22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Dust particle charging in a dry air plasma created by an external ionization
source
A.V. Filippov and I.N. Derbenev
SRC RF Troitsk Institute for Innovation and Fusion Research, Troitsk, Moscow, Russian Federation
Abstract: Dust particle charging in a dry air plasma created by an external ionization
source is studied. It is found that the main positive ion of the plasma is O 4 + and the main
negative ions are O 2 − and O 4 −. The point sink model based on the diffusion-drift approach
shows that the screening potential distribution around a dust particle is a superposition of
four Debye-like exponentials with four different spatial scales.
Keywords: screening, point sink model, air plasma, external ionization
1. Introduction
The study of dusty plasmas created by an external
ionization source in electro-negative gases is important
for atmosphere physics [1,2], for the development of the
dusty plasma based battery [3,4] and for the rare-gas
halide lasers [5]. The microparticle screening in a plasma
generated by an external ionization source in electropositive gases was considered in papers [6-9]. The dust
charging in plasma of electro-negative gases with an
external ionization source was studied numerically in
[10,11] (in the streamer channel in [12]), and
experimentally in [13]. In the present paper the dust
particle charging is studied in a dry air plasma created by
an external ionization source at atmospheric pressure and
room temperature. The ionization rate is varied in the
range 101-1020 cm-3s-1. The lowest ionization rate is the
ionization intensity of the atmosphere over the ground and
the greatest one is representative of the halide lasers.
2. The main ions in a dry air plasma
The ion components of the plasma are obtained by the
analysis of ion-molecular reactions from [14] and the
processes caused by an electron beam:
O 2 + eb → O 2+ + e + eb ,
O2 + eb → O + + O + e + eb ,
N 2 + eb → N 2+ + e + eb ,
N 2 + eb → N + + N + e + eb .
By numerical solution of the full system of kinetic
equations for the above listed processes and processes
from [13] it was revealed that the main positive ion of the
dry air plasma is O 4 + and the main negative ions are O 2 −
and O 4 −. So, hereinafter we will consider only these ions
and electrons as the dry air plasma components.
3. The microparticle charging in a plasma of an
electronegative gas
At atmospheric pressure the microparticle charging is
described by the following continuity and Poisson
equations:
P-I-2-17
∂ne
+ divje =Qion − β ei ne ni − α ne ,
∂t
∂ni
+ divji =Qion − β ei ne ni − β ii 2 ni n2 − β ii 4 ni n4 ,
∂t
∂n2
+ divj2 =α ne − n 24 n2 + n 42 n4 − β ii 2 ni n2 ,
∂t
∂n4
+ divj4 = n 24 n2 − n 42 n4 − β ii 4 ni n4 ,
∂t
supplemented by the Poisson equation:
∆φ =
−4π e ( ni − n2 − n4 − ne ) .
(1)
(2)
(3)
(4)
(5)
Here je =−sign ( es ) µs ns ∇φ − Ds ∇ns are the fluxes of
the corresponding particle species (s = e, i, 2, 4); e s is the
charge of the s -particle, e is the elementary charge (e i =
e, e e = e 2 = e 4 = −e); n s is the number density of
electrons (s = e), of positive ions O 4 + (s = i), of negative
ions O 2 − (s = 2) and O 4 − (s = 4); µ s and D s are the
mobility and diffusion coefficients respectively; β ei is the
recombination coefficient of electrons and ions O 4 +; β ii2
and β ii4 are the recombination coefficients of ions O 4 +
with O 2 − and O 4 − respectively; α is the attachment
coefficient; n 24 and n 42 are the coefficients of conversion
of the two-atom ions into the four-atom ones and vice
versa.
The unperturbed number densities of the main plasma
components are found from the steady-state equation
system
0,
Qion − β ei ne 0 ni 0 − α ne 0 =
Qion − β ei ne 0 ni 0 − β ii 2 ni 0 n20 − β ii 4 ni 0 n40 =
0,
0,
α ne 0 − n 24 n20 + n 42 n40 − β ii 2 ni 0 n20 =
n 24 n20 − n 42 n40 − β ii 4 ni 0 n40 =
0,
(6)
ni 0 − n20 − n40 − ne 0 =
0.
The analytical estimations of the unperturbed number
densities are
Q
Qion
n n
n n
ne 0 ≈ ion , ni 0 ≈
, n ≈ i 0 24 , n ≈ i 0 42 . (7)
α
β eff 20 n 24 + n 42 40 n 24 + n 42
1
Figure 1 shows that the approximations (7) almost
coincide with the numerical solution of system (6). It also
shows that the electron number density is much less than
the ion densities.
cases of the system of Eqs. (1-5). It is revealed, that the
first constant is approximately equal to the inverse Debye
length:
2
k D2 = k De
+ k Di2 + k D2 2 + k D2 4 ,
(10)
= 4π e 2 n 0 T and n s0 is the number density
where k D2 σσσ
of s-component in the undisturbed plasma.
n0 (cm-3)
1012
3
1011
2
ksh (cm-1)
105
4
1010
1
109
104
1
108
107
103
2
6
10 14
10
15
10
16
17
10
10
18
10
3
-3 -1
Qion (cm s )
4
102
Fig. 1. The steady number densities of electrons and ions
in the unperturbed plasma versus the air ionization rate
Q ion . 1 is n e0 , 2 is n i0 , 3 is n 20 , and 4 is n 40 . Curves are the
numerical solution of Eq. (6), symbols are the analytical
estimations from Eq. (7).
The system of (1-5) is solved by linearization and threedimensional Fourier transform. The screening potential
around a charged particle is found to be a superposition of
four Debye-like exponentials:
eq 4
=
φ (r)
(8)
∑ C j exp ( −kshj r )
r j =1
Calculations reveal that complex constants appear in the
limited regions of Q ion values. In this case the potential
takes the form:
eq
φ=
C1 exp ( −k sh1r ) + C2 exp ( −k sh 2 r )
(r)
r
(9)
+ exp ( −k 3r ) Q3 cos (k 4 r ) + Q4 sin (k 4 r )  ,
{
1010
1011
1012
1013
1014
1015
1016
1017
1018
Qion (cm-3s-1)
Fig. 2. The screening constants as a function of the gas
ionization rate. Curve 1 is k sh1 , 2 is k sh2 , 3 is the real part
of k sh3 , and 4 is the real part of k sh4 found by numerical
calculations; symbols represent their approximate values:
○ is k D from Eq. (10), ∆ is k s from Eq. (11), ◊ is k con from
Eq. (12), and □ is k e2 from Eq. (13).
ksh (cm-1)
350
300
}
where k 3 = ½(k sh3 + k sh4 ), k 4 = ½i(k sh4 − k sh3 ), Q 3 = ½(C 3
+ C 4 ), Q 4 = ½i(C 4 – C 3 ), i is the imaginary unit.
The screening constants obtained by numerical solution
of the determinant of the system of linearized Eqs. (1-5)
are shown in Figs. 2-4. The screening constants are
arranged in descending order. Figure 2 shows the curve
crossing in two regions of gas ionization rate:
(1.8−3.4)×1010 and (0.4−1.1)×1014 cm-3s-1. It means that
two of four screening constants have coincident real parts.
In the first region the coincidence of k sh2 and k sh3 does not
yield the imaginary parts of the screening constants as
shown in Fig. 3, and in the second region the coincidence
of k sh3 and k sh4 does yield the complex constants. The
imaginary part of these constants, k 4 , is shown in Fig. 4.
4. Analytical estimations of the screening constants
The physical meaning of the screening constants is
established within the analytical solution of particular
2
250
200
0.0
2.0x1010
4.0x1010
6.0x1010
8.0x1010
Qion (cm-3s-1)
Fig. 3. The dependence of screening constants on Qion in
the crossing region of 2nd and 3rd constants. The solid
line corresponds to ksh2, the dash-dotted line corresponds
to ksh3, ○ is kD from Eq. (10) and □ is ke2 from Eq. (13).
The second one is close to the inverse length passed by
positive and negative ions and electrons due to the
ambipolar diffusion in the characteristic recombination
time:
k s2 ≈ β ei ne 0 ( Di−1 + De−1 ) + β ii 2 n20 ( Di−1 + D2−1 ) +
P-I-2-17
+ β ii 4 n40 ( Di−1 + D4−1 ) .
(11)
The third one is approximately equal to the inverse
diffusion length of negative ions in their conversion time
into each other:
kcon ≈ n 24 D2−1 + n 42 D4−1.
(12)
The electron attachment and recombination of electrons
and diatomic oxygen ions define the fourth constant:
ke22 ≈ (α + β ei ni 0 ) De−1 + β ii 2 n20 D2−1 .
(13)
k4 (cm-1)
ksh (cm-1)
5
350
4
In the steady state the sum of the fluxes of the negative
ions and electrons near a macroparticle surface is equal to
the positive ion flux: J i 0 = J e 0 + J 20 + J 40 . From this
equation for a case of the isothermal plasma
T=
T=
T=
T4 ≡ T we get
e
i
2
q=
Tr0
µi ni 0
ln
.
e 2 µe ne 0 + µ2 n20 + µ4 n40
(15)
Je,i,2,4/q (s-1)
107
15
106
3
300
10
1
250
2
3
5
10
5
2
104
200
4
0
1
0.0
5.0x1013
1.0x1014
1.5x1014
Qion (cm-3s-1)
Fig. 4. The dependence of screening constants on Qion in
the crossing region of 3rd and 4th constants. Curve 1 is
ksh3, 2 is ksh4, 3 is ks from Eq. (11), 4 is ke2 from Eq.
(13), 5 is k4, the imaginary part of the screening constants
ksh3 and ksh4.
The screening constants obtained by the numerical
simulation and from analytical Eqs. (10-13) are shown in
Figs. 2-4. We see that Eqs. (10-13) are a very good fit to
the screening constants obtained by numerical simulation.
According to paper [15] the electron and ion fluxes are
approximately expressed in terms of their undisturbed
densities far from microparticle as follows:
β Lσσσ
n 0z q
Jσ 0 = −
,
(14)
1 − exp ( zσσ
e 2 q T r0 )
= 4π eµ is the coefficient of Langevin
where β Lσσ
recombination of s-type plasma particles due to the sink
to the dust particle with charge q = −z s ; z s = 1 for positive
ions s = i and z s = −1 for s = e,2,4. The comparison of
analytical estimation (14) with results of numerical
calculations is shown in Fig. 5.
Figure 5 shows that Eq. (14) gives underestimated
values due to assumption of flux uniformity that is not
valid because of attachment, recombination and
conversion processes. Nevertheless Eq. (14) gives proper
qualitative character of dependencies. Despite evidence
that the electron number density is much less than the
number density of oxygen negative ions (see Fig. 1) the
electron flux dominates in macroparticle charging at gas
ionization rates higher than 1014 cm-3s-1 due to the high
electron mobility.
P-I-2-17
109 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019
Qion (cm-3s-1)
Fig. 5. The plasma component fluxes on a microparticle
versus air ionization rate Qion. Curves 1 are the electron
fluxes, 2, 3 and 4 are the fluxes of positive ions O4+,
negative ions O2− and O4−, respectively. Solid curves are
data from numerical calculations, dash curves are
analytical estimations (14).
104
−q
103
102
101
109 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019
Qion (cm-3s-1)
Fig. 6. The macroparticle charge versus air ionization rate
Qion. Solid curve is the numerical calculation charge,
dash curve is analytical estimation (15).
The macroparticle steady charges calculated from Eq.
(15) and obtained in the numerical simulations are
presented in Fig. 6. We see that Eq. (15) underestimates
the charge but with variation of the ionization rate gives
the behavior similar to the behavior of the numerical
results.
3
5. The potential distribution around a macroparticle
For the verification of the point-sink model of screening
Eqs. (1-5) are solved numerically using the finitedifference method with the following boundary
conditions:
q
nσ r r =
Er r
0;=
;
=
=
0
0
r02
(16)
= 0; φ
=n ; E
n
σσ
0
r ad
r a=
r ad
=
=
d
Here
nd
is
the
dust
particle
= 0.
number
density,
is the Wigner-Seitz cell radius. The
ad = ( 4π nd 3)
solution of Eq. (5) in a finite cell with boundary
conditions (16) takes the form (see, for example, [16])
q
f fc =
( r )  B1 exp ( −ksh 4 r ) + B2 exp ( ksh 4 r ) + B3 (17)
r
in the case of real screening constants and
q
=
fκκκ
G1 exp ( − 3r ) + G2 exp ( 3r )  cos ( 4 r ) + G3
fc
r
(18)
q
+  K1 exp ( −κκκ
3 r ) + K 2 exp ( 3 r ) 
 sin ( 4 r ) + K 3
r
in the case of complex ones. The coefficients B i , G i , K i (i
= 1, 2, 3) are to be found from the boundary conditions
for the potential and the electric field strength (16).
−1/3
Ψ
0.8
0.6
2
0.4
3
0.2
1
0.0
0.00
0.02
r (cm)
0.04
0.06
Fig. 7. The reduced potential distribution Ψ around a
dust particle at Qion = 1013 cm-3s-1 (curve 1), 1014 cm3s-1 (curve 2) and 1015 cm-3s-1 (curve 3). The solid lines
correspond to the numerical calculations, the dotted ones
with symbols correspond to the sum of (17) with ksh4 and
the Debye exponentials with ksh2 and ksh3 (for curves 1
and 3), and the sum of (18) and the Debye exponential
with ksh2 (for curve 2).
The comparison of the potential obtained from
analytical Eqs. (17-18) and by numerical simulation of
Eqs. (1-5) at three different air ionization rates is shown
in Fig. 7. The reduced potential is defined as
r
k
r−r
(19)
Ψ ( r ) = φ ( r ) × (1 + k sh 4 r0 ) e sh 4 ( 0 ) ,
eq
4
where k sh4 is the smallest screening constant.
This figure reveals that the expressions (17-18) are in a
good agreement with the numerical calculations. Note that
from numerical simulation data we can identify properly
only two smallest constants k sh3 and k sh4 , and the accuracy
of the third-smallest constant k sh2 definition is rather low
although this constant becomes apparent at short distances
(the potential growth at r < 0.01 cm is due to the Debye
exponential with this screening constant), where strong
nonlinearity takes place as well as in the k sh1 appearance
region. Thus the values of k sh1 and k sh2 obtained within
the linear theory are almost physically meaningless.
6. Acknowledgments
The work was supported by the State Atomic Energy
Corporation “Rosatom” (contract no. N.4h.44.90.13.1107)
and a grant from the President of the Russian Federation
(no. NSh-493.2014.2).
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