12
1 Nucleation Theory
the usual entropic component of the free energy. However, formation of a cluster will
deplete the monomer molecules from its surroundings and as such if the monomers
are not replenished from the rest of the system in a timely manner, the local concentration and hence the chemical potential of the monomer molecules could temporarily
fall in the vicinity of the cluster due to this depletion. This effect is generally negligible in a single-component system but may not necessarily be so in nucleation of
clathrate hydrates as we will discuss in Chap. 5.
Another important note to make at this point is that a basic premise of classical
nucleation theory is that a metastable system has no memory of the past. It means
that the nucleation probability density of a system only depends on the driving force
at the moment of interest, regardless of its history. It follows that the nucleation
rate becomes constant (time-invariant) when the driving force is constant. Since this
constant driving force condition is the simplest case and forms the basis for further
analysis, we will spend some time in explanation of this setting.
First, we define the survival probability at a constant subcooling temperature
as F(t), which represents the survival probability of a liquid that diminishes with
time, t. F(t) is a function of time and has a numerical value at any given moment
but, unlike the nucleation rate, is not a probability density. F(t) is a probability,
as opposed to a probability density, and directly measurable for a given system:
Experimentally, if one can repeat an identical measurement 100 times and if only
one of the 100 measurements has experienced nucleation at a particular time, then
the survival probability at that time can be interpreted as 99%. Strictly speaking, no
two measurements can be identical; at the very least either the time or the space in
which said two measurements are carried out must be different (never mind that the
universe itself is expanding at a rapid rate). In reality, more factors other than just the
time or the space are likely different, so the concept of being “identical” is only valid
in a thought experiment. Nevertheless, we phrase two very similar measurements as
“identical” in this book, and in the world in fact, which incidentally highlights the
intrinsically approximate nature of Modern Physics.
We define the nucleation probability density (i.e., nucleation rate) at time t as p,
which, as we saw above, is time-invariant under the constant driving force condition.
The nucleation probability between t and a short time dt later, (t + dt), can be
expressed in terms of the nucleation rate (nucleation probability density), p, as p
· dt. Ab initio considerations show that the survival probability at (t + dt) must be
the survival probability at time t multiplied by the probability that nucleation does
not occur in the subsequent duration dt. Mathematically, this probability can be
expressed as [1 − p · dt]. Thus,
F(t + dt) = F(t) · [1 − p · dt]
(1.2.7)
Then,
[F(t + dt) − F(t)]/F(t) = −p · dt
(1.2.8)
1 Nucleation Theory
the usual entropic component of the free energy. However, formation of a cluster will
deplete the monomer molecules from its surroundings and as such if the monomers
are not replenished from the rest of the system in a timely manner, the local concentration and hence the chemical potential of the monomer molecules could temporarily
fall in the vicinity of the cluster due to this depletion. This effect is generally negligible in a single-component system but may not necessarily be so in nucleation of
clathrate hydrates as we will discuss in Chap. 5.
Another important note to make at this point is that a basic premise of classical
nucleation theory is that a metastable system has no memory of the past. It means
that the nucleation probability density of a system only depends on the driving force
at the moment of interest, regardless of its history. It follows that the nucleation
rate becomes constant (time-invariant) when the driving force is constant. Since this
constant driving force condition is the simplest case and forms the basis for further
analysis, we will spend some time in explanation of this setting.
First, we define the survival probability at a constant subcooling temperature
as F(t), which represents the survival probability of a liquid that diminishes with
time, t. F(t) is a function of time and has a numerical value at any given moment
but, unlike the nucleation rate, is not a probability density. F(t) is a probability,
as opposed to a probability density, and directly measurable for a given system:
Experimentally, if one can repeat an identical measurement 100 times and if only
one of the 100 measurements has experienced nucleation at a particular time, then
the survival probability at that time can be interpreted as 99%. Strictly speaking, no
two measurements can be identical; at the very least either the time or the space in
which said two measurements are carried out must be different (never mind that the
universe itself is expanding at a rapid rate). In reality, more factors other than just the
time or the space are likely different, so the concept of being “identical” is only valid
in a thought experiment. Nevertheless, we phrase two very similar measurements as
“identical” in this book, and in the world in fact, which incidentally highlights the
intrinsically approximate nature of Modern Physics.
We define the nucleation probability density (i.e., nucleation rate) at time t as p,
which, as we saw above, is time-invariant under the constant driving force condition.
The nucleation probability between t and a short time dt later, (t + dt), can be
expressed in terms of the nucleation rate (nucleation probability density), p, as p
· dt. Ab initio considerations show that the survival probability at (t + dt) must be
the survival probability at time t multiplied by the probability that nucleation does
not occur in the subsequent duration dt. Mathematically, this probability can be
expressed as [1 − p · dt]. Thus,
F(t + dt) = F(t) · [1 − p · dt]
(1.2.7)
Then,
[F(t + dt) − F(t)]/F(t) = −p · dt
(1.2.8)