# Dislocation Structure and Mobility in Hcp Rare-Gas Solids: Quantum versus Classical

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## Abstract

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## 1. Introduction

## 2. Methods Outline

#### 2.1. Classical Simulations

#### 2.2. Analysis Methods

#### 2.2.1. Differential Displacement Analysis (DD)

#### 2.2.2. Nearest Neighbor Analysis (NN)

#### 2.2.3. Common Neighbor Analysis (CNA)

## 3. Results and Discussion

#### 3.1. Edge Dislocation Structure

#### 3.2. The Peierls Stress

#### 3.2.1. Method A: Fixed Boundary Conditions

#### 3.2.2. Method B: Periodic Boundary Conditions

#### 3.3. Dislocation Mobility: Finite-T Simulations

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Sketch of a fully relaxed system containing an edge dislocation from different views. Green spheres represent atoms belonging to the dislocation core, blue spheres atoms belonging to the fcc-like stacking fault, and yellow spheres atoms with common hcp atomic coordination features. Red arrows represent the Burgers vectors of the partial dislocations. (

**a**,

**b**) represent different views of the simulation cell.

**Figure 2.**Relative displacement (

**a**,

**b**) and differential displacement (

**c**,

**d**) of the atoms above and below the glide plane of the edge dislocation in the x- and y-directions. The simulation cell contains n = 344,544 atoms and is fully relaxed.

**Figure 3.**Relative displacement (

**a**,

**b**) and differential displacement (

**c**,

**d**) of the atoms above and below the glide plane of the edge dislocation in the x and y directions. The simulation cell contains n = 18,424 atoms. Green and blue lines represent the results obtained in a non-relaxed (${\sigma}_{ij}\ne 0$) and a fully relaxed (${\sigma}_{ij}=0$) simulation cell, respectively.

**Figure 4.**Relative displacement (

**a**,

**b**) and differential displacement (

**c**,

**d**) of the atoms above and below the glide plane of the edge dislocation in the x- and y-directions. The simulation cell contains $n=1368$ atoms and is fully relaxed.

**Figure 5.**The $\gamma $-surface of the analyzed classical hcp rare-gas crystal. Perfect hcp stacking positions correspond to the four corners of the plot while large white spheres indicate metastable fault positions. Iso-$\gamma $ curves are represented with solid black lines at 5 and 10 mJ/m${}^{2}$ intervals.

**Figure 6.**Sketch of the system used to estimate the Peierls stress with Method A. Three main parts are differentiated: the upper part “U”, the lower part “L”, and the region with mobile atoms “M”. “P” indicates application of periodic boundary conditions and the dashed line the orientation of the edge dislocation.

**Figure 7.**Evolution of the shear stress expressed as a function of strain for a simulation cell containing n = 344,544 atoms in which the thickness of the “U” region is taken to be ${d}_{U}=5c$. Results obtained with $\Delta \eta =8\times {10}^{-4}$ and $1\times {10}^{-4}$ are shown in (

**a**,

**b**), respectively (see text).

**Figure 8.**(

**a**) evolution of the shear stress expressed as a function of strain for a simulation cell containing n = 18,424 atoms in which the thickness of the “U” part is equal to ${d}_{U}=2.5c$. Several mechanical strain steps, $\Delta \u03f5$, are considered; (

**b**) position of the corresponding dissociated edge dislocation expressed as a function of strain. “DD” and “NN” stand for the analysis methods of differential displacement and nearest neighbors, respectively.

**Figure 9.**Energy (

**a**) and shear stress (

**b**) expressed as a function of shear strain for a simulation cell containing n = 18,424 atoms in which periodic boundary conditions are applied along the three Cartesian directions.

**Figure 10.**Position of the dissociated edge dislocation expressed as a function of time for a simulation cell containing n = 18,424 atoms. Molecular dynamics simulations have been performed in the three thermodynamic ensembles (

**a**) $NVT$, (

**b**) $NVE$, and (

**c**) $NPT$; the temperature has been fixed to 25 K, the pressure to zero, and the volume to the equilibrium one. In the $NVE$ case, an equilibrium temperature of $12.5$ K is reached. “DD”, “NN”, and “CNA” stand for the analysis methods of differential displacement, nearest neighbors, and common neighbor, respectively.

**Figure 11.**Position of the edge dissociated dislocation expressed as a function of time for a simulation cell containing $N=18424$ atoms. Molecular dynamics simulations have been performed in the three thermodynamic ensembles (

**a**) $NVT$, (

**b**) $NVE$, and (

**c**) $NPT$; the temperature has been fixed to 50 K, the pressure to zero, and the volume to the equilibrium one. In the $NVE$ case, an equilibrium temperature of 25 K is reached. “DD”, “NN”, and “CNA” stand for the analysis methods of differential displacement, nearest neighbors, and common neighbor, respectively.

**Figure 12.**Time-accumulated average displacement of the dissociated edge dislocations expressed as a function of time, temperature, and simulated thermodynamic ensemble (n = 18,424 atoms). Solid lines represent the actual dislocation positions and dashed lines are linear fits performed on regions in which the dislocation motion is not disturbed by the simulation cell boundaries. Dislocation diffusion velocities are deduced directly from the slope of the linear fits.

**Table 1.**Average width of the fcc-like stacking fault, ${\omega}_{sf}$, and edge dislocation diffusion velocity, ${v}_{d}$, expressed as a function of temperature, and simulated thermodynamic ensemble (n = 18,424 atoms). ${\omega}_{sf}$ results are expressed in units of lattice parameter a and the figures within parentheses indicate the corresponding statistical uncertainty.

T = 25 K | ||

$\phantom{\rule{2.em}{0ex}}{v}_{d}\phantom{\rule{3.33333pt}{0ex}}(\mathrm{m}/\mathrm{s})\phantom{\rule{2.em}{0ex}}$ | $\phantom{\rule{2.em}{0ex}}{\omega}_{sf}\phantom{\rule{3.33333pt}{0ex}}\left(a\right)\phantom{\rule{2.em}{0ex}}$ | |

$\mathrm{NVT}$ | $57.2\phantom{\rule{3.33333pt}{0ex}}\left(0.5\right)$ | $12.0\phantom{\rule{3.33333pt}{0ex}}\left(0.3\right)$ |

$\mathrm{NVE}$ | $37.0\phantom{\rule{3.33333pt}{0ex}}\left(0.5\right)$ | $12.1\phantom{\rule{3.33333pt}{0ex}}\left(0.6\right)$ |

$\mathrm{NPT}$ | $53.1\phantom{\rule{3.33333pt}{0ex}}\left(0.5\right)$ | $12.1\phantom{\rule{3.33333pt}{0ex}}\left(0.6\right)$ |

T = 50 K | ||

$\phantom{\rule{2.em}{0ex}}{v}_{d}\phantom{\rule{3.33333pt}{0ex}}(\mathrm{m}/\mathrm{s})\phantom{\rule{2.em}{0ex}}$ | $\phantom{\rule{2.em}{0ex}}{\omega}_{sf}\phantom{\rule{3.33333pt}{0ex}}\left(a\right)$ | |

$\mathrm{NVT}$ | $79.6\phantom{\rule{3.33333pt}{0ex}}\left(0.5\right)$ | $11.9\phantom{\rule{3.33333pt}{0ex}}\left(3.3\right)$ |

$\mathrm{NVE}$ | $56.2\phantom{\rule{3.33333pt}{0ex}}\left(0.5\right)$ | $11.9\phantom{\rule{3.33333pt}{0ex}}\left(3.3\right)$ |

$\mathrm{NPT}$ | $76.1\phantom{\rule{3.33333pt}{0ex}}\left(0.5\right)$ | $11.9\phantom{\rule{3.33333pt}{0ex}}\left(3.3\right)$ |

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**MDPI and ACS Style**

Sempere, S.; Serra, A.; Boronat, J.; Cazorla, C. Dislocation Structure and Mobility in Hcp Rare-Gas Solids: Quantum versus Classical. *Crystals* **2018**, *8*, 64.
https://doi.org/10.3390/cryst8020064

**AMA Style**

Sempere S, Serra A, Boronat J, Cazorla C. Dislocation Structure and Mobility in Hcp Rare-Gas Solids: Quantum versus Classical. *Crystals*. 2018; 8(2):64.
https://doi.org/10.3390/cryst8020064

**Chicago/Turabian Style**

Sempere, Santiago, Anna Serra, Jordi Boronat, and Claudio Cazorla. 2018. "Dislocation Structure and Mobility in Hcp Rare-Gas Solids: Quantum versus Classical" *Crystals* 8, no. 2: 64.
https://doi.org/10.3390/cryst8020064