Theory and Simulation of Dust in Tokamak Plasmas

Theory and Simulation of Dust in Tokamak Plasmas James David Martin Submitted in partial fulfilment of the requirements for the degree of Doctor of Ph...
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Theory and Simulation of Dust in Tokamak Plasmas James David Martin Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy of the University of London and for the Diploma of Membership of Imperial College London Plasma Physics Group The Blackett Laboratory Imperial College London Prince Consort Road London SW7 2BZ July 2006

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Abstract This thesis details the development of the DTOKS code, designed to model the transport of dust in Tokamak plasmas. Beginning with a literature review of work on dust in Tokamaks, and the basic theory of dust-plasma interaction, we discuss the need for such a model. The theory of dust grain charging in a plasma is then reviewed. In Tokamak plasmas, significant secondary and thermionic electron emission can be caused, charging the grain positive. A theory describing this based on the orbit motion limited (OML) model is developed. Following this we develop a dust grain heating model, and predict steady state temperatures and survival times for graphite and tungsten dust in plasmas of different temperatures and densities. The dust grain equation of motion is then discussed, and the algorithms in DTOKS are outlined. Simulations of graphite and tungsten dust transport through MAST and ITER plasmas are then presented. It is shown that tungsten dust transport to the core plasma in ITER is possible for micron radius grains injected at 103 ms−1 , and 100 µm radius grains at 100 ms−1 .

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Contents 1 Introduction 1.1

1.2

1.3

16

Non-Fusion Dusty Plasmas . . . . . . . . . . . . . . . . . . . . . . . . 16 1.1.1

Dust in Atmospheric and Space plasmas . . . . . . . . . . . . 17

1.1.2

Dust in DC and RF Discharges . . . . . . . . . . . . . . . . . 18

Tokamaks: Introduction and Terminology

. . . . . . . . . . . . . . . 21

1.2.1

The Scrape-off Layer (SOL) . . . . . . . . . . . . . . . . . . . 21

1.2.2

Plasma-Surface Interaction . . . . . . . . . . . . . . . . . . . . 23

1.2.3

ELMs and Disruptions . . . . . . . . . . . . . . . . . . . . . . 24

Dust in Tokamaks: A Literature Review . . . . . . . . . . . . . . . . 24 1.3.1

Studies of the Erosion and Deposition of Wall Material . . . . 27

1.3.2

Characterisation of Dust and Flakes . . . . . . . . . . . . . . . 28

1.3.3

Safety and Dust Removal

1.3.4

Dust Production and the Effect on Plasma Operation . . . . . 34

1.3.5

Dust Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.3.6

Afterthoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2 Dust Grain Charging

. . . . . . . . . . . . . . . . . . . . 31

39

2.1

The Maxwellian Distribution . . . . . . . . . . . . . . . . . . . . . . . 40

2.2

The Debye-H¨ uckel Potential . . . . . . . . . . . . . . . . . . . . . . . 41

2.3

Planar Wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.3.1

Presheath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3.2

Sheath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3

2.4

Radial Motion (ABR) Theory . . . . . . . . . . . . . . . . . . . . . . 49

2.5

Orbit Motion (OM) Theory . . . . . . . . . . . . . . . . . . . . . . . 52

2.6

2.5.1

OML theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.5.2

Full OM theory . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Other Charging Mechanisms . . . . . . . . . . . . . . . . . . . . . . . 57 2.6.1

Secondary Emission . . . . . . . . . . . . . . . . . . . . . . . . 57

2.6.2

Thermionic Emission . . . . . . . . . . . . . . . . . . . . . . . 59

2.6.3

Field Emission . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.6.4

Photoemission . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.6.5

Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3 Positively Charged Grains 3.1

3.2

3.3

63

The Effect of Electron Emission . . . . . . . . . . . . . . . . . . . . . 63 3.1.1

Emission Flux ≪ Collection Flux . . . . . . . . . . . . . . . . 64

3.1.2

Emission Flux ≫ Collection Flux . . . . . . . . . . . . . . . . 65

The Intermediate Case . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.2.1

General Equations . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.2

a ≫ λD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2.3

a ≪ λD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A Simplified Charging Model for Fusion . . . . . . . . . . . . . . . . 75

4 Dust Temperature and Survival Times in a Steady Plasma Background 4.1

4.2

78

Heating and Cooling Mechanisms . . . . . . . . . . . . . . . . . . . . 79 4.1.1

Particle Bombardment . . . . . . . . . . . . . . . . . . . . . . 79

4.1.2

Electron Emission . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.1.3

Recombination Processes and Neutral Emission . . . . . . . . 84

4.1.4

Radiative Cooling . . . . . . . . . . . . . . . . . . . . . . . . . 85

Steady State Temperatures and Survival Times . . . . . . . . . . . . 85

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5 Dust Grain Motion 5.1

5.2

5.3

5.4

92

Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.1.1

Electromagnetic Forces . . . . . . . . . . . . . . . . . . . . . . 93

5.1.2

Flow Pressure and Drag Forces . . . . . . . . . . . . . . . . . 93

5.1.3

Pressure Gradient . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.1.4

Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.1.5

The Rocket Force . . . . . . . . . . . . . . . . . . . . . . . . . 95

Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.2.1

Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2.2

Point of Injection . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.2.3

Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Plasma Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.3.1

Introduction to the B2-solps5.0 Code . . . . . . . . . . . . . . 97

5.3.2

B2-solps5.0 Plots . . . . . . . . . . . . . . . . . . . . . . . . . 99

Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.4.1

Data Input - Resampling the B2-solps5.0 Grid . . . . . . . . . 103

5.4.2

Description of the Code . . . . . . . . . . . . . . . . . . . . . 105

5.4.3

MAST Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4.4

ITER Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.4.5

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6 Conclusions 6.1

118

Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

A Sputtering Formulae

121

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List of Figures 1.1

Saturn’s rings; the dark features are “spokes”. . . . . . . . . . . . . . 17

1.2

A typical capacitative discharge setup. . . . . . . . . . . . . . . . . . 19

1.3

The magnetic field configuration of the JET Tokamak in a divertor phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.4

Dust production in MAST observed using an IR camera [1]. . . . . . 29

1.5

A scanning electron microscope image of dust in TEXTOR-94 [2, 3]. . 30

1.6

The effect of surface corrugation on the trajectory of a dust particle [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.1

The ABR plasma solution. . . . . . . . . . . . . . . . . . . . . . . . . 50

2.2

ABR normalised potential Vd as a function of normalised grain radius A for hydrogen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.3

Ion trajectories towards a grain in OML theory. . . . . . . . . . . . . 53

2.4

OML floating potential against θ for different gases. . . . . . . . . . . 54

2.5

Ion trajectories: ions are absorbed if they are within the absorption radius [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.6

Normalised potential against grain radius for OM theory [5]. . . . . . 57

2.7

Plot of secondary electron emission yield against electron temperature for tungsten and graphite. . . . . . . . . . . . . . . . . . . . . . . . . 59

2.8

Magnitude of the current density due to thermionic emission for MAST conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.1

Floating potential for δ < 1. . . . . . . . . . . . . . . . . . . . . . . . 65

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3.2

A positive grain with a negative surface potential, having a potential minimum at a distance r = b. . . . . . . . . . . . . . . . . . . . . . . 69

3.3

Currents to and from a positively charged grain. Both emitted electrons and plasma electrons can be reflected at the minimum. All ions that reach b are assumed to be absorbed. . . . . . . . . . . . . . . . . 69

3.4

Cylindrical coordinates vt , ψ, and vr . . . . . . . . . . . . . . . . . . . 71

3.5

Floating potential for δ > 1 for a ≫ λd . . . . . . . . . . . . . . . . . . 72

3.6

Floating potential V (a) against V (b) with δ > 1 and a ≪ λd for fusion parameters.

3.7

Floating potential V (a) against V (b) with δ > 1 and a ≪ λd for different values of δ.

3.8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

. . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Demonstration of the change in collection area for magnetised and non-magnetised plasma particles. . . . . . . . . . . . . . . . . . . . . 76

3.9

V against C for θ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1

Incoming ions interacting with dust grain material. . . . . . . . . . . 80

4.2

Steady state temperatures for graphite (left) and tungsten (right) with Ti = Te for typical densities. . . . . . . . . . . . . . . . . . . . . 87

4.3

The contribution of each heating component at the steady state temperature for graphite for n0 = 1018 m−3 (left) and n0 = 1019 m−3 (right). The components are as follows: 1. Particle bombardment, 2. Recombination, 3. Secondary Emission, 4. Thermionic emission, 5. Neutral emission and 6. Radiative cooling. . . . . . . . . . . . . . . . 89

4.4

The contribution of each heating component at the steady state temperature for tungsten for n0 = 1018 m−3 (left) and n0 = 1019 m−3 (right). The components are as above. . . . . . . . . . . . . . . . . . 89

4.5

Survival times for carbon and tungsten. . . . . . . . . . . . . . . . . . 90

5.1

The B2-solps5.0 grid for MAST (left) and ITER (right). . . . . . . . 100

5.2

B2-solps5.0 data for electron temperature (eV) in MAST (left) and ITER (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7

5.3

B2-solps5.0 data for electrostatic potential (Vm−1 ) in MAST (left) and ITER (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4

Electron temperature (eV) in MAST after resampling. . . . . . . . . 104

5.5

Trajectories of micron radius dust grains injected at 10 ms−1 viewed in the poloidal (left) and toroidal (right) planes. Circles in the toroidal plane indicate the inner and outer walls of the vessel. The particle tracks are colour coded for comparison. . . . . . . . . . . . . . . . . . 107

5.6

Acceleration by each component of the equation of motion for two of the particles in figure 5.5: one that leaves the plasma (left) and one that evaporated in the plasma (right).

5.7

. . . . . . . . . . . . . . . . . 108

Relative levels of impurity deposition in MAST by a cosine distribution of the particles in figure 5.5. . . . . . . . . . . . . . . . . . . . . 108

5.8

Relative levels of impurity deposition in MAST by a cosine distribution of micron radius graphite dust particles travelling at 100 ms−1 . . 110

5.9

Relative levels of impurity deposition for a cosine distribution of micron radius graphite dust particles in ITER travelling at 10 (left), 100 (middle) and 103 ms−1 (right). . . . . . . . . . . . . . . . . . . . . . . 112

5.10 Trajectories of micron radius graphite dust injected at 103 ms−1 into ITER. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.11 Relative levels of impurity deposition for a cosine distribution of micron radius tungsten dust particles in ITER travelling at 10 (left), 100 (middle) and 103 ms−1 (right). . . . . . . . . . . . . . . . . . . . 115 5.12 Relative levels of impurity deposition for a cosine distribution of 100 micron radius tungsten dust particles in ITER travelling at 10 (left) and 100 ms−1 (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

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List of Tables 1.1

Relevant Tokamak parameters. Magnetic field, plasma current and fusion power data are maximum values. . . . . . . . . . . . . . . . . . 25

1.2

Dust characterisation for various Tokamaks. . . . . . . . . . . . . . . 32

2.1

Relevant data for carbon and tungsten electron emission. . . . . . . . 57

4.1

Particle reflection parameters for hydrogen bombardment of carbon and tungsten. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.2

Carbon and Tungsten thermal properties. . . . . . . . . . . . . . . . . 86

A.1 Data for chemical erosion with hydrogen isotopes [6]. . . . . . . . . . 122

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List of Symbols a

Radius of the dust grain

A

Normalised grain radius

a , λD

Richardson’s constant

(thermionic emission), universal constant (field emission) A1 . . . A6

Constants in backscattering formula

b

Impact parameter (OML), radius of the potential minimum

bc

Critical impact parameter (OML)

B

Universal constant (field emission)

B

Magnetic field

c

Specific heat

cs

Sound speed

C

Proportionality constant, ratio of ion to electron collection area

e

The electronic charge, base of natural logarithms

E

Electric field magnitude, energy of ions (full OM theory), beam energy (secondary electron emission and backscattering)

Emax

Energy at which maximum secondary electron emission occurs

ET F

Thomas-Fermi energy

E

Electric field

f

Particle distribution function

fi

Ion distribution function

F

Force

Fcent

Centrifugal force

Florentz

The Lorentz force

10

Fndrag

Force due to neutral drag

F∇P

Pressure gradient force

Fg

Gravitational force

g

Gravitational acceleration

h

Latent heat, characteristic interpolation length (resampling)

I

Current

Ie

Electron current

Iem

Emitted electron current

in Iem

Returning emitted electron current

out Iem

Outward emitted electron current

Ii

Ion current

j

Current density

je

Electron current density

jem

Emitted electron current density

jf e

Field emission current density

ji

Ion current density

jth

Thermionic electron current density

J

Normalised ion current (ABR theory)

m

Mass

m1

Mass of incoming particle

m2

Mass of substrate

md

Dust grain mass

me

Electron mass

mi

Ion mass

mn

Neutral mass

M

Plasma Mach number, ion Mach number

Ms

Ion Mach number at the sheath edge

n

Particle density, plasma density

n0

Bulk plasma density

11

ne

Plasma electron density

nem

Emitted electron density

ni

Plasma ion density

nn

Neutral density

ns

Plasma density at the sheath edge

nt

Density of tritium atoms (radioactivity)

nt0

Initial density of tritium atoms

Ne

Normalised electron density

Ni

Normalised ion density

Ns

Normalised plasma density at the sheath edge

p

Pressure

pe

Electron pressure

pi

Ion pressure

q

Charge

qd

Charge on the dust grain

Qpd

Energy flux

r

Radial coordinate

rL

Larmor radius

r1

Absorption radius

RN

Fraction of particles backscattered

¯N R

Mean fraction of ions backscattered for a distribution of ion energies

RE

Energy flux of backscattered particles

¯E R

Mean fraction of ion energy backscattered for a distribution of ion energies

Sp

Plasma production rate

ne n0

t(y) Image charge function (field emission) t

Time

tpt

Time taken for a phase transition to occur

T

Temperature, plasma temperature

Td

Dust grain temperature

12

Te

Plasma electron temperature

Tem

Emitted electron temperature

Ti

Plasma ion temperature

v(y)

Image charge function (field emission)

v

Speed, ambipolar plasma speed

v1

Limit on tangential velocity for particles collected

vB

Bohm velocity

vd

Dust grain speed

vdr

r component of the dust grain velocity

vdθ

θ component of the dust grain velocity

vr

Radial velocity

v¯e

Electron thermal speed

v¯em

Emitted electron thermal speed

vi

Ion speed

vi′

Ion speed in the bulk plasma

v¯n

Neutral thermal speed

vs

Plasma speed at the sheath edge

vt

Tangential velocity

vx , vy , vz

x, y and z components of velocity

v

Velocity

vd

Dust grain velocity

vp

Plasma velocity

V

Normalised potential − kBeφTe

Vd , V (a) Normalised potential at the surface of the dust grain VOM L

Normalised OML potential excluding electron emission

Vs

Normalised potential at the sheath edge

Vw

Normalised potential at the planar wall surface

∆V

Normalised potential difference

x

Cartesian coordinate

13

φ(b)−φ(a) kB Te

x , λD

normalised radial coordinate

r λD

X

Normalised x coordinate

y

Dimensionless parameter (field emission), Cartesian coordinate

w

Weight function (interpolation)

Wf

Work function

z

Cylindrical polar coordinate, Cartesian coordinate

Z

Number of charges per ion

Z1

Charge of incoming particle

Z2

Bare nuclear charge of substrate

Zd

Number of charges on the dust grain

α

Material dependent constant (radiative cooling)

β

m 2kB T

βi

mi 2kB Ti

γ

Heat transmission coefficient,

Γe

Electron flux

Γem

Emitted electron flux

Γi

Ion flux

Γpd

Particle flux

δ

Γem Γe

δ(r)

the Dirac delta function

δmax

Maximum secondary electron yield

∆φ

Potential difference φ(a) − φ(b)

ζ

Neutral drag constant

ζΓ

Numerical constant

η

Normalised potential

θ

Polar angle in spherical or cylindrical polar coordinates,

b2 a2

(secondary emission)

qφ kB T

ratio of ion to electron temperature λD , λDe

Electron Debye length

λDd

Dust Debye length

λDem

Emitted electron Debye length

14

Ti Te

λDi

Ion Debye length

Λ

Proportionality constant

µ

Ratio of ion to electron masses

Ξe

Electron energy flux

Ξi

Ion energy flux

Ξn

Neutral energy flux

Ξnet

Net energy flux

ρd

Density of the dust grain material

σ

Ratio of primary to emitted electron temperature

mi me

Te , Tem

Stefan’s constant (radiative cooling) σi

OML ion cross-section

φ

Azimuthal angle in spherical coordinates, electrostatic potential

φd

φ(a), the surface potential of the dust grain

φw

Electrostatic potential at the planar wall surface

χ

Ionisation potential

ψ

Polar angle for cylindrical coordinates in velocity space, angle

ωpe

Electron plasma frequency

ωpi

Ion plasma frequency

15

Chapter 1 Introduction Dust is ubiquitous in plasmas [2]. Its origin varies with environment. For example, in space it may be produced as a result of a collision between asteroids, whereas in a Tokamak it may be produced by large disruptions. Dust is composed of a range of elements depending on the environment, mainly water in Saturn’s rings, whereas it may be hydrocarbons and tungsten compounds in Tokamaks. As dust in a plasma is usually charged, its presence in large quantities will significantly alter the plasma parameters by localising large amounts of charge in the dust grain volume. The interaction between charged grains adds an extra level of complexity, which encourages some authors to coin the term ‘complex plasma’ for the three component dust-ion-electron plasma. I feel this is somewhat self-indulgent, as two component plasmas are also rather tricky! I will therefore refer to ‘dusty plasmas’.

1.1

Non-Fusion Dusty Plasmas

This thesis is primarily concerned with fusion plasmas, specifically Tokamaks, and therefore will not concentrate on the large amount of work conducted on dusty plasma crystals and space plasmas. However, these other main areas of dusty plasma interest will be summarised in the following sections.

16

Figure 1.1: Saturn’s rings; the dark features are “spokes”.

1.1.1

Dust in Atmospheric and Space plasmas

Space plasmas vary hugely, stars are extremely hot dense objects, whereas the interstellar medium has a temperature of only 3 K. There are therefore many interesting environments where dust exists in large quantities: planetary rings, planetary nebula, interstellar space and comet halos to name but a few. The interstellar medium is full of dust, either remnants of collisions between large objects, comet debris or leftovers from stellar and planetary formation. The particles are usually very diffuse and composed mainly of carbon. NASA routinely collects samples in the Earth’s stratosphere, and they are pebbles 5-20 mm in diameter [2]. Planetary rings present different dust environments. All the large gas giants have ring systems, produced either from satellites ripped apart by tidal forces, or meteor impacts. Saturn has the most famous ring system (figure 1.1), because the main rings are usually visible with a telescope. The particles are composed mainly of ice, but vary from microns to metres in size. The largest dust grains exist in Saturn’s main rings, the A and B ring. On the whole, they are uncharged since there is no plasma around them. The sheer amount of material in the A and B rings absorbs any ionised particles produced by photoionisation. The high electron mobility along 17

magnetic field lines perpendicular to the ring plane means that any electrons that escape from the ring plane bounce back from the poles and are absorbed. However, there is evidence of some charged dust particles levitating away from the B ring, from dark features called spokes which are observed to be moving with planet’s magnetic field. These are poorly understood. Clusters of small (nm sized) charged particles may exist in the faint E ring (not visible in figure 1.1), where there is a lower concentration of material. Comets are objects made of various dust materials, mainly carbon, mixed with frozen gases. They are thought to exist in a large cloud far outside the solar system, the Oort cloud. Those that enter the solar system evaporate as they near the sun, forming a coma of diffuse material. They develop two tails, an ion tail and a dust tail. Dust grains in the coma vary from hundredths of microns to microns in size. The Earth’s upper atmosphere is a plasma. Of particular interest to dusty plasma researchers is the mesosphere, where one observes noctiluscent clouds (NLCs). These form in the polar mesopause, the top of the mesosphere where the temperature is lowest, particularly during the summer. A peculiarity of the polar mesopause is that it is colder in the summer than in the winter [2]. NLCs are composed of sub-micron size ice crystals, and shine brightly enough to be observed at night. Interest in these has increased since the discovery of polar mesosphere summer echoes (PMSEs), unexpectedly strong radar echoes from the mesopause. Some theoretical work has been conducted on a possible mechanism for PMSEs involving charged dust. The dust is seen as a major charge carrier, and the charge density of the dust can vary significantly over a few metres [7]. This strong influence of dust on the plasma conditions may alter the wave-plasma interaction enough to explain the radar echo.

1.1.2

Dust in DC and RF Discharges

Discharges are used to generate a plasma for a variety of purposes. For example, they are used in industry for processing silicon wafers to manufacture microchips, or depositing shiny coatings onto crisp packets, in the home in the form of a fluorescent

18

Figure 1.2: A typical capacitative discharge setup.

light, or in specially designed experiments in academia. Discharges can be either dc, for example fluorescent lights, or rf where the frequency of the alternating current is usually 13.56 MHz. The plasma is formed inside a container in which electrodes are arranged to suit the purposes of the discharge. A simple example of a capacitive discharge circuit is shown in figure 1.2. Discharges have a wide range of applications when it comes to plasma processing. Production of integrated circuits involves repeated stages of deposition, masking, etching and stripping to make circuit elements such as transistors and capacitors [8]. The production of hundreds of such chips can be made from a small silicon wafer. The environment in which such fabrications take place is generally devoid of dust due to filtering of the air. However, sputtering events can still create dust particles that can ruin the process. Etching events are encouraged by placing the substrate on a biased electrode to accelerate ions towards it from the plasma. A mask is used if one needs to prevent the whole surface of the substrate being etched. The ion species can be chosen to react with the substrate to enhance the etching, and for this reason Cl and F ions are used for their high reactivity. Redeposition is used to produce thin film coatings to condition tools. Here, the film material is placed on a biased electrode to encourage sputtering, then ions are directed by other electrodes onto a substrate. As most discharges use a biased electrode, sputtering is encouraged. Sputtering 19

is an unpredictable process which produces dust as well as individual ions. It is observed that dust is produced in both rf and dc discharges, but that the former produces significantly more. The dust particles produced are usually sub-micron in size, and are an unwanted by-product. Rf discharges are also used for custom made dusty plasma experiments. These usually involve specially manufactured dust in order that the sizes are closely controlled. The behaviour of an ensemble of dust has been of interest since the discovery that dusty plasma crystals can be formed in experiment [9]. On Earth, where the discharge is orientated vertically, the dust is usually levitated in the sheath area above the lower electrode, as gravity is a significant force. However, there are also experiments conducted on the International Space Station. This produces microgravity conditions, and the dust fills the whole discharge, excluding an eye-shaped void in the centre. The void is thought to be a result of the potential distribution in the discharge. Hexagonal close packed and body centred cubic structures have be observed both on Earth and in microgravity. The interaction between the dust grains is poorly understood, although the analysis for most experiments assumes a Debye-H¨ uckel repulsive potential (section 2.2) around each dust grain. Many in the community also think that there is a attractive force between dust grains. It is extremely hard to design an experiment to look at the potential profile between two grains due to the fact that diagnostics such as Langmuir probes alter the plasma themselves. The kinetic behaviour of the phase transitions of coupled dust systems can also be seen by changing the pressure of the neutral gas in the rf discharge, and therefore the amount of ionisation. This reduces the charge on each dust grain, and therefore the electrostatic coupling, although the temperature of the dust ensemble is the same. The thermal motion overcomes the electrostatic forces, and liquid and gaseous phases can be observed.

20

1.2

Tokamaks: Introduction and Terminology

In Tokamaks, large quantities of dust can be produced during a shot by power flux to surfaces, especially during a disruption. The significance of this has two aspects: efficiency and safety. The first relates to erosion of components which as a result need regular replacement and conditioning, but also to impurities introduced by the possibility of dust evaporating near the core plasma. The safety aspect includes tritium retention in dust as a radioactive hazard, and the large surface area presented by the dust cloud results in an explosive limit. Furthermore, there is the possibility of cracking water molecules, producing explosive hydrogen atoms. Despite these facts, little work has been done on the subject of dust grain evolution and movement though plasma.

1.2.1

The Scrape-off Layer (SOL)

A Tokamak is characterised by its toroidal shape and helical magnetic field configuration. The toroidal part of the magnetic field is produced by external coils, and the poloidal part by currents in the plasma, the result being helical field lines. The mobility of charged plasma particles along the field lines is very high, whereas it is very hard for them to move perpendicularly. The result is that particles are quite efficiently confined, and travel on toroidal surfaces, which are called flux surfaces. However, the particles are not perfectly confined, so a limiter or divertor is used to prevent particle flux to the walls of the torus. A limiter is usually a rail along the outside of the torus which particles hit if they diffuse into the outer flux surfaces. A divertor is an object which has current passing through it, and the magnetic field produced draws the field lines into a configuration where the outer flux surfaces overlap a solid surface. In both of these cases, the solid surface which the particles hit can be specially designed to take the energy flux. The Joint European Torus (JET) uses a divertor which leaves the magnetic field lines in the configuration shown in figure 1.3. There is a zero in the poloidal magnetic field where the field lines cross over each other (the x-point), and 21

Figure 1.3: The magnetic field configuration of the JET Tokamak in a divertor phase.

all the flux surfaces inside these field lines are closed. The plasma inside the closed field lines constitutes the core. The outermost flux surface with field lines that do not touch the solid surface is called the last closed flux surface (LCFS) [10]. Because the particle motion is fast along the field lines and slow perpendicularly, particles that cross the LCFS are very swiftly transported to the divertor surface. This thin area of plasma outside the LCFS and extending to the surface is called the scrape off layer (SOL). The line separating the core from the SOL is known as the separatrix.

Typical divertor surface materials are tungsten, used because it has a low sputtering yield, graphite and carbon fibre composite (CFC). The disadvantage of tungsten is that, as it is a high-Z material, only a very small amount needs to reach the core plasma to radiate large amounts of energy. Carbon does not have this problem, and also has excellent thermal properties [11], but it is very reactive with hydrogen isotopes increasing the amount sputtered. Beryllium is also used for large wall areas being very low-Z, but it is a known health hazard if inhaled [12], and also reacts exothermically with water. This latter issue is important in the risk assessment of the case of a broken cooling line, where steam may be released into the vessel. 22

1.2.2

Plasma-Surface Interaction

The divertor wall acts as a sink for plasma particles. A sheath (section 2.3) is set up at the wall due to electrons collected from the plasma charging its surface. More electrons are collected than ions as they have higher mobility. Ions are then accelerated inwards by the sheath potential until they reach the sound speed. This creates a shock, and results in a thin space charge region next to the wall. Incoming ions recombine with electrons at the wall and are released thermally as neutrals. These then ionise via collisions in the plasma and travel along the field lines back to the wall. In most Tokamaks the particles undergo this process many times over the course of one shot [13]. This is referred to as recycling. Incoming ions gain a significant amount of energy from the sheath potential. The basic recycling mechanism accounts for some of this by adding thermal energy to the solid. However, very energetic ions have enough energy to free one or more atoms from the lattice. This process is called physical sputtering. If the plasma ion species are particularly reactive with the wall material this can enhance the sputtering yield. This process is called chemical sputtering, and is of interest in many Tokamaks which use graphite or CFC divertor plates, as hydrogen is particularly reactive with carbon producing hydrocarbons such as methane. Theoretical models for both chemical and physical sputtering are discussed in section 4.1.1 and Appendix A. Apart from sputtering, the other main dust creation mechanisms are evaporation and arcing. Evaporation requires the temperature of the wall to exceed the boiling/sublimation point of the wall material. Arcing is driven by large potential differences, for example the sheath potential. Usually, an arc is initiated at a sharp point where there is an enhancement of the electric field. This in turn produces a flow of current to a localised point. Joule heating results, and electrons are produced by field emission and thermionic emission [13]. The erosion produced by arcs is caused by evaporation due to Joule heating. Arcing is seen a lot in Tokamaks when the plasma is unstable, for example during disruptions. Dust grains may also grow whilst in the plasma [3]. Large concentrations of

23

particles near the wall increase the chance of multiple ion-molecule reactions and other agglomeration processes.

1.2.3

ELMs and Disruptions

Two features of Tokamak operation that result in large plasma and heat fluxes to surfaces are Edge Localised Modes (ELMs) and plasma disruptions. Any large fluxes result in increased erosion of material from surfaces, and may cause increased dust production. Arcing can also be induced. We now discuss these two mechanisms. Under certain conditions in Tokamaks, one can observe an abrupt transition to a mode of operation where the plasma confinement is much improved. The confinement time is usually increased by a factor of two compared to the low confinement mode (L-mode). Such modes are called high confinement modes (H-modes). Along with this new mode of operation one observes a new edge instability, the ELM. The instability causes the release of plasma in short bursts which decreases the density and temperature in the outer zone of the plasma. ELMs can produce unacceptably large heat fluxes to the divertor [13]. A disruption is a dramatic event in a Tokamak where confinement is destroyed [13]. A large disruption can be terminal, causing a complete loss of plasma current. The causes of disruptions are many and varied. As potentially all of the energy contained in the plasma is thrown against the walls of the device, this must be a major source of dust.

1.3

Dust in Tokamaks: A Literature Review

Having had a general discussion of the relevance of dust in Tokamaks, we can now discuss the experimental and theoretical work that has been published. Whilst this survey is not exhaustive, it should give the reader a good idea of the reality of the problem, and how well the community understands the problem in relation to the development of ITER, the next big experiment in nuclear fusion research.

24

Tokamak

Major Radius Minor Radius Magnetic Field Plasma Current

Fusion Power

25

(m)

(m)

on axis (T)

(MA)

(MW)

ITER

6.2

2.0

5.3

15

500

JET

2.96

2.5/4.2

3.45

4.8

16

MAST

1.0

0.8

0.5

2.0



ASDEX-Upgrade

1.65

0.5

3.1

1.6



TEXTOR-94

1.75

0.47

3

0.8



DIII-D

1.67

0.67

2.2

3



Alcator C-Mod

0.68

0.22

8.1

2.0



TFTR

2.65

1.1

5.5

2.7

10.7

Tore-Supra

2.25

0.70

4.5

1.7



NSTX

0.85

0.67

0.6

1.5



Table 1.1: Relevant Tokamak parameters. Magnetic field, plasma current and fusion power data are maximum values.

We will discuss a number of different Tokamaks in the coming sections. Here is a list to give the reader an idea of the size and location, and any interesting features. See table 1.1 for more information. • ITER (see www.iter.org): The next international reactor. ITER will be built in Cadarache, France, and the first plasma operation is expected in 2016. It will be a huge reactor, and it is hoped that it will achieve ignition (self-sustained heating). ITER means “the way” in Latin, and was referred to at one point as the International Thermonuclear Experimental Reactor, although for some reason the powers-that-be have decreed that this usage should be discontinued. • The Joint European Torus (JET, see www.jet.efda.org): The largest Tokamak in the world at present, based UKAEA Fusion, Culham, Oxfordshire, UK. It has achieved a maximum of 65% ratio of fusion power to total input power. • The MegaAmp Spherical Tokamak (MAST, see www.fusion.org.uk/mast): This is a spherical Tokamak, meaning it has a tight aspect ratio. It is also located UKAEA Fusion, Culham, Oxfordshire, UK. • The Axially Symmetric Divertor Experiment Upgrade (ASDEX-Upgrade): A medium-sized Tokamak at IPP in Garching, Germany. • The Tokamak Experiment for Technology Oriented Research (TEXTOR-94): A medium-sized Tokamak at IPP in J¨ ulich, Germany. • DIII-D: A medium-sized Tokamak at General Atomics, USA. • Alcator C-Mod: A small Tokamak at Massachusetts Institute of Technology (MIT), Boston, USA. It is notable for having a very large magnetic field. • The Tokamak Fusion Test Reactor (TFTR): A large Tokamak at Princeton Plasma Physics Laboratory, New Jersey, USA. Now out of use. • Tore-Supra: A large Tokamak at EURATOM-CEA, Cadarache, France. It has superconducting field coils, and an actively cooled first wall allowing longer discharges. 26

• The National Spherical Torus Experiment (NSTX): A spherical Tokamak at Princeton Plasma Physics Laboratory, New Jersey, USA.

1.3.1

Studies of the Erosion and Deposition of Wall Material

The wall material of a Tokamak can be eroded and transported into the plasma before being redeposited. Listed below are a few studies of this effect. • ASDEX-Upgrade: This was equipped with tungsten divertor plates for an experiment running for about 800 shots [14]. Some of these were then removed for analysis. The main chamber components showed a constant low tungsten contamination level. Peak values of tungsten redeposition, up to 40 times larger than in the main chamber, occurred in the divertor. The positions of the maximum redeposition coincide well with the overall peak values of deuterium flow for the whole experimental campaign. By comparison, the amounts of Fe, from steel components, and Cu redeposited in the same region were more than an order of magnitude higher. Surface analysis of the tungsten plates revealed only a 2% to 25% concentration of tungsten, with large amounts of boron, carbon and oxygen present. This is a result of various chemical reactions between components, the plasma, and the air in which the components are stored. Up to 60% of the deuterium inventory was found in the divertor, thought to be co-deposited with the low-Z material previously discussed. • JET: The JET mark I divertor was analysed after around 3500 shots [15]. It was constructed of carbon tiles, with the vessel walls being made of inconel 600 (75% Ni, 15.5% Cr, 8% Fe) conditioned with beryllium, and limiters made of CFC. A typical JET shot starts with a circular plasma controlled by limiters for around 10 s, followed by 10-15 s in a divertor configuration, and finally back to the limiter configuration before switching off. For analysis, sections of tiles in each area of the Tokamak were cut. Considerable erosion of metals from the inner wall and beryllium from the outer wall were measured, as well as carbon from the limiters. Redeposition of carbon was significant in all areas 27

apart from the inner wall. Less redeposition was observed for beryllium and metals in these areas. • TFTR: Flaking of redeposited layers has been observed in Tokamaks, including TFTR [11]. The plasma facing components are CFC (45%) and graphite tiles. Co-deposited layers of hydrogenated carbon tens of microns thick form on a lot of the plasma facing tiles. Flaking was observed on 15% of the tiles. After baking in air, a significant amount of tritium was recovered. Possible mechanisms proposed for the flaking are all due to catalyzed water absorption causing the layers to swell and detach. • MAST: Heat loads on surfaces in MAST have been calculated using an IR camera [16]. A numerical factor needs to be estimated to account for surface effects. During ELMs, the value of the surface factor is seen to increase, suggesting increased erosion. Indeed, it has been suggested that dust levitating above the surface of the tile could cause the camera to measure a different temperature to what one may expect, due to the size of the dust relative to the camera pixel size. The same camera has also observed dust production during disruptions (see figure 1.4), where it has been estimated that dust could be travelling as fast as 1 kms−1 .

1.3.2

Characterisation of Dust and Flakes

Out of all the environments discussed so far, dust in Tokamaks probably has the least information published. This is probably due to the fact that analysis of the plasma facing components is only possible when the Tokamak has not been operated for a while. However, there are characterisation studies that can give us an idea of the size, shape and composition of the dust. • JET: Dust was collected from JET’s MkIIa divertor approximately 2000 shots after its installation [17]. The divertor materials were inconel and CFC, with

28

Figure 1.4: Dust production in MAST observed using an IR camera [1]. the main chamber also containing beryllium. Loose dust was collected by a cyclone vacuum cleaner, the finer particles wiped of the plates afterwards with a cloth. The median diameter of these particles was 27 µm. Flakes of redeposited material were also collected from areas shielded from the plasma flux. There was too little loose dust to analyse, but the cloth samples and flakes showed that the majority of material present was carbon. The smears were 97% carbon, 1% beryllium and 2% metals by weight. The samples were radioactive, containing tritium and 7 Be. The flakes contained 99% carbon, with half a percent of both beryllium and the metals. There was a deuterium/carbon ratio of 0.4 (deuterium is ignored for the composition given previously). Flakes were around 40 µm in thickness. A significant amount of tritium was also found, with most being in the flakes. JET was again analysed in 2005 [18] specifically to examine the flakes formed. Flakes appear to be efficient absorbers of tritium and other hydrogen isotopes. They were found to have approximately 3-3.3 mg of tritium per gram of flakes. The conclusion was that the bulk of the tritium inventory missing after gas cleaning was in the flakes. This highlights the fact that tritium retention in dust and flakes is a serious problem for a fusion reactor. 29

Figure 1.5: A scanning electron microscope image of dust in TEXTOR-94 [2, 3].

• TEXTOR-94: The Tokamak was opened for analysis 3120 shots into a new experimental campaign [3]. A vacuum cleaner was used to collect dust. A coarse fraction of particles was obtained by shaking them off the vacuum cleaner’s sticky bag. Samples of smaller sizes were obtained by rinsing sections of the bag in 2-propanole. The main part of the coarse fraction contained irregularly shaped dark or whitish particles with typical diameters of 0.1-0.5 mm, shown in figure 1.5. 15% were ferromagnetic. Perfect spheres are found in this size range as well as redeposited low-Z layers up to 0.05 mm in thickness. The smaller particle samples were also observed to be partly ferromagnetic. Many of these particles had structures consistent with plasma-induced growth. • TFTR, DIII-D and Alcator C-Mod: Another study in 2000 [19] looks at dust samples taken from these Tokamaks. Samples were vacuumed from various areas of the plasma-facing surface, but also taken from windows attached to diagnostic pipes in TFTR. The count-based median diameter (CMD) was calculated. The CMD of dust in DIII-D ranged between 0.46 and 1.0 µm, compared to between 0.78 and 2.89 µm in Alcator C-Mod. They quote geometric standard deviations (GSD) of 1.42-4.39 µm, and 1.33-3.43 µm respectively. For TFTR the range was 0.76-3.00 µm, with GSDs ranging between 1.65 and 2.73 µm. All these results suggest particles on the micron scale are most important. The composition of the dust was directly related to the first wall material 30

used in the machine, this was carbon in TFTR and DIII-D, but molybdenum in Alcator C-Mod. The study concluded that dust was primarily found in the lower regions of the Tokamaks on horizontal surfaces protected from the plasma. Measurements of the dust structure concluded that the dust surface area was greater than one would expect for a spherical particle, suggesting particles have irregular shapes, or that they are agglomerates of smaller particles. • Tore Supra: Similar analysis has also been carried out for Tore Supra [20]. The samples were obtained in a number of different ways, vacuum cleaning, using cotton swipes, scratching surfaces with a metallic blade, and locally washing the vessel with water and passing the solution through a filter. The dust was found to be largely located at lower parts of the vessel, presumably due to gravity. The geometrical mean diameter was found to be rather uniform, on average 2.7 ± 2.8 µm (again quoted with the GSD). This is similar to the median size of the dust in the Tokamaks discussed in the previous paragraph. • ASDEX-Upgrade: This has also been the subject of analysis [21]. The particle diameter averaged 3.33 µm in ASDEX-Upgrade, with dust particles composed mainly of the divertor material. A review of dust characterisation [22] summarises data collection for many Tokamaks. See table 1.2.

1.3.3

Safety and Dust Removal

Having discussed the experimental studies, it is worth looking at the theoretical work that has been published on the subject of dust in Tokamaks. Again, there is not a large amount, but there have been a number of interesting publications, which will be discussed in the following sections. J. Winter has written about challenge of dust and its sources and consequences [3, 23, 24]. He points out that the issues of safety have been addressed to some extent, 31

Tokamak

Lower Regions

Middle Regions

Upper Regions

Collected Mass CMD (µm) Collected Mass CMD (µm) Collected Mass CMD (µm)

32

(mg)

± GSD

(mg)

± GSD

(mg)

± GSD

DIII-D

3.34

0.66 ± 2.82

18.5

0.60 ± 2.35

4.20

0.89 ± 2.92

TFTR

32.0

0.88 ± 2.63

67.0

1.60 ± 2.33





Alcator C-Mod

40.2

1.58 ± 2.80

1.73

1.53 ± 2.80

0.48

1.22 ± 2.03

JET



27.0 ± (—)









TEXTOR



5.20 ± (—)









Tore Supra

14.3

2.68 ± 2.89

1.97

2.98 ± 2.94

0.55

3.32 ± 2.94

ASDEX-Upgrade

116

2.21 ± 2.93

2.54

3.69 ± 2.81

1.25

3.59 ± 3.08

Table 1.2: Dust characterisation for various Tokamaks.

whereas plasma performance issues have not. Safety issues highlighted include a dust-bound tritium inventory and radioactive dust becoming mobile in the case of an accident. In the event of a broken cooling line, steam could be cracked on the dust surface liberating H and CO, a possible explosive hazard. Other aspects include dust impeding the heat flow to cooling structures, and blocking gaps left for engineering reasons such as electrical insulation. These things may not be critical for this generation of Tokamaks, but the thermal load predicted in a disruption for ITER is 100 GWm−2 . Significant amounts of materials such as carbon will vaporise under this sort of load. Light particles produced during previous shots may be levitated by their interaction with transient electric or magnetic fields. Winter has also highlighted the role of radioactive decay in both inducing plasmas, and in charging the dust [25]. In an experiment designed to test this theory, radioactive CeO2 dust with a mean diameter of 1 µm was seen to levitate above a biased electrode. The charge on the dust was estimated to be between 200 and 400 electrons. Therefore, an option for the removal of dust, if this charge is acquired, is with a positively biased electrode, or other appropriately shaped electric fields. We will see later that charging by radioactivity is probably only a significant charging mechanism outside the plasma. A method considered for removing dust from surfaces is vibratory conveying (VC) [26]. Any method used needs to be robust and reliable, able to operate in a vacuum and high magnetic fields. VC is a simple method consisting of a trough supported by springs and vibrated at an appropriate frequency. Tests have been conducted, with surrogate particles to avoid handling radioactive materials, and show that the method works well horizontally and downwards, as well as up slopes to a maximum of 15◦ . The implication of the size of dust particles and their surface area has been considered [27]. The study identifies the major safety issues to be the radiological hazard (containing tritium), toxicity and chemical reactivity. The radiological and toxic hazards depend on how much is produced and how well it is confined, the chemical hazard depends on the size of the dust. Smaller dust grains can get into 33

the air more easily, however, they probably contain less of the hazard material than larger grains. Chemical reactivity is dependent on the surface area of the dust particle population. As discussed previously, the actual surface area of the dust can be very different to that expected for a spherical particle. A dust-air mixture can be treated as an industrial aerosol, so that the effective mass of particulate in the air and its overall surface area can be studied by looking at the CMD and GSD of the grain population. The effective surface area is minimised when both the CMD and GSD are large. This simply states that a fine powder has a greater surface area than a coarse powder, whilst a large GSD results in minimising the effect of fine particles in a distribution, as the spread of particle sizes is greater. From the data discussed in the previous paragraphs, the specific surface area of the dust grain distribution is found to be large enough to be a concern with respect to tritium retention etc., but for all these considerations it is clear that one needs to understand the particle distribution.

1.3.4

Dust Production and the Effect on Plasma Operation

In another publication [28] Winter discusses the effect of dust on plasma operation. He points out that heavy and sudden plasma-wall interaction leads to the liberation of particles. For example, disruptions are particularly important, as these result in plasma flux to previously unexposed surfaces where dust has been allowed to accumulate. Redeposited films are particularly susceptible to the stresses of plasma flux, have very poor thermal conductivity and can break away from the wall easily forming flakes of material. Arcing is also listed as a mechanism for the creation of spheres of material. The currents produced by arcing can liberate drops of molten metal, which on cooling would result in spherical dust particles. The conditions at the edge of a fusion reactor can be compared to the conditions produced in a plasma used for materials processing. Thus it is thought that particles can grow from atomic or molecular precursors, in particular from hydrocarbons produced by chemical erosion. In situ laser scattering experiments in TEXTOR-94 have identified

34

dust before and after shots, and it is thought likely that the dust is repeatedly interacting with the plasma. Predictions of dust production in ITER have been made by a comprehensive safety analysis code [29]. The code can calculate plasma dynamics and thermal characteristics in the Tokamak simultaneously. It simulated both beryllium and carbon plasma facing components separately. The results showed that beryllium dust production could be between 7 and 10.3 kg/disruption, whereas for carbon it was 1.9 to 2.4 kg/disruption. The authors comment that carbon production could be between 2 and 5 times larger than predicted due to a large uncertainty in its latent heat. These are unacceptably large amounts.

1.3.5

Dust Dynamics

There has been consideration of dust grain motion in the edge region of a Tokamak [4, 30, 31]. It is worth paying special attention to these publications, as they are the closest equivalent to the work discussed later, and the dust transport model has been developed almost in parallel to ours. The papers use a simple charging model. The vacuum estimate of the dust grain charge (Zd ) from the surface potential φd = Zd e/(4πǫ0 a) is used, along with the fact that we know that the potential on the dust grain will be ≈ kB T /e. This leads to a relation between the number of charges on the dust grain Zd , and the plasma temperature Ti = Te = T

e2 Zd = ΛT, a

(1.1)

where a is the dust grain radius. The proportionality constant Λ is a weak function of the ratio of electron to ion temperature, plasma flow, magnetic field etc. A value of Λ = 3 is chosen for the estimates, which is sensible when using the charging theory discussed in chapter 2. The particle flux (Γpd ) and energy flux (Qpd ) are estimated as Γpd = ζΓ πa2 n

p

T /mi

(1.2)

Qpd = Γpd (γT + χ),

(1.3)

35

where mi is the ion mass, n is the plasma density, χ is the ionisation potential (13.6 eV for a hydrogenic plasma) and ζΓ and γ are inferred from experimental data. ζΓ is a numerical constant ≈ 5-6, and γ is the heat transmission coefficient, which is taken to be 6. The second of these equations has the energy gained by plasma bombardment and the energy gain from recombination of ions and electrons on the grain surface. Neutral bombardment is ignored, as it is expected to be small. By balancing these incoming energy fluxes with cooling processes including evaporation and radiation, the dust grain temperature Td can be calculated. For a T =10 eV plasma with n = 1018 − 1020 m−3 (typical SOL conditions), it is found that for a graphite dust grain, Td ranges between 2000 and 4000 K. The dominant erosion mechanism is chemical sputtering for Td < 1000 K, radiation enhanced sublimation for 1000 K < Td < 3000 K, and evaporation for Td > 3000 K (this transition occurs at around the sublimation point of carbon). We discuss these erosion mechanisms in detail in chapter 4. For n ≤ 1019 m−3 , a micron radius particle can survive for the whole shot at this temperature, but its survival time for densities of around 1020 m−3 is about 10−2 s. This is long enough to travel large distances around the Tokamak. The forces on the particle are then considered to model the motion. We do not cover this in detail, as this is discussed in chapter 5. Among the forces considered are the electric force, gravity, the magnetic force caused by the presence of material such as iron or nickel (not v × B), the rocket force, and the plasma-dust particle frictional force. It is shown that the magnetic force and gravity dominate for large particles, however, for the standard micron-scale particle observed in experiment, these forces are less important. The rocket force is also estimated to be small, so that the dominant forces are friction and electric forces. In a magnetised sheath, the components of the friction and electric forces normal to the wall point in opposite directions, so there exists an equilibrium point some distance from the wall. In the direction parallel to the wall, the friction force can accelerate the particle. While travelling 1 cm, it is estimated that a particle can be accelerated to a velocity of 30 ms−1 in the radial direction, or 100 ms−1 in the 36

Figure 1.6: The effect of surface corrugation on the trajectory of a dust particle [4]. toroidal direction. These are large enough velocities to penetrate the sheath and hit the wall. The motion of a dust particle confined in this potential profile is affected by surface corrugation (see figure 1.6). This may cause scattering such that it is directed towards the plasma. It is found that sinusoidal 2-D surface corrugations can cause large excursions, around 10 cm, of the dust particle from the sheath area into the plasma volume. Assuming that the characteristic scalelength of the wall corrugation is much greater than the ion gyroradius, it is predicted that the dust has to be accelerated by the plasma for a distance of around 20 cm along the surface before a significant excursion can occur. At the velocities discussed previously, if ejected from the sheath into the plasma, a particle could travel a significant distance. Estimates of transport of dust to the separatrix results in a dust density of 104 m−3 , and it is suggested that dust may be a source of core plasma contamination. However, this concentration is too low to verify easily by experiment. In the latest paper, dust transport in NSTX and ITER is modelled [31]. In NSTX, 10 µm radius carbon dust is injected at 10 ms−1 , and is is found to be accelerated considerably by the plasma, up to 1 kms−1 . Reflecting boundary conditions are used, meaning that the particle can bounce off the walls. Particles are observed travelling significantly beyond the separatrix. For injection velocities greater than 100 ms−1 , dust trajectories are observed to be straight, less than this and they are affected by the force of the plasma. In ITER, both 1 and 10 µm radius grains are injected, at 1, 10 and 100 ms−1 . The dust is observed to evaporate mainly near the divertor, although the trajectories are observed to pass the last closed flux surface. Particles are observed to penetrate the plasma more easily from the inboard divertor 37

than the outboard.

1.3.6

Afterthoughts

Work in this field draws on a lot of experience from materials science, dusty plasma physics and Tokamak physics. The field itself has as many questions as you wish to ask. The plasma dynamics in the scrape-off layer are not fully understood, nor is the plasma chemistry, nor the plasma-material interaction. Where does the dust go during the shot? Where will it be found afterwards? How far can dust particles get into the plasma? The dynamics of a single dust grain in a Tokamak plasma is an interesting and difficult problem in itself, which has only recently started to be addressed. But what will happen when there are many? The information presented in the experimental papers that have been discussed earlier have shown that dust is a huge problem. Of course, anything observed in today’s Tokamaks will most probably be magnified for ITER. This point is emphasised by the predictions from the safety code [29]. Winter’s considerations may be interesting now, but for ITER they will probably become critical.

38

Chapter 2 Dust Grain Charging One of the fundamental properties of a dust grain immersed in a plasma is its charge. To obtain this, one usually needs to find the floating potential, the potential at which the electron and ion currents to the grain cancel each other exactly. There is no universally accepted theory for this. However, it is fundamental when calculating the energy fluxes to and from the grain (chapter 4), the interaction between dust grains, and the motion of a grain through electric and magnetic fields (chapter 5). To add a further complication, in a plasma there is no easy relationship between the dust grain potential and the dust grain charge as there is for a charged sphere in a vacuum. The following mechanisms contribute to the charge of a dust grain immersed in a plasma • Collection of charged plasma particles. • Secondary emission. • Thermionic emission. • Field emission. • Photoemission. • Radioactivity. 39

The main charging mechanism is usually the collection of charged plasma particles, as with the plasma-surface interaction discussed in section 1.2.2, although this depends on the plasma environment. Due to their high mobility electrons are collected in greater numbers than ions, resulting in a net negative charge. Ions are accelerated towards the dust grain and a sheath is formed. The dust grain reaches its floating potential when the electron current and the ion current cancel each other exactly. The effect of the electrons defining the charging process means the floating potential is found to be of the order kB Te /e. Assuming this, there are basically two approaches to finding the floating potential of a spherical object immersed in a plasma: radial motion theory (section 2.4) and orbital motion theory (section 2.5). Before explaining these theories it is pertinent to introduce the Maxwellian distribution in spherical geometry, and the concept of the Debye length. The other charging mechanisms will be discussed in section 2.6 with reference to their relevance to fusion, in order to find an appropriate model to use in the later work.

2.1

The Maxwellian Distribution

The Maxwellian velocity distribution is   32  β exp −β(vx2 + vy2 + vz2 ) , f (v) = n π

(2.1)

where

m , (2.2) 2kB T where m is the particle mass, T is the temperature, n is the density and kB is Boltzβ=

mann’s constant. In some cases it is useful to use a speed distribution rather than a velocity distribution. We define speed v in terms of the three velocity components as v = |v| = We can write ZZZ

f (v)dv =

q

vx2 + vy2 + vz2 .

ZZZ 40

f (v)v 2 sin θdvdθdφ.

(2.3)

(2.4)

Assuming spherical symmetry, we integrate across θ and φ to obtain Z Z f (v)dv = f (v)4πv 2 dv.

(2.5)

Using (2.1) and (2.5), this yields the speed distribution   23  β v 2 exp −βv 2 . f (v) = 4πn π

2.2

(2.6)

The Debye-H¨ uckel Potential

One of the most important properties of a plasma is its ability to shield the effect of a charge immersed in the plasma. This means that the potential associated with the charge drops away quickly with distance. We present a derivation of the form of this screened potential. Consider a plasma with a Boltzmann distribution of electrons.   eφ(r) , ne (r) = n0 exp kB Te

(2.7)

where we adopt the conventions that a suffix 0 denotes a property of the bulk plasma, and the subscripts e and i properties of the plasma electrons and ions respectively. e is the magnitude of the electron charge and φ is the potential. We ignore the effect of the charge on the ion population as they are far less mobile than the electrons, and assume the ions have a uniform charge density. Another assumption is quasineutrality n0 = Zni ,

(2.8)

where Z is the number of charges per ion. Poisson’s equation for this situation is (Zni − ne )e ǫ0    eφ n0 e = − 1 − exp . kB Te ǫ0

∇2 φ = −

We make the weak coupling assumption eφ ≪ kB Te , and assume spherical symmetry to yield 1 d r2 dr



r

2 dφ

dr



41

≈−

φ , λ2D

(2.9)

where the characteristic scale λD =

r

ǫ0 kB Te n0 e2

(2.10)

is called the Debye length. For a sphere of radius a, with surface potential φ(a), the solution is the Debye-H¨ uckel potential a φ(r) = φ(a) exp r



 a−r . λD

(2.11)

It is worth mentioning that the weak coupling approximation is not valid for dust, where eφ ≈ kB Te , but that the Debye length is still a useful reference length. Many texts assume a Boltzmann distribution of ions. This results in the dust Debye length λDd 1 1 1 = 2 + 2 , 2 λDd λDe λDi

(2.12)

where λDe is the (electron) Debye length from above, and λDi is the ion Debye length q ǫ0 kB Ti . In most plasmas this results in the shielding length being dominated by n0i (Ze)2 the ion Debye length, as the ion temperature is generally significantly lower. It is

unlikely that ions will have a Maxwellian distribution around a highly charged object, as electrons adjust to changes in potential far more quickly. Therefore they will always be responsible for the length scale of the shielding, unless there is significant depletion of the electron population. λD will refer to the electron Debye length in the following sections. We can estimate the dust grain charge as follows. At the surface of the dust grain, the electric field is E(a) =

qd , 4πǫ0 a2

(2.13)

where qd is the charge on the dust grain. Using the Debye-H¨ uckel potential (equation 2.11) and differentiating, we find φ(a) E(a) = a



a 1− λD



.

(2.14)

From equations 2.13 and 2.14, we find as a/λD → 0 qd ≈ 4πǫ0 aφ(a). 42

(2.15)

For both MAST and ITER, the Debye length is tens of microns. This is due to the fact that ITER has both a greater density and temperature than MAST, which results in the Debye length being around the same as in MAST. As the dust grains we consider are generally a micron radius, this is smaller than the Debye length, and the approximation in equation 2.15 can be justified.

2.3

Planar Wall

Before addressing the question of the floating potential of a dust grain, we will look at what happens to a planar surface in contact with a plasma. The reason for doing this is that we will obtain some useful results to compare with the spherical case. Furthermore, the interaction between a planar wall and a plasma happens at the divertor. We ignore the magnetic field for simplicity, in reality this changes the plasma transport in the presheath, and may change the shape of the sheath. This procedure will also help the proof of the Bohm criterion. −1 On the electron timescale ωpe (the inverse of the electron plasma frequency) the

wall charges up by collection of electrons. The ions respond more slowly (timescale −1 ωpi ) and set up a shielding cloud. The wall reaches a steady state at potential

φw , as the ions are accelerated towards the wall until the ion and electron fluxes balance. It is found that the situation can be split up into two separate regimes on different scalelengths: a large presheath (quasi-neutral plasma) and a very thin space charge region called the sheath [32]. This is a convenient simplification as Poisson’s equation only needs to be solved for the sheath region with easy boundary conditions. The presheath derivation is presented first to demonstrate this.

2.3.1

Presheath

A simple derivation of the steady state isothermal presheath will be presented as done by Stangeby [10]. We assume singly charged ions, quasi-neutral plasma (ni = ne = n), ambipolar outflow (vi = ve = v) and that ions and electrons are produced

43

at the same rate Sp , resulting in the continuity equation d(nv) = Sp . dx

(2.16)

We assume that electron-ion pairs are produced at rest and therefore produce an effective drag on the fluid. From the kinetic energy of a fluid, we see that d d dv (nmv 2 ) = mv (nv) + nmv dx dx dx dv = mvSp + nmv . dx

(2.17)

Therefore, the momentum equations for the electron and ion fluids are dv dpe = −neE − − me vSp dx dx dpi dv = neE − − mi vSp , nmi v dx dx

nme v

(2.18) (2.19)

where pe,i = nkB Te,i are the electron and ion pressures, and we assume the situation is isothermal. This is a 1-D problem, so we only have to consider the direction perpendicular to the wall, in this case the x direction. The terms proportional to me are negligible, so the equation for the plasma fluid is nmv

dn dv = −kB (Ti + Te ) − mvSp dx dx dn − mvSp , = −mc2s dx

(2.20)

where m = mi +me ≃ mi , and c2s = kB (Ti +Te )/mi is the sound speed. We introduce the plasma Mach number M = v/cs and rewrite in terms of

dM dx

using the continuity

equation to give Sp (1 + M 2 ) dM = . dx ncs (1 − M 2 )

(2.21)

Equation 2.21 clearly has a singularity for M = ±1, and this is where the assumptions break down. This places a condition on the speed of the fluid in the presheath, which is v2 ≤

kB (Ti + Te ) = c2s . mi

(2.22)

We show in the next section that only the equality is satisfied (the Bohm criterion). 44

Of course, where the presheath breaks down is in the space charge region, the sheath. The sheath is on a different scale length, so we are effectively ‘zooming in’ on the singularity in the next section. But what is the potential at the edge of the sheath relative to the bulk plasma? The conservative form of the momentum equation in this case is [10] d dpe dpi + + (nmv 2 ) = 0 dx dx dx → pe + pi + mnv 2 = constant.

(2.23)

Inserting pe,i = nkB Te,i , and equating with the pressure in the bulk plasma gives n=

n0 . 1 + M2

(2.24)

At the edge of the presheath M = ±1, and the electrons obey the Boltzmann law in the retarding potential of the wall, so equating (2.7) with (2.24) yields φs = −

kB Te ln (2), e

(2.25)

where the subscript s denotes the sheath edge. The previous results apply to an isothermal presheath, but there are actually a number of different assumptions one could make about the presheath (see [10]), and this results in a different density and potential at the sheath edge. For example, for a collisionless presheath, the potential can be found from conservation of energy using the Bohm criterion. For cold ions this is φs = −

1 kB Te . 2 e

(2.26)

The plasma concentration at the edge of the sheath in the collisionless case is therefore n0 exp(− 12 ). It is perhaps interesting to dwell on these solutions, as we have assumed that the presheath can reach a steady state. Is this assumption valid? If we derive equation 2.21 for a collisionless presheath (Sp =0), we find (M 2 − 1)

dM = 0. dx

45

(2.27)

This implies that either

dM dx

= 0 or M = ±1. From the sheath solution (section

2.3.2), we find that we must select the second choice, and this implies that strictly there is no steady state solution to the presheath unless the whole presheath is moving with at Mach 1 towards the wall. The isothermal case is more realistic, but there must be a condition on the size of the source term such that plasma is produced in the volume at the correct rate to compensate for wall losses. We assume that the source term is proportional to the plasma density (Sp = Cn), which is reasonable if plasma is produced by electron-neutral collisions. We can integrate equation 2.21 Z x Z M C 1 + M2 dM = dx, 1 − M2 0 cs 0

(2.28)

to give 2 tan−1 M − M =

Cx . cs

(2.29)

We now use the knowledge that over the system size l, the plasma must be accelerated to M (l) = 1 (we ignore the size of the sheath) to get Cl π −1= . 2 cs

(2.30)

So in order to reach a steady state, the plasma temperature will adjust such that the sound speed changes to satisfy this relation. This is observed in low temperature discharges.

2.3.2

Sheath

Here we use the ion momentum equation ni m i v i

∂vi ∂φ ∂pi = −ni e − , ∂x ∂x ∂x

(2.31)

where mi is the ion mass, and we have assumed that the ions are singly charged and isothermal. This can be rearranged to give   ∂ 1 2 mi vi + eφ + kB Ti ln (ni ) = 0. ∂x 2 46

(2.32)

At the sheath edge (i.e. the sheath/presheath boundary), we define ni = ne = ns , φ = 0 and vi = vs . Thus integrating 2.31 gives 1 1 mi vi2 + eφ + kB Ti ln (ni ) = mi vs2 + kB Ti ln (ns ). 2 2

(2.33)

We assume the source term Sp is negligible as the sheath is so thin, so ion continuity is simply ni (x)vi (x) = ns vs .

(2.34)

The electrons are assumed to satisfy the Boltzmann law   eφ(x) . ne (x) = ns exp kB Te

(2.35)

Finally we use Poisson’s equation d2 φ (ni − ne )e =− . 2 dx ǫ0

(2.36)

The following normalisations will be used V

= −

eφ kB Te

ne,i,s n0 x X = λD vi M = cs Ti , θ = Te

Ne,i,s =

(2.37)

so the equations become M

2

Ni

2 = + 1+θ N s Ms = M Ms2



V − ln



Ni Ns



(2.38) (2.39)

∂2V = Ni − Ne . ∂X 2

(2.40)

Poisson’s equation in normalised form becomes ∂2V = Ns ∂X 2

 1+

2 (1 + θ)Ms2



V − θ ln 47



Ni Ns

− 21

!

− exp (−V ) .

(2.41)

For cold ions (θ = 0), one can multiply both sides of equation 2.41 by ∂V = 0 to give integrate using the boundary conditions Vs = ∂X s 

∂V ∂X

2

= 2 Ms2

"

2V 1+ 2 Ms

 12

#

!

− 1 + [exp (−V ) − 1] .

∂V ∂X

and

(2.42)

The right-hand side of this equation can be expanded for small values of V around the sheath edge to give 

∂V ∂X

2

≈ Ns V

2



1 1− 2 Ms



+ O(V 3 ).

(2.43)

As the left hand side must be positive, this places a condition that Ms2 > 1. This gives the θ = 0 Bohm criterion vs2 ≥

kB Te . mi

(2.44)

More generally for non-zero θ we can return to equation 2.41 and perform a similar expansion as before to yield     Ni 1 ∂2V + O(V 2 ). V − θ ln ≈ Ns V − 2 2 ∂X (1 + θ)Ms Ns

(2.45)

However, close to the sheath edge Ni ≈ Ne = Ns exp (−V ), which gives the following result ∂2V ≈ Ns V ∂X 2



1 1− 2 Ms



+ O(V 2 ).

(2.46)

The right-hand side must be positive to avoid oscillatory solutions for the potential which are considered unphysical [32]. The Bohm criterion is therefore more generally vs2 ≥

kB (Ti + Te ) . mi

(2.47)

Comparing this with (2.22), we see that the equality must be satisfied at the sheathpresheath interface, i.e. the ions are moving at the sound speed at the sheath edge. The potential at the wall can also be found. From ion continuity, the ion flux reaching the wall is that reaching the sheath edge s kB (Ti + Te ) . Γi = n s v s = n s mi 48

(2.48)

The electrons are Maxwell-Boltzmann all the way to the wall surface, so the one way flux at the wall is 1 Γe = ns exp 4



eφw kB Te



8kB Te πme

 21

.

(2.49)

where φw is the wall potential. Equating these and solving for φw yields [10, 33]    kB Te Ti 2πme φw = 1+ . (2.50) ln 2e mi Te

2.4

Radial Motion (ABR) Theory

We continue our analysis by considering a spherically symmetric case. ABR theory is named after its inventors Allen, Boyd and Reynolds, and considers ions with purely radial motion [5, 34, 35, 36]. In a repulsive potential, as for the planar case, the electrons can be assumed to obey the Boltzmann law. Therefore, the problem is to find ni (r). Making the assumption of collisionless ions, we look at an ion approaching a grain, of radius a centred at r = 0, from the bulk plasma at speed vi′ . Conservation of energy is 1 mi vi′2 = 2 However, we will assume the ions in

1 mi vi2 (r) + eφ(r). (2.51) 2 the plasma are cold, that is vi′ ≈ 0, the ions

have no angular momentum and therefore move radially. Therefore, the velocity at coordinate r can be related to the potential as s 2eφ(r) . vi (r) = − mi

(2.52)

Whereas in the planar case we had ion continuity (constant ion current density), now we have constant ion current. The current through a spherical shell of radius r is Ii (r) = 4πr2 ni (r)evi (r).

(2.53)

Combining equations 2.52 and 2.53 gives ni (r) =

Ii (r) q . 4πr2 e − 2eφ(r) mi 49

(2.54)

1.0 0.8

V

0.6 0.4 0.2 0.0 1.4

1.6

1.8

2.0

2.2 X/J

2.4

2.6

2.8

3.0

0.5

Figure 2.1: The ABR plasma solution.

Poisson’s equation for the spherically symmetric case is   (ni − ne )e 1 d 2 dφ r =− . 2 r dr dr ǫ0

(2.55)

We use the same normalised variables V , X (now = r/λD ) and Ne,i as in the planar case (equation 2.37), and add the following normalised variables Ii q 2 4πλD n0 e 2kmBiTe a A = . λD J =

Poisson’s equation becomes   1 d 2 dV X = JV − 2 − X 2 exp (−V ). dX dX

(2.56)

(2.57)

The quasi-neutral ‘plasma solution’ is obtained by assuming the left-hand side of Poisson’s equation is negligibly small compared to the terms on the right hand side (figure 2.1). The plasma solution has an infinite gradient at V = 0.5, the collisionless presheath edge potential. It is also multi-valued, but the upper branch is considered to be unphysical. To find the potential at the surface of the object we need to integrate inwards from a point on the plasma solution. The plasma solution 50

(ne = ni ) relates X to J, and by differentiating the plasma solution, we can also relate dV /dX to J, giving us the two conditions we require  1 J 2 exp V2 X = 1 V4 3 dV 2XV 2 exp (−V ) . = dX J(V − 12 ) This point at which we begin the integration has to be chosen carefully in order that the plasma solution is valid. The condition can be put into concrete form as [5] 4V (2V − 3)(2V + 1) J ≪ 1. 3 (2V − 1) V2

(2.58)

We are interested in the floating potential, and therefore we need to find the floating condition Ii (a) + Ie (a) = 0. The electron current is  1   eφd 8kB Te 2 2 exp , Ie (a) = −πa n0 e πme kB Te

(2.59)

where φd = φ(a) is the dust grain surface potential. Using this and Ii from earlier and normalising, we obtain J = A2

r

mi exp [−Vd ]. 4πme

(2.60)

One chooses a value of J, finds a value of X for which inequality 2.58 is satisfied from the boundary conditions, and integrates in until the floating condition is satisfied. The normalised potential Vd as a function of grain radius for ABR is plotted in figure 2.2. The potential for large grains correctly tends to the planar case. There are a number of drawbacks to ABR theory. Firstly, ions are assumed to have no angular momentum. For a highly biased spherical probe, this is probably quite accurate, but for a dust grain with a low potential, angular momentum will be important. Secondly, integrating from deep into the plasma is a laborious process, and the thing you want to know is the potential for a given radius, not current. If the potential for a specified radius is required, one has to use a shooting code to try different values of J. Thirdly, solving the boundary conditions and checking they are far enough away from the grain adds an extra unwanted complexity to the solution that does not exist for the planar case. 51

3.5 3 2.5

Vd

2 1.5 1 0.5 0 0.01

0.1

1

10

100

1000

A

Figure 2.2: ABR normalised potential Vd as a function of normalised grain radius A for hydrogen.

2.5

Orbit Motion (OM) Theory

Orbit Motion theory, unlike ABR, treats the case of hot ions. The simplest case is orbit motion limited (OML) theory, which prescribes the limiting ion orbit that is collected. This results in a simple momentum equation allowing one to calculate the potential on the grain without solving Poisson’s equation. The theory itself is deceptively simple, and does have pretty severe limitations, as will be discussed later. Full OM theory, which self-consistently solves Poisson’s equation, is extremely laborious.

2.5.1

OML theory

Consider the same dust grain as in the previous section. The electrons again satisfy the Boltzmann law. We look at an ion approaching a grain from the bulk plasma at the edge of its shielding cloud (r = ∞) at speed vi′ , and hitting a grain with surface potential φd with speed vi . 52

Figure 2.3: Ion trajectories towards a grain in OML theory.

An impact parameter b is defined as in figure 2.3. We assume that there are no collisions with particles in the shielding cloud. Conservation of energy and angular momentum equations are as follows 1 1 mi vi′2 = mi vi2 + eφd 2 2 mi vi′ bc = mi vi a,

(2.61) (2.62)

where we have defined a critical impact parameter bc for which ions have a grazing collision with the grain. Rearranging (2.61) to solve for vi and inserting into (2.62) yields 

2eφd bc = a 1 − mi vi′2

 21

.

The cross section for collection of ions is therefore   2eφd 2 2 . σi = πbc = πa 1 − mi vi′2

(2.63)

(2.64)

For a given speed distribution fi (vi′ ) in the bulk plasma, we can now integrate to find the ion current to the grain Ii = e

Z



σi fi (vi′ )vi′ dvi′ .

0

53

(2.65)

4.5 4.0 3.5

V

d

Argon

3.0

Hydrogen

2.5 2.0 1.5 1.0

-2

-1

10

0

10

10

1

2

10

10

Figure 2.4: OML floating potential against θ for different gases.

We assume Maxwellian distribution. The current can be expressed as follows Ii

3 Z  βi 2 ∞ σi vi′3 exp −βi vi′2 dvi′ = 4πn0 e π 0    23 Z ∞   2eφd ′ βi ′3 2 2 vi exp −βi vi′2 dvi′ . vi − = 4π a n0 e π mi 0 

After integrating, the current is [37] Ii

 23 

  vi′2 1 1 ′2 = 4π a n0 e exp (−βi vi ) − 2 − 2 βi βi ∞ 2eφd 1 − exp (−βi vi′2 ) mi 2βi 0  12    8kB Ti eφd 2 . = πa n0 e 1− πmi kB Ti 

2 2

βi π

(2.66)

We can determine the floating potential of the grain using the results from (2.66) and equating to the electron current from (2.59) 

Te me

 21

exp



eφd kB Te



=

 54

Ti mi

 12 

eφd 1− kB Ti



.

(2.67)

Figure 2.5: Ion trajectories: ions are absorbed if they are within the absorption radius [5]. This can be solved numerically for φd . A plot of normalised potential Vd versus θ = Ti /Te is shown in figure 2.4. Because of its simplicity, OML is the most widely used theory to calculate the floating potential of a spherical grain. However, it has many problems. The potential of the grain is independent of the grain radius, and therefore does not tend to the planar result for large values of grain radius. It has no consideration of the presheath or sheath. It considers a completely collisionless ion trajectory from deep in the plasma. Furthermore, the assumption of the limiting orbit being one that grazes the grain surface breaks down in many cases [38, 39]. This brings us to Langmuir’s concept of the absorption radius [40]. The absorption radius defines a theoretical spherical surface where the grazing orbit is the limiting orbit that an ion of a particular energy will hit the grain surface (figure 2.5). The reason that this may be larger than the radius of the grain will be due to the form of the electric field around the particle. The value of floating potential obtained by finding the ion distribution for this scenario and integrating Poisson’s equation will not match that obtained by the OML solution. Allen, Annaratone and de Angelis [39] have shown that this is always the case for parameters of interest, and therefore OML is not valid. The only cases where it will be correct is 55

for Ti ≫ Te , which is extremely unlikely, although it can happen in a fusion device.

2.5.2

Full OM theory

In order to obtain the correct result for the potential, one needs to solve Poisson’s equation. The theory has been reviewed extensively by Laframboise [37] and Kennedy [5], and only a brief review will be presented here. The concept of the absorption radius is explored further by Bohm, Burhop and Massey [41] who derive the ion density in the case of monoenergetic ions s r !  φ(r) r12 φ(r1 ) n0 φ(r) 1+ , − 1+ − 2 1+ ni = 2 E E r E

(2.68)

where r1 is the absorption radius, and E is the energy of the ions. To find the   eφ(r1 ) potential at r = r1 , the absorption radius, they assume ni = n0 exp − kB Te and find



eφ(r1 ) exp − kB Te



1 = 2

r

1+

φ(r1 ) . E

(2.69)

Bernstein and Rabinowitz [42] derive an expression for the ion density for an arbitrary distribution in the plasma, but then also go on to find the monoenergetic ion solution. They find the full solution to Poisson’s equation using this distribution. Laframboise [37] went on to solve the problem for Maxwellian ions, and tabulated his results for larger objects. However, results for dust grains with radii significantly smaller than the Debye length were not tabulated, presumably due to computational difficulties. These are the sizes we are interested in. Kennedy solves the whole problem [5] (figure 2.6), and shows the important graphs of floating potential versus grain radius. He also comments that the potential asymptotes to the OML solutions for very small grain radius. Therefore, even though OML is strictly incorrect, it is a good approximation to the full OM theory for dust grain sizes of interest. This is good news, as it means that there is little need for the full derivation for small objects.

56

Figure 2.6: Normalised potential against grain radius for OM theory [5]. Material

Z

δmax

Emax (eV) Wf (eV)

C (graphite)

6

1.0

300

4.8

W

74

1.4

650

4.55

Table 2.1: Relevant data for carbon and tungsten electron emission.

2.6

Other Charging Mechanisms

As discussed before, there are a number of other mechanisms that can contribute to the charge of a dust grain immersed in a plasma. These are all electron emission mechanisms. An interesting problem arises when the electron emission flux is greater than the incoming electron flux, as the dust grain charges positive. This will be discussed in chapter 3.

2.6.1

Secondary Emission

Incoming plasma particles with sufficient energy are able to dislodge electrons from the grain material. The dominant component comes from electron bombardment. Secondary electrons are emitted with a range of energies, but the most common emission energies are around a few eV. The secondary electron emission coefficient 57

(δsec ) for an electron beam of energy E at normal incidence to a semi-infinite slab is well modelled by the Sternglass formula [43] "   21 # δsec E E , = (2.72)2 exp −2 δmax Emax Emax where δmax and Emax are material dependent constants (see table 2.1). For a Maxwellian electron distribution, one can integrate numerically to find δsec as a function of Te using Ee = 21 me ve2 δsec (Te ) =

R ∞ q 2Ee 0

f (Ee )δsec (Ee )dEe q . R ∞ 2Ee f (E )dE e e me 0 me

(2.70)

It is worth mentioning that Chow et. al. [44] have developed an improved model for secondary electron emission from a spherical object. However, they still assume that the electrons hit the grain at normal incidence, which is not necessarily true for hot ions in a Tokamak plasma. The theory itself is far more complicated. If incorporated into a code it would significantly increase the computation time for integrations, without necessarily being more accurate. The major conclusion is that for small dust grains (around the nanometre scale), the yield can be significantly increased, as the electrons can escape from the dust grain more easily than the infinite slab. The method of calculating the yields using equation 2.70 with a Maxwellian distribution of incoming electrons allows us to generate an empirical fit for δsec (Te ), which is a suitable approximation for the energy range of interest. Ion induced electron emission is most important at keV energies [43], but does contribute at eV energies as well. However, data in this energy region is almost nonexistent. Extrapolating from data at higher energies, it should be a valid assumption for fusion energies to ignore this component in comparison to the electron-induced component. We can compare the size of the secondary electron current with other mechanisms, for a fusion device such as MAST. The electron current density is  1   eφd 8kB Te 2 1 exp , je = − n0 e 4 πme kB Te 58

(2.71)

1.2 1.0

sec

0.8

0.6

0.4

C W

0.2 0.0

0

100

200 T

300 e

400

500

(eV)

Figure 2.7: Plot of secondary electron emission yield against electron temperature for tungsten and graphite. so the secondary electron current density is simply this multiplied by δsec . Taking a typical SOL density of 1018 m−3 , a δsec (Te ) of between 0.1 and 1, a typical normalised potential V = 2 and an electron temperature of 10 eV, we find that the current density of secondaries is between −103 and −104 m−2 s−1 . It is interesting to note that for Tungsten, the secondary electron yield can exceed 1, that is the dust will charge positive purely by secondary emission. From figure 2.7 we can see that this happens at around 120 eV.

2.6.2

Thermionic Emission

Thermionic emission occurs when a material is hot enough that some electrons have enough thermal energy to escape their bonds. The process therefore depends on the work function (Wf ) of the material used. The current density can be estimated by the Richardson-Dushman equation jth =

−ATd2



 Wf exp − , kB Td

(2.72)

where A = 1.20173 × 106 Amps m−2 K−2 is Richardson’s constant. A plot of jth for MAST conditions against Td , the dust grain temperature is shown in figure 2.8. As 59

9

10

7

10

5

jth(Am

-2

)

10

3

10

1

10

-1

10

-3

10

-5

10 1000

2000

3000

4000

5000

6000

T (K) d

Figure 2.8: Magnitude of the current density due to thermionic emission for MAST conditions. we can see, for temperatures less than around 2500 K, this mechanism is negligible, but above this very rapidly becomes the dominant charging mechanism. Thus, for hot dust grains, it is important to construct a model in which the grain has a positive charge.

2.6.3

Field Emission

Field emission occurs when the electric field close to a material is large enough to rip electrons out. This along with thermionic emission causes arcing. The current density is estimated by the Fowler-Nordheim equation [45]   3 2 BW v(y) A f , E 2 exp − jf e = − 2 Wf t (y) E

(2.73)

where E is the electric field next to the dust grain, A and B are universal constants, q 3 3 e E A = 1.54 × 10−6 Amps eV V−2 , B = 6.83 × 109 eV− 2 V m−2 , y = is a W2 f

dimensionless parameter, and the functions t(y) and v(y) are due to image charge effects. The latter two functions can usually be set equal to 1. 60

For an estimate, we take a work function of 4.5 eV, and find   6.52 × 1010 −7 2 . jf e ≈ 3.42 × 10 E exp − E

(2.74)

From this equation, we can calculate that we require a very high electric field (∼ 109 V m−1 ) to match secondary and thermionic emission. Approximating the electric field next to a dust grain by φd /a and taking the dust grain potential to be -25 V (10 eV plasma, Vd = −2.5), we need a dust grain of radius 10−9 m for this electric field. This is a small dust grain which will probably be evaporating in our simulations, so thermionic emission will be much larger.

2.6.4

Photoemission

Photoemission occurs when an incoming photon has enough energy to excite an electron from the dust grain. The minimum energy for this process to occur is given by the work function of the material. Most photons in Tokamaks will be of energies in the UV range, above the work function of the materials discussed, so only the radiation flux needs to be considered. A photon flux of between 1022 and 1023 m−2 s−1 is needed to match the secondary emission current. This could be possible in areas of high photon flux, but we will ignore it for the SOL.

2.6.5

Radioactivity

As dust can absorb plasma particles, it can end up containing a large amount of tritium. Tritium is radioactive, and decays to helium-3 by beta decay. This will charge the material positive, both by the decay and by any secondaries produced as the electron leaves the grain. The half-life of tritium is 12.3 years (=3.88×108 s). This results in a decay rate of ∂nt = −3.72 × 10−9 nt0 . ∂t

(2.75)

where nt is the density of tritium atoms. In order to compute the current density, we need to estimate the speed of these beta particles. The average energy of a 61

beta particle emitted by tritium is 5.7 keV. This results in an average velocity of 4.5×107 ms−1 , which gives a current density calculated from the decay rate of j = −2.7 × 1020 n0 . This means for densities of 1018 or 1019 , the current will be negligible in comparison with thermionic and secondary emission.

62

Chapter 3 Positively Charged Grains In this chapter we consider the effect of electron emission mechanisms. We consider two simple extreme cases: very small emission flux compared to primary electron and ion fluxes, and the converse scenario. In the latter case, the dust grain charges positive. From the equations for secondary and thermionic electron emission currents (see section 2.6), we expect dust grains to charge positive in some regions of Tokamak plasmas, although no experiment has yet verified this. In the previous chapters we have seen although OML is strictly incorrect for all Ti < Te , it is a reasonable approximation for dust grain radii significantly smaller than the Debye length of the plasma [5] (we are referring to the Debye length of the plasma electrons here, another length will be introduced for the emitted electron population later). The Debye length in a fusion SOL plasma is typically tens of microns, compared to dust grain sizes mostly in the size range of a micron or less. We develop a model based around OML theory, and conclude the chapter by outlining the parts of the model used in our simulations.

3.1

The Effect of Electron Emission

Imagine a dust particle which is at the floating potential in a plasma, when we suddenly turn on a flux of electrons moving away from the dust grain, Γem . In order to understand what happens, we have to consider the relative magnitudes of the 63

plasma flux and the emitted flux. We assume singly charged ions.

3.1.1

Emission Flux ≪ Collection Flux

If Γem ≪ Γe , where Γe is the incoming electron flux, the emitted electrons will not significantly effect the electric field structure around the grain, and they will be accelerated away from the grain into the plasma. The effect is therefore to reduce the electron flux so that the flux balance at the surface becomes (1 − δ)Γe = Γi , where δ = Γem /Γe . If the electron emission is purely secondary emission, this is simply the yield. The planar case has already been discussed [10]. The ion flux is given by the Bohm condition Γi = ns vB with ns being the density at the sheath edge, and the   w v¯e . electron flux to the wall using the Boltzmann relation is Γe = 14 ns exp keφ B Te Along with the modified flux balance, this yields the following equation for the floating potential s

"

Vw = − ln (1 − δ)

# 2π(1 + θ) , µ

(3.1)

where Vw = −eφw /(kB Te ), θ = Ti /Te and µ = mi /me as before. In a similar way, we can modify the OML theory to include this effect, as we have a small reduction in the total electron flux on the grain surface, altering the floating condition. The currents from the plasma are unchanged, as the outgoing flux at r → ∞ is negligible. As we are considering currents at the dust grain surface we only need to look at the fluxes. This results in the following expression for the floating potential    12  Vd θ , (1 − δ) exp(−Vd ) = 1+ µ θ

(3.2)

where Vd = −eφd /(kB Te ) = V (a). We would expect this model to break down as δ gets close to unity, as this is the regime where the emitted electron population becomes significant, and begins to alter the electric field around the dust grain. This invalidates the assumption that all emitted electrons escape into the plasma. The floating potential as a function of

64

3

V

d

2

1 = 1

0

= 0.1 = 0.01

-1 0.0

0.2

0.4

0.6

0.8

1.0

Figure 3.1: Floating potential for δ < 1. δ for different temperature ratios is shown in figure 3.1. As the temperature ratio decreases from unity, the magnitude of Vd also decreases, as in normal OML. The effect of the addition of electron emission is to decrease the magnitude of Vd with increasing yield. This is due to a reduction in the electron current. As expected, the model can be seen to break down as δ approaches unity. The model predicts negative Vd (positive φd ), but this is inconsistent with the assumption of a Boltzmann factor for electrons, as they are now in an attractive potential. As δ approaches unity, Vd approaches θ. This results in the potential becoming positive for lower values of δ when θ is higher, due to a higher thermal ion current. This suggests the model is more reliable for lower values of θ.

3.1.2

Emission Flux ≫ Collection Flux

If we now look at the opposite case where the emitted flux dominates the collected flux, the dust grain will begin to charge positive. In this situation for fusion temperatures, thermionic emission is the most important emission mechanism. One might naively expect the grain to charge up to a large positive potential. However, the emitted electrons do not have a large kinetic energy, as they are usually emitted

65

with energies around that of the thermionic work function of the solid. As with a conventional sheath, the potential difference produced is due to the kinetic energy of the particles forming the sheath. The electric field is usually formed to accelerate ions in to balance the electron current. However, in this case, each emitted electron creates a positive charge on the grain. Thus the outward current creates an electric field which will eventually begin to trap the emitted electrons. When a potential barrier is formed that is equal in magnitude to Wf /e, all electrons will be trapped. Thus we can estimate the normalised potential by Vd = VOM L −

eWf , kB Te

(3.3)

where VOM L is the normalised OML potential excluding electron emission. Having considered the two extreme cases, we turn to the case where the two currents are of the same order. This is more complicated, and various different scenarios need to be considered.

3.2

The Intermediate Case

Here we develop a simple model using OML considerations. Whilst it is not strictly accurate, it is convenient to assume that the emitted electrons have a well defined temperature, so we assume a Maxwellian distribution. The energy of secondary electrons is usually between 1 and 5 eV independent of incoming energy [43], for thermionically emitted particles it will be around Wf . As we have two populations of electrons, we have two Debye lengths, a primary Debye length λD , and an emitted electron Debye length λDem . The two lengths can be defined λD =

r

ǫ0 kB Te ne e2

λDsec =

r

ǫ0 kB Tem . nem e2

(3.4)

We can decide which of the two populations of electron is responsible for the shielding by comparing the magnitudes of their scalelengths. For example, if λDem ≪ λD , this suggests that the emitted electrons are doing almost all the shielding, and therefore must be being trapped near to the dust grain. This is the case where jem ≫ je above, 66

and implies a non-monotonic potential. More specifically, this means a potential well is formed, and this has been observed in simulations for thermionic emission [46]. The converse situation would imply the emitted electrons all escape from the field structure, and corresponds to jem ≪ je . Now we have a situation where jem ≈ je . To estimate how the two Debye lengths are related, we use the relationship between the incoming electron current and the emitted electron current at the grain surface Iem (a) = δIe (a). This relation also applies to the current densities jem (a) = δje (a).

(3.5)

For simplicity, we assume that jem ∝ nem v¯em , and je ∝ ne v¯e where the velocities are the thermal velocities. Substituting into equation 3.5 and rearranging yields r Te . (3.6) nem = δne Tem The secondary Debye length is therefore λDem =

λD 1

3

δ2σ4

,

(3.7)

where σ = Te /Tem . For fusion plasmas σ > 1. We consider δ > 1, and therefore the secondary Debye length is likely to be smaller. If we consider the case where δ > 1 and λDem ≫ λD , the emitted electrons do not shield the dust grain, and the grain continues to charge positive. As the potential of the grain increases, the energy gained by the primary electrons increases. If the electron emission is purely secondary emission, a steady state can still be achieved. There may be a maximum in the secondary yield curve at a high enough incoming electron energy after which the yield decreases. One can estimate where this occurs by looking at the secondary yield curve, and identifying at what primary electron energy it falls back below unity. If we use the Sternglass formula [43], for a Maxwellian distribution of the primary electrons incident on Tungsten, we see the voltage required is 1900 V! This suggests that this ratio of Debye lengths will never be achieved as this situation seems unphysical.

67

3.2.1

General Equations

For our simple model we assume λDem ≪ λD , implying that a potential well exists (figure 3.2). The floating condition is in out Ie (a) + Ii (a) + Iem (a) − Iem (a) = 0,

(3.8)

in out where Iem (a) and Iem (a) correspond to the inward (returning) and outward currents

of emitted electrons respectively (figure 3.3). We assume that there is a minimum in the potential profile at r = b, where b > a. Emitted electrons that are moving away from the grain and reach b will escape. Thus the net current of emitted electrons outwards must equal the outward current of emitted electrons at b, so in out out (a) = Iem (b). (a) − Iem Iem

(3.9)

The emitted electrons are assumed to obey the Boltzmann law between a and b as they are in a repulsive potential, and therefore we can relate the outward currents at a and b out out Iem (b) = γ exp (∆V σ)Iem (a),

(3.10)

where γ = b2 /a2 , σ = Te /Tem and ∆V = (φ(b) − φ(a))/(kB Te ) (hence ∆φ is positive, out whereas ∆V is negative). Using the relation Iem (a) = δIe (a) and combining with

the previous equations, we find Ii (a) = [δγ exp (∆V σ) − 1] Ie (a).

(3.11)

As the emitted electrons usually have a lower temperature than the primary electrons, and hence can generate less of a potential difference from conversion of kinetic energy, we assume that the overall potential is still negative (figure 3.2). Thus we can safely assume that all ions that get to b will have enough energy to get to a. As they have been accelerated radially, we assume that all are collected, hence Ii (a) = Ii (b). Now we come to calculate the plasma electron current to the grain surface. As we are considering a positively charged object, the electrons are in an attractive 68

Figure 3.2: A positive grain with a negative surface potential, having a potential minimum at a distance r = b.

Figure 3.3: Currents to and from a positively charged grain. Both emitted electrons and plasma electrons can be reflected at the minimum. All ions that reach b are assumed to be absorbed. 69

potential for a < r < b. We can use OML considerations, but the electron orbitmotion needs to include the fact that electrons are coming from a finite distance, the potential minimum, rather than infinity. We again assume that the limiting orbit of an incoming particle which is collected grazes the surface of the grain, i.e. the absorption radius is at r = a. For particles of charge q and mass m coming from radius b approaching a grain of radius a the equations are [47] 1 1 m(vr2 (a) + vt2 (a)) + qφ(a) = m(vr2 (b) + vt2 (b)) 2 2 avt (a) = bvt (b),

(3.12) (3.13)

where vr and vt are the radial and tangential components of velocity respectively. Solving for vr (a) and vt (a) gives vr2 (a)

=

vt (a) =

vr2 (b)

a2 + 1− 2 b

b vt (b). a





vt2 (b) −

2qφ(a) m

(3.14) (3.15)

We require that vr2 (a) > 0 for an incoming particle to be collected, so setting vr2 (a) = 0 imposes a limit on the magnitude of vt (b)   b2 2qφ(a) 2 ±v1 = ± 2 vr (b) − . b − a2 m The current of particles collected can now be calculated by the integral Z v1 Z 2π Z ∞ 2 I = 4πb n(b)q vr f (vr , vt sin (ψ), vt cos (ψ))vt dvt dψdvr , −v1

0

(3.16)

(3.17)

0

where we are working in the cylindrical coordinates (vt , ψ, vr ), where ψ defines the direction of the vt (tangential) component in a plane tangential to vr (figure 3.4), f is the distribution function of the particles and n(b) is the density at b. Evaluating this integral for a Maxwellian distribution function gives the current at a, r      8kB T 1 1 η 2 I = 4πb n(b) 1− 1− exp , 4 πm γ γ−1

(3.18)

where η = qφ/(kB T ) and γ = b2 /a2 . The classic OML result (equation 2.66) can be recovered in the limit a/b → 0 (γ → ∞), by performing a Taylor expansion for 70

Figure 3.4: Cylindrical coordinates vt , ψ, and vr . small values of η/(γ − 1). The electron current for the case of a potential minimum is therefore      ∆V 1 exp Ie (a) = πb ne (b)¯ ve (b) 1 − 1 − γ γ−1      1 ∆V = Ie (b) 1 − 1 − exp , γ γ−1 2

(3.19)

where v¯e (b) is the electron thermal speed at b. Inserting equation 3.19 into equation 3.11, and using the fact that Ii (a) = Ii (b), we find      Ii (b) ∆V 1 exp (δγ exp (∆V σ) − 1). = 1− 1− Ie (b) γ γ−1

(3.20)

This is the equation we must solve to find ∆V . To find the potential of the grain relative to the plasma we need to make an assumption about the potential at b, which is discussed shortly. Note that Ii (b)/Ie (b) = ji (b)/je (b) = −Γi (b)/Γe (b). The solution of equation 3.20 depends on the relative magnitude of a, λD and λDem . We take λDem ≪ λD and consider two cases: a ≫ λD and a ≪ λD . We take θ = 1 throughout, which is applicable to fusion plasmas.

71

2.5 2.0 1.5 V

d

= 1

1.0

= 5 = 10 = 15

0.5

= 20

0.0 20

40

60

100

80

Figure 3.5: Floating potential for δ > 1 for a ≫ λd .

3.2.2

a ≫ λD

In this case, we basically have planar geometry, and can set a ≈ b (γ = 1) which eliminates the OML factor for electrons inside a. We can also use the Bohm condition p for ions ji = ens vB , where vB = kB (Ti + Te )/mi is the Bohm velocity. We know from the Boltzmann relation at the edge of the sheath that je = −e 41 ns v¯e . This

results in the equation " 1 1 ∆V = ln σ δ

1−

s

2π(1 + θ) µ

!#

.

(3.21)

This is an equation for the potential difference between the potential minimum and the dust grain. To find the potential relative to the plasma, we require the potential at b. Since we have assumed λDem ≪ λD , the emitted electrons dominate the shielding process close to the dust grain forming a sheath of negative charge around the grain of width ≈ λDem . Most of the emitted electrons are trapped as the charge on the dust grain draws them back. Thus as far as the plasma is concerned, there will be a neutral object of radius a + λDem immersed in it, and it will proceed to shield it in the normal way. Thus an estimate for V (b) will be from the OML potential, which 72

for θ = 1 is 2.5. Figure 3.5 shows the overall potential plotted for various values of σ and δ. We can see that the potential difference increases as σ decreases, this is due to there being more kinetic energy available to the emitted electrons relative to the plasma population. Similarly, it increases for increasing δ, this is as a result of there being more secondary electrons produced. However, one of the most striking things to notice is that as σ increases, ∆V very quickly becomes small. A typical fusion plasma, one may expect Te ≈ 100 eV, Tem ≈ 5 eV, therefore σ = 20. This part of the model deals with values of δ above 1, but not much greater, so we take a value δ = 10, and this results in |∆V | ≈ 0.12, a small potential difference relative to V (b). Note that if λDem ≈ λD , the point b will be at a potential less in magnitude than the planar wall floating potential. It is difficult to estimate this potential.

3.2.3

a ≪ λD

In this case we have to consider the spherical geometry of the problem. The inward electron flux at b can be written in terms of the potential at b using the Boltzmann ve . For simplicity, we will use OML to estimate relation Γe (b) = 14 n0 exp (−V (b))¯ the ion flux from infinity to b, and we find this is Γi (b) = 41 n0 (1 + V (b)/θ)¯ vi from 2.66. Inserting the electron and ion fluxes into equation 3.20 results in the following equation −1

s

(1 + V (b)θ ) exp (V (b))

     θ 1 ∆V = 1− 1− exp (1 − γδ exp (∆V σ)) . µ γ γ−1 (3.22)

We have a number of different parameters that need to be estimated before this equation can be solved. V (b) and γ (= b2 /a2 ) both need to be specified before we can solve for ∆V . γ is probably related to the radius of the dust grain, as it will be determined by λDem . Hence if we assume λDem ≪ λD , we can assume that the plasma sees a neutral particle of radius a + λDem . For this analysis, we will simply choose a value of γ, and proceed. Once we have chosen this parameter, we can test various values of V (b) and solve for ∆V to see if the results are consistent with the 73

2

V(a)

1

0 = 4

-1

= 8 = 13

-2 0.0

0.5

1.0

1.5

2.0

2.5

V(b)

Figure 3.6: Floating potential V (a) against V (b) with δ > 1 and a ≪ λd for fusion parameters. assumptions. For example, we have assumed a negative potential, so we must satisfy the criterion V (a) > 0 (remember, V has the opposite sign to φ). We concentrate on the fusion scenario, so again we take σ = 20 and δ = 10. Figure 3.6 shows the V (a) against V (b) for various values of γ. We can see that the equation yields positive values of V (a) for only selected values of V (b), and that as γ increases in magnitude, V (a) becomes more negative. The curve is singular when V (b) is equal to the OML potential, in this case 2.5. Due to computational difficulties, results for values of γ greater than 13 cannot be computed, however, it seems clear from the graph that if the particle is very small, the assumption that the grain is at a negative potential (positive V ) will be incorrect. If we approximate b as a + λDem , then γ = (1 + a/λDem )2 . Thus γ = 13 corresponds to a/λDem = 0.38, and we wish to find out about far smaller grain sizes than this. In experiments, the dust found is < 0.1λD , hence it is likely to be < 0.01λDem given our assumption λDem ≪ λD . We can also look at the effect of changing δ by a large amount. Figure 3.7 shows the curves for δ = 1.1 and δ = 100 with γ = 8. We see that the curves are identical down to a given value of V (b), but beyond that as V (b) decreases in magnitude 74

1

V(a)

0

= 1.1 = 100

-1

-2 0.0

1.0

0.5

1.5

2.0

V(b)

Figure 3.7: Floating potential V (a) against V (b) with δ > 1 and a ≪ λd for different values of δ. the curves diverge. However, the difference between the curves is actually quite small, suggesting that the geometry is far more important than the magnitude of the current of secondaries. Which point on the curves in figure 3.6 we select is worth some consideration. If we were to select the minima on the curves, we can see that V (b) decreases with increasing γ, as does V (a), which is to be expected as more and more electron orbits will miss the dust grain. We know that V (b) must be at a potential lower in magnitude than the OML potential, as the curve is singular. However, it is difficult to justify the use of a specific point on the curve, and this requires further work.

3.3

A Simplified Charging Model for Fusion

We have to consider both a positive and negative dust. In a fusion plasma, the electron temperature is high enough such that both secondary (for tungsten) and thermionic emission currents can exceed the plasma electron current. This is not the case for most materials in an industrial plasma. Our knowledge of the positive charging regime is such that it is not possible to calculate the intermediate case 75

B

B

Figure 3.8: Demonstration of the change in collection area for magnetised and nonmagnetised plasma particles. reliably. However, as the dust grain heats up, we first calculate an emitted electron current that is primarily due to secondary emission, but as the dust further heats thermionic emission quickly becomes dominant (figures 2.7 and 2.8). Hence the current of emitted electrons very quickly becomes much greater than 1, and the dust grain is in the intermediate regime short enough that we can ignore it. We have two models to use, first for δ < 1 we use equation 3.2. For δ > 1 we pass straight to the regime where the emission flux dominates and therefore will use equation 3.3. In addition to this, we need to consider the effect of a magnetised plasma on the floating potential. In a magnetised plasma, the Larmor radius (rl ) will affect the collection of plasma particles perpendicular to the field. In strong B fields (rl ≪ a), magnetised plasma particles will be effectively confined to one dimension, and the collection area presented by the dust grain is 2πa2 (figure 3.8). In weaker B fields (rl ≫ a), the collection is isotropic (cross section 4πa2 ). For the dust grain radii we are considering this effect is only important for electron collection, as the ion Larmor radius is much bigger. Taking a plasma temperature of 10 eV and an electric field of 1 T we find rL ≈ 10 µm for electrons (≈ 500 µm for ions). Thus we see that strong B fields reduce the electron flux to the grain by up to a factor of 2. We are likely to 76

2.6

V

d

2.4

2.2

2.0

1.8 0.5

0.6

0.7

0.8

0.9

1.0

C

Figure 3.9: V against C for θ = 1. be in a regime where the area of electron collection is between these two extremes. If we define C as the ratio of the electron to ion collection areas, C has a range of 0.5 (fully magnetised electrons) to 1 (unmagnetised electrons). Inserting C into the OML equation (equation 2.67) and solving, we see how Vd varies with C for θ = 1 (figure 3.9). As can be seen, the potential can be decreased by up to 25%. This will introduce errors into the calculations in later chapters.

77

Chapter 4 Dust Temperature and Survival Times in a Steady Plasma Background We know that dust is produced by various processes in Tokamaks, but having been created, what happens to it? The dust can move into the plasma, collecting energy until it evaporates, or could move into a cooler region and reach a steady state. Some grains end up on surfaces and cool down after the discharge. It is important to know whether impurity atoms can be transported into the core by dust, as the concentration of impurities in the core can terminate the discharge. Also, where the dust ends up after the shot is interesting, as it can roughen surfaces which then do not perform as well under high heat flux, or block gaps left for engineering reasons. However, before we can study the transport of dust in the Tokamak (see chapter 5), we need to develop a model for the heating and cooling processes on a dust grain in order to make some quantitative predictions of the temperature and density of a plasma in which a dust grain can survive, and the lifetime of various sized dust grains in a hot plasma. Having developed a model for the floating potential, and the ion and electron currents, we can now attempt to calculate the energy fluxes to and from a dust

78

grain. The only other work on the specific heating and cooling mechanisms has been done by Karderinis et al. [48, 49]. In this section we will review this work, and add significantly to it. Much of this work is published by the authors [50]. The various heating and cooling mechanisms of a dust grain immersed in an isotropic plasma are discussed, with relation to fundamental quantities such as the dust grain potential. From this we calculate the dust grain temperature in a steady-state plasma background for the two main divertor materials, tungsten and graphite. Beryllium, which is used as a wall conditioner, need not be considered, as its low boiling point means it will not survive in the plasma.

4.1

Heating and Cooling Mechanisms

The main heating and cooling mechanisms that act on a dust grain in a plasma are the following: • Particle Bombardment (including backscattering and erosion processes). • Electron Emission (both secondary and thermionic emission). • Recombination Processes (e.g. electron-ion recombination). • Neutral Particle Emission. • Radiative Cooling. Each mechanism will now be discussed separately.

4.1.1

Particle Bombardment

Incoming plasma particles add thermal energy to the dust grain material through collisions. The one-way energy flux for a Maxwellian distribution is 2kB T Γ. Ions also gain energy as they are accelerated through the sheath towards a negatively charged dust grain. The total energy for ions is the one-way flux at the presheath

79

Incoming ion

+

Sputters neutrals from the solid which leave at around grain temperature N

N

Backscattered with a large fraction of incident energy

Thermalises and leaves at grain N temperature

_ _ _ _ _ _ _ _ _ _

Figure 4.1: Incoming ions interacting with dust grain material. edge plus the energy gained in the potential (assuming no collisions). The ion and electron energy fluxes at the grain are therefore Ξi (a) = (2kB Ti + |eφ|)Γi (a) = (2kB Ti + Vd kB Te )Γi (a) Ξe (a) = (2kB Te )Γe (a).

(4.1)

For a positively charged dust grain, we use the approximation that the potential changes to Vd = VOM L − eWf /(kB Te ), as the thermionic emission flux is the major electron flux, and the potential increases to trap the emitted electrons. We assume a potential minimum at radius b, and electrons are accelerated by the field inside this, whilst ions are decelerated. The energy fluxes are therefore Ξi (a) = (2kB Ti + Vd kB Te )Γi (a) Ξe (a) = (2kB Te + eWf )Γe (a).

(4.2)

There may be a neutral population in the plasma, either due to recycling (see section 1.2.2), charge-exchange or molecular dissociation. The latter two populations are likely to be of relatively low concentration. Neutrals thermalise with the dust grain or assist in chemical sputtering (see below). The one-way heat flux for neutrals is calculated in the same way and is 2kB Tn Γn . Neutrals which thermalise with the grain are released taking away a heat flux 2kB Td Γn . The overall contribution is 80

therefore Ξn = 2kB (Tn − Td )Γn , where 1 Γn = n n 4

r

8kB Tn . πmn

As Tn ≪ Te , this term is negligible unless nn ≫ ne , which is unlikely. We ignore this term. In addition to these primary mechanisms we need to consider erosion processes and backscattering as correction terms. Here we only consider hydrogen ions. In a Tokamak, the vast majority of ions hitting the dust grain are hydrogenic, but there are other elements in the plasma as well. For example, if the dust grain is bombarded with ions of the same species as the dust grain material, this may lead to dust grain growth. Therefore strictly, this study is only applicable outside dust nucleation regions. There is experimental evidence that these regions exist in Tokamaks, as dust grains clearly formed by agglomeration processes have been observed (see section 1.3.2). Backscattering Backscattered particles take away a fraction of the incoming bombardment energy. In the case of a negatively charged grain, backscattered ions can be assumed to recombine with a loose electron at the grain surface. For a positively charged grain where there are no free electrons on the surface, backscattered ions will contribute to the charging process. However, as we only consider the regime where the emitted electron current is dominant in comparison to the plasma particle currents, we can neglect this. The coefficient of interest for energy loss is the energy reflection coefficient, the average fraction of energy taken away per particle. Backscattering is energy dependent; higher energy particles are less likely to be reflected as they have enough energy to bury themselves within the solid. An empirical fit to experimental data has been derived as a function of beam energy. The fraction of energy

81

Material

Parameter

C

RN

0.6192 20.01 8.922 0.6669 1.864 1.899

RE

0.4484 27.16 15.66 0.6598 7.967 1.822

RN

0.8250 21.41 8.606 0.6425 1.907 1.927

RE

0.6831 27.16 15.66 0.6598 7.967 1.822

W

A1

A2

A3

A4

A5

A6

Table 4.1: Particle reflection parameters for hydrogen bombardment of carbon and tungsten.

backscattered (RE ) is well described by [51] RE =

A1 ln(A2 ET F + e) , 1 + A3 ETAF4 + A5 ETAF6

(4.3)

where the constants A1 etc. depend on the mass ratio of the incoming particles to the substrate particles, e is the base of natural logarithms, and ET F is the Thomas-Fermi reduced energy. The latter is given by ET F =

0.0325m2 E q , 2 2 3 3 (m1 + m2 )Z1 Z2 Z1 + Z2

where m1 and Z1 are the mass and charge of the incoming particles respectively, and m2 and Z2 are the mass and the bare nuclear charge of the substrate respectively. The fraction of particles reflected (RN ) can also be found from equation 4.3 using a different set of constants. The values of the constants for hydrogen bombardment of carbon and tungsten are shown in table 4.1. Backscattering of electrons is negligible compared with secondary and thermionic electron emission. However, the energy flux of backscattered ions is significant. For a Maxwellian distribution, one needs to integrate over the OML ion energies to find the mean fraction of energy backscattered per ion, RE . We assume that the ion distribution at the grain surface retains the shape that it has in the plasma, but each individual ion gains an energy |eφ|. The energy flux lost is then −RE Ξi (a).

82

Erosion Processes Erosion processes reduce the incoming energy from particle bombardment. Ions are the dominant reactants compared to electrons being both high energy and high mass. The erosion yield has a number of components. Physical sputtering is caused by high energy particles knocking atoms or clusters of atoms out of materials. This can be enhanced by chemical reactivity between the incoming ion species and the material, and is dependent on the material temperature. For tungsten, the threshold energy for physical sputtering is sufficiently high that we can ignore it. There is no chemical component as tungsten does not readily react with hydrogen isotopes. Physical and chemical sputtering processes of carbon materials by hydrogen isotopes are reasonably well understood. These processes have been reviewed by a number of authors [6, 10, 52]. The theory developed by Roth [6] is useful in order to examine the relevance of these mechanisms (see Appendix A). Although the yields are well documented, the energies of the sputtered atoms are not. We can probably assume that the energies of the liberated particles are fairly low, around the grain temperature, as a lot of the incoming energy will be absorbed by lattice vibrations. Thus the energy carried away is negligible, as total chemical erosion yields rarely exceed 6%. We neglect chemical erosion.

4.1.2

Electron Emission

Emitted electrons cool the dust grain by taking away a fraction of the incoming electron bombardment energy. We have already calculated the yield (section 2.6.1), but we need to know the average energy per electron to calculate the total energy loss. Most secondary electrons are emitted in a narrow range of energies, about 1-5 eV, regardless of incoming electron energy. Therefore, it is safe to assume an average energy loss per electron of 3 eV. Thermionically emitted electrons are released with an energy equal to Wf . Therefore using the individual yields δtherm and δsec for thermionic and secondary emission respectively, we have a total energy loss of (Wf eδtherm + 3.0eδsec )Γe . In the case of positively charged grains, this term is ig83

nored, as we are assuming the grain is sufficiently highly charged to trap the emitted electrons.

4.1.3

Recombination Processes and Neutral Emission

Ions incident on the dust grain will form various neutral molecules depending on the material on which they are incident and the incoming ion species. We are only considering hydrogen bombardment, and assume that the sputtering yield is low enough that we can ignore molecules produced by sputtering when calculating the recombination energy. Energy will be liberated after the recombination of an electron and an ion. For hydrogen, the energy per reaction is 13.6 eV. For a negative dust grain, we assume all ions incident on the grain recombine due the number of free electrons loosely bound to the grain surface, and the large electrostatic forces involved. If we assume all the photons produced by electron-ion recombination are absorbed, the energy flux gained is 13.6eΓi for a negative grain. Although heating by radiation from the plasma is difficult to quantify, we expect it to be negligible compared to the flux of photons from recombination which are produced at the grain surface. Similarly, two neutral particles may combine to form a molecule. This process releases 2.2 eV for hydrogen. Again, we assume that all neutrals are released as molecules. There is a factor of a half introduced: two neutrals to one molecule, so the energy flux is 1.1eΓi . These molecules thermalise with the grain and are released. Each have energy 2kB Td , resulting in an energy flux loss of kB Td Γi . This last term is quite small due to the fact Td ≪ Te , but will be included for completeness. For a positively charged grain, backscattered ions do not contribute, so the previous values are modified to (13.5e + 1.1e − kB Td )(1 − RN )Γi , where RN is the mean fraction of particles backscattered, obtained by integrating over the ion distribution.

84

4.1.4

Radiative Cooling

The particle cools as dictated by Stefan’s law. The flux of energy lost is −ασTd4 where α is material dependent (1 for a black body), and σ is Stefan’s constant. This becomes the dominant cooling mechanism for larger grain temperatures. For graphite α ≈ 0.8, and for tungsten it is approximated as 1, although it actually shows a slight temperature dependence.

4.2

Steady State Temperatures and Survival Times

Having identified all the heating and cooling mechanisms, we can consider how quickly the dust grain heats and changes state in various plasma backgrounds. We can find the range of plasma parameters where the dust grain may reach a steadystate temperature, and those for which the dust grain evaporates. The net power flux for a negatively charged grain is Ξnet = [(2kB Ti + Vd kB Te )(1 − RE ) + 13.5e + 1.1e − kB Td ]Γi + (2kB Te − Wf eδtherm − 3.0eδsec )Γe − ασTd4 .

(4.4)

For a positively charged grain, this changes to Ξnet = [(2kB Ti + Vd kB Te )(1 − RE ) + (13.5e + 1.1e − kB Td )(1 − RN )]Γi + (2kB Te + Wf e)Γe − ασTd4 .

(4.5)

If we assume that the specific heat (c) stays constant as the grain heats, then md c

dTd = 4πa2 Ξnet , dt

(4.6)

where md = 4πa3 ρd /3 and ρd is the density of the dust grain material. In reality, c is temperature dependent. We also assume that temperature gradients across the grain are negligible, and the grain heats uniformly. This is probably true for most grains, 85

Material

Melting Point Boiling Point

Latent Heats h

Specific Heat c

Density ρ

86

(K)

(K)

(Jkg−1 )

(Jkg−1 K−1 )

(kgm−3 )

C (graphite)



3925

2.97×107 (of sublimation)

710

2.26×103

W

3680

5930

2.96×105 (of vaporisation),

130

1.93×104

6.94×106 (of fusion) Table 4.2: Carbon and Tungsten thermal properties.

4000

6000

3500

5000

4000

2500 n

0

1500

n

0

1000

18

-3

19

-3

= 10 = 10

d

2000

T (K)

d

T (K)

3000

m

3000 n

0

2000

m

t

n

0

1000

t

Sublima ion poin

500

18

-3

19

-3

= 10 = 10

m

m

t

t

Evapora ion poin

0

0

20

40

60

80

100

0

50

T (eV)

100

150

200

250

300

350

T (eV)

e

e

Figure 4.2: Steady state temperatures for graphite (left) and tungsten (right) with Ti = Te for typical densities. although those which contain layers of two different materials may sustain significant gradients across the interface (c.f. the rocket force, section 5.1.5). Equation 4.6 is easily integrated numerically to find the time taken for the particle to heat from its initial temperature (we take 300 K), up to its evaporation/sublimation point or steady-state temperature in a plasma. We take into account that the electron emission yield depends on the dust grain temperature. We see from equations 4.4 and 4.5 that the energy fluxes are independent of the dust particle radius a. This is because they assume an OML floating potential. Thus, if a steady-state temperature Td is achieved, it too is independent of a. However, it does depend on the material due to the inclusion of backscattering and electron emission. Figure 4.2 shows the predicted steady-state temperatures for graphite and tungsten with Ti = Te . The particle can survive indefinitely (ignoring sputtering) if the predicted steady-state temperature is less that the evaporation/sublimation temperature. Graphite sublimes at 3925 K, but tungsten has a liquid phase above 3680 K and does not evaporate until it reaches 5930 K. We assume that tungsten particles stay intact in the liquid phase. For typical MAST SOL densities (1018 m−3 ), graphite dust reaches a steady state for Te ≤ 79 eV. In contrast, for ITER SOL densities (1019 m−3 ), graphite dust only reaches a steady state in temperatures below 13 eV. Tungsten grains survive at much higher temperatures, up to 306 eV

87

in MAST and 64 eV in ITER. Another feature to notice in figure 4.2 is the effect of the charging assumptions. As the plasma gets hotter, electron emission becomes the dominant charging mechanism and the grain charges positive. This change in the sign of the charge leads to a change in the energy fluxes, resulting in the temperature of the dust grain undergoing a sharp increase. By looking at the energy fluxes at the steady state temperature for graphite (figure 4.3), we can see that at this point secondary and thermionic emission cease contributing to the energy balance, and the other components increase. At the steady state temperature, particle bombardment and radiative cooling are the dominant mechanisms for plasma temperatures higher than a few eV, where recombination is the largest contributor to the incoming energy flux. We see the same phenomenon for tungsten (figure 4.4), although the ranges for which a steady state temperature is achieved are different, tungsten being able to survive at higher temperatures. For higher temperatures, where the dust grain changes state, the latent heat of the material needs to be taken into account. At the melting/sublimation temperature, all the incoming energy will be used to make graphite sublime (latent heat of sublimation) or tungsten melt (latent heat of fusion). Tungsten then evaporates after further heating to its boiling point. In most cases, the time taken to change state is 1-2 orders of magnitude longer than the heating times. As the temperature stays constant during a phase transition, the time taken (tpt ) for the change to occur can be calculated by balancing the incoming energy with that needed to change the state of a spherical shell of material, and then integrating over the particle. The incoming energy is 4πa2 Ξnet dt, and the energy required to evaporate a spherical shell of the material is 4πa2 daρd h, where h is the relevant latent heat. Therefore equating and rearranging, we have Z a 0

da =

Z

tpt

0

Ξnet dt, ρd h

(4.7)

where tpt is the time taken for the phase transition to occur. After integrating, we

88

8

8

10

-2

Energy Flux (Wm )

-2

Energy Flux (Wm )

10

6

10

4

10

2

10

0

20

40

1

2

3

4

5

6

60

80

100

120

6

10

4

10

1

2

3

4

5

6

2

10

140

0

10

20

30

40

50

T (eV)

T (eV)

e

e

Figure 4.3: The contribution of each heating component at the steady state temperature for graphite for n0 = 1018 m−3 (left) and n0 = 1019 m−3 (right). The components are as follows: 1. Particle bombardment, 2. Recombination, 3. Secondary Emission, 4. Thermionic emission, 5. Neutral emission and 6. Radiative cooling. 8

8

-2

Energy Flux (Wm )

10

-2

Energy Flux (Wm )

10

6

10

4

10

1

2

3

4

5

6

2

10

6

10

4

10

1

2

3

4

5

6

2

0

100

200

300

10

400

T (eV)

0

20

40

60

80

100

T (eV)

e

e

Figure 4.4: The contribution of each heating component at the steady state temperature for tungsten for n0 = 1018 m−3 (left) and n0 = 1019 m−3 (right). The components are as above.

89

0

-1

-3

19

-3

= 10

n

= 10

0

-2

18

n

0

10

-2

m

Survival Time (s)

Survival Time (s)

10

m

10

-3

10

-4

10

10

-3

10

-4

10

n

0

n

0

-3

19

-3

= 10

m m

-5

-5

10

18

= 10

0

100

200

300

400

10 500

0

100

200

300

400

500

T (eV)

T (eV)

e

e

Figure 4.5: Survival times for carbon and tungsten. obtain tpt =

ρd ha . Ξnet

(4.8)

The total survival time of a dust grain is therefore the time taken to heat to the phase transition temperatures, plus the time taken to undergo the phase transitions. Figure 4.5 shows the calculated survival times of 1 µm radius graphite and tungsten dust grains respectively using the same plasma backgrounds as before. Graphite evaporates at a lower temperature, but can survive for large fractions of a second at temperatures around the sublimation point. Even at 500 eV the particle can survive for 1 ms in MAST and 0.1 ms in ITER. Tungsten is molten at a lower plasma temperature than that at which graphite starts to evaporate, but survives at higher plasma temperatures due to its high boiling point. Once at its boiling point, it evaporates much more quickly than graphite, with a maximum survival time of around 0.01 s. At higher temperatures, the survival times are similar in magnitude to graphite, around 1 ms in MAST and 0.1 ms in ITER. Dust grains created during ELMs have been observed travelling at speeds of up to 1 kms−1 in MAST experiments. Using this speed along with the calculated survival times, the 1 µm radius graphite grains can travel between ∼ 1-103 m in MAST (minor radius 0.5-0.65 m) compared to ∼ 0.1-10 m in ITER (minor radius 2 m) before completely evaporating. Tungsten grains could travel ∼ 1-10 m in MAST compared to ∼ 0.1-10 m in ITER.

90

As we can see, there is a distinct possibility that dust grains of this size could contribute to impurity transport around the SOL, and very fast grains may even be able to penetrate the core plasma. We now need to use this heating model with a realistic Tokamak plasma background to model the movement of dust around the SOL properly. This is the subject of the next chapter.

91

Chapter 5 Dust Grain Motion “There is a strong need to improve knowledge and understanding on mechanisms leading to dust formation and location in Tokamaks and its survivability on the plasma facing surface [53].” This quote is part of the R & D recommendations with relation to dust in ITER. It is speaking generally about the need for more experimental data for the characterisation of dust in existing devices, and the need for more modelling of dust production and transport for safety predictions. This chapter outlines a numerical model which can track a dust particle through a Tokamak plasma, calculating its temperature, solving the equation of motion and modelling the melting and evaporation. Thus we can calculate the lifetime of the particle, and the impurity deposition by dust in the SOL. The trajectories calculated also shed light on the redistribution of dust produced on one surface to other surfaces in the Tokamak. We make assumptions about the initial conditions such as point of production, size and initial velocity, but do not attempt detailed modelling of dust production. Hopefully the latter topic will be a subject of further work in the field.

92

5.1

Forces

We already have a model to calculate the heating and cooling of the dust material, and therefore need the equation of motion. The rest of this section summarises the components, and discusses the relative importance for various plasma and grain parameters.

5.1.1

Electromagnetic Forces

Dust grains are charged, and are therefore are affected by external electric and magnetic fields according to the Lorentz force FLorentz = qd (E + vd × B),

(5.1)

where qd is the charge on the grain, and vd is its velocity. For a negatively charged grain, we calculate the charge using equation 2.15. The force is dependent on the charge-to-mass ratio 3ǫ0 Vd kB Te 4πǫ0 aVd kB Te /e qd ∼− =− , 3 md 4πa ρd /3 ea2 ρd

(5.2)

where Vd = −eφd /kB Te , and ρd is the density of the dust grain material. This can be seen to increase as a−2 as the particle evaporates. Therefore electromagnetic forces become more important as the particle evaporates in hot plasma. However, in a hot plasma, we know that the particle is charged positive (chapter 3). Can we use the vacuum approximation? We assume that a positively charged grain still has a negative potential, so it appears difficult to estimate the magnitude of the positive charge. However, the electron cloud around the dust grain is significant, and is clustered close to the dust grain, as λDem ≪ λD . Hence we assume that the fields interaction with the grain and shielding cloud is important, and use the same approximation to estimate the overall charge.

5.1.2

Flow Pressure and Drag Forces

The flow of plasma around the dust particle exerts a drag force upon it. This drag force is usually separated into a number of components: neutral drag and ion drag. 93

Electron drag is usually ignored due to the tiny mass of the electron. We consider the separate components now. Neutral drag can be modelled using work done by Epstein [54]. The general formula is 4 Fndrag = ζ πa2 vd nn mn v¯n 3 where a is the dust grain radius, vd its speed, and nn , mn and v¯n are the concentration, mass and thermal velocity respectively of the neutral gas species. ζ is a constant that depends on the assumptions of the model. Neutral drag will be ignored because the concentration of neutrals outside the very edge of the plasma is small compared to the plasma concentration. One should note that some Tokamaks use gas pumping to cool the divertor plasma. In this case neutral drag should be added to the model. In a Tokamak there are large plasma flows throughout. We therefore have an effective flow pressure due to plasma flow, or more specifically ion flow, as they are more massive than electrons. This component will actually be found to be more of an acceleration mechanism than a drag force, as the flows considered can be supersonic. Using the relative velocity vp − vd , the force on the particle is Fdrag = πa2 mi n0 |vp − vd |(vp − vd ),

(5.3)

where vp is the plasma velocity, and we assume a cross-section of πa2 , rather than the OML cross-section. This is more accurate if the plasma moves at high speeds, and any focussing of the ion flow is small.

5.1.3

Pressure Gradient

Due to density and temperature gradients within the plasma, there may be a ∇p force on the dust particle. The plasma pressure is a combination of the various pressures of the plasma components. This force results from the difference in momentum transfer between one side of the particle and the other, resulting in a net force in the opposite direction to the pressure gradient. As ∇p is the force density, 94

for a sphere F∇p = −

4πa3 ∇p. 3

(5.4)

Much dusty plasma literature makes reference to thermophoresis. This is a force due to a temperature gradient in a neutral gas, and we do not consider this as a separate force. It is one component of the ∇p force density.

5.1.4

Gravity

Gravity is usually small compared to the other forces mentioned in this section, but is important for the motion of large dust grains, or the motion of dust grains outside plasma regions. It is Fg = md g, where md is the dust grain mass, which can be estimated from the density of the dust grain material and its radius.

5.1.5

The Rocket Force

This force [4, 30] requires a temperature gradient to be sustained by the particle, so that one side of the particle begins to ablate, and propels the particle in the opposite direction. However, Allen et. al. [49] argue that a micron-sized grain cannot sustain a large enough temperature gradient for this to occur, and we therefore ignore it.

5.2

Initial Conditions

We have discussed the various factors that affect dust particles in the plasma, but we need to have some idea of the initial conditions to complete the dust side of the model. This includes the size of the dust particle, which affects heating and evaporation, the point of injection, which is important for a realistic simulation, and the initial velocity. These factors are discussed in the following paragraphs.

95

5.2.1

Size

We can find the most likely size of the particles by looking at the dust characterisation studies discussed in section 1.3.2. The modal size is around a micron, although nanometre scale dust is also found in large quantities. For modelling dust, the larger sizes are more interesting simply due to the fact that this means the particle will survive for longer. As in previous chapters we assume a spherical dust grain. There are extra effects associated with non-spherical particles which require further study.

5.2.2

Point of Injection

We discussed earlier that we believe most dust is produced during arcing events, ELMs and disruptions which cause evaporation of the plasma surface. Most of the energy is usually deposited in the divertor region. Increased erosion is observed from the outer divertor surface. We launch dust from this surface, as it is a region of high flow, and it is one of the most difficult surfaces from which dust can reach the core plasma. Hence, if dust gets to the core from here, we know that it will also be able to from the other surfaces.

5.2.3

Velocity

As mentioned, dust has been observed travelling in MAST at ∼1 kms−1 . The research conducted by Krashenninikov et al. [4, 30] covered in section 1.3.5 considering a particle moving along a corrugated surface shows the particle can be injected at speeds between 10 and 100 ms−1 . For the larger velocities that may be observed, other mechanisms of injection can be considered. Arcing could be a possible mechanism [55]. A small point on a surface can focus the electric field, causing a concentration of current, and possibly cause an explosion expelling material into the plasma. The speeds produced will be especially high in the event of a disruption. The “rocket” effect is another suggestion (section 5.1.5), although as discussed, the force is probably too small. The distribution of dust injected is taken to be a cosine distribution, which is the 96

standard distribution used when considering rough surfaces [56]. This simply states that the probability of a particle being injected at a given angle is proportional to cos (ψ) where ψ is the angle between the trajectory vector and a normal to the surface.

5.3

Plasma Background

The model that we have developed requires a realistic plasma background for meaningful simulation. Experimental or theoretical data can be used. The parameters required are the densities, temperatures and types of the plasma components, the electric and magnetic fields, and the flow velocity of the plasma. In a predictive sense it is more sensible to use a theoretical data source, as the code can be tested using data for current Tokamaks and comparing with experiment, and then used to simulate the effect of ITER conditions on dust in order to diagnose future problems. The region of plasma we are most interested in is the SOL, and there are two major codes that exist that model edge plasma, B2-solps5.0 [57] which is the European code, and UEDGE [58] which is the US equivalent. We use the former. In this section we discuss the B2-solps5.0 code and show plots of the grid used, and some of the parameters that are relevant for our simulations. We are primarily interested in two Tokamaks, MAST and ITER. MAST because it is a small Tokamak, and allows in situ measurement of dust using, for example, an IR camera [16], and ITER because dust may pose a serious threat for its operation. The MAST IR camera potentially will allow us to produce data to benchmark the code against, and has already recorded some velocity data for dust, but this requires a significant amount of analysis before it can be used.

5.3.1

Introduction to the B2-solps5.0 Code

B2-solps5.0 is a standard code describing plasma transport in the SOL. It is a general model that can be configured to model different Tokamaks worldwide [57].

97

The code is a version of B2-Eirene [59], which itself is a combination of the B2 dual fluid plasma code, and the Eirene Monte-Carlo (PIC) neutrals code. The B2 plasma code uses the two-fluid equations with Braginskii transport. The equations include continuity, momentum and energy equations for both the electrons and ions, equations for electric current density, ion and electron energy fluxes, ionelectron energy exchange, and the Braginskii transport coefficients. As diffusion of plasma perpendicular to the magnetic field is determined by turbulence, the perpendicular diffusion coefficient is usually found by comparison with experiment. Neutrals can be modelled either as a fluid as with the plasma, which is useful when the neutral mean free path is small, or with the Monte-Carlo code, Eirene. Eirene solves the complete 3D Boltzmann transport equations, including complex atomic, molecular and surface physics. Processes modelled include charge exchange, volume recombination, neutral radiation, and particle reflection from surfaces. In B2-solps5.0 the boundary conditions are applied at the divertor plates, and the core plasma/SOL interface. At the divertor, the Bohm condition is applied for the plasma fluid velocity parallel to the magnetic field, and currents and heat fluxes are derived using the wall potential. Impurity production is determined by sputtering models. At the core, boundary conditions are generally inferred from experiment. A B2-solps5.0 run will generally take core plasma profiles measured in experiment, and use the perpendicular diffusivity χ⊥ and diffusion coefficient D⊥ as free parameters. The code is run to find the steady state values for the plasma parameters throughout the simulated region, and χ⊥ and D⊥ are varied until broad agreement is achieved between the code and experimental data. Once this has been achieved, the code can be used to find values for parameters that are not measured experimentally, or check the importance of various processes on the plasma formation by eliminating terms from the various equations. As there is no experimental data for ITER, the profiles taken are those expected from scaling relations. The models and the equations used are discussed in greater detail in the literature (e.g. [57, 60, 59]).

98

5.3.2

B2-solps5.0 Plots

In our simulations, we use a cylindrical coordinate system where r and θ map out the toroidal plane, and z is the vertical direction. B2-solps5.0 uses flux coordinates, meaning that when the grid is transformed to cylindrical coordinates, the grid lines on a poloidal cross-section trace out a poloidal projection of the magnetic field lines (the code is toroidally symmetric). This is useful as it clearly shows the null points, and the last closed flux surface. Figure 5.1 shows the grids for both MAST and ITER, and shows that the volume of plasma modelled is confined mainly to the SOL. The important point to note is that this is an irregular grid, i.e. the cell sizes and orientation are not easily related. This means that it is difficult to use directly when working in cylindrical coordinates. It is interesting to visualise some of the data using contour plots. One of the most important parameters for these simulations is the temperature of the plasma, as plasma collection is a major heating mechanism. Plots of the electron temperature for both Tokamaks are shown in figure 5.2. The plots show that the electron temperature increases as you move towards the core for both devices, but the range is very different. For MAST, in the region simulated, the temperature varies between 0 and 75 eV, whereas for ITER the range is between 0 and 1.5 keV. From the considerations in section 4.2, we do not expect carbon to survive if the temperature exceeds 79 eV in MAST, or 13 eV in ITER. From this we can see that the dust can travel through a large proportion of the simulation area in MAST, but is unlikely to get past even the outer SOL in ITER. Tungsten is not used as a divertor material in MAST. In ITER tungsten will not be able to survive a temperatures over ∼64 eV, and should again be confined to the outer SOL. One of the major forces will be the electrostatic force, so we also include a plot of the potential profiles (figure 5.3). Blue is a negative potential, and red is positive.

99

100 Figure 5.1: The B2-solps5.0 grid for MAST (left) and ITER (right).

101 Figure 5.2: B2-solps5.0 data for electron temperature (eV) in MAST (left) and ITER (right).

102 Figure 5.3: B2-solps5.0 data for electrostatic potential (Vm−1 ) in MAST (left) and ITER (right).

5.4

Simulations

Having discussed both the heating and dust transport model along with the plasma background, we now have all the tools needed to simulate dust grain transport around the SOL. This section first discusses how the B2-solps5.0 data is resampled into cylindrical polar coordinates, and then describes the structure of the dust transport code used for our simulations. Simulations of dust transport through MAST and an ITER are then presented.

5.4.1

Data Input - Resampling the B2-solps5.0 Grid

For ease of use, a square grid of plasma background data is preferred, as it is trivial to locate the cell in which a particle is. The grid produced forms the r, z plane of a set of cylindrical coordinates used to describe dust grain motion. We use an interpolation method used in smoothed particle hydrodynamics (SPH) codes in astrophysics [61]. We use the electrostatic potential (φ) as an example. One starts with a set of points where the trivial identity holds Z φ(r) = φ(r′ )δ(r − r′ )dr′ ,

(5.5)

where δ(r) is the Dirac delta function. One constructs an approximation to this Z hφ(r)i = φ(r′ )w(r − r′ , h)dr′ , (5.6) where w(r, h) is an interpolating kernel and h is the characteristic interpolation or “smearing” length. w(r, h) satisfies the properties Z w(r, h)dr = 1

w(r, h) −−→ δ(r). h→0

(5.7) (5.8)

There are infinitely many kernels one may choose, we use a Gaussian function in 2 dimensions  2 r 1 . exp − w(r, h) = πh2 h2 103

(5.9)

0.5 0.1000

0.0

z(m)

13.85 -0.5

27.60 41.35

-1.0

55.10

-1.5 -2.0 0.4 0.8 1.2 r(m)

Figure 5.4: Electron temperature (eV) in MAST after resampling. As we are using N discrete points, we convert the integral in equation 5.6 to a sum. Then, at any point r, we calculate PN

k=1 P N

φk wk (r − rk , h)

k=1

wk (r − rk , h)

,

(5.10)

where, for example, φ(rk ) = φk . Hence the method amounts to summing over the smoothed values of the plasma parameters, and normalising using the density of points. The method is used to smooth scalar values, as well as components of vectors. We ensure that the new Cartesian grid spacing is smaller than the smallest distance between cells in the B2-solps5.0 grid, and use the SPH method to find values at these new points. h, the smearing length, is found by trial and error to find the best result. For example, if h is too large, the data becomes homogenised, and in the converse situation, clusters of data result. A plot of the electron temperature from a MAST simulation after resampling is shown in figure 5.4. Notice that the values are interpolated throughout the core. However, values outside of the original grid are unreliable. 104

5.4.2

Description of the Code

DTOKS (dust in Tokamaks) is a Tokamak dust transport code which uses the equations derived in the previous chapters and plasma data from B2-solps5.0 to predict the motion and lifetime of dust particles in a given Tokamak. The code has been developed from scratch by the author. We define the radius and material of the dust grain and the initial position and velocity. The grid cell corresponding to the initial particle position is located, and the electric field (from the electrostatic potential) and pressure gradient across the cell are calculated component-wise by finite differencing with adjacent cells (centre differencing is used unless the particle is at the edge of the grid). The dust grain is then coupled to the plasma parameters. Firstly, the modified OML equation is solved (equation 3.2, by Newton’s method) for the potential, unless δ > 1 in which case equation 3.3 is used. From the floating potential, the ion and electron fluxes can be calculated, and the energy fluxes. Following this, we solve the dust grain equation of motion md

4 dvd = qd (E + vd × B) + ρ|vp − vd |(vp − vd )πa2 − πa3 ∇p + md g + Fcent dt 3 dxd = vd , (5.11) dt

where the subscripts d and p denote a parameter of the dust grain and plasma respectively, and 2 Fcent = md (vdθ /r, −vdr vdθ /r, 0)

is the centrifugal force due to the fact we are working in cylindrical coordinates. We do this using a leapfrog scheme vdi+1 = vdi−1 + 2∆tF xdi+2 = xdi + 2∆tvdi+1 ,

(5.12)

where i is the time index and F is the total force. The dust grain temperature is updated by integrating equation 4.6 using a second order Runge-Kutta. Any phase changes are handled by rearranging equation 4.8 to yield a change in radius for 105

evaporation which can be calculated from the incoming energy flux (Ξnet ) and the latent heat (h) as Ξnet dt , ρd h

(5.13)

4πa2 Ξnet dt . h

(5.14)

da = or a change in the mass melted dm =

The mass deposited in each cell is recorded for each dust grain injected. This process is repeated until the dust grain evaporates, or leaves the area of simulation. Unlike Krasheninnikov (see section 1.3.5) we do not consider reflection from surfaces, as we consider this too unpredictable.

5.4.3

MAST Simulation

We begin by considering micron radius graphite grains, as MAST has graphite divertor plates. Using a MAST plasma background, we inject grains at increments of 0.1◦ in the poloidal plane, with an initial speed of 10 ms−1 from the middle of the outboard divertor. Figure 5.5 shows the trajectories of a selection of these particles in the poloidal and toroidal planes. The circles marked on the toroidal plane show the inside and outside wall of the chamber. One can see that the dust is accelerated significantly toroidally, and this causes it to leave the SOL plasma, unless the angle of injection is in towards the x-point. If we look at the acceleration caused by each component of the force (figure 5.6), we can see that the toroidal acceleration comes from the plasma flow pressure. Indeed, this is the dominant force whilst the particle is in the solid state in the plasma. For a particle that is ejected, we see from the figure a peak in the flow pressure, after which the plasma related forces drop away, and then disappear as the particle leaves the plasma. If we compare with a particle that has a track which terminates in the plasma (for example the claret line in figure 5.5) we see that as the grain evaporates, the electromagnetic forces become dominant. This is due to the increasing charge-to-mass ratio (equation 5.2). The mass deposited by each grain is weighted so that by repeatedly injecting grains, we build up a deposition pattern for a cosine distribution, a likely source 106

107 Figure 5.5: Trajectories of micron radius dust grains injected at 10 ms−1 viewed in the poloidal (left) and toroidal (right) planes. Circles in the toroidal plane indicate the inner and outer walls of the vessel. The particle tracks are colour coded for comparison.

8

4

10

E

10

v xB d

gradP -2

)

Flow pressure g

Acceleration (ms

Acceleration (ms

-2

)

3

10

2

10

1

10

0

10

-1

10

E v xB d

gradP 4

10

Flow pressure g

2

10

0

10

-2

-2

10 0.00

6

10

0.02

0.04

0.06

0.08

0.10

10 0.00

0.12

0.04

0.08

0.12

t(s)

t(s)

Figure 5.6: Acceleration by each component of the equation of motion for two of the particles in figure 5.5: one that leaves the plasma (left) and one that evaporated in the plasma (right).

Figure 5.7: Relative levels of impurity deposition in MAST by a cosine distribution of the particles in figure 5.5.

108

term for dust injection from a rough surface. For example, if the grain is injected at angle ψ to the vertical, the mass deposited by it in any grid cell is weighted by the probability of dust being injected at that angle, which is cos ψ. Hence if the particle deposits a mass dm, the mass recorded is dm cos ψ. The deposition pattern is shown in figure 5.7. The units are relative, as we have not specified the quantity of dust injected. We can see that the majority of dust deposition is on the inside of the plasma, with some slight deposition around the x-point. There is an area of high deposition (red) due to particles leaving the plasma, skirting the centre column and re-entering the plasma on the other side of the device (notice the orange line). The trajectories of these particles terminate at the same position poloidally, although they travel different distances toroidally. Particles travelling at this speed are unable to deposit mass in the hot plasma on the outboard side, as they are prevented from doing this by flow pressure ejecting them from the plasma. If we increase the speed to 100 ms−1 , the trajectories are basically straight lines, as a result of the dust grain speed not being significantly affected by acceleration mechanisms in the plasma. Dust travelling vertically can now travel past the seperatrix and evaporate, but when it is injected further to the left, the extra momentum takes it out of the plasma, and it hits the centre column. Therefore, the dust deposition is now primarily on the outside of the Tokamak (figure 5.8), from particles that would have been ejected from the plasma by flow pressure at a lower injection speed. The nodal nature of the plot is almost certainly due to an artefact of the resampling process. The dust is able to penetrate further into the plasma, and the deposition trails are longer, as the dust is travelling faster, and moves further whilst evaporating.

5.4.4

ITER Simulation

We repeat the study for micron radius grains in an ITER plasma background, launched in at 10, 100 and 103 ms−1 . Higher injection speeds will be expected in ITER as there is much more energy in the plasma compared to a small Tokamak

109

Figure 5.8: Relative levels of impurity deposition in MAST by a cosine distribution of micron radius graphite dust particles travelling at 100 ms−1 .

110

like MAST. For graphite, we find that the majority of impurity deposition occurs within a few tens of cm of the injection point even for those particles with an initial velocity of 103 ms−1 (figure 5.9). The slower dust particles deposit most of their mass in the first 5-10 cm. The tracks for these particles actually reach further into the Tokamak than the deposition patterns suggest. We concentrate on 103 ms−1 data, as these particles move furthest. If we look at a selection of these tracks (figure 5.10) we see that the distance travelled depends critically on the angle of injection. Depending on the angle, particles travel relatively unimpeded to near the separatrix (red and green lines), are ejected by flow pressure (pink line), or even evaporate near the wall (blue line). The areas of high deposition do not necessarily reflect the end of the particles tracks due to the fact the temperature in the divertor plasma can increase or decrease depending on which trajectory the particle follows. The plasma flow speed can also change direction and magnitude. Therefore, a dust grain can escape a hot region of plasma and cool down. However, the particle may have already evaporated considerably before this, which is why the deposition is much smaller beyond the region close to the edge of the divertor. Therefore, the results for these particles suggest that graphite dust will be confined primarily to the outer SOL unless the injection speed is large. This is still of interest, as dust may result in an increase in the quantity of impurities at various places throughout the SOL plasma. Now we look at tungsten, which is the preferred divertor material for ITER. Tungsten dust evaporates more quickly than graphite dust when molten, due to a lower latent heat, and this means that the deposition pattern (figure 5.11) is a better indicator of the end of the particle trajectories, as the deposition trail is shorter. At 10 ms−1 , the dust evaporates within a few cm of its injection point. However, if we increase the speed to 100 ms−1 , the dust can travel further into the plasma, and depending on injection angle up to 0.5 m. We can see pockets of deposition with varying injection angle for the dust injected at 100 ms−1 , the best example yet of the importance of the variation of the plasma parameters near the edge of the 111

112 Figure 5.9: Relative levels of impurity deposition for a cosine distribution of micron radius graphite dust particles in ITER travelling at 10 (left), 100 (middle) and 103 ms−1 (right).

Figure 5.10: Trajectories of micron radius graphite dust injected at 103 ms−1 into ITER.

113

plasma on dust transport. At 103 ms−1 , the dust can reach inside the separatrix, a concern for core contamination. What about larger dust grains? The code is capable of handling different dust sizes. If we increase the size of the dust particles, we find that the dust travels further before evaporating. For 100 µm radius tungsten particles at 10 and 100 ms−1 , the deposition patterns are shown in figure 5.12. By increasing the size, we find that we only need a speed of 10 ms−1 for the dust to reach inside the separatrix. This is an important point as tungsten divertors have not been well studied, and the typical size of tungsten dust in ITER is not known. The simulation presented here suggests that the impurity transport by dust will be greatly exacerbated if particles of 100 µm or greater are produced in significant quantities. It should be noted that the simulations for these larger grains involve inaccuracies due to some assumptions breaking down. As the dust grain radius is now significantly greater than the Debye length, the OML theory is probably not accurate. Also, the rocket force will probably become significant, as the grains are now large enough to sustain temperature gradients. This could result in dust attaining a much greater speed than predicted. Furthermore, during the evaporation of large grains, a cloud of material builds up around the particle which affects the heat transmission from the plasma to the particle. The effect of this is to lengthen the survival time, meaning that our results are probably an underestimate.

5.4.5

Summary

We have discussed a number of DTOKS simulations for MAST and ITER. Whilst by no means exhaustive, these results lend some insight into the dust transport issue. In MAST, we have seen that flow pressure causes the dust to travel significant distances toroidally. The trajectories of the dust injected at 10 ms−1 suggest we should be able to see dust being swept around with the plasma. Indeed, some video footage has observed this phenomenon [56]. In ITER, the results produced here show that the fate of dust grains of different

114

115 Figure 5.11: Relative levels of impurity deposition for a cosine distribution of micron radius tungsten dust particles in ITER travelling at 10 (left), 100 (middle) and 103 ms−1 (right).

116 Figure 5.12: Relative levels of impurity deposition for a cosine distribution of 100 micron radius tungsten dust particles in ITER travelling at 10 (left) and 100 ms−1 (right).

materials are vastly different. ITER will use a divertor which initially will have both carbon and tungsten plates. The simulations have shown that the redistribution of this material is acutely related to the position of this material in the vessel, and therefore the design of the divertor. We expect carbon dust particles to be confined to the outer divertor region, while tungsten particles may be able to travel inside the separatrix. However, we have looked at the specific case of dust injected from the outer divertor, when it could easily be injected from surfaces much closer to the core plasma. Therefore it is critical to understand how dust is produced in order to further enhance the model.

117

Chapter 6 Conclusions Dust production and transport is a major operational concern for Tokamak power generation. Dust is already observed in Tokamaks around the world. Tungsten, which will be used in the manufacture of divertor plates for ITER, can terminate plasma operation with concentrations of around 10−5 tungsten atoms per hydrogen atom [52]. Until now, dust has been an unpredictable method of transporting impurities. We have developed a dust transport code (DTOKS), which can model the trajectory of a dust particle through a supplied Tokamak plasma background. It is based on a thorough model of the dust-plasma interaction, including charging based on OML theory, and a carefully considered heating model containing a number of components. This is coupled to an equation of motion. It gives us access to numerous facts about the dust, such as temperature, trajectories, the relative size of the various forces acting on the dust grain, and impurity deposition patterns for a distribution of particles. We have used the B2-solps5.0 code as a plasma background to generate data for cosine distributions of micron radius dust particles launched from the outer divertor in both the MAST and ITER Tokamaks. In MAST, we find that graphite particles moving at both 10 and 100 ms−1 can move quite freely through the plasma, and penetrate the core. Dust moving at 10 ms−1 also moves significantly toroidally. In

118

ITER, these speeds are not sufficient for graphite or tungsten dust to reach the separatrix in large quantities, but if we increase the speed to 103 ms−1 , tungsten is able to move inside the last closed flux surface before evaporating. Our major conclusion has to be that tungsten is able to penetrate inside the separatrix in ITER. A speed of 103 ms−1 seems quite realistic for ITER energies. This is dust launched from the outer divertor, which is quite a distance from the plasma. If dust is produced on surfaces nearer the plasma, we may have a significant problem of core plasma contamination. Our results are qualitatively in agreement with Krasheninnikov et al. [4, 30, 31].

6.1

Further Work

DTOKS needs to be improved in a number of ways: • The main algorithms need to be made more efficient so that the code runs faster, and can be integrated into a plasma code. • The resampling method can be improved. • The charging theory needs to be improved to include the effect of high speed ion flows and high magnetic fields. • Positive charging needs to be revisited, to improve the model and to find a better estimate of the charge whilst the dust grain is positive. • Non-spherical particles need to be considered. • 3-D visualisation of trajectories would be useful. It would be interesting to compare the capabilities of DTOKS with DUSTT [4, 30, 31], the only other dust transport model, and compare data produced with experiment. Integration into a plasma code such as B2-solps5.0 as a set of dust subroutines is a hope for the future.

119

Acknowledgements I would like to extend my heartfelt thanks to the many people who have helped me during the course of my PhD. at both Imperial College London and UKAEA Fusion, Culham Science Centre. There are far too many to mention them all. Special thanks goes to my two supervisors Michael Coppins and Glenn Counsell, whose enthusiasm and expertise, along with endless patience, are the only reason I ever finished this thesis at all! I would also like to thank John Allen for many informative chats, and some considerable help with positive charging theory. Vladimir Rozhansky and his co-workers provided B2-solps5.0 data for MAST, and the ITER data was provided by Xavier Bonnin and co-workers. This work was jointly funded by the United Kingdom Engineering and Physical Sciences Research Council and EURATOM.

120

Appendix A Sputtering Formulae In the case of physical sputtering, an incident particle needs to have enough energy to overcome the surface binding energy of an atom in the solid. For this reason there is a threshold energy Eth , below which no sputtering occurs. For an incident ion energy E0 the sputtering yield is [6] "

Yphys (E0 ) = QSn (E0 ) 1 −



Eth E0

 32 # 

Eth 1− E0

2

,

(A.1)

where Q is the yield factor [10] (depends on the surface binding energy) and Sn (E0 ) =

E0 /ET F

0.5 ln [1 + 1.2288(E0 /ET F )] p , + 0.1728 E0 /ET F + 0.008(E0 /ET F )0.1504

(A.2)

where ET F is the Thomas-Fermi energy.

In addition to this yield, various chemical processes can be quantified. Thermal erosion by thermalised ions (neutrals) at a temperature T and an ion flux Γ is   Etherm 0.033 exp − kB T 3  , Ytherm = csp (A.3) 2 × 10−32 Γ + exp − Ektherm BT with

csp and

3

i h C 2 × 1032 Γ + exp − Ektherm BT i  ,  h = 29 Etherm Erel exp − exp − 2 × 10−32 Γ + 1 + 2×10 Γ kB T kB T C=

1

 . 1 + 3 × 107 exp − 1.4eV kB T 121

(A.4)

(A.5)

Parameter

Hydrogen

Deuterium

Tritium

ET F

415 eV

447 eV

479 eV

Q

0.035

0.1

0.12

Eth

31 eV

27 eV

29 eV

Edam

15 eV

15 eV

15 eV

Edes

2 eV

2 eV

2 eV

Erel

1.8 eV (Pure Carbon)

Etherm

Mean 1.7 eV, σ = 0.3 eV

D

250

125

83

Table A.1: Data for chemical erosion with hydrogen isotopes [6].

This yield is itself enhanced by radiation damage (radiation enhanced sputtering or RES) and the sputtering of radicals. The former yield is "  2 #  2 Edam Edam 3 Ydam (E0 ) = QSn (E0 ) 1 − 1− , E0 E0

(A.6)

and the latter is " 2 #  2  3 Edes csp Edes 3  QSn (E0 ) 1 − . 1− Ysurf (E0 ) =  −65eV E0 E0 1 + exp E040eV

(A.7)

These yields are then combined to produce the following total yield Ytot = Yphys + Ytherm (1 + DYdam ) + Ysurf , where D depends on the hydrogen isotope.

122

(A.8)

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