Environmental Engineering Reference
In-Depth Information
We first consider the case
D
C
D
D
when the average cluster charge is zero and
the distribution function is an even function of
z
and the kinetic equation for the
charge distribution function for clusters takes the form
dz
2
e
z
1
zf
(
z
)
d
dz
[
zf
(
z
)]
x
2
d
2
C
1
C
D
0.
e
z
For small values of
z
that determine the normalization of the charge distribution
function, this kinetic equation takes the form
x
d
2
f
(
z
)
d
[
zf
(
z
)]
dz
C
dz
2
D
0.
Solving this equation taking into account the symmetry of the charge distribution
function
f
(
z
)
D
f
(
z
) and the boundary condition
f
(
1
)
D
0, we obtain
C
exp
.
z
2
2
x
f
(
z
)
D
Returning to the cluster charge
Z
as a variable for the distribution function and
normalizing the distribution function, we obtain
r
x
2
exp
.
Z
2
x
2
f
(
Z
)
D
π
In the general case, when
D
¤
D
C
, we obtain for the charge distribution func-
tion for clusters
r
2
exp
,
Z
0
)
2
x
2
x
(
Z
f
(
Z
)
D
(6.17)
where
Z
0
is the average cluster charge, and since
x
1, a typical particle charge
Z
1.
6.1.5
Charging of Clusters or Particles in a Rare Ionized Gas
We now consider the kinetic regime of cluster charging where criterion
r
0
λ
is fulfilled and the problem of cluster charging is reduced to pairwise collisions of
electrons and ions with the cluster. Assuming that each contact of a colliding ion
or electron with the cluster surface leads to transfer of its charge to the cluster, we
find the cross sections of collisions where the distance of closest approach
r
min
of
colliding particles does not exceed the cluster radius
r
0
. The cross section of ion
contact with the cluster surface, if the cluster charge
Z
has the same sign as that of
colliding ion, in the classical case is
r
0
1
,
Ze
2
r
0
Ze
2
r
0
σ
D
π
ε
.
ε
