2.1 Constant Temperature Method
37
A constant temperature (subcooling) experiment means that the driving force
does not change with time during an experiment. An important conclusion of
Sect. (1.2) was that the survival probability, F, of such a system has an exponential distribution of induction times of the form e
–ckt , where c is a constant,
k is the nucleation rate, and t is time. We derived this conclusion in Sect. (1.2)
and in [5] from ab initio considerations that the survival probability at a time
(t + dt) is the survival probability at a time t multiplied by the probability
that nucleation does not occur in the subsequent duration dt. Essentially, the
same conclusion can be reached from the Poisson distribution once we assume
that a single nucleation event is all that is required to initiate a phase transition
[3, 4, 6, 7]. Thus, if one can measure the survival probability as a function of induction time, a plot of lnF versus t will yield a straight line with a slope of –ck. Since c
= ln2 in our definition, the most probable nucleation rate can be readily calculated
from the best fit to the slope.
The first step of calculating F(t) is to rearrange a chronological histogram of
induction times, such as the one shown in Fig. (2.1), to an ascending order. Three
examples of chronological histograms and the corresponding rearranged histograms
are shown in Fig. (2.2). In Fig. (2.2a), several experimental runs resulted in zero
experimental induction times, which means that a nucleation event took place before
the system reached the target subcooling temperature of interest. This event becomes
increasingly more common as the subcooling of interest becomes deeper (as the set
temperature becomes colder). In Fig. (2.2b), none of the experimental runs resulted in
zero experimental induction times or none of the induction time measurements were
maxed out by an arbitrary cut-off waiting time. This is undoubtedly an ideal case that
is rare in reality. In Fig. (2.2c), a substantial number of experimental runs reached
the maximum waiting time of 15000 s without encountering nucleation events. As
may be expected, this situation occurs more commonly at shallow subcoolings (high
temperatures). Even in this case, the chronological histogram shown on the left panel
can be rearranged in an ascending order, as shown on the right panel.
The survival probability as a function of induction time, F(t), can be calculated if
one can assume equivalency among all experimental runs—i.e., each experimental
run contributes equally to the whole induction time distribution, regardless of the
chronological sequence. For example, when there are a total of 1000 experimental
runs and if none of the 1000 runs has nucleated at a very short induction time of t =
0+, then F(0+) = 1, as may be expected. Likewise, at a very long induction time, all
experimental runs would have experienced a nucleation event by then F(t → ∞) =
0. In between, F(t) at a given intermediate time, t, can be calculated by dividing the
number of experimental runs that have not experienced a nucleation event until t by
the total number of the experimental runs. For the above example of a total of 1000
experimental runs, F(t) will monotonically decrease with t from 1 to 0 in 1000 steps.
Once F(t) is determined, the calculation of lnF(t) is a simple numerical operation of
taking its natural logarithm.
Where an artificial maximum waiting time is set, F(t) may not reach 0 at the
maximum waiting time. Importantly, even though the numerical values of the induction times of these experimental runs that have not encountered nucleation events
37
A constant temperature (subcooling) experiment means that the driving force
does not change with time during an experiment. An important conclusion of
Sect. (1.2) was that the survival probability, F, of such a system has an exponential distribution of induction times of the form e
–ckt , where c is a constant,
k is the nucleation rate, and t is time. We derived this conclusion in Sect. (1.2)
and in [5] from ab initio considerations that the survival probability at a time
(t + dt) is the survival probability at a time t multiplied by the probability
that nucleation does not occur in the subsequent duration dt. Essentially, the
same conclusion can be reached from the Poisson distribution once we assume
that a single nucleation event is all that is required to initiate a phase transition
[3, 4, 6, 7]. Thus, if one can measure the survival probability as a function of induction time, a plot of lnF versus t will yield a straight line with a slope of –ck. Since c
= ln2 in our definition, the most probable nucleation rate can be readily calculated
from the best fit to the slope.
The first step of calculating F(t) is to rearrange a chronological histogram of
induction times, such as the one shown in Fig. (2.1), to an ascending order. Three
examples of chronological histograms and the corresponding rearranged histograms
are shown in Fig. (2.2). In Fig. (2.2a), several experimental runs resulted in zero
experimental induction times, which means that a nucleation event took place before
the system reached the target subcooling temperature of interest. This event becomes
increasingly more common as the subcooling of interest becomes deeper (as the set
temperature becomes colder). In Fig. (2.2b), none of the experimental runs resulted in
zero experimental induction times or none of the induction time measurements were
maxed out by an arbitrary cut-off waiting time. This is undoubtedly an ideal case that
is rare in reality. In Fig. (2.2c), a substantial number of experimental runs reached
the maximum waiting time of 15000 s without encountering nucleation events. As
may be expected, this situation occurs more commonly at shallow subcoolings (high
temperatures). Even in this case, the chronological histogram shown on the left panel
can be rearranged in an ascending order, as shown on the right panel.
The survival probability as a function of induction time, F(t), can be calculated if
one can assume equivalency among all experimental runs—i.e., each experimental
run contributes equally to the whole induction time distribution, regardless of the
chronological sequence. For example, when there are a total of 1000 experimental
runs and if none of the 1000 runs has nucleated at a very short induction time of t =
0+, then F(0+) = 1, as may be expected. Likewise, at a very long induction time, all
experimental runs would have experienced a nucleation event by then F(t → ∞) =
0. In between, F(t) at a given intermediate time, t, can be calculated by dividing the
number of experimental runs that have not experienced a nucleation event until t by
the total number of the experimental runs. For the above example of a total of 1000
experimental runs, F(t) will monotonically decrease with t from 1 to 0 in 1000 steps.
Once F(t) is determined, the calculation of lnF(t) is a simple numerical operation of
taking its natural logarithm.
Where an artificial maximum waiting time is set, F(t) may not reach 0 at the
maximum waiting time. Importantly, even though the numerical values of the induction times of these experimental runs that have not encountered nucleation events