Isoconversional Methodology - Isoconversional Kinetics of Thermally Stimulated Processes - page 56

and should at certain point exceed 1, i.e., 100 %. This means that the absolute er-
ror ∆ t ʱ would exceed the t ʱ value itself that deems the prediction meaningless. For
example, if E ʱ = 200 ﾱ 10 kJ mol − 1 and T ʱ = 400 K, the 100 % error is reached when
predicting to T 0 = 297 K. Although the error in the lifetime depends on the param-
eters of Eq. 2.49, the obtained estimate gives a fair idea about how far the prediction
temperature can be stretched beyond the actual experimental region. Typically, it is
rather difficult to make reasonably precise predictions at temperatures that deviate
from the experimental temperature region by more than several tens of degrees.
The limited accuracy of kinetic predictions is even bigger problem than the lim-
ited precision. This is because the accuracy cannot be evaluated without performing
actual measurements under the conditions to which the prediction is made. Since
kinetic predictions cannot be carried out without first making some kinetic assump-
tion, the resulting predictions are always as accurate as the underlying assumption
made to carry them out. As already mentioned, the underlying assumption of kinetic
predictions is that the rate equations and respective kinetic triplets evaluated within
an experimental range of temperatures would remain the same outside this range.
Extensive experience suggests that most of the time such assumption is fairly ac-
curate at least when the temperature range of the predictions does not extend more
than several tens of degrees beyond the experimental range. However, this should
not be taken as a rule because sometimes even very small temperature can cause a
failure of the underlying assumption. An example of such situation is when the tem-
perature regions of experiment and prediction are separated by a phase transition. As
discussed in Sect. 1.3, the kinetic triplets may change significantly due to melting
[ 83 ] or solid-solid phase transitions [ 84 , 85 ]. In particular, one should be extremely
cautious when trying to predict the thermal stability of a solid material from higher
temperature data obtained above the melting temperature of the solid material.
Another implicit assumption that may affect the accuracy of kinetic predictions
is that a process proceeds to completion, i.e., ʱ changes from 0 to 1, regardless of
the heating rate and/or temperature. This is not always the case. A well-known ex-
ample is the reaction of epoxy curing. At higher temperatures and/or faster heating
rates, the reaction proceeds to completion (i.e., ʱ = 1), yielding a fully cured epoxy
material that is characterized by the limiting glass transition temperature. If cur-
ing is performed isothermally below the limiting glass transition temperature, the
reaction system vitrifies effectively, stopping the process. The resulting material
reaches some ultimate extent of cure that is smaller than 1 (i.e., ʱ < 1). The use of
progressively lower curing temperatures results in progressively smaller ultimate
extents of cure. Then, the use of complete cure data for predicting the curing below
the limiting glass transition temperature would result in inaccurate predictions, with
the ultimate extent of cure equal to 1 as shown in Fig. 2.20 [ 86 ]. Elimination of the
inaccuracy requires introduction of a diffusion correction factor that accounts for
vitrification [ 86 ].
Yet another source of inaccuracy in kinetic predictions is linked to the inaccuracy
of determining ʱ = 0. This is easy to understand upon recognizing that Eq. 2.39 takes
its origin from Eq. 2.6, in which the lower limit of integration is 0. This means that
Eq. 2.39 assumes that the process starts when t and ʱ are zero. However, for all
practical purposes, the process starts when it becomes detectable experimentally.