Geoscience Reference
In-Depth Information
where
d
is the distance between the opposite charges of the polar molecule. There
are strong grounds for using Eq.
3.127
. The point is that the dipole reorientation time
r
is much shorter than the time of flight, the distance of the order of the particle size
a
/
a=
v
T
. Indeed, the dynamics of the dipole rotation is governed by the equation
of motion I
R
D
F ,whereI
/
md
2
is the moment of inertia of the molecule,
m
is its mass, and the rotating moment is F
/
e
2
d
2
=a
3
. The angular acceleration is
R
/
1=
r
. Hence,
r
/
a
3
m=e
2
and
r
=
a
/
am
v
T
=e
2
/
akT=e
2
/
a=l
c
1.
Here, l
c
D
ˇe
2
is the Coulomb length. Hence,
ˇe
2
d
a
2
;
".a/
D
1
C
(3.128)
Equations
3.116
,
3.124
,and
3.125
define the concentration profile of the
condensing molecules
r
1
!
(3.129)
p
e
ˇde
2
=r
2
3=2;ˇde
2
=r
2
n
2
a
2
r
2
C
2
n.r/
D
3.5.4
Charging of Particles
Here we consider the charging of dielectric particles consisting of a material with
the dielectric permeability ". In this case the image potential is not given by a simple
analytical formula. Still, the situation is not hopeless.
3.5.4.1
Image Potential
The expression for the potential of the image force can be found in Stratton (
1941
):
."
1/
X
nD1
q
2
e
2
2
n
n."
1/
C
1
a
2nC1
r
2nC2
P
n
.0/;
U
image
.r/
D
(3.130)
where
e
is the electron charge,
q
is the ion charge in units of
e
, " is the dielectric
permeability of the particle material, and P
n
.cos/ are the Legendre polynomials
of the
n
-th order. At
D
0,allP
n
.0/
D
1. This fact allows for the summation in
Eq.
3.130
:
q
2
e
2
2a
"
1
"
C
1
S.r=a/;
U
image
.r/
D
(3.131)
