Investigations of Physical Properties of XTiH₃ and Implications for Solid State Hydrogen Storage

3.1 Ground State Properties and Mechanical Stability

The unit cells of CaTiH₃ and MgTiH₃ with a perovskite structure in space group Pm3¯m (no.221) are depicted in Figure 1. Titanium (Ti) atoms are placed at the centre (1/2, 1/2, 1/2), Calcium (Ca)/Magnesium (Mg) atoms are positioned at corners (0, 0, 0) and hydrogen (H) atoms are positioned (1/2, 0, 0). The structures are optimised for both materials by computing total energy versus volume and then this is fitted to the Murnaghan’s equation of state [42] to extract equilibrium lattice constants (a) and the Bulk modulus (B). These parameters are presented in Table 1 along with elastic constants and Cauchy pressures of materials. Also, the stabilities of materials are verified by calculating formation energies of the materials using the descriptions in [43], [44]. The obtained values are −32 eV and −33 eV for MgTiH₃ and CaTiH₃, respectively. Obtained formation energies indicate that both materials are stable.

Figure 1:

Crystal structures of MgTiH₃ and CaTiH₃.

Bulk modulus of a material gives information about resistance capacity towards volume change when the materials is under pressure from all its surfaces. From Table 1, it can be said that MgTiH₃ is the hardest compared to CaTiH₃ and creating a deformation on this material requires much energy than CaTiH₃. Unfortunately, there is no values found in literature for comparison to the best of the author’s knowledge for these materials.

Elastic constants are fundamental and crucial properties of materials. They can ensure a link between mechanical and dynamical properties of a material such as internal forces, stability, brittleness, ductility, and stiffness [45]. They are also related to thermodynamic properties through the Debye theory. For a cubic material, there are three independent elastic constants; C₁₁, C₁₂ and C₄₄.

Mechanical stability evaluations of materials are carried out using the computed elastic constants. The well-known Born mechanical stability criteria via elastic constants is given as follows [46];

Equating (9) also restricts bulk modulus of the material as;

By evaluating elastic constants using (8) and (9), it can be seen that both compounds fulfill the mechanical stability criteria. Thus, MgTiH₃ and CaTiH₃ compounds are both mechanically stable. Generally, C₁₁ presents elasticity in length whilst C₁₂ and C₄₄ are shear constants and demonstrate elasticity in shape. Therefore, the resistance to change in length can be predicted using C₁₁. CaTiH₃ has smaller value of C₁₁ than MgTiH₃, showing that CaTiH₃ is more compressible along x-axis. This is consistent with the value of bulk modulus of compounds as MgTiH₃ requires much energy to create a deformation.

Various physical properties including magnetic moments of compounds are presented in Table 2. The shape change of a material is also associated with the Shear modulus (G). The Shear modulus of compounds is slightly different from each other. The value of Shear modulus of CaTiH₃ is slightly higher than that of MgTiH₃.

Further investigation is carried on brittleness and ductility of compounds. The Cauchy pressures given in Table 1 could be also adopted to describe angular characteristic of atomic bonding and ductility of compounds. The negative Cauchy pressure is associated with the directional bonding with angular character and brittleness. A directional character dominates as the pressure becomes more negative. Contrarily, a more positive Cauchy pressure is associated with more metallic and ductile nature [47]. In addition, the ratio of Bulk modulus to Shear modulus (B/G) is also related to ductility and brittleness. Pugh [48] proposed that there is a transition from ductile to brittle at about 1.75. If that ratio is smaller than this value, the material is brittle, otherwise the material is ductile. Based on the Cauchy pressure (6.41) and B/G (1.85) of MgTiH₃ it can be said that this compound has a ductile nature. On the other hand, CaTiH₃ has a negative value of Cauchy pressure (−8.03) and B/G (1.52) ratio, indicating a brittle and directional angular character. In hydrogen storage application, ductility and brittleness of materials is extremely important. Ductile materials will be easy to handle, especially for portable application; however, brittle materials will require extra caution. Thus, ductile materials are preferable for hydrogen storage. Therefore, it can be said that MgTiH₃ is the much preferable material in this study.

In addition to previous properties, Poisson’s ratio of compounds are also investigated. Poisson ratio can reflect the stability against shear and bonding properties of materials. As the Poisson’s ratio goes up, material’s plastic nature rises. A correlation is proposed as about 0.1 for materials with covalent characteristics and about 0.25 for materials with ionic characteristics for Poisson’s ratio in literature [49]. From Table 2, Poisson’s ratio of both the compounds indicate an ionic nature due to about 0.25 Poisson ratio.

A material’s stiffness and tensile elasticity is characterised by the Young’s modulus (E) which is also a ratio between tensile stress to tensile strain. As the Young’s modulus increases, the material gets stiffer. The Young’s modulus compounds in this study are close to each other, meaning that both compounds have similar stiffness.

The anisotropy factor (A) is an indication of anisotropy degree in materials. It is related to micro cracks, precipitation, anisotropic plastic deformation, elastic instability, and internal friction [50], [51]. If A = 1, the material is completely isotropic, deviation from this value shows the anisotropy degree. Table 2 shows that both compounds have anisotropy since A is lower than 1. Since both compounds are elastically anisotropic, 2D directional change of compressibility, Poisson’s ratio, Shear and Young’s modulus of MgTiH₃ and CaTiH₃ have been computed using EIAM code [52] and the results are presented in Figure 2. The maximum and minimum values of Young’s modulus and Poisson’s ratio are presented in Table 3. Figure 2 illustrates that compressibility of compounds are spherical and isotropic whereas Poisson’s ratio, Shear and Young’s modulus of compounds are deviated from spherical shape and become anisotropic at all planes.

Figure 2:

2D directional change of compressibility, Poisson’s ratio, Shear and Young’s modulus of MgTiH₃ and CaTiH₃.

The total magnetic moments of compounds are also computed. MgTiH₃ has a total magnetic moment of 0.41 μ_B per formula unit and CaTiH₃ has 1.54 μ_B total magnetic moment, meaning that both compounds are magnetic.

3.2 Electronic Properties

The electronic band structures of MgTiH₃ and CaTiH₃ have been predicted along the high symmetry directions in the Brillouin zone using the equilibrium lattice constants. The computed electronic band structures and their related partial and total density of states of the compounds are presented in Figures 3 and 4, respectively. As shown in Figure 3 that both compounds show metallic behaviour since the electronic band structures of them have no band gap at the Fermi level (E_F). Both majority and minority spin states determine the metallic nature of compounds by cutting the Fermi level. Also, the valence and conduction bands overlap perfectly that both compounds are expected to demonstrate metallic electrical conduction.

Figure 3:

Electronic band structures of MgTiH₃ and CaTiH₃.

Figure 4:

The total and partial density of states of MgTiH₃ and CaTiH₃.

Density of states (DOS) is one of the critical properties that describe the allowed electron states per unit at a given energy. Figure 4 clearly depicts that both compounds are metallic since DOS is not zero at the Fermi level. For MgTiH₃, between −6 eV and −4 eV the biggest contribution to the conduction comes from the 1s orbital of H atom and the 3d orbitals of Ti atom whereas between −1 eV and 1 eV the major contribution to the DOS are due to 3d orbitals of Ti atom. For CaTiH₃, the majority of contribution to the conduction between −6 eV and −4 eV is due to 1s orbital of H atom and 3d orbitals of Ti atom whilst between −1 eV and 1 eV, the most contribution to the DOS is owing to 3d orbitals of Ti and Ca atoms.

3.3 Thermodynamic Properties

Several thermodynamic parameters for both compounds are estimated within the quasi-harmonic approximation for a range of temperatures from the energy-volume relation. The temperature dependence of specific heat capacities at a constant volume and pressure are predicted and presented in Figure 5. The compounds’ specific heat capacities increase up sharply until 600 K and then the increase slows down and finally they are expected to reach the Dulong–Petit limit [53] where a saturation limit happens. The specific heat capacities are dependent upon temperature greatly up to 600 K and they are zero at 0 K. This is presented as new data to the literature, thus no data has been found in literature for comparison.

Figure 5:

The change in specific heat capacities at constant volume (C_V) and pressure (C_p) of MgTiH₃ and CaTiH₃.

The thermal expansion coefficient gives information about a change in size of a material with temperature. More specifically, it defines the fractional change in size when the temperature changes per unit at a constant pressure. The main contribution to thermal expansion comes from vibrations of lattice. As temperature increases, the thermal energy of atoms increases and so thus vibrations of atoms within the crystal. Vibrations of atoms result in a rise in distance of atoms and thermal expansion coefficient. Figure 6 illustrates variations in thermal expansion coefficients of compounds. A rapid increase is seen on both thermal expansion coefficients up until 500 K and above 500 K, a steady increase is observed in temperature for both compounds.

Figure 6:

The change in thermal expansion coefficient versus temperature for MgTiH₃ and CaTiH₃.

Grüneisen parameter of compounds are also calculated using the quasi-harmonic approximation. It defines the anharmonic impact on the vibrating lattice and could be adopted to estimate a material’s anharmonic properties. Grüneisen parameters for both compounds are given in Figure 7 which clearly depicts that Grüneisen parameters of compounds show little change with temperature. It can be also noted that Grüneisen parameter differs from zero at 0 K, indicating that thermal expansion coefficient and heat capacity becomes zero in the same asymptotic way. Generally, Grüneisen parameter at 0 K is proportional to the logarithmic derivative of T³ coefficient in the heat capacity in regards to volume [54].

Figure 7:

Changes in Grüneisen parameter versus temperature for MgTiH₃ and CaTiH₃.