Martensitic transformation of Ti₅₀(Ni_50−xCu_x) and Ni₅₀(Ti_50−xZr_x) shape-memory alloys

Abstract

Martensitic transformation and phase stability of Ti₅₀(Ni_50−xCu_x) and Ni₅₀(Ti_50−xZr_x) 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(Ni_50−xCu_x) alloys lose the monoclinic martensite in favor of the orthorhombic martensite structure. The martensite monoclinic B19´ structure of Ni₅₀(Ti_50−xZr_x) 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.

Introduction

Among the reported shape memory alloys (SMAs), NiTi alloys are the most attractive ones due to their unique shape memory effect and excellent superelasticity, resulting from the reversible thermoelastic martensite transformation (MT) between the high-temperature austenite phase and low-temperature martensite phases¹. NiTi SMAs have been studied extensively and widely used for a variety of applications in a number of different fields, such as aeronautic, microelectronic and biomedical industries, since 1960s when first discovered by Buehler and Wiley². However, some applications of the NiTi SMAs are greatly limited by its particular hysteresis (ΔT) and the martensite transformation temperature, T_m, is usually lower than 100 °C³. The need of addressing these limitations has led to numerous experimental and theoretical researches^{4,5,6,7,8,9,10,11,12,13,14,15}.

The addition of Fe and Co (substituted for Ni) or Al, V, Cr and Mn (substituted for Ti)⁴ depresses the T_m of NiTi alloys severely even at very small levels, but still results in a monoclinic martensite phase after MT. According to the recent reports^5,6, there exists a strain glass (STG) transition in doped Ti₅₀(Ni_50−xD_x) (D = Co, Cr, Mn) alloys. Wang et al.⁷ discovered the existence of strain glass in Ti₅₀Ni_50−xFe_x, beyond a critical Fe doping level x > x_c (5 < x_c < 6). Alloying NiTi with Cu (Cu substitutes for Ni) reduces the thermal hysteresis⁸ and composition sensitivity of the transformation temperatures⁹, meanwhile, changes the MT pathway^8,16,17.

Transformation temperature increases linearly with the X content for alloys containing greater than approximately 10–15 at.% in ternary NiTi-X (X = Zr, Hf, Pd and Pt) SMAs¹⁰. These high-temperature shape memory alloys (HTSMAs) with the T_m higher than 100 °C have various potential applications. NiTiZr system has drawn much attention owing to the much lower cost compared with the noble metals (Pd, Pt, and Au). Experiments have indicated that, the T_m decreases with small amounts of Zr (X ≤ 3 at.%)¹⁸, and then sharply increases when Zr content exceeds 10 at.%, approaches as high as 170 °C for 20.2 at.% Zr^11,12,13; martensite structure remains monoclinic (B19′)^11,12; and its hysteresis is slightly wider than that of TiNi binary alloy¹³. Frenzel et al.¹⁹ found the strong dependence of T_m on alloy composition in binary NiTi, ternary Ni–Ti–X (X = Cr, Cu, Hf, Pd, V, Zr) and quaternary Ni–Ti–Cu–Y (Y = Co, Pd) SMAs.

From above experimental researches, it can be seen that the alloying elements play an important role in determining the phase transformation temperature, hysteresis as well as STG. Zarinejad and Liu²⁰ conclude that the valence electron concentration is found to be an important parameter influencing the T_m. There requires a better 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 calculated²¹. The martensite crystal structures of Ti₅₀Ni_50−xCu_x SMAs were investigated by using CASTEP code²². Wang’s ab-initio calculations⁶ show the origin of STG transition in Ti₅₀(Ni_50−xD_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 work¹⁵ has investigated the effect of Pd on the martensitic transformation of Ni–Ti–Pd HTSMAs through DFT calculations. Attracted by the small hysteresis width⁸ and the low composition sensitivity of T_m⁹ 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 K²³. 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 annealings²⁴. 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 alloys⁴, thus antisite defects were not discussed in the present work. In this paper, we adopt the Ti₅₀(Ni_50−xCu_x) and Ni₅₀(Ti_50−xZr_x) (x = 4.2, 6.3, 8.4, 10.4, 12.5, 18.8, 25) alloys as our modeling. Considering the symmetry of the cell, a 2 × 2 × 3 supercell containing 48 atoms of the austenite phase is chosen, and X, Y, Z directions are along [001]_B2, ${[\bar{1}10]}_{B2}$ and [110]_B2, respectively. The lattice parameters are ${\rm{a}}={a}_{0},\,{\rm{b}}={\rm{c}}=\sqrt{2}{a}_{0}$, here a₀ 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 Ti₅₀(Ni_50−xCu_x) and Ni₅₀(Ti_50−xZr_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.

Figure 1

Schematic diagram of the 2 × 2 × 3 supercell model of austenite and martensite phase of NiTi alloy: (a) B2, (b) B19 and (c) B19′. Smaller blue spheres are Ni atoms; larger gray spheres are Ti atoms.

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 and B19′ phase is almost unaffected by the doping position. Moreover, the calculated deviations of energy of all possible doping position is about 0.2 eV.

The calculations are performed using DFT as implemented in the Vienna Ab initio Simulation Package^25,26,27 (VASP) together with plane-wave projector augmented wave²⁸ (PAW) pseudopotentials and the PW91²⁹ generalized gradient approximation³⁰ (GGA) for the exchange and correlation effects. The valence electron configurations for Ni, Ti, Cu and Zr are 3d⁸4s², 3d²4s², 3d¹⁰4s¹ and 4d²5s¹, respectively. We tested k-point sampling³¹ 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.

Table 1 Lattice constants of B2, B19 and B19′ phases in NiTi (taken from our previous calculation ref¹⁵), compared with the experimental data and other ab initio calculations.

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 Ti₅₀(Ni_50−xCu_x) and Ni₅₀(Ti_50−xZr_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 Ti₅₀(Ni_50−xCu_x) and Ni₅₀(Ti_50−xZr_x) alloys that contains 48 atoms are constructed. It is calculated (per atom) as

$${E}_{form}^{1}=\frac{{E}_{tot}-[24E\,{{(}_{Ti}}^{hcp})+(24-n)\,E\,{{(}_{Ni}}^{fcc})+nE\,{{(}_{Cu}}^{fcc})]}{48}$$

(1)

$${E}_{form}^{2}=\frac{{E}_{tot}-[24E\,{{(}_{Ni}}^{fcc})+(24-n)E\,{{(}_{Ti}}^{hcp})+nE\,{{(}_{Zr}}^{hcp})]}{48}$$

(2)

where ${E}_{form}^{1}$ and ${E}_{form}^{2}$ are for Ti₅₀Ni_50−xCu_x and Ni₅₀Ti_50−xZr_x alloys respectively, E_tot is the total energy of Ti₅₀Ni_50−xCu_x and Ni₅₀Ti_50−xZr_x alloys, $E\,{{(}_{Ni}}^{fcc})$, $E\,{{(}_{Ti}}^{hcp})$, $E\,{{(}_{Cu}}^{fcc})$ and $E\,{{(}_{Zr}}^{hcp})$ denotes, respectively, the energy of one Ni, Ti, Cu and Zr atom in their bulk states, n is the number of doping atoms (Cu or Zr) in the supercell. The calculated results of E_form as a function of doping concentration are plotted in Fig. 2.

Figure 2

Formation energies, E_form, as a function of the addition content in quasi-equiatomic ternary (a) Ti₅₀(Ni_50−xCu_x) and (b) Ni₅₀(Ti_50−xZr_x) alloys. The trendline is a fit through all data points.

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 results¹⁶.

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 results¹⁷. 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(b).

The partial density of states (PDOSs) of the austenitic phase with different doping concentration for Ti₅₀(Ni_50−xCu_x) and Ni₅₀(Ti_50−xZr_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)).

Figure 3

Partial density of states (PDOSs) of the doping atoms as well as their nearest Ti and Ni atoms in the austenitic phase of different doping concentration for (a) Ti₅₀(Ni_50−xCu_x) and (b) Ni₅₀(Ti_50−xZr_x) alloys. The vertical line denotes the Fermi level, which is located at 0 eV.

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 Ti₅₀(Ni_50−xCu_x) and Ni₅₀(Ti_50−xZr_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.

Table 2 Computed energy differences ΔE = E_B2 − E_B19/B19′ (in meV/atom) in binary NiTi, compared with other first-principles studies.

The previous observation³² 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 Ti₅₀(Ni_50−xCu_x) and Ni₅₀(Ti_50−xZr_x) alloys. For Ti₅₀(Ni_50−xCu_x) alloys with low doping concentrations (Cu < 10.4 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.

Figure 4

Dependency of the energy differences between austenite and martensite ΔE for (a) Ti₅₀(Ni_50−xCu_x) and (b) Ni₅₀(Ti_50−xZr_x) on doped content (0–25 at.%). (c) Composition dependence of ΔE and transformation temperature T_m of Ti₅₀(Ni_50−xCu_x) and Ni₅₀(Ti_50−xZr_x) alloys. The values of T_m refer to the right axis and those of ΔE to the left axis. T_m of Ni-Ti-Cu is cited from refs^16,35, T_m of Ni-Ti-Zr is cited from refs^10,13. Lines are guides for the eyes.

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 Ti₅₀(Ni_50−xCu_x) and Ni₅₀(Ti_50−xZr_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 theory³⁶ of martensite specifies several conditions for SMAs with extremely low hysteresis. The first condition $det(U)={\lambda }_{1}\,{\lambda }_{2}\,{\lambda }_{3}=1$, 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 λ₁ ≤ λ₂ ≤ λ₃ are the ordered eigenvalues of U. The second condition λ₂ = 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

$$U=[\begin{array}{ccc}\beta & 0 & 0\\ 0 & \frac{\alpha -\gamma }{2} & \frac{\alpha +\gamma }{2}\\ 0 & \frac{\alpha +\gamma }{2} & \frac{\alpha -\gamma }{2}\end{array}].$$

(3)

Their values are entirely determined by the lattice parameters of the austenite (a₀) and the orthorhombic martensite phase (a, b, c): ${\rm{\beta }}={{\rm{\lambda }}}_{1}=\frac{a}{{a}_{0}}$, ${\rm{\alpha }}={{\rm{\lambda }}}_{2}=\frac{b}{\sqrt{2}{a}_{0}}$, ${\rm{\gamma }}={{\rm{\lambda }}}_{3}=\frac{c}{\sqrt{2}{a}_{0}}$.

According to the above theory, Cui et al.³⁸ discovered that there is a strong relationship between hysteresis (ΔT) and λ₂, but not with det(U), in Ni–Ti–Cu and Ni–Ti–Pd thin-films. Zhang et al.³⁹ further demonstrated the same correlation for bulk alloys of Ni–Ti–Au, Ni–Ti–Cu, Ni–Ti–Pd, and Ni–Ti–Pt. Low hysteresis with phase compatibility between austenite and martensite was investigated in Ti₅₀(Ni_50−xPd_x) system³⁷. Recently, the direct correlation between thermal hysteresis and functional stability was demonstrated in binary (Ni-Ti), ternary (Ni-Ti-Cu, Ni-Ti-Pd) and quaternary (Ni-Ti-Cu-Pd) SMAs^40,41.

To investigate the influence of alloy elements content on thermal hysteresis (ΔT) in the Ti₅₀(Ni_50−xCu_x) and Ni₅₀(Ti_50−xZr_x) systems, we calculated λ₂ values by ${{\rm{\lambda }}}_{2}^{B19}={b}_{B19}/{b}_{B2}$ for B2 → B19 crystallographic transformation, λ₂ shown as ${\lambda }_{2}^{B19^{\prime} }={b}_{B19^{\prime} }\,\sin \,{\rm{\beta }}/{b}_{B2}$ for B2 → B19′ phase transformation, where b_B2, b_B19 and b_B19′ are the lattice parameters of the B2, B19 and B19′ structures in the [010] direction, respectively, and β is the monoclinic angle of the crystal structure of B19′ martensite. The calculated structural parameters of Ti₅₀(Ni_50−xCu_x) and Ni₅₀(Ti_50−xZr_x) for different doping concentration are listed in Supplementary Tables S3 and S4. The variations of λ₂ vs. doping concentration for these alloys were shown in Fig. 5.

Figure 5

Middle eigenvalue λ₂ vs. doped concentration of (a) Ti₅₀(Ni_50−xCu_x) and (b) Ni₅₀(Ti_50−xZr_x) alloys. (c) Effect of addition content on the |1 − λ₂| and thermal hysteresis ΔT of Ti₅₀(Ni_50−xCu_x) and Ni₅₀(Ti_50−xZr_x) systems. The values of ΔT refer to the right axis and those of |1 − λ₂| to the left axis. ΔT of Ni-Ti-Cu is cited from refs^{10,13,19,35,38,39}. ΔT of Ni-Ti-Zr is cited from refs^10,13.

As indicated in Fig. 5(a), an increase in the percentage of Cu (substituted for Ni) increases the value of λ₂, further approaching 1; for Cu content at 10.4 at.%, λ₂ = 1; Cu > 10.4 at.%, λ₂ is far from 1. In Fig. 5(c), |1 − λ₂| 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 Ti₅₀(Ni_50−xCu_x), and the widths of the thermal hysteresis have a minimum for |1 − λ₂| approaches 0 (Cu = 10.4 at.%). λ₂ quantifies the geometric compatibility of the martensite and the austenite, and the fit between the two phases increases as λ₂ approaches 1^{37,38,39,40,41}. The most stable martensite phase was changed from monoclinic to orthorhombic structure for Ti₅₀(Ni_50−xCu_x) alloys, which increases the phase compatibility, making λ₂ approach 1. For other λ₂ 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 λ₂ 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, λ₂ 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 Ni₅₀(Ti_50−xZr_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 λ₂ for Ti₅₀(Ni_50−xCu_x) and Ni₅₀(Ti_50−xZr_x) systems allow us to draw the following conclusions. For Ti₅₀(Ni_50−xCu_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 λ₂, 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. Ni₅₀(Ti_50−xZr_x) alloy remains the B19′ martensitic phase, which leads the phase compatibility between austenite and martensite reduced and larger lattice deformation involved.