1.2 Classical Nucleation Theory
11
and approaches zero toward the critical point) and consequently the G(r) activation
curve moves lower as the system cools. Then the maximum of the G(r) activation
curve, G
∗
activation , also diminishes with the system subcooling. If the driving force
becomes so large that G
∗
activation < 0, then the formation of clusters of any sizes
will become favorable and, as such, the phase transition will proceed with certainty.
Here, G(r) activation < 0 for all r and its maximum is found at r = 0 where G
∗
activation
= G(0) activation . We also note G(r) activation → −∞ as r → ∞. In reality, though,
the phase transition is likely to proceed before the driving force becomes so large.
The reason is the probabilistic nature of the Boltzmann distribution.
P = A exp
−G
∗
activation /kT
(1.2.5)
where A is a constant. The probability of finding the system above G
∗
activation
increases with the system subcooling because G
∗
activation decreases with the system
subcooling. At some point before G
∗
activation reaches zero, the likelihood of finding
a sufficiently large cluster of r > r* becomes high enough for the phase transition
to proceed from there. Another important attribute of the Boltzmann distribution is
that the size distribution of such clusters becomes sharper as the system cools. This
is the essence of nucleation and the source of stochasticity in nucleation phenomena.
1.2.2 Nucleation Rate
The physical meaning of Arrhenius’s law is almost clear now. The probability distribution of a grand canonical ensemble is the Boltzmann distribution in which the
probability of finding a system in a free energy state G is proportional to exp(−G/kT ).
For nucleation, the probability of finding a cluster that can surmount a free energy
gap diminishes exponentially with the size of the free energy gap. In classical nucleation theory, the free energy gap to be surmounted is given by the activation free
energy barrier that arises from the interfacial free energy cost of cluster formation,
G
∗
activation . Thus,
J ≡ A exp
−G
∗
activation /kT
(1.2.6)
where J is called nucleation rate [7] and A is a constant. The constant A may include
the frequency of attachment of molecules and the concentration of potential nucleation sites. As can be seen from Eq. (1.2.6), the nucleation rate is of central importance
to nucleation phenomena of any system that defines the rate at which critically sized
nuclei form [5]. Nucleation rate is thus equivalent to nucleation probability density
per unit time. Since such nucleation probability density depends on the system size,
it is customary to normalize the nucleation rate to unit system size of a suitable
measure.
An important note: an underlying assumption is that the formation of clusters will
not affect the chemical potential of the surrounding monomer molecules more than
11
and approaches zero toward the critical point) and consequently the G(r) activation
curve moves lower as the system cools. Then the maximum of the G(r) activation
curve, G
∗
activation , also diminishes with the system subcooling. If the driving force
becomes so large that G
∗
activation < 0, then the formation of clusters of any sizes
will become favorable and, as such, the phase transition will proceed with certainty.
Here, G(r) activation < 0 for all r and its maximum is found at r = 0 where G
∗
activation
= G(0) activation . We also note G(r) activation → −∞ as r → ∞. In reality, though,
the phase transition is likely to proceed before the driving force becomes so large.
The reason is the probabilistic nature of the Boltzmann distribution.
P = A exp
−G
∗
activation /kT
(1.2.5)
where A is a constant. The probability of finding the system above G
∗
activation
increases with the system subcooling because G
∗
activation decreases with the system
subcooling. At some point before G
∗
activation reaches zero, the likelihood of finding
a sufficiently large cluster of r > r* becomes high enough for the phase transition
to proceed from there. Another important attribute of the Boltzmann distribution is
that the size distribution of such clusters becomes sharper as the system cools. This
is the essence of nucleation and the source of stochasticity in nucleation phenomena.
1.2.2 Nucleation Rate
The physical meaning of Arrhenius’s law is almost clear now. The probability distribution of a grand canonical ensemble is the Boltzmann distribution in which the
probability of finding a system in a free energy state G is proportional to exp(−G/kT ).
For nucleation, the probability of finding a cluster that can surmount a free energy
gap diminishes exponentially with the size of the free energy gap. In classical nucleation theory, the free energy gap to be surmounted is given by the activation free
energy barrier that arises from the interfacial free energy cost of cluster formation,
G
∗
activation . Thus,
J ≡ A exp
−G
∗
activation /kT
(1.2.6)
where J is called nucleation rate [7] and A is a constant. The constant A may include
the frequency of attachment of molecules and the concentration of potential nucleation sites. As can be seen from Eq. (1.2.6), the nucleation rate is of central importance
to nucleation phenomena of any system that defines the rate at which critically sized
nuclei form [5]. Nucleation rate is thus equivalent to nucleation probability density
per unit time. Since such nucleation probability density depends on the system size,
it is customary to normalize the nucleation rate to unit system size of a suitable
measure.
An important note: an underlying assumption is that the formation of clusters will
not affect the chemical potential of the surrounding monomer molecules more than