Julien Fernandez

Caractérisation des mécanismes d'endommagement de matériaux polymères isolants pour câbles haute-tension.

J'aimerais tout d'abord remercier Florence Clément ainsi que Didier Long, alors directrice et directeur adjoint du Laboratoire des Polymères et Matériaux Avancés (LPMA), qui m'ont permis de réaliser ce travail au sein du laboratoire. Cette thèse a été une expérience à la fois scientifique et humaine extrêmement enrichissante, forte d'une collaboration entre de nombreux partenaires industriels et académique. J'ai pu rencontrer et travailler avec des gens formidables, et c'est en partie grâce à vous. Merci Florence de m'avoir accompagné durant ces années, entre Nexans et Solvay, et d'avoir fait en sorte que je puisse m'exprimer au mieux dans mon travail. Didier, je te remercie de m'avoir accordé ta confiance pour réaliser ce travail, et pour les nombreux échanges que nous avons eu ensemble. J'ai appris énormément de choses, et je suis content d'avoir pu explorer les questions que l'on s'est posé, notamment sur le band gap.

Mes plus sincères et profonds remerciements vont à Paul Sotta, mon directeur de thèse. Paul, merci infiniment pour tout ce que tu m'as apporté: merci de m'avoir fait confiance sur ce « nouveau » sujet, pour ton soutien et ton temps précieux lorsqu'il fallait nous creuser la tête sur ces questions d'électrons, tes conseils et ton expérience qui m'ont permis de progresser et devenir un meilleur chercheur durant ces années, et tes qualités humaines qui m'ont permis de travailler dans un environnement serein. C'est en grande partie en travaillant avec toi que je pu développer cette rigueur dont tu parlais, alors je te remercie pour tout.

Je tiens à remercier Lionel Flandin et Séverine Leroy, pour avoir accepté de rapporter ce travail, et d'avoir pris le temps de l'évaluer de manière claire et pédagogique. Merci Lionel d'avoir accepté de présider le jury lors de ma soutenance, dans ces conditions si particulières. Enfin merci à Eliane Espuche, Sandrine Thérias et Jean-François Larché pour avoir accepté de participer à la lecture du manuscrit et au jury lors de ma soutenance, et pour leurs nombreuses questions qui ont démontré tout leur intérêt pour ce travail. Je souhaite également remercier René Fulchiron, Olivier Lhost et Mathieu Tauban pour m'avoir aidé à la relecture de ce manuscrit et avoir accepté mon invitation pour participer à la soutenance de ma thèse.

Ce travail a été réalisé en grande partie au centre de recherche de Nexans à Lyon, où j'ai pu rencontrer des personnes formidables qui m'ont accueilli, accompagné, et aidé pendant plus de deux ans. Je souhaiterais d'abord remercier Arnaud Allais et Bernard Dalbe pour m'avoir permis de réaliser ce travail dans le cadre du LPSE et au sein de Nexans, pour les nombreux échanges que nous avons eu et leur bienveillance. Je veux remercier chaleureusement Jean-François Larché, et Raphaël Guffond qui m'ont encadré durant ce travail : Jean-François, merci pour tout ce que tu as fais pour moi, pour m'avoir guidé et conseillé pendant ces trois années, m'avoir appris à être plus rigoureux, et m'avoir également permis de gagner en maturité. C'était une réelle chance de pouvoir travailler à tes côtés, et j'ai vraiment beaucoup apprécié. Raphaël, je te remercie de t'être autant investi dans mon travail, de m'avoir appris tant de choses, merci de ta gentillesse et ton écoute. Sans vous deux, beaucoup de ces résultats de travaux expérimentaux et analyses n'auraient jamais pu voir le jour. Je remercie également Dimitri Charrier et Patrick Fornage du laboratoire électrique pour tout le temps précieux qu'ils m'ont accordé, ainsi que l'aide et les conseils qu'ils m'ont apporté. Un grand merci à Jihane Sahyoun, Anne-Gaëlle Alexandre et Jean-Michel Marty pour leur aide et leur temps en caractérisations, en analyses, en formation sur les machines et à la bonne réalisation de mes essais. Je remercie tout particulièrement Linda Lomri Boudiaf, Philippe Rotigni et Charles Barusta pour leur grande gentillesse, leur disponibilité et leur aide précieuse. Enfin, je souhaite remercier toutes les personnes qui ont fait que cette thèse se déroule dans de si bonne conditions et en si bonne compagnie. Merci à mes collègues de bureau n°1, Philippe, Vincent, Anne-Gaëlle, Stéphanie et Nicolas (pour les matchs de tennis bien disputés) ; de bureau n°2, Laurent, Philippe, Linda et Guillaume (pour le tennis aussi, ne sois pas jaloux) ; de bureau n°3 (!), Lucie, Linda, Aurore, Christophe, Didier et Luc (pour nos nombreuses discussions chaleureuses et tes conseils en matière de vins!) ; à la maintenance, Nicolas et Brice ; et au Sud, merci à Alain, Melek, Claire, Serge, Daphné, Cécile, et toutes les personnes qui m'ont accordé du temps et de la sympathie. Enfin, merci pour leur gentillesse et bonne chance aux futurs docteures Camille et Bushra ! J'ai eu la chance durant cette thèse de travailler au sein de deux grandes familles, et je ne pourrais oublier les personnes avec qui l'aventure a commencé, au sein du centre de recherche RICL de Solvay à Saint-Fons. J'aimerais remercier d'abord Mathieu Tauban, qui m'a encadré et épaulé durant ce travail. Mathieu, merci infiniment pour la confiance que tu m'as accordé durant toutes ces années, depuis ce stage qui a été un tournant dans ma vie professionnelle. Merci pour tout ce que tu m'as apporté : ton temps, ton aide, tes conseils, ton professionnalisme, mais surtout ta sympathie, ton honnêteté, et ta bienveillance. Tu le sais déjà mais c'est en grande partie grâce à toi que j'ai décidé de me lancer dans cette aventure, alors oui, un grand merci. Je souhaite remercier les ingénieurs du département PM2D (bien qu'il n'ait changé depuis) avec qui j'ai pu travailler directement ou bien pour les échanges que nous avons eu : merci à Lise Trouillet-Fonti (pour ta sympathie et pour être proche des doctorants.es), Jérôme Bikard (pour l'introduction au calcul numérique!), Alexandra Argoud, Rabih Al Orabi, Olivier Sanseau, Pierre-Yves Lahary, Caroll Vergelati. Je tiens ensuite à remercier tout particulièrement les personnes qui ont rendu ce travail possible. Un grand merci à Sylvie Ghiringhelli pour ton accompagnement et ta gentillesse, merci de prendre soin des étudiants.es et de nous faciliter tant la tâche. Merci à Christine Prebet et Sandrine Rossi, pour votre gentillesse et le temps passé à me former sur les machines et à m'aider au moindre soucis. Je remercie tout particulièrement Pauline Grau, Clémence Abadie, Alexandra Siot et Baptiste Gros (aussi pour les concerts et les sorties culturelles), pour tout le temps et l'énergie passée à me former et me guider dans mes analyses de microscopie. Ce laboratoire est très précieux pour les étudiants.es, et grâce à vous nous avons accès à des analyses de qualité, malgré parfois notre maladresse. . . ! Merci à toutes celles et ceux qui ont participer de près ou de loin à ce travail : Laetitia Trébollet (pour les analyses d'infrarouge très précieuses), Olivier Bronchart (nos discussions et les conseils de pêche), Fabrice Millet, Violaine Ducru, Michel Sorin, Olivier Andres, Vincent Curtil, Tiffanie Bergey (pour avoir fait de v nous des volleyeurs!), Robert Stutz, Thibault Marque (pour tes analyses et connaissances sportives sans failles) et Ilan Villard-Levy (pour ton énergie, tes inspirations musicales et nos matchs de tennis à en mourir de chaud). Merci pour toute votre aide, votre disponibilité et votre générosité.

Je remercie chaleureusement Yves Trollez et Olivier Lhost de Total Petrochemicals, pour nos échanges et leurs conseils. Merci Olivier pour ton investissement, nos discussions scientifiques et ton avis éclairé sur la physique des polymères pour ces applications électriques, j'en garde un bon souvenir.

Lors de ce travail, j'ai eu la chance de pouvoir collaborer avec le milieu industriel mais aussi académique. Notamment, je me souviendrai toujours des échanges que nous avons eu avec les membres de l'équipe DSF du laboratoire LAPLACE à Toulouse, et de ma venue au laboratoire. Je remercie sincèrement Séverine Leroy, Laurent Berquez, Christian Laurent et Gilbert Teyssedre, pour leur accueil, et pour nos échanges si intéressants sur le sujet. Gilbert, merci infiniment pour ta disponibilité, tes explications et conseils si riches, et pour m'avoir fait découvrir ces manips de luminescence si importantes pour moi.

Je remercie très chaleureusement René Fulchiron, du laboratoire IMP, pour son implication dans mon travail, nos échanges avec Paul sur les polymères, et m'avoir apporté un éclairage précieux sur le vieillissement de mes matériaux.

Un très grand merci à Servane Haller, et Xavier Festaz, pour leur accueil au sein du Supergrid Institute, leurs disponibilité et leurs conseils pour les manips réalisées en hautes-fréquences.

Je tiens à remercier également Pierre-Antoine Albouy, du laboratoire LPS, pour les manips de SAXS réalisés sur mes échantillons.

Je ne peux évidemment pas oublier de remercier mes camarades de thèse et amis.es, pièces maîtresses de la réussite de cette thèse, avec qui j'ai passé près de quatre années fortes en intensité : Agathe et Rita, les jeunes mamans qui ont aussi été nos « mamans » au labo en nous guidant alors que l'on était que de jeunes thésards, merci pour votre bienveillance et votre soutien. Rita tes playlists à l'approche de Noël et ton enthousiasme me manquent déjà. Jessica, toujours de bonne humeur et qui m'a aidé et guidé surtout pendant le stage, merci. Éléonore, notre chère déléguée du LPMA, qui a parfaitement tenu son rôle en nous gardant toujours soudés et en apportant une super ambiance, je n'oublierai pas notre expérience scientifique et culturelle en Crête que tu as su organiser brillamment (Flapiakys nous manque). Stéphanie, notre deuxième maman de bureau qui nous a beaucoup aidé et soutenu, toujours disponible et généreuse, et l'aventure continue puisque l'on va à nouveau occuper le même bureau ! Agathe Virfeu, Gatou, partie trop tôt (snif), merci de m'avoir tant apporté, ton sourire, ta bonne humeur et ton énergie m'ont manqués au bureau, mais ce n'est que pour mieux les retrouver (on doit venir à Nancy, ça urge). Héloïse, la reine du MEB (oups), je me souviens quand tu es arrivée en stage ça me paraît être hier (!), tu as depuis pris en assurance, en gardant ta gentillesse et ta bienveillance, tu sais être proche des gens, ne change pas ! Mélissa, notre petite soeur de bureau, et notre soleil avec Nico (littéralement puisque tu bronzais toute l'année à la fenêtre), merci pour m'avoir tant apporté, avoir fait survivre nos plantes, avoir égaillé nos journées de travail et surtout nous avoir supporté dans ce bureau à trois (bon ok deux et demi vu mon temps de présence) ! Anne, notre dernière recrue et grande aventurière, merci de m'avoir inspiré, merci pour ta grande gentillesse et ton écoute, et bonne chance à vous trois pour la fin de thèse ! Thomas, mon grand compagnon d'échec, mon maître à jouer, tu as aussi été une grande source d'inspiration et de motivation, même si cela ne m'a mené qu'à quelques petites victoires par Mat ! Je te souhaite un bon courage pour finir cette thèse, ça ne saurait tarder ! Francesco, le guide de cette belle « compagnie », merci pour tout tes conseils, ton implication, ton énergie infinie, merci pour ton amitié sincère. J'aurais aimé te garder plus longtemps avec nous, mais je ne peux pas lutter contre l'appel de l'Italia ! Enfin Nicolas, mon grand compagnon de tous les instants, merci pour tout ton soutien, tes goûts musicaux hors normes, tes conseils et tes avis éclairés, ta joie de vivre internissable, et bien sûr pour ton amitié si précieuse. A vous tous.tes, pour toutes vos qualités, et vos défauts (aller, il doit bien y en avoir quelques uns), merci d'avoir fait en sorte que ces quatre années deviennent une si belle aventure.

Durant cette thèse j'ai également côtoyé beaucoup de jeunes en stage, qui font partie de l'aventure et que je remercie pour avoir contribué à ce travail, en particulier Renaud Cavailles. Merci Renaud pour ton travail, et pour m'avoir aidé à réaliser autant de manips sur tout ces matériaux, ça été très précieux pour moi et une belle rencontre. Une pensée également pour Carolina, notre soleil brésilien, Océane, notre pile électrique du Sud, désormais parisienne, qui je l'espère viendra nous voir encore souvent (Lyon restera ta ville de coeur!), Emmanuelle, Pauline et son incroyable énergie, Alexandre le grand grimpeur, Farès, Léa et Morgane, merci à vous ! Merci à mes amis.es pour leur grand soutien, pour avoir cru en moi (alors même peut être que notre prof de physique n'aurait pas mis un centime) et pour m'avoir accompagné jusqu'au bout.

Enfin, je ne pourrais remercier assez ma famille, parents, oncles et tantes, frère, soeur et cousins, pour leur soutien infaillible. Merci Papa de m'avoir guidé et encouragé à poursuivre cette voie, qui t'avait autrefois inspiré et qui m'a naturellement attiré ; merci Maman de m'avoir donné l'envie, la motivation et le courage de m'accrocher, et d'être une source d'inspiration si puissante. Merci Alexia et Hugo, mes chers frère et soeur de m'avoir accompagné jusqu'à même la soutenance, c'est aussi grâce à vous que j'ai la motivation de me dépasser pour vous montrer l'exemple ! Merci à tous les quatre d'avoir contribué à faire de moi l'ingénieur et l'homme que je suis devenu aujourd'hui.

Parmi les matériaux utilisés dans les applications d'isolation électrique, comme par exemple les câbles de puissance, les polymères synthétiques sont largement représentés. Notamment, le Polyéthylène (PE) et Polyéthylène réticulé chimiquement (XLPE) constituent une grande part des isolants utilisés pour les applications basse et moyenne tension. Si ces matériaux sont choisis pour leurs bonnes propriétés thermiques et diélectriques, ainsi que leur intérêt économique, la durée de vie du câble en application reste limitée par l'endommagement thermoélectrique de l'isolant. Dans le but d'améliorer la compréhension des phénomènes physiques d'endommagement de ces matériaux sous moyenne tension alternative (MV-AC), une étude expérimentale de l'évolution des propriétés de différentes formulations de matériaux polymères a été réalisée, et ce des prémices de l'endommagement jusqu'à la rupture finale du matériau.

L'étude de l'état de l'art sur les mécanismes d'endommagement thermoélectrique des polymères isolants a permis de mettre en avant les rôles intriqués de nombreux facteurs agissant sur l'endommagement. Le couplage de la température et du champ électrique, les type et la localisation des contraintes, la présence de défauts physiques et chimiques, la morphologie cristalline du matériau, mais aussi sa structure électronique, sont autant de facteurs clés dans la compréhension de l'endommagement.

Nous avons dans un premier temps préparé différentes formulations d'échantillons polymères. Des films de PE et XLPE ont été réalisés dans le but d'étudier l'impact de la formulation et préparation des matériaux sur l'endommagement. Deux grades de Polypropylène (PP) de cristallinités différentes ont également été préparés, afin d'étudier l'impact de la morphologie cristalline. Un banc d'essai MV-AC a été conçu en utilisant une géométrie d'électrode hémisphère-plan, permettant un vieillissement localisé, couplé à des effets de décharges en surface.

Une seconde géométrie d'électrode a également été utilisée sur des échantillons de XLPE, en enrobant le polymère autour de l'électrode afin d'étudier des effets en volume, plus caractéristiques de l'endommagement intrinsèque du matériau (voir figure 1) Les durées de vie des différents matériaux dans ces configurations ont été mesurées à une température d'essai de 90°C, sous différentes valeurs de champs électriques en courant alternatif de fréquence 50 Hz. Afin d'accélérer le vieillissement MV-AC, des valeurs de champ électrique comprises entre 10 et 50 kV/mm ont été utilisées. L'impact de la modification de la fréquence et de la température a également été exploré. Les données de durée de vie obtenue ont permis d'établir des courbes Tension vs Durée de vie pouvant être modélisée par des lois de comportement de type puissance inverse (t ∝ V -n ) . Ces courbes ont montré le rôle important de la structure du matériau, notamment les effets de la réticulation ou de la cristallinité sur l'allongement de la durée de vie.

Sur la base ces données, ces mêmes matériaux ont été soumis à des vieillissements contrôlés permettant d'obtenir des échantillons à des stades d'endommagement croissant du début de la durée de vie à un état proche de la rupture finale. Des caractérisations physico-chimiques et structurales à différentes échelles, comme la Spectroscopie Infrarouge (IR) ou la Microscopie Électronique à Balayage (MEB), ont permis de mettre en évidence des marqueurs d'endommagements caractéristiques ainsi que le caractère progressif de l'endommagement. Par ailleurs, des caractérisations des propriétés électriques, comme la mesure de courant de fuite ou la mesure de la tension claquage ont également été réalisées afin d'affiner la compréhension de l'évolution des propriétés du matériau à différents stades de l'endommagement. Enfin, des mesures de photoluminescence (PL) et luminescence induite par plasma (PIL) sur des échantillons de XLPE vieillis à des stades avancés de la durée de vie ont montrés des effets probables du vieillissement sur la recombinaison de charges au sein des échantillons. Lors de cette étude, une attention particulière a été portée à la décorrélation des effets thermiques et électriques sur l'endommagement. De plus, la comparaison des durées de vie, ainsi que des marqueurs observés lors du vieillissement dans les deux géométries d'électrode, ont permis de montrer les effets du type de contrainte sur les mécanismes d'endommagement. La synthèse des résultats expérimentaux obtenus lors de ces caractérisations multi-échelles a permis une avancée expérimentale notable dans la compréhension des phénomènes de vieillissement des polymères isolants sous différentes contraintes. Ces résultats constitueront une base de travail dans le but d'alimenter un modèle numérique de prédiction de la durée de vie des matériaux polymères pour l'isolation électrique. ix Cette thèse est organisée comme suit. Le premier chapitre a pour but de présenter l'état de l'art concernant les matériaux isolants polymériques utilisés pour le transport électrique, ainsi que les principaux mécanismes et théories connus sur le vieillissement électrique. Plus précisément, il vise à présenter l'importance de contrôler les propriétés physiques et chimiques de l'isolation, ainsi que les conditions de stress appliquées sur le matériau.

Le deuxième chapitre présente les matériaux qui ont été choisis pour les études de durée de vie et vieillissement. Le processus de fabrication et préparation des échantillons dans les deux géométries d'électrodes sera présenté, ainsi que les bancs expérimentaux développés pour les tests électriques et les différentes contraintes appliquées sur les matériaux. En outre, les techniques de caractérisation physique, chimique, structurelle et électrique, ainsi que leur principe de fonctionnement sont décrits.

Le troisième chapitre présente les principaux résultats obtenus sur les mesures de durée de vie. La méthode d'analyse des données de durée de vie sera discutée, ainsi que la signification physique des différents paramètres régissant la durée de vie des matériaux testés. Les paramètres critiques, parmi les propriétés des matériaux ou bien les conditions de contraintes appliquées, responsables de la dégradation majeure des matériaux seront discutés. De plus, l'effet des deux géométries d'électrode sur la durée de vie sera discuté et interprété.

Introduction General Objectives

The development and efficient use of electrical power cables, especially the choice of proper insulation is a topical issue which dates back the nineteenth century [START_REF] William A Thue | Electrical Power Cable Engineering[END_REF]. Insulating materials used for these applications have been various, but today Crosslinked Polyethylene (XLPE) is the most commonly used insulation material for medium (MV) and high voltage (HV) applications. The demand for power cables is growing in these days, at the time when electrification of our society is a key societal and industrial concern for replacement of fossil fuel and limits the risks associated to climate changes [START_REF] Ribbing | Climate Change Leadership -the case for Electrification[END_REF]. Replacement of these energy sources by renewable energy production thus leads to an increased interest for electrical transport, and thus the need of reliable and durable power cables.

The durability of power cable insulation, and the understanding of failure mechanisms under electrical stress is thus a major industrial concern.

The mechanisms of electrical breakdown have been extensively studied : Paschen mechanisms, treeing, avalanche breakdown, etc. [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF]. Nowadays, many improvements have been done on the comprehension of damaging mechanisms of synthetic insulators and researchers proposed universal theories to summarize the identified mechanisms of electrical ageing [START_REF] Morshuis | Degradation of solid dielectrics due to internal partial discharge: some thoughts on progress made and where to go now[END_REF][START_REF] Alexander | The Theory of Everything In The Electrical Breakdown Of Polymeric Dielectrics: Maybe[END_REF]. Well detailed books were also written aiming to unravel the different theoretical and experimental aspects of electrical ageing and breakdown (failure) [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Kao | Dielectric phenomena in solids[END_REF][START_REF] Martinez-Vega | Dielectric materials for electrical engineering[END_REF][START_REF] Ramu | Diagnostic testing and life estimation of power equipment[END_REF][START_REF] Gorur | Dielectrics in electric fields[END_REF]. These mechanisms mostly describe the final failure of a material, that is, the final stage of damage. In such cases, the time scale between the onset of significant or measurable damage, and the failure of the material, is very short.

Describing situations corresponding to very long lifetimes (so-called fatigue) is a different, even more challenging issue. In such situations, the main practical difficulty is to guaranty that the material shall not fail within a certified lifetime, which can be extremely long (such as 20 years, or even 50 years in some cases [START_REF] William A Thue | Electrical Power Cable Engineering[END_REF]). Therefore it is essential to identify the reason why a material shall fail after some time. More precisely, it is a key aspect to determine whether the material evolves during its lifetime. As the lifetime is very long, these situations necessarily correspond to extremely low, if any at all, rates of damage. Besides, predicting material lifetimes over time scales of several years necessarily relies on extrapolating the results of studies performed over much shorter duration. Such extrapolation in turn relies on strong assumptions which can be difficult to justify.

It may seem that studying fatigue under thermo-electrical stress may rely on approaches similar to those used to study thermo-mechanical fatigue. There are notable differences, though. Mechanical damage is directly related to the structure of the material. It was shown that the development of structural defects (such as cavities) directly related to the evolution of the mechanical properties (such as the modulus) may be characterized in various semi-crystalline thermoplastic polymers during fatigue [START_REF] Mourglia-Seignobos | Physical Mechanisms of Fatigue in Neat Polyamide 6,6[END_REF][START_REF] Charvet | Study of mechanical properties and damage mechanisms in plasticized cellulose acetate polymers[END_REF]. Electrical damage may be related to the electronic structure of the material. This may be more difficult to characterize than structural defects which directly impact the mechanical behavior. Of course the electronic structure must be related to the physical and chemical structure of the material (presence of defects such as cavities, presence of impurities or chemical defects, etc) but characterizing electronic defects may not be as direct as for structural defects. Besides, the electrical behavior may be more sensitive to the presence of defects. As concerns the mechanical behavior, a material may still retain its property almost unaltered in the presence of a localized defect (such as a micro-crack) while one single failed element is enough to destroy the insulating property of a dielectric polymer layer.

Nevertheless, an approach somehow similar to what has been done for characterizing mechanical fatigue has been adopted in this work. This consists in establishing curves relating the average lifetime to the amplitude of the applied stress (equivalent to Wöhler curves in the context of mechanical fatigue [START_REF] Mourglia-Seignobos | Physical Mechanisms of Fatigue in Neat Polyamide 6,6[END_REF]), while characterizing the evolution of the material along the duration of the fatigue tests.

Lifetime measurements are a key experimental step, for two reasons: first, among the large number of experimental studies on electrical ageing of polymers, only a few investigated the lifetime of the tested materials. Measuring lifetime implies relatively long experiments, but allows to have a clearer perception of the materials properties. Moreover, the established lifetime curves will allow to be able to predict the statistical end of life of a test sample, and thus to chose appropriate ageing steps as respect to the different moments of the samples life (figure 2). The aim of this work is to provide an experimental investigation of the ageing mechanisms responsible for the electrical failure of polymeric insulating materials under alternative current (AC) voltage. To do so, different polymer materials are prepared and aged under various electrical and thermal stress conditions to replicate and accelerate the conditions seen by the insulation in MV-AC power cables. The lifetime of the tested materials will be measured, and the critical parameters driving the end of life will be identified. Then, the samples will be aged under similar stress conditions, at different ageing states from unaged to near end of life. Different characterization techniques will be used in order to identify ageing markers and follow the evolution of these markers during the different life stages of the materials. Figure 3 illustrates the different steps of this work. In this work, various characterization techniques are used in the intent to identify and follow the evolution of ageing markers, which testify the effect of electrical stress on the materials properties: physical, chemical, structural, electrical ageing markers are tracked, and the results will be discussed to reach a deeper understanding of the ageing mechanisms responsible for the breakdown of these materials.

While identification of the ageing markers is possible through these characterizations, telling which marker is critical or not for the insulation is more difficult. In addition, ageing also strongly depends on the stress conditions. In power cables, the insulation can be subjected to various electrical and thermal stress, sometimes localized in millimetric areas of the insulating material, as for instance in the case of the presence of micro-cavities in the insulation, or metallic protrusion. In order to simulate different stress conditions, two electrode geometries will be used for lifetime measurements and ageing: a sphere on plane and a plane on plane geometries. This choice aims to obtain materials aged in two different stress conditions. In the sphere on plane geometry, stress is concentrated in a small region of the sample. Thus, damage is expected to be concentrated and can be more controlled, which allows to know more precisely where the material has been aged. The plane on plane geometry is used in order to simulate a more uniformly distributed stress, which would be expected in the case of a power cable without asperities or defects.

Structure of the chapters

This thesis is organized as follows. The first chapter aims at presenting the state of the art concerning the polymeric insulating materials used for electrical transport and the main known mechanisms and theories of electrical ageing. More precisely, it aims to present the importance of controlling the physical and chemical properties of the insulation, as well as the stress conditions applied on the material.

The second chapter presents the materials which have been chosen for the lifetime and ageing studies. The manufacturing process for samples preparation in both electrode geometries will be presented, as well as experimental benches for electrical tests and the different stress conditions that have been used. In addition, the physical, chemical, structural and electrical characterization techniques and the basis of their working principle are described.

The third chapter presents the main results of lifetime measurements. The method for lifetime data analysis will be discussed, as well as the physical meaning of the different parameters driving the lifetime of the tested materials. The critical parameters, among the materials properties or the applied stress conditions, responsible for the major degradation and breakdown of the materials will be discussed. Moreover, the effect of the two electrode geometries on lifetime will be shown and interpreted.

In the chapters four and five, the experimental principle of the ageing study and the main results of ageing markers characterization will be presented. The forth chapter regroups the results of the ageing study in the sphere on plane electrode configuration, and aims at identifying ageing markers in the case of a localized stress field. The fifth chapter is focused on the ageing of samples in a plane on plane stress field, and aims at understanding the ageing mechanisms in a more homogeneously distributed electrical stress. The results of these ageing studies in the two different electrode configurations will be compared and the effects discussed.

Finally, the main results of this experimental work, and the interpretation of ageing markers will be summarized in the conclusion, as well as the perspectives coming out this study.

Context

This work has been realised in the context of a scientific collaboration between industrial and academic partners, in the framework of the Lyon Polymer Science Engineering (Lyon-PSE) consortium. The LPSE is a research and innovation cluster created in July 2015, which now brings together ten industrial groups (Arkema, Elkem, Hutchinson, Michelin, Nexans, Saint-Gobain, SNF-Floeger, Solvay, Toray and Total), three SMEs (Activation, Addiplast and Processium), and four research laboratories (IMP, C2P2, LPMA and MATEIS). It was created in the purpose of co-developing innovative research works aiming to answer future industrial needs in the field of polymerization and polymer materials.

The present work has been done in the "Laboratoire Polymères et Matériaux Avancés" (LPMA) in St-Fons, FRANCE, supported by Nexans, Solvay and Total groups, and in collaboration with laboratory "Ingénierie des Matériaux Polymères" (IMP) of Université Lyon Claude Bernard, in Villeurbanne, FRANCE.

The sample manufacturing and the main experimental studies on lifetime and ageing has been realised in Nexans Research Center in Lyon, France (NRC Lyon), as well as the majority of the samples characterizations. A part of the experimental work has also been realised in Solvay Research and Innovation Center of Lyon (RIC Lyon). This work has also been possible thanks to collaboration with partners outside the LPSE consortium. Sample characterizations have been done in collaboration with the "Laboratoire Plasma et Conversion d'Energie" (LAPLACE) of Université Toulouse Paul Sabatier, FRANCE, and the "Laboratoire de Physique des Solides" (LPS) of Université Paris Sud, in Orsay, FRANCE.

Chapter 1

State of the Art

The aim of this chapter is to present the state of the art on the nature and used of polymer materials for electric insulation. The basic principle of a high-voltage cable, the manufacturing processes as well as the main properties of PE and PP materials are presented. Among these properties, some have been identified as critical for the insulation under electric field. For instance, the crystalline structure, effect of impurities, and the role of the electronic structure are presented and discussed.

Secondly, the main dielectric properties of these materials and their behavior under low and high electric fields are presented. Moreover, this chapter aims to provide a theoretical base on the existing physical and chemical ageing mechanisms that affect the lifetime of these materials. The main known electrical breakdown and ageing theories are given. In addition, the physical principles of partial discharges in micro-cavities are presented, in order to show in which way this phenomenon is critical for polymeric insulators. Finally, a summary of the main theoretical ideas and hypothesis rising from this literature study is given, as a prelude to the experimental work.

Generalities on power cables

The purpose of a wire or a cable is to convey the electric current to an intended location, while keeping the current from flowing in unintended paths rather than the conductor. It is the role of electrical insulation to isolate the conductor from these other paths, and prevent current short-circuit. The simplest case of insulated conductor can be found in overhead electrical lines, where conducting wires are isolated by the surrounding air. Although this system is low in cost, this works as long as the ground is sufficiently far away from the conductor, as air is not a very good insulating medium due to its low breakdown strength (about 3kV/mm [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF]). It represents also a constraint due to the relatively high space taken by these power delivery installations, and cannot be used in populated area such as cities for safety reasons.

In order to save space and deliver electric power to inhabited places, conductors have been buried in the ground and air has been replaced by other dielectrics (solids or liquids mainly) to create cables [START_REF] William A Thue | Electrical Power Cable Engineering[END_REF]. Power cables are now used everywhere, buried in ground or submerged under oceans. A typical power cable is composed of the following layers as described in figure 1.1. • The conductor in the cable core (1) can be made of different materials. Copper and aluminium are the most used conducting materials. The oldest is copper; it is denser and heavier, but more conductive than aluminum. The conducting core can be either a solid rod, or a stranded conductor, obtained by assembling single wires, which allows homogeneous electrical transport and prevent skin effect: in AC circuits, the current distribution in the conductor leads to a larger current density near the surface of the conductor than at its core [START_REF] William A Thue | Electrical Power Cable Engineering[END_REF].

• The relatively thick material surrounding the conductor is the extruded insulation [START_REF] Ribbing | Climate Change Leadership -the case for Electrification[END_REF]. Insulation physically encloses and protects the metallic conductor from the ground. Materials are composed of either synthetic or natural polymers, and must have specific dielectric properties depending on the application and needs, mainly on voltage (low, medium or high voltage.

• Due to surface irregularities (for instance surface roughness, protrusions) at the metal/insulation interface, shields around the conductor and insulation [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF] are used in order to homogenize the electrical contact between the insulation and the metal (core and ground). The mainly used material for shield are semiconductive layers made of polymer and carbon black.

• A metallic screen ( 4) is used in addition to the outer insulation shield in order to provide a resistance path for charging current to flow to ground. In other words, the screen is used to handle possible high charging current (over tens of milliampers) that cannot be sustained by the polymeric shielding materials.

• The outer material of the cable is the jacket [START_REF] Alexander | The Theory of Everything In The Electrical Breakdown Of Polymeric Dielectrics: Maybe[END_REF], which is a nonmetallic covering used as electrical and mechanical protection of the underlying layers from external stress: mechanical bending, sunlight, fire, and corrosion. Jackets can be made of many different materials from two categories, thermoplastic and thermosetting materials, depending on the operating temperature and environment.

In some cases, an armor material is placed (or incorporated) into cables and bring an additional protection from physical stress, but greatly enhance the cable weight and cost.

Polymeric insulated power cables are manufactured by extrusion process. The insulating and screening materials are used in the form of pellets or strips, and are melted around the conductor as the material advances down the extrusion screw. In the case of thermosetting insulation which requires cross-linking, such as ethylene propylene rubber (EPR) or cross-linked polyethylene, curing is carried over during the extrusion process. For instance, in the case of chemical cross-linking, a peroxide agent are is to the raw material. After melting, the temperature is increased (typically around 180 to 200 • C) in order to decompose the peroxide and initiate the cross-linking reaction. Pressure is used in the curing tube in order to prevent cavitation in the insulation during the cross-linking reaction. Finally, the cable is cooled in a water trough following extrusion.

The described process for cable manufacturing show that different materials with opposite properties (conductor and insulator) cohabit in the final cable. Thus, the extrusion process has to be as clean and precise as possible in order to avoid intrusion of external chemical or even metallic contaminants, which would drastically change the properties of the cable [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF]. Moreover, the properties of the insulating material drive the lifetime of the cable, as it is insulating barrier which allow current to flow without short-circuits. It is thus important to understand and control the properties of the insulation.

Polymers For High Voltage Applications

Generalities on insulating polymers

Nowadays, the mainly used insulation materials for power cables are synthetic polymers, such as Polyethylene (PE), Cross-linked polyethylene (XLPE), Ethylene based rubber (EPR), Polypropylene (PP). The insulating properties of insulating materials are driven by their physical and chemical structure, and understanding and controlling this structure is a fundamental aspect for ensuring cable insulation with higher lifetime. Polyethylene and Polypropylene are semi-crystalline polymers, which confers them an heterogeneous and multi-scale structure. The present section aims to describe the microstructure of both PE and PP, and the link between structure and their final properties.

Polyethylene: Structure and Properties

PE is a polymer made of one of the simplest monomers: it is produced by the polymerization of Ethylene gas at high pressures. Depending on the process conditions, different PE grades can be obtained with various densities.

Polyethylene grades

Several grades of polyethylene exist and are found in various applications, including power cables. Polyethylene grades are differentiated by their densities: low density polyethylene (LDPE) and high density polyethylene (HDPE) are the most commonly grades. LDPE is a branched polymer with densities comprised between 0.910 and 0.935 g.cm -3 , while HDPE possesses linear chains with no or few branches, which gives it a high density, typically between 0.94 and 0.965 g.cm -3 [START_REF] Louis | Polymer Materials : Macroscopic Properties and Molecular Interpretations[END_REF][START_REF] Pelzer | Etude de vieillissement des isolants synthétiques des câbles moyenne tension[END_REF]. This difference in the polymer densities leads to singular microstructures and properties for LDPE and HDPE: branches that constitute LDPE chains reduce the possibility to crystallize, and thus LDPE has a lower crystallinity than HDPE. However, LDPE is generally used for cross-linking.

Although LDPE is frequently used in cable insulation for low and medium voltage applications, it has a low melting point comparatively to operating temperature at a certain voltage, which leads to melting, creeping and loss of insulating properties. Thus, LDPE is not suitable for medium and high voltage applications, due to its poor thermal properties [START_REF] Füzesséry | Polyéthylènes basse densité PE bd[END_REF][START_REF] Louis | Polymer Materials : Macroscopic Properties and Molecular Interpretations[END_REF]. PE can also be cross-linked chemically, which confers it high thermal and mechanical properties. Indeed, cross-linked PE (XLPE) has similar dielectric properties as compared to LDPE, and has higher thermal and mechanical properties. XLPE is generally obtained by three different processes: by irradiation, by the use of silane compounds or by peroxide compounds.

The use of peroxide as cross-linking agent is the most common process for obtaining XLPE in cable industry.The general mechanism of PE cross-linking is described in figure 1.2:

FIGURE 1.2:
The chemical reactions and by products involved in PE cross-linking using DCP (adapted from [START_REF] Hussin | The Effects of Crosslinking Byproducts on the Electrical Properties of Low Density Polyethylene[END_REF]).

The cross-linking agent, generally a Dicumyl peroxide (DCP), initiates the chemical process by decomposing itself with temperature. The self decomposition can occur at temperatures as low as 120 • C, but temperatures between 180 and 220 • C are usually necessary in order to accelerate the decomposition kinetics [START_REF] Hussin | The Effects of Crosslinking Byproducts on the Electrical Properties of Low Density Polyethylene[END_REF]. This process is known as curing, and is performed during the extrusion process in a curing tube under proper temperature and pressure conditions. DCP dissociates into Dicumyl radicals, which can initiate two other reactions:

• Radicals can react on PE macromolecular chains and form α-Cumyl alcohol (αCA) and a PE radical chain.

• Radicals can split and form a methyl radical and Acetophenone (ACP).

In the first case, the PE radical chain can react with another radical chain and crosslink, while αCA can form α-Methylstyrene (αMS) and Cumene. In the second case, methyl radical can react with a PE macromolecular chain to form Methane and PE radical chain. In the same way as in the first case, PE radical chain can react with another radical chain and cross-link. As more DCP decomposes in presence of PE, cross-linking density increases and chemical by-products (ACP, αCA, αMS, Cumene, Methane) forms in the material.

Among these by-products, ACP, Methane and αCA are known to diffuse in the material [START_REF] Hussin | The Effects of Crosslinking Byproducts on the Electrical Properties of Low Density Polyethylene[END_REF][START_REF] Wutzler | Diffusion of acetophenone in peroxide cross-linked cable compounds[END_REF][START_REF] Suh | Evaluation of organic peroxide decomposition byproducts from incompletely crosslinked high voltage power cables[END_REF][START_REF] Sahyoun | Diffusion mechanism of byproducts resulting from the peroxide crosslinking of polyethylene[END_REF]. Indeed, as the methane gas is forming, one has to pay attention on controlling the pressure applied during cable manufacturing in order to avoid cavitation. Temperature and pressure have to be uniformly and properly controlled during the curing process for this purpose [START_REF] Hussin | The Effects of Crosslinking Byproducts on the Electrical Properties of Low Density Polyethylene[END_REF]. Moreover, studies have pointed out the presence and diffusion of ACP, αMS and αCA in cable insulation, and the role of these by-products on the alteration of the insulation dielectric properties [START_REF] Hussin | The Effects of Crosslinking Byproducts on the Electrical Properties of Low Density Polyethylene[END_REF][START_REF] Wutzler | Diffusion of acetophenone in peroxide cross-linked cable compounds[END_REF][START_REF] Suh | Evaluation of organic peroxide decomposition byproducts from incompletely crosslinked high voltage power cables[END_REF][START_REF] Sahyoun | Diffusion mechanism of byproducts resulting from the peroxide crosslinking of polyethylene[END_REF][START_REF] Aida | Influence of curing by-products on dielectric loss in XLPE insulation[END_REF][START_REF] Anbarasan | Crosslinking of high-density polyethylene in the presence of organic peroxides[END_REF][START_REF] Andrews | The role of degassing in XLPE power cable manufacture[END_REF]. These studies highlight the important role of degassing the insulation in order to get rid of it and improve power cable lifetime [START_REF] Andrews | The role of degassing in XLPE power cable manufacture[END_REF]. Other studies suggest on the contrary that ACP can play a role in retarding electrical treeing and enhancing dielectric strength of XLPE in AC power cables [START_REF] Li | Effect of acetophenone on the breakdown strength of polyethylene[END_REF]. What is generally accepted is that these products, if not degassed, can play a role in trapping and conducting charge carriers, and are considered as chemical heterogeneities which will be discussed later [START_REF] Hussin | The Effects of Crosslinking Byproducts on the Electrical Properties of Low Density Polyethylene[END_REF].

Despite the presence of these by-products, the peroxide cross-linking is the most common process because of the ease of incorporation of the peroxide in the cable extrusion process [START_REF] Duh | Thermal decompositions of dialkyl peroxides studied by DSC[END_REF]. In order to understand main aspects of ageing of this material, it is essential to cover the close link between the structure of the polymer and its final physico-chemical properties.

Microstructure

Polymers properties are mostly defined by their structure. It exists many different polymers, which belong to different classes, namely thermoplastics, thermosets or elastomers. Polyethylene is a semicrystalline thermoplastic polymer, meaning that it has a melting temperature and a microstructure composed of a coexistence between amorphous regions (in which chains are disordered and characterized by a glass transition temperature) and crystralline regions (in which chains are organized and characterized by a melting temperature) [START_REF] Louis | Polymer Materials : Macroscopic Properties and Molecular Interpretations[END_REF]. This class of material is characterized by a temperature dependence of the Young Modulus schematized by the figure 1.3: At low temperature, the polymer is in a rigid so called "glassy state", which is responsible for its highest value of the Young modulus, typically some GPa. At the glass transition temperature, the Young modulus drops by one order of magnitude and reaches a first plateau which corresponds to a "leathery state", until the melting temperature. At the melting temperature, the crystalline regions melt which leads to a drop in the Young modulus. This drop is dependent on the molecular weight of the polymer. Increasing the temperature higher leads to a material which behaves like a viscous fluid. These properties are governed by the specific structure of the polymer, made of amorphous and crystalline regions.

The semicrystalline characteristics of polyethylene confers it a multi-scale structure. Indeed, by looking at an X-ray scattering spectra of an amorphous polymer and of polyethylene, one can notice the difference and similarities of the two systems. Figure 1.4 shows the X-ray scattering Debye-Scherer pattern of an amorphous and semicrystalline polyethylene [START_REF] Schmacke | Investigations of Polyethylene Materials by Means of Xray Diffraction Artificial Ageing of Polyethylene Gas Pipes[END_REF]: The scattering pattern of the amorphous material is a "diffuse halo", characteristic of a disordered state, while the pattern of the semicrystalline polymer shows both a diffuse halo, then several concentric rings, localized and corresponding to the scattering by different (h, k, l) planes. Looking at the scattering curve, the crystalline peaks are superposing the diffusive halo, which indicates that a semicrystalline polymer exhibits scattering behavior of both a crystalline and an amorphous phase. It demonstrates that a semicrystalline material is made disordered amorphous regions and of oriented crystalline regions. The fraction of crystalline phase (in mass or in volume) is generally noted as the crystallinity index χ c .

Amorphous phase

In the amorphous region, or phase, there is no order at large scale. Chains are settled statistically and are linked from one to another by entanglements. The mobility of these chains with respect to each others define the properties of the amorphous phase, and the chain mobility is driven by the temperature. As discussed previously, at low temperature the amorphous material is in a rigid glassystate because the chain mobility is limited. In this state, as there are no or few chain motions, it is not able to accommodate the applied stress, and has the characteristics of a glass, including hardness, stiffness, and brittleness [START_REF] Manas | Introduction to Polymer Science and Chemistry[END_REF][START_REF] Louis | Polymer Materials : Macroscopic Properties and Molecular Interpretations[END_REF] . When increasing the temperature above the glass transition temperature (noted T g ), chain mobility increases which allow free volumes to emerge. The material loses its glassy behavior and its mechanical properties decrease, and it starts to behave like a rubbery solid. Moreover, the T g depends on the cooling rate of the sample: the slower the cooling rate, the lower the glass transition temperature, making the glass transition a kinetic phenomena [START_REF] Louis | Polymer Materials : Macroscopic Properties and Molecular Interpretations[END_REF][START_REF] Humbert | Influence de la topologie moléculaire et de la microstructure sur les propriétés mécaniques des Polyethylènes[END_REF]. The T g can be observed experimentally by measuring the heat transfers using Differential Scanning Calorimetry (DSC) or by measuring the evolution of the storage modulus as function of temperature by Dynamic Mechanical Analysis (DMA) as shown in figure 1.5. Crystalline phase In the crystalline phase, long macro-molecular chains adopt their most stable conformation. In the case of polyethylene, chains adopt the lowest energy conformation which is a symmetrical and regular all-trans conformation known as "Planar Zig-Zag" [START_REF] Louis | Polymer Materials : Macroscopic Properties and Molecular Interpretations[END_REF]. This configuration allows a maximum packing of the chains, and also optimize the stereometry due to hydrogen bonds. The planar zig-zag conformation fits the chains in an orthorhombic unit cell, which is the most commonly encountered cell for PE. Figure 1.6 shows the crystalline unit cell of PE and the Zig-Zag conformation: FIGURE 1.6: Arrangement of the "Zig-zag" polyethylene chains within the orthorhombic cell [START_REF] Louis | Polymer Materials : Macroscopic Properties and Molecular Interpretations[END_REF].

At a larger scale, crystal structures form crystalline lamellae. The thickness of these lamellae is in the range of 5 to 20 nanometers, and their size varies with the crystallization temperature kinetics. Inside the lamella, chains are aligned perpendicularly to the growth direction of the lamella and form the crystalline material, while at the lamella border, chain foldings and endings form the interlamellar amorphous material [START_REF] Fontanille | Structure morphologique des polymères[END_REF]. During melt crystallization, or from solvent evaporation, lamellae grow radially from a nucleation point and form superstructures at an even larger scale, known as spherulites [START_REF] Louis | Polymer Materials : Macroscopic Properties and Molecular Interpretations[END_REF][START_REF] Fontanille | Structure morphologique des polymères[END_REF]. Figure 1.7 shows the schematic representation of a crystalline lamella and a spherulite: FIGURE 1.7: Scheme of chain arrangement within a PE lamella (adapted from [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Louis | Polymer Materials : Macroscopic Properties and Molecular Interpretations[END_REF]).

This microstructure can be obtained during a homogeneous crystallization, without external stress. In the case of crystallization during flow, which is the case for example during extrusion, due to a high degree of shear stress, the molecular orientation can affect the crystalline morphology. Chains may become extended in one specific direction, and in this case the growth of such lamellae is less likely to happen, but rather fibrils-like structure [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Louis | Polymer Materials : Macroscopic Properties and Molecular Interpretations[END_REF].

Cross-linking and stabilization PE grades used in power cables generally involve stabilization with antioxidant agent, and more generally cross-linking for higher voltage applications. However, the addition of additives such as antioxidant can disturb the crystallization of the long macromolecular chain. It was shown in the literature that addition of antioxidant in LDPE base resin counteract the development of spherulites during crystallization [START_REF] Nilsson | Morphology of polyethylene for power cable insulation: effects of antioxidant and crosslinking[END_REF]. Additionally, cross-linking was found to influence the lamellar structure of the LDPE resin, and form very small and diffuse spherulites. Figure 1.8 shows the observed morphology of pure LDPE and stabilized XLPE plates. Initially on pure LDPE, spherulites with diameters between 2-40 μm can be observed, whereas addition of antioxidant leads to a different morphology, without fully rounded spherulites and appearance of parallel lamellae stacks, of 1-2 μm in size. Antioxidant can play a role of nucleation center for crystallization. Crosslinking of LDPE resin shows small spherulites less than 3 μm in diameter. The explanation given to this morphological modification is the reduction of crystallization caused by cross-linking covalant bonds.

Polypropylene

Polypropylene, as Polyethylene and many other polyolefins, is obtain by polymerization of an olefin hydrocarbon gas: Propylene. The first stereospecific polymerization of propylene was performed by Giulio Natta in 1954, using a Ziegler catalyst [START_REF] Lieberman | Propylene polymers[END_REF]. A number of low cost polymerization processes were developed subsequently, and lead to an increase in production of this material. Currently, Polypropylene is one of the most widely used polymers with Polyethylene, due to the low manufacturing costs, and the wide rand of applications.

Polypropylene grades

Contrary to PE, PP grades are subdivided based on the stereochemistry (or spatial conformation of monomers) of the polymer chain during polymerization. As the propylene monomer is asymmetrical, the position of the methyl group (-CH 3 ) relatively to other methyl group in the chain and its regularity determine the stereospecificity, or tacticity, of the finale macromolecule. Three conformations are possible for the PP macromolecule based on this tacticity, and are detailed in figure 1.9.

• Isotactic PP (iPP), which is composed of macromolecules where all methyl groups have the same position with respect to the chain.

• Syndiotactic PP (sPP), where methyl groups occupy alternating positions in the macromolecule.

• Atactic PP (aPP) finally, where methyl groups have random positions along the chain. FIGURE 1.9: Polypropylene stereochemical configurations [START_REF] Lieberman | Propylene polymers[END_REF].

iPP is probably the most commercially significant form of PP, due to its higher mechanical and thermal properties: increase atacticity in polypropylene results in a decrease of its stiffness, moduli, melting temperature and long term heat ageing resistance [START_REF] Maier | Polypropylene: the definitive user's guide and databook[END_REF].

Microsctructure

The tacticity of PP plays a critical role in its crystalline morphology, by influencing the packing conformation of chains in the crystalline regions. The regularity in the chemical structure of iPP and sPP clearly facilitate chain crystallization, thus giving rise to relatively high crystalline ratio. On the contrary in aPP, as there is no regularity in spatial conformation of the side groups, it does not crystallize and is fully amorphous. The PP chains adopt a helical arrangement as it crystallizes, as shown in figure 1.10, due to the pendant methyl group which allows atoms to pack together without bond distortion. The unit cell is different for iPP and sPP: for the dominant form of iPP (α-form), the elementary unit cell is a mono-clinic cell with 4 chains containing 12 monomeric units, with an identity period of 0.65nm. For sPP, the unit cell is orthorhombic with 4 chains and 16 monomeric units, with an identity period of 0.74nm [START_REF] Lieberman | Propylene polymers[END_REF].

FIGURE 1.10: Polypropylene's chain in helical conformation [START_REF] Lieberman | Propylene polymers[END_REF].

The degree of crystallinity can vary from completely amorphous in the case of aPP to crystalline ratio of 70 to 80 % in the case of iPP.

As in the case of other semi-crystalline homo-polymers, PP chains arrange by folding back and forth to form crystalline lamellae, with thickness of 2 to 50 nm. At a larger scale, lamellae aggregates form spherulites, which dimension vary from few to 100 μm (see figure 1.11) [START_REF] Lieberman | Propylene polymers[END_REF][START_REF] Maier | Polypropylene: the definitive user's guide and databook[END_REF].

Commercial forms of PP

It is worth mentioning that PP is produced in various different form depending on the application. Homo-polymers as discussed previously can be replaced by copolymers using ethylene or less commonly 1-butene or 1-hexene [START_REF] Maier | Polypropylene: the definitive user's guide and databook[END_REF]. In ethylenepropylene co-polymers for instance, the final properties of the material can be tuned by changing the amount of ethylene monomers, generally from 1 to 7 wt%. This results in lower crystallinity, lower melting points and glass temperatures but higher strength due to an increased amount of amorphous material. Co-polymers can be polymerized as random or block co-polymers: in random co-polymers the propylene and the ethylene monomers are placed randomly in the polymer chains, whereas in block co-polymers the positions of the two monomers alternates (see figure 1.12.

FIGURE 1.12: Random and block co-polymer using propylene and ethylene monomers [START_REF] Maier | Polypropylene: the definitive user's guide and databook[END_REF].

Defects in polymers

We have seen previously that polymeric materials for insulation come in various compositions and forms. In practice, processing polymeric materials from polymerization to cross-linking and stabilization induce physical and chemical defects in the polymer's structure, as compare to a hypothetically pure and stable polymer. These heterogeneities can distributed in the bulk in small concentration. However, being aware of the nature of these defects is important, as some physical and chemical heterogeneities can trigger bigger changes in the polymer's structure and alter the final properties [START_REF] Chen | Electronic Structure of Polyethylene: Role of Chemical, Morphological and Interfacial Complexity[END_REF].

Physical heterogeneities

In semi-crystalline polymers, as PE and PP, a part of the macromolecules are arranged in highly organized crystalline lamellae. The other part of the material, cannot crystallize and remain pendant outside the lamellae. This implies physical disorder in the structure of the material, such as chain folding (loops), chain endings (or cilia), chain entanglements, branches or cross-links [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Lewis | Polyethylene under electrical stress[END_REF]. Figure 1.13 shows a representation of structural defects originating from polymer chains arrangement. Moreover, empty volumes and voids can be formed in the polymer due to density variations, or during the growth of spherulites (see figure 1.14). Chemical or physical impurities, like pieces of unreacted material, decomposition byproducts, or insoluble matter like water, are known to be pushed outside the crystalline material and stack in the boundaries between spherulites, and form small droplets which eventually evaporate [START_REF] Muccigrosso | The Morphology of Cross-Linked Polyethylene Insulation[END_REF].

These defects can also be formed after the process, during secondary crystallization or chain relaxations, as the amorphous material changes in the action of temperature or mechanical stress [START_REF] Barlow | Possible Mechanism of Microvoid Formation in Polyethylene Insulated High Voltage Cables[END_REF][START_REF] Guffond | Characterization and modeling of microstructure evolution of cable insulation system under HVDC fields[END_REF].

Chemical heterogeneities

Indeed, as discussed previously the synthesis process leads in the formation of chemical heterogeneities in the polymer structure. In PE for instance, small amount of unsaturated C = C functions can be found. Other chemical defects are common in PE, such as conjugated double bonds, vinyl, carbonyl, carboxylate, hydroxyl, ether and epoxy-ring, in concentrations less than 1 mol/L [START_REF] Chen | Electronic Structure of Polyethylene: Role of Chemical, Morphological and Interfacial Complexity[END_REF]. Additionally, and as seen in the case of PE cross-linking by DCP, decomposition by-products can also act as chemical defects and play a role in the material's properties. By-products of antioxidant agent may also have a role of chemical heterogeneities [START_REF] Meunier | Molecular modeling of electron traps in polymer insulators: Chemical defects and impurities[END_REF]. These chemical defects are believed to induce drastic changes in the structure of the insulation, even in its electronic structure as shown by figure 1.15. As for physical disorder, it is believed that chemical heterogeneities can be found in the amorphous part of the insulation material, as they tend to disrupt formation of the crystalline material [ [START_REF] Huzayyin | An overview of impurity states and the basis for hole mobility in polyethylene[END_REF]. Chemical defects are frequently associated to models and calculations as they act directly on the electric properties of the final material, in particular their specific role on charge conduction [START_REF] Huzayyin | An overview of impurity states and the basis for hole mobility in polyethylene[END_REF][START_REF] Chen | Electronic Structure of Polyethylene: Role of Chemical, Morphological and Interfacial Complexity[END_REF][START_REF] Chen | Atomistic mechanisms for chemical defects formation in polyethylene[END_REF]. Figure 1.16 shows the main identified chemical and physical disorders reported in the literature for PE. The main identified chemical disorders are induced by oxygen and insaturations (double bonds), whereas physical disorders are caused by cross-linking and chain irregularities.

Dielectric phenomena in polymeric insulators

Band theory: structure of insulators

The concept of energy levels in solids, describing the processes leading to electrical conduction, is based on the "band theory" [START_REF] Ramu | Diagnostic testing and life estimation of power equipment[END_REF][START_REF] Kittel | Introduction to solid state physics[END_REF][START_REF] Gorur | Dielectrics in electric fields[END_REF]. In a solid constituted of N atoms which are considered sufficiently close, electrons constituting the outer shells, or the valence electrons, interact with each others. Each orbital is associated with an energy level n, and the superposition of orbitals lead to a split of isolated energy levels. The overlapping of these energy bands results in a single broad band which is partially full, as shown in figure 1.17: FIGURE 1.17: Schematic representation of band theory for conduction in metals, semiconductors, and insulators (adapted from [START_REF] Gorur | Dielectrics in electric fields[END_REF]).

The ground state, also known as the valence band, is constituted of the electrons which are still attached to the atoms, and is full. The conduction band is constituted of energy levels which are not directly accessible for electrons in the atoms. Between the two bands resides a forbidden band where no energy levels can be found. The width of this forbidden band, ΔE g , which is the energy difference between the top of the valence band and the bottom of the conduction band is called the "band gap", and is the minimum of energy required for detaching electrons from the atoms. The electric properties of a given material, as discussed earlier, is driven by the position of the Fermi level as compared to the filled valence states and the empty conduction states. In the case of metals, the Fermi level is positioned in the conduction band, which means that part of the conduction states are filled with electrons, making metals good conductors. In the case of semiconductors and insulators, the Fermi level is positioned between the valance and conduction band, in the middle of the forbidden band. All the valence states are filled with electrons, and the conduction band is empty. For semiconductors, the gap is relatively small, in the order of k B T.

A small voltage can trigger a "jump" from the valence state to the conduction state. These materials are used in solar panel for example, where photons acts as triggers.

For insulators however, the forbidden band is several electron volts larger than k B T. Thermal excitation is not sufficient to provide a jump from the valence band to the conduction band. In a perfect dielectric, high electric fields are required for this to happen. However, in most of dielectrics, the presence of impurities and structural disorders creates localized energy levels in the forbidden band, which make intermediates localization of electrons possible [START_REF] Gorur | Dielectrics in electric fields[END_REF].

Polyethylene, as many other synthetic polymers, is not a perfect dielectric. Chemical impurities and physical disorder lead to localized states in the forbidden band. Without these imperfections, polymers would act as perfect insulators. These defects are difficult to identify, although recent calculations based on molecular dynamics and density functional theory allowed to estimate some of the impurity states and quantify energy levels [START_REF] Chen | Electronic Structure of Polyethylene: Role of Chemical, Morphological and Interfacial Complexity[END_REF][START_REF] Meunier | Molecular modeling of electron trapping in polymer insulators[END_REF][START_REF] Unge | Electronic structure of polyethylene: Crystalline and amorphous phases of pure polyethylene and their interfaces[END_REF][START_REF] Chen | Unraveling the luminescence signatures of chemical defects in polyethylene[END_REF][START_REF] Moyassari | First-principle simulations of electronic structure in semicrystalline polyethylene[END_REF]. A schematic representation of the effect of impurity states on PE band gap is shown in figure 1.18: FIGURE 1.18: Electronic structures of bulk Polyethylene with chemical and physical defects, calculated using molecular dynamics and density functional theory [START_REF] Chen | Electronic Structure of Polyethylene: Role of Chemical, Morphological and Interfacial Complexity[END_REF].

Previous researchers estimated theoretically as well as experimentally the band gap energy value of PE between 7 and 9 eV [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Chen | Electronic Structure of Polyethylene: Role of Chemical, Morphological and Interfacial Complexity[END_REF][START_REF] Moyassari | First-principle simulations of electronic structure in semicrystalline polyethylene[END_REF][START_REF] Less | Intrinsic photoconduction and photoemission in polyethylene[END_REF][START_REF] Delhalle | Electronic structure of polyethylene: Theory and ESCA measurements[END_REF][START_REF] Hoffmann | A chemical approach to the orbitals of organic polymers[END_REF][START_REF] Laurent | A discussion on the electronic properties of polyethylene interfaces[END_REF]. However, due to the density of defects distributed in the forbidden band, charge carriers can be trapped at intermediate energy levels thus reducing the energy gap locally [START_REF] Chen | Electronic Structure of Polyethylene: Role of Chemical, Morphological and Interfacial Complexity[END_REF]. The theory stats that the energy required to produce equivalent numbers of electrons up to the conduction state increases successively from metals to semiconductors and to insulators. From another point of view, at a given and equivalent external applied field E, the number of promoted charge carriers to the conduction state is much lower in an insulator than in a metal. Thus, the current density generated by the promoted charge carrier is lower in the insulator. The general expression for current density in a solid a relatively low fields (< 1kV.mm -1 ) is given by local expression of Ohm's law:

J = σ E = nqμ m E (1.1)
where, J is the current density (A.m -2 ), σ is the electrical conductivity (S.m -1 ), n is the number of charge carriers, q (C) is the magnitude of charge carriers and μ m is the charge carrier mobility (m 2 /(V.s)). In metals, the conduction is entirely electronic, so that q can be replaced by the elementary charge (e = 1.6 × 10 -19 C). However, the processes of electrical conduction are more complex in dielectrics, because conductivity depends on other factors than electron conduction only [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF]. Thus, Ohmic conduction is not a proper model for insulators, and other phenomena such as trapping in the forbidden band have to be included. Conduction is considered non-linear with electric field. This aspect is treated later in more details.

Electrical conductivity in dielectrics such as polymeric materials is thus a complex phenomena that depends on the material structure and composition. High fields are required in order to promote electrons from the valence to the conduction states. However, other phenomenon can occur at lower fields, such as polarization, which is discussed in the next section.

Linear dielectric properties of polymers

In order to understand the behavior of polymeric materials under electric field, it is convenient to use a simple geometrical consideration where the dielectric material is placed under two plane electrodes.

Let's consider a system comprised of a generator and two plane electrodes, as illustrated in figure 1.19: Placed under vacuum, a capacitor under a voltage U 0 accumulates a net charge Q 0 at its positive bound (and a charge -Q 0 at its negative bound) which is linearly linked to the applied voltage:

Q 0 = C 0 U 0 (1.2)
When a dielectric is placed between the capacitor bounds, it modifies the generation of charge carriers thus increasing the capacity of the system to C(> C 0 ). The total charge has to be the same, which means that the dielectric will see a lower effective voltage U.

Q 0 = CU (1.3)
Under an applied electric field, the dielectric material get polarized: chemical groups linked to the macromolecular backbone of the dielectric form dipoles that orientates perpendicularly to the capacitor bounds, and neutralize part of the capacitor charges. The ability of the dielectric material to polarize, and provide induced charges) is defined by its relative permittivity r . It is obtained by the ratio of the new capacity over the capacity of the system under vacuum.

r = C/C 0 (1.4)
The relative permittivity is thus an intrinsic measure of the material's ability to polarize under an external applied field. Each dielectric material has a distinct relative permittivity. Table 1.1 shows the relative permittivity values of some dielectric materials: Dielectric polarization is the phenomenon that involves the appearance of electric charges at the surface of the dielectric, which are induced by the generation of dipolar moments in atoms and molecules composing the material [START_REF] Chelkowski | Dielectric Physics[END_REF]. By definition, the polarization field P induced in the dielectric under an external electric field E is given by equation 1.5:

P = χ 0 E (1.5)
where χ is the dielectric susceptibility of the material, and 0 is the vacuum permittivity ( 0 = 8.854 × 10 -12 F/m). The dielectric susceptibility is given by:

χ = r -1 (1.6)
The previous expressions works fine in the case of a perfect dielectric material, without losses. However in practice, dielectric materials are not perfects, and polarization phenomena induces dielectric losses. To understand this in the most practical way, it is convenient to express permittivity by means of complex numbers [START_REF] Chelkowski | Dielectric Physics[END_REF]. Thus, under a sinusoidal alternating field the relative dielectric permittivity can be written as:

r (ω, T) = r (ω, T) -j r (ω, T) (1.7)
with r (ω, T) the real part of permittivity and corresponds to the relative permittivity of the material at a given frequency and temperature. r (ω, T) is the imaginary part of the permittivity and expresses how the material dissipates the energy induced by the polarization process at a given frequency and temperature: it is know as the loss factor. The dielectric properties of a material depends thus on the evolution of these two measures as a function of the voltage frequency and temperature.

Polarization is thus a phenomena driven by the voltage frequency and temperature. A change in the value of the real permittivity and the loss factor traduces a change in the polarization mechanism. It exists four different known polarization mechanisms: electronic, atomic, orientation and inter-facial polarization. Each type contributes to specific values of real permittivity and corresponds to relaxation phenomena expressed by peaks in the values of the loss factor [START_REF] Bureau | Propriétés diélectriques des polymères[END_REF][START_REF] Chelkowski | Dielectric Physics[END_REF][START_REF] Seghier | Ethude des déchagres partielles et leurs influences sur l'apparition des arborescences dans la couche isolante de câbles haute tension[END_REF].

• Electronic polarization is due to the deformation of the electronic cloud of atoms composing the material under the applied electric field. The electronic cloud is shifted apart from the nucleus which generates a dipole. This type of polarization is the major contribution to the relative permittivity and can be observed at frequencies up to 10 15 Hz.

• Atomic polarization takes place in a molecule with two atoms linked by a covalent bound. If there is a difference in electronegativity between the two atoms, the most electronegative atom will attract electrons to its side, which will create an induced negative charge, while the other atom will be charged positively. The molecule will act as a dipole. However if the symmetry of the molecule hinder the dipolar moment, the molecule can act as a dipole. Under external electric field yet some of the atoms nuclei can be shifted away from each others and create an induced dipole. This type of polarization is possible at frequencies up to 10 12 Hz.

• Orientation polarization, also known as "Debye polarization", is the polarization of a molecule under external electric field which already possesses a permanent dipolar moment. The molecule is thus shifted in the same way as the electric field. This type of polarization depends thus a lot on the structural properties of the dielectric and affects only polymers which are polar, which is not the case of PE (symmetry due to the four hydrogen atoms). However, during aging PE macromolecular chains can be affected and chemical groups such as carbonyl can introduce asymmetry in the monomers, and thus dipolar moments. Debye polarization depends also on the ability of the molecules to shift, which is affected by the viscosity of the material, and thus has a major contribution to the loss factor [START_REF] Bureau | Propriétés diélectriques des polymères[END_REF].

• Finally, interfacial polarization, also known as "Maxwell-Wagner-Sillars effect", appears in materials which are heterogeneous and possess different phases. As for example in the case of a polymer containing semi-conductive nanoparticles, an applied electric field generates electrical charges at the polymer-particles interface. This polarization can be observed at relatively low frequencies (10 -1 to 10 -2 Hz).

The different polarization mechanisms discussed above are represented in figure 1.20. The four polarization mechanisms mentioned above are thus responsible for the variation of the value of real permittivity r (ω, T) and loss factor r (ω, T) as a function of the frequency and the temperature. The measurement in frequency of the dielectric permittivity allows to observe the different polarization and the relaxation domains, as can be seen in figure 1.21: At high frequencies, only electronic polarization contributes to a value of real permittivity. At low frequencies, the four contributions superimpose and contribute to a high value of real permittivity. Moreover, it can be seen that for each frequency domain which corresponds to the extinction of a polarization mechanism, a peak appears in the value of the loss factor : these peaks corresponds to relaxation mechanisms which act on the conformation of polymer chains, as for example the glass transition [START_REF]Broadband Dielectric Spectroscopy[END_REF]. The contribution of these relaxation mechanisms on the dielectric permittivity are easier to observe by performing measurements at different isotherms (see for instance figure 1.22). More generally, one uses the ratio of over , which is known as the loss angle tanδ, in order to measure the evolution of the dielectric losses as function of frequency and temperature:

tan(δ) = r r (1.8)
The variation of the loss angle indicates the different types of relaxation occurring in the polymer. These relaxation are not the same for amorphous or semi-crystalline polymers, but generally convey chains reorganization with frequency and temperature. In polyethylene, three main relaxations are generally observed, as shown in figure 1.23: The relaxation α occurs at relatively high temperature, and is attributed to the fusion of the crystalline phase. The second peak β is associated to glass transition, and finally the γ peak is attributed to movements of small chemical groups in the amorphous phase, at low temperatures [START_REF] Bureau | Propriétés diélectriques des polymères[END_REF].

Although these polarization mechanisms are a tool to measure the dielectric properties of polymers, other dielectric phenomena taking place at higher electric fields can directly alter the insulation properties, and thus need to be studied carefully. Therefore, the dielectric phenomena at high fields are discussed in the following section.

Dielectric phenomena at high fields

As discussed in the previous section, due to their electronic structure polymers are not perfect insulators. Charge carriers can be generated in the bulk or at interfaces with electrode and result in electrical conduction. These carriers are of different forms and origins. Intrinsic charge carrier can be generated from impurities such as metallic inclusions, or in the case of XLPE by chemical by-products such as acetophenone. Electrons can be exchange and circulate between these impurities, and participate to conduction. However, it is not likely that electrons can be produced in the bulk by thermal excitation because of the high energy gap [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF]. But extrinsic charge carriers can also be generated by electrical contacts. At electrode-polymer and under relatively high fields (> 1kV/mm), charge carriers can be injected from the metallic electrode into the the polymer. Different charge carriers injection model can be found in the literature. However, as the nature of the electrode-polymer interface is complex, the measured experimental current densities may not fit properly the discussed models [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Chiu | A Review on Conduction Mechanisms in Dielectric Films[END_REF].

Charge injection mechanisms

An electric contact is generally defined as a contact between a metallic and a non metallic material. In the case of a dielectric material, which have a high energy gap and a Fermi level positioned inside the forbidden band, a contact with a metallic electrode leads to the formation of an energy barrier φ inj for injection of electrons [START_REF] Kao | Dielectric phenomena in solids[END_REF].

Let's consider a truly intimate electric contact in order to neglect inter-facial effects due to surface roughness. Without electric field, an electron present in the metallic bulk has to overcome an energy barrier Φ int in order to be injected in the polymer's conduction band. The height of the barrier depends on the interface properties, on the nature of the metal and of the dielectric. In case of a metal-vacuum interface, the barrier height is simply the work function of the metal Ψ m . In the case of a metal-polymer interface however, it depends on the work function of the metallic electrode, on the electronic affinity of the polymer and on local polarization mechanisms at the interface as described by equation 1.9 [START_REF] Chen | Charge injection barriers at metal/polyethylene interfaces[END_REF].

Φ m = Ψ m -E ea -Δφ (1.9)
where Ψ m is the metal work-function (eV), E ea is the polymer electronic affinity (eV), Δφ is the vacuum level shift between the two materials due to polarization of the interface (eV). Experimental values of E ea for PE were found between -1.20 and 0 eV [START_REF] Less | Intrinsic photoconduction and photoemission in polyethylene[END_REF]. The vacuum shift Δφ has been measured for different metal/PE interface and estimated between -0.70 and -0.30 eV [START_REF] Ishii | Energy Level Alignment and Interfacial Electronic Structures at Organic/Metal and Organic/Organic Interfaces[END_REF]. Those values are negative which means that the barrier is theoretically increased. However in practice, the system is much more complex, and considering the small energy variations, the barrier is sometimes taken as the value of the metal work function, as shown in figure 1.24 (a).

On the base of these statements, without electric field electrons have to overcome an energy barrier which value is approximately equal to the metallic electrode work function. Work function of metals like Aluminum or Copper, which are generally used as conductors in power cables are found between 4.25 and 4.4 eV which is larger than 160 × k B T [START_REF]Handbook of Thermionic Properties[END_REF]. This barrier is thus impossible to overcome by selfthermionic emission. But external energy can be brought electrically. After applying an electric field the energy barrier is reduced, due what is a called the "Schottky effect" [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Hussin | The Effects of Crosslinking Byproducts on the Electrical Properties of Low Density Polyethylene[END_REF]. Several mechanisms take place and superimpose which lower the energy barrier for electrons, as represented in figure 1.24 (b), (c) and (d).

FIGURE 1.24: Reduction of injection barrier at the metal-polymer interface by the application on en electric field (adapted from [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Hussin | The Effects of Crosslinking Byproducts on the Electrical Properties of Low Density Polyethylene[END_REF]).

If an electron leaves the metal and is injected in the polymer, it will be subject to Coulombic attraction from the positive charge left by the electron. The deeper the electron is injected in the material, the lower is the attraction force. This phenomenon is known as the "image charge theorem", gives rise to a negative potential barrier which gradually increases with the distance separating the electron from the interface. This potential can be considered as the energy required to take the electron from the metal into a distance x in the polymer. The potential V(x) image is given by the integration over the distance x of the electrical attraction force between the two charges:

V(x) image = ∞ x F(x)dx = - e 2 16π 0 r x (1.10)
A second consideration to make is the effect of potential energy due to the applied electric field. Assuming the electric field is constant across the interface, and of magnitude E, then the potential barrier is a straight line decreasing as the distance from the interface increases:

V(x) f ield = -eEx (1.11)
The total potential barrier at the interface V(x) int due to the applied field is the sum of the two components V(x) image and V(x) f ield . Thus, the total energy barrier for injection as function of distance x is the sum of the work function and the potential energy due to the interface:

V(x) tot = Ψ m + V(x) int (1.12)
The current density generated by the electron flowing through the barrier is given by the Richardson equation applied to Schottky injection [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Hussin | The Effects of Crosslinking Byproducts on the Electrical Properties of Low Density Polyethylene[END_REF][START_REF] Chiu | A Review on Conduction Mechanisms in Dielectric Films[END_REF][START_REF] Crowell | The Richardson constant for thermionic emission in Schottky barrier diodes[END_REF]:

J = A * T 2 exp(- φ inj k B T ) (1.13)
where A * is the Richardson constant (A * ≈ 1.20 × 10 6 A.m -2 .K -1 ) and depends on the effective mass of the electron, as well as Boltzmann and Planck constants.

The Schottky injection allows to fit well the injection regimes in dielectric solids at moderately high fields. But at very high fields (∼ 10 9 V.m -1 ) and at room temperature, other mechanisms can take place due to the potential barrier being very thin.

In classical physics, if the potential barrier is higher than the energy of the incident electrons, then they will be reflected. However particles such as electrons which exhibit a particle-wave duality have a non-zero probability of crossing the barrier even if their energy is not high enough. This is known as the Fowler-Nordheim injection mechanism, also known as injection by tunneling-effect [START_REF] Fowler | Electron emission in intense electric fields[END_REF]. The Fowler-Nordheim injection barrier is schematized in figure 1.25: In this figure, Φ is the potential barrier due to contact only, and Φ e f f is the maximum of the reduced barrier due to the applied electric field. For convenience, image forces are ignored in this representation. The potential barrier is thus considered to be of triangular shape as shown in the figure. The energy of this potential barrier is thus defined by:

V(x) = 0; x = 0 V(x) = Φ -eEx; x = 0 (1.14)
In order obtain the current density obtained from electrons crossing the barrier by tunneling, one has to solve Schrodinger equation in one dimension, along the x direction. Details of the calculation are not presented here. The current density J is given by [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Hussin | The Effects of Crosslinking Byproducts on the Electrical Properties of Low Density Polyethylene[END_REF][START_REF] Guffond | Characterization and modeling of microstructure evolution of cable insulation system under HVDC fields[END_REF][START_REF] Chiu | A Review on Conduction Mechanisms in Dielectric Films[END_REF][START_REF] Fowler | Electron emission in intense electric fields[END_REF]:

J = x 0 0 dj x dx = AE 2 exp( - B E ) (1.15)
where A and B are functions of the energy barrier Φ e f f .

Space charges

Dielectrics, upon application of a low electrical field can polarize and generate an induced field that lead to transport of intrinsic electrical charges, such as ions. However, these process usually extinguish when the external applied field is turned off. Upon application of higher fields, extrinsic charge carriers (electrons) may be injected in the dielectric as explained in the previous section. These charge carriers may be transported to the polymer bulk and participate to electronic conduction. Also, ionic dissociation can take place and generate additional charge carriers which may as well get involved in conduction, accumulate at interfaces, and form what is known as "space-charge" [START_REF] Ahmed | Review of space charge measurements in dielectrics[END_REF]. Quite old experimental investigation on these space charge formation and effects on dielectrics have pointed out that these charges form during long term poling of dielectric samples at high fields, which eventually lead to breakdown [START_REF] Takada | Space Charge Formation in Dielectrics[END_REF][START_REF] Ieda | Carrier Injection, Space Charge and Electrical Breakdown in Insulating Polymers[END_REF]. The concern on the effect of these space charge on the lifetime of polymeric insulation led to lots of research work on the mechanism of formation of space charges and on the relation between charge accumulation, transport, and breakdown [START_REF] Takada | Space Charge Formation in Dielectrics[END_REF][START_REF] Ieda | Carrier Injection, Space Charge and Electrical Breakdown in Insulating Polymers[END_REF][START_REF] Sekii | A study on the space charge formation in XLPE[END_REF][START_REF] Roy | Charge transport and dissipative processes in insulating polymers: experiments and model[END_REF]. Accumulation of these charges is possible due to the disordered states present in the polymer, for instance physical disorders as chain segments or chemical impurities, which act as "trapping site" for charges [START_REF] Laurent | A discussion on the electronic properties of polyethylene interfaces[END_REF][START_REF] Roy | Charge transport and dissipative processes in insulating polymers: experiments and model[END_REF][START_REF] Thomas | Space-charge dynamic in polyethylene: from dc to ac stress[END_REF].

However, these space-charge mechanisms are mainly of concern under DC voltage, as charges accumulate in the material without reverse polarization.

Charge trapping

Localized states in the band-gap of polymers, as described in section 1.3.1, are present in the forbidden band, below and above conduction and valence band respectively. These energy states are introduced by intrinsic structural disorder (physical defects, such as cross-links, kinks, chain segments) as well as by extrinsic chemical disorder (cross-linking by-products, double bonds, oxidation). Such defects are known to act as charge traps, electron traps for acceptors and hole traps for donors. The effect of these conformational defects on the energy level distributions has been calculated using molecular-dynamics simulations, in the case of PE [START_REF] Chen | Electronic Structure of Polyethylene: Role of Chemical, Morphological and Interfacial Complexity[END_REF][START_REF] Huzayyin | An overview of impurity states and the basis for hole mobility in polyethylene[END_REF][START_REF] Laurent | A discussion on the electronic properties of polyethylene interfaces[END_REF][START_REF] Roy | Charge transport and dissipative processes in insulating polymers: experiments and model[END_REF]. Its was predicted that the two kinds of defects led to two different distributions of trap energy levels: physical defects produce shallow trap levels with energies up to 0.3eV and with a density of about 10 20 cm -3 , while chemical impurities led to deep trap levels with energies up to 1.5eV for electrons and holes [START_REF] Laurent | A discussion on the electronic properties of polyethylene interfaces[END_REF]. Figure 1.26 shows the schematic representation of shallow and deep trapping levels in the band gap of a polymeric insulator.

The distribution of trapping sites has been estimated using a decreasing exponential law: N t(e) = N 0 exp (-

Δ e k B T ) (1.16)
where Δ e is the electron trap depth, N t(e) is the electron trap density distribution at a given trap level and temperature, and N 0 is a pre-exponential factor. Thus, the lower the trap depth, the higher the trapping site density. These traps are essential for the comprehension of the conduction and ageing models, as it is believed that charges transport in AC is mainly described by hopping mechanism between trapping site [START_REF] Laurent | A discussion on the electronic properties of polyethylene interfaces[END_REF]. In particular, shallow trapping sites are of high importance as a higher fraction of charges are expected to be found in this energy range.

Charge transport

Following charge injection at high fields, it is necessary to understand how charges move from the metal-polymer interface into the bulk. When the electric field is sufficiently high, electrons and ions acquire an average velocity which is proportional to the field, and the voltage-current relation follow Ohm's law, as defined by equation 1.1. Charges have a given velocity which depends only on the carrier mobility and the applied field: v = μ E.

However, upon application of even higher fields, additional charges are injected through the electrodes and build up at the interfaces, progress in the bulk and modify the local contribution of the field. Different mechanisms and model have been built to describe conduction at high fields, which take into account trapping of the charges at heterogeneities and space charge accumulation. It is relevant to point out that these models have been developed in terms of applied DC fields [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF]. What is important concerning the high field phenomena, especially degradation and breakdown mechanisms, is the field amplitude or peak-to-peak value of the field. So most of the statistical analysis of DC life tests have been made in the same way as for AC. However this is not always valid, and conduction models built up for the DC case may not be appropriate for AC field. The aim of this section is to describe first the main conduction mechanisms presented in the literature, and take a dedicated paragraph at the end for the AC case.

Space charge limited conduction At high DC fields, and under a threshold value, the Ohm's law is not valid anymore and voltage-current characteristics follow a different trend, described by the Space charge limited conduction model (SCLC). This threshold value is the onset of space charge accumulation in the polymer bulk [START_REF] Guffond | Characterization and modeling of microstructure evolution of cable insulation system under HVDC fields[END_REF]. Extrinsic injected carriers become predominant over the intrinsic charge carrier present in the material, making the conduction current not linearly dependent from the field. As these charges accumulate in the material, the charge carrier density increases, and induce electric field modification. To the conduction current induced by the flow of charges under the applied field, a displacement current is added due to the space charge trapped in the bulk, as well as a diffusion current [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Hussin | The Effects of Crosslinking Byproducts on the Electrical Properties of Low Density Polyethylene[END_REF][START_REF] Guffond | Characterization and modeling of microstructure evolution of cable insulation system under HVDC fields[END_REF][START_REF] Chiu | A Review on Conduction Mechanisms in Dielectric Films[END_REF]:

J out = J cond + J sc + J di f f = neμ E + 0 r d E dt + (-eD dn dx ) (1.17)
where, μ is the charge carrier mobility, D is the Fick's diffusion coefficient for carriers and n(x) is the charge carrier number density, which is a function of the depth x in the bulk. At this stage, several assumptions are generally made.

Consequently, in a real dielectric, according to the SCLC model the transition from the Ohmic conduction regime to the space charge limited regime follows two steps: first, at a given threshold voltage V tr , the number density of injected charges n 1 exceeds the number density of intrinsic charges n 0 . Injected charges form space charges in the bulk material, and only a fraction of these charges participate to conduction as some of the carriers are being trapped in the materials by heterogeneities. The current density follows then a square dependence on the applied voltage. Increasing the voltage again at a second threshold voltage, all traps are filled with extrinsic charges, and the new injected charges can fully participate to conduction. This threshold voltage is known as the trap-filled limit voltage V TFL . Above this voltage, the fraction of extrinsic conducting carriers reach unity, and the dielectric is considered as a trap-free dielectric [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Guffond | Characterization and modeling of microstructure evolution of cable insulation system under HVDC fields[END_REF][START_REF] Chiu | A Review on Conduction Mechanisms in Dielectric Films[END_REF]. The different steps of the SCLC model are represented in the figure (1.27): Hopping conduction As described in previous sections, the density of states of a real insulator is not zero in the band gap. Localized energy states due to impurities are present below and above the conduction and valence edge respectively. Electrons can be trapped in these impurity states. These trapping-sites can be shallow (0.3 eV) or deep (1 eV), compared to the edge of conduction and valence band, but are sufficiently close that the potential barrier V(x) between them can be overcome with an energy of few k B T [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Hussin | The Effects of Crosslinking Byproducts on the Electrical Properties of Low Density Polyethylene[END_REF][START_REF] Guffond | Characterization and modeling of microstructure evolution of cable insulation system under HVDC fields[END_REF][START_REF] Chiu | A Review on Conduction Mechanisms in Dielectric Films[END_REF][START_REF] Roy | Charge transport and dissipative processes in insulating polymers: experiments and model[END_REF]. Conduction can be reached by thermally-activated transfer, or tunneling through potential barrier between localized states, or a combination of the two processes. Hopping conduction is also defined elsewhere as the intermediate situation in which a trapped electron is brought by thermal activation to a level which has the same energy as that of an empty neighboring site, which it then moves to by tunneling, as represented in figure 1.28 [START_REF] Roy | Charge transport and dissipative processes in insulating polymers: experiments and model[END_REF].

There is therefore a finite probability for the electron trapped in a localized state i to be raised to an empty neighboring site j at a distance R, by thermal-excitation then by tunneling. The probability of thermal-promotion and of tunneling are given by: Under application of an electric field, the barrier for thermal-assisted tunneling is reduced by a factor ΔE = -1 2 eE(x)R in the opposite direction of the field, and increased by a factor ΔE = 1 2 eE(x)R in the field direction [START_REF] Hussin | The Effects of Crosslinking Byproducts on the Electrical Properties of Low Density Polyethylene[END_REF][START_REF] Guffond | Characterization and modeling of microstructure evolution of cable insulation system under HVDC fields[END_REF][START_REF] Roy | Charge transport and dissipative processes in insulating polymers: experiments and model[END_REF]. This process in represented in figure 1.29. Assuming a randomly uniform distribution of trap levels, the general form of hopping conductivity is given by [START_REF] Mott | Electronic Processes in Ionic Crystals. 2nd[END_REF][START_REF] Anta | Models of electron trapping and transport in polyethylene: Current-voltage characteristics[END_REF]:

P(therm) ∝ exp (-V(x) k B T ) P(tunnel) ∝ exp (-2R 2mE/h 2 ) (1.18)
σ ∝ exp (- V(x) k B T )sinh( eE(x)R k B T ) (1.19)
For high values of electric field E(x) the hopping barrier in the opposite direction of the field is drastically reduced, so the sinh(x) function becomes equivalent to 1 2 exp (x), which gives a simplified expression for the hopping conductivity:

σ ∝ 1 2 exp (- V(x) k B T ) exp ( eE(x)R k B T ) (1.20)
Hopping conduction is considered as a non-band transport, as it does not involve transport in the conduction band (or valence band for holes), but only the transfers between localized states. Band-transport associated mechanisms are described by the Pool-Frenkel mechanism.

Pool-Frenkel effect Detrapping has been described in hopping conduction as a possible mechanism to occur under thermal excitation, leading to conduction from site to site. In the condition of shallow electron traps, near the conduction edge, electrons can be detrapped and directly access to delocalized states in the conduction band. This effect is known as the Pool-Frenkel effect [START_REF] Frenkel | On Pre-Breakdown Phenomena in Insulators and Electronic Semi-Conductors[END_REF]. This effect is similar to the Schottky thermionic emission from metal to polymer conduction band, but from bulk to conduction band. For an electron trapped in localized state below the conduction edge level to be detrapped, it must overcome an energy barrier V which cannot be with application of electric field. Under an applied field, image force and field effect reduce the potential energy required for the electron to reach the conduction band level (see figure 1.30). As described in the Schottky effect, the reduced potential barrier energy is given by:

E pot = V - e 2 4π 0 r x -eEx (1.21)
The conductivity of electrons generated by the Pool-Frenkel effect is of the form:

σ ∝ exp (- E det k B T ) = exp (- V k B T ) exp (- e 3 E π 0 r ) (1.22)
Charge transport under AC case The effect of space charge under DC field on the injection and conduction mechanisms have been extensively studied in the past thirty years as explained in the previous section. However, only limited research on the existence and effect of space charge under AC field is available as compared to DC case. Nevertheless, a bi-polar numerical model has been proposed by Zhao et al. [START_REF] Zhao | Numeric description of space charge in polyethylene under ac electric fields[END_REF]. According this model, the reasons for such limited research are a smaller quantity of charges accumulated in the material and the difficulty to monitor experimentally the behavior of charge carriers. Under DC field, the material sees a maximum voltage U max which is constant over time. Above a threshold voltage, electrons are injected from the cathode and move in the direction of the anode where they can be extracted. Charge carriers follow Shottky and/or tunneling injection. When applying a uniform AC field on a polymer sample, the voltage switches frequently from positive to negative values and is not constant over time. The expression of a typical sinusoidal AC applied voltage can be written as follow:

V app ( f , t) = V peak × sin(2π f t) (1.23)
where V app ( f , t) is the applied voltage and is a function of voltage frequency f (Hz) and time t(sec) and V peak is the maximum value of the voltage at each AC cycle. Thus, for the electrons to be injected, the threshold voltage is reached twice during an AC half-cycle, at a time t 1 during the increase of applied voltage and time t 2 during the decrease of voltage. Charges can be injected in this time period before voltage reaches the zero-crossing point and switches to the inverse polarity. Then during the following half-cycle, the voltage reaches a threshold value for injection of holes, before in reaches back the zero-crossing level. Electrons and holes are, according to this model, injected and extracted by turns. This mechanism implies that both injected charges can participate to conduction during a specific time period Δt before the voltage switch. Part of the injected charges can also be trapped in shallow and deep trapping centers as explained in previous subsections. Thus, charges have limited time to accumulate, participate to conduction, and thus, a limited distance to progress in the bulk. Also, part of the trapped charges can encounter injected charge of opposite polarity after the voltage switch, and undergo charge recombination [START_REF] Zhao | Numeric description of space charge in polyethylene under ac electric fields[END_REF][START_REF] Laurent | Time-resolved space charge and electroluminescence measurements in polyethylene under ac stress[END_REF]. AC space charge build up at the electrode-polymer interface and in the polymer bulk is mainly driven by frequency and threshold voltage. A schematic representation of the injection of charge carriers under AC applied voltage is given in figure 1.31: Different experiments have been developed to measure the build of space charge under AC field, and obtained similar results [START_REF] Laurent | Time-resolved space charge and electroluminescence measurements in polyethylene under ac stress[END_REF][START_REF] Chen | Ac aging and space-charge characteristics in low-density polyethylene polymeric insulation[END_REF][START_REF] Thomas | A New Method for Space Charge Measurements Under Periodic Stress of Arbitrary Waveform by the Pulsed Electro-Acoustic Method[END_REF][START_REF] See | Measurement of space-charge distributions in solid insulators under rapidly varying voltage using the high-voltage, high-speed pulsed electro-acoustic (PEA) apparatus[END_REF][START_REF] Bamji | Phase Resolved PEA Technique to Measure Space Charge Distribution in Polymeric Insulation[END_REF]. For instance, Laurent et al. measured the space charge accumulation and the charge recombination in different PE materials under AC stress [START_REF] Laurent | Time-resolved space charge and electroluminescence measurements in polyethylene under ac stress[END_REF]. As a result, they concluded that it is difficult to detect space charge accumulation under AC stress because of the low amount of charges and the close location of charges as respect to the electrode-polymer interface (only 5 to 10 μm away from the electrode) which lays in the spacial resolution limit of the measuring device. They also detected luminescence signature during injection and extraction of opposite charges at 60 and 80 • . Luminescence upon AC field source indicates a recombination of opposite charges during AC cycle. Also, the phase dependence of the luminescence signature indicates that injected charge recombine with remaining opposite charges from the previous half-cycle, thus proving the presence of space-charge under AC. It was also found that a threshold field exists for space-charge and luminescence measurement. Injection, extraction, accumulation and recombination of charges occur at field values as low as 10 kV/mm in PE. Space-charges under AC are believed to initiate at the vicinity of the electrode due to the frequently reversing field. However, upon application of the field for higher time period (several hours) it has also been found that space-charge accumulates in the material and move towards the polymer bulk [START_REF] Chen | Ac aging and space-charge characteristics in low-density polyethylene polymeric insulation[END_REF]. Above a threshold voltage for space-charge injection, most of the charges are trapped in shallow traps which allows detrapping, conduction by hopping and extraction/recombination of charges. It is explained that with increasing exposure time to AC field, deep trapping centers can be created with ageing. These charges then cause field enhancement in the polymer bulk and can initiates degradation.

Conduction paths for electrons and holes

When dealing with charge transport in polymeric insulation, important theories for conduction and breakdown were established, as described previously and detailed in the next section. But concerning the behavior of charge carriers only, in this type of material several questions were encountered by researchers in the past seventy years. Especially, what are the dominant carriers between electrons and holes, and how do they move in the insulation [START_REF] Mizutani | Behavior of Charge Carriers in Organic Insulating Materials[END_REF]. We have seen that the morphology of semi-crystalline polymers is rather complex and influence numerous physical properties. An important question to answer is how electrons and holes behaves in the crystalline and amorphous regions of the material. Or in other words, what are the possible conduction paths for charge carriers in the material's structure. The wave-function of valence states (holes) is maximum along the C-C bonds whereas the wave-function of conduction states (electrons) is maximum between chains [START_REF] Serra | Interchain electron states in polyethylene[END_REF][START_REF] Ceresoli | Exciton self-trapping in bulk polyethylene[END_REF]. These results suggest that holes exist along PE chains, while electrons are more likely to reside and move between chains. Thus, electrons would prefer staying in amorphous inter-lamellar regions and not in the crystalline part. These conduction behavior are represented in figure 1.33 below: Amorphous material would play a major role in charge conduction of real PE. Band conduction would be expected in the crystalline part and hopping conduction in the amorphous material [START_REF] Mizutani | Behavior of Charge Carriers in Organic Insulating Materials[END_REF]. However, Mizutani highlights these calculations should be supported by experimental results. Nevertheless, these results support the fact that the complex morphology of semi-crystalline materials have a critical influence on the major properties of the final material, including charge carrier transport. The evolution of this morphology during the material's lifetime may have a huge impact on these properties.

Degradation phenomena: breakdown and ageing of polymeric insulation 1.4.1 Breakdown mechanisms 1.4.1.1 Thermal breakdown

Thermal breakdown is one of the oldest known phenomena in solid dielectrics and "perhaps the most obvious form of breakdown mechanism" [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF]. Under an applied field, a solid dielectric acts as an insulator which continuously generates joule heating from electrical conduction and polarization. A balance between heat flowing in the dielectric and power dissipation takes place. If, upon application of an increasing voltage, the equilibrium condition is no longer valid then the dielectric can no longer dissipate heat generation and thermal instability occurs, which can lead to a large increase of the temperature, to physical changes of the polymer and ultimately to breakdown [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Kao | Dielectric phenomena in solids[END_REF]. The general expression of the thermal equilibrium is given by the balance of heat generation rate and heat loss over electrical power dissipation [START_REF] Kao | Dielectric phenomena in solids[END_REF]:

C V × dT dt -K∇ 2 T = σE 2 (1.24)
where C V is the specific heat capacity per unit volume (J.K -1 .m -3 ), K and σ are the dielectric's thermal (W.m -1 .K -1 ) and electrical (S.m -1 ) conductivity respectively. Usually, two cases are taken to solve equation 1.24:

• In the first case, a fast build up of electric field is assumed, and the heat losses are neglected. From these assumption, the temperature can be obtained as a function of the applied voltage. This case is known as the Impulse Thermal Breakdown.

• In the second case, a very thick system is considered, so that all generated heat is carried away to its surroundings, and the therm dT dt can be neglected. In this case, a finite maximum thermal voltage can be obtained for a given temperature, before breakdown. This is known as the Steady State Thermal Breakdown.

In both case, the important parameter is the electrical losses, and they are larger in AC than in DC, that is why it is important to keep the tan δ as low as possible.

Electro-mechanical breakdown

Electro-mechanical breakdown is a mechanism existing due to the electrostatic attraction force of oppositely charged electrodes which generates a compressive stress on the dielectric in between them. Indeed, the attraction of electrically charged electrodes generates an electrostatic force also known as Maxwell compressive stress which depends on the dielectric's permittivity and the square of the applied field [START_REF] Crine | Role of electrostriction on the electrical aging of polymers[END_REF]:

σ Maxwell = 0 r 2 E 2 (1.25)
This compressive stress can lead to damaging and ultimately to breakdown of the system by different possibilities. Under the influence of the compressive stress, the dielectric opposes an elastic stress which is dependent on its mechanical properties.

In the case in which the Maxwell stress exceeds a critical value which cannot be balanced by the dielectric elasticity, breakdown can occur. Stark and Garton proposed a criterion for equilibrium of the two opposite stress [START_REF] Stark | Electric Strength of Irradiated Polyethylene[END_REF]:

0 r 2 E 2 > Yln( d 0 d ) (1.26)
where, Y is the dielectric's Young Modulus (GPa), d 0 and d are the dielectric thickness (m) before and after application of applied field respectively. From this criterion, the maximum voltage before exiting equilibrium can be obtained:

V max = d 2Yln( d 0 d ) 0 r (1.27)
Breakdown is usually not suspected to occur electro-mechanically, because of the relatively high values of Young modulus of polymer dielectrics. However, under long term solicitations and at operating temperatures, softening of the material can enhance this phenomenon and generate mechanical fatigue, creeping or creation of microscopic defects, which will be detailed later in this chapter.

Electronic breakdown

Electronic breakdown regroups the field dependent mechanisms that causes either the energy or the number of charge carrier to reach a critical values so that irreversible changes take place in the material lattice [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF]. It is also the most difficult form of breakdown phenomena to predict as only few measurable markers have been identified. It is usually convenient to consider two main different mechanisms leading to electronic breakdown: "Intrinsic breakdown" and "Avalanche breakdown" or "Hot-carrier impact ionization".

Intrinsic breakdown

In the case in which a free charge, as for example an injected or a detrapped electron, acquires energy from the field faster than it loses it from carrier interactions, then it can reach critical energy magnitudes than lead to local deterioration of the polymer lattice and ultimately to breakdown. This mechanism depends only on the temperature and the intrinsic polymer's properties. It is occurring in very short time period (10 -8 to 10 -7 seconds) and therefore is considered purely electronic. The energy acquired by a free electron in the conduction band can be written as follow:

E kinetic = q e E app λ (1.28)
where q e (C) is the electron's charge, E app (V/m) is the applied field, and λ (nm) is the mean free path of the electron. λ is an intrinsic property of the polymer, and represents the mean distance available for the electron before being trapped, or interact with the lattice or another electron. If the distance is large enough, then the electron can acquire sufficient energy to cause local damage, and ultimately lead to breakdown. Energy values as high a few eV can be sufficient to damage molecules, or to break bounds. However, the theoretical values available for the mean free path are relatively low in solids (about 5 -20 Å) [START_REF] Kwan | New theory of electrical discharge and breakdown in lowmobility condensed insulators[END_REF]. Therefore, single electrons are not able to cause breakdown but local disruption of the lattice instead. Intrinsic breakdown may occur due to a concentration damage, or in low density regions where mean free path can reach bigger values.

Zener breakdown

The earliest theory for intrinsic breakdown was proposed by Zener in 1934 [START_REF] Zener | A theory of the electrical breakdown of solid dielectrics[END_REF]. The general idea is that under strong applied electric field, free electrons can be generated by tunneling from the occupied valence states to the conduction states. The application of an external field lead to a small but finite probability of tunneling from an initial energy state to a higher unoccupied state in the field direction. The process is represented in figure 1.34. However, this mechanism was proposed in order to explain intrinsic breakdown in dielectric solids with lower band gaps and experimental measures of breakdown in polymeric insulators concluded that other mechanisms were acting instead. Zener tunneling can prevail in narrow band gap materials such as doped semi-conductors [START_REF] Kao | Dielectric phenomena in solids[END_REF].

Avalanche criterion If a conducting electron with high energy interacts inelastically with another trapped electron, or with a bound electron from an atom or a molecule, there is a probability to generated a secondary electron which can be accelerated by the applied field. The pair of electrons can acquire then sufficient energy to generate another pair of free electrons. This multiplication process, known as "Avalanche", may lead to a critical density of free electrons which can lead to a high concentration of damage and thus to breakdown. The avalanche criterion lies in the fact that at each collision a higher number of new free electrons are produced, until a critical number of electron is produced at the i th generation to cause breakdown. Considering that at each generation, 2 electrons are produced, then 2 i electrons are produced at the critical i th generation. In the literature, theoretical values of i are reported around 30 to 40 [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF]. As the ionization process is considered to occur very quickly, by the end of it all the electrons will have acquired an energy equal to 2 i q e E app λ, which, even at relatively low applied field can be enough to damage irreversibly the lattice and cause breakdown. However, the threshold field for self electron avalanche is difficult to estimate, since it depends on the local structure and chemistry of the material which influences the mean free path for charge carriers. An early approach for the threshold field estimation has been proposed by Stratton [START_REF] Stratton | The influence of interelectronic collisions on conduction and breakdown in covalent semi-conductors[END_REF]. However this theory was suitable for crystals or semi-conductor materials, and thus hardly adaptable to lowmobility solids such as polyethylene, where other breakdown mechanisms such as thermal runaway may be acting.

Theory of ageing in polymeric insulation

Ageing of electrical insulation is described in the literature as all "occurrence of irreversible, deleterious changes in insulating materials or systems which affect their serviceability, i.e. their ability to satisfy requested performances" [START_REF] Montanari | Aging phenomenology and modeling[END_REF]. More generally, ageing can be defined as a set of progressive and irreversible changes in the insulation structure and physico-chemical properties which decrease its lifetime, and make it unpredictable. Ageing regroups all the prebreakdown mechanisms, and thus is the most important aspect when studying the long term properties of insulating materials. As for breakdown mechanisms, there are several different ageing mechanisms that can act on the material. The causes of ageing are the applied stresses (thermal, mechanical and electrical) and the intrinsic/extrinsic factors that influence the stress. Dissado and Fothergill describe electrical ageing as a combination and acceleration of physical and chemical ageing under application of electric field [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF]. It is thus essential to cover the aspects of physical and chemical ageing of polymeric insulation in the absence of electric field before presenting the common electrical ageing phenomena.

Physical Ageing

In physical ageing, the chemical structure of the polymer chains is not modified.

Ageing is due to a modification of the spatial configuration of the macromolecules composing the chains, which for instance can act on the chain relaxation [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Fayolle | Vieillissement physique des matériaux polymères[END_REF].

Physical ageing can involve changes in the structure of the polymers because of relaxation and/or re-orientation of the polymer chains. It exists in both amorphous and semi-crystalline polymers. The chain relaxation occurs due to the meta-stable state in which the polymer has been put during processing [START_REF] Fayolle | Vieillissement physique des matériaux polymères[END_REF]: when the polymer is cooled down from a molten rubbery state to a state S 1 below its molten temperature, the polymer structure eventually freezes to a meta-stable state because the chain mobility cannot follow the cooling kinetics. If the material is then used at a temperature T use < T melt , the polymer structure tends to slowly relax in order to reach the stable state, giving rise to re-orientation of the polymer chains. This mechanism is represented in figure 1.36 : In this illustrated case, the evolution of a structural parameter S of a polymer is represented as a function of temperature. This structural parameter can represent a variation of free volume or enthalpy in the polymer structure. Free volumes are the difference between the specific volume of the material measured at the meta-stable temperature and the specific volume of the packed material form at equilibrium, at the same temperature [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF]. When cooling down the material, the structural parameter decreases linearly with temperature until it reaches the molten temperature T melt of the material. From this point, further cooling leads to a freezing of the structural parameter at the meta-stable state. Eventually, when using back the polymer at a temperature T use the structural parameter slowly recover its stable equilibrium state represented by the dashed line. This slow recovery is responsible for modification of the polymer chains orientation, density and possibly crystallinity, in the case where the material could not fully crystallize and thus secondary crystallization can take place. Physical ageing is thus the progressive recovery of the polymer to a more stable states, changing its thermal and mechanical properties with time [START_REF] Fayolle | Vieillissement physique des matériaux polymères[END_REF].

Chemical Ageing

Chemical ageing involves the modification of the polymer chemical structure by its surrounding environment, which leads to a decrease in the material's lifetime. Different mechanisms and factors can be at the origin of the degradation, such as temperature, UV-light, water, or other chemically reactive species, interacting with the material [START_REF] Richaud | Vieillissement chimique des polymères -Mécanismes de dégradation[END_REF]. From this interaction, several degradation processes can be expected with irreversible consequences at the macromolecular scale. Among them: oxidation, scission and cross-linking of chains. Oxidation changes the nature of a chemical group and does not affect the macromolecular length, while chain modification can affect the length of the macromolecules. These mechanisms always target a molecule or chemical group, as for instance a methyl group in a polymer chain which can be subject to oxidation. The change in the nature of this chemical group is a marker of chemical ageing [START_REF] Richaud | Vieillissement chimique des polymères -Mécanismes de dégradation[END_REF].

Chain scission can occur due to various causes: hydrolysis, thermo-oxidation, and photo-or radio oxidation. The first case is characterized by the scission of chains initiated by water at specific chemical bonds, like esters or amides. Thermo-oxidation is characterized by chain scissions initiated by temperature, and photo-and radiooxidation by photons or radiations. These chain scission processes are generally statistical, which means that they are randomly distributed on the polymer chains.

Other chain scission mechanisms, such as depolimerization, are selective mechanisms, which attack the end of the chains and occur at relatively high temperatures. Figure 1.37 shows a representation of the chain scission occurring in polymers. Chain cross-linking is characterized by the formation of covalent bonds between the linear polymer chains [START_REF] Richaud | Vieillissement chimique des polymères -Mécanismes de dégradation[END_REF]. It can occur during chemical ageing after thermo or photo-oxidation and formation of chain radicals, which recombine to form a bond. Cross-linking is generally competitive with chain scission, and tend to increase the molecular mass of the final material [START_REF] Richaud | Vieillissement chimique des polymères -Mécanismes de dégradation[END_REF][START_REF] Salem | Oxidation and crosslinking processes during thermal aging of low-density polyethylene films[END_REF]. Figure 1.38 shows the scheme of crosslinking reaction during chemical ageing due to chain radicals. • Oxidation is initiated by the scission of covalent bonds in the polymer chain (R), which leads to the formation of two alkyl radicals (R•). This initiation is initiated preferentially at structural irregularities or impurities, which have weaker bond energy.

• Once oxidation is initiated, it is propagating by radical recombination reactions. The highly reactive formed alkyl radical (R•) form peroxyl radicals with oxygen (RO 2 •), which then attack the polymer macromolecular backbone on secondary (for PE) or tertiary carbons (for PP) [START_REF] Pelzer | Etude de vieillissement des isolants synthétiques des câbles moyenne tension[END_REF][START_REF] Gardette | Photo-and thermal-oxidation of polyethylene: Comparison of mechanisms and influence of unsaturation content[END_REF]. This leads to the formation of hydroperoxydes (ROOH) and new alkyl radicals, which continue to propagate oxidation.

• Termination occurs in the case where two alkyl radicals cross-linked together (R-R), or where two peroxyl radicals react together (ROOR).

During propagation, highly reactive species, namely hydroperoxydes (ROOH), are formed. These hydroperoxydes can dissociate easily and form new radicals: alcoxyls (RO•) and hydroxyls (OH•). The freshly formed radicals can in turn attack hydrogen atoms of the macromolecular chain, which forms new alkyl radical chains (R•), and participate to propagation of oxidation. Oxidation is thus a sustainable reaction supplied by hydroperoxydes [START_REF] Pelzer | Etude de vieillissement des isolants synthétiques des câbles moyenne tension[END_REF]. Additionally, other chemical active species such as ketones, esters, alcohols, carboxyilic acids can be formed during propagation, due to the decomposition of hydroperoxydes (see figure 1.40).

Phenomenological approach of ageing and life models

Under application of thermo-electrical stress, various physical phenomena take place in the insulation from macroscopic to nanometric scale. But the first models that have been proposed aiming to predict the lifetime of insulators were based on a phenomenological approach [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Montanari | Aging phenomenology and modeling[END_REF]. These life models consist basically in looking for adequate relationships between insulation life and applied stress, i.e. electric field, temperature, mechanical stress, and so on. In this section the principle life models used for polymeric insulation are reviewed.

Dakin model

In the early research on ageing of electrical insulation, Dakin proposed a thermal ageing life model based on a chemical process and the rate of deterioration [START_REF] Dakin | Electrical Insulation Deterioration Treated as a Chemical Rate Phenomenon[END_REF]. In this model, a physical property P(C) of an insulator is assumed to be a function of a concentration C(t) of a parameter of the insulation constitution, for instance the average number of molecular links. The deterioration of this property is described as a the rate of change of this concentration with time:

C(t) = C 0 × exp(-Kt) (1.29)
where C 0 is the initial concentration and K is a rate constant of chemical change, which depends on the stress (here temperature) and activation energy of the reaction. Dakin proposes thus an exponential law, based on the Arrhenius model, where the concentration and thus the life of the insulation decreases exponentially. However this model has been criticized on the basis of being dimensionally incorrect and not taking into account the effect of electric stress.

Eyring model and inverse power law

Eyring proposed a model from Dakin theoretical considerations including an additional stress term to the temperature [START_REF] Endicott | Application of the Eyring Model to Capacitor Aging Data[END_REF]. In his model, the rate of degradation R is a function of temperature, applied stress:

R = R 0 V n = k dD dt (1.30)
where R 0 represents the degradation rate in absence of stress, V (kV) is the applied voltage, n is a parameter and a function of temperature, k is a constant and D is the total damage at time t. By rearranging equation 1.30:

D = R 0 k V n t + d 0 (1.31)
where d 0 is the initial damage before applied stress. Eyring assumes that the accumulated damage at a given period of the material's life can be written as a power law with applied multiple stress.

Other power law such as inverse power law have been used to describe insulation life time as a function of applied stress, and are still used nowadays for life data [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Howard | The effect of electric stress on the life of cables incorporating a polythene dielectric[END_REF][START_REF] Yoshimura | Effect of microvoids on V-t characteristics and tree growth in crosslinked polyethylene[END_REF][START_REF] Simoni | A General Approach to the Endurance of Electrical Insulation Under Temperature and Voltage[END_REF].

Crine model Crine et al. have proposed an ageing model based on

Eyring's rate theory of damage, by including the stress induced by macromolecules deformation under electro-mechanical stress [START_REF] Crine | Role of electrostriction on the electrical aging of polymers[END_REF][START_REF] Crine | A molecular model to evaluate the impact of aging on space charges in polymer dielectrics[END_REF][START_REF] Crine | Influence of high field and molecular effects on some electrical properties of polymers[END_REF]. In this model, the main emphasis is on the effect of high field (≈ 60kV/mm) on the deformation of molecular segments: strong electric field generate strain on the C-C bonds of the polymeric insulation. The C-C bond breaking leads to the formation of voids in the insulation. In this voids, the energy barrier for electron injection is lowered, thus initiating electron avalanche and macro molecules ionization, further enlarging the cavities, and thus increasing kinetic energy of carriers. Lifetime in this model is thus a function of stress, temperature as well as an activation energy for C-C bond breaking:

t = h ak b T × exp[ ΔG -1 2 ΔV + 1 2 δv σ 2 Y k b T ] (1.32)
where h (J/s) is the Planck constant, a is a function of the voltage frequency (a ≈ 2 × f in AC; a = 1 in DC), k b (J/K) is the Boltzmann constant, T (K) is temperature, ΔG (J/mol) is the activation energy term related to the bond strength of C-C bonds, ΔV is the stress activation volume, δv is the strained activation volume, σ (Pa) is the Maxwell electrical-mechanical stress (σ = E 2 2 ) and Y (Pa) is the young modulus. Crine's model considers thus that damage is initiated by C-C bond breaking caused by electro-mechanical strain, and propagates locally at nanoscale by ionization in voids and cavities. This model takes into account the action of energetic carrier such as electrons in the propagation of ageing, but does not consider that charges accumulating in the insulation can be the cause of damage.

Lewis model On a similar reflection, Lewis et al. proposed a model for polymer degradation based on the electro-mechanical breakdown proposal by Stark and Garton, or on the electro-fracture breakdown mechanism proposed by Zeller [START_REF] Lewis | Polyethylene under electrical stress[END_REF][START_REF] Stark | Electric Strength of Irradiated Polyethylene[END_REF][START_REF] Zeller | Electrofracture mechanics of dielectric aging[END_REF][START_REF] Jones | The contribution of field-induced morphological change to the electrical aging and breakdown of polyethylene[END_REF]. In his model, Lewis emphases the complex and specific morphology of semicrystalline polymers, and the morphological response of the polymer to a high electrical field. The basic principle of the model is that ageing has an electro-mechanical origin due to the stress set up by a mechanical response to the applied field, and occur independently of electronic processes such as charge injection, conduction or ionization [START_REF] Jones | The contribution of field-induced morphological change to the electrical aging and breakdown of polyethylene[END_REF].

Lewis explains that the morphology of semi-crystalline polymers such as PE is considerably non-uniform regarding mechanical strength: strong intra-molecular covalant bonds dominate the properties of the crystalline lamellae whereas intermolecular bonds between the chains in the lamellae and in the amorphous interlamellae space are much weaker. In addition, inter-lamellar region are more likely to foster defects and impurities as seen previously. Crazes and cracks are known to develop in amorphous inter-lamelllar part of semi-crystalline polymers subject to mechanical stress [START_REF] Mourglia-Seignobos | Physical Mechanisms of Fatigue in Neat Polyamide 6,6[END_REF]. Thus when considering the influence of forces generated by electric field, these amorphous inter-lamellar paths are more likely to undergo ageing.

Effect of space-charges

As most of the previous detailed models were developed in the aim of understanding electrical ageing in the case of HV-AC insulation, the effect of space-charge has been considered in the model developed by Dissado, Montanari and Mazzanti in the DC case [START_REF] Dissado | The role of trapped space charges in the electrical aging of insulating materials[END_REF]. In the later, the aim was to describe quantitatively the effect of trapped space charge in localized regions on the aging mechanisms of polymeric insulation.

In practice, the model is derived from Dakin and Crine models, and the basic concept is based on a the following diagram: A material in a thermodynamic initial state 1 and having a free energy G 1 have to acquire an energy Δ = G 2 -G 1 in order to reach a second stable state state 2. In other words, evolution of an unaged material in a stable state 1 to an aged stable state 2 is not favored at thermodynamic equilibrium. This evolution to a second state describes the formation rate of defects in the insulation with ageing which is thermally activated. However, the model states that space-charge accumulation is responsible for this physical transformation, by lowering the energy barrier required between the two states. This is explained by the effect of space-charge on local electric field enhancement.

Montanari et al. recently proposed a new approach of this DC model by incorporating the role of defects in space-charge accumulation and localization of stress, leading to mechanical strain and formation of crazes [START_REF] Carlo Montanari | Aging mechanisms of polymeric materials under DC electrical stress: A new approach and similarities to mechanical aging[END_REF]. Application of the electric life model to accelerated life date shows relevant results in the case of XLPE under DC stress.

Ageing associated to charge carrier interaction

The models described previously have demonstrated a rather suitable approach for fitting life data of polymeric insulation under AC and DC stress. However, other physical mechanisms have been proposed intending to explain the complex nature of ageing induced by interaction with carriers, involving the generation and conduction of high energy charge carriers, or "hot-electrons" [START_REF] Kwan | New theory of electrical discharge and breakdown in lowmobility condensed insulators[END_REF][START_REF] Zakrevskii | Mechanism of electrical degradation and breakdown of insulating polymers[END_REF]. Frölich was one of the first to describe intrinsic breakdown based on electron-phonon interactions [START_REF] Fröhlich | On the theory of dielectric breakdown in solids[END_REF].

These theories are based on the principle of intrinsic breakdown explained earlier: if the applied electric field is insufficient to cause spontaneous generation of carriers from the valence band, localized degradation events caused by individual ionizations can explain ageing and breakdown. Kao proposed an ageing theory in which injected electrons into the insulator conduction band play a major role in initiation of ageing [START_REF] Kwan | New theory of electrical discharge and breakdown in lowmobility condensed insulators[END_REF].

Let's consider a free electron injected from cathode by tunneling: it is admitted that this charge will quickly be trapped by the material within few scattering, because of the small mean free path involved. However, this charge can still create low density regions with a larger mean free path, and thus increasing the number of scattering and the energy involved for the next carriers. A carrier with a sufficiently high mean free path will acquire kinetic energy high enough, in the order of few eV, to initiate ionization of a molecule as represented in the figure 1.44: The dissociation process can lead to a trapped charge and an energy release from the second electron that can be transferred to a third electron, susceptible of accumulating in its turn kinetic energy and propagate [START_REF] Kwan | New theory of electrical discharge and breakdown in lowmobility condensed insulators[END_REF]. Experimental evidence of this hot electron processes have been published in the literature [START_REF] Pfluger | Hot-Electron Transport in Polymeric Dielectrics[END_REF][START_REF] Sanche | Electronic Aging and Related Electron Interactions in Thinfilm Dielectrics[END_REF][START_REF] Laurent | Hot electron and partial-discharge induced ageing of polymers[END_REF][START_REF] Teyssedre | Experimental Evidence of Hot Electrons in Insulating Polymers[END_REF][START_REF] Teyssedre | Evidence of hot electron-induced chemical degradation in electroluminescence spectra of insulating polymers[END_REF].

Bond dissociation may be followed or accompanied by oxidation or cross-linking as discussed previously. The progressive change in the insulation's chemical structure may be responsible for the weakening of its properties, including the electronic structure. Chen et al. proposed a theoretical approach of the electronic structure of PE by quantum calculations, focusing on the specific role of physical and chemical defects [START_REF] Chen | Electronic Structure of Polyethylene: Role of Chemical, Morphological and Interfacial Complexity[END_REF]. It is shown that the presence of these defects in the insulation results in the formation of localized states in the band gap of PE, which reduce the injection barrier for charges at the electrode/polymer interface.

Treeing

Treeing is a degradation phenomenon that has been observed since the early days of electrical insulation [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Dissado | Understanding Electrical Trees in Solids: from experiment to theory[END_REF]. Different forms of treeing degradation exist, mainly water and electrical trees, but the general aspects are the same. Electrical trees are formed in insulation subjected to locally high electric stress concentration, starting from a nucleation point and resulting in the generation of a connected system of conducting channels, as shown in figure 1.45: Electrical trees are the consequences of a series of ionization of the polymer in conduction paths for carriers, forming the different branches of the tree, without gas discharge. Once developed, an electrical tree can propagate across the insulation and eventually lead to breakdown. They are different stage in the development of treeing. First, there is an inception stage where the first observable damage is found, forming small tubes or cavities [START_REF] Zhu | The Growth Mechanism of Brush-type Electrical Tree in XLPE[END_REF]. Inception of the tree is generally triggered at insulation weak points of stress enhancing geometrical features, such as a point electrode, a protrusion or inclusion, a conducting impurity, or a discharge in a gasfilled cavity. An initiation time at a given applied voltage can be measured. Then the tree propagates, an channels form statistically by choosing the easiest path to ionize. Wiesmann and Zeller proposed a fractal model for the growth of electrical trees, and for which case the tree would develop or eventually stop [START_REF] Wiesmann | A fractal model of dielectric breakdown and prebreakdown in solid dielectrics[END_REF]. The size of this channels can vary, but are generally micrometric, as shown in figure 1.46. Electrical trees can take different typical shapes like branched trees, bush trees or bow-tie trees [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF][START_REF] Dissado | Understanding Electrical Trees in Solids: from experiment to theory[END_REF][START_REF] Zheng | The influence of AC and DC voltages on electrical treeing in low density polyethylene[END_REF]. It was also found elsewhere that electrical trees propagate in the interlamellar regions and at the boundaries of spherulites in LDPE [START_REF] Shi | Investigation on electrical tree propagation in polyethylene based on etching method[END_REF]. It is explained that electrons can be accelerated more easily by electrical field in disordered amorphous regions where the molecular chain density is low the free volume large, therefore propagating more easily electrical trees. Electrical trees are thus a structural marker of pre-breakdown phenomena occurring in the insulation, and are believed to lead to the final breakdown of the insulation.

Ageing associated to partial discharges

The degradation of solid polymeric insulation can be described by many different mechanisms, as explained earlier. But in the case of a presence of a critical concentration of microscopic defects such as voids and cavities, it is generally accepted that the main process driving insulation degradation is the occurrence of partial discharges (PD) [START_REF] Morshuis | Partial Discharge Mechanisms -Mechanisms leading to breakdown, analysed by fast electrical and optical measurements[END_REF][START_REF] Niemeyer | A generalized approach to partial discharge modeling[END_REF][START_REF] Gutfleisch | Measurement and simulation of PD in epoxy voids[END_REF][START_REF] Morshuis | Degradation of solid dielectrics due to internal partial discharge: some thoughts on progress made and where to go now[END_REF][START_REF] Montanari | Model of ageing inception and growth from microvoids in polyethylene-based materials under AC voltage[END_REF][START_REF] Wang | Evolution of pd patterns in polyethylene insulation cavities under AC voltage[END_REF][START_REF] Andres | Lumped mechanism for polymeric dielectric degradation under high electrical fields[END_REF][START_REF] Cavallotti | Thermodynamic analysis of the degradation of polyethylene subjected to internal partial discharges[END_REF].

Cavity-polymer environment

The presence of a micro-volume in the polymer matrix, or at interfaces, brings several changes in the distribution of the electrical field (and thus energy) but also in the behavior of charges. These micro-volumes may be filled with a mixture of air (N 2 + O 2 ) and water [START_REF] Morshuis | Degradation of solid dielectrics due to internal partial discharge: some thoughts on progress made and where to go now[END_REF][START_REF] Serra | Theory of inception mechanism and growth of defect-induced damage in polyethylene cable insulation[END_REF] . Thus, a discharge is likely to happen, considering that free charges are available, if the electric field in the cavity exceeds the breakdown strength of the gas, leading to the formation of an electron avalanche [START_REF] Hazlee A Illias | The influence of spherical cavity surface charge distribution on the sequence of partial discharge events[END_REF]. The system is then susceptible to produce active species such as electrons, ions, free radicals and excited species that would lead to deleterious chemical transformations due to the their prolonged bombardment in the polymer surface [START_REF] Novak | Ionization and excitation behavior in a microcavity[END_REF]. Then degradation grows into damage paths inside the dielectric such as electrical trees [START_REF] Dissado | Understanding Electrical Trees in Solids: from experiment to theory[END_REF][START_REF] Morshuis | Degradation of solid dielectrics due to internal partial discharge: some thoughts on progress made and where to go now[END_REF][START_REF] Cavallotti | Thermodynamic analysis of the degradation of polyethylene subjected to internal partial discharges[END_REF]. In this representation, ageing is believed to be initiated by free electrons injection from the electrode, and discharge in micro-cavities present in the insulation. The size of the cavity and the free charges available for discharge inception are the key parameter controlling the damage mechanism.

Electric field homogenization in the cavity

Considering a cavity filled of air in a dielectric medium such as polyethylene as shown in figure 1.50: the cavity can be considered as a dielectric sphere of permittivity i = air = 1, embedded in a dielectric medium of permittivity r ≈ 2.3 and subject to the background field E p of the polymer matrix. The difference of permittivity will cause a polarization contribution in the sphere, which will cause a field enhancement in the cavity. The enhancement factor is a dimensionless factor that take account of the cavity geometry and of the permittivities of the whole system. A calculation of the enhancement factor was proposed by Cricthon et al., and mentioned later in other works [START_REF] Niemeyer | A generalized approach to partial discharge modeling[END_REF][START_REF] Crichton | Partial discharges in ellipsoidal and spheroidal voids[END_REF][START_REF] Priscillia | Partial Discharges for Diagnosis of Cavity Growth in HVDC XLPE Model Cables[END_REF][START_REF] Daniel | Discharging Microcavities and Electric Field Enhancement inside Cavities in HVDC XLPE Cables[END_REF]. The field inside the cavity is related to the background field by the enhancement factor h as follows:

E 0 = hE p (1.33)
This enhancement factor can be expressed as function of a geometric dimensionless factor K, mentioned as the inverse polarization factor, which depends on the cavity size and geometry [START_REF] Niemeyer | A generalized approach to partial discharge modeling[END_REF]:

h = K r /[ air + (K -1) r ] (1.
K = ⎧ ⎪ ⎨ ⎪ ⎩ ≈ 1 if b a 3 ifb = a r if b a ⇔ h = ⎧ ⎪ ⎨ ⎪ ⎩ ≈ r if b a ≈ 1.23 if b = a ≈ 1 ifb a (1.35)
It can be seen that for a spheroidal cavity, the enhancement factor is h = 1.23. For ellipsoidal cavity, the maximum field enhancement will be obtained for a cavity extended perpendicularly to the field, in which case the cavity field will be at least twice higher than the background field.

In his thesis, Wang Le proposed that under AC field, the electric field in the cavity can be considered being made of two contributions: the background field in the cavity, as presented previously, and a space charge generated field E q [START_REF] Wang | Physical Model of PD Behavior and relevant Damage Grotwh from Micro-Cavities in PE based material under AC voltage[END_REF].

E cav = E 0 + E q (1.36)
The background field E 0 in the cavity under AC is expressed as a function of a periodical component and the peak value of the field. The space charge field is the electric field generated by an amount of charge q sc deployed at polymer/cavity interface during the previous partial discharge. Thus, the cavity field E cav is a function of time. The dependence of E q on q sc is assumed to be linear and accounted by a geometrical factor f c , which is different from the field enhancement factor K discussed previously:

E 0 = h × E p = E peak × sin(ωt) (1.37) E q = f c × q sc (1.38)
The different fields generated by homogenization of charges in the cavity are represented in figure 1.52. As explained by Wang, the space charge generated electric field in equation (6.6) is expressed as a function of q sc the remaining space-charge generated at the polymer/cavity interface at a time interval Δt after a partial discharge. This space charge decays through neutralization processes due to conductivity at the cavity surface, or diffusion in the bulk. This decay process is approximated as follow:

q sc = q sc0 × e -Δt/τ (1.39)
where q sc0 represents the amount of charge at the cavity surface after previous discharge, and τ represents the time constant of the charge neutralization process.

Figure 1.53 represents the time evolution of the cavity field E cav , the background field E 0 and the space charge generated field E q , after the occurrence of a partial discharge at the maximum value of E 0 .

FIGURE 1.53: Time evolution of the total field in the cavity E cav (blue curve), regarding the background field E 0 (black curve) and the space charge generated field E q (red curve), after a partial discharge [START_REF] Wang | Physical Model of PD Behavior and relevant Damage Grotwh from Micro-Cavities in PE based material under AC voltage[END_REF].

Before the occurrence of the partial discharge, the space charge field is null and the cavity field E cav equals the background field E 0 . Immediately after the first discharge 1.4. Degradation phenomena: breakdown and ageing of polymeric insulation 63 occurs, space charge are generated and migrate to cavity walls before being neutralized. These charge induce a field opposite to the others, which reduces the cavity field. As the background field changes its polarity, the cavity field is enhanced by the addition E q and E 0 in the opposite polarity. This mechanism has been illustrated From the numerous works presented above, the partial discharge activity in a cavity embedded in a dielectric is a phenomenon which mainly depends on the cavity geometry, the polymer matrix nature, and background field. The cavity-polymer system is a preferential site for partial discharge inception in AC due to the presence of gas in which the field can reach bigger values compared to the polymer matrix.

The discharge inception is conditioned by the presence of initial free charges and the state of the gas medium, which are discussed in the next section.

PD inception

As introduced in the sections above, electrical discharges are likely to initiate inside a cavity medium. The main conditions for the initiation of a partial discharge are of two nature: the presence of a free initial charge inside the cavity, and an electric field sufficiently high to initiate avalanche discharge.

The presence of a free charge in the cavity has to be taken as a probabilistic approach. Electrons may come from injection at electrodes and travel trough the matrix via thermally-activated hopping conduction to reach the polymer-cavity interface [START_REF] Montanari | Model of ageing inception and growth from microvoids in polyethylene-based materials under AC voltage[END_REF]. There, electrons get trapped and can be injected in the void by a Schottky emission mechanism. The probability P for the presence of a free initiatory charge during a time interval dt is estimated by a cumulative probability function as defined below [START_REF] Wang | Physical Model of PD Behavior and relevant Damage Grotwh from Micro-Cavities in PE based material under AC voltage[END_REF]:

P = 1 -e -h.dt (1.40) h = h bg + λ (1.41)
with h the "hazard" factor for having a initiatory electron. The hazard factor is expressed by the sum of two contributions: h bg accounts for the electron availability from conduction in the polymer and trapping at the interface, while λ represents the detrapping rate of the charges produced by the previous partial discharge and is a function of the number of available charges and the field in the cavity [START_REF] Wang | Physical Model of PD Behavior and relevant Damage Grotwh from Micro-Cavities in PE based material under AC voltage[END_REF].

Once a free charge is extracted from the interface into the cavity, it can initiate a discharge if the electric field inside the cavity reaches a critical value. The breakdown criteria of a gas medium is governed by the critical avalanche criterion, also known as the Paschen's criterion, which for voids can be expressed as follows [START_REF] Niemeyer | A generalized approach to partial discharge modeling[END_REF]:

E crit = E p cr p 1 + B (2pa) n (1.42)
where E p cr , B, and n characterize the ionization process in the gas: in air, E p cr

= 25.2VPa -1 m -1 , B = 8.6m 1/2 Pa 1/2 , n = 1/2. p is the pressure in [Pa], a is the dimen- sion of the void [m].
Thus, if the two conditions are fulfilled (free charge available and critical field for ionization) a partial discharge starts in the cavity.

Impact ionization and damage growth

Under the conditions discussed previously, if a discharge starts in the cavity and initiates an electron avalanche it may trigger ionization and excitation processes of the gas in the cavity media. These processes are leading to the production of new electrons and other excited species such as ions or radicals, which can be deleterious for the insulation due to their bombardment on the cavity surface [START_REF] Cavallotti | Thermodynamic analysis of the degradation of polyethylene subjected to internal partial discharges[END_REF].

The gas composition in the cavity determines the degradation ionization/excitation processes by providing active species to the discharge. According to Morshuis, the degradation process is triggered by the presence of oxygen and nitrogen [START_REF] Morshuis | Partial Discharge Mechanisms -Mechanisms leading to breakdown, analysed by fast electrical and optical measurements[END_REF]. If oxygen is present in the cavity, oxidation byproducts by O 2 ionization can be created: O, O - 2 and O 3 . These active species are responsible for oxidation of the polymer, while discharges in N 2 cause degradation by ion impact on cavity's surface, which produces polymer radicals, and initiates pitting prior to treeing and breakdown. Other chemical reactions in the gas phase are responsible for the production of hydrogen, carbon monoxide, methane and carbon dioxide, which react with polymer radicals. Solid byproducts can thus accumulate on the cavity surface, such as formic, acetic and carboxylic acids. These acids rise the surface conductivity of the cavity/polymer interface as reported by Gamez-Garcia et al. [START_REF] Garcia | Circular degradation patterns on XLPE surfaced electrodes[END_REF][START_REF] Gamez-Garcia | Synthesis Reactions Involving XLPE Subjected to Partial Discharges[END_REF][START_REF] Gamez-Garcia | Modification of XLPE exposed to partial discharges at elevated temperature[END_REF]. These acid by-products have been founded in a solid crystal form on the cavity surface, which increase the surface roughness and lead to local field enhancement.

Luminescence and associated phenomena

When a solid such as PE is submitted to a an excitation source, the material absorb part of the energy of the source, and some atoms or molecules are excited to a different energy level. It can de-excite by two different pathways: radiative (typically photons) and non-radiative (typically phonons) [START_REF] Kao | Dielectric phenomena in solids[END_REF][START_REF] Qiao | Electrical ageing of insulating polymers: approach through electroluminescence and cathodoluminescence analyses[END_REF]. The physical process of luminescence is basically the radiative emission of a photon due to de-excitation of these atoms or molecules to their ground state. Depending on the material and the excitation source, different luminescence mechanisms in insulating polymers are known to occur:

• Photoluminescence (PL) is light emission due to absorption of incident photons.

• Chemiluminescence (CHL) is due to chemical reactions involving excitation of molecules (ex: oxidation).

• Electroluminescence (EL) is due to excitation by an applied electric field (ex: recombination of injected electrons and holes from electrodes).

• Cathodoluminescence (CL) due to high-energy electron impact or irradiation.

• Thermoluminescence (TL) due to the conversion of thermal heating to optical energy.

Thus, these radiative phenomena traduce of an energy transfer in the material on the application of the excitation source. It can provide information on the nature of the transfer, as for instance in the case of electrical degradation. Several research works have been produced on the light emission produced by insulating polymers under electrical solicitations in order to investigate the degradation mechanisms.

Early evidence and models of electroluminescence were obtained during electricaltree initiation using a specific needle tip electrode [START_REF] Laurent | Light detection during the initiation of electrical treeing at room temperature[END_REF][START_REF] Bamji | Evidence of near-ultraviolet emission during electrical-tree initiation in polyethylene[END_REF][START_REF] Bamji | Threshold voltage of luminescence and electrical tree inception in low-density polyethylene[END_REF][START_REF]Dynamic bipolar charge recombination model for electroluminescence in polymer based insulation during electrical tree initiation[END_REF]. It was reported that light is emitted due to bipolar charge injection during full AC cycle [START_REF] Bamji | Evidence of near-ultraviolet emission during electrical-tree initiation in polyethylene[END_REF][START_REF] Lebey | Charge injection and electroluminescence as a prelude to dielectric breakdown[END_REF]. Moreover, existence of threshold voltage for light emission was found, and that degradation of the insulation was not observed below this threshold. It is believed then that EL is a key marker of degradation, because it traduces energy dissipation due to bond-breaking [START_REF] Teyssedre | Evidence of hot electron-induced chemical degradation in electroluminescence spectra of insulating polymers[END_REF]. In other works, the role of space-charge, in both AC and DC, in the emission of light was investigated [START_REF] Teyssedre | Ac and dc electroluminescence in insulating polymers and implication for electrical ageing[END_REF][START_REF] Hozumi | Behaviour of space charge correlated with electroluminescence in cross-linked polyethylene[END_REF][START_REF] Roy | Charge transport and dissipative processes in insulating polymers: experiments and model[END_REF]. Characteristics of PE and XLPE, involving deep-trapping centers have been investigated by molecular modeling and other luminescence techniques, such as PL and Recombination induced Luminescence (RiL) [START_REF] Teyssedre | Spectral analysis of optical emission due to isothermal charge recombination in polyolefins[END_REF][START_REF] Teyssedre | Deep trapping centers in crosslinked polyethylene investigated by molecular modeling and luminescence techniques[END_REF][START_REF] Teyssedre | Characterisation of crosslinked polyethylene materials by luminescence techniques[END_REF]. Figure 1.57 shows a representation of the charge recombination process occurring between energy band levels of the insulation. Light emission spectra and photon energy were also dependent on the material used, indicating different energy transfer mechanisms [START_REF] Zhang | Light emission characteristics during the initiation of the electric tree in insulating polymers[END_REF]. Luminescence techniques have shown the evidence of light emission during electrical degradation of polymeric insulation, indicating that degradation involves energy transfers of a few eV, and can be used as a monitoring technique for pre-breakdown mechanisms [START_REF] Qiao | Electrical ageing of insulating polymers: approach through electroluminescence and cathodoluminescence analyses[END_REF][START_REF] Laurent | Energetics of charge transport in insulating polymers[END_REF].

Summary and Hypothesis of the Study

The state of the art on electrical degradation of polymeric insulation highlights key aspects that need to be investigated experimentally.

We have seen that among many possible candidate materials for electrical insulation, polymers such as PE and PP are materials of choice and are widely used in power cables. The complex morphology of these materials has a tremendous influence of the final properties of the insulation: a highly heterogeneous structure at the microscale, composed of ordered crystalline part arranged in spherulitic structures and disorganized amorphous material in between. This semi-crystalline morphology undeniably drive the behavior of the material under thermo-electrical stress, and thus need to be further investigated.

Moreover, polymers are not pure and perfect insulators. Due to the presence of chemical and physical impurities, their electronic structure is highly modified, introducing local states in the band gap of the material that contribute to charge conduction at low field and to charge injection at higher fields. The effect of such impurities on ageing mechanisms has been studied and reported in the literature, but it is not clear how they influence the final steps prior to breakdown yet.

A large panel of breakdown and ageing models have been developed in order to explained the behavior of polymeric insulation under thermo-electric stress. Lifetime in ageing models dependent mostly on the applied stress, and ageing is believed to be initiated from defects as energy barriers required for degradation are lowered by the presence chemical and physical defects. Experimental evidence of ageing markers are numerous, such as oxidation are structural modification. However, the evolution of these ageing markers from the beginning to the end of the material's lifetime is less described than the nature of these markers.

Charge carrier behavior in the insulation under electric field have been widely studied. Electrons and holes are believed to obey band and hopping conduction models at high fields, and the current density varies non linearly with the applied electric field. The impact of ageing on the conduction states is rather poorly detailed, and should be studied more in details.

An ageing scenario for polymeric insulation under constant thermo-electric stress can be proposed. Under the application of sub-critical field, i.e. below breakdown field, only few charge carriers are responsible for conduction. Eventually, some charge carriers can acquire sufficient energy to break bonds, as for instance in low density regions formed by physical relaxation of polymer chains or in microcavities. These phenomena can either occur individually and with a low probability, or concentrate locally at weak points where energy barriers are lower. Bond breaking readily leads to physical changes such as cross-linking and chemical changes such as oxidation. Thus, the concentration of damage increases with chemical and physical defects. These event result in the formation of local valence and conduction states in the band gap of the material. At a critical concentration of local states, charge injection from electrode is facilitated, increasing the concentration of charge carriers and increasing the probability and the number of charge-polymer interactions. The final steps of ageing are still poorly described in the literature. But one hypothesis would be that due to the increase concentration of local conduction and valence states, deeper in the band gap, promotion of charges from the valence band as described by Zener could be possible. In the case of partial discharge in presence of cavities, a similar ageing scenario can be described, but energies and charge carrier densities involved are much larger due to the avalanche process occurring in the cavity environment.

The aim of this experimental work is to provide evidence of the presence and nature of ageing markers in polymeric materials under thermo-electrical fatigue, and monitor the evolution of these markers at different moment of the sample lifetime, especially in the final steps prior to breakdown. This work also aims to obtain a deeper understanding of the various aspects of electrical ageing, namely the correlation between chemical and physical ageing markers, and be able to tell which ageing mechanisms are the most critical for the insulation.

Chapter 2

Materials and Methods

The purpose of this chapter is to present the experimental approach, materials and set-ups that have been used in this work. As discussed earlier, insulating polymer materials under application of various electrical stresses, either localized or homogeneously distributed in the material, are believed to age following different possible mechanisms. The main identified mechanisms are linked to a progressive degradation of the structural and chemical properties of the material, and possibly its electronic structure. The specific role of crystalline structure also needs to be investigated. The aim of this study is thus to provide experimental evidence of the impact of the material chemical structure and of the degree of crystallinity on ageing mechanisms, as well as the effect of a localized and/or uniform stress field.

To do so, different formulations of PE based materials have been developed, and aged using two electrode configurations, in order to perform an accelerated ageing under a localized electrical stress, or a uniform electrical stress. The working principle and conditions of these ageing tests are detailed. Two PP grades with different degree of crystallinity have also been studied in order to study the role of crystallinity upon degradation mechanisms. After ageing, the samples are being characterized at different ageing steps of their lifetime, in order to monitor the evolution of ageing markers. The characterization techniques and their working principles are also detailed in this chapter.

Experimental approach and set-ups

In this experimental study, different PE and PP based materials were aged under two different electrode configurations: a sphere-on-plane ("sphere-plane" in the following) and a parallel plate ("plane-plane") configurations. These electrode geometries were chosen in order to study ageing mechanisms in a localized heterogeneous electric field, and in a uniform electric field respectively. Figure 2.1 shows a scheme of the designed electrode geometry for ageing tests. In addition to the applied field, the samples were aged at controlled temperatures: the electrode set-ups with the materials were placed in an oven during tests, with a thermal probe for temperature control. In the first part of the study, polymer films were aged under the sphere-plane configuration, using stainless-steel rod electrodes with a hemispherical tip of 4 mm in diameter (Case (1) in figure 2.1). The electrodes are mirror polished. The spherical tip is placed on top of the polymer film, and the ground electrode is made of a steel plate covered with aluminium sheets.

In this electrode configuration, the electric field is localized in a small volume of the sample. In order to have a better insight at this effect, electrical field isolines (|E| (V/m)) where calculated using finite elements FEMM software (Finite Element Method Magnetics) in this specific electrode-polymer configuration. The calculation is done using different values of permittivity for air ( air = 1), polymer ( air = 2. In a second time, polymer tests samples were prepared and aged in a plane-plane configuration. The electrode has been designed using a Rogowski-like profile (Case (2) in figure 2.1): this kind of profile is made of a plane disk with curved edges, which aims to generate a homogeneous electrical field under the plane region and prevent field intensification at the edges [START_REF] Rogowski | Die elektrische Festigkeit am Rande des Plattenkondensators[END_REF][START_REF] Giao | Electrode Design for testing in uniform field gaps[END_REF][START_REF] Espino-Cortes | Numerical Study of the Profile of parallel plate electrodes[END_REF]. The large diameter of the electrode is 34 mm, and the plane region is about 20 mm in diameter. As the electrode is encapsulated in the polymer, the electric field is expected to be homogeneous in the plane region, which allows to age a larger volume of the material. Electrical field isolines were calculated using the same method as for sphere-plane electrode, and the results are shown in figure 2.4. Thus, the chosen electrode configuration are expected to induce two different types of electrical stress in the materials, which is confirmed by electrical field calculations: a localized stress originates from the hemispherical electrode, and a homogeneous stress is expected in the plane-plane configuration.

Test samples have been aged using an MV-AC generator which allows using up to 7 kV AC (zero-to-peak), with 10 independent measurement channels. A specifically designed test cell was used for lifetime measurements and ageing in which the samples are placed horizontally on the ground electrode, and the upper electrodes are placed individually on the top. The test cell is then put in an oven which has a back entry for connection cables. Figure 2.5 shows a schematic representation of the ageing set-up. The same set-up was used for the case of encapsulated electrodes. Using this set-up, the samples support the electrodes weight, which maintains the contact between the samples and both high voltage and ground electrodes.

Materials and processed samples

For the experimental study of ageing mechanisms under AC electric field, different polymers, grades and formulations have been used. The materials formulation and processing are detailed in this section.

First, PE-based polymers with different amount of cross-linking agent were used for polymer film processing, in order to study the impact of cross-linking of the material's performances. Two PP grades, with different crystalline morphology were used as well for polymer film processing, in order to study the impact of crystallinity upon ageing mechanisms under localized field.

In a second time, one grade of cross-linked PE material was reused for encapsulation of uniform-field electrodes, in order to compared the effect of ageing on PE under sphere-plane and plane-plane electrode geometries.

Polymer films for sphere-plane configuration

Polymer films were prepared from LDPE and PP commercial pellets. Antioxidant was used in PE and PP in order to prevent thermal oxidation of the samples during processing and ageing. A cross-linking agent has been used in the PE-based materials in increasing amount in order to obtain materials with different cross-linking densities.

Polyethylene and Cross-linked Polyethylene films

LDPE pellets were used as a base for the formulation of five different materials.

A neat LDPE material has been prepared using the pellets without antioxidant. The pellets are then stabilized using an antioxidant and tert-Butyl cumyl peroxide (tBCP) is used as a cross-linking agent. Increasing amounts of tBCP are used in order to obtain three different formulations of XLPE. The details of the different materials compositions are shown in table 2.1. The polymer films are obtained by extrusion and compression molding. First, antioxidant is added as solid flakes to the LDPE pellets, which are then extruded at 130 • C and granulated to obtain stabilized pellets. For the preparation of the different cross-linked materials, liquid tBCP is added to the stabilized LDPE pellets by soaking at 60 • C. A second extrusion step is performed on the pellets at 130 • C, and the extruded materials are shaped as strips of 300 to 500 μm. Finally, the polymer films are obtained from the strips by compression molding at 190 • C, which also triggers cross-linking by tBCP decomposition. The thickness of the processed films is controlled by using spacers of 100 μm thick.

The processed films are then placed in an oven at 60 • C for 2 hours. This step is performed for degassing of the cross-linking by-products which may have formed during the process. The LDPE films without tBCP are obtained under the same process conditions. Films are cut to obtained disks of 30 mm in diameter. The mean thickness of the processed films is around 95 μm.

Corss-linking of the materials was checked by solid-state Nuclear Magnetic Resonance (NMR), which is described later in this chapter.

Polypropylene films

Pellets of two different PP commercial grades have been used for the formulation of PP films. The two PP grades are ethylene-propylene copolymers, which have been chosen for their different respective crystalline ratios, in order to study the impact of crystalline morphology and ratio on the ageing mechanisms. The relative crystallinity of the prepared films have been calculated by Differential Scanning Calorimetry (DSC), which is also described later in this chapter. Table 2.2 shows the compositions and relative crystallinities of the two different PP grades. The amount of antioxidant is the same as for the PE materials. No cross-linking was performed on the PP samples. The PP films were prepared using the same experimental protocol as for LDPE and XLPE films, except for extrusion temperature which was set to 190 • C.

The processed films were cut to obtained disks of 30 mm in diameter. PP films have been produced with thickness between 50 and 150 μm, thus only those with thickness 90 and 110 μm have been selected and used in the later steps. The obtained test samples are put in an oven at 60 • C for by-products degassing. Thickness of the XLPE layer is checked after the molding process, in order to select only samples of around 100 μm thick. For characterization after electrical tests, XLPE samples can be removed by gently peeling off from the electrode.

Cross-linked Polyethylene for plane-plane configuration

Ageing Conditions

Several ageing conditions were applied to the samples in order to characterize the effect of ageing on the samples properties at different stages of their lifetime. These conditions correspond to accelerated ageing as compared to real conditions seen by MV-AC power cable insulation.

First, the samples were aged until breakdown in order to measure the lifetime as a function of the stress.Tests samples were subjected to AC high voltage between 1500 and 4500 V peak , which is the zero-to-peak value of the AC cycle. Knowing the mean thickness of the samples this corresponds to an average electrical field value between 15 and 45 kV/mm. which is 5 to 15 times higher than MV-AC fields seen in power cables, typically 3 kV/mm [START_REF] William A Thue | Electrical Power Cable Engineering[END_REF][START_REF] Osama | Investigating the effect of cavity size within mediumvoltage power cable on partial discharge behaviour[END_REF]. In addition to electrical stress, tests samples were aged in an oven at 90 • C in order to simulate thermal stress that can be seen intermittently by the insulation during its lifetime.

Under these stress conditions, the tested materials have been aged until breakdown, for different stress fields. From the data collected, reference curves of the materials lifetimes were drawn. In a second step, using the lifetime curves obtained in the previous test, the materials were aged under intermediate stress fields at different ageing steps prior breakdown, corresponding to given fractions of the sample lifetime. For ageing study of the PE based samples, only one grade was used because of the large quantity of tests and characterizations. The sample with the highest expected cross-linking density, XLPE C , was chosen.

Tables 2.3 and 2.4 summarize the electrical stress conditions imposed on the different materials for sphere-plane and plane-plane geometry respectively, for lifetime measurement and ageing steps. E avg (kV/mm)

V 0-peak (kV) Samples LDPE LDPE stab XLPE A XLPE B XLPE C PP A PP B 15 1.4 ⊗ ⊗ ⊗ ⊗ ⊗ 1.7 ⊗ 20 2.1 ⊗ 30 2.8 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ 45 4.2 ⊗ ⊗ ⊗ ⊗ ⊗ 4.5 ⊗ ⊗ 25 2.25 •• 2.4 •• •• 35 3.2 ••

••

The reason why different voltage where applied for a given electric field is that in some cases, samples where thicker, thus a higher voltage was needed.

Characterization methods

Different characterization techniques were used, before the lifetime and ageing tests and after ageing. The aim of these characterizations was mainly to seek for ageing markers and follow their evolution.

Nuclear Magnetic Resonance

Time-domain double-quantum (DQ) proton ( 1 H) NMR spectroscopy is based on the measurement of the residual tensorial interactions between the spins of protons under a static magnetic field. These interactions originate from incomplete motional averaging of network chain segments fluctuating rapidly between topological constraints, such as cross-links or chain entanglements [START_REF] Sotta | Effect of Residual Dipolar Interactions on the NMR Relaxation in Cross-Linked Elastomers[END_REF][START_REF] Saalwächter | 1H multiple-quantum nuclear magnetic resonance investigations of molecular order distributions in poly(dimethylsiloxane) networks: Evidence for a linear mixing law in bimodal systems[END_REF][START_REF] Saalwächter | Proton multiple-quantum NMR for the study of chain dynamics and structural constraints in polymeric soft materials[END_REF][START_REF] Chasse | Precise dipolar coupling constant distribution analysis in proton multiple-quantum NMR of elastomers[END_REF][START_REF] Vieyres | Sulfur-cured Natural Rubber Elastomer Networks: Correlating Crosslink Density, Chain Orientation and Mechanical Response by Combined Techniques[END_REF]. Under specific experimental conditions, which involve excitation of coupled proton spins by sequences of electro-magnetic pulses, quantitative measurements of 1 H dipole-dipole coupling constants (D res ) can be realised [START_REF] Saalwächter | 1H multiple-quantum nuclear magnetic resonance investigations of molecular order distributions in poly(dimethylsiloxane) networks: Evidence for a linear mixing law in bimodal systems[END_REF]. In the case of a cross-linked network the experimentally measured average residual dipolar coupling constant D res is proportional to ν, the number of constraints (cross-links plus entanglements) per unit volume:

D res = kD static P 2 (cosθ) t R ≈ 1 M c ≈ ν (2.1)
where D static is the static proton-proton dipolar coupling (in the solid state), k is a scaling factor of order roughly 0.1 accounting for the spin arrangement and motions within a statistical segment, θ is the time-dependent angle between the local chain direction (segmental orientation) and a reference direction. It involves first a time average inside a chain or sub-chain (subscript t) and then the sum of contributions from all chains (subscript R). M c is the average molecular weight of network chains between constraints. In order to obtain data which can be properly analyzed in terms of fractions of cross-linked chains, the temperature must be above melting temperature and high enough above glass temperature to be in the molten (elastomer) state, with fast molecular motions. Experiments were done at 125 • C.

In this work, NMR spectroscopy was used to check the different levels of crosslinking in XLPE materials before ageing tests. The pulse sequence and phase cycle used were the same as in [START_REF] Saalwächter | Proton multiple-quantum NMR for the study of chain dynamics and structural constraints in polymeric soft materials[END_REF]. The measurements gives two sets of experimental data: the DQ signal I DQ (τ), which is a function of D res , and the reference signal I re f (τ). Both curves are measured as a function of the evolution time τ DQ (the variable duration of the pulse sequence). The contribution of "defects" I de f is first subtracted, then a normalized signal is obtained as :

I nDQ = I DQ I re f + I DQ -I de f (2.2)
I nDQ depends only on D res , thus on the cross-link density ν.

As generally D static and k are not known quantitatively, ν is not known quantitatively (in terms of number of cross-links per unit volume/mass), but series of samples can be compared quantitatively and with high sensitivity. The total number of chains contributing to the reference signal is made of a fraction φ elast of elastically active chains which belong to the cross-linked network and a fraction φ de f of "defects", mainly dangling chains which are not attached at both ends and behave isotropically.

f (τ) = 1 2 [ Ψ(1 -exp(-D (1) res τ) P 1 )) + (1 -Ψ)(1 -exp(-D (2) res )τ) P 2 ))] (2.3)
where Ψ corresponds to the fraction of the signal originating from the cross-linking network formed in the material, P 1 and P 2 are adjustable parameters (see [START_REF] Saalwächter | 1H multiple-quantum nuclear magnetic resonance investigations of molecular order distributions in poly(dimethylsiloxane) networks: Evidence for a linear mixing law in bimodal systems[END_REF]). The obtained fitting parameters are given in table 2.5. P 1 varies from 1.24 to 1.36 and P 2 from 2.46 to 2.18. Thus, these results indicate that increasing the amount of tBCP introduced in the process leads to material with a higher cross-linking density, even if it cannot be quantitatively calculated.

Differential Scanning Calorimetry

Differential Scanning Calorimetry (DSC) consists in measuring the heat flow absorbed (endothermic) or released (exothermic) by a sample when submitted to different ramps of temperature. In the case of polymers it is generally used in order to determine glass transition and other phase transformation phenomena occurring in the material within a temperature range, such as melting of crystalline matter or crystallization [START_REF] Grenet | Analyse calorimétrique différentielle à balayage (DSC)[END_REF].

A reference sample, a material which does not change in the desired temperature range, is placed in the heating chamber as well. Calibration, is done using Sapphire and Indium at a rate of 10 • C/min.

It is worth mentioning that the first heating cycle gives information on the thermal memory of the sample, for example the effect of stress. A second heating cycle after cooling would only give information on the intrinsic properties of the material.

In this work, the DSC measurements were performed on LDPE, XLPE and PP samples in order to measure the evolution of the degree of crystallinity of the materials during thermo-electrical ageing. The sample is heated between -20 and 200 • C with a temperature ramp of 10 • C/min. The degree of crystallinity (χ sample ) of the polymer sample is calculated during the first heating cycle from the ratio of the sample melting enthalpy over the enthalpy of a hypothetical 100% perfect crystalline polymer [START_REF] Louis | Polymer Materials : Macroscopic Properties and Molecular Interpretations[END_REF]:

χ sample = ΔH sample melt ΔH crystal melt × 100 (2.4) ΔH crystal melt
is 290 J/g in PE, and 196 J/g in PP [START_REF] Guffond | Characterization and modeling of microstructure evolution of cable insulation system under HVDC fields[END_REF][START_REF] Thanh | Approche multi-échelles dans les matériaux polymères: de la caractérisation nanométrique aux effets d'échelles[END_REF].

DSC measurements have been done on PP A and PP B before ageing, in order to measure the degree of crystallinity of the two grades. This result shows that two crystalline populations co-exist. As PP A and PP B are ethylene-propylene copolymers, the broad peak in PP A and the two peaks in PP B may be attributed to ethylene and propylene crystalline phases. Small PE lamellae may contribute to the melting peak around 50 • C, whereas the other melting peak is due to the PP crystalline phase.

In order to account only for the degree of crystallinity due to the PP phase, a choice was made to measure the melting enthalpy above 80 • C. Besides, ageing tests are done above this temperature, thus the crystalline phase attributed to PE is expected to melt during ageing.

The degrees of crystallinity calculated in this way are 36.5% for PP A and 11.9% for PP B .

In the next chapters, the properties of unaged and aged XLPE and PP materials measured by DSC will be presented and discussed.

Fourier Transform Infrared Spectroscopy

Fourier Transform Infrared Spectroscopy (FTIR) allows identifying molecular groups by measuring the specific energy resulting from the interaction of the molecules of the sample with an incident infrared radiation. In this study, FTIR has been used mainly to monitor the oxidation state of samples during its lifetime, as for instance the formation of carbonyl or hydroxyl groups. As samples are relatively thin, IR analysis was performed in transmission mode (TR). In some cases, IR spectroscopy was carried out in attenuated total reflectance mode (ATR). This mode uses light reflection from sample's surface on a tip with a known refractive index, usually diamond. This allows to obtain IR spectra of the surface down to a few microns deep.

According to the Beer-Lambert law, the absorbance A of a molecular group, i.e the intensity of the corresponding IR peak, is proportional to its concentration, which thus allows to tell if this concentration changes during ageing.

Measurements were performed using a Nexus 470 FT-IR ESP Spectrometer by Thermo Nicolet, with KBr as capture window and reference, in the range 4000-400 cm -1 , steps of 1.928 cm -1 and 32 scans per sample. Background measurements were done in air.

The resulting spectra of PE and PP materials will be discussed in the next chapters.

Microscopy

In order to study the effect of thermo-electrical ageing the surface aspect of the tested samples, Optical Microscopy (OM) and Scanning Electron Microscopy (SEM) were used.

Imaging by OM (Macroscope Wild M420 and Microscope Leica DMRX) was used prior to SEM in order to map the different regions of the sample that have been damaged. It was performed on unaged samples, using bright and dark field in transmission mode.

Imaging by SEM (Ultra 55 by Zeiss) was performed with an accelerated voltage of 15 kV in high-current mode and magnifications up to 6000x. The analysis was mainly performed on the surface of the samples by secondary electrons. In a few cases, samples were prepared beforehand via cryo-fracture in liquid nitrogen, and the edge of the sample was characterized. Samples were also metallized on their surface by Platinum sputtering.

Small Angle X-Ray Scattering

Small angle X-ray scattering (SAXS) has been used in order to study the material crystalline morphology at smaller scale. Measurements were done in the Laboratoire de Physique des Solides (LPS), Université Paris Sud, in collaboration with P.A. Albouy.

SAXS allows to investigate repeating structures at scales between few to tens of nanometers. In semi-crystalline polymers such as PE and PP, it is used in order to investigate the long period of lamellar stacks. When a sample is subjected to an incident X-ray beam, one can define a scattering vector q [nm -1 ] as:

q = 4πsin(θ) λ (2.5)
with 2θ the angle between the incident X-ray beam and the scattered beam, and λ the wavelength of the incident X-ray source. For a Copper source: λ = 0.154 nm. Figure 2.11 shows a schematic representation of X-ray scattering at small angles. 2D scattering patterns represent the scattered intensity as a function of scattering angle in all directions. In the case of an isotropic sample, in which lamellar stacks have no preferential orientation, the scattered intensity is the same in all directions, which is the case in our materials (see figure 2.12). The scattering pattern of the PP film shows an isotropic pattern, indicating that crystalline lamellae have grown without preferential orientation.

For an isotropic sample, by integrating the intensity as a function of q over all azimuthal directions, the intensity I q as a function of q is obtained, and shows a peak. The long period L p is obtained by plotting q 2 I(q) as function of q, as shows in figure 2.13. FIGURE 2.13: Plot of q 2 I(q) as a function of q for PP A film before ageing.

L p is obtained by the Bragg law:

L p = 2π q max (2.6)
where q max corresponds to the wave vector value for which the signal q 2 I(q) is maximum.

SAXS measurements were done on the unaged and aged PP samples for investigating the change in long period due to ageing.

Electrical Breakdown Strength

The Electrical Breakdown Strength (E B ) of a material is defined as the maximum electrical field it can support before failure. It is obtained by applying an AC voltage ramp until breakdown occurs, which is measured when a short circuit of a 5 to 10 mA is recorded. The measurement is performed at room temperature and in silicone oil in order to avoid flash-overs and partial or surface discharges.

For this specific case, as compared to geometries used for lifetime and ageing tests, samples are placed between two hemispherical stainless-steel electrodes of 20 mm in diameter, mirror polished, as shown in figure 2.14. The contact is ensured using a screw, carefully tightening the sample between the electrodes. A ramp of 500 V.sec -1 and a maximum output current of 10 mA are used for the breakdown test. As seen in the previous chapter, breakdown can be analyzed as a statistical event.

For each material, several samples are broken, giving rise to a distribution of breakdown strengths, which is then analyzed by means of probability of failure. In studies of system failure and reliability, the Weibull distribution is generally used for analyzing failure events [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF]. The cumulative probability of failure P(X) of a two-parameter Weibull distribution is a function of electrical stress field or time in case of an ageing test at a constant stress. In case of breakdown strength measurement P(E) is given by the following equation:

P(E) = 1 -exp - E α β (2.7)
where α is known as the scale parameter and β the shape parameter. The scale parameter α [kV/mm] represents the value of E at which the probability of failure is P(E) = 1exp(-1) = 63.2%. It is generally used as the characteristic breakdown strength of the material. The shape parameter β represents the data dispersion of the Weibull distribution: the higher the β, the sharper the distribution.

During breakdown strength measurements, for practical reasons, only 8 to 10 samples were broken for each material, which is lower than the 20 samples usually recommended [START_REF] Montanari | International Standard -Guide for the statistical analysis of electrical insulation breakdown data[END_REF]. When plotting the measured data of n samples, the data first need to be sorted in increasing order. A cumulative failure probability P(i, n) approaching P(E) should be assigned to discrete value x i , where i is the rank of the ordered data point [START_REF] Sari | Kungliga tekniska högskolan, and Institutionen för fiberoch polymerteknologi[END_REF]. Different approximations for the cumulative probabilities can be used, but the most common is obtained by the median rank approximation, in which P(E) of the i th sample is given by:

P(i, n) = i -0.3 n + 0.4 (2.8)
The values of α and β are estimated from the plot: Breakdown strengths of XLPE and PP materials have been measured before and after different ageing steps in order to monitor the impact of ageing upon resistance to failure.

ln(-ln(1 -P(E))) = βln(E) -βln(α) (2.9) using P(E) = P(i, n).

Leakage Current measurements

As explained in the previous chapter, charge transport under high fields follows different regimes depending on the applied field. At low field, the current density is proportional to the applied field (Ohmic conduction) while at higher fields, injection of charges from electrodes induces a different regime for current density. Measurement of the current density going through a sample as a function of the applied field, for increasing value of electric field give rise to so-called "current-voltage characteristics". Analysis of these curves for samples with increasing ageing time gives an indication on the effect of ageing on conduction states of the materials.

For this purpose, a DC current density measurement bench is used. This bench has been developed in a previous thesis by Guffond [START_REF] Guffond | Characterization and modeling of microstructure evolution of cable insulation system under HVDC fields[END_REF]. It is made of a DC voltage source (± 12. For each voltage step, the current is measured during 300 seconds before voltage is increased. After applying the voltage, the measured current is maximum and starts decreasing with time following a double exponential law:

i = i DC + i C × exp(- t τ C ) + i P × exp(- t τ P ) (2.10) where i C [A] corresponds to charging current, i P [A] to polarization current, τ C [s]
is the time constant of the charging current, and τ P [s] is the time constant for the polarization current [START_REF] Guffond | Characterization and modeling of microstructure evolution of cable insulation system under HVDC fields[END_REF]. The order of magnitude of τ C and τ P are roughly 10 sec and 200 sec respectively. For the current-voltage characteristic curves, a mean value is obtained from the last three values of current around 300 sec. These values are then drawn as a function of the applied field for each material.

Luminescence measurements

Excitation of polymeric materials from various sources leads to light emission, result of photons released when molecules go back to their ground states. This light emission, more precisely the density of emitted photons and their energy, gives crucial information on the nature of emitting centers, which belongs to the polymer chemical and thus electronic structure.

In this study, two different luminescence measurements have been done: Photoluminescence (PL) and Plasma-induced Luminescence (PIL). The aim of these measurements was to investigate the evolution of chemical markers with ageing and their role on the modification of the structure of the material. The luminescence measurements were done in collaboration with G. In this set-up, [1] is the PL excitation device, constituted of a Xenon UV Lamp associated to a monochromator. [2] is the photomultiplier tube (PMT), which allows to count the number of photon per unit time emitted from the sample. [3] is the chamber, comprising a sample holder with a temperature control system, a pumping system, and a rotating mirror with a gap for the PMT. Finally in [4] is the CCD Camera with its own cooling system.

For the PIL measurements the same set-up is used, but instead of the xenon lamp a specific electrode device for plasma generation is used, as shown in figure 2.19. The sample is set on lower ground electrode of diameter 20 mm and is covered with a Quartz-ITO upper electrode connected to a HV-AC generator. The gap between the two dielectrics is set to 5 mm, and filled with helium at atmospheric pressure.

Applying the HV-AC field, a plasma discharge is induced in the gap, and charges are deposited on the surface of the sample and are used as a excitation source. Following discharge, emitted photons pass through the upper transparent electrode to the detector.

The experimental protocols for PL and PIL are the following ones:

For PL, the sample is placed on the holder at room temperature and in air. The detector mode, as well as the mirror inclination is set on for the CCD camera. The chamber is closed so that no other external source can reach the sample. The monochromator filters UV light in order to obtain monochromatic excitation from 220 to 300 nm, and is set manually before light emission. The sample is then exposed for 10 seconds to the xenon lamp, for each wavelength of excitation between 220 and 300 nm. A split of 0.5 mm wide is used to focus the beam on the sample. The CCD camera then capture the PL emission spectra of the sample for each wavelength.

In case of PIL, in order to accurately measure the intensity of light emission, the measurement has to be done in helium atmosphere and at liquid nitrogen temperature. First, vacuum is reached by pumping the chamber down to approximately 10 -3 mbar. Then helium is used to fill the chamber at atmospheric pressure. Liquid nitrogen is introduced under the sample holder in order to cool the sample at roughly -180 • C. For plasma discharge, an AC voltage of about 6.5 kHz and 1.5 kV is switched on for 5 sec. After that period, the voltage is switched off and the mirror is set for the PMT (Photon counting mode). It counts the number of photons with a time step of 2 sec, during a least 300 sec. A second plasma discharge is used for the spectral analysis using the CCD camera. Spectra are collected at different times from 1 to 300 sec after the plasma discharge.

Chapter 3

Lifetime measurements of PE and PP based materials

As described in the literature, a key tool for understanding the different steps of electrical ageing, which is a critical aspect from an industrial point of view, is being able to predict polymeric insulation lifetime. Various life models have been proposed based on physical and chemical mechanisms, and on experimental lifetime measurements, also known as endurance tests [START_REF] Howard | The effect of electric stress on the life of cables incorporating a polythene dielectric[END_REF][START_REF] Yoshimura | Effect of microvoids on V-t characteristics and tree growth in crosslinked polyethylene[END_REF][START_REF] Simoni | A General Approach to the Endurance of Electrical Insulation Under Temperature and Voltage[END_REF][START_REF] V M Montsinger | Breakdown Curve for Solid Insulation[END_REF]. These models predict that the lifetime of polymeric insulators decreases as the applied stress increases, but also that different degradation mechanisms are effective under different levels of electrical stress, leading to different regimes in the lifetime-vs-stress curves. Equivalent tests are usually performed for studying mechanical fatigue of polymers, and the corresponding fatigue curves are known as Wöhler curves [START_REF] Mourglia-Seignobos | Physical Mechanisms of Fatigue in Neat Polyamide 6,6[END_REF].

In this chapter, the prepared PE and PP based materials are aged under different stress conditions in order to estimate their lifetimes, and study the influence of different stress parameters on their endurance.

First, the lifetimes of LDPE and XLPE films are measured using the sphere-plane geometry under different electrical stress levels, and data are discussed in terms of lifetime-vs-stress curves. The different life regimes are discussed in terms of possible degradation mechanisms. The effects of PE formulation on lifetime, i.e antioxidant and cross-linking, are also studied. In addition, the effects of changing temperature and voltage frequency have been studied on some PE formulations.

Secondly, the lifetime of PP films has been measured using the same sphere-plane geometry. In this case, the effect of the degree of crystallinity on lifetime curves is discussed.

Finally, the lifetime of XLPE in plane-plane geometry has been studied, and data were compared to lifetime of XLPE under sphere-plane configuration. The effects of localized or uniform applied electrical stress on lifetime are discussed and degradation mechanisms are proposed.

The measurement of lifetime curves is a first step in this experimental work and will serve as a reference for studying degradation at different ageing steps in the next chapters.

Lifetime of PE and PP in sphere-plane geometry

Effects of materials properties on lifetime

As described in chapter 2, the lifetimes of PE based samples were measured for AC voltage values between 1.4 kV and 4.5 kV (zero-to-peak) corresponding to an applied electric stress between 15 kV/mm and 45 kV/mm (films of 90-100 μm thick).

The tested temperature is fixed to 90 • C.

Times-to-breakdown data were fitted using a two-parameter Weibull distribution P(t) (t [hours]). For the cumulative probability of failure, 10 to 15 samples were used. Values of α and β are recorded for a given material and applied stress.

Figure 3.1 shows the plot of P(t) as a function of t for XLPE C samples (according to equation 2.9 in chapter 2), after breakdown under sphere-plane geometry for different applied voltages. The values of obtained Weibull distribution parameters for the five different materials are represented in table 3.1. From the obtained results, it can be seen that the scale parameter is a function of the applied voltage for all the tested materials. The values of α measured under 15 kV/mm decrease by roughly a factor 3 when the voltage is changed to 30 kV/mm, and by 3 when increasing the electric field to 45 kV/mm. In addition, the values are material dependent: it seems that α is larger for the cross-linked material as compared to the others. In parallel, the influence of applied voltage on the distribution of the breakdown events for a given material, namely the shape parameter β, can be discussed. Table 3.1 shows that, except in the case of one value of XLPE B at 45 kV/mm, β seems to decrease with increasing applied stress. Moreover, β values seem slightly higher in the case of XLPE materials, as compared to LDPE. β appears to increase with increasing α, thus with decreasing applied stress. Decreasing electrical stress leads to a longer lifetime but also to a sharper distribution of breakdown events. The study of the Weibull distribution parameter α and β has highlighted that both voltage and the material's properties have an impact on lifetime under AC voltage in sphere-plane geometry.

In order to have a better insight at the effects of the applied voltage on lifetimes of the different materials, the applied voltage is plotted as a function of the corresponding lifetime, to obtained so-called "lifetime-vs-stress" curves. In the following, I have chosen to plot the voltage as a function of lifetime, as this is the controlled parameter in the measurements. The corresponding average electric field, which is the relevant physical quantity, is indicated as well. It has been estimated by considering the same average thickness for all samples (≈ 95 μm). The chosen life model allows fitting the experimental lifetime data of the different materials. Moreover, it allows estimating the lifetimes of the materials at intermediate voltages by interpolating the data. Voltage appears to have the same effect on the lifetime of the four materials, as the increases in lifetime with decreasing voltage have approximately the same slope.

However, compared to LDPE with antioxidant, the lifetimes of XLPE materials are slightly shifted to longer times as the cross-link amount increases. XLPE C presents the highest lifetime, followed by XLPE B and XLPE A .

Different hypotheses can be made on this effect. Two mechanical effects may contribute to the observed change: creep and electrostriction. The creep during lifetime measurements was estimated to be of order 10 % at most, thus is considered to not significantly impact the the lifetime of PE. The electrostriction, due to the attraction between the oppositely charge electrodes, can be estimated as follows. The Maxwell stress σ M is given by σ M ≈ 0 r E 2

2 [START_REF] Zeller | Electrofracture mechanics of dielectric aging[END_REF]. With E ≈ 10 7 V/m, r = 2.3, this gives σ M ≈ 10 3 Pa, which corresponds to completely negligible deformation as the modulus of PE is of order 10 9 Pa Cross-linking is expected to produce chemical by-products as discussed in chapter 1. Hussin has shown in a previous thesis that the presence of cross-linking byproducts in LDPE, such as acetophenone and cumyl-alcohol, slightly increased the breakdown strength of LDPE [START_REF] Hussin | The Effects of Crosslinking Byproducts on the Electrical Properties of Low Density Polyethylene[END_REF]. Thus, even though the effect can be different during long term ageing, these by-products may act as charge traps and slightly retard electric degradation.

The lifetime of un-stabilized LDPE has also been measured and compared to the stabilized materials in order to study the effect of antioxidant on the lifetime. Figure 3.6 shows the lifetime curves of LDPE, LDPE stab and XLPE C polymer films in sphereplane geometry. The lifetime of LDPE without antioxidant seems to be slightly increased without as compared to the stabilized LDPE (3.6). At the highest voltage both LDPE materials, with or without antioxidant, present the same lifetime. However, when decreasing the voltage the addition of antioxidant tends to shorten lifetime by 20 %. Antioxidants generally used for cable insulators are aromatic compounds. Such molecules can have an effect of "voltage stabilizers" by acting as electron traps, which would increase the lifetime [START_REF] Esbo | Enhanced dielectric AC breakdown strength of polyethylene using voltage stabilizing additives[END_REF][START_REF] Jarvid | High electron affinity: a guiding criterion for voltage stabilizer design[END_REF]. However, antioxidant may also act as chemical and physical defects, and thus deteriorate the materials properties and act on their lifetime [START_REF] Martinez-Vega | Dielectric materials for electrical engineering[END_REF].

From the observed results on the Weibull parameters and the lifetime curves, the effect of voltage on the materials can be discussed physically.

The Weibull analysis of statistical data may refer to very different physical situations.

One of these can be illustrated by, for instance, disintegration of radioactive nuclei.

For N nuclei the number of nuclei disintegrating between t and t + dt is dN = Ndt α ( 1 α is the probability for a nucleus to disintegrate per unit of time). It follows immediately the probability of being still alive (not disintegrated) at time t :

P(t) = exp - t α (3.2)
In this case, disintegration is an intrinsic (external) event, all nuclei are strictly equivalent and the actual state of nuclei does not evolve as a function of time.

The case of the lifetimes of a set of polymer samples, as studied in our case, is very different for two reasons:

• First, samples may be different from one another, due to the presence of defects randomly distributed between them.

• Second, samples may evolve as a function of time due to damaging mechanisms, leading to a probability to fail which is not constant with time.

Usually, the probability to fail per unit time may increase as a sample ages. However the opposite case may in principle be possible as well, though unlikely. A material may perhaps reinforce during its lifetime due to some physical mechanism.

The instantaneous failure rate as a function of time is given by the hazard function [START_REF] Fothergill | Identifying Electrical Ageing in Polymeric Insulation[END_REF]:

h(t) = β α -β t (β-1) (3.3)
This corresponds to a probability to observe a failure event per unit of time expressed as:

P(t) = β α t α (β-1) exp - t α β (3.4)
which in turns corresponds to a lifetime distribution (probability to not have failed) at time t given by:

P(t) = exp - t α β (3.5)
For β = 1, failures occur randomly and spontaneously and the distribution function tends to an exponential decay as described in the case of nuclei disintegration. In this case, there is no possible distinction in terms of breakdown probability between an aged but unbroken specimen from one unaged [START_REF] Fothergill | Identifying Electrical Ageing in Polymeric Insulation[END_REF].

For β > 1, the failure probability increases as the sample ages [START_REF] Fothergill | Identifying Electrical Ageing in Polymeric Insulation[END_REF].

Thus, it is important to realize that a shape parameter β > 1 may be related both to an initial distribution of lifetimes within the set of samples (defective samples shall fail early) and/or to ageing processes within each sample. As the β values observed in the lifetime curves of the PE based materials are larger than 2 under sphere-plane geometry in air and for the range of applied stress, the lifetimes of the different materials are controlled by degradation processes which rate increases with ageing time.

Analysing the inverse power law fits can also help to understand the ageing processes of the different materials. As described previously, the VEC parameter (n) indicates which ageing mechanism predominates based on how fast the lifetime drops with applied voltage. The inverse power law fitting parameters A and n, as well as the coefficient of determination R 2 , are reported in table 3.2. The R 2 values (> 0.9) suggest that the chosen life model fits properly the experimental data. The parameter A is different for the stabilized materials and LDPE without antioxidant by a factor 20. It is however unclear whether this difference is significant or not. The VEC values however are close to n = 2 for all the materials. This suggests that the materials are subjected to similar degradation mechanisms. A value of n = 2 is in the range of 2 < n < 5, which are values reported in the literature for breakdown of polymeric films associated to partial or surface discharges [START_REF] Yoshimura | Effect of microvoids on V-t characteristics and tree growth in crosslinked polyethylene[END_REF]. As the electrode geometry has been chosen to generate a highly localized electrical stress, it is highly probable that in the vicinity of the contact between the electrode and the polymer surface, discharges occur in air and add to the degradation mechanisms induced by the electric field in the bulk of the material. These effects will be discussed later in this chapter.

In addition to lifetime measurements in the range 15 kV/mm to 45 kV/mm, lifetime of XLPE C has been measured under 850 V (≈ 9 kV/mm) in order to study the effect of a lower electrical stress on lifetime. Samples have been aged for over 2100 hours under 850 V and 90 • C. After this period, the test was stopped, as no breakdown Extrapolating the XLPE C lifetime under 850 V would give around 14 days (330 h), which is far shorter than the measured duration of the test, even without any broken samples. An extrapolation of a life model using a value of n = 5 seems to fit better.

Observations made from figure 3.7 indicate that two regions with different lifetimevoltage dependencies can be drawn:

• In the first region, above 1.7 kV, VEC value is n = 2 as described earlier. Ageing is suspected to be driven by localized degradation, related to surface or partial discharges.

• In the second region, below 1.7 kV, the expected lifetime would correspond to a VEC value of n ≥ 5.

In this region, two domains co-exist as well: between n = 2 and n = 5, no breakdown event have been recorded, which traduces the existence of a "safety zone" (purple dashed triangle in figure 3.7) in which the material is not expected to break. But as the samples have not failed at this voltage and this lifetime, the VEC value must be larger. It could for instance be closer to n = 10, which is a value which has been reported in the case of voltage-time breakdown on cross-linked polyethylene cable [START_REF] Yoshimura | Effect of microvoids on V-t characteristics and tree growth in crosslinked polyethylene[END_REF]. Above n = 5, it should thus exists a zone in which the samples are expected to break at any moment (red dashed triangle in figure 3.7). Failure of the samples at 850 V could thus even happen after one year at this voltage.

These observations, showing two different life-vs-voltage dependencies, indicate that it may exist a threshold voltage below which the material lifetime may be driven by slower degradation mechanisms. The existence of this threshold voltage may be directly linked to the electrode geometry used for the lifetime measurement. Applying relatively high AC voltage, localized on the surface of a sample in air and using a spherical electrode tip, may generate surface discharges in the small volume of air surrounding the contact zone between the electrode and the material, when the field in air exceeds its breakdown field. The surface of the material present a opaque region with a ring pattern on its center. This pattern is situated on the contact zone between the electrode and the material.

No such pattern can be seen on unaged samples. This grey disk may be a marker of degradation, which could be linked to surface discharges. A look an samples aged under lower voltages show a different the ring pattern.

Broken and aged XLPE C films after lifetime measurements under voltages between 850 V and 4.5 kV optical microscopy picture are shown in figure 3.9. As discussed in the literature, partial discharge are likely to occur in micro-gaps or voids which may be formed in synthetic insulators. These discharges occur in the presence of free electrons, and under specific conditions of voltage, pressure and distance, which can be described by Paschen's law [START_REF] Loveless | A universal theory for gas breakdown from microscale to the classical Paschen law[END_REF]. It has been reported and is generally accepted that under atmospheric pressure, ionization of air can occur at field as low as 3 kV/mm [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF].

To verify if the presence of the observed pattern is correlated to the effect of surface discharges, electrical field lines have been calculated in the sphere-plane geometry for different applied voltages, as respect to the distance from the center of the electrode-polymer contact. The aim is to quantify the maximum distance from the center at which discharges are expected to occur. Figure 3.10 illustrates this estimation using FEMM software. This representation shows electric field isolines |E| (kV/mm), i.e. lines where the magnitude of electric field is constant. It shows that the magnitude of the applied field overpass the threshold value of 3 kV/mm in the small volume of air between the electrode and the polymer surface. In this gap, ionization of air is very likely to occur.

An approximate calculation of the electric field lines has been realised in order to obtain a more precise estimation of the distance from the center of the material up to which discharge are expected to occur. The details of the calculation in this sphereplane geometry can be found in the appendix. The magnitude of the field has been calculated at a point P on the polymer surface for the four different voltages used in the lifetime measurements of XLPE C . It has then been plotted as a function of the distance X (mm) between the point P and the center of contact. A threshold limit for discharge (E b (air)) has been plotted as well in order to estimate up to which distance the discharges are expected to occur. Results are shown in figure 3.11. Electric field magnitude is maximum at the center of the electrode-polymer contact, then decreases towards zero. The threshold field for discharges is represented by the black horizontal line at 3 kV/mm. It can be seen that the lower the applied voltage, the shorter the distance. At a voltage of 4.5 kV, discharges are likely to occur at distance larger than 4 mm, that is to say 8 mm in diameter. Calculations were limited to a distance of 4 mm due to the electrode diameter. At lower voltages, the distance decreases down to 0.9 mm in the case of 850 V. These values are compared to the diameter of the ring patterns observed on the aged samples. Results are reported in table 3.3. The calculated diameters are in the right order of magnitude as compared to values observed experimentally, for the three highest voltage values. The differences may come from the approximation in the field calculation, and from the over estimation of the threshold field, due to the unknown parameters of the gaseous medium. This value may be affected by air pressure and humidity. In the case of diameter value at 850 V, calculations estimate that discharges can occur at a distance less than 1 mm away from the center. However, no sign of discharges has been observed on the aged film. a possible explanation is that the geometry taken for the calculation involves an ideal one-point contact between the electrode and the polymer. In reality, it is rather expected that the electrode slightly penetrates into the volume of the material by a depth δ (μm), which would lead to a contact with a different geometry, as shown in figure 3.12 FIGURE 3.12: Scheme of electrode indentation by a depth δ on a polymer film of thickness e.

These estimations confirm that the observed patterns are likely to be due to surface discharges in the air medium surrounding the material's surface. These discharges are expected to form active species, such as ozone (O 3 ), as well as free electrons which in turn could be accelerated by the field and hit the sample surface [START_REF] Morshuis | Degradation of solid dielectrics due to internal partial discharge: some thoughts on progress made and where to go now[END_REF][START_REF] Morshuis | Partial Discharge Mechanisms -Mechanisms leading to breakdown, analysed by fast electrical and optical measurements[END_REF]. These discharge are suspected to drastically affect the material's lifetime, as in the case of samples aged under 850 V for which lifetime was drastically increased as compared to others under higher voltage.

At this stage, it is however not sure how much these discharges affect the lifetimes of the polymer samples. Therefore, the threshold voltage between 1.7 kV and 850 V responsible for change in slope of the lifetime curve could be attributed to other mechanisms. If the threshold voltage is sufficiently low to limit any physical or chemical degradation mechanisms, the material lifetime may be much higher than the value discussed above. This hypothesis can be expressed analytically by a modified life model.

The inverse power law model as used in equation 3.1 stipulates that the lifetime tends to infinity when the voltage tends to zero. Thus, samples will always fail unless the applied voltage is zero. Let us consider a threshold voltage V 0 (kV/mm) under which the mechanisms responsible for degradation are negligible, i.e. under which lifetime tends to infinity:

(V -V 0 ) n t = A (3.6)
Thus, equation 3.6 leads to:

V = A 1/n × t -1/n + V 0 (3.7)
In this case, the lifetime tends to infinity when the voltage is equal to the threshold voltage V 0 . This hypothetical threshold stress may however be low, as compared to the stress values used in MV-AC power cables for which the lifetime is not infinite, regardless of other stress factors which are not included in this life model.

Nevertheless, studying the effects of these discharges on the properties of the material during lifetime may lead to a better understanding of these lifetime curves, which is the subject of the next chapter. In addition to the measurement of lifetime of PE based materials under different voltages, the effects of other parameters have been studied: the temperature, the frequency, and the degree of crystallinity.

Effect of temperature and frequency on lifetime

In addition to the lifetime tests mentioned above, the effect of changing temperature and voltage frequency has been studied in PE based materials.

Effect of temperature

LDPE without antioxidant has been aged at 70 Decreasing the thermal stress may have different effects on the material. The mechanical properties of LDPE without antioxidant and cross-linking may be lowered when increasing temperature. Indeed, the young modulus of semi-crystalline polymers such as PE may be drastically affected by the applied temperature. Simulations and experimental studies on PE and PP reported that the Young modulus of such polymers could drop by 15 % to 40 % by increasing temperature from 70 to 90 • C [START_REF] Daniel | Simulation of the temperature dependence of mechanical properties of polyethylene[END_REF][START_REF] Li | Quantification of the Young's modulus for polypropylene: Influence of initial crystallinity and service temperature[END_REF].

Also, the crystalline morphology of LDPE may be impacted by this 20

T m = T m 0 × (1 - 2γ l × ΔH m 0 ) (3.8)
where, T m 0 [K] is the theoretical melting temperature of PE crystal with infinite size, ΔH m 0 [J/m 3 ] is the theoretical melting enthalpy of 100% crystalline PE and γ [J/m 2 ] is the surface energy of lamellae extremity.

Applied to PE, the relation becomes [START_REF] Wunderlich | Thermodynamics and kinetics of crystallization of flexible molecules[END_REF]:

T m = 414.2 × (1 - 0.627 l ) ± 0.8K (3.9)
with l given in nanometers. From this equation, PE crystalline lamellae melting at 90 • C are about 5.6 nm thick, and 3.9 nm thick at 70 • C. This is coherent with the increase in the degree of crystallinity of LDPE when aged at 90 • C.

The presence of these intermediate crystalline lamellae may impact the properties of the polymer under electric stress. Indeed, as discussed in the literature, electron transport is affected by the ratio of crystalline and amorphous domains. It is not possible at this stage to tell if the lifetime is increased due to the presence of thinner crystalline lamellae, but the crystalline morphology is certainly impacted and may influence the degradation mechanism of PE.

In addition to crystalline morphology, increasing the applied temperature may increase the density of charge carriers injected at the electrode, thus leading to an increased probability of local damage induced by the charges. The properties of charge transport will be studied as a function of temperature and ageing steps in the next chapter.

Effect of voltage frequency on lifetimes

The effect of voltage frequency has been investigated on XLPE C samples at 90 • C for different applied voltage amplitudes. Measurements were performed in Super-Grid Institute using an AC generator which allows applying HV-AC voltages with various frequencies. The ageing cell is the same as used before for lifetime measurements at 50 Hz under sphere-plane geometry.

Based on the literature, increasing AC voltage frequency has an accelerating effect on failure, proportional to the applied frequency [START_REF] Guastavino | Medium term aging characterization of enamelled wires for high frequency applications[END_REF]. Voltage is then not a function of time to failure, but instead a function of the number of AC cycles seen by the insulation before breakdown. Failure under a given voltage would thus be determined by the number of AC cycles and not by the time.

Experimental data have been obtained on XLPE C samples at 2.8 kV (30 kV/mm) for a frequency of 1000 Hz, based on Weibull statistics of 8 breakdown events. In complement, individual breakdown events have been measured at 1.7 kV, 2.8 kV and 4.2 kV (≈ 15 kV/mm to 45 kV/mm) for a frequency of 3000 Hz. These experimental data have been compared to estimations of lifetime at these frequencies based on the hypothesis that the samples should break after the same number of AC cycles as compared to f 0 = 50 Hz (see figure 3.15), that is: This figure confirms that at a given voltage, the measured lifetime is roughly inversely proportional to the applied voltage frequency. Referred to the number of cycles, the lifetime tends to increase a little bit as the frequency increases. However, these results should be confirmed by more extensive measurements.

t( f ) = t( f 0 ) f 0 f (3.10)
These results confirm that voltage frequency plays a role in the acceleration of ageing, and that lifetime of XLPE under AC voltage may be controlled by the number of AC cycles seen by the material during its life. The lifetime curves can therefore be expressed in terms of the number of AC cycles C instead of time [START_REF] Guastavino | Medium term aging characterization of enamelled wires for high frequency applications[END_REF]:

C(V, T) = A (T) × V -n (3.11)
where C(V, T) is the number of AC cycles, and is a function of the applied voltage and temperature, not on the frequency, whereas the lifetime depends on the frequency. This life model could be used to predict failure of polymeric insulation at low voltages by increasing applied frequency.

This observation may be explained by a "bipolar-charge" model, in which carriers (electrons) are injected during the first half cycle and recombine with opposite carriers (holes) during the second half cycle, independently of the duration of the cycle [START_REF] Zhang | Light emission characteristics during the initiation of the electric tree in insulating polymers[END_REF]. The increase in the number of probable charge-polymer interactions during these change in polarity of the applied voltage could thus increase the degradation rate.

These results however are only indicative and should be confirmed by additional experimental data. Indeed, as different regimes were identified in the case of V-t lifetime curves, it would be interesting to verify whether such changes can be observed at higher or lower frequencies. Moreover, these effects are observed in a relatively small frequency range, and it could be expected that at very low or very high frequencies, other mechanisms for instance the relaxation of polarization may have a role in the evolution of lifetime.

Effect of crystallinity on lifetime

The effect of crystallinity on the breakdown strength of polymeric insulation have been studied in the literature (see chapter 1). However, to our knowledge, no study on the effect of crystallinity on the lifetime has been reported in the scientific literature yet. A priori the amorphous phase contains more defects or charge carriers, and thus is more susceptible to electrical damaging. On the other hand, electron propagation in crystals may be easier, with a longer average mean free path. Thus, it is of high interest to know which of these mechanisms is predominant.

To study the effect of crystallinity on the evolution of the lifetime in the sphere-plane geometry, two PP materials with different degrees of crystallinity were studied: PP A (χ r ≈ 36.5 %) and PP B (χ r ≈ 11.9 %).

The lifetimes of the two different PP based materials have been studied under the same conditions as PE based films. 10 to 15 samples of each material were aged under three different voltages at 90 For each material and applied voltage, a cumulative probability of failure P(t) was fitted using a two-parameter Weibull distribution. α values were used as characteristic lifetimes, and used to compare the two polymers.

As an example, figure 3.17 shows the plot of P(t) for PP A under different applied voltages: The characteristic lifetime α decreases from 67.4 h under 1.4 kV to 9.57 h under 4.2 kV. The data dispersion (slope β) does not vary much when changing the voltage. The values of the scale (α) and shape parameters (β) of both materials are reported in table 3.5. The inverse power law model fits with a relatively good precision the experimental data of both materials (R 2 > 0.85). Various observations can be made from the measured lifetimes of PP materials. First, the slopes of the fitted lifetime curves of PP A and PP B are significantly different. Eventually, the two curves cross each others around 20 kV/mm.

Two regions can be described on this figure: at relatively high electric stress, the material with the highest degree of crystallinity (PP A , χ r ≈ 36.5 %) has a lifetime twice longer than the material with the lowest degree of crystallinity (PP B , χ r ≈ 11.9 %). Below 2.8 kV, the lifetime curves of the two materials cross each others. At 2.1 kV and below, PP B lifetime is longer than PP A .

The inverse power law model fitting parameters A and n, as well as the coefficient of determination R 2 , are reported in 3.6. The parameter A is much lower for PP A as compared to other fitting parameters obtained previously. The values of parameter A are not discussed in the literature, because it depends on different unknown variables such as initial rate of degradation [START_REF] Endicott | Application of the Eyring Model to Capacitor Aging Data[END_REF]. Therefore, it is difficult to propose a physical explanation for these differences.

However, as discussed earlier the VEC of materials is a good indicator of which degradation mechanism predominates.

The fitted n parameter values of both PP materials are different from each other, and from the PE based materials. n value for PP A is lower than 2. As the R 2 value is not as high as for other fit done previously, a fourth experimental point would be need to have a better approximation of n for this material. For PP B , n is still in the range of values reported for breakdown on polymeric films associated to discharges, even higher than for PE based materials, which suggest eventually that the material could be slightly less subject to discharges than PE.

Based on these observations, similar degradation mechanisms seem to occur in PP and PE based materials, in this specific test configuration. n values indicate a degradation associated to discharges, but the difference in α and n values highlights a difference in the behavior of PP A and B under applied voltage in this sphere-plane geometry. At relatively high voltages, the material with a higher degree of crystallinity shows a higher resistance to failure as compared to the other one. But the tendency switch at lower voltages.

As the two PP based materials have similar chemical structures with different chain architecture, the crystalline morphology and its evolution during ageing may play a role in the behavior of the materials under applied voltage. The main difference in the polymer morphology between the two PP grades is the crystallinity, which is lower in the case of PP B . The role of amorphous and crystalline phases in the charge transport has been discussed in the literature: as electron transport is supposed to be preferentially of inter-chain nature, the lower crystalline ratio of one polymer as respect to the other may favor electron transport and trapping [START_REF] Serra | Interchain states and the negative electron affinity of polyethylene[END_REF][START_REF] Serra | Interchain electron states in polyethylene[END_REF][START_REF] Serra | Self-trapping vs. non-trapping of electrons and holes in organic insulators: polyethylene[END_REF]. This could lead to a higher density of charge in the material, and a higher probability for degradation associated to charge transport to occur, which may explain the shorter lifetime of the less crystalline material at higher fields.

However, the behavior at lower electric stress cannot be explained in the same way.

The trends to the lifetime curve of the material with the lowest degree of crystallinity bring to mind a similarity with XLPE C samples aged under 850 V: a change in the slope indicate that the material may undergo different degradation mechanisms. Thus, it is worth questioning if both materials are subjected to the same degradation mechanisms. Both surfaces show with the naked eye similar ring patterns as found in PE. The reason why lifetime tests at the lowest voltage were performed at 1.4 kV for PP A , and at 2.1 kV for PP B , is that preliminary tests on PP B at this voltage did not lead to any breakdown, nor observation of this degradation marker. Eventually, this means that the degradation mechanisms which were acting on the materials at higher voltage were not triggered in the case of PP B at this voltage. This could be explained geometrically as in figure 3.12. PP B may be more subjected to indentation by the electrode. In that case, the applied voltage may be below the threshold voltage for discharge inception.

Nevertheless, the observed differences in the lifetime of the PP materials at voltage above 2.1 kV (20 kV/mm) indicate that increasing the degree of crystallinity of the insulating polymer leads to better fatigue properties under AC voltage in this electrode configuration. Polymer crystallinity is thus a key parameter in order to achieve polymer insulators for power cables with reliable properties.

The evolution of crystalline morphology with ageing has been studied and will be discussed in the next chapter.

In the next section, the lifetime of XLPE is studied in a different electrode configuration in order to study the impact of electrode geometry on insulation's lifetime.

Lifetime of XLPE in plane-plane geometry

Lifetime of XLPE was measured in plane-plane electrode geometry in order to investigate the effect of a uniform stress on the lifetime of the material. The test samples were made of Rogowski-like electrodes encapsulated in XLPE C material, as described in chapter 2. The samples were aged until breakdown at 90 • C under application of two different voltages. Time to breakdown data were analysed using a Weibull distribution as well. α and β parameters were compared to XLPE C samples aged under the previous sphere-plane configuration.

Weibull analysis of both XLPE materials life data are shown in figure 3.19 below: Based on the time to breakdown data measured at 2.8 kV, it can be seen that the distributions of breakdown events of the two materials are quite different. Under the same applied voltage, the material aged under plane-plane geometry has a characteristic lifetime twice longer as compared to the same material aged under sphereplane. The slopes indicate that the distribution of times to breakdown is sharper in the later case. The parameters α and β were calculated and compared. Table 3.7 shows the scale and shape parameters of both materials for the different applied voltage. This result may indicate that the degradation and breakdown events under planeplane configuration are more randomly distributed but occur with a longer period of applied stress.

The lifetime curves of XLPE under both geometries have been plotted in order to compare the evolution of lifetime with the applied stress (see figure 3.20). From these lifetime curves in the two different electrode geometries, observations seem to indicate that two stress regions govern the properties of these materials under the different electrode geometries.

First, around 4 kV XLPE lifetime seems to be independent of the electrode geometry, as the expected lifetimes are very close in both cases. Below this voltage, a second region appears where the expected lifetime of XLPE under sphere-plane geometry drops faster than the the same material in the plane-plane electrode geometry.

The slope of the fitted lifetime is smaller in the later case, which indicates a larger value of the parameter n, as reported in table 3.8.

Sample

A n R 2 Sphere-plane 1.95E+08 1.97 0.9798 Plane-plane 7.68E+14 3. [START_REF] Crine | Role of electrostriction on the electrical aging of polymers[END_REF] 1

The estimated value of the parameter n in the case of plane-plane geometry is almost twice larger than the one in the case of sphere-plane geometry. This difference in the slope of life data may be explained as follows: at high fields, intrinsic damage (bulk) may predominate over surface discharges, which may explain that lifetimes are similar in both geometries. At low fields, the lifetimes may be limited by surface damages, which may explain that lifetimes are shorter in the sphere-plane geometry.

The VEC value obtained for lifetime in the plane-plane configuration is below n = 5 and is in the range of breakdown associated to discharge as described in the literature. Therefore, even though the use of a plane-plane geometry was expected to generate a uniform field in the bulk of the material, we cannot exclude the possibility that ageing in this geometry may be associated to some discharges event occurring either in the material, or on the counter electrode side. A combination of bulk degradation and discharge could explain the value of n in that case. Indeed, as the Rogowski-like electrode is coated only on one side of the polymer sample, even if the other side of the sample is expected to be plane some surface roughness due to the process may lead to the formation of small gaps, in which discharges may be triggered. Such surface roughness may either come from the metallic surface of the counter electrode, or from the polymer sample. The interface between the HV electrode and the polymer could also be affected if gaseous by-products have formed during cross-linking.

But, we suspect that the discharges originate from the counter electrode side. A schematic representation of such material-electrode interface is shown in figure 3.21. The surface roughness of counter-electrode was not controlled, but as it is covered with aluminium sheets which are neat we can expect the roughness to be of the order of a few ten of nanometers. However, the surface aspect of the polymer on this side is driven by the process conditions during compression molding. Thus, some surface asperities may appear on the surface of the sample, and small gaps (hundreds of nm to a few μm) could form. It was reported in the literature, based on the Paschen's curve, than the expected threshold voltage for a cavity of a few μm wide, filled with air is in the order of a few kV [START_REF] Daniel | Discharging Microcavities and Electric Field Enhancement inside Cavities in HVDC XLPE Cables[END_REF][START_REF] Priscillia | Partial Discharges for Diagnosis of Cavity Growth in HVDC XLPE Model Cables[END_REF]. Thus, if micro-voids are present at the interface between the metal and the polymer, and that the gap is sufficiently wide, discharges can be triggered in it and affect the lifetime of the material. Observations with the naked eye revealed that on some of the samples, a few white spots are present and distributed on the surface of the samples. These spots are smaller than the ring pattern observed on the samples aged under sphere-plane geometry, and are distributed on the samples surface. Nevertheless, the lifetime in this geometry being twice longer at low voltage indicates that if discharges are likely to affect the material's lifetime, the effects are much more important in the case of the sphereplane geometry. The effect of such discharges will be discussed more in details in chapter 5.

To summarize, the lifetime curves of XLPE in these two electrode geometries shows that for the same material, application of a localized and a uniform stress leads to two different expected lifetime trends. The lifetime of a material aged under sphereplane is expected to be shorter than the same material under plane-plane. From what have been observed so far in this study, application of a localized stress using the sphere-plane may trigger partial discharges in air and affect the polymer properties during ageing. Discharges in the other geometry are possible, but would be distributed on a bigger surface, which could explain the longer lifetime of the materials in this geometry.

While the observations seem to indicate two different ageing mechanisms from the slopes of the curves, the fact that both geometries leads to the same lifetime at 45 kV/mm suggests that regardless of the geometry, increasing applied stress may trigger similar degradation mechanisms. The main observed differences between the two geometries are observed at lower stress levels, at which threshold field for other mechanisms could exist. In that case, applying a localized stress is deleterious for the polymer and is expected to shorten its lifetime as compared to a more uniformly distributed electrical stress.

Conclusion

In this chapter, two main studies have been presented. The lifetimes of polymer films have been measured in a sphere-plane electrode configuration, and the impact of different parameters have been studied. Then, a plane-plane geometry was used to measure the lifetime of XLPE, in order to compare the impact of the two geometries.

These studies led to the following conclusions:

• Cross-linking seems to increase lifetime by a few hours independently of the applied voltage. The effect could be attributed to the introduction of chemical impurities such as cross-linking by-products, as they may act as electron traps.

• Lifetime values are associated to partial discharges in the sphere-plane geometry. In this electrode geometry, below a threshold voltage, the lifetime of XLPE samples could not be measured. The existence of this threshold voltage has been correlated to the extinction of the discharge events, which indicate that they may be responsible for the most part of the material degradation.

• Lifetime of PE is also impacted by the applied temperature. Ageing under 70 • C increased the lifetime of the material by up to 70 % at 45 kV/mm. This effect is attributed to the change in the crystalline morphology during ageing. Crystallinity seems to be a key parameter controlling the lifetime of a given material. The lifetime study of two PP materials with different degrees of crystallinity has shown that increasing the relative crystallinity from 11.9 % to 36.5 % may double the expected lifetime.

• Increasing the voltage frequency to shorter lifetime in XLPE. Experimental data confirm that lifetime under AC voltage may be governed by the number of AC cycles seen by the material, instead of the exposure time.

• Finally, measurement of XLPE lifetime in the plane-plane geometry has shown that the applied stress has a drastic influence on its lifetime. Under this geometry, the stress field is expected to be more distributed in the volume of the material, and experimental results showed that lifetime is increased. However, discharges may also be triggered in that case, but could be less critical that in the case of a localized stress.

This lifetime study using both spherical and plane electrode configurations showed that controlling the materials properties, mainly its crystallinity, may help to achieve an enhanced durability in the use of polymers for power cable insulation. The specific role of temperature and applied frequency needs to be further studied, but we have shown how both parameters influence the insulation lifetime.

But above all, it is important to make sure that the insulating materials does not undergo localized electrical stress, which in turn may drastically reduce its lifetime, even though being concentrated on a small region of the sample.

These results allow to understand better the behavior of polymer materials under AC field, and the importance of controlling their properties. However, the different ageing states taken by the material leading to breakdown are still unknown. It is therefore necessary to study the evolution of the materials properties at these different ageing steps until ultimate degradation states prior to breakdown.

These lifetime curves are a key result which allows to predict the ageing time before breakdown of a population of samples, and thus study the materials at these increasing ageing steps. In the following chapters, XLPE and PP samples will be aged using both sphere-plane and plane-plane geometries, in order to achieve a deeper understanding of the degradation steps during the insulation lifetime.

Chapter 4

Study of electrical ageing in sphere-plane electrode geometry

This chapter presents the work done on identifying ageing markers involved in the degradation of polymeric insulation, in the case of a localized electric field or a homogeneous electric field. Indeed, polymeric insulators, used for instance in power cables, are usually subjected to various types of stresses which may be strongly localized on small parts of the material. Understanding the effects of such localized stresses on the properties of the material at different stages of its lifetime is thus a key point in understanding how failure occurs.

Lifetime curves of XLPE and PP materials measured in a sphere-plane electrode geometry have shown that a localized electric stress on polymer films may accelerate degradation (see 3). In this specific configuration, it was shown that the spherical shape of the HV electrode induced sufficiently high fields to trigger partial discharges in air. The role of these discharges on controlling the lifetime of these materials is however still not clearly identified. However the observations indicate that discharges may generate damage on the surface of the polymer films. It was reported in the literature that partial discharges triggered in micro-cavities of polymeric insulators are suspected to generate and concentrate structural and chemical damage, and create a weak point from which treeing may grow [START_REF] Morshuis | Partial Discharge Mechanisms -Mechanisms leading to breakdown, analysed by fast electrical and optical measurements[END_REF][START_REF] Morshuis | Degradation of solid dielectrics due to internal partial discharge: some thoughts on progress made and where to go now[END_REF]. Thus, the discharges triggered in this geometry may be responsible for a significant part of the insulation ageing in this electrode configuration.

In order to study the effect of such stress on the polymer properties, XLPE C samples were aged in the sphere-plane geometry at 90 • C and under two different applied voltages,at various ageing steps from unaged to near breakdown. As crystallinity has been identified to play a major role in the lifetime of the studied materials, ageing of the two PP materials was also studied. The evolution of structural, chemical, physical and electrical properties of the materials has been characterized as a function of ageing time. The whole study is focused on identifying key ageing markers in order to understand the critical aspects of ageing during the samples' lifetime. Specific attention has been made on decorrelating the effects due to electrical and thermal ageing.

After introducing the different ageing steps that have been used, the results of characterizations are presented and discussed. The impact of ageing on the surface aspect of XLPE samples was analysed using SEM. The evolution of the physicochemical properties was then studied by FTIR and DSC. SAXS measurements were realised on PP samples as a complementary characterization to DSC for studying the evolution of the crystalline morphology. The evolution of the electrical properties were studied by measuring the breakdown strength of the samples, and leakage current measurements were also performed on XLPE samples. Finally, Luminescence measurements were realised on XLPE samples in order to study the effect of ageing on the electronic properties of the material. The aim of these characterizations is to highlight the presence and nature of ageing markers during the the materials lifetime.

Ageing steps

Based on the lifetime study of XLPE C in the sphere-plane geometry presented in the previous chapter, the lifetime of the material has been predicted at intermediate applied fields. Two different ageing voltages have been chosen in order to have an expected lifetime of 24h and 48h. These values are obtained by interpolating the lifetime data obtained previously using the inverse power law. Likewise, based on the measured lifetime of PP A and PP B , a suitable ageing voltage has been chosen in order to have lifetimes of 22h and 18h respectively. A few ageing steps (in bold in tables 4.1 and 4.2) longer than the expected lifetime have been performed in order to make sure that the material is close to its statistical end of life, and thus should show the strongest state of damaging. At these ageing stages, the samples are expected to fail at any moment but have not failed yet.

Characterization of ageing markers 4.2.1 Structural and physico-chemical characterization 4.2.1.1 Surface degradation at electrode-polymer interface

The surface degradation of the aged XLPE samples have been characterized by SEM.

As described in the previous chapter the surface exposed to the spherical electrode is likely to be damaged during ageing (see chapter 3). The top surface (HV electrode side) of aged XLPE samples have thus been investigated for different ageing steps and compared to unaged XLPE reference.

First, an unaged XLPE C sample has been characterized and will serve as a reference. As shown in figure 4.4, the aged region is visible in light contrast, centered on the contact zone. The delimitation between aged and unaged region is also visible. Two different regions on interest have been circled in this picture (4.4 (B)): the one in the left is the peripheral of the "aged" region, and the one in the middle of the picture is a zone approximately 1 mm away from the center of the contact zone, in which damage has been found in the highest concentration.

In order to have a better understanding of the observed surface damage, a zoom has been performed in these two different zone of interest. In the peripheral region, the frontier between unaged (non-eroded) and aged (eroded) region is clearly visible. A zoom-in in the aged side shows the formation of craters of a few microns wide. The surface appears etched by application of the ageing voltage.

At the same scale, closer to the center of the aged region, the surface aspect changes. The analysis of the top surface in this area shows a different topology as compared to the peripheral region. Craters have disappeared, and solid pits of 1 to 5 μm wide are visible instead. At the same scale, craters found in the peripheral region changes to smaller pits.

It is likely that the surface in contact with the sphere-electrode has been affected by a degradation caused by the electric field. Both objects (craters and pits) are similar to reported results in the literature in the case of ageing of polymeric insulation associated to partial and surface discharges [START_REF] Morshuis | Degradation of solid dielectrics due to internal partial discharge: some thoughts on progress made and where to go now[END_REF][START_REF] Gamez-Garcia | Synthesis Reactions Involving XLPE Subjected to Partial Discharges[END_REF][START_REF] Gamez-Garcia | Modification of XLPE exposed to partial discharges at elevated temperature[END_REF][START_REF] Leon-Garzon | Experimental Characterization of Polymer Surfaces Subject to Corona Discharges in Controlled Atmospheres[END_REF]. More recently, Leon-Garzon et al. have studied the effects of corona discharges over polymeric slabs and characterized the modifications of the surfaces [START_REF] Leon-Garzon | Experimental Characterization of Polymer Surfaces Subject to Corona Discharges in Controlled Atmospheres[END_REF]. Sample's surface presented some roughness and were covered with granular-like formations when treated with discharges in air.

This surface degradation may thus be caused by the discharges generated in air near the HV electrode. The craters and pits may have formed due to the bombardment of the surface by highly energetic electrons and ions [START_REF] Morshuis | Degradation of solid dielectrics due to internal partial discharge: some thoughts on progress made and where to go now[END_REF].

The difference between the observed surface in the two regions can be explained in terms of distance to the center and exposure to surface discharges. Close to the center, the material may be exposed to a higher number of discharge events, compared to the peripheral region. Far enough away from the center, the discharge eventually stops leading to a clear frontier between aged and unaged region.

In addition to the surface imaging, the edge of aged sample's surface has been characterized. It was prepared using a microtome at - The edge of the degraded surface shows craters of about 500 nm in depth. The bulk of the material below these craters seems unaffected by the degradation. The observable degradation phenomenon seems thus to appear only on the surface as no degradation is observed deeper in the sample.

In order to study the evolution of this surface degradation with ageing, the surface of aged samples have been studied from the beginning until the very end of the expected lifetime. Figure 4.8 shows the surface of aged XLPE at different moments of the material's life, in the aged region 1 mm away from the sample's center. Closer to the center, the surface structure evolves with ageing time. Craters are observed at a lifetime ratio of 4 % (A), and are similar to those observed at the periphery of the aged region. By increasing the ageing time (12.5 % to 100 %, B-D), the surface roughness increases, channels erode progressively, the number of pits decreases and granular-like solid by-products appear. After 48h, only bigger pits remain.

As these pits appears near the material's end of life, the evolution from craters to pits could be an ageing marker in this electrode configuration.

The applied voltage may have an influence on the inset of this damage associated to discharges. Figure 4.9 shows the surface aspect of XLPE C surface, 1 mm away from the center, for 24 hours under 2.25 kV and 3.2 kV, which corresponds to a lifetime ratio of 50 % and 100 % respectively. After 24 hours of exposure to the voltage, the surfaces of the aged samples are quite different: a higher number of pits, with a smaller size, can be seen in the case of the lowest voltage as compared to the other sample. According to the previous observations, the concentration of pits decreases after longer ageing time. The surface aspect of the sample aged under 3.2 kV (4.9 (B)) is characteristic of a longer ageing time.

This surface erosion may be thus accelerated when applying a higher voltage. The surface discharge triggered at a higher voltage should involve a higher energy, which could lead to a faster degradation. It appears from these observations that the discharges highly affect the surface of the samples, and that, either a higher number of discharges, or more energetic discharges, are involved when increasing the ageing voltage. It is important to mention that the diameter of the damaged disk (eroded region) depends on the applied voltage (as discussed in chapter 3) but it does not depend on time during the test. The disk appears after 30 min and does not change in size afterwards. Only the surface aspect inside the disk changes with ageing. This observation confirms that surface damage is effectively due to discharges in air.

In order to quantify the effect ageing on damaging on the sample's surface, a method to measure the surface erosion induced by the discharges has been proposed. The surface erosion could have been measured using Atomic Force Microscopy (AFM) in order to have a better precision. Instead we proceed by an image analysis of the etched surface using the contrast between the eroded surface (black) and the remaining surface (white). Figure 4.10 shows the surface of XLPE aged 48h under 2.26 kV and the same image in red and white contrast using a threshold mask. The white area corresponds to the pits remaining from the top surface, and the red area corresponds to the underlying new surface after erosion. The ratio of red surface over total surface corresponds to the eroded surface due to the surface discharge. A ratio of 100 % would correspond to a surface completely eroded without any remaining pits or craters. This ratio has been measured for the XLPE samples aged for increasing ageing steps. On the opposite, the eroded surface at 0 % of life ratio is zero. Figure 4.11 shows the evolution of the eroded surface ratio as a function of ageing time. It is generally admitted that pits are indicative of discharge-induced degradation as they are due to continuous bombardment of the surface by active species. They correspond to an advanced stage of ageing and eventually lead to treeing and final breakdown [START_REF] Morshuis | Degradation of solid dielectrics due to internal partial discharge: some thoughts on progress made and where to go now[END_REF]. The observed degradation may thus be described by ageing mechanisms occurring due to discharges, for instance in the vicinity of micro-cavities embedded in power cable insulators, or near a metallic protrusion in the presence of air.

In addition to XLPE, PP samples were aged and characterized using SEM. The top surface of aged PP samples has been characterized and compared to unaged reference. Unaged PP samples were first characterized and used as references. The surface of both materials is neat and clean, only slight deformations can be observed when zooming at the micro-scale, which are attributed to the sample's preparation. The aged samples top surfaces have then been characterized and compared to the unaged materials. The surface degradation of PP A evolves from an unaged state to a progressive eroded surface. Few hours after application of the voltage, the surface is covered with craters of a few microns wide. As ageing time increases, the gap between craters slightly increases and goes up to 5 microns. The surface aspect near the end of life shows wide gaps and a dug surface. This hollow surface indicates that treeing may be initiated inside the gap by partial and surface discharges, eventually leading to breakdown in the bulk.

Similarly to XLPE analysis, the surface of PP A aged samples has been measured using an image analysis method. Results are shown in figure 4.15. Measurement of surface erosion by image analysis reveals that the global eroded surface increases just after ageing starts, but does not change significantly overall the sample's life. However the topological features observed on the surface of the material during evolution of ageing indicates that the material has undergo surface degradation, sharing some similarities with XLPE.

Indeed, the procedure used for analysis of surface erosion could be more robust using other characterization techniques such as AFM. Still, these results indicates that surface degradation evolves during the first step of the samples lifetime but seems to not change drastically during the end of life.

Surface analysis has then been done on PP B at different stages of the material's life, as shown in figure 4.16. The surface aspect evolution of PP B is quite different than the other material. Unaged surface aspect evolves to a progressive degraded surface characterized by small pits of hundreds of nanometers big. At 39 % of expected lifetime, the surface erosion increases and hollows appear between pits. At 83 % of expected lifetime, more hollows can be observed, and the topological aspect of the surface has changed. The surface seems to have undergo melting of the polymer due to a local thermal stress.

These observation seem to indicate that the surface has undergo surface discharge as in the case of XLPE, coming with a high deformation of the surface in case of PP B . Studies of surface degradation of PP materials subject to corona discharges have reported in the literature that the degradation was characterized by a rough surface with granular structures and a decrease of surface angle measured by wettability [START_REF] Leon-Garzon | Experimental Characterization of Polymer Surfaces Subject to Corona Discharges in Controlled Atmospheres[END_REF][START_REF] Sellin | Surface composition analysis of PP films treated by corona discharge[END_REF].

A possible explanation to these observation may come from the different in crystallinity between the two samples. As PP B is less crystalline than PP A , and is not cross-linked as XLPE, it is possible than the energy involved in electrical ageing and surface discharges heated up the surface of the sample and led to the melting of the polymer on its surface, and could explain this high surface roughness.

Nevertheless, XLPE and PP materials experienced the same type of surface degradation. This degradation is believed to occur due to the discharges induced by the sphere-plane geometry. SEM analysis revealed that the degradation was concentrated about 500 nm deep in the material's surface, and that the bulk was not affected. It is thus not possible at this stage to tell if this surface degradation is sufficient to cause breakdown. However, it is probable that this surface degradation may be coming with other type of degradation induced by the field and/or the discharges.

In order to correlate the evolution of the surface damage with other ageing mechanisms, the physico-chemical properties of the aged XLPE samples need to be investigated.

Characterization of chemical degradation

FTIR analysis has been performed on reference and aged XLPE and PP samples in order to investigate the effect of ageing on the chemical properties of the materials.

As seen with SEM analysis, the degradation seems to be concentrated inside the ring pattern caused by the discharges, in a sub-region 1 mm away from the contact zone between electrode and the polymer film. Thus, FTIR analysis were done first in transmission mode, and focused on this specific sub-region. As a comparison, an FTIR analysis in reflection mode (ATR) was also performed on an aged XLPE sample in order to characterize and track chemical ageing markers on the different region of the sample's surface.

As the samples are also aged in temperature at 90 • C, and in order to decorrelate the effects of electrical and thermal ageing, the aged samples spectra were compared to samples aged at 90 • C without applied voltage (thermal references). In addition, to be able to compare quantitatively the FTIR data, the unaged and aged XLPE spectra have been normalized with respect to a reference absorption peak near 2019 cm -1 which is commonly used in the literature, associated to polyethylene backbone and proportional to the film thickness [START_REF] Möller | Examination of a low density polyethylene (LDPE) film after 15 years of service as an air and water vapour barrier[END_REF][START_REF] Li | Performance evaluation of XLPE materials for 90°C HVDC applications[END_REF].

The FTIR spectrum of unaged reference XLPE C sample is shown in figure 4.17. The signature of polyethylene is characterized by three main peaks which saturates due to the relatively high thickness of the film. From this spectrum, the main components of the polyethylene macromolecules can be identified [START_REF] Socrates | Infrared and Raman characteristic group frequencies: tables and charts[END_REF][START_REF] Gulmine | Polyethylene characterization by FTIR[END_REF][START_REF] Rapp | Influence of the physical state of a polymer blend on thermal ageing[END_REF]]:

• CH 2 symmetric and asymmetric stretching are known to absorb between 2919 and 2851 cm -1 .

• CH 2 bending deformation between 1473 and 1463 cm -1 .

• CH 3 symmetric deformation at 1377 cm -1 .

• CH 3 rocking deformation between 731 and 720 cm -1 .

Smaller weak peaks can be found in sub regions on the spectrum. These regions are characteristics of hydroxyl groups (3600-3100 cm -1 ), carbonyl groups (1850-1550 cm -1 ) and finally ether groups and insaturations (1400-700 cm -1 ). They may corresponds to the presence of chemical defects or by-products (acetophenone for instance) introduced in the polymer during process and can thus be found in unaged XLPE [START_REF] Rapp | Influence of the physical state of a polymer blend on thermal ageing[END_REF][START_REF] Gulmine | FTIR characterization of aged XLPE[END_REF][START_REF] Kemari | Correlations between structural changes and dielectric behavior of thermally aged XLPE[END_REF].

The effect of electrical ageing is investigated in XLPE by comparing FTIR spectra of unaged and aged XLPE at different ageing steps. Figure 4.18 shows the FTIR spectra of a reference and aged XLPE samples after ageing under 2.25 kV at 90 • C. The FTIR spectra of aged samples share the same general features as the unaged reference XLPE material, except for the peaks present in the three sub-regions mentioned above. The position and size of these peaks suggests the presence of ageing markers, and thus need to be investigated further in details. In the region corresponding to hydroxyl groups and inter-molecular hydrogenbonded -OH groups [START_REF] Socrates | Infrared and Raman characteristic group frequencies: tables and charts[END_REF], two main peaks at 3370 and 3430 cm -1 already present in the unaged XLPE is found in all the aged samples. The absorbance of this peak increases with ageing time, for ageing under both 2.25 kV and 3.2 kV. The absorbance of the peaks in the case of thermally aged samples does not change significantly as compared to the electrically aged samples. In the latter case, absorbance of the main peak in this region increases with ageing time. In the case of advance ageing steps (126 % and 125 % under 2.25 kV and 3.2 kV respectively), the absorbance increases significantly as compared to the unaged sample. This result suggests that ageing induces the formation of additional hydroperoxydes, which generally occur during oxidation reactions as explained previously in chapter 1 [START_REF] Holmström | Thermooxidative Degradation of Polyethylene. I and II. Structural changes occurring in low-density polyethylene, high-density polyethylene, and tetratetracontane heated in air[END_REF]. In addition, it can be noted that in under both applied voltages, the absorbance value is multiplied by two between the penultimate ageing steps and the last ageing steps, indicating an acceleration in the oxidation mechanism. This acceleration will be quantified later in this section.

In the region between 1850 and 1550 cm -1 , the effect of ageing can also be identified:

The spectrum of unaged XLPE shows a main peak between 1750 and 1720 cm -1 , associated to ester groups and ketones respectively which could originate from thermal decomposition during the process [START_REF] Gulmine | FTIR characterization of aged XLPE[END_REF]. It could also be linked to the presence of the antioxidant, which shows a main peak at 1741 cm -1 [START_REF] Möller | Examination of a low density polyethylene (LDPE) film after 15 years of service as an air and water vapour barrier[END_REF].

Thermal ageing performed on the thermal reference XLPE leads to a weak increase in absorbance at these peak positions. The presence of the antioxidant introduced in the material could explain the limited thermal oxidation observed in these samples.

However, the electrically aged XLPE samples show a significant increase in the absorbance of the identified species. A broad peak between 1800 and 1680 cm -1 can be observed, as well as a second peak at 1630 cm -1 . The species formed around 1773 cm -1 can be identified as γ lactones, while esters and ketones are represented by the two peaks at 1736 and 1714 cm -1 [START_REF] Gulmine | FTIR characterization of aged XLPE[END_REF][START_REF] Socrates | Infrared and Raman characteristic group frequencies: tables and charts[END_REF]. The peak at 1630 cm -1 is in the region of vinylene groups (C=C) which can be found between 1560 and 1650 cm -1 [START_REF] Kemari | Correlations between structural changes and dielectric behavior of thermally aged XLPE[END_REF]. As in the case of hydroxyl groups, the absorbance slightly increases during the first ageing steps, and accelerated at the last step. Carbonyl groups like ester and ketones, and insaturated carbon bonds are likely to be formed during electrical ageing, either in the bulk or at the surface due to the discharges. Applied voltage also accelerates the formation of these chemical species. Finally, the spectra of aged XLPE samples have been studied in the 1400-700 cm -1 region:

In this region, several contributions are responsible for absorbance: 1169 and 1414 cm -1 is associated with the vibrations of the -C-O-C-bond, and 800 and 1128 cm -1 are associated to C=C double bond of the unsaturated groups, vinylidene and trans-vinyl [START_REF] Kemari | Correlations between structural changes and dielectric behavior of thermally aged XLPE[END_REF].

Again, the thermally aged samples show only weak changes in this region, as compared to the electrically aged samples. The signature of aged samples shows a slight decrease in absorbance with ageing at first, before increasing significantly. Insaturations have been identified previously with a peak at 1630 cm -1 , thus is correlated with the observed changes in this region.

In all cases, these hydroxyl, carbonyl and insaturations represent an significant ageing marker, as a clear difference can be observed between thermal and thermoelectrical ageing. These chemical groups may have formed during thermo-oxidative reactions accelerated or triggered by the electric field, or the discharges. The specific role of voltage in the evolution of the markers shows that ageing is accelerated in the presence of applied voltage. As described by Beer-Lambert, the increase in the absorbance of these chemical markers traduces an increase in their concentration. The increase of these chemical groups concentration also seems to accelerate at the very end of the material's life. Figures 4.22 and 4.23 shows the absorbance ratio between the aged samples and the reference as a function of ageing time, for the identified peaks corresponding to ageing markers. Absorbance ratio are measured using the peak value of the respective samples at the given wavenumber. Under application of voltage, the concentration of ageing markers increases faster than thermally aged samples. The increase in concentration accelerates for both applied voltage at the last ageing step, at an ageing time longer than the expected material's lifetime. These observations indicate that electrical ageing is associated to the formation of chemical groups which may originate from thermo-oxidation reactions. At advance ageing steps, prior to electrical breakdown, the concentration of these species increase exponentially, which is not the case for thermally aged samples. In his thesis, Pelzer studied the effect of thermal ageing on the properties of PE based materials, in order to investigate ageing markers [START_REF] Pelzer | Etude de vieillissement des isolants synthétiques des câbles moyenne tension[END_REF]. It was reported that the absorbance ratio between samples aged at various temperature and a reference peak of PE corresponding to carbonyl and hydroxyl groups increased exponentially with ageing time after a couple hundreds of hours of ageing. This exponential increase was describe as a strong ageing marker, and was attributed to decomposition of hydroperoxides. Another recent work from Blivet et al. studying the thermal ageing of XLPE reported a similar exponential increase of the absorbance ratio [START_REF] Blivet | Thermal oxidation of cross-linked PE and EPR used as insulation materials: Multi-scale correlation over a wide range of temperatures[END_REF]. Moreover, this exponential increase was correlated to an exponential decrease of other properties of the aged materials, such as elongation at break or ultimate tensile strength.

In our case, this effect appears only on electrically aged samples. It seems that the observed effect under electric field are similar to those observed under long-term thermal ageing. Thus, a strong link between the increase of these chemical groups concentration and electrical ageing can be made.

In addition to the transmission FTIR spectra measured on the aged region of XLPE, a linear tracking of a broken sample has been done using an IR Transmission Microscope. The aim was to track the observed oxidation markers inside and outside the aged region. Figure 4.24 shows the microscope image of a broken XLPE and the representation of the linear tracking made by IR. 10 spectra were measured every 860 μm, starting outside the aged region. The different spectra were compared and analysed. Figure 4.25 shows the spectra corresponding to the spots on the linear tracking of XLPE surface. As can be seen on the spectra, the major contribution comes from the compounds in the carbonyl region (1800-1600 cm -1 ). The increase of absorbance observed on the spectra is due to the spots located in the aged region. In order to have a better insight on the results, absorbance of the main identified peak in this region have been normalized by a reference peak at 2020 cm -1 for PE as used previously, then compared. Figure 4.26 shows the evolution of the absorbance ratio of the different peaks as a function of the spot's distance from the center of the aged region.

The absorbance ratio increases for all peak contributions when entering the aged region and approaching the center, and decreases back when entering back the unaged region. This results confirm that the observed chemical degradation is concentrated inside the aged region near the contact zone between the electrode and the polymer, and that the structural degradation comes along with a chemical thermo-oxidation.

The same sample have been characterized by ATR in order to study the contribution from the surface only of the sample, and compare it to the unaged sample. ATR was performed on a reference XLPE sample surface, and on both side of the broken XLPE sample: electrode side (top) surface and counter-electrode side (bottom) surface. Results are shown in figure 4.27. The main peaks observed on the unaged XLPE spectrum are similar to the one detailed in transmission mode. When comparing unaged and aged sample surfaces, a clear difference appears in the hydroxyl (-OH), carbonyl zones (C = O) and ethers (-C -O -C-). Both aged surfaces shows an increase of absorbance of the bands in these sub-regions, indicating that the thermo-oxidation observed in transmission is mainly due to a surface effect.

Nevertheless, as oxidation is also observed on the bottom of the sample, where the discharges are not expected to occur, this results tells us that oxidation may not only be concentrated on the top surface. At this stage, it cannot be surely affirmed that the discharge induce chemical ageing in the bulk. It can however be supposed, that bulk degradation can also come from the applied field. In addition, hypothetically, chemical ageing may originate from the top surface and reach the bulk of the polymer, which can then be measured on the other side of the sample.

An analysis of the chemical ageing in the bulk would have been needed to tell if the chemical markers observed are mainly due to the discharge, are also bulk associated.

In a second time, FTIR analysis has been performed on reference and aged PP samples in order to investigate the effect of ageing on the chemical properties of the materials, and study the role of crystallinity in chemical ageing. First, the FTIR spectra of unaged reference PP samples were measured, and the results are shown in figure 4.28. All spectra of unaged and aged PP materials presented in the following section have been normalized using a reference band at 973 cm -1 which is generally used for this purpose in the literature [START_REF] Burfield | The use of infrared spectroscopy for determination of polypropylene stereoregularity[END_REF]. The spectra of unaged PP films share common features. One large peak in the range 2955-2840 cm -1 comprises four contributions according to the literature: peaks at 2955 and 2870 cm -1 can be attributed respectively to asymmetric and symmetric stretching mode of (C -H) bonds in -(CH 3 ) groups. Two other peaks at 2922 and 2840 cm -1 are attributed to the same stretching mode but in -(CH 2 )-groups [START_REF] Leon-Garzon | Experimental Characterization of Polymer Surfaces Subject to Corona Discharges in Controlled Atmospheres[END_REF][START_REF] Morent | Comparison between XPS-and FTIR-analysis of plasmatreated polypropylene film surfaces[END_REF]. Two other intense peaks at 1460 and 1375 cm -1 are attributed to deformation vibration modes of -(CH 2 )-and -(CH 3 ) groups respectively. Finally, smaller peaks are observed in the range 1200-750 cm -1 : peaks at 1167 and 998 cm -1 are partly due to rocking vibrations of -(CH 3 ) groups, or asymmetric stretching of C-C bonds.

The main difference that can be observed between the two materials is a peak at 730 cm -1 , which is attributed to rocking vibration modes of -(CH 2 )-groups. The peak is more intense in the case of PP B . As explained in chapter 2, this polymer is copolymerised with ethylene in a higher concentration as compared to PP A , which is why the crystallinity is lower. Thus, the rocking vibration could come from the PE part [START_REF] Socrates | Infrared and Raman characteristic group frequencies: tables and charts[END_REF].

Spectra of aged PP samples were then analysed. It was found that very similar peaks located in the sub-regions already described in the case of XLPE are also present in both aged PP. FTIR spectra of aged samples share the same backbone as the unaged PP material, but as in the case of aged XLPE, present smaller peaks in hydroxyl (3650-3200 cm -1 ) and carbonyl (1850-1500 cm -1 ) groups regions.

As the intensity of the peaks have changed after ageing regarding the unaged samples, these peaks are investigated in details. In the hydroxyl groups region, a slight difference can be observed between the unaged and samples. The absorbance of the peak at 3345 cm -1 increases with ageing time, for both ageing of PP A and PP B. Besides, absorbance in the case of thermally aged samples does not change significantly with ageing time. Thus, as in the case of aged XLPE, electrical ageing induces the formation of additional hydroperoxydes. Thermo-oxidation coming with increase of hydroxyl groups contribution was reported in the literature in both corona discharge and plasma treated PP films [START_REF] Leon-Garzon | Experimental Characterization of Polymer Surfaces Subject to Corona Discharges in Controlled Atmospheres[END_REF][START_REF] Sellin | Surface composition analysis of PP films treated by corona discharge[END_REF][START_REF] Morent | Comparison between XPS-and FTIR-analysis of plasmatreated polypropylene film surfaces[END_REF].

In the region 1850-1500 cm -1 , similar features to aged XLPE were observed: a broad peak near 1745 cm -1 appears after ageing, which is characteristic of the presence of carbonyl compounds, such as lactones, ketones, esters or carboxyilic acids [START_REF] Morent | Comparison between XPS-and FTIR-analysis of plasmatreated polypropylene film surfaces[END_REF][START_REF] Socrates | Infrared and Raman characteristic group frequencies: tables and charts[END_REF]. The contribution from these compounds is larger in the case of electrical ageing as compared to thermally aged samples, which confirms the effect of applied field to thermo-oxidation of PP. Another peak can be observed at 1630 cm -1 , which could come as in case of XLPE from stretching of (C = C) bonds, or to hydrogen bonding between hydrogen of carbonyl compounds [START_REF] Morent | Comparison between XPS-and FTIR-analysis of plasmatreated polypropylene film surfaces[END_REF]. At this stage, it seems that aged PP B which is less crystalline, present a higher response in these regions.

As discussed previously for XLPE, the increase in the absorbance of these chemical markers traduces an increase in their concentration. The ratio of absorbance between aged and unaged PP samples for the peaks discussed above have been calculated in order to quantify the increase in concentration of the identified compounds. Figure 4.31 shows the evolution of the absorbance ratio for both PP materials. As shown in these figures, the concentration of hydroxyl and carbonyl compounds increases with ageing time, and faster than for thermally aged samples. Comparing PP A and PP B , the concentration seem to increase faster for the later. It is found that material with the lowest degree of crystallinity is found to be more affected by chemical ageing.

As chemical defects are more likely to form and concentrate in the inter-phase between crystalline lamellae (amorphous), the material which is less crystalline could thus be more exposed to chemical ageing (see chapter 1).

These observations indicate that electrical ageing of PP can as for XLPE be associated to the formation of chemical groups induced by thermo-oxidation reactions accelerated by the electric stress. This thermo-oxidation is expected to be mainly due to the material being exposed to surface discharge originating from the high voltage electrode.

Structural evolution

The structural properties, namely the crystalline morphology of the XLPE and PP samples have been studied by DSC. Two temperature cycles were used. The first cycle is a heating of the sample from -20 to 200 • C, which gives information on the melting properties of the material. It is characteristic of the thermal history of the material. The second one is a cooling from 200 to -20 • C, and corresponds to the crystallization of the material.

Before studying the aged samples, the thermograms of unaged reference XLPE were measured. Figure 4.32 shows the thermogram of a reference XLPE sample after a first melting cycle. Polyethylene is a semi-crystalline material with a distribution of crystalline lamellae with different sizes [START_REF] Louis | Polymer Materials : Macroscopic Properties and Molecular Interpretations[END_REF]. As the temperature increases, the smallest lamellae start to melt, and this can happen at temperatures close to room temperature. A large melting peak located at 104 • C corresponds to the melting of the largest and the most numerous lamellae. Finally, an intermediate and small endothermic peak can be observed between 60 and 70 • C: this peak corresponds to the melting of crystalline lamellae which have formed during degassing of cross-linking byproducts at 60 • C.

As both thermo-electrically aged samples and thermal reference have been aged at the same temperature and for the same duration, DSC measurements have been done on both. As both aged samples present nearly no difference in their thermal properties, only electrically aged are compared in the following. Figure 4.34 shows the different thermograms of the aged XLPE samples under 2.25 and 3.2 kV. The thermograms of aged XLPE samples show a progressive effect of the thermal treatment on the melting properties of the material. At the beginning of ageing, a melting peak appears at 90 • C due to annealing of the thin crystalline lamellae. After a few hours (2 to 4%), the secondary peak moves to higher temperatures. Two distinct peaks can still be observed, indicating that two different populations of crystalline lamellae co-exist in the material. After about 6 hours (12.5 % for 2.25 kV and 25 % for 3.2 kV), the peaks due to annealing at 90 • C is roughly stabilized.

Based on the melting thermograms of the aged samples, the relative crystallinity of the materials has been measured in order to attest the effect of thermal treatment. Tables 4 As the ageing time increases, the relative crystallinity increases from 41 up to 46 % after 60h. This slight increase in the relative crystallinity can be explained by the thermal treatment which may induce secondary crystallization in the material. It is reported in the literature that during thermo-oxidation processes, short chain segments produced by the chain scission contribute to the linear increase of crystallinity, by two major secondary crystallization mechanisms: modification of existing crystallites and nucleation of new crystallites [START_REF] Salem | Oxidation and crosslinking processes during thermal aging of low-density polyethylene films[END_REF].

New crystallites can form due to these short chain segment created in the amorphous phase with help of the increased temperature. It has been seen with FTIR that electrical ageing induced significant chemical oxidation as compared to the thermal reference, which could lead to the formation of these short chain segments. However, DSC thermograms indicated that both electrically and thermally aged samples presented similar melting enthalpy. Thus, the increase in crystallinity should be explained by the modification of existing crystalline lamellae.

The thermal annealing at 90 • C may have increased the mean thickness of crystalline lamellae. This lamellae thickness has been calculated using the Gibbs-Thomson relation presented earlier for the two populations of lamellae at 90 • C. It has been found that the thickness of the crystalline lamellae of 90 • C melting peak slightly increases from 5.0 to 6.0 nm.

This increase in crystallinity may by linked to the results obtained in FTIR. Indeed, as the relative crystallinity seems to increase at the very last steps of lifetime, the results were compared to the absorbance ration increase in the case of XLPE in the hydroxyl (3370 cm -1 ) groups. The comparison between the increase of XLPE crystallinity and hydroxyl concentration as a function of ageing time shows that both the increase of properties seem to accelerate at the same lifetime ratio (≈ 115 to 120 %). This characteristic lifetime could be associated to an advanced ageing stage at which both crystallinity and chemical ageing increase are accelerating. It can be proposed that one property may drive the evolution of another. An increase of crystallinity, as explained earlier in this section, may originate from chain-scission which contribute to secondary crystallization by thickening of crystalline lamellae.

However, this increase in crystallinity has also been found in thermally aged samples. Thus, it is difficult to tell at this stage if both properties evolution are correlated.

In a second time, the properties of aged PP samples were characterized by DSC. The DSC thermograms of unaged PP samples (see chapter 2) have been then compared to the aged samples. DSC thermograms of aged samples reveal a melting peak forming during ageing and shifting to 90 • C. This effect is attributed to thermal annealing occurring at 90 • C during thermo-electrical ageing. Besides this observation, the position of the peak is only slightly shifted with ageing. As thermal annealing seems to have modified the morphology of PP A and PP B during ageing, the relative crystallinity of both samples may have changed as well and thus have been estimated. Figure 4.37 shows the evolution of the calculated relative crystallinity of both PP materials as a function of ageing time. Let us note that the crystallinity of PP B has been calculated by measuring the melting enthalpy of the propylene phase only, as the ethylene phase is not expected to contribute to the crystalline morphology during ageing at 90 • C (see chapter 2).

The relative crystallinity, as in the case of XLPE, increases with ageing. This effect is attributed to the thickening of crystalline lamellae by thermal annealing at 90 • C. This lamellae thickening is also attributed to secondary crystallization during thermal ageing induced by chain scissions. The thickening of crystalline lamellae has been studied by SAXS.

In addition to DSC measurements, SAXS were carried out on unaged and thermally aged PP samples, in order to investigate the impact of thermal treatment upon the long period of crystalline lamellar stacks. The scattered intensity I(q) was measured for the different samples. The long period of the materials is obtained by plotting q 2 I(q) as a function of q. All signals present a peak between 0.4 and 0.6 nm -1 . The long period is obtained at the wave vector value corresponding to the peak, by the Bragg law (see chapter 2):

L p = 2π q max (4.1)
The long period has thus been calculated for the four samples, and values are reported in table 4.5.

Sample

Ageing state q max (nm Different features can be seen from these data. First, the calculated long period of unaged materials is larger for PP B than for PP A . Thus, PP B is the material with the lowest relative crystallinity but possesses crystalline lamellar stacks with a higher long period. This could be attributed to a thicker amorphous phase between crystalline lamellae.

Secondly, the long period of both PP A and PP B increases with thermal ageing, by 1.71 and 1.48 nm respectively. The thermal treatment thus increases the relative crystallinity by increasing the mean thickness of crystalline lamellae as discussed in the DSC measurements, and the long period is also slightly increased with ageing.

However, SAXS does not allow to distinguish the difference between the lamellae which belong to the ethylene or propylene phase. Thus, the higher long period observed in the case of PP B could be attributed to both PE and PP crystalline lamellae.

The aged PP materials thus present a changed crystalline morphology as compared to the unaged samples, which could affect both physico-chemical and electrical properties. These results are coherent with DSC measurements. In order to investigate the evolution of electrical properties, the breakdown strength of unaged and aged materials have been measured.

Effects of ageing on the electrical properties

Investigating the effect of ageing on the electrical properties can be done by different techniques and experimental ways depending on the targeted markers that need to be studied. In this work, we focused on studying the evolution of the resistance to applied field, namely the breakdown strength, as it is a physical property of the material which evolution may be linked to important changes in its chemical and physical structure. Moreover, studying leakage current through current-voltage characteristics is a way to study the electrical resistivity of the material, thus its physical properties as an insulator. Other characterization techniques could have been used, such as dielectric spectroscopy, which aims to study the dielectric properties of the materials in the linear regime. Preliminary measurements in broadband spectroscopy were performed on aged and unaged XLPE films, however the obtained spectra show low values of loss angle and did not show significant differences between the samples in the sensitivity range of the experiment. Due to difficulty to obtain relevant results, measurements of the breakdown strength and leakage current were preferred.

Evolution of breakdown strength

The AC breakdown strength of the various thermo-electrically aged and reference (thermally aged only) samples have been measured in order to study the effect of thermal and electrical ageing on the ability of the material to resist to electric stress. The breakdown strength has been estimated using a two-parameters Weibull distribution presented earlier in chapter 2. The scale parameter is used as the breakdown strength value (kV/mm) and the shape parameter is the data dispersion. The AC breakdown strength of the unaged XLPE material is estimated to 96.85 kV/mm which is in the range of reported value for XLPE under AC field [START_REF] Ieda | Dielectric Breakdown Process of Polymers[END_REF][START_REF] Li | Effect of acetophenone on the breakdown strength of polyethylene[END_REF][START_REF] Karlsson | Investigation of the dielectric breakdown strength of polymer nanocomposites[END_REF][START_REF] Esbo | Enhanced dielectric AC breakdown strength of polyethylene using voltage stabilizing additives[END_REF]. The calculated breakdown strength of the aged samples does not change significantly during the lifetime of the material. Only at the very end of ageing steps (112.5 %), the breakdown strength starts to decrease to 74.02 kV/mm. At 126 % of expected lifetime, the breakdown strength has decreased to 68.01 kV/mm. The shape parameter also seems to decrease, as the slope of the Weibull distributions is decreasing with increasing ageing stage.

The same experiment has been done on XLPE samples aged under 3.2 kV, and on thermally aged samples as well. The evolution of the breakdown strength of the aged and reference samples as well as the shape parameter have been drawn as a function of ageing time in figures 4.40 and 4.41. The evolution of β is hardly linked to the ageing time. β seems to slightly decrease with ageing time, which indicates that the data dispersion tends to increase.

Figure 4.40 shows that the breakdown strength of thermally aged samples does not change significantly with ageing time. Thermal treatment only, in the applied ageing duration range, thus has nearly no measurable effects on the electrical properties of XLPE. In addition, ageing under 3.2 kV shows no major effect on the breakdown strength of XLPE samples with ageing time. However, pre-stressing at 2.25 kV seems to accelerate the weakening of the material, as the breakdown strength drops steeply in the ultimate moments of the sample's life.

The decrease of breakdown strength can have different origins. The observed changes in the physico-chemical properties of the aged XLPE materials can be correlated to this effect: indeed, the presence of surface erosion on the aged samples is a marker of structural degradation, thus facilitating the final breakdown when submitted to high fields. The increase in the carbonyl and hydroxyl groups concentration may also influence the electric properties of the material, by changing the injection threshold of charges. Newly introduced chemical impurities may also act as defects which weaken the electronic structure of the material [START_REF] Chen | Electronic Structure of Polyethylene: Role of Chemical, Morphological and Interfacial Complexity[END_REF].

To have a better insight at the obtained results, the evolution of XLPE C breakdown strength as a function of ageing time has been plotted and compared with the increase in absorbance of hydroxyls groups from FTIR measurements. It can be seen in figure 4.42 that the decrease in breakdown strength is correlated with the steep increase of hydroxyl concentration with ageing time. It may thus exists a correlation between the measured ageing markers. At the very last ageing step of the insulation, the concentration of active chemical groups increases steeply, while in parallel its breakdown strength decreases steeply. The evolution of both properties seems to accelerate around ≈ 115 to 120 % of the expected lifetime.

According to the observed results, we could thus propose the following ageing scenario, illustrated in figure 4.43: as the material is ageing toward its end of life, the electric field, which in that case is associated with discharges, induces structural and chemical changes in the materials properties, to a critical point at which these properties changes drastically. Before that point, the observed ageing markers indicate that no critical damage may have been formed in the polymer.

After that point, these properties change radically, to such extent that the breakdown strength may drop until reaching the electric stress at which the material was aged.

If the damage concentration has weaken the insulation to a critical point, failure may occur. The critical inset for such damage stage could be linked to a number of AC cycles seen by the insulation (see discussion in chapter 3). Such scenario could also be linked to the decrease of the injection barriers for electrons due to ageing. In order to attest the effect of ageing on the behavior of charge carriers, leakage current measurements have been done on aged samples, and the results are discussed in the next section.

The AC breakdown strength of PP samples has also been measured. The breakdown field of unaged samples has been compared to the field of aged samples as a function of ageing time. The AC breakdown strength of the unaged PP A is estimated to be 156 kV/mm. This value is much lower than AC breakdown strength reported in the literature for PP, which is higher than 300 kV/mm [START_REF] Sari | Kungliga tekniska högskolan, and Institutionen för fiberoch polymerteknologi[END_REF]. This difference may be due to different experimental conditions, as well as possibly different PP grades. The breakdown strength of PP B has been estimated to 114 kV/mm. The breakdown strength of PP A is higher than PP B . The difference is attributed to the crystalline morphology of the two materials: the higher relative crystallinity of PP A may decrease defects concentration as they are known to accumulate in the amorphous regions of the material, which is also where electrons are expected to be involved in conduction under applied field (see chapter 1). Moreover, lifetime of PP B was measured shorter than PP A , and the effect was attributed to the same reason (see chapter 3).

Regarding the effect of ageing, the breakdown strength of both PP materials decreases with ageing time under electrical ageing. The effect of electrical ageing is stronger in the case of PP B , as the initial breakdown strength is nearly divided by two between the first and the last ageing steps. As compared to XLPE, in the case of PP electrical ageing progressively decreases the breakdown strength, but the decrease does not accelerate at the very last ageing steps. The lower degree of crystallinity of PP B seems to accentuate the loss of properties of the material, which again highlights the link between insulating properties and crystallinity.

Effects on leakage current (current-voltage characteristics)

In addition to breakdown strength measurements, the leakage current has been measured in aged XLPE (and not PP materials) samples as a function of ageing time. The current has been measured using a specific measurement bench described earlier in chapter 2. As a reminder, the test samples are films of 100 % in diameter, metallized on both sides by Gold-Paladium sputtering.

The measured current intensity decreases exponentially with time and stabilizes after a few seconds. The current value measured after stabilization is the leakage current value noted for the applied given voltage. For each applied voltage, the current in steady state (after 300 sec relaxation) is considered in what follows.

As described in the state of the art, the leakage current follows different laws depending on the applied electric field. At low fields, i.e. below 10 kV/mm, the current follows Ohm's law and is directly proportional to the applied field. Above a threshold voltage, the current is proportional to the square of the field (injection regime). Depending on the material and on temperature, the transition between these regimes is triggered at different threshold voltages. In order to have a better insight in these different regimes, the leakage current data are plotted as a function of applied field, to give rise to "current-voltage characteristics" (I = f (V)). At even higher fields, the leakage current follows a different trend as the slopes decrease again. The temperature increase also affect this change and facilitate the transition to the new current-voltage regime. For the aged samples, the same effect can be seen when increasing the measurement temperature. However, the first transition from Ohmic behavior to the injection regime is not clearly visible. Ageing may have introduced impurity states in the material as seen in FTIR, which could modify the Ohmic conduction regime. Above roughly 20 kV/mm, the slope of the curves decreases as in the case of unaged XLPE samples.

The transition between Ohmic behavior and intermediate conduction regimes corresponds to the onset of charge injection from the electrode. Injection onset leads to an increase of the charge carrier density in the material, and thus increase the leakage current. As the temperature increases, injection and conduction of these charges is facilitated which in turn increases the current.

The current-voltage characteristics thus inform on the conductivity of the material at a given applied field and temperature. The leakage currents have been measured on aged XLPE samples in order to investigate the effect of thermo-electrical ageing on the charge injection and conduction properties. In the following, the current is measured at 50 • C only. The results of aged samples indicate that ageing has an impact on the leakage current. Above 20 kV/mm, the current values of aged samples are larger than those of the unaged material for a given applied voltage. The value of leakage current at a given voltage increases with ageing time. The transition from medium to high field regimes also occurs at lower applied field for the oldest aged materials.

One can also note that around 60 kV/mm, the current significantly increases for the sample aged 60h (126 %) under 2.25 kV. This current increase precedes breakdown event. It is not observed for every samples as the experimental bench voltage is limited to 12 kV, and the sample are 130 μm thick in average. The leakage current values of the sample aged 30h (125 %) are similar to the one aged 60h (126 %) under 2.25kV: as they both corresponds to an advance ageing state of the material, we can suppose that the charge carrier behavior of the material is directly affect by electric ageing, and that leakage current may increase during its lifetime until breakdown. Current-voltage characteristics have also been measured for thermal reference XLPE samples, in order to decorrelate effects of electrical and thermal ageing. The results showed in figure 4.54 and 4.55 that the leakage current increases exponentially with ageing time in electrically aged samples under 2.25 kV. The measured current for thermally aged samples increases as well but more slowly that electrically aged ones. After ageing under 3.2 kV, the leakage current grows exponentially with ageing time as well, but at the same ageing time the leakage current of thermally aged samples has not increased significantly.

The effect of electrical ageing on the leakage current measurements can be explained by the changes in the chemical and physical structure observed in SEM, FTIR and DSC. The progressive modification of the material, with incorporation of chemical defects and thickening of the crystalline lamellae can have weaken the energy barrier for charge injection, thus increasing the charge carrier density [START_REF] Chen | Electronic Structure of Polyethylene: Role of Chemical, Morphological and Interfacial Complexity[END_REF]. The thermal treatment seems to have an impact at ageing time corresponding to the end of life of the material under 2.25 kV. As thermal treatment should only impact the crystalline morphology, it is rather clear that the modification of the polymer crystalline morphology directly affect its conduction properties. However, the electrical ageing have a more significant impact as it is shown by ageing at a higher applied voltage.

It is also important to mention that the current measurement is done by measuring the average current over the whole surface (100 mm in diameter) of the sample. Yet, the damage zone corresponds to only a small fraction (≈ 10 %) of the total material surface, but the observed results are still significant. This indicates that damage concentrated in a small region of the sample can contribute to measurable changes in the properties of the whole material. Several features can be observed in this figure. First, the intensity of the emitting signal increases with excitation wavelength. As the emitting light intensity does not change, this feature is due to the light absorbed by the material and re-emitted. The light emission efficiency, or quantum yield, is driven by the ratio of emitted light over absorbed light as described in the following equation [START_REF] Qiao | Electrical ageing of insulating polymers: approach through electroluminescence and cathodoluminescence analyses[END_REF][START_REF] Blasse | Luminescent Materials[END_REF]:

Y q = N p em N p abs (4.2)
where, N p em and N p abs are the number of photons emitted and absorbed respectively. Light is emitted through fluorescence by molecules or atoms composing the material. The material absorbs the energy of emitting photons in order to undergo an energy transition from ground state S 0 to an excited state S 1 . If the energy of the absorbed photons is larger than the required energy for S 0 → S 1 , not all the energy is converted to light emission and some of can be involved in non-radiative relaxations (heat for instance). Absorption may thus be more efficient at higher wavelength (lower energy).

The PL spectra also shows several emission peaks. Some peaks result from the materials, and some are features due to emitting source: the peaks at 300 and 290 nm are due to the emitting source at these respective wavelengths. In the same way, emission at these wavelength generates second harmonics which gives the observed peaks at 600 and 580 nm [START_REF] Blasse | Luminescent Materials[END_REF].

The other peaks are due to emission from the material it-self. Two main peaks can be observed, especially at wavelength longer than 270 nm where the signal is better: a peak around 315-325 nm and another around 370-380 nm. Both peaks may be characteristic of XLPE luminescence signature. Teyssedre and Laurent studied the photoluminescence signature of different PE based materials including XLPE and identified reference bands attributed to bulk XLPE emission at 370 and 330 nm [START_REF] Teyssedre | Semi-quantitative analysis of photoluminescence in thermoelectrically aged cables: I-identification of optical signatures[END_REF].

Therefore the observed peak on this reference XLPE sample are attributed to emission from XLPE bulk. The peak at 315 nm also generates second harmonics giving the peaks observed at 630 nm.

In addition, a broad shouldering centered around 450 nm can be observed on the spectra. A shouldering at 440 nm is expected in bulk XLPE according to the literature, but other features seems to be involve in light emission between 550 and 450 nm. Antioxidant (AO) can also act as a chromophore and be involved in light emission: light emission around 400 and 410 nm has been attributed to AO diffusion in XLPE based cables [START_REF] Teyssedre | Semi-quantitative analysis of photoluminescence in thermoelectrically aged cables: I-identification of optical signatures[END_REF]. It is therefore possible that AO be responsible for a part of emitted light in this broad shouldering.

Data analysis of PL spectra is rather complex. However some signatures of bulk XLPE and additives have been identified. Light emission from the reference XLPE sample has then been compared to emission from XLPE samples electrically aged. For more clarity, only three excitation wavelengths have been chosen to compare the features of aged and reference samples: 240, 270 and 300 nm. Results are shown in figures 4.57 and 4.58. Both aged samples are above their expected time for breakdown respectively. As compared to the reference sample, the spectra of aged samples share similar features: the intensity of the main peaks observed at 325 and 380 nm has decreased, whereas the intensity of emission in the shouldering around 450 nm has increased. The decrease in intensity is due to a quenching of quantum yield, indicating either more absorbed or less emitting photons. The increase in quantum yield at around 450 nm indicates that already present emitting species may be involved in light emission. The broad shouldering is centered around 485 nm for the sample aged 60h under 2.25 kV. Another rather small feature is visible around 550 nm as quantum yield seems to have increased with ageing.

In order to have a better insight on the effect of ageing on PL spectra, the ratio of intensity of the different observed peaks over the intensity of the peak at 325 nm are calculated, as this peak has been attributed to bulk XLPE. Spectra obtained at 300 nm makes the interpretation easier, hence the following calculation is based on results observed in figure 4.58. The results are given in table 4.6. The ratio of intensity for the peak at 380 nm confirms the quenching of emitted light at this wavelength for both aged samples. As this peak is also attributed to bulk XLPE, a quenching indicates that the concentration of the main components involved in light emission from the bulk has decreased with electrical ageing.

On the other side, the opposite effect is observed in the case of features at 485 and 550 nm: at 485 nm, the intensity of emitted light first slightly decreases before increasing to its double. At 550 nm the effect is even more important as the intensity ratio increases from 6.5 to 23 %. According to the literature, a shouldering around 450 nm in LDPE and around 475 nm in XLPE has been in attributed to the presence of carbonyls and unsaturated carbonyls of enone type [START_REF] Teyssedre | From LDPE to XLPE: investigating the change of electrical properties. part II: luminescence[END_REF]. Therefore, an increase of light emission in this spectral domain indicates could indicate an increase of concentration of these compounds. As FTIR analysis revealed the effect of ageing in the increase of carbonyl concentration, the observed feature could be attributed to chemical ageing.

The observed increase at 550 nm is rather difficult to interpret, as it is not clear rather it belongs to the shouldering at 485 nm. However, a similar feature has been observed in thermally aged XLPE cable samples at 540 nm and attributed to thermal ageing [START_REF] Teyssedre | Semi-quantitative analysis of photoluminescence in thermoelectrically aged cables: I-identification of optical signatures[END_REF]. In our case, as both electrically and thermally aged samples have been aged at 90 • C, the effect could come from the discharge: bombardment of the surface by highly energetic electrons may have created damage similar to long term thermal ageing such as thermo-oxidation, as observed by FTIR.

Analysis of PL spectra of aged XLPE samples hence gives an insight of the effect of electrical ageing on light emission. PL analysis is not as precise a FTIR for determining species involved in ageing, but is an important complement in order to have a better understanding of the ageing mechanisms involved.

Recombination of charge carriers

In addition to PL, aged XLPE samples have been analysed using plasma or recombination induced luminescence (PIL/RIL). First, the sample is placed in the dedicated room under vacuum. Helium is injected in the chamber at atmospheric pressure, and liquid nitrogen is used in order to cool the sample to - After 5 seconds, the discharge is stopped, the window is shut and the detector is opened in photon counting mode using a PMT. The PMT counts the number of emitted photons every 2 seconds. Few seconds after the end of the plasma discharge, the number of detected is around 10 6 per seconds and starts to decrease with time. The intensity decrease is different at the very beginning of the experiment and a few seconds after. Three main mechanisms are involved in this plasma induced luminescence: phosphorescence, chemiluminescence and recombination induced luminescence. All three have a specific relaxation kinetic which give rise to three different domain for luminescence decay. Data of the photon counting mode have been fitted using a specific double exponential model described in equation 4.2.3.2 [START_REF] Teyssedre | Spectral analysis of optical emission due to isothermal charge recombination in polyolefins[END_REF]:

I(t) = A 1 × exp -t τ 1 + A 2 × exp -t τ 2 + A 3 × t -m (4.3)
with, I(t) (cps) the light intensity at time t (secs) after discharge, τ 1 (sec) is the relaxation time of photo-induced excited states of the material (phosphorescence), τ 2 (secs) describes the kinetics of chemiluminescence. A 1 to A 3 are pre-exponential factors and m is a constant describing the kinetics of light emission from charge recombination.

Phosphorescence is the shortest mechanism involved, in that case τ 1 = 2 secs. After extinction of the first contribution, chemiluminescence quenches as well with τ 2 = 5.6 secs. Finally, after 35 seconds the intensity of the light emitted through charge recombination overpasses the other mechanisms, and remains for several minutes. The measure is stopped after 300 seconds. m is 0.9.

The photo counting measurements tell us that light emitted by the sample after plasma discharge originates from a combination of different emission mechanisms. The same experiment was performed a second time, but this time the PMT was shut and the CCD camera switched on for measuring the spectral components of light emission. Spectra of the emitted photons were measured at different time of light emission after the plasma discharge, from a few seconds to a few minutes. Figure 4.61 shows the spectra of photons emitted between 1 and 5 seconds after plasma discharge. The PIL spectra of the reference XLPE samples shows a broad peak in the near-UV region (450-400 nm). The peak intensity decreases from 1 to 5 seconds, which is directly linked to the photon emission decay observed previously in photon counting mode. Another observable feature is that peak position shifts with time: the peak is positioned at 436 nm after 1 second, and shifts to 410 nm after 5 seconds. As phosphorescence is expected to decay after 2 seconds, this shift is attributed to the onset of chemiluminescence. Other spectra have been measured after longer time decay, as shown in figure 4.62. After 50 seconds, the shape of the luminescence signal has changed: the broad peak positioned around 410 nm has shifted to 500nm with a shouldering around 420 nm. This energy shift coincides with the quenching of chemiluminescence and the onset of luminescence from charge recombination. The shouldering is undoubtedly caused by the remaining photons from chemiluminescence. After 110 seconds, the peak is sharper and luminescence is attributed only to charge recombination. The energy shifts are represented more clearly in figure 4.63, where some of the measured spectra taken at different moments after the discharge have been normalized using peak's maximum. The spectral shift observed in this figure highlighted a shift of light emission mechanism:

• Between 1 and 5 seconds, the main contribution is around 420 nm and is attributed to phosphorescence.

• Then the emitting wavelength shifts towards 400 nm after 15 seconds. Until 40 seconds the main contribution is attributed to chemiluminescence. A shouldering appears at 500 nm indicating the onset for charge recombination.

• After 110 seconds the spectra main contribution is charge recombination with an emitting wavelength of 507 nm

As in practice the contribution from charge recombination can be decorrelated from the others, the spectra measured after 110 seconds of electrically aged samples have been compared to the thermal reference in order to investigate the effect of ageing upon charge recombination. Figure 4.64 shows the normalized spectra of aged XLPE samples under two different voltages taken 110 seconds after the plasma discharge, and compared to the thermal reference. The comparison of RIL spectra of aged and reference XLPE sample shows mainly two features. First, the position of the main luminescence peak is slightly shifted to lower wavelengths with ageing, with a peak positioned at 507 nm for the reference and a peak positioned at 485 nm for the sample aged 60 hours (126 %) under 2.25 kV.

Secondly, the width of the peak (full width at half maximum) slightly changes with ageing, as a shouldering seems to appear at 600 nm after ageing 60 hours (126 %) under 2.25 kV. This two main features are depicted quantitatively in table 4.7. Interpretation of these results are rather complex as the change is may not be considered significant, but a tentative explanation can be proposed as follows.

The peak position shift indicates that the mean energy of the emitted photon upon charge recombination has slightly increased, from 2.45 to 2.56 eV. Values reported in the literature for recombination induced luminescence of PE indicate a characteristic peak at 500 nm with shoulders at 480, 530, 550 and 585 nm [START_REF] Teyssedre | Spectral analysis of optical emission due to isothermal charge recombination in polyolefins[END_REF]. The emitting species in that case, or recombination centers, are usually considered as possibly being the same as in PL. In fact, RIL and PL spectra share a common feature: a broad peak centered around 485 nm in PL and another broad peak at 485 nm for aged XLPE. The light emission attributed to carbonyl or unsaturated compounds in PL could act as recombination centers, as they have been identified as deep trapping centers (near 1 eV from the band edge for both electrons and holes) in PE [START_REF] Chen | Electronic Structure of Polyethylene: Role of Chemical, Morphological and Interfacial Complexity[END_REF][START_REF] Teyssedre | Spectral analysis of optical emission due to isothermal charge recombination in polyolefins[END_REF]. Thus, the slight shift observed with ageing could be attributed to these compounds and the increase in energy could come from the increase in concentration of these compounds and the creation of new trapping centers closer to the band edge.

Both PL and RIL measurements thus open the reflection upon the effect of chemical and physical ageing on recombination centers in the band gap of the insulating material.

Conclusion

In this section, electrical ageing mechanisms of XLPE and PP have been studied using a multi-scale characterization strategy. Samples aged under increasing ageing states towards breakdown were characterized in order to highlight the presence and evolution of specific ageing markers. The main identified markers are the following:

• Surface damage: SEM has shown that ageing leads to a progressive erosion of the sample's surface with ageing, evolving from craters of a few microns wide to pits and eventually treeing. Similar observations were done on PP samples. However, the less crystalline material shows a different degradation induced by electrical discharge which seems to come with a heating of the sample's surface, attributed to the lower relative crystallinity. The diameter of the eroded zone does not change much during ageing.

• Chemical: FTIR has shown that ageing induces an increase in concentration of hydroxyl, carbonyl and unsaturated compounds with ageing, indicative of a thermo-oxidative accelerated process triggered by the electric stress. During the last ageing step prior to breakdown, the concentration increases exponentially. PL showed that the increase in carbonyl compounds comes with a quenching in the emission from the bulk XLPE. The same chemical markers were observed in the case of PP, but the evolution of the thermo-oxidative markers increases faster in the case of PP B , which has a lower crystallinity.

• Structural: DSC measurements showed that the thermal treatment induces a secondary crystallization which increases the relative crystallinity due to an increase of the mean thickness of crystalline lamellae. This may change the crystalline morphology, as SAXS confirmed that the long period of lamellar stacks in PP had increased with ageing time, confirming a thickening of crystalline lamellae. The electric field has no measurable effect in itself.

• Electrical: Breakdown strength measurements have shown that breakdown occurs earlier on pre-stressed XLPE samples, and that at the end of the sample lifetime breakdown strength drops towards the value of the ageing field. Breakdown strength of both PP samples has also been found to decrease linearly with ageing time. Again, a larger effect has been observed in the case of PP B , which may originate from the degradation observed on the physicochemical properties. Current-voltage characteristics have shown that ageing may facilitate charge injection and conduction by reducing the energy barriers involved for these mechanisms.

• Electronic: PIL measurements seem to indicate that the concentration of recombination centers increases during electrical ageing and the energy involved in charge recombination slightly increases. New deep trapping centers may have been created during ageing.

In this chapter, ageing of XLPE and PP materials under a hemispherical-plane electrode configuration revealed the apparition of similar ageing markers at different scales. These ageing markers are characteristic of a progressive degradation, and in some cases an acceleration of degradation at the near end life of the material.

The observed degradation is mainly attributed to the effect of discharges triggered from the HV electrode, which contribute to clear surface damages with characteristic features. Even though, this degradation is believed to induce sufficient damage, as compared to samples aged thermally, to trigger electric and electronic effects, which may precipitate the material to its failure. Also, ageing of two PP samples with different crystalline morphology revealed that a lower relative crystallinity leads to a higher concentration of ageing markers and a faster degradation.

A degradation scenario under localized field coupled with surface discharges can thus be proposed: the surface discharges induced by the field leads to a bombardment of the material's surface with highly energetic ions, which in turn initiates structural damage of the material and thermo-oxidation. The thermal treatment modifies the crystalline morphology of the material during its life, which may help or trigger the formation of chemical and physical defects by allowing the polymer chains to reorganise. This effect may be enhanced in the case of a lower crystalline material. The created defects during ageing are believed to lower the energy barrier for charge carrier injection and conduction, leading to an increase of the charge carrier density in the material. Thus, the electrical breakdown of aged samples is expected to come earlier due to the weakening of the physico-chemical and electric properties of the material.

A parallel with electrical ageing inside embedded cavities of polymeric insulation can be made, and these results highlight the importance of controlling the stress applied on the material, while helping to understand the different ageing steps leading to breakdown.

Chapter 5

Study of electrical ageing in plane-plane electrode geometry

The previous ageing study of XLPE samples in the sphere-plane geometry have shown in which way a localized electric field impacts the lifetime and the properties of the material. Different ageing markers were identified, such as surface degradation, oxidation and increase of leakage current. However, as the surface discharges generated in this geometry are likely to influence the ageing of the material and its lifetime, it is difficult to interpret the impact of the electric field only on the intrinsic ageing inside the material. Studying the evolution of ageing markers in between parallel plane electrodes can thus help us understand the intrinsic breakdown of polymeric insulators.

To do so, XLPE samples in plane-plane electrode geometry (Rogowski-like electrodes), were used as test samples for an ageing study in homogeneous field. In this geometry, the field is expected to be homogeneously distributed in the bulk of the material, without field intensification at edges (see chapter 2). Lifetime curves in this geometry were obtained and discussed in chapter 3. The main results indicated that a longer lifetime is expected for materials in the plane-plane geometry, as the degradation is not supposed to be concentrated but statistically distributed in the bulk of the material. However, it was also shown that discharges may still occur in the vicinity of the counter electrode, due to the imperfect contact between the polymer and the counter-electrode.

The test samples were aged under an applied ageing voltage for different time steps from unaged to near breakdown. After ageing, the samples were characterized in order to study the evolution of ageing markers during ageing. The aim is to identify ageing markers responsible for the degradation of XLPE in this plane-plane configuration, and finally compare them to the ageing markers observed in the sphere-plane geometry from chapter 4.

Ageing steps

Based on the lifetime curves obtained for XLPE under plane-plane electrodes (see chapter 3), the voltage 2.8 kV has been chosen for the ageing study. At this voltage, a relatively well defined lifetime is obtained, which allows to select intermediate ageing steps. At higher voltage values, breakdown events are more distributed, thus it is difficult to obtain several samples aged to the same stage. Figure 5.1 illustrates the lifetime curve of XLPE and the expected lifetime for the chosen ageing voltage. Under this specific applied stress, the expected characteristic lifetime of the samples is around 63 h. Samples have then been aged for different time stages, corresponding to fractions of the expected lifetime of the material. As compared to samples preparation in sphere-plane geometry, processing of XLPE samples in this geometry is more challenging. Moreover, breakdown events are more randomly distributed, which made difficult to obtain the same number of aged samples as in the sphereplane study. Here for 5 to 6 samples have been aged to each ageing stages. In addition, 1 to 2 samples were aged at advanced ageing stages, longer than the expected lifetime. Table 5.1 represents the chosen ageing steps at this ageing voltage. The two advanced ageing stages (in bold) have been used in order to approach the final stage of ageing in the material. For these stages, a fewer number of samples could been aged.

In the next subsection, the results of the different characterizations are presented.

Characterization of ageing in plane-plane geometry

As explained above, it was challenging in this geometry to obtain series of samples aged at the same ageing time, as breakdown events were more difficult to predict, especially at ageing times above the expected lifetime. A choice had to be made on the characterization techniques to use with the limited number of samples.

The impact of ageing on the surface aspect of XLPE samples was analysed using SEM, as it revealed key degradation markers for the sphere-plane geometry, even at short ageing steps. The evolution of the physico-chemical properties was then studied by FTIR and DSC. Breakdown strength could not be measured on the aged samples, as it requires at least 7 to 8 samples to be comparable to the data obtained previously. Leakage current measurements require large surface samples (≈ 100 mm wide), which would have required larger ageing electrodes, and thus were not performed. Luminescence measurements were performed on a few samples which were aged at ageing time longer than the statistical lifetime value.

Samples preparation

After ageing, the aged samples were peeled off the electrodes before characterizations. It was found that the sample was bonded to the test electrode. This might be due to the cross-linking process, during which chemical bonds with the aluminium electrode may form. Peeling is made by gently pulling the sample by hand, which comes out relatively intact.

The peeled sample consists of a disk of approximately 20 mm in diameter, in average 100 μm thick, and is considered to be aged under a homogeneous field. Both the electrode side ("top") and counter-electrode side surfaces ("bottom") were characterized. Unaged (reference) samples were also prepared under this configuration, peeled and characterized identically to the aged samples, but without any previous thermal or electrical treatment. The surface aspect of aged XLPE in plane-plane geometry shows various features. The top surface of aged samples seems not affected by electrical ageing, as the observed microstructure is similar to the reference sample. Surface abrasion, originating from peeling, is visible on every aged sample.

On the bottom surface however, some patterns are visible on the aged samples. In some parts of the surface, craters and pits are found, similar to the one observed on XLPE samples aged under sphere-plane geometry. In other parts, liquid-like features such as lumps are observed (after 21 and 31 hours for instance). These features are concentrated in isolated areas of the aged surface. In other areas, cracks and crazes of a few μm wide can be observed after 50 hours. The surface aspect of samples thermally aged only does not change during ageing, which means that the observed features are formed in the presence of the electrical stress, as shown in figure 5.5. In the figure 5.5, it is clear that the damage observed on the aged sample is characteristic of a degradation induced by the electrical stress. "Lumps" are characteristic of liquid-like material which could be a mixture of oxidized material (carboxylic acids) forming droplets on the surface of the material [START_REF] Morshuis | Partial Discharge Mechanisms -Mechanisms leading to breakdown, analysed by fast electrical and optical measurements[END_REF][START_REF] Morshuis | Degradation of solid dielectrics due to internal partial discharge: some thoughts on progress made and where to go now[END_REF].

Two additional ageing steps have been independently performed on the XLPE samples. These time steps are considered as advanced ageing stages as they are longer than the expected lifetime of the material in this geometry. The surface aspect of aged samples after 72 and 85 hours under 2.8 kV at 90 • C is shown in the figure 5.6. The top surface of aged XLPE samples after 72 and 85 h is not quite different of the preceding ageing steps. The surface abrasion is slightly larger after 85 h, but is still attributed to mechanical peeling of the material from the electrode. The aspect of the bottom surface shows surface erosion characterized by pits of a few μm wide after 72 h. Lumps have been observed as well. For the sample aged 85 hours, a different degradation feature is visible, which looks like surface erosion. The surface appears molten as well. No lumps or crazes are visible in that case.

Oxidation may be due to the strong electric field between the surface of the material and the counter electrode. As the contact is no perfect (see chapter 3) it might be expected that micro-voids filled with air are distributed at the polymer-metal interface. Discharges could be triggered in such micro-voids, which would explain the presence of the observed structural damage.

It has sometimes been proposed that mechanical stress (electrostriction) may have an impact [START_REF] Crine | Role of electrostriction on the electrical aging of polymers[END_REF][START_REF] Lewis | Polyethylene under electrical stress[END_REF]. However, as already discussed in chapter 3, these these mechanisms are negligible at field values considered here. However, a hypothetical explanation is that the observed crazes originate from thermal expansion of the polymer. In this electrode geometry, the polymer is blocked under two parallel plane electrode which constrain the material and limit its lateral expansion under the applied temperature. After ageing, as the material cools down it goes back to its initial size without damage (in the thermal ageing case only), being sufficiently elastic. However, if the electric field has damage the material, and its deformation is no longer completely elastic, strain can cause the material to damage, which would result in such characteristic crazes.

The surface aspect of the XLPE samples aged in this plane-plane geometry shares some features with the samples aged under discharges: craters and pits are characteristic of surface bombardments initiated by surface discharges, as well as lumps and droplets. The prepared samples with the encapsulated electrode are only dropped off on the counter-electrode: the electrical contact is expected to be plane as the sample are prepared by hot pressing on a plane plate. But it might be possible that the sample be subjected to some surface discharges on the counter-electrode side, at some isolated areas where small air gaps can form due to the samples surface roughness. These discharges may be involved in the formation of the observed surface degradation.

The evolution of structural degradation with ageing steps is not as visible as in the case of XLPE aged in hemisphere-plane geometry. No clear evolution in the observed surface damage features between 21, 31 and 50 hours is noticed. The damaged surfaces of samples aged under 72 hours and 85 hours appear slightly different as compared to the shorter ageing steps. All these aspects can be due to the fact that ageing under this electrode configuration is less predictable than under hemisphere electrode. As in the plane-plane geometry the bulk of the material is more likely to be involved in ageing that in the other geometry, ageing is more difficult to predict, and increasing the ageing time may not lead to samples with increased ageing stages.

In order to investigate the effect of electrical ageing on the chemical ageing markers, FTIR measurements have been performed on the XLPE samples at different ageing steps. Results are described in the next section.

Characterization of chemical degradation

The chemical structure of XLPE samples has been characterized by FTIR in transmission mode in order to seek for chemical markers during electrical ageing in the same way as in the other studied materials.

In the following, the spectra were normalized by using the same reference peak for PE as before (2019 cm -1 ).

The FTIR spectrum of an unaged reference XLPE prepared using plane electrodes is shown in figure 5.7. As expected, the spectrum of this XLPE material is the same as the other XLPE materials, with a large contributions from polyethylene backbone (-CH 2 and -CH 3 ) and smaller peaks attributed to hydroxyl (3600-3100 cm -1 ), carbonyls (1850-1550 cm -1 ) and ethers groups (1400-700 cm -1 ).

Aged XLPE samples have then been characterized and compared to the reference material. During the measurements, a strong heterogeneity between the various samples was found. Indeed, on a given sample, different spectra are obtained depending on the location of the IR spot, and different spectra are obtained in different samples aged at the same stage. Figures 5.8 and 5.9 illustrate IR spectra obtained on samples aged for 21 hours under 2.8 kV in the plane-plane geometry. By contrast, the thermal reference samples remain homogeneous, as corresponding spectra are the same. In figure 5.8, the spectra of five different electrically aged XLPE samples are shown, and compared to two thermal references. Even though the samples are aged for the same ageing time, a strong difference is observed between the spectra. On three of the aged samples, broad peaks can be observed between 1800 and 1600 cm -1 . These peaks are located at the exact same wave-numbers as those observed on aged XLPE in the sphere-plane geometry: they are attributed to ketones, esters and lactones, as described in chapter 4. On the two other aged XLPE spectra, these peaks are still visible but with a much lower absorbance.

This result illustrates that ageing in this geometry leads to samples with different degradation stages, even though they have been aged for the same time. There is a clear effect of electrical ageing though, as the peaks attributed to chemical ageing are not observed on the thermally aged samples. It is worth mentioning that the peak at 1630 cm -1 observed even in the case of aged XLPE samples n • 1 and n • 4 (in figure 5.8) is not visible in the case of thermal reference, which tells that they still have experienced electrical ageing. Thus, the results show that even using an electrode geometry designed for a homogeneously distributed field, electrical ageing is heterogeneous in the material. Visual observations confirm the presence of damage zones on the samples (white spots of a few mm big). The aspect of these zones is very similar to the observed patterns previously observed in the sphere-plane geometry. The larger absorbance values are obtained when locating the IR spot on these damage zones.

A possible explanation to the observed results may come from surface discharge events occurring on the counter electrode side (bottom surface in figure 5.3), which may be heterogeneously occurring on the material, depending on the contact between the polymer and the metallic electrode.

In order to rigorously analyse the FTIR data, the spectra were measured on as much different samples as available for each ageing time, and on three different spots for each samples. A mean spectra was calculated for each ageing time and plotted for comparison. Figure 5.10 shows the mean FTIR spectra of aged XLPE samples with different ageing time, compared to an unaged reference sample. The mean spectra of aged samples show a significant increase in absorbance in the sub-regions described previously, especially in the region of carbonyls (1850-1550 cm -1 ). A zoom in this sub-region is done to have a better insight at the effect of ageing. Figure 5.11 illustrates the effect of ageing time on the chemical degradation of the XLPE samples, but also the heterogeneous ageing induced by this geometry. The difference between the mean spectra of the samples aged between 21 and 50 h, and the samples aged above 72 h is indeed significant. The mean absorbance of the samples increase from 21 h to 50 h significantly, but the absorbance of the spectra measured after 72 and 85 h of ageing is much lower. Still, as compared to the thermally aged samples, it appears that the electrically aged samples all show a significant increase in absorbance, which highlights the effect of the applied electric field on ageing.

In order to have a better insight at the effects of electrical ageing on the chemical oxidation of the XLPE samples in this electrode geometry, the evolution of the absorbance ratio of electrically and thermally aged samples in plane-plane geometry (noted "P/P") as compared to the unaged sample have been measured. The evolution of the absorbance ratio of XLPE samples aged under 2.25 kV in sphere-plane geometry (noted "Sph/P") is also plotted for comparison. Figure 5.12 shows the evolution of these absorbance ratio for the identified peaks in the hydroxyls (3600-3100 cm-1) and carbonyls (1850-1550 cm -1 ) sub-regions. FIGURE 5.12: Evolution of absorbance ratio (absorbance of aged sample over absorbance of unaged sample) of electrically and thermally aged XLPE in plane-plane ("P/P") geometry, compared to XLPE aged under 2.25 kV in sphere-plane ("Sph/P") geometry, for the peak at 3370 cm -1 corresponding to hydroxyl groups, and the peaks at 1715 and 1630 cm -1 corresponding to carbonyl groups. A zoom in the absorbance of the peak at 1630 cm -1 at the last ageing stage is also given. Error bars are calculated using standard deviation.

Figure 5.12 shows first that the standard deviation of the mean spectra measured previously is huge. It illustrates the tremendous variation between the absorbance of the spectra measured for samples at the same ageing stage.

Concerning the evolution of the absorbance ratio, which is the suspected ageing marker, it increases with ageing time between 0 and 50 hours (33 to 79 % of lifetime) for the three contributions (3370, 1715 and 1630 cm -1 ). The concentration of the identified chemical groups increases rapidly after 33 %, but then the results indicate that samples aged up to 49 and 79 % of lifetime have nearly the same ratio. For samples aged above 72 h (114 % and 135 %), the absorbance ratio decreases significantly. It reaches levels which are closer to the values measured for XLPE under sphere-plane geometry. A zoom at the end of lifetime ratio for the absorbance ratio measured at 1630 cm -1 is shown in figure 5.12. It shows the the standard deviation of samples aged at 110 and 135 % is much smaller than for the materials aged a shorter lifetime.

In this figure, one can see that the samples aged under plane-plane geometry have an absorbance ratio still two to three times higher than the thermal reference, which is a good indicator of the effect of electrical ageing.

The results obtained in FTIR show that chemical ageing in this plane-plane geometry is localised and not homogeneous, even though this geometry was designed to obtained homogeneous field. The differences observed between the spectra at 21, 31 and 50 h and the spectra at 72 and 85 h could be explained by the fact that at least twice more samples were used in the case first cases. As shown previously, a high disparity is observed between the samples aged at a same ageing stages. It is probable that using the same number of samples for the longer ageing time would increase of the mean absorbance of the spectra. Still, the samples aged for 72 and 85 % show a peak at 1630 cm -1 , which is not present in the unaged sample, nor in the thermal references, meaning that they have experienced electrical ageing due to the applied field. As compared to the spectra measured in sphere-plane geometry, the mean absorbance of the spectra of the samples aged for 31 and 50 h (corresponding to 49 and 79 % of expected lifetime respectively) is twice bigger than the measured absorbance for samples aged at 125 % of lifetime in the sphere-plane geometry (see 4.20 in chapter 4). Thus, in this geometry the measured chemical degradation is higher for a shorter expected ageing stage. Under this plane-plane geometry, a larger volume of the sample is aged under the applied field, which could increase the probability of damage to occur.

Comparing the effect of the two geometries, the kinetics of absorbance ratio in the sphere-plane geometry was indicative how a progressive chemical degradation during the lifetime of the materials. In the plane-plane geometry, the effect is much less significant, but is probably biased due to the heterogeneity of the ageing degradation. The effect observed in both geometries indicate two different mechanisms: in the case of a strongly located field, electrical stress induces a progressive damage of the insulation, which inevitably leads to breakdown after a critical ageing stage which have been discussed in chapter 4. In the second case, electrical damage is driven by highly statistically distributed damage event, partly due to the higher surface and volume of material involved in ageing.

Nevertheless, the use of a plane-plane geometry did not seem to induced a homogeneous ageing as predicted, but in the contrary leads to a material's lifetime less predictable and driven by intrinsic mechanisms. It is worth mentioning that the contact between the polymer and the electrode could have been better by sputtering a metal on the surface of the sample. This preparation have been tried on some of the samples, but instead gave rise to additional problems: the breakdown of the polymer was systematically happening at the edge of the metallization, possibly due to field enhancement. In addition, this process may bring additional charge carriers to the polymer samples. Thus, the samples were used as prepared without metallization.

In addition to the structural and chemical ageing characterizations, the evolution of the thermal properties of XLPE samples aged in plane-plane geometry have been measured by DSC, and the results are discussed in the next section.

Structural modification

The crystalline morphology of the unaged and aged XLPE samples has been studied by DSC. First of all, the unaged reference XLPE material has been studied after degassing at 60 An aged XLPE sample have then been characterized and compared to thermally aged reference. Figure 5.14 shows the thermograms of an XLPE sample aged for 50 hours (79 % of lifetime) under 2.8 kV at 90 • C, and a sample aged for the same time at 90 • C without electric field. In these thermograms, both aged and reference samples show an intermediate melting peak between 94 and 98 • C. This peak corresponds to the melting of crystalline lamellae which have formed due to the thermal ageing at 90 • C. The melting enthalpy of electrically aged XLPE and its thermal reference are nearly equal within experimental uncertainty. Thus, this effect can be attributed to thermal ageing, independently of electrical ageing. In the following, only the thermal properties of electrically aged samples are presented and discussed. Figure 5.15 shows the thermograms of the unaged and aged samples for the different ageing times. The effect of ageing upon crystalline morphology can be expressed by an increase of the relative crystallinity of the material. The crystalline morphology of the different aged samples has been measured as a function of ageing time. Values are reported in table 5.2. The relative crystallinity these samples slightly increases with ageing, from around 44 % to around 47 %. This increase is attributed to the secondary crystallization induced by the thermal treatment at 90 • C. This effect is similar to the one described in chapter 4 for the case of XLPE films. The thermal treatment allows an intermediate population of crystalline lamellae to grow, which leads to an increase of the mean lamellae thickness, and thus relative crystallinity.

The change in the crystalline morphology during ageing is competing with the creation of physical and chemical defects due to thermo-oxidation, which may preferentially locate in the amorphous phase. Thus a change in the relative crystallinity may induce a redistribution of crystalline boundaries, and of the created defects in the bulk, which could affect the electrical properties.

As a complement to these structural and physico-chemical characterizations, the properties of the aged samples under electrical characterization would have been helpful for understanding the effect of ageing in this geometry. Unfortunately, as discussed earlier, we were not able to measure the breakdown strength of XLPE with ageing under sphere-plane geometry, in the time of the thesis, due to the difficult preparation of the samples and the relatively high number of aged samples necessary for the breakdown strength measurements. The main priority was to obtain experimental results on the lifetime of the samples in this configuration, and on characterizations methods available without consuming lots of samples. All together, the DSC results are similar to those observed in the case of the sphere-plane geometry, which confirms that the observed effects are due to thermal ageing.

Chemical and electronic ageing measured by luminescence

As in the case of XLPE films in sphere-plane geometry, the effect of electrical ageing over the light emission has been investigated on XLPE samples aged under planeplane geometry. At the time of measuring the luminescence on XLPE samples, only samples aged for 72 h and 85 h (110 and 135 %) were available as they were realised first. However, as shown in the results of SEM and FTIR characterization, these samples still show signs of structural and chemical degradation as compared to the thermally aged samples. Thus, measurement of light emission properties of these samples can lead to a better understanding of the effect of plane-plane geometry upon electronic properties (transition levels, trapping and recombination centers for instance).

Photoluminescence

First, Photoluminescence (PL) was investigated in a thermal reference XLPE sample, aged for 85 h at 90 • C. The same experimental parameters as for XLPE films in sphere-plane geometry (chapter 4) were used. Figure 5.16 shows the PL spectra of the thermal reference measured under UV light excitation between 220 and 300 nm. The PL spectra of this thermal reference are similar to the spectra of the XLPE reference under sphere-plane geometry (see figure 4.56 in chapter 4). The peaks at 320 and 380 nm are attributed to reference emission bands of bulk XLPE. The broad shoulder between 400 and 550 nm is attributed to the presence of antioxidant, and other oxygen containing chromophores which could be involved in fluorescence (hydroxyls and carbonyls) [START_REF] Teyssedre | From LDPE to XLPE: investigating the change of electrical properties. part II: luminescence[END_REF]. The other peaks correspond to emitting source at 290 and 300 nm, and their respective secondary harmonics at 580 and 600 nm [START_REF] Blasse | Luminescent Materials[END_REF].

In the following, the PL spectra of the thermal reference XLPE measured at 240, 270 and 300 nm have been compared to the PL spectra of aged XLPE samples. Results are shown in figure 5.17. The PL spectra of electrically aged samples share similarities with XLPE samples aged in sphere-plane geometry. The intensity of the main peaks at 320 and 380 nm decreases for the aged materials, which indicates a quenching of light emission from the bulk. Conversely, intensity of light around 550 nm slightly increases for the aged materials as compared to the reference. This may indicate an increase in the concentration of emitting species at this particular wavelength. The intensity ratios of light emitted at intermediate wavelength upon excitation at 300 nm, over the intensity of the reference peak at 325 nm, are calculated for the different samples. Table 5.3 shows the calculated intensity ratios. These results indicate that light emitted from the reference at 380 nm slightly decreases from 58 to 45 % for the aged samples. As explained in the literature, this contribution may be attributed to the bulk XLPE, thus indicating that electrical ageing may have affect on the emitting species composing the XLPE bulk [START_REF] Teyssedre | Semi-quantitative analysis of photoluminescence in thermoelectrically aged cables: I-identification of optical signatures[END_REF]. The light emitted at higher wavelengths, slightly increases for the sample aged for 72 h under applied field. The increase is less significant in the case of the sample aged for 85 h. These contributions may be attributed to carbonyl groups formed during thermo-oxidation of the sample. This effect can be correlated to the chemical ageing observed in FTIR: the spectra of the sample aged for 72 h showed a slightly larger absorbance as compared to the sample aged 85 h.

Thus, the species identified in FTIR may be responsible for the increase in the light emission observed in PL analysis.

Plasma induced luminescence

In addition to PL spectra analysis, Plasma Induced Luminescence (PIL) measurements have been performed on the reference (thermally aged) and electrically aged XLPE samples. The experimental parameters and procedure are the same as those used for XLPE in sphere-plane geometry (chapter 4). As a reminder, the electrode used for PIL covers most of the surface of the sample, thus light emission is an average over the whole sample surface. The luminescence decay follows the same law as in XLPE films under sphere-plane geometry, fitted using a double exponential model as described in chapter 4. Phosphorescence decays after τ 1 = 1.9 secs, whereas chemiluminescence decays after τ 2 = 5.3 secs. The parameter m describing the time dependency of charge recombination is equal to 1. The parameters are quite similar to those calculated for XLPE films in sphere-plane geometry.

As done in chapter 4, a spectral analysis of the emitted light has been done using the CCD camera. The spectra have been taken from 1 to 300 secs after the plasma discharge, and normalized. In some cases, the spectra were also smoothed to have a better appreciation of the results. Figure 5.19 shows some selected emission spectra from the thermal reference XLPE sample in plane-plane geometry, corresponding to different time periods after the discharge. The emission spectra of the thermal references used in the plane-plane geometry and sphere-plane geometry are very similar. Right after the discharge, the wavelength of emitted photons is around 420 nm, then it shifts to around 400 nm. After 110 seconds, which corresponds to the time period where photons are emitted by charge recombination, the wavelength is centered around 500 nm. These features are similar to the XLPE reference sample in sphere-plane geometry. The emission spectra in the case of the thermal reference are centered around 500 nm, whereas the spectra of the electrically aged sample are centered around 550 nm. This shift in emission wavelength corresponds to a significant shift in energy of the photons emitted by charge recombination. For more precision, figure 5.21 shows the evolution of the PIL emission spectra taken 110 secs after the plasma discharge for the different materials. This figure indicates that the emission spectra of the aged materials is, at first, slightly shifted to higher wavelengths with ageing for 72 h, then, is significantly shifted after ageing for 85 h. Moreover, the width of the peak seems to have increased after 85 h as compared to the other materials. A shouldering seems to be visible around 650 nm. This result have not been reported in the literature so far, and indicate that the main species involved in charge recombination may have changed. At least, the mean energy involved in charge recombination has decreased with electrical ageing. The contribution of other species seems to appear at wavelengths longer than 650 nm after 85 h of applied field. Table 5.4 shows the calculated peak position and corresponding mean energy involved in charge recombination. C, the peak position of photons emitted by charge recombination shifts from 501 to 507 nm for one sample, and 513 nm for another sample aged in the same conditions. The peak width (mean width at half maximum) does not change significantly at this stage. However, after 85 h, the peak position shifts to 550 nm, and the peak width is increased from 132 to 215 nm.

The shift in peak position from 501 to 550 nm corresponds to a shift in energy of 2.48 to 2.25 eV. As no significant effect can be seen on the PL spectra of this sample, the same recombination centers may be involved in charge recombination. But the decrease in charge recombination energy may indicate that the recombination centers have changed their position in energy levels. This may perhaps corresponds to structural defects which would affect charge recombination but does not appear clearly in PL or FTIR.

This effect is quite different from the observed results in the sphere-plane geometry, where an increase of 2.45 to 2.56 eV was found. We suppose that ageing under plane-plane geometry, which may have induced intrinsic damage in the bulk of the material, is responsible for the shift of recombination energy, and for the creation of possibly new emitting centers. This effect could indicate that electrical ageing have created new types of defects located within the band gap of the material, at which charge carriers can be trapped and recombine with a lower energy. Let us mention that the PIL emission spectra do not exhibit a structure with well defined wavelengths around the main peak, as observed for instance in the literature [START_REF] Teyssedre | Spectral analysis of optical emission due to isothermal charge recombination in polyolefins[END_REF]. Thus, it is difficult at this point to further investigate the obtained data.

Nevertheless, the observed results and proposed hypotheses need to be confirmed by additional, and more systematic measurements.

Summary and conclusion

In this chapter, the effect of electrical ageing of XLPE under a plane-plane geometry has been studied. Samples aged under increasing ageing stages, from unaged to near breakdown were characterized in order to investigate the presence of ageing markers.

The main identified ageing markers are similar to those observed in the previous chapter, in XLPE samples aged under sphere-plane geometry, but with some differences:

• Surface damage analysis by SEM shows that ageing does not lead to damage on the electrode-polymer side surface, but do lead to surface erosion at the counter-electrode side, which may be attributed to surface discharges occurring at the metal-polymer interface. This surface damage is not homogeneous, the sample surface exhibit localized damage zones which are not systematically present in all the samples, and/or on the whole surface.

• FTIR analysis has shown that ageing under a plane-plane geometry induces a significant increase in concentration of hydroxyl and carbonyl groups with ageing, which indicates a thermo-oxidative process triggered by the electric stress. However, the broad dispersion of the data indicates that damaging is heterogeneous in the experimental conditions which have been used. PL analysis has confirmed the presence of chemical ageing markers, as an increase of light emission was attributed to carbonyl groups.

• DSC measurements show that the thermal treatment induces a secondary crystallization similar to the one observed in XLPE under sphere-plane geometry.

The changes in the crystalline morphology of the material may facilitate the degradation of the sample by concentrating the created defects at crystalline boundaries. Nevertheless, the electric stress in itself has no direct structural effect.

• Finally, PIL measurements show that electrical ageing under plane-plane geometry leads to the creation of new recombination centers which recombine with a lower energy. This result may indicate that ageing in plane-plane geometry induces damage which may affect intrinsic electronic properties of the polymer such as energy transition levels in the band gap.

The degradation mechanisms of XLPE in both sphere-plane and plane-plane geometries thus share similar ageing markers, but a higher lifetime is expected in the case of the plane-plane geometry. Bulk ageing may be less deleterious for the insulation as compared to a stress highly localized on the surface of the material. In both cases, the observed damage which possibly determine the lifetime may actually be due to surface discharges. The way in which surface discharges are concentrated in one zone (sphere-plane geometry) or randomly distributed over the whole surface (plane-plane geometry) may instead be critical.

Conclusion and perspectives

General Conclusion

In this work we have proposed an approach for characterizing the behavior of insulating polymer materials in conditions of long lifetimes, that is, in conditions in which the materials are submitted to electrical stresses quite low as compared to the breakdown field in the material.

The proposed approach was to combine macroscopic measurements of the lifetimes of the materials under controlled stress levels and 'microscopic' characterization of the material along its life under the same stress levels. We have also attempted to discriminate the effects of temperature and electric field.

Two different geometries were chosen. The sphere-plane geometry was first chosen in order to concentrate the electrical stress in a pre-determined region of the material (namely, at the tip of the upper spherical electrode). However, in this geometry the sample is submitted to partial discharges which introduce another degradation mechanism on the vicinity of the surface of the sample. This surface degradation (erosion) may affect the final lifetime of the samples. A geometry with a priori homogeneous electric field was then chosen, in order to get samples in which a relatively large volume of the material has been aged homogeneously.

Measurements of the materials lifetime indicate that the variation of the durability of a given material on the stress level depends on the intrinsic properties of the materials, as well as on environmental parameters. Increasing the amount of peroxide in XLPE, which is expected to increase the cross-link density, was observed to slightly increase the materials lifetimes. The degree of crystallinity and the crystalline morphology however plays a major role in electrical degradation and considerably influence lifetime, as an increased crystallinity was found to improve lifetime by a factor two to three depending on the applied voltage.

Thus, being able to increase the cross-linking density and crystallinity of PE in power cable, while controlling the process parameters in order to avoid other problems such as cavitation, may lead to an improved durability. At the same time, reducing the thermal stress applied on the insulation would improve its durability under AC electrical stress.

The effects of both localized and homogeneous electrical stress on lifetime was also studied. It was shown that using a spherical electrode tip in air, leads to the development of discharges which are deleterious for the insulation. Indeed, the use of a parallel plane-plane geometry, with a Rogowski-like electrode configuration, has shown to significantly improve XLPE lifetime. It was discussed that under this geometry, the electrical field is more distributed over the whole surface and bulk of the material. Still, the imperfect contact between the material and the counter electrode at the polymer/metal interface is suspected to induce surface discharges as well.

This work also demonstrated the difficulty of preparing PE materials aged using a homogeneous electrical stress field. The use of an encapsulated electrode (plane) in the test material allowed to observe significant differences as compared to the samples aged under the spherical electrode, and results are important in order to attest the effect of electrical ageing on the bulk properties of insulating polymers. However, observations indicated that surface discharges are still possibly occurring in this second geometry. The stress condition imposed on the test materials are readily driving the ageing conditions. It is therefore critical to control the stress conditions imposed on the materials by choosing appropriate electrode configurations.

The observed behavior of the materials in both geometries indicates how critical it is to control the stress conditions imposed on the material before characterizing the impact of ageing. Controlling the surface aspect of the metal/insulation interface in power cables is thus of high importance in order to achieve reliable durability.

Lifetime measurements have also allowed to predict the end of life of the materials under a given applied stress and electrode configuration. The materials have then been aged under AC voltage using both electrode geometries, at different ageing steps in order to obtain samples at different ageing states from unaged to near end of life. Characterization of samples at different stages of ageing was performed in order to identify and study chemical and physical ageing markers. The results revealed that some ageing markers do not evolve much along the materials lifetime, for instance the crystallinity. Conversely, other ones follow a continuous evolution during lifetime and show a significant increase during the last steps of ageing as for instance the concentration of hydroxyl and carbonyl groups (chemical ageing).

The later are believed to play a major role in breakdown. It is proposed, from the observed results, that the concentration of ageing markers increases during lifetime until a critical level of degradation (C critical ) is reached (at a critical ageing time τ critical ), at which the damage concentration is high enough for breakdown to occur. The experimental results allow to distinguish between ageing markers which are critical for the insulation, markers which are significant, and other ones which are not believed to play a major role in breakdown. First, the chemical and electrical ageing markers are found to increase exponentially at the end of lifetime, and may be correlated: it has been discussed that thermo-oxidation could increase the leakage current by providing additional charge carriers are decreasing energy barriers for injection, thus weakening the insulation and decreasing breakdown strength. Both chemical and electrical markers are thus thought to be critical for the insulation. Secondly, observations of the surface aspect evolution by microscopy measurements revealed a progressive degradation due to both electrical field and discharges. In the very last ageing steps, the surface erosion could be high enough to have weakened the insulation sufficiently for breakdown to occur. However, as the observed degradation is mainly concentrated at the surface, breakdown may be triggered by other markers discussed above. The increase in crystallinity during ageing is attributed to thermal annealing, and is expected to reach a plateau after some time. It may not contribute to major degradation during electrical ageing.

Other ageing markers which are linked to those mentioned above have been observed using luminescence measurements. It was found that chemical species created during ageing are responsible for a higher light emission in photoluminescence, which confirms the critical role of chemical ageing markers. Plasma induced luminescence revealed that in the last ageing steps, the energy of recombination photon emissions involved in the aged materials have changed. This effect is of particular interest in the case of materials aged under parallel plane-plane geometry, as the recombination energy decreased with ageing, which could indicate the presence of deeper energy levels in the band gap of the aged materials. These observations are believed to be the proof of a significant ageing markers, but additional measurements are needed to confirm the observed effects. To our knowledge, this is the first time luminescence measurements are performed as monitoring characterization on PE samples aged thermo-electrically at different ageing stages prior breakdown.

Perspectives

Characterized ageing markers may correspond to chemical and/or physical modifications of the material. Both types of modifications may be of course correlated. FTIR and PL are generally discussed in terms of chemical modifications, as they directly give access to the identification of chemical groups (carbonyls, hydroxyls) created as a result of ageing. Identifying such groups is of course a necessary step. However, in order to predict the long-term behavior of the materials, it is also essential to elucidate the changes induced in the physical properties of the materials. These changes may originate from the chemical defects mentioned above, or from structural defects, if any, created as well during ageing. As electrical properties are concerned here, the physical properties of interest are electronic properties, in terms of injection threshold, density of charge carriers and energy barrier to inject charge carriers into the conduction band.

In this work we have attempted to characterize both chemical markers (through FTIR and PL) and physical changes in the behavior of the materials, by following changes in the breakdown field, in the leakage current characteristics and in the PIL spectra. In this context one key issue would be to relate the generation of chemical defects to their impact on the physical, and specifically the electronic properties.

A quantification of the relation between chemical and ageing markers would allow to tell after which quantity of chemical defects critical changes are expected in the physical properties. This quantification would allow a more precise prediction of the end of life. We therefore recommend a more systematic characterization of chemical (through FTIR and PL for instance) and physical ageing markers (electrical properties by breakdown strength and current measurements and electronic properties by PIL for instance), and thus highlight the importance of correlating the observed markers. An other physical marker could be studied by characterizing the evolution of the mechanical properties during lifetime under electrical stress, as the electric field may possibly weaken the material, as shown by the crazes observed on XLPE in chapter 5.

It is clear however that the obtained results are only preliminary and need to be completed in a more systematic way and in perhaps better controlled experimental conditions.

On this basis a theoretical model to predict the lifetimes of the materials based on the description of physical mechanisms may be developed. This is indeed envisaged in the continuation of this PhD. Experimental work will serve as a data base used in the development of a model aiming at predicting the lifetime of polymeric insulation under AC field. As discussed in the results, it is believed that ageing induces chemical and physical changes which lead to the modification of energy levels in the band gap of the polymer materials [START_REF] Chen | Electronic Structure of Polyethylene: Role of Chemical, Morphological and Interfacial Complexity[END_REF]. A proposition for the basis of such model is illustrated in the figure 5.23. The effect of incorporating additional energy levels in the band gap of the insulation could be used to simulate the effect of electrical ageing. Simulating the evolution of the chemical and physical properties of the insulation with ageing and comparing it to the experimental results could give additional proof of the proposed mechanisms responsible for the ageing and breakdown of polymeric insulators.

FIGURE 1 :

 1 FIGURE 1: Schéma des géométries hémisphère-plan (gauche) et planplan (droite) utilisées pour les tests de durée de vie et vieillissement.

FIGURE 2 :

 2 FIGURE 2: Scheme of insulation lifetime and the different ageing stages prior to failure. Failure is predicted by establishing the lifetime curves of the tested materials.

FIGURE 3 :

 3 FIGURE 3: Scheme of the different steps of the experimental study.

FIGURE 1 . 1 :

 11 FIGURE 1.1: High-voltage extruded-insulation cable structure for land applications, inspired by [12].

FIGURE 1 . 3 :

 13 FIGURE 1.3: Temperature dependence of the Young Modulus E of two semicrystalline thermoplastic polymers with molecular weight M A and M B . T g is the glass transition temperature and T m is the melting temperature [13].

FIGURE 1 . 4 :

 14 FIGURE 1.4: X-ray scattering of two polymers: (a) Debye-Scherrer pattern of an amorphous polyethylene; (b) pattern of a semicrystalline polyethylene; (c) one-dimensional diffraction pattern of a polyethylene sample with decomposition in amorphous (light gray)and crystalline (grey) contributions[START_REF] Schmacke | Investigations of Polyethylene Materials by Means of Xray Diffraction Artificial Ageing of Polyethylene Gas Pipes[END_REF].

FIGURE 1 . 5 :

 15 FIGURE 1.5: DMA relaxation curve of a MDPE, as function of applied temperature [27]. The glass transition is observed at around 150 K and corresponds to a decrease of the real part of the storage modulus G, and to a relaxation peak in the loss modulus G". It is also associated to γ transition in the literature.

FIGURE 1 . 8 :

 18 FIGURE 1.8: TEM-micrograph of an LDPE plaque without additives (left), LDPE stabilized with antioxidant (middle) and XLPE plaque (right). Taken from [29].

FIGURE 1 . 11 :

 111 FIGURE 1.11: Polypropylene's spherulites in a cross pattern under polarized light [31].

FIGURE 1 . 13 :

 113 FIGURE 1.13: Defects originating from polymer chains in the crystalline/amorphous interface, showing loops (L), entanglements (E), ties (T) and cilia (C) [33].

FIGURE 1 . 14 :

 114 FIGURE 1.14: Spherulite boundaries showing microvoids [34].

FIGURE 1 . 15 :

 115 FIGURE 1.15: Impurity states introduced in PE band gap [38].

FIGURE 1 . 16 :

 116 FIGURE 1.16: Scheme of common chemical and physical disorders found in PE [32].

FIGURE 1 . 19 :

 119 FIGURE 1.19: Schematic representation of a capacitor under vacuumand in presence of a dielectric material (adapted from[START_REF] Bureau | Propriétés diélectriques des polymères[END_REF]).

FIGURE 1 . 20 :

 120 FIGURE 1.20: Schematic representation of the four polarization mechanisms in polymers under electric field (adapted from [49]).

FIGURE 1 . 21 :

 121 FIGURE 1.21: Schematic representation of the different polarization mechanisms contributions on the real part of dielectric permittivity , and imaginary part as function of frequency [49].

FIGURE 1 . 22 :

 122 FIGURE 1.22: Variation of real and imaginary part of permittivity with temperature, in the case of glycerol, measured from 10 -6 to 10 12 Hz: (a) real part of dielectric permittivity , and (b) imaginary part [52].

FIGURE 1 . 23 :

 123 FIGURE 1.23: Dielectric spectra of LDPE and HDPE at 10 4 Hz with varying temperature as function of temperature [49].

FIGURE 1 . 25 :

 125 FIGURE 1.25: Scheme of the Fowler-Nordheim effect at metalpolymer interface [16].

FIGURE 1 . 26 :

 126 FIGURE 1.26: Schematic representation of an insulator wide band gap with shallow and deep trapping levels.

FIGURE 1 . 27 :

 127 FIGURE 1.27: Schematic plot of log(J) as function of log(V) in the SCLC model (inspired by [53]).

FIGURE 1 . 28 :

 128 FIGURE 1.28: Schematic representation of tunneling between two localized states separated by a distance R [63].

FIGURE 1 . 29 :

 129 FIGURE 1.29: Schematic representation of hopping conduction between two localized states separated by a distance R and under application of electrical field.

FIGURE 1 . 30 :

 130 FIGURE 1.30: Schematic representation of Poll-Frenkel effect on the energy band diagram.

FIGURE 1 . 31 :

 131 FIGURE 1.31: Schematic representation of bi-polar charge injection under applied AC field.

  Some years ago, a succession of research works on the subject using molecular modeling were made, aiming at predicting the behavior of electrons and holes in PE[START_REF] Meunier | Molecular modeling of electron trapping in polymer insulators[END_REF][START_REF] Mizutani | Behavior of Charge Carriers in Organic Insulating Materials[END_REF][START_REF] Serra | Interchain states and the negative electron affinity of polyethylene[END_REF][START_REF] Serra | Interchain electron states in polyethylene[END_REF][START_REF] Serra | Self-trapping vs. non-trapping of electrons and holes in organic insulators: polyethylene[END_REF][START_REF] Ceresoli | Exciton self-trapping in bulk polyethylene[END_REF]. The main result of this study was the duality in the behavior of electrons and holes regarding conduction: Serra et al. first have shown that due to the negative electron affinity of PE, electrons have an inter-chain mobility rather than along chains[START_REF] Serra | Interchain states and the negative electron affinity of polyethylene[END_REF]. They also have shown that the highest valence states mobility, occupied by holes, has to be intra-chain. Further studies by Serra et al. have supported these results by calculating the probability of presence of electrons and holes in PE, as shown in figure1.32.

FIGURE 1 . 32 :

 132 FIGURE 1.32: 3D charge density plot of the valence state wavefunction |Ψ val k=0 | 2 (left) and of the conduction state wave-function |Ψ cond k=0 | 2 (right) indicating the most probable locations of holes and electrons respectively. The probability is maximum in indigo color and minimum in red color [76].

FIGURE 1 . 33 :

 133 FIGURE 1.33: Carrier transport in PE crystal (left) and in the crystal/amorphous interface [74].

FIGURE 1 . 34 :

 134 FIGURE 1.34: Schematic representation of Zener tunneling under strong electrical field [6].

FIGURE 1 . 35 :

 135 FIGURE 1.35: Schematic representation of impact ionization mechanism in a solid dielectric up to the i th generation. White dots are neutral atoms, while red crossed dots are cations.

FIGURE 1 . 36 :

 136 FIGURE 1.36: Evolution of a polymer' structural parameter S as function of temperature during cooling [85].

FIGURE 1 . 37 :

 137 FIGURE 1.37: Schematic representation of chain scission mechanism (adapted from[START_REF] Richaud | Vieillissement chimique des polymères -Mécanismes de dégradation[END_REF]). Chains with small dots are free-radical chains.

FIGURE 1 . 38 :

 138 FIGURE 1.38: Scheme of a chain cross-linking during chemical ageing [87].

FIGURE 1 . 39 :

 139 FIGURE 1.39: Scheme of a general oxidation mechanism of linear polyolefins, and formation of hydroperoxydes (in red). Adapted from [88].

FIGURE 1 . 40 :

 140 FIGURE 1.40: Simplified photo-oxidation mechanism of PE [90].

FIGURE 1 . 41 :

 141 FIGURE 1.41: Schematic representation of the response of interlamellar ties to a tensile stress T, from low stress (i) to break up of ties (v) [99].

FIGURE 1 . 42 :

 142 FIGURE 1.42: Schematic representation of craze formation under tensile stress T in the inter-lamellar region [99].

FIGURE 1 . 43 :

 143 FIGURE 1.43: Free energy diagram of thermodynamic states available for a material, and energy barrier to overcome [100].

FIGURE 1 . 44 :

 144 FIGURE 1.44: Schematic representation of molecule dissociation induced by a hot electron. Taken from [81].

FIGURE 1 . 45 :

 145 FIGURE 1.45: Electrical trees developed in LDPE at different applied voltages [110].

FIGURE 1 . 46 :

 146 FIGURE 1.46: The microstructure of electrical trees in XLPE, under SEM analysis [111].

FIGURE 1 . 47 :

 147 FIGURE 1.47: Scheme of electrical tree propagation in LDPE, at spherulites boundaries and in the interlamellar regions [113].

FIGURE 1 . 48 :

 148 FIGURE 1.48: Microcavity in dielectric medium subject to partial discharge [120].

FIGURE 1 . 49 :

 149 FIGURE 1.49: Partial discharge induced ageing scenario in polymeric insulation filled with a microscopic cavity under AC high voltage (personal vision). The different steps of the ageing model are represented as follow: Charge injection/detrapping (1), charge conduction in bulk (2), accumulation of charge at interface and discharge inception (3), impact ionization in cavity gas (4) and damage growth from cavity wall (5).

FIGURE 1 . 50 :

 150 FIGURE 1.50: Field enhancement in a spheroidal cavity within a dielectric matrix.

34 )K

 34 is a dimensionless geometrical factor given by Crichton et al. for different cavity geometries, as a function of two dimensions of the cavity: the radius a (m) parallel to the field E p (V/m), and radius b (m), perpendicular to it (see figure 1.51).

FIGURE 1 . 51 :

 151 FIGURE 1.51: Geometry of ellipsoidal cavities within a dielectric matrix. Case (1) for cavity extended perpendicularly to external field, case (2) for cavity extended alongside to external field.

FIGURE 1 . 52 :

 152 FIGURE 1.52: Contribution of the space charge distribution to the electric field in the cavity.E 0 -background field in the cavity, E q -space charge generated electric field, E p -electric field in the polymer matrix[START_REF] Wang | Physical Model of PD Behavior and relevant Damage Grotwh from Micro-Cavities in PE based material under AC voltage[END_REF].

  in a different work by Illias et al. and is presented in figure 1.54.

FIGURE 1 . 54 :

 154 FIGURE 1.54: Schematic diagram of a PD event in the cavity: (a) before PD, (b) during PD and (c) after PD. The cavity field E cav is reduced by the apparition of surface charge at cavity wall, responsiblefor an electric field E s[START_REF] Hazlee A Illias | The influence of spherical cavity surface charge distribution on the sequence of partial discharge events[END_REF].

FIGURE 1 . 55 :

 155 FIGURE 1.55: Simplified schematic diagram of the processes leading to breakdown [114].

FIGURE 1 . 56 :

 156 FIGURE 1.56: Stages of PD-induced damage at the insulator surface [4].

FIGURE 1 . 57 :

 157 FIGURE 1.57: Scheme of the charge recombination process occurring by the decay of an electron from an excited state (1) to the ground state (2) [143].

FIGURE 1 . 58 :

 158 FIGURE 1.58: Strength of light emission in different insulating materials (epoxy resin, PP and polystyrene) the different emission stages: onset of emission (I), multiplication of emitted photons (II), and saturation before degradation (III). After stage (III), treeing was observed [144].

FIGURE 2 . 1 :

 21 FIGURE 2.1: Scheme of the tests electrode for lifetime and ageing experiments.

FIGURE 2 . 2 :

 22 FIGURE 2.2: Picture of the test electrodes: a rod electrode with hemispherical tip (A) and a "Rogowski-like" electrodes (B) for plane-plane configuration. The scale is given by the square mesh, where one square is 5 mm.

  3) and metal ( air = 1000) as boundary conditions, as well as two different voltage: 3000 V for HV electrode and 0 V for ground. Results are shown in the figure 2.3.

FIGURE 2 . 3 :

 23 FIGURE 2.3: Electrical field isolines (|E|) calculated using FEMM software in the hemisphere-plane configuration, with an input voltage of 3000 V. The field lines where calculated between 0 (blue) and 30 kV/mm (pink).

FIGURE 2 . 4 :

 24 FIGURE 2.4: Electrical field isolines (|E|) calculated using FEMM software in the plane-plane configuration, with an input voltage of 3000 V. The field lines where calculated between 0 (blue) and 30 kV/mm (pink).

FIGURE 2 . 5 :

 25 FIGURE 2.5: Scheme of ageing set-up with the 7 kV AC generator and the test cell.

FIGURE 2 . 6 :

 26 FIGURE 2.6: LDPE preforms with Rogowski-like electrode.

FIGURE 2 . 7 :

 27 FIGURE 2.7: (Left) Picture of an XLPE encapsulated samples right after processing. (Right) Picture of a peeled XLPE sample after electrical breakdown test. The plane region with 100 μm thick material is indicated with the red arrow. The black spot on the sample is a breakdown pit, and is found in the plane region as expected.

Figure 2 .

 2 Figure 2.8 shows the normalized I nDQ signals of a stabilized LDPE sample and the different XLPE samples. Points beyond about 10 ms are noisy and should be discarded.

FIGURE 2 . 8 :

 28 FIGURE 2.8: Normalized DQ signals (I nDQ (τ)) of a stabilized LDPE sample and different XLPE samples processed with increasing amounts of tBCP, as a function of relaxation time τ DQ .

( 2 )

 2 res which does not change. The cross-linked network itself is rather characterized by the larger D[START_REF] William A Thue | Electrical Power Cable Engineering[END_REF] res value, which increases as the amount of peroxide increases. The fraction Ψ of the material belonging to the cross-linked network also increases as the amount of peroxide increases. The different contributions are schematized in figure 2.9.

FIGURE 2 . 9 :

 29 FIGURE 2.9: Scheme of a cross-linked polymer network with dangling chains and the corresponding D res contributions.

Figure 2 .

 2 Figure 2.10 shows the first heating cycle of unaged PP films. Exothermic flow is pointing up.

FIGURE 2 . 10 :

 210 FIGURE 2.10: DSC thermograms of PP films and measurement of melting enthalpy during first heating cycle. Exo up.

FIGURE 2 . 11 :

 211 FIGURE 2.11: Scheme of the SAXS experiment on a semi-crystalline polymer. q is in the detector plane.

FIGURE 2 . 12 :

 212 FIGURE 2.12: Scattering patterns of PP A (A) and PP B (B) reference (unaged) films.

FIGURE 2 . 14 :

 214 FIGURE 2.14: Electrode cell for breakdown strength. It is composed of the test electrodes (1), a screw (2) which is connected to high voltage, and a Teflon cell designed for the test. The whole cell is immersed in a silicone oil bath.

Figure 2 .

 2 [START_REF] Füzesséry | Polyéthylènes basse densité PE bd[END_REF] shows the plotted P(E) for an XLPE C sample at room temperature.

FIGURE 2 . 15 :

 215 FIGURE 2.15: Probability of failure of XLPE C at room temperature, before ageing. Experimental data points give α = 96.85kV/mm and β = 13.5. Weibull distribution plot of P(E) using these two parameter values give the red curve.

  5 kV) connected to 12 different test channels. The test channels are connected to 12 test cells composed of an electrode and a temperature probe (K-type thermocouple), and are placed in ovens. A voltmeter is used with a multiplexer in order to measure the voltage of the 12 test channels. Current value is obtained from the voltage measured across a calibrated resistance of 100 MΩ. The experiment is controlled from a standard PC with a dedicated user interface. Figure shows a scheme of the equivalent circuit for current measurement.

FIGURE 2 . 16 :

 216 FIGURE 2.16: Schematic representation of equivalent electrical circuit for current measurement.

FIGURE 2 . 17 :

 217 FIGURE 2.17: Current decay in an aged XLPE sample measured at 50 • C and 2000 V, and double exponential curve fitting.

  (A) Picture of the experimental set-up used in LAPLACE. (B) Scheme of the experimental set-up (inspired by [QIAO16]).

FIGURE 2 . 18 :

 218 FIGURE 2.18: Experimental set-up for luminescence measurements.

FIGURE 2 . 19 :

 219 FIGURE 2.19: Scheme of the electrode device and sample holder for PIL.

FIGURE 3 . 1 :

 31 FIGURE 3.1: Cumulative probability of failure P(t) of XLPE C samples under sphere-plane geometry in air at 90 • C under different applied voltages, and Weibull distributions plotted using the estimated α and β.

  Figure 3.2 shows a representation of the β as a function of α values for the different materials.

FIGURE 3 . 2 :

 32 FIGURE 3.2: Influence of applied voltage on the shape parameter of Weibull distribution for LDPE and XLPE test samples.

Figure 3 .

 3 Figure 3.3 shows the lifetime curve of XLPE C material.

FIGURE 3 . 3 :

 33 FIGURE 3.3: Lifetime of XLPE C polymer film samples measured at 90 • C for decreasing applied applied stress in air. The dots represent individual broken samples at a given applied voltage.

FIGURE 3 . 4 :FIGURE 3 . 5 :

 3435 FIGURE 3.4: Lifetimes of LDPE stab , XLPE A , XLPE B and XLPE C polymer films, measured in sphere-plane geometry at 90 • C for different applied voltages in air. Dashed lines are the fits using inverse power law. Data points are the α values of Weibull distributions. Linear scale.

FIGURE 3 . 6 :

 36 FIGURE 3.6: Lifetime of LDPE, LDPE stab , XLPE C polymer films, measured in sphere-plane geometry at 90 • C for different applied voltages in air. Lines are the fits using inverse power law. Log-log scale.

  event was recorded among the aged samples. Extrapolation of XLPE C lifetime based on the life model parameters calculated previously was plotted and compared to this ageing at 850 V. In addition, extrapolations of lifetime with VEC values of n = 5 and n = 10 were plotted. Results are shown in figure 3.7.

FIGURE 3 . 7 :

 37 FIGURE 3.7: Extrapolation over a year of XLPE C lifetime, with comparison to samples aged under 850 V. The expected breakdown at this voltage was around 330 hours.

Figure 3 .

 3 Figure 3.8 shows a picture of an LDPE sample after breakdown.

FIGURE 3 . 8 :

 38 FIGURE 3.8: Picture of an LDPE sample after breakdown under 4.2 kV.

FIGURE 3 . 9 :

 39 FIGURE 3.9: Surface of XLPE C samples after ageing and breakdown under 850 V (A), 1.7 kV (B), 2.8 kV (C), and 4.5 kV (D). Sample (A) has been used aged for 2100 hours without breakdown.

FIGURE 3 . 10 :

 310 FIGURE 3.10: 2D electric field isolines (|E|(kV/mm)) calculation in sphere-plane geometry under 4.5 kV in air, illustrating qualitatively the variation of electric field in air.

FIGURE 3 . 11 :

 311 FIGURE 3.11: Evolution of the calculated electric field magnitude as a function of the distance from the center of the sample's surface.

4 and figure 3 .

 3 13 show a clear effect of test temperature on the lifetime of LDPE. The scale and shape parameters of LDPE at 70 • C decrease with increasing applied voltage. The scale parameter increases when decreasing the test temperature from 90 • C to 70 • C. Shape parameter also seems to increase when decreasing the temperature. The slopes of the lifetime curves of both materials are very similar and indicate a VEC parameter of n ≈ 2.

FIGURE 3 . 14 :

 314 FIGURE 3.14: Melting of LDPE broken samples obtained by DSC after lifetime measurements at 70 and 90 • C under 15 kV/mm.

FIGURE 3 . 15 :

 315 FIGURE 3.15: Effect of voltage frequency on XLPE C lifetime for decreasing applied voltage in air. Dash lines are estimated using inverse power law for the same number of AC cycles. Data at 3000 Hz are individual breakdown events, whereas data at 50 Hz and 1000 Hz are α values of the Weibull dispersion.

FIGURE 3 . 16 :

 316 FIGURE 3.16: Evolution of XLPE C lifetime measured under 2.8 kV at 90 • C in sphere-plane geometry as a function of voltage frequency. Points at 50 and 1000 Hz are α values of a Weibull distribution, whereas the point at 3000 Hz is an individual broken sample. The slope of the dashed line is close to -1.

FIGURE 3 . 17 :

 317 FIGURE 3.17: Cumulative probability of failure P(t) of PP A samples under sphere-plane geometry in air at 90 • C under different applied voltages, and Weibull distributions plotted using the estimated α and β.

FIGURE 3 . 18 :

 318 FIGURE 3.18: Lifetime of PP A , and PP B polymer films, measured in sphere-plane geometry at 90 • C for different applied voltages in air. Dash lines are the fits using inverse power law. Lifetime of XLPE C is provided on an indicative basis.

FIGURE 3 . 19 :

 319 FIGURE 3.19: Cumulative probability of failure P(t) of XLPE C measured in sphere-plane and plane-plane geometry in air at 90 • C, measured under 2.8 kV, and corresponding Weibull distributions.

FIGURE 3 . 20 :

 320 FIGURE 3.20: Lifetime of XLPE C samples measured in sphere-plane and plane-plane electrode geometries at 90 • C for different applied voltages in air. Dashed lines are the fits obtained using the model.

FIGURE 3 . 21 :

 321 FIGURE 3.21: Schematic representation of the metal-polymer-metal interface in the case of the plane-plane electrode configuration. The small voids are represented with an arbitrary scale.

Figures 4 .

 4 Figures 4.1 and 4.2 illustrate the choice of the ageing voltages based on lifetime curves.

FIGURE 4 . 1 :

 41 FIGURE 4.1: Lifetime curve of XLPE C measured in sphere-plane geometry at 90 • C, and the chosen ageing steps at intermediate stress voltages.

FIGURE 4 . 2 :

 42 FIGURE 4.2: Lifetime curves of PP A and PP B measured in sphereplane geometry at 90 • C, and the chosen ageing steps at an intermediate stress voltage.

Figure 4 .

 4 [START_REF] Dissado | Electrical Degradation and Breakdown in Polymers[END_REF] shows the SEM picture of the unaged XLPE sample's surface.

FIGURE 4 . 3 :

 43 FIGURE 4.3: Top surface aspect of reference unaged XLPE, analysed by SEM.

FIGURE 4 . 4 :

 44 FIGURE 4.4: Scheme of aged sample and preparation (A), and top surface of aged XLPE (48h under 2.25 kV), analysed by SEM (B). Zoom on the aged region.

Figure 4 .

 4 [START_REF] Alexander | The Theory of Everything In The Electrical Breakdown Of Polymeric Dielectrics: Maybe[END_REF] shows the top surface of aged XLPE after 48h under 2260 V, zooming in the peripheral of the aged region.

FIGURE 4 . 5 :

 45 FIGURE 4.5: Top surface of aged XLPE C (48h under 2.25 kV), analysed by SEM. Zoom on peripheral of aged region.

Figure 4 .

 4 [START_REF] Kao | Dielectric phenomena in solids[END_REF] shows SEM characterization of the same sample surface, 1 mm away from the center of the contact zone.

FIGURE 4 . 6 :

 46 FIGURE 4.6: Top surface of aged XLPE (48h under 2.25 kV), analysed by SEM. Zoom on middle of aged region.

FIGURE 4 . 7 :

 47 FIGURE 4.7: SEM characterization of the edge of an XLPE sample top surface aged under 2.25 kV for 2 hours (t age = 4%).

FIGURE 4 . 8 :

 48 FIGURE 4.8: SEM images of XLPE C surface aged under 2.25 kV at 90 • C for ageing stages corresponding to 4 % (A), 12.5 % (B), 50 % (C) and 100 % (D) of the expected lifetime. Observed surface aspect 1 mm away from the center of the degraded surface.

FIGURE 4 . 9 :

 49 FIGURE 4.9: Surface of aged XLPE C after 24 hours under 2.25 kV (A) and 3.2 kV (B), analysed by SEM.

FIGURE 4 . 10 :

 410 FIGURE 4.10: SEM image of an aged XLPE surface after 48 hours under 2.25 kV (left), and threshold

FIGURE 4 . 11 :

 411 FIGURE 4.11: Evolution of eroded surface ratio of XLPE as a function of sample's lifetime, for different applied voltage. The point at 0 % is not represented but corresponds to a neat surface without erosion.

Figure 4 .

 4 Figure 4.11 shows that at the very beginning of ageing stages, the surface in this aged region is already eroded by 60 to 80%. The eroded surface increases quickly in the first hours of exposure to the applied voltage, before slowing down. In the ultimate steps of its lifetime, the surface is nearly completely eroded. The surface erosion is slightly higher in the case of 2.25 kV. This difference cannot be explained by the SEM analysis only. Lifetime of the samples at 3.2 kV is shorter, thus breakdown can be also accelerated by other mechanisms.

Figures 4. 12

 12

and 4 .

 4 [START_REF] Louis | Polymer Materials : Macroscopic Properties and Molecular Interpretations[END_REF] show SEM pictures of PP A and PP B sample's top surface.

Figure 4 .

 4 [START_REF] Pelzer | Etude de vieillissement des isolants synthétiques des câbles moyenne tension[END_REF] shows the evolution of PP A top surface analysed by SEM at different ageing states from unaged to near end of life. Images are taken 1 mm away from the center of the aged region, as done in the case of XLPE.

FIGURE 4 . 12 :

 412 FIGURE 4.12: SEM characterization of an unaged PP A surface (left).A zoom at a smaller scale is shown (right).

FIGURE 4 . 13 :

 413 FIGURE 4.13: SEM characterization of an unaged PP B surface (left).A zoom at a smaller scale is shown (right)

FIGURE 4 .

 4 FIGURE 4.14: SEM images of PP A surface aged under 2.4 kV at 90 • C for ageing stages corresponding to 0 % (A), 23 % (B), 50 % (C) and 91 % (D) of the expected lifetime. Observed surface aspect 1 mm away from the center of the degraded surface.

FIGURE 4 . 15 :

 415 FIGURE 4.15: Evolution of eroded surface ratio of PP A as a function of sample's lifetime, for different applied voltage.

FIGURE 4 . 16 :

 416 FIGURE 4.16: SEM images of PP B surface aged under 2.4 kV at 90 • C for ageing stages corresponding to 0 % (A), 17 % (B), 39 % (C) and 83 % (D) of the expected lifetime. Observed surface aspect 1 mm away from the center of the degraded surface.

FIGURE 4 .

 4 FIGURE 4.17: FTIR transmission spectrum of unaged XLPE C sample.

FIGURE 4 .

 4 FIGURE 4.18: FTIR transmission spectra of aged XLPE samples at increasing life stages after ageing under 2.25kV and 90 • C. Spectra are compared to an unaged XLPE sample.

Figures 4 .

 4 19 4.20 and 4.21 show a zoom on the FTIR spectra in the main identified sub regions, for the two different applied voltages, with a comparison to XLPE aged thermally (thermal reference).

FIGURE 4 . 19 :

 419 FIGURE 4.19: FTIR transmission spectra thermally aged XLPE C (top) and electrically aged under 2.25 kV (left) and 3.2 kV (right). Zoom in 3600-3100 cm -1 region.

FIGURE 4 . 20 :

 420 FIGURE 4.20: FTIR transmission spectra thermally aged XLPE C (top) and electrically aged under 2.25 kV (left) and 3.2 kV (right). Zoom in 1800-1600 cm -1 region.

FIGURE 4 . 21 :

 421 FIGURE 4.21: FTIR transmission spectra thermally aged XLPE C (top) and electrically aged under 2.25 kV (left) and 3.2 kV (right). Zoom in 1400-700 cm -1 region.

FIGURE 4 . 22 :

 422 FIGURE 4.22: Absorbance ratio of aged vs reference XLPE measured at 3370 cm -1 identified as characteristic of hydroperoxydes.

FIGURE 4 . 23 :

 423 FIGURE 4.23: Absorbance ratio of aged vs reference XLPE measured at 1715 cm -1 (left) and 1630 cm -1 (right), identified as being characteristic of ketones and insaturated C=C bonds respectivaly.

FIGURE 4 . 24 :

 424 FIGURE 4.24: OM image of a broken XLPE sample and representation of the IR linear tracking. Each point corresponds to a tracking spot and an IR spectrum.

FIGURE 4 .

 4 FIGURE 4.25: FTIR spectra of XLPE corresponding to different region of the sample, measured using an IR Transmission Microscope.

FIGURE 4 . 26 :

 426 FIGURE 4.26: Evolution of the absorbance ratio between aged XLPE samples and a reference peak at 2020 cm -1 as a function of the distance from the sample's center.

FIGURE 4 . 27 :

 427 FIGURE 4.27: FTIR spectra of XLPE samples measured in ATR mode. Surface of an unaged XLPE sample, and top and bottom surface of a broken (aged) sample.

FIGURE 4 . 28 :

 428 FIGURE 4.28: FTIR spectrum of unaged PP A (top) and PP B (bottom).

Figures 4. 29

 29

  and 4.30 shows zooming in the FTIR spectra of electrically aged PP samples in the sub-regions corresponding to hydroxyl and carbonyl groups. Electrically aged samples ("Aged") are compared to thermally aged samples ("Tm Aged").

FIGURE 4 . 29 :

 429 FIGURE 4.29: FTIR spectrum of PP thermal references (Top, "Tm aged") and electrically aged PP A and PP B samples (Bottom, "Aged").Zoom in 3650-3200 cm -1 region (hydroxyls).

FIGURE 4 . 30 :

 430 FIGURE 4.30: FTIR spectrum of PP thermal references (Top, "Tm aged") and electrically aged PP A and PP B samples (Bottom, "Aged").Zoom in 1850-1500 cm -1 region (carbonyls).

FIGURE 4 . 31 :

 431 FIGURE 4.31: Absorbance ratio of aged vs reference PP materials measured at 3345 cm -1 (top), 1745 cm -1 (left) and 1630 cm -1 (right), identified as being characteristic of hydroxyls, ketones and insaturated C=C bonds respectively.

FIGURE 4 . 32 :

 432 FIGURE 4.32: Melting peak of an unaged reference XLPE C sample measured by DSC. Exo up.

Figure 4 .

 4 [START_REF] Lewis | Polyethylene under electrical stress[END_REF] shows the thermograms of an XLPE sample aged at a ratio of 126 % of expected lifetime, under 2.26 kV at 90 • C, and the thermal reference aged for the same time at 90 • C.

FIGURE 4 .

 4 FIGURE 4.33: DSC thermograms of an XLPE sample aged 60h (126 %) under 2.25 kV at 90 • C, and its thermal reference. Exo up.

FIGURE 4 . 34 :

 434 FIGURE 4.34: Melting cycle of different XLPE samples, aged under 2.25 kV (top) and 3.2 kV (bottom) at 90 • C for increasing ageing steps.

Figure 4 .

 4 35 shows this comparison.

FIGURE 4 . 35 :

 435 FIGURE 4.35: Evolution of XLPE relative crystallinity measured by DSC, as a function of lifetime ratio (top), compared to the evolution of absorbance ratio of hydroxyl groups as a function of lifetime (bottom). The black dash lines indicate a possible inset time for increase of both properties.

Figure 4 .

 4 [START_REF] Guffond | Characterization and modeling of microstructure evolution of cable insulation system under HVDC fields[END_REF] shows the evolution of PP thermograms with ageing time under 2.4 kV at 90 • C. As in the case of XLPE, the thermally aged samples thermograms are identical to the aged samples, which indicates that the thermal properties are driven by the thermal treatment only, even in the case of PP B . Only the electrically aged samples are considered in the following.

FIGURE 4 . 36 :

 436 FIGURE 4.36: Melting cycle of PP A (top) and PP B samples (bottom), aged under 2.24 kV at 90 • C for increasing ageing steps.

FIGURE 4 . 37 :

 437 FIGURE 4.37: Evolution of the relative crystallinity of PP A and PP B materials, aged under 2.4 kV at 90 • C, as a function of lifetime ratio.

Figure 4 .

 4 [START_REF] Huzayyin | An overview of impurity states and the basis for hole mobility in polyethylene[END_REF] shows the q 2 I(q) signals for the unaged and aged PP A and PP B sample.

FIGURE 4 . 38 :

 438 FIGURE 4.38: Evolution of q 2 I(q) as function of q for aged PP A (left)and PP B (right) measured by SAXS.

Figure 4 .

 4 [START_REF] Chen | Atomistic mechanisms for chemical defects formation in polyethylene[END_REF] shows the calculated Weibull probability of failure as a function of breakdown strength for XLPE samples aged under 2.25 kV at 90 • C.

FIGURE 4 . 39 :

 439 FIGURE 4.39: Weibull statistics of aged XLPE under 2.25 kV at 90 • C.Measurements are done in dielectric oil at room temperature.

FIGURE 4 . 40 :

 440 FIGURE 4.40: Evolution of the breakdown strength of thermal reference and electrically aged XLPE as a function of ageing time for two different applied voltage at 90 • C. The horizontal line plotted at 25 kV/mm represents the stress level at which the samples (2.25 kV only) were aged.

FIGURE 4 . 41 :

 441 FIGURE 4.41: Evolution of shape parameter from breakdown probability distribution of XLPE C as a function of ageing time for two different applied voltage at 90 • C.

Figure 4 .

 4 Figure 4.41 shows that the shape parameters (β) of the breakdown strength measurements are higher than in the case of lifetime measurements (see chapter 3), as it is

Figure 4

 4

FIGURE 4 . 42 :

 442 FIGURE 4.42: Evolution of XLPE breakdown strength with ageing time, compared to the absorbance ratio of hydroxyl groups measured in FTIR.

FIGURE 4 . 43 :

 443 FIGURE 4.43: Schematic representation of the evolution of breakdown strength as a function of lifetime during ageing. The AC cycles of the applied ageing voltage are represented.

Figure 4 .

 4 [START_REF] Moyassari | First-principle simulations of electronic structure in semicrystalline polyethylene[END_REF] shows the results of breakdown strength measurements on PP A samples at different ageing steps.

FIGURE 4 . 44 :

 444 FIGURE 4.44: Weibull statistics of aged PP A under 2400 V and 90 • C.

Figure 4 .

 4 Figure 4.[START_REF] Moyassari | First-principle simulations of electronic structure in semicrystalline polyethylene[END_REF] shows that the breakdown strength of aged PP slightly decreases as compared to the unaged material, but the effect is less important than in the case of XLPE for a similar ageing voltage. The breakdown strength of aged PP B samples was also measured. The breakdown strength values (α), and the shape parameter (β) are plotted as a function of ageing time in figures 4.45 and 4.46.

FIGURE 4 . 45 :

 445 FIGURE 4.45: Evolution of breakdown strength of PP A and PP B as a function of ageing time under 2400 V at 90 • C.

FIGURE 4 . 46 :

 446 FIGURE 4.46: Evolution of shape parameter from breakdown probability distribution of PP A and B as a function of ageing time under 2400 V at 90 • C.

Figure 4 .

 4 47 show the current decay measured for an XLPE sample aged 15 h under 2260 V AC at 90 • C. Leakage current is measured at 50 • C.

FIGURE 4 . 47 :

 447 FIGURE 4.47: Current decay measured in XLPE for different applied voltage and plotted in log scale.

  The data shown in figures 4.48 and 4.49 are the current-voltage characteristics of unaged and aged XLPE measured at different temperatures.

FIGURE 4 . 48 :

 448 FIGURE 4.48: Effect of applied temperature on current vs field measurements for unaged XLPE.

FIGURE 4 . 49 :

 449 FIGURE 4.49: Effect of applied temperature on current vs field measurements for XLPE aged 48h under 2260 V and 90 • C.

Figure 4 .

 4 50 shows the current-voltage characteristic of XLPE samples aged under 2.25 kV at 90 • C.

FIGURE 4 . 50 :

 450 FIGURE 4.50: Current vs electric field curves for XLPE aged under 2.25 kV at 90 • C for different ageing time, measured at 50 • C.

  These results indicate that electrical ageing may affect the charge carrier density, either by facilitating charge injection, or by creating carrier traps. The current-voltage characteristics were also drawn in the case of XLPE samples aged under 3.2 kV at 90 • C. The results are given in figure4.51.

FIGURE 4 . 51 :

 451 FIGURE 4.51: Current vs electric field for XLPE aged under 3.2 kV for different ageing time, measured at 50 • C.

Figure 4 .

 4 [START_REF]Broadband Dielectric Spectroscopy[END_REF] shows the comparison between leakage current measured in electrically aged XLPE sample under 2.25 kV and thermal reference samples.

FIGURE 4 . 52 :

 452 FIGURE 4.52: Current vs electric field for XLPE aged under 2.25 kV at 90 • C for 24 and 48 hours, measured at 50 • C.

FIGURE 4 . 53 :

 453 FIGURE 4.53: Current vs electric field XLPE samples aged under 2.25 kV and 3.2 kV for 15 and 24 hours. Effect of ageing voltage on leakage current.

FIGURE 4 . 54 :

 454 FIGURE 4.54: Evolution of measured leakage current in electrically and thermally aged XLPE samples as a function of ageing stages. Electrical ageing under 2.25 kV.

FIGURE 4 . 55 :

 455 FIGURE 4.55: Evolution of measured leakage current in electrically and thermally aged XLPE samples as a function of ageing stages. Electrical ageing under 3.2 kV.

4. 2 . 3

 23 Chemical and electronic ageing measured by luminescence 4.2.3.1 Chemical ageing: Photoluminescence First, Photoluminescence has been carried out on a thermally aged XLPE reference sample. The sample is excited with a monochromatic UV light between 220 and 300 nm. For each wavelength of excitation, the CCD camera measures the spectrum of light emission. Photoluminescence spectra of the XLPE reference sample is given in figure 4.56. The labelled reference spectrum on the figure is the light emission from the sample without UV light excitation.

FIGURE 4 . 56 :

 456 FIGURE 4.56: Photoluminescence spectra of reference XLPE thermally aged for 30h at 90 • C.

FIGURE 4 . 57 :

 457 FIGURE 4.57: Photoluminescence spectra of reference and aged XLPE measured upon excitation at 240 (left) and 270 nm (right).

FIGURE 4 . 58 :

 458 FIGURE 4.58: Photoluminescence spectra of reference and aged XLPE measured upon excitation at 300 nm.

  C. An AC discharge is triggered at around 1.5 kV on the top of the sample for 5 seconds. A picture of light emitted during the plasma discharge is shown in figure4.59.

FIGURE 4 . 59 :

 459 FIGURE 4.59: Picture of XLPE sample during plasma discharge.

Figure 4 .

 4 [START_REF] Takada | Space Charge Formation in Dielectrics[END_REF] shows the evolution of the number of detected photons as a function of time after discharge, in a thermally aged XLPE reference sample. Noise level is approximately 50 cps.

FIGURE 4 . 60 :

 460 FIGURE 4.60: Plasma induced luminescence measured in an XLPE reference sample aged 30h at 90 • C, taken after plasma discharge and fitted using double exponential decay model.

FIGURE 4 .

 4 FIGURE 4.61: PIL spectra taken few seconds after discharge, during phosphorescence.

FIGURE 4 . 62 :

 462 FIGURE 4.62: PIL spectra taken tens of seconds after discharge, after phosphorescence vanishing.

FIGURE 4 .

 4 FIGURE 4.63: PIL normalized spectra of reference XLPE taken at different time after discharge.

FIGURE 4 .

 4 FIGURE 4.64: PIL normalized spectra of reference and aged XLPE taken at 110 seconds after discharge during charge recombination.

FIGURE 5 . 1 :

 51 FIGURE 5.1: Lifetime of XLPE measured in the plane-plane electrode geometry, and the different ageing steps chosen for the study.

Figure 5 .

 5 Figure 5.2 shows a scheme of samples peeling for characterization.

FIGURE 5 . 2 :

 52 FIGURE 5.2: Scheme of XLPE samples peeling from electrode.

FIGURE 5 . 4 :

 54 FIGURE 5.4: Surface structure evolution of XLPE samples aged for 21, 31 and 50 hours under 2.8 kV at 90 • C in plane-plane geometry, characterized by SEM. Top surfaces (A, B and C), and bottom surfaces (D, E and F) are represented.

FIGURE 5 . 5 :

 55 FIGURE 5.5: Surface structure of the bottom surface of an XLPE sample aged for 50 hours under 2.8 kV at 90 • C (top) and a thermal reference sample aged 50 hours at 90 • C (bottom).

FIGURE 5 . 6 :

 56 FIGURE 5.6: Surface structure evolution of XLPE samples aged for 72 (A and C) and 85 hours (B and D) under 2.8 kV at 90 • C in the planeplane geometry, characterized by SEM. Top (A and B) and bottom surfaces (C and D) are represented.

FIGURE 5 . 7 :

 57 FIGURE 5.7: FTIR spectrum of an unaged reference XLPE sample after peeling off from the electrode.

FIGURE 5 . 8 :

 58 FIGURE 5.8: FTIR spectra of five different XLPE samples aged for 21 hours under 2.8 kV in the plane-plane geometry (top) and two different thermal reference samples (21 hours at 90 • C, bottom), focused in the range 1850-1500 cm -1 .

FIGURE 5 . 9 :

 59 FIGURE 5.9: FTIR spectra taken on three different spots of one XLPE sample aged for 21 hours under 2.8 kV in the plane-plane geometry (top), and on three spots of a thermal reference sample (21 hours at 90 • C, bottom), focused in the range 1850-1500 cm -1 .

FIGURE 5 .

 5 FIGURE 5.10: FTIR spectra of a reference XLPE sample and mean spectra of samples aged under 2.8 kV at 90 • C for different ageing times.

Figure 5 .

 5 [START_REF] Charvet | Study of mechanical properties and damage mechanisms in plasticized cellulose acetate polymers[END_REF] shows the mean spectra of aged XLPE samples as a function of ageing time compared to a unaged reference sample (top), in the carbonyls subregion. By comparison, mean spectra of XLPE thermally aged (thermal references) do not show significant changes as compared to the electrically aged samples and the unaged sample.

FIGURE 5 .

 5 FIGURE 5.11: FTIR mean spectra of XLPE samples aged under 2.8 kV and 90 • C in the plane-plane geometry for different ageing time, compared to an unaged reference XLPE sample (top), and mean spectra of thermall aged samples (bottom). Zoom in the carbonyls (1800-1650 cm-1 sub-region.

FIGURE 5 . 13 :

 513 FIGURE 5.13: Thermogram of an unaged reference XLPE sample prepared for plane-plane geometry, measured by DSC after peeling off the electrode. Exotherm up.

FIGURE 5 . 14 :

 514 FIGURE 5.14: Thermograms of XLPE samples aged electrically for 50 h under 2.8 kV at 90 • C, and a thermal reference aged at 90 • C without field, measured by DSC. Exotherm up.

FIGURE 5 . 15 :

 515 FIGURE 5.15: Evolution of thermograms of electrically aged XLPE samples as a function of ageing time. Exotherm up.

FIGURE 5 . 16 :

 516 FIGURE 5.16: Photoluminescence spectra of reference XLPE thermally aged for 85h at 90 • C. Legend indicates the excitation wavelength.

FIGURE 5 . 17 :

 517 FIGURE 5.17: Photoluminescence spectra of a thermal reference XLPE aged for 85 h at 90 • C and of XLPE samples aged for 72 h and 85 h under 2.8 kV at 90 • C, measured with excitation wavelength of 240 nm (top), 270 nm (left bottom) and 300 nm (right bottom).

Figure 5 .

 5 Figure 5.18 shows the evolution of the number of photons emitted by PIL right after the 5 seconds plasma discharge, in a reference (thermally aged) XLPE sample (85 h at 90 • C).

FIGURE 5 . 18 :

 518 FIGURE 5.18: Plasma induced luminescence measured in a thermal reference XLPE sample aged 85 h at 90 • C, taken after plasma discharge (left) and data fitted using double exponential decay model (right). Measurements at liquid nitrogen temperature.

FIGURE 5 . 19 :

 519 FIGURE 5.19: Normalized and smoothed PIL emission spectra of a thermal reference XLPE sample, taken at different time after plasma discharge.

Figure 5 .

 5 Figure 5.20 shows the emission spectra of the thermal reference XLPE sample and a sample aged for 85 h under applied field and temperature, measured 50, 80 and 110 sec after the plasma discharge.

FIGURE 5 . 20 :

 520 FIGURE 5.20: PIL spectra of a thermal reference XLPE sample (left) and a sample aged for 85 h under 2.8 kV at 90 • C (right) in planeplane geometry.

FIGURE 5 . 21 :

 521 FIGURE 5.21: Normalized and smoothed PIL emission spectra of a thermal reference and electrically aged XLPE samples aged in planeplane geometry, taken 110 secs after the plasma discharge.

Fig- ure 5 .

 5 [START_REF] Andrews | The role of degassing in XLPE power cable manufacture[END_REF] illustrates the evolution of the concentration of the most significant ageing markers as a function of lifetime.

FIGURE 5 . 22 :

 522 FIGURE 5.22: Illustration of the evolution of the different ageing markers concentration or level as a function of materials lifetime: chemical markers, electrical (breakdown strength, current), surface erosion and crystallinity.

FIGURE 5 . 23 :

 523 FIGURE 5.23: Illustration of a proposed model for intrinsic electrical ageing study, and the corresponding experimental steps.

TABLE 1

 1

	.1: Relative permittivities of a some dielectric materials [49].
	Material	r	Material	r
	Air	1.02	Silica (SiO 2 )	3.9 to 4.5
	Polypropylene	2.2	Polyvinylidene fluoride	8.2 to 9.5
			(PVDF)
	Polyethylene	2.3 to 2.65 Alumina	9.5
	Polyethylene terephta-	3.0 to 3.3 Water	78
	late (PET)

TABLE 2 . 1 :

 21 Composition of the PE based materials.

	Components Antioxidant	Samples LDPE LDPE stab XLPE A XLPE B XLPE C ∅ 0.3 phr a 0.3 phr 0.3 phr 0.3 phr
	Peroxide	∅	∅	0.5 phr 1.0 phr 1.5 phr
	phr a : Per hundred rubber (1 phr is 1 g of additive for 100 g of LDPE)

TABLE 2 . 2 :

 22 Composition of the PP based materials.

	Components	Samples PP A PP B
	Antioxidant	0.3 phr 0.3 phr
	Relative crystallinity χ r (wt%)	36.5	11.9

  Impregnated pellets corresponding to XLPE C sample (see table 2.1) have been used. Only one PE grade was chosen as it is more difficult to obtain reliable test samples, and also in order to compare with the XLPE C aged in the sphere-plane configuration. Preforms of thickness 1 mm have been prepared by compression molding at 120 • C (same steps as for XLPE films), in order to melt the polymer without decomposition of tBCP. The Rogowski-like test electrodes are then encapsulated in the LDPE preforms by compression molding at 190

• C, which also induces cross-linking. The thickness of the plane region is controlled by spacers.

TABLE 2 .

 2

3: Summary of stress conditions applied on the materials for lifetime measurement (⊗) and ageing steps (••) in sphere-plane geometry.

TABLE 2 .

 2

	4: Summary of stress conditions applied on the materials
	for lifetime measurement (⊗) and ageing steps (••) in plane-plane
		geometry.
	E avg	V 0-peak	Sample
	(kV/mm)	(kV)	XLPE C
	30	2.8	⊗
	45	4	⊗
	30	2.8

TABLE 2 .

 2 5: Fitting parameters obtained from curves of figure 2.8 with equation 2.4.1. D res is in 10 3 rad/s.

	Sample	Φ de f	Ψ	D res (1)	D res (2)
	LDPE stab 0.359 0.268 3.23 0.179
	XLPE A	0.265 0.279 3.891 0.181
	XLPE B	0.202 0.336 4.048 0.172
	XLPE C	0.134 0.38 4.335 0.173

The presence of two different D res values indicates that, besides the defect fraction Φ de f , there is a fraction of the chains which are very loosely connected to the crosslinked network with a corresponding parameter D

  Teyssedre from Laboratoire Plasma et Conversion d'Energie (LAPLACE) in Toulouse, FRANCE, on a dedicated experimental set-up which is shown in 2.18.

TABLE 3 .

 3 1: Weibull parameters of PE based films for the different applied stress.

	Material	15 kV/mm α (hours) β	30 kV/mm α (hours) β	45 kV/mm α (hours) β
	LDPE	85.91	5.67	30.6	3.24	8.45	3.16
	LDPE Antiox	71.17	5.58	23.85	4.22	8.88	4.26
	XLPE A	80.41	8.52	28.94	4.88	10.58	3.07
	XLPE B	104.21	6.87	27.76	4.91	10.88	8.04
	XLPE C	95.55	13.40	27.90	3.70	13.38	4.29

TABLE 3 .

 3

	2: Inverse power law fitting parameters for the different PE
		base films.
	Sample	A	n	R 2
	LDPE	5.50E+09 2.43 0.926
	LDPE Antiox 2.41E+08 2.05 0.973
	XLPE A	2.46E+08 2.03 0.962
	XLPE B	2.87E+08 2.04 0.998
	XLPE C	1.95E+08 1.97 0.980

TABLE 3 .

 3

	3: Comparison of ring pattern diameters attributed to dis-
	charges obtained by calculation and observed experimentally on
		XLPE C .
	Voltage (kV)	Diameter (mm) Calculation Experimental
	4.5	>8	8.5
	2.8	4.8	6.4
	1.7	3.1	4.6
	0.85	1.8	≈ 0

TABLE 3 .

 3 The measured α of both materials indicate that a higher degree of crystallinity leads to higher lifetime at the same applied voltage. The β values are in the same range as for PE based samples (β < 5), but values are lower for PP B than PP A for equivalent voltage. Thus, a higher crystallinity may lead to a sharper data dispersion. It can be noted that the β parameters are all larger than 2, but close to β = 3, which indicates an accelerating rate of failure driven by degradation mechanisms.The characteristic lifetimes of PP A and PP B were plotted for the different applied voltages, and compared. The lifetime of XLPE

		5: Weibull parameters of PP based films for the different
				applied voltages.
	Material	15 kV/mm α (hours) β	20 kV/mm α (hours) β	30 kV/mm α (hours) β	45 kV/mm α (hours) β
	PP A	67.41	4.03	⊗	⊗	13.64	3.72	9.57	4.34
	PP B	⊗	⊗	30.07	2.46	9.93	2.88	4.11	3.01

C is given as a guideline, and should not be directly compared to the PP materials due the differences in PE and PP structure. Results are presented in figure

3.18:

TABLE 3 .

 3

	6: Inverse power law fitting parameters for the different PP
		based films.
	Sample	A	n	R 2
	PP A	8.33E+06 1.65 0.858
	PP B	2.60E+10 2.71 0.954

TABLE 3 .

 3 7: Weibull parameters of XLPE C materials under sphereplane and bulk configuration. Two main differences between the parameters obtained in the two geometries can be observed: β values are relatively low as compared to values of XLPE C and other PE based materials in sphere-plane configuration. A shape parameter value below β = 2 value in plane-plane geometry may indicate that the rate of degradation in this geometry is lower as compared to ageing under sphere-plane electrodes. α values are in the same range for lifetime measured around 4 kV, but below this voltage, lifetime increases in the plane-plane configuration as compared to sphere-plane for the same applied voltage.

	Material	15 kV/mm α (hours) β	30 kV/mm α (hours) β	45 kV/mm α (hours) β
	Sphere-plane	95.55	13.40	27.90	3.70	13.38	4.29
	Plane-plane	∅	∅	62.72	2.56	17.50	1.51

TABLE 3 .

 3 8: Inverse power law model fitting parameters for XLPE C lifetimes in sphere-plane and plane-plane geometries.

TABLE 4 .

 4 1: Summary of ageing steps for XLPE performed in sphereplane geometry. Values in bold fonts correspond to ageing steps longer than the expected lifetime.

	V peak (kV)	Parameter	t 1	t 2	t 3	Ageing steps t 4 t 5 t 6	t 7	t 8	t 9
	2.25	Time (h) Life ratio (%)	0.5 1	2 4	6 12.5	15 31	24 50	39 81 100 112.5 126 48 54 60
	3.2	Time (h) Lifetime ratio (%)	0.5 2	2 8.5	6 25	15 62.5 100 125 24 30

TABLE 4 .

 4 2: Summary of ageing steps for PP materials performed in sphere-plane geometry. Value in bold font corresponds to an ageing step longer than the expected lifetime.

	V peak (kV) Sample	Parameter	t 1	Ageing steps t 2 t 3 t 4 t 5	t 6
	2.4	PP A PP B	Time (h) Life ratio (%) Time (h) Lifetime ratio (%)	2 9 1.5 3 5 11 15 20 24 23 50 68 91 110 7 15 8 17 39 83

  .3 and 4.4 present the calculated relative crystallinity of the different aged XLPE samples.

	Ageing time (h) Lifetime ratio (%) T melt ( • C) Δ H melt (J/g) χ (%)
	0	0	104.08	120.1	41.41
	2	4	104.82	121.8	42
	6	12.5	104.62	120.7	41.62
	15	31	105.04	122.7	42.31
	24	50	106.32	124.6	42.97
	39	81	106.81	121.8	42
	48	100	105.89	122.4	42.21
	54	112.5	105.41	122.5	42.24
	60	126	105.55	134.6	46.41

TABLE 4 .

 4 3: Relative crystallinity calculated in XLPE samples aged under 2.25 kV at 90 • C as a function of ageing time.

	%)

Ageing time (h) Lifetime ratio (%) T melt ( • C) Δ H melt (J/g) χ (

TABLE 4 .

 4

4: Relative crystallinity calculated in XLPE samples aged under 3.2 kV at 90 • C as a function of ageing time.

TABLE 4 .

 4

	5: Crystalline lamellae long period measured by SAXS for
	unaged and aged PP materials.

TABLE 4 .

 4

	6: Calculated intensity ratio of light emitted at different
	wavelength over reference emission peak at 325 nm, for an XLPE ther-
		mal reference and aged XLPE samples.
	Wavelength (nm)	I/I 325nm (%) Reference 30h (125 %, 3.2 kV) 60h (126 %, 2.25 kV)
	380	74	38	36
	485	18	15	32
	550	6.5	8	23

TABLE 4 .

 4

	7: Peak position, corresponding photon energy and width
		evolution.
	Sample	Position (nm) Mean energy (eV) Width (nm)
	Reference	507	2.45	166
	Aged 30h (125 %, 3.2 kV)	500	2.48	155
	Aged 30h (125 %, 3.2 kV) n • 2	497	2.5	158
	Aged 60h (126 %, 2.25 kV)	485	2.56	189

TABLE 5 .

 5

	1: Summary of ageing steps for XLPE under parallel plane-
	plane geometry. Life ratio in bold corresponds to ageing steps longer
	than the expected characteristic lifetime of the material.
	V peak (kV)	Parameter	Ageing steps t 1 t 2 t 3 t 4	t 5
	2.8	Time (h) Life ratio (%) 33 49 79 114 135 21 31 50 72 85

TABLE 5 .

 5 2: Relative crystallinity calculated in XLPE samples aged in plane-plane geometry under 2.8 kV and 90 • C, as a function of ageing time.Ageing time (h) Lifetime ratio (%)T melt ( • C) Δ H melt (J/g) T cryst ( • C) χ

	(%)

TABLE 5 .

 5 3: Calculated intensity ratio of light emitted at different wavelength over reference emission peak at 325 nm, for an XLPE thermal reference and aged XLPE samples. 300 nm excitation.

	Wavelength (nm)	I/I 325nm (%) Thermal reference Aged 72 h (110 %) Aged 85 h (135 %)
	380	58	44	45
	485	14	23	14
	550	6	16	7.5

TABLE 5 .

 5 4: Evolution of PIL emission peak position and width, and the corresponding photon energy with ageing.

	Sample	Position (nm) Mean energy (eV) Width (nm)
	Reference	501	2.48	132
	Aged 72h @2820 V	507	2.45	130
	Aged 72h @2820 V (n • 2)	513	2.42	132
	Aged 85h @2820 V	550	2.25	215

After 72 h under 2.8 kV at 90

Dans les chapitres quatre et cinq, le principe expérimental de l'étude sur le vieillissement électrique et les principaux résultats concernant la caractérisations des marqueurs de vieillissement sont décrits et discutés. Le quatrième chapitre regroupe les résultats de l'étude utilisant la géométrie sphère-plan, et vise à identifier les marqueurs de vieillissement dans le cas d'une contrainte électrique localisée. Le cinquième chapitre est centré sur le vieillissement des échantillons dans un champ de contraintes plan-plan, et vise à comprendre les mécanismes de vieillissement dans le cas d'une contrainte électrique plus homogène. Les résultats de ces études de vieillissement dans les deux configurations d'électrodes seront comparés et les effets discutés.Enfin, les principaux résultats de ce travail expérimental et l'interprétation des marqueurs de vieillissement sont résumés dans la conclusion, ainsi que les perspectives issues de cette étude.

Acknowledgements

Je souhaite prendre le temps avec ces quelques pages de remercier toutes les personnes qui m'ont accompagné durant cette thèse, qui ont été nombreuses, et sans qui je n'aurais pu accomplir ce travail.

Chapter 5. Study of electrical ageing in plane-plane electrode geometry

Evolution of surface damage

As in the case of samples aged in the sphere-plane electrodes, the surface of aged XLPE samples have been characterized by SEM in this configuration. The "top" and "bottom" surfaces have been characterized in order to compare with samples aged under discharges. Figure 5.3 shows the SEM characterization of a reference XLPE sample after peeling. When comparing top and bottom surfaces, they show a relatively different surface aspect. On the top surface, scratches of a few microns appear. A concentric ring pattern is also observed on this side of the sample due to the electrode machining. The observed scratches are attributed to the sample peeling. The bottom surface is neat without visible damage. Only small marks can be seen and are attributed to the sample preparation and handling.

The aged samples have then been characterized and compared to the unaged reference. Figure 5.4 shows the ageing kinetics of the sample surface aspect at different ageing steps shorter than the expected characteristic lifetime. The top and bottom surfaces are shown for the various ageing steps.

Appendix Electric field line calculation

We make the simplifying geometric hypothesis that field lines in air in between the upper spherical electrode and the surface of the sample are portions of circles, perpendicular to the surface of the electrode and to the surface of the sample (see 5.24). The length of a field line which hits the sample surface at distance X from the center of the electrode is L = α 0 X and α 0 is given by tan α 0 = X/(2R 0 ). Eliminating α 0 in these equations gives:

At a point M of curvilinear abscissa s along the same field line, the relation between s and the cord c is obtained from the relationship tan α = X = (2R), using s = αX and c = 2αR to replace α and R:

The cord c may also be expressed as a function of X and R as:

3)

The relation between R and s is:

The electric field is not uniform along a field line due to conservation of the flux (Gauss theorem). Conservation of the flux gives E ext (s)dA(s) = cst along a field line, with dA(s) = 2πrdc(s) where the differential element dc corresponds to the variation of the cord c when X changes by dX with R being constant. Differentiating equation 5.3 (at R = cst) and using equation 5.4 gives:

The radius r is given by:

and the expression of the electric field E ext (s) (in the external medium, which is air) along a field line at distance X from the center can be written as:

The electrostatics then give the following relationships for any value of X (with L(X) given by equation 5.1:

Note that within the sample film, field lines are supposed to be straight lines perpendicular to the surface of the film and consequently the field within the material at a given X value E i (X) is uniform throughout the film thickness. It follows that in this approximated form the electric field DOES NOT satisfy the basic equation ∇. E = 0 inside the sample film.

Denoting

the previous system of equations determines the K constant as:

The maximum value of the field is obtained at point P on the surface of the upper electrode (see figures 5.25 and 5.26).