# Electronic Structure Calculations of Oxygen Atom Transport Energetics in the Presence of Screw Dislocations in Tungsten

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## Abstract

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## 1. Introduction

## 2. Computational Details

#### 2.1. O Solution and Migration Energies under Strain

**I**is the identity tensor, and $\mathit{a}$ are the undeformed lattice vectors. The undeformed configuration is expressed in a canonical Cartesian reference system with ${\mathit{a}}_{1}\equiv \left[{a}_{0}\phantom{\rule{3.33333pt}{0ex}}0\phantom{\rule{3.33333pt}{0ex}}0\right]$, ${\mathit{a}}_{2}=\equiv \left[0\phantom{\rule{3.33333pt}{0ex}}{a}_{0}\phantom{\rule{3.33333pt}{0ex}}0\right]$, and ${\mathit{a}}_{3}\equiv \left[0\phantom{\rule{3.33333pt}{0ex}}0\phantom{\rule{3.33333pt}{0ex}}{a}_{0}\right]$, where ${a}_{0}=3.168$ Å is the lattice parameter of W calculated with the pseudopotential employed here. Two different strain tensors were applied for the simulations:

- (i)
- Hydrostatic strain:$$\mathit{\epsilon}=\left[\begin{array}{ccc}\u03f5& 0& 0\\ 0& \u03f5& 0\\ 0& 0& \u03f5\end{array}\right]$$
- (ii)
- Volume-conserving shear:$$\mathit{\epsilon}=\left[\begin{array}{ccc}0& \gamma /2& 0\\ \gamma /2& 0& 0\\ 0& 0& 0\end{array}\right]$$

#### 2.2. Screw Dislocation–O Interaction

## 3. Results

#### 3.1. Oxygen Atom Stability in Bulk Tungsten

#### 3.2. Oxygen Migration in Bulk Tungsten

#### 3.3. Screw Dislocation–Oxygen Interaction in Tungsten

## 4. Discussion

## 5. Conclusions

- We have conducted electronic structure calculations of the fundamental energetics of oxygen atoms in tungsten, including the heat of solution, migration energies, activation volumes, and interaction energy with screw dislocation cores.
- Oxygen atoms are preferentially found in tetrahedral lattice sites, with solution energies of $0.70$ eV. The substitutional heat of solution (an O atom associated with a vacant site) was found to be $-0.14$ eV.
- The migration energy for the tetrahedral→tetrahedral transition in the bulk is 0.20 eV. This energy is modified by stress according to activation volumes of $0.02{b}^{3}$ and $0.19{b}^{3}$ for volumetric and shear deformations, respectively.
- The interaction energy between a screw dislocation core and an O atom depends on the relative position of the oxygen, but it is found to be either 1.20 or 1.83 eV, depending on the final configuration. We find that this is due to the accommodation of the large local distortions induced by the O atom in the lattice, which is seen to lead to a core reconstruction from an easy core configuration to a hard core one when O atoms are present.
- In most cases, once absorbed at a screw dislocation core, O atoms are seen to induce a dislocation core transformation from an easy core configuration to a hard core one.
- These calculations will serve to parameterize mesoscale models of material deformation by dislocation slip.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Compilation of DFT Results for Dislocation Core–Oxygen Atom Interaction Energy Calculations

**Table A1.**Details of the calculations of the dislocation–O atom interaction energies for the six tetrahedral configurations considered in Figure 2. The first two columns specify the relative core–O atom positions before and after relaxation. The third and fourth columns indicate the distance between the two dislocation cores before and after relaxation in units of the periodicity of the Peierls potential, ${h}_{0}={a}_{0}\sqrt{6}/3$ (equal to the length of the triangles in Figure 2 and Figure 8). For example, $7.5{h}_{0}\approx 19.5$ Å. Columns 5 and 6 give the distance between the oxygen atoms before and after relaxation, also in units of ${h}_{0}$. The next column gives the structure of the dislocation cores obtained after relaxation (refer to Figure 8 for visual guidance). Finally, the last two columns give the uncorrected and corrected interaction energies in each case.

O-Atom | Distance between | Distance between | Final Core | Uncorrected | Corrected | |||
---|---|---|---|---|---|---|---|---|

Location (NN) | Cores (${\mathit{h}}_{0}$) | Oxygen Atoms (${\mathit{h}}_{0}$) | Configuration | ${\mathit{E}}_{\mathit{i}}$ (eV) | ${\mathit{E}}_{\mathit{i}}$ (eV) | |||

initial | final | initial | final | initial | final | |||

1 | 2 | 7.5 | 7.5 | 7.5 | 6.5 | hard | $-1.90$ | $-1.83$ |

2 | 2 | 7.5 | 8.5 | 7.5 | 7.5 | hard | $-1.81$ | $-1.83$ |

3 | 2/5 | 7.5 | 7.5 | 7.5 | 7.5 | hard/ext.easy | $-1.52$ | $-1.52$ |

4 | 2 | 7.5 | 7.5 | 7.5 | 7.5 | hard | $-1.83$ | $-1.83$ |

5 | 5 | 7.5 | 7.5 | 7.5 | 7.5 | ext. easy | $-1.20$ | $-1.20$ |

6 | 2 | 7.5 | 7.5 | 7.5 | 7.5 | hard | $-1.83$ | $-1.83$ |

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**Figure 1.**Illustration of the distortion of tetrahedral→tetrahedral paths under shear deformation. The undeformed path becomes stretched along two directions and compressed along the other two. $\theta $ represents the shear angle (refer to Section 2).

**Figure 2.**Location of the six nearest tetrahedral sites (blue points) to the geometric center of a screw dislocation core (green cross) before the relaxation of the dislocation–O system with DFT. The dislocation is in an easy core configuration. The arrows represent the amplitude of differential atomic displacements along the $\langle 111\rangle $ direction (normal to the image) induced by the presence of the dislocation. The atoms are color-coded according to their relative position along the $\left[111\right]$ direction (off-plane): white for $z=0$, grey for $z=b/3$ and black for $z=2b/3$.

**Figure 3.**Elementary bcc lattice cell showing octahedral and tetrahedral interstitial sites. As a guide to the eye, the figures show a shaded octahedron and tetrahedron with the interstitial site highlighted in their respective centers. In principle, transitions may occur between any two nearest neighbor interstitial sites.

**Figure 4.**Variation of the heat of solution of tetrahedral O as a function of volumetric, $\epsilon $, and shear, $\gamma $, strain.

**Figure 5.**(

**a**) Tetrahedral→tetrahedral oxygen migration energy path in bulk W calculated with DFT between two neighboring tetrahedral sites under hydrostatic strain. ${a}^{\prime}$ is defined in Equation (1). (

**b**) Migration energy barrier under uniform axial strain. The black line represents a least-squares linear fit, from whose slope (black triangle) the activation volume can be calculated.

**Figure 6.**(

**a**) Tetrahedral→tetrahedral oxygen migration energy path in bulk W calculated with DFT between two neighboring tetrahedral sites under shear strain. (

**b**) Migration energy barrier under shear strain. The black line represents a least-squares linear fit, from whose slope (black triangle) the activation volume can be calculated.

**Figure 7.**Relaxed migration path of an oxygen atom between two tetrahedral sites on a $\left(001\right)$ plane under zero strain showing two neighboring octahedral sites for reference. The trajectory follows a curved path arched towards the closest octahedral site.

**Figure 8.**Stable relaxed configurations of the dislocation core/O-atom system. All other initial configurations (see Figure 2) relax to one of these. The cross marks the location of the dislocation core (blue: initial, red: final), while the blue and red dots mark the initial and final position of the O atoms, respectively. (

**a**) The initial configuration starts with the oxygen atom in a fourth NN tetrahedral position. After relaxation, the core shifts to a hard core configuration, and the oxygen atoms shifts to a second NN position. The dislocation maintains its compact core structure after relaxation. The calculated interaction energy is $-1.83$ eV. (

**b**) The initial configuration starts with the oxygen atom in a fifth NN tetrahedral position. After relaxation, neither the dislocation nor the O atom have shifted; however, the dislocation core becomes asymmetric. The calculated interaction energy is $-1.20$ eV.

**Table 1.**Bond lengths and total and half energies of O–O complexes in the vacuum. The top row corresponds to the stable O${}_{2}$ molecule.

${\mathit{d}}_{\mathbf{O}\u2013\mathbf{O}}($Å) | ${\mathit{E}}_{\mathbf{O}\u2013\mathbf{O}}$ (eV) | $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\mathit{E}}_{\mathbf{O}\u2013\mathbf{O}}$ (eV) |
---|---|---|

1.23 (O${}_{2}$) | $-9.84$ | $-4.92$ |

2.76 ($1b$) | $-3.74$ | $-1.87$ |

5.52 ($2b$) | $-3.18$ | $-1.59$ |

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**MDPI and ACS Style**

Zhao, Y.; Dezerald, L.; Marian, J.
Electronic Structure Calculations of Oxygen Atom Transport Energetics in the Presence of Screw Dislocations in Tungsten. *Metals* **2019**, *9*, 252.
https://doi.org/10.3390/met9020252

**AMA Style**

Zhao Y, Dezerald L, Marian J.
Electronic Structure Calculations of Oxygen Atom Transport Energetics in the Presence of Screw Dislocations in Tungsten. *Metals*. 2019; 9(2):252.
https://doi.org/10.3390/met9020252

**Chicago/Turabian Style**

Zhao, Yue, Lucile Dezerald, and Jaime Marian.
2019. "Electronic Structure Calculations of Oxygen Atom Transport Energetics in the Presence of Screw Dislocations in Tungsten" *Metals* 9, no. 2: 252.
https://doi.org/10.3390/met9020252