Thermodynamics and Kinetics of Martensitic Transformation in Ni-Mn-based Magnetic Shape Memory Alloys

We herein present a review of recent thermodynamic and kinetic studies on Ni-Mn-based magnetic shape memory alloys along with some new data supporting the kinetic discussion. Magnetic phase diagrams and Clausius-Clapeyron relationships are mainly discussed for Ni-Mn-Ga and Ni-Mn-In systems. For the kinetics, a phenomenological model based on Seeger’s model is used to describe the temperature dependence of magnetic field hysteresis, as well as the change of hysteresis under different sweeping rates of magnetic fields.


Introduction
Research on Ni-Mn-based magnetic shape memory alloys dates back to 1984 when Webster et al. investigated the ferromagnetic Ni 2 MnGa Heusler alloys showing martensitic transformation [1]. In 1996, Ullakko et al. reported a 0.2% magnetic field-induced strain (MFIS) in a Ni 2 MnGa Heusler alloy [2]. Since then alloys showing variant rearrangement under a magnetic field in their martensite phase have been categorized as ferromagnetic shape memory alloys (FMSMA) and this field of research has received much attention. Other FMSMAs, such as Fe-Pd [3], Fe-Pt [4], Ni-Mn-Al [5], Ni-Co-Ga [6], Ni-Co-Al [7] and Ni-Fe-Ga [8], have successively been reported. In Ni-Mn-Ga systems, a huge MFIS of about 9.4 % has been reported in 14M [9] and about 12 % in non-modulated [10] martensites, despite the fact that the output stress is limited to several MPa [11,12].
On the contrary, as first reported by Sutou et al., Ni-Mn-X (X = In, Sn, and Sb) alloys generally show a large difference in magnetization between the parent and martensite phases (ΔM) [13]. In the Ni-Co-Mn-In alloy, magnetic field-induced transformation (MFIT) has been realized by reverse martensitic transformation [14]. This group of alloys are called metamagnetic shape memory alloys (MMSMA). Substitutional Ni-Mn-based alloy systems such as Ni-Co-Mn-Sn [15], Ni-Co-Mn-Ga [16] and Ni-Co-Mn-Al [17], ferrous systems such as Fe-Mn-Al [18], Fe-Mn-Ga [19] and Fe-Mn-Al-Ni [20], as well as cobalt based Co-Cr-Ga-Si alloys [21] have been reported. In these alloys, there is a strong output stress during the a Corresponding author's e-mail: kainuma@material.tohoku.ac.jp MFIT [22][23][24] though a strong magnetic field is needed for the realization of MMSMA.
In this article, we review some representative experimental studies on Ni-Mn-based alloys. In the first section, magnetic phase diagrams as well as Clausius-Clapeyron relationships are discussed with a focus on thermodynamic analysis. In the second part, some kinetic studies on Ni-Mn-based alloys are reviewed along with some new data supporting the discussion.  [25] and Ni 50 Mn 50−x Ga x [26]. Martensitic transformation starting temperature T Ms , Curie temperatures of parent T C,P and martensite T C,M phases are indicated. x C,i indicates the intersection composition for T Ms and T C,i (i = P or M).
where x C,i (i = P or M) indicates the intersection composition for T M s and T C,i .
For the Ni 50 Mn 50−x Ga x system, region IV is wide and covers the stoichiometric Ni 2 MnGa composition [1,29]. Moreover, the Curie temperature of the martensite phase (T C,M ) is slightly higher or almost equal to the Curie temperature of the parent phase (T C,P ), and the ΔM in the vicinity of martensitic transformation temperature (T M s ) is very small. Similar phase diagrams can be found for Ni-Mn-Ga systems such as in Ni 2+x Mn 1−x Ga [30][31][32] and other sections [33,34]. Moreover, some substitutional quaternary systems such as (Ni 52.5 Mn 23.5 Ga 24 ) 100−x Co x [35], Ni 2 MnGa 1−x Co x [36], Ni-Mn-Ga-Cu [37][38][39], and Ni-Mn-Ga-Fe [40,41] systems also show similar behaviors, and a magnetically coupled structural transition, i.e., a direct transition from region I to region IV, can be found in a wide range of compositions.
On the contrary, for the Ni 50 Mn 50−x In x system, region IV is narrow and disappears far from the stoichiometric Ni 2 MnIn composition. Another major difference is that the T C,P is generally higher than T C,M and transition from region II to region III is possible where a ferromagneticparent-to-paramagnetic-martensite transformation occurs. However, it is of interest that, as reported by Yu et al., this Ni-Mn-In type transition can also be realized with a proper substitution of Co into Ni-Mn-Ga [16]. A clear phase diagram showing this evolution has been reported by Wang et al. [42]. Besides, except for Ga, the phase diagrams for other ternary alloy systems generally show a T C,P much higher than T C,M , as shown by Sutou et al. [13]. This can also be found in other ternary Ni 89−x Mn x In 11 [43] and quaternary Ni 50−x Co x Mn 50−y In y [25], Ni 2 Mn 1.48−x Fe x Sn 0.52 [44], Ni 50−x Co x Mn 50−y Al y [45][46][47] and Ni 50−x Co x Mn 50−y Sn y [48] systems, where a region II to region III transition can be easily found.
A further comparison between the magnetic phase diagrams of Ni 50 Mn 50−x Ga x and Ni 50 Mn 50−x In x in Fig. 1 reveals that the two systems have a similar behavior for the

Entropy Change during Martensitic Transformation
Along with the determination of phase diagrams, which gives information on the change of thermodynamic equilibrium states with variation of composition, investigation of entropy change during first-order martensitic transformation (ΔS ) is also of great importance, as it reveals the Gibbs energy near the equilibrium state when the temperature is subjected to change.
The simplest direct way of obtaining ΔS is by use of from thermoanalysis, where ΔL is the latent heat or the enthalpy change (ΔH) during the martensitic transformation. A graphical meaning of ΔS is shown in Fig. 3(a). Other ways to determine ΔS by use of Clausius-Clapeyron equations are discussed in Sec. 2.3. Figure 2 shows ΔS for Ni 50 Mn 50−x Ga x [49] and Ni 50 Mn 50−x In x [25] systems. Systematic studies on ΔS in the Ni-Mn-Ga systems have been intensively performed [50][51][52] due to its wide interest. For the Ni 50 Mn 50−x Ga x section, with increasing Ga content, the ΔS increases near x C,M and decreases over x C,P , as shown in Fig. 2. The composition where ΔS starts to increase, which is indicated by a small triangle in Fig. 2, does not coincide with x C,M , which is considered to be the result of short-range ordering of the ferromagnetic martensite phase above its T C,M . Refer to [49] for a detailed discussion. For the Ni 50 Mn 50−x In x system, the most important characteristic is that the ΔS to the left of x C,P shows little change, whereas it abruptly decreases to the right of x C,P . Reports of a similar tendency of ΔS can be found in other ternary Ni-Mn-In alloys [53,54]. The decrease of ΔS below T C,P by direct measurements can also be found in Sb-doped [55] and Co-doped [56] quaternary systems, as well as in the Ni 50−x Co x Mn 50−y Al y system [57]. This common tendency is considered to be the thermodynamic cause of the "thermal transformation arrest phenomenon" [58,59], where the martensitic transformation is interrupted at a certain temperature during the cooling process.

Clausius-Clapeyron Equations
On the other hand, the Clausius-Clapeyron equations are widely used as they are convenient approaches for indirect measurements of ΔS due to the first-order nature of the martensitic transformation. By writing the total derivative of Gibbs energy G as where V is the molar volume, p is the hydrostatic pressure, H is the magnetic field, l is the length of the sample, F is the uniaxial force, and μ i is the chemical potential of component i. Assuming that S , V, M, l and μ i show little change in the parent and martensite phases, this deduces the Clausius-Clapeyron equations as [60] where ε is the martensitic transformation strain, σ is the uniaxial stress, and the quantities with zero in subscript correspond to their thermodynamic equilibrium states. Equation 3 is the most commonly used relationship for the investigations of conventional shape memory alloys, as reported for Ni-Ti alloys [61]. Since the martensite phase is uniaxial stress-favored, the application of uniaxial stress usually induces the martensite phase, which is illustrated in Fig. 3(b). In Ni-Mn-based alloy systems, research studies have been done for Ni-Mn-Ga [62], Ni-Co-Mn-In [14,[63][64][65][66][67] single crystals and Ni-Co-Mn-In [68] and Ni-Co-Mn-Al [69] poly-crystals. On the other hand, this is the phenomenon attributable to the elastocaloric effect [70][71][72], which is of practical importance. Equation 4 is effective for the investigation of ΔS since the samples can be small and polycrystalline, and it is applicable to brittle alloys. Note that hydrostatic pressure stabilizes the phase which has a smaller molar volume. A review article can be found in Ref. [73] and this phenomenon is also utilized in the barocaloric effect [74]. In Ni-Mn-Ga systems, the molar volume change during martensitic transformation is so small that reports for both a decrease [75,76] and an increase [77][78][79] of T M s with increasing pressure can be found. In other systems such as Ni-Mn-In [80,81], Ni-Co-Mn-Ga [82] and Ni-Mn-Fe-Ga [83], positive relationships of dT M s /dp have been reported by many groups, which is the cause of greater molar volume of the parent phase. Equation 5 can be found in many reports in the field of Ni-Mn-based alloys both because of the easy access of moderately strong magnetic fields as well as interest in the magnetocaloric effect [84][85][86]. For Ni-Mn-Ga ternary alloys, the magnetization of martensite phase is greater than that of the parent phase [87], and therefore a strong magnetic field generally raises the T M s [88][89][90]. However, due to the large magnetic anisotropy in Ni-Mn-Ga alloys [2,91], a low magnetic field results in a small decrease of T M s [88,89]. Nevertheless, for Ni-Mn-In alloys, the magnetization of the parent phase is much greater than that of martensite phase, and thus magnetic fields will effectively decrease the T M s [23,[92][93][94][95][96], as is schematically shown in Fig. 2.3(c). In quaternary systems such as Ni-Co-Mn-In [58,65,66,[97][98][99][100][101], Ni-Co-Mn-Sn [15,101,102], Ni-Co-Mn-Sb, [101,103,104], Ni-Co-Mn-Al [17,46,[105][106][107] and Ni-Co-Mn-Ga [16,[108][109][110], the same relationship of ΔM as well as MFIT have been found. It should be noted that in most of the above cited studies, magnetization was monitored for the detection of MFIT. Other methods, such as monitoring the electric resistance [99], the variation of sample temperature [100] or the variation of strain [65], have been used. In situ observation of optical microstructure [98,107,111] as well as X-ray diffraction patterns  [25][26][27]49] were used as in Eq. 6 for the calculation of Δμ. [106,112] have also been utilized as detection methods in the above cited studies.

Chemical Potential Change during the Martensitic Transformation
Figure 3(d) shows another section of the Gibbs energy curves, which are against the composition axis. Following Niitsu et al. [113], from Eq. 2 we have where a pseudo-binary system of NiMn-NiZ (Z=In, Ga) under equilibrium is considered, with x Z 0 being the equilibrium composition and Δμ NiZ and Δμ NiMn being the chemical potential change for the end-member NiZ and NiMn phases, respectively. Therefore, Δμ = Δμ NiZ − Δμ NiMn can be calculated for Ni 50 Mn 50−x In x and Ni 50 Mn 50−x Ga x . The curves shown in Figs. 1 and 2 were traced for the calculation of Δμ and the results are shown in Fig. 4. Note that in Fig. 1 the composition at which T M s occurs was used, therefore dx Z Ms /dT was used instead of dx Z 0 /dT for simplicity. It can be seen that Ni 50 Mn 50−x In x generally has a greater absolute value of Δμ compared with that of Ni 50 Mn 50−x Ga x . Δμ also changes at magnetic transitions, as indicated by T C,P and T C,M in Fig. 4. Table 1 shows a comparison of Δμ for three alloy systems. The data at around 400 K, which is above the magnetic transitions, are summarized. One can see that Ni-Ti has the largest Δμ among the three series because of the large composition dependence of T M s as well as the large ΔS . Ni-Mn-Ga and Ni-Mn-In have comparable values of dT/dx Z 0 , whereas Ni-Mn-In has a larger value of Δμ because of its large ΔS at the martensitic transformation. However, an in-depth discussion on Δμ, especially for situations near the magnetic transitions, is avoided in this study, though the bending behavior of Δμ on crossing x C,i is consistent with the second-order nature of magnetic transitions, as indicated by the small triangle and x C,P in Fig. 4 and dT/dx Z 0 → ∞ especially for the case of T C,P , an experimental determination of Δμ near T C,P is difficult and a theoretical background is needed for greater understanding.

Kinetics of Ni-Mn-based Magnetic Shape Memory Alloys
The kinetics of the martensitic transformation in Ni-Mnbased alloys have been paid less attention than thermodynamic phenomena, whereas different phenomenological approaches from several groups have been developed. Sharma et al. first investigated the relaxation process during martensitic transformation in a Ni-Mn-In alloy [114]. Afterwards, in Ni-Mn-Ga [115], Ni-Co-Mn-In [116][117][118][119], Ni-Co-Mn-Sn [120] and Ni-Co-Mn-Sb [121] systems, isothermal behavior has been found for both the forward and reverse martensitic transformations.
As interpretations of the kinetic phenomena, Kustov et al. introduced equations from the magnetic aftereffect [122], within the framework of which the magnetic viscosity coefficient shows a local minimum at the temperature where the fastest transformation can be observed [117,123]. On the other hand, based on the nucleation model [124], Fukuda et al. have shown that their experimental observations of time-temperature-transformation (TTT) diagrams can be well explained [125]. Recently, our group also proposed the use of a model based on Seeger's model [126,127]. The original model had been used for a phenomenological understanding of the critical resolved shear stress (CRSS), which has also proved valid for the application to diffusionless martensitic transformations [128,129] as well as the case of stress hysteresis in stress-induced martensitic transformation [130]. The following discussions are based on this model.
For the case of MFIT, as in Ni-Mn-In type alloys, the applied magnetic field, H app , which is half of the magnetic field hysteresis H hys = H A f − H M s , is thought to be divided into two parts: the thermally activated term H TA (T ) and the athermal term H μ [99]. This is written as with H TA (0) being the value of H TA (T ) at 0 K, Q 0K being the activation energy, k B being the Boltzmann constant, and m being the kinetic coefficient. p and q are the shape parameters describing the activation barrier, which have  [58].
been obtained as p = 1/2, q = 3/2 for the case of magnetic field-induced martensitic transformation [99]. m is expressed as  Figure 7. Comparison of the applied magnetic fields for Ni 45 Co 5 Mn 36.7 In 13.3 alloy, which is half of the magnetic field hysteresis, between results from steady [58] and pulsed magnetic fields. The black solid line is a fitting curve based on Eq. 7 where the activation energy Q 0K was fixed to be 0.7 eV [133]. The same parameters as for the black curve, except for m as in Eq. 8, was used to draw the red curve.
which is thought to be the thermodynamic equilibrium magnetic field [132] in Eq. 5, is also plotted. It can be seen that H 0 has almost the same values while H M s and H A f show deviation under different sweeping rates.
In Fig. 7, the H app is plotted against the temperature for both steady [58] and pulsed magnetic fields. For H app under steady fields, a fitting against Eq. 7 was conducted. Here, Q 0K was set to be 0.7 eV [133] because they have very close heat treatment conditions. H μ and H TA (0) were obtained to be 0.9 T and 6.4 T, respectively, and m was found to be 35.9 for the steady field. Here, if Eq. 7 is valid, one can estimate the value ofḢ 0 to be 1.9 × 10 13 T/s from Eq. 8 and expect a calculated temperature dependence of H app for the pulsed field to be consistent with the experimental data by only changing the value of m. Therefore m = 23.2 was calculated using Eq. 8 wherė H = 1500 T/s was used, which is a typical sweeping rate for pulsed magnetic fields. This is plotted as the red solid line in Fig. 7 and well reproduces the results of experimental H app under pulsed fields. Therefore Eq. 7 is considered to be a successful phenomenological model which is valid for the interpretation of kinetic phenomena in the current alloy systems. On the other hand, Fig. 7 also shows us that even at room temperature the H hys may increase under high sweeping rates of magnetic fields, thus attention should be given to MMSMAs when they are applied to devices where a high response rate is required.

Conclusion
In summary, a review of the thermodynamics and kinetics of Ni-Mn-based magnetic shape memory alloys was presented.
For thermodynamics: 1. The magnetic phase diagrams of Ni-Mn-Ga and Ni-Mn-In alloys were discussed, along with similar phase diagrams in other quaternary systems.
ESOMAT 2015 01004-p.5 2. Some representative reports from the literature were reviewed, some of which focused on the direct measurement of entropy change during martensitic transformation, whereas others examined the martensitic transformation under a magnetic field, uniaxial stress and hydrostatic pressure.
3. The chemical potential change during martensitic transformation was deduced experimentally for Ni-Mn-Ga and Ni-Mn-In systems, where much smaller values than that of Ni-Ti alloys were found.
For kinetics: 1. A phenomenological model based on Seeger's model was reviewed, which interprets the temperature dependence of magnetic field hysteresis (H hys ).

2.
A comparison of H hys obtained under steady and pulsed magnetic fields was conducted over a wide temperature range. A large difference in the H hys was found under the condition that the supposed equilibrium magnetic field was consistent.
3. The equation based on Seeger's model was used to fit the H hys under steady magnetic fields. By only substituting the actual magnetic field sweeping rate, the predicted H hys showed good agreement with experimental observations.