52
2 Experimental Methods for Determination of Nucleation Rates
expected to become less significant with subcoolings (the growth rate of a nucleus
to detectable sizes after nucleation is expected to be faster at deeper subcoolings).
As for the second factor of thermal lag, no thermal lag exists under a constant
temperature experiment, so a plot of lnF versus t at a constant subcooling (like the
one shown in Fig. (2.3b)) would remain the same. During a linear cooling ramp experiment, the temperature of the linearly cooling system is always slightly colder than
the temperature of the sample, and the thermal lag of the sample (the temperature
differential) likely remains similar over an entire linear cooling ramp. Then a plot of
lnF T (t) versus t would shift to the left by a fixed offset, and consequently the local
slope of d(lnF T (t))/dt at each point would remain largely the same.
As for the third factor of undersaturation of guest gases for a clathrate hydrate
system, the solubility of a guest gas in water is generally not a linear function of
temperature, much less the timescales with which the guest gas diffuses into the
aqueous phase to replenish the undersaturation during a linear cooling, and consequently its effect during a cooling ramp is expected to be highly complex. We cannot
discard the possibility that the observed downward shift of the experimentally determined nucleation curve as a slower cooling rate is used may not be related to any of
the three plausible physical factors considered above.
At the time of this writing, the most plausible explanation is that each experimentally determined nucleation curve by the linear cooling ramp method should be
interpreted to represent the upper bound of the “true” nucleation rate at each system
subcooling. In other words, the “true” nucleation rate must be likely lower but cannot
be higher than the experimentally determined nucleation curve. If we assume this, we
may have a way of explaining the observed trend because the upper bound progressively shifted lower as the experimental cooling rate is reduced [18]. The essence
of the idea is that the gap between the experimentally determined nucleation curve
and the “true” nucleation curve, which is supposed to be lower, would progressively
narrow as the former progressively shifts downward as the slower cooling rate is
used. Then, an experimentally determined nucleation curve would eventually match
the “true” nucleation curve if an infinitesimal cooling rate could be used.
We note that the difference between the experimentally determined upper bound
and the “true” nucleation rates using a typical cooling rate of an HP-ALTA is well
within the scatter of the induction time distributions of the constant temperature
method [5], and therefore one can expect that the location of the “true” nucleation
rates should be largely similar to the experimentally determined nucleation rates.
We expect that the same will hold when an HP-μDSC is used for linear cooling
ramp measurements of lag time distributions in future. Given that a nucleation curve
typically spans over many orders of magnitude, the above uncertainty by no means
lessens the great value of the linear cooling ramp method.
2 Experimental Methods for Determination of Nucleation Rates
expected to become less significant with subcoolings (the growth rate of a nucleus
to detectable sizes after nucleation is expected to be faster at deeper subcoolings).
As for the second factor of thermal lag, no thermal lag exists under a constant
temperature experiment, so a plot of lnF versus t at a constant subcooling (like the
one shown in Fig. (2.3b)) would remain the same. During a linear cooling ramp experiment, the temperature of the linearly cooling system is always slightly colder than
the temperature of the sample, and the thermal lag of the sample (the temperature
differential) likely remains similar over an entire linear cooling ramp. Then a plot of
lnF T (t) versus t would shift to the left by a fixed offset, and consequently the local
slope of d(lnF T (t))/dt at each point would remain largely the same.
As for the third factor of undersaturation of guest gases for a clathrate hydrate
system, the solubility of a guest gas in water is generally not a linear function of
temperature, much less the timescales with which the guest gas diffuses into the
aqueous phase to replenish the undersaturation during a linear cooling, and consequently its effect during a cooling ramp is expected to be highly complex. We cannot
discard the possibility that the observed downward shift of the experimentally determined nucleation curve as a slower cooling rate is used may not be related to any of
the three plausible physical factors considered above.
At the time of this writing, the most plausible explanation is that each experimentally determined nucleation curve by the linear cooling ramp method should be
interpreted to represent the upper bound of the “true” nucleation rate at each system
subcooling. In other words, the “true” nucleation rate must be likely lower but cannot
be higher than the experimentally determined nucleation curve. If we assume this, we
may have a way of explaining the observed trend because the upper bound progressively shifted lower as the experimental cooling rate is reduced [18]. The essence
of the idea is that the gap between the experimentally determined nucleation curve
and the “true” nucleation curve, which is supposed to be lower, would progressively
narrow as the former progressively shifts downward as the slower cooling rate is
used. Then, an experimentally determined nucleation curve would eventually match
the “true” nucleation curve if an infinitesimal cooling rate could be used.
We note that the difference between the experimentally determined upper bound
and the “true” nucleation rates using a typical cooling rate of an HP-ALTA is well
within the scatter of the induction time distributions of the constant temperature
method [5], and therefore one can expect that the location of the “true” nucleation
rates should be largely similar to the experimentally determined nucleation rates.
We expect that the same will hold when an HP-μDSC is used for linear cooling
ramp measurements of lag time distributions in future. Given that a nucleation curve
typically spans over many orders of magnitude, the above uncertainty by no means
lessens the great value of the linear cooling ramp method.