Hydrogen solubility and diffusivity at Σ3 grain boundary of PdCu

Abstract

First principles calculations have been performed to comparatively reveal hydrogen solubility and diffusivity at grain boundaries of BCC and FCC PdCu phases. It is found that the temperature-dependent hydrogen solubility at BCC Σ3 (112) GB of PdCu seems much higher than that in BCC PdCu bulk, while hydrogen solubility in FCC Σ3 (111) GB of PdCu is much lower than that in its corresponding FCC bulk. Calculations also reveal that grain boundary has an important effect on hydrogen diffusion of BCC and FCC PdCu, i.e., hydrogen diffusivities of BCC Σ3 (112) and FCC Σ3 (111) grain boundaries of PdCu seem much smaller and bigger than those of its corresponding bulks, respectively. The predicted results could deepen the comprehension of hydrogen solubility and diffusion of PdCu phases.

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1. Introduction

Hydrogen solution and diffusion at grain boundaries of various metals have raised great research interest in the past decades due to the hydrogen embrittlement, intergranular cracking, increased creep rates, and decreased fatigue lift etc., while the main conclusions available in the literature are not consistent with each other.^1–13 For instance, hydrogen diffusion in grain boundaries of nickel was reported to be much faster than that in the bulk, which would be fundamentally due to a lower activation energy of diffusion in the area of the grain.^3,6 On the other hand, the effects of grain boundary on hydrogen solution and diffusion are related to the grain size and the type of grain boundary, e.g., hydrogen diffusivity in the grain boundary of aluminum and nickel with fine grains is smaller than that in the bulks.^7,10,11,13 Nevertheless, some researchers also reported that grain boundary has no effect on the solution or diffusion of hydrogen.^7,9

As to the hydrogen permeated Pd membranes, Mütschele and Kirchheim found that hydrogen solubility in nanocrystalline Pd is much bigger than that in its single crystalline counterpart.¹⁴ With the decrease of the grain size, hydrogen solution in the α phase of PdH becomes bigger, whereas smaller in the β phase of PdH.^14,15 Interestingly, Stuhr et al. made a quite different conclusion that compared with coarse-grained Pd, no change of hydrogen solubility was discovered for nanocrystalline Pd.¹⁶ In addition, hydrogen diffusion in grain boundaries of Pd seems much faster than that within the grains,⁸ and is also found to be concentration dependent, i.e., hydrogen diffusion coefficients are lower than single crystalline value at low hydrogen contents, and vice versa for higher contents.¹⁴

It is well known that the PdCu membrane is a kind of excellent candidates for hydrogen separation and purification, and compared with pure Pd and other Pd alloys, PdCu possesses excellent hydrogen selectivity, high thermal stability, superior resistance against poisoning, moderate mechanical properties, relatively low price, and wide operating temperature range.^17–20 Regarding hydrogen solubility and hydrogen diffusivity of various PdCu phases, there are already a lot of experimental and theoretical investigations in the literature.^17,19–43 These experimental studies are mainly concentrated on polycrystalline phases of PdCu,^{19–22,24–35,37,39} while the reported theoretical calculations are all related to single crystals of PdCu.^{17,23,36,38,40–42} In other words, the effects of grain boundary on hydrogen behaviors of PdCu membranes need further research investigations.

By means of highly accurate first principles calculations based on density functional theory,^43–47 the present study is aimed to have a comparative investigation of hydrogen solubility and diffusivity at grain boundaries of BCC and FCC PdCu. Both BCC and FCC structures of PdCu are intentionally selected as they are two main phases in the Pd–Cu phase diagram.⁴³ Specifically, the PdCu phases with the stoichiometric ratio of 1 : 1 are intentionally selected due to their superior performance of H permeability.^18,35,37,38 In addition, a lot of investigations have found that the Σ3 (112) grain boundary of BCC metals has the lowest grain boundary energy,^48–52 and the Σ3 (111) grain boundaries of FCC metals are energetically stable structure with the lowest grain boundary energy in the literature.^13,51,53 Accordingly, the Σ3 (112) grain boundaries of BCC PdCu and Σ3 (111) of FCC PdCu are therefore chosen in the present study. And the hydrogen solubility and hydrogen diffusivity of PdCu bulks³⁸ are used for the sake of comparison. The derived results will be compared with available experimental and theoretical evidence in the literature, and could provide a deep understanding of the effects of grain boundary on hydrogen solubility and diffusivity of BCC and FCC PdCu.

2. Method of calculation

The present first principles calculations are implemented based on density functional theory with the projector-augmented wave (PAW) method using Vienna Ab initio Simulation Package (VASP).^44,45,48,49 The generalized gradient approximation (GGA) of Perdew–Burke–Enzerof (PBE) is selected to describe exchange-correlation function,⁵⁰ and the cutoff energy of plane wave basis is 400 eV. The temperature-smearing method of Methfessel–Paxton⁵¹ and the tetrahedron method with the Blöchl corrections⁵² are adopted for structure relaxation and static calculations, respectively. The convergence criteria of relaxation and static calculations are 10⁻⁵ and 10⁻⁶ eV, respectively. The conjugate gradient method is used to minimize the Hellmann–Feynman forces in the ionic relaxations with the force stopping criterion of 0.02 eV Å⁻¹.

Accordingly, the atomic composition of Pd₅₀Cu₅₀ is purposely chosen as a result of its superior hydrogen permeability.^18,35,37,38 The grain boundaries of BCC Σ3 (112) (48 atoms) and FCC Σ3 (111) (48 atoms) of PdCu are constructed based on coincidence site lattice theory (CSL),⁵³ and are shown vividly in Fig. 1. It should be noted that the CSL and structural-unit (SU) models yield the same structure and energy for the Σ3 (112) or Σ3 (111) grain boundary.⁵³ The lattice constants of BCC PdCu (3.026 Å) and FCC PdCu (3.817 Å) obtained recently³⁸ are used in the present study. After the test calculations, the dimensions of BCC Σ3 (112) and FCC Σ3 (111) of PdCu are 5.241 × 4.279 × 29.769 ų and 5.398 × 4.675 × 26.445 ų, respectively. To simulate hydrogen solubility and hydrogen diffusivity, one hydrogen atom is added at the octahedral (O) and tetrahedral (T) sites of each supercell. It should be pointed out that the O1 site is surrounded by four Pd atoms and two Cu atoms, and the O2 site by two Pd and four Cu atoms. During each calculation, periodic boundary conditions are added to the model in three directions. To determine the GB structure with the lowest energy, multiple initial configurations are sampled with a series of rigid body translations (RBTs) in the direction perpendicular to the GB. After the test calculations, the k-meshes of 7 × 7 × 1 and 9 × 9 × 1 are chosen for relaxation and static calculations, respectively.

Fig. 1 Atomic structure of (a) BCC Σ3 (112) and (b) FCC Σ3 (111) grain boundary (GB) of PdCu (c) and (d) are octahedral (O) and tetrahedral (T) sites of H atom in BCC Σ3 (112) and FCC Σ3 (111) GB, respectively. The x-axis is [111̄]_{BCC PdCu} and [011̄]_{FCC PdCu}, the y-axis is [11̄0]_{BCC PdCu} and [11̄0]_{FCC PdCu}, the z-axis is [112]_{BCC PdCu} and [111]_{FCC PdCu}. The gray, blue and red spheres represent Pd, Cu and H atoms, respectively.

For the calculation of H diffusion in BCC Σ3 (112) and FCC Σ3 (111) of PdCu, the climbing image nudged elastic band (CI-NEB) method⁵⁴ is used to search the minimum energy paths. The spring constant and force convergence criterion are 5.0 and 0.05 eV Å⁻¹, respectively. The frequency calculations are performed for the diffusion paths to verify the saddle points with only one imaginary frequency, and the energy barrier is derived as the difference between the total energy of saddle point and initial configuration.

3. Results and discussion

3.1. Hydrogen solubility at Σ3 grain boundary of PdCu

A lot of researches have shown that the hydrogen solubility of FCC PdCu phase is much higher than that of BCC counterpart, which is due to the stronger binding of hydrogen at FCC PdCu phase.^20,34,38,55 As to the effect of grain boundary (GB) on hydrogen solubility at PdCu phases, however, there is not any report so far in the literature. The present first principles calculation is therefore executed to have a comparative study of fundamental influence of GB on hydrogen solubility of PdCu phases.

Accordingly, the binding energy (E_b) and zero point energy (ZPE) of hydrogen at Σ3 GB of PdCu could be calculated as follows:^38,40,42

where E_PdCuH and E_PdCu are total energies of GB of PdCu with and without one hydrogen, respectively; E_H2 is total energy of a H₂ molecule; ν_i is the vibration frequency of interstitial H atom in GB, and is obtained using the harmonic approximation through fixing metal atoms due to their much bigger mass than H atom.

The segregation energy (E_seg) of hydrogen at GB of PdCu is then calculated according to the following formula:

E_seg = E^GB_b − E^bulk_b, (3)

where E^GB_b and E^bulk_b are binding energies of hydrogen at GB and bulk of PdCu, respectively. Fundamentally, the segregation energy is regarded as the efficiency of GB to trap H atoms, i.e., the negative E_seg indicates H atoms are easier to gather in GB, and vice versa.

The hydrogen solubility (C) in GB of PdCu could be then derived by the Sieverts' law:

where P is pressure set as the standard atmospheric pressure (1 atm), and K_S is the Sieverts' constant:⁵⁵where γ_i is the vibration frequency of gaseous H₂, m and I are the mass and molecular moment of inertia of H₂, respectively; ν_i is the vibration frequency of interstitial H atom in GB; k_B is the Boltzmann constant and ℏ is the Planck constant. It should be pointed out that spin polarization is used to calculate the energy, vibration frequency, and molecular moment of inertia of H₂ molecule.

Consequently, the E_b, ZPE, and E_seg values of hydrogen in BCC Σ3 (112) and FCC Σ3 (111) GB of PdCu are obtained and summarized in Table 1. In addition, the temperature-dependent hydrogen solubility of BCC Σ3 (112) and FCC Σ3 (111) GB of PdCu is also derived and shown in Fig. 2. The corresponding bind energies and ZPE of H in PdCu bulks³⁸ are also listed in Table 1, and hydrogen solubility of PdCu bulks³⁸ is included in Fig. 2 for the sake of comparison. Several characteristics could be discerned from Table 1 and Fig. 2.

Table 1 Binding energy (E_b) and segregation energy (E_seg) of H atom at octahedral (O1 and O2) and tetrahedral (T) interstitial sites of BCC Σ3 (112) and FCC Σ3 (111) grain boundaries of PdCu. The relevant zero point energy (ZPE) is added in the parenthesis after E_b. The corresponding E_b values of H in PdCu bulk³⁸ are also listed for comparison

Site	H in BCC PdCu			H in FCC PdCu
	Eb (eV)		Eseg (eV)	Eb (eV)		Eseg (eV)
	Σ3 (112)	Bulk^38		Σ3 (111)	Bulk^38
O1	−0.172 (0.133)	−0.029 (0.138)	−0.136	−0.042 (0.154)	−0.069 (0.089)	0.036
O2	−0.039 (0.127)	0.060 (0.120)	−0.085	0.149 (0.149)	0.136 (0.070)	0.013
T	−0.157 (0.211)	−0.059 (0.192)	−0.119	0.193 (0.205)	0.160 (0.155)	0.04

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Fig. 2 Temperature-dependent hydrogen solubility at BCC Σ3 (112) and FCC Σ3 (111) GB of PdCu. The corresponding values in BCC and FCC PdCu bulks³⁸ are also listed for comparison. H and M represent hydrogen and metal atoms, respectively.

Firstly, the bind energy (E_b) of H in each interstitial site (O1, O2, and T) of BCC Σ3 (112) GB of PdCu is lower than the corresponding value in BCC PdCu bulk,³⁸ suggesting that H should be energetically more favorable in BCC Σ3 (112) GB of PdCu. Consequently, the hydrogen solubility at BCC Σ3 (112) GB of PdCu seems much higher than that in BCC PdCu bulk.³⁸ That is to say, hydrogen energetically prefers to stay at the GB area of BCC PdCu, rather than within the grain. The calculated hydrogen solubility of BCC PdCu bulk³⁸ is slightly lower than the corresponding experimental measurement,³⁷ probably due to the higher hydrogen solubility at the BCC PdCu Σ3 (112) GB. Such a theoretical prediction about hydrogen solubility at BCC Σ3 (112) GB of PdCu from the present study is similar to the experimental observations regarding hydrogen solubility in nanocrystalline Pd and the α phase of Pd.^14,15

Secondly, for each interstitial site of O1, O2, and T, FCC Σ3 (111) GB of PdCu possesses higher binding energy of hydrogen than its bulk. This comparison signifies that hydrogen would be thermodynamically unfavorable in FCC Σ3 (111) GB of PdCu. As shown in Fig. 2, the curve of temperature-dependent hydrogen solubility in FCC Σ3 (111) GB of PdCu is therefore much lower than that in its corresponding FCC bulk. In other words, GB has an important effect to reduce hydrogen solubility of FCC PdCu, which is consistent with similar experimental discovery of the β phase of Pd¹⁵ as well as the theoretical prediction of Ni GB.¹³ It should be noted that the ZPE values of H at BCC Σ3 (112) GB are close to that in bulk, while the ZEP values at FCC Σ3 (111) GB are higher than that in the bulk.

Thirdly, the E_seg values of H at the O1, O2, and T sites of BCC Σ3 (112) GB of PdCu are big and negative values of −0.136, −0.085, and −0.119 eV, respectively, indicating that hydrogen has a strong tendency to segregate in the GB area of BCC PdCu. Such a theoretical prediction from the present study is compatible with similar experimental observations of Pd.¹⁶ On the contrary, hydrogen seems difficult to segregate to the GB area of FCC PdCu due to its slightly positive E_seg values. Interestingly, the above drastic comparison regarding hydrogen segregation in GB of BCC and FCC PdCu is very similar to that reported about Σ3 grain boundary of Fe in the literature.⁵⁶ It should be pointed out that thermal expansion of the lattice and vacancy would have some effects on H binding energy and hydrogen solubility of PdCu, and further studies are welcome to find out these effects.

To have a deep understanding of the effects of grain boundary on hydrogen solubility of PdCu, the electronic structures of H at various interstitial sites of PdCu are calculated and compared with each other. As a typical example, Fig. 3 displays the total density of states of H atom at the T site of BCC and FCC Σ3 GBs and bulk of PdCu. It can be seen from Fig. 3 that the DOSs of H atom BCC Σ3 (112) GB below the Fermi level (E_f) are more centralized than those in BCC PdCu bulk. In addition, the DOS peak value of H atom at BCC Σ3 (112) GB of PdCu is 0.716 states per eV per atom, which is higher than the corresponding value of 0.535 states per eV per atom in BCC PdCu bulk. Similarly, the H atom at the T site of FCC Σ3 (111) GB has a DOS peak of 1.49 states per eV per atom below E_f, and this DOS peak is smaller than the corresponding value (1.85 states per eV per atom) of H atom in FCC PdCu bulk. By means of the Bader analysis, the charge of H atom at the T site of BCC Σ3 (112) GB of PdCu is derived to be 1.15, which is bigger than the value of 1.108 for the corresponding PdCu bulk. The above characteristics of electronic structures indicate that H atom should have formed a stronger bonding at BCC Σ3 (112) GB than that at BCC PdCu bulk, which would therefore bring about higher hydrogen solubility at BCC Σ3 (112) GB of PdCu shown in Fig. 2.

Fig. 3 Comparison of total densities of states (DOS) of H atom at the T site of (a) Σ3 (112) GB and bulk of BCC PdCu (b) Σ3 (111) GB and bulk of FCC PdCu.

3.2. Hydrogen diffusivity at Σ3 grain boundary of PdCu

It is well known that hydrogen diffusion in GB of various metal materials is quite different from that in single crystals or within the grain. Interestingly, some experimental studies reveal that hydrogen diffusion in GB of nickel is much faster than that in bulks due to a lower activation energy,^3,6 whereas the hydrogen diffusivity in GB of aluminum and nickel with fine grains is smaller than that in the bulks.^7,10,11,13 As to hydrogen diffusion in PdCu membranes, the reported experimental results are mainly concentrated on the polycrystalline phases,^{19–22,24–35,37,39} while the theoretical investigations are all related to single crystals.^{17,23,36,38,40–42} It is therefore of importance to find out the effect of GB on hydrogen diffusion of PdCu by means of first principles calculation.

According to lattice symmetry, the BCC Σ3 (112) and FCC Σ3 (111) GBs of PdCu have eight and seven hydrogen diffusion paths, respectively, which are shown clearly in Fig. 4. For BCC Σ3 (112) GB, the four hydrogen diffusion paths along GB are O1 → T3 → O2 → T4 → O1, O2 → T3 → O1 → T4 → O2, O2 → T4 → T4 → O2, and O1 → T3 → T3 → O1, while the four diffusion paths across GB are T1 → T3 → O1 → T3 → T1, T1 → O1 → T1, T2 → T4 → O2 → T4 → T2, and T2 → O2 → T2. Similarly, the hydrogen diffusion paths along GB of FCC Σ3 (111) are O1 → T1 → O2 → T2 → O1, O1 → T2 → O2 → T1 → O1, O1 → T1 → O1, and O1 → T2 → O1, and the diffusion paths across GB are O1 → T1 → T1 → O1, O1 → O1, and O1 → T2 → T2 → O1.

Fig. 4 The three layers atomic structure of the black dotted area in Fig. 1. And hydrogen diffusion through BCC PdCu Σ3 (112) GB along (a) x direction (along GB) (path 1: O1 → T3 → O2 → T4 → O1, path 2: O2 → T3 → O1 → T4 → O2), (b) y direction (along GB) (path 3: O2 → T4 → T4 → O2, path 4: O1 → T3 → T3 → O1), and (c) z direction (across GB) (path 5–8: T1 → T3 → O1 → T3 → T1, T1 → O1 → T1, T2 → T4 → O2 → T4 → T2, and T2 → O2 → T2), respectively; hydrogen diffusion through FCC PdCu Σ3 (111) GB along (d) x direction (along GB) (path 1: O1 → T1 → O2 → T2 → O1, path 2: O1 → T2 → O2 → T1 → O1), (e) y direction (along GB) (path 3: O1 → T1 → O1, path 4: O1 → T2 → O1), and (f) z direction (across GB) (path 5–7: O1 → T1 → T1 → O1, O1 → O1, and O1 → T2 → T2 → O1). The x-axis is [111̄]_BCC and [011̄]_FCC, the y-axis is [11̄0]_BCC and [11̄0]_FCCu, the z-axis is [112]_BCC and [111]_FCC. The gray and blue spheres represent Pd and Cu atoms, respectively. The small balls stand for the interstitial sites.

After a series of calculations, the derived energy barrier (E_d,i) of various diffusion paths are summarized in Table 2. It should be pointed out that the highest diffusion barrier has been regarded as the activation energy (E_a,i) for each diffusion path. Several features could be observed from Table 2. Firstly, for BCC Σ3 (112) GB of PdCu, diffusion path 5 possesses the lowest activation energies of 0.141 eV among the four paths across GB, which is much smaller than the corresponding lowest E_a,i of 0.243 eV of paths 1 and 3 along GB. This comparison suggests that hydrogen is energetically more favorable to diffuse across the BCC Σ3 (112) GB of PdCu, rather than diffusion along GB. On the other hand, the lowest E_a,i along and across FCC Σ3 (111) GB of PdCu are the same value of 0.331 eV, indicating that hydrogen has an equal probability to diffuse along and across FCC Σ3 (111) GB of PdCu.

Table 2 Diffusion barrier (E_d,i) of each hydrogen diffusion path in BCC Σ3 (112) and FCC Σ3 (111) GBs of PdCu and the corresponding imaginary frequency at saddle point is listed in the parenthesis after E_d,i. The unit of imaginary frequency is THz. The corresponding E_a,i of BCC and FCC PdCu bulks are 0.057 and 0.347 eV, respectively³⁸

Direction	Diffusion path	BCC Σ3 (112)	FCC Σ3 (111)
		Ed,i (eV)	Ed,i (eV)
Along GB	1
	2
	3
	4
Across GB	5
	6
	7
	8

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Secondly, the lowest activation energy (0.141 eV) of hydrogen diffusion in BCC Σ3 (112) GB of PdCu seems much bigger than the corresponding value of 0.057 eV of BCC PdCu bulk.³⁸ This characteristic implies that hydrogen diffusion in BCC Σ3 (112) GB should be energetically more difficult than that in BCC PdCu bulk. On the contrary, the lowest activation energy (0.331 eV) of hydrogen diffusion in FCC Σ3 (111) GB is slightly lower than the derived value of 0.347 eV of FCC PdCu bulk,³⁸ suggesting that it should be easier for hydrogen to diffuse in FCC Σ3 (111) GB than in FCC PdCu bulk.

Thirdly, the lowest activation energies (0.243 and 0.141 eV) of hydrogen diffusion along and across BCC Σ3 (112) GB of PdCu are much smaller than the corresponding value of 0.331 eV of FCC Σ3 (111) GB of PdCu. A similar feature has been discovered about the relative magnitude of the lowest activation energies (0.057 and 0.347 eV) of BCC and FCC bulks of PdCu.³⁸ In other words, hydrogen has energetically easier diffusion in BCC Σ3 (112) GB and bulks of PdCu than its FCC counterpart. Interestingly, the difference of lowest activation energies of hydrogen diffusion between BCC and FCC GB of PdCu becomes smaller that the corresponding value between BCC and FCC PdCu bulks. It should be pointed out that the ballistic diffusion may probably occur at the GBs and further studies are welcome to find out this effect on H diffusion at PdCu GB in the future.

We now turn to calculate the diffusivity of hydrogen (D) through the GB of PdCu by means of the following formula:⁵⁷

where C_i is the hydrogen solubility of initial interstitial site on path i; x_i is the number of initial interstitial site; E_a,i is the activation energy of the path i; k_B is the Boltzmann constant. And the pre-factor D_0,i could be calculated by:⁵⁵

f(x) = sinh(x)/x, (13)

where v_m and v_n are vibration frequencies at the initial and transition states, respectively; and λ_i is the jump distance.

The temperature-dependent diffusivity of hydrogen through BCC Σ3 (112) and FCC Σ3 (111) GB of PdCu is thus obtained and shown in Fig. 5. One can discern from this figure that GB has a quite effect on hydrogen diffusivity of BCC and FCC PdCu. For BCC PdCu, the curve of hydrogen diffusivity of GB is well below that of bulk,³⁸ signifying that GB should decrease hydrogen diffusivity of BCC PdCu. Such a feature is consistent with similar experimental observations regarding hydrogen diffusion in GB of nickel and aluminum.^7,9,10,58

Fig. 5 Temperature-dependent hydrogen diffusivity through BCC Σ3 (112) and FCC Σ3 (111) GB of PdCu. The corresponding values of BCC and FCC PdCu bulks³⁸ are also listed for comparison.

On the contrary, hydrogen diffusivity of FCC Σ3 (111) GB of PdCu at each temperature is bigger than that of FCC PdCu bulk.³⁸ It therefore follows that GB has an important effect to increase hydrogen diffusivity of FCC PdCu, which is just opposite to the decrease of hydrogen diffusivity of BCC PdCu as a result of GB. This increase (decrease) of hydrogen diffusivity due to GB is mainly attributed to the lower (higher) activation energies shown in Table 2. The higher hydrogen diffusivity of FCC Σ3 (111) GB of PdCu from the present study is also compatible with similar experimental evidence of nickel in the literature.^3,6,57,59 It should be noted that the calculated hydrogen diffusivities of BCC PdCu bulk³⁸ and FCC PdCu Σ3 (111) GB agree well with the corresponding calculation values in the literature.²⁰

In addition, hydrogen diffusivity through BCC Σ3 (112) GB of PdCu is significantly higher than that of FCC Σ3 (111) GB, and the same statement could be made for hydrogen diffusivity at BCC and FCC bulks of PdCu.³⁸ It can be seen clearly from Fig. 5 that at each temperature, the descending sequence of hydrogen diffusivity of PdCu is as follows: BCC bulk → BCC Σ3 (112) GB → FCC Σ3 (111) GB → FCC bulk. Such a comparison confirms that the BCC PdCu Σ3 (112) GB and bulk have much higher hydrogen diffusivity than its FCC counterpart. It should be pointed out that the hydrogen defect complexes would possibly arise in the GB of PdCu and have some effects on H diffusivities and solubilities.⁶⁰

4. Conclusions

First principles calculations have been performed to comparatively study the effects of grain boundary on hydrogen solubility and hydrogen diffusivity at PdCu phases with BCC and FCC structures. It is found that the hydrogen solubility at BCC PdCu Σ3 (112) GB is higher than that at BCC PdCu bulk. On the contrary, the hydrogen solubility at FCC PdCu Σ3 (111) GB is lower than that at its bulk. Calculations also reveal that hydrogen diffusion across BCC PdCu Σ3 (112) GB is easier than along the corresponding GB, while hydrogen diffusion along and across FCC PdCu Σ3 (111) GB should have an equal probability. Moreover, BCC Σ3 (112) GB could impede hydrogen diffusion and FCC PdCu Σ3 (111) GB would accelerate hydrogen diffusion. The calculated results could provide a deep understanding of the effects of GB on hydrogen solubility and diffusivity of BCC and FCC PdCu.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by Natural Science Foundation of Hunan Province (No. 2018JJ25081), Project supported by the Fundamental Research Funds for the Central Universities of Central South University (No. 2018zzts126), and Project supported by State Key Laboratory of Powder Metallurgy, Central South University, Changsha, China.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0ra10133h

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