2.2 Linear Cooling Ramp Method
51
Second, the use of a slow cooling rate will allow a long time, and hence more
chance of nucleation, at a shallow subcooling (high temperature). Therefore, the
experimentally accessible range of system subcoolings can be expanded at the low
end of system subcoolings. In contrast, the experimental accessibility to the high
end of system subcoolings does not appear to suffer by the use of a slow cooling
rate, in part because rapid quenching that may enable attainment of deep system
subcoolings cannot be carried out orderly, let alone linearly. Thus, the use of a slow
cooling rate can expand the range of system subcoolings over which the nucleation
curve of interest can be determined.
Even though the advantage is clear, the use of a slow cooling rate has an obvious
drawback in which the process becomes time-consuming, especially when one needs
to collect hundreds of data points. Besides, the physical meaning of the impact of the
use of a linear cooling rate is not clear, regardless of the cooling rate. Puzzlingly, it
was experimentally observed that the nucleation rate at a given subcooling became
progressively lower as the experimental cooling rates were lowered [15]. Even though
the effect was rather minor, it was consistently observed and appears real. What makes
the matter worse, the observed trend is difficult to explain.
First, the detection delay due to the finite growth rate of a nucleus is expected
to worsen the faster the cooling rate, because the sample temperature will go lower
for a given growth time of a nucleus. This factor would thus shift a given nucleation
curve to the left. Second, a thermal lag would cause the “true” sample temperature
to be higher (i.e., the subcooling and the driving force to be smaller) for which the
nucleation rate should be lower due to the smaller driving force for nucleation. This
factor would also shift a given nucleation curve to the left. And this factor is also
expected to worsen as the cooling rate becomes faster. Third, the undersaturation of a
guest gas in a clathrate system would cause the “true” driving force to be smaller for
which the nucleation rate should be lower. Again, this effect is expected to worsen
the faster the cooling rate because the rapidly changing temperature would not allow
sufficient time for the system to “catch up”.
In short, all three factors of detection delay, thermal lag, and guest undersaturation for a clathrate system that are expected to be present during a linear cooling
ramp would render the “true” sample temperature higher (“true” system subcooling
smaller) than the experimentally determined face values [18]. Thus, each of these
factors is expected to shift a given nucleation curve to the left and, as the cooling rate
becomes faster, the nucleation curve would shift even more to the left. As Fig. (2.11)
shows, a shift of a nucleation curve to the left would render the nucleation rate at a
given subcooling to be higher, the opposite of the experimentally observed trend.
It has not been possible to decouple these three factors or assess the relative impact
of each factor. As for the first factor of detection delay due to the finite growth rate
of a nucleus, a plot of lnF versus t at a constant subcooling (like the one shown in
Fig. (2.3b)) would shift to the left but the slope of d(lnF)/dt could be very similar.
Then, one may expect that the resulting nucleation rate, k, is unlikely to be materially
affected by the detection delay. We also note that the size of such detection delays is
51
Second, the use of a slow cooling rate will allow a long time, and hence more
chance of nucleation, at a shallow subcooling (high temperature). Therefore, the
experimentally accessible range of system subcoolings can be expanded at the low
end of system subcoolings. In contrast, the experimental accessibility to the high
end of system subcoolings does not appear to suffer by the use of a slow cooling
rate, in part because rapid quenching that may enable attainment of deep system
subcoolings cannot be carried out orderly, let alone linearly. Thus, the use of a slow
cooling rate can expand the range of system subcoolings over which the nucleation
curve of interest can be determined.
Even though the advantage is clear, the use of a slow cooling rate has an obvious
drawback in which the process becomes time-consuming, especially when one needs
to collect hundreds of data points. Besides, the physical meaning of the impact of the
use of a linear cooling rate is not clear, regardless of the cooling rate. Puzzlingly, it
was experimentally observed that the nucleation rate at a given subcooling became
progressively lower as the experimental cooling rates were lowered [15]. Even though
the effect was rather minor, it was consistently observed and appears real. What makes
the matter worse, the observed trend is difficult to explain.
First, the detection delay due to the finite growth rate of a nucleus is expected
to worsen the faster the cooling rate, because the sample temperature will go lower
for a given growth time of a nucleus. This factor would thus shift a given nucleation
curve to the left. Second, a thermal lag would cause the “true” sample temperature
to be higher (i.e., the subcooling and the driving force to be smaller) for which the
nucleation rate should be lower due to the smaller driving force for nucleation. This
factor would also shift a given nucleation curve to the left. And this factor is also
expected to worsen as the cooling rate becomes faster. Third, the undersaturation of a
guest gas in a clathrate system would cause the “true” driving force to be smaller for
which the nucleation rate should be lower. Again, this effect is expected to worsen
the faster the cooling rate because the rapidly changing temperature would not allow
sufficient time for the system to “catch up”.
In short, all three factors of detection delay, thermal lag, and guest undersaturation for a clathrate system that are expected to be present during a linear cooling
ramp would render the “true” sample temperature higher (“true” system subcooling
smaller) than the experimentally determined face values [18]. Thus, each of these
factors is expected to shift a given nucleation curve to the left and, as the cooling rate
becomes faster, the nucleation curve would shift even more to the left. As Fig. (2.11)
shows, a shift of a nucleation curve to the left would render the nucleation rate at a
given subcooling to be higher, the opposite of the experimentally observed trend.
It has not been possible to decouple these three factors or assess the relative impact
of each factor. As for the first factor of detection delay due to the finite growth rate
of a nucleus, a plot of lnF versus t at a constant subcooling (like the one shown in
Fig. (2.3b)) would shift to the left but the slope of d(lnF)/dt could be very similar.
Then, one may expect that the resulting nucleation rate, k, is unlikely to be materially
affected by the detection delay. We also note that the size of such detection delays is