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 E 
that there is a simple relation  p 
Tp
q  E  1
E
. It is demonstrated
RT
between the characteristic crystallization
time τ, the heating rate q and the peak temperature Tp.
Keywords: overall crystallization kinetics, thermal analyses, DSC, DTA
Introduction
It seems that Verhulst first postulated that the transformation fraction (t) , 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 )
 k ( t ) f  
t
(1)
In the following we use the reciprocal of k(t), the characteristic time  
1
. It is
k t 
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 (KJMA) theoretical
model [6-8]. According to it the time dependence of degree of transformation α is:
  1-e
t
- 
 
n
(2)
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) and k(t) 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 (see [13,14]) only when crystallization time is constant, const. In
this case the corresponding derivative form (see Eq.(1)) is:
1
d  t   1  
1 
n
   n 1      ln 1     
dt
  

(3)
Nowadays there are experimental data on the derivative form d/dt (mainly
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, because it is well known that negligible
deviations in integral forms lead to important errors in differentiated ones. For the
isothermal case (τ=const) the most popular approach was to plot the experimental
results in bilogarithmic coordinates, log(  log(1   )) 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(  log(1   )) is very inaccurate for both α→0 and for α→1. Recently,
we proposed [17,18]
derivative
to plot the data in coordinates α against log t. Thus the
d
is determined and through it the n value is obtained in a reliable
d log t
way.
The situation is much more complicated when τ is not a constant (for instance
in non-isothermal conditions). Although there are many approaches (see [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

1      ln 1     n
dt   t 
(4)
So that the transformation rate can be presented as
1
1
d
n
n

1



ln
1



 

dT q (T )
where q 
(5)
dT
is the heating rate. Since at low temperatures the value of (T) is very
dt
large, the process is frozen and we can assume that the heating at constant rate q starts
at T=0 K. The corresponding solution of Eq.(5) is
  T dT  n
  1  exp    

  0 q T  





It is useful to note that the integral in Eq.(6) is
(6)
n
 T dT 

   ln 1   
 0 q T  
(7)
To solve properly Eqs.(6)(7), 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 

 RT 
   o exp 
(8)
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 Tp of the maximums of the
T2 
d
1
against temperature T curves in coordinates ln  p  against
. From the slope
dT
Tp
 q 
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 otherwise, according to Eq.(8), the
RT
corresponding crystallization time would be unreasonably large (see for instance
[10]). Thus, for
E
 37 the exponential term in Eq.(8) is about 1016, so the
RT
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
the
d
against T curves. Therefore here we start here new approach. Let at a given
dT
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
(9)
In Appendix 1 we demonstrate that Eq.(9) is equivalent to assumption made by the
Eq.(8) with E 
E
. The advantage of Eq. (9) is that the integral in Eq.(6) is easily
RTr
solved.
T 
dT
T
0 q T   q r E  1  Tr 
T
E
(10)
So that the solution of Eq.(6) is
E n
 



T
T

  1  exp   
  q r  E  1  Tr  

 




(11)
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 (see 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.
Appendix 2 provides detailed analyses of the peak position of the
d
against
dT
T curve It is shown that at any heating rate q the peak temperature Tp(q) is determines
by the condition
TpE1
q r  E  1 T
E
r
 1 ; or TpE1  q r  E  1 TrE
(12)
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(E  1)  log( r )
The dimensionless activation energy E 
(13)
E
can be determined from the slope of
RTr
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
(14)
In other words, there is an universal relation between the peak temperature Tp , the
corresponding crystallization time τ and heating rate q
Tp
q p
  E  1
(15)
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/min] 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 . The heating rates, the peak temperatures and the corresponding
RTp
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/mol]. The analysis of the same DTA data in [10], using the Kissinger equation,
gave an activation energy of ~400 [kJ/mol]. 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. (11)(12) the degree of crystallinity p at the maximum of
the
d
1
against temperature T curve has an universal value  p  1   0.63 ,
e
dT
independent on the power n. It is useful to determine the dimensionless Avrami
parameter n from the maximal slope of the
d
d ln T

p
d
against ln T curve.
dT
n  E  1
e
(16)
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  = 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
(17)
Conclusions

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.

It is demonstrated that the product
Tp
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
approximation is
Tr
 1, with accuracy of  3% .
T
Therefore,
a
fairly
accurate
Tr
T 
 1  ln  r  so that we can make the following transformation:
T
T 


In this way Eq.(9) is satisfied with E 
E
.
RTr
 E
E T
 E
 T 
r
   o exp 

ln  r   
  1    1  exp 

 RTr RTr  T
 RTr  T   

(18)
Appendix 2
The first derivative of Eq.(11) is
d n  E  1 
T E 1



dT
T
 q 1  E  1 Tr
n
n
 


T E 1
 exp   

  q 1  E  1 TrE 






(19)
Correspondingly, the second derivative is
2
d 2
T E 1

n
E

1
     q E  1 T 
dT 2
r
 r



n
n
n

 

 
 
1
T E 1
T E 1
1 

  exp   
 
 n  E  1  q r  E  1 TrE  
  q r  E  1 TrE  




(20)
Therefore the extreme condition
d 2
dT 2
 0 shows that the temperature of the peak Tp
p
is determines by the condition
n


T E1
 1 ; or TpE1  q r  E  1 TrE

E 

 q r  E  1 Tr 
(21)
As soon as the value of the dimensionless activation energy is about E 
one can neglect the term
E
 30
RTr
1
1

 1 .
n  E  1 30n
Table 1
Crystallization time τ in dependence (Eq.(14)) of the [19] heating rate q and the
corresponding peak temperature Tp
q [K/min]
Tp [K]
τ [min]
5
988
6.4
7.5
1000
4.3
10
1009
3.1
20
1032
1.7
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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