Geoscience Reference
In-Depth Information
kinetic equation. Below I wish to popularize an approximate approach that allows
one to avoid the solution of the full Boltzmann equation and to consider the
aforementioned transport effects from a unified point of view. The final results are
analytical.
3.2
Flux-Matching Theories of Molecular Transport
in the Transition Regime
We introduce the reader 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 existing flux-matching procedures 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 inside which the free
molecule kinetics governs the vapor transport. The concentration profile in the free
molecule zone is considered to be flat. 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 reactant concentration at the particle surface. The third parameter,
the radius of the limiting sphere, cannot be found from such a consideration.
We apply a more sophisticated scheme (Lushnikov and Kulmala
2004a
). We also
introduce a limiting sphere outside of which the density profile of condensing vapor
can be described by the diffusion equation. But inside the limiting sphere we solve
the collisionless Boltzmann equation subject to a given boundary condition at the
particle surface (incomplete sticking in our case) and put an additional condition:
the concentration at the surface of the limiting sphere coincides with that found
from the solution of the diffusion equation. Even in the absence of any potential
created by the particle, the vapor profile in the free molecule zone depends on the
radial coordinate, because the particle surface adsorbs all incoming molecules. 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 limiting sphere.
3.2.1
Flux Matching for Mass and Charge Transport
The steady-state ion flux J.a/ onto a particle of radius
a
can always be written as
J.a/
D
˛.a/n
1
;
(3.1)
