First-Principle Study on Tailoring the Martensitic Transformation of B2 Nb_50−xTi_xRu₅₀ Shape-Memory Alloy for Structural Applications

Abstract

NbRu has a potential as a high-temperature shape-memory alloy (HTSMA) because it has a martensitic transformation temperature above 1000 °C. However, its shape-memory properties could be improved for consideration in the aerospace and automotive industry. The unsatisfactory shape-memory properties could be associated with the presence of a brittle tetragonal L1₀ martensitic phase. Therefore, in an attempt to modify the transformation path from B2→L1₀ in preference of either B2→orthorhombic or B2→monoclinic (MCL), an addition of B2 phase stabiliser, titanium (Ti), has been considered in this study to partially substitute niobium (Nb) atoms. The ab initio calculations have been conducted to investigate the effect of Ti addition on the thermodynamic, elastic, and electronic properties of the Nb_50−xTi_xRu₅₀ in B2 and L1₀ phases. The results showed that the B2 and L1₀ phases had comparable stability with increasing Ti content. The simulated data presented here was sufficient for the selection of suitable compositions that would allow the L1₀ phase to be engineered out. The said composition was identified within 15–30 at.% Ti. These compositions have a potential to be considered when designing alloys for structural application at high temperatures above 200 °C.

1. Introduction

Near-equiatomic NbRu compositions exhibit a cubic B2 phase structure above 1000 °C. During cooling below this temperature, the B2 structure transforms to a tetragonal L1₀ phase at ~1100 °C. Further cooling results in another phase transition from L1₀ to an MCL phase, starting at ~880 °C [1]. This high temperature of martensitic transformation renders this alloy a potential HTSMA. However, the shape-memory properties of NbRu are poor compared to those of the well-known TiNi [2,3,4,5]. Fonda et al. [1] investigated the shape-memory effect of NbRu by performing a bend test at 800 °C, which is in the L1₀ temperature region. It was observed that the test specimens were deformed to a greater extent, leading to cracking. Reheating the sample in the B2 region resulted in a fraction of cracks disappearing, which indicated the potential of this alloy to undergo shape recovery. However, the recovery ratio was very low. The cracking of the specimen is thought to have been due to one of the phases being brittle, thus leading to failure during bending. The recovery ratio of the Nb_50−xRu_50+x increased from 50% for Nb₅₀Ru₅₀ [1,6] to 88.9% for Nb₄₇Ru₅₃ [7], indicating the effect of the reduction of Nb.

The available phase diagram for NbRu shows that a decrease in Nb near-equiatomic compositions leads to promotion of MCL formation at a martensitic transformation temperature closer to 900 °C [1]. The MCL extends between 45–53 at.% Nb region on the phase diagram. Thus, it can be deduced that the previously observed recovery improvement upon the reduction of Nb may be attributed to the B2↔MCL reverse martensitic transformation. A similar martensitic transformation path is observed in TiNi, which is known to have a 100% recovery ratio. The understanding of the martensitic transformation path is important in the study of shape-memory properties. Since the NbRu near-equiatomic compositions involve the B2→L1₀→MCL, it seems the formation of a brittle L1₀ phase which forms easily due to slight tetragonal distortions of the B2 phase is detrimental to the shape-memory properties of this alloy. Therefore, it is of interest to study ways to avoid the formation of the tetragonal phase. At the moment, the promising way to achieve this is by aiming to reduce the Nb content [8,9].

Several studies have been dedicated to improving the shape-memory properties of the NbRu alloy. Chang-Long et al. [10] reported the thermodynamic properties of Nb_50−xRu_50+x using the first-principles method. It was noted that the B2 phase became less stable with decreasing Nb content below 50 at.%, indicating a decrease in the B2→L1₀ martensitic transformation temperature [10]. Additionally, Chang-Long et al. [11] further studied the impact of partially replacing Nb with Fe on the stability of B2 Nb_50−xFe_xRu₅₀. The calculated heats of formation (HOF) showed the instability of the B2 phase as Fe was increased (decreasing Nb) [9]. In both instances, the results indicated that the dissolution of B2 into other phases decreases sharply as Nb is increased until a composition of 58 at.% is reached. Beyond this point, B2 becomes stable at room temperature [1]. However decreasing Nb below 54 at.% shifts the transformation path towards the B2→L1₀→MCL region, where the B2 phase is prone to atomic displacements and transformation. For example, in the Al-containing NbRu alloys with less than 50 at.% Nb, the Nb_34.99Ru_16.03Al_48.98 resulted in the formation of a hexagonal C14 phase which was more stable than a B2 phase with an HOF value of $-$0.613 eV/atom. The Nb_22.93Ru_52.42Al_24.59 composition had an L2₁ cubic structure with an HOF of $-$0.666 eV/atom [12]. This proves that the reduction of Nb and it’s substitution with a ternary element can supress the B2→L1₀ to below the critical temperature where a different phase is formed.

Although SME in NbRu alloys is associated with high-temperature L1₀→B2 phase transition, the tetragonal L1₀ martensitic phase has been found to have a high hardness of 715 HV. Similarly, the L1₀ phase is considered brittle in Co-Ni-Al alloy compared to the β phase and γ-phase, with 632 HV and 428 HV, respectively [13]. Moreover, the presence of a tetragonal L1₀ phase in Co-Ni-Ga alloys has been observed to decrease the compressive strain which leads to poor shape-memory properties and workability [14]. Other alloys, such as Ni-Tb-Mn-Ga [15], Ni-Mn-In [16], and (Ce,Y)-tetragonal Zr polycrystals [17], which contain tetragonal L1₀ martensite, have been found to have high hardness and elastic modulus, poor fracture toughness, and poor shape-memory effect.

The purpose of this study is, therefore, to suppress the formation of the L1₀ phase by substituting Nb with Ti. Most recently, ab initio calculations were used to generate elasticity data and showed that substituting Nb with Ti led to the predicted suppression of B2→L1₀, where the stability of the B2 phase was found to compete with the L1₀, which brought about the possibility that Ti addition may promote either B2→orthorhombic or B2→MCL [18]. In this work, the Nb atoms have been partially substituted by Ti and the effect thereof on the electronic properties of B2 and L1₀ NbRu phases is explored. Furthermore, the thermodynamic and elastic properties of the L1₀ phase have been investigated to establish whether the reduction of Nb composition by the addition of Ti can suppress B2→L1₀ in favour of either a B2→orthorhombic or B2→MCL transformation path.

2. Materials and Methods

All calculations in this study were performed using density functional theory (DFT)-based CASTEP code embedded in the Materials Studio 2020 software package [19]. Robust Vanderbilt ultrasoft pseudopotentials [20] were used to describe the ion–electron interaction within the generalised gradient approximation (GGA) [21] of Perdew–Burke–Ernzerhof (PBE) [22]. The crystal structures of a unit cell and 2 × 2 × 2 supercell of B2 phase with space group no. 221 Pm3m, consisting of 2 and 16 atoms, respectively, were built. Since the L1₀ phase results from slight tetragonal distortions of B2 phase, the crystal structure does not significantly differ visually; thus, the crystal models shown in Figure 1 represent both B2 and L1₀ phases of NbRu and Nb_50−xTi_xRu₅₀. Convergence tests were conducted, and the system was found to converge at the cutoff energy of 700 eV and a k-point set of 16 × 16 × 16 and 8 × 8 × 8 for the unit cell and supercell, respectively. The effect of substituting Nb atoms with Ti atoms on the electronic properties was determined for the B2 phase to complement our previous work on the thermodynamic and elastic properties of the B2 structure [18]. The ternary compositions considered were of stoichiometry Nb_8−xTi_xRu₈ (representing Nb_50−xTi_xRu₅₀, spanning the entire composition range). In addition, the structural, thermodynamic, elastic, and electronic properties of the tetragonal L1₀ phase with space group no. 123 P4mmm (with similar compositions to those of B2) were calculated. The L1₀ systems were found to converge with an energy cutoff of 700 eV and the k-point set of 9 × 9 × 8 and 19 × 19 × 13 for a 2 × 2 × 2 supercell comprising 16 atoms and a unit cell, respectively. The lattice constants were calculated for a small strain of 0.004. All ground-state structures were optimised using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization scheme. The convergence criterion was set to the limit of 1 × 10⁻⁵ eV/atom. The maximum residual forces of 0.03 eV/Å, the maximum residual bulk stress of 0.05 GPa, and the maximum atomic displacement of 1 × 10⁻³ Å were utilised. Lattice dynamic properties such as phonon dispersion were calculated using a supercell defined by a cutoff radius of 3.5 Å employing the finite displacement supercell method within the code [23].

3. Results and Discussion

3.1. Lattice Parameters

Table 1. Calculated lattice parameters for B2 and L1₀ phases of Nb_50−xTi_xRu₅₀ (Å).

Alloy	B2 [18]	L10
	a	a = b	c	c/a
Nb50Ru50	3.170 * 3.176 [24] 3.174 [10] 3.172 [27] 3.164 [25]	3.059 * 3.106 [24] 3.078 [6] 3.062 [25]	3.419 * 3.307 [24] 3.385 [6] 3.431 [25]	1.118 * 1.065 [24] 1.100 [6] 1.121 [25]
Nb43.75Ti6.25Ru50	3.159	3.066	3.364	1.097
Nb37.50Ti12.50Ru50	3.150	3.094	3.260	1.054
Nb31.25Ti18.75Ru50	3.138	3.099	3.219	1.039
Nb25Ti25Ru50	3.127	3.123	3.128	1.002
Nb18.75Ti31.25Ru50	3.115	3.106	3.127	1.007
Nb12.50Ti37.50Ru50	3.104	3.102	3.105	1.001
Nb6.25Ti43.75Ru50	3.092	3.091	3.092	1.000
Ti50Ru50	3.080	3.080	3.081	1.000

* experimental.

shows the previously calculated lattice parameters of the B2 phase [18] and the L1₀ phase. The values for lattice parameter a decrease with decreasing Nb content on the B2 phase, owing to Ti having a slightly smaller atomic radius than Nb and due to Nb having a larger thermal expansion compared to Ti. On the other hand, the lattice parameter a = b value for L1₀ seems to increase with decreasing Nb content, reaches a maximum for Nb₂₅Ti₂₅Ru₅₀, and decreases further for the entire compositions, while the lattice parameter c decreases with increasing Ti content. The calculated lattice parameters for L1₀ deviate from the available experimental values by 1.54% percentage error for a = b and by 3.56% for c [24]. These error margins are very likely to be accepted for computing crystal structure properties because they are less than 5%. The difference in the calculated values of lattice parameters a and c between those in the current study, the experimental value [24], theoretical value [25],and those reported in [6] may be attributed to the generated convergence parameters. This effect of critically determining the convergence parameters has been discussed in our previous work [26], where poor selection of parameters may result in reporting of erroneous data because the structures are not fully converged and, thus, not fully optimised. The transformation from B2 to L1₀ is a result of Nb atoms shifting to preferentially occupy [1/2, 1/2, 1] octagonal sites along the c-axis; hence, the shift from ordered body-centred cubic (B2) lattice to body-centred tetragonal (bct) lattice with a tetragonal ratio c/a > 1 is observed. The calculated c/a ratio is presented in Table 1 below. The c/a ratio approaches 1 upon Ti addition, indicating that the B2 phase becomes more favourable than the L1₀ phase.

3.2. Thermodynamic Stability

The thermodynamic stability for the B2 and L1₀ phases was determined by calculating the enthalpy (heat) of formation. The heat of formation (∆H_f) is the amount of energy either absorbed or released during the formation of a compound and is calculated using the expression shown in Equation (1) for a constant pressure at T = 0 K. The more negative value of ΔH_f suggests that the structural phase is more thermodynamically stable. The heats of formation are presented in a graphical form in Figure 2 as a function of Ti content (x).

$$∆H_f=E_{\mathit{Total}}-{\displaystyle \sum }_i{x}_i{E}_i$$

(1)

where $E_{\mathit{Total}}$ represents the total energy of the system, $E_i$ is the total energies of the element i, and $x_i$ is the concentration of element i in the compound in their respective ground-state structures.

As shown in Figure 2, the B2 phase is less thermodynamically stable at higher concentrations of Nb compared to L1₀. These results are in agreement with the available binary phase diagram of the NbRu which indicates that at 45 at.% Nb the B2→L1₀ phase transition is more thermodynamically favourable with a decrease in temperature. It is observed here that the addition of Ti increases the thermodynamic stability of the B2 phase, as demonstrated by the ${∆H}_f^{{B2\text{-}L1}_0}$ curve, with the most favourable composition to suppress the B2→L1₀ transformation being between 15 and 30 at.% Ti and the highest suppression expected at 25 at.% Ti. Beyond this composition range, the ∆H_f for both phases begins to equalize. ${∆H}_f^{{B2\text{-}L1}_0}=0$ indicates that both phases have the same chance of forming. The current heat of formation data indicates that to avoid the formation of L1₀, Nb content must be decreased by Ti addition to stabilise B2 to about 900 °C, where the transformation to MCL starts to occur [1]. However, to identify a suitable composition for suppression of L1₀ formation without compromising the martensitic transformation also requires consideration of elastic and electronic properties.

3.3. Elastic Properties

The mechanical stability condition of a lattice implies that the energy change ∆E~$V{C}_{ij{e}_i{e}_j}$, upon any small deformation, is positive. The mechanical stability criteria for the B2 phase were reported in our previous work [18]. In the cubic B2 lattice symmetry, there are three independent constants, which are C₁₁, C₁₂ and C₄₄, of which the stability criteria are defined as C₁₁ $-$ C₁₂ > 0 and C₄₄ > 0. In a tetragonal L1₀ lattice symmetry, there are six independent elastic stiffness constants: C₁₁, C₁₂, C₁₃, C₃₃, C₄₄, and C₆₆. Therefore, the mechanical stability criteria for a tetragonal crystal are as follows: C₁₁ > |C₁₂|, C₃₃ > 0, C₄₄ > 0, C₆₆ > 0, (C₁₁ + C₃₃ $-$ 2C₁₃) > 0 (denoted by C″), (2C₁₁ + C₃₃ $-$ 2C₁₂ + 4C₁₃) > 0 (denoted by C′′′) [28].

Table 2. Elastic constants of B2 and L1₀ phases of Nb_50−xTi_xRu₅₀ (GPa).

Alloy	B2 Elastic Constants [18]				L10 Elastic Constants
	C11	C12	C44	C′	C11	C12	C44	C33	C66	C13	C′	C″	C‴
Nb50Ru50	126	287	69	−81	328	204	46	341	58	178	103	310	1412
Nb43.75Ti6.25Ru50	147	271	71	−62	363	150	68	323	64	177	107	333	1457
Nb37.50Ti12.50Ru50	196	244	75	−24	318	177	73	245	68	212	70	138	1376
Nb31.25Ti18.75Ru50	238	217	78	11	331	151	77	242	73	203	90	167	1414
Nb25Ti25Ru50	274	178	80	48	296	195	83	290	83	196	50	195	1274
Nb18.75Ti31.25Ru50	342	160	84	91	361	165	85	348	84	171	100	351	1429
Nb12.50Ti37.50Ru50	396	133	87	132	398	130	87	405	88	132	133	521	1485
Nb6.25Ti43.75Ru50	396	119	86	139	413	129	89	420	87	130	142	574	1506
Ti50Ru50	421	114	88	154	418	121	86	420	87	118	149	602	1488

shows the calculated elastic constants for the B2 and L1₀ phases. It is observed that the B2 phase meets the mechanical stability criteria, except for the first three Nb-rich compositions. On the other hand, the L1₀ phase shows mechanical stability and non-linearity as Ti content is increased, but the elastic constants are always positive.

To compare the mechanical stability of both phases, the tetragonal shear modulus C′ for both phases has been calculated using the expression in Equation (2), and the values are presented in Table 2 and graphically depicted in Figure 3. The L1₀ phase has a more positive tetragonal shear modulus than the B2 phase, which means it is more mechanically stable than the B2 phase for most compositions but has lower C′ values at 12.50 and 25 at.% Ti compositions. The decrease in C′ for L1₀ may occur because at these compositions, the preferred structure is B2; thus, a reduction in mechanical stability takes place.

Previously [18], we indicated that the suitable composition for further investigation of B2 alloys is where C′→0, which is between 12.50 and 18.75 at.% Ti. It is believed that the stress required to induce martensite at these compositions is small, and thus, the shape-memory properties could be improved. On the other hand, the current results show that the C′ of the L1₀ phase remains positive but decreases with an increase in Ti composition, reaching a minimum at 25 at.% Ti. This further validates the claims that the limited shape-memory properties observed in NbRu arise due to the presence of this mechanically stable and brittle L1₀ phase which seems to compromise the possibility of further transition to the MCL phase.

$$C^{\prime }={\displaystyle \frac{C_{11}-C_{12}}{2}}$$

(2)

3.4. Electronic Properties

The electronic origin of the composition dependence on phase stability has been investigated and presented for all compositions in Figure 4, which shows the calculated total density of states (tDOS) of electrons for the B2 and L1₀ phases of Nb_50−xTi_xRu₅₀ compositions. The tDOS can be used to predict the phase stability of a compound by observing the behaviour of energy states near the Fermi level (E $-$ E_f = 0) concerning pseudogap. The Fermi level of the most stable structure can be indicated by the pseudogap being located in the deep valley, whereas the least stable structure has a Fermi level that lies at the shoulder or the peak. Also, if the DOS profiles and the Fermi level values for different structural phases are very similar due to the small energy difference, it means that there is a possibility for the co-existence of both phases [29].

The DOS around the Fermi level of Nb_50−xTi_xRu₅₀ is mainly attributed to d-electrons of the constituent elements, with a small contribution from the s-orbital states. Figure 4a shows the B2 phase of Nb_50−xTi_xRu₅₀. The Fermi level cuts the DOS curve at the shoulder, and the Fermi level seems to shift towards the centre of the pseudogap with increasing Ti content, indicating possible hybridisation between electronic d-orbitals. These d-d interactions are known to form tight bonds and thus exert strong d-d electron interactions; hence, the electrons are transferred to a lower energy range and a pseudogap is produced. This is why the B2 phase becomes more stable as Ti content increases. This indicates that the high symmetry B2 phase of Nb-rich alloys cannot maintain the ordered structure at low temperatures; thus, they readily undergo martensitic transformation to the L1₀ phase.

On the other hand, Figure 4b shows the tDOS for the L1₀ Nb_50−xTi_xRu₅₀. Similar to the B2 phase, the stability of the L1₀ phase seems to improve with increasing Ti content. This agrees well with the experimental data, as Nb₅₀Ru₅₀ has been found to have a high transformation temperature, meaning that it undergoes structural reconfiguration from B2→L1₀. For the Ti-rich systems, it appears that the L1₀ phase follows similar DOS profiles and Fermi level values as those of B2 structures, implying that both phases are likely to occur at T = 0 K, as was earlier demonstrated by the calculated heats of formation.

3.5. Lattice Dynamics

Figure 5a,b show the analysis of vibrational properties of the B2 and L1₀ phases of Nb_50−xTi_xRu₅₀ as a function of Ti concerning the phonon dispersions at 0K. For B2 Nb_50−xTi_xRu₅₀ shown in Figure 5a, the lattice vibrations revealed negative imaginary frequencies along X-R, R-M, G-M, and G-R high-symmetry directions of the Brillouin zone (BZ). Negative frequencies indicate that the B2 phase of NbRu is vibrationally unstable at 0K, confirming that it is a high-temperature phase likely to martensitically transform to another phase upon cooling. The atomic vibrations are stronger along the G-R direction, corresponding to the [111] direction on the lattice, which is predicted to be responsible for the formation of the trigonal R3M phase with space group no. 160, whilst the negative frequencies along the R-M direction are associated with the formation of the tetragonal phase. As the Ti content is increased, the vibrations along the R direction intensify until Nb_31.25Ti_12.50Ru₅₀ but weaken afterwards. These vibrations correspond to the [111] direction of the lattice, which is responsible for the transformation of B2 to an MCL phase variant. The soft modes along G-R disappear as Ti is increased towards 25 at.%, where only positive frequencies are observed, indicating that the B2 structure becomes more stable as Ti is increased.

Similarly, the vibrational stability of the L1₀ phase of NbRu showed negative frequencies along the Z-A, G-Z, Z-R, and R-X directions of the Brillouin zone, representing [110], [001], and [010] respective directions of the L1₀ lattice structure. These vibrations are associated with the formation of a different variant of tetragonal structure with a space group no. 99 (P4MM) along [001], whilst the vibrations along the [110] direction are associated with the transformation to orthorhombic symmetry with space group no. 38 (AMM2); for the Z-R direction, the vibrations along the [010] direction of the lattice lead to the formation of an orthorhombic phase with symmetry space group no. 25 (PMM2). The displayed frequencies are indicative of the further transformation of the L1₀ phase. As the Ti content increases, the negative frequencies are intensified along the A-M and M-G directions, which corresponds to the [001] and [110] direction on the L1₀ lattice structure. Atomic displacement along these directions indicates that the L1₀ symmetry is maintained or the transformation to an orthorhombic phase is likely to occur as Ti is increased. The negative frequencies disappear on the Nb₂₅Ti₂₅Ru₅₀, meaning that the L1₀ phase becomes stable with no possibility of further martensitic transformation. Although L1₀→MCL has been reported previously [24,25], there are still debates about the actual structure of the low-temperature phase of NbRu regarding whether it is orthorhombic or monoclinic. However, this is beyond the scope of the current paper. The results shown on the dispersion curves, therefore, predict that the stability of both the B2 and L1₀ phases is almost the same within the considered compositions. However, the composition where the vibrations along the [111] direction are intensified corresponds to the transformation path alteration to possible B2→orthorhombic or B2→MCL transformations that are of interest, i.e., between 12.50 at.% and 18.75 at.% Ti.

4. Potential Applications of Ti Containing NbRu Alloys

It has been established that NbRu exhibits high martensitic temperature above 1000 °C. However, the martensite formed at this temperature is brittle. In a quest to improve its mechanical properties, Ti has been considered because it is lighter and has good corrosion resistance, making it suitable for structural applications above 200 °C. There is a threshold with adding Ti because it decreases the martensitic transformation to below 1000 °C. The engineering-out of the brittle L1₀ phase entails that the new transformation temperature will be below 800 °C. At this temperature, Ti-containing NbRu will still be considered the HTSMA. Therefore, the proposed alloy may find application in designing materials with high damping capacity for turbomachinery rotor blades operating at temperatures around 300 °C, with a high structural loss factor of at least 0.18 [30]. Furthermore, the alloy could be used for designing actuating materials in aeronautics, such as aircraft wing box structure reinforcements and fuselage structures [31,32], and in aerospace propulsion, with transformation temperatures ranging between 200–1000 °C [32,33,34].

5. Conclusions

The structural, thermodynamic, elastic, and electronic properties of the B2 and L1₀ phases have been investigated and reported in this study. This study aimed to predict a composition range within which potential HTSMA NbRu shape-memory properties can be improved by avoiding the B2→L1₀ and facilitating either the B2→orthorhombic or B2→MCL martensitic transformation. This was achieved by the systematic addition of Ti as a substitute at the Nb sites. The DFT method was then used to generate thermodynamic and mechanical stability data for both B2 and L1₀. The results from calculated HOF and elasticity data indicated that the addition of Ti improves both stabilities of the B2 and L1₀ phases. Thus, properties investigated in this work suggest that Ti compositions between 15 and 25 at.% are worth exploring experimentally in an attempt to improve the shape-memory properties of B2 NbRu for HTSMA applications. This work further predicts that it is possible to suppress B2→L1₀ in favour of either a B2→orthorhombic or B2→MCL martensitic transformation path by substitutional replacement of Nb atoms by Ti atoms. The likelihood of the latter will be considered in future work.