First-Principles Investigation of Structural Stability, Mechanical, Anisotropic, and Thermodynamic Properties of CeT₂Al₂₀ Intermetallics

3.1 Structural Stability

CeT₂Al₂₀ intermetallics can crystallise in face-centred cubic structure with the Fd-3m space group. In the prototypical CeT₂Al₂₀ compounds, Ce atoms occupy the 8a Wyckoff sites, the T atoms occupy the 16d Wyckoff sites, and the Al atoms occupy the 16c, 48f, and 96g Wyckoff sites. The unit cell of CeT₂Al₂₀ intermetallics is shown in Figure 1 [25]. In order to investigate the structural properties in the ground state, the total energy under different volumes was calculated and fitted to the Birch-Murnaghan equation of state (EoS). The obtained lattice constants and bulk moduli are listed in Table 1. In this table, the calculated values are slightly larger than the experimental ones [31], [32] but the error is less than 5 %, which indicates that our computational method is reliable. CeNb₂Al₂₀ has the largest bulk modulus B₀ among the five intermetallics, which indicates that CeNb₂Al₂₀ has the highest resistance against compression and volume change. In order to obtain the structural stability under high pressure, Figure 2 displays the relationship between the values of V/V₀ under different pressures. These curves are smooth and have no inflection points until 50 GPa (Fig. 2), indicating that these intermetallics show structural stability under high pressure. The curves of CeTa₂Al₂₀ and CeV₂Al₂₀ are located at the upper and lower part, respectively, which indicates that CeTa₂Al₂₀ has the highest resistance to compression and CeV₂Al₂₀ has the lowest, among the five intermetallic compounds.

Figure 2:

Equation of state for CeT₂Al₂₀ intermetallics.

Figure 3:

Phonon dispersion curves of Al; the circles correspond to experimental data.

Figure 4:

Phonon dispersion curves and density of states for CeT₂Al₂₀ intermetallics.

Generally speaking, the structural stability of solid materials mainly depends on two factors: the thermodynamic factor, which depends on the formation enthalpy; and the dynamic factor, which depends on the phonon dispersion and density of states [35], [36]. The formation enthalpy of CeT₂Al₂₀ intermetallics was calculated using the following expression [37], [38], [39], [40]:

where E_Total represents the total energy of CeT₂Al₂₀; x, y, and z are the number of atoms in the unit cell; E_Ce, E_T, and E_Al represent the total energy per atom in the solid. The calculated formation enthalpies (ΔH) for CeT₂Al₂₀ are all negative in Table 1, indicating that the CeT₂Al₂₀ compounds are all thermodynamically stable but more inclined to form CeTa₂Al₂₀ because of its lower value of ΔH.

In order to explore their dynamic stability, the phonon dispersion and density of states were calculated, which are shown in Figures 3 and 4. Figure 3 shows the phonon dispersion curves of pure Al. The solid lines are the calculated values and the scattered points are the experimental ones obtained from inelastic neutron scattering measurements [41]. The calculated values are very close to experimental ones, indicating that the present method and the results are reasonable. Figure 4 shows the phonon dispersion curves and density of states for the five CeT₂Al₂₀ intermetallic compounds. No imaginary phonon frequency is present along the Brillouin zone path in Figure 4, which indicates that the five intermetallic compounds have dynamic stability. The masses of the five intermetallics follow the order: CeTi₂Al₂₀ < CeV₂Al₂₀ < CeCr₂Al₂₀ < CeNb₂Al₂₀ < CeTa₂Al₂₀, and the maximum values of the phonon frequency gradually decrease with increasing mass. In the phonon density of states (PHDOS) also, imaginary frequencies are absent, which also confirms that the five intermetallics have dynamic stability. In summary, the calculated formation enthalpies and phonon frequencies confirm that the five CeT₂Al₂₀ intermetallics are structurally stable.

3.2 Elastic Properties

The elastic constants and elastic moduli are important parameters for engineering applications. Table 2 lists the calculated elastic constants (C_ij) and compliance matrix constants (S_ij) for CeT₂Al₂₀ intermetallics and pure Al [42], [43], [44]. The values of the elastic constants of Al are consistent with the experimental ones, indicating that the computational methods are reliable. For these cubic structures, there are three independent constants, namely C₁₁, C₁₂, and C₄₄. For mechanical stability, the elastic constants must satisfy the following conditions: C₁₁ > 0, C₄₄ > 0, C11>|C12|, (C₁₁ + 2C₁₂) > 0 [45], and so the five intermetallics are mechanically stable. Of these elastic constants, the larger constant C₁₁ indicates that the crystalline material has better ability to resist linear compression along the x-axis. Among the five CeT₂Al₂₀ intermetallics, CeTa₂Al₂₀ has the largest elastic constant C₁₁, which indicates that CeTa₂Al₂₀ has the highest ability to resist compression along the x-axis. For these intermetallics, the values of C₁₁ are obviously larger than C₄₄, indicating that they have better resistance to compressibility along the x-axis than along the z-axis. The elastic constant C₄₄ can indirectly determine the hardness of solid materials and their resistance against shear along the (100) plane [45]. CeTa₂Al₂₀ has the largest elastic constant C₄₄, which indicates that CeTa₂Al₂₀ has the highest capacity to resist shear deformation along the [100] plane and therefore corresponds to the highest hardness. In Table 2, both the elastic constants C₁₁ and C₄₄ of CeT₂Al₂₀ are obviously larger than those of pure Al, indicating that the five intermetallic compounds possess higher resistance to compression and shear deformation along the (100) plane than pure Al.

After obtaining the elastic constants C_ij and S_ij, the bulk modulus B, the shear modulus G, Young’s modulus E, and the hardness H_v were calculated by using the VRH method and the following equations [46], [47]:

Figure 5:

Calculated elastic moduli for different CeT₂Al₂₀ intermetallics.

The calculated elastic moduli of CeT₂Al₂₀ and pure Al are listed in Tables 3 and 4 together with the experimental values of Al [42], [44]. The larger values of the bulk modulus B and the shear modulus G correspond to better resistance against volume change under hydrostatic pressure and shape deformation by the shearing force. The values of the elastic modulus B follows the order CeTa₂Al₂₀ > CeCr₂Al₂₀ > CeNb₂Al₂₀ > CeTi₂Al₂₀ > Al > CeV₂Al₂₀ in Table 3, indicating that CeTa₂Al₂₀ has the highest resistance against volume deformation. The values of the elastic modulus G follow the order CeTa₂Al₂₀ > CeNb₂Al₂₀ > CeV₂Al₂₀ > CeTi₂Al₂₀ > CeCr₂Al₂₀ > Al, indicating that CeTa₂Al₂₀ has the highest resistance against shape change under external conditions. In comparison to pure Al, the five CeT₂Al₂₀ intermetallic compounds have better resistance to shear deformation. The larger moduli G and E correspond to higher hardness of solid materials. The values of G and E of the five intermetallics are obviously larger than those of pure Al, indicating that these ternary intermetallics are harder than pure Al. The hardness was calculated by using the semi-empirical equations (4). The values of hardness are 12.138, 12.827, 7.149, 13.114, 13.452, and 2.868 GPa for CeTi₂Al₂₀, CeV₂Al₂₀, CeCr₂Al₂₀, CeNb₂Al₂₀, CeTa₂Al₂₀, and Al. Figure 5 displays the elastic moduli E, G, H_v, and C₄₄ for the different CeT₂Al₂₀ intermetallics. The elastic moduliE, G, and H_v show the same trend for the different CeT₂Al₂₀ intermetallics and pure Al, indicating that the CeT₂Al₂₀ intermetallics are obviously harder than Al and that CeTa₂Al₂₀ has the highest hardness among these ternary intermetallics.

Figure 6:

Calculated B/G ratio and Poisson’s ratio for different CeT₂Al₂₀ intermetallics.

The ductile or brittle behaviour influences the high-temperature applications of solid materials. The values of the ratio B/G and Poisson’s ratio v can determine the brittle and ductile behaviour of solid materials. When the values of B/G and v are smaller than 1.75 and 0.26, respectively, solid materials exhibit brittle behaviour; otherwise, they show ductile behaviour. The calculated values of B/G and v are shown in Table 3 and Figure 6. Except for CeCr₂Al₂₀, the values of B/G and v for the other CeT₂Al₂₀ intermetallics are smaller than the critical values 1.75 and 0.26, indicating that all the CeT₂Al₂₀ intermetallics, other than CeCr₂Al₂₀, are brittle materials. CeCr₂Al₂₀ has larger values of B/G and v, indicating that it is ductile in nature among these intermetallics. The values of the elastic constants (C₁₁–C₁₂) and Young’s modulus are important parameters to estimate the plasticity of solid materials. Generally speaking, if the materials have smaller values of (C₁₁–C₁₂) and E, they have better plasticity [44]. In order to intuitively describe the plastic behaviour, the results for the CeT₂Al₂₀ compounds and the experimental values of pure Al [44] are shown in Figure 7. Al has the best plasticity, whereas the five CeT₂Al₂₀ compounds have worse plasticity than pure Al. In order to present the brittleness of CeT₂Al₂₀ intermetallics, the re-normalised hyperbolic correlations derived by (C₁₂–C₄₄)/E and G/B are shown in Figure 8, including the values of CeT₂Al₂₀ and some typical materials, namely Au, Pd, Al, Si, Ge, BN, diamond, and so on [48]. According to the Pugh criterion and the Pettifor criterion, Al and CeCr₂Al₂₀ intermetallics are ductile materials, whereas the other four CeT₂Al₂₀ intermetallics are all brittle and have high hardness, which is close to that of Si, Ge and BN. Moreover, we also calculated the melting points of CeT₂Al₂₀ compounds using the empirical equation [48]

Figure 7:

Values of C₁₁–C₁₂ and Young’s modulus for pure Al and CeT₂Al₂₀ intermetallics.

Figure 8:

Re-normalised hyperbolic correlation derived by (C₁₂–C₄₄)/E and G/B.

The values of the melting point are between 1300 and 1500 K for the five CeT₂Al₂₀ intermetallics, as shown in Table 3.

3.3 Anisotropic Properties

The anisotropic properties of solid materials are closely related to the micro-cracks. The elastic anisotropic properties are described by the universal and shear anisotropic indices A^u and A. The anisotropic constants are calculated by following equations [48]:

The calculated anisotropic constants are shown in Table 4. When A^u and A approach 0 and 1, respectively, crystalline materials show isotropic properties. According to the anisotropic constants, the indices A^u and A correspond to the same order: CeTi₂Al₂₀ > Al > CeCr₂Al₂₀ > CeNb₂Al₂₀ > CeTa₂Al₂₀ > CeV₂Al₂₀, which indicates that CeTi₂Al₂₀ has the largest anisotropy and CeV₂Al₂₀ the smallest. For CeNb₂Al₂₀, CeTa₂Al₂₀, and CeV₂Al₂₀, the indices A^u and A are very close to the characteristic values of 0 and 1, indicating that they are nearly isotropic.

In order to visually describe the anisotropic properties of CeT₂Al₂₀ intermetellics, the three-dimensional (3D) surface of the Young’s modulus E was used to further explore their anisotropy. The 3D figures are given by the following equation [49]:

Figure 9:

Surface contours of the Young’s modulus for (a) CeTi₂Al₂₀, (b) CeV₂Al₂₀, (c) CeCr₂Al₂₀, (d) CeNb₂Al₂₀, (e) CeTa₂Al₂₀, and (f) Al.

For crystalline materials, if they possess isotropic properties, their 3D surfaces will be spherical; otherwise, they will be anisotropic. The greater the degree of deviation from a sphere, the stronger the anisotropy. The 3D figures of Young’s modulus are shown in Figure 9. The figures display the degree of deviation, which is small, indicating that these compounds are anisotropic but the anisotropy is small. The 3D figures for CeNb₂Al₂₀, CeTa₂Al₂₀, and CeV₂Al₂₀ are close to a sphere and the degree of deviation from the sphere is relatively small among the six figures, suggesting that they are nearly isotropic. Based on the degree of deviation from the sphere, CeTi₂Al₂₀ has the largest anisotropy. In order to further describe the anisotropic properties of the CeT₂Al₂₀ intermetellics, the two-dimensional figures of Young’s modulus on different planes are shown in Figure 10. If the contour of the closed curve on the planes is a circle, the crystalline material will have isotropic properties; otherwise, it will have anisotropic properties. In Figure 10, the contours of CeNb₂Al₂₀, CeTa₂Al₂₀, and CeV₂Al₂₀ are close to circles on the different planes, indicating that these intermetallics have elastic properties closer to isotropic.

Figure 10:

Projections in the (001) and (110) planes of Al and CeT₂Al₂₀ intermetallics.

To further investigate the elastic anisotropy, the acoustic velocities along different directions were calculated by the Brugger equations [50]:

The solid density ρ(g/cm⁻³) and anisotropic acoustic velocities (km s⁻¹) for CeT₂Al₂₀ intermetallics and pure Al are shown in Table 5. According to (8), the elastic constants C₁₁ and C₄₄ determine the longitudinal and transverse acoustic velocities, respectively. The elastic constants C₁₁ are obviously larger than C₄₄ in Table 2, so the longitudinal wave velocities are larger than the transverse ones along the different directions. The calculated results indicate that the acoustic velocities for CeT₂Al₂₀ intermetallics are anisotropic. These CeT₂Al₂₀ intermetallics have the maximum longitudinal sound velocities along the <111> direction, especially. Furthermore, the difference in the acoustic velocities also implies the elastic anisotropy of CeT₂Al₂₀ intermetallics.

Figure 11:

Relationship between the Gibbs free energy and temperature of pure Al and CeT₂Al₂₀ intermetallics.

Figure 12:

Relationship between the bulk modulus and temperature of CeT₂Al₂₀ intermetallics.

Figure 13:

Relationship between the thermal expansion and temperature of pure Al and CeT₂Al₂₀ intermetallics.

Figure 14:

Relationship between the entropy and temperature of pure Al and CeT₂Al₂₀.

Figure 15:

Relationship between the bulk modulus and temperature of pure Al.

Figure 16:

Relationship between the entropy and temperature of pure Al.

3.4 Thermodynamic Properties

For these cubic CeT₂Al₂₀ intermetallics, we investigated the thermodynamic properties by calculating their dynamic properties under 11 lattice parameters (from 0.94a to 1.06a) with the Phonopy code. According to the obtained F (V,T) versus V data, we fitted them to the EoS and obtained the thermodynamic quantities under different temperatures. The relationships between the thermodynamic quantities and temperature for CeT₂Al₂₀ intermetallics and Al are shown in Figures 11–14. Because the melting point of Al is ∼960 K, the thermodynamic quantities between 0 and 900 K are plotted for pure Al. Figure 11 shows the relationships between the Gibbs free energy and temperature for pure Al and CeT₂Al₂₀. In Figure 11, the values of the Gibbs free energy obviously decrease with temperature and become more and more negative, indicating that CeT₂Al₂₀ intermetallics satisfy the condition for thermodynamic stability at elevated temperature. In the meantime, the values of the Gibbs free energy of CeT₂Al₂₀ become more negative than those of pure Al with the increase in temperature, which indicates that CeT₂Al₂₀ intermetallics have better thermal stability than pure Al at high temperatures. Figure 12 shows the relationship between the bulk modulus and temperature of pure Al and CeT₂Al₂₀ intermetallics. The values of the bulk modulus become smaller with higher temperature. But the variation in the case of pure Al is larger than that of CeT₂Al₂₀ intermetallics, indicating that the five intermetallics have better ability to resist volume change than pure Al. Figure 13 shows the relationship between the thermal expansion coefficients and temperature of pure Al and CeT₂Al₂₀. In the low-temperature region, the values of pure Al and CeT₂Al₂₀ intermetallics are close to each other. But in the high-temperature region, the values of pure Al are obviously larger than those of other five CeT₂Al₂₀ intermetallics at the same temperature, indicating that the pure Al deforms more easily at high temperature. Figure 14 shows the relationship between the entropy and temperature of pure Al and the CeT₂Al₂₀ intermetallics. In Figure 14, the values of entropy of pure Al and CeT₂Al₂₀ intermetallics increase rapidly and linearly with temperature. In order to prove the correctness of our calculation methods, we compare the theoretical values and experimental ones [51], [52] of the bulk modulus and entropy for pure Al in Figures 15 and 16. The calculated values are very close to the experimental ones, indicating that our calculation methods and results are reliable.