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).

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

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.

# n-th Order Kinetics Models

Most existing methods were developed by assuming a n-th order reaction:

(2) dα/d*t* = *k* (1-α)^{n}

where, α — 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 dα/d*t*) reaches maxium at the
temperature (*T*_{p}) where DSC curve displays the peak. i.e.

(3) d(dα/d*t*)/d*t* = *A* exp(-*E*_{a}/*R*/*T*)
(*E*_{a}/*R*/*T*^{2}) (1-α)^{n} d*T*/d*t* - *n*
(1-α)^{(n-1)} *A* exp(-*E*_{a}/*R*/*T*) dα/d*t* = 0

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

(4) ln(β/*T*_{p}^{2}) = ln(*AR*/*E*_{a})
- *E*_{a}/(*RT*_{p})

The Kissinger plot thus says that for a given DSC curve with the heating rate, β, 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(β/*T*_{p}^{2}) against 1/*T*_{p} to obtain the Kissinger plot. From the slope of the
Kissinger plot, one in turn obtains the activation energy, *E*_{a}; futher from the intercept one obtains the
pre-exponential factor, *A*, as well.

## 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:

(5) ln(β) = const - 1.052 *E*_{a}/*R*/*T*_{p}

where, β — heating rate; *E*_{a} — activation energy; *R* — gas constant;
*T*_{p} — peak temperature.

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(β) against 1/*T*_{p}, we obtain the Ozawa plot. The activation
energy can be determined from the slope of the Ozawa plot.

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:

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

or

(6b) ln[*k*(*T*)] = ln(*A*) - *E**RT*)

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(α) is used to replace the function of the nth order
(1-α)^{n};

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

## Friedman's method:

Taking logrithms for Eq.(7) leads to:

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

Under the isoconversion assumption, the function f(α) reaches a given value thus a constant. Therefore the plot of
ln(dα/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.

(9) ln(β) = ln[*AE*_{a}/*R*] - G(α) - 5.3305 - 1.052
*E*_{a}/(*RT*)

Under the isoconUnder the isoconversion assumption, the function G(α) reaches a given value thus a constant. Therefore the plot
of ln(β) against 1/*T* results in a straight line with the slope being -1.052 *E*_{a}/(*RT*)

...

## 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.

**An 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: Δ*H* = -205 J/g;

Frequency factor: *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:**

(10) dα/d*t* =
(*k*_{1} + *k*_{2} α* ^{m}*) (1-α)

^{n}where:

(11)

*k*

_{1}=

*k*

_{10}exp(-

*E*

_{a1}/

*R*/

*T*)

(12)

*k*

_{2}=

*k*

_{20}exp(-

*E*

_{a2}/

*R*/

*T*)

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

dα/d

*t*= 12200 exp[-51450/(

*RT*)] α

^{0.6}(1-α)

^{1.45}

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

**Q1: The resin was accidentally left at 70°C for an hour, how much degree of curing it has experienced?**

Many commercially available mathematics software, e.g. MatLab, Mathmatica can be used to solve the differential equation. Alternatively, VisualLab, a universal mathematics software also developed by CaoTechnology, can be useful. Simply type in the equation and parameters in, click Button Solve, one learns that the resin has cured 2.1% by exposing to 70°C for an hour. Following graph is a screenshot showing how easily VisualLab solves this question.

**Q2: The resin is subsequently heated to 120°C at 5°C/min, what is degree of curing?**

Solving the differential equation again by altering the heating condition as shown in following screenshot, one obtains: 7.2%. The curing graph, plot of degree of curing, α, vs. time (seconds), is also shown below.

To visit DSC Curve Solutions (DCS ®) and VisualLab, click links shown in naviation bar.