Journal of Materials Processing Technology 92±93 (1999) 31±34
Relationship between the DTA peak and the phase
diagram: symbiosis between a thermodynamic
database and a DTA curve
M.H. Braga*,1,a, L.F. Malheirosa, J.M.V. Machadob, O.M. Freitasa
a
GMM/IMAT, Dep. Eng MetaluÂrgica, Faculdade de Engenharia UP, Porto, Portugal
b
Dep. Eng QuõÂmica, Faculdade de Engenharia UP, Porto, Portugal
Abstract
The shape of a differential thermal analysis (DTA) curve depends on the sample and the reference, the heating or cooling rates used in the
experiment as also the thermal resistance to heat ¯ow from the heat source to the cell containing the sample (or the reference), Ra (Rr).
Having access to a database that contains the values of the sample heat capacity, Cp, and those of the enthalpy of transformation, H, and
calculating the parameters that depend on the apparatus, the cells and the reference, as well as the heating rate used, it is possible to
determine the T (Ta ÿ Tr) values through the equation
Ta ÿ Tr Ra
dH
dTr
dTa
Rr Cr
ÿ Ra Ca
dt
dt
dt
where Ca is the heat capacity of the cell sample system and Cb the heat capacity of the cell reference system. The comparison between
the calculated and experimental values for T enables an assessment to be made as to whether the modelled parameters are in agreement
with the experimental information. This procedure enables a deeper perception of the thermal analysis in general and of the apparatus used
in particular. # 1999 Elsevier Science S.A. All rights reserved.
Keywords: Differential thermal analysis (DTA); Phase diagram's modelling
1. Introduction
Phase diagrams contain condensed data about the physic
state of materials. Such information is introduced into a
database following the thermodynamic modelling of the
system, developed essentially from experimental data [1].
Differential thermal analysis (DTA) is a technique based
on the temperature difference between a reference cell and
another that contains the sample, for a constant heating or
cooling rate [2]. The solidus and liquidus temperatures are
determined from the DTA curves from the `key values'
associated with the commonly-named endothermic and
exothermic peaks. Furthermore, the con®guration of the
DTA curve also gives some information about the sample,
the reference, the type of cells, the heating and cooling rates
*Corresponding author. Tel: +351-2-204-1790; fax: +351-2-204-1792
E-mail address: mbraga@fe.up.pt (M.H. Braga)
1
PRAXIS XXI's grant holder.
as well as the heat ¯ow from the heat source to the cell
containing the sample.
2. Experimental work
The Cu±Zr system, which has already been modelled by
the authors [3], was chosen for the study of the relationship
between the peaks, the con®guration of the DTA curve and
the phase diagram. Thus, there was already a database of the
equation corresponding to the variation in Gibbs energy with
the temperature and composition [3].
The authors used a SETARAM TGDTA92 apparatus,
which enables operation within the 293±1873 K temperature
range. All the experiments were conducted under an argon
atmosphere, at a constant pressure of 1.12 105 Pa. The
cells were pure alumina, of 100 ml capacity, also supplied by
SETARAM. One of the cells contained the sample to be
tested, and in the other, no sample was introduced. Thus, as
the cells were fully open, the reference used corresponds to
the sample volume in argon.
0924-0136/99/$ ± see front matter # 1999 Elsevier Science S.A. All rights reserved.
PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 2 1 7 - 4
32
M.H. Braga et al. / Journal of Materials Processing Technology 92±93 (1999) 31±34
Right at the beginning, the authors conducted some
experiments with standard samples of pure In, Al and Ni
for the same experimental conditions as the ones to be used
for the Cu±Zr system. The heating rate was 58C/min. Just
before each experiment with the samples, a `baseline' was
obtained for the empty cells for the same experimental
conditions as the ones to be used for the Cu±Zr system.
So, it could be af®rmed that both cells contain the same
reference (argon).
effect
Ta ÿ Tr ÿ Ta ÿ Tr b:l:
dH
dTr
dTa
Ra Cr0 Cr00
ÿ Ra Cr0 Ca00
Ra
dt
dt
dt
(6)
If dTa/dt dTr/dt, being the heating rate, and as
just before and after any transformation, dH/dt 0
Ta ÿ Tr ÿ Ta ÿ Tr b:l: Ra Cr00 ÿ Ca00
(7)
+
Ta ÿ Tr ÿ Ta ÿ Tr b:l:
Cargon ÿ Cstandard
3. Results and discussion
Ra
In DTA, the heat balance in the cells containing the
sample and the reference can be de®ned by the expressions
[4]
The resolution of this equation will give the value for the
thermal resistance to heat ¯ow from the heat source to the
cell containing the sample.
Taking into account the weight of the standard sample, its
density [5] and its heat capacity [6] as functions of the
temperature, and that argon can be considered an ideal gas
(at least for the temperature range and pressure of the
experiments), the value for Ra can be determined as a
function of the temperature, for the standards already mentioned. For the same temperature, the values obtained for the
resistance are quite similar.
Nevertheless, an exception has been registered with the
Im test in which, by experimental dif®culties associated
essentially with the sample oxidation, only a point for
comparison at 373 K has been obtained, a temperature that
is quite near to its melting point (429.6 K [7]). The authors
have also come to the conclusion that the resistance doesn't
vary too much with the temperature, at least in the temperature range of the experiments (298±1473 K). Thus,
it seems quite correct to consider a mean value for that
resistance (see Table 1) over that temperature range.
Also, the thermal resistance to heat ¯ow from the heat
source to the cell containing the sample (Ni standard) has
been compared to that for alumina [8] (the support of the
cells was in pure alumina [9]) in order to establish a
qualitative comparison. The results are presented graphically in Fig. 1.
It should be noted that the DTA curve for the Ni standard
sample could be affected, and consequently, the calculation
of the value for Ra, by the second-order transformation
(ferromagnet ! paramagnet) observed for Ni at 633 K [7].
The oxidation of Ni, perfectly identi®ed from 850 K in
the graph representing the variation in the weight of the
sample with the temperature (see Fig. 1), has probably
dQa
dTa dH
Ca
ÿ
;
dt
dt
dt
dQr
dTr
Cr
dt
dt
(1)
where Ca (Cr) is the heat capacity of the cell sample
(reference) system, Qa (Qr) the heat supplied to the sample
(reference), Ta (Tr) the temperature of the cell containing the
sample (reference), H the enthalpy of transformation and t
the time.
By Newton's law [2]
Ca
dTa
Tf ÿ Ta dH
;
Ra
dt
dt
Cr
dTr
Tf ÿ Tr
Rr
dt
(2)
+
T a ÿ T r Ra
dH
dTr
dTa
ÿ Ra C a
Rr Cr
dt
dt
dt
(3)
where Tf is the temperature of the heat source and Ra (Rr) the
thermal resistance to heat ¯ow from the heat source to the
cell containing the sample (reference).
By tracing the baseline for cells that do not contain a
sample, any transformation in the temperature range of the
experiment (dH/dt 0) will not occur:
Ta ÿ Tr b:l: Rr Cr0 Cr00
dTr
dTa
ÿ Ra Ca0 Ca00
dt
dt
(4)
where Cr0 Ca0 is the heat capacity per unit volume of the
reference cell (equal to that containing the sample) and
Cr00 Ca00 the heat capacity per unit volume of the reference
(sample).
As (i) the cells are hypothetically equal and (ii) the sample
and the reference are the same (argon used as ¯ushing
gas)
Ta ÿ Tr b:l: Rr ÿ Ra Cr0 Cr00
dTr
dt
(5)
It should be emphasized that, in the last two expressions,
it has been considered that the heat capacity per unit
volume as a comparison is established between the sample
and the corresponding volume of argon. For an experiment
with a standard sample and deducting the baseline
(8)
Table 1
Mean values for the thermal resistance to heat flow from the heat source to
the cell containing the sample
Standard sample
Rmean (107) (Km3 s/J)
In
Al
Ni
0.9
3.3
3.3
33
M.H. Braga et al. / Journal of Materials Processing Technology 92±93 (1999) 31±34
values for the heat capacity and the enthalpy of transformation taken from a database [1,3] (see Fig. 2). For the ®rst
reaction (eutectoid), ideally isothermal, we have for the
onset
Ta0 ÿ Tr0 ÿ Ta ÿ Tr b:l: Ra Cr00 ÿ Ca00
Fig. 1. Comparison between the thermal resistance to heat flow from the
heat source to the cell containing the sample (Ni standard) and that for
alumina. (a) DTA curve for the Ni standard sample.
affected the DTA curve, and in consequence, the calculation
of the thermal resistance.
After having determined the values of Ra for the different
standards, some DTA experiments have been conducted
with samples of the Cu±Zr system for x(Zr) 0.536, the
equilibria of which for T 1184.8 K and T 1187.8 K [3],
respectively, are already known as
fcc-A1 Cu5 Zr8 $ CuZr Cu5 Zr8
and
fcc-A1 $ fcc-A1 Cu5 Zr8
This alloy has been chosen in order to test the symbiosis
between the values from a database and the experimental
data obtained from a DTA curve. Thus, the curve obtained
by DTA experiments for the selected alloy of the Cu±Zr
system has been compared to the curve drawn from the
Fig. 2. Comparison between the DTA curves obtained from experimental
data and by calculation from the database. Superimposition with the phase
diagram where the different reactions corresponding to the selected alloy
of the Cu±Zr system are represented.
dTr
dt
(9)
in which Ta0 represents the temperature for the equilibrium
in accordance with the diagram already modelled [3] (see
Fig. 2). Thus, taking into account that, for an isothermal
reaction, dH/dt 0,
dTr
t
Ta ÿ Tr ÿ Ta ÿ Tr b:l: Ta0 ÿ Tr0
dt
dT
dT
r
r
ÿ
t
(10)
ÿ Ta ÿ Tr b:l: Ra Cr00 ÿ Ca00
dt
dt
Just after the reaction, dH/dt is still null, and in consequence (see Eq. (6)),
Ta ÿ Tr ÿ Ta ÿ Tr b:l:
dTr
dTa
Ra Cr0 Ca00
Ra Cr0 Cr00
dt
dt
(11)
This equation has been solved by Laplace's transforms
[10] in order to analyse its resolution as a function of the
different parameters:
t
Ta Ta0 exp ÿ
Ra Cr0 Ca00
Tr0 Tb:l: Ra Cr00 ÿ Ca00
t
t
1 ÿ exp ÿ
Ra Cr0 Ca00
(12)
+
t
Ta ÿTr ÿTb:l: Tmax ÿ Tb:l: exp ÿ
Ra Cr0 Ca00
t
(13)
Ra Cr00 ÿ Ca00 1 ÿ exp ÿ
Ra Cr0 Ca00
where Tr Tr0 t, Cr0 is the heat capacity per unit volume
for alumina [8], Cr00 is the heat capacity per unit volume for
argon, Ca00 is the heat capacity per unit volume for the alloy of
the system Cu±Zr [1,3] and t is the time taken from the peak.
It should be emphasized that the expression Ta0 ÿ
Tr0 Tmax is valid from the peak that corresponds to
the end of the reaction, and in consequence, is valid for
all the systems.
It should also be noted that all the information necessary
for the control of the time span (t) for a given calculation,
governed by a particular equation (for example, for the
eutectoid reaction mentioned earlier, the time span between
the reaction onset and the peak) has been taken from the
experimental data ®le.
For the calculations, Ra has been taken as 3.3
10ÿ7 Km3 s/J and equal to the mean value obtained for
the thermal resistance in the experiments with the Al and
Ni standard samples (see Table 1).
34
M.H. Braga et al. / Journal of Materials Processing Technology 92±93 (1999) 31±34
For the second reaction mentioned earlier for the alloy of
the Cu±Zr system selected (for which dH/dt 6 0), the calculation of (Ta ÿ Tr) has been developed from Eq. (6) by the
application of the Butcher method [11], using a speci®c
routine created in Fortran 77. In this case, the resolution of
Eq. (6) by the application of Laplace's transforms is not
possible as the variation in the heat capacity of the sample
with the temperature during the reaction should have been
taken into account. Thus, the speci®c routine developed by
the present authors has been used, bearing in mind that dH/
dt(Ta(t)), Cr Cr(Tr) and Ca Ca(Ta)dH/dt(Ta(t)) and the
heat capacity per unit volume Ca00 Ca00 (Ta) have been calculated from the database's content [1,3] (data concerning
the thermodynamic functions and the phase diagrams).
It should be noted that all the calculations have taken into
account the expansion of the sample with temperature
because it affects not only the properties of the sample itself
but also the properties of the reference. Effectively, during
the experiments, the volume of the reference has always
been taken as equal to the volume of the sample tested.
Nevertheless, it is observed that such a correction does not
in¯uence the ®nal results too much.
from the experimental results. The difference (2 K)
between the calculated and the experimental values for
the first onset enables the foreseeing of a small adjustment
of the theoretical parameters. On the other hand, the difference between the temperatures for the onset and for the
second peak (5 K in the experimental curve and 3 K for the
theoretical curve) implies a higher slope for the liquidus
curve if the error associated with the atomic fraction of
zirconium in the sample of the Cu±Zr system chosen is less
than 0.3%, which has been totally impossible for the authors
to confirm.
The authors are quite sure that they have proven that
symbiosis could be established between a database and a
DTA curve. By a calculation with the parameters contained
in the database, a rough approach to the system can be
obtained, concerning essentially the peaks and the respective
temperatures. The DTA curve will allow the adjustment, step
by step, of those parameters. On the other hand, the calculation of the thermal conductance allows one to know the
apparatus better, and in consequence, the DTA curves.
Acknowledgements
4. Conclusions
From the calculation of the thermal resistance (Ra) to heat
¯ow from the heat source to the cell containing the sample,
and from the baseline and the DTA curve for standard
samples, a mean value has been obtained that can be used
for other calculations within the temperature range of the
experiments (298±1473 K). This value is dependent on
the apparatus used, but for the present equipment, is quite
independent of the temperature.
The authors have also come to the conclusion, from the
calculation of the theoretical curve from the Ra value and the
parameters included in the database, that the deviation
between the experimental and the calculated results,
observed in the eutectoid reaction for the sample of the
Cu±Zr system, could result from the consideration of an
in®nite thermal conductance between the sample and its cell
(the ideal case [12]). Nevertheless, due to the super®cial
oxidation always detectable at such high temperatures and
that restrains the thermal contact between the sample and its
cell, a different value should be considered for the thermal
conductance. However, as this parameter is very hard to
determine and varies from experiment to experiment, it has
been neglected, although its in¯uence is clearly present in
the experimental curve. Finally, it can be concluded that the
parameters of the database are quite similar to those deduced
The authors are deeply grateful to Prof. Fonseca Almeida
for the loan of the DTA equipment.
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