Environmental Engineering Reference
In-Depth Information
ions crossing a sphere of a radius
R
around a cluster per unit time:
R
2
,
dN
C
dR
C
J
C
D
4
π
D
C
K
C
EN
C
(6.6)
where we ignore the cluster field screening by a surrounding plasma, so the electric
field strength acting on an ion from the charged cluster is
E
Ze
/
R
2
,
N
C
(
R
)is
the current number density of positive ions with electron charge
e
,
D
C
and
K
C
are
the diffusion coefficient and the mobility of positive ions, and
D
e
is the negative
cluster charge. The first term in this expression corresponds to diffusive motion,
the second term relates to ion drift.
If outside the cluster there is no recombination involving positive ions, (6.6) may
be considered as the equation for the ion number density in this region. Assuming
that a positive ion transfers its charge to the cluster as a result of contact with the
cluster surface, we have the boundary condition
N
(
r
0
)
j
Z
j
D
0. Another boundary con-
dition far from the cluster is
N
(
N
0
,(
N
0
is the equilibriumnumber density of
atomic ions in an ionized gas). The latter means that attachment of electrons and
ions to clusters does not violate the ionization balance in this plasma. Assuming
the electric field of the cluster to be relatively small, we use the Einstein relation
K
C
D
1
)
D
eD
C
/
T
between the ion mobility
K
C
and diffusion coefficient
D
C
,where
T
is the ion temperature. This leads to the following equation for the number density
N
C
(
R
) of positive ions:
R
2
D
C
e
dN
C
TR
2
.
Ze
2
N
C
J
C
D
4
π
dR
Solution of this equation with the boundary condition
N
C
(
r
0
)
D
0gives
Z
(
R
0
)
2
exp
Ze
2
r
J
C
dR
0
Ze
2
TR
N
C
(
R
)
D
TR
0
D
C
4
π
r
0
D
C
Ze
2
exp
Ze
2
1
.
Ze
2
TR
J
C
T
D
Tr
0
4
π
Applying the boundary condition
N
C
(
N
0
far from the cluster leads to the
Fuks formula [62] for the total rate of ion attachment to the cluster surface:
1
)
D
D
C
N
0
Ze
2
T
˚
exp
Ze
2
/(
Tr
0
)
4
π
J
C
D
1
.
(6.7)
In the limiting case
Z
0, the Fuks formula (6.7) is transformed into the Smolu-
chowski formula for the diffusive flux
J
0
of neutral atomic particles to the surface
of an absorbed sphere of radius
r
0
[63]:
!
J
0
D
4
π
D
C
N
0
r
0
.
(6.8)
We now consider the case of different signs of the particle and ion, in particu-
lar, the positive ion current goes to a negatively charged particle. Then replacing
