Environmental Engineering Reference
In-Depth Information
4.4.5
Mobility and Diffusion of Large Clusters in a Gas
In considering the behavior of a large cluster in an ionized gas, we use the liquid
drop model, that is, we model a cluster by a spherical ball whose radius
r
0
is con-
nected to the number of its atoms by (2.16). We assume the cluster is large enough
so that its radius is large compared with the distance of action of atomic forces, and
then the diffusion cross section of atom-cluster collision is given by (2.18). Next,
we consider the kinetic and diffusion regimes of cluster drift depending on the re-
lation between the cluster radius
r
0
and the mean free path of atoms
λ
.Namely,
the kinetic regime corresponds to the criterion
λ
r
0
or
g
)
1
r
0
(
N
a
σ
,
(4.109)
where
N
a
is the number density of atoms and
g
is the gas-kinetic cross section.
In particular, in atmospheric air the boundary between these regimes corresponds
to a cluster radius
r
0
σ
100 nm, and in reality both cases may be realized.
We start form the diffusion regime if many atoms simultaneously collide with
the cluster surface. The resistive force for a cluster of radius
r
0
that is moving with
velocity
w
is given in this case by the Stokes formula (4.52). Then we obtain the
mobility
K
of a cluster of charge
e
and the cluster diffusion coefficient
D
in a gas
on the basis of the Einstein relation (4.38) [90]:
w
E
D
e
K
0
n
1/3
,
K
0
e
K
D
η
D
D
η
I
6
π
r
0
6
π
r
W
KT
e
D
T
D
0
n
1/3
T
D
D
η
D
,
D
0
D
,
λ
r
0
.
(4.110)
6
π
r
0
6
π
r
W
η
According to these formulas, the dependence on the number
n
of cluster atoms
for the diffusion coefficient and mobility is
D
,
K
n
1/3
. Next, the only cluster
parameter in (4.110) for the cluster mobility and diffusion coefficient is the cluster
radius
r
0
. Table 4.15 contains values of the reduced diffusion coefficients for metal
clusters in some gases.
We now determine the resistive force that acts on a moving cluster in a gas in
the kinetic regime of cluster drift. In this regime a cluster can collide at each mo-
ment with one atom only. The momentum transferred in atom-cluster collision
is
m
a
g
(1
/
is the scattering angle in the
center-of-mass frame of reference, and
g
D
v
w
is the collision velocity, where
v
is
the atom velocity and
w
is the cluster velocity. Because the cluster mass
M
is large
compared with the atom mass
m
a
, the center-of-mass frame of reference coincides
practically with the frame of reference where a cluster is motionless. Next, we have
a Maxwell velocity distribution function for atoms that has the form
cos
#
), where
m
a
is the atom mass,
#
exp
,
N
a
m
a
2
T
3/2
m
a
v
2
'
(
v
)
D
π
2
T
