Effects of Mo alloying on stability and diffusion of hydrogen in the Nb16H phase: a first-principles investigation

First-principles calculations and the method of climbing-image nudged elastic band were used to investigate the effects of Mo alloying on the structural stability, mechanical properties, and hydrogen-diffusion behavior in the Nb16H phase. The Nb12Mo4H phase (26.5 at% Mo) was found to be the most thermodynamically stable structure, with a low ΔHf value (−0.26 eV) and high elastic modulus. Calculations revealed that the tetrahedral interstitial site (TIS) was the predominant location of H in both Nb16H and Nb12Mo4H phases. The calculated H-diffusion energy barrier and the diffusion coefficient of the Nb12Mo4H phase were 0.153 eV and 5.65 × 10−6 cm2 s−1 (300 K), respectively, which suggest that the addition of Mo would lead to a lower energy barrier and high diffusion coefficients for the Nb16H phase, thus improving the hydrogen-permeation properties of Nb metal.


Introduction
Hydrogen is not only an important raw material for chemical and petrochemical industries, but also a potential clean fuel as well as a good energy carrier. Pure hydrogen does not exist as a natural resource like coal and oil, however. Since it has to be produced from hydrogen-containing compounds, a safe, lowcost, and highly efficient separation and purication technology is always required. Hydrogen-permeable alloy membranes have been well regarded as the most important materials for hydrogen separation and purication. [1][2][3][4][5] Currently, group V metals (vanadium, niobium, and tantalum) have attracted many investigations as promising hydrogenseparation materials owing to their lower price and higher hydrogen permeability than those of currently used Pd-based alloys. 3,6 However, there is still a large barrier to practical application of these metals because of their poor resistance to hydrogen embrittlement. 3,7,8 Experimental studies have veried that alloying the metals is an effective way to solve this problem. [6][7][8][9] Niobium is one of the most promising hydrogen-permeable candidates for membranes because it possesses a good combination of excellent high-temperature mechanical properties and corrosion resistance. [10][11][12] Recent theoretical research performed by Watanabe et al. revealed that the addition of W could decrease the hydrogen solubility in Nb and therefore improve its resistance to hydrogen embrittlement. 6,13 Hu et al. performed a similar study and found that the addition of W can improve the mechanical properties of the Nb 16 H phase, decrease the structural stability of the Nb 15 WH (tetrahedral (T)) phase, lower the diffusion barrier of H, and enhance diffusion paths for H. 14 Both W and Mo are high-Z refractory metals (i.e., refractory metals containing impurities with high atomic numbers (Z)) with similar physical properties. Moreover, Mo has several characteristic properties: compared with W, Mo has a lower melting point (2883 K) and a lower erosion rate, while H has higher diffusivity and lower solubility in Mo, leading to lower H retention. [15][16][17] These characteristics make Mo an important alloying candidate for Nb-based alloy membranes for hydrogen permeation. However, since relevant works have not been reported in the literature, it is necessary to engage in rstprinciples theoretical investigations that are free from any experimental limitations on the effect of Mo addition on the structure and diffusion properties of the NbH phase in a rstprinciples way. Such calculations will also contribute to the understanding and design of H-storage and H-separation materials based on Nb.
The effects of the addition of Mo to the electronic structure, structural stability, H diffusion, and mechanical properties of the NbH phase were investigated by rst-principles calculation based on density functional theory. Nb 16 H was purposely selected and four Mo atoms were added to reach the equivalent of an experimental composition of 25 at% Mo in NbH. 6,7,13,18 The

Computational method
The calculations were carried out by using the Vienna Ab-initio Simulation Package (VASP). 19 The interactions between core and valence electrons were described with the projector augmented wave (PAW). 20 The exchange and correlation functions were generalized gradient approximations (GGAs) developed by Perdew et al. 21 An energy cutoff of 360 eV was used for the planewave basis sets, and the K points set used in our calculations is 5 Â 5 Â 5 grid generated by Monkhorst-Pack schema. 22 During structure relaxation of the lattice parameters, the volume and atomic positions were fully optimized with insymmetry restrictions until the total energy converged to 10 À5 eV in the self-consistent loop, and the criterion of force used in the calculations is 0.01 eVÅ À1 atom À1 .
Accordingly, we built supercell models of basic defect structures in which a H atom was placed at the tetrahedral interstitial site (TIS) and another at the octahedral interstitial site (OIS). A unit cell of 2 Â 2 Â 2 (16 atoms) with a bodycentered cubic (BCC) structure 23 was selected for pure Nb, and a Mo atom was introduced to replace the Nb atom-a series of structures with the composition Nb 16Àx Mo x was thus obtained. One H atom was then added at the TIS and one at the OIS of BCC Nb and Nb 16Àx Mo x . Fig. 1 shows the schematic illustrations of Nb 16Àx Mo x H (x ¼ 4) with TIS and OIS.
To probe the diffusion properties of hydrogen in the bulk of the Nb-Mo alloy, the climbing image nudged elastic band (CI-NEB) method 24 was used to determine the diffusion barriers between the initial and nal positions. Four images were taken and all the images were relaxed until the maximum force on each atom was less than 0.01 eVÅ À1 and the other computational parameters were the same as the above.
The diffusion coefficient (D) is also an important index that determines the diffusion velocity of H. According to the Arrhenius diffusion equation, D can be expressed by D ¼ D 0 exp(ÀE a /kT), where the D 0 , E a , k, and T are the pre-exponential factor, diffusion energy barrier, the Boltzmann constant, and the absolute temperature, respectively. For a metal with a cubic structure, D 0 can be expressed as D 0 ¼ 1 6 r 2 n; where r and n are the jump distance and the vibration frequency, respectively. We calculated n according to Zener and Wert's theory, 25 which is approximately where m is the mass of the impurity atom. As it is already known that the mass of the H atom is 1.67 Â 10 À27 kg, the jumping distance of the TIS H in Nb was set as a=2 ffiffiffi 2 p A: 3. Results and discussion    Aer the series of calculations for the supercell models, the lattice constants (a) of various Nb 16 H and Nb 12 Mo 4 H phases were obtained; the results are listed in Table 1. The calculated lattice constants of pure Nb, Nb 16 H (TIS and OIS) and Nb 12 Mo 4 H (TIS) are 6.60, 6.63, 6.58, and 6.53Å, respectively. The values of pure Nb and Nb 16 H (TIS) match well with the corresponding experimental unit cell values of 6.61 and 6.63 A. [26][27][28] In addition, Mo has a smaller atomic radius than Nb, which may lead to a slight decrease in the lattice constant with the addition of Mo to the Nb 16 H phase.

Mechanical properties of Nb 16 H and Nb 12 Mo 4 H phases
To nd out the effect of Mo alloying on the mechanical properties of Nb hydride, the elastic constants of Nb 16 H (TIS and OIS), Nb 12 Mo 4 H (TIS), and pure Nb were calculated for comparison. The specied elastic constant was obtained by analyzing the difference between the total energy of the original cell and that of the deformed cell under a series of small strains. For the BCC crystal of Nb, there are three independent components of elastic constants: C 11 , C 12 , and C 44 . Nb 16 H has tetragonal symmetry and possesses three more components: C 13 , C 33 , and C 66 . Mo doping reduces the symmetry to orthorhombic and thus adds two components: C 22 and C 23 . 29,30 The calculated elastic constants are listed in Table 1.
According to the equations of elastic moduli and the criteria for mechanical stability, the mechanical stability is dened by the following restrictions for a tetragonal crystal: 31,32 C 11 . 0; C 33 . 0; C 44 . 0; C 66 . 0; ðC 11 À C 12 Þ . 0; For an orthorhombic crystal, the criteria of mechanical stability are given by The results of elastic constants, C ij , indicate that both Nb 16 H and Nb 12 Mo 4 H phases meet the criteria for mechanical stability.
The obtained elastic constants were then used to calculate the bulk modulus (B) and shear modulus (G) from the Voigt-Reuss-Hill approximations. 33,34 The Young's modulus (E) is determined using the equation E ¼ 9BG/(3B + G). 31 The values of B, G, and E of the Nb 12 Mo 4 H (TIS) phase are larger than those of the Nb 16 H (TIS and OIS) phase, indicating that Mo alloying improves the mechanical properties of the Nb 16 H phase, which possibly enhances the resistance against hydrogen embrittlement.  there are three possible paths of H diffusion between TIS and OIS, namely, from TIS to TIS, from TIS to OIS, and from OIS to OIS. Since the diffusion path from OIS to OIS cannot not be realized because the TIS is just located along the path, it was excluded in our study. [38][39][40] We rst investigated the pathways of H in the bulk of pure Nb. Generally, there are two pathways for H to diffuse in a BCC lattice, i.e., TIS / TIS and TIS / OIS. It can be clearly seen that the energy barrier from TIS to TIS in Nb is 0.225 eV, which is much lower than the corresponding value of 0.362 eV from TIS to OIS, suggesting that the diffusion path of H in bulk Nb should be mainly from TIS to TIS instead of TIS to OIS. For H diffusion in the Nb 12 Mo 4 alloy, it can be clearly seen in Fig. 5 and 6 that the energy barrier from TIS to TIS (0.157 eV) is smaller than the corresponding value in Nb, and a similar observation can be seen from TIS to OIS. The above comparison demonstrates that the addition of Mo in Nb can make H diffusion easier with a smaller energy barrier.

Electronic properties of Nb 16 H (TIS) and Nb 12 Mo 4 H (TIS)
The H-diffusion energy barrier calculated for pure Nb and the Nb 12 Mo 4 alloy is 0.225 and 0.157 eV, respectively. The vibration frequency of pure Nb and the Nb 12 Mo 4 alloy is 2.686 Â 10 13 s À1 and 2.345 Â 10 13 s À1 , respectively. According to the Arrhenius diffusion equation, the calculated diffusion   coefficient is 8.87 Â 10 À7 cm 2 s À1 for pure Nb and 5.65 Â 10 À6 cm 2 s À1 for the Nb 12 Mo 4 phase at the standard room temperature of 300 K. From the above analysis, it can be deduced that the addition of Mo should have an important effect on the diffusion of H in Nb, i.e., H diffusion in the Nb 12 Mo 4 phase should become energetically more favorable when the energy barrier is lower. These characteristics would therefore bring about an increase in the H-diffusion coefficient and an improvement in H permeability. In other words, the addition of Mo could lower the diffusion barrier of H, which would fundamentally lead to higher H diffusion and high H permeability in the Nb 12 Mo 4 H phase.

Conclusions
We used rst-principles calculations and the CI-NEB method to perform a comprehensive study on the effects of Mo on the structural stability and mechanical properties of the Nb 16 H phase and the diffusion of hydrogen through the alloy. The calculations revealed that the Nb 12 Mo 4 H phase is the most thermodynamically stable structure with a low DH f of about À0.26 eV and a high elastic modulus. The diffusion paths of H in both Nb and Nb 12 Mo 4 phases should be mainly from TIS to TIS. The calculated H-diffusion energy barrier and the diffusion coefficient are 0.153 eV and 5.65 Â 10 À6 cm 2 s À1 (at 300 K), respectively. The lower energy barrier and higher diffusion coefficient of the Nb 12 Mo 4 phase imply that the addition of a suitable amount of Mo could improve hydrogen permeation in Nb metal.

Conflicts of interest
None.