Geoscience Reference
In-Depth Information
where it is readily derived on solving the diffusion equation or in the free molecule
regime, where it is a consequence of the balance of in- and out-fluxes J
D
J
in
J
out
.
Here J
in
D
a
2
v
T
n
1
and J
out
D
a
2
v
T
n
C
.
To d er ive E q .
3.31
in the transition regime, let us split the distribution function
into two terms, f
D
f
0
C
Jf
J
,wheref
0
is the part of the distribution independent
of the reactant flux
J
and the second term is linear in
J
because of the linearity of the
transport equation with respect to
f
. We rewrite this equation in the integral form:
f
D
f
fm
C
D
1
RŒf;
(3.81)
where D
1
is the inversion of the differential operator standing on the left-hand side
of the transport equation, RŒf is the collision integral, and f
fm
is the solution to
the collisionless transport equation with Maxwell's boundary condition. Let then
the triangle brackets <
> stand for the operation that produces the flux from
f
,
<f >
D
J. Let us apply this operator to both sides of Eq.
3.81
. On introducing
B
D
<D
1
Rf
J
> gives
J
D
a
2
v
T
.n
1
n
C
/
C
BJ
(3.82)
or
a
2
v
T
.n
1
n
C
/
1
B
J
D
:
(3.83)
This is exactly Eq.
3.31
.
The efficiency ˛.a/ has dimension
l
3
=t
. Very simple dimension considerations
allow us to establish a general form of the condensational efficiency. Three
parameters govern the condensation kinetics: th
e particle
radius
a
,thethermal
velocity of the condensable gas molecules
v
T
D
p
8kT=m, and their diffusivity
D
.
Their dimensions are a
D
Œcm,
v
T
D
Œcm=s,andD
D
cm
2
=s
. Because
˛.a/
D
cm
3
=s
, we can write
˛.a/
D
S
p
a
2
v
T
.a
v
T
=D/:
(3.84)
The multiplier normalizes .0/to unity, .0/
D
1 (see Eq.
3.72
). The function
.x/is not yet known. To find this function one should solve the Boltzmann kinetic
equation that describes the time evolution of the coordinate-velocity distribution of
the condensing molecules, then find the flux of the condensing molecules toward the
particle, and then extract ˛.a/. This is not easy to do in general form. However, the
limiting situations are well analyzable (see Seinfeld and Pandis
2006
): .x/
D
1 at
small
x
and .x/
D
4=x as x
!1
.
It is remarkable that all existing approaches give similar dependence on the
sticking probability:
1
1
C
S
p
F.x/
.x/
D
(3.85)
