Electric Field and Hot Spots Formation on Divertor Plates

doi:10.4236/jmp.2011.23020

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Journal of Modern Physics, 2011, 2, 131-135

doi:10.4236/jmp.2011.23020 Published Online March 2011 (http://www.SciRP.org/journal/jmp)

Electric Field and Hot Spots Formation on Divertor Plates

Yuri Igitkhanov, Boris Bazylev

Karlsruhe Institute of Technology, Association EURATOM-KIT, Karlsruhe, Germany

E-mail: [email protected]

Received January 18, 2011; revised February 24, 2011; accepted February 27, 2011

Abstract

In this paper, we consider the intensive erosion of tungsten brush-type armour structures that face the plasma

in divertor fusion reactors. Surface erosion caused by multiple transient events (ELMs, disruption, etc.) could

lead to the formation of a corrugated wedge-type shape. Our analysis shows that the augmentation of surface

roughness increases the electric field at the vicinity of the wedge-type tips, thus enabling the formation of

electric arcs. Specifically, under reactor conditions, the breakdown of the sheath potential may trigger uni-

polar arcs that will strongly contaminate the plasma with the resulting tungsten ions. We show that the ero-

sion caused by arcs is almost two orders of magnitude larger than that caused by DT ion sputtering and

comparable with that caused by self-sputtering.

Keywords: Plasma Facing Components, Erosion, Unipolar Arcs, Dust Formation

1. Introduction

The tungsten macro-brush (W-brush) structure has been

envisaged as a plausible design for the divertor plates of

ITER in order to mitigate surface cracking caused by

energy loads of transient events [1]. Experimental and

numerical investigations have already indicated that dis-

ruptions and giant ELMs power loadings may result in

melting, evaporation, and vapor-shield formation of

W-brush structures under ITER conditions [2,3]. In addi-

tion, it is known that the pressure gradient along the tar-

get surface moves the molten layer and contributes to the

surface roughness. Actually, for the proposed ITER

ELMs size, it has been estimated that, after more than

one thousand ELMs, the molten layer thickness is within

the millimeter scale. During ITER operation, several

hundred disruptions may occur and create a molten sur-

face layer of depth of up to hundred microns per disrup-

tion. This melting process has been investigated numeri-

cally using the MEMOS code [4]. The calculation

showed that ELMs are the main responsible for the tar-

get’s erosion and thus determine the lifetime of the di-

vertor plates.

The cross-section of the model for the W-brush plate

used in the present numerical simulations is shown in

Figure 1.

The typical sizes of the elements of the macro-brush

amour vary within the following ranges: diameter of

brushes, D ~ 0.5 - 1.0 cm; depth of the gaps between the

brushes, h ~ 1 cm; width of the gap, a~0.5-1mm (see

Figure 1). In order to avoid sharp corners, each brush

element is supposed to be rounded with a radius R that

varies from 0.5 to 1 mm. The surface roughness is nor-

mally much less than 1 mm but it may reach 1 mm in the

case of giant ELMs. Yet, as shown by earlier calculations,

splashing of molten droplets does not occur because the

capillary pressure exceeds the centrifugal pressure at the

rounded surface corners, which constitutes the Taylor

criterion limiting the outflowing of the molten layer [4].

However, we will show below that surface roughness can

trigger unipolar arcs that would eject a substantial

amount of tungsten atoms into the plasma. This mecha-

nism may contaminate the plasma much more than

plasma sputtering and evaporation.

In Figure 2, we show the evolution of the corrugated

surface under recurrent impact of ELM-pulses with a

heat load of Q = 1.6 MJ/m2 and an exposition time t =

0.5 ms, as reported in [4].

The corrugated shape of the surface starts to form with

the melting of the originally flat W-brush elements

shown in Figure 1. The molten layer is then pushed to

the left under the pressure of the incoming plasma. Since

the molten material re-solidifies in between two ELMs, a

peak starts to appear on the left edge of each brush ele-

ment. As this mechanism repeats itself, the right edge of

the brush element is shadowed by the corresponding

peak of the neighboring brush element, thus protecting it

from plasma exposure. This process eventually leads to

Y. IGITKHANOV ET AL.

132

Figure 1. Schematic representation (cross-section) of the W

macro-brush target. The geometrical parameters are given

in [4].

Figure 2. Side view of the corr ugated W-brush target under

recurrent ELM-like plasma heat loads Q = 1.6 MJ/m2 and

exposition time t = 0.5 ms [4].

the appearance of the second peak. The details of the

resulting wedge-like shape of a brush element after ex-

posure are shown in Figure 3 for various numbers of

pulses. In the following section, we will show that such

an augmentation of surface roughness increases the elec-

tric field at the vicinity of the wedge’ tips.

2. Simulation of the Sheath Electric Field

In this work, we will consider the voltage drop across the

Langmuir sheath as the primary driving force for unipo-

lar arc ignition. The electric field in the vicinity of cor-

rugated W-wedges, as those shown in Figures 2 and 3,

will be estimated. The electric field profile can be found

by solving the 2D Poisson’s equation:









 (1)

222

4e ii ie e

nT nT





 





 (2)

where ψ ≡ φ / φf is the electrostatic potential normal

ized to the floating potential φf, i is the ion density,

including tungsten impurities in

charge state, is

Figure 3. View of a single W-brush after melting and dis-

placement of the molten layer for various numbers of pulses

[4].

the electron density, e and i are the electron and ion

temperatures, respectively. The coordinate along and

normal to the surface are denoted by

T T

and , respec-

tively. The electric potential is considered to be averaged

in time so that the right side of Equation (1) is linearized

and



is inversely proportional to the Debye length.

We solve Equation (1) within the SOL region, which is

bounded from the bottom by the corrugated metallic sur-

face at which, necessarily, ψ = 0; and from the top by an

imaginary flat boundary (at y→∞) at which we set arbi-

trarily ψ = 1. We also assume that ψ increases as a linear

function of y along the lateral sides of the SOL. The

standard variation procedure of a finite element method

is applied here for solving the 2D Poisson’s equation in

the case of systems with sudden changes in the boundary

shape, such as rectangular corners.

The numerical grid used for solving Equation (1) for

the surface after 300 ELMs exposition was generated

with triangles. Such a grid in the area adjacent to the

corrugated surface is shown in Figure 4. A set of sup-

plementary functions is used to model the rectangular

corners along with the mesh refinement (see [5] for de-

tails).

Our results for the electric potential are presented as a

contour plot in Figure 5, which clearly shows that the

equipotential lines follow the shape of the corrugated

surface in its vicinity and smoothen away from it (top

region in Figure 5). On the overall, as expected, the elec-

tric potential increases from the surface to upstream re-

gion. Most importantly, our calculation shows that the

electric field (see Figure 6) has components along y and

x directions and that, at the wedge’ tips, it reaches values

as large as 7

max 35 10E~.V/cm. Such large values of

the electric field can trigger intensive field electron

emission (see Figure 8). Since electron field emission is

extremely sensitive to the actual value of the electric

field and is crucial for arcs ignition, the calculation of the

Y. IGITKHANOV ET AL.133

Figure 4. Grid for solving Equation (1) in the case of a cor-

rugated W-brush-type divertor plate after 300 ELMs

pulses.

Figure 5. Equipotential lines ( ψ = const ) graduated ac-

cording to the column shown on the right side. The lines are

smoothened by using the cubic spline interpolation.

Figure 6 Electric field contour lines based on a normaliza-

tion unit of 1 × 104 keV/cm. The electric field is graduated

according to the column shown on the right side.

singular electric field at the wedge’ tips requires a much

higher precision than that achieved by only refining the

numerical mesh. For that reason, the behaviour of the

electric field near the metallic wedge’ tip is estimated

analytically by assuming that the metallic 3D tip has a

wedge-like shape with an aperture



(see picture em-

bedded in Figure 7) [6] and by solving the correspond-

ing Laplace’s equation in spherical coordinates.

The electric field components on the metallic wedge

behave like |E|~ const./r α(θ), where α(θ) is a function of θ

(see Figure 7), and const.= |Emax|



α(θ). The latter constant

can be determined from the numerical calculations (dis-

cussed above) for some mesh size

in the vicinity of the

tip. The electric field diverges for 0



 and very sharp

wedge’s tips (



→ 0). For small θ,



characterizes the

degree of singularity and behaves as



~1–π/(2π–θº). This

allows one to estimate the electric field at the wedge tip

for arbitrary values of the cone angle: E(r,



)=const.

(1-



)·sin(



//2)/2r







[7].

3. Unipolar Arc Ignition and Stationary

Burn

The large electric field found at the wedge tips (Figure 6)

strongly enhance the electron field-emission. Emitted

electrons accelerate within the sheath potential and can

easily acquire a kinetic energy of ~100 eV. At that en-

ergy the ionization cross section for tungsten atoms has

its maximum. Ionized tungsten atoms accelerate towards

the tips of the wedge. Such a tungsten bombardment

leads to heating of some spots, augmenting the electron

thermal emission and vaporization. The initial electron

field-emission breaks the sheath potential and eventually

drops itself. However, because of the high temperature at

the spot, an arc current can be sustained by increased

thermal electron emission and ejection of tungsten atoms

from the hot spot. The requirement for arc ignition is that

the initial current density from the tip to the plasma

(dominated by field emission) must exceed some thresh-

old value~1 A/cm2 for tungsten [8]. For stationary burn-

ing, the arc voltage and current must exceed a~15 V

and A, respectively [9]. Calculations show that

10~

Figure 7. The electron field emission J (E) as a function of

the electric field E.

Y. IGITKHANOV ET AL.

134



the electric field of ~3.5  107 V/cm at the tips vicinity is

sufficient for triggering the field-emission current on the

level of ~1A/cm2 (see Figure 8). The arc current density

Ja needed to maintain a stationary hot spot can be esti-

mated as [10-12]:







  (3)

Here



D = e



f/Te is the floating potential

f, normal-

ized on the electron temperature Te,



a=Ua /Te and Ii is

the ion current from the plasma. The electron tempera-

ture is lower than the ion temperature due to cold elec-

tron emission from the arc spot and can be estimated

from (Ti–Te)/Ti ≤ Ja/Ii [13]. From Equation (3) one can

determine the ratio Ja/Ii as a function of the ion tempera-

ture Ti, as shown in Figure 9. It is seen that the arc cur-

rent is a few times larger than the ion current for pure DT

plasma and can be one order of magnitude larger in

presence of tungsten impurities. Assuming that the

plasma leakage from the vapour cloud to the plate is

cm–2·s–1 and that the spot area is

cm2, one can estimate the ion current Ii

~1 A and for expected ion energies ≤ 100 eV (see Figure

8) the arc current from one spot Ja ≤ 20 A. Such a current

density exceeds the minimum arc current needed for

sustaining a unipolar arc [8, 9] and the associated surface

heating by ions can result in the explosive formation of

hot spots and a strong tungsten impurity ejection.

610

j~ 

200S~ r. 1

4. Discussion

In order to initiate an arc on even tungsten surfaces, in

the quiescent operation stage, the floating voltage should

exceed a critical value of ~24 V and the arc current must

be at least Ia ~10 A [9]. In the presence of thermal elec-

tron emission and tungsten impurities in the vapour

cloud surrounded the plate, the sheath floating potential

was calculated in [12] and is within the range



f ≈

Figure 8. The degree of singularity



as a function of the

wedge aperture





Figure 9. Arc current Ja as a function of plasma ion tem-

perature Ti compared to that in the DT plasma.

(2.5 − 0.5) Te. To reach the critical voltage drop for arc

stationary burning, the minimum electron temperature

needed is in the range of 10 - 50 eV. Such high electron

temperatures, however, are unlikely at the first-wall sur-

face of existing machines, therefore arcs are not expected.

Even so, at the divertor plate, the arcs could ignite.

However, the magnetic field intersecting the surface of

the divertor plates prohibits arcs formation because it

limits the plasma volume which can supply the electrons

needed to close the arc current circuit.

In contrast, a corrugated surface surrounded by a va-

pour cloud facilitates the triggering of arcs and the clos-

ing of the arc current. Namely, the large value of the

electric field (> 3.5 × 107 eV/cm) yields a strong electron

field emission that allows to achieve the minimum re-

quired arc current. The effect of the magnetic field does

not appear explicitly in this case because the plasma it-

self has a large capacity to supply sufficient electrons to

the plates, and a current loop can be formed through the

plasma due to surface electric conductance, associated

with electron scattering on the potential in homogeneity

[11].

The creation of an arc is also facilitated by the exis-

tence of a dense vapour cloud and high ion density in

front of the cathode spot. Only then will the electrons

from field-emission be able to ionize the neutral atoms in

the vapour cloud. The local heating derived from the

consequent tungsten-ion bombardment can lead to metal

evaporation. Considering that tungsten atoms are emitted

at the melting temperature (3695 K), they will leave the

surface with a thermal velocity of 4  105 cm/s. For a

sheath potential of 20 eV and an electron density of

1013cm–3, the sheath width is about 1  10–3cm and the

time of flight for W atoms to enter the plasma would be

3  10–8 s - well within the duration of the ELM pulse. If,

for instance, 10% of a monolayer of typically 3.6  1018

cm–2 is suddenly released from a surface spot, the neutral

density in the sheath would locally increase up to 1020

Y. IGITKHANOV ET AL.

135

cm–3. Assuming an ionization cross section of 210–16

cm2, a small fraction of the neutral atoms (10–3) would be

ionized within the sheath, i.e., iw1017cm–3. This

would effectively increase the existing plasma density

near the target. For this example, a W+ ion produced in

the sheath would need about several nanoseconds to fall

back onto the surface.

Arcs erosion of tungsten according to vacuum data [9]

is 0.62  10–4 g/coulomb and is proportional to the cur-

rent flowing through the plate. Since the electron current

forms a considerable part of the whole arc current, about

0.62  10–4  20 A = 1.2 mg per second of tungsten ma-

terial will be released into plasma from one spot. Arcs

are triggered at each tip and one arc occurs from 1cm2

wedge surface (Figure 1), therefore, the contamination

rate from the all divertor plates (~100 m2 in ITER) may

reach the level of 12 grams of tungsten per second or ~ 4

 1022 tungsten impurity ions per second.

5. Conclusions

In this paper, we have demonstrated that repetitive ELM

events in ITER discharges increase the probability of

arcing on corrugated wedge-shape armour surfaces. Al-

though tungsten is a refractory material and the probabil-

ity of arcing is low in quiescent plasma, in the case of

transients, unipolar arcs can ignite and strongly contrib-

ute to the erosion of the armour by ejecting tungsten im-

purities (neutral vapor, molten and solid droplets) into

the plasma. Moreover, arcs at the wedge’s tips may grow

and eventually create hot spots. These have dominant

thermal or burst-type emission that also releases a sub-

stantial amount of tungsten impurities into the plasma.

It is shown that the geometric enhancement of the

electric field on the corrugated surface facilitates the arc

triggering. We find that the electric filed at the wedges’

tips reaches a value sufficient to originate intensive field

electron emission. The tungsten arc erosion rate per each

spot is estimated as 1.2 mg per second. We note that

since sputtering of the tungsten armour by DT ions is ~

10-2 tungsten atoms per incident DT ion (for energies

≤100eV), the arc erosion according to Figure 9 is almost

two orders of magnitude larger than that of sputtering

yield, and comparable with self-sputtering yield.

6. Acknowledgments

This work, supported by the European Communities un-

der the contract of Association between EURATOM and

Karlsruhe Institute of Technology, EURATOM and

CCFE, was carried out within the framework of the

European Fusion Development Agreement. The views

and opinions expressed herein do not necessarily reflect

those of the European Commission

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