Geoscience Reference
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i.e., the flux is proportional to the ion density n
1
far away from the particle. The
proportionality coefficient ˛.a/ is known as the charging efficiency. The problem is
to find ˛.a/. Nothing, however, prevents us from generalizing Eq.
3.1
as follows:
J.a;R;n
R
/
D
˛.a;R/n
R
;
(3.2)
where n
R
is the ion concentration at a distance
R
from the particle center. It is
important to emphasize that n
R
is (still)
an arbitrary value
introduced as a boundary
condition at the distance
R
(also arbitrary) to a kinetic equation which is necessary
to solve for defining ˛.a;R/. The flux defined by Eq.
3.1
is, thus,
J.a/
D
J.a;
1
;n
1
/ and ˛.a/
D
˛.a;
1
/:
(3.3)
The value of ˛.a;R/ does not depend on n
R
because of the linearity of the
problem.
Let us assume for a moment that we know the exact ion concentration profile
n
exact
.r/ corresponding to the flux J.a/ from infinity (see Eq.
3.1
). Then, using
Eq.
3.2
we can express J.a/ in terms of n
exact
as follows:
J.a/
D
J.a;R;n
exact
.R//
D
˛.a;R/n
exact
.R/:
(3.4)
Now let us choose
R
sufficiently large for the diffusion approximation to
reproduce the exact ion concentration profile
n
exact
.R/
D
n
.J.a//
.R/;
(3.5)
with n
.J/
.r/ being the steady-state ion concentration profile corresponding to a
given total ion flux
J
. The steady-state density of the ion flux j.r/ is the sum of
two terms
j.r/
D
D
dn
.J/
.r/
dr
B
dU.r/
n
.J/
.r/;
(3.6)
dr
where
D
is the ion diffusivity, U.r/ is a potential (here the ion-particle interaction),
and
B
is the ion mobility. According to the Einstein relationship, kTB
D
D.On
the other hand, ion flux density is expressed in terms of total ion flux as follows:
j.r/
D
J=4r
2
, with J>0. Equation
3.6
can be now rewritten as
dr
n
.J/
.r/e
ˇU.r/
D
e
ˇU.r/
d
J
4Dr
2
;
where ˇ
D
1=kT . The solution to this equation is
n
.J/
.r/
D
e
ˇU.r/
0
1
Z
e
ˇU.r
0
/
dr
0
r
0
2
J
4D
@
n
1
A
:
(3.7)
r
