Geoscience Reference

In-Depth Information

Fig. 3.20
Same as in

Fig.
3.19
, but for repulsive

potential. In contrast to

attraction, in this case the free

molecule limit works even

when the particle size is

comparable to the ion mean

free path

1.0

0.8

0.6

0.4

l
c
=1

l
c
=3

0.2

0.5

1.0

1.5

2.0

2.5

3.0

PARTICLE RADIUS
a
(in units of
l
)

where ˛
fm
.a/
D
˛
fm
.a;R
D1
/.

Let us analyze the expression for charging efficiency (Eq.
3.185
) in the free

molecule limit a
l. First, we notice that at a
l the denominator in Eq.
3.185

can be always replaced by unity. Then for the recombination rate Eq.
3.175
gives

˛.a/
a
2
v
T
Œ1
C
ˇ
j
U.a/
j C
ˇ
j
U.4D=
v
T
/
j
e
ˇjU
.
4D=
v
T
/
j

(3.187)

To obtain the widely cited free molecule limit ˛.a/
D
a
2
v
T
Œ1
C
ˇU.a/,the

term ˇU .4D=
v
T
/ should be small compared to 1. The inequality ˇU .4D=
v
T
/
1

does not hold for the Coulomb potential at ambient conditions. Indeed, l
c
D
ˇe
2

6
10
6
cm
l. For the attractive potential given by Eq.
3.1
one finds

1
C
qQ
l
c

a

1
C

e
qQl
c
v
T
=4D
:

q

2Q

"
1

"
C
2

C
qQ
l
c
v
T

4D

˛.a/
a
2
v
T

(3.188)

Figure
3.19
clearly demonstrates the role of the Coulomb distance in the case

of the Coulomb attraction. It is seen that even at small particle sizes a
l the