Chemistry Reference
In-Depth Information
0 the integration
of (3.230) results in the well-known solution for the space charge limited ion current,
the Child-Langmuir
With the boundary conditions ϕ
(
z
=
s
)
=−
ϕ
s
and ϕ
(
z
=
0
)
=
ϕ
3
/
2
law
2
1
/
2
4
9
·
e
m
+
·
ϕ
3
/
2
s
s
2
j
+
,
s
=
ε
0
·
·
.
(3.231)
By usage of the Bohm current density at the sheath edge
,itfollows
an expression for the thickness of the space charge sheath, scaled with the electron
Debye length
(
j
+
,
s
=
j
Bohm
)
2
3
/
4
2
1
/
2
3
·
·
e
·
ϕ
s
s
Child
=
λ
De
·
.
(3.232)
k
B
·
T
e
In comparison with the matrix sheath thickness, the Child-Langmuir sheath
thickness has a larger extension as shown by the following ratio
√
2
3
·
2
1
/
4
ϕ
s
[
1
/
4
s
Child
s
matrix
=
·
e
·
ϕ
s
V
]
≈
5.8
·
.
(3.233)
k
B
·
T
e
T
e
[
K
]
10
16
m
−
3
, the ratio amounts to
about 3.26 with a Child-Langmuir sheath thickness of about 11 mm compared with
about 3 mm for the matrix thickness. For the potential, the electric field and the ion
density in the sheath it is obtained
For example, at ϕ
s
=
1000 V,
T
e
=
10
4
K and
n
e
=
z
s
Child
4
/
3
ϕ
(
z
)
=−
ϕ
s
·
,
(3.234)
z
s
Child
1
/
3
4
3
·
ϕ
s
s
Child
·
E
(
z
)
=
,
(3.235)
z
s
Child
−
2
/
3
4
9
·
ε
0
e
·
ϕ
s
s
Child
·
n
+
(
z
)
=
.
(3.236)
3.6.2.3 Plasma Sheath with Ion Collisions
The description of the plasma sheath with ion collisions represents a challenge
because of several reasons:
1. Most of the nonthermal plasmas are characterized by a transition regime
λ
+
≈
s
.
2. The usual definition of the macroscopic quantity for the ion mobility
in
strong inhomogeneous electric fields of the plasma sheath is a critical point.
3. On the microscopic scale the different elementary collision processes of ions
with the neutrals have to be included, for example elastic collisions, charge
transfer collisions and in molecular plasmas further ion-molecule reactions.
(
b
+
)
