Geoscience Reference
In-Depth Information
F.R/
D
e
ˇE
0
r
1
a
2
R
2
The equation for determining
R
takes the form (nonsingular case)
R
2
ˇ
j
U.R/
j
0
R
C
.1
C
2ˇ.
j
U.a/
j
j
E
0
j
/
;
˛
o
.a;R/
2DR
D
1
q
1
e
ˇ
a
2
a
2
R
2
(3.172)
where
a
2
D j
U.R/
E
0
j D
a
2
.
j
U.a/
j j
U.R/
j
/:
R
2
Now our task is to find ˛
o
. A trivial but tedious algebra yields
˛
o
.a;R/
D
a
2
v
T
e
ˇ
1
C
ˇ
j
U.a/
j C
ˇ
j
E
0
j C
1
ˇ
R
2
a
2
e
ˇ
Once the matching distance is known as a function of the particle size, it is easy to
find the charging efficiencies for any potential. We therefore begin with the analysis
of the dependencies of R
D
R.a/ and then present the results on the dependence of
the charging efficiencies on particle sizes for the potentials given by Eq.
3.102
.
The equation describing the dependence of the matching distance on the particle
size for U.r/
D
0 has the structure,
q
R
o
.a/
C
a
2
;
R.a/
D
(3.173)
with
2D
v
T
:
R
0
.a/
D
(3.174)
The value of R
o
.a/ is independent of
a
, so at very small particle size the matching
distance is of the order of the molecular mean free path, as has been expected. At
large particle size a
l the difference R.a/
a
/
l.
When the ion-particle interaction is turned on, the analysis becomes more
complex. It can be done only numerically, but first we analyze the behavior of the
function R.a/ at small a
l;l
c
. In our analysis we assume that U.a/
!1
as
a
!
0.
Let us begin with the attractive potentials. At small particle size, ˛
0
a
2
v
T
ˇ
j
U.a/
j
(the leading term in U.a/ is retained). The term of the same order of
magnitude on the right-hand side of Eq.
3.172
is 2ˇ
j
U.a/
j
a
2
=R
2
. Equation
3.172
then gives
4D
v
T
:
R.a/
(3.175)
