Classical Nucleation Theory - Aerosols: Measurement, Dosimetry, and Health Effects - page 505

where
K is the translation-rotation free energy correction factor (arising due to the so-called replace-
ment free energy [8,10,32])
W crit is the reversible work required to form a critical drop from vapor molecules (it is assumed
as if the drop is at rest in space)
Z in Equation 19.1 is the so-called Zeldovich factor taking into account the fact that the drop is
able not only to grow due to the vapor condensation but also to decrease in size due to the
molecule evaporation:

Y

Z = (19.3)

2 k B T

In CNT the quantity Y is identiied with [1]:

⎛
⎜

⎞
⎟

d W
dn

2

Y = − (19.4)

2

n n crit =

where
W is the reversible work of formation of a non-critical droplet
n and n crit are, respectively, the numbers of molecules in the drop and the critical drop (i.e., the
numbers of molecules enclosed by the equimolar dividing surfaces)

In Equation 19.4 the derivation is made for the gaseous pressure being constant.
Our modern understanding of interfacial thermodynamics has its origins in the Gibbs the-
ory of surface tension [12]. This theory considers a luid maternal phase with i components
and another luid phase (with the same i components within it) being in equilibrium with the
maternal one. This system is compared with a hypothetical reference system composed by two
homogeneous bulk phases (maternal phase α and another phase β) and a mathematical dividing
surface between them, at a certain position. The key parameter in the Gibbs theory is the surface
tension σ S attributed to the so-called surface of tension. In the case of spherical symmetry the
radius of this surface is denoted as R S . The chemical potential μ i of molecules in the bulk phases
is the same as that of the real system and the difference in pressure and composition between the
drop and the phase β is incorporated in the value of surface tension. The critical nucleus (i.e.,
the embryo which is in equilibrium with the maternal phase) is often extremely small in size so
that the homogeneous bulk properties are not attained even at its center, but Gibbs interfacial
thermodynamics remains valid. Thus, to understand the surface thermodynamics of a two-phase
system one should only know the surface tension and the location of the surface of tension.
As follows from the Gibbs theory the surface tension is a function of curvature. In the case of
spherical symmetry for the single component system this dependence on curvature is governed by
the Gibbs-Tolman-Koenig-Buff (GTKB) differential equation [33,34]:

[ ]

d ln σ ( R ) ( 2 δ ( R R ) / )[ 1 + ( ( δ R R ) / ) + ( 1 3 / )( ( δ R R ) / ) ]

2

S S

S S S S S S

= (19.5)

[ ]

2

d ln R 1 + ( 2 δ R R ( ) / )[ 1 + ( ( δ R R ) / ) + ( 1 3 / )( ( δ R R ) / ) ]

S S S S S S S

where δ( R S ) is the so-called Tolman length which is equal to

δ( R ) = R − R (19.6)

S e S