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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 0 .a/ :
".a/ D
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.
Solution of the Kinetic Equation
Below we solve the kinetic equation assuming that no ions escape from the particle
f 1 .a;E;L/ D 0:
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/:
dE s
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
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