# Atomic Simulations of U-Mo under Irradiation: A New Angular Dependent Potential

^{*}

## Abstract

**:**

## 1. Introduction

^{S}→ body-centered tetragonal (bct) γ

^{0}→ monoclinic α’’ → orthorhombic α’ [5,6]. The addition of Mo was suggested to stiffen the U lattice against shear, thus hindering and also complicating the transition progress. However, during service life, Mo has been observed to be depleted from grain boundaries (GBs) among a wide range of compositional banding [7], which could lead to an onset of phase decomposition. In particular, the elemental redistribution near GBs could also be implicated in the generation of irradiation-induced recrystallization (IIR) [8], where fuel grains are subdivided into nano-sized grains from the GBs during service life. Meanwhile, IIR is suggested as an important culprit behind accelerated swelling behavior of nuclear fuel alloys [9], which enhances the reach of GBs into the fuel grains, destroying low swelling intra-granular fission gas bubbles (FGBs) and producing high swelling inter-granular FGBs.

## 2. Materials and Methods

## 3. Results

#### 3.1. Fitting Results

_{2}Mo has previously been found in experiments.

#### 3.2. Cascade Simulations

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The energetics of the supercell as a function of the ratio between the distance of the nearing dimers and the lattice constant of bcc U. In particular, nearing atom pairs include (

**a**) U–U in <110>, (

**b**) U–Mo in <110>, (

**c**) Mo–U in <110>, (

**d**) Mo–Mo in <110>, (

**e**) U–U in <100>, (

**f**) U–Mo in <100>, (

**g**) Mo–U in <100> and (

**h**) Mo–Mo in <100>. The black and red dash lines represent calculation results from DFT and MD using ADP. In the configuration diagrams, blue and orange circles represent U and Mo atoms, respectively, and the original positions of displaced atoms are indicated by dotted circles.

**Figure 2.**Angular distribution of launch directions sampled in the fundamental orientation zone of bcc lattice.

**Figure 3.**Potential functions of the newly developed ADP. Embedding function $F$, electron density function $\rho $ and pair potential $\varphi $ are shown in (

**a**–

**c**), respectively; angular-dependent terms of ${\psi}^{\mathrm{u}},{\psi}^{\mathrm{v}}$ and ${\psi}^{\mathrm{w}}$ are grouped together and plotted for elemental pairs of (

**d**) U–U, (

**e**) Mo–Mo and (

**f**) U–Mo.

**Figure 6.**Statistical results of (

**a**) residual defect population and (

**b**) peak values of defect population during cascade simulations with different Mo contents.

**Figure 7.**(

**a**) Average distances between residual defects and the initial position of PKA as a function of Mo content and (

**b**) Average atomic fractions of Mo among atoms with displacement more than 1 Å as a function of Mo content. The corresponding Mo atomic fractions in the total simulation bulk are also shown for comparison.

Parameter | U–U | Mo–Mo | U–Mo |
---|---|---|---|

${a}_{0}$(eV∙Å^{−3}) | −1.41881 | 1.75876 | 6.095612 |

${a}_{1}$ (eV∙Å^{−4}) | 2.499597 | −3.74227 | −11.667 |

${a}_{2}$ (eV∙Å^{−5}) | −1.40956 | 2.276131 | 7.291495 |

${a}_{3}$ (eV∙Å^{−6}) | 0.279328 | −0.42748 | −1.48751 |

α (eV∙Å^{−2}) | 0.86737 | 0.255577 | - |

β | −0.88394 | 2.017747 | - |

${c}_{\mathrm{u}}$ (eV^{1/2}) | 0.083778 | 0.369529 | −0.2 |

${d}_{\mathrm{u}}$ (Å) | 2.825854 | 2.590854 | 3.173449 |

${\lambda}_{\mathrm{u}}$ (Å) | 0.151763 | 0.62249 | 0.312741 |

${c}_{\mathrm{v}}$ (eV^{1/2}) | 0.015177 | 0.225399 | 0.2 |

${d}_{\mathrm{v}}$ (Å) | 3.153919 | 2.791784 | 3.208256 |

${\lambda}_{\mathrm{v}}$ (Å) | 0.149995 | 0.186653 | 0.409074 |

${c}_{\mathrm{w}}$ (eV^{1/2}) | 0.133803 | - | −0.2 |

${d}_{\mathrm{w}}$ (Å) | 3.402176 | - | 2.952746 |

${\lambda}_{\mathrm{w}}$ (Å) | 0.359213 | - | 0.167571 |

${r}_{\mathrm{c}}$ (Å) | 4.7 | 4.7 | 4.7 |

${r}_{0}$ (Å) | 2.7408 | 2.7387 | - |

**Table 2.**Reproduced values of lattice constants, cohesive energies, and elastic constants of pure uranium.

Structure | Properties | Present Work | MEAM [22] | ADP [15] | Experiment | FP |
---|---|---|---|---|---|---|

$\alpha $-U | ${E}_{\mathrm{coh}}$ (eV/at) | 5.46 | 5.547 | 4.23 | 5.550 [23] | - |

a (Å) | 2.881 | 2.721 | 2.849 | 2.836 [24] | - | |

b (Å) | 5.486 | 6.381 | 5.841 | 5.867 [24] | - | |

c (Å) | 5.221 | 4.858 | 4.993 | 4.935 [24] | - | |

y | 0.108 | 0.093 | 0.103 | 0.102 [24] | - | |

B (GPa) | 153 | 143 | 147 | 136 [25] | - | |

${E}_{\mathrm{v}}$ (eV) | 1.17 | 2.597 | - | - | 1.95 [26] | |

$\gamma $-U | Δ${E}_{\mathrm{bcc}\to \mathrm{ort}}$ (eV/at) | 0.01 | 0.15 | 0.09 | - | 0.278 |

a (Å) | 3.479 | 3.463 | 3.52 | 3.47 [27] | 3.455 | |

C_{11} (GPa) | 128.2 | 144.0 | 183.6 | - | 103.0 | |

C_{12} (GPa) | 124.1 | 49.0 | 92.8 | - | 142.0 | |

C_{44} (GPa) | 38.2 | −36.2 | 79.9 | - | 46.0 | |

${E}_{\mathrm{v}}$ (eV) | 1.34 | - | - | - | 1.38 | |

${E}_{i}$ (eV) | 1.08 | - | - | - | 0.9 |

**Table 3.**Reproduced values of lattice constants, cohesive energies, and elastic constants of pure molybdenum.

Structure | Properties | Present Work | FP | Experiment [28,29,30] |
---|---|---|---|---|

bcc-Mo | ${E}_{\mathrm{coh}}$ (eV/atom) | 6.349 | 6.290 | - |

$\mathrm{a}$ (Å) | 3.171 | 3.162 | 3.147 | |

C_{11} (GPa) | 473.9 | 488.8 | 465 | |

C_{12} (GPa) | 143.3 | 146.6 | 176 | |

C_{44} (GPa) | 67.0 | 108.3 | - | |

${\mathrm{E}}_{\mathrm{v}}$ | 2.72 | 2.723 | 2.6–3.2 | |

fcc-Mo | Δ${E}_{\mathrm{bcc}\to \mathrm{fcc}}$ (eV/atom) | 0.439 | 0.327 | - |

$\mathrm{a}$ (Å) | 4.156 | 4.004 | - | |

hcp-Mo | Δ${E}_{\mathrm{bcc}\to \mathrm{hcp}}$ (eV/atom) | 0.628 | 0.328 | - |

$\mathrm{a}$ (Å) | 2.842 | 2.948 | - | |

$\mathrm{c}$ (Å) | 4.212 | 4.263 | - |

**Table 4.**Reproduced values of lattice constants, cohesive energies, and elastic constants of several U–Mo binary phases.

Structure | Properties | Present Work | FP |
---|---|---|---|

tetra-U_{2}Mo | ${E}_{\mathrm{coh}}$ (eV/at) | 6.044 | 6.361 |

$\mathrm{a}$ (Å) | 3.304 | 3.427 | |

$\mathrm{c}$ (Å) | 9.921 | 9.833 | |

y | 0.329 | 0.328 | |

bcc-U_{15}Mo | Δ${E}_{\mathrm{tetra}\to \mathrm{bcc}}$ (eV/at) | 0.391 | 0.116 |

$\mathrm{a}$ (Å) | 6.861 | 6.84 | |

C_{11} (GPa) | 159.3 | 124.4 | |

C_{12} (GPa) | 134.7 | 140.6 | |

P6-U_{2}Mo | Δ${E}_{\mathrm{tetra}\to \mathrm{P}6}$ (eV/at) | 48.4 | 34.9 |

$\mathrm{a}$ (Å) | −0.119 | −0.035 | |

$\mathrm{c}$ (Å) | 4.823 | 4.818 |

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**MDPI and ACS Style**

Ouyang, W.; Lai, W.; Li, J.; Liu, J.; Liu, B.
Atomic Simulations of U-Mo under Irradiation: A New Angular Dependent Potential. *Metals* **2021**, *11*, 1018.
https://doi.org/10.3390/met11071018

**AMA Style**

Ouyang W, Lai W, Li J, Liu J, Liu B.
Atomic Simulations of U-Mo under Irradiation: A New Angular Dependent Potential. *Metals*. 2021; 11(7):1018.
https://doi.org/10.3390/met11071018

**Chicago/Turabian Style**

Ouyang, Wenhong, Wensheng Lai, Jiahao Li, Jianbo Liu, and Baixin Liu.
2021. "Atomic Simulations of U-Mo under Irradiation: A New Angular Dependent Potential" *Metals* 11, no. 7: 1018.
https://doi.org/10.3390/met11071018