Martensitic transformation of Ti50(Ni50−xCux) and Ni50(Ti50−xZrx) shape-memory alloys

Martensitic transformation and phase stability of Ti50(Ni50−xCux) and Ni50(Ti50−xZrx) shape memory alloys are investigated based on density functional theory (DFT). According to the results of formation energy we calculated, upon substitution of Ni by Cu at levels of about 10.4 at.%, Ti50(Ni50−xCux) alloys lose the monoclinic martensite in favor of the orthorhombic martensite structure. The martensite monoclinic B19´ structure of Ni50(Ti50−xZrx) becomes more stable with increasing of the Zr content. The energy difference between austenite and martensite decreases when Cu < 10.4 at.%, and then increases slightly, which suggesting that Cu addition reduces the composition sensitivity of martensitic transformation temperature comparing with binary NiTi alloys. The energy difference decreases slightly firstly when Zr < 10.4 at.% and then increases sharply, which indicates that Zr addition increases martensitic transformation temperature dramatically. Furthermore, a geometric model is used to evaluate the thermal hysteresis. More interestingly, it is found that the lowest thermal hysteresis is achieved at 10.4 at.% for Cu-doped NiTi; whereas the thermal hysteresis increases with increasing of Zr. The electronic structures of austenite phase are also discussed in detail.

understanding of how they affect the phase structure and properties and the strong composition dependence of T m .
The first-principles has been used to study the microscopic origin of NiTi MT further. Recently, the site preference for ternary additions in NiTi has been systematic calculated 21 . The martensite crystal structures of Ti 50 Ni 50−x Cu x SMAs were investigated by using CASTEP code 22 . Wang's ab-initio calculations 6 show the origin of STG transition in Ti 50 (Ni 50−x D x ) (D = Cr, Mn, Fe and Co) alloys, and indicate that Cu and Pd cannot induce the STG transition. Hu et al. 14 investigated the alloying effect of Zr on the elastic properties of Ni-rich Ni-Ti-Zr HTSMAs. Our previous work 15 has investigated the effect of Pd on the martensitic transformation of Ni-Ti-Pd HTSMAs through DFT calculations. Attracted by the small hysteresis width 8 and the low composition sensitivity of T m 9 in Ni-Ti-Cu alloys, the relatively low materials cost and large change of T m in Ni-Ti-Zr HTSMAs [10][11][12][13] . Consequently, we devote this paper to study the phase stability and transition behaviors of Ni-Ti-Cu and Ni-Ti-Zr alloys based on DFT, further rationalize the experimental findings about MT, and explain the strong content dependence of the phase stability of martensite crystal structure, transformation temperature and thermal hysteresis.
Generally, the DFT calculations are implemented at the temperature of 0 K 23 . As we are just concerned with the relative variations of the energy between austenite and martensite phases, the energy differences are almost unaffected by temperature. Transmission electron microscopy shows the presence of precipitates in Ni-Ti-Cu films after aging, precipitations vary with different annealings 24 . The effect of precipitation is beyond of the scope of the current work.

Computational Models and Methods
We focus only on the Ti-deficient Ni-Ti-Zr and Ni-deficient Ni-Ti-Cu alloys 4 , b c 2 0 0 , here a 0 is the lattice constants of cubic B2 phase. We focus on the orthorhombic B19 and monoclinic B19′ structures for the martensite phase observed experimentally in Ni-Ti-Cu and Ni-Ti-Zr shape memory alloys 11,12,16 . Figure 1 shows the supercell model of austenite and martensite phase of NiTi alloy. Geometry optimizations were implemented for all possible doping structures. More details of site preference for Ti 50 (Ni 50−x Cu x ) and Ni 50 (Ti 50−x Zr x ) alloys were shown in Supplementary Fig. S1, Supplementary Tables S1 and S2. The site occupation of B19 and B19′ phase corresponds to that of B2.
We calculate the most stable position of B2 phase with the principle of lowest energy first, and the site occupation of doped atoms of B19 and B19′ phase is set according to the B2. The energy difference between B2, B19 The calculations are performed using DFT as implemented in the Vienna Ab initio Simulation Package 25-27 (VASP) together with plane-wave projector augmented wave 28 (PAW) pseudopotentials and the PW91 29 generalized gradient approximation 30 (GGA) for the exchange and correlation effects. The valence electron configurations for Ni, Ti, Cu and Zr are 3d 8 4s 2 , 3d 2 4s 2 , 3d 10 4s 1 and 4d 2 5s 1 , respectively. We tested k-point sampling 31 and an energy cutoff convergence for all supercells. As a result of the convergence tests, Brillouin zone sampling was performed using 4 × 4 × 4 special k-point mesh, the plane-wave cutoff energy was set to 500 eV. In the calculations, the total energies were converged up to 10 −4 eV/atom, and the atomic positions in our models were fully relaxed until the force of every atom was less than 0.01 eV/Å. We relaxed the structure with ISIF = 7 option (not change cell shape, change cell volume and not relax ions) first, and then, with ISIF = 2 option (not change cell shape, not change cell volume and relax ions) in VASP.
Structural parameters adopted in the present work are summarized in Table 1. The calculated lattice parameters are in line with each other and with the experimental result.

Results and Discussion
Effects of additions (Cu or Zr) on phase stability of ternary NiTi-based alloys. In order to analyze the effect of Cu or Zr additions on phase stability, we evaluate the formation energies (E form ) for B2, B19 and B19′ phases of Ti 50 (Ni 50−x Cu x ) and Ni 50 (Ti 50−x Zr x ) alloys, respectively. E form is defined as the total energy of the alloy minus the concentration weighted average of the pure elements' total energies at their equilibrium volumes [32][33][34] . As previously mentioned, the 2 × 2 × 3 supercell of B2, B19 and B19′ structures of Ti 50 (Ni 50−x Cu x ) and Ni 50 (Ti 50−x Zr x ) alloys that contains 48 atoms are constructed. It is calculated (per atom) as  15 ), compared with the experimental data and other ab initio calculations. www.nature.com/scientificreports www.nature.com/scientificreports/ E form of B2 phase is much higher than that of B19 and B19′ phases in NiTi alloy (x = 0), and, E fB19′ < E fB19 < E fB2 < 0, which reveals that the B19 and B19′ phases are more stable compared with the B2 phase, and, B19′ phase is the lowest temperature phase, which agrees with the experimental results 16 .
The E form of each phase increases almost linearly with the increasing of Cu content which indicates that the phase stabilities of three structurers become worse (see Fig. 2(a)). This is in accordance with the experimental results 17 . There is a crossover at the formation energy curve for Cu = 10.4 at.% concentration. For Cu < 10.4 at.%, the B19′ phase is more stable than B19 phase; for Cu > 10.4 at.%, the B19 phase is more stable than B19′ phase, which indicates the phase transition path changed. For Cu < 10.4 at.%, phase transition takes place from B2 → B19′, while for Cu ≥ 10.4 at.%, the MT path thus becomes B2 → B19. From the calculation of formation energies, the most stable martensitic phase was changed from monoclinic to orthorhombic (see Fig. 2(a)), which is also in accordance with previous experimental works 8,16 .
For the Zr-doped, however, the formation energies of three phases decrease with the doping concentration, which means the stability for all phases increases with Zr doping. Moreover, the formation energy of B19′ is the lowest one. The B19′ phase is the most stable structure all the time with increasing x, which consistent with the previous results 11,12 , see Fig. 2 The partial density of states (PDOSs) of the austenitic phase with different doping concentration for Ti 50 (Ni 50−x Cu x ) and Ni 50 (Ti 50−x Zr x ) alloys have been calculated and plotted in Fig. 3. We can see that the PDOSs of Ni and Ti atoms have a sharp peak located at −4.4 and −3.5 eV, respectively, for pure NiTi alloy. The PDOSs of Cu atoms are similar to the Ni atoms, and, there is no obvious hybridization with the nearest neighbor Ti atoms. Cu doping weakens the overlap between Ti and Cu electronic states at −4 ~ −1.5 eV. With the increasing of the Cu concentration, the peak positions of Ti atoms and Ni-Cu resonated atoms shift to the higher energy slightly. This suggests that B2 phase become unstable with the increasing of Cu content, in accordance with the results of E form (see Fig. 2(a)).
It can be seen from Fig. 3(b) that due to the Zr doping, the PDOSs of Zr atoms are similar to the Ti atoms, and there is obvious hybridization with the nearest neighbor Ti and Ni atoms. With the increase of Zr addition, there are the increasing resonant states between the Zr and its nearest neighbor Ti atoms at energy region −4 ~ −3 eV, which enhances the interaction between Ti and Zr atoms. The peak of Ni atoms and Ti-Zr resonated atoms shift to the lower energy slightly. This indicates that strong bonding states and thus increases the stability of the austenitic phase, which is in consistent with the formation energy result shown in Fig. 2

(b).
Composition dependence of transformation temperature in ti 50 (Ni 50−x Cu x ) and Ni 50 (ti 50−x Zr x ) alloys. The transformation behaviors can be described by the energy differences (ΔE) between the austenitic B2 structure and martensitic B19 and B19′ structure, ΔE = E B2 − E B19/B19′ . Table 2 summarizes the energy differences for all three phases in NiTi. The calculated results indicate that B19 phase is lower in energy than B2 by 29.95 meV/atom, and, the monoclinic B19′ phase is the lowest in energy (ΔE = E B2 − E B19′ = 40.39 meV/atom) in NiTi alloys. The present calculations are in good accordance with other DFT studies.
The previous observation 32 shows that martensite transformation temperature is closely related to the value of ΔE, and, T m usually increases with increasing ΔE. Figure 4 presents the concentration dependence of ΔE for the at.%), as 0 < ΔE B19 < ΔE B19′ (see Fig. 4(a)), the most stable phase is B19′ phase. The ΔE B19′ decreases slightly with increasing x (see Fig. 4(a)), thus the lower energy is needed for the martensitic transformation. The lower ΔE corresponds to a lower T m , therefore, the T m decreases slightly. At high doping concentrations (Cu > 10.4 at.%), as 0 < ΔE B19′ < ΔE B19 , the most stable phase is B19, and ΔE B19 increases slightly with increasing Cu content, thus a little higher T m occurs.
It is clearly seen from Fig. 4(b) that, for Zr < 10.4 at.%, ΔE B19′ decreases slightly; with x further increasing, ΔE B19′ increases significantly. Lager ΔE indicates more energy is needed for the martensitic transformation thus corresponds to a higher T m .
In Fig. 4(c), we plot the total energy differences between B2 and martensite phases (ΔE) as well as T m as a function of the doping concentration of Ti 50 (Ni 50−x Cu x ) and Ni 50 (Ti 50−x Zr x ) systems. T m of Ni-Ti-Cu is cited from refs 16,35 , and T m of Ni-Ti-Zr is cited from refs 10,13 . Addition of Cu decreases the T m , and then increases slightly with further increasing of Cu content. This makes the T m less sensitive to composition changes comparing with the binary NiTi alloys, which is in accordance with previous experimental works 8,9,16,17,35 . Zr addition raises the T m obviously, which is fully in line with the experimental observations that the Zr additions increase the T m dramatically 10,13 . Influence of Cu or Zr on the hysteresis. Hysteresis (ΔT) in shape memory alloys (SMAs) is the macroscopic presentation of the energy dissipation of the martensitic phase transformation, which plays an important role in their thermomechanical behavior. The geometric non-linear theory 36    www.nature.com/scientificreports www.nature.com/scientificreports/ conditions for SMAs with extremely low hysteresis. The first condition λ λ λ = = U det( ) 1 1 2 3 , represents no volume change during phase transformation, where U is the transformation stretch tensor that maps the martensite lattice to the austenite lattice, detU is its determinant and λ 1 ≤ λ 2 ≤ λ 3 are the ordered eigenvalues of U. The second condition λ 2 = 1 represents phase compatibility between austenite and martensite. The transformation stretch tensor between a bcc austenite and one particular lattice correspondence variant of an orthorhombic martensite is 37,38 β α γ α γ Their values are entirely determined by the lattice parameters of the austenite (a 0 ) and the orthorhombic martensite phase (a, b, c): β = λ =  www.nature.com/scientificreports www.nature.com/scientificreports/ As indicated in Fig. 5(a), an increase in the percentage of Cu (substituted for Ni) increases the value of λ 2 , further approaching 1; for Cu content at 10.4 at.%, λ 2 = 1; Cu > 10.4 at.%, λ 2 is far from 1. In Fig. 5(c), |1 − λ 2 | was calculated in the present study while the corresponding ΔT values were taken from the literature 10,13,19,35,38,39 . One can see a remarkable collapse of the data onto two lines shaped like a V for Ti 50 (Ni 50−x Cu x ), and the widths of the thermal hysteresis have a minimum for |1 − λ 2 | approaches 0 (Cu = 10.4 at.%). λ 2 quantifies the geometric compatibility of the martensite and the austenite, and the fit between the two phases increases as λ 2 approaches 1 [37][38][39][40][41] . The most stable martensite phase was changed from monoclinic to orthorhombic structure for Ti 50 (Ni 50−x Cu x ) alloys, which increases the phase compatibility, making λ 2 approach 1. For other λ 2 value which is lower or higher than 1, wider thermal hysteresis is observed. Adding Cu lower the width of the thermal hysteresis, ΔT appears to be minimized when Cu content at about 10.4 at.%at.%.
The value of λ 2 decrease slightly for Zr < 10.4 at.%, and then nearly stable, keeping further from 1 (see Fig. 5(b)). This is because phase compatibility is reduced. With increasing of Zr content, λ 2 deviates from 1 leading higher thermal hysteresis temperature for martensitic transformation. Therefore, replacement of Ti by Zr in the NiTi alloy raises the ΔT slightly in Ni 50 (Ti 50−x Zr x ) SMAs, which are in line with other literatures 10,13 .

Conclusions
Our first-principles calculations on formation energies E form , energy differences between austenite and martensite ΔE and the middle eigenvalue of transformation stretch tensor λ 2 for Ti 50 (Ni 50−x Cu x ) and Ni 50 (Ti 50−x Zr x ) systems allow us to draw the following conclusions. For Ti 50 (Ni 50−x Cu x ) alloys, when the doping Cu concentrations are less than 10.4 at.%, phase transformation takes place from B2 to monoclinic B19′ phase, while the doping concentration ≥10.4 at.%, phase transformation happens from B2 to orthorhombic B19 phase. According to the results of ΔE and λ 2 , for Cu < 10.4 at.%, adding Cu slightly decreases T m and thermal hysteresis ΔT, moreover, gets a minimum of the widths of thermal hysteresis; with further doping, T m and ΔT increase slightly. In short, the addition of Cu results in a decrease of T m and ΔT. The present investigation exhibits that the replacement of Ti by Zr rises transformation temperature significantly, and increase the hysteresis slightly. Ni 50 (Ti 50−x Zr x ) alloy remains the B19′ martensitic phase, which leads the phase compatibility between austenite and martensite reduced and larger lattice deformation involved.

Data Availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.