# Digital Synthesis of Realistically Clustered Carbon Nanotubes

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Concepts

#### 2.1. Carbon Nanotube Structure

^{2}bonds which give graphene such a stable two-dimensional structure. If the carbon atoms are spread in the x–y plane, the remaining carbon valence electron’s orbital, the 2p

_{z}orbital, comes out of the plane in the perpendicular direction to the graphene sheet. This layer of probable electron density provides the relatively weaker bonds between separate sheets of graphene that make up graphite. It is therefore intuitive to imagine that a carbon nanotube is a rolled-up graphene sheet that will tend to focus the interaction potential to act normally to the curved surface of the nanotube. Either van der Waals forces between atoms or transient delocalization of the lingering 2p

_{z}orbital from the carbon atoms contributes to how carbon nanotubes attract and repel matter around them, including other nanotubes (inter-CNT attraction) and even different parts of the same nanotube (intra-CNT attraction) [17].

#### 2.2. Multi-Scale Coupling

**Phase I:**Use a random walk and coarse-grained molecular model to create a template that serves as the scaffold upon which atomistic nanotubes will be built.

**Phase II:**Use differential geometry to refine the coarse-grained representations into fully atomistic carbon nanotubes, forming a cluster.

**Phase III:**Relax the atomistic nanotube cluster using classical molecular dynamics with an accurate potential model that allows for dynamic bonding and bond-breaking, and inter-and intra-nanotube attraction.

#### 2.3. Phase Ia: Random Walk to Generate an Initial CG-CNT

#### 2.4. Phase Ib: Coarse-Grained Carbon Nanotube Modeling

#### 2.5. Phase II: Differential Geometry

#### 2.6. Phase III: Atomistic Carbon Nanotube Modeling

_{ij}between atoms i and j, shown respectively in Equations (9) and (10) which includes both attractive Equation (11) and repulsive Equation (12) potential energy terms, as well as a smoothing function Equation (13) that implements a cut-off for interactions that become vanishingly small with increased separations.

## 3. Results

#### 3.1. Coarse-Grained Cluster Generation

#### 3.2. Differential Geometry with Molecular Dynamics Relaxation

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) A straight section of an atomically resolved (5,5) singled-walled carbon nanotube, and (

**b**) a simplified coarse-grained model based on [21] that serves as the coarsest representation of the presented hierarchical modeling approach.

**Figure 2.**(

**a**) Orthographic view of a small section of a (5,5) SWNT, (

**b**) Isolated view of two levels of concentric carbon rings that form a unit cell, colored by individual ring, and (

**c**) View down the central axis of this straight example SWNT, where the central axis is denoted by the cross symbol.

**Figure 3.**A three-dimensional space curve (red line) with consistently oriented finite frames along the curve. The red line is representative of the coarse-grained nanotube’s central axis and the consistency in the finite frames defined by planes spanned by vectors $\overline{N}$ and $\overline{B}$ normal to the curve’s tangent vector $\overline{T}$ is required to enforce the appropriate rotations and re-orientations of the atomic carbon rings for building the atomistic nanotubes.

**Figure 4.**(

**a**) Example of a cluster of random walk realizations for N = 10 nanotubes of varying lengths sampled from specifications provided by Carbon Solutions Inc. [35]. (

**b**) Highly maligned and energetically unfavorable configuration shown in the magnified section.

**Figure 5.**(

**a**) TEM image from Liu et al. [19] (

**b**) slice of the relaxed coarse-grained configuration from the present research, and (

**c**) the energy relaxation from the canonical MD simulation of the bead spring filament system showing convergence in kinetic and potential energies.

**Figure 6.**(

**a**) Images of 4 carbon nanotubes differentiated by color from a large cluster on the order of 100 nm and (

**b**) a close-up view of one representative loop to show atomistic resolution.

**Figure 7.**100 nm cluster of 10 relaxed atomically resolved carbon nanotubes (

**e**) with insets (

**a**–

**d**) showing greater detail from different locations in the cluster. Different colors denote individual carbon nanotubes.

**Figure 8.**Transient records of the average kinetic (red, right axis) and average potential energy (black, left axis) for the 5-stage relaxation of the 100 nm cluster of 10 atomically resolved carbon nanotubes where the ensemble is noted graphically in concurrence with Table 3.

**Figure 9.**Atomically resolved nanotubes exhibit features and interactions coarse-grained models under-represent. Examples shown here include (

**a**) nanotube fracture, (

**b**) bending-buckling, and (

**c**) fusing.

L-J Cutoff Distance | ${\mathit{k}}_{\mathit{s}}$ | ${\mathit{b}}_{0}$ | ${\mathit{k}}_{\mathit{\theta}}$ | ${\mathit{\theta}}_{0}$ |
---|---|---|---|---|

$9.35\text{}\AA $ | $43\text{}\frac{eV}{{\AA}^{2}}$ | $10\text{}\AA $ | $614.9\text{}\frac{eV}{ra{d}^{2}}$ | $\pi $ |

δ | $\mathit{\theta}$ |
---|---|

$1.228\text{}\AA $ | $36\xb0$ |

Stage 1 | Stage 2 | Stage 3 | Stage 4 | Stage 5 | |
---|---|---|---|---|---|

Damping Coefficient, $\gamma $ | 1.0 | 0.5 | 0.1 | 0.01 | 0.0 |

Simulated Duration | 0.001 ns | 0.001 ns | 0.001 ns | 1 ns | 2.5 ns |

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Susi, B.T.; Tu, J.F.
Digital Synthesis of Realistically Clustered Carbon Nanotubes. *C* **2022**, *8*, 34.
https://doi.org/10.3390/c8030034

**AMA Style**

Susi BT, Tu JF.
Digital Synthesis of Realistically Clustered Carbon Nanotubes. *C*. 2022; 8(3):34.
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**Chicago/Turabian Style**

Susi, Bryan T., and Jay F. Tu.
2022. "Digital Synthesis of Realistically Clustered Carbon Nanotubes" *C* 8, no. 3: 34.
https://doi.org/10.3390/c8030034