Environmental Engineering Reference
In-Depth Information
means that the form of the distribution function is conserved in the course of par-
ticle evolution, but the average particle size varies in time as
n
D
k
s
N
b
t
,
(6.80)
and we assume the total number density of bound atoms to be conserved in time.
This version of coagulation is valid in a dense buffer gas, in particular, for particle
growth in an aerosol plasma with micrometer-sized aerosol particles. In the case
of nanosized clusters, the case of a gas rareness (4.109) is realized. Assuming each
contact of two colliding clusters leads to their joining, we have the cross section
of the association process for large clusters, that is, consisting of many atoms, as
σ
D
π
r
2
)
2
,where
r
1
and
r
2
are the radii of colliding clusters. This leads to
a size-dependent expression for the rate constant for cluster joining, which for the
liquid drop model of colliding atoms gives
(
r
1
C
s
8
T
m
1/3
)
2
r
n
m
nm
C
π
m
a
k
0
(
n
1/3
r
2
W
k
(
n
,
m
)
D
C
k
0
D
,
,
(6.81)
where
r
W
is the Wigner-Seitz radius and
m
a
is the mass of a cluster atom. This
leads to a nonlinear dependence of the average cluster size on time that has the
form [118, 119]
6.3(
N
b
k
0
t
)
1.2
.
n
D
(6.82)
Since we used the assumption
n
1, this formula is valid under the condition
k
0
N
b
t
1.
The coalescence mechanism of cluster growth or Ostwald ripening [116] results
from an equilibrium between clusters and their atomic vapor through processes of
atom attachment to the cluster surface and cluster vaporization. This mechanism
is realized at high temperatures where the pressure of an atomic vapor is high
enough. Let the critical cluster size be such that the rates of the attachment and
evaporation processes are identical for this size. Then clusters of larger size will
grow, whereas clusters of smaller size will evaporate. As a result, the average cluster
size will increase.
A strict theory of coalescence [120-124] was elaborated for the case of grain
growth in solid solutions. In this case atoms evaporated from the grain surface
propagate inside the solid and attach to another grain. As a result, the number of
small grains decreases, the number of large grains increases, and the average grain
size increases in time. This process is restricted by slow diffusion of atoms inside
the solid. In the case of a cluster plasma with small clusters according to (4.109),
the processes of atom attachment to the cluster surface involving free atoms pro-
ceeds fast by analogy with growth of drops in a supersaturated vapor [125, 126].
Hence, in this case the coalescence process results from equilibrium between gas
of clusters and an atomic vapor [41, 43], and other parameters determine the rate
of the coalescence process.
