1.2 Classical Nucleation Theory
13
It follows that
d(ln F(t))/dt = −p
(1.2.9)
Equation (1.2.9) shows that the nucleation probability density, p, at a given
moment t is given by the negative of the derivative of lnF with respect to t at that
moment. Solving the differential equation (Eq. 1.2.9) while neglecting the constant
of integration yields
F(t) = exp(− p · t)
(1.2.10)
If we define the most probable (expected) survival time,, as the time when it is
equally likely that a randomly selected sample has nucleated or not (experimentally,
when half of the samples are found to have nucleated), then
< t >≡ ln 2/ p(t) ≈ 0.7/ p(t)
(1.2.11)
The validity of Eq. (1.2.11) can be verified by substituting t = ln2/p(t) into F(t)
= exp[−p · t], which will yield F(t) = 2
−1
= 0.5. This most probable (expected)
survival time is what induction time in the relevant literature is referring to, which is
the time elapsed from the attainment of subcooling of interest to the eventual moment
of nucleation.
We neglected the constant of integration when we solved the differential equation (Eq. 1.2.9). While the nucleation rate is physically equivalent to the nucleation
probability density, we still have to account for the constant of integration for these
two properties to match. For now, let us employ a different symbol, k, to express
the nucleation rate so that we can distinguish the nucleation rate from the nucleation
probability density, p. The nucleation rate, k, can be defined as the inverse of the
average (most probable) induction time:
k ≡< t >
−1
(1.2.12)
Then, from Eq. (1.2.11),
k = p(t)/ ln 2 ≈ p(t)/0.7
(1.2.13)
Thus, strictly speaking, the nucleation rate, k, differs from the nucleation probability density, p, by a constant, which has a numerical value of ln2. Substituting
Eq. (1.2.13) into Eq. (1.2.10) reveals an important attribute of a nucleation rate
under a constant driving force condition: the survival probability of the system, F(t),
diminishes exponentially with time as e
−ckt , where c is a constant (c = ln2 in our
definition) [5, 8–12]. This exponential distribution of induction times can be verified
from the expected fact that the most probable induction time,, should coincide
with the time at which the integrated area of F from 0 to becomes equal to the
13
It follows that
d(ln F(t))/dt = −p
(1.2.9)
Equation (1.2.9) shows that the nucleation probability density, p, at a given
moment t is given by the negative of the derivative of lnF with respect to t at that
moment. Solving the differential equation (Eq. 1.2.9) while neglecting the constant
of integration yields
F(t) = exp(− p · t)
(1.2.10)
If we define the most probable (expected) survival time,
equally likely that a randomly selected sample has nucleated or not (experimentally,
when half of the samples are found to have nucleated), then
< t >≡ ln 2/ p(t) ≈ 0.7/ p(t)
(1.2.11)
The validity of Eq. (1.2.11) can be verified by substituting t = ln2/p(t) into F(t)
= exp[−p · t], which will yield F(t) = 2
−1
= 0.5. This most probable (expected)
survival time is what induction time in the relevant literature is referring to, which is
the time elapsed from the attainment of subcooling of interest to the eventual moment
of nucleation.
We neglected the constant of integration when we solved the differential equation (Eq. 1.2.9). While the nucleation rate is physically equivalent to the nucleation
probability density, we still have to account for the constant of integration for these
two properties to match. For now, let us employ a different symbol, k, to express
the nucleation rate so that we can distinguish the nucleation rate from the nucleation
probability density, p. The nucleation rate, k, can be defined as the inverse of the
average (most probable) induction time:
k ≡< t >
−1
(1.2.12)
Then, from Eq. (1.2.11),
k = p(t)/ ln 2 ≈ p(t)/0.7
(1.2.13)
Thus, strictly speaking, the nucleation rate, k, differs from the nucleation probability density, p, by a constant, which has a numerical value of ln2. Substituting
Eq. (1.2.13) into Eq. (1.2.10) reveals an important attribute of a nucleation rate
under a constant driving force condition: the survival probability of the system, F(t),
diminishes exponentially with time as e
−ckt , where c is a constant (c = ln2 in our
definition) [5, 8–12]. This exponential distribution of induction times can be verified
from the expected fact that the most probable induction time,
with the time at which the integrated area of F from 0 to