Environmental Engineering Reference
In-Depth Information
The factor Φ
LP
is to substitute the Frenkel factor in Equation 19.26. As to the nucleation rate, it
is to be multiplied by the free energy correction factor Φ
LP
/
N
1
(the appearance of the denominator
N
1
will be discussed later). One should note that it is assumed in the Lothe and Pound theory that
Equation 19.32 refers to the absolutely incompressible cluster because the relative coordinates of the
n
molecules in the cluster are ixed.
Approximating
q
rep
by exp(
s/k
B
) (where
s
≈ 5
k
B
is the entropy of a single molecule in the bulk
liquid) Lothe and Pound have estimated the magnitude of Φ
LP
/
N
1
to be 10
17
for the water cluster
containing about 100 molecules [8]. A correction factor this large appeared excessively large to
many. Thus, a serious controversy has developed since the beginning of Lothe-Pound theory. Reiss
et al. introduced the concept of the so-called stationary cluster [9,10,32]. The partition function of
the stationary cluster is given by
1
∫
st
n
−
U
/
k T
q
=
d
r
e
(19.33)
n
B
n
3
n
Λ
n
!
n
st
r
∈
V
where
n
particles are all conined to some volume
V
st
, which in turn is held ixed in space
U
n
is particle's interaction potential
Λ is the thermal wavelength of a particle
It was assumed that the reversible work of formation of the stationary cluster is equal to
W
crit
(Equation 19.30). The partition functions
q
n
and
q
st
are linked by the factor Φ
R
(which we refer to
as the Reiss factor):
st
q
= Φ
R n
(19.34)
n
When deriving the expression for the factor Φ
R
Reiss et al. deactivated the rotational motion inside
the volume
V
st
and the translational motion resulting from the luctuation of the position of the
center of mass of the stationary cluster prior to activating the free rotation corresponding to
Q
rot
and the free translation corresponding to
Q
tr
. Since the rotational motion of the
n
particles inside
the volume
V
st
is essentially a free rotation, no explicit account needs to be taken for the rotational
partition function [32]. Therefore, the Reiss factor is the ratio between the partition function for the
free translations in the volume
V
and that for the translations of the center of mass in the volume
V
st
and proves to be [10,32]
V
Φ =
(19.35)
R
3 2
/
3
(
2
π
)
σ
where
σ is the standard deviation in any of the three Cartesian coordinates of the center of mass
2
(
)
is the volume in which the center of mass of the drop luctuates
3
πσ
The evaluation of σ in the framework of the model of rigid spheres gives [10,32]
1/3
1/3
σ ≅
0 2
.
v
0 2
.
v
n
l
=
(19.36)
1 2
/
1/6
n
n
where
v
n
is the drop volume. Under the capillarity approximation
v
n
=
nv
l
, where
v
l
is the volume per
one molecule in the bulk liquid phase. If one assumes that
V
st
=
v
n
, Equations 19.35 and 19.36 lead to the
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