Geoscience Reference

In-Depth Information

The first successful attempt to apply the free molecule approximation for

calculating the charging efficiencies of small aerosol particles was undertaken

by Natanson (
1959
,
1960
). Later this problem was considered by many authors

(Marlow and Brock
1975
; Natanson
1959
,
1960
; Gentry and Brock
1967
; Keefe

et al.
1968
;Hahn
1997
; Huang et al.
1991
). None of these works could avoid the

difficulty related to the very inconvenient expression for the ion-dielectric particle

potential. The latter has been replaced by the ion-metal particle potential modified

by the multiplier ."
1/=."
C
1/, with " being the dielectric permeability of the

particle material.

Attempts to consider the transition regime using as the zero approximation the

solution of the collisionless kinetic equation have been done (Smith et al.
1999
;

Huang et al.
1990
,
1991
; Hoppel and Frick
1986
) and also fairly recently (Lushnikov

and Kulmala
2004a
,
b
,
2005
). The analysis of these authors clearly demonstrated

the significance of the ion-carrier gas interaction in calculating the efficiency of

the particle-ion recombination. The point is that the ion can be captured by the

charge particle from the bound states with negative energies. This effect has been

considered (Hoppel and Frick
1990
) by taking into account a single ion-molecular

collision in the Coulomb field created by the charged particle. A new version of

flux-matching theory (Natanson
1959
,
1960
) has been applied by Lushnikov and

Kulmala (
2004a
) to take this effect into account explicitly.

Commonly accepted theories (Reist
1984
; Hidy and Brock
1971
) of particle

charging apply a purely mechanical approach and include two steps:

First, one calculates the dependence of the impact parameter of the ion capture

on the ion velocity.

Second, one averages the cross section thus found over the Maxwell distribution

of ions and finds the charging efficiency.

Of course, there is nothing wrong in this approach, and it gives right results.

Still, it has some disadvantages compared to an alternative, kinetic approach that

we apply below. This approach also includes two (very different) steps:

First, one solves the collisionless Boltzmann equation in the external field created

by the charged particle or by the image forces and finds the ion distribution over

coordinates and velocities.

Second, the ion distribution thus found is used for calculating the ion fluxes

toward the target particle.

There are strong arguments in favor of the second approach. Any attempts

to consider the correction from ion-carrier gas interaction demand a solution

(approximate, of course) of the Boltzmann equation. The second approach gives

a good starting platform for this, whereas the mechanical approach gives nothing

in this respect. Next, the kinetic approach allows one to find the ion concentration

profiles for different boundary conditions at the particle surface. And finally, the

kinetic approach is simpler than the mechanical one. We thus apply the second

route for deriving rather simple and well observable
exact
expressions for charging