Geoscience Reference
In-Depth Information
Again, in the limit of small
a
the matching distance is of the order of the ion
mean free parts and independent of the particle size. Moreover, it is independent of
the ion-particle potential.
In the limit of large particles a
l the left-hand side of Eq.
3.172
becomes large,
which can happen because of the growth of the expression under the square root on
the right-hand side of this equation when
R
approaches
a
. Our numerical analysis
showed that the solution to Eq.
3.172
can be well approximated by the formula of
the type of Eq.
3.173
, with
2D
v
T
1
C
2ˇ
j
U.a/
j
1
C
ˇ
j
U.a/
j
R
0
.a/
D
:
(3.176)
Let us now analyze the repulsive potentials. In the limit of sm
a
ll
a
the
leading
term on the right-hand side of the equation for
R
is small as 1
j
p
ı
/
1
j
p
ˇU.a/.
Then at small a.q
D
Q
D
1/,
a
1=4
2."
C
2/
"
C
5
1=4
R
1=4
.2D=
v
T
/
1=2
l
1=4
c
:
(3.177)
The dependence of the matching distance
R
(
a
) on the particle size is shown in
Fig.
3.18
for the potentials given by Eq.
3.102
with "
D
4 and q
D
Q
D
1.
In contrast to zero or attractive potentials, where the matching distance has the
order of the ion mean free path and does not depend on the particle size, ion-particle
repulsion leads to the matching distances decreasing with diminishing particle size.
From first sight this fact is very unpleasant, for the diffusion approximation cannot
work at distances much smaller than the ion mean free path. On the other hand, the
dependence of
R
(
a
) is very weak, and even for a
D
1nm
R
is comparable with the
ion mean free path.
3.6.3.2
Very Small Particles
Let us first analyze Eq.
3.172
in the limit of small
a
for the Coulomb attraction,
ˇ
j
U.r/
j D
l
c
=r,wherel
c
D
ˇe
2
is the Coulomb distance .l
c
/
l/.Asisseenfrom
Eq.
3.172
˛
o
al
c
:
Hence,
al
c
v
T
2DR
D
2
al
c
R
2
or
4D
v
T
R
D
