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quantity Φ
R
/
N
1
≈ 10
6
for a water cluster of 100 molecules [32]. This value is considerably less than the
estimation of Lothe and Pound for the ratio of translational partition functions
Q
10
⊕10 . The
difference between the evaluation of Reiss and that of Lothe and Pound is due to the fact that in
deriving Equation 19.35 the assumption of the incompressibility of the liquid was not used.
Reiss notes reasonably [32] that in the theory of Lothe and Pound the translational degrees of
freedom are to be related to the motion of the center of mass of the embedded (compressible) cluster
with respect to the ixed spherical boundary of the cluster but not to the vibrational translation of
the rigid cluster as a whole.
On the other hand, Reiss believes that in the theory of Lothe and Pound (as well as in the case of
stationary cluster) the ratio
Q Q
rot
l
/
N
Q
tr
tr
1
/
l
is about unity assuming that the rotation of a drop which is a
part of a bulk liquid is essentially the same as the free rotation [32], that is, no explicit account needs
to be taken for the rotational partition function (as well as in the case of stationary cluster). However,
one can agree with the statement that
Q Q
rot
l
is about unity only in the case of high temperatures.
But at the temperature near the melting point (typical temperatures for homogeneous nucleation
experiments) the mechanical behavior of the viscous liquid is to be more similar to that of the solid
in the case of quick processes (for the time shorter than the Maxwell relaxation time) [1].
rot
/
rot
19.5 THEORY OF KUSAKA AND ANALYTICAL FORMULA
FOR THE CORRECTION FACTOR
As is seen from the preceding section, the theory of Lothe and Pound considers the Gibbs extru-
sion process but uses rather rough and ill-founded approximations. Therefore, it seems to be clear
that the approximate formula of Lothe and Pound (Equation 19.31) cannot be used as a basis for
the calculation of correction factor. To derive a formula suitable for such a calculation, Kusaka has
developed recently a rigorous statistical-mechanical approach [11] based on the Gibbs extrusion
process considered in the previous section. Let us look at the milestones of the Kusaka's theory.
The isothermal-isobaric partition function of the bulk liquid held at constant (
T
,
P
l
,
N
) is [11]
dV
a
exp(
−
PV k T
/
)
⎛
⎜
H
k T
⎞
⎟
∫
∫
l
B
N
N
N
Y T P N
( ,
,
)
=
d
p
d
r
exp
−
(19.37)
l
3
N
h
N
!
B
where
p
N
collectively denotes the momentum of each of the
N
particles
h
is Planck's constant
H
N
is the system Hamiltonian
The constant
a
arises from the mechanical degrees of freedom of a piston imposing the constant
pressure
P
l
Then a cluster embedded in the bulk liquid phase is deined by taking a spherical region of vol-
ume
V
S
, which contains
m
particles. The phase points embraced by Equation 19.37 are partitioned
according to the number
m
of molecules inside the spherical region:
N
dV
a
⎛
⎜
PV
k T
⎞
⎟
1
⎛
⎜
H
k T
⎞
⎟
∑
−
=
∫
∫
l
N
−
m
N
−
m
N
−
m
Y T P N
( ,
,
)
=
exp
−
×
d
p
d
r
exp
−
l
3
(
N m
)
h
(
N m
−
)!
B
B
m
3
N m
−
r
∈ −
V
V
S
1
⎛
⎜
H
k T
⎞
⎟
⎛
⎜
U
B
⎞
⎟
∫
m
m
m
int
×
d
p
d
r
exp
−
exp −
(19.38)
3
m
h m
!
k
B
m
r
∈
V
S
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