Chemistry Reference
In-Depth Information
In the case of the mobility-dominated ion transport in the space charge sheath we
use for the ion current density
d
ϕ
dz
j
+
=
e
·
n
+
·
v
+
=
e
·
n
+
·
−
b
+
·
(3.237)
with the ion mobility
b
+
·
const
. The problem is the appropriate definition
of the ion mobility. Using the ion mobility defined above and the special condition
λ
+
p
=
b
+
0
=
s
, which may be realized in high-pressure plasmas as well as neglecting the
electron density in the sheath, we can insert the ion density from (3.237) into the
Poisson equation for a plane geometry. The integration over
z
with the boundary
condition at the surface ϕ
(
=
s
coll
)
=−
z
ϕ
s
results in the following ion current density
at the surface
9
8
·
ϕ
s
s
coll
j
+
,
s
=
ε
0
·
b
+
·
.
(3.238)
Using the Bohm current density as an approximation of the ion current density
in (3.238), the sheath thickness can be estimated
9
8
·
1
/
2
1
/
3
m
+
k
B
·
ε
0
·
b
+
s
coll
∝
n
pl
·
·
ϕ
2
/
3
s
∼
ϕ
2
/
3
s
·
p
−
1
/
3
.
(3.239)
e
·
0.61
·
T
e
The more interesting case is the definition of the ion mobility in the transition regime
in nonthermal plasmas at lower pressure for the condition λ
+
≤
s
.
Without ionization processes in the sheath the continuity of the current density
in the sheath is given by (3.198)
n
+
(
z
)
·
v
+
(
z
)
=
n
+
(
0
)
·
v
+
(
0
)
(3.240)
with
v
+
(
.
Taking into calculation a mean free path length of ions independent on the ion
velocity and v
+
(
z
)
=
b
+
(
v
+
)
·
E
(
z
)
≈
2
·
e
·
λ
+
/(
π
·
m
+
|
v
+
|
)
·
E
(
z
)
0, the solutions for the electric field, potential, and current
density at the surface follows according to Lieberman [36]
z
)>
2
/
3
3
·
e
·
n
+
(
0
)
·
v
+
(
0
)
E
(
z
)
=
·
z
2
/
3
,
(3.241)
1
/
2
2
·
ε
0
·
(
2
·
e
·
λ
+
/
π
·
m
+
)
3
2
2
/
3
2
/
3
3
5
·
(
e
·
n
+
(
0
)
·
v
+
(
0
))
ϕ
(
z
)
=−
·
·
z
5
/
3
,
(3.242)
·
ε
0
1
/
3
(
2
·
e
·
λ
+
/
π
·
m
+
)
5
3
3
/
2
2
1
/
2
ϕ
3
/
2
sf
s
5
/
2
.
2
3
·
·
e
·
λ
+
j
+
,
sf
=
·
ε
0
·
·
(3.243)
π
·
m
+
