Environmental Engineering Reference
In-Depth Information
K
Finally, the quantity
Q
rot l
,
is
⎛
⎞
ʹ
S
k
K
c
Q
=
ς ς ς
1 2 3
exp
(19.52)
⎜
⎟
rot
,
l
⎝
⎠
B
l
As was shown previously the quantity
S
c
� in Equation 19.46 (as well as in Equation 19.52) has a
physical sense of the entropy related to the rotational movement of embedded cluster. Therefore,
Q
rot l
K
(due to the averaging in Equation 19.52) has a sense of the rotational partition function for the
embedded spherical drop of volume
V
S
. Following Frenkel's views one can hope that the rotational
movement of such a drop with respect to the environmental viscous liquid at low enough tempera-
ture near the melting point will be like the rotational vibrations of a solid body. The necessary
condition for this kind of vibrations is that the period of vibrations
T
v
be small with respect to the
Maxwell relaxation time τ
M
of shear stress which occurs during the drop rotations [1]:
T
v
<< τ
M
or
ντ
M
>>1 (where ν = 1/
T
v
is the vibrational frequency). One may expect that the last inequality is
valid for the vibrational processes with the frequency ν equal to the Debye frequency ν
max
(or even
less) [38].
There is a series of publications devoted to the evaluation of the vibrational frequency of the
solid cluster [40-42]. These evaluations have shown that the frequency of rotational vibrations ν of
small clusters with the number of molecules
n
≈ 100 is about the Debye frequency ν
max
. Taking into
account the fact that at the ordinary temperatures the partition function of an oscillator depends
weakly enough (only linearly) on frequency, one can use the simplest formula for the vibrational
frequency [40]
ν
,
−
1 6
/
≅
3
n
ν
max
. One can see this formula gives ν ≅ ν
max
for
n
= 27. We suppose that
the equation ν ≅ ν
max
is valid for all the small clusters. Thus, we assume that the rotational partition
function
Q
rot l
K
,
can be written as the vibrational partition function for three degrees of freedom:
−3
⎧
⎫
⎡
⎢
⎤
⎥
⎛
⎜
h
ν
max
⎞
⎟
⎛
⎜
h
k T
ν
⎞
⎟
⎪
⎪
⎪
⎪
K
3
max
Q
≅
(
Q
)
≅
exp
1
−
exp
−
(19.53)
rot l
,
v
2
k T
B
B
19.6 ASSEMBLY OF DROPS: THE CORRECTION
FACTOR FOR THE NUCLEATION RATE
The Kusaka factor considered in the previous section is to give a connection between the partition
function (
q
rest
) of the drop at rest and the partition function (
q
n
) of the drop in motion:
⎛
⎜
⎞
⎟
f
k T
rest
rest
n
q
=
Φ
q
=
Φ
exp
−
(19.54)
n
K
n
K
B
Note that the drop at rest is implied here to be such a drop the work of formation for which
W
is
governed by the expression Equation 19.8 derived by Gibbs;
f
n
rest
is the Helmholtz free energy for
this drop. Strictly speaking, Equation 19.8 is valid only for the critical nucleus. However, the rate of
nucleation is proportional to the number
N
crit
of the critical nuclei [1,11,32]. Therefore, to evaluate
the nucleation rate one should know just the work of formation of these critical nuclei.
To know the number of critical nuclei, one should derive the equilibrium drop size distribution.
We will use Reiss's approach [31] to obtain this distribution. To determine the equilibrium size dis-
tribution, Reiss has considered an ensemble of physical clusters as an ideal gas mixture [32]. It was
assumed in this model that these clusters did not interact with the molecules of the environmental
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