Environmental Engineering Reference
In-Depth Information
6.2
Particle Fields in an Ionized Gas
6.2.1
Self-Consistent Particle Field in a Rare Ionized Gas
If a micrometer-sized particle is inserted in an ionized gas, it creates a large field in
some region, so the particle potential energy
U
(
R
) in this region exceeds a thermal
energy of electrons and ions. Since ions provide the main contribution to screening
of the particle field, we define the dimension
l
of the particle field as
j
U
(
l
)
j
T
i
,
(6.25)
where
T
i
is the ion temperature. We now consider a rareness gas when this dimen-
sion is small compared with the mean free path of ions
λ
in this gas:
l
λ
.
(6.26)
We assume that the presence of particles in a gas does not influence the ionization
balance in the gas, and the number density of electrons and ions far from the par-
ticle is equal to the equilibrium number density
N
0
that is realized in the absence
of particles. Next, the number density of electrons in the particle field is lower than
that far from the particle, whereas near the particle the ion number density ex-
ceeds the equilibrium one
N
0
remarkably. Hence, we ignore below the influence
of electrons on particle field screening. In addition, the screening of the particle
field influences weakly the particle charge, which follows from the equality of the
electron and ion fluxes to the particle surface, and these fluxes originate far from
the particle where its field does not act.
In considering the distribution of free ions in the particle field, we use the er-
godic theorem [67-69]: the probability
dP
i
for a particle to be in a distance range
between
R
and
R
dR
from the particle is proportional to the time
dt
during
which an ion is found in this region. We have from the ion motion equation [70]
C
dR
v
dR
dt
D
R
D
v
p
1
,
2
/
R
2
U
(
R
)/
ε
R
(
R
) is the normal component of the ion velocity in the particle field at
adistance
R
from the particle center,
v
where
v
is the ion velocity far from the particle,
ε
D
is the impact parameter
for ion motion with respect to the particle. From this within the framework of
statistical mechanics [71-73] we have that the probability of ion location in a given
region is proportional to a time of ion location in this region (
dP
i
m
i
v
2
/2 is the ion energy far from the particle, and
dt
)andis
proportional to the number density of ions
N
i
(
R
)
dP
i
.Fromthiswehaveforthe
number density of ions
C
R
(
R
)
Z
d
dP
i
d
N
i
(
R
)
D
R
2
dR
D
N
0
q
1
,
(6.27)
4
π
2
R
2
U
(
R
)
ε
0
