2.2 Linear Cooling Ramp Method
45
difference between the blue area in the panel (b) and the green area in the panel (a).
The difference in the areas is equal to the area enclosed by F = 1, the red survival
curve, F = 0 and the black survival curve in the panel (c). The net inhibition effect
of the KHI can thus be expressed with the area labeled with blue in the panel (c).
For finite numbers of linear cooling ramps, Eq. (2.2.8) becomes a summation
instead of an integral. Then, the enclosed area in Fig. 2.6 is calculated by adding
up the areas of horizontal stripes in which each horizontal stripe corresponds to the
contribution from a single linear cooling ramp to the expected T, and the summation
is from F = 0 to F = 1.
The obtained is sometimes called the expected or the most probable
metastable zone width (MSZW). The method described here is statistically far superior to the use of simple numerical averages or medians commonly found in the
literature, because all data points contribute equally. For example, a few outlier data
points out of 100 total data points could pose undue influence on simple numerical
averages but will be systematically accounted for in the present method.
2.2.3 Analysis in the Time Domain
The above analysis in the subcooling domain is useful in quantitative comparisons
of most probable subcoolings between different samples or systems, but does not
involve any element of time and hence is powerless in the determination of nucleation
rates or nucleation curves. The basic theoretical framework that underpins simultaneous determination of an entire nucleation curve over the whole range of experimentally accessible subcoolings from an experimentally obtained survival curve is
documented in [5, 12].
Equation (1.2.7) shows the basic logic behind the nucleation rate at a constant
driving force. The situation becomes more complex in the linear cooling ramp mode
than in the constant temperature mode, because the survival probability function,
F(t), itself changes with subcooling, T. Still, a reasonably tractable protocol can
be established when an experiment uses a constant cooling rate. The driving force
linearly keeps increasing during a linear cooling ramp, so the nucleation probability
density depends on the system subcooling which in turn depends on time: p =
p T (t). Then, the nucleation probability between t and (t + dt) becomes p T (t) ·
dt. Analogous to Eq. (1.2.7), the survival probability at (t + dt) can be expressed in
terms of the survival probability at t, F T (t), and the probability that nucleation does
not occur in the subsequent duration dt, which is [1–p T (t) · dt]. Then
F T +d((T ) (t + dt) = F T (t) · [1 − p T (t)dt]
(2.2.9)
Unlike in Eq. (1.2.7), though, T is not a constant but changes with t. During a
short period, dt, T changes by αdt where α is the experimental cooling rate, which
is a constant for a linear cooling ramp. At the same time, F T (t) will change to
a different function, F T + αdt (t + dt). The key approximation we use here is that,
45
difference between the blue area in the panel (b) and the green area in the panel (a).
The difference in the areas is equal to the area enclosed by F = 1, the red survival
curve, F = 0 and the black survival curve in the panel (c). The net inhibition effect
of the KHI can thus be expressed with the area labeled with blue in the panel (c).
For finite numbers of linear cooling ramps, Eq. (2.2.8) becomes a summation
instead of an integral. Then, the enclosed area in Fig. 2.6 is calculated by adding
up the areas of horizontal stripes in which each horizontal stripe corresponds to the
contribution from a single linear cooling ramp to the expected T, and the summation
is from F = 0 to F = 1.
The obtained
metastable zone width (MSZW). The method described here is statistically far superior to the use of simple numerical averages or medians commonly found in the
literature, because all data points contribute equally. For example, a few outlier data
points out of 100 total data points could pose undue influence on simple numerical
averages but will be systematically accounted for in the present method.
2.2.3 Analysis in the Time Domain
The above analysis in the subcooling domain is useful in quantitative comparisons
of most probable subcoolings between different samples or systems, but does not
involve any element of time and hence is powerless in the determination of nucleation
rates or nucleation curves. The basic theoretical framework that underpins simultaneous determination of an entire nucleation curve over the whole range of experimentally accessible subcoolings from an experimentally obtained survival curve is
documented in [5, 12].
Equation (1.2.7) shows the basic logic behind the nucleation rate at a constant
driving force. The situation becomes more complex in the linear cooling ramp mode
than in the constant temperature mode, because the survival probability function,
F(t), itself changes with subcooling, T. Still, a reasonably tractable protocol can
be established when an experiment uses a constant cooling rate. The driving force
linearly keeps increasing during a linear cooling ramp, so the nucleation probability
density depends on the system subcooling which in turn depends on time: p =
p T (t). Then, the nucleation probability between t and (t + dt) becomes p T (t) ·
dt. Analogous to Eq. (1.2.7), the survival probability at (t + dt) can be expressed in
terms of the survival probability at t, F T (t), and the probability that nucleation does
not occur in the subsequent duration dt, which is [1–p T (t) · dt]. Then
F T +d((T ) (t + dt) = F T (t) · [1 − p T (t)dt]
(2.2.9)
Unlike in Eq. (1.2.7), though, T is not a constant but changes with t. During a
short period, dt, T changes by αdt where α is the experimental cooling rate, which
is a constant for a linear cooling ramp. At the same time, F T (t) will change to
a different function, F T + αdt (t + dt). The key approximation we use here is that,