Environmental Engineering Reference
In-Depth Information
this case is
K
10
32
cm
6
/s, whereas a typical rate constant for the attachment
process is
k
0
3
10
11
cm
3
/s for metal atoms in a buffer gas. Below we consider
10
21
cm
3
,
The number density
N
m
of free metal atoms, the number density
N
cl
of clusters,
the number density
N
b
of bound atoms in clusters, and the number of cluster
atoms
n
are given by the following set of balance equations according to scheme
(6.68):
the regime
G
1 of cluster growth that takes place at
N
a
3
Z
N
m
k
0
n
2/3
f
n
dn
dN
b
dt
D
dN
m
dt
D
dN
cl
dt
D
KN
m
N
a
,
KN
m
N
a
,
C
(6.69)
where
f
n
is the size distribution function for clusters and is introduced such that
f
n
dn
is the number density of clusters with a size range between
n
and
n
C
dn
,
and this distribution function is normalized by the condition
Z
Z
nf
n
dn
.
N
cl
D
f
n
dn
,
N
b
D
Because the ratio of the first and second terms on the right-hand side of the first
balance equation (6.69) is of the order of
Gn
2/3
, and since a typical number
n
of
cluster atoms and the cluster parameter
G
are large, one can ignore the second
term in the first equation of the balance equations (6.69).
The set of balance equations (6.69) must be supplemented with the equation for
cluster growth
dn
dt
D
k
0
n
2/3
N
m
,
(6.70)
and the boundary condition
N
b
(
τ
)
D
N
m
for this model
N
m
D
const, where
τ
is
the time for conversion of an atomic vapor into a gas of clusters.
The solution of the last equation gives for a cluster size
n
max
(
t
)attime
t
after the
this process has begun
N
m
k
0
t
3
3
n
max
(
t
)
D
.
Correspondingly, the maximum cluster size
n
max
(
τ
) at the end of the growth pro-
cess is
N
m
k
0
3
τ
n
max
(
D
τ
)
.
(6.71)
3
Next, according to the nature of this process, the value
f
n
dn
is proportional to
thetimerange
dt
of formation of diatomic molecules which are the condensation
nuclei for the clusters. This leads to the following size distribution function for
clusters:
C
n
2/3
,
f
n
D
n
<
n
max
(
t
) ,
(6.72)
