Doctoral thesis Aix-Marseille University

Doctoral School: Physique et Sciences de la Matière Speciality: Énergie, Rayonnement, Plasma

Dust transport in tokamaks

Presented by: Adrien Autricque

Board of examiners for public Ph.D. defense on the 18th of October 2018

Dr. Thierry Belmonte Referee – IJL, Université de Lorraine Pr. Khaled Hassouni Referee – LSPM, CNRS Dr. Cécile Arnas Examiner – PIIM, Aix-Marseille Université Dr. Nicolas Fedorczak CEA supervisor – IRFM, CEA Cadarache Dr. François Gensdarmes Examiner – IRSN, CEA Saclay Dr. Christian Grisolia PhD co-director – IRFM, CEA Cadarache Dr. Sergey Khrapak Examiner – DLR, German Aerospace Center Pr. Jean-Marc Layet PhD director – PIIM, Aix-Marseille Université

Laboratory: Institut de Recherche sur la Fusion par confinement Magnétique CEA – Cadarache 13108 Saint-Paul-lez-Durance, France

November 2015 – November 2018

Thèse de doctorat Université d’Aix-Marseille

École doctorale : Physique et Sciences de la Matière Spécialité : Énergie, Rayonnement, Plasma

Transport des poussières dans les tokamaks

Presentée par : Adrien Autricque

Thèse soutenue publiquement le 18 octobre 2018 devant le jury composé de :

Dr. Thierry Belmonte Rapporteur – IJL, Université de Lorraine Pr. Khaled Hassouni Rapporteur – LSPM, CNRS Dr. Cécile Arnas Examinatrice – PIIM, Aix-Marseille Université Dr. Nicolas Fedorczak Responsable CEA – IRFM, CEA Cadarache Dr. François Gensdarmes Examinateur – IRSN, CEA Saclay Dr. Christian Grisolia Co-directeur de thèse – IRFM, CEA Cadarache Dr. Sergey Khrapak Examinateur – DLR, German Aerospace Center Pr. Jean-Marc Layet Directeur de thèse – PIIM, Aix-Marseille Université

Laboratoire d’accueil : Institut de Recherche sur la Fusion par confinement Magnétique CEA – Cadarache 13108 Saint-Paul-lez-Durance, France

Novembre 2015 – Novembre 2018

Remerciements

À mes directeurs de thèse, Jean-Marc Layet et Christian Grisolia, pour avoir émis ce sujet de thèse passionnant, pour m’avoir accueilli au sein de leurs équipes respectives, et pour avoir su orienter mon travail et me conseiller, toujours avec sagesse. Je remercie également Nicolas Fedorczak pour avoir toujours été présent, et ce depuis 2013, et à qui je dois une grande partie de mes connaissances en matière de fusion. À mes collaborateurs de Corée du Sud (NFRI/KAIST), Soohyun Son, Hyunyong Lee, Wonhoe Choe, ainsi qu’aux thésards de KAIST. J’adresse une mention particulière à Suk-Ho Hong et Inwoo Song, dont l’accueil et la gentillesse furent et, oserai-je, resteront, inégalés. À mes collaborateurs du PIIM, Sergey Khrapak, Lénaïc Couëdel, Boris Klumov, Cé- cile Arnas et Ning Ning, dont les qualités de physiciens m’impressionnèrent et qui surent me sortir de nombreuses impasses. À mes collaborateurs de l’IRSN Saclay, Samuel Peillon, François Gensdarmes et Ma- madou Sow, pour les discussions passionnantes sur le thème de l’adhésion des particules. À l’équipe de MIGRAINe, Svetlana Ratynskaia et Ladislas Vignitchouk, ainsi qu’à Frédéric Brochard et les auteurs de l’article [Shalpegin 2015] pour nos échanges fructueux lors de conférences, mais aussi pour m’avoir transmis les données de l’expérience d’injection de poussière dans TEXTOR qui me furent d’une grande utilité. À mes parents, Alex et Véronique, mon frère Flo, mon grand-père Francis (alias GP), mes tantes Aude et Anne, Annick et Régis, ainsi qu’au reste de ma famille, pour leur soutien sans faille. À mes amis, notamment de l’époque Centrale, pour leurs encouragements constants, avec une mention particulière à Clément « Gwaka » Cocquempot, pour les weekends de décompression bien arrosés, ainsi qu’à mes autres comparses de pérégrinations : Laure « Louloute » Ferry, Nicolas « Dodo » Marshall, Ghislain « Adibou » Gandolfi, Bruno « Boobz » Thomann, Maud « Mood » Duforest, Thierry « Thity » Law-Hine, Sabine Barets, et tous les autres. À tous mes collègues de l’IRFM, aux groupes de plasma de bord et matériaux, et plus particulièrement à la jeune génération : Sarah Breton, Etienne Hodille, Didier Vezinet, Jorge Morales, Laurent Valade, Olivier Février, Julien Denis, Julie Joly, Camille Baudoin, Nicolas Nace, Anastasia Dvornova, aux habitants de Little Italy (Alberto Gallo, Federico « Nespo » Nespoli, Matteo « Nuzzi » Valentinuzzi, Gorgio « Gorgio Giorgiani » Giorgiani, Serafina Baschetti, Raffaele « Waglio » Tatali, Cristian Sommariva, Elisabetta Caschera) et aux Théoriciens (Guillaume Brochard, Peter Donnel, Nicolas Bouzat). ii

À Pomone, ses occupants passés, présents et futurs : Axel Jardin, Davide Galassi, Clément Nguyen, Mylène Plumier, Damien Colette, Camille Gillot. Et Jerry. Résumé détaillé

La consommation mondiale en énergie est en forte augmentation depuis des décennies, passant de 6000 millions de tonnes équivalent pétrole (Mtoe) à 13000 Mtoe entre 1973 et 2013. Parmi les nombreuses sources utilisées, les énergies fossiles sont les plus répandues dans le monde, représentant environ 80% de l’apport global en énergie primaire. Comme toute source d’énergie, elles offrent avantages (haute concentration en énergie, flexibilité, sécurité, portabilité, . . . ), mais aussi inconvénients (rejet de gaz à effet de serre). Les conséquences liées à l’utilisation des énergies fossiles attirent de plus en plus l’attention : augmentation du niveau et acidification des océans, réchauffement climatique, problèmes de santé publique, . . . De plus, leur raréfaction peut poser des problèmes économiques et énergétiques. La nécessité de développer des énergies alternatives, pauvres en émissions de gaz à effet de serre (GES) et durables se fait de plus en plus pressante. Dans ce contexte, la fusion thermonucléaire (le processus ayant lieu au coeur des étoiles) pourrait jouer un rôle clé. La réaction de fusion choisie pour la production d’électricité sur Terre, car elle est la plus efficace en terme d’énergie libérée, est la suivante

2 + 3 + 4 2+ 1 1D + 1T → 2He (3.5 MeV) + 0n (14.1 MeV). (1)

Le deutérium est naturellement présent en abondance dans l’eau des océans. Le tritium, quant à lui, n’est pas présent dans la nature à cause de sa faible demi vie (environ 12 ans). Il peut cependant être généré à partir du lithium, qui, quant à lui, est présent en abondance. Le produit de la réaction, l’hélium, est inoffensif pour l’homme et la nature. Seules les composants structurels de la machine, activés par les neutrons de fusion ou piégeant du tritium, formeraient les déchets radioactifs de la filière. Les noyaux des atomes étant chargés positivement, ils se repoussent les uns les autres. Afin de pouvoir fusionner, il faut qu’ils se rapprochent jusqu’à franchir la barrière Coulom- bienne (par effet tunnel). Il existe plusieurs manières de réaliser ce franchissement : chauffer le milieu et/ou augmenter sa densité. Dans le cœur des étoiles, la compression induite par la force de gravitation est suffisante pour que les réactions de fusion puissent avoir lieu. Un tel effet n’étant pas reproductible sur Terre, l’approche “température” est privilégiée. La température maximisant l’efficacité des réactions de fusion est formidable- ment élevée (de l’ordre de 200 millions de degrés). Dans un tel environnement, les atomes constituant la mixture d’isotopes d’hydrogène utilisée comme combustible perdent leurs électrons. Le gaz passe à l’état de plasma, c’est-à-dire un fluide où les particules sont électriquement chargées. iv

Le problème principal de la filière fusion réside dans le confinement de ce plasma chaud. Puisqu’aucun matériau ne peut soutenir la température du plasma, une solution consiste à le confiner dans une “bouteille magnétique”. La configuration tokamak utilise cette méthode. La chambre contenant le plasma a une forme torique, et une collection de bobines crée un champ magnétique hélicoïdal qui confine le plasma tout en minimisant autant que possible les dérives. La configuration du champ magnétique sépare naturelle- ment la chambre du tokamak en deux régions très différentes : le plasma confiné (ou cœur) où les lignes de champ magnétique sont fermées, et le plasma de bord où elles sont ouvertes. Si l’on souhaite que le plasma soit le plus chaud possible dans le cœur, on veut, à l’inverse, le moins de plasma possible dans le bord, puisque toute particule plasma s’y trouvant suivra les lignes de champ jusqu’à impacter la paroi de la machine, y déposant au passage son énergie. Le confinement n’est évidemment pas parfait. A cause, entre autres, des collisions entre particules, de la turbulence ou des dérives, un peu de plasma s’échappe contin- uellement du cœur vers le bord, induisant un flux de chaleur continu sur les parois de la machine. Ce flux est concentré dans la région du divertor, où des cibles spécialement conçues pour soutenir ces flux sont placées. Ces flux peuvent également être significative- ment augmentés lors d’évènements transitoires dans le plasma : Edge-Localized Modes (ELMs), disruptions, impacts d’électrons découplés, . . . Dans le futur tokamak ITER (In- ternational Thermonuclear Experimental Reactor, en cours de construction à Cadarache), des flux de l’ordre de 10 MW/m2 sont attendus en régime permanent, avec des excur- sions transitoires pouvant atteindre 100 MW/m2. Le matériau choisi pour le divertor d’ITER est le tungstène (W), et ce en raison de son faible taux de sputtering, haut point de fusion, faible pression de vapeur saturante ainsi que sa faible tendance à piéger du tritium. Les interactions plasma-paroi sont l’un des plus importants challenges physiques et technologiques auquel les scientifiques font face dans le domaine de la fusion à l’heure actuelle. Comme évoqué plus haut, le mur du tokamak subit de l’érosion, c’est-à-dire qu’il perd de la matière. Les matériaux constituant les surfaces face au plasma d’un tokamak sont en général des métaux ou du carbone. Dans tous les cas, ces matériaux sont plus lourds que les espèces principales présentes dans le plasma. Les atomes libérés par les parois et transportés dans le plasma sont appelés “impuretés”, puisqu’ils ont des conséquences prin- cipalement néfastes : augmentation des pertes par radiation (donc baisse de l’efficacité du réacteur), création d’instabilités pouvant aller jusqu’à la disruption. Pour assurer le bon fonctionnement d’un réacteur à fusion, la concentration en impuretés doit être contrôlée et maintenue en dessous d’un certain seuil (environ 10−5 pour le W dans ITER). A titre v

Figure 1: Concentration maximale en W pour un plasma devant atteindre l’ignition. Le temps de confinement de l’hélium est supposé valoir 5τE. Source: [Pütterich 2010]

d’exemple, la concentration maximale en W admissible dans un plasma de fusion devant atteindre l’ignition (lorsque le chauffage du plasma peut être assuré par la réaction de fusion elle-même) est tracée sur la Fig.1, en fonction de la température du plasma T et le triple produit nT τE, où n est la densité du plasma et τE le temps de confinement de l’énergie. Une autre conséquence importante des interactions plasma-paroi est la création de poussière, littéralement des morceaux du mur de la machine qui peuvent en être détachés par divers phénomènes physiques et être transportés dans le plasma. Par opposition aux impuretés, qui sont des atomes ou ions isolés, les poussières sont des particules composées d’un grand nombre d’atomes, ayant des tailles comprises entre ∼ 1 nm et ∼ 1 mm (voir un exemple sur la Fig.2). Les procédés de création de ces poussières sont variés, mais tous en lien avec les interactions plasma-paroi : fonte locale de la surface (à cause d’impacts d’ELMs, d’électrons découplés, d’arcs électriques), accrétion du matériel érodé dans le plasma de bord, rupture de couches co-déposées instables, . . . La taille, vitesse et morphologie des poussières dépend de leur procédé de création. Lorsqu’elles sont transportées dans le plasma, les poussières collectent et émettent des particules (ions, électrons, neutres) qui participent à sa charge électrique, son chauffage, son érosion et son transport dans la machine. En se déplaçant dans le plasma, la poussière libère (par érosion et vaporisation/sublimation) des impuretés, dont les conséquences ont été évoquées plus haut. Les poussières ayant, dans certains cas, une taille faible ou de l’ordre de la longueur d’écrantage électrique du plasma (longueur de Debye), les particules chargées se trouvant dans le voisinage d’un grain ont un mouvement orbital. Le calcul de la section efficace de vi

Figure 2: Distribution en taille de poussières collectées dans le tokamak AUG. Source: [Rondeau 2015]

collection de particules chargées par une poussière sphérique, en l’absence de collisions et de champ magnétique, peut être effectué grâce à la théorie OML (Orbital Motion Limited). Cette théorie présente l’indéniable avantage de lier la surface de la poussière avec le plasma par des expressions simples ne nécessitant pas de résoudre l’équation de Poisson. L’OML permet ainsi de développer des codes numériques simulant le transport de poussières dans un plasma donné, le but étant de pouvoir, à terme, prédire leur transport pour mieux le contrôler. DUMBO (DUst Migration in plasma BOundaries), un tel code, implémente ce modèle. Les quatre équations présente dans le code peuvent être résumés ainsi :

charge électrique = électrons collectés − électrons émis + ions collectés taille = −sputtering − vaporisation/sublimation température = plasma collecté − électrons émis − radiation (2) − vaporisation/sublimation + recombinaison accélération = friction des ions + force de Lorentz + gravité DUMBO permet ainsi de simuler le transport d’une poussière jusqu’à sa complète vaporisation dans le plasma, ou jusqu’à ce qu’une collision avec le mur de la machine ait lieu. En parallèle de ce travail de modélisation, des données expérimentales doivent être obtenues. Malheureusement, les diagnostics dédiés au mesures de poussières sont rares, et le meilleur d’entre eux reste l’observation des trajectoires par caméra visible ou infrarouge. Des routines de traitement d’image permettant de détecter les poussières sur des films vii

Figure 3: Durée de vie des 123 grains de poussières détectés lors de l’expérience d’injection de TEXTOR, ainsi que le temps de vie des poussières simulées avec DUMBO dans les mêmes conditions. Le rayon initial des poussières dans les simulations est fixé à 10 µm.

et de reconstruire des trajectoires depuis une séquence d’images sont développées. Les données obtenues par le biais de ces routines peuvent être directement comparées avec des simulations DUMBO. La difficulté rencontrée pour effectuer des comparaisons valables réside dans l’estimation des paramètres initiaux des poussières : taille, position, vitesse, température, matériau. Ces données ne sont pas accessibles pour les poussières intrinsèques de tokamaks. La méthode utilisée pour réduire la quantité de variables inconnues est l’injection de pous- sières pré-caractérisées. De telles expériences devraient fournir des données de suffisam- ment bonne qualité pour valider ou invalider le modèle implémenté dans DUMBO. Lors de cette thèse, deux expériences d’injection de poussières ont été utilisées, réal- isées dans les tokamaks TEXTOR et KSTAR. Dans les deux cas, des poussières de W de taille ∼ 10 µm ont été injectés dans des plasmas stables. Les trajectoires expérimentales sont obtenues, et des simulations DUMBO sont réalisées pour tenter de reproduire les mesures en matière de longueur de trajectoire et de temps de vie des poussières. Le constat est sans appel : les trajectoires simulées sont, dans tous les cas, significativement plus courtes que les observations, comme le montre la Fig.3. La théorie OML conduit donc à une sous-estimation du temps de vie des poussières, due à une surestimation du chauffage de la particule. Le problème du surchauffage est répandu et a été observé dans d’autres travaux visant à comparer des trajectoires expérimentales avec des résultats de codes de transport de viii

Figure 4: Courants d’électrons collectés (Je) et émis (Jthe) en fonction du potentiel de la poussière, sans (OML) et avec (SCL) prise en compte des puits de potentiel.

poussières. Différentes manières de résoudre ce problème ont été proposées: diviser le flux de chaleur impactant la poussière par un coefficient ad hoc (DUSTT) ou utiliser des tailles de poussières plus grandes dans les simulations (DTOKS). Mais aucune de ces deux solutions ne peut être entièrement satisfaisante. Côté théorie, de nombreux progrès ont été effectués. La théorie OML classique néces- site le respect de nombreuses hypothèses contraignantes : plasma non magnétisé et non collisionnel, absence de barrières dans l’énergie potentielle effective des particules collec- tées. Cette série d’hypothèse peut être résumée ainsi : la poussière doit avoir une taille faible devant les principales longueurs caractéristiques du plasma environnant (longueur d’écrantage électrique, libre parcours moyen et rayon de giration). Dans des plasmas de bord typiques, la théorie OML n’est plus applicable pour des poussières plus grandes qu’environ 10 µm. Depuis une décennie, les progrès suivants ont été effectués : - Introduction des puits de potentiels. Lorsqu’une surface est en contact avec un plasma, une gaine électrostatique se forme au voisinage de cette surface. Elle se caractérise par un profil de potentiel électrique, classiquement monotone. Le potentiel de la surface est inférieur à celui du plasma car les électrons (chargés négativement), sont plus mobiles que les ions grâce à leur faible masse. Dans certains cas (sonde de Langmuir émettrices, poussières chaudes), le flux d’électrons émis est tel que le potentiel de la surface croît (pouvant atteindre des valeurs positives) et un puit de potentiel se crée dans la gaine. Ce puit de potentiel agit comme une barrière qui réduit les flux d’électrons collectés et émis par la poussière, comme on peut le voir sur la Fig.4. La théorie OML + propose de prendre en ix

compte l’effet de tels puits sur le courant d’électrons thermioniques. Une partie du travail de thèse a consisté à étendre l’OML+ en incluant les effets des puits de potentiel sur les électrons collectés, ainsi qu’en simplifiant les calculs. - Effets du champ magnétique. En présence d’un champ magnétique, les partic- ules chargées ont un mouvement hélicoïdal dont le rayon dépend de leur énergie. Les particules ayant l’énergie et la masse les plus faibles (les électrons émis) ont le rayon de giration le plus petit. C’est donc sur elles que les effets du champ magnétique seront les plus grands. Dû à ce mouvement de giration, une partie des électrons émis retombent promptement sur la surface de la particule. Le champ magnétique induit donc une réduction de l’émission d’électrons par un grain de poussière, ce qui conduit à des potentiels plus faibles, comme on peut le voir sur la Fig.5. Cet effet est cependant difficile à calculer, c’est pourquoi il est pris en compte par le biais de simulations Monte Carlo. - Régime collisionnel. Pour l’instant, les effets de la matière perdue par la pous- sière lorsqu’elle subit de l’érosion sur les interactions plasma-poussière ne sont pas pris en compte. Cette matière érodée forme deux régions distinctes autour de la particule : (i) un nuage de vapeur neutre, de forme sphérique ; (ii) un plasma sec- ondaire qui s’étend le long des lignes de champ magnétique. Les particules plasma impactant la poussière doivent circuler au travers de ces deux régions, perdant de l’énergie au passage, avant d’atteindre leur cible. Cet écrantage vapeur devrait donc résulter en des flux de chaleur plus faible sur la poussière. Si de nombreux modèles existent pour prendre en compte cet effet dans le cas des glaçons (composés d’hydrogène), ils ne sont pas applicables pour des matériaux au numéro atomique élevé, comme le W. Dans le cas des poussières, la littérature sur l’écrantage vapeur reste mince, et les efforts futurs doivent être fournis dans cette direction. Les progrès présentés plus haut (puits de potentiels et champ magnétique) tendent à réduire le problème du surchauffage car ils induisent des réductions de potentiel des poussières, et donc un chauffage électronique plus faible. Cependant, leurs effets ne sont pas suffisants pour que les simulations correspondent aux observations. L’écrantage vapeur, quant à lui, n’a pas été implémenté à cause du manque de modèles simples dans la littérature. Il manque donc encore certaines améliorations du modèle implémenté dans DUMBO avant que ce code puisse être utilisé de manière prédictive. Un dernier aspect primordial concernant le transport des poussières n’a pas été men- tionné jusqu’à présent : le problème des sources et puits de particules. En effet, même si le transport d’une poussière pouvait être calculé à l’aide de DUMBO, il faudrait connaître avec précision la quantité de poussière crée ou remise en suspension (remobilisée) ainsi x

Figure 5: Potentiel d’uns poussière en fonction de sa température sans et avec prise en compte des effets du champs magnétique sur les électrons émis.

Figure 6: Types d’impacts possibles en fonction de la vitesse d’impact de la poussière ⊥ sur une surface vi . que leurs paramètres initiaux (position, taille, vitesse, morphologie, . . . ), mais aussi com- ment ces poussières sont détruites, afin de pouvoir estimer la contribution des poussières au bilan impuretés du plasma. Les poussières peuvent être détruites de deux principales manières : vaporisation totale (calculée par DUMBO) ou collision avec la paroi du toka- mak résultant en la destruction ou l’adhésion de la poussière. Des modèles de mécanique du contact, tels que Johnson-Kendall-Roberts (JKR) et Thornton and Ning (T&N) per- mettent d’estimer les conditions sous lesquelles une poussière impactant une surface peut ou non y rester adhérée, le tout dépendant principalement de la vitesse d’impact de la particule. La Fig.6 montre schématiquement les types d’impacts possibles en fonction de la vitesse d’impact. Si l’impact est adhésif, il convient de connaître la force d’adhésion de la particule sur la paroi en vue d’estimer la possibilité pour cette poussière d’être remobilisée. Dans le cas du W, il est montré dans le travail de thèse que cette force d’adhésion dépend de la vitesse d’impact de la particule, et ce à cause de la déformation plastique irréversible ayant lieu lors de l’impact. xi

Figure 7: Forces d’adhésion de particules de tungstène sur différentes surfaces en fonction de la taille des particules. Le modèle de Rabinovich prédit très efficacement les valeurs mesurées. Mesures effectuées par S. Peillon et al. à l’IRSN Saclay.

Des mesures directes de forces d’adhésion réalisées par AFM (Atomic Force Mi- croscopy) par S. Peillon et al. à l’IRSN Saclay montrent que le modèle de Rabinovich, basé sur la force de van der Waals et incluant les effets de la rugosité de la surface, prédit la force d’adhésion de manière efficace (voir Fig.7). Ayant validé le modèle de Rabinovich, des extrapolations au cas tokamak peuvent être effectuées. Un particule collée à une surface en contact avec un plasma sera soumise à une force d’adhésion, une force électrique due à la présence de la gaine électrostatique et à la force de friction du plasma. La possibilité de remettre en suspension ces poussières dépend de leur matériau, leur taille, ainsi que des paramètres plasma locaux (température et densité). La Fig.8 (a) représente, en fonction des paramètres plasma, le rayon critique de particules de W à partir duquel les poussières sont remises en suspension. La Fig.8 (b) montre alors la vitesse des particules de 5 µm à la sortie de la gaine, i.e., à leur entrée dans le plasma. xii

Figure 8: (a) Contours du rayon critique pour remobilisation (en µm) pour des particules de tungstène sur une surface de tungstène rugueuse. (b) Contours de la vitesse de sortie de gaine (en m/s) pour les particules de 5 µm. List of Symbols

- me and mi: electron and ion masses - e: elementary charge

- ε0: vacuum permittivity - c: light speed - h: Planck constant

- kB: Boltzmann constant

- B: magnetic field - E: electric field

- Vi: ion flow velocity

- nα: number density of the species α

- Tα: temperature of the species α

- φ: electric potential - J: current density - I: current - Q: heat flux - F : force - σ: stress

- rd: dust radius

- Td: dust temperature

- md: dust mass

- Vd: dust velocity

- Qd: dust electric charge

- t: time - τ: characteristic time scales

List of Acronyms

- AFM: Atomic Force Microscopy - BW: Black and White - DEP: DiElectroPhoretic - DMT: Derjaguin-Muller-Toporov - DTOKS: Dust in TOKamakS - DUMBO: DUst Migration in plasma BOundaries - DUSTT: DUST Transport - DUSTTRACK: DUST-TRACKing - EBS: Electron BackScattering - ELM: Edge-Localized Mode - HFS: High Field Side - IBS: Ion BackScattering - IR: InfraRed - JKR: Johnson-Kendall-Roberts - LFS: Low Field Side - MIGRAINe: MIgration of GRAINs in fusion devides - NBI: Neutral Beam Injection - OM: Orbital Motion - OML: Orbital Motion Limited - PFC: Plasma-Facing Component - RMS: Root Mean Square - SCL: Space-Charge Limited - SEE: Secondary Electron Emission - SEM: Scanning Electron Microscopy - SOL: Scrape-Off Layer - THE: Thermionic Emission - T&N: Thornton and Ning - TPB: Trapped-Passing Boundary - VC: Virtual Cathode

List of fusion devices mentioned in the manuscript

- AUG: ASDEX (Axially Symmetric Divertor EXperiment) UpGrade (tokamak lo- cated in Garching, Germany) - COMPASS: COMPact ASSembly (tokamak located in Prague, Czech Republic) - DIII-D: Doublet III-D (tokamak in San Diego, USA) - DITE: Divertor Injection Tokamak Experiment (located in Culham, UK, in oper- ation until 1989) - EAST: Experimental Advanced Superconducting Tokamak (located in Hefei, China) - ITER: International Thermonuclear Experimental Reactor (located in Cadarache, France, under construction) - JET: Joint European Torus (located in Culham, UK) - JT-60U: Japan Tokamak Upgrade (located in Naka, Japan) - KSTAR: Korea Superconducting Tokamak Advanced Research (located in Daejeon, South Korea) - LHD: Large Helical Device (located in Toki, Japan) - MAST: Mega Ampere Spherical Tokamak (located in Culham, UK) - NSTX: National Spherical Torus Experiment (located in Princeton, USA) - TCV: Tokamak à Configuration Variable (located in Lausanne, Switzerland) - TEXTOR: Tokamak Experiment for Technology Oriented Research (located in Jülich, Germany) - TFTR: Tokamak Fusion Test Reactor (located in Princeton, USA, in operation until 1997) - Tore Supra: Tokamak located in Cadarache, France, in operation until 2011 - WEST: Tungsten (W) Environment Steady-state Tokamak (located in Cadarache, France)

Contents

Introduction1

1 Magnetic confinement fusion and generalities on dust3 1.1 Fusion facing global energy needs...... 4 1.1.1 Energy consumption and projections...... 4 1.1.2 Viability of fusion as an energy source...... 7 1.2 The tokamak configuration...... 8 1.2.1 Magnetic confinement...... 8 1.2.2 Plasma-wall interactions...... 10 1.3 Dust in magnetic fusion devices...... 13 1.3.1 Dust creation processes...... 14 1.3.2 Impacts of dust on plasma performances and stability...... 15 1.3.3 Consequences for the safety of fusion devices...... 19 1.3.4 Diagnostics for dust-related measurements...... 19 1.4 Goal of the thesis...... 21

2 Dust injection experiments and image processing 23 2.1 Introduction...... 24 2.2 Dust injection experiments...... 24 2.2.1 KSTAR injection experiment...... 24 2.2.2 TEXTOR injection experiment...... 26 2.3 Image processing for dust tracking...... 28 2.3.1 Observability of dust using cameras...... 28 2.3.2 State of the art: existing dust tracking codes...... 29 2.3.3 Dust detection on videos from CCD cameras...... 29 2.3.4 Trajectory reconstruction over a frame sequence...... 33 2.3.5 Deductions from slow camera data...... 35 2.4 Design of a gun-type powder injector for the WEST tokamak...... 38 2.4.1 Design and location on WEST ...... 38 2.4.2 Collimation tests...... 38 2.5 Conclusion on dust injection and image processing...... 44 xx Contents

3 Dust-plasma interactions 45 3.1 Introduction...... 46 3.2 Generalities on sheath physics...... 46 3.2.1 The Maxwellian distribution...... 46 3.2.2 Poisson’s equation...... 47 3.2.3 1D sheath model without electron emission...... 47 3.2.4 1D sheath model with electron emission...... 50 3.3 Behavior of a particle in a central force field...... 53 3.3.1 The Yukawa potential...... 53 3.3.2 Energy and angular momentum...... 54 3.3.3 Bounded and unbounded particles...... 54 3.3.4 Effective potential energy...... 55 3.4 The Orbital Motion Limited approach...... 57 3.4.1 Assumptions and collection cross-section...... 57 3.4.2 2D sheath model with electron emission in the OML framework. 59 3.5 The general Orbital Motion theory...... 60 3.6 Models for other parameter regimes...... 62 3.6.1 Thin sheath regime...... 62 3.6.2 Dense and strongly magnetized plasmas...... 62 3.7 Conclusion on dust-plasma interactions...... 63

4 Dust transport in the tokamak vacuum vessel 65 4.1 Generalities...... 67 4.1.1 State of the art: existing dust transport codes...... 67 4.1.2 General aspects on DUMBO ...... 68 4.2 The plasma background...... 69 4.2.1 Background from plasma modelling codes...... 69 4.2.2 Background from experimental profiles...... 71 4.3 Dust charging in the OML regime...... 73 4.3.1 Main plasma collection...... 74 4.3.2 Electron emission...... 76 4.3.3 Current balance and dust electric charge...... 81 4.4 Vaporization and sputtering...... 82 4.4.1 Sputtering...... 82 4.4.2 Vaporization/sublimation...... 88 4.4.3 Dust mass equation...... 88 4.5 Heat collection...... 88 Contents xxi

4.5.1 Heat fluxes from plasma collection, THE and SEE...... 89 4.5.2 Electron backscattering...... 90 4.5.3 Ion recombination and backscattering...... 91 4.5.4 Radiative cooling...... 94 4.5.5 Vaporization/sublimation and sputtering...... 95 4.5.6 Heating equation...... 95 4.5.7 Phase changes...... 96 4.6 Dust motion...... 98 4.6.1 Forces acting on a dust grain...... 98 4.6.2 Equation of motion...... 101 4.7 Numerical aspects...... 102 4.8 Comparison between dust transport codes...... 102 4.9 Conclusion on dust modelling...... 103

5 Confronting the model with measurements 105 5.1 Comparison between simulated and experimental dust trajectories.... 106 5.1.1 Applicability of DUMBO ...... 106 5.1.2 TEXTOR injection experiment...... 107 5.1.3 KSTAR injection experiment...... 108 5.1.4 Qualitative comparison with data from Tore Supra ...... 111 5.2 The overheating issue...... 114 5.3 Recent progress...... 115 5.3.1 Influence of the magnetic field on electron emission...... 115 5.3.2 Vapor shielding...... 123 5.3.3 Effects of potential wells on thermionic electrons...... 125 5.3.4 Full Space-Charge Limited theory...... 127 5.4 Conclusion...... 142

6 Dust-wall interactions in tokamaks 145 6.1 Introduction...... 146 6.2 Dust-wall collisions...... 146 6.2.1 Adhesion of dust onto a planar surface upon collision...... 147 6.2.2 Bouncing collisions...... 152 6.2.3 Accounting for the roughness of the wall...... 153 6.2.4 Dust and/or wall local damaging...... 154 6.3 Remobilization/resuspension...... 155 6.3.1 Adhesion of dust on rough planar surfaces...... 156 xxii Contents

6.3.2 The particular case of W: importance of the impact velocity... 158 6.3.3 Experimental investigation of the adhesion force...... 165 6.3.4 Remobilization condition and exit velocity in tokamaks...... 167 6.3.5 Conclusions on dust remobilization studies...... 173 6.4 Conclusion on dust-wall interactions...... 173

Conclusion 175

Appendices 187

Bibliography 193 Introduction

Global energy consumption is on the rise. Various sources of energy are currently being used, from fossil fuels to renewables to nuclear, each of which presenting advantages and shortcomings. The quality of an energy source is measured through its adaptation to the problem it is brought to solve. Fossil fuels remain the most widely used source of primary energy, reaching about 80% of global supply. Yet, the cons of coal, oil and gas are becoming increasingly prominent: greenhouse gas emissions, leading to global warming, rise of ocean levels, and so on. The need for alternative, carbon-free energy sources is therefore deepening. In this framework, nuclear fusion can become a major player in the near future. Used as a centralized power generator for the electricity grid of a country (in the same way as current fission power plants), it would present many advantages: stable power production, unprecedented fuel abundance, marginal amounts of wastes produced, low risks (i.e., low probability and low severity) of accidents, just to name the main ones. Yet, the fusion fuel used, an isotopic mixture of hydrogen (H), must be brought to awfully high temperatures in order for the fusion reactions to occur, since the atomic nuclei are made of protons which repel each other due to their positive electric charges. At these temperatures of hundreds of millions of degrees, the gaseous medium reaches the plasma state, where atoms are stripped of their electrons. No known material would sustain being in contact with such a hot burning plasma. Thankfully, plasma particles being electrically charged, it is possible to confine the plasma in a magnetic “bottle”. This is the central idea of the magnetic confinement fusion principle, whom the tokamak is the most advanced finial. The word itself, tokamak, is a Russian acronym standing for “toroidal chamber with magnetic coils”, which perfectly describes the device. It was invented in the 1950s by soviet physicists Igor Tamm and Andrei Sakharov, inspired by a letter by Oleg Lavrentiev. If the tokamak is the perfect energy provider, why are we still burning oil? one might ask. The answer is simple: creating a plasma is one thing, achieving a burning plasma at reasonable cost and sustaining it is a completely different challenge. Even after half a century of studies, and even without focusing on the unanswered question of the price of fusion electricity, this process is still not ready for the energy market. This is due to the technical and physical complexity of such a reactor, which, to this day, still challenges scientists. It has even been said that fusion will be ready in 20 years, and forever will be. Yet, this is an oversimplification of the problem, as plenty of progress has been achieved since the 1950s: tokamaks TFTR (Tokamak Fusion Test Reactor) and 2

JET (Joint European Torus) achieved the first controlled release of fusion power in the 1990s, the latter holding the record for the highest peak of fusion power (about 15 MW). The Tore Supra tokamak made the longest stable low confinement mode discharge, also in the 1990s, lasting 6 minutes, while the EAST (Experimental Advanced Superconducting Tokamak) tokamak holds the record for the high confinement mode with about 100 s. The upcoming ITER (International Thermonuclear Experimental Reactor) project, currently under construction in Cadarache, France, will be the biggest experiment to date, in every sense. Largest tokamak ever built with a major radius of more than 6 m, but also the most expensive, its price being initially of the order of 10 billion euros. It is also the widest collaboration to date in the field, bringing together the European Union, the United States, Russia, China, India, South Korea and Japan. ITER aims at pretty big numbers: 500 MW of fusion power with 50 MW of heating power, bringing its amplification factor to about 10, and continuous plasma discharges up to 400 s. Facing these numbers, it cannot be supported that fusion is not making any progress. Among the main problems encountered today are plasma instabilities, impurities and plasma-surface interactions, the latter being the focus of the present manuscript. Like two faces of the same coin, plasma-surface interaction has severe consequences for both: the plasma damages the surface (via erosion, melting, tritium implantation, . . . ) whilst the surface pollutes the plasma with impurities. A major phenomenon occurring at this interface is the production of dust, small particles of wall material that are, for various reasons, transported into the plasma. Dust is mainly harmful since it is a source of impurities that can be located deep inside the plasma. For long pulse operation of a fusion power plant, the amount of impurities in the plasma must be knowable and controllable to some degree, thus it is mandatory to study dust transport as well as dust sources and sinks in fusion plasmas. All of this raises numerous questions: how is dust created in tokamaks? How can we acquire data on dust? What physics plays a role in the transport of dust in edge plasmas? What happens when a dust grain hits the tokamak walls? If a dust grain is resting on the wall, can it be resuspended and reach the plasma again? The present manuscript aims at bringing some elements of answers to these questions. To do so, we will dive into the physics of dust transport and dust adhesion on surfaces. But fist of all, let us start by introducing the notions of fusion, plasmas, plasma-wall interactions and some generalities on dust. Chapter 1 Magnetic confinement fusion and generalities on dust

Contents 1.1 Fusion facing global energy needs...... 4 1.1.1 Energy consumption and projections...... 4 1.1.2 Viability of fusion as an energy source...... 7 1.2 The tokamak configuration...... 8 1.2.1 Magnetic confinement...... 8 1.2.2 Plasma-wall interactions...... 10 1.3 Dust in magnetic fusion devices...... 13 1.3.1 Dust creation processes...... 14 1.3.2 Impacts of dust on plasma performances and stability...... 15 1.3.3 Consequences for the safety of fusion devices...... 19 1.3.4 Diagnostics for dust-related measurements...... 19 1.4 Goal of the thesis...... 21 4 Chapter 1. Magnetic confinement fusion and generalities on dust

1.1 Fusion facing global energy needs

1.1.1 Energy consumption and projections

Since the industrial revolution, human societies have been relying on the predominant use of machines to produce everything they consume, from goods (clothing, food) to transportation and the electricity lighting people’s homes. This is easily understood as they are way more efficient than humans to generate power. Contrary to us, machines are commonly said to “consume” energy, even though the term “transform” would be more adequate. This is why the discovery of the machine along with the abundance of energy resources such as fossil fuels lead to a grand increase in global energy consumption over the last century that keeps going on today, as can be seen in Fig. 1.1. Fossil fuels have been the most successful source of energy so far, owing to their high energy concentration, low extraction cost, relative safety of use and good capability of storage and transport. Yet, the renewal time scale of fossil fuels is so large compared to the typical time scale of the evolution of human societies that their initial quantities can be assumed to be constant. Thus global reserves of fossil fuels are irredeemably decreasing. In the last few decades, the consideration of environmental constraints on energy policies has become more and more important. Indeed, the combustion of fossil fuels produces greenhouse gases such as carbon dioxide (CO2), the accumulation of which in the atmosphere leads to an increase in the average temperature and ocean levels. This is already being observed, see Fig. 1.2. The alteration of Earth atmosphere’s composition also has important social and hu- man consequences, such as famines, respiratory diseases or migrations of climate refugees. Since the report by the Club of Rome in 1972, it is known that a society ignoring the limits in the stocks of non-renewable energy sources and the capacity of planetary outlets (i.e., pollution storage pools) would run towards economic, demographic and ecological collapse [Meadows 1972] (see Fig 1.3). Thus it is compulsory to replace fossil fuels with low carbon energy sources if the human carbon footprint is to be reduced. Several alternative energy sources exist and are being increasingly used (see Fig. 1.1), all of which presenting the undeniable advantage of emitting low amounts of greenhouse gases. Hydroelectricity is the most developed, owing to its high energy concentration and fast responsiveness. Yet, the total amount of hydropower is limited and far from being sufficient to satisfy today’s global energy demand [WEC 2016]. Biomass requires large soil areas that can compete with agriculture lands. Solar and wind are the most rapidly expanding, yet their intermittent nature is an 1.1. Fusion facing global energy needs 5

Figure 1.1: Global energy consumption in recent years. Source: [BP 2017]

Figure 1.2: Global temperature increase over the past millennium. Source: [Chen 2011, Mann 1999] 6 Chapter 1. Magnetic confinement fusion and generalities on dust

Figure 1.3: Projections of relevant indicators calculated with the World3 code in 1972. “Business as usual” scenario: no major change in physical, economic or social relation- ships. Source: [Meadows 1972]

issue for human electricity networks that require stable input. Thus large energy storage facilities would be required to mitigate their irregularity. Nuclear fission is also widely developed (∼ 5% of total consumption), particularly in rich countries, and uses highly concentrated fissile material as fuel for a stable elec- tricity production at relatively low and stable cost. Despite these advantages, fission produces long-life radioactive wastes that require adequate management and storage so- lutions. Also, the multiple accidents endured by the field, along with the risks of military weaponry proliferation, participate to its unpopularity. Finally, the current estimates of worldwide uranium pools yield ∼ 100 years of nuclear power with current reactor technologies and consumption rate, even though new solutions are under extensive study (Generation IV, Thorium, . . . ). 1.1. Fusion facing global energy needs 7

Figure 1.4: Reactivity of different fusion reactions against the plasma ion temperature. Source: [Chen 2011]

1.1.2 Viability of fusion as an energy source

Nuclear fission of heavy nuclei and fusion of light ones release a fraction of the binding energy of the initial nucleus whilst generating products tending towards iron (Fe), which is the most stable chemical element on the periodic table. If the first is more commonly used than the latter to generate electricity on Earth, fusion is ubiquitous in the Universe as stars are fusion powered. Concerning fusion on Earth, the fuel used is a 50 − 50 mix of deuterium (D) and tritium (T), which produces a helium (He) nucleus and an energetic neutron with the following reaction

2 + 3 + 4 2+ 1 1D + 1T → 2He (3.5 MeV) + 0n (14.1 MeV). (1.1)

This specific reaction has been chosen due to its high efficiency compared to others, as can be seen in Fig. 1.4. The reagents and products of this reaction enclose all the advantages of fusion as an energy source. Indeed, D is abundant and homogeneously distributed on Earth since it can be extracted from the oceans (up to 33 g/m3 of sea water). T, on the other hand, is an unstable nucleus with a half-life of 12.6 years, so it is not present in significant quantities on the planet. Yet, it can be produced from lithium (Li), which can be extracted from salt flats, using the following reaction

1 6 4 3 0n + 3Li → 2He (2.1 MeV) + 1T (2.7 MeV). (1.2) 8 Chapter 1. Magnetic confinement fusion and generalities on dust

At the current energy consumption rate, the estimation of the amount of fusion fuel could power humanity for ∼ 108 years. In addition to virtually no greenhouse gas emis- sion, the radioactive wastes produced would be limited to the irradiated material struc- tures of the reactor itself. Thus fusion conserves the advantages of a fission power plant (centralization, manageability, emission-free), without the issues linked with the chain reaction process. This explains why tremendous efforts are put into solving the techno- logical limits and increasing the physical knowledge in order to make nuclear fusion a viable commercial energy source. This is particularly the case through the ITER project, dedicated to the demonstration of the scientific and technical feasibility of fusion as a source of energy.

1.2 The tokamak configuration

1.2.1 Magnetic confinement

Fusion is the process where two light nuclei come close enough to form a heavier one. Since nuclei are positively charged, they repel one another via Coulomb forces. This Coulomb barrier must be overcome for the reaction to occur. To be more precise, it must be almost overcome, since the quantum tunneling effect helps. Practically, this is done by increasing the pressure of the system. In stars, the pressure is high thanks to the gravitational forces which induce high density in the core. On Earth, such forces are not easily reproduced and another solution is to focus on a high temperature medium. In such conditions, nuclei are very fast and their kinetic energy is used to overcome the Coulomb barrier. One other way to look at the problem is to use the Lawson criterion, which gives a lower boundary of plasma parameters for the plasma to reach ignition, i.e., the point where fusion reactions release enough heat to sustain the plasma temperature without the need for external heating systems [Lawson 1957]. According to Lawson, a plasma reaches ignition if the following condition is respected

21 3 niTiτE > 5 × 10 keV.s/m , (1.3) where ni and Ti are the plasma ions density and temperature, respectively, and τE is the plasma energy confinement time, which is the typical time it takes for the plasma to lose its energy to its environment. Temperatures are usually expressed in electronvolts (eV), with 1 eV corresponding to about 11600 K. As said above, gravitational confinement fusion benefits from high density and confinement time to compensate for its relatively low 1.2. The tokamak configuration 9

Figure 1.5: Helical motion of a charged particle around a magnetic field line. Source: [Eurofusion 2017]

temperature (∼ 1 keV). Fusion occurring on Earth is characterized by high temperature. Inertial confinement fusion associates low confinement time with high density, whereas magnetic confinement fusion focuses on the opposite. From Fig. 1.4, we can see that the maximum in D-T reaction efficiency is reached at a very high temperature (20 keV in ITER conditions). Such temperatures exceed the binding energy of electrons to the D and T nuclei, so the fuel is in the state of plasma, i.e., a fluid-like mixture of positive ions (nuclei) and electrons. A crucial issue with fusion is the high temperature of the plasma and how to confine it, since no known material could withstand such temperatures. Thankfully, due to being composed of charged particles, a plasma is sensitive to electromagnetic forces. Indeed, a charged particle immersed in a homogenous magnetic field will have a helical motion around a field line, as can be seen in Fig. 1.5. One solution to making fusion on Earth is thus to use magnetic fields to confine the plasma. This is magnetic confinement fusion. Aligning a series of coils will result in the plasma having a single degree of freedom coinciding with the axis of symmetry of the coil system. This cylindrical configuration presents issues at the two edges of the cylinder, so it seems natural to “close” it in order to form a torus. Then, in theory, charged particles should be perfectly confined on circular field lines thanks to the now called toroidal magnetic field. However, other forces at play can challenge this ideal view, such as collision between particles, electric fields and magnetic field inhomogeneities. The latter presents the most difficulties in a torus-shaped reactor: the coils used to generate the magnetic field are closer to one another at the center of the torus than on the outer side, meaning that the magnetic field intensity will be higher on the inner side. This magnetic field gradient, inherent to the toroidal shape of the machine, induces a vertical drift motion on the charged particles composing the plasma, which will be of opposite direction for ions and electrons (due to their opposite electric charge). Such a charge separation is a major source of plasma instabilities. A 10 Chapter 1. Magnetic confinement fusion and generalities on dust

Figure 1.6: Left: magnetic elements allowing for the tokamak configuration. Right: Nested magnetic flux surfaces. Source: [Jardin 2017, Chen 2011]

solution to this problem is to “mix” the plasma by bending the magnetic field lines into a helical shape so that, on average, this drift is largely reduced. This can be achieved by inducing an electrical current in the plasma itself, in the toroidal direction. This current in turn induces a secondary magnetic field in the poloidal direction, hence its name: poloidal magnetic field. The sum of these two components generate the desired helical field lines. Practically, the plasma current (and, as a consequence, the poloidal magnetic field) is generated by a coil placed vertically at the center of the torus. It is called the central solenoid and creates the plasma current by induction, i.e., using the same method as an electric transformer. Thus the torus is intrinsically a pulsed device, except if the plasma current can be generated by external heating systems (cyclotron radio-frequency, lower hybrid current drive, . . . ), self-generated (bootstrap current), or avoided by using a proper magnetic configuration (which is the principle of stellarators) [Wesson 1997]. See Fig. 1.6 for a schematic view of a tokamak.

1.2.2 Plasma-wall interactions

Due to the presence of material surfaces (the wall), the magnetic configuration described above can be separated into two drastically different regions: that where the field lines close on themselves and that where they are open (i.e., pass through the wall at some point). The first region is called the core as it is where the confined plasma lies. As mentioned in the previous Section, despite the presence of magnetic fields, perfect confinement is nonsense and the plasma tends to expand in its accessible volume. This can be seen as the Second Law of thermodynamics, collisions between particles, drift 1.2. The tokamak configuration 11 velocities, turbulence, instabilities, . . . Due to all of these, some plasma is also present in the open field lines region of the vacuum chamber [Stangeby 2000]. In this region, called the Scrape-Off Layer (SOL), plasma particles are accelerated towards the points where the field lines encounter wall surfaces. In the SOL, there is a competition between the parallel velocity and perpendicular drifts (“parallel” and “perpendicular” refer to the field lines). Since the parallel motion is much faster than the perpendicular one, the SOL is quite thin (typically a few cm in current tokamaks and ITER projections down to 1 cm). The surface separating the core from the SOL is the last closed flux surface, commonly called the separatrix. In Fig. 1.7, the plasma is represented in a poloidal cross-section in two configurations: - Limiter configuration. The limiter is the plasma-facing surface the separatrix grazes with. The plasma has a roughly circular shape. - Divertor configuration. An additional coil in the lower part of the machine allows the creation of this plasma shape. The separatrix is twisted and presents a so-called X-point. The plasma touches the wall on two particular locations (strike points) in the divertor region. Initially, the limiter configuration was used, owing to its simplicity. Yet, since the confined plasma is directly in contact with the limiter, material eroded from the wall would penetrate the plasma and pollute it easily. There lies the advantage of the divertor configuration: the plasma-wall interaction regions are further away from the hot plasma core. Plasma-wall interactions are one of the most important challenges of fusion devices, because of the high heat fluxes the wall is subjected to. In ITER, the heat fluxes close the strike points are expected to be ∼ 10 MW/m2 in steady-state, with transients up to ∼ 100 MW/m2. These fluxes induce erosion of the wall, sometimes local melting (depending on the material chosen), important local damages due to plasma transient events, activation of plasma-facing components due to the trapping of radioactive T and neutrons produced by the fusion reactions, and others. The focus of the present manuscript is made on another important consequence of plasma-wall interaction: dust, which are particles composed of wall material. They can be transported into the plasma, thereby jeopardizing its stability and reducing its performances due to the release of large quantities of impurities. 12 Chapter 1. Magnetic confinement fusion and generalities on dust

Figure 1.7: Left: the limiter configuration, where the last closed flux surface (separatrix) is, roughly, a cylinder in contact with a limiter on the Contact Point (CP). Right: the di- vertor configuration, where there is an X-point and the separatrix touches the wall on two particular locations, the Inner and Outer Strike Points (ISP and OSP), thereby secluding plasma-wall interactions from the core. Between the two strike points, the Private Flux Region (PFR) lies, where there is little to no plasma at all. Source: [Gallo 2018] 1.3. Dust in magnetic fusion devices 13

Figure 1.8: Observation of “UFOs” in an unstable discharge of the DITE tokamak, in 1982. Source: [Goodall 1982]

1.3 Dust in magnetic fusion devices

The first observations of dust in tokamaks trace back to the early 1980s. A good example is from an unstable discharge in the DITE tokamak [Goodall 1982], where “UFOs” were observed via high speed cine photography, using 100 ft of film to record 0.2 s of a plasma discharge. See Fig. 1.8 for an example of observation. In the 1990s, the attention of the scientific community on the possible consequences of the presence of dust in tokamak plasmas grew significantly. Using diagnostics dedi- cated to dust and samples harvested during the shutdown periods of various machines, the mechanisms at play for their creation were revealed and the consequences for plasma oper- ations started to be discussed, mainly concerning T retention [Winter 1998, Winter 1999]. In the life cycle of a dust grain, several primordial events can be identified, as can be seen in Fig. 1.9. A dust grain is created by phenomena linked with plasma-wall interaction that will be discussed in the next Section. The interaction of the dust grain with the surrounding plasma (Chapter3) leads to various forces inducing its transport in the tokamak vacuum vessel (Chapter4). Whilst being transported, the grain releases impurities that can have harmful consequences for the plasma itself (as will be seen at the end of this Chapter). Another possible outcome of a dust trajectory is its collision with plasma-facing compo- nents resulting in either the dust sticking to the surface or the dust and/or the wall being locally damaged. In the first case, the grain can be remobilized later due to plasma-wall interaction (Chapter6). 14 Chapter 1. Magnetic confinement fusion and generalities on dust

Figure 1.9: Schematic view of the life cycle of a dust grain in a tokamak plasma.

1.3.1 Dust creation processes

As said above, dust is created by plasma-wall interaction via different phenomena. The production of dust is assured by a steady-state plasma and can be enhanced by transient plasma events.

Steady-state plasma. When in steady-state, the plasma ions located in the SOL are accelerated towards the divertor plates near the strike points. This is due to the presence of a so-called Debye sheath, which will be explained in detail in Chapter3. The sheath is characterized by an important electric field near the wall which accelerates ions up to supersonic velocities. Such high energy species impacting a solid surface result in atoms or molecules being ejected from the wall. This phenomenon, called sputtering, is a source of dust. The large heat fluxes (steady-state and transient) impacting the surfaces can also lead to vaporization/sublimation which, similarly to sputtering, result in the emission of matter from the wall. Eroded material can be redeposited on wall surfaces, forming co-deposited layers, which are highly unstable due to internal stresses and poor thermal conductivity, and can easily be broken by transient plasma events. However, in future ITER operations, the plasma properties will be completely different than in current tokamaks. The plasma will be operated in detachment mode (low plasma temperature and high ion density) in order to reduce the conducted power on the divertor targets as much as possible. Also, the plasma discharges will last longer, and injection of a radiator (such as argon) will enhance target sputtering. In that case, the dusty plasma laboratory conditions could be at play, leading to the creation of smaller particles by accretion [Couëdel 2014]. The expected size of the produced particles is much smaller than those currently observed, ranging from tens to hundreds of nanometers. Due to this creation process, the surface morphology of the particles is complex (with high fractality). In terms of toxicity, they will be the most active particles due their high specific surface 1.3. Dust in magnetic fusion devices 15 area.

Transient plasma events. Amongst the many instabilities encountered in tokamak plasmas, two are of main interest here since they can be major sources of dust: Edge- Localized Modes (ELMs), disruptions and impacts of runaway electrons. ELMs occur when the plasma is in high confinement mode (or H-mode), a regime where the pressure in the core region is high and presents a strong gradient near the separatrix (the pedestal). In H-mode, sudden and periodic bursts of plasma particles and energy, ELMs, are released from the core region into the SOL, leading to heat fluxes on the targets much higher compared with the inter-ELM periods. ELMs lead to an increase in wall erosion and can induce local melting of metallic wall surfaces. This splashing liquid metal droplets are also dust. Disruptions are, on the other hand, particular events that lead to the brutal termination of a plasma discharge. Due to some plasma instability, plasma confinement is lost as the magnetic field becomes stochastic. Similarly to ELMs, disruptions lead to important heat and particle fluxes on the wall, thus have the same consequences in terms of dust creation. During disruptions, electric arcs can occur where the wall surface presents spikes. The high current carried by the arc heats up the wall by Joules effect, also leading to the creation of dust. Runaway impacts are highly energetic electrons that may be generated during disruptions. They appear when the electric fields accelerates plasma electrons up to relativistic velocities without the plasma resistivity to slow them down. A runaway beam impacting the tokamak wall is an important source of dust particles, especially droplets. See Fig. 1.10 for an image of runaway impact. Finally, these transients events can break the co-deposited layers created by erosion, leading to the expulsion of flaky dust particles. All the creation processes discussed above imply that tokamak dust can have very dif- ferent sizes and morphologies. Accreted dust grains are usually rather small (∼ 10 − 100 nm), flakes are larger (∼ 0.1−10 µm) and spheroids are even bigger (∼ 1−100 µm). This can be seen, for example, in Fig. 1.11, where W dust was collected during a shutdown pe- riod of the AUG (ASDEX UpGrade: Axially Symmetric Divertor EXperiment) tokamak. Two distinct dust populations (flakes and spheroids) of different sizes are observed.

1.3.2 Impacts of dust on plasma performances and stability

Dust grains present several issues for the operation of tokamaks, the main one being linked with impurities. In a fusion plasma, any ion species other than main ions (reagents and products of the fusion reaction) is considered as an impurity. In this sense, dust grains, which are made out of materials present in the tokamaks walls, are impurity vectors. 16 Chapter 1. Magnetic confinement fusion and generalities on dust

Figure 1.10: Runaway impact on the wall of the Tore Supra tokamak, shot #43493.A large amount of dust, appearing as white dots due to their high temperature, is created.

Figure 1.11: Size distribution of W dust collected in the AUG tokamak. Two populations are identified: flakes at 0.6 µm and spheres at 1.8 µm. Source: [Rondeau 2015] 1.3. Dust in magnetic fusion devices 17

Figure 1.12: Maximal W concentration allowing ignition of the plasma. It is assumed that the confinement time of He is 5τE. Source: [Pütterich 2010]

A dust grain releases large amounts of these impurities in the plasma since they are subjected to high erosion and vaporization/sublimation rates. Even worse, they can be transported deep in the core. The consequences of impurities on the plasma have been extensively studied. Mainly, impurities are, compared to main ions, high Z and cold species that ionize and ther- malize with the plasma, inducing an increase in density and decrease in temperature. Also, impurities tend to radiate a non-negligible part of the plasma energy due to ioniza- tion, radiative recombination, line emission and Bremsstrahlung [Jardin 2017], thereby significantly reducing plasma performances. The presence of a too high concentration of impurities in a plasma can even prevent it from reaching the ignition point. This can be seen in the case of W in Fig. 1.12, where the ignition contours are plotted against the triple product of the Lawson criterion and the plasma temperature for different W concentrations, cW . Since, for a given cW , ignition is permitted inside each contour, one can see how attaining ignition becomes increasingly difficult in a polluted plasma. The effects of impurities are also widely observed experimentally. As an example, Fig. 1.13 shows the evolution of different plasma parameters in a JET discharge as a dust grain penetrates the plasma. Data from camera KL11 shows that, at t = 47.3 s, dust grains are emitted from the divertor region and reach the plasma, which leads to a large increase in the radiative power of the plasma. Such mechanisms can lead, in extreme cases, to disruptions, which may create even more dust. 18 Chapter 1. Magnetic confinement fusion and generalities on dust

Figure 1.13: Data from JET pulse #86529. (Top) Images from camera KL11 at three times. (Bottom) Radiative power measurement. 1.3. Dust in magnetic fusion devices 19

1.3.3 Consequences for the safety of fusion devices

Plasma-facing materials tend to trap light elements like H isotopes through various phys- ical processes [Hodille 2016]. This can be an issue in the case of T because of its radiotox- icity. Recent studies have shown that the migration of T in plasma-facing components is critical for ITER as it could reach the cooling system [Hodille 2017]. Dust grains tend to trap larger quantities of T than bulk materials because of their large specific surface area. This means that dust could play an important role in the ITER T management scenarios [El-Kharbachi 2014, Grisolia 2015]. As a reminder, the maximum amount of T in the ITER vessel cannot exceed 700 g [Roth 2009]. Another effect is linked with the natural radioactive decay of T, following

3 3 + − 1T → 2He + e +ν ¯e. (1.4)

If the reaction occurs in a dust grain near its surface, the electron can escape, leading to a net positive electric charge. Thus tritiated dust can acquire a time increasing positive charge. This can have important effects on adhesion and resuspension of dust, either by plasma-induced forces or during Loss Of Vacuum Accidents (LOVA). In the unlikely case of a LOVA, tritiated dust can be easily resuspended by air flows and released in the atmosphere, thereby making dust an important vector of radioactive contamination for tokamak accidents.

1.3.4 Diagnostics for dust-related measurements

Dust grains being very small and moving rather rapidly (∼ 1 − 103 m/s), it is difficult to design a diagnostic to obtain precise measurements. The amount of data available on dust is usually rather sparse. Still, there exist a wide variety of diagnostics allowing measurements of different dust-related quantities. From the reviews in [Rudakov 2008, Annaratone 2009, Ratynskaia 2011], they can be separated into two main groups: in plasma and on surface diagnostics.

In plasma diagnostics. The most widely used tool to obtain dust measurements is CCD imaging. As a dust grains penetrates the plasma, it heats up rapidly and soon emits light in both the visible and infrared (IR) spectra. Among all the diagnostics available in tokamaks for dust-related measurements, the most reliable and data providing one is fast visible or IR CCD cameras. Most tokamaks host several cameras and at least one wide angle view that allows the observation of a large portion of the vacuum vessel. Nevertheless, the videos obtained give access to the dust trajectory projected on the 20 Chapter 1. Magnetic confinement fusion and generalities on dust camera sensor plane only, which can be considered as rather poor data, considering that the dust transport is a complex function of the local plasma conditions and dust parameters (as we will see in Chapter4). Still, this technique can provide valuable data for the benchmarking of dust transport codes. The quality of the data depends on the CCD camera resolution and frame rate, as most tokamak cameras have a frame rate of 60 frames/s. In some cases, when two cameras simultaneously observe the same region of the vacuum vessel, 3D dust trajectories can be reconstructed. Another diagnostic is laser scattering, which is also commonly used in dusty plasmas. Analysis of the light originating from a laser and scattered by dust grains allows the estimation of the dust size distribution and density. Also, when high velocity dust approaches Langmuir probes, the dust velocity can be inferred. Collection of dust using highly porous materials such as aerogel allows the determination of the dust velocity and size distributions on wide ranges of velocity and size. The main drawback of this technique is the high sensitivity of such materials to high temperatures.

On surfaces diagnostics. The main way to obtain measurements on dust on surfaces is to collect dust samples during shutdown periods and perform postmortem analysis. Analysis of these samples provide the dust size distribution, composition, morphology and estimations of the total dust inventory of the machine. The main shortcoming comes from the fact that measurements are integrated over a large periods of time, after nu- merous plasma discharges, and no correlation between the measurements and the plasma conditions can be drawn. Many other on-surfaces diagnostics exist (electrostatic detec- tors, capacitive diaphragm microbalance, infrared thermography, laser-induced break- down scattering, . . . ).

Possibility to use dust as a plasma diagnostic. Using a different approach, some have proposed to use dust to their advantage, i.e., as part of a plasma diagnostic. In [Wang 2003], it is proposed to inject small and high velocity dust grains in a plasma for magnetic field mapping. Due to erosion and vaporization, the grains lose matter that quickly ionizes. A cloud of ablated material forms and extends along the magnetic field lines, somewhat like a comet plume. Using fast CCD images, the local direction of the magnetic field can be inferred. The main shortcoming of this technique is linked with impurities, which was already discussed. 1.4. Goal of the thesis 21

1.4 Goal of the thesis

From the discussions of this Chapter, it appears that poor understanding of the physics related to dust transport in tokamak plasmas could result in operation and safety issues in tokamaks. Even though tremendous work has already been done on this subject (as we will see later on), precise quantitative predictions on dust sources and sinks, as well as accurate dust transport models are still lacking. In this light, the main goal of this thesis is to improve existing dust-plasma and dust- surfaces models in order to help predict (therefore, control) the transport of dust in fusion plasmas. In detail, several objectives can be drawn out: - Develop a simple and fast-calculating dust transport code in order to test existing models and improvements. This code could be used later on to predict tokamak impurity inflows related to a known source of dust. - Compare experimental data with simulations to verify the acuity of the code. In order to do so, we have to: – Perform pre-characterized dust injection experiments. – Develop image processing routines to obtain good quality experimental data, since videos are the best dust diagnostic in tokamaks. – Design a dust injector for the WEST (W Environment Steady-state Tokamak) tokamak to obtain experimental data on long-pulse discharges. - Study the case of dust-wall collisions to estimate the possibility for dust to stick to surfaces and be remobilized.

Chapter 2 Dust injection experiments and image processing

Contents 2.1 Introduction...... 24 2.2 Dust injection experiments...... 24 2.2.1 KSTAR injection experiment...... 24 2.2.2 TEXTOR injection experiment...... 26 2.3 Image processing for dust tracking...... 28 2.3.1 Observability of dust using cameras...... 28 2.3.2 State of the art: existing dust tracking codes...... 29 2.3.3 Dust detection on videos from CCD cameras...... 29 2.3.4 Trajectory reconstruction over a frame sequence...... 33 2.3.5 Deductions from slow camera data...... 35 2.4 Design of a gun-type powder injector for the WEST tokamak. 38 2.4.1 Design and location on WEST ...... 38 2.4.2 Collimation tests...... 38 2.5 Conclusion on dust injection and image processing...... 44 24 Chapter 2. Dust injection experiments and image processing

2.1 Introduction

We saw in Chapter1 that videos obtained from CCD cameras of intrinsic dust trajec- tories, while being one of the best diagnostics available for dust studies, remain rather poor in data quality, since only 2D trajectories are observed. A way to diminish the amount of unknown parameters is to perform pre-characterized dust injection experi- ments. Knowing the dust material, initial size, location and injection velocity allows for better comparison with available models. Dust injection experiments have been performed on different fusion devices during the last decade: DIII-D (Doublet III – D), TEXTOR (Tokamak Experiment for Technology Oriented Research), NSTX (National Spherical Torus Experiment), MAST (Mega Ampere Spherical Tokamak), KSTAR (Korea Superconducting Tokamak Advanced Research), among others. In the framework of this thesis, the data from two experiments performed on KSTAR and TEXTOR were used to compare with the model presented in Chapter4. Experimental dust trajectories from TEXTOR were kindly provided by the authors of [Shalpegin 2015]; the materials and methods will only be briefly recalled here. In the case of KSTAR, the experiment will be presented in more extended details, along with the image processing routines developed to obtain the trajectories.

2.2 Dust injection experiments

2.2.1 KSTAR injection experiment

In the 2015 KSTAR experimental campaign, dust injection was performed using a powder injector placed slightly below the midplane on the low field side of the machine. Dust was injected horizontally towards the center of the plasma with low initial velocities (∼ 1 m/s). Fig. 2.1 (a) and (b) show the location and design of the injector used. The chosen powder falls from the storage reservoir into the canon by gravity and is propelled into the plasma by a piston, which is put into motion by a piezoelectric motor. More details on the KSTAR gun-type injector can be found in [Lee 2014]. The injected amount was ∼ 2 mg of W powder per shot. The grains size distribution was wide, ranging from ∼ 10 µm up to ∼ 100 µm. A Scanning Electron Microscopy (SEM) image on Fig. 2.1 (c) shows that dust grains are mostly accreted into clusters of ∼ 100 µm size and of irregular shape. Injected dust trajectories were recorded by the fast wide angle CCD visible camera installed in KSTAR. Fig. 2.2 shows a few snapshots of the video obtained. Two distinct dust behaviors were observed. First, a large white cloud falls from the dust injection point towards the divertor region, while being exposed to little plasma. Labeled as case 2.2. Dust injection experiments 25

Figure 2.1: Setup for the KSTAR dust injection experiment. Location of the injection point (a), schematic view of the gun-type injector (b) and SEM image of the W dust used (c). 26 Chapter 2. Dust injection experiments and image processing

1, this main trajectory corresponds to the powder that just left the injector and falls downwards due to gravity. Little to no toroidal motion is seen on the video. At the end of the case 1 trajectory, dust gets closer to the wall and cools down enough to stop emitting light in the visible spectrum, and they disappear from the video. During the end of the case 1 trajectory, other dust grains are observed, being more isolated and having a toroidal motion. These dust trajectories will be labeled as case 2. Assuming that they have smaller radii, the dominant force acting on them will be the ion drag, which is roughly oriented in the toroidal direction. The dust grains from case 2 can either be the result of grains from case 1 having experienced a bouncing dust/wall collision, or grains that were isolated from the dust cluster of case 1 at some point during its falling towards the divertor. In both cases it is not illegitimate to consider that the trajectories in case 2 are isolated ∼ 10 µm dust grains since the ∼ 100 µm size clusters from case 1 could be broken upon the eventual dust/wall collision or simply due to internal forces. Since the end of the case 1 trajectory cannot be observed with the CCD camera due to too low dust temperature, no conclusions can be made on this point and cases 1 and 2 will be treated separately later on. It is not certain that the end of the case 2 trajectories correspond to full vaporization of the dust grains, since it is possible that the centrifugal acceleration drives them to colder, outer regions of the plasma where they cool down, stop emitting light in the visible spectrum and become undetectable with the camera we used. Even though the word “lifetime” will be used to characterize the durations of the experimental trajectories, it must be kept in mind that it corresponds to a lower boundary. In this experiment, the lifetimes measured were comprised between 15 and 75 ms for the case 2 trajectories.

2.2.2 TEXTOR injection experiment

During the 2012 and 2013 TEXTOR campaign, pre-characterized C and W dust was injected in the SOL plasma from the top of the machine, downwards, using a dust dropper [Shalpegin 2015]. The size of the W individual dust grains was below 5 µm, with a size of ∼ 1.5 µm. Yet, the dust grains were agglomerated into clusters of size up to 20 µm, which corresponds to the mesh size of the dust injector. The injected mass was of the order of 0.1 mg in stable low confinement mode (L-mode) plasmas. Dust trajectories were obtained using image processing routines. The area of the vac- uum vessel where dust grains were injected was observed by two CCD cameras, allowing for 3D trajectory reconstruction. More details on the trajectories obtained will be done in Chapter4. 2.2. Dust injection experiments 27

(a) (b)

(c) (d)

(e)

Figure 2.2: Image sequence from camera TV2 in KSTAR shot #13101. Injection is triggered in a stable plasma (a). The dust begins to emit visible light upon interacting with the plasma (b). A large cluster falls downwards (c). No more dust is visible (d). Some isolated grains are accelerated in the toroidal direction (e). Dust grains are circled in red on images when observable. 28 Chapter 2. Dust injection experiments and image processing

Figure 2.3: The dependence of minimal size of dust visible with the camera at distance 1 m on (a) plasma temperature for the various plasma densities and on (b) plasma density for various plasma temperatures assuming 1 ms frame exposure time and 100 counts sensitivity threshold. Source: [Smirnov 2009a]

2.3 Image processing for dust tracking

Cameras provide with RGB (Red-Green-Blue) videos that need further processing for dust events to be detected. In this section, after a brief state of the art, are detailed the DUMPRO (DUst Movie PROcessing) routines developed to detect dust events on videos and reconstruct dust trajectories.

2.3.1 Observability of dust using cameras

It seems obvious that the smaller dust grains will not be detectable with a camera. There exist a minimal dust radius, below which observation is impossible. It depends on the camera parameters (exposure time, sensitivity threshold), the dust material, size and temperature, and the distance between the dust grain and the camera. In [Smirnov 2009a], this minimal dust size was estimated for carbon particles located at 1 m from a camera with 1 ms of exposure time and 100 counts sensitivity threshold, for various values of plasma parameters. Values are available in Fig. 2.3. Dust of size 1 µm or larger is observable in typical SOL plasmas. Note that very small grains (. 0.1 µm) will tend to vaporize completely in the plasma faster than the camera exposure time used in Fig. 2.3. 2.3. Image processing for dust tracking 29

2.3.2 State of the art: existing dust tracking codes

Several dust detection and tracking algorithms have been developed through the years. Early work performed on the NSTX tokamak allowed trajectory reconstructions in 3D thank to the use of two cameras [Roquemore 2007, Boeglin 2008]. Algorithms for single camera observation were also developed for the DIII-D tokamak [Yu 2009] where the dust size were inferred by measuring the trajectory lengths and using a theoretical size- dependent ablation rate. On the Tore Supra and AUG tokamaks, our team performed analysis on low frame rate CCD camera data allowed the detection of dust events only [Hong 2009, Hong 2010]. This lead to the temporal distribution of dust events during numerous plasma discharges. Later on, the TRACE (TRAcking and Classification of pinpoints Events) code was developed [Endstrasser 2011, Bardin 2011]. It allows for dust events to be detected and trajectories to be reconstructed in 2D or 3D, depending on the experimental configuration. Presenting a detailed list of similarities and differences in all of the aforementioned codes is beyond the scope of this manuscript. We will only briefly recall their main common features. First, the main step allowing dust event detection on frame consists in identifying the mobile pinpoints from the background image, composed of light emitted from the wall and the plasma itself. This background suppression is generally performed by subtracting to a current frame the anterior one. Then, trajectory reconstruction is usually performed by associating two detected dust events from two adjacent frames if their distance is below a given threshold.

2.3.3 Dust detection on videos from CCD cameras

The first step is to isolate the dust events appearing on frame. A black and white (BW) video is computed from the raw RGB data using an operation sequence similar to that described in [Hong 2009]: gray-scale conversion, logical filtering (for pixel-size noise reduction), background removal, BW conversion. The latter preprocessing step differs from the usual pixel intensity thresholding used to isolate dust events in previous works [Endstrasser 2011, Boeglin 2008]. DUMPRO includes other features, such as frame vibration compensation.

Pre-processing: greyscale conversion and reshaping. Videos are rather heavy files to handle, and only a small fraction of the information they contain is of interest when extracting dust trajectories. In order to reduce computational time, the file size must be reduced as much as possible. This is why the input video can be reshaped to keep only a region of the frame and a selected time window. In addition, DUMPRO uses 30 Chapter 2. Dust injection experiments and image processing greyscale images only. This allows to avoid issues with the calibration of the cameras, and further reduces the file size since only a single channel (instead of three) is necessary. To convert a RGB image in greyscale, the red (R), green (G) and blue (B) channels are combined using a standard set of coefficients

Greyscale frame = 0.2 × R + 0.7 × G + 0.1 × B. (2.1)

This set of coefficients allows to preserve the perceived luminance by the human eye, which is more sensitive to green. Note that any other relation could be used, and there may exits some better set of coefficients to use in Eq. (2.1) for higher contrast of the dust grains. The optimization of the greyscale conversion formula has yet to be explored.

Filtering. Videos from tokamak plasmas can be subjected to rather high amounts of noise. This noise can be due to, for example, energetic particles impacting the camera sensor, or numeric noise appearing when the sensor sensitivity (ISO) is increased above a certain level. To eliminate such pixel-size noise, a high-pass space filter is applied to the video.

Image stabilization. In some instances, e.g., during plasma disruptions, the footage can be shaky due to intense forces applied to the vacuum vessel. If required by the user, an image stabilization step can be switched on in DUMPRO. This routine allows for compensation of movement in the plane defined by the camera sensor. A reference frame is constructed as the mean of all the frames of the video, and all frames are translated in the sensor plane in order to match the reference frame. The matching is estimated using a correlation calculus: displacing the current frame along both the X and Y directions allows the calculation of a correlation surface, which presents a maximum at some location named (X0,Y0). Then, the current frame is shifted by X0 and Y0 using a 2D spline interpolation. The result is a much more stable video, which makes the further steps, especially the background removal, more accurate.

Background removal. Background removal is the key step of any dust detection algorithm. In the literature, it is usually performed by subtracting to the current frame an adjacent one, or an average of a few adjacent frames. This technique works perfectly fine for relatively stable movies with still backgrounds. However, when the plasma emission quickly varies, e.g., during ELMs or sudden radiative events, if a hot spot appears, if the footage is shaky, or if the frame rate is too low, this technique does not yield satisfactory background suppression. 2.3. Image processing for dust tracking 31

Figure 2.4: Temporal signal from a pixel of a video containing a dust event. The baseline resolves the background evolution while ignoring the peak.

In DUMPRO, a more local method is implemented. The temporal signal of a given pixel, noted s(t, x), where x is the pixel location on frame and t is the time, presents structures evolving at two different time scales: (i) eventual dust events are recognizable as one or two points large spikes; (ii) background evolution occurs on a much larger time scale. To remove the background, the baseline of s, defined as the signal with slowly evolv- ing structures only, is computed and subtracted to s. This baseline is constructed by averaging several traces that are interpolated from s with a time step corresponding to k > 1 images. Usually, k = 3 suffices. Fig. 2.4 shows an example of baseline calculation. It is clear that the background structures are conserved when the baseline is computed, meaning that the peak will be perfectly resolved. Note that this background removal technique is much more time consuming than simple frame to frame subtraction.

Dust events detection. Once the background is removed, dust detection is usually performed by applying a threshold in intensity to each pixel and ignoring everything below this threshold. This presents the limitation that some low light emitting dust events are lost while some strongly emitting artifacts can be kept. In DUMPRO, a peak detection method is applied to each pixel temporal signal s.A 32 Chapter 2. Dust injection experiments and image processing

Figure 2.5: Temporal signal from a pixel of a video containing a dust event. The original signal can exceed the shifted one only if a sharp peak, corresponding to a dust event, is present.

shifted signal ssh(t, x) is defined by

ssh(t, x) = s(t + dt, x + dx) + ds, (2.2) where dx is of the order of a few pixels, dt a few time steps and ds > 0 is a fraction of the peak intensity. Peaks are located where and/or when s > ssh. This method shows better results on movies with varying backgrounds since only sudden and local events are detected, whereas a brutal threshold could keep some long lasting elements the background suppression step could not delete properly, such as hot spots apparition or plasma emission changes. Fig. 2.5 shows an example of peak detection in the case dt = 0.

Removing artifacts. The technique presented above allows the detection of dust events on a video while ignoring most of the noise and background evolution. Still, some artifacts can be present at this stage. They can be due to single frame emitting events from the plasma or the wall, but are usually much larger than typical dust grains. These artifacts can be ignored by setting a maximal dust size in DUMPRO, which is usually ∼ 100 pixels. The different steps detailed up to here (from reshaping to binary movie) are shown in Fig. 2.6, applied to, as an example, a movie taken in the COMPASS tokamak. One should note that dim dust trajectories are lost at the end of the process. Indeed, we 2.3. Image processing for dust tracking 33

Figure 2.6: Dust detection performed on a movie taken in COMPASS. The original movie (a) is filtered (b). Then, the background is removed (c) and dust events detected (d). In this case, no frame reshaping was used.

prefer to delete all non-dust light emitting phenomena from the film, even if this is done at the expense of some dust information.

2.3.4 Trajectory reconstruction over a frame sequence

The second step consists in associating the previously detected dust events together to reconstruct trajectories. The algorithm works using a recurrence method over time: given a dust trajectory reconstructed until the frame t0, a probability is associated to every dust detected on the next frame t0 + dt to be the following point. If the most probable dust on frame (t0 + dt) has a probability over a given threshold, the point is added to 34 Chapter 2. Dust injection experiments and image processing the trajectory, making this method fully automatic. Later on, two successive points on a dust trajectory will be referred to as parent and child, respectively. In DUMPRO, the probability formula that drives the parent/child association depends on two parameters: - The distance between potential parent and child: since a dust motion is mostly inertia driven, its velocity vector norm and orientation changes rather slowly with respect to the frame rate of a fast CCD camera (about 200 Hz in the case of the TV2 camera in KSTAR). Thus the distance between two consecutive points on a trajectory recorded by a CCD camera must not change too drastically. - The difference in apparent size of the potential parent and child: similarly to the previous point, dust temperature and size evolutions are rather slow processes com- pared to the frame rate of a fast CCD camera. Here is where the algorithm differs from previous works [Endstrasser 2011, Boeglin 2008], which did not take into ac- count the dust apparent size. Let us consider a BW video containing dust events and a dust trajectory reconstructed until frame t0. Let i be the final point of the trajectory, on frame t0, and j a dust located on frame t0 + dt. The probability for j to be the child of i is written as follows:

cos θ + 1! P (i, j) = α × i,j × G (d ) + α × G (s ) , (2.3) 1 2 dist i,j 2 size i,j where αi are weights, di,j and si,j are the distance and apparent size difference between i and j, respectively, Gk are Gaussian functions with parameters to be chosen (center and width), and θi,j is the angle between the vectors linking the parent of i to i and i to j. The centers of Gdist and Gsize are the average distance and apparent size difference between two successive points of the dust trajectory up to frame t0, respectively. The widths of

Gdist and Gsize are parameters depending on the camera resolution, usually a few pixels.

The weights αi are usually set as α1 = 5/6 and α2 = 1/6, as it shows better trajectory reconstruction capabilities. Fig. 2.7 gives an overview of the probability computation in DUMPRO: an example of trajectory is plotted over probability values for each pixel of the frame.

Application to the KSTAR dust injection experiment. Results of DUMPRO rou- tines are given on Fig. 2.8 (b) for a movie from the 2015 KSTAR dust injection experiment. Blue circles representing the detected dust events on the whole video are plotted over the superimposed frame and blue lines show the trajectories reconstructed by the algo- rithm. Both the case 1 and case 2 trajectories are successfully reconstructed. Also, due to the high dust density on frame in this video, some obviously impossible trajectories 2.3. Image processing for dust tracking 35

Figure 2.7: Map of the probability to find a dust position on frame t0 + dt, given its position at times t0 (i) and t0 − dt (Parent of i).

are found by the code. This can happen when two trajectories cross each other with a grazing angle, and there is no way to distinguish them in DUMPRO at this point.

2.3.5 Deductions from slow camera data

When the frame rate of a camera is low, the trajectory reconstruction cannot be per- formed since it is not clear how to associate dust events on adjacent frames. In this case, dust grains on frame no longer appear as bright pinpoints but as elongated lines because the exposure time is longer. This is illustrated in Fig. 2.9, which was taken in the Tore Supra tokamak with a frame rate of 25 Hz. The first step of the DUMPRO code can be applied in order to detect the dust events. The shape of the elongated lines give a portion of the dust trajectory. Then, knowing the sensor exposure time, which is a fraction of the inverse of the frame rate, one can deduce the dust average velocity during this time window. An example is shown on Fig. 2.10, where the dust velocity distribution function from Tore Supra shot #46313 is estimated. Most dust grains are very slow, with velocities below ∼ 10 m/s. Yet, some fast grains are observable (cases similar to the ones observable on Fig. 2.9), with projected velocity reaching 50 m/s. 36 Chapter 2. Dust injection experiments and image processing

Figure 2.8: Results of the DUMPRO algorithm applied on the video from camera TV2 on KSTAR shot #13101, where dust injection was performed. (b) corresponds to a zoom of (a) in the DUMPRO region of interest. Rebuilt trajectories are plotted in blue over the superimposed frame. 2.3. Image processing for dust tracking 37

Figure 2.9: Still frame from a slow (25 Hz) camera in the Tore Supra tokamak. Dust grains appear as elongated lines.

Figure 2.10: Dust parallel (to the camera sensor plane) velocity distribution function estimated using DUMPRO on Tore Supra shot #46313. 38 Chapter 2. Dust injection experiments and image processing

2.4 Design of a gun-type powder injector for the WEST tokamak

2.4.1 Design and location on WEST

It is important to investigate how dust is transported in various plasma conditions, there- fore to multiply experiments. This is why a dust injector has been designed for the WEST tokamak. Similarly to the TEXTOR experiment, the injector is placed in an upper port of the machine, next to the upper divertor and injecting dust downwards. The setup is represented in Fig. 2.11. The injection port is in the field of view of a fast wide angle visible camera. No binocular view will be available, which will prevent from doing 3D trajectory reconstruc- tion. This is not a problem if W is to be the material injected, since we can expect the dust to behave in the same way as in the TEXTOR experiment, where dust fell straight downwards. In WEST, assuming straight trajectories will allow to estimate the trajectory length and dust lifetime, which are the two main features of interest here. The injector design is identical to that of KSTAR, i.e., a gun-type injector, which has been described in Section 2.2.1. It presents the same advantages, the main one being its insensitivity to magnetic fields, thanks to the use of a piezoelectric motor. Since the gun-type injector uses gravity to load its cannon, it cannot be simply tilted down for a vertical shot. A cap was added at the end of the cannon, enclosing a reflector plane making an angle of 45◦ with the axis of the cannon and the vertical direction. The injector and holding structure are represented in Fig. 2.12.

2.4.2 Collimation tests

The injector is placed in a port usually occupied by a reciprocating Langmuir probe, which is removed for dedicated experiments. When in parked mode, the probe is protected from the plasma heat fluxes by a protection tube, called the sheath (not to be mistaken with the Debye sheath already mentioned), that cannot be disassembled from the port. The dust shot by the injector must then travel through this probe sheath, which can be assimilated to a 1.2 m long and 5 cm in diameter cylinder. Thus the powder flow leaving the injector must be collimated to ensure that no dust is deposited on the guiding sections of the probe sheath. Collimation tests were performed in NFRI for various powder types using a vacuum chamber. The test setup is schematically represented on Fig. 2.13. The injector is located 2.4. Design of a gun-type powder injector for the WEST tokamak 39

Figure 2.11: Location of the gun-type injector on the WEST tokamak: queusot (port) Q4Bh, in place of a reciprocating Langmuir probe. The injected powder must flow through the 1.2 m long probe sheath. 40 Chapter 2. Dust injection experiments and image processing

Figure 2.12: Design of the gun-type dust injector for the WEST tokamak. L-shaped vacuum chamber not represented.

at the top of the vacuum chamber, 1.2 m from a plane target and 5 cm above a collimator plate. The collimators used were simple plates with circular holes of diameter 5, 7.5, 10, 12.5 and 15 mm. A light and camera are placed inside the chamber to take snapshots of the powder pattern on the target before breaking vacuum. The working pressure was ∼ 10−6 Torr. For each powder type and collimator size used, several tens of shots were performed to maximize the contrast on the images (see Fig. 2.16 for an example). The powder used is W 12 µm (except for one test performed using Cu 10 µm) since it is the powder that is planned to be used for injection experiments on WEST. The test with Cu 10 µm was made to check whether or not the material mass plays an important role in the spreading. The size distribution of the dust was estimated from 8 SEM images. q The dust radii were estimated using rd = area/π, even though the shapes of the grains do not appear spherical. An example of SEM image is shown in Fig. 2.14. The size distribution of the powder is shown in Fig. 2.15. One can see that the distribution is not monodisperse and ranges from ∼ 8 µm (isolated dust grains), where a peak appears, up to ∼ 25 µm (clusters). The SEM images and the size distribution measured provide evidence that the W powder used tends to form rather large clusters, which will change drastically its behavior in a tokamak plasma. For each image, three profiles are taken on the collector and fitted using a Gaussian 2.4. Design of a gun-type powder injector for the WEST tokamak 41

Figure 2.13: Experimental setup for the collimation experiment: (a) sketch and (b) pictures of the injector chamber (top left), the collector plate (bottom left) and the whole system (right).

function. The dust spreading is approximated by the mean full width at tenth of max- imum of the three Gaussian functions computed. Fig. 2.16 illustrates the processing of an image from the collimation experiment. The processing was applied for each of the five collimator sizes. The diameter of the powder spot is plotted against the collimator size in Fig. 2.17. It appears that the powder spot size increases linearly with the collimator diameter and that it is not affected by the powder mass. The 50 mm upper limit for spreading (set by the size of the reciprocating probe sheath) is reached for the 10 mm collimator. Thus we conclude that any collimator size below 10 mm is suitable for integration of the injector in WEST. The choice of the size of the collimator for a dedicated experiment will be dictated by the desired injected amount, which will depend on the powder type. In addition to the collimation experiment, the injected amount was measured for the usual 12 µm W powder and for each of the five collimation sizes. To perform this measurement, the same setup as presented in Fig. 2.13 was used and a petri box of known mass was attached below the collimator. After numerous shots, the petri box was retrieved and weighted. Fig. 2.18 shows the results of these measurements. The linear fit deduced from the measurements can be used as a first order estimate for 42 Chapter 2. Dust injection experiments and image processing

Figure 2.14: SEM pictures of the 12 µm W powder used for the collimation tests.

Figure 2.15: Size distribution of the 12 µm W powder shot by the WEST injector. 349 dust grains counted. 2.4. Design of a gun-type powder injector for the WEST tokamak 43

Figure 2.16: Processing the collimation experiment data: (left) image for 12 µm W with collimation size 10 mm, the image scale and the location of the three profiles selected; (right) plot of the three profiles and their Gaussian fits.

Figure 2.17: Dust spreading for different collimation sizes and powder types (12 µm W and 10 µm Cu). The green line represents the 50 mm spreading limit for the integration in WEST. 44 Chapter 2. Dust injection experiments and image processing

Figure 2.18: Measurement of the injected amount per shot for W 12 µm powder and dif- ferent collimator sizes along with a linear fit. Unfortunately, the error was not estimated.

the quantity of W injected in the plasma, which is an important quantity for the impurity balance. Moreover, since the dust is injected from the top of the plasma, downwards, virtually all the injected matter could penetrate the plasma as little would be lost in the SOL.

2.5 Conclusion on dust injection and image process- ing

Experimental dust data (trajectories) have been obtained from the KSTAR and TEX- TOR injection experiments. Trustworthy image processing routines were developed and applied in several cases to obtain good quality experimental data. In both cases, the dust material, initial size, location and velocity is known. This will allow for better comparison with models than intrinsic dust trajectories. A dust injector for the WEST tokamak was designed, built and tested. Even if the installation of the dust injector was not finalized at the moment this manuscript is being written, its design is ready and the task should be easily performed. Chapter 3 Dust-plasma interactions

Contents 3.1 Introduction...... 46 3.2 Generalities on sheath physics...... 46 3.2.1 The Maxwellian distribution...... 46 3.2.2 Poisson’s equation...... 47 3.2.3 1D sheath model without electron emission...... 47 3.2.4 1D sheath model with electron emission...... 50 3.3 Behavior of a particle in a central force field...... 53 3.3.1 The Yukawa potential...... 53 3.3.2 Energy and angular momentum...... 54 3.3.3 Bounded and unbounded particles...... 54 3.3.4 Effective potential energy...... 55 3.4 The Orbital Motion Limited approach...... 57 3.4.1 Assumptions and collection cross-section...... 57 3.4.2 2D sheath model with electron emission in the OML framework.. 59 3.5 The general Orbital Motion theory...... 60 3.6 Models for other parameter regimes...... 62 3.6.1 Thin sheath regime...... 62 3.6.2 Dense and strongly magnetized plasmas...... 62 3.7 Conclusion on dust-plasma interactions...... 63 46 Chapter 3. Dust-plasma interactions

3.1 Introduction

When immersed into a plasma, a dust grain collects and emits charged and neutral particles, all of which contribute to its electric charge, heat balance and transport in the vacuum vessel. In tokamak applications, the main charged particles interacting with a dust grain are plasma electrons and ions. A dust grain in contact with a plasma behaves in a similar way as a Langmuir probe in the sense that a so-called Debye sheath forms around it and the total current it collects is dictated by its electric potential φd, with the difference that the probe’s potential is usually fixed by the experimenter, whilst in the case of dust the potential is floating. A grain’s floating potential is a key parameter to understanding its behavior. By convention, the potential reference is taken to be the plasma potential itself, which means that φd is actually the potential difference between the dust surface and the plasma. In order to understand what happens near a dust grain, it is mandatory to explore a little of sheath physics. In this Chapter, sheath models of increasing complexity are presented. After presenting the basics, we will start with one of the simplest sheath models possible (monodimensional, only electrons and ions), then electron emission will be added along with 2D orbital effects. Obviously, estimating the current collected by a charged particle in a fusion plasma is not an easy task, as numerous assumptions are to be made. At the end of the Chapter, the widely used Orbital-Motion Limited (OML) theory is recalled and theories applying to other regimes of parameters are presented. By the end of the Chapter, we will have expressions for the cross-sections of collection of plasma species by a spherical dust grain.

3.2 Generalities on sheath physics

When a plasma is put into contact with a piece of material, the electrons it contains reach the surface quickly due to their light mass (therefore, high mobility). Thus the surface accumulates negative charges and, by doing so, starts to repel electrons and attract positive ions so much that, if the potential of the surface if maintained free, electron and ion currents will balance out while the surface has a negative charge and negative potential with respect to the plasma potential [Stangeby 2000].

3.2.1 The Maxwellian distribution

Before any sheath model can be presented, it is important to sidetrack to the distribu- tion function predominantly used in plasma physics (amongst many other fields): the 3.2. Generalities on sheath physics 47

Maxwellian distribution. It corresponds to the way particles are distributed in a gaseous medium is a stationary container, where particles move freely and collide with one an- other, having reached its thermodynamic equilibrium. It is also widely used in plasma physics, where all the plasma species can be considered Maxwellian-distributed. The Maxwellian distribution of a given plasma species k is written

3/2 2 !  mk  mkv fk(v) = nk exp − , (3.1) 2πTk 2Tk where nk, mk and Tk are the density, mass and temperature of the species k. The quantity fk(v) corresponds to the number of particles of the species k having a velocity comprised between v and v + dv.

3.2.2 Poisson’s equation

In order to assess the behavior of the plasma parameters around a dust grain, one must solve Poisson’s equation

ρ ∇2φ = − , (3.2) ε0 where φ is the potential, ρ is the volume charge density and ε0 is the vacuum permittivity. X In a plasma, the volume charge density is written ρ = e zknk, where zk and nk are the k charge number and density of the plasma species k. Since different types of gases can be used to make a tokamak plasma (from hydrogen isotopes to helium), the number of charges of plasma ions is usually 1 or 2. This is without accounting for impurities, which have higher charge and mass, but significantly lower density. In the following, we will consider the plasma to be composed of background electrons and singly charged ions, so that Eq. (3.2) becomes

2 e ∇ φ = (ne − ni) . (3.3) ε0

3.2.3 1D sheath model without electron emission

In order to get a representative idea of the evolution of the plasma parameters around a collective body like a dust grain, solving Poisson’s equation in a simple case is quite useful. The surface is considered flat with infinite dimensions so that the problem reduces to a single dimension named z. We assume the plasma far from the surface to be unperturbed, i.e., quasineutral (ne = ni = n0) and without electric field. Also, for the sake of simplicity, electron and ion temperatures are assumed to be equal everywhere (Te = Ti). In the 48 Chapter 3. Dust-plasma interactions absence of magnetic field, the two forces acting on plasma particles are the electric and pressure gradient forces. Here we introduce the Debye length and electron (ion) thermal velocity, respectively defined by

s s ε0Te Te λD = 2 and vthe,i = . (3.4) e ne me,i

λD actually corresponds to the characteristic electrostatic screening length of the plasma. We also introduce the following normalizations that will be used throughout the Chapter

z vi ne,i eφ ξ = , M = , n˜e,i = and ϕ = , (3.5) λD vthi n0 Te where vi is the ion fluid velocity. We note that M corresponds to the definition of the Mach number. The energy conservation law for the plasma electrons in the parallel direction yields

d  ve  dn˜e dϕ men˜e = − +n ˜e , (3.6) dt Te dξ dξ where ve is the electrons velocity. It is common to assume that the electrons have a mass low enough for its energy is conserved (adiabatic electrons). We obtain

1 dn˜ dϕ e = . (3.7) n˜e dξ dξ After integration one finds the well-known Botlzmann expression

ϕ n˜e = e . (3.8)

We assume the ions to be monokinetic, which allows us to write their energy conser- 2 2 vation between the unperturbed plasma and a given position as mivi /2 + eφ = mivi,0/2, where vi,0 is the ions velocity far from the surface. Solving for vi and naming M0 the 2 2 Mach number far from the surface, one finds M = M0 − 2ϕ. We also use the ion flux conservation, which is written n˜iM = M0 to obtain the following expression for the ion density

" 2ϕ #−1/2 n˜i = 1 − 2 . (3.9) M0 Eqs. (3.8-3.9) can be substituted in Poisson’s equation, Eq. (3.2), which can be inte- 3.2. Generalities on sheath physics 49

Figure 3.1: Results of the simple 1D sheath model presented in Section 3.2.3 using M0 = 1 (Bohm criterion). Profiles for (a) the plasma potential, (b) electron and ion densities and (c) ion Mach number against the distance normalized to the Debye length.

grated once. One finds

!2 s ! 1 dϕ ϕ 2 2ϕ = e − 1 + M0 1 − 2 − 1 . (3.10) 2 dξ M0 Eq. (3.10) can be easily solved to find the potential profile, knowing the surface potential ϕd. Eqs. (3.8-3.9) can then be used to recover the profiles for the electron and ion densities and Mach number.

The main difference between a dust grain and a Langmuir probe is ϕd. Indeed, a probe has a given bias voltage which, when scanned, allows the determination of plasma parameters. A dust grain does not have a fixed surface potential, and the fluxes of all charged particles on and from the grain end up balancing out. This is the floating condition. In the model presented in this Section, the floating potential ϕd is obtained, in steady-state, by equating the electron and ion fluxes on the surface

1 s 8T s T e ϕd i n0 e = n0 , (3.11) 4 πme mi which can be solved to obtain

1  me Ti  ϕd = ln 2π . (3.12) 2 mi Te

In the case of deuterium ions and if Ti = Te, Eq. (3.12) gives ϕd ≈ −3.19. Fig. 3.1 shows the profiles calculated. In typical tokamak SOL plasmas, the Debye length is about ∼ 0.1 − 10 µm. Thus the region where plasma parameters are significantly altered by the presence of the surface 50 Chapter 3. Dust-plasma interactions is very thin (compared to the size of the plasma itself). Hence its name: the Debye sheath. In the sheath, the quasineutrality is lost, ions become supersonic, the surface is negatively charged (with respect to the plasma) and all the profiles are monotonic. This model can be trustfully applied to surfaces exposed to a plasma, but not to dust grains. This is for several reasons, one of which being the fact that we neglected the effects of emitted electrons.

3.2.4 1D sheath model with electron emission

Due to several phenomena that will be described later on, dust grains immersed in plas- mas can emit large amounts of electrons. The effects induced by the presence of this new electron population in the sheath must be accounted for. Monodimensional sheath models accounting for electron emission have been developed [Takamura 1998, Takamura 2004, Ye 2000]. They introduce a new population into Poisson’s equation, i.e., the emitted elec- trons, which have a different temperature than the background (or primary) electrons. Usually, considering typical plasma temperatures and the electron emission processes at play (which will be detailed in Chapter4), the temperature of emitted electrons is lower than that of primaries. The assumptions made in Section 3.2.3 on primary electrons (Boltzmann density) and ions (flux and energy conservation) remain. As the surface emits electrons, it loses negative charges and its charge will rise towards zero, and possibly above. The presence of cold emitted electrons will generate a well, or Virtual Cathode (VC), in the potential profile. This is called the Space-Charge Limited (SCL) regime. Electrons leaving the surface can either have enough energy to pass the VC and exit the sheath, or are recollected by the surface. Thus the typical depth of the VC is of the order of the temperature of emitted electrons [Martin 2006]. We will neglect the fraction of primary incoming electrons that are reflected back towards the sheath edge because their energy is usually much higher than the depth of the well. This allows us to conserve Boltzmann’s expression for the primary electron density, Eq. (3.8), throughout the sheath. A schematic view of the sheath structure in the presence of an emitted electron population is displayed in Fig. 3.2, where the fluxes are represented by arrows. The problem being monodimensional, the domain can be divided into two regions where Poisson’s equation will take different forms: - Region α: between the bottom of the VC and the unperturbed plasma. - Region β: between the surface and the bottom of the VC. Emitted electrons are assumed to be Maxwellian-distributed with an associated tem- perature Tse. In order to reach region α, emitted electrons are required to have a minimum 3.2. Generalities on sheath physics 51

휑

푧

Region 훼 Region 훽

휑d Surface

휑vc

Ions

Electrons

Emitted electrons

Figure 3.2: Sheath structure in the presence of electron emission. Arrows represent the fluxes of the three plasma species considered.

initial velocity, thus the density of emitted electrons, n˜se, is obtained by integrating the distribution function over the part of the velocity space where the electrons have enough energy to override the VC potential. This critical velocity is obtained using the energy 2 2 conservation law for an emitted electron: vse,crit = 2vthe (ϕd − ϕvc). Takamura et al. obtained

B × G  q  n˜ = 1 − erf C (ϕ − ϕ ) exp (C (ϕ − ϕ )) , (3.13) se,α 1 + A × G vc vc where ϕvc is the VC potential. A, B and G are parameters depending on ϕvc detailed in

Appendix 6.4, and C = Te/Tse. In region β, the density n˜se is given by Boltzmann’s law

B × G n˜ = exp (C (ϕ − ϕ )) . (3.14) se,β 1 + A × G vc Poisson’s equation becomes

d2ϕ =n ˜ +n ˜ − n˜ . (3.15) dξ2 e se i where n˜e, n˜i and n˜se are given by Eq. (3.8), (3.9), (3.13) and (3.14), respectively. Given the simple density expressions, Eq. (3.15) can be integrated once in each of the two 52 Chapter 3. Dust-plasma interactions regions. In region α, we assume that the potential and electric field at the sheath edge are zero and obtain

1 dϕ!2 eϕ − 1 = + 2 dξ 1 + A × G B × G " q  2 q # 1 − erf C (ϕ − ϕ ) exp (C (ϕ − ϕ )) + √ C (ϕ − ϕ ) + D C (1 + A × G) vc vc π vc s ! 2 2ϕ + M0 1 − 2 − 1 , (3.16) M0 where D is another parameter depending on ϕvc detailed in Appendix 6.4. The equation for region β is obtained by using the boundary conditions ϕ (ξvc) = ϕvc and dϕ/dξ (ξvc) = 0

!2 1 dϕ eϕ − eϕvc B × G = + [exp (C (ϕ − ϕ )) − 1] 2 dξ 1 + A × G C (1 + A × G) vc s s ! 2 2ϕvc 2ϕ + M0 1 − 2 − 1 − 2 . (3.17) M0 M0

Finally, the value of ϕvc is found by solving the transcendental equation

1 G √ −πϕ = exp (C (ϕ − ϕ )) , (3.18) 2δ 1 + A × G vc vc d where δ is a parameter related to the number of emitted electrons per incident electron (electron emission yield). Then, similarly to Section 3.2.3, Eqs. (3.16-3.17) are easily solved for a given value of ϕd. Profiles for the plasma potential, densities and ion Mach number are plotted in Fig. 3.3. The profiles are similar to the case without electron emission from the sheath edge up to the VC, located at ξvc ≈ 0.5 with a depth ϕvc ≈ −1.71. Between the VC and the surface, the electric field is oriented away from the surface, thereby slowing down incoming plasma ions. Both ne and ni increase with φ and nse decreases with z because of the presence of the VC that cuts off part of the emitted electron distribution function. This model presents the interest of allowing the linkage of the parameters at the surface with those at the sheath edge without the necessity of solving Poisson’s equation. Mainly, knowing the current of emitted electrons leaving the surface, one can compute in a self-consistent way the current leaving the sheath. This model has been used to estimate 3.3. Behavior of a particle in a central force field 53

Figure 3.3: Results of the 1D sheath model presented in Section 3.2.4 using ϕd = −1, C = 10 and δ = 1000. Profiles for (a) the plasma potential, (b) primary and secondary electron and ion densities and (c) ion Mach number against the distance normalized to the Debye length.

the current-voltage characteristics of cylindrical emissive Langmuir probes [Ye 2000]. The main drawback in the context of dust modelling is its monodimensional aspect. Indeed, the ratio of the collecting body size to the Debye length is much larger for probes than in the case of dust grains because (i) probes are usually larger (∼ 1 mm) than dust grains and (ii) close to the tokamak walls the Debye length is the smallest. Actually, typical dust dimensions are smaller than the sheath thickness. This leads to plasma particles having orbital motions around the body. The angular momentum plays an important role in the outcome of the trajectory. In this light, the necessity to move to two dimensional models is obvious.

3.3 Behavior of a particle in a central force field

3.3.1 The Yukawa potential

First, let us consider a simple case: a spherically symmetric potential profile with Boltz- mannian electrons and a constant ion density. Then, Poisson’s equation is written, with the Laplace operator in spherical coordinates,

d2ϕ 2 dϕ + = eϕ − 1 ≈ ϕ, (3.19) dξ2 ξ dξ under the condition that ϕ is small. This model is much too simplistic for our purpose, but it presents the undeniable advantage of having an analytical solution, called the 54 Chapter 3. Dust-plasma interactions

Yukawa potential. It is written [Whipple 1981]

rd  r − rd  φ(r) = φd exp − . (3.20) r λD This profile is monotonic and is plotted in Fig. 3.6. As we stressed above, Eq. (3.20) cannot be used to model all the complexity of the dust-plasma interactions. Yet, it is useful to know this expression as it will be referred to throughout the rest of this Chapter as it provides valuable examples.

3.3.2 Energy and angular momentum

Let us consider a spherical dust grain with a given value of potential and a charged plasma particle in its vicinity. We note m the mass, z the number of charges and v the velocity at infinity of the plasma particle (subscript e (i) for electrons (ions) will be added when required). We neglect the effects of other plasma particles, such as collisions, onto the one considered. We also still neglect the effects of an eventual magnetic field. This way, the charged particle evolves freely in a central force field and its potential energy is U(r) = zeφ(r), where r is the distance to the center of the grain. Under these conditions, the trajectory of the particle will be contained in a plane defined by its initial velocity vector and the vector linking the particle to the center of the grain. The problem thus naturally reduces to two dimensions and the energy and angular momentum for the charged particle are conserved. They can be written as follows

1 2 1 2 E = mvr + mvθ + U(r), 2 2 (3.21) M = mrvθ = mρv,

where ρ is the impact parameter of the particle, vr and vθ are the radial and tangential components of the particle velocity, respectively. Fig. 3.4 shows the geometry of the problem.

3.3.3 Bounded and unbounded particles

Depending on the behavior of φ and the particle initial parameters, there are three categories of trajectories the particle can have. - Scattering. If the particle has an impact parameter and velocity sufficiently high, its kinetic energy will not be significantly altered by the presence of the grain and it will be simply scattered. Scattered particles do not participate to the dust charging or heating, but they can have an effect on the forces applied on the grain. 3.3. Behavior of a particle in a central force field 55

Figure 3.4: Configuration of interest: a charged particle in the central force field generated by the presence of a charged spherical dust grain.

- Collection. Contrarily to the previous case, if the particle has low kinetic energy and impact parameter, it is likely to be collected onto the dust surface, thereby giving momentum, heat and its electric charge to it. - Bounded motion. In an unlikely case, the particle can have just the right initial velocity and impact parameter to end up having a closed orbit around the grain

(vr = 0). These are called bounded particles by opposition to the two previous cases where the particles are unbounded. Bounded particles have no effect on the grain since they do not bring any charging or heat, and the overall momentum they transfer to the grain is canceled over one revolution. Bounded particles are commonly neglected, as we will do in the following.

3.3.4 Effective potential energy

When rearranging Eq. (3.21), one can write the dust tangential kinetic energy as a func- tion of the radial coordinate r, i.e.,

1 1 M2 mv2 + m + U(r) = cst. (3.22) 2 r 2 m2r2 This equation resembles an energy conservation law in the radial direction. The second and third term in the left-hand side of Eq. (3.22) are regrouped under the name effective potential energy [Al’pert 1965, Fortov 2005]. When normalized to the particle total kinetic energy, mv2/2, the effective potential energy is written

ρ2 2U(r) U (ρ, r) = + . (3.23) eff r mv2 56 Chapter 3. Dust-plasma interactions

Assuming that, at infinity, U = 0, the conservation of the charged particle total 2 energy between infinity and a given position is equivalent to writing Ueff = 1 − (vr/v) .

Thus Ueff ≤ 1 and the distance of closest approach of the particle to the center, given by vr = 0, is obtained by solving the equation Ueff (ρ, r) = 1.

The difficulty of the problem will depend on how many solutions the equation Ueff (ρ, r) = 1 has. If there are more than one, a so-called barrier has emerged in the effective po- tential, and the largest value of r verifying Ueff (ρ, r) = 1 corresponds to the distance of closest approach. In the end, if this distance of closest approach is smaller than the radius of the body, the particle is collected. Otherwise, it is simply elastically scattered. Let us assume for now that the potential profile is monotonic. Then, it keeps the same sign throughout the sheath, which is the sign of the dust surface potential φd. For incoming particles with a charge of the same sign as φd (zφd > 0), we observe that Ueff is positive and monotonically decreasing with r. In this case, there are no barriers in the effective potential energy and the equation Ueff (ρ, r) = 1 has one solution and a critical impact parameter ρc (above which no particle can be collected) can be easily determined from the particle velocity v and the dust radius rd and potential φd

 s s  2zeφd 2zeφd  r 1 − if v ≥ ,  d mv2 m ρc = s (3.24)  2zeφd  0 if v < .  m

This equation is usable for electrons if φd ≤ 0 and ions if φd ≥ 0.

For incoming particles with an opposite sign to φd (zφd < 0), the behavior of Ueff with r is not so obvious, even when φ is monotonic. The equation Ueff (ρ, r) = 1 can have several solutions outside the grain, at r = rm > rd (effective potential barriers). rm is called absorption radius. Thus, when the potential profile is monotonic, this effect only appears for attracted particles (zφd < 0). As dust grains are mostly negatively charged, absorption radii exist for ions. Accounting for the presence of the absorption radii would significantly complicate the picture, this is why they are commonly neglected. The re- maining question is: under what conditions are the absorption radius effect negligible? Assessing the effect of effective potential barriers is complicated. In the following, we study the case of a Yukawa potential profile, Eq. (3.20), which is monotonic. Since we are looking at attracted particles, U = zeφ < 0. The existence of eventual barriers at ∗ location rm > rd in the effective potential is obtained by solving Ueff (ρ , rm) = 1, where

s 2U(r ) ρ∗ = r 1 − m (3.25) m mv2 3.4. The Orbital Motion Limited approach 57 is called the transitional impact parameter. Using a Yukawa potential, absorption radii are given by the transcendental equation

rmexp(rm) = β(rm − 1), (3.26) where β is the scattering parameter, defined by

zeφd rd  rd  β = − 2 exp . (3.27) mv λD λD Eq. (3.26) has no solution (i.e., there are no barriers) if β is lower than a critical value q βc ≈ 13.6. Using the definition of β and v ≈ T/m, this condition is equivalent to

r  r  β d exp d < c . (3.28) λD λD zeφd/T

In the worst case, zeφd/T ≈ 3, and we find that there are no barriers in the effective potential energy for rd/λD < 1.3 in the case of a Yukawa potential profile. Thus we conclude that the correction due to absorption radii effects remains small as long as the dust size is small with respect to the Debye length. Adding this last simplifying assumption makes us reach the domain of the Orbital- Motion Limited (OML) theory, which is the focus of Section 3.4. In Fig. 3.5 are plotted ion trajectories in an attractive Yukawa potential and corresponding profiles of effective potential energy for β = 30 > βc. In this case, a barrier exists, located at rM ≈ 1.1λD

(dashed red circle in Fig. 3.5 (a)). The barrier can be seen in Fig. 3.5 (b) for ρ/λD = 3 ∗ and 4.2. The transitional impact parameter is ρ ≈ 4.8λD.

3.4 The Orbital Motion Limited approach

The OML theory is the most commonly used approach for the determination of the collected currents onto a small spherical body. It has the immense advantage to lead to simple self-consistent expressions for these currents while not requiring the knowledge of the full potential profile around the dust. Indeed, computing the φ profile would require to solve Poisson’s equation, which is time consuming for a CPU, especially if we need to solve it many times (which is the case in a dust transport code).

3.4.1 Assumptions and collection cross-section

In order to use Eq. (3.24) for the critical impact parameter, we recall the assumptions we made so far: 58 Chapter 3. Dust-plasma interactions

Figure 3.5: (a) Ion trajectories in a Yukawa potential profile with β = 30 and (b) corresponding profiles of effective potential energy, for various impact parameters. The dashed-red circle in (a) represents rM .

- The magnetic field effects are negligible for all the plasma species considered. This

translates into the condition on the dust radius: rd  ρLe, ρLi, ρLse, where ρLe,

ρLi and ρLse are the Larmor radii of plasma electrons, ions and emitted electrons.

Usually, the smallest of the three is ρLse since emitted electrons have a much lower energy than main plasma species. - The collisions between any plasma species are negligible. This is equivalent to the

constraint rd  λe, λi, λse, where λe, λi and λse are the electron, ion and emitted electron mean free paths. - There are no barriers in the effective potential energy for incoming plasma particles. This assumption is not straightforwardly translated into a constraint on the dust radius. If the potential profile around the dust follows the Yukawa expression, it

was shown that there are no barriers in Ueff in the thick sheath regime, i.e., rd  λD. Implicitly, we also assumed the potential profile to be monotonic, which is verified when the dust electron emission yield is low. The first two allow us to write the conservation of energy and angular momentum for an incoming plasma particle, while the third gives the simple expression for the critical impact parameter, Eq. (3.24). We note that all the OML assumptions can be summarized by the constraint the dust radius must be much smaller than all the main characteristic lengths of the surrounding plasma. Typical values in tokamak SOL plasmas lead to the radius upper limit of ∼ 10 µm, above which no dust can be trustfully modelled using the OML approach. 3.4. The Orbital Motion Limited approach 59

2 Then the dust collection cross-section is defined by σOML = πρc . It can be used to compute the collection currents, heat fluxes and drag forces acting on the grain.

3.4.2 2D sheath model with electron emission in the OML frame- work

In the framework of the OML theory, the densities of plasma species (main plasma electrons and ions and emitted electrons) can be determined by integrating the velocity distribution function of the species considered in the portion of the velocity space where particles are: (i) collected, if we are looking at the collection of main plasma particles; (ii) ejected out of the sheath, is we are interested in electron emission. It is commonly assumed that all the plasma species are Maxwellian-distributed. The expressions for the density profiles were initially derived in [Al’pert 1965], then improved in [Tang 2014, Delzanno 2014a, Delzanno 2015]. The primary electron density is

s eϕ " √ q ϕ − ϕ ϕ − ϕ !# n˜ = 1 + erf ϕ − ϕ + 1 − ξ−2erfc d exp d , (3.29) e 2 d 1 − ξ−2 ξ2 − 1 where the complementary error function is defined by erfc = 1 − erf. The ion density is

! r η ϕ s ϕ e−ηiϕ √ n˜ = − i 1 + 1 − d + erfc −η ϕ i π ϕξ2 2 i √ −2 1 − ξ √ ϕd + e−ηiϕt erfc −η ϕ for ϕ ≤ , (3.30) 2 i t ξ2 √ r −ηiϕ −2 ηiϕ e √ 1 − ξ ϕd n˜ = − + erfc −η ϕ + e−ηiϕt for ϕ > , i π 2 i 2 ξ2 where ηi = Te/Ti and

ϕ − ϕ /ξ2 ϕ = d . (3.31) t 1 − ξ−2 Finally, the emitted electron density is given by

rπη  q n˜ = d J exp (η (ϕ − ϕ )) erfc η (ϕ − ϕ ) se 2 emis d d d d " # s  (3.32) ηd(ϕ − ϕd) q ηd(ϕ − ϕd) −exp 1 − ξ−2erfc  , ξ2 − 1 1 − ξ−2 where ηd = Te/Tse and Jemis is the emitted current normalized to en0vthe. The Poisson 60 Chapter 3. Dust-plasma interactions

Figure 3.6: Potential profile calculated in the OML framework, with rd = λD, ηi = 1, ηd = 33 and for Jemis = 0 and 0.31.

equation is then written

2 2 d ϕ 2 dϕ  rd  2 + = [˜ne − n˜i +n ˜se] . (3.33) dξ ξ dξ λD

The Poisson equation is solved numerically with the boundary conditions ϕ(rd/λD) = 1 and dϕ/dξ ∼ −2ϕ/ξ as ξ → ∞ (which mimics a 1/r2 shielding potential). The potential profile is plotted against the distance normalized to the dust radius in Fig. 3.6, for Jemis = 0 (i.e., no electron emission at all) and 0.31. The Yukawa potential profile is also plotted. We observe that the OML potential profiles remain close to the Yukawa expression, thereby confirming the utility to estimate the condition for the absence of absorption radii with the Yukawa profile. As the dust begins to emit large amounts of electrons, the potential profile is extended and the electric field at the dust surface tends to zero. We can expect that, similarly to the 1D model from Section 3.2.4, if the electron emission exceeds a certain threshold, a potential well will form in the sheath. Unfortunately, this is outside of the framework of the OML theory.

3.5 The general Orbital Motion theory

There are a certain number of improvements to OML and alternative approaches to dust-plasma interactions. The Allen, Boyd and Reynolds (ABR) theory [Allen 1957] is somewhat similar to the OML, but considers ions with a purely radial motion. This makes the ABR approach more applicable to Langmuir probes than to dust grains, which 3.5. The general Orbital Motion theory 61 usually have a lower potential [Martin 2006]. The behavior of charged particles in central force fields can be resolved in the general case (i.e., without neglecting bounded particles and accounting for barriers in the effective potential energy): this is the framework of the general Orbital Motion (OM) theory [Delzanno 2005a]. The expressions for the density are very complex and will not be displayed here. The Poisson equation can be numerically solved to compute the potential profile, which can present a well when electron emission is important. In [Delzanno 2005a], a simplification of the sheath model called Radial Model (RM) is proposed and is shown to be a good approximation for the full OM theory. In the RM, plasma electron and ion densities are assumed to be given by the Boltzmann law, Eq. (3.8), and electrons emitted from the surface are considered. The existence of a well of depth φmin in the potential profile located at rmin is assumed a priori and the emitted electron density profile is

  ! 2 s eφ rd  e(φ − φmin) nse(r) = n∞th exp 1 + (2H(rmin − r) − 1) erf  , (3.34) Tse r Tse where H is the Heaviside function and

2 s ! 4πmeTse πTse eφd + Wf n∞se = 3 exp − , (3.35) h 2me Tse where Wf is the material work function and h is the Planck constant. This expression is similar to the one obtained by Takamura et al., Eqs. (3.13-3.14), with the difference that 2 there is a (rd/r) coefficient in Eq. (3.34), which originates from the spherical symmetry of the problem (whilst Takamura’s model was 1D). The potential profile calculated with the RM is plotted in Fig. 3.7, in the case of a positively charged dust, against the distance normalized to the linearized Debye length, defined by

λD λDlin = q . (3.36) 1 + Te/Ti Note that, in this model, contrarily to what was assumed in the early sheath models of this Chapter, the Bohm criterion (ion Mach number equal to unity at the sheath edge) is no longer used. The OM theory is very accurate to compute dust-plasma interactions in the collision- less and unmagnetized regime since it shows good agreements with Particle In Cell (PIC) simulations [Delzanno 2005a]. Yet, it is not used in dust transport codes owing to its 62 Chapter 3. Dust-plasma interactions

Figure 3.7: Potential profile calculated with the RM with Tse = 0.1 eV, Wf = 2.2 eV and rd = 2λDlin.

complexity.

3.6 Models for other parameter regimes

3.6.1 Thin sheath regime

The most constraining assumption of the OML theory is usually the thick sheath regime one (rd  λD), so it seems normal that early efforts to generalize the OML went in the direction of characterizing plasma collection when the sheath is thin. The so-called MOML (for Modified OML) is based on a simple idea [Willis 2012a, Bacharis 2014]: if the dust is negatively charged (attracts ions) and the sheath is thin (rd  λD), every ion penetrating the sheath will eventually be collected by the grain. This is because the dust tends to a flat surface with respect to the sheath size and orbital motion of the ion cannot be sufficient for it to be scattered. This is why, in the MOML, the ion current is calculated at the sheath edge instead of at the dust surface. This leads to lower ion flux (because the acceleration in the sheath is neglected), thus to lower dust floating potentials.

3.6.2 Dense and strongly magnetized plasmas

A recent work proposes a model for dust-plasma interactions in the regime where plasma electrons are strongly magnetized and the sheath thickness is small, i.e., ρLe, λD  rd . ρLi [Vignitchouk 2017]. In the strongly magnetized regime (rd  ρLe), the presence of a potential overshoot is assumed a priori [Sanmartin 1970], and the saturation current 3.7. Conclusion on dust-plasma interactions 63

is taken from Bohm et al. [Guthrie 1949]. In the thin sheath regime (rd  λD), ion collection is assumed to occur at the sheath entrance, similarly to the MOML approach. The main conclusions in term of dust heating in this regime is that heat collection is roughly independent on the dust electron emission yield, while the overall dust lifetime is increased by up to a factor of two for W and Be.

3.7 Conclusion on dust-plasma interactions

In this Chapter, the Debye sheath was introduced and sheath models of increasing com- plexity were presented. The most interesting model for dust-plasma interactions to be implemented in a dust transport code remains the OML. The beauty of OML is that it links the sheath edge with the dust surface with simple expressions, and Poisson’s equa- tion needn’t be solved, which would be dramatically time consuming for a dust transport code. It is important to recall, once again, that, in order to use the OML expressions pre- sented in this Chapter, it is assumed that the local plasma is unmagnetized, collisionless, that the sheath is thick and that the potential profile in the sheath is monotonic.

Chapter 4 Dust transport in the tokamak vacuum vessel

Contents 4.1 Generalities...... 67 4.1.1 State of the art: existing dust transport codes...... 67 4.1.2 General aspects on DUMBO ...... 68 4.2 The plasma background...... 69 4.2.1 Background from plasma modelling codes...... 69 4.2.2 Background from experimental profiles...... 71 4.3 Dust charging in the OML regime...... 73 4.3.1 Main plasma collection...... 74 4.3.2 Electron emission...... 76 4.3.3 Current balance and dust electric charge...... 81 4.4 Vaporization and sputtering...... 82 4.4.1 Sputtering...... 82 4.4.2 Vaporization/sublimation...... 88 4.4.3 Dust mass equation...... 88 4.5 Heat collection...... 88 4.5.1 Heat fluxes from plasma collection, THE and SEE...... 89 4.5.2 Electron backscattering...... 90 4.5.3 Ion recombination and backscattering...... 91 4.5.4 Radiative cooling...... 94 4.5.5 Vaporization/sublimation and sputtering...... 95 4.5.6 Heating equation...... 95 4.5.7 Phase changes...... 96 66 Chapter 4. Dust transport in the tokamak vacuum vessel

4.6 Dust motion...... 98 4.6.1 Forces acting on a dust grain...... 98 4.6.2 Equation of motion...... 101 4.7 Numerical aspects...... 102 4.8 Comparison between dust transport codes...... 102 4.9 Conclusion on dust modelling...... 103 4.1. Generalities 67

4.1 Generalities

Dust transport models are indispensable as they are the first step to the prediction of the effect s of dust on a tokamak plasma. In a tokamak, a dust trajectory can end in the following ways: - The grain can journey towards hot and dense regions of the plasma and be de- stroyed by vaporization/sublimation. This means that all the matter the grain once enclosed is released as impurities in the plasma. - The grain can hit a plasma-facing surface. Such dust-wall collisions can have several outcomes depending on the dust impact velocity, material properties and matter state: the dust can bounce on the wall and be transported in the plasma again, the dust can remain stuck onto the wall surface or can be damaged/destroyed upon collision. During its journey in the plasma, the grain quickly becomes electrically charged. This affects the collection of plasma species, which in turn determines the forces and heat fluxes onto the dust, along with its erosion rate, which is tightly linked to the amount of impurities deposited in the plasma. Thus dust transport is at the crossroads of different physical phenomena that can be encapsulated into the broad names of plasma-surface interaction and material physics. In the present Chapter, the detailed model implemented in dust transport codes are presented via the example of the DUMBO (DUst Migration in plasma BOundaries) code [Autricque 2017b].

4.1.1 State of the art: existing dust transport codes

Several dust transport codes have been developed in the fusion community since the 2000s. In all these codes, the implemented model is very similar and based on the OML theory described in Chapter3. The differences lie in some of the expressions used, the physical phenomena taken into account, the materials available for simulations and some numerical aspects (programming language and integration schemes). At the time this manuscript is being written, there exist 5 main dust transport codes to the knowledge of the author. They are listed below in chronological order of publication. - DUSTT (DUST Transport): University of California (USA) [Pigarov 2005, Smirnov 2007, Martin 2008a]. - DTOKS (Dust in TOKamakS): Imperial College (UK) [Bacharis 2010]. - DUSTTRACK (DUST-TRACKing): Istituto di Fisica del Plasma (Italy) [Lazzaro 2012]. - MIGRAINe (MIgration of GRAINs in fusion devides): Royal Institute of Technology 68 Chapter 4. Dust transport in the tokamak vacuum vessel

Table 4.1: Material parameters for W, Be and C used in the DUMBO code and given at room temperature. Whether the dependency on the dust temperature Td is taken into account or not is also displayed.

Name Symbol W Be C Td-dependent Atomic number Z 74 4 6 No Molar mass (g/mol) M 183.85 9.012 12.011 No Density (g/cm3) ρ 19.3 1.85 2.25 Yes Heat capacity (J/K/kg) cp 130 1825 1709 Yes Melting point (K) Tmelt 3695 1560 No Boiling/sublimation point (K) Tboil 5930 2742 3925 No Heat of fusion (MJ/kg) Hfus 0.192 1.350 No Heat of vaporization (MJ/kg) Hvap 4.009 32.4 29.7 No Emissivity ε 0.05 0.18 0.85 Yes Work function (eV) Wf 4.55 4.98 5 No Maximum SEE yield δm 0.45 0.45 0.45 No Energy of max. SEE yield (eV) Em 200 200 200 No Fitting parameter in SEE law k 2.86 1.69 1.53 No Surface energy (J/m2) γ 4.36 1.83 4.36 No Young modulus (GPa) E 400 300 70 Yes Poisson ratio ν 0.28 0.05 0.1 Yes Yield strength (MPa) σy 500 400 110 Yes Ultimate strength (MPa) σu 1500 600 600 Yes

(Sweden) [Ratynskaia 2013, Vignitchouk 2014]. - DUMBO (DUst Migration in plasma BOundaries): Institut de Recherche sur la Fusion par confinement Magnétique, CEA Cadarache (France) [Autricque 2017b]. The point of this manuscript is not to detail the (numerous) similarities and (sparse) differences between each of these simulation tools. In the following, we will focus on the DUMBO code. The three materials available for dust modelling are tungsten (W), beryllium (Be) and carbon (C), since these are the most widely used materials for plasma- facing surfaces. All the material parameters used in the code are taken from bulk values and are listed in Table 4.1 (at room temperature). Note that some of them are strongly dependent on the temperature of the material.

4.1.2 General aspects on DUMBO

DUMBO, like the other codes, allows the modelling of small spherical dust grains of a given material. The adjective small refers to dust radii much less than the plasma Debye length, electron and ion mean free paths and Larmor radii. This, associated with the 4.2. The plasma background 69 spherical shape of the dust, allows us to use the OML theory described in Chapter3. The dust grain is assumed to be 0D, which means that all the dust parameters (tem- perature, electric charge, erosion rates, . . . ) are homogenous in its volume. No dust-dust interactions are considered as dust grains are simulated one at a time. Thus DUMBO cannot be used to model complex (dusty) plasmas, where the dust density is high and dust grains strongly interact with one another. Every dust transport code is divided into four sections, where one of the four main equations is solved. First, the dust surface potential and electric charge are calculated. Second, the erosion and vaporization/sublimation rates are calculated to describe the evolution of the dust mass (or size). Third, the heat equation is solved to obtain the dust temperature. Finally, the equation of motion allows the actualization of the dust position and velocity. The dust grain is simulated over a plasma background that remains unaffected by the presence of the grain, i.e., only plasma-on-dust interactions are considered. In this Chapter, we will first focus on the plasma background. Then the model implemented in DUMBO will be described in details.

4.2 The plasma background

As we saw in Chapter3, the knowledge of the plasma parameters close to the dust grain (i.e., local) is mandatory to describe the sheath. In dust transport codes, maps of plasma parameters are imported for this purpose. These maps represent the plasma background, above which the dust grain is transported without perturbing it. They can come from codes dedicated to plasma modelling or experimental measurements, if we are to reproduce a particular plasma discharge. The plasma parameters required to run a simulations are: ion and electron densities and temperatures, electric and magnetic fields, and ion flow velocity.

4.2.1 Background from plasma modelling codes

Using a plasma background calculated by a plasma code presents data of unparalleled quality, since plasma parameters are accessible in the entire simulation volume. An example in Fig. 4.1 shows the the main species temperatures and densities, along with the ions Mach number, in a poloidal cross-section of the WEST tokamak, calculated using the SOLEDGE-2D code [Bufferand 2011]. Since SOLEDGE-2D is an edge plasma code, the core region is not simulated. This will not be much of a problem for dust modelling because most dust grains will completely 70 Chapter 4. Dust transport in the tokamak vacuum vessel

Figure 4.1: 2D maps for the plasma electron (a) and ion (b) temperatures, electron density (c) and Mach number (d), computed by the SOLEDGE-2D code for the WEST tokamak. 4.2. The plasma background 71 vaporize before reaching the inner limit of the simulation domain.

4.2.2 Background from experimental profiles

In the latter case, very little information is available as plasma diagnostics often give access to simple profiles for the plasma temperature and density. From these profiles, one can estimate the value of the plasma parameter in any region of the plasma by making some some symmetry considerations. The main assumption here is that temperatures and densities remain constant along the magnetic field lines. This can be justified as parallel transport is much faster than the perpendicular. Due to the helical structure of the magnetic field, this means that these quantities are toroidally and poloidally symmetric. The minimum amount of information required to use the DUMBO code is the knowl- edge of a temperature and a density profile, along with the magnetic equilibrium. Then, if unknown, the temperature and density profiles of the other species are set equal to the given ones. Note that this might not be the case, particularly close to the divertor target, where ionization of large quantities of neutrals emitted from the wall takes place, which can lead to Te 6= Ti. The profiles are extended to a whole poloidal cross-section assuming constancy of the parameters on the magnetic flux surfaces. If not measured, the other parameters required for the simulations are estimated as follows.

Electric field. The electric field E depends on the plasma potential following E =

−∇φp, and can be divided into its parallel and perpendicular components. Let us first derive simple equations for the radial electric field in the core. Writing the energy con- servation equation for ions, and assuming that the ions have no poloidal velocity in the core, one can obtain

∇⊥(niTi) E⊥,core = . (4.1) ni In the SOL, a different picture applies. When the sheath model from Section 3.2.3 is applied to the wall of the tokamak, it appears that every magnetic field line in the

SOL has a potential of ΛTe (with Λ ≈ 3) with respect to the wall (which is grounded).

Hence, in the SOL, φp = ΛTe, which yields the electric field E⊥ = Λ∇⊥Te. In the SOL, the electron temperature profile follows an exponential with characteristic decay length named λT , which usually is 1 − 10 cm. Thus

ΛTe E⊥,SOL = . (4.2) λT 72 Chapter 4. Dust transport in the tokamak vacuum vessel

In the parallel direction, the equation for the energy conservation of electrons, Eq. (3.6), assuming the electrons are adiabatic (me ≈ 0), yields

∇k(neTe) Ek = −∇kφp = − . (4.3) ne

In the core, the parallel gradients can be neglected, leading to Ek,core = 0. In the SOL, parallel gradients exist due to the presence of the sheath at the plasma-facing surfaces.

The potential along a field line decreases of about 0.5Te along a length equivalent to half the connection length Lc (length of the field line). Thus Ek,SOL ≈ Te/(2Lc). The ratio of the radial to parallel electric fields in the SOL is ≈ 2ΛLc/λT . Given that Lc ∼ 1 − 10 m, we conclude that parallel electric fields can be neglected in the SOL as well. In the end, only radial electric fields are considered, given by Eq. (4.1) in the core and Eq. (4.2) in the SOL. The electric field can thus be determined from the electron and ion density and temperature profiles. Note that these two equations give values of opposite sign. This means that the electric field shears at the separatrix, which might be of important consequences for dust transport.

Ion flow velocity. The ion flow velocity, Vi, is the sum of the parallel and perpendic- ular motions, the latter being significantly lower than the first. It is important to note that this quantity is not constant along the field lines in the SOL: ions are accelerated in the parallel direction, towards the targets. At the sheath entrance, they reach their acoustic velocity (Bohm criterion). The parallel motion is expressed Vi,k = MkCs, where

Mk is the parallel Mach number, and Cs is the ion acoustic velocity, defined by

s Te + Ti Cs = , (4.4) mi for singly charged ions. The parallel Mach number can be written [Fedorczak 2010]

Mk 1 2 = s − , (4.5) Mk + 1 2 where s ∈ [0, 1] is a normalized curvilinear distance along the field line. In the core plasma, Mk = 0.1 is assumed. The perpendicular motion of ions is dictated by drift velocities. Many kinds of drifts exits due to a variety of phenomena. In our case, only the E × B and pressure gradient drifts are accounted for. The total ion flow velocity is

B E × B ∇(neTe) × B Vi = MkCs + 2 − 2 . (4.6) B B neB An example of 2D maps created from experimental profiles is presented in Fig. 4.2 4.3. Dust charging in the OML regime 73

Figure 4.2: 2D maps for the plasma ion temperature (a), electron density (b) and toroidal ion flow velocity (c) in the KSTAR shot #13101. The first wall contour appears in white.

for the KSTAR tokamak. The ion temperature profile was obtained by charge exchange spectroscopy, while the line integrated density allowed the determination of the density profile, assuming its shape. The ion flow velocity map was estimated with Eq. (4.6). These maps will be used in Chapter5 to make comparisons with the dust injection experiments described in Chapter2.

4.3 Dust charging in the OML regime

Now that the plasma background has been addressed, let us dive into the DUMBO model, starting with the dust charging. The collection and emission of charged particles by the dust grain drives its floating potential, which in turn forms its electric charge. The currents accounted for are: - Plasma electrons and ions. Like in the simple sheath models presented in Chapter3, the collection of plasma species drives the potential of the surface in the absence of other effects. - Electron emission. These processes are important in the case of dust grains. Of the many existing emission processes, three are found to be significant for dust in tokamak plasmas [Krasheninnikov 2011, Tolias 2014a]: Secondary Electron Emis- sion (SEE), Thermionic Emission (THE) and Electron Backscattering (EBS). 74 Chapter 4. Dust transport in the tokamak vacuum vessel

4.3.1 Main plasma collection

Here we follow the discussion in Chapter3, where we described the critical impact param- eter for an incoming charged particle to be collected by the spherical dust, Eq. (3.24). The corresponding collection current is obtained by integrating this expression on the velocity distribution function of the considered species, named f. Explicitly, the current density J of a given plasma species α onto the dust is defined by

Z 1 2 Jα = 2 e vfα(v)σOML(v)d v. (4.7) 4πrd 2 The current Iα is linked to the current density with Iα = 4πrdJα. For the sake of readability of the following expressions, we define the random electron/ion current densities by

s 0 1 8Te Je = − ene , 4 πme s (4.8) 0 1 8Ti Ji = eni . 4 πmi

These expressions correspond to the current collected by an uncharged grain (φd = 2 0, i.e., σOML = πrd). Concerning plasma electrons, fe is commonly assumed to be a Maxwelian, Eq. (3.1). Substituting into Eq. (4.7), the expression can be easily integrated to obtain

! 0 eφd Je = Je exp for φd ≤ 0, Te ! (4.9) 0 eφd Je = Je 1 + for φd > 0. Te In the case of ions, if we also consider a Maxwellian distribution, one finds a very similar expression (for singly charged ions)

! 0 eφd Ji = Ji 1 − for φd ≤ 0, Ti ! (4.10) 0 eφd Ji = Ji exp − for φd > 0. Ti Yet this latter expression has the weakness of neglecting the mean flow velocity of ions, Vi. This velocity can be important in SOL plasmas, close to the divertor targets, as ions are accelerated towards the wall and reach supersonic velocities in the sheath. The core region of the plasma can also be the set of an important ion flow if heated via Neutral Beam Injection (NBI). Accounting for the ion flow can be done by using 4.3. Dust charging in the OML regime 75

Figure 4.3: Dust steady-state floating potential against the ion flow velocity for ni = ne and various values of the Ti/Te ratio.

a so-called shifted Maxwellian in plasma of the stationary Eq. (3.1). This leads to the Shifted-OML (SOML) [Willis 2012b], where the ion current density is

"√ ! # 0 π 2 eφd 1 2 Ji = Ji 1 + 2u − 2 erf(u) + exp(−u ) for φd ≤ 0, 4u Ti 2 "√ π   J = J 0 1 + 2u2 − 2u2 (erf(u − u )erf(u + u )) i i 8u m m m 1  u    1  u    + 1 + m exp −(u − u )2 + 1 − m exp −(u + u )2 for φ > 0, 4 u m 4 u m d (4.11) where

s |Vi − Vd| eφd u = and um = , (4.12) vthi Ti and Vd is the dust velocity. Note that the electron flow velocity can always be neglected because their thermal motion is much faster than that of ions. In Eq. (4.11), the different terms can be attributed to thermal motion and others to the grain potential drop. The dominance of one term over the others depends on u and is complex. An extensive discussion, detailed in [Willis 2012a], will not be reproduced here. As an example, the variation of the steady-state dust potential, obtained by solving

Je + Ji = 0, is plotted in Fig. 4.3 in the case ni = ne and various Ti/Te. 76 Chapter 4. Dust transport in the tokamak vacuum vessel

4.3.2 Electron emission

In addition to electron and ion collection, the dust grain may emit electrons via different phenomena. When electron emission is dominant, the dust potential will be shifted towards positive values, thereby affecting considerably its heat collection and lifetime. Hence the importance of taking into account such mechanisms. The three emission processes implemented in DUMBO are SEE, THE and EBS. In fusion applications, other processes such as photoelectron emission are not considered since they are characterized by marginal fluxes compared to THE and SEE.

Thermionic emission. When heated up to high temperatures, materials emit elec- trons via THE. This is because the thermal energy of the bulk is transmitted to the charge carriers so that their energy exceed the material work function Wf . The THE current is given by the Richardson-Dushman formula [Dushman 1930]

2   0 4πmeTd Wf Jthe = e 3 exp − . (4.13) h Td In the framework of the commonly used OML theory, the potential profile in the sheath surrounding a dust grain is assumed to be monotonic. In this case, the current is given by Eq. (4.13) when φd ≤ 0, since all emitted electrons are repelled from the body and escape the sheath. When φd > 0, part of the emitted electrons (corresponding to the low energy tail of their distribution function) are attracted back, forming a return current that effectively reduces electron emission. Electrons that manage to escape the sheath are named passing electrons, while those that return to the body are named trapped. Then, the THE current density accounting for the potential drop is obtained by integrating the velocity distribution function of THE electrons, named fthe, over the passing electrons population

ZZ 2 Jthe = e vfthe(v)d v. (4.14) pas. In the following, we assume the THE electrons to follow a Maxwellian distribution associated with the temperature Td

3 2 ! me  mev Wf fthe(v) = v exp − − . (4.15) h 2Td Td This distribution ensures that Eq. (4.13) is recovered when it is integrated over the whole velocity space. The distinction between trapped and passing electrons is named Trapped-Passing Boundary (TPB) and is given by the energy conservation for one elec- 4.3. Dust charging in the OML regime 77

tron. In polar coordinated, vr and vθ designate the radial and tangential velocities of an electron in the sheath around the spherical body. In the OML case (monotonic potential profile), the TPB is, when φd > 0

2 2 2e vr + vθ = φd. (4.16) me In the velocity space, the OML TPB is a circle and the trapped population is located inside. Thus the OML THE current is obtained by integrating in the velocity space outside of the TPB. We obtain the well-known expression [Sodha 1961]

0 Jthe = Jthe for φd ≤ 0, ! ! (4.17) 0 eφd eφd Jthe = Jthe 1 + exp − for φd > 0. Td Td

Secondary electron emission. Secondary electrons are electrons from the material that are ejected upon interacting with incoming (primary) electrons. SEE is characterized by the SEE yield δsee(E, θ), defined as the number of secondary electrons per incident electron, and which depends on the energy of the primaries E and the angle of impact

θ. These two dependencies are commonly assumed to be separable, i.e., δsee(E, θ) =

δE(E)δθ(θ).

A wide variety of models exist for the energy dependence of δsee [Tolias 2014b]. The two most used expressions are the Sternglass and Young-Dekker formulas. The Sternglass formula is the most commonly used due to its simplicity, yet it was shown to overestimate

δE at relatively low energies (1−100 eV) relevant for tokamak SOL plasmas [Walker 2008]. Thus, in DUMBO, the Young-Dekker formula is used

1−k " k#! δm  E   E  δE(E) = −r 1 − exp −rm , (4.18) 1 − e m Em Em where Em and δm are the position and amplitude of the maximal SEE yield, 1 < k ≤ 2 is

rm a fitting parameter and rm is the nonzero solution of the equation rm = [1−1/k][e −1]. −β The angular dependence of δsee is commonly assumed to be δθ(θ) = (cos θ) , where β is a material dependent parameter [Vignitchouk 2014]. The SEE yield of a dust grain is obtained by

Z 1 2 δsee = 2 e vσsee(v)fe(v)d v, (4.19) 4πrdJe where fe is the primary electrons distribution function and σsee is the OML cross-section 78 Chapter 4. Dust transport in the tokamak vacuum vessel

Figure 4.4: F (Te, x) against Te for different relevant values of x in the case of W and using the Young-Dekker formula for δE.

for SEE, which is [Vignitchouk 2014]

2 1  σ (v) = δ m v2 + eφ σ (v). (4.20) see 2 − β E 2 e d OML Finally, the SEE yield is obtained by

2 δ = F (T , 0) for φ ≤ 0, see 2 − β e d ! ! (4.21) 2 1 + 3eφd/Wf eφd eφd δsee = 3 exp F Te, for φd > 0, 2 − β (1 + eφd/Te) (1 + eφd/Wf ) Te Te where

Z ∞ −z F (Te, x) = ze δE(zTe)dz. (4.22) x

We note that δsee is independent on φd when φd ≤ 0. This is because, when the dust is negatively charged, every SEE electron is repelled by it and leaves the sheath: there is no return current. When φd > 0, the normalized dust potential remains below

1, thus F (Te, x) will only be evaluated for 0 < x < 1. We note that, for a range of Te representative of tokamak plasmas (0.1 − 1000 eV), F (Te, x) marginally depends on x, as can be seen in Fig. 4.4.

Thus we conclude that F (Te, x) ≈ F (Te, 0) for 0 ≤ x < 1 and for tokamak-relevant plasma temperatures. This is especially true for cold plasmas (Te ≤ 100 eV), which correspond to tokamak SOLs, where most grains are found. This allows us to use a fit to estimate the values F in DUMBO, which is way less expensive in CPU time than 4.3. Dust charging in the OML regime 79

Table 4.2: Fit parameters for the function F in the SEE yield.

Material c0 c1 c2 c3 c4 W −2.1205 0.9984 0.0180 0.0017 −0.0156 Be −2.0522 1.0535 0.0222 −0.0671 −0.0044 C −1.8865 1.0279 0.0217 −0.0352 −0.0101

calculating integrals. A polynomial fit of degree 4 shows satisfying agreement with the exact computations, as can be seen in Fig. 4.4. Using Te in eV, it is written

4 X i log (F (Te, 0)) = ci(logTe) . (4.23) i=0

The four fitting parameters ci are given in Table 4.2 for each of the materials available in the DUMBO code. The error between Eqs. (4.23-4.22) remains below 10%, even when x = 1.

Finally, the SEE current is given by Jsee = δseeJe.

Electron backscattering. Part of the incoming plasma electrons can be inelastically reflected in the grain and escape to the ambient plasma. This electron emission phe- nomenon has been neglected in early versions of dust transport codes because backscat- tered electrons are less numerous than secondary or thermionic electrons. Yet, they have much higher energy than electrons emitted via other processes, so we expect the associated cooling rate to be significant (see Fig 4.13 for an example).

Similarly to SEE, electron backscattering is characterized by the yield δebs(E, θ), which depends on the energy of primary electrons E and their angle of incidence θ. The values of the backscattering yield at normal incidence are obtained from [Tolias 2014a], where empirical formulas based on experimental data are proposed.

∗ δebs(E, 0) = a [1 − exp(−bE)] H(E − E ) for Z ≥ 20, (4.24) ∗ δebs(E, 0) = a [1 + bEexp(−cE)] H(E − E ) for E ≤ 14, where Z is the target atomic number and E∗ = 50 eV is the threshold energy below which it is experimentally impossible to distinguish between backscattered and secondary electrons. a, b and c are material dependent fitting parameters tabulated in [Tolias 2014a].

The angle dependence of δebs is given by the Fitting formula [Fitting 2004], δebs(E, θ) = cos θ δebs(E, 0) .

Then, in order to find the backscattered electron current, one must integrate δebs on the primary electron Maxwellian distribution function and on the OML collection 80 Chapter 4. Dust transport in the tokamak vacuum vessel

Figure 4.5: SEE, THE and EBS emission yields from a W dust grain with Td = 3200 K 19 -3 and eφd/Te = 0. We used ne = 10 m .

cross-section. The global backscattering current is

! 0 eφd Jebs = Je exp fch(Te), (4.25) Te where

Z ∞ −x 1 − δebs(E, 0) + δebs(E, 0)lnδebs(E, 0) fch(Te) = 2 xe dx. (4.26) ∗ 2 E /Te (lnδebs(E, 0)) For the convenience of implementing in a dust transport code, another fitting formula for fch is used

h −g i fch(Te) = d 1 − fTe exp(−hTe) H(Te − 10) for Z ≥ 20, (4.27) −f h h i fch(Te) = dTe 1 − gTeexp(−Te ) H(Te − 10) for Z ≤ 14, where d, e, f, g, h, d0, e0, f 0, g0 and h0 are material dependent fitting parameters tabulated in [Tolias 2014a].

Global electron emission yield and comparison. The yields of all three electron emission processes, defined as the ratio of the emission current to the primary electron current, are plotted against the electron temperature in Fig. 4.5 for a constant dust 19 temperature (Td = 3200 K) and potential (eφd/Te = 0) and electron density (ne = 10 m-3). The THE yield appears to decrease even though the current should be constant be- 4.3. Dust charging in the OML regime 81 cause the primary electron current increases. It is also observed that, even if EBS has a smaller yield than the other two, it cannot be neglected.

4.3.3 Current balance and dust electric charge

Once all the currents are defined, the variation of the dust charge Qd is given by the sum of the currents onto the grain

dQ d = X I. (4.28) dt The characteristic time for the dust charging can be obtained, considering only pri- mary electrons and ions with non-shifted Maxwellian distributions, as [Smirnov 2007] √ λDi 2 τch = , (4.29) rd ωpi(1 + Ti/Te − eφd/Te) where λDi and ωpi are the ion Debye length and plasma frequency, defined by

s s 2 ε0Ti 4πnie λDi = 2 and ωpi = . (4.30) e ni mi 19 -3 For typical plasma and dust parameters, ni = 10 m , Te = Ti = 10 eV and rd = 1

µm, one finds τch ∼ 1 − 10 ns, which is much faster than most other plasma transport mechanisms. Thus, in dust transport codes, instead of solving Eq. (4.28), we solve dQd/dt = 0, which is rewritten, with the considered currents,

Je + Ji − Jsee − Jthe − Jebs = 0. (4.31)

Solving Eq. (4.31) allows the determination of the dust floating potential φd. An example of current balance solving is performed in Fig. 4.6 for a W dust grain immersed 19 -3 in a homogenous plasma with Te = Ti = 50 eV and ne = ni = 10 m . As the dust temperature increases, THE becomes the dominant electron emission pro- cess and leads to an increase in the dust floating potential. When φd > 0 (corresponding to Td ≥ 3200 K in this case), return currents appear: the SEE yield decreases, the THE yield still increases (due to the rising Td), but not as sharply as when φd < 0. Only the EBS yield remains unaffected, because the energy of backscattered electrons is too high for the return current to be non marginal. The second quantity of interest concerning dust charging is the dust electric charge 82 Chapter 4. Dust transport in the tokamak vacuum vessel

Figure 4.6: (a) SEE, THE and EBS emission yields and (b) floating potential of a W 19 -3 dust grain against its temperature. We used Te = Ti = 50 eV and ne = ni = 10 m .

Qd. It can be obtained using the Gauss law

I Qd = ε0 E.dS (4.32) Sd where Sd is the dust surface and E = −∇φ is the electric field at the dust surface. Once again, using a Yukawa potential profile, Eq. (3.20), we have [Whipple 1981]

 rd  Qd = 4πε0rdφd 1 + ≈ 4πε0rdφd, (4.33) λD since, in the OML framework, rd  λD.

4.4 Vaporization and sputtering

As the dust grain is transported in the plasma and exchanges matter with it, the dust mass and size evolve. The dust can gain mass through ion and neutral deposition and loses mass via erosion, vaporization/sublimation and sputtering.

4.4.1 Sputtering

When energetic ions impact the dust material, some initially bound atoms can be ejected by sputtering. The global sputtering yield depends on the materials of the target and the projectile material, energy and angle of incidence. It is usually divided into two major 4.4. Vaporization and sputtering 83 processes: (i) physical sputtering, where projectile ions impacting the grain with enough energy remove target atoms and (ii) chemical sputtering which can enhance erosion due to chemical reactivity. In DUMBO, chemical sputtering is accounted for carbon only. The sputtering yield is divided into its energy and angle dependencies: Ytot = Y⊥(E)Ψ(E, θ).

Physical sputtering at normal incidence. The physical sputtering yield, Yphys, is obtained by fitting experimental data, using Eckstein’s work [Behrisch 2007]. The yield at normal incidence is given by

µ KrC (E/Ethe − 1) Yphys(E) = qsn (ε) µ , (4.34) λ/ω(ε) + (E/Ethe − 1) where

√ ω(ε) = ε + 0.1728 ε + 0.008ε0.1505, (4.35)

KrC sn is the nuclear stopping power for the Kr-C interaction potential (which was found to be in good agreement with experiment in many cases [García-Rosales 1995]) and is defined by

0.5 ln (1 + 1.2288ε) sKrC = , (4.36) n ω(ε) ε is the reduced energy defined by

mat aLε0 ε = E 2 , (4.37) mi + mat ZiZate where aL is the Lindhard screening length (electronic screening in metals) defined by

!1/3 9π2  −1/2 a = a Z2/3 + Z2/3 , (4.38) L 128 B i at where aL ≈ 0.0529 nm is the Bohr radius. In Eq. (4.34), Ethe is called the threshold energy and q, λ and µ are fitting parameters tabulated in [Behrisch 2007].

Chemical sputtering (for carbon only). When chemistry can take place between species from the target and projectiles, the sputtering is enhanced. To the physical sputtering yield calculated above, one must add the chemical sputtering yield. For the tokamak relevant materials accounted for in DUMBO, only carbon is subjected to chem- ical sputtering since it tends to form hydrocarbons. The chemical sputtering yield of carbon under hydrogen isotopes at normal incidence 84 Chapter 4. Dust transport in the tokamak vacuum vessel

Figure 4.7: f, b and c parameters against the ion energy for D ions on W. Markers represent the experimental data [Behrisch 2007] and solid lines are the exponential and linear fits.

is given by [Roth 1999]

Ychem = Ytherm (1 + DYdam) + Ysurf , (4.39) where D is a parameter depending on the hydrogen isotope, Ytherm is the thermal ero- sion yield, Ydam accounts for damage production and Ysurf for surface erosion processes. Definitions of these yields and values for D are available in Annex Appendix 6.4.

Angle dependence of the sputtering yield. The angle dependence of the sputtering yield is obtained by

" !c# ! −f θ b Ψ(E, θ) = cos exp b − c , (4.40) θ0 cos [(θ/θ0) ] where

   2 1 θ0 = π − arccos q  , (4.41) π 1 + E/Esp where Esp is a binding energy. b, c and f are energy dependent fitting parameters. b and f show an exponentially decaying behavior with E, while c is roughly linear. Thus the energy dependence of these three parameters can be fitted with an exponential for b and f and a first order polynomial for c. An example for D ions on W is shown in Fig. 4.7. 4.4. Vaporization and sputtering 85

Figure 4.8: Energy (a) and angle (b) dependencies of the sputtering yield of D on W. In this case, only physical sputtering takes place. The choice of 4 keV in (b) is arbitrary.

The energy and angle dependencies of the sputtering yield are plotted in Fig. 4.8 in the case of D ions on W. The characteristic threshold of 230 eV in the sputtering of W is clearly observable on Fig. 4.8 (a), which participated in making it a material of choice for tokamak plasma-facing surfaces. In the end, the total sputtering yield for a given projectile energy and angle of inci- dence is Ytot = Y⊥(E)Ψ(E, θ), where

 Yphys + Ychem for C, Y⊥ = (4.42) Yphys otherwise.

Sputtering yield of a spherical dust grain in a plasma. All the yields presented above are obtained for monoenergetic ions impacting a flat target at a given angle. To obtain the global sputtering yield of a spherical dust grain immersed in a plasma, one must integrate the yield Ytot on the ions distribution function (shifted Maxwellian) and OML cross-section. Following the calculation in [Vignitchouk 2014], we have

! 0 Z xma Z π/2 Ii eφd < Ytot >= xHi , x Ytot(xTi, θ) sin (2θ)dθdx, (4.43) Ii xmi 0 Te 86 Chapter 4. Dust transport in the tokamak vacuum vessel

Figure 4.9: Sputtering yield of D on Be against the ion temperature, for various dust potentials and ion flows. In (a), the case eφd/Ti = 0 coincides with eφd/Ti ≥ 0.

where xmi = max(Ethe/Ti, −eφd/Ti), xma = Emax/Ti and

 q   T  sinh 2u x + zTe/Ti H (z, x) = exp −z e − x − u2 , (4.44) i T q i 2u x + zTe/Ti where we recall the definition of u:

|V − V | u = i d . (4.45) vthi c−1 c Emax is obtained by solving (π/2) = θ0. Results are plotted in Fig. 4.9 for D ions on Be for various dust potentials and ion flows. We used Te = Ti.

We observe that Ytot, though depending strongly on Ti, is almost independent on u.

The dependence on eφd/Ti is also rather small, considering the fact that vaporization/- sublimation usually dominates the dust mass loss equation over sputtering. Neglecting these two dependencies will allow us to use a simple and fast-calculating fit function, similarly to the SEE yield. The fit is performed in the case u = 0 (whom Ytot appears to be independent on) and eφd/Ti = 0 (which is an average value). We use the following fit function

 !2  Ti  d3 ln(Ti) − d4 < Ytot >= d1tanh + exp −  , (4.46) d2 Ti d5 4.4. Vaporization and sputtering 87

Table 4.3: Fit parameters for the sputtering yield for various tokamak-relevant ion/target material pairs.

Ion/target material pair d1 d2 d3 d4 d5 H/W 2.9975 × 10−5 1000 0.78940 8.0172 0.98850 D/W 1.3351 × 10−5 1000 0.0674 6.5006 0.99930 T/W −2.2547 × 10−5 1010 0.40420 6.4404 1.046 H/Be 4.6157 × 10−3 332.32 32.226 7.576 1.8147 D/Be 8.215 × 10−2 224.02 7.1296 5.0354 1.0782 T/Be −2.8351 × 10−3 10876 38.688 6.6280 1.3461 H/C 1.7493 × 10−4 163.88 39.439 8.7204 1.9149 D/C 1.1870 × 10−1 11391 48.974 8.1129 1.7508 T/C 2.0500 × 10−2 403.64 28.219 7.4755 1.4329

Figure 4.10: Sputtering yield of D on Be (a) and W (b) against the ion temperature, for eφd/Te = u = 0 (solid lines). The fits used in DUMBO are plotted in dashed red.

where Ti is expressed in eV and the parameters di depend on the ion/target material pair and are summarized in Table 4.3. The final data from [Behrisch 2007] and the fit used in DUMBO are plotted in Fig. 4.10.

Some error is being made at very low Ti, but this is of little importance because < Ytot > is so small that sputtering will be negligible compared to vaporization/sublimation. In the end, the total sputtering rate of the dust grain is

J Γ = 4πr2 i < Y > . (4.47) spu d e tot 88 Chapter 4. Dust transport in the tokamak vacuum vessel

4.4.2 Vaporization/sublimation

A dust grain is continuously subjected to a vaporization/sublimation rate. Materials that sublime, such as C, only experience a sublimation rate, while materials that melt (such as metals) undergo sublimation (when solid) and vaporization (when liquid). This rate is calculated using the Hertz-Knudsen formula [Marek 2001]

2 Pvap(Td) Γvap = 4πrd √ , (4.48) 2πTdmat where mat is the dust material atom mass and Pvap is the vapor pressure. Pvap exhibits a strong dependence on Td, and fits of experimental data are available in the literature [Plante 1972].

4.4.3 Dust mass equation

In DUMBO, a dust grain can only lose mass due to the two phenomena described above (sputtering and vaporization/sublimation). The global mass equation is

dm d = −m (Γ + Γ ) , (4.49) dt at spu vap where md is the dust mass. In terms of dust radius, this equation becomes

drd mat = − 2 (Γspu + Γvap) , (4.50) dt 4πrdρd where ρd is the dust material density.

4.5 Heat collection

All the particle fluxes discussed so far also participate to the dust heating/cooling. In this Section is described the DUMBO heating model. The heat fluxes considered are associated with: - The collection of charged particles (plasma electrons and ions) - Electron emission (SEE, THE and EBS) - The interaction of plasma ions with the dust grain (backscattering and recombina- tion) - Radiative cooling - Vaporization cooling 4.5. Heat collection 89

4.5.1 Heat fluxes from plasma collection, THE and SEE

Similarly to the charged particle currents, the heat fluxes linked with the collection of background plasma species are calculated using

Z 3 Qα = Evfα(v)σOML(v)d v, (4.51) where E = mv2/2 is the particle energy, while the heat fluxes associated with the emission of particles are given by

Z 2 Qα = Evfα(v)d v, (4.52) pass. where “pass” still designates the passing population. Similarly to the notations used for the currents, it is convenient to introduce the following heat fluxes

|J 0| Q0 = 4πr2T e , e d e e |J 0| Q0 = 4πr2T i , (4.53) i d i e |J 0 | Q0 = 4πr2T the . the d d e

Primary and THE electrons are Maxwellian distributed with temperatures Te and Td, respectively. The corresponding heat fluxes are [Smirnov 2007]

! ! 0 eφd eφd Qe = Qe 2 − exp for φd ≤ 0, Te Te ! (4.54) 0 eφd Qe = Qe 2 + for φd > 0, Te and

0 Qthe = 2Qthe for φd ≤ 0,  !2 ! (4.55) 0 eφd eφd eφd Qthe = Qthe 2 + 2 +  exp − for φd > 0. Td Td Td It is assumed that SEE electrons follow the Chung-Everhart distribution function [Chung 1974]. The calculation of the SEE heat flux, performed in [Vignitchouk 2014], 90 Chapter 4. Dust transport in the tokamak vacuum vessel yields

! 0 Wf 2 eφd Qsee = 3Qe F (Te, 0)exp for φd ≤ 0, Te 2 − β Te ! (4.56) 0 Wf 2 1 + 2eφd/Wf eφd Qsee = 3Qe F (Te, 0)exp 2 for φd > 0. Te 2 − β (1 + eφd/Te)(1 + eφd/W ) Te

The case of plasma ions is slightly more complex due to the mean flow velocity. The final expression for the ion heat flux is [Delzanno 2014b]

" ! 0 1 5 2 eφd 2 Qi = Qi + u − exp(−u ) 2 2 Ti √ ! # π 3 2 4 eφd 2 + + 3u + u − (1 + 2u ) erf(u) for φd ≤ 0, u 4 2Ti √ (4.57) 1 " π 3 Q = Q0 A + A + + 3u2 + u4 i i 2 + − 2u 4 u2 ! # − m (1 + 2u2) (erf(u − u ) + erf(u + u )) for φ > 0, 2 m m d where

! 5 u2 3 + 2u2 h i A = + ∓ u exp −(u ± u )2 . (4.58) ± 4 2 4u m m

4.5.2 Electron backscattering

The electron backscattering cooling rate is not computed the same way as Qsee and

Qthe because the energy of backscattered electrons strongly depends on the energy of primaries. Indeed, the mean energy of backscattered electrons is given by the empirical formula [1+δebs(E, θ)]E/2 [Fitting 2004]. In this light, the backscattered electron cooling rate is given by [Tolias 2014a]

! 0 eφd Qebs = 2Qeexp fcr(Te), (4.59) Te where

Z ∞ 2 2 2 −x 5 − 4δ − δ + (4δ + 2δ )lnδ fcr(Te) = x e dx ∗ 2 E /Te 8(lnδ) W Z ∞ 1 − δ + δlnδ + d xe−x dx, (4.60) ∗ 2 Te E /Te (lnδ) 4.5. Heat collection 91

where δ = δebs(E, 0) for convenience. Once again, a fitting formula is used for fcr

0 h 0 −g0 0 i fcr(Te) = d 1 − f Te exp(−h Te) H(Te − 10) for Z ≥ 20, (4.61) 0 −f 0 h 0 h0 i fcr(Te) = d Te 1 − g Teexp(−Te ) H(Te − 10) for Z ≤ 14, where d0, e0, f 0, g0 and h0 are material dependent fitting parameters tabulated in [Tolias 2014a].

4.5.3 Ion recombination and backscattering

When a plasma ion impacts the dust surface, two possible outcomes arise: - The ion can neutralize with an electron and recombine with another hydrogen atom. The newly formed molecule thermalizes with the dust material and can be released in the plasma. - The ion can be backscattered on the dust surface whilst also having neutralized with an electron. It is important to note that both of these phenomena lead to no net current from the grain. This is why they do not appear in the dust charging equation.

Ion backscattering. The particle and energy fractions of backscattered ions were fit- ted against experimental and Monte Carlo data in [Eckstein 2009] for various ion/target material pairs. For ions of energy E and angle of incidence θ, the particle and energy backscattered fractions (noted RN and RE, respectively) are fitted with the same expres- sion. Due to a misprint in [Eckstein 2009], the expression is recalled here

RN,E(E, θ) = c1(E) + c2(E)tanh [c3(E) × θ + c4(E)] . (4.62)

Then, Eq. (4.62) must be integrated on the ion shifted Maxwellian distribution func- tion and OML cross-section. We define the global particle and energy backscattered fractions by < RN >= Γibs/Γi and < RE >= Qibs/Qi, where Γibs and Γi are the backscat- tered and primary ion particle fluxes (Γi = Ii/e) and Qibs is the backscattered ion heat

flux. < RN > and < RE > depend on the ion and dust materials, Ti, u and φd. Following the calculation in [Vignitchouk 2014],

0 Z ∞ Z π/2 ! Ii eφd < RN > = xHi , x RN (xTi, θ) sin (2θ)dθdx, I x 0 T i mi e (4.63) 0 Z ∞ Z π/2 ! Qi 2 eφd < RE > = x Hi , x RE(xTi, θ) sin (2θ)dθdx, Qi xmi 0 Te where xmi = max(0, −eφd/Ti). < RN > and < RE > are plotted against Ti for D ions on

W for various values of u and φd in Fig. 4.11 (we used Te = Ti). 92 Chapter 4. Dust transport in the tokamak vacuum vessel

Figure 4.11: < RN > (top) and < RE > (bottom) against the ion temperature for W and for various values of dust potential and ion flow velocity. It is observed that < RN > and < RE > are roughly independent on u and eφd/Ti for tokamak-relevant values of these parameters. 4.5. Heat collection 93

Table 4.4: Fit parameters for the particle ion backscattering yields for various tokamak- relevant ion/target material pairs.

Ion/target material pair d1 d2 d3 d4 d5 H/W 0.3595 3.1132 0.0246 20.8472 42.1102 D/W 0.3578 3.1340 0.0294 20.9594 34.7901 T/W 0.3623 3.1112 0.0242 25.6273 37.7753 H/Be 0.1937 2.5751 0.0812 7.0877 34.4468 D/Be 0.1681 2.2678 0.0797 2.7603 37.8269 T/Be 0.1435 2.4337 0.0683 5.3882 37.9359 H/C 0.2212 2.9872 0.0677 9.4632 37.1804 D/C 0.1944 2.8985 0.0653 8.9178 35.5254 T/C 0.1709 2.8690 0.0632 8.3735 35.3074

Table 4.5: Fit parameters for the energy ion backscattering yields for various tokamak- relevant ion/target material pairs.

Ion/target material pair d1 d2 d3 d4 d5 H/W 0.4988 2.0444 0.1274 −3.2503 46.0212 D/W 0.5058 2.0545 0.1338 −8.0746 56.9926 T/W 0.4881 2.0780 0.1406 −8.1651 63.4269 H/Be 0.1925 1.8087 0.1796 −8.2687 42.8748 D/Be 0.1588 2.3135 0.1133 1.8021 35.5110 T/Be 0.1299 1.5142 0.1069 −6.0928 42.3663 H/C 0.2285 2.0437 0.1661 −3.7915 40.9874 D/C 0.1927 2.0727 0.1287 −1.4340 37.8700 T/C 0.1619 1.9973 0.1070 −2.2705 38.6665

We observe that these quantities are roughly independent on u and φd, which allows us to use a fit in the case u = 0 and φd = 0 for faster calculations. The three main features we want the fit function to reproduce are: (i) < RN,E > (Ti = 0) = 0, (ii) < RN,E > presents a maximum and (iii) < RN,E > tends to a positive finite value for high Ti. We opt for a hyperbolic tangent summed with a Gaussian for a correct reproduction of the maximum. The chosen fit function is identical for < RN > and < RE > and is

 !2  Ti  Ti − d4 < RN,E >= d1tanh + d3exp −  , (4.64) d2 d5 where Ti is expressed in eV and di depend on the ion/target material pair and are sum- marized in Table 4.4 and 4.5.

< RN > and < RE > from Eq. (4.63) for u = eφd/Ti = 0 are plotted along with the 94 Chapter 4. Dust transport in the tokamak vacuum vessel

Figure 4.12: In DUMBO, a fit of < RN > (a) and < RE > (b) at u = eφd/Ti = 0 is used.

fit used in DUMBO in Fig. 4.12. Finally, the ion backscattered heat flux is simply given by

Qibs =< RE > Qi. (4.65)

Ion recombination. Considering hydrogen isotopes plasmas only, each ion neutraliza- tion releases 13.6 eV, while each dihydrogen molecule formation releases an additional 2.2 eV (i.e., 1.1 eV per incident ion). We also assume that the molecule thermalizes with the dust before being re-emitted in the plasma. This means that every ion that undergoes recombination brings a net energy of 13.6e + 1.1e − Td to the dust. Finally, since only a fraction equal to 1− < RN > of incoming ions is re-emitted in the plasma via recombination, the net heating rate due to recombination is [Bacharis 2010]

0 14.7e − Td Ji Qrec = (1− < RN >)Qi 0 . (4.66) Ti Ji

4.5.4 Radiative cooling

When heated up to high temperatures, the cooling rate due to radiation emission becomes important. The radiative cooling rate is given by the black body law

2  4 4 Qrad = 4πrdσεd Td − Tw , (4.67) where Tw is the tokamak wall temperature, εd is the dust emissivity and σ is the Stefan- 4.5. Heat collection 95

Boltzmann constant. In dust transport codes, Tw is usually fixed between 300 K and 400

K. In DUMBO, we use Tw = 300 K.

A difficulty arises as εd cannot be considered as a constant. Indeed, the emissivity of any material increases with the temperature. In the case of a dust grain size of the order or smaller than the light wavelength, the emissivity decreases [Rosenberg 2008]. The dependence of εd on Td and rd can be estimated in the framework of the Mie theory. In

DUMBO, a fixed value for εd is used. This was shown to lead to slight underestimations of the dust lifetime [Tanaka 2008]. The implementation of the full Mie theory for the calculation of the dust emissivity could be made in future works.

4.5.5 Vaporization/sublimation and sputtering

The vaporization/sublimation rate of the dust participates to the dust cooling in addition to being a mass loss phenomenon. Using the definition of the vaporization rate from Section 4.4.2, the vaporization cooling rate is [Vignitchouk 2014]

 hsub if solid Qvap = Γvapmat (4.68) hvap if melted where hsub and hvap are the dust material heat of fusion and heat of vaporization, respec- tively. Concerning sputtering, the typical energy of sputtered atoms is less than the binding energy, i.e., . 1 eV. Thus the contribution of sputtering to the heat balance is expected to be negligible.

4.5.6 Heating equation

Having defined all the heat fluxes a dust grain is subjected to in a plasma, the temperature evolution of the dust grain is dictated by an equation that takes the form

dT m c d = Q + Q − Q − Q − Q − Q − Q − Q + Q , (4.69) d p dt e i see the ebs ibs vap rad rec where md and cp are the dust mass and heat capacity, respectively. In typical tokamak plasmas, the main source of dust heating is from background electrons. The main cooling rate is usually due to electron emission at low dust temperature and vaporization at high temperature. Fig. 4.13 shows the different heat fluxes against the dust temperature from a typical DUMBO simulation. 96 Chapter 4. Dust transport in the tokamak vacuum vessel

Figure 4.13: All heat fluxes implemented in DUMBO against the dust temperature. Data extracted from a DUMBO simulation of a W dust grain with rd = 1 µm initially and 19 -3 immersed in a homogenous plasma with Te = Ti = 50 eV and ne = ni = 10 m . Heat 0 fluxes are normalized to Qe.

The sudden change observed at Td ≈ 3000 K corresponds to the point where the dust potential changes sign and becomes positive, due to enhanced thermionic electron emission. Then, all heat fluxes concerning electron are increased while those concerning ions are reduced. It is also noted that, even though the EBS particle flux is low, the associated cooling rate is far from being negligible due to the high energy of backscattered electrons.

4.5.7 Phase changes

When heated by the plasma, the dust temperature can increase from room temperature all the way to the material vaporization/sublimation point. For metals such as W, there exist a boiling point above which the material is in the liquid state. Such phase changes must be accounted for in a dust transport code. In DUMBO, all materials exhibit a boiling temperature except C, which sublimates.

Solid/liquid transition. When the dust temperature reaches the boiling point of the material, noted Tmelt, the dust undergoes a solid/liquid transition. An additional heating, corresponding to the heat of fusion, must be brought to actually perform the transition.

In DUMBO, as soon as Td ≥ Tmelt, all the incoming heating power is directed to the phase transition until the integrated power equals the heat of fusion of the material. Then, the 4.5. Heat collection 97

Figure 4.14: Dust temperature against time. Data extracted from a DUMBO simulation of a W dust grain with rd = 1 µm initially and immersed in a homogenous plasma with 19 -3 Te = Ti = 50 eV and ne = ni = 10 m .

dust is fully liquid and the temperature can rise again. When liquid, the dust is assumed to keep its spherical shape so that OML expressions for currents, heat fluxes and forces remain accurate.

Vaporization/sublimation. The vaporization/sublimation is accounted for via the heat flux Qvap. For this phase change, no temperature saturation is required since the presence of Qvap in the heating equation induces a “natural” saturation of Td at the boiling/sublimation temperature Tvap. In Fig. 4.14, the temperature of a W dust grain from a typical DUMBO simulation is plotted against time. The Td plateau at the melting point is clearly observable, as well as the saturation at the boiling point. The oscillations observed at the end of the curve are due to numerical noise. In Fig. 4.15 are plotted contours of the temperature reached by a W dust of initial radius rd = 1 µm in an homogeneous plasma of given temperature and density at the end of a simulation made with the DUMBO code. If the dust is fully vaporized, we plot the last computed value. If an equilibrium is reached, the corresponding temperature is used. In this instance, the word “equilibrium” does not correspond to an actual steady- state because the sputtering and vaporization heat fluxes are never zero, so the dust lifetime calculated by DUMBO is always finite if the plasma temperature is not zero.

Here “equilibrium” means that critical dust parameters, φd, Td and rd, vary by less than 10−4 % each between two subsequent time steps, which is as close to a steady-state 98 Chapter 4. Dust transport in the tokamak vacuum vessel

Figure 4.15: W dust equilibrium or final temperature against the plasma temperature and density for an initial radius of 1 µm.

as we can get, and gives interesting orders of magnitude. As obviously expected, the hotter/denser the plasma, the hotter the dust. In Fig. 4.16 are plotted contours of the dust lifetime with the same parameters as in Fig. 4.15. It appears that the dust are mostly fully vaporized with a wide range of lifetimes (0.1 − 100 ms). Also, full vaporization can occur even at relatively low dust temperature (Td ≈ 3500 K). This is because the vaporization flux, given by Eq. (4.48), is always present. What was called “equilibrium” above corresponds to the bottom left- hand corner of Fig. 4.16.

4.6 Dust motion

The final section of the DUMBO code concerns dust motion. Forces are estimated and the equation of motion is solved to actualize the dust position and velocity in the tokamak vacuum vessel.

4.6.1 Forces acting on a dust grain

Among the different forces acting on a dust grain in a tokamak plasma, only three were shown to be dominant: the ion drag, Lorentz and gravitational forces [Krasheninnikov 2011]. 4.6. Dust motion 99

Figure 4.16: W dust lifetime against the plasma temperature and density for an initial radius of 1 µm.

Of course, drag forces for every other plasma species (electrons, impurity ions and neu- trals) exist. Yet, they are usually much smaller in most plasma regions, because of (i) much lower density of impurities and neutrals and (ii) much lower mass of electrons. However, objections to this argument can be made in the case of neutrals close to the divertor targets, where recycling occurs. This is why some of the dust transport codes implement this force. Also, it was shown that the electron drag force can be non negligible in some dusty-plasma conditions [Khrapak 2004].

Ion drag force. The ion drag force is somewhat the friction force applied by the plasma ions on the dust grain. There are several ways to estimate this force, the two main ones being the Binary Collision and Linear Response approaches [Khrapak 2005]. The first is the most widely used in the fusion community. The drag force is calculated by integrating the momentum transferred by a single ion over the ion shifted Maxwellian distribution function and OML cross-section, i.e.,

Z Fid = mini vvfi(v)σOMLvdv. (4.70)

Fid is divided into two components: - The first concerns collected ions that directly transfer their momentum to the grain, col Fid . sca - The second is due to ions that are scattered in the sheath around the grain, Fid . 100 Chapter 4. Dust transport in the tokamak vacuum vessel

0 2 In the following, we note Fid = πrdminivthi(Vi − Vd). The collected ion drag force is [Delzanno 2014b]

" 1 Fcol = F0 √ (1 + 2w )exp(−u2) id id 2u2 π − 1  1 − 2w   + 1 + 2w − + erf(u) for φ ≤ 0, 2u − 2u2 d 1 " 1 − 2u2 ! Fcol = F0 √ 1 + 2u2 + u × exp[−(u + u )2] (4.71) id id 4u2 π u m m 1 − 2u2 ! # + 1 + 2u2 − u × exp[−(u − u )2] u m m 1  1 − 2w  + 1 + 2w − + × [erf(u + u ) + erf(u − u )] for φ > 0, 4u − 2u2 m m d

2 where w± = u ± eφd/Ti. The scattered ion drag force is [Delzanno 2014b]

!2 sca 0 eφd G (u) Fid = 2Fid logΛ, (4.72) Ti u √ where G is the Chandrasekhar function defined by G (u) = [erf(u)−2uexp(−u2)/ π]/(2u2) and logΛ is the Coulomb logarithm, given by

2 2 2 1 b90 + ηfitλs logΛ = log 2 2 for φd ≤ 0, 2 b90 + rd   ∞ !2 Z λlin Ti logΛ = exp(−x)log 1 + 4 x  dx (4.73) 0 rd eφd

∞ ! Z Ti − 2 exp(−x)log 2 x − 1 dx for φd > 0, eφd/Ti eφd where

rd eφd b90 = − 2 , 3 + 2u Ti λD λs = , q 3Te 1 + 2 Ti(3+2u ) (4.74) s ! rd Te ηfit = 1 + 1 + , λs 6Ti 1 1 1 2 = 2 + 2 . λlin λDe λDi

b90 is an impact parameter, λs is the screening length and λlin is the linearized Debye length. 4.6. Dust motion 101

In the end, since the ion flow is usually faster than dust grains and ions are confined by the magnetic field, the ion drag force is roughly oriented along the magnetic field lines, i.e., about in the toroidal direction. It is important to note that, in the SOL of a diverted plasma, the plasma ions flow in opposite directions close to the inner and outer strike points. This is due to the magnetic field topology and the fact that ions are accelerated along the field lines towards the divertor targets. Due to this phenomenon, dust will be accelerated in opposite directions, depending if they are located in the High of Low Field Side (HFS/LFS).

Lorentz force. Due to the strong electromagnetic fields present in a tokamak plasma, and because dust grains are electrically charged, they are subjected to the Lorentz force, which is written

Flo = Qd (E + Vd × B) , (4.75) where E and B are the local electric and magnetic fields. In tokamak plasmas, the electric

field is mainly oriented in the direction perpendicular to the magnetic field, thus Flo is roughly oriented in the poloidal direction.

Gravitational force. Finally, the dust motion can be dominated bu gravity

Fg = mdg, (4.76) where g is the acceleration of gravity.

4.6.2 Equation of motion

The equation of motion is written

d (m V ) = F + F + F . (4.77) dt d d id lo g Developing the derivative in the left-hand side of Eq. (4.77), a rocket force due to the dust mass loss appears. Since the vaporized particle flux strongly depends on Td

(through the vapor pressure), inhomogeneities in Td originating in non isotropic heating can result in non isotropic vaporized particle flux that is the source of the rocket force. In [Krasheninnikov 2010], it was shown that, in the most favorable case for the rocket force, it barely reaches the same order of magnitude as the ion drag force. Also, the smaller the dust grain, the lower the inhomogeneities in Td, thus the lower the rocket force. In this light, this force is not accounted for in DUMBO. 102 Chapter 4. Dust transport in the tokamak vacuum vessel

The equation of motion simplifies to

dV m d = F + F + F . (4.78) d dt id lo g Usually, in typical tokamak SOL plasmas, the ion drag force is dominant, meaning that dust grains tend to be accelerated in the toroidal direction. In regions of the plasma with low temperature and density, or with heavy dust grains, gravity can be dominant.

4.7 Numerical aspects

DUMBO is implemented using the MATLAB programming language. Each of the four main equations driving the dust transport require different integration schemes since they are not equally numerically demanding. The numerical methods used are: - The current balance, Eq. (4.31), is not a differential equation but merely a root- finding problem. The unique root of the current-voltage characteristic must be found. DUMBO uses Brent’s root-finding algorithm. It combines the bisection method, the secant method and inverse quadratic interpolation. It associates the reliability of the first with the quickness of the other two. - The heating equation, Eq. (4.69), is the most unstable one. It is solved using a 4th order Runge-Kutta method. - The equation of motion and mass equation, Eqs. (4.78-4.50), are more numerically stable than the heating equation. They are solved using a simple forward Euler scheme. The time step of a DUMBO simulation is not fixed. At each given step, the charac- teristic times of the four equations are estimated and the next time step is fixed as the largest value possible, while remaining below these characteristic times. This method, associated with the numerical schemes detailed above and the fast calculating fit functions used for SEE, EBS, ion backscattering and sputtering allow for very fast calculations. The calculation of a typical dust trajectory using DUMBO takes no longer than ∼ 10 − 100 s.

4.8 Comparison between dust transport codes

The equations and physical phenomena described in this Chapter correspond to the DUMBO code. As mentioned, other dust transport codes implement a similar model, but with slight differences. In Table 4.6 are listed the processes and physical phenomena that are taken into account in a different way in the DUSTT, DTOKS, DUSTTRACK, 4.9. Conclusion on dust modelling 103

Table 4.6: Comparison between the different dust transport codes. S and YD stand for the Sternglass and Young-Dekker formulas, respectively. Question marks when the information is not available in the literature.

DUSTT DTOKS DUSTTRACK MIGRAINe DUMBO Materials C, metals C, W Fe Be, W C, Be, W SOML Yes No ? Yes Yes (Ion flow velocity) SEE S S ? YD YD EBS No No No Yes Yes Currents from Yes No No Yes No impurities Neutral drag Yes No No Yes No force Magnetic force No Yes Yes Yes Yes Magnetic dipole No No Yes No No interaction Rocket force No No Yes No No Ion backscattering Yes Yes No Yes Yes

MIGRAINe and DUMBO codes. Everything that is not mentioned in this Table is im- plemented in the same way as in DUMBO. Note that this information is valid at the moment this manuscript is being written. Since such codes are quickly evolving, the reader is invited to refer to the latest publications of each code for better accuracy. Concerning SEE, S and YD stand for the Sternglass and Young-Dekker formulas, respectively. Question marks are used when the information is not available in the liter- ature.

4.9 Conclusion on dust modelling

The OML-based dust transport code DUMBO was developed, incorporating various phys- ical phenomena to describe the dust charging, heating and transport over an unperturbed plasma background in the OML framework, that is when the grain is rather small. The main results to remember from this Chapter are: (i) dust in plasma is negatively charged at first, then the charge increases with the dust temperature towards positive values due to thermionic emission; (ii) dust mostly loses mass due to vaporization/sub- limation; (iii) dust is heated primarily by plasma electrons, and cools down via electron emission and vaporization heat fluxes; (iv) dust is transported in the toroidal direction due to the ion drag force. 104 Chapter 4. Dust transport in the tokamak vacuum vessel

This drives the following conclusions. First, when the dust becomes positively charged, attraction of primary electrons is enhanced, which increases heating and, in turn, raises electron emission even more. This vicious circle accelerates dust destruction via vaporiza- tion, inducing a diverging behavior in DUMBO simulations. Second, due to the toroidal shape of the tokamak, the ion drag force will accelerate dust towards the tokamak walls, resulting in dust-wall collisions. A model for such events is required. Chapter 5 Confronting the model with measurements

Contents 5.1 Comparison between simulated and experimental dust trajec- tories...... 106 5.1.1 Applicability of DUMBO ...... 106 5.1.2 TEXTOR injection experiment...... 107 5.1.3 KSTAR injection experiment...... 108 5.1.4 Qualitative comparison with data from Tore Supra ...... 111 5.2 The overheating issue...... 114 5.3 Recent progress...... 115 5.3.1 Influence of the magnetic field on electron emission...... 115 5.3.2 Vapor shielding...... 123 5.3.3 Effects of potential wells on thermionic electrons...... 125 5.3.4 Full Space-Charge Limited theory...... 127 5.4 Conclusion...... 142 106 Chapter 5. Confronting the model with measurements

5.1 Comparison between simulated and experimen- tal dust trajectories

Comparison of observed dust trajectories with simulations has already been performed on MAST [Temmerman 2010], LHD [Shoji 2015] and TEXTOR [Shalpegin 2015], using stereoscopic observations in the latter case. Here, the focus is made on the two experi- ments described in Chapter2 and try to reproduce the observed dust trajectories with DUMBO.

5.1.1 Applicability of DUMBO

First of all, since the theoretical background DUMBO is built on is based on certain assumptions, we must verify that we respect them before attempting any simulations with the code. In more details, we must check that the dust radii we intend to use remain below the Debye length, gyroradii and mean free paths of plasma species. Note that the collisionless assumption of the OML theory will rarely be the most constraining one, since, in most plasma conditions, the mean free path is much larger than the Debye length and Larmor radii. Indeed, collision times can be estimated to be

τe ∼ 0.1 − 10 µs for electrons and τi ∼ 0.01 − 1 ms for ions [Wesson 1997], and the mean free paths are λα ≈ ταvthα. This gives values of the order of, at least, ∼ 10 cm, which is much larger than any dust size we intend to simulate with DUMBO. These figures can be challenged in divertor plasmas with strong recycling, where the collisionality is increased due to high density and low temperature of the plasma, but this is out of the framework of the experiments to be reproduced in this Chapter.

Similarly, in plasmas with Te = Ti, the ion Larmor radius is much larger than the q electron one, because of much higher ion mass (ρLi/ρLe ≈ mi/me  1). Thus we will focus on comparing the dust radii with the electron gyroradius. Finally, one must keep in mind that the dust radius evolves during the simulation, and only downwards. Thus, even if the initial dust radius is not much smaller than λD or ρLe, rd will quickly reduce to much smaller values and the OML assumptions will be respected during most of the simulation.

TEXTOR experiment. The experimental profiles for the electron density and tem- perature during the dust injection experiment were measured and are plotted in Fig. 5.1. Dust grains were located in the SOL, exposed to a plasma with T ≈ 80 eV and n ≈ 9×1018 m-3.

Thus the Debye length is λD ≈ 22 µm and the Larmor radii are ρLe ≈ 21 µm, with 5.1. Comparison between simulated and experimental dust trajectories 107

Figure 5.1: Te (solid blue) and ne (dashed green) profiles, measured as described in [Ratynskaia 2013].

a magnetic field of 1 T. We conclude that DUMBO may be used with dust sizes up to 20 µm, which conveniently corresponds to the mesh size of the dust injector used (so, to the maximal dust size possible in the experiment).

KSTAR experiment. Profiles for ne and Ti are provided on Fig. 5.2. Dust grains that reached the plasma were exposed to T ≈ 200 eV and n ≈ 7 × 1018 m-3 in a 3 T magnetic

field, which yields λD ≈ 40 µm and the Larmor radii are ρLe ≈ 11 µm. We conclude that we can use DUMBO up to initial radii of 10 µm in this case.

5.1.2 TEXTOR injection experiment

Attempts to reproduce experimental C and W dust trajectories in the TEXTOR injection experiment using the MIGRAINe code is reported in [Shalpegin 2015]. Plasma parameters profiles and the 3D experimental dust trajectories were kindly provided by the authors of [Shalpegin 2015] to allow a similar attempt for the W trajectories with the DUMBO code. For each experimental trajectory of W dust, a DUMBO simulation was performed. The initial dust position and velocity vector was the same as in the experiment, and the initial size was set at 20 µm, which is the mesh size of the dust injector used, i.e., the maximum dust size possible. All experimental and simulated trajectories are plotted in Fig. 5.3. First of all, it must be noted that the experimental trajectories obtained are inertial. It means that this data cannot be used to validate the force model implemented in 108 Chapter 5. Confronting the model with measurements

Figure 5.2: Ti (red) and ne (green) profiles at t = 6.4 s from charge exchange spectroscopy and line integrated density, respectively. Te profile (blue) obtained by fitting the Ti one.

DUMBO. Still, since the full length of the trajectories are measured, the heating model can be tested by comparing experimental and theoretical dust lifetimes. It appears that, even though we used the maximum dust initial size possible, the trajectories are shorter in the simulation when compared with the experiment. Indeed, the average experimental trajectory length is 15.45 ± 3.63 mm and the simulated one is 11.97 ± 2.71 mm. The dust lifetime is, on average, 6.70 ± 1.52 ms in the experiment and 5.81 ± 1.16 ms in the simulation. It is important to note that a comparison attempt was also performed using the MIGRAINe code in [Shalpegin 2015], showing a similar trend of lifetime underestimation.

5.1.3 KSTAR injection experiment

Since no binocular view is available in KSTAR, the dust trajectories given by DUMPRO are 2D, result of 3D trajectories projected in the camera sensor plane. In order to compare with simulated dust trajectories generated with DUMBO, 3D trajectories are recreated from the measured 2D ones by assuming the following: (i) for case 1, since the dust are heavy (rd ∼ 100 µm) and have a gravity driven motion, we assume the trajectory to remain at a chosen toroidal angle (no toroidal motion). (ii) Concerning case 2, dust grains are lighter (rd ∼ 10 µm) and have an ion drag force driven motion, which is roughly oriented along the magnetic field lines. Thus we assume the dust to remain in a chosen flux tube (no perpendicular motion). The toroidal angle where the case 1 trajectory was placed was chosen in a way that it remains mostly in the SOL without crossing the wall 5.1. Comparison between simulated and experimental dust trajectories 109

Figure 5.3: All 123 experimental and simulated W dust trajectories from the TEX- TOR dust injection experiment. 110 Chapter 5. Confronting the model with measurements

Figure 5.4: KSTAR dust injection experiment - Comparison between dust experimental trajectories, reconstructed with DUMPRO, and simulated ones made with DUMBO: (a) In a poloidal cross-section, above the ion flow velocity map, with the first wall geometry in white; (b) View from the top of the machine, with the first wall geometry at the mid plane in black.

surface. The case 2 trajectories were placed on flux tubes as far as possible from the plasma core while ensuring the existence of a solution. Note that in order to make this 2D-to-3D extrapolation some features of the CCD camera must be known: position in the vessel, focal length, sensor size, among others. A simple pinhole camera model was used, and the camera parameters were chosen to match the background (wall) frame as accurately as possible. Results of the 2D-to-3D extrapolation process are shown for the case 1 trajectory and three trajectories from case 2 on Fig. 5.4. For each of the four trajectories extrapolated in 3D from the DUMPRO routines results, simulations were made using the DUMBO code. The plasma background was determined using several diagnostics on discharge #13101: EFIT data for the magnetic equilibrium and poloidal magnetic field, charge exchange spectroscopy for the ion tem- perature (Ti) profile, line integrated density for the electron density (ne). Profiles were extended in the SOL using exponential decays respecting a C1 match with the core pro- 5.1. Comparison between simulated and experimental dust trajectories 111

files. The ne profile was determined from the integrated density using a square root profile in the core, and we assumed Te = Ti. Finally, quantities were assumed to remain constant over flux surfaces. The toroidal magnetic field was ≈ 3 T, and the plasma ions flow velocity map can be seen in the background of Fig. 5.4 (a). The ion flow is predom- inantly parallel, even though E × B and pressure gradient drift velocities are taken into account. In the simulations, the dust grains were initiated at the same location and with the same velocity vector as the first point of each experimental trajectory. The initial dust radii were 100 µm for case 1 and 10 µm for case 2. The large size used to reproduce case 1 is not a problem in terms of OML applicability because the dust remains in the far SOL for the most part of the trajectory and is exposed to little plasma. Results are plotted on Fig. 5.4 along with the experimental trajectories extrapolated in 3D. One can see that the agreement between experimental and simulated trajectories is satisfying in case 1, since they are both dominated by gravity. Discrepancy can be seen on the toroidal trajectory, since the ion drag force, which is dominated by gravity yet not negligible, pushes the simulated dust in the toroidal direction, counter-clockwise. Concerning case 2, if simulated trajectories seem close at first, they end up to be much shorter than the experimental ones. Once again, it seems that DUMBO leads to lifetime underestimations.

5.1.4 Qualitative comparison with data from Tore Supra

As described in Chapter2, videos from slow cameras do not allow the determination of complete dust trajectories, thereby preventing any lifetime comparison with simulations. Nevertheless, some dust particles in Tore Supra exhibit sudden trajectory changes, as can be seen in Fig. 5.5. These sudden changes, while they cannot be explained by dust-wall collisions, can be understood in terms of ion drag force orientation. This force is usually the strongest one acting on a dust grain, and thus drives its motion. Similarly to a friction force, its orientation is given by Vi −Vd. Since the dust velocity is usually much lower (a few m/s) than the ion flow velocity (∼ 104 m/s), and considering that the parallel ion velocity is much higher than any drift velocity, we can assume the ion drag force to be oriented along the magnetic field lines. This implies the main force driving a dust grain motion to be oriented toroidally. In the core plasma, the plasma ions have a parallel motion in the same direction as the plasma current, with a Mach number of roughly 0.1 [Asakura 2007]. In the SOL, plasma ions follow the field lines from a stagnation point up to the plasma facing compo- 112 Chapter 5. Confronting the model with measurements

Figure 5.5: Still image from a slow camera in Tore Supra, shot #41406 (left) and #46313 (right). Dust trajectories exhibiting a sudden change of direction are circled in red.

nents (limiters or divertor) whilst accelerating to supersonic velocities (Bohm criterion [Wesson 1997]), regardless of the magnetic field orientation. Thus in the SOL, close to the main targets, plasma ions have a fast toroidal motion in opposite directions in the HFS and the LFS, as can be seen on Fig. 5.6 (results are the same for diverted or lim- ited plasmas). There is one field side (LFS on Fig. 5.6) where a shear in ions toroidal velocity occurs at the separatrix. Whether this shear is located on the LFS or on the HFS depends on the plasma scenario, i.e., the orientation of the toroidal magnetic field

(forward/reverse B) and the sign of the plasma current Ip. In Fig. 5.6 the shear is located on the LFS: the ions toroidal velocity is positive (has the same orientation as Ip) in the core and becomes negative in the SOL. The separatrix is plotted in blue, the first wall surface in black. Thus, a dust grain crossing the separatrix at this location will experience a sudden change of sign of the main force acting on it (the ion drag force). If the grain has a sufficient low mass (with a small size and/or if it is made of a light material such as C), its trajectory can be strongly affected, creating the sudden changes observed in Fig 5.5. Simulations made using the DUMBO code confirm these results: a C dust grain with a radius of 100 µm was generated in a typical WEST plasma. Its initial location was just inside the separatrix, close to the shear region, with an initial poloidal velocity oriented outwards (1.5 m/s) and no initial toroidal velocity. Poloidal and toroidal trajectories are plotted in Fig. 5.7. As the dust grain crosses the separatrix, the expected change in its motion is observed. A 3D view of the trajectory from the WEST baffle is shown in Fig 5.8. This work hints that the motion model implemented in DUMBO is adequate. 5.1. Comparison between simulated and experimental dust trajectories 113

Figure 5.6: Ions toroidal flow velocity in a typical WEST plasma.

Figure 5.7: Poloidal (a) and toroidal (b) trajectory of a C dust grain with initial radius rd = 100 µm simulated using DUMBO in a typical WEST plasma. 114 Chapter 5. Confronting the model with measurements

Figure 5.8: 3D representation of the trajectory from Fig. 5.7, in blue. The wall and separatrix surfaces appear in gray and red, respectively.

5.2 The overheating issue

As was observed in the two first comparisons with the experiment detailed above, the DUMBO code tends to underestimate the dust lifetime. This is because dust grains vapor- ize too fast in the simulations, due to overheating. Similar conclusions were drawn with other codes: DTOKS with experiments from MAST [Temmerman 2010], DUSTT with experiments from NSTX [Smirnov 2011] and MIGRAINe with experiments from TEX- TOR [Shalpegin 2015]. This is to be expected, since the models implemented in these codes are all similar. It is known that the OML approach used in DUMBO (and other dust simulation codes) presents severe limitations, since it assumes the absence of barriers in the effective potential energy of incoming charged particles. Effective potential barriers can trap a non negligible part of the slow incoming ions if rd gets to the order of the screening length. On the other hand, if the emitted electron flux gets close to the incoming one, potential wells can form and reduce the electron emission itself [Krasheninnikov 2011]. Another

OML limitation appears whilst plasma electrons become magnetized with respect to rd: their gyration motion induces a reduction in the incoming electron flux [Autricque 2017a]. These three effects are not accounted for in DUMBO and impact the dust charging and heating. Second, in the present version of the code, the material ablated or vaporized from the grain does not affect it nor the surrounding plasma. To be accurate, the ablated material 5.3. Recent progress 115 can form a cloud shielding the grain from plasma heat fluxes. In [Brown 2014], the dust radius above which vapor shielding effects become non negligible was shown to be ∼ 1 µm (for W and under the plasma parameters relevant in this study), which is below the dust sizes used in our simulations. Vapor shielding models have shown a reduction of the evaporation rate up to an order of magnitude [Krasheninnikov 2015]. The DUSTT and DTOKS codes incorporate a compensation method for overheat- ing. In the DUSTT code an ad hoc empirical reduction coefficient to the incoming heat flux is included [Smirnov 2011]. Concerning DTOKS, larger dust sizes are used in the simulations to reproduce accurately the experimental dust lifetimes [Temmerman 2010]. Still, the overheating issue remains the main problem encountered by OML-based dust transport codes.

5.3 Recent progress

The overheating tendency of DUMBO must come from the simplifying assumptions made in the OML theory. As a reminder, OML assumes collisionless, unmagnetized plasma and no barrier in the effective potential energy of collected particles. Each of these assumptions must be thoroughly discussed as the solution to our measurement-modelling discrepancy hides behind, at least, one of them. We will first focus on effects of the magnetic field, then mention a few words on the collisional regime and finish with effects of barriers in the effective potential energy.

5.3.1 Influence of the magnetic field on electron emission

Generalities and assumptions. Emitted electrons are subject to both the sheath potential profile and the magnetic field. The reduction of the electron emission yield from a positively charged grain has already been investigated and shown to be non negligible mainly because the mean energy of emitted electrons is of the order of the kinetic energy required to overcome the potential drop around the positive dust. The effect of the magnetic field has been investigated in the case of plasma collection by a spherical probe [Patacchini 2007], but never in the case of electron emission in connection to dust transport codes in fusion related conditions. Emitted electrons have rather low −2 temperatures: ∼ 1 eV for SEE electrons and the dust surface temperature Td (∼ 10 − −1 10 eV) for THE electrons. Consequently, their gyroradii ρLse are one to two orders of magnitude smaller than that of typical SOL plasma electrons (ρLe & 1 µm). Moreover, THE electrons are generally magnetized with respect to micron and sub-micron dust in standard SOL plasmas. 116 Chapter 5. Confronting the model with measurements

The gyromotion can lead to prompt recollection of emitted electrons on the dust surface regardless of their energy. This effect has been studied on Plasma-Facing Com- ponents (PFCs), considered as planar surfaces oriented with a given angle to the magnetic field (using PIC modelling [Komm 2017], analytical models [Tskhakaya 2000, Igitkhanov 2001, Berezina 2014] and in experimental measurements [Takamura 1998]) but not in the case of a spherical emitting body, for which the ratio rd/λD might play an important role. Since analytical expressions for the prompt recollection of emitted electrons around a spherical body in the presence of both electric and magnetic fields cannot be easily obtained, a Monte Carlo approach is used in the particular case of THE, since it is the emission process that plays a more important role in dust floating potential calculations [Autricque 2017a]. Main results and effects on dust charging, transport, and heating are discussed for conditions relevant to fusion-related plasmas. In the following paragraph, Monte Carlo simulations of the kinetics of thermionic elec- trons emitted by a spherical dust grain in the presence of a permanent and homogenous magnetic field B are presented. In addition to B, the other main factor leading to electron recollection is the sheath electric field E, which depends largely on the dust surface potential φd and λD. Since the sheath potential profile around the dust particle is much steeper than typical SOL potential gradients, we choose E to have only a radial dependance: E = −∇φ where φ is the Yukawa potential, Eq. (3.20). By choosing this potential profile, potential wells are either absent or their effects are negligible and the emitted electron population does not affect the potential profile (it remains fixed throughout the simulation) [Autricque 2017a]. The THE electron population is generated with random positions on the dust surface. The initial angle distribution function of THE electrons exit velocities is random. The velocity distribution is assumed to be Maxwellian with a temperature of Td. The emitted electrons trajectories are computed using the forward Euler method until all of them have either reached the sheath edge given by the condition r > 5λD or have been recollected, i.e., r < rd. All types of electron collisions are ignored. The dust grain is assumed to be spherical and made of W, the effects of eventual roughness at the surface are not considered. Thus hereby presented results can only be applied to spheroids such as solidified droplets and not to aggregated or flaky dust which have a more complex shape. The dust parameters are varied for a parametric study in the following ranges: 0 ≤ eφd/Te ≤ 0.2, rd ≤ 1 µm and 3500 K ≤ Td ≤ 6000 K. Plasma parameters that are used in simulations and throughout this Section are summarized in Table 5.1. They ensure that rd, ρLse  ρLe. As an example, Fig. 5.9 shows a view of the simulation domain along 5.3. Recent progress 117

Table 5.1: Typical SOL plasma parameters used in the simulations. The dust surface potential, radius and temperature are varied for the parametric study shown in Fig. 5.11.

Name Symbol Value

Electron temperature Te 10 eV 20 -3 Background density n0 10 m Debye length λD 2.4 µm Magnetic field B 3 T

with 50 electron trajectories. Results presented below are obtained from the analysis of 10000 electron trajectories.

No electric field case. Electron trajectories can be analytically recovered in the case

φd = 0, i.e., without the electric field. The equation of motion is solved for one electron in Cartesian coordinates with B oriented along the x axis. The condition for the electron to be recollected, r ≤ rd, is equivalent to, after some straightforward algebra, the existence of at least one root to the function

2 R (t) = At + Bt + C (cos ωet − 1) + D sin ωet, (5.1) where

2 A = ωevx,0/2,

B = ωex0vx,0, (5.2)  2 2  C = y0vz,0 − z0vy,0 − vy,0 + vz,0 /ωe,

D = y0vy,0 + z0vz,0. The function R has a minimum at t = τ > 0 and it is obvious that an electron emitted from position x0 = (x0, y0, z0) with velocity v0 = (vx,0, vy,0, vz,0) will be recollected if R (τ) ≤ 0, i.e.,

 0 if R (τ) ≤ 0, δE,B (x0, v0) = (5.3) 1 if R (τ) > 0. Thus, the effective emission yield, defined as the number of emitted electrons that escape the grain over the whole emitted electron population, is obtained by integrat- ing δE,B (x0, v0) over the initial position and velocity distribution functions of emitted 118 Chapter 5. Confronting the model with measurements

Figure 5.9: 50 emitted electron trajectories around a dust grain (grey circle) with rd = 1 µm, eφd/Te = 0.02 and Td = 6000 K. Trajectories are plotted in red for recollected electrons and in black for escaped electrons. B is oriented along the red arrow. In this case, the effective emission yield is δE,B = 46.7%. 5.3. Recent progress 119

Figure 5.10: Time evolution of the fraction of recollected electrons for rd = 1 µm, Td = 6000 K and two values of dust potential: eφd/Te = 0 and 0.02. The analytical recollected fraction in the case φd = 0 computed from Eq. (5.4) gives 17.5%. The final value of the recollected fraction in the case eφd/Te = 0.02 is 52.3%.

electrons, respectively named fx and fv

Z δE,B (φd = 0) = δE,B (x0, v0) fx (x0) fv (v0) dx0dv0. (5.4)

The method used is similar to that of [Tskhakaya 2000], but in the case of a spherical emitting body, and can be used whenever the sheath electric field effects can be neglected.

δE,B from Eq. (5.4) is estimated in the case represented in Fig. 5.11 (b) and is in good agreement with the Monte Carlo results.

Monte Carlo results. The time evolution of the fraction of promptly recollected elec- trons (i.e., 1 − δE,B) is plotted in Fig. 5.10 for two values of dust normalized potential chosen to be eφd/Te = 0 and 0.02. The time is normalized to the inverse of the electron gyrofrequency ωe = eB/me. As expected, higher recollection occurs on positively charged grains due to the at- tracting electric field. Two distinct phases can be identified: −1 - In the order of one period of gyration (t ∼ ωe ), the recollection quickly increases in both cases due to the gyration motion. This will be labeled as magnetic recollection. −1 - Electrons that managed to complete at least one gyromotion (t > few ωe ) escape the uncharged grain while some low energy electrons are still recollected by the 120 Chapter 5. Confronting the model with measurements

positively charged one. This will be referred to as electrostatic recollection. Electrons are more likely to undergo electrostatic recollection if they are emitted at a grazing angle rather than perpendicularly to the dust surface, whereas magnetic recollection preferentially occurs if electrons are emitted perpendicularly to the magnetic field and from regions where the magnetic field makes a grazing angle with the dust surface, as can be seen in Fig. 5.9. This can be intuitively understood since that, in these areas, the velocity of an emitted electron is mostly perpendicular to the magnetic field, leading to it being reflected back to the surface of the grain. This was also shown in studies performed on planar surfaces (see [Komm 2017] and references therein). Fig. 5.11 shows the Monte Carlo simulations results in terms of effective emission yield δE,B for different values of dust temperature and potential. The two phenomena cannot be decoupled since E also plays an important role in the first gyration motion of the electron, thus on the magnetic recollection.

Contours of δE,B are plotted in Fig. 5.11 (a) for a fixed value of the dust potential, eφd/Te = 0.1, and varying dust temperature and size. The grey shaded area is the region −2 where the gyroradius of secondary electrons is comparable to rd, i.e. 10 ≤ rd/ρLse . 1. When rd  ρLse, secondary electrons are not magnetized and δE,B increases with Td since electrons have more energy to escape the sheath potential drop. When rd & ρLse, δE,B becomes independent on rd since the scale length of an electron trajectory remains small with respect to the dust curvature. In Fig. 5.11 (b), δE,B is plotted against the dust size for different values of the dust potential and Td = 5000 K, which corresponds to ρLse/λD ≈ 0.22. When rd/λD  0.1, emitted electrons are unmagnetized and only electrostatic recollection is observed on the positively charged grain (eφd/Te = 0.1) while

δE,B ≈ 1 for the uncharged grain. As expected, the effective emission yield decreases as rd gets to the order of ρLse due to magnetic recollection. The effect of the prompt recollection on any electron emission process satisfying the condition ρLse  ρLe can be estimated using the method presented above, given the distribution functions for their velocity and angle of emission. The overall yield of a given electron emission process including prompt recollection effects can be recovered by multiplying δE,B by the emission yield of the process considered, i.e., the number of emitted electrons per incident electron. In the case of SEE, one should use the Chung- Everhart distribution function [Chung 1974] and an angle distribution that follows a cosine of the angle with the normal to the surface. This is because SEE electrons are created in the bulk with random velocity orientation and their probability to reach the dust surface is inversely proportional to the length of their path. This work is easily performed and redundant results will not be presented here. 5.3. Recent progress 121

Figure 5.11: Effective emission yield δE,B presented as: (a) contours against Td/Te and rd/λD for eφd/Te = 0.1 with the region corresponding to magnetized emitted electrons in grey and (b) plots against rd/λD for Td = 5000 K and eφd/Te = 0 and 0.1 in red along with the solution from Eq. (5.4) in black.

Consequences for dust transport. Fig. 5.12 shows the dependency of a W dust

floating potential on its temperature Td, computed by solving the current balance using the usual OML expressions for collected currents and taking into account both SEE and THE. In the first case (dashed black line), only electrostatic recollection is accounted for, using the expressions implemented in the MIGRAINe code whilst in the second one (solid red line) δE,B is incorporated for both emission processes. The SEE current was com- puted from the Young-Dekker formula and integrated over the primary electrons energy distribution function (assumed to be Maxwellian) and the impact angle distribution. The Richardson-Dushman formula was used for the THE current. Both currents were then multiplied by the yield δE,B computed for each emission process in the form of tabulated values. The background density used and displayed in Table 5.1 ensures that rd  λD.

Yet, it has been shown that OML is accurate up to rd ∼ 10λD [Delzanno 2015]. Thus hereby presented results hold for denser plasmas, e.g., up to ∼ 1022 m-3 for micron size dust.

Differences are observed for φd ≥ 0, where the floating potential is reduced by up to

40% as Td approaches 6000 K. The values of φd computed with the inclusion of magnetic recollection are consistent with the range described above (0 ≤ eφd/Te ≤ 0.2) and for which potential well effects can be neglected. Incorporating magnetic recollection effects in the dust transport code DUMBO con- 122 Chapter 5. Confronting the model with measurements

Figure 5.12: W dust floating potential as a function of its temperature computed by solving the current balance accounting for electrostatic recollection only in dashed black and both electrostatic and magnetic recollection in solid red. The dust radius is fixed at rd = 1 µm.

firmed that the dust lifetime can be increased due to reduced electron heating at high dust temperature. In Fig. 5.13, a W dust grain with initial radius of 1 µm was simulated in a plasma with varying temperature (keeping Te = Ti) and density. The most signifi- cant lifetime increase is observed in the coldest and least dense plasma (Te = 10 eV and 18 -3 n0 = 5 × 10 m ) where it reaches 23%, which is not negligible yet not sufficient to ex- plain lifetime discrepancies between dust transport simulations and camera observations of injected dust.

Conclusion. The prompt recollection of emitted electrons on a spherical dust grain due to the presence of a magnetic field has been investigated through Monte Carlo simulations. The gyration motion of electrons leads to a reduction of the emission yield. It was shown that dust charging can be significantly affected by this effect, especially at high dust temperature (where the thermionic emission is strong and the dust is positively charged). In this regime, the dust floating potential is reduced. Consequently, dust heating is also reduced and the dust lifetime increases at temperatures and densities relevant for tokamak SOL plasmas. Yet, the lifetime increase due to this effect is not sufficient to explain the discrepancies observed in Section. 5.1. When the sheath electric field can be neglected, Eq. (5.4) can be used to estimate prompt recollection. Results presented in the fusion- related context can be applied to any dusty plasma application where magnetic fields are 5.3. Recent progress 123

Figure 5.13: W dust lifetime in a pure D homogeneous plasma with varying electron/ion temperature and density. The initial dust radius is rd = 1 µm.

present.

5.3.2 Vapor shielding

As we saw, a dust grain immersed in a tokamak plasma is subjected to high heat and particle fluxes, thereby inducing erosion and vaporization/sublimation. In models im- plemented in classical dust transport codes, the effects of the ablated material on the dust-plasma interactions is neglected. In reality, when the dust loses matter at a high rate, the density of ablated material around it can be high enough to increase the electron density locally, increase collisionnality and reduce the energy of plain plasma particles impacting it. This is out of the framework of the OML theory that neglects collisions in the sheath around the dust. First vapor shielding studies concerning dust in fusion devices attempted to deter- mine the critical dust radius above which vapor shielding effects become non-negligible [Krasheninnikov 2009, Brown 2014], see Fig. 5.14. This critical radius, which depends on the dust material and the plasma temperature and density, is of the order of typical 19 -3 tokamak dust for relatively hot and dense plasmas, i.e., Te & 10 eV and ne & 10 m . A wide variety of vapor shielding models exist in the case of fuel pellets, which are ice cubes of cold hydrogen isotopes, thrown at high velocity in the plasma to refuel it [Rozhansky 2005]. In these models, the pellet ablation rate is determined by assuming that the ablated material cloud is divided into two regions: (i) close to the pellet, the 124 Chapter 5. Confronting the model with measurements

Figure 5.14: Contours of the critical dust radius over which vapor shielding effects become non negligible, for different dust materials and varying the plasma background density and temperature. Source: [Krasheninnikov 2009, Brown 2014] 5.3. Recent progress 125 ablated material forms a neutral vapor cloud that expands with a spherical shape; (ii) further from the pellet, the ablated material ionizes and forms a secondary plasma that expands along the magnetic field lines. Primary plasma particles impacting the pellet have to travel though both the vapor cloud and the secondary plasma and lose energy in the process. Yet, it was argued that, in the case of high-Z materials (like W), the heat flux transported to the grain is mainly due to electron heat conduction, and not attenuation of free streaming electrons [Marenkov 2014]. This makes the pellet vapor shielding models unusable in the case of dust grains. Models dedicated to tokamak dust vapor shielding exist and show that the dust ablation rate can sometimes be reduced by up to an order of magnitude for sufficiently large dust grains, while, in some other cases, increased when compared with models without vapor shielding [Marenkov 2014, Krasheninnikov 2015]. It was also shown that dust grains having a high cross-field velocity (& 100 m/s) have a reduced vapor shielding effect [Smirnov 2017]. This is because the secondary plasma tends to remain confined by the magnetic field while dust grains are non magnetized. Yet, these models are not easily usable in dust transport codes, because they are incomplete: the effects of vapor shielding on dust charging and transport has yet to be assessed in a self-consistent way.

5.3.3 Effects of potential wells on thermionic electrons

The electron emission reduction in the presence of a VC (Virtual Cathode) has been studied in the so-called OML+ theory [Delzanno 2014a, Tang 2014]. It was observed that ∗ above a critical value of dust potential φd, corresponding to a certain electron emission current, a VC forms, the dust becomes positively charged and the OML theory is no longer applicable since we enter the SCL (Space-Charge Limited) regime. ∗ φd is found by solving Poisson’s equation in the OML case (see Section 3.4.2) with the additional boundary condition dφ/dr = 0 at the dust surface, which corresponds to ∗ the onset of potential wells. φd is shown to depend on the local plasma parameters and the dust size and temperature. Contours are plotted in Fig. 5.15. It assumed that, when in the SCL regime, only the THE current is significantly reduced (due to a return current). This is justified in plasmas where the background temperature is much higher than the emitted electron temperature (which corresponds to that of the dust grain in general). Then, to account for the reduction of the THE current due to the VC, the following expression is proposed:

" ∗ # " ∗ # 0 e(φd − φd) e(φd − φd) Jthe = Jthe 1 + exp − . (5.5) Td Td 126 Chapter 5. Confronting the model with measurements

∗ Figure 5.15: Contours of eφd/Te against the dust size and temperature. Source: [Delzanno 2014a]

Also, the expression fot the THE heat flux, Eq. (4.55), must be accordingly altered following

Qthe ∗ 0 = 2 for φd ≤ φd, Qthe  ∗ " ∗ #2 " ∗ # (5.6) Qthe e(φd − φd) e(φd − φd) e(φd − φd) ∗ 0 = 2 + 2 +  exp − for φd > φd. Qthe Td Td Td

In [Delzanno 2014a], it is argued that Eq. (5.5) shows better agreement with PIC simulations that the classical OML one. Once the Poisson equation has been solved in order to produce the data of Fig. 5.15, this correction effect has been easily implemented in the dust transport code MIGRAINe. When implemented in DUMBO, we can generate Fig. 5.16, where the dust lifetimes are plotted for various sizes in homogeneous plasmas. As expected, OML+ leads to lifetime increase. When applied to the TEXTOR experiment case with initial dust size equal to 5 µm, the lifetime increase appears to be only of ≈ 1% on average, which is way insufficient to reproduce the observed dust lifetimes. 5.3. Recent progress 127

Figure 5.16: W dust lifetime in a homogeneous plasma with Ti = Te = 5 eV and ni = 21 -3 + ne = 10 m against the dust size, with OML and OML theories.

The main drawback of the OML+ is that it is required to solve a priori the Poisson ∗ equation for various dust sizes and temperatures in order to obtain tables of φd that can be interpolated at each time step in a dust transport code. It also presents the limitation of considering the reduction of the THE flux only. This is a problem because (i) the primary electron current is also expected to be significantly reduced in plasmas where the background temperature is of the order of the dust temperature and (ii) the SEE flux is also expected to be significantly reduced in relatively hot plasmas. Note that effects of potential wells on EBS are marginal due to the high energy of backscattered electrons. Thus the OML+ presents some undeniable problems that prevent it from being implemented in a self-consistent way in DUMBO. A more complete theory of the effects of potential wells must be developed.

5.3.4 Full Space-Charge Limited theory

A recent work aims at broadening the idea of the OML+ theory by incorporating the effects of the VC on collected electrons as well as overcoming the need for solving Poisson’s equation a priori [Autricque 2018b]. We saw in Section 3.2.4 that the presence of a VC in the sheath results in lower collected and emitted electron fluxes, using a one dimensional sheath model. This effect is conserved when orbital motions are at play. 128 Chapter 5. Confronting the model with measurements

Figure 5.17: Effective potential energy Ueff of incoming electrons (z = −1) versus the distance r for ρ = rd/2 and different particle velocities. The largest solution of the equation Ueff = 1 corresponds to the distance of closest approach, since particles are not allowed to exist in the region Ueff > 1. A double Yukawa potential, Eq. (5.9), is used, with eφd/Te = −1, eφ0/Te = 10, λ/rd = 2 and ξ = 2.

Barriers in the effective potential energy. When in the SCL regime, the non- monotonicity of φ may induce the emergence of barriers in Ueff for incoming particles, which make the situation significantly more complex when compared to the OML case.

Barriers in Ueff is a well-known problem that has been extensively studied in the case of incoming ions in monotonic potential profiles under the name of absorption radius effect [Laframboise 1966, Bernstein 1959, Al’pert 1965, Allen 2000, Kennedy 2003]. In the case of a non-monotonic potential profile, barriers are expected to emerge for both electrons and positive ions. Fig. 5.17 shows profiles of Ueff for electrons computed with a double Yukawa potential (see Eq. (5.9) below) [Autricque 2018b].

The largest solution to the equation Ueff (ρ, r) = 1 corresponds to the actual critical impact parameter. Barriers, located at rM (which depends on the particle velocity v), are found by solving Ueff = 1 and dUeff /dr = 0, which is equivalent to

dU r3 (r ) = mv2ρ2, (5.7) M dr M ∗ where ρ∗ is named the transitional impact parameter and is defined by

s 2zeφ(r ) ρ (v) = r 1 − M . (5.8) ∗ M mv2 It is obvious that knowledge of the full potential profile is required to find the barrier. 5.3. Recent progress 129

In the following, we study the special case of a double Yukawa profile

r  r − r  r r − r  φ(r) = (φ + φ ) d exp ξ d − φ d exp d , (5.9) d 0 r λ 0 r λ where φ0, λ and ξ are parameters [Delzanno 2005b]. It was shown that this expression can fit accurately results from the OM theory [Delzanno 2005a]. Note that λ is the characteristic screening length of the second term in Eq. (5.9), which is the one that decays the slowest (since we use ξ > 1). Hence λ should correspond to the plasma Debye length. Two examples of double Yukawa profiles are plotted in Fig. 5.18 for λ = 2rd,

ξ = 2, eφd/Te = −1, eφ0/Te = 10 (a) and eφd/Te = 0.5, eφ0/Te = 5 (b). In the following, T designates the temperature of the plasma species, with a subscript e (i) for electrons (ions) when required.

The position of the barrier, rM , is found by solving Eq. (5.7), which is equivalent to (after some straightforward algebra) solving the transcendental equation

φ !  r   r − r   r  r − r  r mv2 1 + d 1 − ξ M exp ξ d M − 1 − M exp d M = M . (5.10) φ0 λ λ λ λ rd zeφ0

On the other hand, the location of the minimum of φ (due to the VC), named rmin, 0 is found by solving φ (rmin) = 0, i.e.,

φ !  r   r − r   r  r − r  1 + d 1 + ξ min exp ξ d min = 1 + min exp d min . (5.11) φ0 λ λ λ λ

A fundamental difference between rM and rmin is that the latter is a constant defined by the potential profile, while the first depends on the velocity of the charged particle v. The important assumption we bring here is that, in the case of electrons (z = −1), we can approximate the location of the barrier in the effective potential energy by the

VC itself, i.e., rM ≈ rmin. If verified, the currents can be calculated for known location and depth of the VC. The exact values of rM for electrons were calculated for various shapes of the double Yukawa profile, varying φd, φ0, λ and ξ. The two extreme cases, corresponding to the largest deviation between rM and rmin, are plotted in Fig. 5.19. For all the cases tested, the ratio r /r remains in the range 0.5 − 3. Keeping in M min √ mind that electrons are Maxwellian distributed, the most probable velocity will be vth 2 q (vth = T/m is the thermal velocity), where rM /rmin is very close to unity. Thus we conclude that this approximation is reasonable. 130 Chapter 5. Confronting the model with measurements

Figure 5.18: Double Yukawa potential profile for λ = 2rd, ξ = 2, eφd/Te = −1, eφ0/Te = 10 (a) and eφd/Te = 0.5, eφ0/Te = 5 (b). The VCs are located at rmin ≈ 1.5rd (a) and rmin ≈ 2rd (b). OML critical impact parameter, transitional impact parameter and ρVC against the particle velocity normalized to the thermal velocity, in the case of electrons (c) and (d). (c) and (d) correspond to the potential profiles plotted in (a) and (b), respectively. 5.3. Recent progress 131

Figure 5.19: Ratio of the location of the barrier in Ueff from Eq. (5.10) to the location of the VC from Eq. (5.11) against the particle velocity v. rM /rmin remains in the range 0.5 − 3. We used eφd/Te = −1.

Derivation of the collection cross-section. We now define the so-called VC impact parameter ρVC in the same way as ρ∗ but replacing rM with rmin, i.e.,

s 2zeφ ρ (v) = r 1 − min , (5.12) VC min mv2 where φmin = φ(rmin). Fig. 5.18 (c) and (d) show ρ∗ and ρVC for electrons (z = −1) plotted versus the particle velocity normalized to vth for the potential profiles plotted in Fig. 5.18 (a) and (b), respectively. Both impact parameters are actually very close in the OML interval of velocity where they are both below the OML critical impact parameter ρc ,

Eq. (3.24). When a barrier exists in Ueff , particles with ρ(v) ≥ ρ∗(v) cannot be collected, so the critical impact parameter for collection departs from the OML one and saturates OML at ρ∗(v) when ρc (v) ≥ ρ∗(v). Thus the critical impact parameter for collection in the SCL regime is

 q 2zeφ q 2zeφ r 1 − d if r 1 − d ≤ ρ (v)  d mv2 d mv2 ∗  q 2zeφd ρc(v) = ρ∗ if rd 1 − 2 > ρ∗(v) . (5.13)  mv  q  2zeφ(rM ) 0 if v ≤ m

Since ρ∗ and ρVC are actually close (see Fig. 5.18 (c) and (d)), we can approximate 132 Chapter 5. Confronting the model with measurements

ρc in the SCL regime by substituting ρ∗ by ρVC in Eq. (5.13). We find

 q 2zeφ r 1 − d if v > v  d mv2 m  q ρSCL(v) = 2zeφmin , (5.14) c rmin 1 − mv2 if vc < v ≤ vm   0 if v ≤ vc where

v s u ! 2zeφmin u2ze χφd − φmin v = and v = t , (5.15) c m m m χ − 1

2 and χ = (rd/rmin) . Again, the last lines in Eqs. (5.13-5.14) should be taken into account only when z and φ(rM ) (or φmin) are of the same sign (i.e., for ions). This impact parameter can be understood as follows: (i) if v ≤ vc, the incoming electron does not have a sufficient kinetic energy to overcome the VC and cannot be collected; (ii) if vc < v ≤ vm, a barrier in the effective potential energy exists but particles have a sufficiently high kinetic energy to pass the VC and are collected; (iii) if v > vm, particles have such a high kinetic energy that they do not see the VC and the critical impact parameter is identical to the OML one.

We note that when φmin → φd, vm → vc, which leads to the disappearance of the second line in Eq. (5.14). In this case, we recover the classic OML impact parameter. 2 The final cross-section for collection is σ = πρc . The slight overestimation of the cross-section obtained with ρVC instead of ρ∗ is due to the fact that rM < rmin in this regime of velocities. The error in the interval vc ≤ v ≤ vm corresponds to the area where OML ρ∗ ≤ ρ ≤ ρVC, ρc in Fig. 5.18 (c) and (d) and reaches ∼ 30% in this case. In all calculations performed in the case of ions using the double Yukawa profile, we found the barrier to be located inside the grain, rM < rd, meaning the exact cross-section for collection is equal to σOML with Eq. (3.24). In this case, the ion current collected by the dust grain is given by the well-known OML expression. In the following, we will focus SCL on electrons and approximate the exact critical impact parameter ρc with ρc since the VC is easier to compute than the barrier in the effective potential.

Electron collection current. The electron current density is obtained by integrating the collection cross-section with the velocity distribution function fe of the species. When SCL2 substituting σ = πρc from Eq. (5.14) into Eq. (4.7), we obtain, using the change of q variable v = u 2Te/me,

! ! Z um Z ∞ Je 2 3 eφmin −u2 3 eφd −u2 0 = u + u e du + 2 u + u e du, (5.16) Je χ uc Te um Te 5.3. Recent progress 133 where

v s u " # eφmin u 1 e(χφd − φmin) uc = − and um = t . (5.17) Te 1 − χ Te After integration, we obtain the final expression for the electron current in the SCL regime

! Je 1 eφmin h ϕ˜i 0 = exp 1 + (χ − 1)e , (5.18) Je χ Te where

χ e(φ − φ ) ϕ˜ = min d . (5.19) 1 − χ Te This expression is also valid when the VC vanishes, in which case the OML expression is recovered. Indeed, the disappearance of the VC is obtained by rmin = rd and φmin = φd if φd ≤ 0, and by rmin → ∞ and φmin = 0 if φd > 0. Moreover, this more general expression allows us to extend the validity domain of the theory to larger collectors, since the assumption rd  λD made in the OML that ensures the negligible role of barriers in Ueff can be dropped. However, the collisionless and unmagnetized plasma assumptions still require rd  λe,i, ρLe,i. Hence the maximum body size that can be used depends on the plasma background temperature, density and the magnetic field.

Thermionic current. The THE current is obtained by integrating the distribution function, Eq. (4.15), on the passing population. The OML TPB (Trapped-Passing Bound- ary), Eq. (4.16), is a circle shown in the velocity space in Fig. 5.20, where the trapped population is located inside the dotted area. In the SCL regime, emitted electrons can experience potential barriers in the effective potential energy, identically to collected electrons. The TPB in the SCL regime is not easily defined. Similarly to [Delzanno 2014a] and the case of primary electrons treated above, we assume that the barrier in the effective potential energy due to the presence of the VC is located at the minimum of the VC, i.e., rmin. This means that the SCL TPB can be approximated by

2 2 2 vr + (1 − χ)vθ = vp, (5.20) q where we introduce vp = 2e(φd − φmin)/me. In the velocity space, this contour is an ellipse elongated along the vθ axis and is represented in Fig. 5.20. The emission current 134 Chapter 5. Confronting the model with measurements

푣휃

OML TPB

푣푟

SCL TPB

Figure 5.20: Integration domains for the determination of the THE currents, delimited by the TPBs for the OML and SCL cases.

is obtained by integrating outside of the TPB. We use the following change of variables

s 2Td vr =u cos α, me s (5.21) 2Td sin α vθ =u √ . me 1 − χ Writing the current as the integral of the distribution function over the passing elec- tron population, one finds

SCL J K 1   2 the 1 2 −up 0 = 1 − √ + √ K2up + K3 e , (5.22) Jthe 1 − χ 1 − χ q where up = e(φd − φmin)/Td and

2 Z π/2 dα K1 = χ 2 , π 0 1 + 1−χ sin α Z π/2 ! 2 2 χ 2 K2 = exp −up sin α dα, (5.23) π 0 1 − χ Z π/2 ! 2 1 2 χ 2 K3 = χ 2 exp −up sin α dα. π 0 1 + 1−χ sin α 1 − χ 5.3. Recent progress 135

Figure 5.21: f and g functions. g(up, ·) can be approximated with f for up & 1.

One can find that √ K1 = 1 − χ, u2 χ ! u2 χ ! K =exp − p I p , 2 2 1 − χ 0 2 1 − χ (5.24) s χ ! K =g u , u , 3 p p 1 − χ where I0 is the modified Bessel function of the first kind and g is a function defined as

2 1 Z y 1 e−t2 g(x, y) = 2 q dt. (5.25) π y 0 1 + (t/x) 1 − (t/y)2 Since the typical VC depth is of the order of the dust surface temperature, we expect 2 2 up & 1. In this case, g(up, y) can be approximated by f(y) = exp(−y /2)I0(y /2), as can be seen in Fig. 5.21. On this ground, the THE current can be expressed

SCL 2 ! 2 ! J 1   2 u χ u χ the 2 −up p p 0 = √ 1 + up e exp − I0 . (5.26) Jthe 1 − χ 2 1 − χ 2 1 − χ

We observe that, since I0(0) = 1, Eq. (5.26) recovers the expression proposed in the OML+ theory, Eq. (5.5), for χ  1, i.e., when the VC is far from the grain. The expression from Eq. (5.26) is compared with exact calculations of Eq. (5.22) in Fig. 5.22 SCL (a), where Jthe is plotted for eφd/Te = 0, Td/Te = 0.2 and χ = 1/4 as a function of the

VC depth φmin. The agreement is nearly perfect for large φmin because up & 1. As φmin → 0, Eq. (5.26) 136 Chapter 5. Confronting the model with measurements

Figure 5.22: (a) THE current from Eq. (5.22) (solid line) and Eq. (5.26) (dashed line). + ∗ (b) THE currents from OML (dotted line), OML with eφd/Te = −0.125 (dashed line), Eq. (5.26) with χ = 0.05 (solid line) and χ = 0.8 (dash-dotted line).

leads to overestimations and the current calculated can exceed the saturation value. We conclude that the THE current in the SCL regime can be reasonably estimated by 0 SCL 0 Eq. (5.26) with a forced saturation at Jthe when Eq. (5.26) gives Jthe > Jthe. The new expression for the THE current is then compared with the OML+ theory. + Identically to OML , we assume that the VC appears when φd exceeds the critical value ∗ φd which depends on Td/Te and rd/λD (see Fig. 5.15). Once the VC is formed, we ∗ use reasonable values for its depth, i.e., φmin = φd, and vary the VC position χ. The currents are plotted in Fig. 5.22 (b) for rd = λD and Td/Te = 0.2, corresponding to ∗ + eφd/Te = −0.125 according to OML . It is confirmed that the new expression for the THE current in the SCL regime recovers the OML+ when the VC is located far from the grain (χ  1). When χ ∼ 1, the THE current is reduced. This can be understood using 2 2 2 the SCL TPB: electrons are passing if v ≥ vp + χvθ , meaning that there are less passing electrons as χ → 1.

Collected and emitted electron heat fluxes. The heat fluxes corresponding to the currents calculated above are obtained using Eq. (4.51) and the SCL cross-section. We 5.3. Recent progress 137 obtain

SCL ! Qe 1 eφmin h 2 2 ϕ˜i 0 = exp 2 + uc + (χ − 1)(2 + um)e , Qe χ Te SCL 2 ! 2 ! (5.27) Q 1   2 u χ u χ the 2 4 −up p p 0 =√ 2 + 2up + up e exp − I0 . Qthe 1 − χ 2 1 − χ 2 1 − χ

Once again, these expressions recover the ones found in the OML framework, Eqs. (4.54- 4.55) when the VC vanishes and the potential profile becomes monotonic.

Description of the VC. Describing the VC is easier than the barriers in the effective potential energy, yet it is still not straightforward. To be perfectly accurate, one should solve the Poisson equation in the OM framework, as was done in [Delzanno 2005a]. Since we are looking for a fast and simple way to estimate the currents in the SCL regime, we need a direct expression for rmin and φmin. In the following we will use results from [Delzanno 2005a] that allow an estimate of the VC parameters within 15% accuracy when compared with exact calculations made with the OM theory. The transcendental equation for φmin is

 2  q  ˜2 n˜the rd ˜ φmin(1 + β) = 1 − erf −δφmin 4 λD  ˜ ˜ ˜  × exp(φmin) − exp(−βφmin) + Hexp(δφmin) , (5.28)

˜ where φmin = eφmin/Te, β = Te/Ti, δ = Te/Td,

q ˜  q  exp(δφ˜ ) −δφ˜ /π −1 (1 + δφmin) ˜ min min H = ˜ erf −δφmin − 1 − ˜ , (5.29) φmin(1 + β) φmin(1 + β) and

0 ! Jthe eφd n˜the = 0 exp − . (5.30) 2Je Td Finally, the VC position is obtained with

H χ = . (5.31) n˜the Another argument states that the depth of the VC is of the order of the energy of emitted electrons that is, in the case of THE, the dust temperature Td [Martin 2006]. 138 Chapter 5. Confronting the model with measurements

+ ∗ Finally, the OML theory allows the estimation of the dust critical potential φd above which the VC appears [Delzanno 2014a]. In Fig. 5.23 (a) and (b) are plotted the VC parameters using OM results against the dust potential for a W dust grain with rd = 1

µm and Td = 4500 K and for two different electron temperatures (0.5 and 2 eV). In both ∗ + cases, eφd/Te ≈ −0.125 according to OML . This value differs from the estimations made in the figure, where the VC appears at eφd/Te < −2, according to the OM theory.

This link between φd and φmin allows the determination of the dependence of the

OML and SCL currents on φd, for a given set of plasma parameters. This presents an improvement to the OML+ theory, where Poisson’s equation had to be solved a priori ∗ for the value φd to be known. In Fig. 5.23 (c) and (d) are plotted the OML and SCL currents against the normalized SCL body potential. We observe that Jthe starts decreasing as the VC appears and soon OML becomes significantly lower than Jthe , though having the same qualitative behavior. This result differs significantly from what one would obtain using the OML+. Indeed, the OML+ estimates the VC depth to be lower, resulting in a THE current closer to the OML value (as in Fig. 5.22). The primary electron current is also lower in the SCL than in the OML, because the VC acts as a filter that cuts off the low velocity tail of the distribution function. Yet the discrepancy is significant only when Te is of the order of (or lower than) Td, i.e., in

Fig. 5.23 (c). In Fig. 5.23 (d), for Te = 2 eV, primary electrons have, for the most part, enough energy to pass the well (because Te  Td), and the SCL current is very close to the OML result. The much lower electron temperature used in Fig. 5.23 (c) induces a much lower electron current onto the spherical body because most of the impinging electrons bounce back on the VC.

Consequences for the dust lifetime. The expressions presented in this Section were implemented in DUMBO and simulations were run to compare with OML and OML+ results. In Fig. 5.24, a 1 µm W dust grain is simulated in a homogenous plasma with 21 -3 Ti = Te = 5 eV and ni = ne = 10 m . To calculate the VC parameters in the SCL ∗ expressions, we use φmin = φd and Eq. (5.31) for χ. The dust floating potential appears to be significantly reduced, in a similar way to + the OML , and the lifetime is slightly increased. Additional reduction of φd comes from the fact that Eq. (5.26) accounts for the location of the VC, while the OML+ expression assumes rmin → ∞. Also, the present SCL theory accounts for the reduction of the incoming electron flux, which (i) makes the theory self-consistent, which was not the case with OML+, and (ii) increases the dust lifetime even more due to reduced electron heating. 5.3. Recent progress 139

Figure 5.23: (a) and (b) VC depth and location from OM radial model approximation and (c) and (d) OML and SCL electron and THE currents and OML ion current against the dust potential. The dust is made of W with the radius rd = 1 µm and temperature 20 -3 Td = 4500 K. Background plasma parameters are n0 = 10 m and Te = Ti = 0.5 eV (left) and Te = Ti = 2 eV (right). 140 Chapter 5. Confronting the model with measurements

Figure 5.24: Dust floating potential against the temperature using OML, OML+ and 21 -3 SCL theories. The plasma is homogeneous with Ti = Te = 5 eV and ni = ne = 10 m , and the initial dust size is 1 µm. The step observable at Td = 3695 K corresponds to the solid-liquid transition.

Dust electric charge in the thick sheath regime. The dust particle electric charge

Qd is among the most important dust parameters since it dictates particle transport in the plasma via the Lorentz forces, as well as others (ion and electron drag, thermal, etc.). The charge is related to the electric potential through Gauss’s law

I Qd = ε0 ∇φ.dS, (5.32) Sd 2 0 where Sd is the dust surface area. This expression simplifies to Qd = −4πrdε0φ (rd) in our case. Using a Yukawa potential profile, one obtains Eq. (4.33). As pointed out in

[Delzanno 2014a], in the SCL regime, Qd can no longer be obtained from this classical expression since the dust electric charge can be positive even whilst the dust potential is negative. Hence there is a need for a new expression for Qd. 0 Using the double Yukawa profile in Eq. (5.9), along with φ(rmin) = φmin and φ (rmin) = 0, we find

      0 φd rd φmin rmin rmin − rd φ (rd) = − 1 + ξ + 1 + ξ exp . (5.33) rd λ rd λ λ This expression can be conveniently simplified if we place ourselves in the thick sheath 5.3. Recent progress 141

Figure 5.25: Dust electric charge against the Debye length in the SCL and OML regimes. A W dust grain with rd = 0.1 µm, Td = 6000 K and eφd/Te = −0.01 is used in the calculations.

regime, rd, rmin  λ/ξ. In this case

"  # SCL φmin rmin − rd Qd = 4πε0rdφd 1 − exp . (5.34) φd λ

SCL The calculated charge Qd is plotted in Fig. 5.25 along with the OML result for a W dust grain with rd = 0.1 µm, Td = 6000 K and eφd/Te = −0.01. We used Eqs. (5.28-5.31) to estimate the values of rmin and φmin, and assimilated λ to the Debye length λD. As expected, the charge sign is changed from the OML result. Moreover, the presence of the VC induces a much higher electric field at the dust surface, leading to a charge more than two times higher in the SCL regime. This could drastically alter the dust transport in tokamak vacuum vessels, since the electric force is directly proportional to 2 Qd, while the ion and electron drag forces are proportional to Qd.

Conclusion. New expressions for the collection and emission of electrons by a spherical body in the SCL regime have been derived. They are based on the assumption that the barrier in the effective potential energy is located close to the VC. These expressions can be applied to any type of strongly emissive spherical body immersed in a weakly or non-magnetized and collisionless plasma. The thick sheath assumption made in the OML theory is no longer required for using the new expressions. In the SCL regime, the collection current is significantly reduced when the primary electron temperature is of the order of (or lower than) the body temperature. The emission current is always strongly reduced due to the presence of the VC because the average energy of emitted electrons is of the order of the VC depth. 142 Chapter 5. Confronting the model with measurements

The association of the current expressions presented in this Section and the equations for the VC parameters detailed above form an important progress in comparison with the OML+ theory since it is less numerically demanding and the correction to the electron correction current is accounted for. The determination of the VC parameters (location and depth) remains an important challenge even though some estimates are available. An expression for the dust electric charge is proposed and can be used when the thick sheath regime applies, which is the case for small grains and/or hot plasmas (since √ λD ∝ Te). It leads to changes in the dust charge sign and magnitude that are carried forward to the electric and plasma drag forces a dust grain experiences when transported in a tokamak plasma. The SCL expressions proposed in this Section present significant improvements to the previously used OML+ theory and are accompanied with dust lifetime increase in DUMBO simulations, but this increase does not seem to be sufficient to reproduce exper- imental lifetimes.

5.4 Conclusion

Results from the dust transport code DUMBO described in Chapter4, implementing the OML theory presented in Chapter3, have been compared with experimental data from Chapter2. The model leads to overestimations of the dust heating, leading to underestimations of its lifetime. The search for the solution to the overheating issue lead us to consider several effects neglected in current models: the effects of the magnetic field on emitted electrons, that of potential wells around the dust, and vapor shielding. All of these effects incline towards less dust heating. It is likely that a predictive dust transport code would require all of the above to be implemented in its model. On this ground, some work, especially on the vapor shielding effect, is still required before this milestone is achieved. As a means of summarizing all the improvements proposed to the OML theory in the second part of this Chapter, see Table 5.2 for the domains of applicability and Table 5.3 for the major consequences for the DUMBO simulations. 5.4. Conclusion 143

Table 5.2: Summary of the domains of applicability of the improvements to the OML theory presented in this Chapter.

Improvement to OML Domain of applicability

rd, ρLse  ρLe,i (only secondaries are magnetized), Magnetized electron emission rd  λe,i (collisionless) and rd  λD (thick sheath) OML+ None δ  δ (cold plasmas), r  ρ (Unmagnetized), SCL see the d L rd  λe,i and rd  λD (if Eq. (5.34) used)

Table 5.3: Summary of the consequences of the improvements to the OML theory pre- sented in this Chapter.

Improvement to OML Consequences for dust lifetime Magnetized electron emission Lifetime slightly increased OML+ Lifetime increased SCL Lifetime increased Reduced ablation rate =⇒ lifetime increase Vapor shielding But: complete theory still lacking

Chapter 6 Dust-wall interactions in tokamaks

Contents 6.1 Introduction...... 146 6.2 Dust-wall collisions...... 146 6.2.1 Adhesion of dust onto a planar surface upon collision...... 147 6.2.2 Bouncing collisions...... 152 6.2.3 Accounting for the roughness of the wall...... 153 6.2.4 Dust and/or wall local damaging...... 154 6.3 Remobilization/resuspension...... 155 6.3.1 Adhesion of dust on rough planar surfaces...... 156 6.3.2 The particular case of W: importance of the impact velocity.... 158 6.3.3 Experimental investigation of the adhesion force...... 165 6.3.4 Remobilization condition and exit velocity in tokamaks...... 167 6.3.5 Conclusions on dust remobilization studies...... 173 6.4 Conclusion on dust-wall interactions...... 173 146 Chapter 6. Dust-wall interactions in tokamaks

6.1 Introduction

An important aspect of dust transport in tokamaks was not yet addressed: that of dust sources and sinks. The first provide valuable inputs for dust modeling codes such as DUMBO, while the second dictates the simulation ending conditions. Main dust sources are dust creation, remobilization and injection, even though the latter brings a marginal amount of dust. Additional sources of dust exist during shutdown periods, when the vacuum vessel is put to atmospheric pressure and opened. Dust total vaporization due to plasma heating was mentioned earlier and acts a main sink. Also, dust samples harvesting during shutdown phases is a common thing. But the amount of dust sampled is generally too low to play any significant role in the dust amount balance of a current tokamak, though it might not be the case for ITER. Other important dust sinks are linked with dust-wall collisions. The transport of dust grains in tokamaks is mainly inertial, due to their high mass compared to plasma drag forces. Because of the toroidal shape of fusion devices, dust grains are centrifugally accelerated towards the outer walls. This is why dust-wall col- lisions is a crucial part of overall dust transport. A dust-wall collision can have several outcomes, depending on the materials at play, the impact velocity, the dust size and temperature: bouncing, sticking, or dust/wall damaging. When dust adheres on the tokamak wall, it is crucial to estimate the adhesion force in order to predict the resuspension (and remobilization) capacity of the plasma. In this Chapter, we propose to explore two major aspects of dust sources and sinks in tokamaks: dust-wall collisions and dust remobilization.

6.2 Dust-wall collisions

We consider a spherical dust of radius rd impacting a flat surface with velocity vi = ⊥ k vi × n + vi × t, where n and t are the normal and tangential unity vectors to the surface ⊥ k and vi and vi are the normal and tangential impact velocities, respectively. The outcome of a collision between a dust particle and a surface depends mainly on the impact velocity, as can be seen in Fig. 6.1. The most important characteristic velocity is named the adhesion velocity vadh. ⊥ If the dust normal impact velocity vi is below vadh, it will stick to the surface as adhesion forces are strong enough to prevent rebound. For impact velocities above this threshold, rebound is possible and usually comes with significant energy loss because of plastic deformations and irreversible work of adhesion forces. Even faster impacts can induce dust/wall damaging that can result in the particle destruction and possibly 6.2. Dust-wall collisions 147

Figure 6.1: Sketch of the possible outcomes and deformation types for a dust-wall collision versus the impact velocity. Two aspects are to be taken into account: (i) the types of deformation at play (blue part) and (ii) the outcome of the impact (red part). The ranking of vy and vadh depends on the particle size and the materials at play.

⊥ creation of new grains. For vi < vadh, the adhesion force can depend on the impact velocity as irreversible plastic deformations increase the contact area. The quantification of the adhesion force is critical to predicting dust remobilization. The first attempt of modelling dust-wall collisions in tokamaks was made in the DUSTT code, where constant restitution coefficients (defined as the ratio of a given quantity before and after a given event) for the dust size and temperature were used [Pigarov 2005]. Then, constant restitution coefficients for the normal velocity were com- bined with a constant probability for adhesion [Smirnov 2007]. Collisions were also stud- ied using a finite element code coupled to simulations performed with the DUSTT code ⊥ in the case of high velocity impacts (vi ∼ 0.1 − 1 km/s) only [Smirnov 2009b]. The MIGRAINe code implements the most complete analytical dust-wall collisions model [Ratynskaia 2013]. It is based on the Johnson-Kendall-Roberts (JKR) model to de- termine the adhesion velocity vadh, which depends on dust and wall material param- eters (Young moduli, Poisson coefficients and surface energies) and the dust radius [Johnson 1971]. The velocity restitution coefficients are calculated using the Thornton and Ning (T&N) model [Thornton 1998].

6.2.1 Adhesion of dust onto a planar surface upon collision

The kinetics of a dust-wall impact can be divided into two distinct phases: ⊥ - Loading phase. The dust impacts the surface with a given velocity vi . It is deformed so that the initial kinetic energy is converted into deformation energy, and some dissipation phenomena can play a role (irreversible plastic deformation, viscoelastic dissipation, adhesion work, . . . ). - Unloading phase. The accumulated elastic deformation energy is released into adhesion work, dissipation effects and, if high enough, output dust kinetic energy. The models presented hereafter differ by the type of dissipation mechanisms they include, which is why they lead to different values of the adhesion velocity. This sticking 148 Chapter 6. Dust-wall interactions in tokamaks criterion depends on both the normal and tangential components of the dust impact velocity (since grazing impacts are more likely to result in the grain bouncing). First, let us focus on the simple case of normal impacts to derive estimates of the adhesion velocity.

Sticking criterion for elastic adhesive spheres. In the framework of the JKR theory, only elastic deformation is considered (dissipation phenomena such as plastic deformation or viscoelastic effects are neglected), and the adhesion work is taken into account during the unloading phase only. Full development of the JKR theory leads to the following expression for the adhesion velocity

s 2/3 2 5 !1/6 JKR 3(1 + 6 × 2 ) π γ vadh = 3 2 5 , (6.1) 5 2ρdE∗ rd √ where γ = γdγs, γi being the surface energy and E∗ is the reduced Young modulus defined by

1 1 − ν2 1 − ν2 = d + s , (6.2) E∗ Ed Es where Ei and νi are the Young moduli and Poisson ratios, respectively. Subscripts d and s are used for parameters of the dust grain and the substrate, respectively.

Sticking criterion accounting for viscoelastic effects. A recent work proposes to incorporate viscoelastic dissipation into the picture [Chen 2015], as well as the angle ⊥ VE dependence of the adhesion velocity. When the impact is normal and vi = vi = vadh, the output dust kinetic energy is zero and the kinetic energy associated with the adhesion velocity equals the adhesion work plus the viscoelastic dissipation energy

1 2 1 2 m vVE = m vJKR + ∆E , (6.3) 2 d adh 2 d adh diss where ∆Ediss is the viscoelastic dissipation. Following [Chen 2015], the viscoelastic dissi- pation energy is equal to the work of dissipation forces, which are assumed to be propor- tional to the rate of change of the material deformation with a given dissipation coefficient (see Fig. 5 and 6 in [Chen 2015]). It comes that the adhesion velocity at normal inci- VE dence, vadh, accounting for viscoelastic dissipation, is obtained by solving the following equation

VE VE !4/5 vadh vadh q JKR − αk1 JKR = 1 + αk2, (6.4) vadh vadh 6.2. Dust-wall collisions 149 where √ 67/30 5 k = ≈ 2.68, 1 (1 + 6 × 22/3)1/10 s (6.5) 35/6 5 k = ≈ 0.77, 2 27/6 1 + 6 × 22/3 and α is the dissipation coefficient, calculated by the 6th order polynomial α = 1.2728 − 2 3 4 5 6 4.2783 × e0 + 11.087 × e0 − 22.348 × e0 + 27.467 × e0 − 18.022 × e0 + 4.8218 × e0, where e0 is the restitution coefficient set at 0.6 to account for the damping effects [Marshall 2009]. In this light, Eq. (6.4) exhibits a solution that does not depend on any dust or surface parameter. We have

VE JKR vadh = k × vadh . (6.6) where the coefficient k ≈ 2.514 is solely due to viscoelastic dissipation.

Sticking criterion accounting for plasticity. Incorporating irreversible plastic de- formation further increases the adhesion velocity, since a greater part of the dust initial kinetic energy is dissipated. First of all, there exist a threshold impact velocity below which only elastic deformation occurs (see Fig. 6.1). This yield velocity, noted vy, is written [Thornton 1998]

π2 p5 !1/2 √ y vy = 4 , (6.7) 2 10 ρdE∗ where py is called the limiting contact pressure, which is proportional to the yield strength of the material, σy, with a multiplying constant ranging between 1.6 and 2.8 due to the ambiguity of the elastic-plastic transition [Ratynskaia 2013]. In DUMBO, py = 2σy JKR is used. The ranking between vadh and vy depends on the dust radius. Following JKR [Ratynskaia 2013], the threshold dust radius for which vadh = vy is defined

3/5 2 2 × 6  2/33/5 γE∗ rth = 2 1 + 6 × 2 3 . (6.8) π py

Calculations give rth = 507 µm for W/W, rth = 197 µm for Be/Be and rth = 1.25 mm for C/C impacts. These values are much higher than the dust size we are able to model using an OML-based transport code as well as being irrelevantly high for tokamaks. JKR Thus, in DUMBO, it is assumed that rd < rth, i.e., vy < vadh , which means that plastic effects are always present for the three materials usable in the code. In this case, the JKR and Chen et al. theories, which neglect plasticity, are inapplicable. The dissipation 150 Chapter 6. Dust-wall interactions in tokamaks

Figure 6.2: Adhesion velocities against the dust radius for W/W impacts from the JKR model, which only considers elastic and adhesive effects (dashed red), the model by Chen et al., which adds viscoelastic dissipation (dash-dotted black), and the T&N model, which incorporates plastic dissipation (solid black). The yield velocity is also plotted. The material used is W. θ is the angle with the normal to the surface.

due to plastic deformation further increases the actual sticking velocity up to a value named vadh. It was determined in [Thornton 1998, Ratynskaia 2013] as the solution of the equation

√  1/2 "  2# JKR !2  3/2 s  2 6 3 1 vy vadh vy vy 2 vy 1 − =  + √ 6 −  . (6.9) 5 6 vadh vy vadh vadh 5 vadh

The three models for the adhesion velocity presented above, with increasing complex- ity and incorporated physics, lead to increasing values of the adhesion velocity, as shown in Fig. 6.2 in the case of W. Note that there are other dissipation mechanisms that are still not accounted for, such as viscoplasticity. Moreover, the adhesion criteria derived above are for normal impacts only.

Effects of the impact angle. The adhesion criteria derived up to now only consider k normal impacts (vi = 0). Yet, it is known that the adhesion velocity strongly depends on the impact angle. Oblique impacts have been extensively studied in the literature in terms of experiments [Brach 2000], models and simulations [Konstandopoulos 2006]. The existence of a critical impact angle above which sticking cannot occur was found 6.2. Dust-wall collisions 151

Figure 6.3: Adhesion velocities for a 1 µm W dust grain impacting a W surface from the JKR model (dashed red), independent on the impact angle, and the T&N model with dependency on the impact angle (black). The dash-dotted curve represents the impact velocity averaged over the surface roughness.

in [Konstandopoulos 2006]. In the work by Chen et al., the overall adhesion velocity is written

vadh(θ) = vadh × g(θ), (6.10) where θ is the impact angle with the normal to the surface. The dependency to the impact angle is obtained by fitting discrete element method simulations

−0.6 g(θ) = 1 − 0.0011 (1 − | sin θ|) if | sin θ| ≤ b0, (6.11) −0.4 g(θ) = 1 − 0.0454 (1 − | sin θ|) if | sin θ| > b0, where b0 = 0.955 is the critical impact parameter dividing two distinct regimes. Fig. 6.3 shows the impact angle dependency of the adhesion velocity for W/W impact and rd = 1 µm. The consequence is that the adhesion criterion does not only depend on the normal impact, but also on the tangential velocity. Indeed, the dust adheres to the surface when ⊥  k ⊥  vi ≤ vadh(θ) = vadh × g arctan(vi /vi ) . 152 Chapter 6. Dust-wall interactions in tokamaks

6.2.2 Bouncing collisions

⊥ When the dust-wall collision results in the dust bouncing (vi > vadh), part of the dust kinetic energy is lost into adhesion work and other dissipation effects. Thus we need to compute a restitution coefficient, defined as the ratio of the rebound and impact velocities. The dust rebound velocity is written

k ⊥ vr = ekvi × n + e⊥vi × t, (6.12) where ek and e⊥ are the tangential and normal restitution coefficients, respectively.

Perpendicular restitution coefficient. Similarly to the MIGRAINe code, DUMBO im- plements the T&N model for the normal impact of elastic perfectly plastic adhesive spheres [Thornton 1998], which allow the determination of simple expressions for the perpendicular restitution coefficient e⊥ [Ratynskaia 2013]. Then, the normal restitution ⊥ coefficient is given by, if vi ≥ vadh(θ) [Ratynskaia 2013]

 v −1/2 √  !2 s u !2 !2 6 3 1 v 6 v⊥ u 1 v vJKR 2 y  i t y  adh e⊥ = 1 − ⊥  1 + 2 1 − ⊥  − ⊥ , (6.13) 5 6 vi 5 vy 6 vi vi

⊥ and e⊥ = 0 if vi ≤ vadh(θ). Eq. (6.13) accounts for dissipation from irreversible plastic deformation and adhesion work as well as the effect of the impact angle. e⊥ is plotted ⊥ against vi for a 1 µm W dust grain in Fig. 6.4. We observe that the values taken by e⊥ remain low (∼ 0.1), meaning that a large part of the energy is lost. This can be understood as follows: in the case of W, vy is very low (7.5 mm/s) compared to typical dust velocities, and in particular vy  vadh. This means that, for bouncing impacts, plastic deformations are always at play, and are actually dominant.

Tangential restitution coefficient. If the impact is oblique (θ 6= 0), the tangential restitution coefficient must be known. ek is very difficult to calculate because of the many possible regimes the collision can have (rolling, sliding, . . . ), which means that normal and tangential physics are tightly linked. It was argued in [Ratynskaia 2013] that it can be written, when in the sliding regime,

⊥ vi ek = 1 − µ k (1 + e⊥) , (6.14) vi where µ is the Coulomb friction coefficient, which is an empirical property of the con- 6.2. Dust-wall collisions 153

Figure 6.4: Normal restitution coefficient against the normal impact velocity for a 1 µm W dust grain.

tacting materials, fixed at µ = 0.15 in DUMBO.

6.2.3 Accounting for the roughness of the wall

All the considerations above were made for spherical dust impacting flat and smooth surfaces. In tokamak applications, the wall roughness must be taken into account when computing a dust-wall impact. In [Ratynskaia 2013], the surface roughness was divided into two distinct contributions at different scales: (i) the large-scale roughness, which only affects the orientation of the surface with respect to the dust velocity vector; (ii) the small-scale roughness, which does not alter the impact geometry but the adhesion velocity itself, since it tends to reduce the contact area. Here the adjectives “large” and “small” refer to the roughness size with respect to the dust radius.

Large-scale roughness. The first can be modeled by introducing an effective normal vector to the smooth surface and varying its angle α with the normal to the perfect surface with a Gaussian law [Ratynskaia 2013]. We note that α is constrained between to values

αmin and αmax that depend on the impact geometry, since the dust cannot encounter a surface at angles larger than grazing incidence. The angle extremes are defined by

αmin = −π/2+θ ×H(θ) and αmax = π/2−θ ×H(−θ), where H is the Heaviside function. 154 Chapter 6. Dust-wall interactions in tokamaks

Small-scale roughness. The small-scale roughness can be accounted for by intro- ducing a random variable χ taking values between 0 and 1 and using χ × vadh as the adhesion velocity at normal incidence. This effect can be easily implemented in dust transport codes and the immediate consequence is the reduction of the sink effect of the wall. In this light, the adhesion velocity at a given impact angle and averaged over both scales of surface roughness, noted < vadh >r, is written

α vadh Z max < vadh >r (θ) = g(α − θ)fα(α)dα. (6.15) 2 αmin where fα is the Gaussian distribution function for the angle α (large-scale roughness) and g is given by Eq. (6.11). The 1/2 factor comes from the small-scale roughness. < vadh >r is plotted in Fig. 6.3. We observe that the adhesion velocity is significantly reduced, by about 50%, at grazing incidence, while still remaining about two times higher than the JKR result.

6.2.4 Dust and/or wall local damaging

When the normal impact velocity is very high, the dust can be destroyed and/or the wall locally damaged. This was shown using a discrete element code for velocities in the range 0.1 − 1 km/s [Smirnov 2009b]. In such cases, the outcome of the impact is dictated by the internal compressive stresses due to the shock waves in both the dust and the wall, noted σd and σw, respectively. These stresses can be estimated using the Hugoniot equations [Zukas 1990]

⊥ σd ≈ ρdcd(v − κ), i (6.16) ⊥ σw ≈ ρwcwvi , where

v⊥ κ = i . (6.17) 1 + ρwcw ρdcd and cd and cw are the speeds of sound in the respective materials, expressed

v u u Ei(1 − νi) ci = t . (6.18) ρi(1 + νi)(1 − 2νi)

If the compressive stresses exceed the ultimate tensile strengths of the materials, σu, the DUMBO simulation is stopped as the dust or the wall experience damage. Thus 6.3. Remobilization/resuspension 155 there exist another threshold impact velocity, above which the impact is destructive.

This velocity is about vmax ≈ σu/(ρici), which is 17.3 m/s, 25.5 m/s and 47.3 m/s for W/W, Be/Be and C/C impacts, respectively. It results that rebounds occur in a quite ⊥ narrow interval of normal impact velocities (5 ≤ vi ≤ 20 m/s for micron W dust on a W surface). In addition to this, if a dust grain bounces in the lower region of the vacuum vessel, it is likely to go back towards the surface due to gravity and bounce again until sticking. This means that the tokamak wall can be considered, in first approximation, as a sink for dust grains. All the considerations of this Section allows us to estimate the contribution of the wall as a sink for dust particles, either because of sticking or destructive collisions. Now if a dust grain remains stuck to the wall, can it be unstuck (remobilized) by external forces and be transported again in the plasma?

6.3 Remobilization/resuspension

Correct depiction of the forces acting on a dust grain stuck onto a planar surface is critical to the understanding of dust remobilization. When a dust particle lays on a surface, the adhesion force implies that a threshold external force is required to remobilize it. Among the most common external forces encountered in tokamaks are electric forces due to the Debye sheath potential drop, drag forces due to collected and scattered plasma particles and gravity, which can act either as part of the external force or add up to the adhesion force depending on the orientation of the surface. Dust remobilization has been studied by Tomita et al. [Tomita 2006b, Tomita 2006a, Smirnov 2006, Tomita 2007, Tomita 2008], but the adhesion forces were neglected, lead- ing to a net underestimation of the release threshold. This problem was recently pointed out in a recent work by Tolias et al. [Tolias 2016], where the adhesion forces from the JKR and Derjaguin-Muller-Toporov (DMT) models [Johnson 1971, Derjaguin 1975] were incorporated in the picture. The authors concluded that 10 µm W dust remobilization during steady-state plasmas in tokamaks is impossible. Yet the JKR and DMT mod- els used are known to overestimate the adhesion forces because they neglect the surface roughness [Biasi 2001]. In this Section, classical adhesion force models are presented, compared against ad- hesion force measurements for tokamak-relevant situations, and the possibility for dust to be remobilized in tokamak conditions is discussed. 156 Chapter 6. Dust-wall interactions in tokamaks

Table 6.1: Hamaker constants for the materials used in DUMBO [Tolias 2018].

Material pair A (J) C on C 2.55 × 10−19 Be on Be 3.48 × 10−19 W on W 4.98 × 10−19

6.3.1 Adhesion of dust on rough planar surfaces

When close or in contact with a surface, a dust particle is subjected to an adhesion force. An extensive literature exists on adhesion of spherical particles on planar surfaces. In the framework of van der Waals forces, the adhesion force is due to the fields generated by the atoms dipole moments (bottom up approach) [Hamaker 1937]. The well-known van der Waals force expression is obtained for spherical particles at close range with a planar surface.

Calculation of the van der Waals force. The van der Waals interaction potential between a planar surface and a sphere is written [Hamaker 1937]

2 2 A Z 2rd r − (x − r ) U(r) = − d d dx, (6.19) 6 0 (H + x)3 A is the Hamaker constant, which depends on the material pair. Values for the Hamaker constants of materials used in DUMBO are available in Table 6.1.

After integration, the van der Waals adhesion force is obtained by FVdW = −dU/dr, i.e.,

3 2Ard FVdW = − 2 2 . (6.20) 3H (H + 2rd)

When the dust is in contact with the surface (H = H0 ≈ 0.36 nm  rd), we recover the well-known expression for the van der Waals force

Ard F0 = − 2 . (6.21) 6H0

Accounting for the surface roughness. In real life, the surfaces a dust particles impacts is never smooth as different scales of roughness exist. Efforts have been made to incorporate this effect in Eq. (6.21)[Rabinovich 2000a]. It is considered that the dust particles interacts with an asperity and the underlying plane. The modified Rumpf model offers the simplest approach, since the asperity is assumed to be hemispherical and with 6.3. Remobilization/resuspension 157

Figure 6.5: Roughness picture as considered by the Rumpf (a) and Rabinovich (b) models. Source: [Rabinovich 2000a]

its center coinciding with the underlying plane, see Fig. 6.5 (a). The total adhesion force is the sum of the particle/asperity (sphere/sphere) and par- ticle/plane (sphere/plane) contributions. It yields

" 1 1 # Fa = F0 + 2 , (6.22) 1 + rd/(1.485rms) (1 + 1.485rms/H0) where rms is the Root Mean Square roughness. A more complete picture was proposed by Rabinovich: the roughness is composed of two scales. Each scale presents hemispherical asperities and their center is allowed not to coincide with the underlying plane. They are characterized by a Root Mean Square roughness rmsi and a peak-to-peak distance λi, as can be seen in Fig. 6.5 (b). The resulting expression for the adhesion force is

" 1 1 F = F + a 0 2 2 2 1 + 58rms2rd/λ2 (1 + 58rms1rd/λ1) (1 + 1.82rms2/H0) 2 # H0 + 2 . (6.23) (1 + 1.82 (rms1 + rms2))

In all cases, roughness reduces adhesion because it reduces the contact area between the particle and the surface and pulls the particle away from the surface by inserting an asperity between the two. For very low values of roughness (rmsi ∼ 10 nm and lower), the Rabinovich model gives similar values to the classical van der Waals expression. 158 Chapter 6. Dust-wall interactions in tokamaks

푓 Elastic deformation energy Plastic deformation energy Adhesion work

푓푚 C

D B 푓푦 A 훿

훿푦 훿푓 훿푚

Figure 6.6: Sketch of the force (f) - overlap (δ) curve for the impact of an elastic perfectly- plastic adhesive sphere. Three phases are distinguishable: elastic (AB) and plastic (BC) loading and elastic unloading (CD).

Adhesion forces from contact mechanics models. Another approach, based on the Hertz theory, considers the dust deformation and uses the global energy conservation to determine the adhesion force. This top-down approach leads to the JKR and DMT theories, where the adhesion force is expressed as

JKR,DMT Fa = ξaπrdγ, (6.24) where ξa = 3/2 or 2 for the JKR and DMT models, respectively. These two models present the limit that the surface roughness is ignored, and are commonly known to overestimate the adhesion force [Biasi 2001].

6.3.2 The particular case of W: importance of the impact ve- locity

In the case of a W/W impact, we have seen that the yield velocity is vy = 7.5 mm/s, which is small compared to typical dust velocities in tokamaks [Smirnov 2009b]. In the ⊥ following, we focus on the case of W/W impacts and in the regime where vy  vi ≤ vadh, where the deformation energy is dominated by plasticity, meaning that the contact radius is a function of the impact velocity [Autricque 2018a]. 6.3. Remobilization/resuspension 159

Estimation of the contact radius. When a spherical dust particle impacts and sticks to a flat surface, it is elastoplastically deformed. The kinetics of an adhesive dust- surface normal impact for an elastic-perfectly plastic adhesive sphere is divided into the loading and unloading phases, as can be seen in Fig. 6.6. δ designates the overlap 2 (indentation depth), with δ = a /rd (see also Fig. 6.7). During the loading phase, the dust initial kinetic energy is converted into elastic deformation energy (AB) until the yield ⊥ (elastoplastic transition) is reached (if vi ≥ vy). Then, the remaining energy is converted into irreversible plastic deformation (BC). If the material is perfectly plastic, which is the case for W [Wirtz 2017], the force f increases linearly with δ since the pressure saturates at py (and the force increase is solely dictated by the contact radius increase). When all the dust initial kinetic energy is converted into deformation energy, δ has reached its maximal value δm. During the unloading phase (CD), the material elastically unloads until the remaining elastic deformation energy equals the adhesion work. At the end of the cycle, there is a residual overlap δf due to both adhesion work and irreversible plastic deformation. Since the amount of plastic deformation depends on the impact velocity, so does the final overlap. In this Section, the final contact radius is determined as a function of the impact velocity. First, if the impact velocity is exactly vy, the dust kinetic energy equals the full elastic deformation energy, which can be approximated by

1 1 m v2 ≈ k δ2, (6.25) 2 d y 2 e y where ke is the elastic stiffness of the material. The yield force can be linked to the yield pressure with fy = keδy = pyπrdδy, and we can deduce the yield overlap

2 rd πpy  δy = √ . (6.26) 30 E∗ In the regime of impact velocities considered, the elastic deformation energy (in the loading phase) is negligible with respect to the plastic deformation energy, which in turn equals the dust initial kinetic energy

1 2 1 1 m v⊥ ≈ (δ − δ )(f + f ) ≈ δ (f + f ), (6.27) 2 d i 2 m y m y 2 m m y since δm  δy. Also, since W is perfectly plastic [Wirtz 2017], the force-overlap law in the plastic domain (BC) is linear and the evolution of the force is solely dictated by the 160 Chapter 6. Dust-wall interactions in tokamaks

increase of the contact radius, i.e., fm = pyπrdδm. We obtain

δ !2 δ δ !2 v⊥ !2 m + m ≈ m = i . (6.28) δy δy δy vy

This expression is valid as long as the deformation remains small, i.e., δm  rd, which q is equivalent to the impact velocity being small compared to vmax = 3py/(4ρd). For W, vmax = 198 m/s, which is much higher than adhesion velocities for typical dust. Thus the deformation should remain small as long as the dust sticks to the surface. The elastic unloading (CD) is also negligible due to the high elastic stiffness of W. √ Thus the final contact radius is close to the maximal value am = rdδm. In this light

1/4  ⊥2  4ρdvi a = rd   . (6.29) 3py

⊥ For a 1 µm dust with vi = 1 m/s, a = 71 nm.

Van der Waals force on a deformed dust grain. Due to adhesion forces and irreversible plastic deformation accumulated during the impact, a once spherical particle exhibits a non-zero contact area with the surface. Elastic deformation due to the surface energy of the system is the framework of the JKR theory. Plasticity has yet to be accounted for in these models. When a dust-wall impact results on the dust sticking to the surface, an adhesion force exists and must be quantified. A spherical dust grain impacting a planar surface loses its shape because of elastoplastic deformation as a flat contact area is created. Here we attempt to describe the van der Waals force acting on such a deformed dust grain. Let us consider a deformed W dust located at a distance H from the surface. We assume the dust is deformed only at the contact area, and keeps its spherical shape elsewhere. This assumption is valid of the deformation remains small (δ, a  rd). A schematic view is shown in Fig. 6.7. The van der Waals potential between a planar surface and the truncated sphere represented in Fig. 6.7 is written

2 2 A Z 2rd r − (x − r ) U(r) = − d d dx, (6.30) 6 δ (r + x)3 where r = H − δ. After integration, we obtain the van der Waals adhesion force by

FVdW = −dU/dr, i.e., 6.3. Remobilization/resuspension 161

푟푑

푎

훿 퐻

Figure 6.7: Schematic of the deformation of the dust grain. a is the contact radius, δ the overlap and H the dust-surface distance. Dimensions are deliberately exaggerated for the sake of representativeness.

" 2 Ard H + δ  H  FVdW = − 2 + 6H H − δ H − δ + 2rd δ − 2r δ H − δ + 2r # +H d − − δ2 d . (6.31) rd(H − δ + 2rd) rd rdH(H − δ)

In the case of a spherical (non-deformed) particle (δ = a = 0) at close range (H  rd), we recover the well-known expression for the van der Waals force, Eq. (6.21). In the general case (δ 6= 0), we make two simplifying assumptions. The first one is that the dust is in contact with the surface, i.e., H = H0  rd. This first assumption is compulsory, since we consider the dust to have had a sticking impact with the surface. This allows us to simplify Eq. (6.31) to

2a2 ! FVdW = F0 1 + . (6.32) rdH0 Second, for the deformed dust grain to remain spherical everywhere but at the contact area and in order to assimilate the deformed dust radius to be equal to the dust radius prior to the collision, we must assume that the deformation is small, i.e., a  rd =⇒

δ  rd, which is consistent with our calculation of the contact radius. Eq. (6.32) implies that the adhesion force is increased from the non-deformed case. Note that we recover the expression proposed in [Alvo 2010].

FVdW/F0 is plotted against a/rd for a W dust grain of radius rd = 1 µm on a W substrate in Fig. 6.8. It appears that the adhesion force can be enhanced by up to two orders of magnitude 162 Chapter 6. Dust-wall interactions in tokamaks

Figure 6.8: Van der Waals force on a deformed W dust grain against the contact radius for rd = 1 µm.

when the dust is significantly deformed. Still, Eq. (6.32) does not consider the fact that actual plasma-facing surface are rough, which in turn diminishes adhesion.

Accounting for the surface large-scale roughness. Here we attempt to incorporate the effects of the roughness on Eq. (6.32), which should tend to reduce the adhesion force by reducing the contact area. The roughness is represented, in a similar way to the Rumpf model modified by Rabinovich et al. [Rabinovich 2000a], by a hemisphere of radius r and an underlying plane. The maximum height of the asperity (with respect to the underlying plane) is called ym, and λ is the peak-to-peak distance. As both the dust grain and the asperity are plastically deformed and share the same contact radius a, the total van der Waals force is the sum of two components: (i) the force between the deformed dust and the deformed asperity and (ii) the force between the deformed dust and the underlying plane. A schematic view of the problem is represented in Fig. 6.9. The exact same calculation as above can be done in the case of two deformed spheres of radii R1 and R2 sharing the same contact area of radius a. The van der Waals force between these two spheres is, when H0, a  R1,R2,

2 ! A R1R2 2a FV dW = 2 + . (6.33) 6H0 R1 + R2 H0 This calculation is easily performed using the same method as in [Hamaker 1937]. Also, the contact radius between these two spheres can be calculated the same way as 6.3. Remobilization/resuspension 163

푟푑

2푎

퐻0 푦푚

푟

휆

Figure 6.9: Schematic of the deformation of the dust grain and the asperity when the surface is rough. Dimensions are deliberately exaggerated for the sake of representative- ness.

above. One finds

2E R R !1/4 a = 0 1 2 , (6.34) πpy R1 + R2 where E0 is the initial kinetic energy of one of the two bodies, the other being considered ⊥2 immobile. Eq. (6.34) can be rewritten, with R1 = rd, R2 = r and E0 = mdvi /2,

1/4  ⊥2  4ρdvi a = rd   . (6.35) 3py (1 + rd/r) Using the same calculation as in [Rabinovich 2000a], the total van der Waals force is

" 2 # " 2 # Ard r 2a Ard 2a FVdW = − 2 + − 2 1 + , (6.36) 6H0 rd + r rdH0 6(H0 + ym) rd(H0 + ym) where the two terms correspond to the contributions (i) and (ii) detailed above, respec- tively. It is more convenient to express the force in terms of rms roughness and peak-to- 2 peak distance λ. From [Rabinovich 2000a], we have r ≈ λ /(58rms) and ym ≈ 1.82rms. Rearranging, we find 164 Chapter 6. Dust-wall interactions in tokamaks

Figure 6.10: Van der Waals force on a W dust grain on a rough W surface against the rms for rd = 1 µm and for varying impact velocities. The value given by the modified Rumpf model, Eq. (6.37) with a = 0, is shown as a comparison.

" 1 1 FVdW = F0 2 + 2 1 + 58rdrms/λ (1 + 1.82rms/H0) 2a2 1 !# + 1 + 3 . (6.37) rdH0 (1 + 1.82rms/H0)

Note that we recover the expression from Rabinovich et al. for a non-deformed dust (a = 0) and the modified Rumpf model in the particular case where the center of the asperity coincides with the underlying plane (ym = r). FVdW is plotted in Fig. 6.10 against rms in the case ym = r. Eq. (6.37) is valid for relatively rough surfaces, i.e., when r & rd, because the defor- mation of the dust due to contact with the underlying plane is not considered. In this regime, the sphere/sphere interaction dominates over the sphere/plane. It appears that the adhesion force can be significantly increased when compared to the modified Rumpf ⊥ model, even at low impact velocities, as vi ∼ 1 m/s corresponds to a free fall in vacuum of a ∼ 10 cm height. This highlights the high sensitivity of the adhesion force on the way the dust was deposited on the substrate. Moreover, the values of forces obtained are in quantitative agreement with adhesion force measurements performed on W dust [Riva 2017]. 6.3. Remobilization/resuspension 165

Figure 6.11: AFM images of the three surfaces, glass (a), W smooth (b) and W rough (c), used by S. Peillon et al. for the adhesion force measurements.

Table 6.2: Surface roughness parameters measured on the different surfaces and used in the Rabinovich adhesion force model.

Surface λ1 (µm) rms1 (nm) λ2 (µm) rms2 (nm) Glass 0.48 4.23 0.24 0.63 W smooth 2.78 9.6 1.5 3.2 W rough 2.13 852 0.3 0.3

6.3.3 Experimental investigation of the adhesion force

Measurements, performed at Institut de Recherche sur la Sûreté Nucléaire (IRSN) Saclay by S. Peillon et al., allowed direct measurements of the W/W adhesion force for different surface roughness and particle sizes [Peillon 2018]. This Section describes these measure- ments and comparisons with adhesion force models cited above.

Experimental setup. In the experiment, three substrate were used: one of glass and two of W of different roughness. The roughness of the three samples was measured using Atomic Forme Microscopy (AFM). AFM images can be seen in Fig. 6.11. The W rough surface was exposed to H and He plasmas, as described in [Stancu 2017]. This induced temperature increase of the surface up to about 1000◦C and high erosion, which lead to inhomogeneous mass loss and roughness increase. The roughness parame- ters measured by AFM are summarized in Table 6.2. The procedure for the adhesion force measurement is as follows. A spherical W dust particle of given (and measured) radius is stuck to the tip of the AFM, as can be seen in 166 Chapter 6. Dust-wall interactions in tokamaks

Figure 6.12: Example of sherical dust stuck on the AFM tip.

Fig. 6.12. Then, a squarish section of the substrate is scanned with the AFM tip, with a resolution of 128 × 128 points. Each measurement is repeated three times. The bending of the AFM cantilever allows the deduction of the adhesion force. Dust particles of W with radii ranging from 3.5 to 10.5 µm were used.

Results. Measured adhesion force is plotted in Fig. 6.13 along with result from the Rabinovich model, Eq. (6.23), using the roughness parameters measured by AFM on the −19 substrates (see Table 6.2). For W/glass, A = 1.3×10 J[Israelachvili 2015]. H0 = 0.36 nm is commonly used, and this value fits the experimental measurements nicely for glass.

In the case of W, we found that H0 = 0.56 nm fits better with the measurements for both the smooth and rough surfaces. These results are in agreement with adhesion force measurements performed on sim- ilar samples using another technique. In [Riva 2017], dust is stuck on a surface and a perpendicular electric field is applied and increased until detachment of the particles. Even though the roughness of a tokamak plasma-facing surface is difficult to estimate, it is usually high because of plasma-induced erosion of the surface. Moreover, dust remobilization is expected to occur where the plasma pressure is the highest, i.e., close to the strike points, which is also where there is the most erosion, so a high surface roughness. In this light, of the three samples used, the rough W is the closest to the surface state we can expect from a tokamak plasma-facing surface, because it has been exposed to plasma. Moreover, due to its high roughness, the adhesion force is very low (∼ 10 nN), thus we can expect dust to be remobilized by plasma-induced forces. 6.3. Remobilization/resuspension 167

Figure 6.13: Adhesion force, measured by AFM, against the dust radius on different surfaces. The force given by the Rabinovich model, using the parameters in Table 6.2, is also plotted and fits the experiment nicely.

6.3.4 Remobilization condition and exit velocity in tokamaks

Let us now estimate the remobilization condition for typical intrinsic tokamak dust laying on PFCs. In tokamak environment, exposure to oxygen might occur during shutdown periods (when the vacuum vessel is open), leading to the formation of insulating oxide layers on the dust grains. In the following, the effect of oxidation are not considered, as we use the well-known expression for the electric charge of a conducting sphere laying on a conducting surface [Tomita 2006b]

2π3 Q = r2ε E. (6.38) d 3 d 0 First, we derive a simple sheath model in order to estimate the electric field E and the ion flow velocity Vi on the dust.

Electric field and ion flow velocity. Here we re-use the simple sheath model from section 3.2.3, where the plasma electron density ne follows the Boltzmann law and ions ni are monokinetic combined with the flux conservation. In order to obtain the electric field and ion fluid velocity at the surface, the Poisson equation, Eq. (3.10), is integrated once, assuming zero potential and electric field at the sheath edge. We find

√ 1/2 √ T h  ϕw i E = − 2 e + 1 − 2ϕw − 2 and Vi = vthi 1 − 2ϕw, (6.39) eλD 168 Chapter 6. Dust-wall interactions in tokamaks

Figure 6.14: Trajectories of ions reaching the sheath, from the side (a) and the front (b).

The wall normalized potential ϕw is obtained by equating the electron and ion fluxes to the wall: ϕw = ln(2πme/mi)/2 ≈ −3.19 for deuterium ions. The typical thickness of the sheath is ∼ 10λD. In the following, we neglect the parallel component of the ion velocity, which may induce rolling and sliding of the dust on the surface, to consider only the perpendicular component. Indeed, in tokamak plasmas, the magnetic field makes a grazing angle θ ≈ 3◦ with the wall surfaces in order to reduce the perpendicular heat fluxes to the PFCs. Ions reaching the surface are initially following the magnetic field lines with a Larmor motion, making their initial parallel and perpendicular (to the surface) velocities difficult to determine. Moreover, when passing through the sheath, their potential energy is converted into more perpendicular velocity (see Fig. 6.14). In the following, we will consider that the ion velocity is oriented purely perpendicularly to the surface. This will result in an overestimation of the ion drag force on the dust grain.

Electric force. In the case of conducting particle and substrate, the total electric force applied on the grain is the combination of the Coulomb and image forces. The derivation of the image force is rather complex [Jones 2005], but yields a simple expression when the particle and surface are in contact (and the fluid in which the system is plunged is a 6.3. Remobilization/resuspension 169 perfect insulator). The total electric force is

2 2 Fe = κ4πε0rdE , (6.40) where κ = ζ(3) + 1/6 ≈ 1.37, ζ(·) is the Riemann zeta function [Sow 2013]. Eq. (6.40) is often called the Lebedev formula. One could argue that the formation of insulating oxide layers on the particles, due to the presence of oxygen in the vacuum vessel, could challenge this ideal view. The electric force in a dielectric particle is the sum of the image, Coulomb and multipolar dielectrophoretic forces

2 Qd 2 2 Fe = −α 2 + βQdE − γ4πrdε0E , (6.41) 16πε0rd where, in our case,

1 3 α ≈ 1, β = 1 + δ, γ = δ2, (6.42) 2 8 with δ = (εd − 1)/(εd + 2), εd being the particle dielectric constant. There does not exist an analytical expression for the electric charge Qd of a dielectric particle laying on a surface. We can use the expression of the saturation charge of a spherical dielectric particle [Lackowski 2003]

εd 2 Qd = 12πε0 rdE. (6.43) εd + 2 The comparison between the electric forces on a dielectric and conducting particle of radius 1 µm is shown on Fig. 6.15. For the dielectric case, we used εd = 30, which is a typical value for W oxide (WO3). We can see that the difference is rather small, especially when compared to the wide variations of adhesion forces with the surface roughness. In this light, we choose to focus on the conducting case in the following.

Contribution of the ion drag force. In tokamak conditions, the other main actor impacting dust remobilization from PFCs is the ion drag force. As ions are accelerated from the sheath edge towards the wall, this force is oriented towards the wall and adds up to the dust adhesion. When computed for a dust grain isolated in a plasma, it is divided into the contributions from ions collected and scattered by the dust grain. Yet, in our case, the dust is located on a surface and scattered ions will transfer their momentum to the surface after little scattering by the grain, thereby making the contribution of collected ions the only significant one to the drag force. The ion drag force depends on the collection cross-section of the dust. In [Tolias 2016], the classical expression for isolated dust grains [Hutchinson 2006] in Maxwellian-distributed 170 Chapter 6. Dust-wall interactions in tokamaks

Figure 6.15: Electric force against the electric field for a 1 µm conducting or dielectric (εd = 30) particle.

ions was used. To remain consistent with the sheath model described above, we prefer to use a simpler expression derived from monokinetic ions. In [Tomita 2006a], the OML cross-section was used, but it is computed for ions in a central force field and makes little sense for a dust particle laying on a planar surface. To remain consistent with the 2 dust charge expression, Eq. (6.38), we preferentially use the dust projected surface πrd to approximate its cross-section. This is equivalent to assuming that the electric field is oriented perpendicularly to the substrate. In the case of monokinetic ions, the component of the ion drag force perpendicular to the wall, noted Fid, is written

2 2 Fid = πrdminvthiu⊥, (6.44) where u⊥ = Vi/vthi is the component of the ion velocity perpendicular to the wall. As discussed above, we will consider the best case scenario and assume that the ion velocity is oriented perpendicularly to the wall. Thus the remobilization thresholds obtained √ below are an underestimation of the reality. In this case, u⊥ = 1 − 2ϕw.

Condition for dust remobilization. Using Eqs. (6.22), (6.40) and (6.44), a dust grain is resuspended if the electric force exceeds the sum of the adhesion and ion drag forces, i.e.,

Fe > Fa +Fid. According to our model, this condition varies with the background plasma conditions, n and T , the dust material, radius rd and the surface roughness parameters. The behavior of the remobilization condition is not straightforward since both the electric 6.3. Remobilization/resuspension 171

Figure 6.16: (a) Contours of the W dust critical radius for remobilization (in µm) with the roughness parameters of the rough W substrate. Remobilization occurs above each line. Typical micron-size W dust appears to be remobilized by plasma forces. (b) Contours of the W dust perpendicular exit velocity vout (in m/s) with rd = 5 µm.

2 and ion drag forces increase in hotter and denser plasmas and are proportional to rd. The adhesion force is independent on the plasma conditions and varies as ∝ rd. The surface roughness only acts on Fa. A scan of n and T is shown in Fig. 6.16 (a) for a W dust grain laying on a surface with the roughness parameters of the rough W surface displayed in Table 6.2. For each plasma condition, the critical radius for remobilization is determined and displayed on the contours in Fig. 6.16 (a) in µm. Hence the contour plot in Fig. 6.16 (a) can be understood in two ways: (i) each line represents a critical dust radius above which, for a set of plasma conditions, all dust is remobilized; (ii) dust of a given dust radius will be remobilized for plasma conditions above the corresponding line. Note that the validity of our results breaks down for large grains, as their size becomes of the order of the sheath thickness. The quantity 10λD varies between 2 µm (top left- hand corner of Fig. 6.16) and 700 µm (bottom right-hand corner of Fig. 6.16) for the parameter ranges used. We observe that remobilization occurs preferentially in hot and dense plasmas. Due to the many assumptions we made and the simplicity of our model, the precise behavior of the curves in Fig. 6.16 is less important than the following conclusions in steady-state plasmas: (i) dust smaller than ∼ 0.1 µm are never remobilized; (ii) dust of ∼ 10 µm are always remobilized. Contrarily to what was stated in [Tolias 2016], typical values of tokamak sheath elec- 172 Chapter 6. Dust-wall interactions in tokamaks tric fields (∼ 30 kV/cm) can lead to micron-size dust remobilization in tokamaks from relatively smooth wall surfaces. This is because, in the work by Tolias et al., the electric force was compared with the adhesion force from contact mechanics models, which ne- glect the effects of the surface roughness and give adhesion forces of the order of ∼ 105 nN.

Dust exit velocity. In this paragraph, we propose an estimation of the perpendicular dust velocity when exiting the sheath, which is a critical input for dust transport codes. In order to estimate this perpendicular exit velocity, the following assumptions are made. First, if the dust is detached from the surface, the adhesion and image forces are neglected, so that only the Coulomb and ion drag forces are considered. This is a reasonable −6 −2 assumption since Fa ∝ r and the image force ∝ r , where r the distance between the dust and the surface. Second, at the sheath edge, the ions are flowing along the magnetic field lines that c make an angle grazing enough to write u⊥ = 0. This way, the two remaining forces, F √ e and Fid, are zero at the sheath edge. In this case, u⊥ = −2ϕ throughout the sheath. Following these assumptions, the perpendicular energy conservation of the dust grain between the surface and the sheath edge yields

1 m v2 = −Q φ − πr2m nv2 λ K, (6.45) 2 d out d w d i thi D where md is the dust mass, vout is the dust perpendicular exit velocity and

1 Z ∞ √ K = −2ϕdr ≈ 18.2 (6.46) λD 0 is obtained by solving the Poisson equation using the sheath model presented above. Rearranging and using Eq. (6.38), we obtain

2 " 2 # 2 2πrd 2π vout = λDnT ϕwEn − K , (6.47) md 3 where En is the electric field normalized to T/(eλD). First of all, it is important to note that the quantity between the brackets in Eq. (6.47) is independent on dust and plasma parameters, and is always positive. This means that every grain that is detached from the surface will eventually leave the sheath and enter the main plasma. Also, it is obvious that the exit velocity does not depend on the surface properties of the wall. While the roughness does not affect the exit velocity, it plays a crucial role on the resuspension threshold.

Fig. 6.16 (b) shows contours of vout for a 5 µm W dust grain laying on a surface with 6.4. Conclusion on dust-wall interactions 173 the roughness parameters of the rough W surface (see Table 6.2). The velocity does not vary much with the plasma parameters scanned and remains in the order of ∼ 0.1 − 1 −1/2 m/s. Since vout ∝ rd , it is expected that smaller grains (rd ∼ 0.1 µm) will exit the sheath with higher velocities (∼ 10 m/s), if remobilized. On the other hand, if we showed that grains with rd > few µm are always remobilized, their exit velocities will be very low (∼ 0.1 m/s). Keeping in mind that, contrarily to what we assumed, the ion flow has a non-zero velocity component oriented perpendicularly to the wall at the sheath edge, and that gravity can also play an important role for large W grains, such large grains are likely to return to the surface quickly. Of course, no conclusion can be drawn for very large grains since our work lies in the regime rd . 10λD. Another limitation comes from the expression used for the ion drag force. It was argued that Eq. (6.44) is usable when the dust is lying on the surface, but, when at the sheath edge, the collection cross-section of the dust will be closer the OML expression. In addition, the scattered ions contribution should increase as the dust is lifted through the sheath. Accounting for these effects is not possible in the framework of the simple model derived here, thus we conclude that Eq. (6.47) might lead to overestimations.

6.3.5 Conclusions on dust remobilization studies

Electric fields applied on dust grains stuck onto planar surfaces can lead to remobilization under certain tokamak conditions. The van der Waals-based Rabinovich model for the adhesion force was used since it showed better agreement with measurements performed by Peillon et al. It was shown that typical sheath electric fields can lead to micron-size dust remobilization in tokamaks. It was also demonstrated that, if a grain is detached from the surface, it will eventually reach the sheath edge and potentially enter the main plasma. The remobilization conditions and perpendicular dust exit velocities were cal- culated for various edge plasma parameters. We showed that typical tokamak dust (W, rd ∼ 1 µm) can be remobilized by plasma-induced forces. The exit velocities remain relatively small (∼ 0.1 − 1 m/s). This effect can cause important impurity seeding in the divertor region of a fusion plasma, especially in the case of a sweeping strike point.

6.4 Conclusion on dust-wall interactions

As we saw in Section 6.2, dust-wall collisions can have several outcomes, depending on the amplitude of the normal impact velocity. For relatively low velocity impacts, the dust sticks to the surface until it is eventually remobilized by external forces. For high 174 Chapter 6. Dust-wall interactions in tokamaks velocity impacts, the dust can be damaged or destroyed, which can lead to the injection of several smaller dust grains into the plasma. In between, the grain bounces and returns to the plasma. Thus the wall acts as a dust sink that cuts off the low and high sections of the dust velocity distribution functions. Yet, we saw that proper rebound occurs in a quite narrow range of normal impact ⊥ velocities (5 ≤ vi ≤ 20 m/s for micron W dust on W surfaces). This means that, as a first approximation, dust-wall impacts remove dust from the plasma dust content, either by sticking or destructive impacts. The creation of new dust under destructive dust-wall impacts was not considered here due to its complexity. When dust is adhered onto a surface, the adhesion force depends on the impact velocity, especially in the case of W because of its very low yield velocity. Thus the total history of a grain is necessary to predict its remobilization probability. It was shown that the Rabinovich adhesion force model predict accurately the adhesion force for tokamak relevant materials, particle sizes and surface roughness. Extrapolations to tokamak-relevant cases show that micron size W dust can be remobilized when close to the strike points, upon which they enter the plasma with relatively low velocities. All the work presented in this Chapter can provide valuable inputs to dust transport codes like DUMBO and help predict the contribution of dust to the impurity content of the plasma. Conclusion

In this thesis, we attempted to bring some answers to today’s hottest questions about dust transport, sources and sinks in tokamak plasmas. Dust injection experiments were performed on the KSTAR tokamak and image processing routines for the recovery of dust trajectories were successfully developed and tested. They are now being used to track dust in the WEST tokamak, especially during runaway electron impacts. Experimental dust trajectories from KSTAR and TEXTOR were compared with results from the dust transport code DUMBO, confirming the tendency of OML-based codes to underestimate the dust lifetime. This prevents current codes from being fully predictive yet, as simulated trajectories remain significantly shorter than experimental ones. Leads for solving this issue were addressed, some of which presenting promising but not sufficient improvements: magnetic field effects on electron emission, potential well effects on electron fluxes, vapor shielding. To this day, more experimental data is required before any conclusion can be drawn on the accuracy of DUMBO, especially concerning the dust motion model. Indeed, comparisons reported in this manuscript focus on the heating model predominantly. This could be done performing dust injection experiments in WEST using the dust injector developed and tested during this thesis. Unfortunately, due to lack of time, the device has not yet been installed on the machine. Finally, considerations on dust-wall collisions and dust resuspension were made to bring some light on dust sources and sinks. The role of the tokamak wall as a W dust sink was emphasized, and W dust adhesion force measurements allowed to estimate the effectiveness of dust remobilization by plasma forces, which can be an important input for DUMBO simulations. The effect of the impact velocity on the adhesion force of W dust confirmed the importance of well characterizing the deposition method used when performing experiments with this material. Adhesion force measurements for tokamak-relevant materials and surface roughness were performed and confirmed the accuracy of the Rabinovich van der Waals-based ad- hesion model. When extrapolated to tokamak scenarios, it was shown that W dust can be remobilized by plasma-induced forces, leading to an additional dust source. At the end of this manuscript, I hope the reader is convinced of the importance of dust in the overall fusion field. Still, some elements of problematic were not solved during this thesis and must be focused on in the future: mainly, the overheating issue of DUMBO and the installation and usage of the WEST dust injector. Other important dust-related topics were not addressed in this manuscript, not due to their lack of interest, but due to lack of time: dust creation and the effect of the 176 impurities generated by dust erosion on the plasma itself. Both these aspects consists of PhD subjects on their own, and the latter could be numerically studied by coupling DUMBO to a plasma code as soon as the overheating issue is solved. When all these tasks are carried out, prediction of the effects of dust on the operation of fusion devices will be possible, and the development of detailed dust-safe plasma scenarios will be achievable. Fusion is said to be the Holly Grail of energy. In order for it to be found, current challenges evoked in the introduction of this manuscript must be overcome. Plasmas with harmless amounts of dust in it would be a good start as it would help sustaining stable and efficient plasma discharges. List of Figures

1 Concentration maximale en W pour un plasma devant atteindre l’ignition.

Le temps de confinement de l’hélium est supposé valoir 5τE. Source: [Pütterich 2010]...... v 2 Distribution en taille de poussières collectées dans le tokamak AUG. Source: [Rondeau 2015]...... vi 3 Durée de vie des 123 grains de poussières détectés lors de l’expérience d’injection de TEXTOR, ainsi que le temps de vie des poussières simulées avec DUMBO dans les mêmes conditions. Le rayon initial des poussières dans les simulations est fixé à 10 µm...... vii

4 Courants d’électrons collectés (Je) et émis (Jthe) en fonction du potentiel de la poussière, sans (OML) et avec (SCL) prise en compte des puits de potentiel...... viii 5 Potentiel d’uns poussière en fonction de sa température sans et avec prise en compte des effets du champs magnétique sur les électrons émis.....x 6 Types d’impacts possibles en fonction de la vitesse d’impact de la poussière ⊥ sur une surface vi ...... x 7 Forces d’adhésion de particules de tungstène sur différentes surfaces en fonction de la taille des particules. Le modèle de Rabinovich prédit très efficacement les valeurs mesurées. Mesures effectuées par S. Peillon et al. à l’IRSN Saclay...... xi 8 (a) Contours du rayon critique pour remobilisation (en µm) pour des par- ticules de tungstène sur une surface de tungstène rugueuse. (b) Contours de la vitesse de sortie de gaine (en m/s) pour les particules de 5 µm.... xii

1.1 Global energy consumption in recent years. Source: [BP 2017]...... 5 1.2 Global temperature increase over the past millennium. Source: [Chen 2011, Mann 1999]...... 5 1.3 Projections of relevant indicators calculated with the World3 code in 1972. “Business as usual” scenario: no major change in physical, economic or social relationships. Source: [Meadows 1972]...... 6 1.4 Reactivity of different fusion reactions against the plasma ion temperature. Source: [Chen 2011]...... 7 1.5 Helical motion of a charged particle around a magnetic field line. Source: [Eurofusion 2017]...... 9 178 List of Figures

1.6 Left: magnetic elements allowing for the tokamak configuration. Right: Nested magnetic flux surfaces. Source: [Jardin 2017, Chen 2011]..... 10 1.7 Left: the limiter configuration, where the last closed flux surface (sepa- ratrix) is, roughly, a cylinder in contact with a limiter on the Contact Point (CP). Right: the divertor configuration, where there is an X-point and the separatrix touches the wall on two particular locations, the Inner and Outer Strike Points (ISP and OSP), thereby secluding plasma-wall interactions from the core. Between the two strike points, the Private Flux Region (PFR) lies, where there is little to no plasma at all. Source: [Gallo 2018]...... 12 1.8 Observation of “UFOs” in an unstable discharge of the DITE tokamak, in 1982. Source: [Goodall 1982]...... 13 1.9 Schematic view of the life cycle of a dust grain in a tokamak plasma... 14 1.10 Runaway impact on the wall of the Tore Supra tokamak, shot #43493.A large amount of dust, appearing as white dots due to their high tempera- ture, is created...... 16 1.11 Size distribution of W dust collected in the AUG tokamak. Two popu- lations are identified: flakes at 0.6 µm and spheres at 1.8 µm. Source: [Rondeau 2015]...... 16 1.12 Maximal W concentration allowing ignition of the plasma. It is assumed

that the confinement time of He is 5τE. Source: [Pütterich 2010]..... 17 1.13 Data from JET pulse #86529. (Top) Images from camera KL11 at three times. (Bottom) Radiative power measurement...... 18

2.1 Setup for the KSTAR dust injection experiment. Location of the injection point (a), schematic view of the gun-type injector (b) and SEM image of the W dust used (c)...... 25 2.2 Image sequence from camera TV2 in KSTAR shot #13101. Injection is triggered in a stable plasma (a). The dust begins to emit visible light upon interacting with the plasma (b). A large cluster falls downwards (c). No more dust is visible (d). Some isolated grains are accelerated in the toroidal direction (e). Dust grains are circled in red on images when observable...... 27 2.3 The dependence of minimal size of dust visible with the camera at distance 1 m on (a) plasma temperature for the various plasma densities and on (b) plasma density for various plasma temperatures assuming 1 ms frame exposure time and 100 counts sensitivity threshold. Source: [Smirnov 2009a] 28 List of Figures 179

2.4 Temporal signal from a pixel of a video containing a dust event. The baseline resolves the background evolution while ignoring the peak.... 31 2.5 Temporal signal from a pixel of a video containing a dust event. The orig- inal signal can exceed the shifted one only if a sharp peak, corresponding to a dust event, is present...... 32 2.6 Dust detection performed on a movie taken in COMPASS. The original movie (a) is filtered (b). Then, the background is removed (c) and dust events detected (d). In this case, no frame reshaping was used...... 33

2.7 Map of the probability to find a dust position on frame t0 + dt, given its

position at times t0 (i) and t0 − dt (Parent of i)...... 35 2.8 Results of the DUMPRO algorithm applied on the video from camera TV2 on KSTAR shot #13101, where dust injection was performed. (b) cor- responds to a zoom of (a) in the DUMPRO region of interest. Rebuilt trajectories are plotted in blue over the superimposed frame...... 36 2.9 Still frame from a slow (25 Hz) camera in the Tore Supra tokamak. Dust grains appear as elongated lines...... 37 2.10 Dust parallel (to the camera sensor plane) velocity distribution function estimated using DUMPRO on Tore Supra shot #46313...... 37 2.11 Location of the gun-type injector on the WEST tokamak: queusot (port) Q4Bh, in place of a reciprocating Langmuir probe. The injected powder must flow through the 1.2 m long probe sheath...... 39 2.12 Design of the gun-type dust injector for the WEST tokamak. L-shaped vacuum chamber not represented...... 40 2.13 Experimental setup for the collimation experiment: (a) sketch and (b) pictures of the injector chamber (top left), the collector plate (bottom left) and the whole system (right)...... 41 2.14 SEM pictures of the 12 µm W powder used for the collimation tests.... 42 2.15 Size distribution of the 12 µm W powder shot by the WEST injector. 349 dust grains counted...... 42 2.16 Processing the collimation experiment data: (left) image for 12 µm W with collimation size 10 mm, the image scale and the location of the three profiles selected; (right) plot of the three profiles and their Gaussian fits. 43 2.17 Dust spreading for different collimation sizes and powder types (12 µm W and 10 µm Cu). The green line represents the 50 mm spreading limit for the integration in WEST...... 43 180 List of Figures

2.18 Measurement of the injected amount per shot for W 12 µm powder and different collimator sizes along with a linear fit. Unfortunately, the error was not estimated...... 44

3.1 Results of the simple 1D sheath model presented in Section 3.2.3 using

M0 = 1 (Bohm criterion). Profiles for (a) the plasma potential, (b) electron and ion densities and (c) ion Mach number against the distance normalized to the Debye length...... 49 3.2 Sheath structure in the presence of electron emission. Arrows represent the fluxes of the three plasma species considered...... 51

3.3 Results of the 1D sheath model presented in Section 3.2.4 using ϕd = −1, C = 10 and δ = 1000. Profiles for (a) the plasma potential, (b) primary and secondary electron and ion densities and (c) ion Mach number against the distance normalized to the Debye length...... 53 3.4 Configuration of interest: a charged particle in the central force field gen- erated by the presence of a charged spherical dust grain...... 55 3.5 (a) Ion trajectories in a Yukawa potential profile with β = 30 and (b) corresponding profiles of effective potential energy, for various impact pa-

rameters. The dashed-red circle in (a) represents rM ...... 58

3.6 Potential profile calculated in the OML framework, with rd = λD, ηi = 1,

ηd = 33 and for Jemis = 0 and 0.31...... 60

3.7 Potential profile calculated with the RM with Tse = 0.1 eV, Wf = 2.2 eV

and rd = 2λDlin...... 62

4.1 2D maps for the plasma electron (a) and ion (b) temperatures, electron density (c) and Mach number (d), computed by the SOLEDGE-2D code for the WEST tokamak...... 70 4.2 2D maps for the plasma ion temperature (a), electron density (b) and toroidal ion flow velocity (c) in the KSTAR shot #13101. The first wall contour appears in white...... 73

4.3 Dust steady-state floating potential against the ion flow velocity for ni = ne

and various values of the Ti/Te ratio...... 75

4.4 F (Te, x) against Te for different relevant values of x in the case of W and

using the Young-Dekker formula for δE...... 78

4.5 SEE, THE and EBS emission yields from a W dust grain with Td = 3200 19 -3 K and eφd/Te = 0. We used ne = 10 m ...... 80 List of Figures 181

4.6 (a) SEE, THE and EBS emission yields and (b) floating potential of a

W dust grain against its temperature. We used Te = Ti = 50 eV and 19 -3 ne = ni = 10 m ...... 82 4.7 f, b and c parameters against the ion energy for D ions on W. Markers represent the experimental data [Behrisch 2007] and solid lines are the exponential and linear fits...... 84 4.8 Energy (a) and angle (b) dependencies of the sputtering yield of D on W. In this case, only physical sputtering takes place. The choice of 4 keV in (b) is arbitrary...... 85 4.9 Sputtering yield of D on Be against the ion temperature, for various dust

potentials and ion flows. In (a), the case eφd/Ti = 0 coincides with

eφd/Ti ≥ 0...... 86 4.10 Sputtering yield of D on Be (a) and W (b) against the ion temperature,

for eφd/Te = u = 0 (solid lines). The fits used in DUMBO are plotted in dashed red...... 87

4.11 < RN > (top) and < RE > (bottom) against the ion temperature for W and for various values of dust potential and ion flow velocity. It is observed

that < RN > and < RE > are roughly independent on u and eφd/Ti for tokamak-relevant values of these parameters...... 92

4.12 In DUMBO, a fit of < RN > (a) and < RE > (b) at u = eφd/Ti = 0 is used. 94 4.13 All heat fluxes implemented in DUMBO against the dust temperature.

Data extracted from a DUMBO simulation of a W dust grain with rd = 1

µm initially and immersed in a homogenous plasma with Te = Ti = 50 eV 19 -3 0 and ne = ni = 10 m . Heat fluxes are normalized to Qe...... 96 4.14 Dust temperature against time. Data extracted from a DUMBO simulation

of a W dust grain with rd = 1 µm initially and immersed in a homogenous 19 -3 plasma with Te = Ti = 50 eV and ne = ni = 10 m ...... 97 4.15 W dust equilibrium or final temperature against the plasma temperature and density for an initial radius of 1 µm...... 98 4.16 W dust lifetime against the plasma temperature and density for an initial radius of 1 µm...... 99

5.1 Te (solid blue) and ne (dashed green) profiles, measured as described in [Ratynskaia 2013]...... 107

5.2 Ti (red) and ne (green) profiles at t = 6.4 s from charge exchange spec-

troscopy and line integrated density, respectively. Te profile (blue) ob-

tained by fitting the Ti one...... 108 182 List of Figures

5.3 All 123 experimental and simulated W dust trajectories from the TEX- TOR dust injection experiment...... 109 5.4 KSTAR dust injection experiment - Comparison between dust experimental trajectories, reconstructed with DUMPRO, and simulated ones made with DUMBO: (a) In a poloidal cross-section, above the ion flow velocity map, with the first wall geometry in white; (b) View from the top of the machine, with the first wall geometry at the mid plane in black...... 110 5.5 Still image from a slow camera in Tore Supra, shot #41406 (left) and #46313 (right). Dust trajectories exhibiting a sudden change of direction are circled in red...... 112 5.6 Ions toroidal flow velocity in a typical WEST plasma...... 113 5.7 Poloidal (a) and toroidal (b) trajectory of a C dust grain with initial radius

rd = 100 µm simulated using DUMBO in a typical WEST plasma..... 113 5.8 3D representation of the trajectory from Fig. 5.7, in blue. The wall and separatrix surfaces appear in gray and red, respectively...... 114 5.9 50 emitted electron trajectories around a dust grain (grey circle) with

rd = 1 µm, eφd/Te = 0.02 and Td = 6000 K. Trajectories are plotted in red for recollected electrons and in black for escaped electrons. B is oriented

along the red arrow. In this case, the effective emission yield is δE,B = 46.7%.118

5.10 Time evolution of the fraction of recollected electrons for rd = 1 µm,

Td = 6000 K and two values of dust potential: eφd/Te = 0 and 0.02.

The analytical recollected fraction in the case φd = 0 computed from Eq. (5.4) gives 17.5%. The final value of the recollected fraction in the

case eφd/Te = 0.02 is 52.3%...... 119

5.11 Effective emission yield δE,B presented as: (a) contours against Td/Te and

rd/λD for eφd/Te = 0.1 with the region corresponding to magnetized emit-

ted electrons in grey and (b) plots against rd/λD for Td = 5000 K and

eφd/Te = 0 and 0.1 in red along with the solution from Eq. (5.4) in black. 121 5.12 W dust floating potential as a function of its temperature computed by solving the current balance accounting for electrostatic recollection only in dashed black and both electrostatic and magnetic recollection in solid

red. The dust radius is fixed at rd = 1 µm...... 122 5.13 W dust lifetime in a pure D homogeneous plasma with varying electron/ion

temperature and density. The initial dust radius is rd = 1 µm...... 123 List of Figures 183

5.14 Contours of the critical dust radius over which vapor shielding effects be- come non negligible, for different dust materials and varying the plasma background density and temperature. Source: [Krasheninnikov 2009, Brown 2014]124 ∗ 5.15 Contours of eφd/Te against the dust size and temperature. Source: [Delzanno 2014a]126

5.16 W dust lifetime in a homogeneous plasma with Ti = Te = 5 eV and 21 -3 + ni = ne = 10 m against the dust size, with OML and OML theories. 127

5.17 Effective potential energy Ueff of incoming electrons (z = −1) versus the

distance r for ρ = rd/2 and different particle velocities. The largest so-

lution of the equation Ueff = 1 corresponds to the distance of closest ap-

proach, since particles are not allowed to exist in the region Ueff > 1.A

double Yukawa potential, Eq. (5.9), is used, with eφd/Te = −1, eφ0/Te =

10, λ/rd = 2 and ξ = 2...... 128

5.18 Double Yukawa potential profile for λ = 2rd, ξ = 2, eφd/Te = −1,

eφ0/Te = 10 (a) and eφd/Te = 0.5, eφ0/Te = 5 (b). The VCs are located at

rmin ≈ 1.5rd (a) and rmin ≈ 2rd (b). OML critical impact parameter, tran-

sitional impact parameter and ρVC against the particle velocity normalized to the thermal velocity, in the case of electrons (c) and (d). (c) and (d) correspond to the potential profiles plotted in (a) and (b), respectively.. 130

5.19 Ratio of the location of the barrier in Ueff from Eq. (5.10) to the location

of the VC from Eq. (5.11) against the particle velocity v. rM /rmin remains

in the range 0.5 − 3. We used eφd/Te = −1...... 131 5.20 Integration domains for the determination of the THE currents, delimited by the TPBs for the OML and SCL cases...... 134

5.21 f and g functions. g(up, ·) can be approximated with f for up & 1..... 135 5.22 (a) THE current from Eq. (5.22) (solid line) and Eq. (5.26) (dashed line). + ∗ (b) THE currents from OML (dotted line), OML with eφd/Te = −0.125 (dashed line), Eq. (5.26) with χ = 0.05 (solid line) and χ = 0.8 (dash- dotted line)...... 136 5.23 (a) and (b) VC depth and location from OM radial model approximation and (c) and (d) OML and SCL electron and THE currents and OML ion current against the dust potential. The dust is made of W with the radius

rd = 1 µm and temperature Td = 4500 K. Background plasma parameters 20 -3 are n0 = 10 m and Te = Ti = 0.5 eV (left) and Te = Ti = 2 eV (right). 139 184 List of Figures

5.24 Dust floating potential against the temperature using OML, OML+ and

SCL theories. The plasma is homogeneous with Ti = Te = 5 eV and 21 -3 ni = ne = 10 m , and the initial dust size is 1 µm. The step observable

at Td = 3695 K corresponds to the solid-liquid transition...... 140 5.25 Dust electric charge against the Debye length in the SCL and OML regimes.

A W dust grain with rd = 0.1 µm, Td = 6000 K and eφd/Te = −0.01 is used in the calculations...... 141

6.1 Sketch of the possible outcomes and deformation types for a dust-wall collision versus the impact velocity. Two aspects are to be taken into account: (i) the types of deformation at play (blue part) and (ii) the

outcome of the impact (red part). The ranking of vy and vadh depends on the particle size and the materials at play...... 147 6.2 Adhesion velocities against the dust radius for W/W impacts from the JKR model, which only considers elastic and adhesive effects (dashed red), the model by Chen et al., which adds viscoelastic dissipation (dash-dotted black), and the T&N model, which incorporates plastic dissipation (solid black). The yield velocity is also plotted. The material used is W. θ is the angle with the normal to the surface...... 150 6.3 Adhesion velocities for a 1 µm W dust grain impacting a W surface from the JKR model (dashed red), independent on the impact angle, and the T&N model with dependency on the impact angle (black). The dash- dotted curve represents the impact velocity averaged over the surface roughness...... 151 6.4 Normal restitution coefficient against the normal impact velocity for a 1 µm W dust grain...... 153 6.5 Roughness picture as considered by the Rumpf (a) and Rabinovich (b) models. Source: [Rabinovich 2000a]...... 157 6.6 Sketch of the force (f) - overlap (δ) curve for the impact of an elastic perfectly-plastic adhesive sphere. Three phases are distinguishable: elastic (AB) and plastic (BC) loading and elastic unloading (CD)...... 158 6.7 Schematic of the deformation of the dust grain. a is the contact radius, δ the overlap and H the dust-surface distance. Dimensions are deliberately exaggerated for the sake of representativeness...... 161 6.8 Van der Waals force on a deformed W dust grain against the contact radius

for rd = 1 µm...... 162 List of Figures 185

6.9 Schematic of the deformation of the dust grain and the asperity when the surface is rough. Dimensions are deliberately exaggerated for the sake of representativeness...... 163 6.10 Van der Waals force on a W dust grain on a rough W surface against the

rms for rd = 1 µm and for varying impact velocities. The value given by the modified Rumpf model, Eq. (6.37) with a = 0, is shown as a comparison.164 6.11 AFM images of the three surfaces, glass (a), W smooth (b) and W rough (c), used by S. Peillon et al. for the adhesion force measurements..... 165 6.12 Example of sherical dust stuck on the AFM tip...... 166 6.13 Adhesion force, measured by AFM, against the dust radius on different surfaces. The force given by the Rabinovich model, using the parameters in Table 6.2, is also plotted and fits the experiment nicely...... 167 6.14 Trajectories of ions reaching the sheath, from the side (a) and the front (b).168 6.15 Electric force against the electric field for a 1 µm conducting or dielectric

(εd = 30) particle...... 170 6.16 (a) Contours of the W dust critical radius for remobilization (in µm) with the roughness parameters of the rough W substrate. Remobilization occurs above each line. Typical micron-size W dust appears to be remobilized by

plasma forces. (b) Contours of the W dust perpendicular exit velocity vout

(in m/s) with rd = 5 µm...... 171

List of Tables

4.1 Material parameters for W, Be and C used in the DUMBO code and given at room temperature. Whether the dependency on the dust temperature

Td is taken into account or not is also displayed...... 68 4.2 Fit parameters for the function F in the SEE yield...... 79 4.3 Fit parameters for the sputtering yield for various tokamak-relevant ion/- target material pairs...... 87 4.4 Fit parameters for the particle ion backscattering yields for various tokamak- relevant ion/target material pairs...... 93 4.5 Fit parameters for the energy ion backscattering yields for various tokamak- relevant ion/target material pairs...... 93 4.6 Comparison between the different dust transport codes. S and YD stand for the Sternglass and Young-Dekker formulas, respectively. Question marks when the information is not available in the literature...... 103

5.1 Typical SOL plasma parameters used in the simulations. The dust sur- face potential, radius and temperature are varied for the parametric study shown in Fig. 5.11...... 117 5.2 Summary of the domains of applicability of the improvements to the OML theory presented in this Chapter...... 143 5.3 Summary of the consequences of the improvements to the OML theory presented in this Chapter...... 143

6.1 Hamaker constants for the materials used in DUMBO [Tolias 2018].... 156 6.2 Surface roughness parameters measured on the different surfaces and used in the Rabinovich adhesion force model...... 165 3 Parameters for the chemical sputtering of pure carbon by deuterium ions. 192

Details on the 1D sheath model with electron emission

Poisson’s equation as written in Chapter3 involves parameters detailed below

 q  A =B 1 − erf −Cϕvc exp(−Cϕvc), q B = −πCϕvc, (48) s A −Cϕ D = − − 2 vc . B π 3 2 G is the only positive root of the polynomial β3G + β2G + β1G + β0 = 0, where

2 2 3 β3 =EF − 2A F + 2ϕvcA ,

2 ϕvc 2 ϕvc 2 β2 =F + 2EF (e − 1) − 2[A (e − 1) + 2AF ] + 6ϕvcA , (49) ϕvc ϕvc 2 β1 =2(F − 2A)(e − 1) + E(e − 1) − 2F + 6ϕvcA,

ϕvc 2 ϕvc β0 =(e − 1) − 2(e − 1) + 2ϕvc, where E = C(A − 1) and

A r πϕ F = − + 2ϕ + − vc . (50) C vc C

Finally, the value of ϕvc is found by writing the relation between the electron current emitted by the surface Js and the emitted current leaving the sheath Jse

Jse = Jsexp (C (ϕvc − ϕd)) . (51)

The current Js depends on the electron emission process considered. To keep things general, we use in the following the notion of electron emission yield, δ, defined as 0 0 the number of emitted electron per incident electron, i.e., δ = Js/Je , where Je = q en0 8Te/(πme) is the random electron current. Then, using Eq. (51), ϕvc is the so- lution of the transcendental equation

1 G √ −πϕ = exp (C (ϕ − ϕ )) . (52) 2δ 1 + A × G vc vc d

Theoretical aspects on chemical sputtering

The chemical sputtering yield of carbon under hydrogen isotopes at normal incidence is given by [Roth 1999]

Ychem = Ytherm (1 + DYdam) + Ysurf , (53) where D is a parameter depending on the hydrogen isotope,

sp3 0.033exp (−Etherm/T ) Ytherm = c −32 , (54) 2 × 10 Φ + exp (−Etherm/T ) is the thermal erosion yield, with

−32 sp3 C[2 × 10 Φ + exp(−Etherm/T )] c = −32 29 , (55) 2 × 10 Φ + [1 + 2 × 10 /Φ × exp(−Erel/T )]exp(−Etherm/T ) and

1 C = , (56) 1 + 3 × 107 × exp(−1.4/T ) where the flux, Φ, is given in ions/m2s. If for high ion fluxes a possible influence of the hydrogenation time is taken into account, the correction term C could be replaced by

1 C = . (57) 1 + 3 × 10−23Φ

" E 2/3#  E 2 Y (E) = qs 1 − dam 1 − dam (58) dam n E E accounts for damage production and

3 Y (E) Y (E) = csp des (59) surf  E−65 eV  1 + exp 40 eV accounts for surface erosion processes, where

" E 2/3#  E 2 Y (E) = qs 1 − des 1 − des . (60) des n E E

The quantities q, Etherm, Eth, Edam, Edes, Erel and D are given in Table3 for pure carbon and deuterium ions. 192

Table 3: Parameters for the chemical sputtering of pure carbon by deuterium ions.

Parameter Value q 0.1 Etherm 1.7 eV Eth 27 eV Edam 15 eV Edes 2 eV Erel 1.8 eV D 125 Bibliography

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[Zukas 1990] J.A. Zukas. High velocity impact dynamics. Wiley, 1990. (Cited in section 6.2.4.) Résumé Depuis des décennies, la consommation énergétique mondiale est en forte augmentation. Principale source d’énergie utilisée depuis la Révolution Industrielle, les combustibles fossiles posent aujourd’hui des problèmes grandissants à cause de leur progressive raréfaction et la pollution atmosphérique qu’ils engendrent. Parmi les nombreuses alternatives étudiées, la fusion thermonucléaire pourrait jouer un important rôle, grâce à la configuration magnétique du tokamak. Les nombreux avantages que présenteraient cette filière (abondance des réserves de réactifs, risque faible d’accidents, peu de déchets) en font un candidat idéal en vue de la transition énergétique. Cependant, un certain nombre de difficultés technologiques et physiques restent à résoudre avant que l’étape d’une centrale électrique à fusion puisse voir le jour. La production de poussières est l’une des principales difficultés rencontrées dans les tokamaks. Ces petites particules composées de matériaux présents dans les parois de la machine sont créées par l’érosion de ces parois par le plasma dans lequel les réactions de fusion doivent avoir lieu. Les poussières peuvent être transportées dans le plasma et y libérer de grandes quantités d’impuretés, ce qui a pour conséquence de baisser les performances de la machine (en augmentant les pertes radiatives et en créant des instabilités), et qui peut mettre en danger les composants face au plasma. Dans le but de comprendre le transport de ces poussières, des expériences d’injection sont réalisées sur le tokamak coréen KSTAR. Les trajectoires des poussières dans le plasma sont observées par des caméras rapides et sont extraites des films à l’aide de routines de traitement d’images. Un code numérique implémentant les derniers modèles d’interactions plasma-poussières est développé, et des comparaisons avec les données expérimentales sont faites, confirmant la tendance générale de ces modèles à la sous-estimation de la longueur des trajectoires des poussières. Des pistes d’amélioration sont présentées. Concernant les sources et puits de poussières, l’accent est porté sur l’adhésion et remise en suspension de particules sur les parois de la machine. Abstract For decades, global energy consumption has been on the rise. Fossil fuels, thought having been the main energy source used since the Industrial Revolution, now present issues of increasing importance because of increasing rarefaction and air pollution. Thermonuclear fusion could play an important role amongst the numerous alternative energy sources, especially though the tokamak configuration. It could be a prime candidate for the energy transition, owing to its significant advantages (fuel abundance, low amount of wastes generated, low risks of accidents). However, a certain amount of technological and physical challenges require solving before any fusion power plant can be built. Dust production is one of the major difficulties encountered in tokamaks. These small particles, made out of wall material, are created by erosion of the plasma-facing components by the plasma, where the fusion reactions occur. Dust particles can be transported in the plasma, thereby unleashing large amounts of impurities, which in turn reduces the plasma performances (by raising radiative losses and generating instabilities) and can even jeopardize plasma-facing components. Aiming to understand dust transport, injection experiments are performed on the Korean tokamak KSTAR. Trajectories are recorded on film via fast cameras and are extracted by image processing routines. A numerical tool implementing the latest models for dust-plasma interactions is developed, and comparisons with experimental data is made, confirming the overall tendency of these models to underestimate the trajectory lengths. Leads of improvements are presented. Concerning dust sources and sinks, the focus is made on dust adhesion and resuspension of dust on the machine walls.