46
2 Experimental Methods for Determination of Nucleation Rates
when there are hundreds of data points like in a typical HP-ALTA study, [F T + αdt (t
+ dt) – F T (t)]/F T (t) may be approximated by the numerical average of [F T + αdt (t
+ dt) – F T +αdt (t)]/F T (t) and [F T (t + dt) – F T (t)]/F T (t). For a first approximation, it may suffice to approximate [F T +αdt (t + dt) – F T (t)]/F T (t) with [F T (t
+ dt) – F T (t)]/F T (t). The assumption here is that the data density is high enough
so that the impact of the difference in the functional forms of dF T +αdt (t) and dF T (t)
between two neighboring data points can be neglected over a short duration dt.
We saw in Chap. 1 that F T (t) at a given constant subcooling becomes an exponential function of the form exp(–ckt). Likewise, the functional form of another F T (t)
at another constant T is another exponential function of the form exp(–ckt). As we
will see in the Tea Time break at the end of this chapter, exponential functions form
a set of orthogonal base vectors in a Hilbert space. Therefore, there is unique one-toone correspondence between an exponential function and its exponent, which is the
nucleation rate here (i.e., its uniqueness is assured). Since we defined the constant
of integration as ln2, and t is the time, the difference between the two exponential
functions that belong to two different subcoolings boils down to the difference in
the two nucleation rates (i.e., the values of k) that are specific and unique to each
subcooling. To make clear that the nucleation rate is a function of subcooling and
that each subcooling during a given cooling ramp has a unique and specific value of
k, we may denote each k as k T . For two particular subcoolings of T and (T +
αdt), F T (t) = exp(–ck T t) and F T +αdt (t) = exp(–ck T +αdt t). We also note, as we
saw in Chap. 1, that the nucleation rate becomes independent of time at a constant
subcooling, regardless of the sample history.
A thought experiment might be in order at this stage. We may envision two
identical systems of A and B that are exact replica of each other. We then suppose
that we first start an induction time measurement of the system A at a constant
subcooling of T 1 . We then suppose, sometime later, we start another induction time
measurement of the system B at a slightly greater constant subcooling of T 2 . The
survival probability of the system B, which is supposed to be at a colder temperature,
should fall faster than that of the system A. Hence, there will come a time when
the survival probabilities of the two systems coincide. We may take that particular
moment as time zero. We then wish to know the error that is caused by jumping from
one constant subcooling survival curve of the system A (at T 1 ) to another constant
subcooling survival curve of the system B (at T 2 ) at a short time dt after time zero.
We can further suppose that T 1 happens to have the same numerical value as T
at a certain time during a linear cooling ramp and that T 2 happens to have the same
numerical value as (T + αdt) of the same linear cooling ramp. We also selected the
time zero to be the moment when F T +αdt (t) and F T (t) coincide, as schematically
depicted in Fig. 2.7. Since dt is small, the difference between T 1 and T 2 is small
for a slow cooling rate. This is a type of thought experiment which was once popular
in the early twentieth century.
Now, what should be the suitable measure of the error that most appropriately
describes the difference that arises from jumping from one exponential curve to the
other? If one stayed on the original curve, the change in the survival probability
after the duration dt would have been dF T , as shown in Fig. 2.7. Instead, because
2 Experimental Methods for Determination of Nucleation Rates
when there are hundreds of data points like in a typical HP-ALTA study, [F T + αdt (t
+ dt) – F T (t)]/F T (t) may be approximated by the numerical average of [F T + αdt (t
+ dt) – F T +αdt (t)]/F T (t) and [F T (t + dt) – F T (t)]/F T (t). For a first approximation, it may suffice to approximate [F T +αdt (t + dt) – F T (t)]/F T (t) with [F T (t
+ dt) – F T (t)]/F T (t). The assumption here is that the data density is high enough
so that the impact of the difference in the functional forms of dF T +αdt (t) and dF T (t)
between two neighboring data points can be neglected over a short duration dt.
We saw in Chap. 1 that F T (t) at a given constant subcooling becomes an exponential function of the form exp(–ckt). Likewise, the functional form of another F T (t)
at another constant T is another exponential function of the form exp(–ckt). As we
will see in the Tea Time break at the end of this chapter, exponential functions form
a set of orthogonal base vectors in a Hilbert space. Therefore, there is unique one-toone correspondence between an exponential function and its exponent, which is the
nucleation rate here (i.e., its uniqueness is assured). Since we defined the constant
of integration as ln2, and t is the time, the difference between the two exponential
functions that belong to two different subcoolings boils down to the difference in
the two nucleation rates (i.e., the values of k) that are specific and unique to each
subcooling. To make clear that the nucleation rate is a function of subcooling and
that each subcooling during a given cooling ramp has a unique and specific value of
k, we may denote each k as k T . For two particular subcoolings of T and (T +
αdt), F T (t) = exp(–ck T t) and F T +αdt (t) = exp(–ck T +αdt t). We also note, as we
saw in Chap. 1, that the nucleation rate becomes independent of time at a constant
subcooling, regardless of the sample history.
A thought experiment might be in order at this stage. We may envision two
identical systems of A and B that are exact replica of each other. We then suppose
that we first start an induction time measurement of the system A at a constant
subcooling of T 1 . We then suppose, sometime later, we start another induction time
measurement of the system B at a slightly greater constant subcooling of T 2 . The
survival probability of the system B, which is supposed to be at a colder temperature,
should fall faster than that of the system A. Hence, there will come a time when
the survival probabilities of the two systems coincide. We may take that particular
moment as time zero. We then wish to know the error that is caused by jumping from
one constant subcooling survival curve of the system A (at T 1 ) to another constant
subcooling survival curve of the system B (at T 2 ) at a short time dt after time zero.
We can further suppose that T 1 happens to have the same numerical value as T
at a certain time during a linear cooling ramp and that T 2 happens to have the same
numerical value as (T + αdt) of the same linear cooling ramp. We also selected the
time zero to be the moment when F T +αdt (t) and F T (t) coincide, as schematically
depicted in Fig. 2.7. Since dt is small, the difference between T 1 and T 2 is small
for a slow cooling rate. This is a type of thought experiment which was once popular
in the early twentieth century.
Now, what should be the suitable measure of the error that most appropriately
describes the difference that arises from jumping from one exponential curve to the
other? If one stayed on the original curve, the change in the survival probability
after the duration dt would have been dF T , as shown in Fig. 2.7. Instead, because