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in the range 10 7 -10 9 for all metals except for Cs for which it is about 10 11 . The high values of Q ro K for
Cs are due to the high radius R S of the critical nucleus which is a few times larger for Cs than that
for other metals. The vibrational partition function Q rot l
K
, is not less than 0.91 for all the cases which
gives the difference between the quantum and classical forms of the partition function not higher
than 4% and it looks quite reasonable to substitute the quantum form to the classical one. Thus, in
the case of metals the total correction factor is very large being 10 15 -10 19 . It is necessary to note once
again that this correction factor takes into account the translational and rotational degrees of free-
dom of the critical nucleus and, therefore, the quantity σ S ( R S ) presented in Table 19.2 is referred to
the critical nucleus at rest and, hence, has the same nature as the capillary meniscus surface tension.
The last column in Table 19.2 gives the ratio between the nucleation rate measured experimentally
( I exp ) and that predicted by CNT ( I CNT ). The CNT prediction was calculated by Equation 19.9 with
R e estimated by the classical form of the Kelvin formula:
ρ ln
m
R
=
(19.69)
e
k T
S
B
One should note that in general case Equation 19.69 is applicable only for large drops ( R e > 10-100 nm);
however, for the Lennard-Jones systems it was found to be correct even for 1 nm size drops [24,25].
As seen from Table 19.2 the monovalent metals demonstrate the ratio between the experimental
measurements and CNT predictions of the nucleation rate in the range 10 4 -10 −12 . On the other
hand, for bivalent metals the CNT nucleation rate is many tens of orders of magnitude lesser than
the experimental rate. This difference between mono- and bivalent metals is related to the fact that
the CNT formula neglects both the correction factor (underestimating the nucleation rate) and the
dependence of σ S on the drop radius (overestimating the nucleation rate in the case of monovalent
metals and underestimating it in the case of bivalent metals). Thus, the neglect of Φ/ N 1 and the
size dependence of σ S in the case of monovalent metals gives a partial compensation resulting in
decrease of the total error in the CNT formula, but in the case of bivalent metals both these factors
act in concert increasing the total error.
Thus, the considerations of this section show that the formula for the nucleation rate is a useful
tool to study the thermodynamic properties of the nano-sized systems. The approximate formula
Equation 19.61 can be used in calculations of the nucleation rate giving high enough accuracy.
19.9  CONCLUSION
An analytical formula for the translation-rotation correction factor is proposed. The formula is
based on the Reiss approach considering the contribution from the clusters translational degrees of
freedom, the Frenkel's kinetic theory of liquids, and Kusaka's theory. The formula for Zeldovitch
factor that takes into account the size dependence of the critical nucleus surface tension is also
derived. With the help of these two formulas the rigorous expression Equation 19.60 for the rate
of vapor-to-liquid homogeneous nucleation is obtained. However, despite being rigorous the
expression cannot be used for direct calculation of the nucleation rate as it includes the unknown
quantity: the surface tension of critical nucleus σ S . At the same time the expression 19.60 is a
useful tool in the solution of reverse problem, that is, the investigation of the thermodynamic
properties of interface surface which are fully characterized by the state function σ S . Such an
investigation was carried out in this paper for some mono- and bivalent metals. Using Equation
19.60 (or simpliied Equation 19.61) we determined the critical drops' surface tension and radius
of the surface of tension R S . It was found that the metals considered can be divided in accordance
with their valency: for monovalent metals σ S > σ and for bivalent ones σ S < σ . It is important
to note that for all the metals σ S differs essentially from the lat interface surface tension σ .
In advance this large difference between σ S and σ could be expected only for bivalent metals as
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