Kinetic analysis of crystallization processes in amorphous materials

Article abstract of DOI:10.1016/S0040-6031(00)00449-4

Thermal analysis methods are widely used to study crystallization kinetics in amorphous solids. The experimental data is frequently interpreted in terms of the Johnson-Mehl-Avrami (JMA) nucleation-growth model. This paper discusses the limits of such an approach. A simple and convenient method is proposed to verify the applicability of the JMA model as well as the basic assumptions of kinetic analysis. It is shown that the two parameter autocatalytic model includes the JMA model and that is a plausible description of the crystallization kinetics. The main advantage of this model is its flexibility in describing quantitatively the kinetics of complex crystallization processes. The experimental data for crystallization of a chalcogenide glass and zirconia gel analyzed in this paper clearly demonstrate the rather complex nature of these processes. As a consequence, it is very difficult to explore the real mechanism of the crystallization unless some complementary studies are made. (C) 2000 Elsevier Science B.V.

Full text of DOI:10.1016/S0040-6031(00)00449-4

Thermochimica Acta 355 (2000) 239±253
Kinetic analysis of crystallization processes in amorphous materials
*
ÏÂ Â
Jirõ Malek
Joint Laboratory of Solid State Chemistry, Academy of Sciences of the Czech Republic
and University of Pardubice, 532 10 Pardubice, Czech Republic
Received 10 June 1999; received in revised form 13 September 1999; accepted 26 September 1999
Abstract
Thermal analysis methods are widely used to study crystallization kinetics in amorphous solids. The experimental data is
frequently interpreted in terms of the Johnson±Mehl±Avrami (JMA) nucleation-growth model. This paper discusses the limits
of such an approach. A simple and convenient method is proposed to verify the applicability of the JMA model as well as the
basic assumptions of kinetic analysis. It is shown that the two parameter autocatalytic model includes the JMA model and that
is a plausible description of the crystallization kinetics. The main advantage of this model is its ¯exibility in describing
quantitatively the kinetics of complex crystallization processes. The experimental data for crystallization of a chalcogenide
glass and zirconia gel analyzed in this paper clearly demonstrate the rather complex nature of these processes. As a
consequence, it is very dif®cult to explore the real mechanism of the crystallization unless some complementary studies are
made. # 2000 Elsevier Science B.V. All rights reserved.
Keywords: Kinetics; Crystallization; Nucleation-growth model; Autocatalytic model; Zirconia; Chalcogenide glass
1. Introduction
morphology [4]. The rate equation can be obtained
from Eq. (1) by differentiation with respect to time:
ꢀ
ꢁ
Thermal analysis (TA) methods such as DTA or
DSC are quite popular for kinetic analysis of crystal-
lization processes in amorphous solids. The crystal-
lization kinetics based on these data is usually
interpreted in terms of the standard nucleation-growth
model formulated by Avrami [1±3]. This model
describes the time dependence of the fractional crys-
tallization a, usually written in the following form:
da
dt
1
1=m
ˆ Kmꢀ1 a†‰ lnꢀ1 a†Š
(2)
Eq. (2) is usually referred to as the JMA equation, and
it is frequently used for the formal description of TA
crystallization data. It should be emphasized, how-
ever, that validity of the JMA equation is based on the
following assumptions [5±7]:
m
ꢁ
ꢁ
Isothermal crystallization conditions,
Homogeneous nucleation or heterogeneous
nucleation at randomly dispersed second-phase
particles,
a ˆ 1 exp ‰ ꢀKt† Š
where K and m are constants with respect to time, t.
(1)
The kinetic exponent m depends on the crystal growth
ꢁ
ꢁ
Growth rate of new phase is controlled by tem-
perature and is independent of time and
Low anisotropy of growing crystals.
* Fax: ‡420-40-603-6011.
Â
E-mail address: jiri.malek@upce.cz (J. Malek)
0040-6031/00/$ ± see front matter # 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 0 4 0 - 6 0 3 1 ( 0 0 ) 0 0 4 4 9 - 4

Â
J. Malek / Thermochimica Acta 355 (2000) 239±253
240
Henderson [5,6] has shown that the validity of the
JMA equation can be extended in non-isothermal
conditions if the entire nucleation process takes place
during the early stages of the transformation, and it
becomes negligible afterward. The crystallization rate
is de®ned only by temperature and does not depend on
the previous thermal history. Fundamental kinetic
equations for non-isothermal crystal growth from
preexisting nuclei have been developed by Ozawa
[8] and a simple method of kinetic analysis of TA
data for these processes has been proposed.
change associated with the crystallization process.
The fractional conversion a can be easily obtained
by partial integration of isothermal or non-
isothermal TA curve.
2. The rate of the kinetic process can be expressed
as a product of temperature dependent rate
constant K(T) and a dependent kinetic model
function f(a):
ꢀ
ꢁ
da
dt
ˆ KꢀT†fꢀa†
(4)
It is clear that all the above mentioned assumptions
should be carefully considered before the JMA equa-
tion is used for the description of isothermal or non-
isothermal TA data and any conclusions concerning
the growth morphology are made. Although the limits
of applicability of the JMA equation are well known,
in practice it is not so easy to verify whether they are
ful®lled or not. Thus, in view of the key role of the
JMA model in crystallization research, it is important
to address this problem. The structure of the paper is as
follows. First, we summarize several additional
assumptions inherently included in the kinetic analy-
sis of TA data. Second, we propose a consistent
method to test all these assumptions as well as the
applicability conditions of the JMA model for both
isothermal and non-isothermal TA data. Finally, this
method is demonstrated using experimental data for
crystallization of a chalcogenide glass and zirconia gel.
3. The rate constant in Eq. (4) follows Arrhenius
form:
ꢀ
ꢁ
Ea
RT
KꢀT† ˆ A exp
(5)
where the pre-exponential factor A and activation
energy Ea are kinetic parameters that should not
depend on the temperature T and the fractional
conversion a. This assumption is not necessary,
e.g. in the case of non-parametric kinetic method
described by Serra et al. [10,11].
Taking into account these assumptions, the kinetic
equation for the JMA model can be written as
ꢀ
ꢁ
Ea
f ˆ DHcA exp
fꢀa†
(6)
RT
where the function f(a) is an algebraic expression of
the JMA model
1
1=m
fꢀa† ˆ mꢀ1 a†‰ lnꢀ1 a†Š
(7)
2. Theory
The f(a) function should be invariable with respect to
procedural parameters such as sample mass and heat-
ing rate (non-isothermal conditions) or temperature
(isothermal conditions).
Fig. 1 shows isothermal and non-isothermal DSC
curves calculated by using Eqs. (6) and (7) for the
kinetic exponent mˆ2. The fractional conversion is
obtained by integrating Eq. (3) in isothermal condi-
tions:
2.1. Basic assumptions in kinetic analysis
In spite of the importance of the applicability test of
the JMA model mentioned above, it is also essential to
verify three basic assumptions inherent in any kinetic
treatment of isothermal and non-isothermal TA data
[9]. These assumptions are formulated below:
1. The rate of the kinetic process da/dt is propor-
tional to the measured speci®c heat ¯ow f,
normalized per sample mass (W/g):
Z
and in non-isothermal conditions
t
1
a ˆ
f dt
(8)
DHc
0
da
dt DHc
f
ˆ
(3)
Z
T
1
a ˆ
f dT
(9)
DHcb
where DHc corresponds to the total enthalpy
T0

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J. Malek / Thermochimica Acta 355 (2000) 239±253
241
(see Fig. 1). Moreover, any direct comparison of the
DSC crystallization curves f(T) and f(t) is compli-
cated by the fact that they strongly depend on the
heating rate and temperature, respectively. Neverthe-
less, the kinetic model f(a) should be invariant with
respect to these procedural variables. Similar behavior
is expected also for the function de®ned as
R
fꢀa† 0ada=fꢀa† (see Appendix A). It can be shown
[12,13] that these two functions are proportional to the
y(a) and z(a) functions that can easily be obtained by a
simple transformation of DSC data (see Appendix A).
In isothermal conditions these functions are de®ned as
yꢀa† ˆ f
zꢀa† ˆ ft
(10a)
(10b)
In non-isothermal conditions these functions are
de®ned as follows [12,14]:
ꢀ
ꢁ
Ea
RT
yꢀa† ˆ f exp
zꢀa† ˆ fT2
(11a)
(11b)
For practical reasons the y(a) and z(a) functions are
normalized within the (0 1) range. These functions can
be written in a more general form using the general-
ized time concept introduced by Ozawa [8,15] (see
Appendix A).
It is evident that the assumption (3) is not necessary
for analysis of isothermal experiments, and both the
y(a) and z(a) functions directly follow from the experi-
mental data. However, this assumption is vital for
analysis of non-isothermal data and the value of
apparent activation energy Ea is needed to calculate
the y(a) function. The value of Ea can be determined
by an isoconversional method without assuming the
kinetic model function [16,17]. From the logarithmic
form of Eq. (6) it follows that Ea is obtained from the
slope of the ln f vs.1/T plot for a constant a:
Fig. 1. (a) Isothermal DSC curve calculated for the JMA model
(mˆ2, Eaˆ100 kJ/mol, ln(A/s)ˆ50); (b) Non-isothermal DSC
curve calculated for the same set of kinetic parameters.
where b is the heating rate and T0 corresponds to the
beginning of the baseline approximation (i.e.
f(T0)ˆ0). The crystallization enthalpy DHc corre-
sponds to the total peak area.
The validity of the assumptions (1)±(3) is not given
a priori, and it should be thoroughly veri®ed before
any attempt at kinetic analysis of the isothermal or
non-isothermal data is made. Thus, it seems to be
necessary to develop a simple and reliable testing
method. This task will be carried out in the following
section.
ꢂ
ꢃ
d ln f
dꢀ1=T†
Ea
R
ˆ
(12)
a
This method can be applied to both isothermal and
non-isothermal DSC data. The Ea should be practi-
cally independent of the fractional conversion in the
0.3ꢂaꢂ0.7 range (i.e. within ca. 10%). Some changes
may be expected for lower and higher values of a Ð
particularly for fast processes Ð because of higher
errors in the baseline interpolation for peak tails.
2.2. Tests of the basic assumptions
The shape and symmetry of the DSC curve that
corresponds to the same kinetic model may be quite
different in isothermal and non-isothermal conditions

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J. Malek / Thermochimica Acta 355 (2000) 239±253
242
The validity of the assumptions (1) and (2) can
very sensitive to subtle changes to its argument.
Therefore, one can expect that the plots of
ln [ ln (1 a)] versus ln t or 1/T may be linear even
in the case that the JMA model is not ful®lled [14].
A more reliable test is based on the properties of the
y(a) and z(a) functions. These functions exhibit max-
ima at aM and a1P , respectively. These maxima are
de®ned by Eqs. (A.3) and (A.9). The maximum of the
y(a) function for the JMA model depends on the value
of the kinetic exponent:
easily be veri®ed by checking the invariance of the
y(a) and z(a) plots with respect to procedural variables
such as heating rate (non-isothermal conditions) and
temperature (isothermal conditions). These functions
should also be identical for both isothermal and non-
isothermal DSC data provided that the kinetic model
f(a) does not change. In case that there are consider-
able differences or pronounced in¯uence of the pro-
cedural parameters affecting the shape of the y(a) and
z(a) functions then, some of these assumptions are
probably not ful®lled. Such behavior can be caused by
many factors. One possibility is that the baseline has
not been drawn correctly due to substantial change of
sample heat capacity during the measurement.
Another possible explanation is that the measured
data corresponds to a complicated process (parallel
or consecutive processes, branching, etc.) due to a
more complex reaction scheme than expressed by Eq.
(6). Thermal inertia effects caused by lower thermal
contact between the sample and temperature sensor or
low thermal conductivity of amorphous material can
also play an important role.
aM ˆ 0 for m ꢂ 1
aM ˆ 1 exp ꢀm
1
1† for m > 1
(15)
The value of aM is always lower than the maximum of
the z(a) function a1P (see Appendix A). The latter is a
constant for the JMA model
a1P ˆ 0:632
(16)
This value is a characteristic `®ngerprint' of the JMA
model; and, according to our experience, it can be used
as a simple test of the applicability of this model [14].
The value of the kinetic exponent can then be esti-
mated from the position of the maximum of the y(a)
function. This is illustrated in Fig. 2 where the DSC
data from Fig. 1 transformed by using Eqs. (10) and
(11) are shown. The maximum of the z(a) function is
located at a1P 0.632 and, therefore, the curves in
Fig. 1 evidently correspond to the JMA model. This
is con®rmed also by the shape of the y(a) function
[12,13] which exhibits a maximum at aM0.393. This
value corresponds to the kinetic exponent mˆ2, as
follows from Eq. (15). It should be pointed out,
however, that the typical error limit of aM and a1P ,
determined from typical TA data, is ca. Æ0.02. There-
fore, the validity of the kinetic model should always be
considered within these limits of experimental uncer-
tainty.
An interesting testing method has been proposed by
Ozawa [18]. It is based on the plot of log [ ln (1 a)]
as a function of log b at a given temperature. The slope
of such plot should be equal to the kinetic exponent. If
the secondary crystallization is negligible then these
plots can be superimposed by longitudinal shifts and
the master curve is obtained. In contrast, if there is
appreciable secondary nucleation, the superposition
cannot be made and thus such process can easily be
detected [18].
2.3. Tests of the applicability of the JMA model
Probably the most popular testing method for iso-
thermal data is an inspection of the linearity of the
Avrami plot; i.e. the dependence of ln [ ln (1 a)] as a
function of logarithm of time. From logarithmic form
of Eqs. (A.5) and (A.10), it follows that the kinetic
exponent m is obtained from the slope of this plot:
d ln ‰ lnꢀ1 a†Š
ˆ m
(13)
d ln t
A similar testing method has also been developed for
non-isothermal TA data [9]. From logarithmic form of
Eqs. (A.10) and (A.11), it follows that a plot of
ln [ ln (1 a)] as a function of reciprocal temperature
1/Tshouldbelinear,providedthatthetermln [ATp(x)/b]
is a constant (see Appendix A). The slope of this plot is
expressed as
dln ‰ lnꢀ1 a†Š
dꢀ1=T†
mEa
R

(14)
The value of the kinetic exponent can then be calcu-
lated if the Ea is known. Nevertheless, it is well known
that a double logarithmic function, in general, is not

Â
J. Malek / Thermochimica Acta 355 (2000) 239±253
243
Fig. 2. Normalized y(a) and z(a) function obtained by transformation of DSC data from Fig. 1 by using Eq. (10) for isothermal data (&) and
Eq. (11) for non-isothermal data (&). The broken lines show the theoretical aM value for mˆ2 and the typical a1P value for the JMA model.
3. Experimental
The amorphous nature and purity of prepared materi-
als were checked by X-ray diffraction and energy
dispersive microanalysis.
DSC experiments presented in this paper were
performed by using a Perkin±Elmer DSC-7 instrument
on samples of ca. 10 mg encapsulated in standard
aluminum sample pans in an atmosphere of dry nitro-
gen. The instrument was calibrated with In, Pb and Zn
standards. Non-isothermal DSC curves were obtained
with selected heating rates 2±20 K/min in the range
30±6008C. Isothermal DSC experiments were
initiated by raising the temperature of the DSC cell
from room temperature to the selected temperature, Ti,
at 200 K/min. The temperature of the cell was then
maintained constant at Ti, and the exothermic heat
¯ow f out of the sample was measured as a function of
time.
4. Results and discussion
4.1. Crystallization kinetics of (GeS2)0.3 (Sb2S3)0.7
glass
The crystallization kinetics of chalcogenide glass of
(GeS2)0.3(Sb2S3)0.7 composition has been described
Ï
Â
previously [19±23]. Rysava et al. [20,21] have found
that the crystallization of a bulk sample can be inter-
preted within the JMA model and reported the kinetic
Â
exponent mˆ2. On the other hand, Malek et al. [22,23]
have shown that the crystallization of a powder sample
cannot be described by this model. These conclusions
seem to be con®rmed in the present study. Fig. 3 shows
isothermal crystallization data for a powder sample
and non-isothermal crystallization data for a bulk
sample of (GeS2)0.3 (Sb2S3)0.7 glass. The enthalpy
change associated with the crystallization process is
DHcbulkˆ56Æ1 J/g for the bulk sample, and
DHcpowderˆ50Æ2 J/g for the powder sample. The
difference of 6 J/g corresponds approximately to the
temperature difference of ca. 25 K between isothermal
The (GeS2)0.3(Sb2S3)0.7 glass was prepared by
synthesis from pure elements (5 N purity) in an evac-
uated silica ampoule by melting and homogenization
at 9508C for a period of 12 h. Amorphous zirconia gel
(hydrous ZrO2) was prepared by precipitation from
ZrOCl28ÁH2O (analytically pure reagent, Kanto,
Japan) 0.1 M solution using an excess amount of
1 M NH4OH added slowly to a stirred aqueous solu-
tion. The precipitated gel was washed ®ve times in
distilled water for removal of chloride ions, ®ltered
and dried in a vacuum at room temperature for 3 days.

Â
J. Malek / Thermochimica Acta 355 (2000) 239±253
244
Fig. 4. The apparent activation energy as a function of fractional
conversion calculated from DSC data for the crystallization of bulk
(&) and ®ne powder sample (~) of (GeS2)0.3 (Sb2S3)0.7 glass (see
Fig. 3). Broken lines correspond to an average value of Ea
calculated in the 0.3ꢂaꢂ0.7 range.
tion of surface nuclei and the crystals grow at a higher
viscosity of supercooled melt. Consequently, the appar-
ent activation energy increases. A more pronounced
variation of Ea versus a probably indicates a more
complex mechanism of the crystallization process.
As discussed in Section 2.3, one of the most popular
tests of the applicability of the JMA equation is the
linearity of the double logarithmic plots vs. ln t or 1/T.
Such plots obtained from isothermal and non-isother-
mal DSC data for crystallization of (GeS2)0.3
(Sb2S3)0.7 glass (see Fig. 3) are shown in Fig. 5. It
seems that these plots are linear. The kinetic exponent
obtained from isothermal data by using Eq. (13) was
found to be mˆ3.0Æ0.2. The same value has been
obtained for non-isothermal data by using Eq. (14) and
Eaˆ163 kJ/mol. This result would imply that the JMA
model is valid for crystallization kinetics of the bulk
and powder sample. Nevertheless, the y(a) and z(a)
functions shown in Figs. 6 and 7 clearly reveal that the
JMA model is valid only for the non-isothermal
crystallization of the bulk sample, and it is not ful®lled
for isothermal crystallization of a powder sample.
There are small but noticeable differences among
the curves for different heating rates as well as for
isothermal temperatures. These differences can prob-
ably be attributed to lower thermal contact between
the bulk sample and the temperature sensor (non-
isothermal data) or to errors in baseline approximation
(isothermal data). Such problems are expected for this
Fig. 3. DSC data for the crystallization of the (GeS2)0.3 (Sb2S3)0.7
glass: (a) isothermal data for ®ne powder sample; (b) non-
isothermal data for the bulk sample.
and non-isothermal experiments taking into account
the heat capacity change between glass and under-
cooled liquid in the glass transition range
(DCp0.24 J/gK).
Fig. 4 shows the activation energy Ea as a function
of fractional conversion calculated by isoconversional
method from the slope of the ln f vs. 1/T plots for
DSC data shown in Fig. 3. The activation energy for
the bulk sample is practically constant in the 0.3<a<0.7
range, and its value was found to be Eaˆ163Æ11 kJ/
mol. Within the limits of experimental errors, this
Ï
Â
value is close to 159 kJ/mol, reported by Rysava et al.
[21]. In contrast, the activation energy for a powder
sample is ca. 70% higher, being Eaˆ276Æ7 kJ/mol.
Observed differences can be explained by the fact that
the crystallization of a powder sample takes place at
lower temperatures due to non-negligible concentra-

Â
J. Malek / Thermochimica Acta 355 (2000) 239±253
245
Fig. 6. Normalized y(a) and z(a) function obtained by transforma-
tion of isothermal DSC data for the crystallization of ®ne powder
(GeS2)0.3 (Sb2S3)0.7 glass (Fig. 3a). The temperature is shown by
points: 3408C (&); 3308C (*); 3208C (~). Solid lines show the
typical interval of aP1 values for the JMA model.
Fig. 5. The double logarithmic plots obtained from DSC data
shown in Fig. 3.
material as the crystallization enthalpy DHc is rela-
tively low. Nevertheless, the basic assumptions for-
mulated in Section 2.1 seem to be ful®lled. The
maximum of the z(a) function falls in the range
predicted for the JMA model. The value of the kinetic
exponent mˆ3 can be obtained from the maximum of
the y(a) function. Somewhat lower value m2.8 has
been calculated using the method proposed by Ozawa
[18] (see Section 2.3). Any tendency for secondary
crystallization has not been observed in this case. This
value of the kinetic exponent corresponds to three-
dimensional crystal growth of an orthorhombic Sb2S3
formed during the crystallization process. Such crys-
talline phase exhibits relatively high entropy of fusion;
and, therefore, one can expect three-dimensional crys-
tal growth [24]. These conclusions were con®rmed by
microscopy and X-ray diffraction analysis. Fig. 8
shows scanning electron microscope photographs of
crystallized powder and bulk sample of (GeS2)0.3
(Sb2S3)0.7 glass. The crystallites in powder sample
exhibit a rod-like morphology with relatively low
compactness which reveals low dimensionality of
crystal growth from numerous nucleation sites. In
contrast, there are relatively big three-dimensional
crystals grown in the bulk sample suggesting the value
of the kinetic exponent should be close to 3.
The crystallization behavior of different samples
can easily be compared in the aM±a1P plot as shown in
Fig. 9. The present results are combined with some
previously published data [22,23]. The value of a1P for
a ®ne powder sample (<1 mm) is considerably lower
than corresponds to the JMA model. There are notable
differences between isothermal and non-isothermal
experiments. Such behavior clearly indicates a more

Â
J. Malek / Thermochimica Acta 355 (2000) 239±253
246
Fig. 7. Normalized y(a) and z(a) function obtained by transforma-
tion of non-isothermal DSC data for the crystallization of bulk
(GeS2)0.3 (Sb2S3)0.7 glass (Fig. 3b). The heating rates are shown by
points: 2 K/min (&); 5 K/min (*); 10 K/min (~); 15 K/min (Â);
20 K/min (&). The broken line corresponds to the theoretical aM
value for the JMA model (mˆ3). Solid lines show the typical
interval of a1P values for the JMA model.
Fig. 8. Scanning electron microscope photograph showing mor-
phology of Sb2S3 crystals grown in (GeS2)0.3 (Sb2S3)0.7 glass: (a)
powder sample; (b) bulk sample. A fresh fracture of crystallized
sample has been etched in an aqueous solution of NaOH.
complex crystallization mechanism. The coarse pow-
der sample (<100 mm) exhibit a slightly higher value
of a1P , but it is still below the value typical for the JMA
model. However, the values of aM and a1P for the bulk
sample correspond well to the crystallization of three-
dimensional crystals, and the kinetic exponent m is
close to 3, which is consistent with the value obtained
from double logarithmic plot and the maximum of the
y(a) plot.
As mentioned above, the JMA model is valid in
non-isothermal conditions provided that a new crystal-
line phase grows from a constant number of nuclei and
all nucleation is completed before the macroscopic
crystal growth started. This so-called site saturation is
an important condition for the isokinetic crystalliza-
tion process where the crystallization rate is de®ned
only by temperature and does not depend on the
previous thermal history [6]. In the light of these
facts, it seems that the nucleation and growth pro-
cesses are probably overlapped for a ®ne powder
sample. Therefore, the overall crystallization cannot
be described by the JMA model. Mutual overlapping
of the nucleation and growth phases is obviously lower
for a coarse powder sample but, still, the JMA model is
not valid in this case. It seems that in the bulk sample
the nucleation is completed before the growth phase is

Â
J. Malek / Thermochimica Acta 355 (2000) 239±253
247
Fig. 9. The a1P ±aM plot for the crystallization of (GeS2)0.3
(Sb2S3)0.7 glass. Full lines correspond to the limits of applicability
of the JMA model. Points correspond to non-isothermal data (&)
and isothermal data (&). Other data are taken from Ref. [22] (*),
and Ref. [23] (~). The dotted line is drawn as a guide to the eye.
The broken line shows the theoretical limit aMˆaP1
.
started and, therefore, the condition of the validity of
the JMA model in non-isothermal conditions is ful-
®lled. A ®ne powder sample exhibits a lower value of
a
M than corresponds to the bulk sample, which reveals
a lower in¯uence of the crystallized phase due to
dominant surface nucleation. These conclusions were
con®rmed by direct microscopic observations.
Fig. 10. Non-isothermal DSC data for the crystallization of ZrO2
gel: (a) as prepared sample; (b) partially crystallized sample after a
heat treatment at 3708C for 60 min (42% crystallinity).
4.2. Crystallization kinetics of zirconia gel
It is known that the metastable tetragonal poly-
morph crystallizes during the heating of hydrous
zirconia gel [25]. Although the crystallization of t-
ZrO2 phase have been studied by many authors [26±
32] the detailed mechanism of its formation is still not
clear. Aronne et al. [30] and Ramanathan et al. [31]
interpreted their experimental data within the JMA
model and the calculated kinetic exponent (mˆ2.7 and
m<1, respectively) has been attributed to the dimen-
sionality of crystal growth. However, the experimental
data have not been compared directly with the theo-
retical model or the z(a) function and, therefore, it is
dif®cult to verify whether such an interpretation is
consistent or not. Recently, it has been found that the
crystallization kinetics of t-ZrO2 nanocrystals in par-
tially dried zirconia gel is strongly affected by the
water content retained in dried gel [32]. The crystal-
lization process starts when practically all the water is
removed (at the onset of the DSC crystallization peak
the corresponding weight loss is ca. 13.4% and the
®nal weight loss is 13.7% of the initial sample weight).
Thus, in this case, the isothermal measurements can-
not be used because the exothermic crystallization is
partially overlapped by the endothermic effect of
water evaporation.
Fig. 10 shows non-isothermal crystallization data
for as prepared zirconia gel and a partially crystallized
sample obtained after heat treatment at 3708C for
60 min. The enthalpy change DHc associated with
these crystallization processes was found to be
157Æ4 and 91Æ4 J/g for as prepared and partially
crystallized sample, respectively. Therefore, the crys-
tallinity of the partially crystalline sample is ca. 42%.
The activation energy Ea calculated by isoconver-

Â
J. Malek / Thermochimica Acta 355 (2000) 239±253
248
Fig. 11. The apparent activation energy as a function of fractional
conversion calculated from DSC data for the crystallization of
ZrO2 gel (see Fig. 10), for as prepared sample (~) and partially
crystallized sample (~). Broken lines correspond to an average
value of Ea calculated in the 0.3ꢂaꢂ0.7 range.
sional method using these DSC data is plotted as a
function of fractional conversion in Fig. 11. The
activation energy is practically constant in the
0.3<a<0.7 range, being Eaˆ264Æ11 kJ/mol for an
as prepared sample and Eaˆ234Æ4 kJ/mol for the
partially crystallized sample. The activation energy
slightly decreases with increasing crystallinity of
the zirconia gel [32]. The difference of 37 kJ/mol
is probably associated with the nucleation process
taking place in the as prepared sample which is no
longer operative in the case of the partially crystalline
sample.
Fig. 12 shows the double logarithmic plots vs. 1/T
obtained from non-isothermal crystallization data of
the as prepared zirconia gel and the partially crystal-
lized sample (see Fig. 10). The plots are clearly non-
linear for the as prepared sample, indicating that the
JMA model cannot be applied. On the other hand, the
double logarithmic plots are linear for the partially
crystallized sample. The kinetic exponent obtained
from the slope of these plots by using Eq. (14) and
Eaˆ234Æ4 kJ/mol was found to be mˆ1.05Æ0.04. A
very similar value of m1.07Æ0.04 has been obtained
by using the method proposed by Ozawa [18] (see
Section 2.3). Practically identical conclusions can also
be obtained from the y(a) and z(a) functions. Fig. 13
shows the y(a) and z(a) plots for the crystallization of
as prepared zirconia gel. It is evident that these
Fig. 12. The double logarithmic plots obtained from DSC data that
are shown in Fig. 10.
functions are nearly invariant with respect to heating
rate. The maximum of the z(a) function is consider-
ably lower than the value predicted for the JMA
model. However, there is a shoulder that appears close
to a0.632. This rather complicated shape of the z(a)
function can be explained assuming that the nuclea-
tion and growth processes are partially overlapped at
the beginning of the DSC peak [32]. However, it seems
that the nucleation process is negligible for a>0.5 and
zirconia crystals are growing from a practically con-
stant number of nuclei. Fig. 14 shows the y(a) and z(a)
plots for the crystallization of partially crystalline
(42%) zirconia gel. The maximum of the z(a) function
is located close to the value a1P 0.63 predicted for the
JMA model. The y(a) function decreases almost lin-
early; i.e. y(a)/(1 a) corresponds to the kinetic
exponent mˆ1.

Â
J. Malek / Thermochimica Acta 355 (2000) 239±253
249
Fig. 13. Normalized y(a) and z(a) function obtained by transfor-
mation of non-isothermal DSC data for the crystallization of as
prepared ZrO2 gel (Fig. 10a). The heating rates are shown by
points: 2 K/min (&); 5 K/min (*); 10 K/min (~); 15 K/min (Â);
20 K/min (&). Solid lines show the typical interval of a1P values
for the JMA model.
Fig. 14. Normalized y(a) and z(a) function obtained by transfor-
mation of non-isothermal DSC data for the crystallization of
partially crystallized ZrO2 gel (Fig. 10b). The heating rates are
shown by points: 2 K/min (&); 5 K/min (*); 10 K/min (~); 15 K/
min (Â); 20 K/min (&). The broken line corresponds to the ®rst-
order kinetic model for mˆ1. Solid lines show the typical interval
of a1P values for the JMA model.
The crystallization behavior of as prepared and
partially crystallized zirconia samples can be easily
visualized in the aM±aP1 plot, shown in Fig. 15. The
samples containing lower content of crystalline phase
(<25%) do not correspond to a simple crystallization
kinetics as expressed by the JMA model and the value
of a1P is considerably lower than 0.6. Such behavior is
a consequence of the fast increase of the initial crystal-
lization rate. This acceleration can be due to a sec-
ondary nucleation induced by the crystal growth. An
alternative explanation is associated with the adiabatic
temperature drift in the amorphous gel as the crystal-
lization heat is released. In the latter case, however, the
shape of the z(a) function should be sensitive to
heating rate, which was not observed. The crystal-
lization rate is lower for partially crystallized samples
(crystallinity >40%) and the maximum of the z(a)
function is a1P 0.63 and the y(a) function linearly
decreases. Therefore, the JMA model can be used for
the description of the crystallization rate, and the value
of the kinetic exponent should be close to unity as has
been anticipated above. This value is also expected
[33] when crystallite dimensions are suf®ciently small
and the rate of crystallization is controlled by nuclea-
tion in assemblage of similar particles. Therefore, one
can expect that numerous and rather small crystals will
be formed in the zirconia gel during the crystallization
process. The experimental results support these expec-
tations. Electron microscopy observations revealed

Â
J. Malek / Thermochimica Acta 355 (2000) 239±253
250
isothermal conditions should be critically examined,
in particular, the meaning of the kinetic exponent m.
In Sections 4.1 and 4.2, we have shown that it is
convenient to discuss crystallization kinetics in terms
of aM±a1P plot based on the maxima of the y(a) and
z(a) function. The validity of the JMA can easily be
veri®ed checking the maximum a1P of the z(a) func-
tion. If the maximum falls into the 0.61ꢂa1P ꢂ0.65
range then the experimental data probably correspond
to the JMA model. If the maximum is shifted to lower
values of fractional conversion (a1P <0.6) the condi-
tions of validity of the JMA model are not ful®lled.
Such a displacement indicates increasing complexity
of the process and can be caused, for example, by the
in¯uence of surface nucleation. Nevertheless, the
complex behavior can also be observed when the
temperature distribution within the sample is affected
considerably by liberation of the crystallization heat at
the growth interface [6]. If the dimensions of the
crystallized phase are small, the rate of the process
is controlled by nucleation in assemblage of similar
particles [33], and the kinetic exponent is mˆ1. The
maximum of the y(a) function is then close to zero, i.e.
aMˆ0. However, if the maximum of the y(a) function
is shifted to higher values, it clearly indicates an
increasing in¯uence of the product (i.e. crystallized
phase) to the overall crystallization kinetics. A typical
example is spherulite crystal growth morphology
where the spatial constrains in highly viscous media
play an important role and the crystallized phase
further increases the rate of the process in the direction
preferred.
Fig. 15. The a1P ±aM plot for the crystallization of the ZrO2 gel.
Full lines correspond to the limits of applicability of the JMA
model. Points (&) are taken from Ref. [32] and they correspond to
non-isothermal data for different crystallinity shown as a number
next to the points. The dotted line is drawn as a guide to the eye.
The broken line shows the theoretical limit aMˆaP1
.
that t-ZrO2 nanocrystals (ꢃ13 nm in size) are formed
after heat treatment of as prepared zirconia gel [32].
Similar average crystallite size was also estimated
from corrected halfwidth of (0 1 1) X-ray diffraction
peak. The nanocrystallization of the zirconia gel is
consistent with the mechanism of hydrolytic polymer-
ization of zirconyl species suggested by Clear®eld
[34].
4.3. Autocatalytic behavior of the crystalization
kinetics
Such autocatalytic behavior has been observed for
many crystallization processes [12,22,23,32] and can
be described by means of an empirical two parameter
model [9,35]:
The crystallization kinetics of amorphous solids is
usually interpreted in terms of the JMA model. How-
ever, strictly speaking, this model is valid in isother-
mal conditions and it can be rigorously applied to
transformations involving nucleation and growth only
in a limited number of special cases in non-isothermal
conditions. Under the restrictions outlined by Hender-
son [5,6] an example of a system which allows the
non-isothermal application of the JMA model is one in
which entire nucleation process takes place during
early stages of the transformation and becomes neg-
ligible afterward. In this case, the crystallization rate is
de®ned only by temperature and does not depend on
the previous thermal history. Nevertheless, even in this
case the applicability of the JMA model under non-
N
fꢀa† ˆ aMꢀ1 a†
(17)
where the parameters M and N de®ne relative con-
tributions of acceleratory and decay regions of the
kinetic process. It was shown [36] that this two para-
meter autocatalytic model is physically meaningful
only for M<1. The maxima of the y(a) and z(a) depend
on the value of the kinetic exponents M and N. The
maximum aM of the y(a) function can be expressed as
M
aM
ˆ
(18)
M ‡ N

Â
J. Malek / Thermochimica Acta 355 (2000) 239±253
251
evidence, including microscopic observations
together with all other relevant information.
5. Conclusions
The presented results clearly indicate that the John-
son±Mehl±Avrami (JMA) model has limited validity
both in isothermal and non-isothermal conditions, and
it is strongly recommended to always test its applic-
ability for a particular crystallization process. In addi-
tion to this validity test, there are several basic
assumptions implicitly involved in kinetic treatment
of isothermal and non-isothermal TA data which should
also be checked before any kinetic analysis is made.
A uni®ed approach based on two functions y(a) and
z(a) obtained by a simple transformation of experi-
mentaldataisproposed,allowingatestingofthevalidity
of the JMA model as well as the basic assumptions in
kinetic analysis. The y(a) and z(a) functions exhibit
maxima at aM and aP1, respectively (aM<a1P ). Their
position and shape should be invariant with respect
to procedural variables such as heating rate (non-
isothermal conditions) and temperature (isothermal
conditions). The shape of these plots should also be
identical for isothermal and non-isothermal data. The
validity of the JMA model can easily be veri®ed by
checking the maximum aP1 of the z(a) function. If the
maximum falls into 0.61ꢂa1P ꢂ0.65 range then experi-
mental data probably correspond to the JMA model.
It is convenient to describe crystallization kinetics
in terms of aM±a1P plot. This diagram helps to visua-
lize the complexity of the crystallization process as
well as to verify the applicability of the JMA model.
The two parameter autocatalytic model includes the
JMA model as a special case and, therefore, is a
plausible mathematical description for the nucleation
and growth processes in amorphous solids. The main
advantage of the two parameter model is its potential
to describe quantitatively even complex crystalliza-
tion processes.
Fig. 16. The a1P ±aM plot for the two parameter autocatalytic model
de®ned by Eq. (17). The dotted lines show the in¯uence of kinetic
exponents M and N. Full lines correspond to the limits of
applicability of the JMA model.
Nevertheless, the maximum a1P of the z(a) function
cannot be expressed in an analytical form and has to be
obtained by the numerical solution of Eq. (A.9).
Fig. 16 shows aM a1P diagram calculated using
Eqs. (A.3) and (A.9) for a constant value of the
parameters M and N of the autocatalytic model. It
is evident that the JMA model is a special case of the
Ï
Â
two-parameter model as already pointed out by Sestak
[9]. Therefore, the model de®ned by Eq. (17) is a
plausible mathematical description for the nucleation
and growth processes in non-crystalline solids. The
data shown in Figs. 9 and 15 can be also described by
this model [23,32]. It was shown that TTT diagrams
predicted for this model correspond well to experi-
mental data [37]. The increasing value of the kinetic
exponent M indicates a more important role of the
crystallized phase on the overall kinetics. It seems that
a higher value of the kinetic exponent N>1 means
increasing complexity. However, the temptation to
equate parameters M and N with a de®nite crystal-
lization mechanism should be avoided, and too much
physical signi®cance should not be attached to the
numerical values of what is essentially a phenomen-
ological convenience. In particular, this warning
applies to processes exhibiting a complex behavior.
Therefore, it seems that meaningful conclusions con-
cerning the real mechanism of the process should
always be based on other types of complementary
Acknowledgements
This work was supported by the grant Agency of the
Czech Republic under grants No. 203/96/0184, 104/
97/0589 and 202/98/K002.

Â
J. Malek / Thermochimica Acta 355 (2000) 239±253
252
Appendix A. Transformation of DSC data
where CiˆDHc is a constant. Differentiating Eq. (A.7)
with respect to a, we obtain
A.1. The y(a) function
z0ꢀa† ˆ Ci‰ f 0ꢀa†gꢀa† ‡ 1Š
(A.8)
In isothermal conditions the y(a) function can be
expressed from Eq. (6) as
By setting Eq. (A.8) equal to zero, we obtain the
condition that must be ful®lled by a1P at the maximum
of the z(a) function [13]:
yꢀa† ˆ f ˆ Bi fꢀa†
(A.1)
f0ꢀa1P †gꢀa1P † ˆ 1
(A.9)
where BiˆDHcA exp ( Ea/RT) is a constant. Thus, by
differentiation of Eq. (A.1) with respect to a, we obtain
The g(a) function can be written for the JMA model as
y0ꢀa† ˆ Bi f 0ꢀa†
By setting Eq. (A.2) equal to zero, we obtain the
condition which must be ful®lled by aM at the max-
imum of the y(a) function
(A.2)
1=m
gꢀa† ˆ ‰ ln ꢀ1 a†Š
(A.10)
Comparing Eqs. (A.9) and (A.3), it follows that
a1P ˆaM holds only for an in®nite g(a). This condition
can be considered as a theoretical limit of any kinetic
model; and, for the JMA model, it is ful®lled for an
in®nite value of the kinetic exponent m. Therefore, the
maximum of the z(a) function is always located at
higher values of a than the maximum of the y(a)
function.
f0ꢀaM† ˆ 0
(A.3)
Therefore, the shape of the y(a) function is formally
identical with the kinetic model function f(a), and the
value of aM can be obtained from Eq. (A.3).
In non-isothermal conditions, the exponential term
exp ( Ea/RT) in Eq. (A.1) is not constant and, there-
fore, the y(a) function is de®ned as [12]:
If the temperature rises at a constant rate bˆdT/dt,
then, after the integration of Eq. (4), one can obtain the
following equation
ꢀ
ꢁ
ꢀ
ꢁ ꢀ
ꢁ
Ea
RT
AT
Ea
Ea
RT
yꢀa† ˆ fexp
ˆ Bn fꢀa†
(A.4)
gꢀa† ˆ
exp
p
(A.11)
b
RT
where BnˆDHcA is a constant. The y(a) is usually
normalized within the (0 1) range and, therefore, its
shape should be identical both in isothermal and non-
isothermal conditions.
where p(Ea/RT) is an approximation of the tempera-
ture integral which has to be introduced because the
K(T) term cannot be integrated analytically [9]. Com-
bining Eqs. (A.11) and (6), we can obtain the follow-
ing expression for the z(a) function in non-isothermal
conditions
A.2. The z(a) function
ꢀ
ꢁ
Ea
By integration of Eq. (4) in isothermal conditions,
the following equation is obtained
zꢀa† ˆ fTp
ˆ DHc bfꢀa†gꢀa†
(A.12)
RT
ꢀ
By inserting this equation into Eq. (2), one can obtain
ꢁ
Z
a
da
fꢀa†
Ea
It was shown [14] that in the case of the z(a) function
the approximation pRT/Ea is suf®ciently accurate.
Then the z(a) function can be expressed as
gꢀa† ˆ
ˆ A exp
t
(A.5)
RT
0
zꢀa† ˆ fT2 ˆ Cn fꢀa†gꢀa†
where CnˆDHc bEa/R is a constant and, therefore, the
value of the a1P characteristic for any kinetic model
can be obtained from Eq. (A.9).
The z(a) function is usually normalized within the
(0, 1) range and its shape, as well as the condition for
the maximum, should be identical both in isothermal
and non-isothermal conditions.
(A.13)
the expression
ꢀ
ꢁ
da
dt
1
ˆ
ꢀa†gꢀa†
(A.6)
t
Combining Eqs. (A.6) and (3), we obtain the following
expression for the z(a) function in isothermal condi-
tions
zꢀa† ˆ ft ˆ Ci fꢀa†gꢀa†
(A.7)

Â
J. Malek / Thermochimica Acta 355 (2000) 239±253
253
[5] D.W. Henderson, J. Therm. Anal. 15 (1979) 325.
[6] D.W. Henderson, J. Non-Cryst. Solids 30 (1979) 301.
A.3. The generalized y(a) and z(a) functions
[7] M.P. Shepilov, D.S. Baik, J. Non-Cryst. Solids 171 (1994)
141.
A very useful concept of generalized time intro-
duced by Ozawa [8,15] can be used to obtain the
generalized y(a) and z(a) functions. For Arrhenius
type rate constant the generalized time, y, is de®ned as
[8] T. Ozawa, Bull. Chem. Soc. Jpn. 57 (1984) 639.
Ï
Â
[9] J. Sestak, Thermophysical Properties of Solids. Their
Measurements and Theoretical Analysis, Elsevier, Amster-
dam, 1984, Chapters 8 and 9, pp. 172±254.
[10] R. Serra, J. Sempere, R. Nomen, Thermochim. Acta 316
(1998) 37.
ꢀ
ꢁ
Z
t
Ea
y ˆ
exp
dt
(A.14)
RT
0
[11] R. Serra, J. Sempere, R. Nomen, J. Therm. Anal. 52 (1998)
933.
Differentiating Eq. (A.14) with respect to time we
obtain
Â
[12] J. Malek, Thermochim. Acta 138 (1989) 337.
Â
[13] J. Malek, Thermochim. Acta 200 (1992) 257.
ꢀ
ꢁ
ꢀ
ꢁ
Â
[14] J. Malek, Thermochim. Acta 267 (1995) 61.
dy
dt
Ea
Rt
ˆ exp
(A.15)
[15] T. Ozawa, Thermochim. Acta 100 (1986) 109.
[16] H.L. Friedman, J. Polym. Sci., Part C. Polym. Lett. 6 (1964)
183.
Combining Eqs. (4), (5) and (A.15) we obtain the
following general expression for the y(a) function:
[17] T. Ozawa, J. Therm. Anal. 31 (1986) 547.
[18] T. Ozawa, Polymer 12 (1971) 150.
ꢀ
ꢁ
Ï
[19] N. Rysava, L. Tichy, C. Barta, A. Trõska, H. Ticha, Phys. Stat.
Ï
Â
Â
ÏÂ
Â
da
dy
yꢀa† ˆ
ˆ Af ꢀa†
(A.16)
Sol. (a) 87 (1985) K13.
Ï Â
[20] N. Rysava, T. Spasov, L. Tichy, J. Therm. Anal. 32 (1987)
Â
1015.
Ï
[21] N. Rysava, C. Barta, L. Tichy, J. Mat. Sci. Lett. 8 (1989)
The function (da/dy) has already been used for the
description of thermal decomposition processes
[38,39].
Ï
Â
Â
91.
 Ï
[22] J. Malek, V. Smrcka, Thermochim. Acta 186 (1991) 153.
Ï
Ï
Ï
Â
Ï
Â
Â
[23] J. Malek, E. Cernoskova, R. Svejka, J. Sestak, G. Van der
Plaats, Thermochim. Acta 280-281 (1996) 353.
[24] K.A. Jackson, D.R. Uhlmann, J.D. Hunt, J. Crystal Growth 1
(1967) 1.
By integrating of Eq. (4) the following equation is
obtained:
Z
By combining Eqs. (A.17) and (A.16), we obtain the
a
da
gꢀa† ˆ
ˆ Ay
(A.17)
 Ï
[25] J. Malek, L. Benes, T. Mitsuhashi, Powder Diffraction 12
(1997) 96.
fꢀa†
0
[26] G. Gimblett, A.A. Rahman, K.S.W. Sing, J. Chem. Tech.
Biotechnol. 30 (1980) 51.
following general expression for the z(a) function:
[27] P. Colomban, E. Bruneton, J. Non-Cryst. Solids 147-148
(1992) 201.
ꢀ
ꢁ
da
dy
zꢀa† ˆ
y ˆ fꢀa†gꢀa†
(A.18)
[28] T. Mitsuhashi, M. Ichihara, U. Tatsuke, J. Am. Ceram. Soc.
57 (1974) 97.
Â
[29] J. Livage, K. Doi, C. Mazieres, J. Am. Ceram. Soc. 51 (1968)
Eqs. (A.16) and (A.18) are the generalized expression
for the y(a) and z(a) function. These functions depend
on two fundamental variables only; i.e. the fraction
crystallized and the generalized time, and they can
easily be applied for the analysis of both isothermal
and non-isothermal processes.
349.
[30] A. Arone, P. Pernice, A. Marotta, J. Mat. Sci. Lett. 10 (1991)
1136.
[31] S. Ramanathan, N.C. Sini, R. Prasad, J. Mat. Sci. Lett. 12
(1993) 122.
 Â
[32] J. Malek, T. Mitsuhashi, J. Ramõrez-Castellanos, Y. Matsui, J.
Mater. Res. 14 (1999) 1834.
[33] M.E. Brown, D. Dollimore, A.K. Galwey, Reactions in Solid
State, Elsevier, Amsterdam, 1982, Chapter 3, pp. 41±113.
[34] A. Clear®eld, J. Mater. Res. 5 (1990) 161.
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Ï
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Â
Â
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