Environmental Engineering Reference
In-Depth Information
In the case of different signs of charges for the colliding ion and cluster, this cross
section is
r
0
1
,
C
j
Z
j
e
2
σ
D
π
ε
0.
r
0
ε
For a Maxwell distribution function for ions this gives for the rate constant for
cluster charging when each contact leads to charge transfer and the same sign of
cluster and ion charges
Z
1
t
)
1
t
x
D
T
,
k
0
e
x
k
Dh
v
σ
iD
k
0
tdt
exp(
D
,
t
x
s
8
T
π
e
2
r
0
T
D
j
Z
j
r
0
,
x
,
k
0
D
m
i
π
(6.18)
where
m
i
is the ion mass,
k
0
is the rate constant for ion collision with a neutral
cluster when this collision leads to their contact, and
T
is the ion temperature. For
different signs of charges of colliding particles the rate constant for their contact
averaged over the Maxwell distribution function for ions is given by
Z
1
t
)
1
t
x
k
D
k
0
tdt
exp(
C
D
k
0
(1
C
x
) .
(6.19)
0
One can combine these formulas for the rates of cluster charging by introducing
the probability
that the ion transfers its charge to the cluster as a result of their
contact [65]:
J
>
D
k
0
N
i
(1
C
x
),
J
<
D
k
0
N
i
exp(
x
) .
(6.20)
Here
N
i
is the number density of ions and the rates
J
>
and
J
<
relate to different
and identical signs of ion and cluster charges.
Let us find on the basis of these rates the equilibriumcluster charge in a nonequi-
librium plasma consisting of electrons and ions with different electron
T
e
and ion
T
i
temperatures. Then equalizing the rates of attachment of electrons and ions to
the cluster surface, we obtain for the equilibrium cluster charge
Z
Dj
Z
j
of a
quasineutral plasma
ln
"s
m
i
T
e
m
e
T
i
1
#
,
1
e
2
r
0
T
i
r
0
T
e
e
2
C
j
Z
j
j
Z
jD
where
m
e
and
m
i
are the electron and ion masses. Let us rewrite this formula in
terms of the reduced cluster charge
x
e
2
/(
r
0
T
e
)
e
2
/(
r
W
n
1/3
T
e
)(
r
W
is
Dj
Z
j
Dj
Z
j
the Wigner-Seitz radius):
"
ln
s
m
i
T
e
m
e
T
i
#
,
ln
1
x
T
e
T
i
x
r
0
T
e
e
2
x
D
C
j
Z
jD
.
(6.21)
