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Modelling the multifaceted physics of metallic dust and droplets in fusion plasmas LADISLAS VIGNITCHOUK Doctoral Thesis Stockholm, Sweden 2016 TRITA-EE 2016:084 ISSN 1653-5146 ISBN 978-91-7729-041-4 KTH School of Electrical Engineering SE-100 44 Stockholm SWEDEN Akademisk avhandling som med tillstånd av Kungliga Tekniska högskolan framlägges till oﬀentlig granskning för avläggande av doktorsexamen i Electrical Engineering måndag, 13 juni 2016, klockan 13.30 i E3, Osquars backe 14, Kungliga Tekniska högskolan, Stockholm. © Ladislas Vignitchouk, June 2016 Tryck: Universitetsservice US AB iii Abstract Plasma-material interaction constitutes one of the major scientiﬁc and technological issues aﬀecting the development of thermonuclear fusion power plants. In particular, the release of metallic dust and droplets from plasmafacing components is a crucial aspect of reactor operation. By penetrating into the burning plasma, these micrometric particles act as a source of impurities which tend to radiate away the plasma energy, cooling it down below the threshold temperatures for sustainable fusion reactions. By accumulating in the reactor chamber, dust particles tend to retain fuel elements, lowering the reactor eﬃciency and increasing its radioactivity content. Dust accumulation also increases the risk of explosive hydrogen production upon accidental air or water ingress in the vacuum chamber. Numerical dust transport codes provide the essential framework to guide theoretical and experimental dust studies by simulating the intricate couplings between the many physical processes driving dust dynamics in fusion plasmas. This thesis reports on the development and validation of the MIGRAINe code, which speciﬁcally targets plasma-surface interaction processes and the physics of dust particles impinging on plasma-facing components to address long-term dust migration and accumulation in fusion devices. iv Sammanfattning Plasma-material interaktion är en av de viktigaste tekniska problemen som påverkar utvecklingen av fusionskraftverk. Produktionen av metalldamm och droppar från komponenter i kontakt med plasman är särskilt en kritisk aspekt av reaktordrift. Genom att tränga in i den brinnande plasman, blir dessa mikrometerpartiklar en källa av föroreningar, som kyler plasman nedanför tröskeltemperaturer för hållbara fusionsreaktioner. Dammpartiklar kan också ackumuleras i reaktorn och behålla bränsleelement; därigenom sänker de reaktorens eﬀektivitet och ökar de dess radioaktivitet. Dessutom utgör ackumulerade damm en risk för explosiv vätgasproduktion vid oavsiktliga luft- eller vattenläckor i vakuumkammaren. Numeriska simuleringsprogram är ett nödvändigt verktyg för att styra teoretiska och experimentella studier av dammsdynamik i fusionsplasma, eftersom de reproducerar de intrikata kopplingarna mellan de relevanta fysikaliska mekanismerna. Denna avhandlingen rapporterar om utvecklingen av MIGRAINe koden, som speciﬁkt riktar sig mot plasma-yta interaktionsprocesser och kollisioner mellan dammpartiklar och reaktorväggen, i syfte att förutsäga deras långsiktiga migration och ackumulering. v Acknowledgments First and foremost, I would like to express my most sincere gratitude to my Dearest Supervisor, She of Many Titles, Tsarina of Complex Plasmas, oqi strastnye nauki, Throned Feline and Generous Dispenser of Semicolons, Prof. Svetlana Ratynskaia, for guiding me during those three years of rewarding hard work and for helping me grow as an independent researcher. In deﬁance of all pedagogical clichés, you didn’t let me do all of my own mistakes, but I shall bet that learning how to plan ahead and avoid most of them proves to be far more valuable. I am very thankful to Panagiotis Tolias for his co-supervision. It surely wouldn’t have been the same without all those sometimes heated physics discussions — I will forever remember the few times when I was right — or those many pieces of advice delivered, coﬀee cup in one hand and cigarette in the other, in front of the lab. My heartfelt thanks go to Gian Luca Delzanno for welcoming me twice at the Los Alamos National Laboratory, introducing me to the endless mysteries of PIC and showing me ﬁrsthand the richness of simple models. I also wish to express my appreciation to Savino Longo for hosting me at the University of Bari, and of course to Carla Coppola for giving me the opportunity of that ﬁrst and unforgettable taste of home-made panzerotti. I am grateful to Jean-Philippe Banon, with whom I made my ﬁrst research steps in dust studies, for letting me access The Mie Code and for keeping in touch. I am much obliged to Gregory De Temmerman and Steve Lisgo for insightful discussions on safety issues in ITER and for sharing their research data. Many thanks to Nickolay Ivchenko, Tomas Karlsson, Per-Arne Lindqvist, Thomas Jonsson and Göran Marklund for always being helpful, be it to solve science problems or tackle more earthly concerns. And I couldn’t forget to acknowledge my family and friends, for being so supportive and understanding, for accepting my peculiar work schedule even on ‘vacation time’ and for actually taking an interest in my work. Last but not least, I would like to mention Sir Terry Pratchett and his numerous works, which have been one of my favorite sources of quality insomniac-night entertainment and practical wisdom on experimental dust handling. Use your dustpan wisely, and remember: a small brush for the corners. — Lu-Tze’s Way of the Sweeper, in Thief of Time, by Terry Pratchett. Contents Contents vi List of Papers ix List of Figures xi List of Tables xii Introduction 1 I Dust-plasma interaction 3 1 Fundamentals of dust modelling in fusion 1.1 OML theory . . . . . . . . . . . . . . . . . 1.2 Particle currents and heat ﬂuxes . . . . . 1.2.1 Collection of plasma species . . . . 1.2.2 Particle emission . . . . . . . . . . 1.3 The ion drag force . . . . . . . . . . . . . 1.4 Dust heating and phase transitions . . . . 1.4.1 Thermal radiation . . . . . . . . . 1.4.2 Sublimation and vaporization . . . 1.5 Dust modelling equations . . . . . . . . . 2 3 The 2.1 2.2 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6 7 7 8 10 12 13 13 14 MIGRAINe model Dust transport codes and their speciﬁcities . Thermodynamic properties . . . . . . . . . Plasma-surface interactions . . . . . . . . . 2.3.1 Electron emission and backscattering 2.3.2 Ion backscattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 17 18 18 18 22 Benchmarking and validity 25 3.1 Experimental studies . . . . . . . . . . . . . . . . . . . . . . . . . . 25 vi CONTENTS 3.2 3.1.1 Qualitative code validation 3.1.2 Temperature discrepancies . Non-OML regimes . . . . . . . . . 3.2.1 Absorption radius eﬀects . . 3.2.2 Strong electron emission . . 3.2.3 Magnetization . . . . . . . . 3.2.4 Ablation clouds . . . . . . . vii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II Collisions with plasma-facing components 4 5 6 The elastic contact 4.1 Theoretical framework . . . . . . . . 4.1.1 Linear elasticity . . . . . . . . 4.1.2 Problem statement . . . . . . 4.2 Hertz’s theory . . . . . . . . . . . . . 4.3 Normal elastic sphere-surface impact 25 26 27 27 29 31 31 33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 35 35 36 36 39 Dissipative impacts 5.1 Plasticity . . . . . . . . . . . . . . . . . . 5.1.1 Loading stage and yield velocity . 5.1.2 Elastic recovery and collision time 5.2 Adhesion . . . . . . . . . . . . . . . . . . . 5.2.1 JKR theory . . . . . . . . . . . . . 5.2.2 Sticking velocity . . . . . . . . . . 5.2.3 Elastic-adhesive collision time . . . 5.3 The Thornton and Ning model . . . . . . 5.3.1 Normal restitution coeﬃcients . . . 5.3.2 Characteristics of sticking impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 41 41 43 47 47 48 50 52 52 53 Dust-wall collisions in MIGRAINe 6.1 Collision detection . . . . . . . . . . . . . . 6.2 Oblique impacts . . . . . . . . . . . . . . . . 6.3 Surface roughness and randomization . . . . 6.4 Variations of the mechanical properties . . . 6.4.1 Temperature dependence . . . . . . 6.4.2 Size dependence of the yield strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 57 60 61 62 62 62 . . . . . . . . . . Contribution to publications 67 Outlook 69 Bibliography 71 List of Papers This thesis is based on the work reported in the following papers: Paper I Dust-wall and dust-plasma interaction in the MIGRAINe code L. Vignitchouk, P. Tolias and S. Ratynskaia Plasma Phys. Control. Fusion 56 095005 (2014) Paper II Migration of tungsten dust in tokamaks: role of dust-wall collisions S. Ratynskaia, L. Vignitchouk, P. Tolias, I. Bykov, H. Bergsåker, A. Litnovsky, N. den Harder and E. Lazzaro Nucl. Fusion 53 123002 (2013) Paper III Transport asymmetry and release mechanisms of metal dust in the reversedﬁeld pinch conﬁguration I. Bykov, L. Vignitchouk, S. Ratynskaia, J.-P. Banon, P. Tolias, H. Bergsåker, L. Frassinetti and P. R. Brunsell Plasma Phys. Control. Fusion 56 035014 (2014) Paper IV Elastic-plastic adhesive impacts of tungsten dust with metal surfaces in plasma environments S. Ratynskaia, P. Tolias, A. Shalpegin, L. Vignitchouk, M. De Angeli, I. Bykov, K. Bystrov, S. Bardin, F. Brochard, D. Ripamonti, N. den Harder and G. De Temmerman J. Nucl. Mater. 463 877 (2015) Paper V Fast camera observations of injected and intrinsic dust in TEXTOR A. Shalpegin, L. Vignitchouk, I. Erofeev, F. Brochard, A. Litnovsky, S. Bozhenkov, I. Bykov, N. den Harder and G. Sergienko Plasma Phys. Control. Fusion 57 125017 (2015) ix x LIST OF PAPERS Paper VI Interaction of adhered metallic dust with transient plasma heat loads S. Ratynskaia, P. Tolias, I. Bykov, D. Rudakov, M. De Angeli, L. Vignitchouk, D. Ripamonti, G. Riva, S. Bardin, H. van der Meiden, J. Vernimmen, K. Bystrov and G. De Temmerman Nucl. Fusion 56 066010 (2016) Paper VII Beryllium droplet cooling and distribution in the ITER vessel after a disruption L. Vignitchouk, S. Ratynskaia, P. Tolias, G. De Temmerman, M. Lehnen and S. W. Lisgo Submitted to Nucl. Fusion List of Figures 1.1 1.2 Collection of a plasma particle attracted by a spherical dust grain. . . . Orbital collision between a charged particle and a dust grain. . . . . . . 7 11 2.1 2.2 Electron reﬂection yield of polycrystalline tungsten. . . . . . . . . . . . Angle-integrated backscattering energy yield of hydrogen ions. . . . . . 21 23 3.1 Electron Debye length and Larmor radius proﬁles in the edge plasma of ITER. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dust heating rate in the thin sheath limit. . . . . . . . . . . . . . . . . . Predictions of the lifetime of tungsten dust under OML and OML+ assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CPIC simulation of the electric potential around an emitting dust particle in a ﬂowing magnetized plasma. . . . . . . . . . . . . . . . . . . . . 32 4.2 4.3 Simpliﬁed visualization of the contact radius and the relative approach for a sphere-plane contact. . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic view of the initial and deformed contacting surfaces. . . . . . Force-displacement diagram for a normal elastic impact. . . . . . . . . . 37 38 40 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Stress-strain curve for an elastic-perfectly plastic material. . . . . . . Collision times for a normal elastic-perfectly plastic impact. . . . . . Force-displacement diagram for an elastic-perfectly plastic impact. . Schematic plot of the interaction energy between two surfaces. . . . Force-displacement diagram for an elastic-adhesive impact. . . . . . Collision times for a normal elastic-adhesive impact. . . . . . . . . . Normal restitution coeﬃcient for an elastic-perfectly plastic collision. Normal restitution coeﬃcient for an elastic-adhesive collision. . . . . . . . . . . . . . . . . . . . . 42 45 46 48 50 51 53 54 6.1 6.2 6.3 6.4 6.5 Four triangles in a plane and their representation using a BVH. . Example of a triangulated surface and its bounding volumes. . . Yield strength of bulk tungsten as a function of temperature. . . Schematic view of edge and screw dislocations in a cubic lattice. Size-depedent yield strength of room-temperature tungsten. . . . . . . . . . . . . . 58 59 63 65 66 3.2 3.3 3.4 4.1 xi . . . . . . . . . . 28 29 30 List of Tables 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Thermodynamic data for beryllium and tungsten. . . . . . . . . . . Fit coeﬃcients for the enthalpy-temperature relationships. . . . . . . Fit coeﬃcients for the temperature-dependent vapour pressures. . . Material parameters for the secondary electron emission yields. . . . Fit coeﬃcients for the elastic electron reﬂection yield of tungsten. . . Fit coeﬃcients for the backscattering energy yield of hydrogen ions. Fit coeﬃcients for the self-backscattering particle yields. . . . . . . . 6.1 Fit coeﬃcients for the temperature-dependent solid tungsten. . . . . . . . . . . . . . . . . . Fit coeﬃcients for the temperature-dependent solid beryllium. . . . . . . . . . . . . . . . . . 6.2 xii . . . . . . . . . . . . . . mechanical properties of . . . . . . . . . . . . . . . mechanical properties of . . . . . . . . . . . . . . . 19 19 19 20 21 23 23 63 64 Introduction Thermonuclear fusion power is regarded as one of the most promising energy sources to address the ever-growing energy consumption worldwide. Compelling arguments in its favor include its high power yield, its independence to rare fuel sources, its intrinsic safety in comparison with current-day nuclear ﬁssion reactors, and its low waste production. Out of the two main design routes currently pursued in the hope to harness fusion energy in a sustainable manner, magnetic conﬁnement fusion is the most developed. It relies on conﬁning a burning plasma by strong magnetic ﬁelds set up in toroidal chambers. Although the evolution of magnetic conﬁnement technologies over the past decades is impressive, it is impossible to completely prevent the conﬁned plasma to enter in contact with the material walls of the reactor. The resulting plasma-wall interaction has a number of undesirable consequences, including but not limited to the production of dust upon material damage. Once released from the reactor wall, these micrometric particles may penetrate into the burning plasma and contaminate it, thereby smothering fusion reactions and likely extinguishing the plasma. The accumulation of dust in an operating fusion reactor also poses safety issues. Dust and droplets produced from tungsten and beryllium, which have been chosen as the main plasma-facing materials of the international thermonuclear experimental reactor ITER, are not only susceptible to pollute the burning plasma, but also to retain radioactive fuel elements. Moreover, in the event of an accident involving loss of vacuum or water ingress in the reactor chamber, accumulated dust may react chemically to form highly explosive hydrogen or be released, along with its radioactive content, in the environment. Understanding the dynamics of dust immersed in a fusion plasma is essential to form reliable predictions of the location of potential dust accumulation sites and the characteristics of the deposited population. Those can then be used to guide the design of eﬃcient dust monitoring and removal techniques, which are an integral part of the safety systems present in a thermonuclear reactor. However, dust-plasma interaction is the result of the coupling of many diﬀerent physical processes, which greatly complicates its experimental study. Moreover, dust studies in actual fusion environments are hindered by a very limited diagnostics capability; fast cameras and collector probes are the main tools available to the experimentalist, and they are generally unable to provide detailed information on some key properties of the 1 2 INTRODUCTION dust particles, such as their temperature and electric charge. For these reasons, numerical simulations have emerged as the principal framework of dust research, and various dust transport codes have been developed. This thesis work focuses on the development of the MIGRAINe dust dynamics code, whose main distinguishing features are the implementation of state-of-the-art plasma-surface interaction models and a physically transparent treatment of the mechanical collisions between dust and plasma-facing components. Part I Dust-plasma interaction 3 Chapter 1 Fundamentals of dust modelling in fusion As any physical object immersed in a plasma, dust grains acquire an electric charge by absorbing ﬂuxes of ambient plasma particles. Due to their smaller mass, electrons are typically collected at a higher rate than ions and, if no other sources of electric currents are present, dust eventually reaches a steady state — so-called ‘ﬂoating’ — with negative charge and negative potential with respect to the background plasma. While this generic picture is a useful reference, a variety of other physical mechanisms greatly complicates the actual response of dust particles to fusion plasmas. The following considerations are of particular importance: • In addition to their electric charge, collected plasma species also transfer their kinetic energy to the dust, thereby heating it, possibly to melting or vaporization temperatures. • Under most fusion-relevant conditions, electron emission processes result in signiﬁcant positive currents to the dust. Those strongly impact charging dynamics, and the dust potential may become positive with respect to the plasma potential. • Dust particles typically sample a wide range of plasma conditions along their trajectory, so that their evolution is driven by strongly time-dependent sources. The combination of these eﬀects makes dust-plasma interaction a truly interdisciplinary topic, drawing from classical electrodynamics, thermodynamics and even solid-state physics. Nevertheless, the core of any dust model is a theoretical framework describing the collection of plasma species. The most widely used of such frameworks is known as the orbital-motion-limited (OML) theory, which can be 5 6 CHAPTER 1. FUNDAMENTALS OF DUST MODELLING IN FUSION dated back to Mott-Smith and Langmuir [1]. Despite its reliance on rather stringent hypotheses, OML theory is highly versatile and readily adapted to incorporate the various couplings governing dust dynamics. 1.1 OML theory OML theory is in fact an approximated form of the more general orbital-motion (OM) theory [1–4]. The latter is a single-particle kinetic description of the plasma in contact with an immersed body, which in the context of dust modelling is a spherical grain of radius Rd . Assuming that the plasma is collisionless and unmagnetized, i.e. that Rd is much smaller than the mean free paths and gyroradii of the background species, plasma dynamics are dictated by the conservation of energy and angular momentum. Quantitatively, the motion of a plasma particle of species α (in the following, α = e, i for electrons and ions, respectively) with mass mα and charge qα satisﬁes E≡ 1 mα vr2 + vθ2 + qα φ(r) = const 2 L ≡ mα rvθ = const , (1.1) (1.2) where r is the radial distance from the center of the dust grain, vr and vθ are the radial and orthoradial velocity components, respectively, and φ(r) is the central electric potential ﬁeld, set to zero for the unperturbed plasma, φ(r → ∞) = 0. Combining equations (1.1) and (1.2) reduces plasma dynamics to the one-dimensional Hamiltonian motion 1 mα vr2 + Ueﬀ (r) = const , 2 (1.3) in which the eﬀective potential energy is deﬁned as Ueﬀ (r) = L2 + qα φ(r) . 2mα r2 (1.4) While the centrifugal term in equation (1.4) is always positive, the electrostatic term can be positive or negative. As a result, even if φ(r) is assumed to be monotonic, as is the case of the classical Coulomb and Debye-Hückel potentials [5], Ueﬀ may feature potential barriers conﬁning the particle orbits to some limited domain in space. Local extrema of the eﬀective potential have to satisfy qα r3 φ (r) = L2 , mα (1.5) where the prime denotes the derivative with respect to r. If φ(r) is monotonic, then clearly Ueﬀ is a decreasing function of r for repelled particles, i.e. those 1.2. PARTICLE CURRENTS AND HEAT FLUXES mα , qα , E 7 γ b Rd φd Figure 1.1: Collection of a plasma particle attracted by a spherical dust grain. with qα φ(r) > 0. Attracted particles may however encounter potential barriers depending of their energy and angular momentum [6–8]. OML theory is obtained by assuming that there are no eﬀective potential barriers, so that a particle coming from inﬁnity with given angular momentum L and energy E is allowed to reach the grain if and only if E ≥ Ueﬀ (Rd ) . 1.2 1.2.1 (1.6) Particle currents and heat fluxes Collection of plasma species Considering a plasma particle coming from inﬁnity with energy E and angular momentum L as sketched in ﬁgure 1.1, its impact parameter b can be deﬁned as b= √ L . 2mα E (1.7) Equation (1.6) states that the particle reaches the surface of the dust grain if and only if E ≥ qα φd , with φd the dust potential, and qα φd . (1.8) b ≤ b0 ≡ Rd 1 − E It follows that the collection of plasma species by the dust particle can be understood via the OML cross-section 8 CHAPTER 1. FUNDAMENTALS OF DUST MODELLING IN FUSION qα φd σα (E, φd ) = πb20 = πRd2 1 − H (E − qα φd ) , E (1.9) where H is the Heaviside step function. Provided a kinetic description of the unperturbed plasma, far away from the dust, in terms of velocity distribution functions fα (v), the electric current Iα received by the dust grain is Iα = qα σα (E, φd ) vfα (v) d3 v , (1.10) with E = mα v 2 /2 the kinetic energy at inﬁnity. Similarly, assuming that the absorbed particles transfer their kinetic energy as heat to the dust, the heating rate Qα is Qα = σα (E, φd ) (E − qα φd ) vfα (v) d3 v . (1.11) If the plasma species are Maxwellian, possibly drifting with a ﬂow velocity vα , as is often assumed in fusion-relevant settings, then fα (v) = nα mα 2πTα 3/2 mα 2 , exp − |v − vα | 2Tα (1.12) where nα and Tα are the density and temperature of species α, respectively. The integral in equation (1.10) can then be evaluated analytically, yielding the wellknown OML currents and heat ﬂuxes, which can be found in Paper I or [9, 10]. 1.2.2 Particle emission The power of OML theory is its ability to relate the local plasma properties at any two positions with simple formulas; in particular, the OML cross-section (1.9) essentially connects the conditions at the dust surface to those at inﬁnity. This allows a self-consistent treatment of all secondary emission processes, i.e. particle emission that is triggered when plasma species hit the grain surface. Those include: • Electron- or ion-induced electron emission, wherein an electron is expelled from the dust. • Electron and ion backscattering, wherein a plasma particle is reﬂected — elastically or not — at the dust surface. • Physical sputtering, wherein dust material is ejected upon energetic ion impact. Secondary emission processes are typically described by an energy- and angledependent yield Y (E , γ), corresponding to the number of emitted particles per impinging particle of kinetic energy E and incidence angle γ. Taking into account 1.2. PARTICLE CURRENTS AND HEAT FLUXES 9 that an incident particle with impact parameter b hits the grain at an angle given by sin γ = b/b0 , with b0 as deﬁned in equation (1.8), the emitted electric current can be expressed as (1.13) Iβ = −qβ Y (E , γ) σα (E, φd ) sin (2γ) vfα (v) d3 vdγ , where the minus sign denotes emission, E = E − qα φd is the kinetic energy of the particles hitting the dust, and the incidence angle γ spans (0, π/2). Moreover, if an energy distribution fβ (E, E , γ) can be associated with the emission process, in which E is the energy released upon emitting one particle, the dust cooling rate Qβ due to emission can be written Qβ = − Efβ (E, E , γ) Y (E , γ) σα (E, φd ) sin (2γ) vfα (v) d3 vdγdE . (1.14) In fusion environments, another process of importance is thermionic emission, i.e. heat-induced over-the-barrier electron emission. Thermionic emission depends only on the properties of the dust particle — its temperature Td , chieﬂy — and results in a net positive electric current ITE given by the Richardson-Dushman formula [11], 4πTd2 eme Wf ITE = 4πRd2 λR exp − , (1.15) h3 Td where h is the Planck constant, e is the elementary charge, Td is in energy units and Wf is the work function of the dust material, i.e. the minimum thermodynamic work required to extract one electron to inﬁnity. The material-dependent numerical factor λR accounts for the approximated linear temperature dependence of Wf and deviations from the free-electron theory of metals, such as internal electron reﬂection at the surface potential barrier and non-parabolic features of the electron density of states [12]. Moreover, when qd < 0, the electric ﬁeld at the surface of the dust grain lowers the surface potential barrier, resulting in extra emission known as Schottky emission. The overall eﬀect is an apparent reduction of the work function by [13] e3 Esurf , (1.16) ΔWf = 4πε0 where ε0 is the permittivity of free space and Esurf is the electric ﬁeld at the dust surface, which can be expressed as a function of qd following Gauss’s law. To treat charged particle emission properly, particular attention must be given to so-called return currents, which occur in cases where the emitted particles are attracted by the dust, and will therefore be recaptured if they lack the required energy to reach inﬁnity — under the OML assumptions, recapture is possible if and 10 CHAPTER 1. FUNDAMENTALS OF DUST MODELLING IN FUSION only if qβ φd < 0. For secondary processes, the return current Iβ can be expressed as Iβ = qβ E≤Ec fβ (E, E , γ) Y (E , γ) σα (E, φd ) sin (2γ) vfα (v) d3 vdγdE , (1.17) where Ec is the critical energy required to escape the dust attraction. In the case of thermionic emission, the emitted electrons are generally assumed to be Maxwellian with the dust temperature Td , so that the net thermionic current when φd ≥ 0 becomes [13, 14] eφd Wf + eφd 4πTd2 eme φd ≥0 2 1+ exp − . (1.18) ITE = 4πRd λR h3 Td Td Corrections to the cooling ﬂuxes can be incorporated in a similar fashion, leading to the ﬁnal expressions reported in Paper I and [9, 10]. 1.3 The ion drag force In addition to charge and energy transfer, plasma species also exchange momentum with dust grains as they are scattered in its electrostatic potential. If the plasma has a signiﬁcant directed velocity, which is typically the case in fusion devices, this results is a net force on the dust particle, often called the drag force since it is directed along the ﬂow velocity. Given their small mass and high thermal velocity, electrons do not produce a drag force. The ion drag, however, is a major source of momentum for dust grains embedded in fusion plasmas. It can be split into two components: the collection drag Fi,coll , resulting from ions impacting the grain surface, and the orbital drag Fi,orb , stemming from ion deﬂection in the dust potential. While the former can be expressed with the help of the OML cross-section as (1.19) Fi,coll = mi vσi (E, φd ) vfi (v) d3 v , the latter cannot be estimated by using only the OML assumptions. Indeed, considering an ion coming from inﬁnity with impact parameter b > b0 and velocity v as in ﬁgure 1.2, the momentum δp transferred along the initial direction of motion is δp = mi v (1 − cos ψ) , (1.20) where the deﬂection angle ψ depends on the full structure of the potential around the grain. Note that momentum transfer perpendicularly to v cancels out by rotational symmetry around the horizontal axis of ﬁgure 1.2. It follows that the orbital drag force is given by 1.3. THE ION DRAG FORCE 11 v b Rd v ψ φd Figure 1.2: Orbital collision between a charged particle and a dust grain. Fi,orb = b≥b0 (v) 2πbmi v (1 − cos ψ) vfi (v) d3 vdb , (1.21) in which the expression of ψ(b, v) depends on the radial potential proﬁle φ(r) [15,16]. Under the assumption of a Coulomb potential, the scattering motion can be described analytically, e.g. by using the Lenz vector, leading to |qi φd | Rd b90 ψ = , (1.22) = 2 mi bv 2 b is the impact parameter for right-angle scattering. The resulting drag tan where b90 force is Fi,orb = 2πmi vvfi (v) b≥b0 2b290 bdb b2 + b290 d3 v , (1.23) in which the integral over b diverges logarithmically. Similarly to the treatment of Coulomb collisions between plasma species, an ad hoc maximum impact parameter bm has to be introduced [17, 18], giving rise to a velocity-dependent Coulomb logarithm 2 bm + b290 1 ln Λi = ln . (1.24) 2 b20 + b290 The orbital drag force can then be expressed as q 2 φ2 v fi (v) ln Λi d3 v Fi,orb = 4πRd2 i d mi v3 q 2 φ2 v = 4πRd2 i d ln Λi fi (v) d3 v , mi v3 (1.25) 12 CHAPTER 1. FUNDAMENTALS OF DUST MODELLING IN FUSION where the average Coulomb logarithm ln Λi can be interpreted as a measure of the momentum contribution of deﬂected ions relative to collected ions. For a drifting Maxwellian ion distribution, the velocity integral can be carried out analytically, yielding Fi,orb √ = 8πRd2 ni mi Ti mi qi φd Ti 2 G (ui ) (vi − vd ) ln Λi , (1.26) where vd is the dust velocity, ui = |vi − vd |/ 2Ti /mi is the normalized ion ﬂow velocity and √ 2 πerf u − 2ue−u . (1.27) 2u3 While the orbital drag force arising from ion scattering in an arbitrary central force ﬁeld cannot be derived analytically, equation (1.26) is assumed to remain valid for screened potentials, provided that the value of ln Λi is properly adapted. Following the work of Khrapak et al [19–21], expressions of the form λs + beﬀ 90 ln Λi = ln (1.28) Rd + beﬀ 90 G (u) = have been ﬁtted against self-consistent numerical simulations of plasma dynamics around a spherical grain by Hutchinson [22]. The free parameters λs and beﬀ 90 appearing in equation (1.28) can respectively be interpreted as a characteristic screening length — expected to be comparable to the Debye length — and an eﬀective right-angle scattering impact parameter for the dust potential. Since ln Λi quantiﬁes the eﬀect of deﬂection versus collection, it can be noted that the orbital drag force is dominant for small grains Rd λs , whereas the collection drag force is dominant in the opposite limit. It is important to point out that the ion drag models applicable to dust dynamics in fusion environments diﬀer from those often used by the dusty plasma community. Indeed, in most dusty plasma discharges, ions have much lower temperatures than electrons, so that eφd Ti . This induces a strong ion-dust coupling and nonlinear screening eﬀects that must be accounted for in the evaluation of the drag force [23, 24]. 1.4 Dust heating and phase transitions The dust heating equation can be written in its most generic form as dHd = Qtot , (1.29) dt where Hd is the dust enthalpy and Qtot is the total heating power received by the grain. Using the enthalpy instead of the temperature to describe dust heating 1.4. DUST HEATING AND PHASE TRANSITIONS 13 allows for a simpler treatment of the melting transition, during which dTd /dt is discontinuous due to latent heat eﬀects. In addition to particle heating described in section 1.2, contributions to Qtot include thermal radiation and phase transitions. 1.4.1 Thermal radiation The radiative cooling of dust grains is modelled according to the Stefan-Boltzmann law, 4 , Qrad = −4πRd2 εd σSB Td4 − Twall (1.30) where σSB is the Stefan-Boltzmann constant, Twall is the temperature of the reactor wall, and εd is the thermal emissivity of the dust grain, which accounts for deviations from Planck’s black-body law. It is formally deﬁned as the average of the grain’s emission cross-section per unit area Σem over the black-body spectrum: ∞ εd (Rd , Td ) = Σem (Rd , Td , λ) P (Td , λ) dλ , (1.31) 0 where P denotes the normalized Planck distribution, P (Td , λ) = σSB Td4 λ5 2πc2 h , (exp (hc/λTd ) − 1) (1.32) with c the speed of light in vacuum. Assuming that the emission and absorption cross-sections are equal — as is the case at thermal equilibrium according to Kirchhoﬀ’s law — εd can be computed from the Mie scattering theory, which requires the complex, frequency-dependent refractive index of the dust material as an input [25]. The refractive index can be evaluated from classical theories of optical constants, such as the Drude conduction model, which is generally chosen [26–28] since it relies only on the metal plasma frequency and temperature-dependent resistivity [29, 30]. However, tungsten has been extensively studied for spectroscopy calibration purposes, so that experimental measurements of its optical constants at various temperatures are available in the literature [31–33], allowing for better estimates of the emissivity of tungsten dust. 1.4.2 Sublimation and vaporization Vaporization is the main mass loss mechanism aﬀecting dust particles in fusion environments; it controls their lifetime and penetration depth into the conﬁned plasma. In fact, since the partial pressure of metallic species in the plasma is essentially zero, dust atoms continuously escape the grain’s surface, albeit at a typically negligible rate for temperatures below the melting point. The HertzKnudsen formula [34] quantiﬁes the atomic ﬂux evaporating oﬀ the dust surface into vacuum, leading to the rate of change of the dust mass Md , 14 CHAPTER 1. FUNDAMENTALS OF DUST MODELLING IN FUSION dMd dt vap mat = −4πRd2 Pvap (Td ) , 2πTd (1.33) where mat is the mass of one dust atom and Pvap is the vapour pressure of the dust material. The associated cooling rate Qvap is related to the speciﬁc heat of vaporization Δhvap via Hd dMd Qvap = + Δhvap . (1.34) Md dt vap Vaporization is assumed to occur uniformly over the dust surface, so that the grains remain spherical and the time-evolving dust radius can be directly deduced from Md . 1.5 Dust modelling equations The physical processes aﬀecting the dynamics of non-interacting dust particles can be formally modelled as source terms in a system of coupled ordinary diﬀerentialalgebraic equations describing the time evolution of the dust position rd , velocity vd , mass Md , enthalpy Hd and potential φd : drd = vd dt dvd Δp = Fdrag + qd (E + vd × B) + Md g + Md dt Δt coll dMd dMd dMd = + dt dt dt vap dHd = Qtot dt Itot (φd ) = 0 , (1.35) (1.36) (1.37) i (1.38) (1.39) which can be solved by standard numerical schemes once appropriate initial and termination conditions are provided. The subscript ‘tot’ in equations (1.35–1.39) refers to the sum over all physical processes detailed in this chapter, and the extra term dMd /dt i in equation (1.37) accounts for mass variations due to heavy impurity ion collection, backscattering and sputtering. Gravity Md g and the Lorentz force qd (E + vd × B) due to the external electromagnetic ﬁeld (E, B) are almost always negligible with respect to the ion drag force Fdrag . The latter mainly acts in the toroidal direction due to the strong toroidal ﬂows in fusion devices. As a result of centrifugal eﬀects, any particle surviving 1.5. DUST MODELLING EQUATIONS 15 the plasma heat ﬂuxes is quickly expelled outwards and collides with plasma-facing components. This is formally included in equation (1.36) as an impulsive force term Δp/Δt coll . While dust charging could be treated dynamically via dqd /dt = Itot , equation (1.39) states that dust grains are always ﬂoating, i.e. in electrostatic equilibrium. This assumption relaxes the stiﬀness of the model equations and is justiﬁed by the fact that characteristic dust charging times are a fraction of the ion plasma period [7] or the electron diﬀusion time [35], both of which are negligible with respect to the millisecond-range time scales associated with dust heating and dynamics [9, 10]. Equation (1.39) also removes the need for a relation between qd and φd , for which widely-used formulas have been shown to be signiﬁcantly inaccurate [36, 37]. Chapter 2 The MIGRAINe model While all dust codes operate within a common theoretical framework based on OML theory, they tend to focus on diﬀerent physical processes and are usually designed to address speciﬁc issues. After a brief overview of the main dust codes developed by the scientiﬁc community, this chapter provides a detailed description of the dust-plasma interaction models implemented in MIGRAINe. 2.1 Dust transport codes and their specificities Although dust-plasma interactions in fusion environments have been extensively studied both theoretically and experimentally over the past decades — see e.g. [9,38] and references therein — the development of simulation codes implementing the whole range of relevant physical processes started to attract signiﬁcant attention after dust trajectory reconstructions were made possible from multiple simultaneous fast camera recordings [39, 40]. The ﬁrst dust transport code to be developed, DUSTT, initially targeted the dynamics of carbon dust in NSTX and DIII-D [41]. DUSTT simulations provided essential insights on major aspects of dust dynamics in fusion plasmas, such as the importance of electron emission and the identiﬁcation of diﬀerent dust charging and heating regimes [10]. It is mainly aimed at predicting impurity production from dust vaporization and has been recently coupled with the plasma ﬂuid code UEDGE, allowing self-consistent simulations of time-evolving tokamak discharges [42]. Following the pioneering work of DUSTT, the DTOKS code introduced a numerical treatment of ion-surface interactions into a simpliﬁed version of the DUSTT model [43–45]. It has been recently updated to account for deviations from OML theory caused by large dust sizes [46] and its predictions have mainly been used in synergy with experimental observations in MAST [47] and JET [48]. Whereas DUSTT and DTOKS focused mainly on dust as a potential impurity source, the development of MIGRAINe was initially motivated by the need to quantify the eﬀect of dust-wall collisions on long-range dust migration, as detailed 17 18 CHAPTER 2. THE MIGRAINE MODEL in Paper II. While the occurrence of mechanical impacts with plasma-facing components had already been pointed out as a consequence of the prevalence of the toroidal ion drag force [9,38,49,50], this aspect of dust dynamics was either treated on a simpliﬁed phenomenological basis [10, 41, 51], or ignored entirely [45]. Following the adaptation of contact-mechanical models to dust migration simulations, MIGRAINe was further updated by implementing state-of-the-art descriptions of plasma-surface interactions. The following sections provide a detailed account of those updates; an analysis of their quantitative impact on simulation results can be found in Paper I. 2.2 Thermodynamic properties As explained in section 1.4, the dust heating equation (1.29) is most generally written in terms of variations of the dust enthalpy Hd . It is therefore necessary to be able to relate Hd to the dust temperature Td . While the assumption of a constant heat capacity, i.e. Hd ∝ Md Td , is often invoked [44], it is unsuitable for dust in fusion plasmas since Td may vary by several thousands K along a single trajectory and heat capacities cannot be used to treat the melting transition. MIGRAINe uses polynomial enthalpy-temperature relationships for materials in single phase, T = cn h n , (2.1) n where T is expressed in K and h is the molar enthalpy in kJ mol−1 . During melting, the temperature remains constantly equal to Tm while h lies between two critical enthalpies corresponding to the solid and liquid phases at Tm . The numerical values used in MIGRAINe are given in tables 2.1 and 2.2. The temperature dependence of the vapour pressure Pvap , required to model dust vaporization, is estimated by analytical ﬁts to experimental data using a generalized form of the Antoine equation: log10 Pvap Patm =a+ b + c log10 (T ) , T (2.2) where Patm = 1.013 × 105 Pa is the atmospheric pressure and T is expressed in K. The ﬁt coeﬃcients for equation (2.2) are given in table 2.3, while the sublimation and evaporation enthalpies are given in table 2.1. 2.3 2.3.1 Plasma-surface interactions Electron emission and backscattering The quantitative description of secondary electron emission is reviewed extensively in [56]. A main distinguishing feature of MIGRAINe with respect to other 2.3. PLASMA-SURFACE INTERACTIONS Be W solid liquid solid liquid 19 Tm K h (Tm ) kJ mol−1 Δhvap kJ mol−1 1560 33.048 47.725 116.78 169.09 323.20 308.52 858.60 806.29 3695 Table 2.1: Thermodynamic data for beryllium and tungsten [52–55]. Be W solid liquid solid liquid c0 c1 c2 c3 c4 2.982e+2 -2.147e+2 2.982e+2 5.574e+2 5.578e+1 4.025e+1 4.005e+1 1.864e+1 -1.117e+0 -7.240e-2 -8.080e-2 -9.354e-4 3.090e-2 2.000e-4 -2.505e-4 3.165e-6 -4.000e-4 -6.000e-7 1.187e-6 -4.054e-9 Table 2.2: Fit coeﬃcients for the enthalpy-temperature relationship (2.1) of beryllium and tungsten [52]. The ﬁts are valid up to a temperature of 3000 K for Be and 6000 K for W, beyond which linear extrapolations may be used. Be W solid liquid solid liquid a b c 8.042e+0 5.786e+0 1.496e+2 7.355e+0 -1.702e+4 -1.573e+4 -9.851e+4 -4.286e+4 -4.440e-1 -3.566e+1 Table 2.3: Fit coeﬃcients for the temperature-dependent vapour pressure (2.2) of beryllium and tungsten [53, 54]. 20 CHAPTER 2. THE MIGRAINE MODEL Be W δm Em eV rm k β 0.45 0.93 200 600 1.5862 2.1934 1.69 1.38 1.3 0.8 Table 2.4: Material parameters for the secondary electron emission yield (2.3–2.4) of beryllium and tungsten [56, 60–63]. dust codes is its use of the Young-Dekker formula [57, 58] to describe the energydependent secondary electron emission yield δ at normal incidence, δm δ (E, γ = 0) = 1 − e−rm E Em 1−k 1 − exp −rm E Em k , (2.3) for which ﬁtting parameters are given in table 2.4. Equation (2.3) is more accurate than the widely-used Sternglass formula [59], which can overestimate the secondary electron current by a factor 2 for incident energies far from the maximum Em . The angular dependence of δ is modelled by [60] δ (E, γ) = δ (E, γ = 0) β (cos γ) . (2.4) While electron backscattering and secondary electron emission are diﬀerent physical processes, they are indistinguishable experimentally for incident electron energies below ∼ 50 eV, as the emitted electrons then have similar energies [58]. As a consequence, the electron backscattering model used in MIGRAINe — described in full details in [64] — only applies to electron temperatures above ∼ 10 eV. At low energies, impinging electrons may be reﬂected elastically from metal surfaces with appreciable probability. Available measurements of the electron reﬂection yield display a complex energy dependence, with local maxima echoing the structure of the interaction potentials inside the metal [60]. There is no experimental data on low-energy electron reﬂection on beryllium, but such measurements have been carried out on tungsten surfaces [60] and are plotted in ﬁgure 2.1, along with an analytical ﬁt of the reﬂection yield r at normal incidence, 2 E − μn rn exp − , (2.5) r (E, γ = 0) = σn n for which the ﬁt coeﬃcients are given in table 2.5. It can be seen that elastic reﬂection dominates secondary electron emission at low energies, implying that it may signiﬁcantly aﬀect dust charging and heating in low-temperature plasmas. 2.3. PLASMA-SURFACE INTERACTIONS 21 n rn μn eV σn eV 1 2 3 4 4.311e-2 1.753e-1 2.884e-2 1.114e-1 3.718e+0 9.405e+0 1.801e+1 2.468e+1 1.294e+0 7.531e+0 2.437e+0 1.146e+1 Table 2.5: Fit coeﬃcients for the energy-dependent elastic electron reﬂection yield of tungsten (2.5). 0.2 Bronstein and Fraiman Fit Young-Dekker formula Yield 0.15 0.1 0.05 0 0 5 10 15 20 25 30 35 40 Normal incident energy [eV] Figure 2.1: Fitted electron reﬂection yield of polycrystalline tungsten as a function of the normal incident electron energy (2.5) and comparison with experimental data [60]. The secondary electron emission yield according to the Young-Dekker formula (2.3) is plotted for reference. 22 2.3.2 CHAPTER 2. THE MIGRAINE MODEL Ion backscattering In relatively low-energy plasmas, such as can be encountered in the far scrape-oﬀ layer or the divertor region of tokamaks operating in the so-called detached regime, dust heating is typically governed by the absorption of ambient particle ﬂuxes. In that case, the contributions of plasma electrons and ions are approximately equal, implying that ion backscattering may result in signiﬁcant cooling. The ion backscattering model implemented in MIGRAINe relies on the results of Monte-Carlo ion transport simulations performed by Eckstein [65]. Those simulations provide the backscattering particle RN and energy RE yields for hydrogen isotopes impinging on fusion-relevant metals, as well as for self-bombardment cases. While the data mainly concerns normal incidence, values of RN and RE are also given as a function of the incidence angle γ for a few energies, allowing twodimensional ﬁts to be carried out. In fact, as can be seen in (1.13–1.14), the angular dependence can be averaged out under OML assumptions by using angle-integrated yields R, deﬁned by π/2 R(E) = R(E, γ) sin (2γ) dγ . (2.6) 0 Backscattered species are assumed to be neutral, so that ion backscattering does not contribute to dust charging. Therefore, hydrogen backscattering is only included as a cooling mechanism, while self-backscattering is only included as a mass loss mechanism compensating impurity ion absorption by the grain. The angleintegrated yields can be approximated with satisfactory accuracy by analytical expressions of the form α E RE (E) = R0 exp − (2.7) E0 for hydrogen isotopes and RN (E) = exp (c1 εα1 ) 1 + exp (c2 εα2 ) (2.8) for self-bombardment, where ε = E/εL and εL is the Lindhard energy. The ﬁt coeﬃcients used in MIGRAINe are given in tables 2.6 and 2.7. As shown in ﬁgure 2.2, low-energy hydrogen ions typically retain more than half of their initial kinetic energy after they are backscattered. 2.3. PLASMA-SURFACE INTERACTIONS 23 R0 E0 eV α H → Be D → Be T → Be 8.663e-1 1.000e+0 1.000e+0 4.070e+1 7.220e+0 2.558e+0 2.589e-1 1.870e-1 1.640e-1 H→W D→W T→W 1.000e+0 1.000e+0 1.000e+0 1.312e+3 2.162e+3 2.544e+3 2.002e-1 1.844e-1 1.729e-1 Table 2.6: Fit coeﬃcients for the backscattering energy yield (2.7) of hydrogen ions. Angle-integrated energy yield 1 H → Be D → Be T → Be H→W 0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 Incident ion energy [eV] Figure 2.2: Angle-integrated backscattering energy yield RE for hydrogen ions impinging on beryllium and tungsten. Isotopic diﬀerences over the plotted energy range are only appreciable when Be is the target material, due to its low atomic mass. Be → Be W→W εL eV c1 c2 α1 α2 2.208e+3 1.999e+6 -2.396e-4 -7.476e+0 2.409e+0 6.983e-5 -1.778e+0 3.338e-1 2.214e-1 -9.981e-1 Table 2.7: Fit coeﬃcients for the self-backscattering particle yields (2.8). Chapter 3 Benchmarking and validity As any simulation software, dust transport codes need to be benchmarked and validated against experimental data so that their predictions are deemed trustworthy. However, a remarkable feature of dust studies in fusion environments is the very limited range of usable diagnostics. In fact, the large majority of experimental results consists in statistical analyses of collected particles and fast camera observations of dust trajectories. While indubitably valuable, neither of those diagnostics can provide detailed information on the numerous physical processes at play. Moreover, dust codes require substantial input — chieﬂy plasma parameter proﬁles — which are subject to experimental uncertainties. For those reasons, validation experiments must be carefully designed to allow for meaningful feedback and help improving dust models. This chapter brieﬂy summarizes the main applications of MIGRAINe to concrete experimental scenarios, how they provide information on the validity of existing dust-plasma interaction models, and which lines of research are being pursued in order to further the current understanding of dust dynamics in fusion plasmas. 3.1 3.1.1 Experimental studies Qualitative code validation As detailed in Paper II, the ﬁrst comparison between MIGRAINe and experimental data was carried out in the context of a controlled dust injection and collection campaign in TEXTOR. Given the large level of uncertainty in the experimental input to dust codes, this particular setup was designed to restrict the various unknowns as much as possible: • The locations of the dust injector and collector were ﬁxed during the entire campaign. 25 26 CHAPTER 3. BENCHMARKING AND VALIDITY • As TEXTOR was a carbon-wall tokamak, the injected tungsten particles were distinguishable from intrinsic dust. • The size distribution of the injected dust population was calibrated before the experiment. MIGRAINe simulations successfully reproduced major qualitative features of the experimental observations, namely the relatively narrow size range of collected particles, attributed to the competition between the higher inertia of large grains and the poor survivability of small ones, and the low particle count observed on toroidally exposed silicon collectors, interpreted as a signature of the bouncing impacts resulting from the large toroidal dust velocities acquired through ion drag. Intrinsic molybdenum dust collection data in EXTRAP-T2R, while diﬃcult to interpret owing to the unknown initial conditions, was used to postulate reasonable dust release scenarios in agreement with MIGRAINe simulations. As explained in Paper III, asymmetries between upstream- and downstream-facing collectors could be traced back to estimate some characteristics of the dust source, which were found to be consistent with the remobilization of pre-existing particles adhered on the inner side of the vacuum vessel. Finally, further trust in MIGRAINe was gained from injection experiments performed in the linear plasma generator Pilot-PSI and reported in Paper IV. Despite its short residence time in the narrow Pilot-PSI plasma column, tungsten dust was predicted to get heated up to incandescence temperatures, while acquiring enough horizontal velocity from the plasma ﬂow to reach a target plate located more than 30 cm away from the injection point. 3.1.2 Temperature discrepancies In spite of the overall qualitative agreement with experimental results, signiﬁcant inconsistencies were found when comparing the MIGRAINe heating model to camera recordings of tungsten dust trajectories in TEXTOR. Those observations, described in Paper V, yielded unambiguous evidence of heating overestimation by the numerical model. Indeed, any set of realistic input parameters resulted in predictions contradicting experimental observations, i.e. dust lifetimes signiﬁcantly shorter than the time during which particles could be seen glowing on camera. Overheating trends in dust transport simulations can actually be found in published validation experiments for DUSTT and DTOKS [47, 66], wherein dust survivability had to be artiﬁcially enhanced to match the measurements. In the case of DUSTT the collected electron heat ﬂux was reduced by an arbitrary factor with respect to the baseline model [66], while the choice was made in DTOKS to use dust size as an unconstrained ﬁtting parameter to achieve reasonable agreement [47]. As detailed in the following sections, discrepancies in dust heating are likely a manifestation of the breakdown of OML theory in fusion plasmas. 3.2. NON-OML REGIMES 3.2 27 Non-OML regimes Aside from external factors such as input uncertainty, the main source of inaccuracy in dust codes is the limited validity of OML theory to describe dust-plasma interaction in fusion environments. As mentioned in chapter 1, OML theory is valid if: • The plasma is collisionless, such that energy and momentum conservation applies to single particles. • The plasma is unmagnetized, such that plasma motion can be described by a one-dimensional eﬀective potential proﬁle. • No eﬀective potential barriers aﬀect the trajectories of plasma particles. The OML hypotheses can be rephrased by ascribing characteristic spatial scales to the neglected physical processes. Those are the mean free path λc for collisions, the Larmor radius RL for magnetic ﬁelds, and the Debye length λD for deviations of the eﬀective potential from the vacuum limit φ (r) ∝ 1/r. In general, such characteristic lengths can be deﬁned for each plasma species; OML predictions are expected to be trustworthy when the dust size is much smaller than all of them: Rd λcα , RLα , λDα . (3.1) As magnetic conﬁnement fusion research progresses, the densities, temperatures and magnetic ﬁelds encountered in fusion plasmas continuously increase, allowing new physical regimes to be explored. As far as dust-plasma interaction is concerned, those regimes lie beyond the scope of OML theory and call for the development of new models. As shown in ﬁgure 3.1, λDe and RLe in ITER plasmas can be less than 1 μm, making OML theory inadapted to the treatment of micrometric dust particles or larger liquid droplets. The accurate description of non-OML regimes is a major focus of dust research. While a variety of plasma-object interaction models has been developed in other contexts, e.g. probe or pellet physics [35, 67], they generally lack the formal versatility of OML, making them diﬃcult to implement self-consistently in dust codes. The following sections outline the main non-OML eﬀects relevant to fusion settings and the modelling challenges they represent. 3.2.1 Absorption radius eﬀects As explained in chapter 1, eﬀective potential barriers may aﬀect the motion of the attracted species even when φ (r) is monotonic. While this is not an issue for the Coulomb potential, it can be shown that potential barriers always exist for Maxwellian species in the Debye-Hückel potential, and that they become more important as the dust size increases [4, 6]. When dust charging is governed only by plasma collection, the dust potential is negative and the reduction of the positive ion 28 CHAPTER 3. BENCHMARKING AND VALIDITY Figure 3.1: Simulated spatial proﬁles of the electron Debye length (a) and Larmor radius (b) for a typical ITER edge plasma (PSOL = 100 MW), SOLPS data courtesy of S. Lisgo. current due to eﬀective potential barriers is often referred to as the ion absorption radius eﬀect [6], since the radial position of the barrier eﬀectively replaces the dust radius in the collection condition (1.6). While the full treatment of absorption radius eﬀects requires in principle to solve the OM model [3, 6, 68, 69], Willis et al [70] have derived a simple approximation valid for singly charged ions in the thin sheath limit, i.e. when Rd λDe . The main assumption behind their model is that ions are eﬀectively collected at the sheath edge r = rse Rd , so that the ion OML cross-section (1.9) is replaced by eφse 2 σi (E, φd ) = πRd 1 − , (3.2) E with φse = φ (rse ) the sheath edge potential. Assuming furthermore that the sheath can be treated as planar and that the ion velocity at its edge is equal to the sound speed [71], φse can be expressed as [70] 2πme (1 + γTi /Te ) 1 Te ln φse = φd − , (3.3) 2 e mi where γ = 5/3 is the heat capacity ratio. The total dust heating rate predicted by equations (3.2–3.3) is plotted in ﬁgure 3.2 for fusion-relevant values of Ti /Te , showing a reduction by a factor up to 2 with respect to OML theory. Nevertheless, ion absorption radius eﬀects are only signiﬁcant when electron emission is negligible, which strongly limits the applicability of the above thin 3.2. NON-OML REGIMES 29 0.7 Q/Q OML 0.65 0.6 0.55 0.5 0.45 0.5 1 1.5 2 T i/T e Figure 3.2: Dust heating rate Q for a large ﬂoating grain in a nonﬂowing deuterium plasma according to the thin sheath model by Willis et al [70] normalized to the OML prediction QOML as a function of the ion-to-electron temperature ratio. sheath model to fusion environments. Although some corrections have been suggested to account for electron emission [46], they involve the average current yield as an external parameter, and therefore cannot be employed self-consistently to treat charging and heating. 3.2.2 Strong electron emission Emitted electron populations diﬀer from the background plasma electrons in terms of density and temperature, implying that electron emission can add new spatial scales to the problem and alter the validity of OML assumptions. The spacecharge-limited sheath is a well-known example of a new physical regime arising in the presence of electron emission [72, 73]. In that regime, the electrons emitted by a plasma-facing surface develop an electrostatic potential well in its vicinity, which in turn modiﬁes background plasma collection. In terms of length-scale analysis, the emitted electrons introduce short-range screening which aﬀects the structure of the potential near the surface. A thorough investigation of dust-plasma interaction with strong thermionic emission has recently been undertaken by Delzanno and Tang [74]. By comparing the results of ﬁrst-principle particle-in-cell (PIC) simulations with OML predictions, they were able to estimate the critical values of the emission parameters 30 CHAPTER 3. BENCHMARKING AND VALIDITY 8 OML 7 OML + Lifetime [ms] 6 5 4 3 2 1 0 0 2 4 6 8 10 Initial dust radius [µm] Figure 3.3: MIGRAINe predictions of the lifetime of tungsten dust immersed in a uniform unmagnetized deuterium plasma with ne = 1021 m−3 and Te = Ti = 5 eV as a function of the initial grain radius under OML and OML+ assumptions. beyond which OML theory breaks down. This led to the development of the socalled OML+ theory, which accurately describes dust charging and heating in the presence of strong thermionic emission while retaining a simple enough formalism to be readily incorporated in dust codes. Indeed, formally, OML+ only diﬀers from OML when the dust potential is larger than some critical negative value φ∗d , which is a function of the dust size and temperature. When φd ≥ φ∗d , a potential well forms close to the dust surface and reﬂects a fraction of the emitted electrons, leading to a reduced thermionic current [74] φ ≥φ∗ d d ITE = 4πRd2 λR 4πTd2 eme h3 e (φd − φ∗d ) Wf + e (φd − φ∗d ) 1+ exp − , (3.4) Td Td hence a lower dust potential and less heating by background electrons. The OML+ theory is implemented in MIGRAINe via externally provided tables for φ∗d and can result in signiﬁcantly longer dust lifetimes, as shown in ﬁgure 3.3. In its current state, however, OML+ cannot treat secondary emission processes due to their coupling with the background plasma. Therefore, it can only be applied to relatively cold plasmas, such as those encountered in divertor regions, wherein thermionic emission is expected to be dominant. 3.2. NON-OML REGIMES 3.2.3 31 Magnetization The motion of magnetized plasmas is inhibited perpendicularly to the ﬁeld lines, causing immersed absorbing bodies to collect plasma particles preferentially in the ﬂux tube they intersect. Consequently, a depleted plasma region forms in the vicinity of the collector and extends along the magnetic ﬁeld. This phenomenon is a key aspect of scrape-oﬀ layer physics in fusion devices, wherein unconﬁned plasma ﬂows rapidly along the ﬁeld lines towards material surfaces [73,75]. In steady state, cross-ﬁeld transport is required to replenish the depleted ﬂux tube; it is typically provided by anomalous diﬀusion processes in the case of scrape-oﬀ layers or large probes, but classical collisional transport is eﬃcient enough over the spatial scale of the dust and causes the perturbed region to extend over distances of the order of Rd λce /RLe [76]. Magnetic ﬁelds also aﬀect emitted electrons, causing them to gyrate back to the dust grain, thereby short-circuiting emission currents across the ﬁeld lines [77]. Although plasma-object interaction in magnetic ﬁelds has been the subject of numerous studies — see e.g. [35, 76, 79–81] — those were mainly conducted with probe measurements in mind, so that their application is often limited to the collection of a certain species, or to a certain part of the current-voltage characteristic. In any case, while all the models available in the literature agree qualitatively on a net reduction of plasma collection and particle emission, they only evaluate the integrated currents instead of the statistical distributions required for a self-consistent assessment of dust charging and heating. A ﬁrst step towards the development of adequate models for dust dynamics in strongly magnetized plasmas can be undertaken by numerical investigations using PIC simulations, similarly to the strong emission studies mentioned above. Due to their reliance on ﬁrst principles, PIC codes provide a framework for numerical experiments, making it possible to test analytical results and to explore new parameter regimes beyond the capabilities of any realistic laboratory setting. However, the need to resolve all the temporal and spatial scales of interest in order to achieve meaningful results introduces high computational costs, requiring careful design and high-performance computing facilities. The CPIC code developed at the Los Alamos National Laboratory [78] has recently been updated for this purpose. 3.2.4 Ablation clouds Large dust particles penetrating into energetic plasma regions are likely to vaporize at a high rate, releasing a dense cloud of neutrals susceptible to get ionized by the background electrons. This phenomenon is ubiquitous in the ﬁeld of fuel and impurity seeding by pellet injection [67, 82], wherein the interaction between the ablation cloud and the surrounding plasma dictates the pellet vaporization rate and the spatial features of material deposition. Secondary pellet plasmas are often highly collisional, acting as an energy sink for fast particles and a heat shield for their core regions. 32 CHAPTER 3. BENCHMARKING AND VALIDITY Figure 3.4: CPIC simulation [78] of the electric potential around an emitting dust particle in a ﬂowing magnetized plasma. Note that the characteristic ﬂux tube structure extending in the direction of the magnetic ﬁeld is sharply cut oﬀ near the top and bottom outer boundaries. This is a numerical artifact due to the mean free path being much larger than the simulation domain. Dust and pellets belong to diﬀerent physical regimes due to the wide size gap separating them. Nonetheless, the basic ingredients of pellet modelling [83–86] can be adapted to dust studies. Recent attempts to quantify how ablation clouds aﬀect dust-plasma interaction have led to the estimation of critical dust radii above which vapour shielding cannot be neglected [87] and the identiﬁcation of crucial diﬀerences between the heat transfer mechanisms governing dust and pellet shielding [88,89]. Part II Collisions with plasma-facing components 33 Chapter 4 The elastic contact The purpose of this chapter is to give a general presentation of contact mechanics, and in particular the fundamental description of elastic contacts initially formulated by Hertz [90], from which recent models are derived. While the focus is set on sphere-plane contacts, the theories summarized hereafter are readily adapted to solids with arbitrary smooth surfaces; extended discussions and mathematical derivations can be found in reference textbooks [91, 92], which serve as the main source material for the following sections. 4.1 4.1.1 Theoretical framework Linear elasticity The complete description of two contacting solids is most naturally formulated in the language of continuum mechanics, in which the state of each material is quantiﬁed locally by the stress tensor σ, the strain tensor ε and the displacement vector u. These quantities are related by Newton’s second law ∇ · u + F = ρü , (4.1) where F is the volumetric body force and ρ is the local mass density, and the strain-displacement equation ε= 1 T . ∇u + (∇u) 2 (4.2) The behaviour of the materials is furthermore described by constitutive equations, which are required to close the system. For elastic materials, Hooke’s law provides a relation between stresses and strains, σ=C:ε, 35 (4.3) 36 CHAPTER 4. THE ELASTIC CONTACT via the fourth-order stiﬀness tensor C, which can be thought of as an elaborate spring constant. In the case of homogeneous isotropic elastic materials at equilibrium, the system can be simpliﬁed into the Navier-Cauchy equation (λ + μ) ∇ (∇ · u) + μ∇2 u + F = 0 , (4.4) where λ and μ are the material-dependent Lamé parameters. 4.1.2 Problem statement Despite the numerous simplifying assumptions used in its derivation, the NavierCauchy equation (4.4) remains too detailed, for it carries information on the solids’ behaviour at every point in space. It is however possible to achieve a simpler, macroscopic understanding of the contact that is compatible with the point-particle description of dust grains by using three quantities: • The contact radius a, deﬁned as the radius of the circular contact area between the dust and the planar wall. • The relative approach α, deﬁned as the diﬀerence between the sphere radius Rd and the distance separating the sphere’s center from the unperturbed wall surface. • The contact force P , obtained by integrating the surface stresses over the contact area. A schematic representation of a and α is given in ﬁgure 4.1. It is worth insisting that the actual geometry of the contacting surfaces is more complicated than what is shown, as both bodies are deformed by internal stresses. Within this global formalism, the normal sphere-plane impact can be described by Newton’s second law Md α̈ + P = 0 , (4.5) where, by convention, P is chosen to be positive when the net force results in compression, i.e. tends to draw the solids away from each other. In order to solve equation (4.5), the contact phenomena of interest must be accounted for by closedform expressions involving P , α and a. Deriving such expressions is the crucial step in bridging microscopic and macroscopic contact models. 4.2 Hertz’s theory Assuming that the size of the deformed region is small enough with respect to the other spatial scales, i.e. a Rd , the shape of the unperturbed sphere’s surface can be described by a Taylor expansion around the ﬁrst contact point. In the situation 4.2. HERTZ’S THEORY 37 Rd a α Figure 4.1: Simpliﬁed visualization of the contact radius a and the relative approach α for a sphere-plane contact. described in ﬁgure 4.2, with z1 and z2 lying on the unperturbed surface of the sphere and the wall, respectively, such expansion yields z2 − z1 = r2 . 2Rd (4.6) Considering now a point z0 inside the bulk of the sphere, far enough from the contact area for its motion to be treated as rigid, the positive vertical displacement uz1 of z1 due to compression against the wall is given by uz1 = (z1 − z0 ) − (z1 − z0 ), where the primes denote the displaced positions. Similarly, uz2 = z2 − z2 , so that the total displacement can be written uz = uz1 + uz2 = (z0 − z0 ) − (z2 − z1 ) + (z2 − z1 ) = α − r2 , 2Rd (4.7) where α = z0 − z0 is the displacement of the bulk of the sphere towards the wall and z1 = z2 during contact. Equation (4.7) expresses a geometrical constraint that is to be used as a boundary condition in the solution of the Navier-Cauchy equation (4.4). Equation (4.4) was solved by Boussinesq in the case of a normal point force applied to an elastic half-space [93], thereby providing a Green’s function Gi for the solid i as a function of its Young modulus Ei and Poisson ratio νi , 38 CHAPTER 4. THE ELASTIC CONTACT z0 z0 z1 r z1 z2 z2 z Figure 4.2: Schematic view of the initial and deformed contacting surfaces. Gi (r) = 1 − νi2 , πrEi (4.8) from which the vertical displacement of the surface can be expressed as a function of the contact pressure proﬁle p(r): Gi (r − r )p(r )dA(r ) . uzi (r) = (4.9) contact area Assuming that the contact area is a disk of radius a, the pressure proﬁles leading to displacements that satisfy the geometrical constraint (4.7) were found by Hertz to be of the form [90] r 2 1/2 r 2 −1/2 p = p0 1 − − p1 1 − , a a (4.10) where p1 is set to zero since it would otherwise lead to either inﬁnite compressive stress at the contact boundary, or tensile (negative) stresses, none of which can be supported by smooth, purely elastic solids. Pressure proﬁles with non-zero values of p1 are however encountered when adhesive eﬀects are considered, as shown in chapter 5. Equations (4.7–4.10) can now be combined to ﬁnd p0 = 3P/2πa2 = 2aE ∗ /πRd , leading to the elastic contact equations: α= a2 Rd (4.11) 4.3. NORMAL ELASTIC SPHERE-SURFACE IMPACT a P = 2π p(r)rdr = 0 4E ∗ a3 , 3Rd 39 (4.12) where E ∗ is the reduced Young modulus, also called plane-strain modulus, given by 1 − ν12 1 − ν22 1 = + . ∗ E E1 E2 (4.13) These equations are in fact valid for the elastic contact of two solids of revolution, provided that 1/Rd is replaced by the total curvature of the unperturbed surfaces and that strains remain small enough to justify the treatment of each solid as a half-space. In particular, the contact between two spheres of radii R1 and R2 is equivalent to that between a planar wall and a sphere of radius R∗ = R1 R2 /(R1 + R2 ). 4.3 Normal elastic sphere-surface impact The elastic contact equations (4.11–4.12), together with (4.5), provide a complete macroscopic description of the normal impact of an elastic spherical dust particle on an elastic planar wall. It is useful to analyze the impact from the energetic point of view, as this lays the foundation for the development of more realistic models taking dissipative processes into account. The energy budget of the impact is expressed by integrating equation (4.5), 1 Md α̇2 + 2 α P (α )dα = 0 , (4.14) where the integral is the compressive work W (α) done by the contact force between the initiation of the contact and the current time. From the elastic contact equations, 4 ∗ E Rd α3 3 (4.15) 8 ∗ E Rd α5 . 15 (4.16) P (α) = W (α) = The impact can then be visualized on a force-displacement diagram as in ﬁgure 4.3. The shaded area represents the initial kinetic energy of the dust particle, which is converted into elastic energy during compression and back into kinetic energy as the particle moves away from the wall. Unsurprisingly, since only elastic deformations are considered, the grain recovers all of its initial kinetic energy by the end of the impact. Equation (4.14) can also be used to derive the duration of the impact as 40 CHAPTER 4. THE ELASTIC CONTACT Figure 4.3: Force-displacement diagram for a normal elastic impact. tcoll = αm αi dα + v02 − M2d W (α) αm αf dα , v02 − M2d W (α) (4.17) where v0 is the initial velocity of the dust particle and the impact is described by a loading stage αi → αm followed by an unloading stage αm → αf . For the elastic impact, αi = αf = 0 and αm is given by Md v02 /2 = W (αm ), so that dα 2 αm elastic tcoll = (α) v0 0 1 − WW(α m) 1 dx 2αm (4.18) √ = v0 0 1 − x5/2 ρ 2/5 1/10 Γ(7/5) d −1/5 Rd v0 = 2500π 9 , Γ(9/10) E∗ where the numerical prefactor is approximately equal to 5.09. Chapter 5 Dissipative impacts The Hertzian elastic impact model detailed in chapter 4 is insuﬃcient to describe realistic dust-wall collisions, for it does not include any dissipation channel for the dust kinetic energy and hence cannot predict ﬁnite migration times. This chapter introduces plastic deformations and adhesive forces as the most relevant dissipative contact phenomena, culminating in the elastic-perfectly plastic adhesive impact model developed by Thornton and Ning [94]. 5.1 Plasticity Elastic models, in which the stress tensor depends linearly on the strain, are only valid for small deformations. Real materials cannot support arbitrarily large stresses and deform plastically above a threshold quantiﬁed by the yield strength σy of the material. The simplest model for plastic behaviour is perfect plasticity, in which the stress remains constantly equal to σy beyond the plastic limit, independently of the strain. Once the stress applied to the material is removed, it recovers elastically until it reaches a ﬁnal irreversibly deformed state, as shown in ﬁgure 5.1. 5.1.1 Loading stage and yield velocity Although σy is usually measured by tensile tests on cylindrical samples, perfect plasticity can be applied to spherical bodies by truncating the Hertzian pressure proﬁle (4.10) at a limiting value py [94, 95]. Finite-element simulations of sphereplane impacts suggest that py /σy lies between 1.6 and 2.8 [96, 97], but even higher values have been measured experimentally [98]; MIGRAINe simulations assume py = 2.8σy by default. Pressure truncation implies that, once yield is initiated, the contact area consists of a central disk of radius ap , inside which the pressure is maintained constant at py , surrounded by an annular region over which the pressure follows the Hertzian proﬁle. The contact force can then be written as [94] 41 42 CHAPTER 5. DISSIPATIVE IMPACTS σ σ y 0 p Figure 5.1: Stress-strain curve for an elastic-perfectly plastic material subject to a traction-compression cycle. The residual plastic strain εp measures the irreversible deformation of the material. P (a) = PHertz (a) 1 − a 2 3/2 p a + πa2p py , (5.1) where PHertz is the Hertzian contact force as given by equation (4.12). The pressure at the boundary of the yield region is 3PHertz (a) 2πa2 1− a 2 p a = py , (5.2) which can be equated to the pressure at the center of the contact area when yield is initiated, at a certain value ay of the contact radius, py = 3PHertz (ay ) , 2πa2y (5.3) so that a2 = a2p + a2y . Equation (5.1) can then be rewritten as (5.4) 5.1. PLASTICITY 43 Py P = Py + πpy a2 − a2y = Py + πpy Rd (α − αy ) = 2 3α −1 αy , (5.5) where Py = PHertz (ay ) is the contact force at yield and equation (4.11) is assumed to remain valid when plastic eﬀects are included. Considering py as a given material property, Py and αy can be expressed as follows: Py = π 3 Rd2 p3y 6E ∗2 (5.6) π 2 Rd p2y . (5.7) 4E ∗2 Recalling that for a Hertzian impact, the relative approach at maximum compression is related to the initial sphere velocity v0 by Md v02 /2 = WHertz (αm ), it follows that plastic deformations are triggered only when v0 is larger than the socalled yield velocity vy [94] αy = π2 vy = √ 2 10 p5y ρd E ∗4 1/2 . (5.8) When this is the case, which will be assumed in the following calculations, the compressive work done by the contact force during plastic loading is found by integrating equation (5.1), so that the total loading work up to a deformation state α is 2 α Py αy α 15 − 10 +3 , (5.9) Wload = 20 αy αy and the maximum relative approach is given by Md v02 /2 = Wload (αm ): αy v02 αm = 5 + 2 30 2 − 5 . 15 vy 5.1.2 (5.10) Elastic recovery and collision time Plastic deformations cause the surface of the dust particle to ﬂatten, thus its elastic recovery occurs with an increased curvature radius Rp , which is found by equating the contact area πa2m supporting the actual contact force at maximum compression P (am ) over the curvature 1/Rp with that supporting the equivalent Hertzian contact force PHertz (am ) over the curvature 1/Rd [94, 95], Rp P (am ) = Rd PHertz (am ) , (5.11) 44 CHAPTER 5. DISSIPATIVE IMPACTS which leads to 3/2 v2 5 + 2 30 v02 − 5 y Rd Rp = √ . 2 3 15 v 30 v02 − 5 (5.12) y The force-displacement relationship during elastic unloading is then written as 4 Punload = E ∗ Rp (α − αp )3 , (5.13) 3 where αp measures the residual plastic ﬂattening of the grain surface after the impact and is found by imposing the continuity of P (α) across loading and unloading, Rd (5.14) αp = 1 − αm . Rp The total compressive work done since the start of the impact up to a deformation state α during the unloading stage can ﬁnally be expressed as α Punload (α )dα 5/2 5/2 α − α R 1 8 ∗ d p 2 = Md v0 − E Rp α5m − . 2 15 Rp αm Wunload = Wload (αm ) + αm (5.15) Equations (5.9) and (5.15) can now be substituted into equation (4.17) to derive the elastic-perfectly plastic collision time te-p coll . Splitting the impact into elastic p e loading, plastic loading and elastic unloading, one can write te-p coll = tload + tload + tunload and calculate the duration of each stage separately: 1 αy dα e tload = 2 5/2 v0 0 v α 1 − vy0 αy (5.16) √ −2 −1 2 1 7 v0 10 ρd v0 , Rd , ; ; = 2 F1 2 py vy 5 2 5 vy where 2 F1 is the Gaussian hypergeometric function. The plastic loading time is tpload = √ αm 20 ρd dα Rd 2 2 αy py αy αm αm α − 10 15 − αy αy αy − α αy , in which the integral can be evaluated by substituting x = (αm − α)/αy , (5.17) 5.1. PLASTICITY 45 5 Elastic loading Plastic loading Unloading Total t coll / τp 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 v0 /v y Figure 5.2: Collision times as a function of the impact velocity v0 for a normal elastic-perfectly plastic impact. Time is normalized to τp (5.21) and speed is normalized to the yield velocity vy (5.8). tpload where κ = ααm −1 √ y ρd dx = 20Rd py 0 10 3 ααmy − 1 x − 15x2 ρd 3αm − 3αy 4 = √ Rd arcsin py 6αm − 2αy 3 ρd κ−5 4 , = √ Rd arcsin py 2κ 3 (5.18) 30(v0 /vy )2 − 5. The unloading time is given by tunload = 15Md 1/2 16E ∗ Rp αm αp dα , 5/2 (αm − αp ) − (α − αp )5/2 (5.19) which can be rewritten, after substituting x = (α − αp )/(αm − αp ) and plugging in equations (5.10) and (5.14), 46 CHAPTER 5. DISSIPATIVE IMPACTS Figure 5.3: Force-displacement diagram for an elastic-perfectly plastic impact. tunload = 15Md 1/2 1/2 16E ∗ Rd αm 1 0 dx √ 1 − x5/2 −1/4 αm 5πρd Γ(7/5) Rd 2py αy Γ(9/10) 1/4 375π 2 Γ(7/5) ρd −1/4 Rd (5 + 2κ) . = 4 Γ(9/10) py = (5.20) Beyond the many numerical constants involved, equations (5.16–5.20) provide valuable insight into the impact: • The relevant time scale for elastic-plastic impacts is given by τp = Rd ρd /py . (5.21) • If plastic yield occurs, the total duration of the impact cannot exceed 5τp . elastic scales asymptotically as (v0 /vy )1/5 for large speeds, • The ratio te-p coll /tcoll implying that equation (4.18) is a reliable order-of-magnitude estimate even elastic ≤ 2 for v0 /vy ≤ 1000. for very plastic collisions; in fact, te-p coll /tcoll 5.2. ADHESION 47 Figure 5.2 details the contributions of the three stages to te-p coll as a function of the impact velocity. As in chapter 4, the energy budget of the impact can be visualized on the force-displacement diagram shown in ﬁgure 5.3, wherein the hysteresis loop marks the irreversibility of the process. The energy denoted as W1 is dissipated into plastic deformations and W2 is the kinetic energy recovered by the spherical particle as it rebounds. 5.2 Adhesion Despite their incorporation of irreversible energy losses, elastic-plastic models are insuﬃcient to treat low-velocity impacts, during which adhesive processes can play a major role. While several models exist to treat adhesion between surfaces in contact, the derivations presented below focus on the so-called JKR theory, originally formulated by Johnson, Kendall and Roberts in 1971 [99]. The reader is referred to the available literature for extended discussions on other adhesion models and their domains of validity [100–102]. 5.2.1 JKR theory At the microscopic scale, adhesion is a consequence of the inter-atomic and intermolecular forces, such as the Van der Waals interactions. Those consist of a shortrange repulsive part and a long-range attractive part, hence the existence of an equilibrium separation distance between surfaces, as sketched in ﬁgure 5.4. The surface energy γ of a material is then deﬁned as half the energy required to separate two identical unit surfaces from their equilibrium position to inﬁnity. Considering two diﬀerent materials with surface energies γ1 and γ2 in adhesive contact, the energy Γ required to separate them is known as the work of adhesion, or sometimes interface energy, and is generally assumed to be well estimated from the LorentzBerthelot combining rules [103–105], √ Γ = 2 γ1 γ2 . (5.22) JKR theory is obtained as a generalization of Hertz’s formalism, in which a negative surface contribution −πa2 Γ is added to the contact energy U , and the tensile stress term p1 in equation (4.10) is allowed to be nonzero. Considering that, at ﬁxed penetration α, ∂U/∂a has to vanish at equilibrium, the value of p1 is obtained as [91] 2ΓE ∗ , (5.23) p1 = πa from which the JKR contact equations are derived: a2 2πaΓ α= − Rd E∗ (5.24) 48 CHAPTER 5. DISSIPATIVE IMPACTS V(z) zeq 0 z Figure 5.4: Schematic plot of the interaction energy V (z) as a function of the distance z separating two surfaces. P = 4E ∗ a3 √ − 8πE ∗ a3 Γ . 3Rd (5.25) A fundamental feature of these equations is their prediction of a nonzero contact radius at zero relative approach; the surface of the sphere stretches towards the wall and forms a neck, which was in fact observed experimentally and motivated the need for an extended theory [99, 100]. 5.2.2 Sticking velocity Since adhesion forces act over typical inter-atomic distances, which are much smaller than the dust sizes of interest, the formation of the neck eﬀectively results in a discontinuous jump of the contact radius at α = 0, from 0 to a0 = 2πΓRd2 E∗ 1/3 , (5.26) while the contact force instantaneously drops to P0 = −8Pc /9, where Pc = 3 πΓRd 2 (5.27) 5.2. ADHESION 49 is known as the pull-oﬀ force [94]. The JKR contact equations (5.24–5.25) can be rewritten by letting u = a/a0 , α= a20 4 u −u Rd (5.28) 8Pc 6 2u − 3u3 , (5.29) 9 and integrated to provide an expression of the work done by the contact forces upon loading up to u ≥ 1: P = πa20 Γ 16u10 − 40u7 + 15u4 + 9 . (5.30) 15 Similarly to the Hertzian case, the loading stage extends until u = um such that Md v02 /2 = W (um ) and elastic unloading proceeds back to the initial state with α = 0 and u = 1. At this point, however, the two surfaces are still in contact and energy is required to separate them. As the spherical particle draws away from the wall, α and a decrease continuously while the contact force works against the motion. If v0 is high enough, the point u = ud = 4−1/3 is reached, where the situation becomes unstable as equation (5.28) forbids further separation while maintaining contact. Therefore, the contact breaks abruptly and a jumps down to 0. The whole impact process is summarized in ﬁgure 5.5, where W3 = W (ud ) represents the energy loss due to irreversible adhesive work; it can be rewritten as W3 = Md vs2 /2, where √ 1/6 Γ5 3 1/3 1 + 6 × 22/3 vs = π (5.31) 2 5 ρ3d E ∗2 Rd5 W = is deﬁned as the sticking velocity. If v0 < vs , the dust grain does not have enough kinetic energy to overcome adhesion forces and it remains attached to the surface, eventually reaching an equilibrium state characterized by P = 0 and a contact radius aeq = a0 2/3 3 . 2 (5.32) Note that due to the scaling of vs with Rd , sticking is more easily achieved by smaller particles. Once at equilibrium, detaching the grain from the surface requires an external operator to apply a force at least equal to Pc , hence the appellation ‘pull-oﬀ force’. It is worth insisting at this point that irreversibility arises from an asymmetry between the attachment and detachment processes. Due to the short-range nature of Van der Waals forces, the energy released in forming the initial contact area πa20 is spent to deform the surfaces into a neck shape while the sphere remains globally immobile. On the contrary, as the sphere bounces away from the wall, energy has 50 CHAPTER 5. DISSIPATIVE IMPACTS Figure 5.5: Force-displacement diagram for an elastic-adhesive impact. to be provided both to reduce the size of the contact area and to increase the strain on the neck structure. 5.2.3 Elastic-adhesive collision time Splitting the impact into the reversible loading-unloading stage with α > 0 and the detachment stage with α < 0, the total collision time te-a coll can be written as the sum trev + tdet , where αm dα (5.33) trev = 2Md W (α 0 m ) − W (α) Md 0 dα tdet = . (5.34) 2 αd W (αm ) − W (α) Substituting u allows all material properties to be gathered in a characteristic adhesive collision time τa , given by τa = and such that ρ3d Rd7 ΓE ∗2 1/6 , (5.35) 5.2. ADHESION 51 8 Reversible Detachment Total 7 6 t coll / τa 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 v0 /v s Figure 5.6: Collision times as a function of the impact velocity v0 for a normal elastic-adhesive impact. Time is normalized to τa (5.35) and speed is normalized to the sticking velocity vs (5.31). 1/3 trev = (16π) √ 10τa um 1 √ tdet = (2π)1/3 10τa 1 ud 3 4u − 1 du (5.36) 10 7 7 4 4 16 (u10 m − u ) − 40 (um − u ) + 15 (um − u ) 3 4u − 1 du , (5.37) 10 7 7 4 4 16 (u10 m − u ) − 40 (um − u ) + 15 (um − u ) in which the integrals depend only on um , that is on v0 /vs since the deﬁning relationship Md v02 /2 = W (um ) can be rewritten as v 2 0 7 4 −5/3 16u10 − 40u + 15u + 9 = 9 + 3 × 2 . m m m vs (5.38) The integrals in equations (5.36–5.37) have to be evaluated numerically; the resulting collision times are plotted in ﬁgure 5.6 as a function of v0 /vs and the following observations can be made: • The duration of a bouncing impact cannot exceed 8τa . 52 CHAPTER 5. DISSIPATIVE IMPACTS elastic • The ratio te-a is always less than 2 for bouncing impacts and tends coll /tcoll towards unity for large speeds, implying once again that equation (4.18) can be used as a reliable order-of-magnitude estimate of the collision time. 5.3 5.3.1 The Thornton and Ning model Normal restitution coeﬃcients The impact models developed in this chapter allow the entirety of the collision processes to be encompassed into a single ﬁgure of merit, the so-called normal restitution coeﬃcient e⊥ , deﬁned as the ratio of the sphere’s normal rebound speed vr to its normal incident speed: e⊥ = vr . v0 (5.39) It can be expressed as a function of v0 from energy dissipation analysis. In the e-a elastic-perfectly plastic and elastic-adhesive cases, the resulting values ee-p ⊥ and e⊥ are given by Thornton and Ning [94] and are plotted in ﬁgures 5.7 and 5.8, ee-p ⊥ ⎡ √ 1/2 2 1/2 ⎢ 6 3 1 vy ⎢ = 1− ⎣ 5 6 v0 vy v0 ee-a ⊥ = 1− vs v0 ⎤1/4 +2 vy v0 6 5 − 1 5 vy v0 ⎥ ⎥ 2 ⎦ (5.40) 2 , (5.41) where vy and vs appear as the only parameters of the model. In order to describe the combined eﬀects of plasticity and adhesion, Thornton and Ning have extended JKR theory by introducing the limiting pressure py as an upper limit for the compressive contact stresses, similarly to the their treatment of elastic-perfectly plastic impacts. The resulting model is based on the JKR contact equations (5.24–5.25) and a new expression for the contact force after yield has been initiated, obtained by assuming that equation (5.4) remains valid when adhesion is present [94]: P = √ 4E ∗ a3y − ay 8πΓE ∗ a + πpy a2 − a2y , 3Rd where the contact radius at yield ay is given by [94] 2ΓE ∗ 2E ∗ ay py = − . πRd πay (5.42) (5.43) 5.3. THE THORNTON AND NING MODEL 53 1 0.8 e e-p ⊥ 0.6 0.4 0.2 0 10 -1 10 0 10 1 10 2 v0 /v y Figure 5.7: Normal restitution coeﬃcient as a function of the normalized impact velocity (5.8) for an elastic-perfectly plastic sphere-plane collision. As in section 5.1, the recovery stage is assumed to occur with a modiﬁed curvature radius Rp , found from equation (5.11). Numerical solutions of the full impact process within this model are given by Ning [95], whose results show that the resulting values of e⊥ are in good agreement with those obtained by assuming that no coupling exists between plasticity and adhesion, that is [94] 2 2 e-a e2⊥ = (ee-p ⊥ ) + (e⊥ ) − 1 . 5.3.2 (5.44) Characteristics of sticking impacts While the analysis given by Thornton and Ning is aimed at obtaining expressions for the normal restitution coeﬃcient, their impact model can be used to derive other quantities of interest for dust studies in fusion environments. Considering a yielding impact and adopting the notations of the previous sections, the contact force during plastic loading is 3 4Pc P = u4 − 6u2y u − 2u6y + 3u3y , (5.45) 6u2y − 9 uy where uy satisﬁes 54 CHAPTER 5. DISSIPATIVE IMPACTS 1 0.8 e e-a ⊥ 0.6 0.4 0.2 0 10 -1 10 0 10 1 10 2 v0 /v s Figure 5.8: Normal restitution coeﬃcient as a function of the normalized impact velocity (5.31) for an elastic-adhesive sphere-plane collision. 2u2y − 1 py = , uy pc (5.46) in which pc is the compressive stress at the center of the initial contact area, E ∗ a0 pc = = πRd 2ΓE ∗2 π 2 Rd 1/3 . (5.47) A remarkable consequence of equation (5.46) is that py /pc can serve as an intrinsic measure of the plasticity of a spherical particle made from a given material. More precisely, three regimes can be delimited as follows: • If py /pc ≤ 1, equation (5.46) has no solution larger than 1, implying that uy = 1 and plastic deformations are triggered instantaneously when contact is initiated. • If 1 < py /pc ≤ p̂crit 2.22, the solution of equation (5.46) is such that P (uy ) ≤ 0. As a consequence, the spherical particle yields at some point during impact, regardless of its initial velocity. • If py /pc > p̂crit , then P (uy ) > 0 and there exists some critical impact velocity — not necessarily equal to vy — below which yield does not occur. 5.3. THE THORNTON AND NING MODEL 55 Integrating equation (5.45) from u = 1 to u ≥ uy gives the contact work done during plastic loading as πa20 Γ 1 1 2 8 2 15 2uy − W = u − 6 10uy − u5 − 10 2u6y − 3u3y u4 15 uy uy 6 2 2 3 10 7 4 + 30uy u + 10 2uy − 3uy u + 6uy − 15uy + 9uy + 9 , (5.48) which is equal to the initial kinetic energy of the dust particle at maximum compression u = um . The increased curvature radius Rp due to irreversible ﬂattening is then given by Rp = Rd 3 2u2y − 1 uy 4u6m u4m + 6u3m − 6u2y um − 2u6y + 3u3y , (5.49) from which the post-recovery equilibrium contact radius and the pull-oﬀ force of a sticking impact can be derived: aeqr = a0 3Rp 2Rd 2/3 (5.50) 3 πΓRp . 2 The contact force during the unloading stage is Pcr = Punload 4E ∗ a3 √ 8Pc = − 8πΓE ∗ a3 = 3Rp 9 (5.51) 2Rd 6 u − 3u3 Rp , (5.52) which can be integrated to ﬁnd the contact work Wunload = Md v02 /2 − ΔW , where ΔW = 4 Rd 7 πa20 Γ R2 10 7 4 − 40 u + 15 u , − u − u − u 16 2d u10 m 15 Rp m Rp m (5.53) and detachment occurs at u = udr , with udr = Rp 4Rd 1/3 . (5.54) Chapter 6 Dust-wall collisions in MIGRAINe This chapter delves in more detail into the implementation of dust-surface impacts in MIGRAINe. It focuses not only on the algorithmic aspect of collision detection, but also on the incorporation of physical eﬀects with particularly high importance in fusion plasma environments. Before proceeding, it is worth pointing out that while plasma forces govern the dynamics of dust particles during the majority of their lifetime, their work over the characteristic impact time scales derived in chapter 5 is negligible with respect to the dust kinetic energy. This justiﬁes the separate treatment of dust-wall collisions within a purely mechanical framework. 6.1 Collision detection In order to provide reliable predictions of long-range dust migration in a large variety of plasma devices, MIGRAINe must be able to process particle impacts on complex, highly detailed tridimensional wall surfaces in a computationally eﬃcient way. Granted triangulation is the most widely used method to represent arbitrary 3D surfaces for simulation purposes, at the fundamental level, detecting the collision between a simulated dust particle and a given surface reduces to testing whether the straight-line path of the particle between two consecutive time steps intersects a triangle, that is solving a 3 × 3 linear system, a trivial task from the algorithmic point of view. However, the high number of triangles required to capture the detailed features of plasma-facing components, such as grooves and castellations, forbids the use of the naive approach — repeating the test for every wall triangle at every time step — owing to its prohibitive computational cost. Cost-eﬀective treatment of complex geometrical objects is a commonly encountered issue in the ﬁeld of computer graphics, where elaborated 3D scenes must often be rendered in real time. Many eﬃcient algorithms have therefore been developed over the past decades to suit various rendering problems. The approach selected in MIGRAINe is to organize the data describing the wall surface into a bounding volume hierarchy (BVH). More speciﬁcally, the BVH used in MIGRAINe is a 57 58 CHAPTER 6. DUST-WALL COLLISIONS IN MIGRAINE A G A C 4 B C 3 F D E F G 1 2 3 4 B D1 2 E Figure 6.1: Four triangles in a plane and their representation using a BVH. binary tree structure representing a set of axis-aligned bounding boxes (AABB), i.e. rectangular parallelepipeds whose sides are parallel to the axes of the cartesian coordinate system used in the simulation. Each wall element is enclosed into an AABB stored in a leaf of the BVH. The leaves are then recursively clustered in pairs into larger AABBs stored in the internal nodes of the BVH, as shown in ﬁgure 6.1. This structure can easily be adapted to any dimension, in particular to axisymmetric wall surfaces which can be represented in two dimensions via their poloidal cross-section. Given a BVH representing the wall, collision detection can be split into two phases. The so-called broad phase consists in traversing the tree from its root and testing for collisions against the current node, whose children are visited only if the collision test for that node is positive. The broad phase has a short execution time since collision tests against AABBs are computationally cheap — they consist in checking some inequalities without any need for other calculations — and are only performed if they have a chance to yield a positive result. Once the broad phase is over, the list of BVH leaves having passed the test is used as an input for the narrow phase, during which accurate collision tests against the corresponding wall elements are executed. Whereas the naive approach requires N expensive dusttriangle tests for a wall consisting of N elements, the algorithm presented above results in O(log2 N ) cheap dust-AABB tests and only a few dust-triangle tests. Out of the many possible BVHs representing a given surface, those allowing for optimized collision detection are built in a way that each branch node is split evenly by its children, ensuring that the tree traversals are kept as short as possible. While several algorithms constructing such balanced BVHs are available in the literature, MIGRAINe relies on fast agglomerative clustering [106]. This approach consists in building the BVH from its leaves by pairing the AABBs together in order to 6.1. COLLISION DETECTION 59 Figure 6.2: Example of a triangulated surface and its bounding volumes at diﬀerent levels of the hierarchy. 60 CHAPTER 6. DUST-WALL COLLISIONS IN MIGRAINE minimize the volume of the resulting cluster. An example of a concrete BVH is shown in ﬁgure 6.2. In addition to the geometric structure incorporated in the BVH, each wall element may carry additional information such as the material it is made of, or any other potentially useful data. This gives signiﬁcant freedom in the level of detail of the reactor wall description and enables dedicated studies of dust-wall interaction phenomena. 6.2 Oblique impacts Due to the strong toroidal component of the ion drag force in fusion plasmas, dustwall impacts typically occur with a large tangential velocity component. Therefore, normal impact models such as Thornton and Ning’s cannot a priori be applied, and a tangential restitution coeﬃcient e|| must be introduced to describe a generic oblique impact via vr = e|| vi − e|| + e⊥ (vi · n̂) n̂ , (6.1) where the subscripts i and r refer to the incident and rebound velocity, respectively, and n̂ is the local unit normal vector to the wall surface. The reference literature on contact mechanics includes a large corpus of studies regarding the importance of tangential stresses on oblique impacts, starting from seminal works carried out around 1950, in which extensions of Hertz’s theory are proposed [107, 108]. While the microscopic behaviour of the contacting surfaces under oblique loads may become rather involved and strongly dependent on impact history [109, 110], both experimental and numerical studies [110–113] show that tangential restitution coeﬃcients of oblique sphere-plane impacts are well described by rigid body theory. According to this model, e|| = 1 − f (1 + e⊥ ) cot θ , (6.2) where θ is the impact angle with respect to n̂ and f is the ratio of the tangential impulse over the whole impact duration to the normal impulse. In the sliding regime, which occurs for large θ, f is simply equal to the Coulomb friction coeﬃcient μ between the materials in contact. Despite its simplicity, equation (6.2) introduces an inconsistency in the treatment of dust dynamics in MIGRAINe. Indeed, sliding friction forces, whose eﬀects are described by μ, can cause the bouncing particle to spin after an oblique impact [110, 113]. This may then aﬀect subsequent impacts, possibly resulting in scenarios with restitution coeﬃcients above unity depending on the energy exchange between rotational and translational motion. Moreover, since the rotational dynamics of dust particles in plasmas are not treated in MIGRAINe, self-consistency requires the assumption of frictionless impacts, that is f = 0 and e|| = 1. As for e⊥ , ﬁnite-element simulations [110] have demonstrated that the Thornton and 6.3. SURFACE ROUGHNESS AND RANDOMIZATION 61 Ning model remains adequate for oblique impacts, implying in particular that e⊥ depends weakly on the tangential component of vi . Finally, it is noteworthy that while experimental and numerical evidence of the accuracy of the Thornton and Ning model almost exclusively concerns the impacts of millimeter-sized particles in air or vacuum, recent experimental observations of bouncing impacts of metallic micrometer-sized dust particles in Pilot-PSI divertor-like plasmas demonstrate near-unity values for e|| [114]. 6.3 Surface roughness and randomization The unavoidable roughness of real surfaces has long been recognized as a major issue in contact mechanics, especially in the study of adhesion, wherein the responsible inter-molecular forces can be shown to induce local stresses orders of magnitude larger than the atmospheric pressure, such that any two perfectly smooth macroscopic surfaces would be practically impossible to separate [105]. The actual contact between two rough solids is limited to the tips of surface asperities, resulting in a strong reduction of adhesion forces in the case of stiﬀ materials [115]. Roughness is typically modelled by superposing a random height proﬁle to the smooth nominal surface. The surface proﬁle is often assumed to consist of asperities of given shape — traditionally hemispherical — leaving only a few free parameters in the model, such as the ﬁrst moments of the underlying probability distribution [116– 120]. In MIGRAINe, the distinction is made between micrometric and nanometric roughness, respectively referring to spatial scales larger and smaller than the dust radius Rd . Micrometric roughness is treated by randomizing the geometry of the wall surface under the assumption of normally distributed triangular corrugations, as described in Paper II. This approach eﬀectively reduces to a redistribution of the dust kinetic energy into normal and tangential components, which in turn signiﬁcantly enhances transport as the randomized normal impact velocity tends to be larger than its nominal counterpart. The model relies on only one physical parameter, the standard deviation of the surface slope, which can be obtained experimentally from proﬁlometry measurements. Nanometric roughness is treated by randomizing the work of adhesion Γ between the surfaces, or, equivalently, the sticking velocity vs . Since nano-scale surface characteristics are quite diﬃcult to measure experimentally, especially in the case of plasma-exposed materials, the choice has been made to avoid the dependence on unknown physical parameters. Therefore, MIGRAINe simulations involving nano-scale roughness uniformly draw the value of vs between 0 and its theoretical value. In that regard, nanometric roughness is included as a way to estimate the uncertainty of the collision model. 62 6.4 CHAPTER 6. DUST-WALL COLLISIONS IN MIGRAINE Variations of the mechanical properties While the mechanical properties available in the literature typically concern materials at room temperature, dust particles in fusion environments are readily heated beyond their melting point, making it necessary to take temperature dependences into account. Another, less known phenomenon is the size dependence of material properties — especially the yield strength — at sub-micrometer scales. The eﬀects of such variations on normal restitution coeﬃcients are detailed in Paper I; the purpose of this section is to compile the data used in MIGRAINe to model tungsten and beryllium. 6.4.1 Temperature dependence In most cases, numerical ﬁts describing the temperature variations of mechanical properties are directly available in the literature. Polynomial ﬁts for the property X, X= cn T n , (6.3) n where T is expressed in kelvins, are reported in tables 6.1 and 6.2. Fits of the linear thermal expansion coeﬃcient α are used to estimate the temperature-dependent density ρ as ρ = ρ0 exp −3 T T0 α (T ) dT , (6.4) where the subscript 0 refers to room-temperature values, i.e. ρ0 = 1850 kg m−3 for Be and ρ0 = 19250 kg m−3 for W [53]. The exception to equation (6.3) concerns the yield strength σyb of bulk tungsten, for which experimental data is scarce. Measurements from mechanical tests reported in [121,122] and plotted in ﬁgure 6.3 may however be ﬁtted with satisfactory accuracy by assuming exponential decay below 1000 K and linear decrease at higher temperatures, σyb = c0 + c1 T + a exp (−bT ) , (6.5) with c0 = 9.498 × 107 Pa, c1 = −2.273 × 104 Pa K−1 , a = 3.179 × 109 Pa and b = 6.138 × 10−3 K−1 . 6.4.2 Size dependence of the yield strength The size-dependent mechanical behaviour of small samples — with characteristic dimensions of the order of a few micrometers and below — has been investigated extensively over the last decade. In particular, micropillar compression experiments 6.4. VARIATIONS OF THE MECHANICAL PROPERTIES 63 600 Raffo Luo et al Fit Yield strength [MPa] 500 400 300 200 100 0 500 1000 1500 2000 2500 3000 3500 Temperature [K] Figure 6.3: Fitted yield strength of bulk tungsten as a function of temperature according to equation (6.5) and comparison with experimental data [121, 122]. α E ν γ −1 K GPa J m−2 c0 c1 c2 c3 4.1471e-6 4.1687e+2 2.8100e-1 4.4350e+0 1.1723e-9 -3.2314e-2 4.4628e-6 -2.3667e-4 -4.2166e-13 -5.8710e-6 3.1620e-9 1.9668e-16 Table 6.1: Fit coeﬃcients for the temperature-dependent mechanical properties (6.3) of solid tungsten [123–125]. 64 CHAPTER 6. DUST-WALL COLLISIONS IN MIGRAINE α E ν γ σyb −1 K GPa J m−2 MPa c0 c1 c2 8.4300e-6 3.1353e+2 4.3600e-2 1.8340e+0 5.2873e+2 1.1473e-8 -5.6430e-2 3.3810e-5 -2.5000e-4 -4.7663e-1 -2.9758e-12 -1.3670e-9 1.0000e-4 Table 6.2: Fit coeﬃcients for the temperature-dependent mechanical properties (6.3) of solid beryllium [126–129]. For surface energy, c1 was assumed equal to the temperature coeﬃcient of the surface tension of liquid beryllium [128] and c0 was chosen according to the results of ﬁrst-principle calculations at zero temperature [129]. indicate that the yield strength increases with smaller sizes [130–133]. This phenomenon can be explained by examining the microscopic origin of plasticity, i.e. the motion of dislocations in the crystal lattice [134, 135]. Dislocations are classiﬁed in two main categories, namely edge dislocations and screw dislocations, as schematized in ﬁgure 6.4. In the absence of external stress, dislocations lie on local minima of the inter-atomic potential, separated by potential barriers preventing the atomic planes from gliding with respect to each other [134]. The external stress required to move a dislocation across such a potential barrier, known as the Peierls stress, can be thought of as the microscopic counterpart of the yield strength and strongly depends on dislocation geometry. The strong temperature dependence of the yield strength of tungsten below 1000 K, clearly visible in ﬁgure 6.3, is in fact a characteristic feature of bodycentered-cubic (bcc) metals [131, 136]. Theoretical justiﬁcations of this behaviour involve the existence of two regimes of dislocation mobility in the bcc lattice. The Peierls stress is indeed much larger for screw dislocations than for edge dislocations, so that screw dislocations are essentially immobile at low temperatures. As the temperature rises, the thermally activated screw dislocations become increasingly more mobile, which results in a steep decrease of the yield strength [131, 137]. When the temperature reaches some critical value Tc , the thermal energy of screw dislocations overcomes the Peierls stress and their motion dominates plastic deformations. From this point on, the temperature dependence of the yield strength is reduced and bcc systems behave essentially like face-centered-cubic (fcc) crystals [131]. It is widely accepted that the size dependence of the yield strength should obey a power law of the form d−m , where d is the characteristic length scale on which the deformation occurs and 0 ≤ m ≤ 1. Although theoretical attempts to derive such a scaling law result in either m = 1/2 or m = 1 — see [138] and references therein — the values of m observed experimentally appear to depend on the crystal structure, with 0.6 ≤ m ≤ 1 for fcc crystals and 0.2 ≤ m ≤ 0.5 for bcc crystals [131]. 6.4. VARIATIONS OF THE MECHANICAL PROPERTIES Edge dislocation 65 Screw dislocation Figure 6.4: Schematic view of edge and screw dislocations in a cubic lattice, adapted from [134]. Due to the absence of experimental data on high-temperature plastic deformation at the micron scale, MIGRAINe may only rely on extrapolations of the existing measurements at room temperature. A possible extrapolation strategy beneﬁts from the results of Schneider et al [131] on bcc metals with diﬀerent critical temperatures, hinting at a possibly universal behaviour of the power-law exponent m as a function of T /Tc . However, Dunstan and Bushby have demonstrated that the method typically used to estimate m is contestable, as many diﬀerent values may lead to an excellent agreement with the same data set [138]. In their opinion, since the theoretical values m = 1/2 and m = 1 also produce predictions in agreement with experimental results, m should simply not be ﬁtted. In the end, the approach chosen in MIGRAINe is to assume that the size dependence of the yield strength of a spherical particle of radius Rd and temperature Td is analog to the scaling of Hall-Petch grain-boundary strengthening [139, 140], which relates the yield strength of polycrystalline materials to their crystallite size: ky σy (Td , Rd ) = σyb (Td ) + √ . Rd (6.6) Comparing equation (6.6) with numerical values estimated from stress-strain measurements plotted in [131, 132] gives a best ﬁt with ky = 0.91 MPa m1/2 for W, see ﬁgure 6.5. In the absence of available measurements for Be, the Hall-Petch constant ky = 0.41 MPa m1/2 is chosen [134]. Evidently, owing to the lack of experimental studies concerning the coupled size and temperature dependence of σy , these values should not be considered as precise quantitative estimates, but rather as educated guesses meant to capture the generic size scalings. 66 CHAPTER 6. DUST-WALL COLLISIONS IN MIGRAINE 3500 Schneider et al Kim et al Fit Yield strength [MPa] 3000 2500 2000 1500 1000 0 0.5 1 1.5 2 2.5 3 Micropillar radius [µm] Figure 6.5: Fitted size-depedent yield strength of room-temperature tungsten micropillars according to equation (6.6) and comparison with experimental data [131, 132]. Contribution to publications While experimental aspects constitute a large part of several of the papers appended hereafter, the following paragraphs focus exclusively on the contribution of the modelling work presented in this thesis to the published results. Paper I — Dust-wall and dust-plasma interaction in the MIGRAINe code A detailed overview of the physical models implemented in MIGRAINe is provided. Particular attention is given to the implementation of state-of-the-art models to treat surface processes such as secondary electron emission, as well as electron and ion backscattering. The inﬂuence of these phenomena on dust survivability in fusion environments is analyzed quantitatively and compared with the predictions of other dust transport codes. Moreover, the eﬀect of size and temperature on normal restitution coeﬃcients is studied. Paper II — Migration of tungsten dust in tokamaks: role of dust-wall collisions The ﬁrst comparison between MIGRAINe simulations and tungsten dust injection and collection experiments in TEXTOR is presented. The main experimental results, concerning the size and spatial distributions of collected particles, are shown to be successfully reproduced by MIGRAINe. The crucial role of collision models on the predicted dust migration is demonstrated. Paper III — Transport asymmetry and release mechanisms of metal dust in the reversed-field pinch configuration MIGRAINe simulations of intrinsic molybdenum dust transport in EXTRAP-T2R are confronted to experimental data. The inﬂuence of initial conditions on the simulated trajectories is exploited to infer the main characteristics of intrinsic dust release mechanisms. The results suggest that the mobilization of pre-existing dust particles is dominant with respect to the production of new dust particles. 67 68 CONTRIBUTION TO PUBLICATIONS Paper IV — Elastic-plastic adhesive impacts of tungsten dust with metal surfaces in plasma environments The paper reports on the ﬁrst direct evidence of the adequacy of the Thornton and Ning model when applied to dust-wall collisions in fusion environments. Further evidence of dust heating discrepancies between numerical predictions and experimental observations is presented. Paper V — Fast camera observations of injected and intrinsic dust in TEXTOR Tridimensional trajectories reconstructed from fast camera observations of carbon and tungsten dust injections in TEXTOR are compared with MIGRAINe predictions. Major issues of dust code validation are raised and examined in the context of successful benchmarking claims concerning pre-existing dust transport codes. Unambiguous evidence of the inaccuracy of the core dust heating models on which all codes rely is presented. Paper VI — Interaction of adhered metallic dust with transient plasma heat loads The response of metallic dust particles adhered on plasma-facing components to fusion-relevant transient heat loads is studied. Experimental results of dust exposure to DIII-D and Pilot-PSI plasmas are analyzed in the light of ﬁnite-element heat transfer simulations. The results are shown to depend strongly on particle clustering and the details of the dust-wall contact. Paper VII — Beryllium droplet cooling and distribution in the ITER vessel after a disruption MIGRAINe simulations are used to conduct the ﬁrst predictive study of beryllium droplet migration in ITER following disruption-induced melt ejection events. The location of preferred re-deposition sites is identiﬁed and characteristic cooling times are reported. The initial droplet size is stressed as the main governing parameter to be targeted by future melt ejection studies. Outlook While MIGRAINe addresses most of the processes of importance for dust modelling in fusion plasmas, the dust-plasma interaction regimes encountered in large-scale fusion devices such as ITER remain partially unexplored, and fully satisfactory theories do not exist as yet. The scope of open issues to be targeted by future work in collaboration with international research teams includes: • The self-consistent modelling of the charging and heating of large, strongly emitting particles in magnetized plasmas, for which OML assumptions are not satisﬁed. PIC simulations can be used to provide essential input in deriving approximated analytical descriptions of dust-plasma interaction in those regimes. • The appropriate treatment of thermionic emission from liquid metals, for which no models are currently available. First-principle calculations of the electronic structure of liquid metals are envisioned as a means to achieve a detailed understanding of the processes at play. • The eﬀect of thermal forces arising from ambient plasma temperature gradients. While theoretical studies of thermal forces have been undertaken, the unprecedented temperature gradients expected in ITER call for a reassessment of their role in dust transport. • Plasma-surface interactions are treated with analytical formulas containing coeﬃcients that correspond to pristine materials. In realistic reactor conditions, the surface of plasma-facing components and dust grains are saturated by fuel elements. 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