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derived and can be found in several studies (Seinfeld and Pandis 2006 ; Friedlander
2000 ; Williams and Loyalka 1991 ; Li and Davis 1995 ; Fuchs and Sutugin 1971 ).
An important step directed to reconciliation of these two limiting cases was taken in
the prominent topic of N. A. Fuchs ( 1964 ), who invented the flux-matching theory.
The general consideration of the kinetic problems in the transition regime can be
found in Cercignani's topic ( 1975 ).
The flux-matching theories are well adapted for studying mass transfer to aerosol
particles in the transition regime. Although these theories mostly had no firm
theoretical basis, they successfully served for systematizing numerous experiments
on growth of aerosol particles, and until now these theories remain rather effective
and very practical tools for studying kinetics of aerosol particles in the transition
regime (see Seinfeld and Pandis 2006 ; Friedlander 2000 ; Williams and Loyalka
1991 ). On the other hand, these theories are always semi-empirical ones; that is,
they contain a parameter that should be taken from somewhere else, not from the
theory itself.
We introduce the readers to the ideology of the flux-matching theories by
considering the condensation of a nonvolatile vapor onto the surface of an aerosol
particle. The central idea of the flux-matching procedure is a hybridization of
the diffusion and the free molecule approaches. The concentration profile of a
condensing vapor far away from the particle is described by the diffusion equation.
This profile coincides with the real one down to the distances of the order of the
vapor molecule mean free path. A limiting sphere is then introduced wherein the
free molecule kinetics governs the vapor transport. The equality of the fluxes in both
zones and the continuity of the concentration profile at the surface of the limiting
sphere define the flux and the condensing vapor concentration at the particle surface.
The third parameter, the radius of the limiting sphere, cannot be found from such a
We apply the more sophisticated approach of Lushnikov and Kulmala ( 2004a ).
This approach starts with an exact expression for the trapping efficiency. This step,
however, does not solve the whole problem. The point is that this exact expression
contains two unknown functions that should be found from the solution of respective
transport equation. However, this formal step is of great use, because some ideas
arise as to how to introduce efficient approximations.
We also introduce a limiting sphere outside of which the density profile of con-
densing vapor can be described by the diffusion equation. Inside the limiting sphere
we solve the collisionless Boltzmann equation subject to a given boundary condition
at the particle surface and put an additional condition: the vapor concentration at
the surface of the limiting sphere coincides with that found from the solution of
the diffusion equation. This condition has also been applied in older theories. The
next step forward was done by Lushnikov and Kulmala ( 2004a ), where the authors
noticed that even in the absence of any potential created by the particle the vapor
profile in the free molecule zone depends on radial coordinates. We thus gain the
possibility to call for the continuity of the first derivatives of the profile on both
sides of the limiting sphere. This additional condition defines the radius of the
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