1. Introduction
Ferromagnetic shape memory alloys (FSMA) show, along with the properties of conventional shape memory alloys, such as shape memory effect and superelasticity, other unique capabilities arising from the coupling between the magnetic and structural phase transitions. The most studied FSMAs are Heusler-type, particularly near-stoichiometric Ni
2MnGa alloys, that undergo thermoelastic martensitic transformation (MT) from the cubic L2
1 parent phase (austenite) to a less symmetric phase (martensite). Reorientation of the martensite variants under magnetic field produces magnetic field-induced strain (MFIS) [
1,
2,
3], which is the most characteristic property of FSMAs. Other multifunctional properties, such as magnetic shape memory, magnetocaloric effect, and magnetoresistance [
4,
5], appear even more promising for application than MFIS.
The main factors that affect the magnetostructural properties of Ni−Mn−Ga type alloys are composition and atomic order. Both can change the MT (TMT) and Curie (TC) temperatures. If these temperatures can be modified in such a way that the TC be placed, as required, above or below the MT, it is possible to achieve that the high and low temperature phases have the desired magnetic order, thus favouring the magnetostructural coupling and related properties.
Regarding composition, in the case of Ni−Mn−Ga alloys, the greatest achievements have been obtained by Co or Cu-doping [
6,
7,
8,
9,
10,
11]. In that line, Co often replaces Ni or Mn, and it has been found that both T
MT and the austenite T
C (
) increase, while the martensite Curie temperature decreases, with increasing electron to atom ratio. As a result, Ni
2-xCo
xMn
1.2Ga
0.8 alloys with
x = 0.2–0.36 show ferromagnetic austenite to low magnetization martensite MT [
6,
7]. In the case of Cu-doping, Cu usually replaces Mn, and it is found that Cu-content increases the T
MT while both the austenite and martensite T
C decrease [
8,
9,
10,
11]. Hence, the structural and magnetic transitions can be tuned in such a way that paramagnetic austenite to ferromagnetic martensite transition occurs for Ni
2Mn
1−xCu
xGa with
x = 0.22–0.3 [
10].
On the other hand, the change of long-range atomic order also affects the MT and the magnetic properties, due to the modification of the electronic structure and the lattice site occupancy by the magnetic atoms. Close to stoichiometric Ni–Mn–Ga alloys solidify from the melt to a cubic B2 structure, from which the austenitic L2
1 structure is formed through a B2→L2
1 ordering reaction that takes place on further cooling. Due to the diffusionless character of the MT, the martensite inherits the austenite atomic order [
12,
13,
14]. The degree of long-range atomic order can be easily modified applying thermal treatments [
14,
15,
16,
17,
18,
19]. As a general rule, quenching from temperatures around the B2↔L2
1 transition (which is approximately 1100 K for ternary Ni–Mn–Ga [
12]) results in partial retention of the low L2
1 atomic order present at high temperatures. The equilibrium order degree in as-quenched alloys can be progressively restored by post-quench ageing at temperatures at which atomic diffusion is possible, as proven by neutron diffraction experiments [
14,
20,
21].
The effect of the atomic order on the MT and magnetic characteristics has been studied both in ternary Ni–Mn–Ga alloys [
14,
18,
19] as well as for Co and Cu-doped Ni−Mn−Ga [
6,
7,
22]. As a general result, it is found that Curie temperatures increase with L2
1 order, as a consequence of the variation of the magnetic moment of the alloys depending on the position of the Mn atoms, which couple antiferromagnetically when they are nearest neighbours (Mn atoms on the Ga positions) and ferromagnetically when they are next-nearest neighbours (Mn atoms in the Mn sublattice) [
23,
24]. Moreover, the MT temperatures are modified by changes in atomic order, although there is no unique trend. MT temperatures respond to atom site occupancy through electronic structure (covalent bond strength and Jahn−Teller effects) but through magnetic structure as well, and they have been observed to increase (as in ternary and Cu-doped Ni−Mn−Ga [
14,
18,
19,
22]) or to decrease (as in Co-doped Ni−Mn−Ga [
6,
7]). It is worth mentioning that in the latter case, the opposite evolution of the Curie and MT temperatures with increasing atomic order can lead to changes in the magnetic order of austenite and martensite, with the concurrent modification of magnetostructural properties [
6,
7]).
As mentioned above, the low L2
1 order retained after quench can be progressively improved up to the equilibrium value by post-quench ageing. A manifestation of this reordering process is the appearance of a broad exothermic calorimetric peak at temperatures well above the MT, which appears only in the first post-quench heating run [
6]. In a recent work, the influence of atomic order on the martensitic and magnetic transformations undergone by Ni
2Mn
1−xCu
xGa ferromagnetic shape memory alloys was investigated [
22]. Both the MT and Curie temperatures were found to increase during post-quench aging, but these temperatures evolve at different rates: the MT temperatures increase from the start of aging while the Curie temperature begins to rise at a later stage in the aging process. Furthermore, during post-quench heating of Ni
2Mn
1−xCu
xGa alloys, two consecutive DSC exothermic peaks were observed, indicative of two thermally activated processes.
This behaviour is exclusive to Ni
2Mn
1−xCu
xGa alloys, since it was proven that Ni–Mn–Ga ternary alloys, as well as Co-doped Ni–Mn–Ga and, surprisingly, Ni
2−xCu
xMnGa alloys, display a single exothermic peak. Additionally, it was observed that the peak that is common to all alloys is suppressed after quench from temperatures well below the B2↔L2
1 transition. Based in the above results, the process undergone by all alloys, and that is responsible for the change in Curie temperature, is attributed to the improvement of L2
1 order, due to the exchange of Mn and Ga atoms located at antisites after quench. For the other process, which only occurs in Ni
2Mn
1−xCu
xGa alloys and underlies the rise of martensitic transformation temperatures, the diffusion of Cu atoms, misplaced in the Ni sublattice after quench, towards their most favourable sites in the Mn sublattice is proposed as the responsible mechanism. A preliminary evaluation of the kinetics of the two post-quench aging processes was performed in [
22] using Kissinger’s method, yielding different activation energies: 1.16 eV for the process that is common to all alloys and 1.35 eV for the one that is solely observed in Ni
2Mn
1−xCu
xGa alloys.
Beyond the evaluation of activation energies and their comparison with the diffusion energies of the components, the analysis and quantification of the kinetics of the reordering processes can shed light on the mechanisms that govern these processes and the factors that affect them. Despite the crucial influence of the changes in the degree of atomic order on the properties of Ni−Mn−Ga-based alloys, there are hardly any detailed studies of the ordering and reordering kinetics in these alloys.
This work presents the results of the in-depth analysis of the kinetics of the reordering processes in Ni2Mn0.8Cu0.2Ga alloy. The quantification of the fundamental kinetic parameters has been carried out based on the calorimetric curves obtained during post-quench heating at constant rates. Isoconversional methods have been used to determine not only the activation energy, but also the pre-exponential factor and the reaction model that best fit the two post-quench reordering processes described above. The kinetic analysis has been extended to samples quenched from different temperatures, highlighting the crucial role of the vacancies retained by high temperature quenching. Thus, the atomic diffusion that gives rise to the atomic reordering underlying the observed processes is enhanced by vacancies.
2. Materials and Methods
The study has focused on a Ni2Mn0.8Cu0.2Ga alloy, (hereinafter called Cu5), although for comparison purposes, the ternary Ni2.1MnGa0.9(Cu0) has also been analysed. All used alloys are polycrystalline, prepared by induction melting in argon atmosphere using high purity elemental metals. The ingots have been melted several times and homogenized for 24 h at 1170 K in a vacuum quartz tube, followed by quench in water at room temperature. Long range atomic order has been modified by means of additional thermal treatments, consisting of annealing for 1 h at temperatures TWQ = 570–1170 K, followed by quench into water at room temperature, and subsequent post-quench ageing. The latter has been carried out by continuous heating up to 670 K.
The martensitic and magnetic transitions, as well as the reordering processes, have been monitored by differential scanning calorimetry measurements (DSC 2920, TA Instruments Inc., New Castle, DE, USA). Runs have been performed at temperature rates ranging between 1 and 20 K/min. The broad exothermic DSC peaks observed at temperatures well above the MT during the first post-quench heating run are considered to be proportional to the rate of progress of the thermally activated process that takes place at the corresponding temperature interval. Thus, the conversion fraction,
x, is calculated as:
where Δ
Q(
T) is the heat flow integrated up to
T, after the proper baseline correction, and Δ
Qtot is the overall heat released in the whole process. To minimize the errors that could arise from the choice of the baseline, the DSC curve corresponding to a second heating has been used as guide for baseline selection. Although calibration with Pb (melting at 600.61 K) and In (melting at 429.75 K) at different heating rates proves that thermal inertia is rather small (error below 1 K), these standards have been used to correct the measured temperatures.
The basis of the kinetic analysis is the rate equation:
which gives the conversion rate as a function of the separable variables
x and
T. The kinetic factor
K(
T) follows the Arrhenius equation:
where
ko and
Eac are the pre-exponential factor and the activation energy, respectively, and R is the universal gas constant.
is the reaction model, which can take a number of forms depending on the nature and steps of the studied process.
The combined equation:
is valid for any temperature program [
25], including constant heating rate for which
For each temperature rate
βi, the fraction
x is reached at a temperature
, at which conversion rate has a value
. Two different approaches have been used to quantify the kinetic parameters [
25]. First, the integral isoconversional method called Kissinger−Akahira−Sunose (KAS), is expressed by:
The plots of for different x should be straight lines whose slope would give the activation energy associated with each x, Eac(x). It is worthwhile to mention that this method reduces to the classic Kissinger’s when the temperature of the maximum of (dx/dt) is used as Tx.
Second, the most common differential isoconvertional method, developed by Friedman, is based on the equation:
which again provides
Eac(
x) from the plots of
for different
x. It is worth keeping in mind that the above expressions allow to obtain activation energies without the need to set a reaction model.
Even though (6) and (7) provide activation energies dependent on the fraction
x, the principle of separation of variables implicit in (2) requires that
Eac be constant. To advance in the determination of the kinetic parameters, an activation energy valid for the entire process must be set. Should that be done,
ko and
f(
x) can be determined choosing a generic type of reaction model. In our case, owing to the sigmoidal shape of the
x(
T) curves and according to the literature [
25,
26,
27], a
n-order reaction model,
is assumed. Therefore, Equation (7) becomes:
and the straight lines obtained by plotting
for each temperature rate yield the corresponding values of ln(
ko) and the exponent
n.
3. Results and Discussion
Figure 1 shows the DSC curve obtained on cooling down to 210 K and subsequent heating up to 670 K, at the temperature rate of 5 K/min, of a Cu
5 sample quenched from 1020 K. The exothermic and endothermic peaks observed at low temperatures correspond to the forward (cooling) and reverse (heating) MT, respectively, whereas the anomaly around 300 K is associated with the ferromagnetic transition taking place in the austenitic phase. The temperatures taken as characteristic for the structural (
MP and
AP) and magnetic transitions (
) are labelled in
Figure 1. As stated in
Section 1, a low degree of L2
1 atomic order is retained after water quench from temperatures around the B2↔L2
1 transition (1020 K for alloy Cu
5 [
22]), and the equilibrium order degree can be progressively restored by post-quench ageing at temperatures at which atomic diffusion is possible. In
Figure 1, complete reordering occurs during continuous heating up to 670 K, and in the course of that process two broad exothermic peaks are observed, labelled as P
1 and P
2. The MT and Curie temperatures change due to order improvement, but with different kinetics. To highlight and summarize those changes,
Figure 2 shows the DSC curves obtained during cooling and subsequent heating up to 500 K just after quench (blue curve); subsequent cooling after overcoming P
2 and heating up to 670 K (red curve), and cooling followed by heating after passing P
1 (black curve). In the latter curve, heating continues up to 670 K (dotted black line) to show that the exothermic peaks P
1 and P
2 are irreversible, since they do not appear in subsequent heating runs. On heating above P
2, a significant change in the MT temperatures (
MP and
AP) is observed (compare blue and red curves), but
does not change, as shown in the inset in
Figure 2. By additional heating up to 670 K (above P
1), the Curie temperature increases as well (compare red and black curves). More details about the evolution of the MT and Curie temperatures during progressive ageing can be found in [
22], but from
Figure 2 it is easy to relate each one of the broad exothermic peaks with the evolution of the MT and Curie temperatures: P
2 accounts for a process which produces an increase in the MT temperatures, while P
1 accounts for a second process that results in an increase of Curie temperature. The time evolution of the transition temperatures during isothermal ageing [
22] proves that both processes are thermally activated.
All tested Ni
2Mn
1−xCu
xGa alloys (
x = 0.12–0.36) present these two post-quench exothermic peaks, with the interesting feature that the magnitude of the lowest temperature peak (P
2) decreases with decreasing copper content. Instead, the ternary Ni−Mn−Ga, Co-doped and Cu-doped Ni−Mn−Ga alloys where Cu replaces Ni display a single exothermic peak (P
1). The role of Cu, specifically Cu replacing Mn, is therefore decisive for the phenomenon associated with the peak as-called P
2 [
22].
Figure 3 shows the DSC curves obtained during heating up to 670 K of Cu
5 and ternary Cu
0 samples previously quenched from 1020 K.
Another relevant characteristic of the studied processes is their evolution when the quenching temperature (
TWQ) changes. In this sense, the most relevant observation is the suppression of P
1 for
TWQ below 770 K, the lower temperature exothermic peak P
2 being the only one left after quench from temperatures at which L2
1 order is complete. However, it is also interesting to note that the peak temperatures for both P
1 and P
2 shift towards higher temperatures as
TWQ is lowered. These features are shown in
Figure 4, where the DSC curves obtained during heating Cu
5 samples previously quenched from the indicated
TWQ temperatures are displayed.
Based in the above results, the process associated to P
1, that all alloys undergo and that is responsible for the change in Curie temperature, is attributed to the improvement of L2
1 order mainly due to Mn diffusion. For the other process causing P
2, which is only observed in Ni
2Mn
1−xCu
xGa alloys and underlies the rise of martensitic transformation temperatures, the diffusion of Cu atoms, misplaced in the Ni sublattice after quench, towards their most favourable sites in the Mn sublattice is proposed as the responsible mechanism [
22].
Figure 5a shows the DSC curves for Cu
5 samples as-quenched from 970 K, corresponding to P
2 after baseline correction, and recorded during heating at different rates between 1 and 20 K/min. According to the conversion fraction given by Equation (1), these curves provide (
dx/
dt), and their normalized integral up to temperature
T gives
x(
T).
Figure 5b shows the
x(
T) curves obtained for the different heating rates. Application of the Kissinger−Akihara−Sunose method expressed by Equation (6) is shown in
Figure 5c, from which an activation energy
is calculated for
x = 0.1−0.9. Similarly, Friedman’s method based on Equation (7) is applied, giving rise to the straight lines shown
Figure 5d, which yield activation energies
for each
x.
Figure 5e shows the obtained apparent activation energies as a function of the conversion fraction. Both
and
vary with
x. A systematic dependence of
Eac on
x indicates the occurrence of several stages with different activation energies, that is, a multi-step process. This being the case, we must not forget that the beginning and end of the conversion are much more affected by the errors that could be made in the baseline correction, so the activation energy values provided at low and high
x will undoubtedly be less precise. In
Figure 5e, a star symbol indicates the
Eac that was obtained in [
22] from the Kissinger’s method, fitting, as expected, with KAS method around
x = 0.5. The average activation energies are 1.3 eV (130 kJ/mol) for KAS method and 1.2 eV (118 kJ/mol) for Friedman’s. Since Equation (8) derives from Friedman’s method, the last one will be used to calculate the other kinetic parameters. The straight lines that according to Equation (8) result from the plots of
for each temperature rate are shown in
Figure 5f, and the obtained values of
n and ln(
ko) are given in the inset.
The same steps have been taken with DSC curves corresponding to P
1 after baseline correction, recorded during heating at different rates between 1 and 20 K/min, giving rise to
Figure 6a,f, the last obtained using an average activation energy of 1.1 eV (107 kJ/mol).
For comparison purposes, the DSC curves obtained during heating at chosen temperature rates for samples of the ternary alloy Cu
0, corresponding to the single exothermic peak (P
1) after quench from 970 K, have been analysed along the same lines, and the results are shown in
Figure 7a,f (average activation energy 1.2 eV = 117 kJ/mol).
Table 1 summarizes the average values of the kinetic parameters obtained for the thermally activated processes underlying P
1 (for Cu
5 and Cu
0 samples) and P
2. When interpreting these parameters, several aspects have to be considered. Atomic diffusion, as needed to restore quenched-in disorder in the studied alloys, is mediated by vacancies [
28]. We can assume that the rate of atom diffusion in a particular process at a given temperature is proportional to the atomic vibration frequency (
ν), the number of present vacancies
cv(
T), and the rate of successful jumps of the energy barrier (
Eac) for that process, exp(−
Eac/
RT). For quenched samples, the retained excess vacancy concentration is much higher than the equilibrium concentration at T. Thus,
cv(
T) can be approached by the vacancy concentration at the quenching temperature
TWQ,
cv(
TWQ). Finally, a factor (α) accounting for the correlation factor for diffusion, the lattice parameter, and the number of jumps has to be included. Thus, we can assume that
3.4. Effect of the Quenching Temperature on the Kinetic Parameters
The kinetic analysis has been extended to the DSC curves obtained during heating samples of alloy Cu
5 quenched from different temperatures
TWQ. These curves are shown in
Figure 4, where it can be noted that the peak temperatures for both P
1 and P
2 shift towards higher temperatures as
TWQ is lowered, indicating that the processes require less energy after quenching from higher temperatures.
In this case, to obtain the kinetic parameters from the available curves, a first order reaction model has been assumed, so that Equation (8) results
and the plots of
for each quenching temperature should be straight lines that yield the corresponding values of ln(
ko) and
Eac. Clearly, the kinetic parameters obtained from a single curve will be less precise than those gathered
Figure 5,
Figure 6 and
Figure 7, but they could still be compared with each other in order to elucidate the effect of the quenching temperature.
Figure 8 presents the
lines corresponding to P
2 (full lines) and P
1 (dotted lines) for different
TWQ.
The corresponding activation energies and pre-exponential factors are displayed in
Figure 9a,b as a function of
TWQ. The changes in
Eac are rather small, as anticipated by the minor slope changes in
Figure 8, yet they are important since they show a tipping point around 1000 K, where the B2−L2
1 order-disorder transition takes place [
22]. Indeed, it is well known that the activation energy depends on the order degree [
31,
34], since the atomic environment of the jumping atoms varies with it. Obviously, the L2
1 order frozen by quench is absolutely
TWQ dependent, as is the activation energy. However, the evolution of
Eac for P
1 and P
2 differs, which can be imputed to the different jumping walks required in each case. Furthermore, P
1 is directly related to the L2
1 order and P
2 is not. Of course, the effect of quenched excess vacancies on the activation energy cannot be underestimated: for higher
TWQ, more vacancies are frozen, lowering, as a general rule, the energy barriers.
The effect of quenched vacancies becomes much more evident when examining the evolution of pre-exponential factors with
TWQ. The equilibrium vacancy concentration at
TWQ calculated from [
28] is also shown in
Figure 9b, allowing to observe a parallel evolution. Corroborating this parallelism, the plots of ln(
ko) vs. 1/
TWQ (
Figure 10) yield activation energies of 0.8 eV (P
1) and 1.0 eV (P
2), very close to the formation energy of vacancies in Ni
2MnGa given in [
28].