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 The latter has been replaced by the ion-metal particle potential modified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