8
1 Nucleation Theory
1.2 Classical Nucleation Theory
The definition of the term “classical nucleation theory” is not clear or universal in
the literature. In a broad sense, classical nucleation theory may refer to a theoretical
framework that allows the formation of spatially non-uniform patches of different
densities (e.g., clusters of a thermodynamically stable phase within a metastable
parent phase), as opposed to a simultaneous and spatially uniform transition of
macroscopic phases (the density of the metastable parent phase uniformly and simultaneously changes to that of the thermodynamically stable phase). In a narrow sense,
the term may refer to a particular aspect of said broad theoretical framework.
There are excellent textbooks on classical nucleation theory, such as Kashchiev’s
book [5], so here we only cover the essence of classical nucleation theory that is
necessary for an understanding of nucleation of clathrate hydrates. The readers are
also referred to Feynman [2] and Callen [4] for thermodynamics and Reif [3] for
statistical mechanics.
1.2.1 Source of Fluctuations
According to the theory of canonical ensemble of classical statistical mechanics,
the probability that one finds a particle in a certain energy state is described by the
Boltzmann distribution [3]:
P = A exp(−E/kT )
(1.2.1)
where P is the probability of finding a particle in an energy state, E, A is a constant, k is
the Boltzmann constant, and T is the absolute temperature. An important conclusion
of the Boltzmann distribution is that there is always a finite (non-zero) probability
that a particle can occupy a very high energy state, no matter how unstable. In other
words, if one waits long enough, one will sooner or later encounter a moment such
an event occurs. This fluctuation is central to nucleation phenomena.
But, physically, where did such fluctuations come from? A salient point is that
a transition of an elementary particle (photon, electron, etc.) between two energy
states can occur not only between two real energy states but also between a real state
and a virtual one. The uncertainty principle of quantum mechanics states that the
lifetime of such a virtual state is inversely proportional to the energy gap between the
virtual state and the nearest real state. In other words, a particle can assume virtually
any energy state, real or virtual, but it becomes progressively less likely to find the
particle in a virtual state that is far away from a real state. This is the source of
fluctuation embedded at the heart of quantum mechanics. In addition to the non-zero
temperatures on earth, high energy cosmic radiation and environmental background
radiation could provide the necessary energy to surmount an activation barrier for
nucleation.
1 Nucleation Theory
1.2 Classical Nucleation Theory
The definition of the term “classical nucleation theory” is not clear or universal in
the literature. In a broad sense, classical nucleation theory may refer to a theoretical
framework that allows the formation of spatially non-uniform patches of different
densities (e.g., clusters of a thermodynamically stable phase within a metastable
parent phase), as opposed to a simultaneous and spatially uniform transition of
macroscopic phases (the density of the metastable parent phase uniformly and simultaneously changes to that of the thermodynamically stable phase). In a narrow sense,
the term may refer to a particular aspect of said broad theoretical framework.
There are excellent textbooks on classical nucleation theory, such as Kashchiev’s
book [5], so here we only cover the essence of classical nucleation theory that is
necessary for an understanding of nucleation of clathrate hydrates. The readers are
also referred to Feynman [2] and Callen [4] for thermodynamics and Reif [3] for
statistical mechanics.
1.2.1 Source of Fluctuations
According to the theory of canonical ensemble of classical statistical mechanics,
the probability that one finds a particle in a certain energy state is described by the
Boltzmann distribution [3]:
P = A exp(−E/kT )
(1.2.1)
where P is the probability of finding a particle in an energy state, E, A is a constant, k is
the Boltzmann constant, and T is the absolute temperature. An important conclusion
of the Boltzmann distribution is that there is always a finite (non-zero) probability
that a particle can occupy a very high energy state, no matter how unstable. In other
words, if one waits long enough, one will sooner or later encounter a moment such
an event occurs. This fluctuation is central to nucleation phenomena.
But, physically, where did such fluctuations come from? A salient point is that
a transition of an elementary particle (photon, electron, etc.) between two energy
states can occur not only between two real energy states but also between a real state
and a virtual one. The uncertainty principle of quantum mechanics states that the
lifetime of such a virtual state is inversely proportional to the energy gap between the
virtual state and the nearest real state. In other words, a particle can assume virtually
any energy state, real or virtual, but it becomes progressively less likely to find the
particle in a virtual state that is far away from a real state. This is the source of
fluctuation embedded at the heart of quantum mechanics. In addition to the non-zero
temperatures on earth, high energy cosmic radiation and environmental background
radiation could provide the necessary energy to surmount an activation barrier for
nucleation.