Design and Development of B2 Ti₅₀Pd_50−xM_x (M = Os, Ru, Co) as Potential High-Temperature Shape Memory Alloys

Abstract

In this chapter, the stability and phase transformation of B2 Ti₅₀Pd_50-xM_x (M = Os, Ru, Co) alloys are investigated using density functional theory. TiPd alloy can be suitable for high-temperature shape memory applications due to its martensitic transformation capability from B2 to B19 at 823 K. It was reported that the binary Ti₅₀Pd₅₀ alloy is mechanically unstable at 0 K. A partial substitution of Pd with Os, Ru, or Co is investigated to determine which alloy will have the best properties. The heat of formation, density of states, and mechanical properties were determined to check the stability. The heat of formation was found to decrease with an increase in Ru and Os concentrations (condition of stability), consistent with the density of states trend. This is in contrast to Co addition, which indicates that the thermodynamic stability is not enhanced (heat of formation increases). It was also found that an increase in Os, Ru, and Co content stabilizes the Ti₅₀Pd₅₀ with a positive elastic shear modulus ($C^{\prime }$>0) above 18.25, 20, and 31 at.%, respectively. This book chapter will provide valuable insights to guide experiments in the design and development of alloys.

1. Introduction

Shape-memory alloys (SMAs) are known as smart materials which exhibit unusual elastic and mechanical behaviors, such as shape memory effect and superelasticity [1]. These materials are capable of restoring their shape after being excessively deformed at low or high temperatures. SMAs undergo a reversible martensitic phase transformation upon the influence of temperature or stress field, giving rise to the shape memory effect and superelasticity [1]. Recently, SMAs are being developed to suit many applications in many fields, especially for engineering properties. The most commonly known SMAs include nickel-titanium (NiTi), nickel-titanium copper (NiTiCu)), and many other metallic alloy systems where the application temperatures do not exceed 373 K [2]. There is an increasing demand for high-temperature shape memory alloys (HTSMAs) in actuators for automobile, pipe couplings and aircraft engines, and other applications. Recently, new alloy systems which can work above that temperature have been investigated such as TiPt, TiAu, and TiPd binary systems as possible new HTSMA materials [3]. As their transformation temperatures are above 800 K and have martensitic transformations from B2 in the austenite phase to B19 in the martensite phase [4].

Ti₅₀Pd₅₀ systems are considered as one of the potential high-temperature shape memory alloys (HTSMA’s) due to their high martensitic transformation temperature [5, 6]. The Ti₅₀Pd₅₀ has two stable phases—the high-temperature phase called austenite and the low-temperature phase called martensite [7]. The Ti₅₀Pd₅₀ has a simple cubic CsCl-type structure (cP2, B2) at high temperatures [8], and an orthorhombic AuCd-type structure (oP4, B19) at ambient temperatures [9, 10]. Besides being lightweight and oxidation resistant, these alloys are also ductile at 823 K [11, 12]. Previously, B2 Ti₅₀Pd₅₀ was reported to be mechanically unstable due to negative elastic shear modulus [4, 11, 13, 14]. Furthermore, it was also reported that the strength of these alloys drop above 823 K. As a result, there is a need to establish ternary alloys to improve the properties of the binary alloy Ti₅₀Pd₅₀ that can be used for actuators and the aeronautics industry [15, 16]. The collapse is due to a possible phase transformation from body-centered cubic (bcc) to other tetragonal and orthorhombic phases, such as L1₀ and B19, similar to those reported in TiPt alloy [17]. Nonetheless, their transformation behavior has not been ascertained explicitly.

In order to enhance the transformation temperature and performance of the Ti₅₀Pd₅₀ SMA’s ternary alloying has been suggested. The elements such as Au, Ni, Ru, Rh, Ir, Pt, Zn, Rc, Tc, Os, and Co were reported as the best site preference for both Ti and Pd substitution with less than 50 atomic percentage (at. %), while Ag and Cd prefer the Ti substitution site in B2 Ti₅₀Pd₅₀ structure [18]. A high work output was shown and good workability was demonstrated by the addition of the third elements such as Ru, Ir, Pt, Co, and Ni. [19]. Previously, the addition of Ni to Ti₅₀Pd₅₀ has shown improvement in shape memory characteristics for Ti₅₀Pd₃₀Ni₂₀ composition [11]. Furthermore, another study was conducted on the ternary alloying of TiPd with Ru addition using density functional theory (DFT)) [20]. The findings showed that the mechanical stability of Ti₅₀Pd₅₀ is enhanced above 25 at. % Ru. A cluster expansion approach was also used in our other study to investigate the phase stability of TiPd_1-xRu_x and Ti_1-xPdRu_x shape memory alloys [21]. As a result of the research, it was found that Ru prefers Ti-site compared to Pd-site, which showed better thermodynamic stability. In a previous study [22], DFTB+ code was used to develop a set of potential parameters of Ti₅₀Pd_50-xRu_x shape memory alloys. A set of potential parameters were developed in order to investigate the stability and transformation temperature of investigated alloys. It was found that the addition of 6.25, 18.75, and 25 at. % Ru on Ti₅₀Pd₅₀ reduce the transformation temperature from B19 to B2 phase.

Cobalt and other PGMs elements have been in use for the production of various components such as vanes or combustion chambers in gas turbines for their exceptional heat-resistant properties [23]. In previous studies, it was discovered that Ru increased the mechanical stability of high-temperature single-crystal superalloys and titanium-based alloys for use in jet engines [24, 25]. Due to their better temperature capabilities and reduced creep rate, these alloys will boost aircraft efficiency and increase the durability of the actuator [24]. Osmium is regularly utilized as an alloying element with other PGMs in order to enhance electricity and stiffness in medical devices and other applications [26]. The ternary alloying with Os, Ru, and Co may improve the stability and the transformation temperature to above 1000 K [27].

In this chapter, the ternary alloying of Ti₅₀Pd₅₀ with Os, Ru, and Co has been performed using the density functional theory approach to investigate the thermodynamic, electronic, and mechanical stability. Furthermore, the effect of Os, Ru, and Co on the ductility/brittleness has been deduced from the anisotropy ratio, which confirms the strength of the systems. The findings will provide very valuable information and practical guidance on the development of these alloys in the future.

2. Computational methodology

2.1 Density functional theory (DFT)

Density functional theory (DFT) is a quantum mechanical theory used in physics and chemistry to study the electronic structure and ground state properties of many-body systems, particularly molecules, atoms, and condensed phase. DFT was first formulated by Hohenberg and Kohn in 1964 [28], then secondly developed by Kohn and Sham in 1965 [29]. Using DFT, independent particle methods have been developed that take into account particle correlations and interactions. The ground state properties of a many-electron system are determined by an electron density that is dependent on the three spatial coordinates as follows:

$$E=E\left[\rho \left(\overrightarrow{r}\right)\right],$$E1

where $E$ is the total energy and $\rho$ is the density.

Kohn and Sham further derived different sets of differential equations which enable the calculation of ground state density ${\rho }_0\left(\overrightarrow{r}\right)$ to be found. The ground state energy of the electronic structure is calculated from the following equation:

$$E\left[\rho \left(\overrightarrow{r}\right)\right]=T_s\left[\rho \left(\overrightarrow{r}\right)\right]+\frac{1}{2}\int \int \frac{\rho \left(\overrightarrow{r}\right)\rho \left({\overrightarrow{r}}^{\text{'}}\right)}{\mid \overrightarrow{r}-{\overrightarrow{r}}^{\text{'}}\mid }d\overrightarrow{r}d{\overrightarrow{r}}^{\text{'}}+E_{\mathit{XC}}\left[\rho \left(\overrightarrow{r}\right)\right]+\int \rho \left(\overrightarrow{r}\right)V_{\mathit{ext}}\left(\overrightarrow{r}\right)d\overrightarrow{r}.$$E2

The kinetic energy of non-interacting electron gas with density $\rho \left(\overrightarrow{r}\right)$ is represented by $T_s\left[\rho \left(\overrightarrow{r}\right)\right]$ as follows:

$$T_s\left[\rho \left(\overrightarrow{r}\right)\right]=-\frac{1}{2}\sum _{i=1}^N\int {\psi }_i^{\ast }\left(\overrightarrow{r}\right){\nabla }^2{\psi }_i\left(\overrightarrow{r}\right)d\overrightarrow{r},$$E3

and Eq. (2) is defined as exchange-correlation energy functional $E_{\mathit{XC}}\left[\rho \right]$. Introducing a normalization constraint on the electron density, $\int \rho \left(\overrightarrow{r}\right)d\overrightarrow{r}=N,$ we get:

$$\frac{\delta }{\mathit{\delta \rho }\left(\overrightarrow{r}\right)}\left[E\left[\rho \left(\overrightarrow{r}\right)\right]-\mu \int \rho \left(\overrightarrow{r}\right)d\overrightarrow{r}\right]=0,$$E4

$$\Rightarrow \frac{\mathit{\delta E}\left[\rho \left(\overrightarrow{r}\right)\right]}{\mathit{\delta \rho }\left(\overrightarrow{r}\right)}=\mu .$$E5

The above equation can be rewritten in terms of an effective potential, $V_{\mathit{eff}}\left(\overrightarrow{r}\right)$:

$$\frac{\delta T_s\left[\rho \left(\overrightarrow{r}\right)\right]}{\mathit{\delta \rho }\left(\overrightarrow{r}\right)}+V_{\mathit{eff}}\left(\overrightarrow{r}\right)=\mu ,$$E6

where

$$V_{\mathit{eff}}\left(\overrightarrow{r}\right)=V_{\mathit{ext}}\left(\overrightarrow{r}\right)+\int \frac{\rho \left({\overrightarrow{r}}^{\text{'}}\right)}{\mid \overrightarrow{r}-{\overrightarrow{r}}^{\text{'}}\mid }d{\overrightarrow{r}}^{\text{'}}+V_{\mathit{XC}}\left(\overrightarrow{r}\right)\mathrm{\text{and}}{V}_{\mathit{XC}}\left(\overrightarrow{r}\right)=\frac{\delta E_{\mathit{XC}}\left[\rho \left(\overrightarrow{r}\right)\right]}{\mathit{\delta \rho }\left(\overrightarrow{r}\right)}$$E7

The V_XC is the exchange-correlation potential, then the one electron Schr$\ddot{\mathrm{O}}$dinger equation takes the form written as follows:

$$\left(-\frac{1}{2}{\nabla }_i^2+V_{\mathit{eff}}\left(\overrightarrow{r}\right)-{\in }_i\right){\psi }_i\left(\overrightarrow{r}\right)=0$$E8

So, solving $\rho \left(\overrightarrow{r}\right)$ we get:

$$\rho \left(\overrightarrow{r}\right)=\sum _{i=1}^N{\left|{\psi }_i\left(\overrightarrow{r}\right)\right|}^2,$$E9

The self-consistent solution is required due to the dependence of $V_{\mathit{eff}}\left(\overrightarrow{r}\right)$ on $\rho \left(\overrightarrow{r}\right)$. Calculations of electronic structures are generally approximated through local density approximations or generalized gradient approximations [30].

2.2 Approximations to exchange-correlation functional

The exchange-correlation functionals used in DFT are categorized into two namely; the local density approximation (LDA) [31] and the generalized gradient approximation (GGA) [32]. These functionals are discussed in sections 2.2.1 and 2.2.2.

2.2.1 Local density approximation

The local density approximation (LDA) is an approximation in which the exchange-correlation (XC) energy function depends upon the value of the electronic density at each point in density functional theory (DFT) [31]. It was first discovered by Kohn and Sham in the context of DFT which can be expressed as:

$$E_{\mathit{XC}}\left[\rho \left(r\right)\right]\cong \int \mathit{dr\rho }\left(r\right){\epsilon }_{\mathit{XC}}\left(\rho \left(r\right)\right),$$E10

where ${\epsilon }_{\mathit{XC}}\left(\rho \right)$ is the exchange-correlation energy per electron in a uniform electron gas of density $n$ [31]. In the uniform electron gas, electrons are distributed in interacting systems with an arbitrary spatial density $\rho$ which acts as a parameter. It has been demonstrated that LDA delivers accurate results even if the electron density in the system is not gradually varied [31]. The function ${\epsilon }_{\mathit{XC}}\left(\rho \right)$ is a combination of exchange and correlation contributions of ${\epsilon }_{\mathit{XC}}\left(\rho \right)$ = ${\epsilon }_X\left(\rho \right)$ +${\epsilon }_C\left(\rho \right)$. It is possible to calculate the exchange energy per particle of a uniform electron gas as follows:

$${\epsilon }_{\mathit{XC}}\left(\rho \right)=c_x{\rho }^{1/3},$$E11

where $c_x=-\left(3/4\right){\left(3/\pi \right)}^{1/3}$.

2.2.2 Generalized gradient approximation

The GGA is known to be semi-local approximation which means that the function does not use the local density $\rho \left(r\right)$ value but its gradient $\nabla \rho \left(r\right)$. Perdew and Wang [32] developed GGA which improves the total energies, atomization energies, energy barriers, and also the difference in structural energies. GGA takes the form:

$$E_{\mathit{XC}}^{\mathit{GGA}}\left[\rho \right]=\int \left(\rho \left(r\right),\nabla \rho \left(r\right)\right)\mathit{dr},$$E12

The spin-independent form is considered in GGA but practically functional is more generally formulated in terms of spin densities $\left(\rho \uparrow ,\rho \downarrow \right)$ and their correspondence gradients of ($\nabla \rho \uparrow ,\nabla \rho \downarrow$).

There are several GGA-based functionals that are the PBE [33], PBEsol [34], RPBE [35], BLYP [36], and AM05 [37]. PBEsol functional is a simple modification of PBE that differs only with two parameters. It is designed to improve the equilibrium properties of bulk solids and their surfaces of PBE in physics and surface science communities. The revised version of the PBE, such as the RPBE functional, is widely used in catalysis to improve the performance of PBE. In the case of AM05, it gives the best performance for applications of catalysis. The GGA-BLYP functional is widely used in the chemistry environment. Other known GGA-based functionals are meta-GGA [38], hyper-GGA, and generalized random-phase approximation. An extension of the GGA, the meta-GGA uses the kinetic energy density and its gradient as inputs to the function and gradient along with the functional density. Hyper-GGA offers an accurate treatment of correlation that goes beyond the level of LDA or GGA when using exact exchange (EXX) to deal with exchange correlation. The generalized random-phase approximation uses EXX and exact partial correlation.

In this chapter, the GGA-PBE [33] functional was used to optimize the Ti₅₀Pd_50-xM_x systems as it provides accurate parameters for this material.

2.3 Computational code and implementation

2.3.1 VASP code

The Vienna Ab initio Simulation Package (VASP) code [39] was used to calculate structural, thermodynamic, electronic, and mechanical properties of ternary Ti₅₀Pd_50-xM_x (M = Ru, Co, Os) alloys. VASP [39] is a computer program for carrying out ab initio quantum mechanical calculations by making use of a plane wave basis along with pseudopotentials or projector-augmented-wave (PAW) [40]. VASP can compute an approximate solution by solving the Kohn-Sham equations within DFT. In VASP, central quantities, such as one-electron orbitals, electron change density, and local potential, are expressed in the form of plane-wave basis sets. The ultra-soft pseudopotentials (US-PP) [41] or the PAW method [40] is used to describe the interactions between the electrons and an ion in VASP. The US-PP method (and the PAW method) are effective in reducing the number of plane waves per atom in transition metals and first-row elements. The code consists of two main loops namely: the outer and inner loop, where the outer loop optimizes the charge density while the inner loop optimizes the wavefunction. VASP code uses a wide range of exchange-correlation functionals such as LDA and GGA as well as Meta- and hyper-GGA and hybrid functionals. All functionals found in VASP have spin-degenerate and also spin-polarized versions.

2.3.2 Implementation

A convergence test was done before calculating any properties, this is to ensure that proper convergence is attained. As such precision was set at “accurate” to minimize errors in the calculation. Importantly, the structures were subjected to full geometry optimization (by allowing both lattice parameters and volume to vary) until the atomic forces were less than 0.01 eV/Å for the unit cell. This was done in order to prepare the structures to be at their ground state energy before determining any properties, such as elastic constants and electronic structures. The effects of exchange-correlation interaction are treated with the generalized gradient approximation (GGA) [29] of Perdew-Burke = Ernzerhof (PBE) [33] and were used with the PAW potential [40]. The strain value of 0.005 was chosen for the deformation of the lattice when calculating elastic properties. A plane-wave cutoff energy of 500 eV and a k-spacing of 0.2 were found to converge the total energy of the systems.

The input structure has a B2 phase with the space group Pm-3 m (as shown in Figure 1a, it is cubic and stable at high temperature. The positions of B2 atoms are denoted by the Pearson symbol cP2 and the prototype is CsCl with all angles being 90°. B2 TiPd experimental observed unit cell parameters are a = b = c = 3.180$\mathring{A}$[4]. In the case of ternary Ti₅₀Pd_50-xM_x, the calculations were carried out using a 2×2×2 supercell with 16 atoms (as shown in Figure 1b). The substitutional search tool embedded within the Medea software platform was used to substitute Pd with Ru, Os, and Co atoms, which provided the most stable compositions.

Figure 1.

(a) The B2 Ti₅₀Pd₅₀ structure with 2 atoms per unit cell and (b) a 2×2×2 supercell with 16 atoms per unit cell structures.

2.4 Theoretical background on calculated properties

2.4.1 Heat of formation

The heat of formation ($∆H_f$) plays a crucial role in determining the thermodynamic stability of the different crystal structures where one can evaluate if the desired alloy can form or not beforehand. It is calculated according to Hess’s law, which states that the standard enthalpy of an overall reaction is the sum of the standard enthalpies of individual reactions into which the reaction may be divided [42]. It provides insight into phase diagram construction and stabilities. The heat of formation is estimated by the following expression:

$$∆H_f=E_C-{\sum }_i{x}_i{E}_i,$$E13

where $E_C$ is the total energy of the respective structure calculated using first principle and $E_i$ is the calculated total energy of the element i in the compound. A structure is considered stable if the value of the heat of formation is negative ($∆H_f$ < 0) otherwise it is unstable. The heat of formation was used to determine the stability trend of Ti₅₀Pd_50-xM_x alloys (M = Ru, Co, Os) as shown in Section 3.1.

2.4.2 Density of states

The density of states (DOS) can be used to predict the electronic stability of metal alloys. It is described by a function, g (E), as the number of electrons per unit volume and energy with electron energies near E. At a specific energy level, a high DOS means many states are open for occupation. In the case of states with DOS of zero, there is no state that can be occupied.

The electrical behavior of a material is determined by the location of E_f within the DOS. Metal alloys’ stability can be predicted using the density of states (DOS). Any material’s electronic density of states can be viewed as a qualitative measure of its electronic structure. DOS is then calculated as the sum of atomic contributions. The DOS is calculated by using the following expression:

$$n\left(\epsilon \right)=2\sum _{n,k}\delta \left(\epsilon -{\epsilon }_n^k\right)=\frac{2}{V_{\mathit{BZ}}}\sum _n\int \delta \left(\epsilon -{\epsilon }_n^k\right)\mathit{dk}$$E14

where $\delta$ is the Dirac delta function and the k is integral extends over the BZ. The number of the electron in the unit cell is given by:

$$\underset{-\infty }{\overset{\mathit{\epsilon f}}{\int }}n\left(\epsilon \right)\mathit{d\epsilon }.$$E15

2.4.3 Mechanical properties

The elastic constants (C_ij) contain information regarding the strength of the materials against an externally applied strain and also act as stability criteria to study structural transformations from ground-state total-energy calculations. For a structure to exist in a stable phase, certain relationships must be observed between the elastic constants. There are various criteria established to deduce the mechanical stability of crystals for different lattice crystals. Accuracy in determining the elasticity of a compound is vital in understanding its mechanical stability and elastic properties. The B2 cubic crystal system has the simplest form of a stiffness matrix, with only 3-independent elastic constants c₁₁, c_12, and c₄₄ [43]. The mechanical stability criteria as outlined in Ref. [43] for the B2 Cubic and its alloyed structures can be expressed as follows:

$$c_{44}>0;c_{11}>c_{12}\text{and}{c}_{11}+2c_{12}>0$$E16

The stability criterion for the elastic constants must be completely satisfied for the structure to be stable. The positive C′ = ((1/2(c₁₁-c₁₂) > 0) indicates the mechanical stability of the crystal, otherwise, it is unstable.

3. Results and discussion

3.1 Structural and thermodynamic properties

In Figure 2, the calculated equilibrium lattice parameters for the B2 Ti₅₀Pd_50-xM_x (M = Os, Ru, Co) systems are shown. It is observed that the partial substitution of Pd with Ru reduces the lattice parameters of the Ti₅₀Pd_50-xRu_x minimally (see Figure 2). This may be attributed to the small atomic radius of Ru as compared to that of Pd.

Figure 2.

The lattice parameter, a (Å) of the B2 Ti₅₀Pd_50-xM_x (M=Os, Ru, Co) (0 ≤ x ≤ 50) ternary SMAs.

Furthermore, the lattice parameters of the Ti₅₀Pd_50-xM_x system decrease as the Co and Os content is increased. This can be understood since the atomic radius of Pd is larger in size than Co, and Os. Recall that the lattice parameter of binary Ti₅₀Pd₅₀ was predicted to be 3.170 Å, which is larger than those calculated for the Ti₅₀Pd_50-xM_x systems.

The heats of formation for the B2 Ti₅₀Pd_50-xM_x systems are shown in Figure 3. As discussed in Section 2.4.1, the heat of formation is calculated to check the thermodynamic stability of the system. We observe that the ∆H_f decreases as Ru is increased this implies that the structure becomes stable at high Ru concentration (thermodynamically stable). Similar behavior was observed for Os as their values decrease with an increase in concentration, indicating thermodynamic stability. Furthermore, the addition of Co concentrations becomes less stable since the values of heat of formation increases as the content is increased. It is seen that Co substitution shows less stability, while the addition of Os and Ru enhances the stability of the Ti₅₀Pd₅₀ system at high concentration (0 $\le x\le$50).

Figure 3.

Heats of formation of the B2 Ti₅₀Pd_50-xM_x (M=Os, Ru, Co) (0 ≤ x ≤ 50) ternary SMAs.

3.2 Electronic total density of states (tDOS)

In order to better examine the differences in electronic structures of Ti₅₀Pd_50-xM_x alloys, it is important to analyze the behavior of the total density of states (DOS) near the Fermi level (E_f) with respect to the pseudogap. It is also known from previous studies that the DOS of structures of the same composition can be used to mimic the stability trend with respect to their behavior at the E_f [17, 44, 45]. A structure is considered the most stable if it has the lowest density of state at E_f. The DOS is expressed as the number of states per atom per energy interval.

Figure 4 shows the total DOS for Ti₅₀Pd_50-xRu_x alloys. The structures show an overlapping peak from the valence band (VB) to the conduction band (CB) suggesting a metallic behavior since there is no visibility of a bandgap. A shift of DOS is also observed when Ru is added toward CB. As the composition of Ru is added, the pseudogap moves toward the E_f, indicating that the electronic stability is enhanced in particular for composition above 20 at. % Ru. It is noted that Ti₅₀Ru₅₀ has the lowest DOS at the E_f, which suggests that it is the most stable, while the Ti₅₀Pd_43.75Ru_6.25 is the least stable compared to other compositions. Furthermore, it was observed that the E_f coincides with the pseudogap. This observation suggests that Ti₅₀Pd_50-xRu_x is electronically stable at the high content of Ru. The predicted DOS analysis is consistent with the stability trend as predicted by the ∆H_f.

Figure 4.

Comparison of the total density of state for Ti₅₀Pd_50-xRu_x systems (0 ≤ x ≤ 50) against energy. The Fermi level is taken as the energy zero (E–E_f = 0).

In Figure 5, we plot the total DOS for B2 Ti₅₀Pd_50-xOs_x (0 ≤ x ≤ 50) alloys. As Os content is added, the DOS for Ti₅₀Pd_43.75Os_6.25 hits the peak at the shoulder near the E_f. Furthermore, at above 18.75 at. % Os, the pseudogap moves toward the E_f, which may suggest that the system starts to stabilize. It is also observed that at 50 at. % Os (Ti₅₀Os₅₀) has the lowest DOS near the E_f. The result suggests that Ti₅₀Pd_50-xM_x is electronically stable at higher content of Os, consistent with the predicted ∆H_f.

Figure 5.

Comparison of the total density of state for Ti₅₀Pd_50-xOs_x systems (0 ≤ x ≤ 50) against energy. The Fermi level is taken as the energy zero (E–E_f = 0).

Figure 6 shows and compares the calculated tDOS for Ti₅₀Pd_50-xCo_x alloys. It is noted that as the Co content is increased the DOS hit the shoulder of the peak at E_f. The highest DOS along the E_f is observed at the high composition of Co (50 at. %), while 6.25 at. % Co has the lowest DOS peak. This observation indicates that Co is not preferable to enhancing the electronic stability of Ti₅₀Pd₅₀ in good agreement with the predicted ∆H_f.

Figure 6.

Comparison of the total density of state against energy for Ti₅₀Pd_50-xCo_x systems (0 ≤ x ≤ 50). The Fermi level is taken as the energy zero (E–E_f = 0).

3.3 Elastic properties

The elastic constants (C_ij) are important parameters, which can be used to predict mechanical stability. To further investigate the stability of Ti₅₀Pd_50-xM_x, elastic constants were calculated. The stability criterion for the elastic constants must be satisfied for the structure to be defined as stable. The stability conditions for the cubic system are outlined in Section 2.4.3.

In Figure 7, the calculated elastic constants of the Ti₅₀Pd_50-xRu_x alloys (0 $\le x\le$50) are shown. Recall that in order for the structure to be stable, it must satisfy certain stability criteria as discussed in Chapter 4. The positive $C^{\prime }$((1/2(c₁₁-c₁₂) > 0) indicates the mechanical stability of the crystal, otherwise, it is unstable. As indicated in Chapter 1, the binary B2 Ti₅₀Pd₅₀ alloy is mechanically unstable at 0 K due to negative elastic shear modulus ($C^{\prime }$= − 5.37 GPa). So, the addition of a third element has been suggested in order to stabilize the B2 Ti₅₀Pd₅₀ alloy. In this case, the addition of Ru shows that the elastic constants$c_{11}$, $c_{12}$, and $c_{44}$ are positive for the entire concentration range (0$\le x\le$50) Figure 7. The c₁₁ and c₄₄ increase with the addition of Ru content while c₁₂ decreases suggesting that the structure is becoming mechanically stable (satisfying the stability condition, c₁₁ > c₁₂).

Figure 7.

The elastic constants (GPa) as a function of the atomic % Ru composition of Ti₅₀Pd_50-xRu_x SMAs.

However, the predicted C_ij does not satisfy the stability criteria for Ti₅₀Pd_50-xRu_x (when x = 6.25 and 18.75) since c₁₁ is less than c₁₂ which resulted in negative elastic shear modulus ($C^{\prime }$<0). Furthermore, it is observed that the elastic constants satisfy the stability criterion above 20 at. % Ru indicating elastic stability (since $C^{\prime }$> 0) of Ti₅₀Pd_50-xM_x alloys. This may suggest that the addition of Ru reduces the martensitic transformation temperature of the TiPd due to an increase in $C^{\prime }$ above 20 at. % Ru. This observation has also been discussed in the previous study [46].

The calculated elastic properties of the Ti₅₀Pd_50-xOs_x alloys (0 $\le x\le$50) are also shown in Figure 8. It is noted that all the independent elastic constants c₁₁, c_12, and c₄₄ are positive in the entire range of Ti₅₀Pd_50-xOs_x alloys (0 $\le x\le$50). It is observed that c₁₁ < c₁₂ which resulted in negative $C^{\prime }$below 18 at. % Os indicating mechanical instability at this concentration. Interestingly, the elastic constants satisfy the stability criterion above 18.75 at. % Os indicating elastic stability (since $C^{\prime }$> 0) of Ti₅₀Pd_50-xM_x alloys.

Figure 8.

The elastic constants (GPa) as a function of the atomic % Os composition of Ti₅₀Pd_50-xOs_x SMAs.

Figure 9 shows the comparison of elastic constant c₁₁ and the c₁₂ for the Ti₅₀Pd_50-xCo_x at different concentrations. From the results, it is noted that the c₁₁ is less than c₁₂ below 25 at. % Co, which suggests instability at those compositions. It is clearly seen that the $C^{\prime }$ curve is below zero ($C^{\prime }$< 0) at lower concentrations. However, the elastic constants satisfy the stability criterion above 31.25 at. % Co indicating elastic stability (since $C^{\prime }$> 0) of Ti₅₀Pd_50-xM_x alloys.

Figure 9.

Elastic constants (GPa) as a function of atomic percent Co for Ti₅₀Pd_50-xCo_x where 0 ≤ x ≤ 50.

3.4 Anisotropy ratio

This section focuses on the anisotropy ratio to describe isotropic behavior and transformation as well as the ductility of the Ti₅₀Pd_50-xM_x systems.

3.4.1 Isotropic and anisotropy behavior

It is important to study the elastic anisotropy of the systems in order to understand material properties and improve their mechanical durability. The anisotropy can be calculated as:

$$A=\frac{c_{44}}{C^{\prime }}$$E17

The factor A = 1 indicates that the material is isotropic and any value greater or smaller than 1 indicates an anisotropic behavior.

The anisotropic plot depicts anisotropic behavior below 25 at. % for Ru, Os, and Co additions. However, A approaches unity (A$\approx$1) for both Ru and Os between 25 and 50 at. % composition. These alloy systems have isotropic behavior at this composition range. The Co additions are highly anisotropic in the entire composition range (as shown in Figure 10).

Figure 10.

Anisotropy as a function of atomic percent M (M=Os, Ru, Co) for Ti₅₀Pd_50-xM_x where 0 ≤ x ≤ 50.

3.4.2 Anisotropy and martensite transformation

An anisotropy ratio (A) plays an important role in predicting the martensitic transformation of the material. A martensitic transformation from B2 to B19 is observed when there is a higher value of A, while a good correlation between c₄₄ and $C^{\prime }$ can be observed when A is smaller which leads to the transformation from B2 to B19′ [47].

In Figure 10, a higher value of A is observed at 25 at. % Ru, which suggests transformation from B2 to the B19 martensite phase. There is a coupling of c₄₄ and $C^{\prime }$ at high concentration above 37 at. % Ru, which may suggest a possible phase transformation from B19 to the B19′ martensite phase. Thus, their addition results in a reversible martensitic transformation that is from B2 to B19, and B19 to B2. In the case of Os addition, A is higher for 18.75 at. % Os, which may suggest the transformation from B2 to B19 (see Figure 10). A strong coupling is observed between the c₄₄ and $C^{\prime }$ at 43.75 at. % Os, which leads to the transformation from B19 to B19′ (since A is small). It can be concluded that Ti₅₀Pd_50-xOs_x alloys transform from B2 to B19 (18.75 at. % Os) and then B19 transform into B19′ (43.75 at. % Os) due to a coupling of the c₄₄ and $C^{\prime }$ at 0 K. This is a similar observation with TiNi and TiNi-based alloys [47]. In the case of Co addition, it was observed that A is negative below 18.75 at. % Co indicating that the martensite transformation is suppressed and the B2 phase is preserved (see Figure 10). As the composition of Co is increased to 31.25 at. %, A is higher than other compositions showing the transformation from B2 to B19. It can be deduced that B2 Ti₅₀Pd_50-xCo_x alloys transform to B19 phase above 31 at. % Co.

Previously, Yi et al. [48] indicated that the c₄₄ and $C^{\prime }$ can be used to predict the change of martensitic transformation (Ms) at the Zener anisotropy factor A$<$10. This factor measures the degree of anisotropy in solid and is calculated using Eq. (17). In a cubic crystal, $C^{\prime }$ is used to measure the basal-plane shear along the direction of {110} <1–10>, while the c₄₄ is along direction {001} <100> shear (non-basal plane shear) which is equal to {001} <1–10 > shear. Hence, the c₄₄ is playing an important role in controlling the transformation temperature of B2 to B19′ [47]. Recall that the formation of the B19′ phase is attributed to the coupling between c₄₄ and $C^{\prime }$ as proposed by Ren et al. [47]. Interestingly, as the Os is added the $C^{\prime }$ decreases and becomes negative below 6 at. % and positive above, which suggest stability (as shown in Figure 11). This observation indicates a possible increase in the transformation temperature below 6 at. % due to negative $C^{\prime }$. It is noted that the entire Zener anisotropy factors are less than 10 for Os and Ru at. % composition, which indicates the possibility and reliability of the prediction of Ms. (see Figure 10). Furthermore, it was observed that the Zener anisotropy ratio is less than 10 for the addition of 6.25, 18.75, 43.75, and 50 at. % Co (except 25 and 31.25 at. % Co) as shown in Figure 10.

Figure 11.

The elastic constants c₄₄ and C′ against the composition of Ti₅₀Pd_50-xM_x alloys.

3.4.3 Anisotropy and ductility

The calculated A can also be used to check the ductility in metals. Thus, for a material to be considered ductile, the anisotropy ratio should be greater than 0.8 otherwise brittle [49]. An anisotropy values were found to be greater than 0.8 for 25 and 31.75 at. % Ru (Figure 10). The results imply that the alloy becomes ductile at this composition‘s ranges and brittle elsewhere. It was also found that the anisotropy ratio is greater than 0.8 above 18.75 at. % Os, which reveals ductile behavior. Furthermore, the Co addition is favorable above 25 at. % and 43.75 at. % (condition of ductility).

4. Conclusions

Ab initio DFT approach was used to study the equilibrium lattice parameters, heat of formation, and elastic properties of Ti₅₀Pd_50-xM_x as potential HTSMAs. Our results of lattice parameters were found to be in good agreement to within 5% with the available experimental and theoretical values. Ternary alloying with Os, Ru, and Co were investigated. The results suggested that the addition of Ru and Os stabilizes the Ti₅₀Pd₅₀ structure since the heat of formation decrease with composition ($∆H_f<0)$. This was confirmed from the DOS analysis. It was found that the states are shifted at E_f as the concentration is increased. For example, as the composition of Ru is added, the pseudogap moved toward the E_f, indicating electronic stability especially above 20 at. % Ru. A similar trend was observed with Os addition.

The effect of ternary addition revealed that Ti₅₀Pd_50-xOs_x alloys are mechanically stable above 18.75 at. % Os according to the criteria of mechanical stability. Furthermore, the C′ was found negative below 25 at. % Ru (C′ < 0, condition of instability) and becomes positive above this composition (C′ > 0, condition of stability). The c₁₁, c_44, and C′ for Ti₅₀Pd_50-xRu_x increase with an increase in Ru concentrations, while c₁₂ decreases above 25 at. % Ru. It was found that the C′ is negative below 31 at. % Co (instability characteristics) and becomes positive above this composition (condition of stability). This suggests that a possible HTSMAs material could be achieved above 31 at. % Co. This analysis has a direct impact on the transformation temperature. For example, a decrease in C′ suggests that the Ms. is likely to increase.

An elastic anisotropy ratio (A) was used to describe the isotropic and anisotropic behavior of the Ti₅₀Pd_50-xM_x systems. The analysis of A proves that the B2 Ti₅₀Pd_50-xCo_x alloys displayed elastic anisotropy behavior (as A is less and greater than 1). It was found that A approaches unity (A$\approx$1) for both Ru and Os between 25 and 50 at. % composition, suggesting isotropic behavior. Furthermore, we have also evaluated how alloying can impact the transformation temperature. The results suggest that the addition of Co, Ru, and Os on Ti₅₀Pd₅₀ alloy reduces the transformation temperature as indicated by positive C′. It was also found that the addition of Ru on Ti₅₀Pd₅₀ can result in transformation from B2 to B19 phase below 25 at. % Ru due to negative C′.

The ductile nature of Ti₅₀Pd_50-xM_x alloys was confirmed by the value of anisotropy. It was revealed that increasing Os and Ru above 6.25 at. % could effectively improve the ductility of the compound. Furthermore, the anisotropy ratios were found greater than 0.8 above 18.75 at. % Os and Co, which reveal ductility behavior.

Acknowledgments

The computational work was carried out using computer resources at the Materials Modeling Center, University of Limpopo. The authors acknowledge the High-Performance Computing (CHPC) in Cape Town for their computing resources. The support of the South African Research Chair Initiative of the Department of Science and Technology is highly appreciated.