Geoscience Reference

In-Depth Information

efficiencies of small metallic and dielectric particles. To this end we use the

method developed by Lushnikov and Kulmala (
2004a
) for studying the charging

efficiencies of metallic particles in the transition regime. The decisive step allowing

for a consideration of dielectric particles is found in (Lushnikov and Kulmala

2005
) where a simple integral representation of the image potential is derived that

immediately solves the problem. We repeat this derivation here.

Our final goal is to find the ion flux J.a/ as a function of the particle size, its

charge, and dielectric permeability. Actually, we look for the enhancement factor

".a/ defined as

J.a/

J
0
.a/
:

".a/
D

(3.102)

Here J
0
.a/ is the ion flux when the ion-particle interaction is switched off (e.g.,

the condensation of neutral molecules on a neutral particle).

Below we use the free molecule approximation, which normally assumes the

smallness of the particle size as compared to the ion mean free path. If, however, the

particle charge exerts the charge transport, an additional parameter characterizing

the kinetic process appears: the ratio of the ion Coulomb energy to its thermal

energy. Then, the free molecule approximation also assumes the smallness of this

parameter, which is of the order of unity at the Coulomb distance, l
c
D
e
2
=kT

0:6
10
5
cm, comparable to the mean free path at normal pressure; this means that

the free molecule limit correctly describes the ion-particle recombination only at

very low pressure of the carrier gas.

3.5.2

Solution of the Kinetic Equation

Below we solve the kinetic equation assuming that no ions escape from the particle

surface:

f
1
.a;E;L/
D
0:

(3.103)

Because the total flux
J
is independent of
r
,Eq.
3.29
can be rewritten as

4
2

m
3
s

dL
2
f
1
.a;E;L/:

J
D

dE
s

(3.104)

3.5.2.1

Free Molecule Distribution

In what follows we consider the free molecule regime, which means that we ignore

the collisions of the incident ions with the carrier gas molecules. The solution to

Eq.
3.23
is constructed as follows: the function f
s
is a non-zero constant (as the