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Non-isothermal kinetics for crystallization, curing and reactions has being been a research topic for more than half a century. This is because non-isothermal kinetics involves an equation termed Arrhenius Equation, that states the reaction rate constant is an exponential function of the minus activation energy divided by the product of the gas constant and temperature, as shown in Equation (1).

*k* = *A* Exp[-*E*_{a}/(*RT*)] Eq. (1)

where, *k* - the rate constant; *A* - pre-exponential factor also
known as frequency factor; *E*_{a} - activation energy; *R* -
gas constant (8.314 J/(mol K) and *T* - temperature

Equation (1) appears simple but there is no analytical solution that can be devised so far in spite of constant challenging endeavours. On the other hand, the kinetics parameters are an absolute necessity in order to make the degree of reaction, curing and crystallization calculable.

nth Order Reaction Models

Most existing methods were developed by assuming a nth order reaction:

da/d*t* = *k* (1-a)^{n} Eq.
(2)

where, a - degree of reaction; *t* - time; *n* - order of reaction

The Kissinger Plot:

Kissinger took the deritive of eq.(1) and eq(2) and assumed that
the reaction rate (namely da/d*t*) reaches maxium at the temperature (*T*_{p}) where DSC
curve displays the peak.

i.e. d(da/d*t*)/d*t* = *A* exp(-*E*_{a}/*R*/*T*) (*E*_{a}/*R*/*T*^{2}) (1-a)^{n} d*T*/d*t* - *n* (1-a)^{(n-1)} *A*
exp(-*E*_{a}/*R*/*T*) da/d*t* = 0
Eq. (3)

By further assuming *n* (1-a)^{(n-1)} is a number close to unit, and d*T*/d*t* =
* b* (heating rate) is a constant, Kissinger reached:

ln(* b*/

The Kissinger plot thus says that for a given DSC curve with the heating rate,
*b*, one observes the maxium reaction rate at the peak temperature,
*T*_{p}; for a set
of DSC curves with different heating rates, one can plot the quantity
of ln(* b*/

*
*

The Ozawa Plot:

Prof. Ozawa made an assumption that the degree of reaction is a constant value independent of the heating rate when a DSC curve reaches its peak, and derived following equation:

ln(* b*) = const - 1.052

where, * b* - heating rate;

According to Eq. (5), we can run several DSC experiements with different heating
rates; observe the peak and determine the peak temperature for each DSC curves. Ploting ln(* b*)
against 1/

ASTM E698 Thermal Stability is based on the theory of the Ozawa plot. Typically, three or more experiments are required with different heating rates between 1 and 10°C/minute.

An example how the Ozawa plot is obtained from the DSC curves for an epoxy resin curing is shown below:

*
*

The Borchardt and Daniels (B/D) Method:

From Eq. (1) and Eq. (2), Borchardt and Daniels simply took logarithms leading to:

ln(da/d*t*) = ln(*A*) - *E*_{a}/(*RT*) +
*n* ln(1-a) Eq. (6a)

or
ln[*k*(*T*)] = ln(*A*) - *E*_{a}/(*RT*)
Eq. (6b)

One then determines *a* and da/dt with a tentative *n* to obtain ln[*k*(*T*)] from a DSC curve for 20 curve segments evenly spaced by
temperature starting at 10 percent of the peak height and ending at 50 percent
of the peak area. A plot of ln[*k*(*T*)] versus 1/*T*
should be a straight line by adjusting the tentative value for *n*.
The activation energy, *E*_{a}, and pre-exponential factor, *A*,
are obtained from the slope and intercept of this plot respectively.

Model Free Kinetics (MFK)

Minding of the restriction of the nth order reaction models, researchers have made
attempts to establish model free kinetics analysis methodology, in which a generic form
function f(a) is used to replace the function of
the nth order (1-a)^{n};

da/d*t* = *k*(*T*) f(a) = *A* exp[-*E*_{a}/(*RT*)] f(a) Eq. (7)

Friedman's method:

Taking logrithms for Eq.(7) leads to:

ln(da/d*t*) = ln[f(*a*)] + ln(*A*) -
*E*_{a}/(*RT*) Eq. (8)

Under the isoconversion assumption, the function f(a) reaches a given value thus
a constant. Therefore the plot of ln(da/d*t*) against 1/*T* results in a
straight line with the slope being -*E*_{a}/*R*.

Ozaw-Flynn-Wall nethod:

Ozawa, Flynn and Wall trieOzawa, Flynn and Wall tried to rewrite Eq. (7) in an integral form, followed by a replacement of the integrant by an approximation function. This treatment has led to establishment of the following equation.

ln(* b*) = ln[

Under the isoconUnder the isoconversion assumption, the function G(a) reaches a given value thus
a constant. Therefore the plot of ln(* b*) against 1/

...

Vyazovkin and Wight's Method:

This method obtains the activation energy and the pre-exponential factor by computing the minimum of Arrhenius integrals ...

...

DSC Curve Solutions (DCS) Approach:

**What do we need?** We need to know the activation energy,
*E*_{a}, and
pre-exponential factor, *A*, and all the relevant parameters as well as the reaction model so that we can calculate
the degree of reaction for any given thermal history - the usefulness of a
methodology lies in its predictive power !

**Beyond the nth order model:** Though the nth order model
does describe a mechanism for a number of reactions, it is not good enough for
many other cases. For instance, we often prefer to use the autocatalytic model
for expoxy curing; the Avrami model for crystallization, and so on and so forth.

**DSC Curve Solutions (DCS) ** deals with non-isothermal
kinetics in a different approach, a trial and error approach - it gives
tentative figures of all the kinetic parameters first to enable computation of
the whole DSC curves, compares these simulated DSC curves with the experimental
curves, then gives new tentative figures until you are satisfied with curve
fitting of the experimental DSC curves with DCS simulated DSC curves. In
this manner, DCS determines all the kinetic parameters from any single DSC run.
When you have several DSC runs with different heating rates, you will find that
the kinetic parameters determined from any particular single run fit for other
runs as well -astonishingly amazing ! superior predictive power ! ** **

**Why and how can DCS achieve this?**

**
* ** DSC Curve Solutions takes advantage of all
information contained in a DSC curve (i.e. all the points on curve) to capture
the kinetic and model parameters; in comparison with the conventional methods that use
only a few featured points such as peak temperature and isoconversion points to
extract kinetic parameters.

* Secondly, DSC Curve Solutions integrates non-isothermal kinetics analysis and DSC curve simulation into one; this means, DCS deals with kinetics as one thermal event only, a DSC curve is the combined results of all the thermal events involving in a DSC experiment.

Example:

Adam wants to know the kinetic parameters for an epoxy resin. He heats the
epoxy to 290 °C at 5 °C/min, and obtains a curing exotherm DSC curve shown in
black. Using the autocatalytic model, Adam readily determines the kinetic
parameters of the epoxy resin by fitting this SINGLE run DSC curve (black) with
the DCS curve (red) to satisfaction. Curing enthalpy, ΔH = -205 J/g; *k*_{10} = 0
(Activation energy, *E*_{a1} can be any in this case); Frequency factor,
*k*_{20} = 12200
s^{-1}; Activation energy, *E*_{a2}* = *51450 J/mol;
Exponent, m* = *0.60; Exponent*
n *= 1.45

Autocatalytic model:
da*/*d*t *= (*k*_{1}* +
k*_{2}* a*^{m}) (1-*a*)* ^{n}
*Eq. (10)

where:
*k _{1} = k*

**
k_{2} = k_{20} exp(-E_{a2}/R/T) Eq. (12)**

**
**

Now, using DSC Curve Solutions, we have leant that the resin's curing behaviour is described: *
*

**
d a/**d

This equation is exactly what we want, no more no less.