Environmental Engineering Reference
In-Depth Information
We now consider the general character of coalescence in a cluster plasma as-
suming this plasma to consist of a rareness buffer gas in accordance with criteri-
on (4.109), an atomic (metal) vapor, and a gas of (metal) clusters. The rate constant
for attachment of an atom to a cluster consisting of
n
1 atoms is given by (6.67)
k
0
n
2/3
,
k
n
D
and the rate of atom evaporation from the cluster surface is determined by the
principle of detailed balance and is
k
0
n
2/3
N
sat
(
T
)exp
ε
,
ε
0
n
ν
D
n
T
where
N
sat
(
T
) is the number density of atoms for the saturated vapor pressure at a
given temperature and
n
is the difference between the atom binding energies
for a bulk particle and a cluster consisting of
n
atoms. Taking the atom binding
energy in a cluster as the sum of the volume and surface parts [127], we obtain
the difference given above for the atom binding energies due to the surface cluster
energies in the form
ε
ε
0
D
Δ
ε
n
1/3
ε
ε
,
0
n
and then the critical cluster size
n
cr
for which the rates of atom attachment to a
cluster and cluster evaporation become identical follows from the relation [40]
N
sat
(
T
)exp
!
,
Δ
ε
Tn
1/3
N
m
D
(6.83)
cr
where
N
m
is the equilibrium number density of free atoms. On the basis of the
above rates of cluster growth and evaporation which proceed step by step as a re-
sult of addition of one atom to or release of one atom from a cluster, we have the
kinetic equation for the size distribution function for clusters
f
n
in the linear ap-
proximation as a flux of atoms in a space of cluster sizes [40]:
k
0
n
2/3
f
n
N
m
N
sat
(
T
)exp
Δ
ε
Tn
1/3
.
@
f
n
@
t
D
@
j
n
,
j
n
D
(6.84)
@
n
The kinetic equation (6.84) describes evolution of the size distribution function
for the coagulation character of cluster growth. We note the characteristics of this
process. The cluster system may be divided into two groups with sizes above and
below the critical size (6.83), and clusters of one group cannot transfer to other
group [40]. Therefore, cluster growth in the coalescence process is accompanied by
an increase of the cluster critical radius that almost conserves the ratio between the
numbers of clusters for the first and second groups. Next, the rate of cluster growth
according to (6.84) depends on the parameter
D
Δ
ε
Tn
1/3
a
.
(6.85)
