150 8C-200 8C range, that is, just above the contribution of the melting endotherm centered around 128 8C.

Figure 3 .Figure 4 .

 34 Figure 3. Examples of UV spectra of virgin and thermo-oxidized PE films after sodium iodide treatment. Thermal ageing conditions: 100 8C in air for 216 h. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] Figure 4. Examples of non-reversible MDSC thermograms of virgin and thermo-oxidized PE films. Thermal ageing conditions: 100 8C in air for 408 h. The arrows indicate the maxima of the decomposition exotherm of POOH. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 5 .Figure 6 .

 56 Figure 5. Example of IR spectra of thermo-oxidized PE films before (dash lines) and after (solid lines) NH 3 treatment. Thermal ageing conditions: 100 8C in air for 216 h. The inset shows the deconvolution of the composite absorption band relative to carbonyl groups not having reacted with NH 3 . [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 7 .

 7 Figure 7. Changes in POOH concentration during the thermal ageing of PE films in air at 100 8C (a), 110 8C (b), 120 8C (c), and 140 8C (d). Comparison of the results obtained by three different titration methods: SO 2 -FTIR (᭡), iodometry (᭜), and MDSC (1). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 8 .

 8 Figure 8. Changes in POOH concentration determined by MDSC before (1) and after (᭹) SO 2 treatment of thermo-oxidized PE films. Thermal ageing conditions: 100 8C (a), 110 8C (b), 120 8C (c), and 140 8C (d) in air. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

  displays the changes in the total concentration of double bonds (i.e., including vinylene, vinyl, and vinylidene unsaturations) versus exposure time in air between 100 8C and 140 8C. It appears clearly that this concentration is of the same order of magnitude than the absolute value of the gap between the POOH concentrations previously determined by iodometry and MDSC (or SO 2 -FTIR), i.e., [POOH] iodometry 2 [POOH] MDSC .

Figure 9 .Figure 10 .

 910 Figure 9. Arrhenius graphs of the maximal POOH concentration (left) and oxidation induction time (right) of PE films in air between 100 8C and 140 8C. Comparison between experiment (points) and theory [eq. (8), solid lines]. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 11 .

 11 Figure 11. Changes in the total concentration of carbonyl groups during the thermal ageing of PE films in air at 80 8C ( ), 90 8C (᭹), 100 8C (᭜), 110 8C (), 120 8C (᭡), and 140 8C (1). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 12 .Figure 13 .

 1213 Figure 12. Changes in the concentration of ketones (᭜), aldehydes (), carboxylic acids (᭡), and esters (1) during the thermal ageing of PE films in air at 100 8C (a), 110 8C (b), 120 8C (c), and 140 8C (d). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 14 .Figure 15 .

 1415 Figure 14. Changes in weight average molecular mass (top) and polydispersity index (bottom) during the thermal ageing of PE films in air at 100 8C (᭜), 110 8C (), 120 8C (᭡), and 140 8C (1). [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 16 .Figure 17 .

 1617 Figure 16. Arrhenius graphs of the maximal rate of chain scissions in PE films in air between 100 8C and 140 8C. Comparison between experiment (points) and theory [eq. (15), solid lines]. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

DH 1 5

 5 DH OAO 1 DH OOAH 1 DH HOAH 5 148 1 376 -498 5 26 kJ=mol (A1)Then, the b scission of one alkoxy radical (IIa) leads to the breaking of one >CHACH 2 A bond and the formation of one CAO bond, so that:DH 2a 5 DH >CHACH2A 2 DH CAO 5 343 -341 5 2 kJ=mol (A2)Moreover, the hydrogen abstraction by one radical (IIb and III) leads to the breaking of one CHAH bond and the formation of one OAH bond. This latter belongs to an alcohol in the case of an alkoxy radical, and a hydroperoxide in the case of a peroxy radical, so that: DH 2b 5 DH >CHAH 2 DH OAH 5 393 -460 5 267 kJ=mol (A3) DH 3 5 DH >CHAH 2 DH OOAH 5 393 -376 5 217 kJ=mol (A4)

Table I .

 I Arrhenius Parameters of Some Rate Constants of PE Thermal Oxidation

	Rate constant	Pre-exponential factor	Activation energy
	(L/mol/s)	(L/mol/s)	(kJ/mol)
	k 1b	2.8 3 10 9	105
	k 3	1.5 3 10 10	73
	k 6	3.0 3 10 9	0

Table A1 .

 A1 Bond Dissociation Energies[START_REF] Calvert | Photochemistry[END_REF][START_REF] Morrison | Organic Chemistry[END_REF][START_REF] Kelter | Chemistry: The Practical Science[END_REF]

	Chemical bond	DH (kJ/mol)
	OAO	148
	OOAH	376
	CAO	341
	OAH	460
	HOAH	498
	>CHAH	393
	>CHACH 2 A	343
	>CHACH<	328
	ACH@CHA	(23) 303