Influence of β-Stabilizing Nb on Phase Stability and Phase Transformation in Ti-Zr Shape Memory Alloys: From the Viewpoint of the First-Principles Calculation

Abstract

In the present study, the effect of the Nb element on the lattice parameters, phase stability and martensitic transformation behaviors of Ti-Zr-based shape memory alloys was extensively investigated using the first-principles calculation. The lattice parameters of both the β parent phase and α′ martensite phase gradually decreased with Nb content increasing. For the α″ martensite phase, the lattice constant (a) gradually increased with the increase in Nb content, whereas the lattice constants (b and c) continuously decreased due to the addition of Nb. Based on the formation energy and density of state, β→α′ martensitic transformation occurred, as the Nb content was not more than 12.5 at.%. However, the Ti-Zr-Nb shape memory alloys with a Nb content higher than 12.5 at.% possessed the β→α″ martensitic transformation. However, both the largest transformation strain and sensitivity of critical stress to temperature (dσ/dT) can be optimized by controlling 12.5 at.% Nb in the Ti-Zr-Nb shape memory alloy, which was favorable to obtaining the largest elastocaloric effect.

1. Introduction

In recent years, metastable β-type Ti-based shape memory alloys have attracted more attention in biomedical implant fields owing to the lower Young’s modulus, non-toxicity, as well as shape memory effect and superelasticity [1,2,3]. To date, typical Ti-based biomedical shape memory alloys are mainly focused on Ti-Zr, Ti-Nb, Ti-V and Ti-Mo systems, which mainly have good biocompatibility and high corrosion resistance, in addition to the shape memory effect and superelasticity [4,5,6,7,8,9,10]. Their functional performances, especially superelasticity stemming from the stress-induced martensitic transformation, are closely related to the chemical compositions [11]. Ti-Zr binary shape memory alloys are completely solid solutions for both the high-temperature β phase and low-temperature α phase (or α′ martensite as well as α′′ martensite) [12,13]. Moreover, the shape memory effect of the Ti-Zr binary shape memory alloy originates from the reverse transformation between the hexagonal α′ martensite phase and bcc β austenite phase at the higher temperature of 800 K, which leads to the poor shape memory effect strain of 1.4% [14,15]. For comparison, the shape memory effect strain originating from the martensitic transformation between the cubic β phase (bcc) and orthorhombic a″ martensite phase is superior to that of martensitic transformation between the hexagonal α′ martensite phase and bcc-β austenite phase [15]. Hence, Ti-Zr shape memory alloys can also be used for higher-temperature applications such as the aerospace and automotive fields, etc.. This is owing to the higher martensitic transformation temperatures and excellent shape memory effect, in addition to the biomedical applications [16,17]. The types of phase constituents in the β-type Ti-based shape memory alloy can be adjusted by changing chemical compositions. The lower-temperature α″ martensite phase can be obtained by adding β stabilizing elements including Nb, Sn, Al, Mo, etc., into Ti-Zr binary shape memory alloys, which can promote to the improvement of shape memory effect strain [18,19,20,21]. For instance, the phase constituents of Ti-Zr-Mo shape memory alloys evolve from the coexistence of the β phase and α′ martensite phase into the coexistence of the β phase and α″ martensite phase [18]. In proportion, the recoverable strain is also enhanced. Similarly, the maximum shape memory strain of 2.5% can be achieved in the Ti-20Zr-based shape memory alloy by optimizing 10 at.% (atom percentage) Nb, as the lower-temperature martensite phase changes from the α′ phase to α″ phase through the introduction of the Nb element [20,21,22]. In addition, the recoverable strain of the Ti-Zr-Nb-based shape memory alloy can be improved to 4.3% by aging treatment [23]. However, the superior shape memory effect strain is as large as 3.3% and 4.1% in quaternary Ti-20Zr-10Nb-4Ta and Ti-19.5Zr-10Nb-0.5Fe shape memory alloys by tailoring the chemical composition [24,25,26].

It is well accepted that the first-principles method is an effective technology to determine the martensitic transformations of shape memory alloys based on the calculations of energies and energy derivatives at 0 K, which further predicts the shape memory effect and superelasticity as well as other functional performances [27,28,29,30,31]. In addition, the first-principles method can be used to calculate the lattice structure, the optimization of chemical composition for obtaining superior performances. For instance, the lattice constant of a_o continuously increases with Nb content increasing in the Ti-Nb shape memory alloy; the lattice constant of a′ is also monotonously increased. However, the lattice constant of b′ and c′ show the opposite trends, continuously decreasing. In proportion, the lattice constant ratio (b/a) is almost equal and the lattice constant ratio (c/a) is slightly increased in Ti-Nb shape memory alloys within the Nb content range from 0 at.% to 12 at.%. Upon the Nb content further increasing, both the b/a and c/a ratios dramatically decrease. Meanwhile, the lattice strain of η₁ is almost linearly increased and the lattice strain of η₂ and η₃ are almost linearly decreased with the Nb content increasing from 10 at.% to 30 at.%. However, it is revealed that the addition of Al, Be, Ca, Cu, Ga, Ge, Hf, La, Mg, Sc, Si, Sn, Sr, Y, Zn and Zr into a Ti-Nb shape memory alloy can increase the transformation strains [29]. In addition, it is found that the equiatomic Ti₂₅Nb₂₅Zr₂₅Sc₂₅ HE-SMA has the larger lattice strain and the higher martensitic transformation temperature (Ms), and Al-containing T_i24Nb₂₅Zr₂₄Sc₂₄Al₃ HE-SMA almost has the maximum recoverable strain in theory, accompanied by the lower Ms (~370 K), on the basis of the first-principles methods [27]. Nevertheless, no literature regarding the effect of Nb content on Ti-Zr-based shape memory alloys are reported.

In short, the recoverable strain of the β-type Ti-Zr-based shape memory alloys is dependent on its type of phase transformation and phase stability. Moreover, it has been reported that the recoverable strain of β-Ti-based shape memory alloys gradually increases, whereas both the martensitic transformation temperatures and yielding stress continuously decrease with the content of β-stabilizing Nb alloying element increasing (Nb content < 30 at.%) [32]. In the present study, it was found that the transition path in the present Ti-Zr-Nb shape memory alloys is largely dependent on the Nb content. The β→α′ martensitic transformation occurred in the Ti-Zr-Nb shape memory alloy with a Nb content within the range between 0 at.% and 12.5 at.%. Upon the Nb content exceeding 12.5 at.%, Ti-Zr-Nb shape memory alloys possessed the reversible β→α″ martensitic transformation. Moreover, in order to optimize the performances of the present β-type Ti-Zr shape memory alloys, the phase transformation and phase stability of Ti-Zr-based shape memory alloys with different Nb contents were evaluated by the first-principles calculation in the present study. This allowed further prediction of the effect of Nb content on the phase constituents, martensitic transformation and recoverable strain as well as the elastocaloric effect of Ti-Zr-based shape memory alloys.

2. Calculation Methods

In the present study, the Cambridge Serial Total Energy Package (CASTEP) mode in Material Studio software was adopted to carry out this work, which can be used to optimize geometries and calculate the band structures and elastic properties [33]. The plane-wave pseudopotential method based on density functional theory was selected. Crystal wave function was developed on the plane-wave basis. The exchange correlation function adopted the Perdew–Burke–Ernzerhof (PBE) method in GGA to select the plane-wave ultra-soft pseudopotential [27]. The interactions between ions and electrons were represented by Vanderbilt-type pseudopotentials for Ti, Zr and Nb atoms [34]. The cell models of the parent phase and martensite phase of Ti-Zr-Nb shape memory alloys were established. Moreover, the Broyden–Flecher–Goldfarb Shanno (BFGS) method was used to optimize the initial crystal structure [35]. During the optimization of the structure, the convergence value of the total energy of the system was set as 1.0 × 10⁻⁵ eV/atom, and the convergence value of the total energy for the electronic self-consistent computing system was 2.0 × 10⁻⁶ eV/atom. The force required on each atom was less than 0.01 eV/A, the tolerance deviation was less than 5.0 × 10⁻⁴ A, and the stress deviation was less than 50 MPa. The plane-wave energy cutoff was maintained at 600 eV. For the B2 parent cell, the Monkhorst–Pack grid with the k point of 8 × 8 × 8 was employed to sample the first Brillouin zone. The corresponding k spot of the remaining supercells was selected according to the same k-spot interval (0.041) as the B2 phase singlet. The supercells were determined by the lowest energy principles.

The crystal structure of the β parent phase, α′ martensite phase and α″ martensite phase for Ti-Zr-Nb shape memory alloys with the different Nb contents was optimized. The atomic positions were further optimized for all phase structures until the change in the total energy was smaller. Finally, the optimal crystal structures of the Ti-Zr-Nb shape memory alloys with the different Nb contents are illustrated in Figure 1.

3. Results and Discussions

In order to investigate the effect of the Nb content on the phase stability of Ti-Zr-Nb shape memory alloys, the formation energy of the α′ and α″ martensite phases as well as the β parent phase was calculated and the corresponding results are displayed in Figure 2. The results reveal that the formation energy of the various phase states for the Ti-Zr-Nb shape memory alloy is different, as the Nb content changes. It is generally accepted that the lower formation energy corresponds to significantly increased stability of the phase. For comparison, the formation energy of both the α′ martensite phase and α″ martensite phase is lower than that of the β phase, regardless of the Nb content, which indicates that the martensite phase is more stable in the ground state. Moreover, the Nb content is not more than 15.0 at.%, and the formation energy of the α′ martensite phase in the Ti-Zr-Nb shape memory alloy is less than that of α″ martensite phase, which means that the α′ martensite phase is more stable within the range of 6.25 to 15.0 at.%. It can be concluded that the martensite transformation of β→α′ occurs, as the Nb content is not more than 15.0 at.%. The Nb content is more than 15.0 at.%, and the formation energy of α″ martensite phase is much lower, which indicates that the α″ martensite phase is more stable in contrast to the α′ martensite phase. In proportion, the β→α′ martensite phase is changed into a β→α″ martensitic transformation, when the Nb content exceeds 15.0 at.% in Ti-Zr-Nb shape memory alloys. This means that the phase constituents in Ti-Zr-Nb shape memory alloys are mainly composed of the α′ martensite phase, as the Nb content is less than 17.5 at.%. The Nb content exceeds 17.5 at.%, and the phase constituents mainly consist of the α″ martensite phase. However, with the Nb content increasing, the formation energy of the β parent phase is decreased continuously, which is very consistent with the existing results that the Nb element belongs to a β-stabilizing element [36]. The evolution of formation energy can be attributed to the total density of state at the nearby Fermi level, which will discussed below.

However, we have calculated the electron density of the electronic state of the β parent phase, α′ martensite phase and α″ martensite phase of Ti-Zr-Nb shape memory alloys with the different Nb contents to evaluate their phase stability. In addition, the density of states at the Fermi level for the different phases including the β parent phase, α′ martensite phase and α″ martensite phase is obtained from the total state density in Figure 3, Figure 4 and Figure 5, respectively. It can be found that all Ti-Zr-Nb shape memory alloys have similar electronic densities of states (DOSs) for the same phase structure, irrespective of the Nb contents. Meanwhile, for the different phases containing the β parent phase, α′ martensite phase and α″ martensite phase, the DOS structure is insensitive to the Nb content in Ti-Zr-Nb shape memory alloys. The total DOSs of Ti-Zr-Nb shape memory alloys with various Nb contents is almost dependent on the partial DOS of the Ti element, regardless of the phase structure, implying that the Ti element plays significant roles in the total DOS. Moreover, with the Nb content increasing, the role of Nb content on the total DOS is becoming more and more important. For the β phase structure, the total DOS splits into two peaks near the E_F, and some of the electrons near the E_F move towards the higher energy level, which can be explained by the Jahn–Teller splitting effect [37,38]. Differing from the β phase, the total DOS near the E_F of Ti-Zr-Nb shape memory alloys with a α′ martensite structure is not affected by the Nb content. However, the total DOS near E_F for the Ti-Zr-Nb shape memory alloys with the α″ martensite phase are significantly influenced by the Nb content, especially for 6.25 at.% Nb.

In addition, the effect of the Nb content on the total DOS at the E_F of Ti-Zr-Nb shape memory alloys possessing the different phase structures is extracted from Figure 3, Figure 4 and Figure 5. and the corresponding results are shown in Figure 6. It can be seen that the total DOS at E_F for all Ti-Zr-Nb shape memory alloys with the β phase is the highest, compared with a Ti-Zr-Nb shape memory alloy with α′ and α″ martensite phases. However, as the Nb content is not more than 15 at.%, the DOS at E_F in Ti-Zr-Nb shape memory alloys with the α′ martensite phase is lower than that of Ti-Zr-Nb shape memory alloys with the α″ martensite phase. This means that the α′ martensite phase is more stable and there is a trend in β→α′ martensitic transformation for the Ti-Zr-Nb shape memory alloy with a lower Nb content. Where the Nb content is more than 15 at.%, the DOS at E_F in Ti-Zr-Nb shape memory alloys with the α′ martensite phase is higher than that of Ti-Zr-Nb shape memory alloys with the α″ martensite phase, indicating the occurrence of β→α″ martensitic transformation.

The transformation strain is closely dependent on the lattice constants of the martensite and parent phases. Thus, in order to gain Ti-Zr-Nb shape memory alloys with a larger recovery strain, it is necessary to investigate the composition dependence of lattice constants of the α″martensite and β parent phases. It can be found from Figure 7 that the lattice constant (a_o) of the β parent phase with a bcc structure in Ti-Zr-Nb shape memory alloys slightly decreases with the Nb content increasing from 6.25 at.% to 25.0 at.%. For the α′ martensite phase with a hexagonal structure, the lattice constant (a_α′) of Ti-Zr-Nb shape memory alloys also continuously decreases, as the Nb content increases from 6.25 at.% to 25.0 at.%. In contrast, the lattice constant (a_α′) of Ti-Zr-Nb shape memory alloys almost remains unchanged. However, the evolution of the lattice constant of orthorhombic martensite (α″) is different from that of the α′ martensite phase and β parent phase. With the Nb content increasing, the lattice constant (a_α″) of the α″ martensite phase also increases, while the lattice constant of (b_α″ and c_α″) shows an obviously opposite trend.

The principal lattice strain along [100]α″, [010]α″ and [001]α″ are defined as η₁, η₂ and η₃, respectively. Moreover, they are calculated by using the following equation [35]:

$${\mathsf{\eta }}_1={\displaystyle \frac{{\mathrm{a}}_{{\mathsf{\alpha }}^{″}}-{\mathrm{a}}_{\mathsf{\beta }}}{{\mathrm{a}}_{\mathsf{\beta }}}}$$

(1)

$${\mathsf{\eta }}_2={\displaystyle \frac{{\mathrm{b}}_{{\mathsf{\alpha }}^{″}}-\sqrt{2}{\mathrm{a}}_{\mathsf{\beta }}}{{\sqrt{2}\mathrm{a}}_{\mathsf{\beta }}}}$$

(2)

$${\mathsf{\eta }}_2={\displaystyle \frac{{\mathrm{c}}_{{\mathsf{\alpha }}^{″}}-\sqrt{2}{\mathrm{a}}_{\mathsf{\beta }}}{{\sqrt{2}\mathrm{a}}_{\mathsf{\beta }}}}$$

(3)

In the above equation, a_β and a_α″, b_α″ and c_α″ represent the lattice parameters of the parent and martensite phases in Ti-Zr-Nb shape memory alloys with different Nb contents.

The calculated results in Figure 8a reveal that the principal lattice strain continuously decreases with the Nb content increasing from 6.25 at% to 25 at.%. In contrast, the principal lattice strains η₁ and η₂ possess the maximum value, regardless of Nb content. However, the value of the principal lattice strain η₃ is relatively lower. In short, the Ti-Zr-Nb shape memory alloy with 6.25 at.% Nb possesses the maximum principal lattice strain.

However, the transformation strain of β→α″ martensitic transformation in Ti-Zr-Nb shape memory alloys is calculated based on the lattice correspondence between martensite variants and the parent phase as well as the lattice distortion matrix and lattice parameters of Ti-Zr-Nb shape memory alloys. The corresponding lattice relationships between the martensite variants and the parent phase for Ti-Nb-Zr shape memory alloys are listed in

Table 1. Lattice correspondence between α″ martensite variant and β parent phase [33].

	CV1	CV2	CV3	CV4	CV5	CV6
[100]α″	[100]β	[100]β	[010]β	[010]β	[001]β	[001]β
[010]α″	[011]β	[0$\overline{1}$1]β	[101]β	[10$\overline{1}$]β	[110]β	[$\overline{1}$10]β
[001]α″	[0$\overline{1}$1]β	[0$\overline{11}$]β	[10$\overline{1}$]β	[$\overline{1}$0$\overline{1}$]β	[$\overline{1}$10]β	[$\overline{11}$0]β

[39]. And, the lattice distortion matrix is shown in Equation (4). The corresponding results reveal that the transformation strain along the [011]_β crystal orientation is largest, regardless of the Nb content. Moreover, the transformation strain along the [001]_β crystal orientation is larger than that of [$\overline{1}$11]_β crystal orientation, as the Nb content is not more than 12.5 at.%. Where the Nb content exceeds 12.5 at.%, the transformation strain along the [$\overline{1}$11]_β crystal orientation is larger than that of the [001]_β orientation. Irrespective of the crystal orientation, the transformation strain firstly increases slightly and then decreases with the Nb content increasing, which is very consistent with the evolution of the transformation strain in β-type Ti-Zr-Nb-based shape memory alloys [32,40]. The calculated results are very consistent with the existing studies demonstrating that the transformation strain of β→α″ martensitic transformation along the [011] _β crystal orientation is largest in Ti-Zr-Nb shape memory alloys, while the transformation strain along the [$\overline{1}$11]_β crystal orientation is lowest [39].

$$\mathrm{T}=⌈\begin{array}{ccc}{\displaystyle \frac{{\mathrm{a}}_{{\mathsf{\alpha }}^{″}}}{{\mathrm{a}}_{\mathsf{\beta }}}}& 0& 0\\ 0& {\displaystyle \frac{{\mathrm{b}}_{{\mathsf{\alpha }}^{″}}+{\mathrm{c}}_{{\mathsf{\alpha }}^{″}}}{2\sqrt{2}{\mathrm{a}}_{\mathsf{\beta }}}}& {\displaystyle \frac{{\mathrm{b}}_{{\mathsf{\alpha }}^{″}}-{\mathrm{c}}_{{\mathsf{\alpha }}^{″}}}{2\sqrt{2}{\mathrm{a}}_{\mathsf{\beta }}}}\\ 0& {\displaystyle \frac{{\mathrm{b}}_{{\mathsf{\alpha }}^{″}}-{\mathrm{c}}_{{\mathsf{\alpha }}^{″}}}{2\sqrt{2}{\mathrm{a}}_{\mathsf{\beta }}}}& {\displaystyle \frac{{\mathrm{b}}_{{\mathsf{\alpha }}^{″}}+{\mathrm{c}}_{{\mathsf{\alpha }}^{″}}}{2\sqrt{2}{\mathrm{a}}_{\mathsf{\beta }}}}\end{array}⌉$$

(4)

Figure 9 shows the influence of applied stress on the free energy difference between the β parent phase and α″ martensite phase of Ti-Zr-Nb shape memory alloys, which can be employed to evaluate the phase stability. It can be found that the increasing applied stress leads to the reduction in free energy difference between the β parent phase and α″ martensite phase, regardless of Nb content. For comparison, the free energy difference between the β parent phase and α″ phase for the Ti-Zr-Nb shape memory alloy containing 6.25 at.% Nb is obviously higher than that of the Ti-Zr-Nb shape memory alloy with 12.5 at.% Nb under the same applied stress condition. This means that the β phase of the Ti-Zr-Nb shape memory alloy containing 6.25 at.% Nb is more stable compared with the Ti-Zr-Nb shape memory alloy containing 12.5 at.% Nb. Meanwhile, with increasing applied stress, the reduction in free energy difference (ΔE) in the Ti-Zr-Nb shape memory alloy containing 12.5 at.% Nb is relatively larger, which implies that the martensitic transformation temperature is more sensitive to variation in applied stress. It can be concluded that the sensitivity of critical stress to temperature (dσ/dT) of the Ti-Zr-Nb shape memory alloy with 12.5 at.% Nb is higher, which is suitable for obtaining the larger elastocaloric effect and becoming the candidate for elastocaloric refrigeration. Nevertheless, it has been reported that the strain glass state can be induced in the cold-rolled Ti-Zr-Nb-Sn shape memory alloy with subsequent lower annealing, which results in the disappearance of martensitic transformation [41]. Meanwhile, it shows the smaller dσ/dT and the maximum recoverable strain of 4% as well as the superior elastocaloric effect with the largest adiabatic temperature change of 6.5 K [41].

4. Conclusions

The phase stability and martensitic transformation behaviors of Ti-Zr-Nb shape memory alloys with various Nb contents were evaluated by adopting the first-principles density functional theory approach. The main conclusions can be obtained as follows:

• From the electronical density of state viewpoint, the β→α′ martensitic transformation occurred in the Ti-Zr-Nb shape memory alloy with the Nb content within the range of 6.25 at.% to 12.5 at.%. Upon the Nb content exceeding 12.5 at.%, Ti-Zr-Nb shape memory alloys possessed the reversible β→α″ martensitic transformation.
• The transformation strain along the different crystal orientations in Ti-Zr-Nb shape memory alloys firstly increased and then decreased with Nb content increasing. For comparison, the transformation strain along the [011]_β orientation can be obtained as the maximum value. Moreover, the maximum value of the transformation strain can be achieved in the Ti-Zr-Nb shape memory alloy by optimizing 12.5 at.% Nb.
• The increased hydrostatic stress led to the reduction in the difference in free energy. The difference in the free energy was largest in the Ti-Zr-Nb shape memory alloy with 12.5 at.% Nb. This indicated that it had the largest dσ/dT, which was suitable for obtaining the largest elastocaloric effect.
• The calculated results revealed that the Ti-Zr-Nb shape memory alloy with 12.5 at.% Nb possessed the perfect combination of a larger recoverable strain and higher elastocaloric effect, making it the most promising candidate for biomedical materials. Moreover, the specific thermo-mechanical treatment can be adopted to optimize its performance.