# Effects of Monovacancy and Divacancies on Hydrogen Solubility, Trapping and Diffusion Behaviors in fcc-Pd by First Principles

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Computational Methodology

^{2}and the force convergent criteria is −0.03 eV/Å. According to our first principles computations, the equilibrium lattice parameters of fcc-Pd are a = b = c = 3.940 Å, are well agreed with the experimental values [39] (a = b = c = 3.908 Å) and previous calculations (a = b = c = 3.854 Å) [32].

_{pd,int,h}, E

_{pd}and ${E}_{{H}_{2}}$ represent the total energy of one H dissolve in the interstitial of super-cell Pd, the total energy of super-cell Pd and the energy of H

_{2}molecule, respectively; ${E}_{\left(N-x\right)pd,V,H}$ and ${E}_{\left(N-x\right)Pd,V}$ are the total energy of one H dissolved in super-cell Pd containing vacancies [x = 1 (monovacancy) or x = 2 (divacancy)] and the total energy of super-cell Pd containing vacancies, respectively.

_{trap}) of H atoms at the vacancy in Pd was calculated by

## 3. Results and Discussion

#### 3.1. Dissolution Behavior of H in Interstitial Sites, Monovacancies and Divacancies

Sites | E_{S} (DFT) (eV) | E_{S} (Experiments) (eV) | |
---|---|---|---|

This Work (eV) | Others ^{a} (eV) | ||

O-site | −0.129 | −0.11 | −0.10 ^{b} |

T-site | −0.099 | −0.09 | 0.31 ^{c} |

1vac-3f | −0.352 | −0.20 | - |

1vac-4f | −0.295 | −0.16 | - |

1vac-top | −0.109 | 0.04 | - |

1vac-NOS | −0.133 | - | - |

2vac-3f−1 | −0.382 | −0.23 | - |

2vac-3f−2 | −0.382 | −0.23 | - |

2vac-3f−3 | −0.377 | −0.22 | - |

2vac-4f−1 | −0.291 | −0.16 | - |

2vac-4f−2 | −0.324 | −0.16 | - |

2vac-4f−3 | −0.416 | −0.28 | - |

2vac-NOS | −0.133 | - | - |

#### 3.2. Multiple H Atoms Trapping in Monovacancies and Divacancies

_{trap}) as a function of H number (n). According to the definition of H-trapping energy, the negative trapping energy means a stable nH-vacancy complex, that is, H atoms are more inclined to be trapped by the vacancies than dispersed at different O-sites. The trapping energies are negative initially, indicating that H atoms are inclined to be trapped by the vacancies at this stage. As a whole, the trapping energies increase with more implanted H atoms for both monovacancy and divacancy defects, and finally turns from negative to positive, indicating that there is a limit to the number of hydrogen atoms that can be trapped by the vacancies. The number of H atoms that can be trapped in a monovacancy and divacancy are 8 and 12, respectively. More H atoms can be trapped at a divacancy defect, indicating that larger vacancy defects in the lattice could accommodate more H impurities, which may be because the larger vacancy can provide a larger optimal electron density isosurface for H atoms. Notably, a single vacancy can trap up to eight H atoms, and each vacancy in a divacancy defect can trap up to six H atoms on average. That is, although the divacancy defect possesses a stronger H-trapping ability, its H-trapping efficiency is reduced, which may be due to its larger Coulomb repulsion interaction.

#### 3.3. The Behavior of H Diffusion in Interstitial Sites, Monovacancies and Divacancies

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Visualization of relevant interstitial sites at the (

**a**) monovacancy and (

**b**) divacancy, such as 1vac_top (yellow diamond), 1vac_3f (purple triangle), 1vac_4f (red square), 2Vac_3f−x (x = 1−3) (purple triangle), 2vac_4f−x (x = 1–3) (red square), as well as the vacancy-nearing O-site (NOS) (orange circle) and T-site (NTS) (blue circle). The red arrow shows the diffusion path of H atom. The path of (1vac_NOS → 1vac_3f) and (1vac_NOS → 1vac_NOT → 1vac_4f → 1vac_3f) is considered a monovacancy, and the path of (2vac_NOS → 2vac_TOS → 2vac_4f−3) is considered a divacancy.

**Figure 2.**Structure diagram of H dissolved in monovacancy (

**a**) and divacancy (

**b**) defects of fcc-Pd. The sequence of H bonding positions is shown.

**Figure 3.**Trapping energies as a function of the number of H atoms embedded in monovacancy and divacancy defects.

**Figure 4.**Diffusion pathways from (

**a**) T-site to T-site (T-T) and (

**b**) O-site to O-site (O-O) in bulk Pd. The corresponding diffusion barrier energy of (

**c**) T-T and (

**d**) O-O.

**Figure 5.**The corresponding diffusion barrier energies of (

**a**) 1vac_NOS → 1vac_3f and (

**b**) 1vac_NOS → 1vac_NOT → 1vac_4f → 1vac_3f.

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**MDPI and ACS Style**

Ma, B.-L.; Wu, Y.-Y.; Guo, Y.-H.; Yin, W.; Zhan, Q.; Yang, H.-G.; Wang, S.; Wang, B.-T.
Effects of Monovacancy and Divacancies on Hydrogen Solubility, Trapping and Diffusion Behaviors in fcc-Pd by First Principles. *Materials* **2020**, *13*, 4876.
https://doi.org/10.3390/ma13214876

**AMA Style**

Ma B-L, Wu Y-Y, Guo Y-H, Yin W, Zhan Q, Yang H-G, Wang S, Wang B-T.
Effects of Monovacancy and Divacancies on Hydrogen Solubility, Trapping and Diffusion Behaviors in fcc-Pd by First Principles. *Materials*. 2020; 13(21):4876.
https://doi.org/10.3390/ma13214876

**Chicago/Turabian Style**

Ma, Bao-Long, Yi-Yuan Wu, Yan-Hui Guo, Wen Yin, Qin Zhan, Hong-Guang Yang, Sheng Wang, and Bao-Tian Wang.
2020. "Effects of Monovacancy and Divacancies on Hydrogen Solubility, Trapping and Diffusion Behaviors in fcc-Pd by First Principles" *Materials* 13, no. 21: 4876.
https://doi.org/10.3390/ma13214876