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Effect of pressure on structural, elastic and mechanical properties of transition metal hydrides Mg7TMH16 (TM = Sc, Ti, V, Y, Zr and Nb): First-principles investigation
Accepted Manuscript Effect of pressure on structural, elastic and mechanical properties of transition metal hydrides Mg7TMH16 (TM = Sc, Ti, V, Y, Zr and Nb): First-principles investigation Kamel Benyelloul, Larbi Seddik, Youcef Bouhadda, Mohamed Bououdina, Hafid Aourag, Khadidja Khodja PII:
S0022-3697(17)30326-8
DOI:
10.1016/j.jpcs.2017.08.001
Reference:
PCS 8155
To appear in:
Journal of Physics and Chemistry of Solids
Received Date: 20 February 2017 Revised Date:
3 May 2017
Accepted Date: 1 August 2017
Please cite this article as: K. Benyelloul, L. Seddik, Y. Bouhadda, M. Bououdina, H. Aourag, K. Khodja, Effect of pressure on structural, elastic and mechanical properties of transition metal hydrides Mg7TMH16 (TM = Sc, Ti, V, Y, Zr and Nb): First-principles investigation, Journal of Physics and Chemistry of Solids (2017), doi: 10.1016/j.jpcs.2017.08.001. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Effect of pressure on structural, elastic and mechanical properties of transition metal hydrides Mg7TMH16 (TM= Sc, Ti, V, Y, Zr and Nb): First-
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principles investigation
Kamel Benyelloula,c, Larbi Seddika, Youcef Bouhaddaa, Mohamed Bououdinac, Hafid
a
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Aouragb, Khadidja Khodjaa
Unité de Recherche Appliquée en Energies Renouvelables, URAER, Centre de
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Développement des Energies Renouvelables, CDER, 47133, Ghardaïa, Algeria Department of Physics, College of Science, University of Bahrain, P.O. Box 32038, Bahrain
c
Department of Physics, University of Tlemcen, 13000, Algeria
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b
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ACCEPTED MANUSCRIPT Abstract The effect of pressure on structural stability, elastic properties and Debye temperature of face centred cubic Mg7TMH16 (TM= Sc, Ti, V, Y, Zr and Nb) hydrides, was investigated by firstprinciples calculations based on density functional theory (DFT) with the generalized gradient
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approximation (GGA). The obtained equilibrium lattice parameters and elastic properties at zero pressure for MgH2 and Mg7TMH16 hydrides, are in good agreement with other experimental and theoretical values. The calculations of the bulk modulus and the ductility
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factors (B/G) show that mixing (MgH2) with small amount of transition metal (TM= Sc, Ti, V, Y, Zr and Nb) can enhance the resistance to volume change and transform it from a brittle
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to a ductile material (brittle ductile). The elastic constants, bulk modulus, shear modulus, Young’s modulus, anisotropy factor and hardness have been studied under pressure. These mechanical quantities are found to increase with increasing pressure. While the B/G and Poisson’s ratios (v) undergo an inverse behavior. In addition to that, the studied hydrides are
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found stable with a ductile behaviour under a pressure between 0 and 20 GPa. Furthermore, the effect of pressure on Debye temperature and sound velocity, was also investigated and
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discussed.
Keywords: Ab initio calculations; Magnesium-transition metal hydrides; Elastic properties,
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high pressure.
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ACCEPTED MANUSCRIPT 1. Introduction Recently, extensive researches were devoted toward hydrogen as clean and sustainable energy source. It could potentially replace rich-carbon energies (such as oil, wood, natural gas, etc…) in the short time. Unlike fossil fuel, it is environmentally friendly, non polluting and abundant
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fuel [1], which can offer a new prospective economy. However, the transition from a fossil economy to hydrogen economy must resolve the mobile application challenges especially the on-board hydrogen storage. Several approaches for hydrogen storage are well-known
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including compressed gas and liquid hydrogen. But the storage of hydrogen in the “solid” form as metallic hydrides is considered the most encouraging. Indeed, the metal hydrides
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offer a safer alternative and have a higher volumetric density than liquid and gaseous forms [2].
Due to its high hydrogen content (7.6 Wt%) [3-4], lightweight, low cost, abundance and nontoxicity, magnesium hydrides (MgH2) is considered a very promising and potential candidate
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thereby has been so far extensively investigated. Nevertheless, its high thermodynamic stability and slow absorption/desorption kinetics [4,5], limits its use for mobile applications in particular fuel cells for electric cars. Therefore, Mg-based hydrides are usually destabilized,
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for example, by 3d transition elements incorporation [3-6] or surface modification [7] to improve its de-/hy-driding performance. In current decades, Mg-transition 3d- metal hydrides
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have received much attention and have widely investigated both experimentally [4,8,9] and theoretically [3,10]. Experimentally, Takeichi et al. [11] found that Mg6VNaxHy (0 ≤ x ≤ 1) reversibly desorb and absorb hydrogen at 620–630 and 590–600 K, respectively, under 0.5 MPa (H2) which are 70 and 120 K respectively, lower than those of MgH2. However, the reaction enthalpies calculated from the Van’t Hoff relation from PCIs (Pressure Composition Isotherms) do not significantly differ from that of MgH2.
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ACCEPTED MANUSCRIPT The Mg-transition metal hydrides Mg7TiHx and Mg7NbHx hydrides with a face-centered cubic (FCC) super-lattice structure with Ga7Ge-type (Space group 225) have been synthesized at 8 GPa and 873 K (for Mg7TiH16) and 600 K (for Mg7NbH16) using ultra-high pressure techniques [8,9]. Moser et al. [12] synthesized (at 600°C and 4 GPa) a series of hydrogen rich
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Mg6–7TMH14–16 (TM= Ti, Zr, Hf, V, Nb and Ta) hydrides. Theoretically, Song et al. [13] have used first principles calculation to investigate the influence of Al, Ti, Fe, Ni, Cu and Nb on MgH2 hydride. They found that the alloying elements decrease the heat of formation of
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(Mg,X)H2 and can enhance its kinetics. In other hand, Dai et al. [7] have also employed DFT calculations to study the preferential sites occupancy of some dopants such as Al, Ti, Mn and
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Ni on MgH2 onto (110) and (001) surfaces.
Face-centered cubic (FCC) magnesium–transition metal (TM) hydrides Mg7TMH16 (TM= Sc, Ti, V, Y, Zr, Nb) were also investigated [3,10,14,15] using first-principles calculations based on DFT method. Indeed, Xiao et al. [3] have calculated the enthalpies of formation and
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analyzed the stability of Mg7TMH16 using both cohesive energies and electronic densities of state. Whereas, Shelyapina et al. [16] have discussed the stability of Mg7TMH16 using both the calculated electronic structure and heat of formation. For these hydrides, the mechanical
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properties have not yet been investigated experimentally or theoretically (Except for Mg7TiH16 [10] and Mg7NbH16 [16] where the theoretical investigations were done at ambient
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pressure).
Indeed, the knowledge of the mechanical properties under pressure, has a great importance in the design of hydride reservoir. The hydrogen absorption causes a significant increase in volume (up to 25%) and creates important stresses within the storage vessel. However, the initial volume could be recovered after desorption. The successive expansion/contraction of volume after several cycles, with a pressure variation, have a great effect on the morphological and microstructural transformations and therefore can reduce the life-time
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ACCEPTED MANUSCRIPT stability of the storage vessels. Consequently, an accurate knowledge of mechanical properties of hydrides, especially their pressure dependence, is of considerable importance in both fundamental science and engineering technology [17,18]. The availability of reliable and assessed information about the effect of pressure is often not
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accessible or rare. The quantum mechanical simulation based on the first principles calculation can be alternative to experimental process with a reduced cost. Moreover, it is a powerful approach and proficient means for solving different problems of material science
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field and may be used in the determination of elastic properties, under different pressures. In the current study, the structural stability, elastic and mechanical properties as well as
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Debye temperature of cubic Mg7TMH16 (TM= Sc, Ti, V, Y, Zr and Nb) hydrides, under pressure are investigated through first principles calculation within the density functional theory (DFT).
The rest of the paper is organized as follows. In section 2, details and computational methods
present work is given.
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are presented. In section 3, the obtained results were analysed. Finally, a conclusion of the
2. Computational methods
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All compounds Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) and MgH2 were investigated by the density functional theory (DFT) [19]. The projector augmented wave (PAW) method
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[20,21] as implemented in the Vienna Ab-initio Simulation Package (VASP) code [22,23] was employed. Also, the exchange-correlation interaction was treated and described using the generalized gradient approximation (GGA) as parameterised by Perdew-Wang 91 (PW91) [24]. The convergence is achieved when the cut-off energy is 450 eV and 375 eV for MgH2 and Mg7TMH16, respectively. For the Brillouin zone integrations we use the Monkhorst-Pack scheme [25] and the k-point were sampled by 7 × 7 × 7 and 7 × 7 × 9 for Mg7TMH16 and TMgH2, respectively. The crystal structure of the tetragonal MgH2 and the stable Ga7Ge-type
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ACCEPTED MANUSCRIPT phase of Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) are illustrated in Fig.1. For rutile (tetragonal) MgH2 with space group P42/mnm [26], Mg atoms occupy the 2a (0,0,0) sites, while H atoms occupy the 4f (0.304, 0.304, 0) sites. However, for Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) have a face-centered cubic (fcc) with space group Fm-3m [3,4,8]. The unit cell
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of Mg7TMH16 contains 96 atoms, where Mg atoms occupy 4b (1/2,1/2,1/2) and 24d (0,1/4,1/4) sites, respectively, TM atoms occupy the 4a (0,0,0) sites and H atoms are localized in two tetrahedral sites 32f (0.094,0.094,0.094) and (0.365,0.365,0.365) sites.
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For an all-electron description the full-potential linear augmented plane wave (FP-LAPW) method [27] implemented on Wien2k code was used [28]. The cut-off parameters RMT.KMax
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were taken as 7.0 and 6.0 Ryd for MgH2 and Mg7TMH16, respectively. The value of 20 for Gmax was used on a Fourrier expansion of potential in the interstitial region. The respective Brillouin zone integration is performed using a k-mesh of 8 × 8 × 13 and 11 × 11 × 11 , equivalent to 105 k-points and 56 k-points for MgH2 and all Mg7TMH16, respectively. The muffin-tin
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(MT) radius RMT around each atom which are taken to be 2 and 1.5 for Mg and H, respectively in MgH2 and 1.45, 1.53, 1.55, 1.57, 1.63, 1.61, 1.65 and 1.22 for Mg, Sc, Ti, V, Y, Zr, Nb and H respectively in Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb).
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The elastic constants are computed by applying the strain energy-strain curve method [29,30]
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using both VASP and Wien2k code [31]. Indeed, there are six independent elastic constants (C11, C12, C13, C33, C44, C66) for tetragonal structure and three independent elastic constants (C11, C12, C44) for the cubic structure, and were deduced from the variation of total energy calculations (and the second-order derivative energy) by applying five small dimensionless strains δ with ( δ = 0,±0.01,±0.02 ). More details can be found in Ref [29] and Ref [30].
3. Results and discussion 3.1. Structural and elastic properties at zero pressure
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ACCEPTED MANUSCRIPT The calculated equilibrium lattice parameters, bulk modulus (B) and its pressure derivative (B’) for MgH2 and Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) are displayed in Table 1. Compared to the experimental values, the optimized lattice parameters for MgH2 hydride are consistent with the literature [32]. In the case of Mg7TiH16, Mg7VH16 and Mg7NbH16, the
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obtained lattice constants are reasonable in accordance with the experimental data [4, 9] and available theoretical values [3, 14-16] (the corresponding relative errors are -2.04 % and 1.69 %, for Mg7TiH16 and Mg7NbH16, respectively). From the remaining hydrides Mg7TMH16
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(TM= Sc, Y and Zr), the obtained structural parameters are also in an excellent agreement with previous theoretical results reported by Xiao-Bing Xiao et al. [3]. It was also found that
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the obtained bulk moduli by fitting the total energy as a function of volumes [33], are in good agreement between our calculated and theoretical values mentioned in Refs [10,14,15] for Mg7TMH16 and Refs [34-37] for MgH2.
It should be noted that the values (see Table 1) calculated by Xiao-Bing Xiao et al. [3] (>419
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GPa) for Mg7TMH16, differ completely from both our calculated values as well as other values reported in the literature [10, 14, 15]. In fact, the values found by Xiao-Bing Xiao et al. [3] are larger and comparable to diamond's values.
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As shown in Table 1, we can notice that the calculated values of bulk modulus for Mg7TMH16 hydrides are larger than (MgH2) value, which indicates that mixing MgH2 by small amount
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(12.5%) of transition metal (TM) can enhance the resistance to volume change in the hydrides.
As mentioned earlier, the elastic constants are very important, since they help us to better understand and measure the resistance of the hydrogen tank to the developed stress during charge/discharge cycles. Therefore, we have calculated at zero pressure the elastic constants as reported in Table 2. It is clear that our calculated elastic constants for MgH2, Mg7TiH16 and Mg7NbH16 using both PAW and FP-LAPW methods, are generally in reasonable accordance
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ACCEPTED MANUSCRIPT with the available theoretical calculations [14-15, 36-37]. However, for the other studied hydrides (Mg7TMH16 with TM= Sc, V, Y and Zr ), no experimental or theoretical elastic constants are available. The as-obtained values can be considered as reference for future experimental and/or theoretical research works. Also, because of the lack of any experimental
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values of elastic constants, we have used the more accurate FP-LAPW approach [38] as a benchmark as suggested by many experts and authors [38-40] in order to confirm our results. The values are reported in Table 2. The FP-LAPW calculations are in good agreement with
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our PAW calculations.
Based on the obtained elastic constants, we can observe that the independent Cij’s satisfy the
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classical Born criteria for tetragonal and cubic structures given in Ref [41]. Therefore, the calculated structures are mechanically stable at zero pressure.
According to Voigt-Reuss-Hill (VRH) approximation, the arithmetic average shear modulus (GV, GR, GH), bulk modulus (BV, BR, BH), Young’s modulus (EH) and Poisson’s ratio (ν) can
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be estimated from the elastic constants (Cij) by applying the relations given in Refs [42-45]. The obtained elastic moduli are summarized in Table 3. We found that the values of bulk modulus (B) obtained from the universal equation of state (EOS) [33] are close to the values
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calculated using the elastic constants. To the best of our knowledge, only the values for MgH2, Mg7TiH16 and Mg7NbH16 are available in the literature [14,15,36] and are in good
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agreement with our calculated values. In order to predict the brittleness or the ductility behaviour of our studied hydrides, we calculated the ratio between bulk modulus and shear modulus (B/G) as introduced by Pugh [46]. Our calculation shows that the ratio B/G of MgH2 hydride (1.46) is lower than the critical value (1.75), thereby the material has a brittle behaviour. On the other hand, the ratio B/G of Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) hydrides is found to be greater than the critical value, which indicates that these hydrides are ductile. In addition, the universal
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ACCEPTED MANUSCRIPT Poisson’s ratio (ν) can also be applied to characterize materials: a material is ductile if ν is higher than 0.26 (ν ≥ 0.26), otherwise the material is brittle [47]. From our results listed in Table 3, we can notice that the value of ν is less than 0.26 for MgH2 and larger than 0.26 for Mg7TMH16, which is consistent with the results found earlier using the ratio B/G. So, it can
also induce mechanical property brittle-ductile transformation. 3.2. Elastic properties under pressure
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be concluded that mixing (MgH2) with a small amount (12.5%) of transition metal (TM) can
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In the following, we studied the behaviour of structural and mechanical properties under pressure variation between 0 and 20 GPa. Hence, we used the third order Birch Murnaghan
3 V P = B 2 V 0
−
7 3
V − V0
−
5 3
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equation (Eq. 1) to plot the pressure vs. unit cell volume [33]: 1 + 3 ( B ' − 4 ) V V 0 4
−
2 3
− 1
(1)
where V is the unit cell volume at external pressure, V0, B and B’ is the equilibrium volume,
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bulk modulus and pressure derivative B’, respectively, at zero pressure. Fig.2 displays the variation of unit-cell volume of Mg7TMH16 with pressure. For Mg7YH16, the ratio V/V0 decreases more significantly compared to the other hydrides. Moreover, for
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Mg7NbH16 compounds the ratio V/V0 decrease more slowly. Based on the bulk modulus (B)
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definition, we can notice from Table 1 that the hydrides having low bulk modulus (Mg7YH16, B = 60.71 GPa) are easily compressed while hydrides with high bulk modulus (Mg7NbH16, B = 73.63 GPa) are difficult to compress. Following the fitting of pressure-volume curves of second-order polynomial expressions shown in eq. 1, the predicted ratio V/V0 values and their corresponding models found to be in good agreement with coefficients of determinations R2 ≈ 1. The mentioned expressions for Mg7ScH16, Mg7TiH16, Mg7VH16, Mg7YH16, Mg7ZrH16 and Mg7NbH16 are reported in Table 4.
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ACCEPTED MANUSCRIPT Based on the corresponding optimized structures at several applied pressures, we calculated the three independent elastic constants for the studied hydrides. The obtained results are depicted in Fig. 3. The first remark is that all the hydrides are stable in the pressure range between 0 and 20 GPa (all Cij are positive). Also, we found that the C11, C12 and C44 increase
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monotonically with the applied external pressure, meanwhile the curve of the elastic constant C11 is located far above the other elastic constants and the value significantly increases with increasing pressure compared to C44 and C12 (the effect of pressure is slight). Knowing that,
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C11 is related to the unidirectional compression along the principle crystallographic direction [100], we can conclude that Mg7TMH16 hydrides become more and more incompressible
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along the principle [100] direction with increasing pressure.
It is well known that the hardness of a material is measured by the bulk modulus (B), shear modulus (G) and Young’s modulus (E) [48]. The bulk modulus is a measure of the resistance of a material to volume change by applying pressure. However, shear modulus can be used to
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measure the resistance to reversible deformations upon shear stress [49] and Young’s modulus measures the stiffness of a material [50]. The evolution of B, G and E of Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) hydrides under pressure is shown in Fig. 4. Starting from the
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initial values of B, G and E at 0 GPa, the representative curves are found to increase with increasing pressure. Also, we can observe that under pressure, the values of bulk modulus are
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larger than that of shear modulus which means that the resistance to volume change is increased. Therefore, the hydrides tend to resist to volume change better than shape change under pressure. The larger the value Young’s modulus is, the stronger of stiffness is. Based on the available standard relation, the hardness (H) of compounds can be determined using the Young’s modulus (E) and Poisson’s ratio (ν) by means of the following expression [48]: H=
( 1 − 2ν)E 6( 1 + ν)
(2)
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ACCEPTED MANUSCRIPT At different pressures, the obtained values of hardness (H) of the studied hydrides are presented in Fig. 5. Starting from the corresponding initial values of H at zero pressure, we can see that the hardness increases with increasing pressure, indicating that the hardness of Mg7TMH16 hydrides can be increased by increasing pressure.
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Fig. 6 shows the calculated ductility factors (B/G and ν) for the studied hydrides under several pressures. It is clearly seen that for pressure range (0-20 GPa), the values of the ductility factors (B/G and ν) are higher than 1.75 and 0.26, respectively, revealing that the studied
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hydrides have a ductile nature (B/G >1.75 and ν ≥ 0.26). Also, it can be seen that the ratios (B/G and ν) varied through three steps before it becomes stationary. Within the range (0-2
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GPa), the ratios are decreasing rapidly, then decrease slightly within the range (2-12 GPa) until reaching a stable regime after pressure (P) = 12 GPa. Surprisingly, Mg7VH16 hydride shows an insignificant increase of ratios at P = 4 GPa. We deduced that the studied hydrides become less ductile when the pressure increases. In addition, the Poisson’s ratio is usually
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used to quantify the stability of crystal structure against shear, and the larger the Poisson ratio the better the plasticity [50]. Accordingly, it can be deduced that the applied external pressure caused weakness plasticity of the mentioned hydrides. The curves of Fig. 6 show that the
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hydrides Mg7TiH16, Mg7NbH16 and Mg7ScH16 reach the same B/G and ν ratios (2.10, 0.295)
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at P = 8 GPa, whereas the hydrides Mg7ZrH16 and Mg7YH16 have the same level values (2.2, 0.30) at P = 10 GPa.
The elastic anisotropy A is also an important elastic property, especially under external conditions like high pressure. At zero pressure, the estimations of the anisotropic factor A for the cubic phase using the standard relation A = 2C44 /(C11 − C12 ) [29] are 0.22, 0.31, 0.36, 0.14, 0.28 and 0.33 for Mg7ScH16, Mg7TiH16, Mg7VH16, Mg7YH16, Mg7ZrH16 and Mg7NbH16 respectively. For isotropic crystal, A is equal to 1 while any values smaller or larger than 1 indicate an anisotropy [29]. From the obtained results, it is clear that all hydrides are
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ACCEPTED MANUSCRIPT anisotropic. From Fig. 7, the values of A increase with increasing the external pressure ranging up to 20 GPa for Mg7TMH16 hydrides, indicating that their anisotropy is less apparent when the pressure is increased. 3.3. Debye temperature under pressure
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The Debye temperature ( TDebye ) is usually related to many physical properties such as specific heat, elastic constants, and melting point [50,51]. Debye temperature can be determined from bulk and shear moduli, using the average sound velocity ( vm ) by means of the following
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equation [52]:
(3)
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1/ 3
h 3n N ρ TDebye = ( A ) vm k 4π M
where h is Planck’s constant, k is Boltzmann’s constant, N A is Avogadro’s number, ρ is the density of molecule, M is the molecular weight and n is the number of atoms in the molecule.
)
−1 / 3
(4)
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1 2 1 vm = ( 3 + 3 3 vt vl
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The average sound velocity ( vm ) in polycrystalline materials is given by [52]:
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4 where vt = G / ρ and vl = (B + G) / ρ are the transverse and longitudinal elastic wave 3 velocities of the polycrystalline materials, respectively [52]. The calculated transversal and longitudinal wave velocities, average sound velocity and Debye temperature of MgH2 and Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) hydrides at normal pressure and zero temperature are listed in Table 5. The obtained Debye temperature of Mg7TiH16 and Mg7NbH16 are 643 and 596 K, respectively, which are relatively close to the theoretical values 618.7 [14] and 619.7 K [15], respectively. However, we found a value of 770 K for MgH2 which differs to the value 854.8 K calculated by Hector et al. [36]. This can
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ACCEPTED MANUSCRIPT be explained by the computational convergence (different cut-off energy) and different PAWGGA parameterization. Unfortunately, no other available experimental data that could be used for further comparison. The Debye temperature of the hydrides gradually decreases in the following order: TDebye
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(Mg7VH16)> TDebye (Mg7TiH16)> TDebye (Mg7ScH16) and TDebye (Mg7NbH16)> TDebye (Mg7ZrH16)> TDebye (Mg7YH16). It is noticed that the Mg-transition metal hydrides namely [Mg7ScH16, Mg7TiH16, Mg7VH16] and [Mg7YH16, Mg7ZrH16, Mg7NbH16], having high bulk
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modulus have a large Debye temperature, respectively.
The computed results of Debye temperature (TDebye), transversal (vt) and longitudinal wave
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velocities (vl) as well as the average sound velocity (vm) of Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) hydrides under different pressures are shown in Figs. 8 and 9, respectively. To describe the relationship between Debye temperature (TDebye) and applied pressure, the calculated data points were fitted by using a third-order polynomial, the obtained equations
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are reported in Table 6.
We observe that the values of Debye temperature, transversal, longitudinal and average sound velocities increase with the increase of pressure. In Fig. 8, it can be seen that the TDebye of
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Mg7VH16, Mg7TiH16, Mg7ScH16 series and Mg7NbH16, Mg7ZrH16, Mg7YH16 series is more sensitive to pressure variation when their bulk and shear moduli are important. This can be
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explained by the direct correlations that exist between vt, vl, and vm and elastic moduli (B and G) (Eqs.3 and 4). 4. Conclusion
Based on density functional theory with generalized gradient approximation, the structural, elastic properties and Debye temperature of Mg7TMH16 (TM= Sc, Ti, V, Y, Zr and Nb) hydrides under pressure, have been investigated.
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ACCEPTED MANUSCRIPT At zero pressure, the obtained structural parameters, bulk modulus and elastic properties determined by strain energy-strain curve method for MgH2 and Mg7TMH16 are in good agreement with previous theoretical results and experimental data. Analysis of single-crystal and polycrystalline elastic parameters revealed that MgH2 and Mg7TMH16 hydrides are
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mechanically stable. The calculated bulk modulus and ratio B/G of Mg7TMH16 are larger than MgH2, thus suggesting that the mixing magnesium hydrides by small amount (12.5%) of transition metal (TM) can enhance the resistance to volume change and can also convert
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MgH2 from brittle to ductile material (brittle ductile).
It was found that the applied external pressure causes weakness of ductility and plasticity of
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Mg7TMH16 hydrides. The obtained elastic anisotropy A values were lower than 1. Also, the anisotropy of the studied hydrides is less apparent when the pressure is increased. Using the elastic moduli the transversal (vt), longitudinal (vl), and average sound velocities (vm) as well as Debye temperature (TDebye) for the studied hydrides, are calculated at zero pressure and
behaviour of TDebye.
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enhance with increasing pressure. In other hand, polynomials have been used to describe the
It is noticed, that some of our theoretical results for the different hydrides Mg7TMH16 where
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TM= Sc, Ti, V, Y, Zr and Nb, are obtained for the first time such as elastic properties (elastic constants, bulk modulus, shear modulus, Young’s modulus, Debye temperature) and the
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corresponding variation under pressure. For our knowledge, the elastic constants have not yet been calculated, except for the Mg7TiH16 and Mg7NbH16 hence our results can serve as a prediction for future research investigation.
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D. Moser, D. J. Bull, T. Sato, D. Noreus, D. Kyoi, T. Sakai, N. Kitamura, H. Yusa, T. Taniguchi, W. P. Kalisvaart, P. Nottene. Structure and stability of high pressure
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synthesized Mg-TM hydrides (TM=Ti, Zr, Hf, V, Nb and Ta) as possible new hydrogen rich hydrides for hydrogen storage. J. Mater. Chem.19 (2009) 8150–8161. Y. Song, Z. X. Guo, and R. Yang. Influence of selected alloying elements on the stability
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Investigation of the elastic properties of Mg7NbH16 hydride. Revue des Energies
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Renouvelables Vol. 18 N°3 (2015) 503-512. M.G. Shelyapina, D. Fruchart, P. Wolfers. Electronic structure and stability of new FCC magnesium hydrides Mg7MH16 and Mg6MH16 (M=Ti, V, Nb): An ab-initio study. Inter.
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A.H. Reshak, M. Jamal, DFT calculation for elastic constants of orthorhombic structure within WIEN2K code: A new package (ortho-elastic). J.Alloys Comp. 543 (2012) 147-
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2647. F. Birch, J. Geophys. Res: Solid Earth 83 (1978) 1257.
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L.G. Hector Jr, J.F. Herbst. Ab Initio thermodynamic and elastic properties of alkalineearth metals and their hydrides. Phys. Rev B 76 (2007) 014121-18. A. Junkaew, B. Ham, X. Zhang, R. Arroyave. Ab-initio calculations of the elastic and
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[37]
finite-temperature thermodynamic properties of niobium and magnesium hydrides. Inter.
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J. Hydrogen Energy 39 (2014) 15530-15539. K. Lejaeghere, V. Van Speybroeck, G. Van Oost, S. Cottenier. Error estimates for solids-sate density-functional theory predictions: An overview by means of the groundstate elemental crystals. Critical Reviews in Solid State and Materials Sciences, 39 (2014) 1-24.
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S.F. Pugh, Relation between elastic moduli and plastic properties of polycrystalline pure metals. Phylos. Mag. 45 (1954) 823-43.
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elastic behaviour of dense C3N4 under high pressure: First-principles calculations. J. Phys. Chem. Solids. 75 (2014) 1324-1333. [48]
Yu. Zhao, H. Hou, Y. Zhao, P. Han. First-principles study of nikel-silicon binary
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compounds under pressure. J. Alloys Comp. 640 (2015) 233-239. A.F. Young, C. Sanloup, E. Gregoryanz, S. Scandolo, R.J. Hemley, H.-K. Mao. Phys.
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Rev. Lett. 96(2006)155501. Y. Zhao, L. Qi, Y. Jin, K. Wang, J. Tian, P. Han. The structural, elastic, electronic properties and Debye temperature of DO22-Ni3V under pressure from first-principles. J. Alloys Comp. 647 (2015) 1104-1110.
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ACCEPTED MANUSCRIPT OL. Anderson. A simplified method for calculating the debye temperature from elastic
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constants. J. Phys. Chem. Solids 24 (1963)909-917.
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[52]
20
ACCEPTED MANUSCRIPT Table 1. Lattice parameter a, c (in Å), bulk modulus B (in GPa) and derivative (B’) for MgH2
Methods of
System
a
c
B
B’
4.519
3.011
52.78
3.71
4.517 a
3.020 a
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and Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) at zero pressure and zero temperature.
3.66
calculations *
Present
Mg7ScH16
Expt. Others Present
PAW-GGA
9.496
55b, 45±2c, 49d ,54.2e 63.94
Others
PAW-GGA
9.5188 f
424.8f
Present
PAW-GGA
9.368 9.564 g
Expt.
USP-GGA
Others
PP-GGA
Present
PAW-GGA
Others
PAW-GGA
Present Others
Mg7NbH16
66.17j 71.22
460.8f
PAW-GGA
9.2938f 9.38 k 9.653
60.71
PAW-GGA
9.6677 f
419.4f
PAW-GGA
9.508
69.28
Others
PAW-GGA
9.5188 f
459.0f
9.381
73.63
PAW-GGA
Expt.
3.73
3.70
3.73
3.65
9.543 h
9.3958 f 488.8f 9.32 k USP-GGA 9.5376l 68.57l PAW: Projector Augmented Wave, USP: Ultra Soft Potential, PP: Pseudopotential, GGA: Others
*
56.46 i
Present
Present
3.64
445.5f
9.268
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Mg7ZrH16
Others
9.3956 f 9.707 k 9.3804 i
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Mg7YH16
PAW-GGA
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Mg7VH16
Others
68.46
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Mg7TiH16
PAW-GGA
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MgH2
PAW-GGA
Generalized Gradient Approximation. a
Ref [32], b Ref [34]. c Ref [35]. d Ref [36]. e Ref [37]. f Ref [3]. g Ref [4]. h Ref [9]. i Ref [14], j
Ref [10]. k Ref [16]. l Ref [15].
21
ACCEPTED MANUSCRIPT Table 2. Elastic constant Cij (in GPa) of MgH2 and Mg7TMH16 (TM=Sc, Ti, V, Y, Zr, Nb) at zero pressure and zero temperature.
Mg7VH16
Mg7YH16
Mg7ZrH16
C44
C66
PAW GGA
69.37
36.67
34.48
143.38
42.62
52.08
FP-LAPW GGA
73.82
34.07
31.98
139.24
39.93
52.11
Others
74.4a
38.8a
31.4a
136.0a
37.6a
53.0a
Others
73.1b
33.9b
20.4b
131.9b
38.5b
52.3b
PAW GGA
141.74
23.54
FP-LAPW GGA
136.06
21.93
PAW GGA
145.35
29.29
FP-LAPW GGA
142.20
Others
12.99
19.49 18.04
27.07
17.42
130.02c
16.48 c
19.68 c
PAW GGA
153.64
29.78
22.41
FP-LAPW GGA
146.67
34.26
14.31
PAW GGA
132.77
20.26
7.631
FP-LAPW GGA
136.61
24.52
12.54
PAW GGA
137.62
30.41
14.83
140.86
32.07
20.00
148.25
32.14
19.13
FP-LAPW GGA
138.90
28.80
20.06
Others
157,25d
24.23d
19.59d
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FP-LAPW GGA Mg7NbH16 PAW GGA
a
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C33
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Mg7TiH16
C13
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Mg7ScH16
C12
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MgH2
C11
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System
Ref [37], b Ref [36], c Ref [14].d Ref [15].
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ACCEPTED MANUSCRIPT Table 3: Bulk modulus BH (in GPa), shear modulus GV, GR, GH (in GPa), Young’s modulus EH (in GPa), Poisson ratio ν and ratio BH/GH of MgH2 and Mg7TMH16 (TM=Sc, Ti, V, Y, Zr, Nb) at zero pressure and zero temperature. BR
BH
GV
GR
GH
EH
PAW GGA
54.82
50.32
52.57
39.23
32.69
35.96
87.85
0.221
1.46
FP-LAPW GGA
53.66
50.21
51.93
38.98
34.63
36.80
89.31
0.213
1.41
36.1a
87.3a
25.16
60.60
0.3236
2.50
26.46
30.49
78.21
0.28264 1.96
24.91
29.47
77.24
0.3106
24.16
28.82
75.39
0.30801 2.27
Other work
Mg7TiH16
PAW GGA
62.94
62.94
62.94
31.43
FP-LAPW GGA
59.97
59.97
59.97
34.52
PAW GGA
67.97
67.97
67.97
34.03
FP-LAPW GGA
65.45
65.45
65.45
33.48
Other work
Mg7ZrH16
PAW GGA
71.06
71.06
2.31
FP-LAPW GGA
71.73
71.73
PAW GGA
57.77
57.77
FP-LAPW GGA
61.88
PAW GGA
71.06
38.22
30.09
34.15
88.31
0.29288 2.08
71.73
31.07
20.39
25.73
68.94
0.33981 2.79
57.77
27.08
11.67
19.37
52.27
0.3492
61.88
61.88
29.94
18.19
24.06
63.91
0.32788 2.57
66.15
66.15
66.15
30.34
20.87
25.60
68.03
0.3286
2.58
68.33
68.33
68.33
33.76
26.77
30.26
79.11
0.3070
2.26
70.84
70.48
70.84
34.70
26.14
30.42
79.83
0.3122
2.33
65.50
65.50
65.50
34.05
26.90
30.48
79.15
0.2986
2.15
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Mg7YH16
BH/GH
56.46b 31.96b 22.91b 27.43b 70.82b 0.29 b
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Mg7VH16
18.88
SC
Mg7ScH16
50.2a
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MgH2
ν
BV
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System
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FP-LAPW GGA
Mg7NbH16 PAW GGA
FP-LAPW GGA
2.98
68.57c 68.57c 68.57c 38.35c 27.27c 32.81c 84.89c 0.29c
a
Ref [36], b Ref [14], c Ref[15].
23
ACCEPTED MANUSCRIPT Table 4. Second order polynomial expressions of the fitting volume-pressure (V/V0(P)) for Mg7TMH16 hydrides. System
V/V0 0.99768 − 0.01356 P + 2.05805 × 10−4 P 2
Mg7TiH16
0.99769 − 0.01268 P + 1.79667 × 10 −4 P 2
Mg7VH16
0.99836 − 0.01272 P + 1.83835 × 10−4 P 2
Mg7YH16
0.99301 − 0.01502 P + 2.66587 × 10 −4 P 2
Mg7ZrH16
0.99775 − 0.0129 P + 1.88422 × 10−4 P 2
Mg7NbH16
0.99735 − 0.01218 P + 1.68094 × 10 −4 P 2
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Mg7ScH16
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Table 5. Molecular mass (M), density (ρ), longitudinal wave velocity (vl), transverse wave velocity (vt), average elastic wave velocity (vm) and Debye temperature (TDebye) at normal pressure and temperature. M (g/mol)
MgH2
26.32
ρ (g/cm3) 1.42
vl (m/s) 8394
vt (m/s) 5067
vm (m/s)
TDebye (K)
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System
5568
770
854.8a
1.797
7322
Mg7TiH16
234.15
1.875
7559
234.13b
1.884b
Mg7VH16
237.20
2.029
Mg7YH16
275.17
2.082
Mg7ZrH16
277.49
2.197
Mg7NbH16
279.17
2.303
4192
602
3965
4434
643
7027.15b
3815.72b
4257.21b
618.67b
7576
4103
4578
679
6333
3050
3429
488
6752
3414
3827
553
6950
3634
4065
596
3819.37c
4262.736c
619.753c
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7166.383c
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Ref [36], b Ref [14], c Ref [15].
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a
3741
SC
231.22
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Mg7ScH16
25
ACCEPTED MANUSCRIPT
Table 6. Third order polynomial expressions of the fitting TDebye-pressure (TDebye (P)) for
System
TDebye
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Mg7TMH16 hydrides.
603.23 + 23.79 × P − 0.723 × P 2 + 0.012 × P 3
Mg7TiH16
646 + 18.396 × P − 0.369 × P 2 + 0.0034 × P 3
Mg7VH16
685.13 + 23.305 × P − 0.787 × P 2 + 0.0105 × P 3
Mg7YH16
502.71 + 36.74 × P − 2.0743 × P 2 + 0.0485 × P 3
Mg7ZrH16
554.59 + 25.63 × P − 1.123 × P 2 + 0.0232 × P 3
Mg7NbH16
600.24 + 22.112 × P − 0.78 × P 2 + 0.013 × P 3
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Mg7ScH16
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Fig 1. Unit cell of (a) MgH2 (b) Mg7TMH16 (TM=Sc, Ti, V, Y, Zr, Nb) (b) hydrides. Color
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scheme: yellow, green and blue balls represent Mg, TM and H atoms respectively.
27
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Fig. 2. The ratio V/V0 of Mg7TMH16 hydrides as function of pressure.
28
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Fig.3. The elastic constant C11, C12 and C44 of Mg7TMH16 hydrides under pressure
29
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Fig.4. The elastic moduli B, G, E of Mg7TMH16 hydrides under pressure
30
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Fig.5. The hardness H of Mg7TMH16 hydrides under pressure
31
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Fig.6. The B/G and ν ratios of Mg7TMH16 hydrides under pressure
32
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ACCEPTED MANUSCRIPT
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Fig.7. The anisotropy A of Mg7TMH16 hydrides under pressure
33
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Fig.8. Debye temperature TDebye of Mg7TMH16 hydrides under pressure.
34
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Fig. 9. Longitudinal, transverse, average wave velocity of Mg7TMH16 hydrides under pressure
35
ACCEPTED MANUSCRIPT
•
Structural and mechanical properties of MgH2 and Mg7TMH16 were estimated at 0 GPa. Adding transition metal to MgH2 turns the brittleness of hydrides material.
•
The mechanical properties of Mg7TMH16 were calculated under pressure.
•
Debye temperature and sound velocities are calculated under pressure.
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•
S0022-3697(17)30326-8
DOI:
10.1016/j.jpcs.2017.08.001
Reference:
PCS 8155
To appear in:
Journal of Physics and Chemistry of Solids
Received Date: 20 February 2017 Revised Date:
3 May 2017
Accepted Date: 1 August 2017
Please cite this article as: K. Benyelloul, L. Seddik, Y. Bouhadda, M. Bououdina, H. Aourag, K. Khodja, Effect of pressure on structural, elastic and mechanical properties of transition metal hydrides Mg7TMH16 (TM = Sc, Ti, V, Y, Zr and Nb): First-principles investigation, Journal of Physics and Chemistry of Solids (2017), doi: 10.1016/j.jpcs.2017.08.001. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Effect of pressure on structural, elastic and mechanical properties of transition metal hydrides Mg7TMH16 (TM= Sc, Ti, V, Y, Zr and Nb): First-
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principles investigation
Kamel Benyelloula,c, Larbi Seddika, Youcef Bouhaddaa, Mohamed Bououdinac, Hafid
a
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Aouragb, Khadidja Khodjaa
Unité de Recherche Appliquée en Energies Renouvelables, URAER, Centre de
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Développement des Energies Renouvelables, CDER, 47133, Ghardaïa, Algeria Department of Physics, College of Science, University of Bahrain, P.O. Box 32038, Bahrain
c
Department of Physics, University of Tlemcen, 13000, Algeria
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b
1
ACCEPTED MANUSCRIPT Abstract The effect of pressure on structural stability, elastic properties and Debye temperature of face centred cubic Mg7TMH16 (TM= Sc, Ti, V, Y, Zr and Nb) hydrides, was investigated by firstprinciples calculations based on density functional theory (DFT) with the generalized gradient
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approximation (GGA). The obtained equilibrium lattice parameters and elastic properties at zero pressure for MgH2 and Mg7TMH16 hydrides, are in good agreement with other experimental and theoretical values. The calculations of the bulk modulus and the ductility
SC
factors (B/G) show that mixing (MgH2) with small amount of transition metal (TM= Sc, Ti, V, Y, Zr and Nb) can enhance the resistance to volume change and transform it from a brittle
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to a ductile material (brittle ductile). The elastic constants, bulk modulus, shear modulus, Young’s modulus, anisotropy factor and hardness have been studied under pressure. These mechanical quantities are found to increase with increasing pressure. While the B/G and Poisson’s ratios (v) undergo an inverse behavior. In addition to that, the studied hydrides are
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found stable with a ductile behaviour under a pressure between 0 and 20 GPa. Furthermore, the effect of pressure on Debye temperature and sound velocity, was also investigated and
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discussed.
Keywords: Ab initio calculations; Magnesium-transition metal hydrides; Elastic properties,
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high pressure.
2
ACCEPTED MANUSCRIPT 1. Introduction Recently, extensive researches were devoted toward hydrogen as clean and sustainable energy source. It could potentially replace rich-carbon energies (such as oil, wood, natural gas, etc…) in the short time. Unlike fossil fuel, it is environmentally friendly, non polluting and abundant
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fuel [1], which can offer a new prospective economy. However, the transition from a fossil economy to hydrogen economy must resolve the mobile application challenges especially the on-board hydrogen storage. Several approaches for hydrogen storage are well-known
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including compressed gas and liquid hydrogen. But the storage of hydrogen in the “solid” form as metallic hydrides is considered the most encouraging. Indeed, the metal hydrides
M AN U
offer a safer alternative and have a higher volumetric density than liquid and gaseous forms [2].
Due to its high hydrogen content (7.6 Wt%) [3-4], lightweight, low cost, abundance and nontoxicity, magnesium hydrides (MgH2) is considered a very promising and potential candidate
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thereby has been so far extensively investigated. Nevertheless, its high thermodynamic stability and slow absorption/desorption kinetics [4,5], limits its use for mobile applications in particular fuel cells for electric cars. Therefore, Mg-based hydrides are usually destabilized,
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for example, by 3d transition elements incorporation [3-6] or surface modification [7] to improve its de-/hy-driding performance. In current decades, Mg-transition 3d- metal hydrides
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have received much attention and have widely investigated both experimentally [4,8,9] and theoretically [3,10]. Experimentally, Takeichi et al. [11] found that Mg6VNaxHy (0 ≤ x ≤ 1) reversibly desorb and absorb hydrogen at 620–630 and 590–600 K, respectively, under 0.5 MPa (H2) which are 70 and 120 K respectively, lower than those of MgH2. However, the reaction enthalpies calculated from the Van’t Hoff relation from PCIs (Pressure Composition Isotherms) do not significantly differ from that of MgH2.
3
ACCEPTED MANUSCRIPT The Mg-transition metal hydrides Mg7TiHx and Mg7NbHx hydrides with a face-centered cubic (FCC) super-lattice structure with Ga7Ge-type (Space group 225) have been synthesized at 8 GPa and 873 K (for Mg7TiH16) and 600 K (for Mg7NbH16) using ultra-high pressure techniques [8,9]. Moser et al. [12] synthesized (at 600°C and 4 GPa) a series of hydrogen rich
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Mg6–7TMH14–16 (TM= Ti, Zr, Hf, V, Nb and Ta) hydrides. Theoretically, Song et al. [13] have used first principles calculation to investigate the influence of Al, Ti, Fe, Ni, Cu and Nb on MgH2 hydride. They found that the alloying elements decrease the heat of formation of
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(Mg,X)H2 and can enhance its kinetics. In other hand, Dai et al. [7] have also employed DFT calculations to study the preferential sites occupancy of some dopants such as Al, Ti, Mn and
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Ni on MgH2 onto (110) and (001) surfaces.
Face-centered cubic (FCC) magnesium–transition metal (TM) hydrides Mg7TMH16 (TM= Sc, Ti, V, Y, Zr, Nb) were also investigated [3,10,14,15] using first-principles calculations based on DFT method. Indeed, Xiao et al. [3] have calculated the enthalpies of formation and
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analyzed the stability of Mg7TMH16 using both cohesive energies and electronic densities of state. Whereas, Shelyapina et al. [16] have discussed the stability of Mg7TMH16 using both the calculated electronic structure and heat of formation. For these hydrides, the mechanical
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properties have not yet been investigated experimentally or theoretically (Except for Mg7TiH16 [10] and Mg7NbH16 [16] where the theoretical investigations were done at ambient
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pressure).
Indeed, the knowledge of the mechanical properties under pressure, has a great importance in the design of hydride reservoir. The hydrogen absorption causes a significant increase in volume (up to 25%) and creates important stresses within the storage vessel. However, the initial volume could be recovered after desorption. The successive expansion/contraction of volume after several cycles, with a pressure variation, have a great effect on the morphological and microstructural transformations and therefore can reduce the life-time
4
ACCEPTED MANUSCRIPT stability of the storage vessels. Consequently, an accurate knowledge of mechanical properties of hydrides, especially their pressure dependence, is of considerable importance in both fundamental science and engineering technology [17,18]. The availability of reliable and assessed information about the effect of pressure is often not
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accessible or rare. The quantum mechanical simulation based on the first principles calculation can be alternative to experimental process with a reduced cost. Moreover, it is a powerful approach and proficient means for solving different problems of material science
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field and may be used in the determination of elastic properties, under different pressures. In the current study, the structural stability, elastic and mechanical properties as well as
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Debye temperature of cubic Mg7TMH16 (TM= Sc, Ti, V, Y, Zr and Nb) hydrides, under pressure are investigated through first principles calculation within the density functional theory (DFT).
The rest of the paper is organized as follows. In section 2, details and computational methods
present work is given.
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are presented. In section 3, the obtained results were analysed. Finally, a conclusion of the
2. Computational methods
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All compounds Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) and MgH2 were investigated by the density functional theory (DFT) [19]. The projector augmented wave (PAW) method
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[20,21] as implemented in the Vienna Ab-initio Simulation Package (VASP) code [22,23] was employed. Also, the exchange-correlation interaction was treated and described using the generalized gradient approximation (GGA) as parameterised by Perdew-Wang 91 (PW91) [24]. The convergence is achieved when the cut-off energy is 450 eV and 375 eV for MgH2 and Mg7TMH16, respectively. For the Brillouin zone integrations we use the Monkhorst-Pack scheme [25] and the k-point were sampled by 7 × 7 × 7 and 7 × 7 × 9 for Mg7TMH16 and TMgH2, respectively. The crystal structure of the tetragonal MgH2 and the stable Ga7Ge-type
5
ACCEPTED MANUSCRIPT phase of Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) are illustrated in Fig.1. For rutile (tetragonal) MgH2 with space group P42/mnm [26], Mg atoms occupy the 2a (0,0,0) sites, while H atoms occupy the 4f (0.304, 0.304, 0) sites. However, for Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) have a face-centered cubic (fcc) with space group Fm-3m [3,4,8]. The unit cell
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of Mg7TMH16 contains 96 atoms, where Mg atoms occupy 4b (1/2,1/2,1/2) and 24d (0,1/4,1/4) sites, respectively, TM atoms occupy the 4a (0,0,0) sites and H atoms are localized in two tetrahedral sites 32f (0.094,0.094,0.094) and (0.365,0.365,0.365) sites.
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For an all-electron description the full-potential linear augmented plane wave (FP-LAPW) method [27] implemented on Wien2k code was used [28]. The cut-off parameters RMT.KMax
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were taken as 7.0 and 6.0 Ryd for MgH2 and Mg7TMH16, respectively. The value of 20 for Gmax was used on a Fourrier expansion of potential in the interstitial region. The respective Brillouin zone integration is performed using a k-mesh of 8 × 8 × 13 and 11 × 11 × 11 , equivalent to 105 k-points and 56 k-points for MgH2 and all Mg7TMH16, respectively. The muffin-tin
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(MT) radius RMT around each atom which are taken to be 2 and 1.5 for Mg and H, respectively in MgH2 and 1.45, 1.53, 1.55, 1.57, 1.63, 1.61, 1.65 and 1.22 for Mg, Sc, Ti, V, Y, Zr, Nb and H respectively in Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb).
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The elastic constants are computed by applying the strain energy-strain curve method [29,30]
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using both VASP and Wien2k code [31]. Indeed, there are six independent elastic constants (C11, C12, C13, C33, C44, C66) for tetragonal structure and three independent elastic constants (C11, C12, C44) for the cubic structure, and were deduced from the variation of total energy calculations (and the second-order derivative energy) by applying five small dimensionless strains δ with ( δ = 0,±0.01,±0.02 ). More details can be found in Ref [29] and Ref [30].
3. Results and discussion 3.1. Structural and elastic properties at zero pressure
6
ACCEPTED MANUSCRIPT The calculated equilibrium lattice parameters, bulk modulus (B) and its pressure derivative (B’) for MgH2 and Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) are displayed in Table 1. Compared to the experimental values, the optimized lattice parameters for MgH2 hydride are consistent with the literature [32]. In the case of Mg7TiH16, Mg7VH16 and Mg7NbH16, the
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obtained lattice constants are reasonable in accordance with the experimental data [4, 9] and available theoretical values [3, 14-16] (the corresponding relative errors are -2.04 % and 1.69 %, for Mg7TiH16 and Mg7NbH16, respectively). From the remaining hydrides Mg7TMH16
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(TM= Sc, Y and Zr), the obtained structural parameters are also in an excellent agreement with previous theoretical results reported by Xiao-Bing Xiao et al. [3]. It was also found that
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the obtained bulk moduli by fitting the total energy as a function of volumes [33], are in good agreement between our calculated and theoretical values mentioned in Refs [10,14,15] for Mg7TMH16 and Refs [34-37] for MgH2.
It should be noted that the values (see Table 1) calculated by Xiao-Bing Xiao et al. [3] (>419
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GPa) for Mg7TMH16, differ completely from both our calculated values as well as other values reported in the literature [10, 14, 15]. In fact, the values found by Xiao-Bing Xiao et al. [3] are larger and comparable to diamond's values.
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As shown in Table 1, we can notice that the calculated values of bulk modulus for Mg7TMH16 hydrides are larger than (MgH2) value, which indicates that mixing MgH2 by small amount
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(12.5%) of transition metal (TM) can enhance the resistance to volume change in the hydrides.
As mentioned earlier, the elastic constants are very important, since they help us to better understand and measure the resistance of the hydrogen tank to the developed stress during charge/discharge cycles. Therefore, we have calculated at zero pressure the elastic constants as reported in Table 2. It is clear that our calculated elastic constants for MgH2, Mg7TiH16 and Mg7NbH16 using both PAW and FP-LAPW methods, are generally in reasonable accordance
7
ACCEPTED MANUSCRIPT with the available theoretical calculations [14-15, 36-37]. However, for the other studied hydrides (Mg7TMH16 with TM= Sc, V, Y and Zr ), no experimental or theoretical elastic constants are available. The as-obtained values can be considered as reference for future experimental and/or theoretical research works. Also, because of the lack of any experimental
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values of elastic constants, we have used the more accurate FP-LAPW approach [38] as a benchmark as suggested by many experts and authors [38-40] in order to confirm our results. The values are reported in Table 2. The FP-LAPW calculations are in good agreement with
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our PAW calculations.
Based on the obtained elastic constants, we can observe that the independent Cij’s satisfy the
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classical Born criteria for tetragonal and cubic structures given in Ref [41]. Therefore, the calculated structures are mechanically stable at zero pressure.
According to Voigt-Reuss-Hill (VRH) approximation, the arithmetic average shear modulus (GV, GR, GH), bulk modulus (BV, BR, BH), Young’s modulus (EH) and Poisson’s ratio (ν) can
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be estimated from the elastic constants (Cij) by applying the relations given in Refs [42-45]. The obtained elastic moduli are summarized in Table 3. We found that the values of bulk modulus (B) obtained from the universal equation of state (EOS) [33] are close to the values
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calculated using the elastic constants. To the best of our knowledge, only the values for MgH2, Mg7TiH16 and Mg7NbH16 are available in the literature [14,15,36] and are in good
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agreement with our calculated values. In order to predict the brittleness or the ductility behaviour of our studied hydrides, we calculated the ratio between bulk modulus and shear modulus (B/G) as introduced by Pugh [46]. Our calculation shows that the ratio B/G of MgH2 hydride (1.46) is lower than the critical value (1.75), thereby the material has a brittle behaviour. On the other hand, the ratio B/G of Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) hydrides is found to be greater than the critical value, which indicates that these hydrides are ductile. In addition, the universal
8
ACCEPTED MANUSCRIPT Poisson’s ratio (ν) can also be applied to characterize materials: a material is ductile if ν is higher than 0.26 (ν ≥ 0.26), otherwise the material is brittle [47]. From our results listed in Table 3, we can notice that the value of ν is less than 0.26 for MgH2 and larger than 0.26 for Mg7TMH16, which is consistent with the results found earlier using the ratio B/G. So, it can
also induce mechanical property brittle-ductile transformation. 3.2. Elastic properties under pressure
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be concluded that mixing (MgH2) with a small amount (12.5%) of transition metal (TM) can
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In the following, we studied the behaviour of structural and mechanical properties under pressure variation between 0 and 20 GPa. Hence, we used the third order Birch Murnaghan
3 V P = B 2 V 0
−
7 3
V − V0
−
5 3
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equation (Eq. 1) to plot the pressure vs. unit cell volume [33]: 1 + 3 ( B ' − 4 ) V V 0 4
−
2 3
− 1
(1)
where V is the unit cell volume at external pressure, V0, B and B’ is the equilibrium volume,
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bulk modulus and pressure derivative B’, respectively, at zero pressure. Fig.2 displays the variation of unit-cell volume of Mg7TMH16 with pressure. For Mg7YH16, the ratio V/V0 decreases more significantly compared to the other hydrides. Moreover, for
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Mg7NbH16 compounds the ratio V/V0 decrease more slowly. Based on the bulk modulus (B)
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definition, we can notice from Table 1 that the hydrides having low bulk modulus (Mg7YH16, B = 60.71 GPa) are easily compressed while hydrides with high bulk modulus (Mg7NbH16, B = 73.63 GPa) are difficult to compress. Following the fitting of pressure-volume curves of second-order polynomial expressions shown in eq. 1, the predicted ratio V/V0 values and their corresponding models found to be in good agreement with coefficients of determinations R2 ≈ 1. The mentioned expressions for Mg7ScH16, Mg7TiH16, Mg7VH16, Mg7YH16, Mg7ZrH16 and Mg7NbH16 are reported in Table 4.
9
ACCEPTED MANUSCRIPT Based on the corresponding optimized structures at several applied pressures, we calculated the three independent elastic constants for the studied hydrides. The obtained results are depicted in Fig. 3. The first remark is that all the hydrides are stable in the pressure range between 0 and 20 GPa (all Cij are positive). Also, we found that the C11, C12 and C44 increase
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monotonically with the applied external pressure, meanwhile the curve of the elastic constant C11 is located far above the other elastic constants and the value significantly increases with increasing pressure compared to C44 and C12 (the effect of pressure is slight). Knowing that,
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C11 is related to the unidirectional compression along the principle crystallographic direction [100], we can conclude that Mg7TMH16 hydrides become more and more incompressible
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along the principle [100] direction with increasing pressure.
It is well known that the hardness of a material is measured by the bulk modulus (B), shear modulus (G) and Young’s modulus (E) [48]. The bulk modulus is a measure of the resistance of a material to volume change by applying pressure. However, shear modulus can be used to
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measure the resistance to reversible deformations upon shear stress [49] and Young’s modulus measures the stiffness of a material [50]. The evolution of B, G and E of Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) hydrides under pressure is shown in Fig. 4. Starting from the
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initial values of B, G and E at 0 GPa, the representative curves are found to increase with increasing pressure. Also, we can observe that under pressure, the values of bulk modulus are
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larger than that of shear modulus which means that the resistance to volume change is increased. Therefore, the hydrides tend to resist to volume change better than shape change under pressure. The larger the value Young’s modulus is, the stronger of stiffness is. Based on the available standard relation, the hardness (H) of compounds can be determined using the Young’s modulus (E) and Poisson’s ratio (ν) by means of the following expression [48]: H=
( 1 − 2ν)E 6( 1 + ν)
(2)
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ACCEPTED MANUSCRIPT At different pressures, the obtained values of hardness (H) of the studied hydrides are presented in Fig. 5. Starting from the corresponding initial values of H at zero pressure, we can see that the hardness increases with increasing pressure, indicating that the hardness of Mg7TMH16 hydrides can be increased by increasing pressure.
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Fig. 6 shows the calculated ductility factors (B/G and ν) for the studied hydrides under several pressures. It is clearly seen that for pressure range (0-20 GPa), the values of the ductility factors (B/G and ν) are higher than 1.75 and 0.26, respectively, revealing that the studied
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hydrides have a ductile nature (B/G >1.75 and ν ≥ 0.26). Also, it can be seen that the ratios (B/G and ν) varied through three steps before it becomes stationary. Within the range (0-2
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GPa), the ratios are decreasing rapidly, then decrease slightly within the range (2-12 GPa) until reaching a stable regime after pressure (P) = 12 GPa. Surprisingly, Mg7VH16 hydride shows an insignificant increase of ratios at P = 4 GPa. We deduced that the studied hydrides become less ductile when the pressure increases. In addition, the Poisson’s ratio is usually
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used to quantify the stability of crystal structure against shear, and the larger the Poisson ratio the better the plasticity [50]. Accordingly, it can be deduced that the applied external pressure caused weakness plasticity of the mentioned hydrides. The curves of Fig. 6 show that the
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hydrides Mg7TiH16, Mg7NbH16 and Mg7ScH16 reach the same B/G and ν ratios (2.10, 0.295)
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at P = 8 GPa, whereas the hydrides Mg7ZrH16 and Mg7YH16 have the same level values (2.2, 0.30) at P = 10 GPa.
The elastic anisotropy A is also an important elastic property, especially under external conditions like high pressure. At zero pressure, the estimations of the anisotropic factor A for the cubic phase using the standard relation A = 2C44 /(C11 − C12 ) [29] are 0.22, 0.31, 0.36, 0.14, 0.28 and 0.33 for Mg7ScH16, Mg7TiH16, Mg7VH16, Mg7YH16, Mg7ZrH16 and Mg7NbH16 respectively. For isotropic crystal, A is equal to 1 while any values smaller or larger than 1 indicate an anisotropy [29]. From the obtained results, it is clear that all hydrides are
11
ACCEPTED MANUSCRIPT anisotropic. From Fig. 7, the values of A increase with increasing the external pressure ranging up to 20 GPa for Mg7TMH16 hydrides, indicating that their anisotropy is less apparent when the pressure is increased. 3.3. Debye temperature under pressure
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The Debye temperature ( TDebye ) is usually related to many physical properties such as specific heat, elastic constants, and melting point [50,51]. Debye temperature can be determined from bulk and shear moduli, using the average sound velocity ( vm ) by means of the following
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equation [52]:
(3)
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1/ 3
h 3n N ρ TDebye = ( A ) vm k 4π M
where h is Planck’s constant, k is Boltzmann’s constant, N A is Avogadro’s number, ρ is the density of molecule, M is the molecular weight and n is the number of atoms in the molecule.
)
−1 / 3
(4)
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1 2 1 vm = ( 3 + 3 3 vt vl
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The average sound velocity ( vm ) in polycrystalline materials is given by [52]:
AC C
4 where vt = G / ρ and vl = (B + G) / ρ are the transverse and longitudinal elastic wave 3 velocities of the polycrystalline materials, respectively [52]. The calculated transversal and longitudinal wave velocities, average sound velocity and Debye temperature of MgH2 and Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) hydrides at normal pressure and zero temperature are listed in Table 5. The obtained Debye temperature of Mg7TiH16 and Mg7NbH16 are 643 and 596 K, respectively, which are relatively close to the theoretical values 618.7 [14] and 619.7 K [15], respectively. However, we found a value of 770 K for MgH2 which differs to the value 854.8 K calculated by Hector et al. [36]. This can
12
ACCEPTED MANUSCRIPT be explained by the computational convergence (different cut-off energy) and different PAWGGA parameterization. Unfortunately, no other available experimental data that could be used for further comparison. The Debye temperature of the hydrides gradually decreases in the following order: TDebye
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(Mg7VH16)> TDebye (Mg7TiH16)> TDebye (Mg7ScH16) and TDebye (Mg7NbH16)> TDebye (Mg7ZrH16)> TDebye (Mg7YH16). It is noticed that the Mg-transition metal hydrides namely [Mg7ScH16, Mg7TiH16, Mg7VH16] and [Mg7YH16, Mg7ZrH16, Mg7NbH16], having high bulk
SC
modulus have a large Debye temperature, respectively.
The computed results of Debye temperature (TDebye), transversal (vt) and longitudinal wave
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velocities (vl) as well as the average sound velocity (vm) of Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) hydrides under different pressures are shown in Figs. 8 and 9, respectively. To describe the relationship between Debye temperature (TDebye) and applied pressure, the calculated data points were fitted by using a third-order polynomial, the obtained equations
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are reported in Table 6.
We observe that the values of Debye temperature, transversal, longitudinal and average sound velocities increase with the increase of pressure. In Fig. 8, it can be seen that the TDebye of
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Mg7VH16, Mg7TiH16, Mg7ScH16 series and Mg7NbH16, Mg7ZrH16, Mg7YH16 series is more sensitive to pressure variation when their bulk and shear moduli are important. This can be
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explained by the direct correlations that exist between vt, vl, and vm and elastic moduli (B and G) (Eqs.3 and 4). 4. Conclusion
Based on density functional theory with generalized gradient approximation, the structural, elastic properties and Debye temperature of Mg7TMH16 (TM= Sc, Ti, V, Y, Zr and Nb) hydrides under pressure, have been investigated.
13
ACCEPTED MANUSCRIPT At zero pressure, the obtained structural parameters, bulk modulus and elastic properties determined by strain energy-strain curve method for MgH2 and Mg7TMH16 are in good agreement with previous theoretical results and experimental data. Analysis of single-crystal and polycrystalline elastic parameters revealed that MgH2 and Mg7TMH16 hydrides are
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mechanically stable. The calculated bulk modulus and ratio B/G of Mg7TMH16 are larger than MgH2, thus suggesting that the mixing magnesium hydrides by small amount (12.5%) of transition metal (TM) can enhance the resistance to volume change and can also convert
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MgH2 from brittle to ductile material (brittle ductile).
It was found that the applied external pressure causes weakness of ductility and plasticity of
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Mg7TMH16 hydrides. The obtained elastic anisotropy A values were lower than 1. Also, the anisotropy of the studied hydrides is less apparent when the pressure is increased. Using the elastic moduli the transversal (vt), longitudinal (vl), and average sound velocities (vm) as well as Debye temperature (TDebye) for the studied hydrides, are calculated at zero pressure and
behaviour of TDebye.
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enhance with increasing pressure. In other hand, polynomials have been used to describe the
It is noticed, that some of our theoretical results for the different hydrides Mg7TMH16 where
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TM= Sc, Ti, V, Y, Zr and Nb, are obtained for the first time such as elastic properties (elastic constants, bulk modulus, shear modulus, Young’s modulus, Debye temperature) and the
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corresponding variation under pressure. For our knowledge, the elastic constants have not yet been calculated, except for the Mg7TiH16 and Mg7NbH16 hence our results can serve as a prediction for future research investigation.
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K. Benyelloul, Y. Bouhadda, M. Bououdina, H.I. Faraoun, H. Aourag, L. Seddik. The effect of hydrogen on the mechanical properties of FeTi for hydrogen storage applications. Inter. J. Hydrogen Energy. 39(2014)12667-12675.
[30]
P. Ravidran, L. Fast, PA. Korzhavyi, B. Johanson, Density functional theory for
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ACCEPTED MANUSCRIPT calculation of elastic properties of orthorhombic crystals: application to TiSi2. J.Appl.Phys.84 (1998)4891-4904. [31]
A.H. Reshak, M. Jamal, DFT calculation for elastic constants of orthorhombic structure within WIEN2K code: A new package (ortho-elastic). J.Alloys Comp. 543 (2012) 147-
[32]
RI PT
151.
FH. Ellinger, CE. Holley, BB Jr. McInteer, D. Pavone, RM. Potter, E. Staritzky. The preparation and some properties of magnesium hydride. J. Am. Chem. Soc 77 (1955)
SC
2647. F. Birch, J. Geophys. Res: Solid Earth 83 (1978) 1257.
[34]
P. Yu, P.K. Lam, Electronic and structural properties of MgH2. Phys. Rev. B 37 (1988)8730-8737.
[35]
M AN U
[33]
P. Vajeeston, P. Ravidran, B.C. Hauback, H. Fjellvag, A. Kjekshus, S. Furuseth, M. Hanfland. Structural stability and pressure-induced phase transition in MgH2. Phys. Rev.
[36]
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B 73 (2006) 224102-8.
L.G. Hector Jr, J.F. Herbst. Ab Initio thermodynamic and elastic properties of alkalineearth metals and their hydrides. Phys. Rev B 76 (2007) 014121-18. A. Junkaew, B. Ham, X. Zhang, R. Arroyave. Ab-initio calculations of the elastic and
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[37]
finite-temperature thermodynamic properties of niobium and magnesium hydrides. Inter.
[38]
AC C
J. Hydrogen Energy 39 (2014) 15530-15539. K. Lejaeghere, V. Van Speybroeck, G. Van Oost, S. Cottenier. Error estimates for solids-sate density-functional theory predictions: An overview by means of the groundstate elemental crystals. Critical Reviews in Solid State and Materials Sciences, 39 (2014) 1-24.
[39]
J. Hafner, C. Wolverton, G. Ceder. Toward computational materials design: The impact of density functional theory on materials research. MRS Bulletin 31 (2006) 659-665.
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ACCEPTED MANUSCRIPT [40]
K. Lejaeghere, G. Bihlmayer, T. Björkman, P. Blaha, S. Blügel, V. Blum, D. Caliste, I. E. Castelli, S.J. Clark, A. Dal Corso, S. de Gironcoli, Th. Deutsch et al. Reproducibility in density-functional theory calculations of solids. Science 351 (2016) 6280. M. Born, On the stability of crystal lattices I math. Proc. Camb. Philos. Soc. 36(1940) 160.
RI PT
[41]
R. Hill. Proc. Phys. Soc. London A65 (1952) 349
[43]
W. Voigt. Ann. Phys 38 (1889) 573
[44]
A. Reuss, Z. Angew. Math. Phys. 9(1929) 49.
[45]
E. Schneider, O.L. Anderson, N. Soga. Elastic constants and their measurement,
[46]
M AN U
McGraw-Hill, New York, 1994.
SC
[42]
S.F. Pugh, Relation between elastic moduli and plastic properties of polycrystalline pure metals. Phylos. Mag. 45 (1954) 823-43.
[47]
L.-W. Ruan, G.-S. Xu, H.-Y Chen, Y.-P. Yuan, X. Jiang, Y.-X. Lu, Y.-J. Zhu. The
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elastic behaviour of dense C3N4 under high pressure: First-principles calculations. J. Phys. Chem. Solids. 75 (2014) 1324-1333. [48]
Yu. Zhao, H. Hou, Y. Zhao, P. Han. First-principles study of nikel-silicon binary
[49]
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compounds under pressure. J. Alloys Comp. 640 (2015) 233-239. A.F. Young, C. Sanloup, E. Gregoryanz, S. Scandolo, R.J. Hemley, H.-K. Mao. Phys.
[50]
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Rev. Lett. 96(2006)155501. Y. Zhao, L. Qi, Y. Jin, K. Wang, J. Tian, P. Han. The structural, elastic, electronic properties and Debye temperature of DO22-Ni3V under pressure from first-principles. J. Alloys Comp. 647 (2015) 1104-1110.
[51]
X-J. Chen, Z-H.Mo, R-N.Wang, M-X. Zang, B-Y. Tang, L-M. Peng, W-J. Ding. Elastic and electronic properties of the Ti5X3 (X=Si, Ga, Sn, Pb) compounds from firstprinciples calculations. J. Solid state Chem. 194 (2012) 127-134.
19
ACCEPTED MANUSCRIPT OL. Anderson. A simplified method for calculating the debye temperature from elastic
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constants. J. Phys. Chem. Solids 24 (1963)909-917.
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[52]
20
ACCEPTED MANUSCRIPT Table 1. Lattice parameter a, c (in Å), bulk modulus B (in GPa) and derivative (B’) for MgH2
Methods of
System
a
c
B
B’
4.519
3.011
52.78
3.71
4.517 a
3.020 a
RI PT
and Mg7TMH16 (TM=Sc, Ti, V, Y, Zr and Nb) at zero pressure and zero temperature.
3.66
calculations *
Present
Mg7ScH16
Expt. Others Present
PAW-GGA
9.496
55b, 45±2c, 49d ,54.2e 63.94
Others
PAW-GGA
9.5188 f
424.8f
Present
PAW-GGA
9.368 9.564 g
Expt.
USP-GGA
Others
PP-GGA
Present
PAW-GGA
Others
PAW-GGA
Present Others
Mg7NbH16
66.17j 71.22
460.8f
PAW-GGA
9.2938f 9.38 k 9.653
60.71
PAW-GGA
9.6677 f
419.4f
PAW-GGA
9.508
69.28
Others
PAW-GGA
9.5188 f
459.0f
9.381
73.63
PAW-GGA
Expt.
3.73
3.70
3.73
3.65
9.543 h
9.3958 f 488.8f 9.32 k USP-GGA 9.5376l 68.57l PAW: Projector Augmented Wave, USP: Ultra Soft Potential, PP: Pseudopotential, GGA: Others
*
56.46 i
Present
Present
3.64
445.5f
9.268
AC C
Mg7ZrH16
Others
9.3956 f 9.707 k 9.3804 i
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Mg7YH16
PAW-GGA
EP
Mg7VH16
Others
68.46
M AN U
Mg7TiH16
PAW-GGA
SC
MgH2
PAW-GGA
Generalized Gradient Approximation. a
Ref [32], b Ref [34]. c Ref [35]. d Ref [36]. e Ref [37]. f Ref [3]. g Ref [4]. h Ref [9]. i Ref [14], j
Ref [10]. k Ref [16]. l Ref [15].
21
ACCEPTED MANUSCRIPT Table 2. Elastic constant Cij (in GPa) of MgH2 and Mg7TMH16 (TM=Sc, Ti, V, Y, Zr, Nb) at zero pressure and zero temperature.
Mg7VH16
Mg7YH16
Mg7ZrH16
C44
C66
PAW GGA
69.37
36.67
34.48
143.38
42.62
52.08
FP-LAPW GGA
73.82
34.07
31.98
139.24
39.93
52.11
Others
74.4a
38.8a
31.4a
136.0a
37.6a
53.0a
Others
73.1b
33.9b
20.4b
131.9b
38.5b
52.3b
PAW GGA
141.74
23.54
FP-LAPW GGA
136.06
21.93
PAW GGA
145.35
29.29
FP-LAPW GGA
142.20
Others
12.99
19.49 18.04
27.07
17.42
130.02c
16.48 c
19.68 c
PAW GGA
153.64
29.78
22.41
FP-LAPW GGA
146.67
34.26
14.31
PAW GGA
132.77
20.26
7.631
FP-LAPW GGA
136.61
24.52
12.54
PAW GGA
137.62
30.41
14.83
140.86
32.07
20.00
148.25
32.14
19.13
FP-LAPW GGA
138.90
28.80
20.06
Others
157,25d
24.23d
19.59d
AC C
FP-LAPW GGA Mg7NbH16 PAW GGA
a
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C33
SC
Mg7TiH16
C13
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Mg7ScH16
C12
EP
MgH2
C11
M AN U
System
Ref [37], b Ref [36], c Ref [14].d Ref [15].
22
ACCEPTED MANUSCRIPT Table 3: Bulk modulus BH (in GPa), shear modulus GV, GR, GH (in GPa), Young’s modulus EH (in GPa), Poisson ratio ν and ratio BH/GH of MgH2 and Mg7TMH16 (TM=Sc, Ti, V, Y, Zr, Nb) at zero pressure and zero temperature. BR
BH
GV
GR
GH
EH
PAW GGA
54.82
50.32
52.57
39.23
32.69
35.96
87.85
0.221
1.46
FP-LAPW GGA
53.66
50.21
51.93
38.98
34.63
36.80
89.31
0.213
1.41
36.1a
87.3a
25.16
60.60
0.3236
2.50
26.46
30.49
78.21
0.28264 1.96
24.91
29.47
77.24
0.3106
24.16
28.82
75.39
0.30801 2.27
Other work
Mg7TiH16
PAW GGA
62.94
62.94
62.94
31.43
FP-LAPW GGA
59.97
59.97
59.97
34.52
PAW GGA
67.97
67.97
67.97
34.03
FP-LAPW GGA
65.45
65.45
65.45
33.48
Other work
Mg7ZrH16
PAW GGA
71.06
71.06
2.31
FP-LAPW GGA
71.73
71.73
PAW GGA
57.77
57.77
FP-LAPW GGA
61.88
PAW GGA
71.06
38.22
30.09
34.15
88.31
0.29288 2.08
71.73
31.07
20.39
25.73
68.94
0.33981 2.79
57.77
27.08
11.67
19.37
52.27
0.3492
61.88
61.88
29.94
18.19
24.06
63.91
0.32788 2.57
66.15
66.15
66.15
30.34
20.87
25.60
68.03
0.3286
2.58
68.33
68.33
68.33
33.76
26.77
30.26
79.11
0.3070
2.26
70.84
70.48
70.84
34.70
26.14
30.42
79.83
0.3122
2.33
65.50
65.50
65.50
34.05
26.90
30.48
79.15
0.2986
2.15
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Mg7YH16
BH/GH
56.46b 31.96b 22.91b 27.43b 70.82b 0.29 b
EP
Mg7VH16
18.88
SC
Mg7ScH16
50.2a
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MgH2
ν
BV
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System
AC C
FP-LAPW GGA
Mg7NbH16 PAW GGA
FP-LAPW GGA
2.98
68.57c 68.57c 68.57c 38.35c 27.27c 32.81c 84.89c 0.29c
a
Ref [36], b Ref [14], c Ref[15].
23
ACCEPTED MANUSCRIPT Table 4. Second order polynomial expressions of the fitting volume-pressure (V/V0(P)) for Mg7TMH16 hydrides. System
V/V0 0.99768 − 0.01356 P + 2.05805 × 10−4 P 2
Mg7TiH16
0.99769 − 0.01268 P + 1.79667 × 10 −4 P 2
Mg7VH16
0.99836 − 0.01272 P + 1.83835 × 10−4 P 2
Mg7YH16
0.99301 − 0.01502 P + 2.66587 × 10 −4 P 2
Mg7ZrH16
0.99775 − 0.0129 P + 1.88422 × 10−4 P 2
Mg7NbH16
0.99735 − 0.01218 P + 1.68094 × 10 −4 P 2
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Mg7ScH16
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ACCEPTED MANUSCRIPT
Table 5. Molecular mass (M), density (ρ), longitudinal wave velocity (vl), transverse wave velocity (vt), average elastic wave velocity (vm) and Debye temperature (TDebye) at normal pressure and temperature. M (g/mol)
MgH2
26.32
ρ (g/cm3) 1.42
vl (m/s) 8394
vt (m/s) 5067
vm (m/s)
TDebye (K)
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System
5568
770
854.8a
1.797
7322
Mg7TiH16
234.15
1.875
7559
234.13b
1.884b
Mg7VH16
237.20
2.029
Mg7YH16
275.17
2.082
Mg7ZrH16
277.49
2.197
Mg7NbH16
279.17
2.303
4192
602
3965
4434
643
7027.15b
3815.72b
4257.21b
618.67b
7576
4103
4578
679
6333
3050
3429
488
6752
3414
3827
553
6950
3634
4065
596
3819.37c
4262.736c
619.753c
TE D
7166.383c
EP
Ref [36], b Ref [14], c Ref [15].
AC C
a
3741
SC
231.22
M AN U
Mg7ScH16
25
ACCEPTED MANUSCRIPT
Table 6. Third order polynomial expressions of the fitting TDebye-pressure (TDebye (P)) for
System
TDebye
RI PT
Mg7TMH16 hydrides.
603.23 + 23.79 × P − 0.723 × P 2 + 0.012 × P 3
Mg7TiH16
646 + 18.396 × P − 0.369 × P 2 + 0.0034 × P 3
Mg7VH16
685.13 + 23.305 × P − 0.787 × P 2 + 0.0105 × P 3
Mg7YH16
502.71 + 36.74 × P − 2.0743 × P 2 + 0.0485 × P 3
Mg7ZrH16
554.59 + 25.63 × P − 1.123 × P 2 + 0.0232 × P 3
Mg7NbH16
600.24 + 22.112 × P − 0.78 × P 2 + 0.013 × P 3
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Mg7ScH16
26
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ACCEPTED MANUSCRIPT
Fig 1. Unit cell of (a) MgH2 (b) Mg7TMH16 (TM=Sc, Ti, V, Y, Zr, Nb) (b) hydrides. Color
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scheme: yellow, green and blue balls represent Mg, TM and H atoms respectively.
27
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Fig. 2. The ratio V/V0 of Mg7TMH16 hydrides as function of pressure.
28
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Fig.3. The elastic constant C11, C12 and C44 of Mg7TMH16 hydrides under pressure
29
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Fig.4. The elastic moduli B, G, E of Mg7TMH16 hydrides under pressure
30
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Fig.5. The hardness H of Mg7TMH16 hydrides under pressure
31
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Fig.6. The B/G and ν ratios of Mg7TMH16 hydrides under pressure
32
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ACCEPTED MANUSCRIPT
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Fig.7. The anisotropy A of Mg7TMH16 hydrides under pressure
33
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ACCEPTED MANUSCRIPT
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Fig.8. Debye temperature TDebye of Mg7TMH16 hydrides under pressure.
34
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Fig. 9. Longitudinal, transverse, average wave velocity of Mg7TMH16 hydrides under pressure
35
ACCEPTED MANUSCRIPT
•
Structural and mechanical properties of MgH2 and Mg7TMH16 were estimated at 0 GPa. Adding transition metal to MgH2 turns the brittleness of hydrides material.
•
The mechanical properties of Mg7TMH16 were calculated under pressure.
•
Debye temperature and sound velocities are calculated under pressure.
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•