# First-Principles Study of Oxygen in ω-Zr

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Computational Methods

^{−1}. The selected calculation parameters were tested to ensure that the energy convergence was less than 1 meV per atom. To avoid oxygen atom behavior being limited by the cell size, a 3 × 3 × 4 supercell containing 108 atoms was used. Before calculating the properties, all structures were fully optimized, including the relaxation atom positions and supercell volumes, until the force between two atoms was less than 0.01 eV/Å to ensure the accuracy of the calculation. The diffusion barriers of oxygen were calculated using the climbing image nudged-elastic band (CI-NEB) [33,34] method.

## 3. Results and Discussion

#### 3.1. Oxygen in ω-Zr

_{2}molecule. According to our calculations, we found that when an O atom is located at the 2e positions, it relaxes to the nearby tetrahedral (Tetra) interstitial position, while the 3f position is the octahedral (Octa) interstitial position. As shown in Table 2, the most stable occupancy for an O atom in ω-Zr is an Octa position, corresponding to a formation energy of −5.97 eV, and the second most stable occupancy is a Tetra position, corresponding to a formation energy of −4.62 eV. Our calculations are in agreement with the values obtained by You et al. [37].

_{1}→ O

_{2}, with an O atom jumping from an Octa position into a first-nearest neighbor (1 nn) Octa position; O

_{1}→ O

_{3,}with an O atom jumping from the Octa position into a 2 nn Octa position; T

_{1}→ T

_{2}, with an O atom jumping from a Tetra position into a first-nearest neighbor (1 nn) Tetra position; T

_{1}→ T

_{3,}with an O atom jumping from a Tetra position into a 2 nn Tetra position. For inter-plane diffusion, we assumed two diffusion paths: O

_{1}→ T

_{1}, with an O atom jumping from an Octa into a Tetra position; and T

_{1}→ O

_{1,}with an O atom jumping from a Tetra position into an Octa position. This is shown in Figure 2.

_{1}→ T

_{1}path) position is 2.00 eV, while the diffusion barrier of jumping from a Tetra to an Octa position (T

_{1}→ O

_{1}path) is 0.66 eV. For intra-plane diffusion, wefound a diffusion barrier of only 0.014 for the T

_{1}→ T

_{2}path, which seems to indicate a very fast diffusion of the O atom involved. A similar situation was found in ZrO

_{2}[38], where the diffusion barrier between two O-sharing bonds with an Zr atom is 0.06 eV. Moreover, for the O

_{1}→ O

_{2}path, the diffusion barrier of O is up to 2.39 eV. The diffusion potentials of the other two intra-plane diffusion paths, O

_{1}→ O

_{3}and T

_{1}→ T

_{3}are 2.40 eV and 0.014 eV, respectively, which are the same as the paths O

_{1}→ O

_{2}and T

_{1}→ T

_{2}. The initial diffusion path is set by the interpolation points between the initial and final states. However, the CI-NEB calculation shows that an O atom cannot jump directly to the 2 nn site. The O atom always jumps to a 1 nn site first and then to a 2 nn site, as shown by the solid line path in Figure 2b.

#### 3.2. Oxygen-Vacancy Interaction in $\omega $-Zr

_{vac}). Therefore, we mainly consider the following diffusion paths for O atoms in the ω-Zr-vacancy system: T

_{vac}→ T

_{3}, O

_{1}→ T

_{vac}, and O

_{1}→ O

_{2}. The diffusions pathways are shown in Figure 4.

_{1}→ O

_{2}and O

_{1}→ T

_{vac}paths decrease to 1.50 eV and 0.50 eV, respectively, while the the value for T

_{vac}→ T

_{3}path increases to 1.03 eV. This indicates that the vacancy reduces the diffusion barriers of the O atom near itand hinders the diffusion of the O atom the distant interstitial position from the vacancy. This indicates that the vacancy favorably reduces the diffusion barriers for the O atoms near a the vacancy and increases the diffusion difficulty of the O atoms to interstitial positions farther away from the vacancy. In Ni [41], the diffusion barrier of O is also significantly influenced by the vacancy due to the interaction between O and vacancy.

#### 3.3. Effect of Alloying Elements (Nb, Sn) on Oxygen in ω-Zr

_{Nb}) position when it is located in a traditional Tetra position near Nb. We further investigated the diffusion of O atoms in ω-Zr with alloying elements. For ω-Zr with Nb, we assumed three paths: O

_{1}→ T

_{Nb}, O

_{1}→ O

_{2}, and T

_{Nb}→ T

_{3}. For ω-Zr with Sn, we constructed O

_{1}→ T

_{1}, O

_{1}→ O

_{2}, and T

_{1}→ T

_{2}diffusion paths.

_{1}→ T

_{1}is 1.32 eV, and for O

_{1}→ O

_{2}is 1.05 eV, which is reduced of 0.68 eV and 1.34 eV, respectively, compared to the diffusion barrier in pure ω-Zr. Our calculated diffusion potential of O in the ω-Zr -Nb system is close to that of O in pure Nb calculated by Chen et al. [44]. This may be due to the stronger Nb-O bond formed in both the ω-Zr -Nb system and pure Nb. For ω-Zr with Sn, compared to pure ω-Zr, the diffusion of an O atom near Sn becomes difficult, and the diffusion barrier of the O

_{1}→ O

_{2}path increases from 2.40 eV to 3.36 eV. For O

_{1}→ T

_{1}, it increases from 2.00 eV to 2.86 eV. Thus, the Sn atom hinders the diffusion of O atom. The experiments [23] on the effect of oxygen content on the corrosion resistance of Nb–Sn–Zr alloys showed that an increase in Sn content can slower down the oxidation of zirconium alloys, which is consistent with our findings on Sn. Our results can also provide a microscopic explanation for their conclusions.

#### 3.4. O Atoms Clustering in ω-Zr

_{x}clusters in ω-Zr. In pure ω-Zr, the most stable occupancy for an O atom is Octa. Therefore, we placed one O atom in the Octa position and the others in different near-neighboring Octa positions in turn. The clusters’ binding energy ${E}^{b}\left(x\right)$ can be obtained as follows:

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The five high-symmetry interstitial positions in ω-Zr. Light blue spheres represent Zr atoms and yellow spheres represent the interstitial positions.

**Figure 2.**The paths of intra-plane (

**a**) and inter-plane (

**b**) diffusion in ω-Zr. The blue, red, and dark blue balls denote Zr, Octa position, and Tetra position. For intra-plane diffusion, the O atom always jumps via a solid line path, O

_{1}→ O

_{2}→ O

_{3,}and T

_{1}→ T

_{2}→ T

_{3}.

**Figure 3.**Comparison of the Tetra position of O in different configurations: (

**a**) in pure ω-Zr; (

**b**) in ω-Zr with vacancy; (

**c**) in ω-Zr with Nb; and (

**d**) in ω-Zr with Sn. Light blue spheres represent Zr atoms, dark blue spheres represent O atoms, yellow spheres represent vacancy, pink spheres represent Nb atoms, and purple spheres represent Sn atoms.

**Figure 5.**The local geometry of Sn and Nb in ω-Zr. (

**a**) The alloying element in the interstitial position. (

**b**) The alloying element in the substitution position.

Initial Position | Coordinates | Final Position | Formation Energies | Ref. [37] |
---|---|---|---|---|

1b | $0,0,\frac{1}{2}$ | 1b | −2.76 | - |

2c | $\frac{2}{3}$$,\frac{1}{3}$, 0 | Octa | −5.97 | - |

2e | 0, 0, z | Tetra | −4.62 | −4.51 |

3f | $0,\frac{1}{2}$, 0 | Octa | −5.96 | −5.44 |

3g | $\frac{1}{2}$$,\frac{1}{2}$$,\frac{1}{2}$ | 3g | −2.11 | - |

Migration Path | ω-Zr | α-Zr [39] |
---|---|---|

O_{1} → O_{2} | 2.39 | 2.94 |

O_{1} → O_{3} | 2.40 | - |

O_{1} → T_{1} | 2.00 | 1.82 |

T_{1} → T_{2} | 0.01 | - |

T_{1} → T_{3} | 0.01 | - |

T_{1} → O_{1} | 0.66 | - |

Paths | With Vacancy | Pure |
---|---|---|

T_{vac} → T_{3} | 1.03 | 0.01 |

O_{1} → T_{vac} | 0.50 | 2.00 |

O_{1} → O_{2} | 1.50 | 2.40 |

Elements | Substitution | Interstitial | ||
---|---|---|---|---|

1b | 2d | Octa | Tetra | |

Nb | 0.78 | 0.42 | 3.90 | 3.45 |

Sn | −0.97 | −0.74 | 3.24 | 3.38 |

Systems | Octa | Tetra |
---|---|---|

Pure | −5.96 | −4.62 |

With vacancy | −5.49 | −5.53 |

With Nb | −11.04 | −9.77 |

With Sn | −12.12 | −9.84 |

Path | with Nb | with Sn | Pure | Ref. [44] |
---|---|---|---|---|

O_{1} → T_{Nb} | 1.32 | - | - | 1.65 |

T_{Nb} → T_{3} | 0.38 | - | - | - |

O_{1} → O_{2} | 1.05 | 3.36 | 2.40 | 0.96 |

O_{1} → T_{1} | - | 2.86 | 2.00 | - |

T_{1} → T_{2} | - | 0.88 | 0.01 | - |

Number of O Atoms | System | 1 nn | 2 nn | 3 nn |
---|---|---|---|---|

O_{2} | Pure | −0.32 | −0.08 | 0.01 |

With Vacancy | −1.32 | −1.38 | −1.19 | |

With Nb | −0.50 | −0.06 | 0.04 | |

With Sn | −0.52 | −0.45 | −0.05 | |

O_{3} | Pure | −0.33 | −0.08 | −0.33 |

With Nb | −0.51 | −0.06 | −0.30 |

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**MDPI and ACS Style**

Chen, Y.; Liu, Z.; Wang, D.; Zhao, Y.
First-Principles Study of Oxygen in ω-Zr. *Metals* **2023**, *13*, 1042.
https://doi.org/10.3390/met13061042

**AMA Style**

Chen Y, Liu Z, Wang D, Zhao Y.
First-Principles Study of Oxygen in ω-Zr. *Metals*. 2023; 13(6):1042.
https://doi.org/10.3390/met13061042

**Chicago/Turabian Style**

Chen, Yonghao, Zhixiao Liu, Dong Wang, and Yi Zhao.
2023. "First-Principles Study of Oxygen in ω-Zr" *Metals* 13, no. 6: 1042.
https://doi.org/10.3390/met13061042