Investigations of Structural, Elastic, Electronic and Thermodynamic Properties of X₂TiAl Alloys: A Computational Study

3.1 Structural and Elastic Properties

The crystal structures of Au₂TiAl, Ru₂TiAl, and Zr₂TiAl alloys are shown in Figure 1. The total energy for different volumes around the equilibrium volume is calculated for the L2₁ phase. The calculated total energy is then fitted to Murnaghan’s equation of state [32] in order to obtain their equilibrium lattice constants, the bulk modulus B, and its pressure derivative B′. The optimised lattice constant (a), bulk modulus (B), its pressure derivative B′, elastic constants (C_ij), and Cauchy pressures (C₁₂–C₄₄) for Au₂TiAl, Ru₂TiAl, and Zr₂TiAl alloys are presented in Table 1. The lattice constants of all alloys are in a good agreement with theoretical and experimental values [16], [17], [21], [22]. The deviations of the computed lattice constant values of Au₂TiAl and Zr₂Tial alloys are about 1.14% and 0.16% from the experimental values [16], [20], which indicates that a satisfactory level of precision can be achieved in the density functional theory calculations. The computed bulk modulus of Zr₂TiAl is only ∼2.93%, inconsistent with the calculated result of [22].

Figure 1:

The crystal structure of the full-Heusler X₂TiAl (X = Au, Ru, and Zr) alloys.

Elastic constants are known as numbers that quantitatively reflect the elastic or non-elastic response of the material when a stress is applied on it. The elastic properties of a cubic crystal are generally characterised by three independent constants, namely, C₁₁, C₁₂, and C₄₄. The obtained values of C₁₁, C₁₂, and C₄₄ for three alloys are given in Table 1, along with the available data in literature. To the best of our knowledge, the experimental or theoretical elastic constants of Au₂TiAl and Ru₂TiAl alloys have not yet been presented in the literature. Therefore, the results of two alloys in the L2₁ phase may serve as a base for future investigations. The three elastic constants for Zr₂TiAl are found to be larger than that of the results of Reddy and Kanchana [22] by about 7.80%, 0.35%, and 9.35%, respectively. The mechanical stability evolution of alloys is also carried out based on their elastic constants. The necessity of mechanical stability in a cubic crystal makes the following constraints on the elastic constants [33]:

Table 1 exhibits that all alloys fulfil the stability condition of (12), including the fact that the values of C₁₁ is higher than that of C₁₂. Equation (12) also leads to a restriction that the equilibrium bulk modulus should have a value between C₁₁ and C₁₂ (C₁₁ < B > C₁₂), which is confirmed for all three alloys. Thus, it can be concluded that X₂TiAl alloys are stable in the L2₁ phase. The value of C₁₁ also provides information about compressive resistance along the x-axis [32]. The value of C₁₁ of Zr₂TiAl is lower than that of Au₂TiAl and Ru₂TiAl, indicating that Zr₂TiAl is more compressible along the x-axis than the other alloys. Cauchy pressures of three alloys, which describe the nature of bonding at the atomic level in metals and compounds, are calculated from the elastic constants. If Cauchy pressure has a value of more negative, the bonding is directional with angular character and shows a characteristic of a covalent material [34], [35]. In the opposite case (positive), the bonding is more metallic and non-directional [36], [37]. Table 1 indicates that Cauchy pressures of alloys are positive, suggesting a metallic bonding for all alloys.

For further analysis of the mechanical properties of X₂TiAl alloys, the brittleness and ductility properties are also investigated via the bulk modulus to shear modulus (B/G) ratio and the Poisson ratio (σ), which are obtained by using C_ij. In general, the bulk modulus of a solid represents the resistance to a volume change and a measure of the average bond strength of atoms, whereas shear modulus represents the resistance to plastic deformation. According to Pugh’s [35] criteria, X₂TiAl alloys are ductile because the B/G values of the three alloys are bigger than 1.75 as shown in Table 2. It can be also noticed that the Poisson’s ratios of the three alloys are around 0.33; hence, all alloys have a ductile nature. The conclusions of both B/G and Poisson’s ratio support each other. Additionally, the Poisson’s ratio is used to estimate the bonding properties within the material and measure the stability of crystal against shear. As the Poisson’s ratio increases, the plasticity of the crystal becomes better. The Poisson ratio is suggested as 0.1 for covalent materials and 0.25 for ionic materials in the literature [38]. The Poisson’s ratios for Au₂TiAl, Zr₂TiAl, and Ru₂TiAl alloys are found to be 0.39, 0.34, and 0.31, respectively, which implies that the alloys have ionic-metal interaction. These results are in well accordance with Cauchy pressures of alloys mentioned previously. The total magnetic moment calculations indicate that Zr₂TiAl has a total magnetic moment of 2.02 μ_B per formula unit. This is in well accordance with the results of Reddy and Kanchana [22]. However, Au₂TiAl and Ru₂TiAl alloys do not show any magnetic moment; thus, these alloys are nonmagnetic materials.

The Young’s modulus (E) of a material, which is a ratio of tensile stress to tensile strain, defines the stiffness and tensile elasticity of a material. From the calculations, Young’s modulus of Au₂TiAl is 86.774 GPa, which is smaller than that of Zr₂TiAl (92.326 GPa) and Ru₂TiAl (243.266 GPa). As the value of E increases, the material becomes stiffer. It can be seen from Table 2 that Ru₂TiAl has the largest Young’s modulus value among the three alloys; hence, Ru₂TiAl is the stiffest alloy among them. The anisotropy factor A is obtained as a measure of the degree of the elastic anisotropy of the crystals and presented in Table 2. For a complete isotropic material, the anisotropy factor value is 1. Deviation from this value defines the elastic anisotropy degree. As Table 2 illustrates, the calculated anisotropy factors of all alloys are higher than unity, showing that the studies alloys are elastically anisotropic. However, Ru₂TiAl’s anisotropy factor is very close to unity.

According to Table 2, Au₂TiAl and Zr₂TiAl possess anisotropy; thus, two-dimensional directional dependence of Young’s modulus has been computed for all alloys using EIAM code [39] and presented in Figure 2. It is known that for an isotropic material, the shape is spherical. Deviation from this shape indicates degree of anisotropy [40], [41]. It is clear from Figure 2 that Ru₂TiAl is isotropic at all planes, whereas Au₂TiAl and Zr₂TiAl are anisotropic.

Figure 2:

Two-dimensional directional dependence of the Young’s modulus of X₂TiAl (X = Au, Ru, and Zr) alloys.

3.2 Electronic and Phonon Properties

The electronic band structures of X₂TiAl alloys are presented in Figure 3, along the high-symmetry directions in the Brillouin zone.

Figure 3:

The electronic band structure of X₂TiAl (X = Au, Ru, and Zr) alloys.

Figure 4:

The total and partial densities of states of X₂TiAl (X = Au, Ru, and Zr) alloys.

Figure 3 illustrates that the band lines and finite DOS cut the Fermi energy level for all alloys; therefore, these alloys have a metallic character. It is shown that there is no gap at the Fermi level, and total density of states are 1.329 and 2.25 states eV⁻¹ cell for Au₂TiAl and Ru₂TiAl, respectively. The bands that cross the Fermi level can be seen in both majority and minority spin states determining the metallic nature of Zr₂TiAl alloy (see Fig. 4).

The band states for these alloys are determined by calculating the total and partial densities of states. The main characters of the computed results for the three alloys are in well accordance with the previous reported studies of electronic structure [22], [42]. There are peaks above and below the Fermi level from the total and partial densities of Au₂TiAl and Ru₂TiAl, as shown in Figure 4. The main contributions to the bands below the Fermi level are due to Au 5d states for Au₂TiAl and Ru 4d states for Ru₂TiAl, whereas above the Fermi level, they are mainly due to Ti 3s states for both alloys. For Zr₂TiAl, it is seen that the majority and minority spin contributions of the DOS in the Fermi level are due to the ‘d’ electrons of the Zr and Ti atoms.

The phonon dispersion curves along the Γ-K-Γ-W-L directions in the first Brillouin zone are computed for X₂TiAl alloys and shown in Figure 5. There are no imaginary phonon frequencies in the whole Brillouin zone for X₂TiAl alloys. This suggests that three alloys are dynamically stable in the L2₁ phase. As X₂TiAl alloys have four atoms in their unit cell, the phonon spectrum should have 12 branches; three of them is being acoustic (lowest three) and nine of them is being optical. The frequencies of the optical phonon modes of X₂TiAl at the zone centre are calculated as 2.567, 5.750, and 6.754 THz for Au₂TiAl; 5.812, 6.395, and 9.876 THz for Ru₂TiAl; and 3.312, 4.625, and 7.723 THz for Zr₂TiAl. It can be seen that the optical phonons of Au₂TiAl overlap and fall in the frequency region between 4.74 and 7.6 THz. On the other hand, for Ru₂TiAl and Zr₂TiAl alloys, optical phonon frequencies are separated. Examination of the atomic masses of Au, Ru, Zr, Ti, and Al allows us to understand this difference between the alloys. The phonon partial density of states of alloys shows that the upper optical frequency is due to the lighter Al atoms, whereas the lower frequency region is due to the heavier Au (Ru and Zr) atoms. Ti atoms vibrate in the middle phonon frequency region for all alloys.

Figure 5:

Full phonon dispersion spectra and their total and projected densities of states of X₂TiAl (X = Au, Ru, and Zr) alloys.

Figure 6:

Variations in the Debye temperature for X₂TiAl (X = Au, Ru, and Zr) alloys.

Figure 7:

The temperature dependence of the specific heat capacities at constant volume (C_V) of X₂TiAl (X = Au, Ru and Zr) alloys.

Figure 8:

Variations of thermal expansion coefficient with temperature for X₂TiAl (X = Au, Ru, and Zr) alloys.

Figure 9:

Grüneisen parameter versus temperature for X₂TiAl (X = Au, Ru, and Zr) alloys.

3.3 Thermodynamic Properties

The thermodynamic properties are obtained within the quasi-harmonic approximation for different temperatures from the energy-volume relation. One of the important thermodynamic property for a solid is the Debye temperature, which provides information about many other quantities such as specific heat, melting temperature, and elastic stiffness. It is also related to the maximum thermal vibration frequency of a solid. Figure 6 demonstrates the dependence of the Debye temperature on temperature. As can be seen from the figure, the Debye temperature decreases slowly over the temperature range of 0–1500 K for all alloys. This variation in Debye temperature with temperature indicates that the thermal vibration frequency of the particles changes with temperature. The slow variation reflects the small effect of temperature on Debye temperature. In addition, vibrational frequency is proportional to the square root of the stiffness within the harmonic approximation; the ‘stiffness’ of solids can also be estimated from Debye temperature and called as ‘Debye stiffness’ [43]. Based on that, it can be said that Debye stiffness of Ru₂TiAl is higher than that of the Au₂TiAL and Zr₂TiAl alloys.

The vibrational contributions due to temperature change to the specific heat capacities of X₂TiAl alloys at a constant volume (C_V) is estimated using ab initio calculation with the quasi-harmonic approximation, too. The results are presented in Figure 7. A rapid increase is observed up to 200 K, and then saturation is reached, which is called Dulong-Petit limit as seen in the figure. The values of specific heat capacities are close to those reported by Dulong-Petit [44] at high temperatures. However, below 200 K, heat capacity is strongly dependent on the temperature. The heat capacity reaches to zero quickly as the temperature reaches to absolute zero. Unfortunately, there is no experimental data in the literature to compare these findings for these alloys to the best of our knowledge.

The variations in volume thermal expansion coefficients of all the three alloys versus temperature are shown in Figure 8. As it is clearly seen from the figure, thermal coefficients of all alloys rapidly increase up to 300 K. After this point, it increases gradually and becomes flat. Another important parameter that is obtained from the quasi-harmonic approximation is the Grüneisen parameter, which describes the anharmonic effects in the vibrating lattice and can be used to estimate the anharmonic properties of a solid. Grüneisen parameters of all alloys are presented in Figure 9. It seems that Grüneisen parameters seem to stay flat and show little sensitivity to temperature. One can notice that the Grüneisen parameter is not zero at 0 K, suggesting that thermal expansion coefficient and heat capacity approach zero in the same asymptotic way. In fact, the Grüneisen parameter at 0 K is proportional to the logarithmic derivative of T³ coefficient in the heat capacity with respect to volume [45].