This study investigated the effects of a variety of point defects on the monoclinic angle of the B19″ martensite in NiTi, including Ni and Ti vacancies, Ni–Ti anti-sites in the lattice, third element substitution, and Ni and Ti enrichments in the B2 stoichiometry. Their effects were studied by density functional theory calculations on the monoclinic B19″ phase as a proxy for the experimentally observed B19′ phase. It was found that vacancies and Ni–Ti anti-sites reduce the monoclinic angle of B19″, whereas third element substitution of Ni and Ti may affect the monoclinic angle in both directions depending on the third element and the host element it replaces. Enriching Ni or Ti content to deviate from the B2 stoichiometry decreases the monoclinic angle. A change in the monoclinic angle of the B19″ phase implies variation in the characteristic strains associated with the transformation. Thus, these findings may provide a new angle on possible interpretations of the variations of recoverable transformation strains experimentally observed of NiTi shape memory alloys.
Notes
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Introduction
Shape memory alloys (SMAs) are a unique class of materials due to their unique properties of the shape memory effect and pseudoelastic behaviour [1]. These properties originate from their thermoelastic martensitic transformations. The most common SMAs are the NiTi-based alloys, which exhibit a B2 austenite to monoclinic B19′ martensite transformation. The B2 → B19′ transformation is associated with a crystalline lattice distortion, which may manifest as global strains that are recoverable upon the reversal of the phase transformation. This self-enabled shape strain, associated with either the forward B2 → B19′ transformation or the reverse B19′ → B2 transformation, is the underlying mechanism of these two unique properties that have rendered them suitable for many applications, e.g., as actuators and sensors [2, 3].
The performance of these alloys in such applications is obviously dependent on the magnitude of the transformation-enabled recoverable strain, as well as the forces they generate in doing so and the critical temperature at which the transformation occurs. For polycrystalline NiTi commercial alloys, the recoverable strain is typically 5–7% [1]. The magnitude of the recoverable strain is directly dictated by the crystallographic lattice distortion of the transformation, of which the monoclinic angle of the B19′ martensite is the most important single parameter [4].
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The general knowledge of the B19′ phase is that it has a monoclinic angle of 97.8°, as has been experimentally determined [5]. However, some studies have shown that the monoclinic angle of the B19′ phase in NiTi-based alloys can be influenced by various factors. Prokoshkin et al. [6] determined the lattice constants of the B19′ martensite of binary NiTi alloys from X-ray diffraction (XRD) analysis. They found that the monoclinic angle decreased from 97.9° to 96.4° when increasing the Ni content from 50.0 at. % to 50.7 at. %. Similarly, Zarinejad et al. [7] studied the crystal structures of the B19′ martensite of NiTiHf SMAs by means of XRD analysis and found that the monoclinic angle of pseudo-equiatomic Ni(Ti50−xHfx) alloys increased with the increase of Hf content, from 98.0° in Ni50Ti45Hf5 to 100.1° in Ni50Ti30Hf20. It was also reported that increasing the Cu content in pseudo-equiatomic Ti(Ni50−xCux) system decreased the monoclinic angle of B19′ from 96.2° at 5 at. % Cu to 95.2° at 10 at. % Cu [8]. Ahadi et al. [9] measured the monoclinic angle of binary NiTi by means of in-situ neutron diffraction and found that the angle increased by ~ 1° when the temperature was decreased from 300 to 50 K.
In addition to these experimental observations, the monoclinic angle has also been found to change by various factors in theoretical analyses, including density functional theory (DFT) calculations [10], molecular dynamics (MD) simulations [11], and crystallographic analysis [4]. Bakhtiari et al. [10] investigated the change in monoclinic angle and lattice constants of B19″ martensite phase in Ti50Ni50−xCux alloys by means of DFT calculation and showed that the monoclinic angle decreases with increasing Cu content, which is consistent with the experimental observations reported by Nam et al. [8, 12]. Mansouri Tehrani et al. [11] studied the effects of vacancy and anti-site defects on equiatomic NiTi B19′ structure by means of MD simulation and showed that these defects can reduce the monoclinic angle to 92.5°. A similar finding was also reported by Ko [13], who examined the martensite structure by using harmonic approximation within the DFT calculation.
Echoing the findings of the variations of the monoclinic angle of the B19′ phase, the shape memory strain of NiTi-based SMAs has also been reported to vary under various conditions in the literature. For example, pseudoelastic deformation cycling [14‐16] and two-way shape memory [17, 18] thermal cycling have been found to reduce the transformation strain. Increasing the amount of cold work decreases the transformation strain [19, 20]. Increasing the temperature of recovery anneal (prior to recrystallization) after cold working increases the transformation strain [20, 21]. Increasing the level of stress of deformation (e.g., by increasing the testing temperature) increases the pseudoelastic strain [22, 23]. Furthermore, Ni-rich NiTi alloys appear to have smaller transformation strains than near-equiatomic NiTi alloys [6, 24]. It is also common knowledge that third element alloying may change the magnitude of the strain output of the B2 → B19′ transformation. For example, substitution of 5 at. % Cu for Ni in NiTi reduces the B2 → B19′ transformation strain to 3.3% [12], and the maximum recoverable strain of (Ni47Ti44)100−xNbx decreases with increasing Nb content [25]. Saghaian et al. [26] observed that the recoverable strain decreased with increasing Ni content in Ni-rich NiTi-20Hf SMAs.
Whereas there may be many reasonable factors, both intrinsic to the transformation crystallography and extrinsic as embodied within the microstructure, that can change the macroscopic transformation strain, there is no reason to exclude the monoclinic angle of the B19′ phase as a contributing factor. To investigate this, the effects of structural and compositional point defects on the monoclinic angle of the B19′ phase in NiTi-based alloys were investigated in this study by means of DFT calculation. Four types of point defects were studied, including vacancies, Ni–Ti anti-site swapping, third element substitution, and Ni or Ti enrichment in the B2 stoichiometry.
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DFT Calculations
The effects of the four types of point defects on the structure of the B19′ martensite in NiTi was studied by means of DFT calculation. It is known that the monoclinic B19′ phase is observed experimentally but does not exist as a minimum energy state in DFT calculation for the Ni–Ti system, and that the monoclinic B19″ phase is often used as a proxy for the B19′ phase [10]. Given that both B19′ and B19″ are product phases of martensitic transformations of the B2 phase, they have identical atomic position and unit cell configuration except the monoclinic angle. This gives the validity for using B19″ in DFT calculation as a proxy for the experimentally observed B19′. Figure 1a shows a comparison of the unit cells selected for the DFT calculation of the B2, B19′, and B19″ phases. The line drawing presents the B2 structure. The blue dashed lines define the unit cell selected. The three sphere model structures show the selected B2, B19′, and B19″ unit cells. For the calculations, supercells of three different sizes of 2 × 2 × 2 B19″ unit cells with 32 atoms, 3 × 3 × 3 B19″ unit cells with 108 atoms, and 4 × 4 × 4 B19″ unit cells with 256 atoms were used. Figure 1b shows the 2 × 2 × 2 B19″ supercell.
The DFT calculation was performed using the Vienna ab initio Simulations Package (VASP) [27]. The PBE (Perdew–Burke–Ernzerh) exchange–correlation functionals [28] and the projector-augmented wave method (PAW) [29] with a cut-off energy of 500 eV were employed to perform the simulations. The k-point mesh density was at least 50 per Å−1, the convergence criterion of the total energy was 10–6 eV, and the maximum force on each atom was 5 × 10–3 eV Å−1. The simulation temperature was 0 K. Each structure was fully relaxed in all degrees of freedom. After relaxation, if the other two angles (α and γ) of the unit cell are within the range of 90° ± 1°, the structure is considered monoclinic and vice versa.
Results and Discussion
The fully relaxed B19″ unit cell used for DFT calculation has a monoclinic angle of β = 101.6° and lattice constants of a = 2.934 Å, b = 4.022 Å, and c = 4.753 Å. This B19′′ unit cell is then transformed to 2 × 2 × 2 or 3 × 3 × 3 or 4 × 4 × 4 supercells for calculation.
Effects of Vacancy Defects
Figure 1 shows the effect of vacancy defects on the lattice parameters of the B19″ martensite, including the monoclinic angle (β) and cell edge constants. Three types of vacancies were created, including Ni-vacancy, Ti-vacancy, and Pair-vacancy. The Ni vacancies and Ti vacancies were created by randomly removing the respective atoms within a 2 × 2 × 2 supercell (except for 1 at. % when a 3 × 3 × 3 supercell was used). The Pair vacancies were created by randomly removing an equal number of Ni and Ti atoms together each time within a 3 × 3 × 3 supercell. In this case, the two atoms of the pair were treated as two individual atoms for random selection and removal (not as an adjacent pair), and the concentration of the Pair-vacancy is calculated by taking the Ni–Ti pair as two atoms. Naturally, the Ni-vacancy and Ti-vacancy lead to changes in the composition of the alloy, whereas the Pair-vacancy does not alter the alloy composition. Figure 1b shows a 2 × 2 × 2 B19″ supercell in the relaxed state and the random removal of one Ni atom within the supercell to create a 3.1 at. % vacancy concentration as an example.
Fig. 1
Effects of vacancy defects on the lattice parameters of the B19″ phase of binary equiatomic NiTi. a Schematic of the unit cell selected for DFT calculation and comparison of the B2, B19′, and B19″ structures. b Schematic of the 2 × 2 × 2 supercell used for DFT calculation. c Effects of Ni-vacancy, Ti-vacancy, and Pair-vacancy concentrations on the monoclinic angle of the B19″ structure. d Effect of Ni-vacancy concentration on the unit cell edge constants of the B19″ structure. e Effect of Ti-vacancy concentration on the unit cell edge constants of the B19″ structure. f Effect of Pair-vacancy concentration on the unit cell edge constants of the B19″ structure
×
The effects of the three types of vacancies on the B19″ monoclinic angle are shown in Fig. 1c. The monoclinic angle decreases when increasing the concentrations of all three vacancy types. It is apparent that the effect of the Ti-vacancy is the strongest. At > 6 at. % of Ti-vacancy, the monoclinic angle (β) of the B19″ structure is reduced to close to 90°, i.e., the structure is becoming an orthorhombic B19-like structure. The effects of the vacancies on the other two angles of the unit cell, a and g, are not shown. These two angles remain within the range of 90 ± 1° for the entire range of the vacancy concentration studied here. In such case the angles are considered right angles and unchanged. For such a case, no further comment is made in the following presentation, unless otherwise.
An alloy containing a high Ti-vacancy concentration may be considered similar to a Ni-rich alloy. Thus, the effect of Ti vacancies suggests that Ti-depleted (or Ni-enriched) NiTi prefer to transform to B19 than to the more distorted B19″. This is consistent with the experimental observation of the effect of increasing the Ni content on lowering the B2 → B19′ transformation temperature [30] and the occurrence of the pre-martensitic phenomena [31]. The Ni-vacancy (Ti–rich alloys) appears to be the least effective in lowering the monoclinic angle. However, the same trend (in reducing the monoclinic angle) suggests that Ti–rich NiTi alloys, which are prohibited in practice according to the experimental phase diagram, also have lowered B2 → B19′ transformation temperatures relative to the equiatomic NiTi. The effect of the Pair-vacancy is somewhat in between that of the Ti-vacancy and that of the Ni-vacancy.
For the unit cell coordinates used (Fig. 1a), the lattice correspondences of the B2 ↔ B19″ transformation may be expressed as \({[100]}_{{B19}^{{\prime}{\prime}}}-{\left[100\right]}_{B2}\), \({[010]}_{{B19}^{{\prime}{\prime}}}-{\left[011\right]}_{B2}\) and \({[001]}_{{B19}^{{\prime}{\prime}}}-{\left[0\overline{1 }1\right]}_{B2}\) [32, 33], i.e., \({a}_{{B19}^{{\prime}{\prime}}}-{a}_{B2}\) (or \(\sqrt{2}{a}_{{B19}^{{\prime}{\prime}}}-{\sqrt{2}a}_{B2}\)), \({b}_{{B19}^{{\prime}{\prime}}}-{\sqrt{2}a}_{B2}\) and \({c}_{{B19}^{{\prime}{\prime}}}-{\sqrt{2}a}_{B2}\). Thus, condition \(\sqrt{2}{a}_{{B19}^{{\prime}{\prime}}}={b}_{{B19}^{{\prime}{\prime}}}={c}_{{B19}^{{\prime}{\prime}}}\) and \(\beta =90^\circ\) represents the B2 structure. Figure 1d shows the effect of Ni-vacancy concentration on the unit cell edge constants of the B19″ phase. The Ni vacancies showed little influence up to 6 at. %. With further increasing the vacancy concentration, \({c}_{{B19}^{{\prime}{\prime}}}\) reduces towards \({b}_{{B19}^{{\prime}{\prime}}}\) whilst \(\sqrt{2}{a}_{{B19}^{{\prime}{\prime}}}\) increases to be above the other two. Figure 1e shows the effect of Ti-vacancy concentration on the unit cell edge constants of the B19″ phase. Ti-vacancy appears to have more noticeable influence on the lattice constants, with \(\sqrt{2}{a}_{{B19}^{{\prime}{\prime}}}\) and \({c}_{{B19}^{{\prime}{\prime}}}\) decreasing and \({b}_{{B19}^{{\prime}{\prime}}}\) increasing with the increase of the Ti-vacancy concentration. Together with the reduced monoclinic angle, these changes indicate an evolution of a B19-like structure. The influence of the Pair-vacancy concentration is presented in Fig. 1f. In this case, \({c}_{{B19}^{{\prime}{\prime}}}\) decreases rapidly whilst \({b}_{{B19}^{{\prime}{\prime}}}\) and \(\sqrt{2}{a}_{{B19}^{{\prime}{\prime}}}\) remain relatively unchanged with the increase of the vacancy concentration. At about 9.38 at. %, the lattice structure is approaching the B2 structure with practically equal unit cell edges (and close to 90° unit cell angles).
In addition to the lattice parameters, unit cell volume, unit cell energy, and a lattice vector tensile strain between \({\left[1\overline{1 }1\right]}_{B2}\) and the corresponding \({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\) directions, defined as \({\varepsilon }_{\left[111\right]B2/\left[101\right]B1}=\frac{{{L}_{\left[101\right]B19"}-L}_{\left[111\right]B2}}{{L}_{\left[111\right]B2}}\), are also calculated, as presented in Fig. 2. This lattice vector strain is also the largest crystallographic tensile strain of the B2 ↔ B19″ transformation [32]. Figure 2a plots the effects of vacancy concentration on the unit cell volume of the B19″ phase. It is seen that all three types of vacancies cause a progressive decrease of the unit cell volume. This is reasonable to understand because the presence of vacancies causes inwards contraction of the lattice. Amongst the three types of vacancies, Ni-vacancy has the weakest effect, similar to its effects on the monoclinic angle and the lattice constants. Figure 2b shows the effect of vacancy concentration on the unit cell energy of the B19″ phase. The B19″ unit cell energy increases (less negative) with increasing the vacancy concentration for all three vacancies. Figure 2c plots the effect of vacancy concentration on the tensile lattice strain of the B2 → B19″ transformation along the \({\left[1\overline{1 }1\right]}_{B2}\)/\({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\) direction (with aB2=3.015 Å [34]). It is seen that the transformation strain decreases with increasing the concentration of all three types of vacancies, with the effect of the Ti-vacancy being the most potent. This is apparently directly related to the reduction of the monoclinic angle seen in Fig. 1c. It is to be noted that the small negative strain for the pair-vacancy at − 1.35% is caused by the changes of the unit cell edge constants and does not imply an acute monoclinic angle. It is also to be noted that the concentration levels of the vacancies calculated here are much higher than what are possible in actual engineering alloys. Whereas the trends of the effects illustrated may present some principles, caution must be taken when using such principles to interpret actual alloy behaviour or guide designs of alloys.
Fig. 2
Effects of vacancy defects on the B19″ lattice characteristics of equiatomic NiTi. a Effects of Ni-vacancy, Ti-vacancy, and Pair-vacancy concentrations on the unit cell volume. b Effects of Ni-vacancy, Ti-vacancy, and Pair-vacancy concentrations on the unit cell energy. c Effects of Ni-vacancy, Ti-vacancy, and Pair-vacancy concentrations on the tensile lattice strain along \({\left[1\overline{1 }1\right]}_{B2}\)/\({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\) direction for the B2 → B19″ transformation
×
Effects of Anti-site Defects
Anti-site defect was created by swapping the positions of a randomly selected Ni atom and a randomly selected Ti atom in the ordered B19″ structure. Figure 3 shows the effect of anti-site defect on the monoclinic angle of the B19″ phase. Figure 3a shows a B19″ 3 × 3 × 3 supercell containing 108 atoms in relaxed state. Figure 3b shows the supercell when one Ni and one Ti atom are randomly selected and swapped in position to create a pair of anti-site defect. This is registered as 2/108 = 1.85% of the anti-site concentration in the B19″ structure. For the lower anti-site defect concentration of 0.78 at. %, a 4 × 4 × 4 supercell was used. An increase in the anti-site concentration represents a decrease in the degree of ordering of the structure. Figure 3c plots the effect of anti-site concentration on the monoclinic angle of the B19″ phase. Similar to the effects of the vacancies, the monoclinic angle decreases progressively with the increase of the anti-site concentration.
Fig. 3
Effect of anti-site defect on the monoclinic angle of the B19″ structure. a Schematic of a 3 × 3 × 3 supercell of the B19″ lattice in relaxed state. b Schematic of the random swapping of one Ni atom and one Ti atom to create a pair of anti-site defect. c Effect of the anti-site concentration on the monoclinic angle of the B19″ structure
×
Figure 4 presents the effects of anti-site defects on the unit cell edge constants, unit cell volume, unit cell energy, and the transformation strain in the \({\left[1\overline{1 }1\right]}_{B2}\)/\({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\) direction. Figure 4a shows the effect of anti-site concentration on the unit cell edge constants. With the increase of the anti-site concentration, the unit cell edge constants somewhat converge towards one another. The effect of anti-site concentration on the unit cell volume is shown in Fig. 4b. The volume decreases with increasing the anti-site concentration, by 5.5% at 9.26 at. % anti-site concentration. Figure 4c shows the effect on the unit cell energy of the B19″ structure. It is seen that the energy increases (less negative) continuously with increasing the anti-site concentration, implying a continuous lowering of the B2 → B19″ transformation temperature. Figure 4d shows the effect on the transformation strain along \({\left[1\overline{1 }1\right]}_{B2}\)/\({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\). It decreases with increasing the anti-site concentration, following the same trend as the monoclinic angle.
Fig. 4
Effects of anti-site defect on the lattice parameters of the B19″ structure. a Effect of anti-site concentration on the unit cell edge constants. b Effect of anti-site concentration on the unit cell volume. c Effect of anti-site concentration on the unit cell energy. d Effect of anti-site concentration on the tensile transformation strain in the \({\left[1\overline{1 }1\right]}_{B2}\)/\({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\) direction
×
Effects of Substitutional Elements
A third type of point defect investigated is the substitution by a third element for either Ni or Ti in the equiatomic NiTi. A good number of elements have been studied and five elements commonly used in NiTi are presented below, including Cu, Fe, Hf, Nb, and Pt. It is generally understood that none of these five elements have exclusive site occupancy within the equiatomic NiTi lattice but have different levels of site preferences [35, 36].
Figure 5 shows the effect of substitution by Cu, Fe, Hf, Nb, and Pt on the monoclinic angle of the B19″ phase in the equiatomic NiTi. Figure 5a shows their effects of substitution for Ni. Such substitution preserves the (Ni,X)–Ti pseudo-equiatomic stoichiometry. Similarly, Fig. 5b shows their effects of substitution for Ti, which preserves the Ni–(Ti,X) pseudo-equiatomic stoichiometry. It is seen that for Ni substitution, Cu, Fe, Hf, and Nb reduce the monoclinic angle, whereas Pt increases the monoclinic angle. For Ti substitution, Cu, Fe, and Pt reduce the monoclinic angle, whereas Nb and Hf appear to increase slightly the monoclinic angle. This echoes with the findings of the increase of the B19′ monoclinic angle by Hf substitution for Ti in NiTi [37].
Fig. 5
Effects of third element substitution on the monoclinic angle of the B19″ structure. a Effect of third element substitution for Ni. b Effect of third element substitution for Ti
×
Figure 6 shows the effects of third element substitution on the unit cell volume, unit cell energy, and the transformation strain in the \({\left[1\overline{1 }1\right]}_{B2}\)/\({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\) direction. The main observations are:
(1)
Substitution for Ni by Cu, Hf, Nb, and Pt increases the unit cell volume, whereas Fe decreases it, as seen in Fig. 6a. Substitution for Ti by Cu, Fe, and Pt decreases the unit cell volume, whereas Nb and Hf increase it, as seen in Fig. 6b. This can be roughly attributed to the difference in atomic radius of the elements relative to those of Ni and Ti. Fe and Cu have the closest atomic sizes to Ni, thus they have the weakest effect on unit cell volume when substituting for Ni. Other elements are larger in atomic size than Ni. Similarly, for Ti substitution, only Hf is larger than Ti. Nb has the closest atomic size to Ti, thus it has the weakest effect on the unit cell volume.
(2)
The unit cell energy is shown in Fig. 6c and d. It is seen that Hf and Nb decrease the unit cell energy for both Ni and Ti substitutions, whereas Cu increases the unit cell energy in both cases. Fe and Pt are moderately negative (decrease) on the unit cell energy when substituting for Ni and mildly positive (increase) it when substituting Ti. Such information may provide some reference for designing NiTi-based SMAs for different transformation temperature.
(3)
Effects of the elements on transformation strain in the \({\left[1\overline{1 }1\right]}_{B2}\)/\({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\) direction are shown in Fig. 6e and f. The dependencies on element concentration for each of the 5 elements follows very closely the same trend as that for the effect on the monoclinic angle shown in Fig. 5. This implies the \({\left[1\overline{1 }1\right]}_{B2}\)/\({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\) transformation strain is largely determined by the monoclinic angle of the B19″ phase.
(4)
One stand-out case is Pt. It causes the opposite effect when substituting for Ni or Ti on all the parameters, including unit cell volume, unit cell energy, transformation strain along \({\left[1\overline{1 }1\right]}_{B2}\)/\({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\), as well as the monoclinic angle. This may be related to the strong preference of Pt for Ni site in the NiTi lattice [38, 39].
Fig. 6
Effects of third element substitution in equiatomic NiTi on unit cell characteristics of the B19″ phase. a and b Effects on the unit cell volume of substitution for Ni and Ti, respectively. c and d Effects on the unit cell energy of substitution for Ni and Ti, respectively. e and f Effects on the transformation strain in the \({\left[1\overline{1 }1\right]}_{B2}\)/\({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\) direction of substitution for Ni and Ti, respectively
×
Effects of Non-stoichiometry Composition of Ni–Ti
The effects of non-stoichiometric compositions for binary NiTi, i.e., Ti–rich and Ni-rich alloys, are also investigated, as presented in Fig. 7. Figure 7a shows the effect of Ni and Ti enrichment on the monoclinic angle of the B19″ phase. It is seen that deviating from the equiatomic B2 stoichiometry on either side decreases the monoclinic angle. These appear to be similar to the effects of the Ni-vacancy and Ti-vacancy seen in Fig. 1c. The effect on unit cell volume is shown in Fig. 7b. The unit cell volume decreases continuously with increasing Ni content in both the Ti–rich and Ni-rich side of the equiatomic stoichiometry, apparently related to the smaller atomic size of Ni relative to Ti. Figure 7c shows the unit cell energy. The energy increases continuously with increasing the Ni content. This is an interesting comparison with the effects of Ni and Ti vacancies shown in Fig. 2b. The increase of the unit cell energy with increasing Ni content on the Ni-rich side seen in Fig. 7c is similar to the effect of increasing Ti-vacancies (Ni-rich) seen in Fig. 2b. However, the decrease of the unit cell energy with increasing the Ti content on the Ti–rich side is opposite to the effect of increasing Ni-vacancies (Ti–rich) seen in Fig. 2b. This demonstrates that the effect of the vacancies is truly due to the vacancy defects and not the associated change of Ni/Ti ratio. The evidence shown in Fig. 7c also suggests that the driving force for B2 → B19′ transformation is lower for Ni-rich alloys, thus lower transformation temperature, but higher for Ti–rich alloys. The former prediction is consistent with experimental observations [40] and the latter cannot be verified because of the composition prohibition on the Ti–rich side for actual alloys. Figure 7d shows the effect on the B2 → B19″ transformation strain along the \({\left[1\overline{1 }1\right]}_{B2}\) direction. It is seen that the transformation strain decreases in both directions for Ti–rich and Ni-rich compositions. This is obviously influenced by the trend of the monoclinic angle. This is consistent with experimental observations for Ni-rich alloys [6, 24].
Fig. 7
Effects of non-stoichiometry composition on the unit cell characteristics of the B19″ phase of binary NiTi. a Effect on the monoclinic angle of the B19″ phase. b Effect on the unit cell volume. c Effect on the unit cell energy. d Effect on the transformation strain in the \({\left[1\overline{1 }1\right]}_{B2}\)/\({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\) direction
×
It should be pointed out that the necessary point defect concentrations required for causing the observed reduction of the monoclinic angle as predicted by the DFT calculation are far higher than what may be possible in real alloys. For example, to reduce the monoclinic angle to 94° requires Pair-vacancy concentration of ~ 9 at% and anti-site concentration of 10 at%.
Another point worth to be reiterated is that the idea that the B19′ monoclinic angle may vary under different thermomechanical or chemical conditions provides a new possibility for the interpretation of the variations of the B2-B19′ transformation strains observed of actual NiTi alloys [6, 12, 19‐26]. A further projection of this concept is the explanation of the much lower-than-expectation Young’s modulus of B19′ martensite [23, 41], though it is not analysed or discussed in this work. For example, one possible explanation of the low apparent Young’s modulus and the apparent non-linearity of the unloading curve preceding the onset of the reverse Lüders band stress plateau of pseudoelasticity [23] could be due to continued “elastic” reduction of the monoclinic angle of the stress-induced B19′ martensite upon unloading. The same may also be applied to the continuous increase of the recoverable strain upon deformation in “stage III” beyond the end of the forward Lüders band stress plateau [42, 43], during which the monoclinic angle continues to increase with the increase of stress [44].
Conclusion
The DFT calculations and the TEM analysis on the lattice structure of the B19″/B19′ martensite in NiTi presented above allow the following main conclusions to be reached:
(1)
Vacancy defects, including Ni-vacancy, Ti-vacancy, and Ni/Ti pair-vacancy, in binary equiatomic NiTi decrease the monoclinic angle of the B19″ phase, decrease the unit cell volume, and increase the unit cell energy. They also reduce the lattice distortion strain of the B2 → B19″ transformation along the \({\left[1\overline{1 }1\right]}_{B2}\)/\({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\) direction. Ti-vacancy appears to be the most potent in its effects on these parameters and Ni-vacancy appears to be the weakest amongst the three defect types.
(2)
The anti-site defects formed by swapping Ni and Ti atoms decrease the B19″ monoclinic angle, the unit cell volume, and the lattice distortion strain of the transformation along \({\left[1\overline{1 }1\right]}_{B2}\)/\({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\) direction. The unit cell energy increases with increasing the defect concentration.
(3)
Third element substitution in NiTi alters the monoclinic angle of the B19″ phase. For Ni substitution, Cu, Hf, Fe, and Nb decrease the monoclinic angle, whereas Pt increases the monoclinic angle. For Ti substitution, Cu, Fe, and Pt decrease the monoclinic angle, whereas Nb and Hf have very week effects on the monoclinic angle. The effect on the unit cell volume roughly follows the expectation for the size change of the substitutional atoms relative to the host atom it replaces. A substitutional element may increase or decrease the unit cell energy. The origin for such influences of the elements is not investigated in this study. The effect on the lattice distortion strain of the transformation along the \({\left[1\overline{1 }1\right]}_{B2}/{\left[101\right]}_{{B19}^{{\prime}{\prime}}}\) direction generally follows the effect on the monoclinic angle for each element, and the exceptions observed are clearly related to the changes of the lattice unit cell edge constants.
(4)
Deviation from the B2 stoichiometry on both sides (Ni-rich and Ti–rich) reduces the monoclinic angle of the B19″ phase and the lattice distortion strain of the transformation along \({\left[1\overline{1 }1\right]}_{B2}\)/\({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\) direction. Increasing the Ti content increases the unit cell volume, whereas increasing the Ni content decreases the volume. This is apparently related to relative atomic sizes of the two elements. The unit cell energy increases monotonically with increasing the Ni content for both Ni-rich and Ti–rich compositions.
(5)
The effect of the point defects on the lattice distortion strain along the \({\left[1\overline{1 }1\right]}_{B2}\)/\({\left[101\right]}_{{B19}^{{\prime}{\prime}}}\) direction largely follow the same trend as the effect on the monoclinic angle, though some influences of the lattice unit cell edge constants may also exist. The variation of the B19′ monoclinic angle has direct implications to the interpretation of the observed variations of B2 → B19′ transformation strain of polycrystalline NiTi alloys.
Acknowledgements
The authors acknowledge the financial support from the Australian Research Council in grants DP180101744 and DP180101955 for this study. The authors acknowledge the facilities and the scientific and technical assistance of Microscopy Australia at the Centre for Microscopy, Characterisation & Analysis, The University of Western Australia, a facility funded by the University and the State and Commonwealth Governments. This work was supported by resources provided by The Pawsey Supercomputing Centre (project pawsey0393) with funding from the Australian Government and the Government of Western Australia.
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