The oxidation mechanisms of Cu54Ni45Mn1 (Constantan) tapes: Kinetic analysis

Article abstract of DOI:10.1016/j.tca.2008.06.017

The kinetic parameters, namely the triplet activation energy E_A, model function f(α) or g(α) and pre-exponential factor A of the oxidation of Constantan tapes in 1 atm of oxygen have been determined from both isothermal and non-isothermal thermogravimetry. For isothermal experiments, with temperatures ranging from 650 °C to 900 °C, the results from direct conversion of the weight increase as a function of the time and curve fitting, are compared with the isoconversion method. For the non-isothermal experiments, with heating rates from 1 °C/min to 20 °C/min, comparison is made between the Friedman differential method and the integral methods of Kissinger, Ozawa and Li and Tang. All methods give apparent activation energies with relative standard deviations as low as 3%. The results converge to the identification of three stages in the oxidation behaviour. A parabolic law for reaction extents α below 15% with E_A = 246 ± 7 kJ mol^-1, ln A = 14.3, is followed by two linear stages with E_A = 244 ± 4 kJ mol^-1 and ln A = 15.3 for 0.18 < α < 0.35 and E_A = 228 ± 15 kJ mol^-1, ln A ≈ 13 for α > 45%, respectively.

Full text of DOI:10.1016/j.tca.2008.06.017

Thermochimica Acta 475 (2008) 44–52
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Thermochimica Acta
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The oxidation mechanisms of Cu54Ni45Mn1 (Constantan) tapes: Kinetic analysis
T.K. Jondoa,b, Ph. Galezb, J.-L. Jordab,∗, J. Le Royb, J.C. Martyb, J.L. Soubeyrouxc
a Faculté des Sciences, Université de Lomé, BP1515 Lomé, Togo, France
b SYMME, Université de Savoie, BP 80439, 74944 Annecy le Vieux Cedex, France
c Institut Néel/CRETA/CNRS, 25 Avenue des Martyrs, BP166-38042 Grenoble Cedex, France
a r t i c l e i n f o
a b s t r a c t
Article history:
The kinetic parameters, namely the triplet activation energy EA, model function f(˛) or g(˛) and pre-
exponential factor A of the oxidation of Constantan tapes in 1 atm of oxygen have been determined from
both isothermal and non-isothermal thermogravimetry. For isothermal experiments, with temperatures
ranging from 650 ◦C to 900 ◦C, the results from direct conversion of the weight increase as a function
of the time and curve fitting, are compared with the isoconversion method. For the non-isothermal
experiments, with heating rates from 1 ◦C/min to 20 ◦C/min, comparison is made between the Fried-
man differential method and the integral methods of Kissinger, Ozawa and Li and Tang. All methods give
apparent activation energies with relative standard deviations as low as 3%. The results converge to the
identification of three stages in the oxidation behaviour. A parabolic law for reaction extents ˛ below
15% with EA = 246 7 kJ mol−1, ln A = 14.3, is followed by two linear stages with EA = 244 4 kJ mol−1 and
ln A = 15.3 for 0.18 < ˛ < 0.35 and EA = 228 15 kJ mol−1, ln A ≈ 13 for ˛ > 45%, respectively.
Received 15 March 2008
Received in revised form 18 June 2008
Accepted 22 June 2008
Available online 10 July 2008
Keywords:
Ni–Cu alloy
Oxidation
Isoconversion
Arrhenius parameters
Model fitting
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Bertrand [9] was also devoted to high temperature corrosion of
Constantan alloys. It is reported that the oxidation kinetics follow
The transport of current in high-temperature superconductors is
conditioned to the deposition of the superconducting phase on a bi-
axially textured metallic substrate [1]. Nickel tapes are widely used
but for AC applications, the ferromagnetism of this metal intro-
duces hysteretic losses which is a major drawback. For this reason,
Ni–Cu substrates with copper content from 50 at.% to 90 at.% have
been produced [2,3]. In this copper composition range, Girard et al.
[4] have shown the ability to perform an adequate bi-axial texture
on industrial Constantan alloy. During the coating process at high
temperature, oxidation of the substrate may occur thus modifying
the surface relation between the tape and the superconductor. The
knowledge of the oxidation kinetics and mechanism of the metal-
lic tape is then of major importance to try to prevent such surface
oxidation which is detrimental to the epitaxial growth of the super-
conductor. Recent studies on the oxidation of copper-rich NiCu
alloys at high temperatures and various oxygen pressures [5–7]
focussed on the formation and the description of the oxide scale and
support an approximately parabolic oxidation behaviour. Brückner
et al. [8] found that the oxidation of Constantan films in air and
argon atmosphere (with about 3 ppm oxygen) starts at 300–350 ◦C
forming, in the final state NiO, CuO or mixed oxide. The work of
a parabolic law with however three steps in the scale formation
having different activation energies.
To investigate the oxidation mechanisms of Constantan tapes in
1 atm of oxygen, we conducted a kinetic analysis based on thermo-
gravimetry. We followed as close as possible the procedures used
in the ICTAC Kinetic Project for the study of the calcium-carbonate
decomposition [10–14]. This paper presents the computational
aspects of a study carried out to reach the characteristic triplet of
the oxidation reaction: activation energy EA, model function f(˛)
or g(˛) of the reaction extent ˛ and the pre-exponential factor A.
It constitutes the first part of a general study on the mechanisms
which govern the oxidation of Constantan tapes, with a comparison
between the behaviour of as rolled and re-crystallized samples and
a discussion on the role of impurities on the oxidation kinetics.
2. Theoretical considerations
The kinetics of solid-state reaction processes is usually governed
either by nucleation and growth or by grain boundary reactions or
by diffusion of the reacting species. The theoretical support for the
kinetic analysis may be found in textbooks [15,16] and numerous
recent reviews [17–22]. Let us summarize here the methods used
in this work. By definition, the experimental part of the study of
a reaction kinetics consists in determining the extent of the reac-
tion ˛ (0 ≤ ˛ ≤ 1) as a function of the time. This is described by the
∗
Corresponding author. Tel.: +33 450 096 518; fax: +33 450 096 543.
E-mail address: jean-louis.jorda@univ-savoie.fr (J.-L. Jorda).
0040-6031/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.tca.2008.06.017

T.K. Jondo et al. / Thermochimica Acta 475 (2008) 44–52
45
reaction rate (d˛/dt) which contains information on both the mech-
anism and the probability of occurrence of the reaction. Indeed, the
reaction rate is generally expressed as a product of two functions:
All these methods are efficient to compute EA which is expected
to be independent from the reaction extent ˛, in the case of a single
reaction mechanism [29,30]. For a known EA and a given f(˛) or g(˛)
function, the pre-exponential factor A can be calculated from one
of the expressions given in Table 1.
The experiments also give access to the reaction model by fitting
g(˛) on the basis of the known kinetic functions [24]. For isothermal
analysis, the solution of the integral form of the kinetic equation is
ꢀ
ꢁ
da
dt
= f (a) · k(T)
(1)
•
f(˛) describes the analytic form of the limiting mechanism of
the kinetic model; it is associated to the corresponding integral
ꢂ
ꢅ
t
function g(˛) = d˛/(f (˛)). Different mathematical expressions
g(˛) − g(˛0) =
k(T) dt = k(T) · (t − t0)
(3)
account for the various mechanisms [23].
t
•
k(T), usually set as an Arrhenius-type function k(T) =
A exp(−EA/RT), is temperature-dependent and is related to
the pre-exponential factor A and to the apparent activation
energy EA. R is the gas constant. Theoretical support for this
assumption has been given by Galwey and Brown [24].
0
in which ˛0 and t0 are the extent of reaction and the time at the
beginning of the temperature plateau, respectively. As already men-
tioned, g(˛) reflects the limiting mechanism for instance species
diffusion for which parabolic, linear and logarithmic laws have been
found to satisfactory describe the oxidation kinetics of many metals
and alloys at least in part of the reaction extent range.
To find f(˛) and g(˛) from non-isothermal experiments, it is
necessary to use complementary model functions y(˛) and z(˛)
introduced by Malek et al. [21] and derived from the ˛(t) data.
In the case of thermogravimetric measurements, ˛(t) is defined
as ˛(t) = (m(t) − m )/(m∞ − m ) where m(t), m and m are the sam-
∞
0
0
0
ple masses at time t, at the beginning of the experiment and when
the reaction is completed, respectively. Our study concerns the
oxidation behaviour of Ni–Cu–Mn alloys; m was calculated con-
∞
sidering that metals are transformed into Cu(II), Ni(II) and Mn(II)
oxides. This assumption may be discussed with the possibility to
form the Cu(I) oxide but we shall see in the second part of this
series of papers that it is largely supported by our optical and elec-
tronic metallographies as well as by XRD which show that Cu2O
represents less than 20% of the total amount of copper oxide after
oxidation whatever the ˛-value. The error on ˛ is then less than
0.5%.
3. Experimental
The Cu54Ni45Mn1 alloy (Constantan) was purchased from Good-
fellow as 8-mm diameter cylinders. Besides copper and nickel, the
manufacturer indicates 7500 ppm Mn and 2500 ppm Fe, present as
impurities. Cold rolling was realised on a Redex rolling mill to a
final thickness of 150 ␮m which corresponds to a deformation of
98.4%.
Isothermal experiments have been considered for long time to
be more accurate than non-isothermal heating but the increase in
sensibility of the temperature and weight sensors allows the use
of small samples in which heat exchange may be considered as
instantaneous so that non-isothermal experiments are now widely
used. Both processes may be associated with an isoconversional
analysis of the data which gives access to EA even if the reaction
mechanism f(˛) is not known. Eq. (1) may be written as
Fifty metal tapes were cut into 1 cm × 2 cm samples about
260 mg in weight. These samples were degreased in ethanol prior to
oxidation. The thermogravimetric data were recorded at PO2 = 1 bar
using a SETARAM TG 92 apparatus. The samples were hung in the
working zone of the apparatus with a platinum wire. For isother-
mal experiments, the sample temperature was rapidly (50 ◦C/min)
raised to the desired plateau value between 650 ◦C and 900 ◦C and
then kept constant for a maximum of 10 h. Because the tempera-
ture was raised in 1 bar O2, the early stage of the oxidation – 0.1–2%
in ˛ depending on the plateau temperature – could not be anal-
ysed. For non-isothermal experiments, the temperature was raised
to 900 ◦C with heating rates ˇ between 1 ◦C/min and 20 ◦C/min. A
gold tape used as blank was monitored for buoyancy corrections,
found to be less than 0.3% in ˛. The results expressed in ˛ terms are
reported in Fig. 1 for isothermal from 650 ◦C to 900 ◦C (Fig. 1a) and
non-isothermal (Fig. 1b) experiments.
ꢀ
ꢁ
d˛
dt
EA
ln
= ln[A · f (˛)] −
(2)
RT
For a given ˛-value, this quantity changes linearly with the inverse
of temperature 1/T and EA is readily deduced from the slope −EA/R.
For isothermal analysis the approach is straightforward. For non-
isothermal experiments, the temperature increases as T = Ta + ˇt
with Ta, the ambient temperature and ˇ the heating rate. The
ln (d˛/dt) versus 1/T analysis for a given ˛ and different ˇ exper-
iments is known as the Friedman method [25]. To calculate the
ꢂ
4. Kinetic analysis
integral form of the kinetic equation g(˛) = k(T) · dt, it is usual
to expand the Arrhenius function. We have used in this work two
approximations known as the Kissinger [26] and the Ozawa [27]
models and also the direct integral isoconversion method proposed
by Li and Tang [28]. The underlying mathematics for these proce-
dures are summarized in Table 1.
4.1. Isothermal process
4.1.1. The ˛(t) behaviour
The oxidation of Ni–Cu alloys has been claimed to imply diffu-
sion through a Cu2O/NiO scale [31]. A glance on the ˛(t) curve after
an isothermal oxidation is a simple way to start the analysis. An
example is given in Fig. 2 which represents the reaction extent ˛(t)
as a function of the time at T = 875 ◦C.
Table 1
Summary of relations used in this work for the isoconversional treatment of non-
isothermal reaction kinetics
It may be seen that parabolic oxidation is only acceptable in the
domain (I) of the graph which corresponds to ˛ < 15%. For higher ˛
values, a linear behaviour is observed with a change of slope k(T)
at ˛ = 35–45%, considered as a transition region.
Method
Equation
Ref.
ꢃ
ꢄ
E
RT
d˛
dt
A
Friedman
ln
= ln[f (˛) · A] −
[25]
ꢀ ꢁ
ꢃ ꢄ
ˇ
E
A
AR
A
Kissinger
Ozawa
ln
= ln
A
R
ꢄ
− ln[g(˛)] −
[26]
[27]
[28]
2
E
RT
˛
T
˛
lnˇ = ln AE − ln[g(˛)] − 2.315 − 1.0516 ·
˛ ln
E
RT
A
˛
4.1.2. Apparent activation energies
In the parabolic domain (region I in Fig. 2), the isothermal
oxidations of tapes with a given surface area S are represented
ꢂ
ꢃ
ꢂ
˛
Li and Tang
d˛ = − E
+ G(˛)
d˛
dt
d˛
T
A
R
0
0

46
T.K. Jondo et al. / Thermochimica Acta 475 (2008) 44–52
Fig. 3. Isothermal oxidation of Constantan in the parabolic regime.
From these results, the apparent activation energy EA in the
parabolic regime of the oxidation of Constantan is found to be
EA = (251 5 kJ mol−1).
The curves in Fig. 3 were obtained once the temperature plateau
was reached, 12–18 min after the heating started. The correspond-
ing minimum ˛ values have been reported in Table 2. Since the
parabolic oxidation of Constantan samples is not a general rule in
the whole ˛-range, isoconversion analysis has been used to better
define the limits for which a given model may be considered. As
explained in Section 2, in such model-free method, the activation
energy for each ˛-value is deduced from ln (d˛/dt) versus 1/T plots.
The result is shown in Fig. 4.
We can clearly distinguish two main domains with different
activation energies. The first one extends up to about ˛ = 35% with
EA = (240 5 kJ mol−1). It corresponds to the parabolic and the first
linear regime of the oxidation. The second linear behaviour, above
˛ = 35%, is characterized by a slightly but significantly lower activa-
tion energy EA = (220 5 kJ mol−1).
Fig. 1. Oxidation of Constantan in oxygen (PO = 1 bar): (a) isothermal and (b) non-
isothermal.
2
4.1.3. Curve fitting: the kinetic model and the frequency factor
We now reconsider the isothermal ˛(t) change such as the one
reported in Fig. 2, from which we have roughly delimited the dif-
ferent domains for the oxidation. The succession of regimes as time
and reaction extent increase is even more evidenced with a log–log
plot of d˛/dt versus time (Fig. 5; T = 850 ◦C).
The parabolic regime is characterized by reaction rates d˛/dt
varying as t−1/2. Of course, d˛/dt remains constant in the case of
Fig. 2. Isothermal oxidation of Constantan at 875 ◦C.
in Fig. 3. The results are given as (ꢀm/S)2 versus t from which
the Arrhenius constant kp(T) may be computed (Table 2). In this
table, the maximum ˛ values for parabolic behaviour have been
indicated.
Table 2
Arrhenius constant kp(T) for the isothermal oxidation of Constantan
Temperature (◦C)
˛ Min %
˛ Max %
kp(T) (mg2 cm−4 s−1
)
650
700
750
775
800
825
850
0.56
0.84
1.07
1.25
1.52
1.90
2.43
4
9
8.2 × 10−6
6.4 × 10−5
2.4 × 10−4
4.8 × 10−4
1.0 × 10−3
1.7 × 10−3
2.9 × 10−3
18
18
19
18
20
Fig. 4. Activation energies from the isoconversion analysis of isothermal data
against the extent of reaction ˛.

T.K. Jondo et al. / Thermochimica Acta 475 (2008) 44–52
47
Fig. 5. Log–log plot of d˛/dt vs. time at 850 ◦C showing the three different regimes:
parabolic below 7000 s (˛ < 15%), linear between 7000 s and 12 000 s (15% < ˛ < 35%)
and linear with a lower k(T) value above 20 000 s (˛ > 45%).
a linear regime. Below 7000 s (˛ < 15%), the curve shows a lin-
ear decrease of log (d˛/dt) with log (t) with a slope close to −1/2.
Then d˛/dt remains constant, first between 7000 s and 12 000 s
(15% < ˛ < 35%) and also above 20 000 s (˛ > 45%) which denotes
linear behaviours with however slightly different k(T) values. It
should be stressed for further analysis of the oxidation mecha-
nism that the transition from the first linear regime to the second
one is not abrupt but takes about 2 h at 850 ◦C, i.e. occurs in
the 35–45% ˛-range. Moreover, for all investigated temperatures
(650 ◦C < T < 900 ◦C), the transitions from one regime to the other
were found to occur for the same ˛ values so that we have:
•
a parabolic oxidation below ˛ = 15–18%,
a first linear oxidation between ˛ = 15–18% and 35%,
a second linear oxidation above ˛ = 45%,
a very smooth transition between the two latter regimes.
•
•
•
For each temperature and each domain, curve fitting has been
used to estimate k(T) and then EA and A from ln (k(T)) versus 1/T
plots (Fig. 6 and Table 3).
We have also summarized in Table 3 the values for the couple
EA–A obtained from isoconversion analysis. The consistency of the
two mathematical treatments has to be noticed. Similarly it is worth
pointing out the low values of the standard deviations generated
by the calculations, excepted for the second linear domain in the
curve fitting method, which is covered with only four temperatures
(Fig. 6c)
Once the model functions f(˛) and their validity range have been
ascertained, the value of the pre-exponential factor can be deduced
from isoconversion analysis for each reaction extent. The corre-
sponding results are shown in Fig. 7 where the two main domains
noticed in Fig. 4 for the apparent activation energies may be recog-
nized.
Fig. 6. Ln (k(T) against 1/T plots and linear fits for the calculation of EA and ln (A) in
the parabolic (a), first linear (b) and second linear (c) domains.
4.2. Non-isothermal methods
In non-isothermal methods the extent of the reaction depends
on the heating rate ˇ for a given final temperature. For the material
studied here, coupled differential thermal analysis and thermo-
gravimetry shown in Fig. 8 reveals that significant mass gain does
not start below 450 ◦C. Complete oxidation of Constantan, when
heated with ˇ = 1 ◦C/min occurs at 1100 ◦C.
Table 3
EA and ln (A) values deduced from ˛(t) fits
Domain
Curve fitting method
Isoconversional method
EA (kJ mol−1
)
Ln (A)
EA (kJ mol−1
)
Ln (A)
Parabolic
Linear I
Linear II
242
241
230 19
6
4
14.4 0.6
15.2 0.5
244
244
220
7
4
7
14.7 0.9
15.3 0.5
12.8 0.7
For lower temperatures, complete oxidation cannot be obtained
(see Fig. 1b). In our experiments, the maximum ˛-value was about
40% for a heating rate ˇ = 1 ◦C/min and a maximum temperature
14
2

48
T.K. Jondo et al. / Thermochimica Acta 475 (2008) 44–52
Fig. 7. Pre-exponential factor A as a function of reaction extent ˛ as deduced from
isoconversion for isothermal data.
of 900 ◦C. From the ˛(t) behaviour with various ˇ, isoconversional
analysis was performed. The results are displayed in Fig. 9 for
the Friedman differential method [25] (Fig. 9a), the generalised
Kissinger [26] (Fig. 9b) and the Ozawa [27] (Fig. 9c) integral meth-
ods for heating rates ˇ from 1 ◦C/min to 20 ◦C/min and selected
˛-values from 0.5% to 12.5%.
Despite some scattering, the expected linear behaviour of the
various functions versus 1/T is worth being pointed out. Using inte-
gral methods of Kissinger and Ozawa, the problem of the expansion
of the temperature integral is raised and the “exact” value of this
term is continuously under discussion [32]. For comparison, the
Li and Tang method [28], free from integral approximation has
been used. An example of ln (d˛/dt) and 1/T curves versus ˛ with
ˇ = 5 ◦C/min is given in Fig. 10.
From such plots at various heating rates, numerical integration
ꢂ
allows to represent ln t[ln(d˛/dt) = ˛ ln (d˛/dt) d˛ as a function
0
ꢂ
of ln t[1/T] = ˛(1/T) d˛. The results are straight lines with slope
0
−EA/R shown in Fig. 11. Each group of points represents a reaction
extent from 1% to 10% for linear heating rates ranging from 1 ◦C/min
to 15 ◦C/min.
The activation energies for the oxidation of the Constantan tapes
for each ˛ value computed from all the above-mentioned non-
isothermal methods are represented in Fig. 12. It may be seen that
in the 1–15% range and within error limits, about 8 kJ mol−1 in each
Fig. 9. Non-isothermal–isoconversion oxidation of Constantan. Differential Fried-
man method (a), integral Kissinger (b) and Ozawa (c) methods; ˛ = 0.005 (ꢀ); 0.01
(ꢀ); 0.02 (ꢁ); 0.05 (ꢁ); 0.1 (ꢂ) and 0.12 (+).
case, the apparent activation energy does not depend on the reac-
tion extent. The mean value of all reported EA is 243 kJ mol−1 with
a standard deviation ꢂ = 15 kJ mol−1
.
The agreement with the results obtained from isothermal exper-
iments in the same ˛-range, as well as the accuracy for each method
underline the coherence of the measurements and, moreover, sug-
gest that oxidation proceeds with a single limiting mechanism.
In our isothermal studies we found that below ˛ = 15%, the oxida-
tion follows a parabolic law, suggesting a D1 diffusion mechanism
for which the f(˛) function is f(˛) = 1/(2˛) and the integral func-
tion g(˛) = ˛2. In non-isothermal studies, it is possible to compute
Fig. 8. Coupled differential thermal analysis (dashed curve)–thermogravimetry
(bold) of the oxidation of Constantan in oxygen atmosphere with a heating rate
ˇ = 1 ◦C/min.

T.K. Jondo et al. / Thermochimica Acta 475 (2008) 44–52
49
Fig. 10. Ln (d˛/dt) and 1000/T as a function of ˛ for a heating rate ˇ = 5 ◦C/min used
for the Li and Tang integral method.
Fig. 13. Fitting of the non-isothermal oxidation curve by the Malek method [21]. For
ˇ ranging from 1 ◦C/min to 20 ◦C/min, g(˛) is proportional to ˛n with n = 1.9 0.1.
the ratio z(˛)/(By(˛)) = g(˛)/A, where z(˛) and y(˛) are the func-
tions introduced by Malek et al. [21]. Fig. 13 shows this ratio as a
function of ˛ for heating rates ˇ between 1 ◦C/min and 20 ◦C/min.
All curves could be fitted as g(˛)/A = a˛n with n = 1.9 0.1 and R2
values larger than 0.999, in excellent agreement with the isother-
mal experiments, extending the model function for D1 mechanism
downwards ˛ = 0.01.
If, as suggested by all results for 1% < ˛ < 15%, this diffusion
mechanism assumption is accepted, it is possible to calculate the
pre-exponential factor A using the relations given in Table 1. In
Fig. 14, the ln A values from all methods are reported as a function
of ˛.
The mean value of ln A from this series of experiments is
ln A = 13.5 1. If we exclude the values computed at low ˛ for some
methods, in particular for the Li and Tang method, the result in the
parabolic range is ln A = 14.3 0.8, exactly the same value as that
determined from isothermal experiments.
5. Discussion
Fig. 11. Li and Tang analysis for the oxidation of Constantan; ˛ = 0.01 (ꢀ); 0.02 (ꢀ);
0.03 (ꢁ); 0.05 (ꢁ); 0.08 (ꢂ) and 0.1 (+).
For the first time, a complete set of the kinetic parameters for the
oxidation of Constantan have been determined following a proce-
Fig. 12. Summary of the activation energies EA from Friedman (ꢀ), Kissinger (+),
Ozawa (ꢁ)and Li and Tang (ꢂ) methods.
Fig. 14. The pre-exponential factor A obtained by Friedman (ꢀ), Kissinger (+) Ozawa
(ꢁ) and Li (ꢂ) methods.

50
T.K. Jondo et al. / Thermochimica Acta 475 (2008) 44–52
Table 4
Summary of methods, models and Arrhenius kinetic parameters of the oxidation of Constantan
˛-Range
EA (kJ mol−1
)
Model f(˛)
(2˛)−1
Ln A
Isothermal process
ꢀm/S = [k(T)t]n
˛ < 0.18
251
5
Isoconversion
0.02 < ˛ < 0.18
0.18 < ˛ < 0.35
0.35 < ˛
244
244
220
7
4
7
(2˛)−1
1
1
14.7 0.9
15.3 0.5
12.8 0.7
Curve fitting
0.02 < ˛ < 0.18
0.18 < ˛ < 0.35
˛ > 0.45
242
241
230 19
6
4
(2˛)−1
1
1
14.4 0.6
15.2 0.5
14
2
Non isothermal process
Isoconversion
Friedmann
˛ < 0.20
˛ < 0.20
˛ < 0.20
˛ < 0.20
241
244
260
7
8
7
(2˛)−1
(2˛)−1
(2˛)−1
(2˛)−1
14.5 0.9
14.6 0.9
14.4 0.7
Kissinger
Ozawa
Li and Tang
Curve fitting (Malek) g(˛)
241 12
12
2
g(˛) = C˛n with n = 1.9 0.1
dure described in the ICTAC kinetic project-data [10]. The samples
for both isothermal and non-isothermal oxidation were taken from
the same rolled batch. They were cut in similar dimensions and
their weight did not differ by more than 3% from the mean value.
For the computational step, the known kinetic models – see for
instance Table 1 in [24] – have been tested in isoconversional or
fitting methods. The coherence between the isothermal and non-
isothermal Arrhenius parameters computed from 10 isothermal
runs (between 650 ◦C and 900 ◦C) or 10 different heating rates (from
1 ◦C/min to 20 ◦C/min) give credibility to the results summarized in
Table 4.
growth can be regarded as a limiting case amongst others, such
as inverse logarithmic and cubic laws. In addition, for oxidation
reactions, it is current to observe a linear kinetic law due to oxy-
gen transfer through cracks in the oxide layer and voids in the
oxide sublayer [35]. The change that we have observed between a
parabolic and a linear law after the growth of the external oxide
scale can be understood on this basis. More striking is the sec-
ond change between the two linear regimes at about ˛ = 35–45%.
It could be due crack closure after re-crystallization of the oxide
grains.
The thermogravimetric study of the oxidation of bulk cop-
per at high temperature under an oxygen pressure from 5 × 10−3
to 0.8 atm by Mrowec and Stoklosa [36] lead to an activation
energy for oxidation of 200 kJ mol−1. Kofstad [35] reports values
ranging from 125 kJ mol−1 to 155 kJ mol−1. For electro-deposited
copper, Bertrand [9] determined EA = 126 kJ mol−1. For pure nickel
the activation energies for the oxidation reported in the literature
are lower, about 100 kJ mol−1 [9]. Recently, Haugsrud [37] distin-
guished a change in the oxidation kinetics of nickel depending on
the temperature range, with EA = 150 kJ mol−1 for T < 800 ◦C and
EA = 200 kJ mol−1 for T > 1000 ◦C. For Cu–10 wt.% Ni and Cu–15 wt.%
Ni alloys, the oxidation at T > 900 ◦C is not perfectly parabolic [6] but
the apparent activation energy is in the range 180–190 kJ mol−1. For
rolled commercial Constantan with a thickness of 50 ␮m, Bertrand
[9] distinguished three successive steps in the oxidation: the for-
mation of NiO, the formation of a copper oxide external layer and
the internal oxidation by oxygen diffusion from the surface to the
alloy with activation energies of 217 kJ mol−1, 214 kJ mol−1 and
160 kJ mol−1, respectively with standard deviations in the order of
16%. Qualitatively, our isothermal results are in agreement with
these values. Moreover, the high accuracy of our results on the
apparent activation energies ascertains the existence of three oxi-
dation regimes and delimits the corresponding reaction extents. It
is noteworthy that the apparent activation energies for the oxida-
tion of Constantan are significantly higher than those reported for
Cu–Ni alloys. The presence of Mn as impurity in Constantan has to
be pointed out. To our knowledge no intermediate phase is reported
in the Cu–Ni–Mn–O system but Cu(1+) and Mn(2+) have compara-
ble ionic radii, 0.96 Å and 0.91 Å, respectively. The presence of Mn on
some Cu sites in the Cu2O phase formed in the oxide sublayer can-
not be excluded, thus modifying the metallic vacancy distribution
responsible of copper diffusion. This mechanism is thought to be
the limiting mechanism in the first two steps for the oxidation of the
alloy. Parabolic and linear behaviours only differ by the shortened
All the results converge to the following conclusions:
(i) The oxidation of the Constantan alloy is governed by a particle
transport mechanism. However, a classical parabolic behaviour
is not observed for the entire range of the reaction extent. It is
restricted to ˛ < 15–18%. For higher ˛ values, two successive
linear laws are observed.
(ii) The Arrhenius parameters for each domain have been deter-
mined with a reasonable precision. A mean apparent activation
energy EA = 246 7 kJ mol−1 is found for the parabolic domain.
A very close value is found using isothermal experiments for
the first linear oxidation. For the second one, EA is slightly
decreased down to 228 kJ mol−1. The distinction is not so clear
for the frequency factor. Ln A ranges between 12 and 15.2 but
in the parabolic field, both isothermal and non-isothermal
experiments converge to ln A = 14.3 except for the Li and Tang
method.
(iii) For non-isothermal experiments, we did not find any signif-
icant difference between the tested methods. All model-free
differential or integral methods give similar Arrhenius param-
eters with similar accuracies. In addition, the different
expansions of the temperature integral do not significantly
affect the results.
These computational results have to be associated to mech-
anisms for the oxidation of copper–nickel alloys. This will be
developed in a forthcoming paper but, for further discussion, it
is necessary to keep in mind some results of the classical mecha-
nism for the oxidation. Since the work of Wagner [33], it is accepted
that the growth of an external oxide scale at the surface of metals
and alloys follows a parabolic equation rate due to lattice diffusion.
Hurlen [34] applied the absolute reaction rate theory to oxidation
and derived a “general oxidation equation” in which the parabolic

T.K. Jondo et al. / Thermochimica Acta 475 (2008) 44–52
51
diffusion path when cracks are generated. In the last linear step
6. Conclusion
of the oxidation, the change of the oxygen pressure inside internal
pores may interfere with the copper diffusion, thus decreasing EA
by about 10%.
The kinetic parameters for the oxidation of Constantan tapes
have been determined from both isothermal and non-isothermal
experiments using thermogravimetry. Direct mass measurements
(ꢀm/S)2, isoconversional and curve fitting methods have been
applied to isothermal data for temperatures ranging from 650 ◦C
to 900 ◦C. For non-isothermal studies, we used 10 different heating
rates ˇ for isoconversion analysis with one differential (Friedman)
and three integral (Kissinger, Ozawa and Li and Tang) methods.
All the measurements and calculations converge to the following
results:
Concerning the frequency factor A, its evaluation depends both
on the activation energy and the reacting mechanism defined by
f(˛) or g(˛). The scattering of the ln A values is obviously related
to the logarithmic and inverse temperature scales inherent to the
Arrhenius fits. In addition, the dimension of A (s−1) indicates that
the pre-exponential factor has something to do with the reactiv-
ity [38,39], i.e. crystal-defects distribution and/or change of the
reaction interface. Therefore, the possible change, during the oxi-
dation of the sample morphology may affect A. However, the
perfect agreement between the isothermal and non-isothermal val-
ues for ln A in the parabolic domain validates the results for the
two subsequent linear domains. From the rate constants measured
by Bertrand [9] for a similar alloy oxidised under PO2 = 1 atm in
the three regions she identified, we can calculate ln A = 17, 17.3
and 12, respectively. The first two values are higher than ours,
as expected, because of the lower values of the apparent acti-
vation energies she determined. The discrepancy could also be
due to different sample thickness (50 ␮m for Ref. [9] compared
to 160 ␮m in this study). For the third step of the oxidation,
i.e. for ˛ > 45%, ln A ≈ 13 seems to be a reasonable value (see
Table 4).
•
The oxidation of Constantan tapes in 1 atm oxygen implies three
different steps characterized by different evolutions of the reac-
tion extent ˛ with time. The first stage, up to ˛ = 15–18% is
the well-accepted parabolic law which describes a 1D diffu-
sion mechanism. The two subsequent stages (˛ = 18–35% and
˛ > 40–45%) are both characterized by a linear increase of ˛ with
time with however different kinetic parameters.
•
The activation energies have been determined with a relative
standard deviation below 3%. In the parabolic and the first lin-
ear domains of ˛(t), the couple EA − ln A is 246 kJ mol−1 − 14.7.
Both values are slightly lower in the second linear regime.
•
Both derivative and integral isoconversional methods applied to
For isoconversional analysis of non-isothermal experiments, our
aim was also to compare the errors introduced in the evalua-
tion of the kinetic parameters by the methods usually employed,
non-isothermal data provide the same results. When integral
methods are used, the results for the activation energies are only
slightly affected by the degree of expansion of the temperature
integral.
differential or integral, and, in the last case, by the two approxima-
ꢂ ∞
((exp(−x))/x2) dx where
•
tions of the temperature integral p(x) =
The characteristic regimes and the convergence of all mathe-
x
x = E/RT. Because of low oxidation kinetics, the comparison was
only possible in the parabolic domain. Kissinger [26] approximated
p(x) as p(x) = exp(−x)/x2 and deduced an expression of ln (ˇ/T2) as
a function of E/RT. Ozawa [27] used p(x) = 0.0048 exp (−1.0516x)
from which ln (ˇ) is derived. Considering activation energies of
about 200 kJ mol−1 and temperatures of the order 1000 K, i.e. x ≈ 20,
the two expansions differ by more than 30%. Some influence on
the values of the kinetic parameters is then expected. In reality,
Figs. 12 and 14 show that EA and A calculated from the two meth-
ods are comparable. In addition, it is worth noting that the standard
deviations over the results obtained from isoconversion are more
dependent on the statistics (number of ˇ values) than on the analyt-
ical model. In a critical review on the temperature integrals, Flynn
[40] observed that the approximation used by Ozawa results in
“large percentage deviations” from the correct value of E/RT. It is
also the view of Starink [41] even if a correction algorithm has been
developed by Opfermann and Kaisersberger [42]. In a more recent
work, Gao et al. [32] discussed the errors introduced by the use of
Kissinger and Ozawa approximations comparing with an “exact”
temperature integral which in fact introduces a correction function
to p(x). Because we found similar results for all methods, we esti-
mated that the use of such developments was not justified in this
work. More questionable is the deviation observed at low ˛ values
for EA and A. In fact, Figs. 12 and 14 show that for ˛ < 0.02 − 0.03,
EA and ln A for all integral methods are significantly lower than
the mean values. It is particularly the case for the values obtained
from the Li and Tang method. We guess that the discrepancy may
be due to a phase diagram effect, a miscibility gap observed at
T < 354 ◦C in the Cu–Ni system [43], or by stress relaxation of the
laminated tapes influencing the initial stage of the oxidation. The
Li and Tang method which uses a direct numerical integration of
the experimental ln (d˛/dt) − ˛ and (1/T) − ˛ curves may be more
sensitive than analytical methods. The background noise observed
in the curve ln (d˛/dt) versus ˛ for ˛ < 0.03 in Fig. 10 supports this
assumption.
matical treatments to the same kinetic parameters give credit to
the existence of a specific limiting reaction mechanism for each
domain. This first analytical stage is a prerequisite to the deter-
mination of the oxidation mechanism, which, on the basis of the
physical and chemical characterizations of the samples, will be
discussed in a forthcoming publication.
•
From a methodological point of view, it is worth mentioning that
both isothermal and non-isothermal techniques give very similar
kinetic parameters when the recommendations and procedures
suggested in the ICTAC project reports are respected.
Acknowledgement
The authors acknowledge financial support from the National
Agency for Research (ANR) through the MADISUP program.
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