Classical Nucleation Theory - Aerosols: Measurement, Dosimetry, and Health Effects - page 504

was proposed to be a factor of 10 3 -10 6 . Recently Kusaka [11] has derived a rigorous formula for
the correction factor within the framework of the Gibbs process of drop formation and calculated
numerically this factor for the Lennard-Jones system. The calculated values ranged from 10 9 to
10 13 . The numerical calculation is probably the most direct way to determine the correction factor.
However, the calculations of this kind are only possible for simple systems and, therefore, an ana-
lytical expression for the correction factor applicable to a wide range of real systems is necessary.
In this paper the derivation of the analytical formula for the translation-rotation correction factor
will be given.
Another source of error in CNT is the so-called capillarity approximation which assumes that
the drop surface tension is that of a lat interface. It is evident that the rigorous formula for the
nucleation rate must take into account the dependence of surface tension on the drop radius. The
problem of the surface tension of curved interfaces was irst analyzed by Gibbs in his thermody-
namic theory of interface [12] taking into account the dependence of surface tension on radius.
One should note that the theory of Lothe-Pound and Kusaka [11] for the correction factor takes
into account the dependence of surface tension on radius automatically. However, the drop surface
tension appears also in the Zeldovich factor in the formula for the nucleation rate. It is hardly
possible to measure directly the small drop surface tension. However, there are many theoretical
contributions in the literature where the surface tension of small drops was calculated as a func-
tion of radius [13-30]. These calculations have shown that the surface tension of small drops with
a size of about 1 nm may be a rather sharp function of radius and differs essentially from that of
a lat interface.
In this chapter we will present a rigorous derivation of the analytical formula for the nucle-
ation rate including both the translation-rotation correction factor and Zeldovich factor taking into
account the dependence of surface tension on the radius of critical drop. Finally, using this formula
we will determine the surface tension of critical drops from the nucleation rate and supersaturation
measured experimentally for some metals taken as an example.

19.2 HOMOGENEOUS NUCLEATION FROM SUPERSATURATED VAPOR
In the classical theory of single component systems the nucleation is considered as a process of
formation of so-called critical nuclei, that is, the drops being in an unstable equilibrium with the
supersaturated vapor. The nucleation rate I (the number of critical nuclei formed per unit time per
unit volume) is [1]

N
V

crit 2

I = β π 4 R Z (19.1)

e

where
N crit is the equilibrium number of critical nuclei in the system of volume V
4≠ R 2 is the surface area of critical nucleus
R e is the radius of equimolar surface
β =

1 / /2 m is the collision frequency for the vapor molecules with the unit surface,
k B is the Boltzmann constant, T is the absolute temperature, m is the mass of molecules, N 1 is
the number of monomeric molecules in the system

( N V k B ) π

The statistical mechanical analysis by Lothe-Pound [8], Reiss et al. [10,31,32], Kusaka [11], and
others in the last decade has shown that N crit is to be deined by the following equation [11]:

⎛
⎜

W
k T

⎞
⎟

crit
B

N = KN

1 exp

− (19.2)

crit