16
1 Nucleation Theory
It can be seen that the first-order approximation led to a conclusion that the nucleation probability density per unit time scales linearly with the system size, whatever
the measure of the system size might be. This is an expected conclusion and has
indeed been traditionally done in the relevant literature; normalize the experimentally measured nucleation probability density per unit time to the unit system size of
a relevant measure.
Then, under this first-order approximation, the expected induction time (time
at which it is equally likely that a given sample has nucleated or not) scales with
the system size as follows. Noting that P(V ) is for unit time (i.e., P(V ) equals the
nucleation rate), the survival probability F(t) for each of the two otherwise identical
systems except for the different sizes of V 1 and V 2 is
F 1 (t) = e
−ct P 1
(1.2.21)
F 2 (t) = e
−ct P 2 = e
−ct (V 2 /V 1 )P 1
(1.2.22)
Equation (1.2.22) shows that the second system behaves as if it has an effective
nucleation probability density of (V 2 /V 1 )P 1 . Therefore, one can deduce the nucleation probability density of a system of interest (P 2 of the system size of V 2 ) if one can
measure the nucleation probability density of a similar system (P 1 of a presumably
more experimentally convenient system of size V 1 ). We defined above the nucleation
rate, k, so that τ ≡ 1/k gives the expected induction time (time at which it is equally
likely that a given sample has nucleated or not) [8]. Then, the expected induction
time of System 2, τ 2 , in terms of the experimentally measurable expected induction
time of System 1, τ 1 , becomes
τ 2 = 1/[(V 2 /V 1 )P 1 ] = (V 1 /V 2 ) · τ 1
(1.2.23)
Thus, under this first-order approximation, the expected induction time of the
second system is merely the expected induction time of the first system (which
presumably is experimentally more convenient to measure) multiplied by the relative
ratio of the two system sizes of interest.
We can apply essentially the same logic to the time domain. We may consider
how the nucleation probability of a unit size of a sample scales with time under a
constant driving force condition. Cumulative nucleation probability of a unit size
over a duration t, P(t), may be subdivided to smaller durations of dt that add up to t.
Then, analogous to Eq. (1.2.18),
P(t) = 1 − [1 − P(dt)]
(t/dt)
(1.2.24)
It can be readily verified from Eq. (1.2.24) that P(0) = 1–1 = 0 for an infinitesimally short time and P(∞) = 1 − 0 = 1 for an infinitely long time, as expected. It is
important to note that P(t) is no longer for unit time and therefore no longer equals
the nucleation rate.
1 Nucleation Theory
It can be seen that the first-order approximation led to a conclusion that the nucleation probability density per unit time scales linearly with the system size, whatever
the measure of the system size might be. This is an expected conclusion and has
indeed been traditionally done in the relevant literature; normalize the experimentally measured nucleation probability density per unit time to the unit system size of
a relevant measure.
Then, under this first-order approximation, the expected induction time (time
at which it is equally likely that a given sample has nucleated or not) scales with
the system size as follows. Noting that P(V ) is for unit time (i.e., P(V ) equals the
nucleation rate), the survival probability F(t) for each of the two otherwise identical
systems except for the different sizes of V 1 and V 2 is
F 1 (t) = e
−ct P 1
(1.2.21)
F 2 (t) = e
−ct P 2 = e
−ct (V 2 /V 1 )P 1
(1.2.22)
Equation (1.2.22) shows that the second system behaves as if it has an effective
nucleation probability density of (V 2 /V 1 )P 1 . Therefore, one can deduce the nucleation probability density of a system of interest (P 2 of the system size of V 2 ) if one can
measure the nucleation probability density of a similar system (P 1 of a presumably
more experimentally convenient system of size V 1 ). We defined above the nucleation
rate, k, so that τ ≡ 1/k gives the expected induction time (time at which it is equally
likely that a given sample has nucleated or not) [8]. Then, the expected induction
time of System 2, τ 2 , in terms of the experimentally measurable expected induction
time of System 1, τ 1 , becomes
τ 2 = 1/[(V 2 /V 1 )P 1 ] = (V 1 /V 2 ) · τ 1
(1.2.23)
Thus, under this first-order approximation, the expected induction time of the
second system is merely the expected induction time of the first system (which
presumably is experimentally more convenient to measure) multiplied by the relative
ratio of the two system sizes of interest.
We can apply essentially the same logic to the time domain. We may consider
how the nucleation probability of a unit size of a sample scales with time under a
constant driving force condition. Cumulative nucleation probability of a unit size
over a duration t, P(t), may be subdivided to smaller durations of dt that add up to t.
Then, analogous to Eq. (1.2.18),
P(t) = 1 − [1 − P(dt)]
(t/dt)
(1.2.24)
It can be readily verified from Eq. (1.2.24) that P(0) = 1–1 = 0 for an infinitesimally short time and P(∞) = 1 − 0 = 1 for an infinitely long time, as expected. It is
important to note that P(t) is no longer for unit time and therefore no longer equals
the nucleation rate.