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
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:
22
2
22
xy




 (1)
222
4e ii ie e
i
Z
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
n
i
Z
charge state, is
e
n
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
x
and , respec-
tively. The electric potential is considered to be averaged
in time so that the right side of Equation (1) is linearized
and
y

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
U
10~
Figure 7. The electron field emission J (E) as a function of
the electric field E.

Y. IGITKHANOV ET AL. 134

1
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]:

e
Da
ai
JI
 
  (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.
20
610
i
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.
n~
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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