Environmental Engineering Reference
In-Depth Information
where
C
is the normalization factor, and we use only half of the trajectory when an
ion moves far away from the particle. In the case of free ion motion
U
(
R
)
D
0and
N
0
.
If we divide the ion trajectories into two groups and account for the absence of
the removed part for
(
R
)
D
R
,thisformulagives
N
i
D
c
is the boundary impact parameter for ion
capture by this particle, we obtain for the number density of free ions in the particle
field [60]
c
,where
"
r
1
#
,
r
1
N
0
2
U
(
R
)
ε
C
c
R
2
U
(
R
)
ε
N
i
(
R
)
D
(6.28)
where the boundary impact parameter for capture
c
is connected to the particle
radius
r
0
by the relation [70]
r
0
1
.
U
(
r
0
)
ε
2
c
D
Let us average the ion number density over the Maxwell velocity distribution
function far from the particle, that is,
exp
,
Z
f
(
2
ε
1/2
T
i
1/2
d
f
(
ε
)
D
N
0
ε
)
ε
ε
D
N
0
,
p
π
T
3/2
i
where
T
i
is the ion temperature expressed in energy units. In the spatial region
j
U
(
R
)
j
T
i
(the potential energy
U
(
R
)isnegative)wehave
2
4
3
5
.
s
s
j
U
(
R
)
j
j
U
(
R
)
jj
U
(
r
0
)
j
r
0
/
R
2
N
i
(
R
)
D
N
0
T
i
C
(6.29)
π
π
T
i
In particular, near the particle, this formula gives
N
0
s
N
0
s
4
j
U
(
r
0
)
j
j
U
(
R
)
j
N
i
(
R
)
D
,
R
r
0
r
0
I
N
i
(
R
)
D
,
R
r
0
.
π
T
i
π
T
i
In the other limiting case,
U
(
R
)
T
i
, which corresponds to large ion distances
from the particle,
R
!1
,wehave
N
(
R
)
D
N
0
. At distances far from the parti-
cle, (6.29) takes the form
N
0
s
1
4
j
U
(
R
)
j
N
i
(
R
)
D
C
.
(6.30)
π
T
i
As is seen, in a region of action of the particle field the number density of ions
exceeds the equilibrium number density
N
0
, whereas the number density of elec-
trons is below the equilibrium number density. Hence, screening of the particle
field is determined mostly by ions.
