Environmental Engineering Reference
In-Depth Information
For the difference between the ion and electron number densities, formulas (6.50)
give
1
exp(
X
X
0
)
C
exp(
X
0
)
exp(
X
0
)
Δ
N
N
i
(
R
)
N
e
(
R
)
D
N
0
.
1
exp(
X
0
)
(6.51)
This quantity is symmetric with respect to transformation
X
!
X
0
X
and has a
maximum at
X
D
X
0
/2, where this quantity is
1
exp(
X
0
/2)
Δ
N
D
N
0
.
1
C
exp(
X
0
/2)
In addition from equality of electron and ion attachment rates we have
X
0
D
ln(
D
/
D
C
).
The electric potential
'
(
R
)
D
U
(
R
)/
e
that is created by the particle follows from
the Poisson equation:
d
2
(
R
1
R
)
dR
2
D
'
Δ
'
4
π
e
Δ
N
.
(6.52)
If
X
D
U
(
R
)/
T
1, we have
Δ
N
D
N
0
X
and the Poisson equation has the form
s
d
2
(
R
1
R
dR
2
D
r
D
,
'
)
T
r
D
D
ZN
0
e
2
.
4
π
Its solution is
R
exp
,
Ce
R
r
D
'
(
R
)
D
(6.53)
and in the Fuks approach
C
D
Ze
2
/
R
is valid for weak shielding of the particle charge by surrounding electrons
and ions, which leads to the criterion
D
Z
. One can see that the Fuks approach
U
(
R
)
N
0
r
0
1 .
(6.54)
N
between the
number densities of electrons and ions, and the Poisson equation (6.52) has the
form
In a general case we use expressions (6.51) for the difference
Δ
d
2
[
RX
(
R
)]
dR
2
1
R
D
Δ
N
N
0
r
D
,
where
X
ln(
D
/
D
C
). From this one can formulate the general
character of variation of the particle potential
X
0
and
X
0
D
'
(
R
) and the number density differ-
N
for ions and electrons on approaching the particle from infinity. At large
distances from the particle surface, the particle potential and the number density
difference drop exponentially with distance from the particle because of the Debye
screening. At the particle surface the number density difference
ence
Δ
Δ
N
is zero and
