18
1 Nucleation Theory
Thus, the nucleation probability of System 2 when the three variables of V 1 , V 2 ,
and τ 1 are known is given as a function of time by Eq. (1.2.33). In short, when one
cannot directly measure the nucleation probability of a system of interest but knows
the size of the system, and if one can measure the time evolution of another system
of a different scale, then one can indirectly deduce the nucleation probability of the
first system.
Finally, a simpler equation can be derived when one is only interested in the most
probable or expected induction time of System 2, τ 2 , and not in its full-time profile.
Substituting P(V 2 , τ 2 ) = F(V 2 , τ 2 ) = 0.5 into Eq. (1.2.33) yields
τ 2 = (V 1 /V 2 ) · τ 1
(1.2.34)
This is the same result as Eq. (1.2.23) obtained from the first-order approximation.
In the end, we found that nucleation rate scales proportionately, both with respect
to time and with respect to system size. These conclusions might have been intuitively
expected before all these calculations. Nevertheless, it is reassuring to know that
these conclusions can be deduced from ab initio calculations that do not depend
on any system-specific knowledge or assumptions. Still, the use of Eq. (1.2.33) or
Eq. (1.2.34) depends on one’s ability to specify the appropriate system sizes that can
be used for comparison, in addition to one’s ability to measure the most probable
induction time in a relevant system of a different scale. How one can identify an
appropriate measure of the system size of a given system is another matter that we
will deal with later, and the matter will be deferred to Chap. 5 for clathrate hydrates.
1.2.4 Nucleation Work of Homogeneous Nucleation
and Heterogeneous Nucleation
The nucleation work (and the activation barrier) of homogeneous nucleation and the
nucleation work (and the activation barrier) of heterogeneous nucleation are related
to each other. Here, we only consider a representative case of a spherical cap-shaped
nucleus on a flat foreign substrate (Fig. 1.5). The question is, how the presence of a
foreign surface impacts the nucleation work and the activation barrier?
The activation barrier due to the interfacial free energy for a spherical cap-shaped
nucleus consisting of n molecules on a flat surface of a foreign substrate is given by
Eq. (3.60) of Ref [5]:
γ heterogeneous =
1/3 aγn
2/3
(1.2.35)
where
Ψ = (1/4)(2 + cos θ)(1 − cos θ)
2
(1.2.36)
1 Nucleation Theory
Thus, the nucleation probability of System 2 when the three variables of V 1 , V 2 ,
and τ 1 are known is given as a function of time by Eq. (1.2.33). In short, when one
cannot directly measure the nucleation probability of a system of interest but knows
the size of the system, and if one can measure the time evolution of another system
of a different scale, then one can indirectly deduce the nucleation probability of the
first system.
Finally, a simpler equation can be derived when one is only interested in the most
probable or expected induction time of System 2, τ 2 , and not in its full-time profile.
Substituting P(V 2 , τ 2 ) = F(V 2 , τ 2 ) = 0.5 into Eq. (1.2.33) yields
τ 2 = (V 1 /V 2 ) · τ 1
(1.2.34)
This is the same result as Eq. (1.2.23) obtained from the first-order approximation.
In the end, we found that nucleation rate scales proportionately, both with respect
to time and with respect to system size. These conclusions might have been intuitively
expected before all these calculations. Nevertheless, it is reassuring to know that
these conclusions can be deduced from ab initio calculations that do not depend
on any system-specific knowledge or assumptions. Still, the use of Eq. (1.2.33) or
Eq. (1.2.34) depends on one’s ability to specify the appropriate system sizes that can
be used for comparison, in addition to one’s ability to measure the most probable
induction time in a relevant system of a different scale. How one can identify an
appropriate measure of the system size of a given system is another matter that we
will deal with later, and the matter will be deferred to Chap. 5 for clathrate hydrates.
1.2.4 Nucleation Work of Homogeneous Nucleation
and Heterogeneous Nucleation
The nucleation work (and the activation barrier) of homogeneous nucleation and the
nucleation work (and the activation barrier) of heterogeneous nucleation are related
to each other. Here, we only consider a representative case of a spherical cap-shaped
nucleus on a flat foreign substrate (Fig. 1.5). The question is, how the presence of a
foreign surface impacts the nucleation work and the activation barrier?
The activation barrier due to the interfacial free energy for a spherical cap-shaped
nucleus consisting of n molecules on a flat surface of a foreign substrate is given by
Eq. (3.60) of Ref [5]:
γ heterogeneous =
1/3 aγn
2/3
(1.2.35)
where
Ψ = (1/4)(2 + cos θ)(1 − cos θ)
2
(1.2.36)