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51

Kinetics of Non-Isothermal Crystallization

Science Diliman (January-June 2003) 15:1, 51-56

Kinetics of Non-Isothermal Crystallization of Coconut-based
Cholesteryl Ester: Avrami and Ozawa Approaches

J.F. Joson*, L.T. Davila, and Z.B. Domingo
Liquid Cystals Laboratory, National Institute of Physics
College of Science, University of the Philippines Diliman

1101 Quezon City, Philippines
E-mail: joihren@nip.upd.edu.ph

ABSTRACT

Kinetics of non-isothermal crystallization of coconut-based cholesteryl ester was performed by differential
scanning calorimetry under various heating rates. Different analysis methods were used to describe the
process of non-isothermal crystallization. The results showed that the Avrami equation could describe the
system very well. However, the Ozawa analysis failed. A probable reason is the difference in the crystallization
kinetics at high and low relative crystallization. The phase transitions of the coconut-based cholesteryl
ester were also observed through optical polarizing microscopy.

INTRODUCTION

Coco-based cholesteryl ester is a thermotropic liquid
crystalline material synthesized from esters of fatty
acids extracted from locally available coconut oil. It is
waxy and powdery at room temperature (Cureg, 2000).
Cholesteryl ester is also known to exhibit two
mesophases between the solid state and isotropic liquid
state, namely the smectic phase and the cholesteric
phase. This polymorphism of cholesteryl ester causes
three phase transitions: crystalline to smectic, smectic
to cholesteric, and cholesteric to isotropic. Unlike other
liquid crystals, cholesteryl ester can exist as crystalline
solid at room temperature. This will help in the
investigation of crystalline to mesophase transitions,
which is hard to achieve from most liquid crystals.

The most common approach used to describe the overall
isothermal crystallization kinetics is the Avrami equation
(Spruiell & Supaphol, 1999):

(1)

where X
t 
is the relative degree of crystallinity, t is the

duration of the crystallization process, n is the Avrami
exponent, and k is the crystallization rate constant. The
Avrami parameters n and k are constants typical of a
given crystalline morphology and type of nucleation.

The constant k in Eq. (1) is given by:

(2)

where U is the rate of lateral growth, I is the rate of
nucleation, and ξ is the average vertical extension of
the crystals (Van Antwerpen, 1971).

Alternatively, Eq. (1) can be written in its natural
logarithmic form

(3)

Using Eq. (3), parameters k and n can be calculated
from the least-square line fit of the double logarithmic
plot of ln[-ln(1-X

t
)] against ln t for each heating or

2

3
k U I

π
ξ⎛ ⎞= ⎜ ⎟

⎝ ⎠

* Corresponding author

1 exp( )ntX kt− = −

( )ln ln 1 ln lntX k n t− − = +⎡ ⎤⎣ ⎦



52

Joson, Davila, and Domingo

cooling rate, where k is the anti-logarithmic value of
the y-intercept and n is the slope (Ziru He et al., 1997).

In the study of non-isothermal crystallization using
Differential Scanning Calorimetry (DSC), the energy
released during the crystallization process is a function
of temperature rather than time as in the case of
isothermal crystallization (Ziru He et al., 1997).
Therefore, the relative crystallinity, X

t
 is given by:

(4)

where ∆H
T
 is the enthalpy of crystallization released

during a temperature change and, ∆H
C
 is the overall

enthalpy of crystallization, which is equal to the area
enclosed by the crystallization peak in the DSC plot.

In order to use Eqs. (1) & (3) in non-isothermal
conditions, the thermal lag of the sample and the DSC
furnace is assumed to be minimal, such that

(5)

where T
o
 and T are the onset and the arbitrary

temperature, respectively; t is the crystallization time;
and φ is the heating or cooling rate (Ziru He et al.,
1997).

Ozawa proposes another kinetic expression as an
extension to Avrami’s equation. Assuming that the non-
isothermal process may be composed of infinitesimally
small isothermal crystallization steps, the following
equation has been derived:

(6)

where K(T) (in °C/min) is the cooling/heating function,
φ is the heating and cooling rate, and m is the Ozawa
exponent, which depends on the dimension of the crystal
growth (Ziru He et al., 1997).

To analyze the non-isothermal crystallization data, the
natural logarithmic form of Eq. (6) is derived:

(7)

With different heating/cooling rates and plotting
ln[-ln(1-X

t
)] against ln φ at a given temperature, a

straight line should be obtained. K(T) and m are
determined from the y-intercept and slope, respectively
(Yuxian An et al., 1998).

Understanding the kinetics of crystallization, or any
phase transition, gives further insight on what
appropriate conditions a phase transition would occur.
This would include the temperature at which transition
occurs, heating or cooling rates, the amount of energy
needed for a transition to happen, the ordering of
molecules, and other parameters.

This research work aims to investigate the kinetics of
the crystal to smectic phase transition of cholesteryl
ester under non-isothermal conditions because it is
usually under this condition that liquid crystals are used.
Differential Scanning Calorimetry will be used with
various heating rates, supplemented by microscope
observations.

METHODOLOGY

The coconut-based cholesteryl ester used in this study
was taken from the Liquid Crystals Laboratory of the
UP Diliman Institute of Chemistry.

For calorimetric measurements, cell samples were
prepared using solid crimp cells. Cholesteryl ester, in
its solid form, was weighed in the sample cells with a
net weight of 2.40 ± 0.01 mg. This sample weight is
enough for the sensitivity of the Shimadzu DSC-50
equipment for thermal analysis. The crimped samples
were heated using the DSC from 25°C to 100°C with
various heating rates (1, 2, 3, 4, and 5°C/min). Analysis
was done using the Thermal Analysis Program of DSC.

For microscopic investigations, a liquid crystal cell was
prepared using two glass plates. The sample was then
introduced through a gap by capillary method. The
textures of cholesteryl ester at different temperatures

T
t

C

H
X

H

∆
=

∆

oT Tt
φ
−

=

( )
1 expt m

K T
X

φ
−⎡ ⎤

− = ⎢ ⎥
⎣ ⎦

( ) ( )ln ln 1 ln lntX K T m φ− − = −⎡ ⎤⎣ ⎦



53

Kinetics of Non-Isothermal Crystallization

were observed and photographed using the BH-2
Olympus Polarizing Microscope equipped with crossed
polarizers and camera attachment. Control of
temperature was made using the Mettler Toledo FP90
Central Processor connected to a hot stage that can be
mounted on the microscope stage.

RESULTS AND DISCUSSION

Non-isothermal crystallization kinetics based on
Avrami equation

The DSC thermograms of cholesteryl ester at various
heating rates are presented in Fig. 1. Table 1 shows a
summary of the non-isothermal data gathered from the
DSC.

The data presented in Table 1 reveal that the exothermic
peak temperature T

p
, which corresponds to the crystal

to smectic phase transition, shifts to a higher temperature
as the heating rate increases.

Moreover, the transition is faster at higher heating rates
(Table 1) as shown by the peak crystallization time t

max
,

which is the amount of time elapsed from the onset of
transition up to the peak temperature. This was taken
using DSC TA analysis. The relative crystallinity at the
peak temperature, X

tp
 ranges from 52.25% to 60.06%.

The relative crystallinity is determined by dividing a
crystallization peak into small strips. Each small strip
corresponds to the amount of heat involved in the
crystallization process from the onset of crystallization
up to a certain temperature, thus obtaining the relative
crystallinity as a function of temperature. The results
are shown in Fig. 2. All the plots of relative degree of
crystallinity versus temperature have essentially the
same pattern. At approximately 52.55% to 60.06% X

t
,

the shift to higher temperature of the transition peak as
the rate increases is evident. In order for these data to
be analyzed using the Avrami equation, the horizontal
temperature scale must be transformed into the time
domain using Eq. (5).

Fig. 3 shows the relative crystallinity as a function of
time. The plots indicate that the faster the heating rate,
the shorter the time needed for the completion of the

transition. The sigmoidal shape of the curves in Fig. 3
suggests that the Avrami analysis might be applicable
to the non-isothermal data from the DSC.

Non-isothermal DSC scans of cholesteryl ester

Fig. 1. DSC thermograms of cholesteryl ester at various
heating rates.

100

90

80

70

60

50

40

20

30

10

40 50 60 70
0

X
tp

Temperature (oC)

1 C/min

2 C/min

3 C/min

4 C/min

5 C/min

Fig. 2. Relative crystallinity as a function of temperature.

Table 1. Parameters of cholesteryl ester during non-
isothermal crystallization process.

φφφφφ
(oC/min)

T
p

(K)
t

max
(min)

X
tp

(%)
∆∆∆∆∆H

C
(J/g)

1
2
3
4
5

327.94
328.09
328.46
329.49
329.63

13.15
8.55
5.83
4.54
3.86

59.94
54.19
55.27
52.55
60.06

-50.81
-49.82
-59.38
-64.08
-57.64



54

Joson, Davila, and Domingo

Fig. 4 presents the Avrami analysis of the data shown
in Fig. 3. It shows the plots of ln[-ln(1-X

t
)] versus ln t at

various heating rates. The linearity maintained from the
initial stages of transition until very high degree of
conversion indicates that the Avrami equation correctly

describes the non-isothermal crystallization process
of cholesteryl ester. From the slopes and intercepts
of the lines in Fig. 4, the Avrami exponent n and rate
constant k can be determined. Moreover, the half-
time for crystallization t

1/2
 , which is analogous to t

max

has been taken directly from Fig. 3. This is calculated
as a function of parameters k and n. All the data are
listed in Table 2.

Values of rate parameters show that the crystallization
rate k increases with the increase of heating rate.
This might have been caused by the effect of
superheating as heating rate is increased (Yuxian An
et al., 1998). These values of k, together with the
values of t

1/2
-1, show that the rate of non-isothermal

crystallization is proportional to the heating rate. The
Avrami exponent is found to be ~3.0, which suggests
either a three-dimensional sporadic nucleation or a
two-dimensional spontaneous nucleation. However,
according to Price and Wendorff, the crystallization
from a smectic mesophase is three-dimensional
sporadic nucleation (Ziru He et al., 1997). Therefore,
the first case is more probable.

Non-isothermal crystallization kinetics based
on Ozawa equation

For comparison, Ozawa’s model is used to deal with
the data of non-isothermal crystallization. The plots
at temperatures from 316 K to 336 K are shown in
Fig. 5. Table 3 gives a summary of the slope and y-
intercept values of the plots. The changing slopes
indicate that m is not constant with temperature. One
probable reason is the assumption of the Ozawa
model. It must be noted that the X

t
 values chosen at a

given temperature include the values selected from
the earliest stage of the transition at high heating rate
and the values from the end state at lower rate (http:/
/ w w w. s h o d o r. o r g / U N C h e m / a d v a n c e d / k i n /
arrhenius.html). The kinetics of transition should be
different at low- and high-relative crystallization. Also,
shoulder transitions occur near the onset of the heating
curves for rates 1, 2, 3, and 4°C/min. These shoulder
transitions may be due to difference in the molecular
dynamics of the sample when undergoing phase
transitions. Thus, Ozawa analysis cannot adequately
describe the non-isothermal crystallization of
cholesteryl ester.

Fig. 3. Relative crystallinity as a function of time.

Time (min)

100

90

80

70

60

50

40

20

30

10

0

X
tp

5 10 15 200 25

1 C/min

2 C/min

3 C/min

4 C/min

5 C/min

Fig. 4. Avrami analysis of the data shown in Fig. 3.

-1.5 -0.5 0.5 1.5 2.5 3.5

3

1

-1

-3

-5

-7

-9

ln (time)

ln
[-

ln
(1

-X
t)
]

1 C/min
2 C/min
3 C/min
4 C/min
5 C/min

Table 2. Parameters k and n for non-isothermal crystallization
of cholesteryl ester.

φφφφφ
(oC/min)

k n t1/2(min)
= (ln2/k)1/n

t
1/2

-1

(min)-1

1
2
3
4
5

5.67 x 10-4

2.27 x 10-3

1.69 x 10-2

2.21 x 10-2

5.12 x 10-2

2.99
3.09
2.95
3.05
2.97

10.76
6.33
3.52
3.08
2.40

0.0929
0.1579
0.2839
0.3247
0.4167



55

Kinetics of Non-Isothermal Crystallization

Fig. 6. Cholesteryl ester in its (a) crystalline state, (b) smectic phase,
(c) cholesteric phase, and (d) isotropic phase.

Increasing temperature

(a) (b) (c) (d)

5.61 5.615 5.62 5.625 5.63

ln (rate)

3

2

1

0

-1

-2

-3ln
[-

ln
(1

-X
t)
]

-4

-5

-6

316 K
318 K
320 K
322 K
324 K
326 K
328 K
330 K
332 K
334 K
336 K

Fig. 5. Ozawa analysis of the data shown in Fig.3 at
different temperatures.

Optical microscopy

At room temperature, the cholesteryl ester is observed
in its crystalline state (Fig. 6a). Upon heating, focal conic

texture (Fig. 6b) is observed before changing to the
grandjean planar texture (Fig. 6c). This change in texture
corresponds to the smectic phase and cholesteric phase,
respectively. Further heating results in an isotropic phase
(Fig. 6d).

CONCLUSION

Non-isothermal crystallization of coconut-based
cholesteryl ester shows rate-dependent characteristics.
The Ozawa analysis, when applied to this system, failed
to provide a description of the non-isothermal
crystallization. This failure is due to the assumption of
this treatment. An alternative method based on the
Avrami analysis was used. The non-isothermal
crystallization data could be suitable over a wide range
of relative crystallinities by using only two adjustable
parameters. This method enables the calculations of
crystallization rate parameters, which increase with
increased heating rates, and half-time of crystallization,
which decreases with heating rate. Observations using
a polarizing microscope confirmed the crystal to smectic
phase transition. The smectic phase was exhibited by
the focal conic texture.

ACKNOWLEDGMENTS

The samples given by the Liquid Crystals Laboratory
UP Institute of Chemistry and the facilities of the Liquid
Crystals Laboratory of UP National Institute of Physics
are deeply acknowledged.

Table 3. Slopes and intercepts for the Ozawa analysis.

Temperature (K) Slope Intercept

316
318
320
322
324
326
328
330
332
334
336

7.4012
0.4740
6.0744
6.1609
4.4005
8.8955

21.1370
39.2660
26.7010
36.3120
99.0850

-40.0370
-1.2953

-33.0340
-33.7850
-24.1760
-49.8170

-119.1500
-221.6700
-151.9200
-206.9200
-561.3500

REFERENCES

Cureg, G., 2000. Cholesteric liquid crystals
formulations: Coco-based cholesteryl ester and
TM74A. University of the Philippines Diliman,
Quezon City.

Spruiell, J. E. & P. Supaphol, 1999. Kinetics of non-
isothermal crystallization of syndiotactic
polypropylene: Avrami, Ozawa, and Kissinger
approaches. In Benedikt, G.M. & B.L. Goodall (eds.)
Metallocene technology and modern catalytic



56

Joson, Davila, and Domingo

methods in commercial applications. Knoxville, University of
Tennessee. 263-273.

Van Antwerpen, F., 1971. Kinetics of crystallization
phenomenon of spherulites in poly(ethylene terephthalate).

Yuxian An et al., 1998. Non-isothermal crystallization kinetics
of poly(β-hydroxybutyrate). Journal of Polymer Science:
Polymer Physics. 36(B): 1305-1312.

Ziru He et al., 1997. A kinetics study of crystallization from
discotic mesophases. Liquid Crystals. 23(3): 317-325.

h t t p : / / w w w. s h o d o r. o r g / U N C h e m / a d v a n c e d / k i n /
arrhenius.html