1.2 Classical Nucleation Theory
9
In a system that contains multiple particles, the Gibbs free energy, G, takes the
place of E in Eq. (1.2.1) when the presence of one particle is not independent of the
presence of the others. Still, the essence remains the same: the theory of grand canonical ensemble states that such particles follow a free energy probability distribution
of the Boltzmann form [3]:
P = A exp(−G/kT )
(1.2.2)
where P is the probability of finding the system that contains multiple particles in a
free energy state G, A is a constant, k is the Boltzmann constant, and T is the absolute
temperature.
Consider a one-component system at the melting point of the component. The
solid phase and the liquid phase can coexist at the melting point. If one heats the
solid at the melting point, it will melt due to the presence of the pre-melting layer (this
is the topic of Chap. 4). If one cools the liquid at the melting point, however, it does
not freeze immediately. Instead of releasing the latent heat at the melting point, the
liquid can cool beyond the melting point as if nothing has happened. This remarkable
attribute of a liquid to cool beyond the melting point (subcooling or supercooling)
suggests that the free energy of the system does not remain constant at a constant
temperature but somehow increases during the freezing process. This increase in the
free energy during the freezing process at a given temperature is called an activation
barrier or activation free energy barrier.
Where did such an activation barrier come from? Kashchiev showed that the
probability that a phase transition proceeds through a spatially uniform change in
density throughout a system is vanishingly small [5]. Instead, a spatially non-uniform
pathway is much less taxing [5]. This spatially non-uniform pathway is the basis of
classical nucleation theory, which allows the formation of not only the monomers
but also clusters of various sizes in a supposedly spatially uniform metastable parent
phase. Given that the presence of such clusters contradicts the spatial uniformity
of a phase (definition of a phase in Sect. 1.1), the formation of such entities is
considered transient in nature. Classical nucleation theory further postulates that,
given the transient nature of such clusters, the population of such clusters follows
the Boltzmann distribution of the form Eq. (1.2.2).
In classical nucleation theory, the formation of a cluster will create an interface
around it and it costs interfacial free energy to do so. This free energy cost is proportional to the interfacial area. We may denote this free energy cost as G interfacial . In
addition, as two molecules come together, the system as a whole will gain intermolecular potential energy [6] but lose entropy [3]. The same consideration applies
when multiple molecules come together. These effects are proportional to the mass
or the volume of the cluster. We may then combine the two effects in the form of
G bulk . Then
G activation = G interfacial + G bulk
(1.2.3)
9
In a system that contains multiple particles, the Gibbs free energy, G, takes the
place of E in Eq. (1.2.1) when the presence of one particle is not independent of the
presence of the others. Still, the essence remains the same: the theory of grand canonical ensemble states that such particles follow a free energy probability distribution
of the Boltzmann form [3]:
P = A exp(−G/kT )
(1.2.2)
where P is the probability of finding the system that contains multiple particles in a
free energy state G, A is a constant, k is the Boltzmann constant, and T is the absolute
temperature.
Consider a one-component system at the melting point of the component. The
solid phase and the liquid phase can coexist at the melting point. If one heats the
solid at the melting point, it will melt due to the presence of the pre-melting layer (this
is the topic of Chap. 4). If one cools the liquid at the melting point, however, it does
not freeze immediately. Instead of releasing the latent heat at the melting point, the
liquid can cool beyond the melting point as if nothing has happened. This remarkable
attribute of a liquid to cool beyond the melting point (subcooling or supercooling)
suggests that the free energy of the system does not remain constant at a constant
temperature but somehow increases during the freezing process. This increase in the
free energy during the freezing process at a given temperature is called an activation
barrier or activation free energy barrier.
Where did such an activation barrier come from? Kashchiev showed that the
probability that a phase transition proceeds through a spatially uniform change in
density throughout a system is vanishingly small [5]. Instead, a spatially non-uniform
pathway is much less taxing [5]. This spatially non-uniform pathway is the basis of
classical nucleation theory, which allows the formation of not only the monomers
but also clusters of various sizes in a supposedly spatially uniform metastable parent
phase. Given that the presence of such clusters contradicts the spatial uniformity
of a phase (definition of a phase in Sect. 1.1), the formation of such entities is
considered transient in nature. Classical nucleation theory further postulates that,
given the transient nature of such clusters, the population of such clusters follows
the Boltzmann distribution of the form Eq. (1.2.2).
In classical nucleation theory, the formation of a cluster will create an interface
around it and it costs interfacial free energy to do so. This free energy cost is proportional to the interfacial area. We may denote this free energy cost as G interfacial . In
addition, as two molecules come together, the system as a whole will gain intermolecular potential energy [6] but lose entropy [3]. The same consideration applies
when multiple molecules come together. These effects are proportional to the mass
or the volume of the cluster. We may then combine the two effects in the form of
G bulk . Then
G activation = G interfacial + G bulk
(1.2.3)