22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Formation of a trap in the dusty plasma of photovoltaic atomic battery
A.V. Filippov, A.F. Pal, A.N. Starostin and V.E. Cherkovets
SRC RF Troitsk Institute for Innovation and Fusion Research, Troitsk, Moscow, Russia
Abstract: Formation of a dusty plasma trap in a non-self-sustained gas discharge
controlled by an electron beam is studied. Dust particle charge is defined by taking into
account the nonlocality of the electron energy distribution function (EEDF). It is obtained
that for the given gas ionization rate and the applied discharge voltage dust particles have a
critical size above which their sedimentation on the cathode occurs.
Keywords: non-self-sustained gas discharge, electron beam, dusty plasma, cathode sheath
1. Introduction
The interest to the dusty plasma under pressures of ~1
bar has been raised quite recently and is basically
connected with studying the possibility of developing
autonomous energy sources on the basis of plasma-dusty
structures [1, 2]. The operation of such an energy source
bases on plasma created by the products of dust
radionuclides particles decay and requires that the ordered
structure from dusty particles be created. The non-selfsustained discharge controlled by an external electron
beam appeared to be most suitable tool for experimental
modeling of such plasma. Therefore, the present paper
reports the results of the work on studying the dusty
plasma excited by a fast electron beam under atmospheric
or higher pressures.
2. Self-consistent model of the non-self-sustained
discharge
The self-consistent model of the non-self-sustained
discharge from [3] by taking into account the processes of
electron and ion losses owing to the sink to dust particles
as well as hydrodynamic equations for the neutral
component was developed. Such model is described by
the following equations:
∂ne
+ ∇ ⋅=
je Qion + n ion ne − β ei ni ne − β de Z d nd ne ,
∂t
∂ni
+ ∇=
⋅ ji Qion + n ion ne − β ei ni ne − β di Z d nd ni ,
∂t
∂nd
+ ∇ ( nd Vd ) = 0,
∂t
∂Vd
∇pd 6pη r0
+ ( Vd ∇ ) Vd =
−
−
( Vd − Vg )
md nd
md
∂t
+g +
ln ( µe ne µi ni )
,
ln ( µe µi )
=
β di 4=
π eµi , β de β di ,
Z d0 is the charge of a dust particle having a 1 µm radius
with taken into account the EEDF non-locality [4], r 0 is
the radius of dust particles in micrometre.
Let us apply the following boundary conditions to
system (1-3) (the cathode is at z = 0, the anode is at
z = H, U dis is the discharge voltage applied to the
electrodes, v Te is the thermal velocity of electrons, γ is the
coefficient of the secondary ion-electron emission from
the cathode):
(1)
ne vTe
z =0
=
( ke ne + γ ki ni ) E z =0 , ne
z=H
=
0;
=
ni =z H 0;=
Vd =z 0,=z H 0,=
nd =z 0,=z H 0;
=
φ
φ
U dis ,
0,=
=z 0=z H
Let us assume the following initial conditions:
(2)
(3).
U dis
, φ= z E , Vd=
0,
=t 0=t 0 =
t 0
H
1
=VNdisd 256 ( y 3 − 83 y 2 + 128
)θ ( 14 − y ) , y =Hz − 43 ,
n= n= =
0, E=
t 0
e t 0=
i t 0
=
nd
− De ∇ne − µe ne E , ji = µi ni E , n e , n i are the
Here je =
electron and ion number densities, E is the electric field
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Z d ( ne , ni , r0 ) = Z d 0 r0
1
4
eZ d
E,
md
∆φ = −4π e ( ni − ne + Z d nd ) , E = −∇φ ,
strength, µ e , µ i are the mobilities of electrons and ions
respectively, D e is the electron diffusion coefficient,
β de Z d , β di Z d  are the recombination coefficients of
electrons and ions on dust particles with the charge Z d ,
Q ion is the bulk gas ionization rate by an external source,
n ion is the frequency of the gas ionization by plasma
electrons, β ei is the coefficient of electron-ion
recombination, V d and V g are the velocities of the directed
movement of dust particles and gas, g is the acceleration
of gravity,
t =0
where V dis is the discharge volume, N d is the number of
dust particles injected into the discharge; θ(x) is the step
function. According [3], the gas ionization in transverse
direction is rather homogeneous. Therefore, it is possible
to consider the problem within one-dimension
approximation along the z-axis. We can also assume that
1
electron transfer coefficients and the coefficient of
electron-ion recombination are constant (this is justified
since the field in the positive column under the considered
regime is weak [3], and the electron number density in the
near-cathode layer is small). The z-axis is directed
upwards.
In the steady regime the gas velocity is equal to zero.
The typical velocity relaxation time of dust particles
considerably exceeds their typical number density time
setting, therefore, it is possible to neglect their inertia and
to define their velocity from the stationary equation of
their movement.
Let us solve the equations in system (1-3) in the
following way. Firstly, we solve the electron number
balance equation by the iteration method (by the sweep
method according to the Crank Nicolson scheme) and the
Poisson equation (by the sweep method) with using the
electron number densities which were determined at this
iteration. Then we solve the number balance equation for
ions according to the explicit scheme with using the
directed differences to determine the drift members. After
this, we solve the equation of continuity for a dusty
component.
3. Numerical simulation results and discussions
The simulation results of dusty trap formation presented
in Figs. 1-5. It is seen in Fig. 1 how a dusty particle trap
for charged dusty particles forms. Fig. 1b shows the
distributions for dusty particle charge Z d and the electric
field in the levitation area in a larger scale. The ionization
rate for xenon at atmospheric pressure and room
temperature was assumed to be equal to Q ion =1014 cm-3s-1,
the voltage U dis = 1 kV was applied to the discharge gap.
N d = 5⋅105 particles of r 0 = 5 µm were injected into the
discharge area.
It is seen in Fig. 1 how a disk-like dusty particle
structure where all the plasma parameters remained
practically constant is formed. In Fig. 2 there are
analogous distributions for a non-self-sustained discharge
in argon.
Comparison of Figs. 1 and 2 shows considerable
difference in the behavior of positive ion number density
in the near-cathode layer: in xenon n i is greater in the
near-cathode layer than in the positive column, and in
argon the ion number density is significantly lower in the
near-cathode layer than in the positive column. This is
due to the following fact.
Balance equation of positive ions (1) in the steady
regime results in the following:
∇=
⋅ ji Qion − β ei ne ni − β id Z d nd ni
(4)
Taking into consideration the equality of the ion number
density gradient in the discharge gap to zero practically
everywhere, we can express the divergence of the positive
ion flux in the following way:
2
ne,i⋅10-10 (cm-3)
E (kV/cm)
1.0
-6.0
ni
0.8
-4.8
−ndZd
0.6
E
-3.6
ne
0.4
-2.4
0.2
-1.2
0.0
0.0
0.0
0.5
a)
1.0
z (cm)
Zd
E (V/cm)
-7500
-150
Zd
E
-5000
-100
-2500
-50
0
0
0.0
0.5
b)
1.0
z (cm)
Fig. 1. Steady distributions for electron (n e ) and ion
(n i ) number densities, a dusty component charge
(n d Z d ) in elementary charge units and the electric
field strength (E) in the discharge gap in xenon at
Q ion = 1014 cm-3s-1, U dis = 1 kV, N d = 106, r 0 = 3 µm,
ρ d = 2.9 g/cm3.
10-10n (cm-3)
E (kV/cm)
1.2
-4.8
1.0
-4.0
E→
0.8
0.6
-3.2
-2.4
ni
0.4
-1.6
ne
0.2
-0.8
-ndZd
0.0
0.0
0.2
0.0
0.4
0.6
0.8
1.0
z (cm)
Fig. 2. Steady distributions for electron (n e ) and ion
(n i ) concentrations, a dusty component charge (n d Z d )
in elementary charge units and the electric field
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strength (E) in the discharge gap in argon at Q ion =
1014 cm-3s-1, U dis = 1 kV, N d = 3⋅105, r 0 = 5 µm, ρ d =
2.9 g/cm3.
P-II-7-23
3
Comparison of Figs. 1 and 2 shows considerable
difference in the behavior of positive ion number density
in the near-cathode layer: in xenon n i is greater in the
near-cathode layer than in the positive column, and in
argon the ion number density is significantly lower in the
near-cathode layer than in the positive column. This is
due to the following fact.
Balance equation of positive ions (1) in the steady
regime results in the following:
∇=
⋅ ji Qion − β ei ne ni − β id Z d nd ni
(4).
Taking into consideration the equality of the ion number
density gradient in the discharge gap to zero practically
everywhere, we can express the divergence of the positive
ion flux in the following way:
∇ ⋅ ji ≈ ∇ ⋅ ( ki ni E=
) ki ni ∇ ⋅ E + ki E ⋅ ∇ni
ki ni ∇ ⋅ E 4π eki ni ( ni − ne )
≈=
We can see from the Table 1 that in xenon electron-ion
and Langevin recombination coefficients are very close to
each other, therefore, the ion number densities in the
positive column and in the near-cathode layer are also
close. In argon β id is more than four times greater than
β ei , therefore, the ion number density in the positive
column is approximately 2 times greater than that in the
near-cathode layer.
In Fig. 3 there is distribution of the electron and ion
number densities, dusty particle charges and electric field
strength in the non-self-sustained discharge in xenon
under injecting dusty particles of two sizes. In this case
two equations of continuity and motion were solved for
the particles of each size.
ne,i⋅10-10 (cm-3)
-6.0
ni
(5).
0.8
The Poisson equation (3) was used in the final step in (5).
Using (5) with the negativity of the charge of levitating
dusty particles, we have from (4) the following:
-3.6
-2.4
0.4
−nd1Zd1
(6).
0.2
-1.2
0.0
0.0
0.0
a)
In the near-cathode layer n e ≈ 0, therefore,
E (V/cm)
-75
E
(7).
Zd2
-5000
-2500
13
Ar
Kr
Xe
4
10
1014
1015
1013
1014
1015
1013
1014
1015
8.5⋅10-7
8.2⋅10-7
7.2⋅10-7
1.2⋅10-6
1.1⋅10-6
1.1⋅10-6
1.5⋅10-6
1.4⋅10-6
1.4⋅10-6
3.8⋅10-6
2.1⋅10-6
1.3⋅10-6
b)
n i,sh ,
cm-3
1.6⋅109
5.1⋅109
1.6⋅1010
2.2⋅109
6.9⋅109
2.2⋅1010
2.7⋅109
8.6⋅109
2.7⋅1010
n i,pc ,
cm-3
3.4⋅109
1.1⋅1010
3.7⋅1010
2.9⋅109
9.4⋅109
3.1⋅1010
2.6⋅109
8.4⋅109
2.7⋅1010
-25
Zd1
0
0.0
β id ,
cm3/s
-50
0
Table 1. Parameters of the dusty plasma created
by e-beam.
β ei ,
cm3/s
1.0
Zd
In Table 1 the electron-ion (β ei ) and Langevin (β id )
recombination coefficients, ion number densities in the
near-cathode layer (n i,sh ) from (7) and in the positive
column (n i,pc ) according to (6) in the non-self-sustained
discharge in heavy inert gases at atmospheric pressure are
presented.
1
0.5
z (cm)
-7500
ni , sh = Qion β id .
Q ion ,
cm-3s-
ne
E
In the positive column n e ≈ n i , therefore, from (5) we
obtain
Газ
-4.8
−nd2Zd2
0.6
Qion − β ei ne ni − β id ni ( ni − ne ) =
0.
ni , pc = Qion β ei .
E (kV/cm)
1.0
0.5
1.0
z (cm)
Fig. 3. Distribution of the electron and ion number
densities (a), dusty particle charge (b) and electric
field strength in the non-self-sustained discharge for
r 01 = 1 µm, N d1 = 106, r 02 = 3 µm, N d2 = N d1 /3, ρ d1 =
ρ d2 = 2.9 g/cm3.
Fig.3 illustrates how the separation of different sized
particles in height occurs in the cathode layer. Let us note
an interesting feature in the distribution of the electric
field strength and the number density of dusty particles: in
the levitation area of dusty particles they appear to be
practically constant. It is seen in Fig. 3 that larger
particles levitate in the area with the greater field than
P-II-7-23
small ones.
In Fig. 4 there is distribution of the modified nonideality parameter in the discharge gap in the non-selfsustained discharge in argon at various applied voltages.
Calculation shows that the critical radius also increases
when the applied voltage to the discharge is increased.
Nd×10-6
1
1.00
20
Γ∗
0.80
15
1
2
3
4
10
2
0.60
3
0.40
4
5
0.20
0
0.00
0.1
1
10
t (s)
0.0
0.2
0.4
0.6
0.8
1.0
z (cm)
Fig. 4. Modified nonideality parameter Г* in argon at
Q ion = 1014 cm-3s-1 as a function of the height over the
cathode. Curve 1 is for U dis = 500 V, 2 is for 1000 V,
3 is for 1500 V, 4 is for 2000 V.
The modified non-ideality parameter Γ* is defined by
the following expression [5]:
Γ* =Γ (1 + κ + 12 κ 2 ) e −κ
where Γ is the non-ideality parameter: Γ = e 2 Z d 2/aT d , T d
is the temperature of the dusty component in energetic
units, κ is the structural parameter: κ = k sh a, k sh is the
screening constant, a is the average interparticle distance:
a = n d -1/3, n d is the density of dust particles.
Fig. 4 illustrates that in the disk-like structure the dusty
component is a liquid, the nonideality parameter
decreasing with the growth of the voltage, which is due to
the decrease in dusty cloud density (it is seen in Fig. 4
that when the applied voltage grows the width of the
dusty cloud increases, which leads to the decrease in the
density in case the total number of particles is constant).
The total number of dusty particles in the discharge area
considerably depends on the radius of dusty particles. It
appears that for the given dust material density, gas
ionization rate and applied voltage the dusty particles
have a critical radius above which all the particles
injected into the discharge area deposit on the cathode.
As an example, Fig. 5 shows that when the radius grows
there remains less and less particles in the discharge gap
in the stationary regime, while the particles with the
radius of 11 µm and larger cannot levitate in the non-selfsustained discharge in xenon at the given parameters at
all.
Table 2 shows the sizes of particles still levitating in the
discharge area while the particles whose radius is 0.5 µm
larger deposit upon the lower electrode within about 1 s.
It is seen that the growth of the gas ionization rate is
accompanied by the growth of the critical radius.
P-II-7-23
Fig. 5. Time evolution of the total number of dust
particles N d in xenon, Q ion = 1015 cm-3s-1, U dis = 1 kV.
Curve 1 – r 0 = 10 µm, 2 – 10.5 µm, 3 – 10.75 µm,
4 - 11 µm.
Table 2. The largest size (in µm) of levitating dust
particles with ρ d = 2.9 g/cm3 at U dis = 1000 V.
Q ion cm-3s-1
Argon
Krypton
Xenon
1013
3
4.5
4
1014
5
7
6.5
1015
8
11
10.5
4. Conclusions
It is shown on the basis of the self-consistency model of
the non-self-sustained discharge in inert gases how a trap
for dust particles forms. It is obtained that for the given
gas ionization rate and the applied discharge voltage the
dust particles have a critical size above which their
sedimentation on the cathode occurs.
5. Acknowledgments
The work was supported by the State Atomic Energy
Corporation “Rosatom” (contract no. Н.4х.44.90.13.1107)
and a grant from the President of the Russian Federation
(no. NSh-493.2014.2).
6. References
[1] V.Yu. Baranov, A F. Pal’, A.A. Pustovalov, et al.
in: Isotopes: Properties, Production, Application.
(V.Yu. Baranov; Ed.) (Moscow: Fizmatlit), Vol. 2,
259 [in Russian] (2005)
[2] A.V. Filippov, A.F. Pal', A.N. Starostin, et al. Ukr.
J. Phys., 50, 137 (2005)
[3] A.V. Filippov, V.N. Babichev, N.A. Dyatko, et al.
JETP, 102, 342 (2006)
[4] A.V. Filippov, N.A. Dyatko and A.S. Kostenko.
JETP, 119, 985 (2014)
[5] O.S. Vaulina and S.A. Khrapak. JETP, 90, 287
(2000)
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