Non isothermal kinetics in phase transformations

Seminar( MT60002)
Bhagyashree (10MT30005)
Munesh Matwa (10MT30012)
Rishabh Singh (10MT30016)
Shashank Kumar Singh (10MT30021)
Hari Shankar Mandi (10MT30026)
Sumeet Khanna (10MT30030)
15-04-2015 1

 Introduction (Kinetics)
 Isothermal process and Non-isothermal process
 Methods of thermal analysis
 Overview of DTA and DSC
 Ozawa method to study the kinetics
 Kissinger method to study the kinetics
 Application of analysis methods
 References
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 What is Kinetics......??
 Study about movement or to move
 In Physics- the study of motion and its causes
 In Chemistry- the study of chemical reaction rates
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Kinetic theory- describes
a gas as a large number of
small particles (atoms or
molecules), all of which
are in constant, random
motion

 It is a change of a system, in which the temperature remains constant:
ΔT = 0. This typically occurs when a system is in contact with an outside
thermal reservoir (heat bath), and the change occurs slowly enough to
allow the system to continually adjust to the temperature of the reservoir
through heat exchange. In contrast, an adiabatic process is where a
system exchanges no heat with its surroundings (Q = 0). In other words,
in an isothermal process, the value ΔT = 0 and therefore ΔU = 0 (only for
an ideal gas) but Q ≠ 0, while in an adiabatic process, ΔT ≠ 0 but Q = 0.
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Applications:- Isothermal processes
can occur in any kind of system,
including highly structured machines,
and even living cells. Various parts of
the cycles of some heat engines are
carried out isothermally and may be
approximated by a Carnot cycle.
Phase changes, such as melting or
evaporation, are also isothermal
processes

 In practice, no reaction is strictly isothermal (all reactions are
accompanied by a heat change)
 It may not possible to attain a thermal equilibrium with the
environment without significant pre-reaction.
 It may not be possible to reproduce the physical characteristics
of the sample from run to run.
 Isothermal conditions are non-existent in actual practice. (do
not apply to real systems)
 Less significance for heterogeneous solid state reactions
 It is inadequate when the product of a reaction becomes the
reactant in another reaction at a higher temperature
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where temperature does not remains constant: ΔT ≠ 0
Advantages of non-isothermal processes
 Less cumbersome and yield more useful data with
less experimentation
 Non-isothermal conditions are nearer to the real
conditions existing in industrial practice
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 It is a group of analytical techniques that studies the property
or properties changes of materials as a function of
temperature
 Purpose of thermal analysis is to provide quantitative and
qualitative information about exothermic, endothermic and
heat capacity changes as a function of temperature and time
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Methods of thermal analysis techniques:
Thermogravimetric analysis (TGA): mass
Differential thermal analysis (DTA): temperature difference
Differential scanning calorimetry (DSC): heat difference

 Sample and reference are symmetrically placed in a Furnace
 Sample dimension and mass should be small
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Conditions for references
It does not undergo thermal events
It does not react with any component
Similar thermal conductivity
Similar heat capacity

 DTA measures the difference in temp between the sample and the
reference when they are both put under the same heat flow
 When the sample and reference are heated identically phase changes
and other thermal processes cause a difference in temperature
between the sample and reference
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 Changes in the sample which lead to the absorption or evolution of
heat can be detected relative to the inert reference
 This differential temperature is then plotted against time, or against
temperature
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 An instrument is designed to measure the heat flow difference
between sample and reference (keep both the reference and sample
at the same temp)
 The Differential Scanning Calorimeter is at constant pressure, heat
flow is equivalent to enthalpy changes: (dq/dt)p = dH/dt
here dH/dt is the heat flow measured in mcal sec. The heat flow
difference between the sample and the reference is
ΔdH/dt = (dH/dt)sample – (dH/dt)reference
15-04-2015 11
This differential heat flow
is then plotted against time,
or against temperature

 Information from dynamic heating mode
◦ Transformation temperature (e.g. onset, peak)
◦ Transformation enthalpy (e.g. area under the peak)
◦ Order of reaction (e.g. Shape of peak)
◦ Activation energy for transformation (e.g. Kissinger analysis)
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Kinetics parameters
rate constant
activation energy
order of reaction
depends on shape, size, position and area of peak of the curve

 Important methods
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Kissinger method
Ozawa method

Objective:
 To find activation energy of non isothermal
reaction
 Find order of reaction
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• Amount of undecomposed reactant decreases as n
decreases
• As n decreases DTA peak becomes asymmetric

Shape index
• Where I and 2 are points of inflection, at these points
Where, i denotes point
of inflection

• Substituting
• Average of T2/T1 ~ 1.08
• Hence S is dependent only on n
From eq.( ), we get
Also (1 – 2RT/E) ~1

The late Dr. Takeo Ozawa (February 14, 1932–October 2nd, 2012)

Two methods for obtaining kinetic parameter using derivative
thermoanalytical curves, suggested by T. Ozawa:
 One method utilizes the linear relation between peak temperature
and heating rate in order to estimate the activation energy, and
only the information of the rate of conversion versus the
temperature is necessary.
 The other method needs the information of both the conversion
and the rate of conversion versus the temperature, and the
Arrhenius plot is made for an assumed kinetic mechanism.

He showed that in rising temperature thermogravimetry, the activation
energy can be graphically obtained.
Theoretical consideration:
In order to estimate the kinetic parameter with thermal analysis the
properties measured should be independent of the temperature and the
experimental time scale, and only depend on the structural quantity of
the sample(x).
The above assumption holds when,
𝑇 = 𝑇𝑜 + α𝑡

Conversion, C ,which is equal to (P- Po)/(P~ -Po), is a function of x:
𝐶 = 𝑓 𝑥 (1)
The structural quantity is assumed to change following ordinary reaction
kinetics :
𝑑𝑥
𝑑𝑡
= 𝐴 𝑒𝑥𝑝 −
Δ𝐸
𝑅𝑇
𝑔 𝑥 (2)
where A, ΔE and R are frequency factor, activation energy and gas
constant , respectively.
Integrating above eq.
𝑑𝑥
𝑔 𝑥
= 𝐴 exp −
∆𝐸
𝑅𝑇
𝑑𝑡 (3)
𝑡
0
𝑥
0
𝐺 𝑥 = 𝐴 𝜃 (4)
Where, θ is reduced time.

When the temperature is increased at a constant rate and the reaction
barely occurs at the initial temperature, θ is given by the following
equation:
𝜃 =
∆𝐸
𝑎𝑅
𝑝(
∆𝐸
𝑅𝑇
)
Where, p-function is given by,
𝑝 𝑦 = −
exp −𝑦
𝑦2
𝑑𝑦
𝑦
0
for y≥ 15, approximation applied,
log 𝑝 𝑦 = −2.315 − 0.4567𝑦
From Eqs. (1) and (4), we can derive the relation between P and Aθ,
because the relation between C and g(x) is equal to the relation between
C and Aθ. The relations between C and Aθ are dependent only on the
functions f and g, i.e. the mechanism of the reaction and the relation of P
with x, and they can be theoretically derived.
(5)
(6)
(7)

For the weight change in the random degradation of a high
polymer, C is given by Simha and Wall [1],
1 − 𝐶 = (1 − 𝑥)(𝐿−1) 1 +
𝑥 𝑁 − 1 𝐿 − 1
𝑁
where x, N and L are the fraction of bonds broken, the initial
degree of polymerization and the least length of the polymer
not volatilized, respectively.
𝑑𝑥
𝑑𝑡
= 𝐴 exp(−
∆𝐸
𝑅𝑇
)(1 − 𝑥)
For, L<<N,
1 − 𝐶 = (1 − 𝑥)(𝐿−1)
[1 + 𝐿 − 1 𝑥]
[1] R. Simha and L. A. WALL, J. Phys. Chem., 56 (1952) 707
(8)
(9)
(10)

 The relations between C and x are obtained by resolving the
equation for a given conversion by using Newton‘s method of
approximation; they are shown in Fig. 3.
 Utilizing the relations in Fig. 3, the curves in Fig. 2 can be drawn.
Thus, we can obtain the theoretical integral types of curve for given
values of the kinetic parameters of A and ΔE and a given mechanism
of f and g, by using these theoretical relations and Eq. (8)
𝑑𝐶
𝑑𝐴θ
= 𝑔(𝑥)
𝑑𝑓 𝑥
𝑑𝑥
Thus, the theoretical relations between dC/dAθ and C are derived from
Eqs (1) and (11). The theoretical relations between dC/dAθ and Aθ are
derived from Eqs (4) and (11).
These relations are the theoretical ones depending only on the
functions f and g.
(11)

Fig.1 The relation between the C
and Aθ for the reactions of the n-th
order
a) 0th, b) 0.5th, c) 1st, d) 1.5th, e)
2nd, f)3rd
Fig.2 The relation between the
conversion, C, and Aθ for the
random degradation of high
polymers; from right to left, L = 2,
3, 4, 5, 6, 7 and 8

Fig. 3. The relation between C and x for the random degradation of high polymers

Fig. 4. The relation between dC/dAθ
and C for the reaction of the order
indicated in the Figure
Fig. 5. The relation between dC/dAθ and C for
the random degradation of high polymers with
L indicated in the Figure

The derivative curves are similarly obtained for a particular set of the
kinetic parameters, because,
𝑑𝐶
𝑑𝑇
=
𝐴
𝑎
exp −
∆𝐸
𝑅𝑇
𝑔 𝑥
𝑑𝑓(𝑥)
𝑑𝑥
And, at peak,
𝑑2 𝐶
𝑑𝑇2
= 0
Fig.6. The relation between
dC/dAθ and Aθ for the reaction of
the order indicated in the Figure
(12)
(13)

For x=xm,
∆𝐸
𝑅𝑇2
𝑑𝑓(𝑥)
𝑑𝑥
+
𝐴
𝑎
exp −
∆𝐸
𝑅𝑇
𝑑𝑔 𝑥
𝑑𝑥
𝑑𝑓 𝑥
𝑑𝑥
+
𝑑2 𝑓 𝑥
𝑑𝑇 𝑑𝑥
= 0
Using approximation and equation (4), (5) and (6) , we get,
𝑑𝑓 𝑥
𝑑𝑥
+ 𝐺 𝑥
𝑑𝑔 𝑥
𝑑𝑥
𝑑𝑓 𝑥
𝑑𝑥
+ 𝑔 𝑥 𝐺 𝑥
𝑑2𝑓 𝑥
𝑑𝑥2
= 0
Here, x is approximately independent of heating rate.
𝐺 𝑥 𝑚 = 𝐴𝜃 𝑚 =
𝐴∆𝐸
𝑎𝑅
𝑝(
∆𝐸
𝑅𝑇
)
Taking log both the sides and using approximation (7), we get,
log 𝑎 = −0.4567
∆𝐸
𝑅𝑇 𝑚
− 2.315 + 𝑙𝑜𝑔
𝐴∆𝐸
𝑅
− log 𝐺(𝑥𝑚)
Thus, log a is in linear relation with 1/Tm, and the "Ozawa plot“..
(14)
(15)
(16)
(17)

If the reaction observed is consistent with Eqs (1) and (2), we can
estimate the activation energy of the reaction by using the linear
dependency of the reciprocal absolute peak temperature on the
logarithm of the heating rate.
In order to obtain a more correct value of the activation energy, the
following correction proposed by Flynn and Wall is applicable:
∆𝐸 𝑐 = ∆𝐸
log 𝑝(𝑦 − 0.5) − log 𝑝(𝑦 + 0.5)
0.4567
Where, ΔEc is corrected activation energy and 𝑦 is the average value of
∆E
RT
.
log
𝑎
𝑇2
𝑚
= −
∆𝐸
𝑅𝑇 𝑚
+ 𝑙𝑜𝑔
𝐴𝑅
∆𝐸
− log 𝐺(𝑥𝑚)
This method of obtaining the activation energy is equivalent to that
proposed by Kissinger for the DTA data, but it is applicable not to DTA
but to the TG and MTA types of thermal analysis and to wider varieties
of reaction.
(18)
(19)

𝐶 = 𝑥
𝑔 𝑥 = (1 − 𝑥) 𝑛
the following equations are derived from Eq. (16):
1 − 𝐶𝑚 = 𝑓 𝑥 =
1
𝑒
, 𝑛 = 1
𝑛1/(1−𝑛)
, 𝑛 ≠ 1
Since,
𝑑𝑓 𝑥
𝑑𝑥
= 1
𝑑2 𝑓 𝑥
𝑑2 𝑥
= 0
And
𝑑𝑔 𝑥
𝑑𝑥
= −𝑛 1 − 𝑥 𝑛−1
Gives,
𝐴𝜃 = 1
In this case following two equation can be derives from above condition
log
𝑎
𝑇2
𝑚
= −
∆𝐸
𝑅𝑇 𝑚
+ 𝑙𝑜𝑔
𝐴𝑅
∆𝐸
log 𝑎 = −0.4567
∆𝐸
𝑅𝑇 𝑚
− 2.315 + 𝑙𝑜𝑔
𝐴∆𝐸
𝑅

Two methods are derived from the above theoretical consideration.
 The one is based on the approximate relation between the peak
temperature and the heating rate, and
 The other is applicable to the case when we can obtain the conversion as
well as the rate of conversion.
In the first method, the logarithm of the heating rate or the logarithm of the
heating rate divided by the square of the absolute peak temperature is plotted
against the reciprocal absolute peak temperature, and from the slope of this
plot we obtain the activation energy by using Eqs (17), (18) and (19).
Then, unless the conversion can be obtained, we cannot normalize the rate
with the total amount of the change, P~ - Po, and since the height of the curve
is proportional to dC/dt and
𝑑𝐶
𝑑𝜃
=
𝑑𝐶
𝑑𝑡
exp
∆𝐸
𝑅𝑇
1
𝑎
the rate proportional to dC/dθ is derived. We can also calculate θ by using Eq,
(5).

 The second method is the generalized one of the method proposed by
Sharp and Wentworth [16], and applied to the case in which both the
rate of conversion and the conversion are obtained as a function of the
temperature.
 If we can obtain the activation energy from the plot of the logarithm of
the heating rates against the reciprocal absolute peak temperature, we
can obtain dC/dθ as a function of C and T.
 The relation of dC/dθ with C is also the experimental master curve, and
it is the generalized form of the relation of dC/dt vs. C proposed by
Simha and Wall [1] to distinguish the kinetic mechanisms. Then, we can
compare it with the theoretically obtained curve of dC/dA θ against C.
Furthermore, an Arrhenius plot can be made, since for a given
conversion
ln
𝑑𝐶
𝑑𝑡
− ln
𝑑𝐶
𝑑𝐴𝜃
= ln 𝐴 −
∆𝐸
𝑅𝑇
where the first and second terms of the left side are the experimental rate
and the rate calculated theoretically for the assumed kinetics. If the
assumed kinetics are correct, the Arrhenius plot should be quite linear and
we obtain a set of reasonable values for the kinetic parameters.

Fig.8. Typical plots of the logarithm of the
heating rate versus the reciprocal absolute
peak temperature a) the random degradation
of a high polymer with A=1010 sec-1 ΔE=40
kcal mole-1 L=6 b) the random degradation of
a high polymer with A=102 sec-1 ΔE=20kal
mole-1 & L=2
Accuracy of the Methods of Obtaining the Activation energy
Fig.7. Typical plots of the logarithm of the
heating rate versus the reciprocal absolute peak
temperature a) Second order reaction with
A=1020 sec-1 and ΔE=80 kcal mole-1 b) First
order reaction with A=1015 sec-1 and ΔE=60
kcalmole-1, c) 0.5th order reaction with 1010
sec-1 and ΔE=40 kcal mole-1

Typical examples of plots of the logarithm of the heating rate against the
reciprocal absolute peak temperature are shown in Figs7 and 8. The
activation energies and the frequency factors are estimated by using Eqs
(16), (17) and (18), and some of them are tabulated in Tables 3A and 4A;

The differential curve is not the derivative curve since in
DTA the heat evolved or absorbed during the process is
supplied in the following two forms:
1. The heat flowing into or out of the sample along the
temperature gradient is different from the steady state,
which appears as the difference of the differential
temperature from the base line of the steady state;
2. The heat stored or consumed in the form of the
temperature gradient is different from the steady state,
which causes the tailing of the differential curve.
Discussion

If one can draw a smooth experimental master curve, this is
evidence of the validity. In order to obtain the real kinetic
parameters, it is necessary, at least to compare the kinetic
analyses of the curves of the different heating
rates, as Flynn and Wall pointed out
false results are obtained if the process is treated
as a whole according to the overall isothermal kinetics
The second point to be considered in the kinetic analysis of
thermoanalytical data is the effect of the temperature
gradient within the sample.

In this study, a theoretical treatment of
Johnson-Mehl-Avrami (JMA) equation has been
performed to determine the grain growth
activation energy and other kinetic parameters
under isokinetic condition and a comparison of
obtained values is performed with those
obtained from Kissinger and Ozawa kinetic
models.

During a non-isothermal DSC experiment, the temperature of the furnace
is
changed linearly with time, t at a given rate β (degree per minute)
T = To + βt …..(1)
where To is the initial temperature.
Assuming that the dynamic case is a close succession of isothermal ones,
then the
JMA equation for the fraction of transformed material can be used to
describe the
non-isothermal continuous heating process, as in DSC measurement with a
constant heating rate, i.e.
x(t) = 1 - exp(-ktn) .....(2)
where x is the fraction of the transformation completed at time t,
k is a constant associated with nucleation rate and growth rate, both being
dependent on temperature,
and n is a constant which reflects the nucleation and growth morphology.

The JMA equation is generally not a valid description of transformation
kinetics
occurring under non-isothermal conditions due to the independent
variations
of nucleation and growth rates with temperature.
But under certain conditions, the JMA equation can still be valid-
 All nucleation occurs in the early stage of process and for rest of the
time, only growth is significant
 The growth rate depends only on instantaneous temperature and is
independent of time
 The nucleation is random
The first condition also implies that the transformation rate depends
only on state variables x and T and not on thermal history.
Avrami concluded that eqn (2) can be applied to non isothermal
transformation if the temperatures and concentrations are within
isokinetic
range, in which the ratio of growth rate to the probability of formation
of
growth nuclei per germ nucleus per unit time becomes a constant.

Taking into account that the temperature is a function of time in
non-
Isothermal experiments, the transformation rate for an isokinetic
reaction,
ẋ = dx/dt, can be obtained from equation (2) by differentiating
x with
respect to t,
ẋ = Km-1 - tn-1 (nK+ mTḰ) (1 -x) ......(3)
where k = Km, and
K, a function of temperature, has the Arrhenius type:
K = Ko exp( - E/RT) .....(4 a)
and
Ḱ = K βE/RT2 ......(4 b)
where Ko is a constant associated with the jump frequency of
atoms,
E the activation energy,
R the universal gas constant,
T the absolute temperature, and
m is a constant which reflects the grain growth geometry.

When a reaction occurs during a DSC measurement, the
change in heat content and in the thermal properties of the
sample is indicated by a heat flow deflection, or a peak in the
plot of heat flow from the reference to the sample vs
temperature.
From Kissinger model, we know that the temperature of
maximum deflection (Peak Temperature) in a DSC plot is also
the temperature at which the transformation rate reaches
maximum. The peak temperature on the DSC curve can be
found by setting the second order derivative of x with respect
to t equal to zero, i.e.

The Activation Energy is obtained by measuring the peak
temperature at different constant heating rates and working out
the slope of a plot of either log(β/Tp2) , log(β) or log(β/(Tp-To))
against 1/Tp .
In this study, a series of DSC measurements at different heating
rates were performed in order to determine the activation
energy from grain growth. All the DSC traces obtained are
represented in the Fig below. The peak temperatures for
different heating rates are listed in Table 1.

After fitting the data to eqn (11) using least square method, it is
found that the activation energy for the grain growth of
nanocrystalline nickel deposits is 131.5 kJ/mol.

For comparison, these data are also represented in Fig. below using
Kissinger and Ozawa methods. It is seen that the activation energies
derived from the different
plots using the same data are found to be very close to each other
and the relative differences are within 5%.

Non isothermal kinetics in phase transformations

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