Molecular Dynamics Study on Vibrational Characterization of Single-Walled Carbon Nanotubes

Abstract

In recent years, significant attention has been given to the physical properties of low-dimensional materials. Carbon nanotubes (CNTs), a prime example of such materials, are emerging as a promising next-generation candidate material for sensor components, including yoctogram (10⁻²⁴ g) measurement devices and antennas capable of handling large-scale digital data. CNTs exhibit a variety of atomic arrangements due to their chirality. However, even 30 years after their discovery, controlling the chirality of CNTs remains challenging, and the specifics of their physical properties still require clarification. Understanding the vibration characteristics of carbon nanotubes (CNTs) is crucial for designing nanoscale structures and devices. In this study, we analyzed the effects of tube diameter, push ratio, and length of CNTs on vibration using molecular dynamics simulation. This method allows for the modeling of ideal geometric structures at the atomic level and the tracking of their behavior. The findings are as follows: For armchair carbon nanotubes, the resonance frequency decreased with an increase in the length of the CNTs. It was observed that the thermal energy generated during vibration tends to decrease with an increase in tube diameter. The full width at half maximum increases with an increase in the push ratio.

Fig. 3 Relationship between CNT length and resonance frequency in Y-axis direction.

1. Introduction

In recent years, much attention has been paid to the physical properties of low-dimensional materials. Carbon nanotubes (CNTs) [1–4], a representative example of such materials, are attracting attention as a next-generation candidate material for sensor components such as yoctogram (10⁻²⁴ g) [5–7] and antennas capable of receiving large-scale digital data [8–10]. However, although CNTs are capable of various atomic arrangements due to their chirality, it is still difficult to control the chirality even today, some 30 years after their discovery, and there is a need to clarify the details of their physical properties.

Understanding the vibration characteristics of carbon nanotubes (CNTs) is crucial for designing nanoscale structures and devices. Specifically, it is essential to investigate the effects of tube diameter, push ratio, and length on the vibration characteristics of double-clamped CNT beams in detail. Firstly, the tube diameter directly affects the mechanical properties, resonance frequencies, and vibration modes of CNTs. Selecting the appropriate diameter contributes to improving device performance. Secondly, the initial push ratio is considered to induces nonlinear effects and significantly influences vibration characteristics, providing critical information for structural stability and initial condition settings. Finally, the length of the beam directly impacts resonance frequencies and vibration modes, playing a key role in design and performance evaluation at the nanoscale through scale effects. Thus, detailed investigation of these parameter dependencies is crucial for optimizing the vibration characteristics of double-clamped CNT beams. This will yield valuable insights for the design and control of CNT-based nanodevices and structures. However, because of the small size of single-walled carbon nanotubes and the uncertain definition of their thickness, classical continuum mechanics cannot be easily applied to predict the resonant frequency of single-walled carbon nanotubes [11].

In this study, we analyzed the effects of tube diameter, push ratio, and length of double-clamped CNT beams on vibration using molecular dynamics (MD), which can model ideal geometric structures at the atomic level and track their behavior [12–17].

2. Simulation Model

2.1 Simulation method

In this study, we used LAMMPS [18] to perform MD simulations dealing with all atoms. The basic flow of the MD method is that atoms are assumed to be mass points with masses, and potentials based on interatomic interactions, initial coordinates, and initial velocities of all atoms are given. The potential is then differentiated by the positional coordinates of each atom to calculate the force received by each atom, and a numerical analysis based on Newton’s equation of motion is applied to each particle to confirm the time variation of its coordinates and velocity. This is the most important feature of the MD method and the reason why MD simulations were introduced in this study. A numerical analysis method called Verlet method! was used to accommodate small atoms with fast thermal oscillations during the simulation.

OVITO [19] was also used to visualize atoms and molecules, which is an open source application that uses 3D graphics and embedded scripts to display atomic and molecular structures. Winmostar [20] was used to create simulation models.

2.2 Calculation conditions

In this study, armchair CNTs of various tube diameter were utilized. Figure 1 presents the simulation model for an armchair-type CNT with a chirality of (5, 5). The CNT was divided into three regions: a vibrating section, a heat bath section, and a fixed section. The vibrating section was left unrestrained, the heat bath section had its temperature maintained at 300 K using the Langevin thermostat, and the fixed section was immobilized through velocity control.

Fig. 1

Schematic diagram of simulation model. (a) Armchair type (5, 5) before tilt (b) Armchair type (5, 5) after tilt.

To induce vibration, the indenter depicted in Fig. 1 was moved along the Y-axis under a velocity constraint, causing the CNTs to tilt by adjusting the push ratio within the elastic region. Following the removal of the indenter, the vibration was analyzed over a duration of 1 ns. The push ratio was determined by the formula: indentation depth divided by the length of the vibrating section L [Å]. Push ratios of 6, 11, and 16% were applied. CNTs with armchair chirality (5, 5), (8, 8), and (11, 11) and lengths of 100, 200, and 300 Å, respectively, were subjected to this analysis.

2.3 Potential functions

In MD simulations, it is crucial to select appropriate potentials to accurately model the interactions between atoms. For this simulation, the following potentials were selected for each atom. The adaptive intermolecular reactive empirical bond order (AIREBO) potential [21] was applied for interactions between carbon atoms within the carbon nanotube, while the Lennard-Jones (LJ) potential [22] was utilized for interactions between the indenter and the carbon nanotube.

The AIREBO potential is well-suited for accurately simulating the mechanical properties of carbon materials and is represented by a function as shown in eq. (1). The first term on the right side denotes the reactive empirical bond order (REBO) potential [23], which accounts for covalent bonding. The second term corresponds to the LJ potential, addressing non-covalent interactions. Finally, the third term signifies the potential related to the dihedral angle.

  

\begin{equation} E_{\textit{tot}} = \sum_{i} \sum_{j > i} \left[E_{ij}^{\textit{REBO}} + E_{ij}^{\textit{LJ}} + \sum_{k \neq i,j} \sum_{l \neq i,j,k} E_{kijl}^{\textit{tors}} \right] \end{equation} (1)

The REBO potential is a function of eq. (2). The first term on the right side represents isotropic repulsive interactions, while the second term represents attractive interactions with consideration for the bond angle.

  

\begin{equation} E_{ij}^{\textit{REBO}} = V_{ij}^{R} + b_{ij}V_{ij}^{A} \end{equation} (2)

The LJ potential, which indicates non-covalent interactions, is represented by a function as shown in eq. (3). Based on eq. (3), the most stable distance between atoms is $2^{\frac{1}{6}}r_{0}$. At distances closer than this, repulsive forces act, while at greater distances, attractive forces act.

  

\begin{equation} E = 4\varepsilon \left[\left(\frac{\sigma}{r}\right)^{12}{} - \left(\frac{\sigma}{r} \right)^{6} \right] \end{equation} (3)

3. Results

3.1 Relation between push ratio and force (Y-axis direction)

The relationship between the push ratio and the force in the y-axis direction, which is the direction of vibration, was explored to enable vibration within the elastic region. Figure 2 indicates that across all CNT lengths, increasing chirality, i.e., increasing tube diameter, correlates with a greater force relative to the push ratio. This is considered to be because the number of bonds perpendicular to the indentation direction increases with increasing tube diameter, leading to increased stiffness in relation to the push ratio. Furthermore, it was verified that the region was elastic when the push ratio was between 6 and 16%.

Fig. 2

Relationship between push ratio and force in Y-axis direction. (a) CNT length L = 100 Å (b) CNT length L = 200 Å (c) CNT length L = 300 Å.

3.2 Relationship between CNT length and resonance frequency (Y-axis direction)

Figure 3 illustrates how the resonance frequency is influenced by CNT length in the Y-axis direction, given a chirality of (5, 5), a push ratio of 16% and a vibration time from 0 to 0.2 ns. It reveals that the resonance frequency generally decreases as the CNT length extends. This trend is believed to stem from the increased number of couplings perpendicular to the vibration direction with longer CNTs, leading to reduced stiffness in that direction.

Fig. 3

Relationship between CNT length and resonance frequency in Y-axis direction.

3.3 Relationship between chirality and resonance frequency (Y-axis direction)

Figure 4 illustrates the relationship between chirality and resonance frequency in the Y-axis direction, given a push ratio of 16% and a vibration time from 0 to 0.2 ns. It indicates that at a CNT length of 100 Å, the resonance frequency progressively increases as the chirality level increases, i.e., the tube diameter increases. This phenomenon is attributed to the augmented number of couplings perpendicular to the vibration direction and the enhanced stiffness in the vibration direction as the tube diameter increases. Additionally, it has been observed that the resonance frequency remains consistent regardless of the tube diameter as the CNT length grows.

Fig. 4

Relationship between chirality and resonance frequency in Y-axis direction.

3.4 Relationship between push ratio and resonance frequency (Y-axis direction)

Figure 5 presents how resonance frequency varies with push ratio in the Y-axis direction for CNT lengths of 100 Å and vibration times ranging from 0 to 0.2 ns. It reveals that for chirality (8, 8) and (11, 11), the resonance frequency remained consistent when the push ratio was between 6 to 16% in this study. This consistency is attributed to the almost linear and stable slope of the stiffness ratio within the 6 to 16% push ratio range, as depicted in Fig. 2(a). Conversely, for chirality (5, 5), there was a noticeable increase in resonance frequency as the push ratio increased. This increase is linked to the variable slope of the stiffness ratio, which intensifies with higher push ratios, resulting in augmented stiffness.

Fig. 5

Relationship between push ratio and resonance frequency in Y-axis direction.

3.5 Time variation of amplitude spectrum

Figure 6 illustrates the temporal changes in amplitude spectra along the X and Y axes for CNTs of 100 Å in length, with a chirality of (11, 11) and a push ratio of 16%. FFT analyses were conducted for vibration time intervals of 0 to 0.2 ns, 0.4 to 0.6 ns, and 0.8 to 1 ns in this study. The figure reveals a gradual reduction in the amplitude spectra over time in both axes, suggesting a diminution of vibration energy. This reduction is believed to stem from the transformation of vibration energy into thermal energy, a process facilitated by the friction between atoms during vibration.

Fig. 6

Temporal changes in amplitude spectra in X and Y axes directions. (a) X-axis direction (b) Y-axis direction.

3.6 Temperature time variation of CNTs during vibration

Figure 7 illustrates the temperature changes over time in CNTs with a length of 100 Å, chirality of (11, 11), push ratio of 16%, and vibration time from 0 to 0.2 ns. To monitor the temperature across different vibration locations, the simulation model’s vibrating region, as depicted in Fig. 1, was evenly divided along the Z-axis into upper, middle, and lower sections. In order to remove the center-of-gravity velocity when calculating temperature, the kinetic energy due to tube vibration is not converted as temperature. The figure demonstrates that the temperature within the vibrating area declines over time, eventually stabilizing near 300 K, the preset constraint temperature. This phenomenon is likely due to the conversion of a portion of the vibration energy into thermal energy, which is then absorbed by the heat bath area. As the vibration energy and amplitude diminish over time, the generated thermal energy consequently reduces.

Fig. 7

Temperature changes of CNTs during vibration in chirality (11, 11). (a) Vibration time 0.1 to 0.12 ns (b) Vibration time 0.5 to 0.52 ns (c) Vibration time 0.9 to 0.92 ns.

3.7 Relationship between push ratio and CNT temperature

Figures 8 and 9 illustrate how temperature correlates with push ratio in CNTs of 100 Å length, for chiralities of (5, 5) and (11, 11), during a vibration time of 0.1 to 0.12 ns. The figures reveal that temperature escalates with an increasing push ratio, a trend observed across all vibrational sections. This rise in temperature is attributed to the augmented friction energy between atoms, which accompanies the heightened amplitude of vibration as the push ratio grows. Notably, a lower tube diameter results in a more pronounced temperature increase in response to push ratio adjustments. This effect is believed to arise because lower tube diameter enhances the change in vibration amplitude relative to adjustments in the push ratio.

Fig. 8

Relationship between push ratio and CNT temperature in chirality (5, 5). (a) 6% of push ratio (b) 16% of push ratio.

Fig. 9

Relationship between push ratio and CNT temperature in chirality (11, 11). (a) 6% of push ratio (b) 16% of push ratio.

3.8 Relationship between push ratio and full width at half maximum (FWHM)

Figure 10 illustrates the influence of the push ratio on FWHM in the Y-axis direction for CNT lengths of 100 Å and chiralities of (5, 5) and (11, 11), over a vibration time from 0 to 0.2 ns. The figure reveals that FWHM generally escalates with an increase in the push ratio. Specifically, at a push ratio of 6%, the temperature in the vibrating area is higher for chirality (5, 5) compared to (11, 11), yet the FWHM is lower. This suggests that FWHM’s variation is influenced not only by temperature but also by additional factors. Notably, the rise in FWHM associated with an increase in push ratio is more significant for chirality (5, 5).

Fig. 10

Relationship between push ratio and full width at half maximum in Y-axis direction.

3.9 Relationship between push ratio and structural deformation of CNTs

Figure 11 illustrates the impact of push ratio on the structural behavior of CNTs with lengths of 100 Å for chiralities of (5, 5) and (11, 11). It demonstrates that at a 6% push ratio, CNTs with chirality (5, 5) exhibit bending, and vibration along the indentation direction. On the other hand, CNTs with chirality (11, 11) at the same push ratio show vibrations both radially and along the indentation direction, attributable to partial structural distortion. As the push ratio increases, there is a noticeable increase in vibration along the indentation direction for chirality (5, 5). While for chirality (11, 11), radial vibrations diminish while vibrations in the indentation direction intensify, indicating a transition towards more pronounced structural distortion.

Fig. 11

Relationship between push ratio and structural deformation of CNTs. (a) 6% of push ratio (b) 16% of push ratio.

3.10 Relationship between push ratio and amplitude spectrum

Figure 12 shows the dependence of the amplitude spectrum on the push ratio in the Y-axis direction for a CNT length of 100 Å and chirality of (11, 11) at vibration times from 0 to 0.2 ns. In Fig. 12(a), the maximum peak near 280 GHz is the spectrum corresponding to the indentation direction vibration, and the peak near 360 GHz is the spectrum corresponding to the radial direction vibration. As shown in Figs. 12(b) and 12(c), as the indentation ratio increases, the radial direction vibration spectrum tends to decrease and becomes closer to the indentation direction vibration spectrum. It is considered that an increase in indentation ratio causes not only a part of the structure but the entire structure to be distorted, resulting in an increase in the indentation direction vibration spectrum and a decrease in the radial direction vibration spectrum. The reason why the FWHM of (11, 11) was larger than that of (5, 5) when the indentation ratio was small in Fig. 10 is thought to be because some nonlinear effect of the radial vibration mixing with the indentation direction vibration was more pronounced.

Fig. 12

Relationship between push ratio and amplitude spectrum in chirality (11, 11). (a) 6% of push ratio (b) 11% of push ratio (c) 16% of push ratio.

4. Conclusion

Our investigation focused on the variation in vibration characteristics by adjusting the push ratio and CNT length in armchair-type structures with different tube diameters. The findings include:

1. (1)    Regardless of CNT length, the force relative to the push ratio consistently increased with greater tube diameter. This trend is attributed to the rise in the number of bonds perpendicular to the indentation direction, enhancing stiffness in relation to the push ratio.
2. (2)    The Full Width at Half Maximum (FWHM) in the indentation direction generally expanded with an increase in the push ratio. Notably, at a push ratio of 6%, the vibrating section’s temperature was higher for chirality (5, 5) compared to (11, 11), yet the FWHM was smaller. This suggests that FWHM variation depends not solely on temperature but also on other factors. For chirality (11, 11), within the push ratio range of 6 to 16%, a radial vibration spectrum was observed at 6%, which then diminished and eventually vanished as the push ratio rose. This phenomenon likely occurs because a higher push ratio induces comprehensive structural distortion, reducing radial vibrations instead of affecting only part of the structure.

REFERENCES

• 1)   S.  Iijima: Helical microtubules of graphitic carbon, Nature 354 (1991) 56–58. doi:10.1038/354056a0
• 2)   D.S.  Bethune,  C.H.  Kiang,  M.S.  de Vries,  G.  Gorman,  R.  Savoy,  J.  Vazquez and  R.  Beyers: Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer walls, Nature 363 (1993) 605–607. doi:10.1038/363605a0
• 3)   R.H.  Baughman,  A.A.  Zakhidov and  W.A.  de Heer: Carbon Nanotubes--the Route Toward Applications, Science 297 (2002) 787–792. doi:10.1126/science.1060928
• 4)   E.T.  Thostenson,  Z.  Ren and  T.W.  Chou: Advances in the science and technology of carbon nanotubes and their composites: a review, Compos. Sci. Technol. 61 (2001) 1899–1912. doi:10.1016/S0266-3538(01)00094-X
• 5)   H.Y.  Chiu,  P.  Hung,  H.W.C.  Postma and  M.  Bockrath: Atomic-Scale Mass Sensing Using Carbon Nanotube Resonators, Nano Lett. 8 (2008) 4342–4346. doi:10.1021/nl802181c
• 6)   B.  Lassagne,  Y.  Tarakanov,  J.  Kinaret,  D.  Garcia-Sanchez and  A.  Bachtold: Coupling Mechanics to Charge Transport in Carbon Nanotube Mechanical Resonators, Science 325 (2009) 1107–1110. doi:10.1126/science.1174290
• 7)   J.  Chaste,  A.  Eichler,  J.  Moser,  G.  Ceballos,  R.  Rurali and  A.  Bachtold: A nanomechanical mass sensor with yoctogram resolution, Nat. Nanotechnol. 7 (2012) 301–304. doi:10.1038/nnano.2012.42
• 8)   P.J.  Burke,  S.  Li and  Z.  Yu: Quantitative theory of nanowire and nanotube antenna performance, IEEE Trans. Nanotechnol. 5 (2006) 314–334. doi:10.1109/TNANO.2006.877430
• 9)   C.  Rutherglen and  P.  Burke: Carbon Nanotube Radio, Nano Lett. 7 (2007) 3296–3299. doi:10.1021/nl0714839
• 10)   K.  Jensen,  J.  Weldon,  H.  Garcia and  A.  Zettl: Nanotube Radio, Nano Lett. 7 (2007) 3508–3511. doi:10.1021/nl0721113
• 11)   C.  Li and  T.W.  Chou: Single-walled carbon nanotubes as ultrahigh frequency nanomechanical resonators, Phys. Rev. B 68 (2003) 073405. doi:10.1103/PhysRevB.68.073405
• 12)   J.  Tersoff: New empirical approach for the structure and energy of covalent systems, Phys. Rev. B 37 (1988) 6991. doi:10.1103/PhysRevB.37.6991
• 13)   D.W.  Brenner: Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films, Phys. Rev. B 42 (1990) 9458. doi:10.1103/PhysRevB.42.9458
• 14)   B.I.  Yakobson,  C.J.  Brabec and  J.  Bernholc: Nanomechanics of Carbon Tubes: Instabilities beyond Linear Response, Phys. Rev. Lett. 76 (1996) 2511. doi:10.1103/PhysRevLett.76.2511
• 15)   P.  Zhang,  Y.  Huang,  P.H.  Geubelle,  P.A.  Klein and  K.C.  Hwang: The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials, Int. J. Solids Struct. 39 (2002) 3893–3906. doi:10.1016/S0020-7683(02)00186-5
• 16)   M.  Yasuda,  Y.  Kimoto,  K.  Tada,  H.  Mori,  S.  Akita,  Y.  Nakayama and  Y.  Hirai: Molecular dynamics study of electron-irradiation effects in single-walled carbon nanotubes, Phys. Rev. B 75 (2007) 205406. doi:10.1103/PhysRevB.75.205406
• 17)   K.  Tada,  M.  Yasuda,  T.  Mitsueda,  R.  Honda,  H.  Kawata and  Y.  Hirai: Molecular dynamics study of electron irradiation effects on mechanical properties of carbon nanotubes, Microelectron. Eng. 107 (2013) 50–53. doi:10.1016/j.mee.2013.03.009
• 18)   A.P.  Thompson,  H.M.  Aktulga,  R.  Berger,  D.S.  Bolintineanu,  W.M.  Brown,  P.S.  Crozier,  P.J.  in ’t Veld,  A.  Kohlmeyer,  S.G.  Moore,  T.D.  Nguyen,  R.  Shan,  M.J.  Stevens,  J.  Tranchida,  C.  Trott and  S.J.  Plimpton: LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales, Comp. Phys. Comm. 271 (2022) 108171. doi:10.1016/j.cpc.2021.108171
• 19)   A.  Stukowski: Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool, Model. Simul. Mater. Sci. Eng. 18 (2010) 015012. doi:10.1088/0965-0393/18/1/015012
• 20)  Winmostar V9, X-Ability Co. Ltd., Tokyo, Japan, (2019).
• 21)   S.J.  Stuart,  A.B.  Tutein and  J.A.  Harrison: A reactive potential for hydrocarbons with intermolecular interactions, J. Chem. Phys. 112 (2000) 6472–6486. doi:10.1063/1.481208
• 22)   J.E.  Jones: On the determination of molecular fields.—I. From the variation of the viscosity of a gas with temperature, Proc. R. Soc. London, Ser. A 106 (1924) 441–462. doi:10.1098/rspa.1924.0081
• 23)   D.W.  Brenner,  O.A.  Shenderova,  J.A.  Harrison,  S.J.  Stuart,  B.  Ni and  S.B.  Sinnott: A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons, J. Condens. Matter Phys. 14 (2002) 783–802. doi:10.1088/0953-8984/14/4/312