# Effect of Domain Size, Boundary, and Loading Conditions on Mechanical Properties of Amorphous Silica: A Reactive Molecular Dynamics Study

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Approach

#### 2.1. Reactive Molecular Dynamics Simulation

#### 2.2. Development of a-SiO${}_{2}$

#### 2.3. Cases Studied and Computations

- T2—simulation domain is fully periodic and subjected to tensile loading on both sides in the x-direction, cf. Figure 2b.
- C2—simulation domain is fully periodic and subjected to compressive loading on both sides in the x-direction, cf. Figure 2c.
- T1—simulation domain is periodic in the y- and z-directions, while in the x-direction, the tensile loading is applied to the top surface as the bottom surface is fixed, cf. Figure 2d.
- C1—simulation domain is periodic in the y- and z-directions, while in the x-direction, the compressive loading is applied to the top surface as the bottom surface is fixed, cf. Figure 2e.

## 3. Results and Discussion

#### 3.1. Global Stress-Strain Curve

#### 3.2. Young’s Modulus and Bimodularity

#### 3.3. Poisson’s Ratio and Isotropy

#### 3.4. Distribution of Si-O Bond

#### 3.5. Computational Cost and Accuracy

## 4. Conclusions

- Mechanical properties converge with increasing domain size.
- With the presence of free surfaces in semi-periodic cases, the impact of domain size is much more significant than full-periodic cases.
- Amorphous silica exhibits strong bimodular behavior and slight anisotropy at the atomic level. Young’s modulus in tension is higher than in compression, while Poisson’s ratio in x-y plane and x-z plane are slightly different from each other.
- A “safe zone” defined as a zone where accuracy and computational cost are balanced. Defining such a zone is necessary for multiscale models, as well as defining RVE at nanoscale. In this zone, bulk properties can be reproduced with good accuracy.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MD | Molecular dynamics |

RMD | Reactive molecular dynamics |

BCs | Boundary conditions |

LCs | Loading conditions |

RDF | Radial distribution function |

## Appendix A. Simulation Data

Simulation Set | N $\times {10}^{4}$ | ${\mathit{E}}_{\mathbf{xx}}$ (GPa) | ${\mathit{\nu}}_{\mathbf{xy}}$ | ${\mathit{\nu}}_{\mathbf{xz}}$ |
---|---|---|---|---|

T2 | 0.24 | 73.052 | 0.377 | 0.395 |

0.96 | 73.319 | 0.376 | 0.396 | |

2.16 | 73.359 | 0.377 | 0.394 | |

3.84 | 73.500 | 0.377 | 0.395 | |

6.00 | 73.300 | 0.377 | 0.395 | |

15.36 | 73.318 | 0.377 | 0.395 | |

C2 | 0.24 | 61.231 | 0.350 | 0.347 |

0.96 | 61.677 | 0.355 | 0.348 | |

2.16 | 61.782 | 0.355 | 0.350 | |

3.84 | 61.779 | 0.355 | 0.350 | |

6.00 | 61.786 | 0.354 | 0.349 | |

15.36 | 61.791 | 0.355 | 0.349 | |

T1 | 0.24 | 92.412 | 0.324 | 0.346 |

0.96 | 83.484 | 0.350 | 0.366 | |

2.16 | 79.549 | 0.362 | 0.375 | |

3.84 | 77.850 | 0.364 | 0.382 | |

6.00 | 76.802 | 0.368 | 0.384 | |

15.36 | 75.500 | 0.370 | 0.389 | |

C1 | 0.24 | 65.246 | 0.370 | 0.368 |

0.96 | 64.832 | 0.323 | 0.339 | |

2.16 | 64.056 | 0.315 | 0.332 | |

3.84 | 63.270 | 0.317 | 0.327 | |

6.00 | 63.092 | 0.314 | 0.324 | |

15.36 | 62.581 | 0.311 | 0.322 |

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**Figure 1.**(

**a**) Density of a-SiO${}_{2}$ in the relaxation stage (circles). The dashed line denotes the average value (2.202 g/cm${}^{3}$). (

**b**) Radial distribution function (RDF) analysis for all pairs of a-SiO${}_{2}$ model. The results of O-O and Si-Si pairs are shifted 10 and 20 up, respectively, for better view.

**Figure 2.**(

**a**) Dimensions of a simulation domain of a-SiO${}_{2}$, where ${L}_{x}={L}_{y}$ and ${L}_{z}=$ 1.51 nm. ${L}_{x}$ is varied to study the effect of simulation domain. (

**b**–

**e**) Different simulation sets are employed in this study. Specifically, (

**b**,

**c**) the simulation domain is fully periodic and subjected to mechanical loading on both sides. T2 and C2 denote tensile and compressive loading on both sides, respectively. (

**d**,

**e**) Simulation domain is semi-periodic with free surfaces in the x-direction, where the bottom surface is fixed and the top surface is subjected to mechanical loading. T1 and C1 denote tensile and compressive loading on one side, respectively.

**Figure 3.**Global stress-strain curves for all scenarios investigated in this study including (

**a**) T2, (

**b**) C2, (

**c**) T1, and (

**d**) C1 simulations.

**Figure 4.**(

**a**) Young’s modulus in the x-direction, ${E}_{xx}$, and (

**b**) bimodular factor ${\alpha}^{\mathrm{bi}}$ from all simulations. The domain-size effect is considered by both number of atoms in the domain N and simulation domain volume ${V}_{\mathrm{d}}$. ${\alpha}_{2}^{\mathrm{bi}}$ and ${\alpha}_{1}^{\mathrm{bi}}$ describe the bimodularity of T2 and C2, and of T1 and C1 simulations, respectively. The gray shaded area describes the "safe zone", which is discussed in Section 3.5.

**Figure 5.**Poisson’s ratio (

**a**) ${\nu}_{xy}$ and (

**b**) ${\nu}_{xy}$ obtained from all simulations. The domain-size effect is considered by both number of atoms in the domain N and simulation domain volume ${V}_{\mathrm{d}}$. The gray shaded area describes the “safe zone”, which is discussed in Section 3.5.

**Figure 6.**Characterization of isotropy of a-SiO${}_{2}$ through the parameter ${\alpha}^{\mathrm{iso}}$ from all simulations. The black solid line indicates ${\alpha}^{\mathrm{iso}}=1$. The gray shaded area describes the “safe zone”, which is discussed in Section 3.5.

**Figure 7.**Radial distribution function RDF ${g}_{ij}\left(r\right)$ of Si-O bond distribution under deformation (solid lines) compared to undeformed state (solid line with solid circles). All the data of undeformed state are obtained at 20% strain of (

**a**) T2, (

**b**) C2, (

**c**) T1, and (

**d**) C1 simulations. The compressive and tensile regions are described by $\sigma <0$ and $\sigma >0$, respectively. An inset in (

**a**) describes the Si-O bond length r.

**Figure 8.**The relative deviation in Young’s modulus from each simulation to the largest simulation. The gray shaded area describes the “safe zone” where the errors are reasonably small.

**Table 1.**RDF first and second peak positions of all pairs in a-SiO${}_{2}$. The data includes both the results from our simulation and experiment from literature [36], which are in good agreements.

Structural Parameters | Our Simulation Results | Experimental Results [36] |
---|---|---|

Si-Si RDF 1st peak position (nm) | 0.3071 | 0.3077 |

Si-Si RDF 2nd peak position (nm) | 0.5193 | 0.5182 |

O-O RDF 1st peak position (nm) | 0.2538 | 0.2626 |

O-O RDF 2nd peak position (nm) | 0.4896 | 0.5097 |

Si-O RDF 1st peak position (nm) | 0.1633 | 0.1608 |

Si-O RDF 2nd peak position (nm) | 0.3969 | 0.4061 |

**Table 2.**Number of atoms N and simulation domain volume ${V}_{\mathrm{d}}$ corresponds to every simulation domain-size studied. The initial simulation box thickness is always a constant, ${L}_{z}=1.51$ nm.

${\mathit{L}}_{\mathit{x}}={\mathit{L}}_{\mathit{y}}$ (nm) | ${\mathit{V}}_{\mathit{d}}$ (nm${}^{3}$) | $\mathit{N}\times {10}^{4}$ |
---|---|---|

4.902 | 36.285 | 0.24 |

9.804 | 145.139 | 0.96 |

14.706 | 326.562 | 2.16 |

19.608 | 580.555 | 3.84 |

24.510 | 907.118 | 6.00 |

39.216 | 2322.221 | 15.36 |

**Table 3.**Example of Young’s modulus, E, and Poisson’s ratio, $\nu $, of a-SiO${}_{2}$ from experiment and simulational studies reported in literature. The bound refers to the maximum and minimum values of reported experimental and numerical data.

Reference Study | E (GPa) | $\mathit{\nu}$ | |
---|---|---|---|

Experiment | Freund and Suresh [39] | 80 | 0.22 |

Deschamps et al. [40] | 71.5 | 0.176 | |

Wiederhorn [41] | 72.1 | ... | |

Wallenberger et al. [42] | 69 | ... | |

Bansal and Doremus [43] | 72.9 | ... | |

Bound | 69–80 | 0.176–0.22 | |

ReaxFF simulation | Hao and Hossain [15] | 69.1 | 0.25–0.32 |

Chowdhury et al. [22] ${}^{\mathrm{a}}$ | 75.4–76.68 | ... | |

Chowdhury et al. [44] | 74 | 0.39 | |

Rimsza et al. [28] | ... | 0.31 | |

Yu et al. [45] | 80.4 ± 1.9 | ... | |

Mei et al. [35] | 60 | 0.296 | |

Bound | 60–82.3 | 0.25–0.39 |

Simulation Set | N $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{4}$ | UU | Simulation Set | N $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{4}$ | UU |
---|---|---|---|---|---|

T2 | 0.24 | 0.164 | T1 | 0.24 | 0.411 |

0.96 | 0.323 | 0.96 | 0.420 | ||

2.16 | 0.575 | 2.16 | 0.804 | ||

3.84 | 1.089 | 3.84 | 1.105 | ||

6.00 | 1.553 | 6.00 | 1.704 | ||

15.36 | 3.107 | 15.36 | 3.129 | ||

C2 | 0.24 | 0.140 | C1 | 0.24 | 0.140 |

0.96 | 0.241 | 0.96 | 0.241 | ||

2.16 | 0.368 | 2.16 | 0.365 | ||

3.84 | 0.539 | 3.84 | 0.532 | ||

6.00 | 0.745 | 6.00 | 0.740 | ||

15.36 | 1.561 | 15.36 | 1.565 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vo, T.; Reeder, B.; Damone, A.; Newell, P.
Effect of Domain Size, Boundary, and Loading Conditions on Mechanical Properties of Amorphous Silica: A Reactive Molecular Dynamics Study. *Nanomaterials* **2020**, *10*, 54.
https://doi.org/10.3390/nano10010054

**AMA Style**

Vo T, Reeder B, Damone A, Newell P.
Effect of Domain Size, Boundary, and Loading Conditions on Mechanical Properties of Amorphous Silica: A Reactive Molecular Dynamics Study. *Nanomaterials*. 2020; 10(1):54.
https://doi.org/10.3390/nano10010054

**Chicago/Turabian Style**

Vo, Truong, Brett Reeder, Angelo Damone, and Pania Newell.
2020. "Effect of Domain Size, Boundary, and Loading Conditions on Mechanical Properties of Amorphous Silica: A Reactive Molecular Dynamics Study" *Nanomaterials* 10, no. 1: 54.
https://doi.org/10.3390/nano10010054