# Effective Elastic Modulus of Wavy Single-Wall Carbon Nanotubes

## Abstract

**:**

## 1. Introduction

## 2. Approach

#### 2.1. Analytical Model

_{0}is given by

_{CNT}and I are the nominal elastic modulus and second moment of inertia, respectively.

_{CNT}is the nominal shear modulus, and γ is the Timoshenko shear coefficient (approximately 0.9).

_{f}is the Poisson’s ratio, which can be taken to be around 0.3.

#### 2.2. Finite Element Analysis

_{eff}is the effective modulus of the SWCNT-reinforced composite, E

_{m}is the modulus of the matrix, and V

_{CNT}is the SWCNT volume fraction.

#### 2.3. Monte Carlo Simulation

#### 2.4. Waviness

_{CNTeffi}is the effective modulus of the i-th SWCNT section, which can be determined from L

_{i}and h

_{i}/L

_{i}.

## 3. Results

#### 3.1. Effect of Curvature

#### 3.2. Results from FEA

#### 3.3. Effect of Statistical Distributions

_{CNTeff}/E

_{CNT}= 0.3478. As the range of h/L increases, the maximum probability tends to occur when the effective modulus approaches zero. Figure 9 also shows the mean and median of the effective moduli vs. the range of h/L. When the range of h/L increases, both the mean and median of the effective moduli decrease.

#### 3.4. Effect of Waviness

## 4. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Set | Simulation | h/L | SWCNT Length (nm) | β | η | E_{CNTeff}/E_{CNT} | |
---|---|---|---|---|---|---|---|

Mean | Median | ||||||

1 | 1 | U(0, 0.0025) | N(500, 100) | 1.77307 | 0.55544 | 0.49432 | 0.45171 |

2 | U(0, 0.005) | N(500, 100) | 0.92055 | 0.27978 | 0.29086 | 0.18789 | |

3 | U(0, 0.01) | N(500, 100) | 0.56502 | 0.07486 | 0.12238 | 0.03913 | |

4 | U(0, 0.02) | N(500, 100) | 0.31703 | 0.01327 | 0.09695 | 0.00418 | |

5 | U(0, 0.01) | N(1000, 100) | 0.35652 | 0.01508 | 0.07119 | 0.00539 | |

6 | U(0, 0.01) | N(300, 100) | 0.75143 | 0.24804 | 0.29487 | 0.15230 | |

7 | U(0, 0.01) | N(1000, 50) | 0.34914 | 0.01340 | 0.06799 | 0.00469 | |

8 | U(0, 0.01) | N(1000, 200) | 0.35343 | 0.01524 | 0.07411 | 0.00540 | |

2 | 1 | N(0.005, 0.001) | N(500, 100) | 1.40284 | 0.05015 | 0.04570 | 0.03862 |

2 | N(0.01, 0.001) | N(500, 100) | 0.63766 | 0.00343 | 0.00478 | 0.00193 | |

3 | N(0.02, 0.001) | N(500, 100) | 1.00000 | 0.00180 | 0.00180 | 0.00125 | |

4 | N(0.01, 0.002) | N(500, 100) | 0.63626 | 0.00410 | 0.00574 | 0.00230 | |

5 | N(0.01, 0.003) | N(500, 100) | 0.45245 | 0.00463 | 0.01134 | 0.00206 | |

6 | N(0.01, 0.001) | N(300, 100) | 0.93528 | 0.03499 | 0.03609 | 0.02365 | |

7 | N(0.01, 0.001) | N(1000, 100) | 1.00000 | 0.00033 | 0.00033 | 0.00023 | |

8 | N(0.01, 0.001) | N(1000, 50) | 1.00000 | 0.00033 | 0.00033 | 0.00023 | |

9 | N(0.01, 0.001) | N(1000, 200) | 1.00000 | 0.00033 | 0.00033 | 0.00023 | |

3 | 1 | U(0, 0.0025) | U(300, 700) | 1.7050 | 0.5424 | 0.48384 | 0.43747 |

2 | U(0, 0.005) | U(300, 700) | 0.8992 | 0.2718 | 0.28611 | 0.18081 | |

3 | U(0, 0.01) | U(300, 700) | 0.5054 | 0.0827 | 0.16220 | 0.04006 | |

4 | U(0, 0.02) | U(300, 700) | 0.3146 | 0.0124 | 0.09352 | 0.00387 | |

5 | U(0, 0.01) | U(100, 500) | 0.7780 | 0.2521 | 0.29160 | 0.15742 | |

6 | U(0, 0.01) | U(800, 1200) | 0.3289 | 0.0131 | 0.08298 | 0.00431 |

## References

- Iijima, S. Helical microtubules of graphitic carbon. Nature
**1991**, 354, 56–58. [Google Scholar] [CrossRef] - Iijima, S. Single-shell carbon nanotubes of 1-nm diameter. Nature
**1993**, 363, 603–605. [Google Scholar] [CrossRef] - Bethune, D.S.; Kiang, C.H.; Devries, M.S.; Gorman, G.; Savoy, R.; Vazquez, J.; Beyers, R. Cobalt-catalyzed growth of carbon nanotubes with single-atomic-layerwalls. Nature
**1993**, 363, 605–607. [Google Scholar] [CrossRef] - Thostenson, E.T.; Chou, T.-W. On the elastic properties of carbon nanotube-based composites: Modelling and characterization. J. Phys. D Appl. Phys.
**2003**, 36, 573. [Google Scholar] [CrossRef] - Deng, L.; Eichhorn, S.J.; Kao, C.-C.; Young, R.J. The Effective Young’s Modulus of Carbon Nanotubes in Composites. ACS Appl. Mater. Interfaces
**2011**, 3, 433–440. [Google Scholar] [CrossRef] - Shi, D.-L.; Feng, X.-Q.; Huang, Y.Y.; Hwang, K.-C.; Gao, H. The Effect of Nanotube Waviness and Agglomeration on the Elastic Property of Carbon Nanotube-Reinforced Composites. J. Eng. Mater. Technol.
**2004**, 126, 250–257. [Google Scholar] [CrossRef] - Li, C.; Thostenson, E.T.; Chou, T.-W. Effect of nanotube waviness on the electrical conductivity of carbon nanotube-based composites. Compos. Sci. Technol.
**2008**, 68, 1445–1452. [Google Scholar] [CrossRef] - Bartels, J.; Jürgens, J.-P.; Kuhn, E.; Ploshikhin, V. Effects of curvature and alignment of carbon nanotubes on the electrical conductivity of carbon nanotube-reinforced polymers investigated by mesoscopic simulations. J. Compos. Mater.
**2018**, 53, 1033–1047. [Google Scholar] [CrossRef] - Fisher, F.T.; Bradshaw, R.D.; Brinson, L.C. Fiber waviness in nanotube-reinforced polymer composites—I: Modulus predictions using effective nanotube properties. Compos. Sci. Technol.
**2003**, 63, 1689–1703. [Google Scholar] [CrossRef] - Bradshaw, R.D.; Fisher, F.T.; Brinson, L.C. Fiber waviness in nanotube-reinforced polymer composites—II: Modeling via numerical approximation of the dilute strain concentration tensor. Compos. Sci. Technol.
**2003**, 63, 1705–1722. [Google Scholar] [CrossRef] - Tandon, G.P.; Weng, G.J. Average stress in the matrix and effective moduli of randomly oriented composites. Compos. Sci. Technol.
**1986**, 27, 111–132. [Google Scholar] [CrossRef] - Maghsoudlou, M.A.; Barbaz Isfahani, R.; Saber-Samandari, S.; Sadighi, M. Effect of interphase, curvature and agglomeration of SWCNTs on mechanical properties of polymer-based nanocomposites: Experimental and numerical investigations. Compos. Part B Eng.
**2019**, 175, 107119. [Google Scholar] [CrossRef] - Ginga, N.J.; Chen, W.; Sitaraman, S.K. Waviness reduces effective modulus of carbon nanotube forests by several orders of magnitude. Carbon
**2014**, 66, 57–66. [Google Scholar] [CrossRef] - Shao, L.H.; Luo, R.Y.; Bai, S.L.; Wang, J. Prediction of effective moduli of carbon nanotube–reinforced composites with waviness and debonding. Compos. Struct.
**2009**, 87, 274–281. [Google Scholar] [CrossRef] - Zhu, L.; Narh, K.A. Numerical simulation of the effect of nanotube orientation on tensile modulus of carbon-nanotube-reinforced polymer composites. Polym. Int.
**2004**, 53, 1461–1466. [Google Scholar] [CrossRef] - Shima, H. Buckling of Carbon Nanotubes: A State of the Art Review. Materials
**2011**, 5, 47–84. [Google Scholar] [CrossRef] - Li, C.; Zhu, C.; Lim, C.W.; Li, S. Nonlinear in-plane thermal buckling of rotationally restrained functionally graded carbon nanotube reinforced composite shallow arches under uniform radial loading. Appl. Math. Mech.
**2022**, 43, 1821–1840. [Google Scholar] [CrossRef] - Chen, G.; Seki, Y.; Kimura, H.; Sakurai, S.; Yumura, M.; Hata, K.; Futaba, D.N. Diameter control of single-walled carbon nanotube forests from 1.3–3.0 nm by arc plasma deposition. Sci. Rep.
**2014**, 4, 3804. [Google Scholar] [CrossRef] - Fagan, J.A.; Hároz, E.H.; Ihly, R.; Gui, H.; Blackburn, J.L.; Simpson, J.R.; Lam, S.; Hight Walker, A.R.; Doorn, S.K.; Zheng, M. Isolation of >1 nm Diameter Single-Wall Carbon Nanotube Species Using Aqueous Two-Phase Extraction. ACS Nano
**2015**, 9, 5377–5390. [Google Scholar] [CrossRef] - Charlier, J.C.; Lambin, P. Electronic structure of carbon nanotubes with chiral symmetry. Phys. Rev. B
**1998**, 57, R15037–R15039. [Google Scholar] [CrossRef] - Park, K.H.; Chhowalla, M.; Iqbal, Z.; Sesti, F. Single-walled carbon nanotubes are a new class of ion channel blockers. J. Biol. Chem.
**2003**, 278, 50212–50216. [Google Scholar] [CrossRef] [PubMed] - Shi, Z.; Lian, Y.; Zhou, X.; Gu, Z.; Zhang, Y.; Iijima, S.; Zhou, L.; Yue, K.T.; Zhang, S. Mass-production of single-wall carbon nanotubes by arc discharge method11This work was supported by the National Natural Science Foundation of China, No. 29671030. Carbon
**1999**, 37, 1449–1453. [Google Scholar] [CrossRef] - Jinno, M.; Ando, Y.; Bandow, S.; Fan, J.; Yudasaka, M.; Iijima, S. Raman scattering study for heat-treated carbon nanotubes: The origin of ≈1855cm−1 Raman band. Chem. Phys. Lett.
**2006**, 418, 109–114. [Google Scholar] [CrossRef] - Jorio, A.; Saito, R. Raman spectroscopy for carbon nanotube applications. J. Appl. Phys.
**2021**, 129, 021102. [Google Scholar] [CrossRef] - Chen, J.; Xu, X.; Zhang, L.; Huang, S. Controlling the Diameter of Single-Walled Carbon Nanotubes by Improving the Dispersion of the Uniform Catalyst Nanoparticles on Substrate. Nano-Micro Lett.
**2015**, 7, 353–359. [Google Scholar] [CrossRef] - Navas, H.; Picher, M.; Andrieux-Ledier, A.; Fossard, F.; Michel, T.; Kozawa, A.; Maruyama, T.; Anglaret, E.; Loiseau, A.; Jourdain, V. Unveiling the Evolutions of Nanotube Diameter Distribution during the Growth of Single-Walled Carbon Nanotubes. ACS Nano
**2017**, 11, 3081–3088. [Google Scholar] [CrossRef]

**Figure 8.**Histogram and cumulative percentage of effective moduli of SWCNTs when h/L follows U(0, 0.01) and L follows N(500, 100).

**Figure 11.**Effect of SWCNT length standard deviation on Weibull distribution for effective SWCNT moduli.

Set | Simulation | h/L | SWCNT Length (nm) |
---|---|---|---|

1 | 1 | U(0, 0.0025) | N(500, 100) |

2 | U(0, 0.005) | N(500, 100) | |

3 | U(0, 0.01) | N(500, 100) | |

4 | U(0, 0.02) | N(500, 100) | |

5 | U(0, 0.01) | N(1000, 100) | |

6 | U(0, 0.01) | N(300, 100) | |

7 | U(0, 0.01) | N(1000, 50) | |

8 | U(0, 0.01) | N(1000, 200) | |

2 | 1 | N(0.005, 0.001) | N(500, 100) |

2 | N(0.01, 0.001) | N(500, 100) | |

3 | N(0.02, 0.001) | N(500, 100) | |

4 | N(0.01, 0.002) | N(500, 100) | |

5 | N(0.01, 0.003) | N(500, 100) | |

6 | N(0.01, 0.001) | N(300, 100) | |

7 | N(0.01, 0.001) | N(1000, 100) | |

8 | N(0.01, 0.001) | N(1000, 50) | |

9 | N(0.01, 0.001) | N(1000, 200) | |

3 | 1 | U(0, 0.0025) | U(300, 700) |

2 | U(0, 0.005) | U(300, 700) | |

3 | U(0, 0.01) | U(300, 700) | |

4 | U(0, 0.02) | U(300, 700) | |

5 | U(0, 0.01) | U(100, 500) | |

6 | U(0, 0.01) | U(800, 1200) |

Effective Modulus, E_{CNTeff}/E_{CNT} | |||
---|---|---|---|

L (nm) | Analytical Model | FEA | Relative Difference |

20 | 0.2390 | 0.2205 | 8.37% |

50 | 0.2414 | 0.2495 | −3.24% |

100 | 0.2417 | 0.2534 | −4.63% |

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Dong, C. Effective Elastic Modulus of Wavy Single-Wall Carbon Nanotubes. *C* **2023**, *9*, 54.
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Dong C. Effective Elastic Modulus of Wavy Single-Wall Carbon Nanotubes. *C*. 2023; 9(2):54.
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Dong, Chensong. 2023. "Effective Elastic Modulus of Wavy Single-Wall Carbon Nanotubes" *C* 9, no. 2: 54.
https://doi.org/10.3390/c9020054