Useful Method to Analyze Data on Overall Transformation Kinetics
I. Avramov, K. Avramova
Institute of Physical Chemistry, Bulg. Acad. Sci., 1113 Sofia, Bulgaria
E-mail for correspondence: avramov@ipc.bas.bg
Keywords: crystallization, glass-ceramics, relaxation
ABSTRACT
We propose an easy and reliable method for analyzing data from DTA and
DSC studies of crystallization at constant heating rate, q. It is demonstrated that
important information can be obtained in log-log coordinates of plot of heating rate,
log q , against peak temperature , log Tp . The slope of the corresponding straight line
is proportional to the dimensionless activation energy
p 
that there is a simple relation
E
E
RT . It is demonstrated
Tp
q  E  1
between the characteristic crystallization
time τ, the heating rate the the heating rate heating the heating rate rate the heating rate q the heating rate and the heating rate the the heating rate peak the heating rate temperature the heating rate Tp. the heating rate Keywords: overall crystallization kinetics, thermal analyses, DSC, DTA
Introduction
It seems that Verhulst first postulated that the transformation fraction (t) , int) , in
a media being subject to exhaustion, can be describes by sigmoid function [1,2]. Later
on, it was assumed [3-5] that the transformation rate is a product of two functions, one
of them representing the reaction mechanism, f(), being explicitly dependent on 
while the other one, k(t) is reliant on time t. So, the rate equation is
 (t) , int )
k (t) , int ) f   
t
11\*
MERGEFORMAT (t) , in)

In the following we use the reciprocal of k(t) , int), the characteristic time
1
k  t
. It is
more appropriate as the dimension of  is [time]. The overall crystallization rate
kinetics has a long history of research. Usually, the experimental results are analyzed
in the framework of the Kolmogorov-Johnson-Mehl-Avrami (t) , inKJMA) theoretical
model [6-8]. According to it the time dependence of degree of transformation α is:
 1-e
t
- 
 
n
22\*
MERGEFORMAT (t) , in)
Here n is the dimensionless power which depends on crystallization mechanism as
well as on wherever nucleation takes place or crystals are developing around existing
active centers. It is easily seen that Eq.2 is solution of Eq.1 for the properly chosen
explicit form of the f(α) the heating rate and the heating rate k(t) the heating rate functions. The crystallization process, that takes
place during the widespread methods of thermal analyses, DTA and DSC, starts
frequently from the surface [9-11]. Although the KJMA approach is discussed in
many books and review articles [12-16], the interpretation of experimental results is
relatively easy (t) , insee [13,14]) only when crystallization time is constant, const. In
this case the corresponding derivative form (t) , insee Eq.1) is:
1
d  t   1  
1 
   n  1      ln  1     n 
dt
  

33\*
MERGEFORMAT (t) , in)
Nowadays there are experimental data on the derivative form d/dt (mainlydt (t) , inmainly
from DSC and DTA studies) [15,16]. Earlier the experiments were focused on the
integral forms, i.e. the dependencies of  on time t. It was considered unreliable to
interpret the differential form, d/dt (mainlydt, because it is well known that negligible
deviations in integral forms lead to important errors in differentiated ones. For the
isothermal case (t) , inτ=constconst) the most popular approach was to plot the experimental
results in bilogarithmic coordinates, log(t) , in  log(t) , in1   )) against log t . The slope of the
expected straight line is the dimensionless power n. The problem is that the estimation
of the term log(t) , in  log(t) , in1   )) is very inaccurate for both α→0 and for α→1.
Recently, we proposed [17,18] to plot the data in coordinates α against log t. Thus
d
the derivative d log t is determined and through it the n value is obtained in a reliable
way.
The situation is much more complicated when τ is not a constant (t) , infor instance
in non-isothermal conditions). Although there are many approaches (t) , insee [13,19-21]
and literature cited there), and all of the treatments are based on the formal theory of
transformation kinetics, they differ greatly in their assumptions. This leads to
contradictions in results interpretation. The aim of the present study is to find a way to
interpret experimental data in an easy and reliable way.
New method of interpretation of experimental data
If  is not constant, Eq.2 is no more valid. However, we can still use an
expression similar to Eq. 3 for the transformation rate
1
1
d
n
n

1



ln
1



 

dt   t 
44\* MERGEFORMAT (t) , in)
So that the transformation rate can be presented as
1
1
d
n
n

1



ln
1



 

dT q (t) , inT )
55\*
MERGEFORMAT (t) , in)
where
q
dT
dt is the heating rate. Since at low temperatures the value of (t) , inT) is very
large, the process is frozen and we can assume that the heating at constant rate q starts
at T=const0 K. The corresponding solution of Eq.5 is
  T dT  n
 1  exp    

  0 q  T  





66\*
MERGEFORMAT (t) , in)
It is useful to note that the integral in Eq.6 is
n
 T dT 

  ln  1   
 0 q  T  
77\*
MERGEFORMAT
(t) , in)
To solve properly Eqs.67, we need a sufficiently accurate, and in the same time
friendly to use, formula for the temperature dependence of characteristic time τ.The
most popular hypothesis is that it is a result of thermally activated process; i.e.
 E 
  o exp 

 RT 
88\*
MERGEFORMAT
(t) , in)
Introduction of Eq.8 yields to rather complicated solution of the integral in Eq.6
because an exponential integral function, Ei, appears. The attempts to find suitable
method to interpret the experimental result lead to the so-called “Kissinger”
coordinates [10,16,17], namely to plot the temperatures T p of the maximums of the
 Tp2 
1
d
ln  
dT against temperature T curves in coordinates  q  against Tp . From the slope
of the corresponding straight line the activation energy E is estimated. Unfortunately,
the value of the activation energy obtained in this way is always overestimated. We
expect that the upper limit is
E
37
RT
otherwise, according to Eq.8, the
corresponding crystallization time would be unreasonably large (t) , insee for instance
E
37
[10]). Thus, for RT
the exponential term in Eq.8 is about 1016, so the
characteristic time τ should be too large as compared with the expected experimental
ones, no matter how small is the preexponential constant τo. Actually, the expected
crystallization time is of the order of minutes at the temperatures of the maximum of
d
the dT against T curves. Therefore here we start here new approach. Let at a given
reference heating rate qr the characteristic time is τr at the temperature of the
maximum of the crystallization peak Tr. We assume that at any temperature T the
characteristic time is
T
  T   r  
 Tr 
E
99\*
MERGEFORMAT
(t) , in)
In Appendix 1 we demonstrate that Eq.9 is equivalent to assumption made by the Eq.8
E
with
E
RTr . The advantage of Eq. 9 is that the integral in Eq.6 is easily solved.
T
T 
dT
T

 

q  T  q r  E  1  Tr 
0
E
1010\*
MERGEFORMAT (t) , in)
So that the solution of Eq.6 is

 1  exp  


E

T  
T


 q r  E  1  Tr  


n




1111\*
MERGEFORMAT (t) , in)
Discussion
Note that Eqs.8,9 are oversimplifications, despite they accounts for the
temperature dependence of the crystallization time. Additionally, the activation
energy for crystallization could depend on the degree of transformation as the
composition of the amorphous phase at the crystallization front changes (t) , insee for
instance [22.23]). So, the present treatment is just a step in solving the problem.
Further steps are needed as the limits of validity of the present approach are
experimentally tested.
d
Appendix 2 provides detailed analyses of the peak position of the dT against
T curve It is shown that at any heating rate q the peak temperature Tp(q) is determines
by the condition
TpE1
E
r
q r  E  1 T
1 ; or TpE1 q r  E  1 TrE
1212\*
MERGEFORMAT (t) , in)
A straight line is expected if the heating rate q is plotted against the peak temperature
Tp in log-log coordinates.
log q  E  1 log Tp  E log Tr  log(t) , inE  1)  log(t) , in r )
1313\*
MERGEFORMAT (t) , in)
E
The dimensionless activation energy
E
RTr can be determined from the slope of
this line. The characteristic crystallization time τr remains the only unknown and is
easily determined from the intercept. Since there are no limitations for the choice of
the reference heating rate qr the characteristic crystallization time τp(q) is determined
through Eq.12 as
 p  q 
Tp
q  E  1
1414\*
MERGEFORMAT (t) , in)
In other words, there is an universal relation between the peak temperature T p , the
corresponding crystallization time τ and heating rate q
Tp
q p
 E  1
1515\* MERGEFORMAT (t) , in)
To illustrate the use of this method we analyze crystallization results of one of the
samples studied in [10]. Powder samples of composition 5.2Al2O3 19.6 CaO 17.9
MgO 53.5 SiO2 with fraction size between 75 and 125 [m] were used. The heating
rates of 5, 7.5, 10 and 20 [K/dt (mainlymin] were applied. The relationship [10] between the
heating rate lg q and the peak temperature lg Tp is demonstrated in Fig.1 in log-log
coordinates. From the corresponding slope the dimensionless activation energy is
E
E
31
RTp
. The heating rates, the peak temperatures and the corresponding
crystallization times calculated according to Eq.14 are summarized in Table 1. With
E 31 and τ≈4 [min] the Eq.8 will be satisfied for τo≈1 10-13 [min]≈ 9 10-12 [s].
Thus, for the first time, the value of the preexponential constant τo is well within the
expected limits.
The proposed method of analysis gives an activation energy of around 250 [kJ/dt (mainly
mol]. The analysis of the same DTA data in [10], using the Kissinger equation, gave
an activation energy of ~400 [kJ/dt (mainlymol]. The reason for overestimated values obtained
from analyses of Kissinger equation is because the activation energy is not constant.
In Kissinger coordinates the corresponding slope depends on the activation energy
plus its first derivative in respect to temperature. Some examples could be seen for
instance in [24]. The present method estimates the activation energy without
involving its temperature derivative.
According to Eqs. 1112 the degree of crystallinity p at the maximum of the
d
dT
against temperature T curve has an universal value
 p 1 
1
0.63
e
,
independent on the power n. It is useful to determine the dimensionless Avrami
d
parameter n from the maximal slope of the dT against ln T curve.
d
d ln T

p
n  E  1
e
1616\*
MERGEFORMAT (t) , in)
with e2.718. One can draw a tangent to the highest slope of the experimentally
determined sigmoid curve. This straight line will intercept abscissa at a given onset
point ln T1. The intercept with  =const 1 gives the end point ln T2 of the process.
According to Eq.16 , the parameter n is determined as follows
n
e
 ln T2 -ln T1   E  1
MERGEFORMAT (t) , in)
Conclusions
1717\*

The activation energy can be obtained from the slope of the dependence of
heating rate , log q, against the temperature of the maximal crystallization
rate , log Tp.

The characteristic crystallization time τp(q) can be determined either from
Eq.14 or from the intercept of the mentioned above straight line.
Tp

It is demonstrated that the product
q p
is constant depending on the
activation energy.
Appendix 1
Note that crystallization appears in a relatively narrow temperature interval,
so, if the reference heating rate is chosen in a way that Tr is about the middle of the
temperature interval in which the transformation takes place, the dimensionless
temperature
is
Tr
1, with accuracy of 3%
T
.
Therefore,
a
fairly
accurate
Tr
T 
 1 ln  r 
 T  so that we can make the following transformation:
approximation is T

 E
 E
E  Tr

 T 
  o exp 

ln  r   
  1   1  exp 

 RTr RTr  T
 RTr  T   

1818\*
MERGEFORMAT (t) , in)
E
In this way Eq.9 is satisfied with
E
RTr .
Appendix 2
The first derivative of Eq.11 is
d n  E  1 
T E1



dT
T
 q 1  E  1 Tr
MERGEFORMAT (t) , in)
n
n
 


T E1
 exp   

  q 1  E  1 TrE 






1919\*
Correspondingly, the second derivative is
2
d 2
T E1

n
E

1



  q  E  1 T 
dT 2
r
 r



n
n



 
1
T E1
1 



exp

 n  E  1  q r  E  1 TrE  






T E1

E 

 q r  E  1 Tr 
2020\* MERGEFORMAT (t) , in)
Therefore the extreme condition
d 2
dT 2
0
p
shows that the temperature of the peak T p
is determines by the condition
n


T E1
1 ; or TpE1 q r  E  1 TrE

E 

q

E

1
T


r 
 r
2121\*
MERGEFORMAT (t) , in)
E
As soon as the value of the dimensionless activation energy is about
one can neglect the term
1
1

 1
n  E  1 30n
E
30
RTr
.
Table 1
Crystallization time τ in dependence (Eq.14) of the [19] heating rate q and the
corresponding peak temperature Tp
q [K/dt (mainlymin]
5
7.5
10
20
Tp [K]
988
1000
1009
1032
τ [min]
6.4
4.3
3.1
1.7
n




1,3
1,2
lg q [K/min]
1,1
1,0
0,9
0,8
0,7
2,995
3,000
3,005
3,010
3,015
lg Tp
Fig.1 Dependence between the heating rate lg q and the peak temperature lg Tp.
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