# Machine Learning-Assisted High-Throughput Molecular Dynamics Simulation of High-Mechanical Performance Carbon Nanotube Structure

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## Abstract

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## 1. Introduction

^{3}interwall bonding, Xia et al. [19] found that the interwall sp

^{3}coupling in MWCNTs enhanced the load transfer among the walls, facilitating their complete mechanical participation; this prevented the telescoping problem during tensile testing and improved the MWCNT strength. Alternately, Peng et al. [15] reported that the fracture strength value of MWCNTs treated with controlled electron irradiation could reach as high as 80% of the value expected in defect-free SWCNTs; this is considered to be the result of crosslinking between the walls. They further confirmed through molecular mechanics approaches that the interwall load transfer improves on increasing Frenkel-pair-type crosslinks, and only a small percentage of these crosslinks is necessary to achieve optimal load transfer. In contrast with their experiment [15], the MD results obtained by Byrne et al. [20] showed that defective MWCNTs with sp

^{3}interwall bonding exhibited strength values exceeding those of SWCNTs (of the same size) with crack-like defects, and concluded that composites with suitably designed MWCNTs would perform better than most SWCNT-based composites. Furthermore, Shirasu et al. [18] performed theoretical calculations and experiments to prove that the nominal tensile strength is the key factor in determining the mechanical properties when designing CNT-reinforced composites, rather than the effective tensile strength. The effective tensile strength is given by the total force divided by the area that bears the load, while the nominal tensile strength is given by the total force divided by the total area, including the hollow core. It is also highlighted in the study [18] that an effective method to improve the nominal tensile strength is to introduce crosslinks between the MWCNT walls.

## 2. Molecular Dynamics Models and Computational Methods

^{9}s

^{−1}and time step at 0.5 fs. Please note that the fixed part also experienced elongation along the tube axis as the load increased, while all the atoms in the mobile part including the middle portion of the outermost wall and all inner walls could move freely, as shown in Figure 1. To avoid non-physical increases in stress values during tensile loading, the AIREBO potential cutoff distance was modified to 2.0 Å [27,28,29]. The resultant tensile strength of the AIREBO potential–based tensile test verification simulation of the zigzag-type CNTs obtained in this study was 120 GPa. This was slightly higher than the experimental result, and consistent with those of the quantum calculation obtained by B. Peng et al. [15], which were approximately 100 GPa and 120 GPa. Moreover, the strain-stress relation for single-walled fracture of MWCNTs was in good agreement with their experiment sample 1, 2 and 3. Because the distribution pattern of the crosslinks may influence the result and the interwall crosslink density has at most 3% margin of error after equilibration, at least three calculations were performed for each MWCNT model, and the average values were considered to be the final results.

^{®}SOMine 5.2.2 Expert Edition [43], was employed to produce SOMs. In addition to the general SOM algorithm, a Kohonen’s batch map based on an advanced unsupervised neural network [44] was deployed inside the software. For parameters in SOM, the number of nodes and topology were set to 1000 and hexagonal, respectively.

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic diagram of the computational model. (

**a**) Entire 2WCNT with Frenkel-pair crosslinks and distortions on the wall with a half front slice on the left side, where the two red parts (on the outermost wall) indicate the regions of load application during the tensile test and are named fixed parts. The blue color (including all inner walls) represents the parts where atoms can move freely, and are thus named mobile parts; (

**b**) Magnified view of the middle mobile part of 2WCNT; (

**c**) Front slice of (

**b**) showing randomly distributed crosslinks; (

**d**,

**e**) Structure of Frenkel-pair crosslink. Light blue balls represent carbon atoms on the inner walls, and yellow balls are atoms on the outer walls.

**Figure 2.**Flowchart of the optimization procedure. (

**a**) Bayesian optimization; (

**b**) genetic algorithm.

**Figure 3.**Strength results of the machine learning–predicted structure with respect to the repetition of the prediction procedure. Error bar shows the maximum and minimum values obtained from the three repetitions of calculations.

**Figure 4.**Fracture patterns of the zigzag-type 5WCNTs assigned with different crosslink densities. (

**a**) “Sword-in-sheath” fracture on the 0.5% crosslink model (in every wall) with nominal strength of 47 GPa; (

**b**) “Sword-and-sheath” fracture on the 0.75% crosslink model with nominal strength of 50 GPa; (

**c**,

**d**) “Near-clean-break” and “clean-break” fracture on the 1.5% crosslink model with nominal strength of 53 GPa and 52 GPa, respectively.

**Figure 5.**SOM results including the cluster map and heat map. (

**a**) Cluster map with 7 clusters; (

**b**) Heat map of nominal tensile strength: values range from 20.1 GPa to 62.6 GPa; (

**c**) Heat map of nominal Young’s modulus: values range from 147 GPa to 694 GPa; (

**d**–

**f**) Heat map of chirality, diameter, and number of walls, respectively; (

**g**–

**j**) Heat maps of crosslink density 1–4, where crosslink density 1 is the density corresponding to the innermost wall, and crosslink density 4 is the density corresponding to the outermost wall.

**Figure 6.**Fracture behavior difference between zigzag-type and armchair-type 2WCNTs. (

**a1**) Approximately 30° fracture for armchair-type CNT; (

**b1**) Nearly linear fracture for zigzag-type tube; (

**a2**,

**b2**) Atom arrangement of armchair-type CNT and zigzag-type CNT, respectively.

**Table 1.**Detailed information for the representative values in Figure 4. “Crosslink density 1” through “Crosslink density 4” represent the crosslink densities between each adjacent wall from the inner tube to the outer tube. The average value is given, and the range is indicated in parentheses.

Model | Outer Diameter (Å) | Number of Walls | Chirality | Crosslink Density 1 (%) | Crosslink Density 2 (%) | Crosslink Density 3 (%) | Crosslink Density 4 (%) | Nominal Tensile Strength (GPa) |
---|---|---|---|---|---|---|---|---|

① | 101.70 | 5 | Armchair | 0.01 (0.01–0.01) | 2.32 (2.31–2.32) | 2.36 (2.35–2.37) | 2.26 (2.25–2.28) | 27.79 (27.53–28.04) |

② | 44.75 | 5 | Armchair | 0.91 (0.89–0.92) | 0.75 (0.74–0.77) | 0.97 (0.95–0.97) | 1.38 (1.35–1.41) | 60.53 (57.60–62.81) |

③ | 44.62 | 5 | Zigzag | 0.02 (0.02–0.02) | 0.83 (0.81–0.85) | 0.05 (0.04–0.05) | 1.70 (1.69–1.71) | 37.58 (36.28–40.02) |

④ | 43.39 | 5 | Armchair | 2.85 (2.81–2.90) | 1.19 (1.18–1.21) | 1.47 (1.45–1.48) | 1.83 (1.81–1.84) | 57.19 (55.80–59.88) |

⑤ | 43.39 | 5 | Armchair | 0.22 (0.20–0.23) | 0.44 (0.44–0.45) | 2.11 (2.10–2.12) | 1.20 (1.20–1.21) | 60.03 (56.62–62.12) |

⑥ | 43.39 | 5 | Armchair | 0.05 (0.04–0.05) | 2.03 (2.01–2.05) | 1.32 (1.30–1.35) | 1.32 (1.31–1.33) | 58.50 (54.77–61.42) |

⑦ | 43.39 | 5 | Armchair | 1.98 (1.97–2.00) | 0.84 (0.82–0.86) | 0.59 (0.57–0.60) | 1.01 (1.00–1.03) | 58.26 (56.25–59.52) |

⑧ | 43.39 | 5 | Armchair | 0.90 (0.89–0.92) | 1.39 (1.36–1.41) | 1.40 (1.40–1.41) | 1.25 (1.23–1.28) | 58.83 (58.02–60.26) |

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**MDPI and ACS Style**

Xiang, Y.; Shimoyama, K.; Shirasu, K.; Yamamoto, G.
Machine Learning-Assisted High-Throughput Molecular Dynamics Simulation of High-Mechanical Performance Carbon Nanotube Structure. *Nanomaterials* **2020**, *10*, 2459.
https://doi.org/10.3390/nano10122459

**AMA Style**

Xiang Y, Shimoyama K, Shirasu K, Yamamoto G.
Machine Learning-Assisted High-Throughput Molecular Dynamics Simulation of High-Mechanical Performance Carbon Nanotube Structure. *Nanomaterials*. 2020; 10(12):2459.
https://doi.org/10.3390/nano10122459

**Chicago/Turabian Style**

Xiang, Yi, Koji Shimoyama, Keiichi Shirasu, and Go Yamamoto.
2020. "Machine Learning-Assisted High-Throughput Molecular Dynamics Simulation of High-Mechanical Performance Carbon Nanotube Structure" *Nanomaterials* 10, no. 12: 2459.
https://doi.org/10.3390/nano10122459