Download Modelling the multifaceted physics of metallic dust and

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
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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 . . . . . . . . . .
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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
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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
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35
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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
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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
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57
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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. . . . .
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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. . . .
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3.2
3.3
3.4
4.1
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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
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mechanical properties of
. . . . . . . . . . . . . . .
mechanical properties of
. . . . . . . . . . . . . . .
19
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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. This will not only alter the yield coefficients but possibly
even basic material properties such as the mass density.
• Several aspects of the MIGRAINe mechanical collision model require further
experimental validation and some important parameters, such as the py /σy
ratio, must be calibrated. To this end, controlled dust injection experiments
supplemented by high-resolution camera observations have been carried out
in Pilot-PSI.
69
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