The Use of DFT-Based ab-initio Technique to Determine the Stability Difference in B2 Ti-PGM Compounds

Abstract

In this chapter, the density functional theory (DFT) based first-principles approach is used to predict the underlying lattice properties associated with the phase transformation and stability of B2 phase in titanium-platinum group metal (Ti-PGM) compounds. This ab- initio technique provides a good platform to accurately explore phase stability variation between the successful Ti-PGM shape memory alloys (SMAs) (Ti50M50, M = Rh, Pd, Ir, Pt) and other B2 Ti-PGM compounds that do not show any shape memory effect (SME), such as Ti50Os50 and Ti50Ru50. The B2 TiFe, TiNi and TiAu have also been considered in this chapter in order to draw similarities and differences. Amongst the predicted results, the heat of formation was calculated to determine the thermodynamic stability, whereas the total densities of states were used to evaluate the electronic stability of these compounds. Insights on the mechanical stability of the B2 crystals were derived from the calculated elastic constants. Mechanical instability was revealed in some compounds, indicative of a possible phase transition responsible for the intrinsic shape memory character. Although an attempt to correlate this mechanical instability with imaginary frequencies established from the phonon dispersion curves is made, the correlation is not yet conclusive due to some discrepancies observed in TiNi.

1. Introduction

An interest in PGM-containing SMAs continues to rise due to their unique functional properties desirable for use in various high-temperature applications such as aerospace, automotive, power plants and chemical industries [1]. Ti-PGM intermetallic compounds such as TiPt [2, 3] and TiPd [4] have gained attention as promising candidates for development of high-temperature shape memory alloys (HTSMAs). This is so because they exhibit a martensitic transformation (MT) from cubic CsCl-type B2 to orthorhombic AuCd-type B19 phase at temperatures higher (above 373 K) than that of a well-known commercial TiNi. TiPt has attracted significant interest in various industries due to its high MT from B2 to B19 at approximately 1273 K [2], whilst the MT for TiPd has been observed at 823 K [5, 6]. Similarly, for TiIr, its B2 phase undergoes MT at 2023 K to monoclinic phase [7], whereas the MT for TiRh binary system occurs at 1118 K from B2 to tetragonal L1₀ [8, 9, 10]. Similarly, the MT of TiAu from B2 to B19 occurs at 900 K [11].

On the other hand, some of the Ti-PGM compounds that have been investigated were previously found to have a highly stable B2 phase to room temperature, with no sign of martensitic transformation occurring, thus no shape memory behavior was observed. Amongst those are TiOs [12] and TiRu [13, 14, 15, 16], which showed similar behavior to TiFe [17, 18]. So far, it is not known, at least from experimental studies, if these compounds may undergo martensitic transformation at cryogenic temperatures. However, such useful information can be generated using DFT computational tools. Moreover, from scientific point of view, it is very important to establish factors that suppress MT.

It is clear that the incorporation of PGMs in Ti influences both the crystal and electronic structure and consequently, the stability of austenite phase and formation of martensite phases. These phases are thoroughly studied as they are key in the design and functionality of SMAs for specific applications [19]. Most researchers have used computational calculations to gain some insight into the underlying shape memory properties of TiPt [20], TiPd [21] and TiRh [10]. Computational studies usually include structural, elastic and electronic properties. First principle calculations have been proven to be a useful and reliable tool for studying ground-state properties of these compounds [22, 23, 24, 25].

In addition to structural, elastic and electronic properties, there has been an increase in the use of lattice vibration properties to predict the dynamic stability for a particular phase by calculating phonon dispersions. For example, Haskins et al. [26] recently used phonon dispersion calculations to resolve discrepancies associated with determination of ground-state phases for TiNi [26, 27]. Also, phonon dispersion calculations have been used by several researchers in order to determine the lattice vibration properties and predict the phase transformation path for various alloy systems [10, 20, 27].

Although DFT predictions may not agree perfectly with experimental observations, the accuracy of predicted results can still be scrutinized against the available experimental data in order to validate the accuracy of DFT calculations. In predicting phase stability, there is a link between the predicted thermodynamic, electronic and mechanical and lattice dynamic stability, which in turn can be used to predict the expected phases at a particular temperature. Some researchers [26, 28, 29] often report these stabilities separately and although that is acceptable, we observed that these stabilities can be used in connection to gain clear understanding of phase transformations for the investigated Ti-PGM compounds.

Furthermore, in order to improve shape memory properties or induce MT in TiOs and TiRu, factors influencing the shape memory behavior must be understood and this can be done by studying the predicted mechanical, electronic and dynamic stabilities. Thus, the main aim of this chapter is to demonstrate the versatile use of ab-initio methods based on DFT to track the shape memory behavior of these well-known Ti-PGM compounds by evaluating their ground-state stability of the high-temperature austenite phase (B2) with reference to mechanical, electronic and dynamical stability. This will assist the reader in identifying the underlying fundamental factors that drive the existence of shape memory behavior and further give an insight towards the design of new and improvement of existing SMAs.

2. Computational methodology

2.1 Density functional theory (DFT)

Computational modeling techniques offer an alternative way of investigating the properties of materials using computers, whereby the simulator builds a model of a real system and explores its behavior. One of the computational techniques that have received immense attention over the last few decades is the first-principles calculations, also known as ab-initio calculations. This interest is attributed to significant usefulness and insightfulness of its calculated data in the materials design.

First-principles methods are based on DFT formalism in which properties of materials, that is, the values of the fundamental constants and the atomic numbers of the atoms present can be calculated using the Schrödinger equation. Due to its improved accuracy achieved over the past years in predicting properties of real solids, an ab-initio approach is adopted in the current work to predict the ground-state and structural properties of several B2 intermetallic compounds at equiatomic compositions. Computing total energies of any system is a necessary starting point for first-principle calculations.

DFT is a quantum-mechanical method used for calculating ground-state properties of condensed matter systems without dealing directly with many electron states [30]. It was first formulated by Hohenberg and Kohn in 1964 [31] and then secondly developed by Kohn and Sham in 1965 [32]. DFT has helped in the development of independent-particle methods that take into account the particle’s correlations and interactions. Hohenberg-Kohn demonstrated the first theorem that the ground-state properties of a many-electron system are uniquely determined by an electron density that depends on only three spatial coordinates [30].

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

Where $E$ is the total energy and n is the density. Within the Kohn-Sham scheme [30], consideration of an interacting electron gas moving in an external potential $v_e\left(r\right)$, as a variational principle leads to the effective single-electron Schrödinger equations,

$$\left\{-{\nabla }^2+v\left(\left[n\right];\overrightarrow{r}\right)\right\}{\psi }_j\left(\overrightarrow{r}\right)={\in }_j{\psi }_j\left(\overrightarrow{r}\right)$$E2

Kohn-Sham electrons in an effective potential, $v_{\mathit{eff}}$ for a system of non-interacting, is solved as follows:

$$V_{\mathit{eff}}\left(\overrightarrow{r}\right)=v\left(\overrightarrow{r}\right)+\int \frac{n\left({\overrightarrow{r}}^{\text{'}}\right)}{\mid \overrightarrow{r}-{\overrightarrow{r}}^{\text{'}}\mid }d{\overrightarrow{r}}^{\text{'}}+\frac{\delta E_{\mathit{xc}}\left[n\left(\overrightarrow{r}\right)\right]}{\mathit{\delta n}\left(\overrightarrow{r}\right)}$$E3

Where $v\left(\overrightarrow{r}\right)$ is the external potential and $E_{\mathit{xc}}\left[n\left(\overrightarrow{r}\right)\right]$ is the exchange-correlation density functional [30].

2.2 Approximations to exchange-correlation functional

The two main types of exchange-correlation functionals used in DFT are the local density approximation (LDA) [33] and the generalized gradient approximation (GGA) [34], which have been discussed in the sub-sections below.

2.2.1 Local density approximation

The local density approximation (LDA) is an approximation in which the exchange-correlation (XC) energy functional in density functional theory (DFT) depends upon the value of the electronic density at each point in space. It was first discovered by Kohn and Sham, which is expressed in the Eq. (4):

$$E_{\mathit{xc}}^{\mathit{LDA}}\left[n\left(r\right)\right]=\int \mathit{drn}\left(r\right){\epsilon }_{\mathit{xc}}\left(n\left(r\right)\right)$$E4

Where ${\epsilon }_{\mathit{XC}}\left(n\right)$ is the exchange-correlation energy per electron in a uniform electron gas of density n [33]. The uniform electron gas represents a group of systems of interacting electrons with an arbitrary spatially constant density n, which acts as a parameter. The local density approximation quantity is known for the limit of high density and can be calculated accurately at densities of interest by the use of Monte Carlo methods. LDA has been proven to give accurate results for many atomic, molecular and crystalline interacting electron systems, even though in these systems, the density of electrons is not slowly varied.

2.2.2 Generalized gradient approximation

The GGA is known to be a semi-local approximation, which means that there is no use of local density $n\left(r\right)$ value but its gradient $\nabla n\left(r\right)$. Perdew and Wang developed generalized gradient approximation (GGA), which is based on a real-space cut-off of the spurious long-range components for the second-order gradient expansion for the exchange-correlation hole [34]. GGA improves total energies, atomization energies, energy barriers and also the difference in structural energies. GGA takes the form:

$$E_{\mathit{xc}}^{\mathit{GGA}}\left[n\left(\overrightarrow{r}\right)\right]=\int {\epsilon }_{\mathit{xc}}^{\mathit{GGA}}\left(n\left(\overrightarrow{r}\right),|\nabla n\left(\overrightarrow{r}\right)|\right)n\left(\overrightarrow{r}\right)d\overrightarrow{r}$$E5

There are several GGA-based functionals, that is, the PBE [35], PBEsol [36], RPBE [37], BLYP [38] and AM05 [39]. Other known GGA-based functionals are meta-GGA [40], hyper-GGA and generalized random phase approximation.

In this chapter, the GGA-PBE [35] functional was used to optimize the Ti₅₀M₅₀ (M = PGMs, Ni, Fe, Au) systems as it provides accurate parameters for these materials.

2.3 Computational code and implementation

2.3.1 CASTEP code

In this book chapter, the plane-wave Cambridge Serial Total Energy Package (CASTEP) [41, 42] code was used to investigate the properties of the B2 structures. CASTEP is a module embedded within the Materials Studio software package. It is a first-principle quantum mechanical code based on DFT formalism, used for performing electronic structure calculations. It can be used to simulate a wide range of materials, including crystalline solids, surfaces, molecules, liquids and amorphous materials; the properties of any material that can be thought of as an assembly of nuclei and electrons can be calculated with the only limitation being the finite speed and memory of the computers being used. This approach to simulation is extremely ambitious given that the aim is to use no experimental data but to rely purely on quantum mechanics.

Aiming to calculate any physical property of the system from first principles, the basic quantity is the total energy from which many other quantities are derived. For example, the derivative of total energy concerning atomic positions results in the forces and the derivative concerning cell parameters gives stresses. To do this, the total-energy code, on CASTEP code, performs a variational solution to the Kohn-Sham equations by using a density mixing scheme to minimize the total energy and also conjugate gradients to relax the ions under the influence of the Hellmann-Feynman forces. CASTEP uses fast Fourier transforms (FFTs) to provide an efficient way of transforming various entities (wave functions and potentials) from real to reciprocal space and back, as well as to reduce the computational cost and memory requirement for operating with the Hamiltonian on the electronic wave functions, a plane-wave basis for the expansion of the wave functions. These are then used to perform full geometry optimizations [41, 42].

A summary of the methodology for electronic structure calculations as implemented in CASTEP is as follows: a set of one-electron Schrödinger (Kohn-Sham) equations are solved using the plane-wave pseudopotential approach. The wave functions are expanded in a plane wave basis set defined by the use of periodic boundary conditions and Bloch’s Theorem. The electron-ion potential is described employing ab initio pseudopotentials within both norm-conserving and ultrasoft formulations [41, 42].

Direct energy minimisation schemes are used to obtain self-consistent electronic wave functions and corresponding charge density. In particular, the conjugate gradient and density mixing schemes are implemented. Also, the robust electron ensemble DFT approach can be used for systems with partial occupancies, in particular, metals [41, 42].

2.3.2 Implementation

Figure 1 illustrates the equiatomic B2 crystal geometry used to carry out all the calculations reported in this chapter.

Figure 1.

Schematic representation of B2 crystal structure of Ti₅₀M₅₀ (M = Ru, Rh, Pd, Os, Ir, Pt, Ni, Fe, Au) intermetallic compounds.

The resulting geometry-optimized crystal structure was used to carry out all the calculations, including structural, thermodynamic and elastic properties, of all considered compounds. Only valence electrons were considered through the use of ultrasoft pseudopotentials [41, 42]. All of our calculations were performed with pseudo-potentials in generalized gradient approximation (GGA) [32] refined by Perdew, Burke and Ernzerhof (PBE) [35]. Before any calculation could be performed, a convergence test was also conducted in this code to determine the suitable cut-off energy and k-point mesh parameter for systems. A plane wave cut-off energy of 500 eV was found to be sufficient enough to converge the total energy of the systems. The Brillouin zone (BZ) sampling was performed using the k-point mesh of 13x13x13 according to the Monkhorst–Pack method [43]. A full geometry optimization was performed to determine the ground-state parameters for the binary systems. To obtain the stable structure of Ti₅₀M₅₀ with minimum total energy, the Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization scheme [44] was performed in geometry optimization. The maximum ionic Hellmann-Feynman force was given to be 0.03 eV/Å. Furthermore, the elastic constants of a solid were calculated by an efficient strain–stress method through a linear least-square fit of first-principles calculation results. The maximum stress set is below 0.05 GPa. The phonons dispersion curves were also calculated.

2.4 Theoretical background on calculated properties

2.4.1 Thermodynamic stability

The heat of formation ($∆H_f$) is the enthalpy change when one mole of a compound is formed from the constituent elements in their stable states and is essential in determining the structural stabilities of the different crystal structures. The heat of formation is estimated by the following expression:

$$\mathrm{\Delta }{H}_f=E_C-\sum _i{x}_i{E}_i$$E6

where $E_C$ is the calculated total energy of the compound and $E_i$ is the calculated total energy of the constituent element in the compound. For a structure to be stable, the heat of formation must have the lowest negative value ($∆H_f$ < 0). The heat of formation is used to determine the stability trend of the B2 systems.

2.4.2 Electronic stability

The electronic stability is determined by the density of states (DOSs), which refer to the occupancy and density of the electronic states in a crystalline solid. It is described by a function, g (E), as the number of electrons per unit volume and energy with electron energies near Fermi-level E_F. In the case of states with DOS being zero implies no state has been occupied (empty orbital). In general, a DOS is an average of all available spaces multiplied by the number of states occupied by the system. The local density of states (LDOSs) is a measure of variation due to distortion of the original system. LDOS can locally be non-zero if the DOS of an undisturbed system is zero due to the presence of local potential. In that case, the DOSs are the total number of states that are available in the system within the plane-wave framework of DFT. It is possible to calculate each orbital’s contribution (partial DOS) to determine which orbitals are occupied or involved in bonding. The electronic behavior of a material is determined by the location of E_F within the DOS. Alloys’ stability can be predicted using the DOS.

In the case of partial density of states (PDOSs), the states are attributed to the basic functions and then to the atoms constituting the unit cell. DOS is then calculated as the sum of atomic contributions. The DOS is calculated by using the following expression:

$$n\left(\epsilon \right)=2\sum _{n,k}\delta \left(\epsilon -{\epsilon }_n^k\right)=\frac{2}{V_{\mathit{BZ}}}\sum _n\int \delta \left(\epsilon -{\epsilon }_n^k\right)\mathit{dk}$$E7

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

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

2.4.3 Mechanical stability

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 elastic constants depend on the type of lattice i.e. for the cubic, there are three (c₁₁, c₁₂, c₄₄) independent elastic constants [20, 45]. For example, applying two types of strains $\left({\epsilon }_1\right.\left.{\epsilon }_4\right)$ to the cubic system gives stresses relating to three elastic coefficients, this is a useful method for obtaining elastic constants. The mechanical stability condition for the cubic system as outlined in Ref. [45] is given as follows:

$$c_{44}>0;c_{11}>c_{12}\text{and}{c}_{11}+2c_{12}>0$$E8

According to Born-Huang’s lattice dynamical theory [46, 47], 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.

2.4.4 Lattice dynamic stability

A phonon dispersion curve along a high symmetry direction is calculated by using interplanar force constants [45], as every plane perpendicular to this direction is displaced within an elongated supercell. Generally, lattice dynamics are analyzed with the ab-initio evaluation of forces on all atoms resulting from finite displacements of few atoms within otherwise perfect crystals. It is usually necessary to construct supercells of the appropriate size to ensure that interactions of the perturbation with all its translational symmetry equivalent copies are small. The techniques for selecting suitable supercells and atomic displacements, assembling force constant matrices from the calculated forces and calculating phonon dispersion relations via Fourier transform are well documented.

Using one of the 230 crystallographic space groups, phonon constructs a crystal structure, calculates the Hellmann-Feynman force constant, builds the dynamical matrix, diagonalizes it and calculates phonon dispersion relations [45].

In phonon dispersion calculations, polarization vectors and irreducible representations (Gamma points) of phonon modes are found, and the total and partial phonon densities are calculated. It plots the internal energy, free energy, entropy, heat capacity and tensor of mean square displacements (Debey-Waller factor). Phonons calculate the dynamical structure factor of coherent inelastic neutron scattering and incoherent doubly differential scattering in single crystal and polycrystalline systems [45].

The properties of phonons can be determined using a harmonic approximation with one fundamental quantity, the force constants matrix [45]:

$$D_{\mathit{\mu \nu }}\left(R-R^{\text{'}}\right)=\frac{{\delta }^{2}E}{\partial u_{\mu }\left(R\right)\partial u_{\nu }}\left|{}_{u=0}\right.$$E9

Where u represents the displacement of a given atom and E is the total energy in the harmonic approximation. This matrix of force constants can also be represented in reciprocal space and is known as a dynamic matrix:

$$D_{\mathit{\mu \nu }}\left(q\right)=\frac{1}{N_R}\sum _R{D}_{\mathit{\mu \nu }}\left(R\right)exp\left(-\mathit{iqR}\right)$$E10

Classical equations of motion can be written as eigenvalue problems with each atomic displacement in the form of plane waves:

$$u\left(R,t\right)=\epsilon e^{\left(\mathit{qR}-\omega \left(q\right)t\right)}$$E11

Where ε is the 3N-dimensional eigenvector of the eigenvalue problem is:

$$M{\omega }^2\left(q\right)\epsilon =D\left(q\right)\epsilon$$E12

The ω on the wave vector is well-known as the phonon dispersion. A guide to the basic theory of phonons has been described in detail by Born and Huang (1954) and Ashcroft and Mermin (1976) [48, 49]. In this chapter, we use the CASTEP code [41, 42] to calculate phonon dispersion curves and their density of states (PHDOS). It is well documented that compounds that radiate real (only positive) vibrational modes along high symmetry directions in the Brillouin zone are considered to be vibrationally stable with no possibility to undergo a martensitic phase transition at lower temperatures [10, 13]. On the other hand, compounds that radiate both positive and negative vibrational modes turn out to be vibrationally unstable with high chances to undergo a martensitic phase transition at lower temperatures, a primary feature of alloys with shape memory effect [20, 23, 50].

3. Results and discussion

3.1 Structural and thermodynamic stability

The stability of the investigated B2 compounds is first discussed on the basis of heat of formation. Table 1 presents the ground-state lattice parameters and the heat of formation of the investigated B2 phases. The reported lattice parameters were obtained from the geometrically relaxed structures, whereby their respective volumes and unit cells were allowed to change to obtain their ground state.

Crystal structures	Lattice parameters, a (Å)		Volume (Å^3)	Density (g.cm^3)	Heat of formation, -ΔHF (eV/atom)
	This work	Literature			This work	Literature
Ti50Ru50	3.08	3.07, 3.09	29.21	8.47	0.750	0.798, 0.743, 0.770 [14, 51]
Ti50Rh50	3.12	3.13	30.41	8.23	0.852	0.741, 0.749, 0.715 [51]
Ti50Pd50	3.17	3.18	31.74	8.07	0.508	0.519, 0.530 [14, 21, 51]
Ti50Os50	3.09	3.09, 3.10	29.58	13.37	0.705	0.710, 0.714, 0.683 [51]
Ti50Ir50	3.12	3.12	30.35	13.14	0.913	0.876, 0.847, 0.845 [51]
Ti50Pt50	3.18	3.19, 3.21	32.18	12.54	0.885	0.824, 0.795 [51]
Ti50Fe50	2.95	2.99	25.73	6.70	0.745	0.321 [52]
Ti50Ni50	3.01	3.02, 2.97	27.34	6.47	0.382	0.358 [53]
Ti50Au50	3.27	3.25	34.96	11.63	0.368	0.442 [51]

Table 1.

Lattice parameters and heat of formation of the investigated binary B2 compounds.

The lattice parameters were found to be in good agreement with those reported previously by other authors [10, 12, 13, 19, 22, 54, 55]. The lattice parameters of the investigated Ti₅₀M₅₀ compounds increase with the position of the M atom along the groups or the periods of the periodic table of elements. This is in line with their respective densities and volume changes observed, as presented in Table 1.

Furthermore, Table 1 also presents the thermodynamic stability of the investigated B2 compounds that were determined by calculating their respective heat of formation using Eq. (6). Heat of formation provide primary insight into the existence of the phase. All the investigated B2 phases reported in this research work were found to be negative (ΔH_F < 0), an indication that they are all thermodynamically stable. Amongst the investigated B2 compounds, the heat of formation value of Ti₅₀Ir₅₀ was found to be the most thermodynamically stable indicated by the lowest value of $-$0.913 eV/atom. The reported heat of formation results were found to be in accordance with other data in literature [51, 52, 55, 56, 57, 58].

3.2 Electronic stability

Figure 2 shows the total density of states (tDOS) of all the investigated B2 compounds reported in this chapter. It is noted that the DOS values g (E) of the investigated compounds were found to be non-zero across the Fermi level (E_F) indicating that all the investigated B2 compounds were mainly characterized by metallic bonds.

Figure 2.

The total density of states (tDOS) of the nine investigated austenite compounds.

Generally, the density of states’ spectra of CsCl-type B2 compounds consists of two peaks [25] that are separated by a pseudogap (deep valley) at the Fermi level (E_F). With the lower energy peak representing the anti-bonding region and the higher energy peak representing the bonding region. Therefore, the position of E_F at the pseudogap provides insight into the phase stability of a compound at the ground state. Such that, if E_F is found to fall at the centre of the pseudogap, that signifies phase stability and if E_F falls at the peak or shoulder of the bonding region. Figure 2 shows that Ti₅₀Fe₅₀, Ti₅₀Ru₅₀ and Ti₅₀Os₅₀ will retain their stable high-temperature austenite phase as their ground-state as their E_F cuts at the centre of the deep valley, while the other investigated B2 compounds (Ti₅₀M₅₀, M = Ni, Rh, Pd, Ir, Au and Pt) will certainly undergo a phase transition from the high-temperature B2 to unstable phases at 0 K because their pseudogap shifted towards the anti-bonding region and enabled phase transition.

The aforesaid provides insight information about the phase transition of the investigated compounds from high-temperature austenite (B2) to low-temperature martensite phase, a gauge for shape memory effect observed on alloys with shape memory properties.

3.3 Dynamic phase stability

The phonon-dispersion curves are used to gain insight into the underlying lattice vibrations that may influence the ability of the crystal to transform to martensite phase on cooling. The information on phonons is very useful for accounting variety of properties and behaviors of crystalline materials, such as thermal properties, phase transition, and superconductivity [59].

As detailed in Section 2.4.4, B2 compounds that show only positive frequencies remain stable with no prospect of undergoing martensitic phase transition, while those that show both positive and negative frequencies are prone to become unstable and undergo martensitic phase transition at lower temperatures.

Figure 3 represents the sets of phonon-dispersion curves of the investigated compounds. It can be shown that Ti₅₀Fe₅₀, Ti₅₀Ru₅₀ and Ti₅₀Os₅₀ consist of only the real positive vibrational modes, while the rest of the investigated compounds were found to consist of both positive and negative vibrational modes. This observed behavior agrees well with the DOS results reported in the previous section as well as the results reported by other authors [20, 23, 50].

Figure 3.

Phonon-dispersion curves of the investigated B2 phases plotted along selected Brillouin zone directions.

Furthermore, Figure 3 gives the calculated B2 phonons density of states (PHDOS). It is evident that the vibrational stable B2 compounds (Ti₅₀M₅₀, M = Fe, Ru and Os) consist of a valence gap between the positive and negative frequency modes. Such observed behavior renders the aforementioned compounds to be brittle-metal-compounds, a kind of semi-conductive material properties. While those austenite phases that were found to be vibrationally unstable (Ti₅₀M₅₀, M = Ni, Rh, Pd, Ir, Pt and Au) show a clear interaction of valence and conduction electrons with a non-zero DOS.

3.4 Mechanical stability

Elastic constants are the primary output parameters of most DFT codes, and they determine the response of the crystal to external forces. They are crucial in predicting the mechanical properties of the crystal structure and their subsequent modulus of elasticity.

Table 2 presents the calculated elastic constants of the investigated alloys reported in this chapter. Furthermore, the trend of the elastic constants plotted against different B2 compounds is shown in Figure 4.

Parameters	TiFe	TiNi	TiRu	TiRh	TiPd	TiOs	TiIr	TiPt	TiAu
C11	357.43	206.68	396.10	166.15	148.46	466.03	−55.28	141.65	131.78
C12	101.88	138.67	122.55	206.86	163.98	140.21	376.01	203.15	139.44
C44	71.64	47.08	82.72	59.41	51.51	125.20	79.08	48.27	43.08
C′	127.78	34.00	136.78	−19.85	−7.76	162.91	−215.65	−30.75	−3.83

Table 2.

The calculated elastic constants (C_ij) of the investigated Ti₅₀M₅₀ compounds.

Figure 4.

The calculated elastic constants (GPa) of the investigated compounds.

Based on the mechanical stability criterion for cubic crystals as outlined in Section 2.4.3, as well as very high value of C′, Ti₅₀Fe₅₀, Ti₅₀Ru₅₀ and Ti₅₀Os₅₀ completely satisfy the criterion, rendering the austenite phase to be mechanically stable in these compounds, thus unlikely to undergo any phase transition at lower temperatures.

Although a similar stability criterion was observed (C′ > 0) for Ti₅₀Ni_50, which is a well-known SMA [14, 60], it should be noted that its C₄₄ was found to be larger than C′, resulting in anisotropic factor A greater than 1. This is another important elastic property factor worth attention in design of SMAs and can be assessed on the relationship between the tetragonal shear modulus (C′) and its corresponding monoclinic shear constant (C₄₄). It is reported that the mechanical instability of a high-temperature phase decreases when nearing the martensitic transition temperature during heating [61], this is the case where C′ becomes close to zero. Furthermore, if C′ < 0, the corresponding B2 phase is unstable at 0 K, signaling a great potential for this high-temperature phase to undergo a martensitic phase transition on cooling, yielding SME character. Since this MT occurs at much higher temperatures, such B2 alloys are of interest for high-temperature structural applications. Table 2 shows that Ti₅₀Os₅₀ and Ti₅₀Ir₅₀ are the most and least mechanically stable amongst the investigated B2 compounds, and the same results are graphically presented as shown in Figure 4.

4. Conclusions

The thermodynamic, electronic, lattice dynamic and mechanical stability of the investigated B2 compounds have been calculated using a first-principles approach and are reported in this chapter. The calculated heat of formation, elastic constants, densities of states and vibrational properties have provided a much-needed clarity in determining the differences in stabilities of various Ti-PGM B2 compounds. This was made possible by choosing the correct model, resulting in reliable data that compares well with results reported in literature.

From all the calculated properties, it was shown that the higher-temperature austenite phase of Ti₅₀Fe₅₀, Ti₅₀Ru₅₀ and Ti₅₀Os₅₀ remains stable with no prospect of martensitic phase transition that is associated with materials with shape memory effect.

This was evident by the electronic and dynamic stability of the investigated compounds. Furthermore, their corresponding tDOS spectra were found to coincide with the pseudogap at the E_F, rendering electronic stability with no phase transition. This was further substantiated by phonon curves, which possessed only positive vibrational frequency indicating that they are not likely to undergo a phase transition. The predicted mechanical stability of the Ti₅₀Fe₅₀, Ti₅₀Ru₅₀ and Ti₅₀Os₅₀ B2 compounds was found to be very high, strongly signaling absence of any possible phase transformation.

On the other hand, using the electronic and elastic properties, this work has shown that other B2 compounds (Ti₅₀M₅₀, M = Ni, Rh, Pd, Ir, Pt and Au) considered in this study are unstable at 0 K, thus predicting the possibility to undergo phase transition to martensite phase on cooling, resulting in shape memory effect character if the transformation is diffusionless.

Acknowledgments

This book chapter is published with the permission of MINTEK. The authors would like to thank the Advanced Metals Initiative (AMI) of the Department of Science and Innovation (DSI) for financial support. The gratitude is also extended to the Centre for High-Performance Computing (CHPC) in Cape Town for allowing us to carry out the calculations using their remote computing resources.