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)