Environmental Engineering Reference
In-Depth Information
Because in this case the average cluster charge
is large, in spite of the stepwise
character of attachment of electrons and ions to the cluster surface, this process
may be considered as a continuous one.
Below we find the rate of charge relaxation to the average charge and the particle
charge distribution function in the stationary case assuming a typical cluster charge
to be large. We assume that the plasma consists of positive and negative ions with
similar diffusion coefficients
D
C
and
D
j
Z
j
in a buffer gas. Introducing the mean
diffusion coefficient of ions
D
D
(
D
C
C
D
)/2 and the difference
Δ
D
Dj
D
C
D
j
, which is relatively small,
Δ
D
D
, we reduce (6.12) for the average particle
charge to the form
r
0
T
e
2
Δ
D
D
Z
0
D
.
The equation of cluster charging has a simple form in this case,
dZ
dt
D
Z
0
Z
e
(
J
J
C
)
D
,
τ
and the time
τ
for establishment of the equilibrium cluster charge is given by
N
0
e
2
D
T
1
τ
D
4
π
D
2
πΣ
,
(6.14)
2
N
0
e
2
D
/
T
is the conductivity coefficient of the ion-
ized gas. The solution of the charging equation in this case at the initial condition
Z
(0)
where
Σ
D
N
0
e
(
K
C
C
K
)
D
D
0hastheform
e
t
/
τ
),
Z
D
Z
0
(1
and as is seen, a typical time for establishment of the equilibrium cluster charge
does not depend on the cluster radius.
6.1.4
The Charge Distribution Function for Clusters
We now find the charge distribution function for clusters
f
(
Z
) taking into account
that as a result of ion attachment, the charge varies by one unit. This leads to the
following kinetic equation for the charge distribution function:
f
(
Z
)[
J
C
(
Z
)
C
J
(
Z
)]
D
f
(
Z
1)
J
C
(
Z
1)
C
f
(
Z
C
1)
J
(
Z
C
1) ,
where
J
C
(
Z
) is the rate (the probability per unit time) for the cluster charge to
change from
Z
to
Z
1 as a result of attachment of positive ions to a cluster
and, correspondingly,
J
(
Z
) is the rate for the cluster charge to change from
Z
to
Z
C
1 due to attachment of negative ions to a cluster. It is convenient to reduce (6.7)
and (6.9) for charging rates to the form
e
2
r
0
T
.
4
π
D
C
N
0
r
0
Zx
exp(
Zx
)
4
π
D
N
0
r
0
Zx
J
C
(
Z
)
D
J
(
Z
)
D
x
D
,
,
1
1
exp(
Zx
)
