Geoscience Reference

In-Depth Information

3.5.5.3

Similar Polarities

For dielectric particles and the repulsive Coulomb potential, Eq.
3.113
gives

Z

e
u

2

".a/
D

.
u
/d
u
;

(3.157)

0

where
u
"
and .
u
"
/ are the roots of the set of two simultaneous equations,

2

C
S./
D
0

u

1

1

"
C
1
S./:

u
D

1/
2
C

(3.158)

.
2

At large ,Eq.
3.157
gives

".a/
e
u
"

2

.
u
"
/:

(3.159)

So the problem is just to find
u
"
and
2

.
u
"
/. It is easy to do. The results are

presented in Table
3.1
.

3.5.6

Results and Discussion

We have proposed a comprehensive kinetic approach for studying the charging of

dielectric aerosol particles in the free molecule regime. A compact exact expression

is derived for the enhancement factor ".a/ that shows how strong is the influence

of the particle and ion charges on the efficiency of particle charging. In addition

to the exact result (Eq.
3.146
), we derived simplified asymptotic expressions for

the enhancement factor. Our numerical analysis has shown that these expressions

can be used up to particle sizes of the order of the Coulomb length l
c
D
ˇe
2
with

precision better than 5%. The most simple result has been obtained for charging a

neutral particle. In this case the asymptotic result looks as follows (Eq.
3.153
):

s
ˇq
2
e
2

2a

"
C
2

2"
C
3
C

"
1

"
C
2
:

".a/
2

The dependencies of ".a/ on the particle size at different dielectric permeabilities

" are displayed in Fig.
3.8
. It is seen that the smaller the particle, the stronger the

influence of the dielectric permeability.