14
1 Nucleation Theory
integrated area of F from to infinity:
t
0 Fdt =
∞
t Fdt:
t
∫
0
exp(−ckt)dt =
∞
∫
t
exp(−ckt)dt
(1.2.14)
Solving Eq. (1.2.14), one will yield
exp(−ck < t >) = 0.5
(1.2.15)
A salient point to make here is that these attributes do not depend on the nature of
the system or specific mechanisms of nucleation involved in a given system, as our
ab initio derivation shows [8]. A nucleation event generally involves multiple steps.
Even for the simplest case of freezing of a single-component liquid, a nucleation
process must at the very least involve formation of transient clusters of various sizes
that, when time-averaged, have the Boltzmann distribution [5]. The formation of a
critically sized nucleus (sometimes called “supernucleus”) could consist of a certain
number of molecules (or other units of building blocks), n*. A number of different
paths exist that a cluster of n*-mers can form from a given ensemble under a set of
initial conditions; one path might be progressive cascading of monomers, dimers,
trimers, etc., that eventually form a small population of clusters of (n* − 1)-mers
and then n*-mers. In some other cases, such a path might involve a larger leap or
two in between. But the salient point is that, regardless of the nature of such specific
nucleation pathways or the complexity of the system, a nucleation event can only be
realized when a continuous path forms from the beginning to the end (i.e., all the
pieces of the puzzle are in place). And the nucleation rate represents the rate at which
such occurrence is realized, over a whole nucleation pathway, in a unit system size.
We note at this stage that another basic premise of classical nucleation theory
is that a typically large driving force (excess chemical potential) that is required
to overcome an activation barrier would lead to an irreversible overgrowth of the
nuclei to macroscopic and experimentally detectable sizes once the activation barrier
has been surmounted [5]. In other words, the differential in the chemical potential
between the driving force and the thermodynamically stable phase is so large that
any countering factor like the release of the latent heat would be insufficient to
deter the growth of the supernucleus. If this assumption holds, then it becomes
possible to experimentally determine nucleation rates from measurable nucleation
probability density per unit system size per unit time once such detection delay can
be accounted for [9]. This second approximation becomes less accurate as the driving
force becomes small, and the nucleation rate low; the smaller the driving force more
wrong we will be.
An experimental determination of a nucleation probability density per unit system
size and per unit time requires a suitable measure of unit system size. The traditional wisdom has been that a suitable measure of the system size for homogeneous
nucleation is the system volume because the number of potential nucleation sites is
1 Nucleation Theory
integrated area of F from
t
0 Fdt =
∞
t Fdt:
t
∫
0
exp(−ckt)dt =
∞
∫
t
exp(−ckt)dt
(1.2.14)
Solving Eq. (1.2.14), one will yield
exp(−ck < t >) = 0.5
(1.2.15)
A salient point to make here is that these attributes do not depend on the nature of
the system or specific mechanisms of nucleation involved in a given system, as our
ab initio derivation shows [8]. A nucleation event generally involves multiple steps.
Even for the simplest case of freezing of a single-component liquid, a nucleation
process must at the very least involve formation of transient clusters of various sizes
that, when time-averaged, have the Boltzmann distribution [5]. The formation of a
critically sized nucleus (sometimes called “supernucleus”) could consist of a certain
number of molecules (or other units of building blocks), n*. A number of different
paths exist that a cluster of n*-mers can form from a given ensemble under a set of
initial conditions; one path might be progressive cascading of monomers, dimers,
trimers, etc., that eventually form a small population of clusters of (n* − 1)-mers
and then n*-mers. In some other cases, such a path might involve a larger leap or
two in between. But the salient point is that, regardless of the nature of such specific
nucleation pathways or the complexity of the system, a nucleation event can only be
realized when a continuous path forms from the beginning to the end (i.e., all the
pieces of the puzzle are in place). And the nucleation rate represents the rate at which
such occurrence is realized, over a whole nucleation pathway, in a unit system size.
We note at this stage that another basic premise of classical nucleation theory
is that a typically large driving force (excess chemical potential) that is required
to overcome an activation barrier would lead to an irreversible overgrowth of the
nuclei to macroscopic and experimentally detectable sizes once the activation barrier
has been surmounted [5]. In other words, the differential in the chemical potential
between the driving force and the thermodynamically stable phase is so large that
any countering factor like the release of the latent heat would be insufficient to
deter the growth of the supernucleus. If this assumption holds, then it becomes
possible to experimentally determine nucleation rates from measurable nucleation
probability density per unit system size per unit time once such detection delay can
be accounted for [9]. This second approximation becomes less accurate as the driving
force becomes small, and the nucleation rate low; the smaller the driving force more
wrong we will be.
An experimental determination of a nucleation probability density per unit system
size and per unit time requires a suitable measure of unit system size. The traditional wisdom has been that a suitable measure of the system size for homogeneous
nucleation is the system volume because the number of potential nucleation sites is