A reduced form for fundamental transverse modal frequency of carbon nanotubes

Determination of fundamental coupled torsional—radial frequency of single-walled carbon nanotubes

“…Depending on the direction of rolling a graphene sheet, three major types, namely, “zigzag,” “armchair,” and “chiral” SWCNTs are obtained. Figure 1 shows the structures of these SWCNT types and the schematic of their formation from graphene sheets (Singh and Dutt, 2020). As observed from the figure, zigzag and armchair have symmetrical structures as the carbon hexagonal rings are arranged symmetrically about the tube axis, whereas chiral SWCNTs have asymmetrical structure as the hexagonal rings are arranged helically about the tube axis.…”

“…Therefore, SWCNT geometry is fully defined by the length of the parent graphene sheet and an important parameter called the chiral vector boldC→ The direction of trueC→ is the direction along which graphene sheet is rolled to form an SWCNT and its magnitude is the cross-sectional circumference of the SWCNT. The chiral vector is further defined in terms of the chirality indices (n, m) and the fixed principal lattice vectors (a 1 , a 2 ) of the parent graphene sheet as given in equation (1) and Figure 1

Figure 1.Structures of single-walled carbon nanotubes with different chirality (Singh and Dutt, 2020).

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“…Here, n = m for armchair SWCNTs, m = 0 for zigzag SWCNTs, and n≠m≠0 for chiral SWCNTs. The SWCNT diameter D in terms of chirality indices and C–C bond length (lc−c) is given by equation (2) (Singh and Dutt, 2020). The chiral angle θ is the angle between the vectors C→ and a1true^, which varies in the range of 0°–30° and is given by equation (3) (Zaporotskova et al, 2016).…”

“…In MSMA, the SWCNT is represented as a three-dimensional space frame structure where every inter-carbon covalent bond is represented as a beam with uniform circular cross section, and every carbon atom is represented as the connecting pin between the beams. Based on MSMA calculations, the elastic properties of the beam representing the inter-carbon covalent bond in an SWCNT are given in Table 2 (see details in Li and Chou (2003); Singh and Dutt (2020); Tserpes and Papanikos (2005)). First, geometries of SWCNT samples were created using nanotube modeler and extracted into ANSYS APDL 17.0.…”

“…In the area of CNT vibrations, most of the research has been conducted over the last decade toward experimental or analytical determination of transverse vibrational modes. In this regard, researchers have been trying to establish relationship between transverse vibration frequencies and the SWCNT geometrical parameters (length, diameter, chiral angle, length-to-diameter ratio, thickness-to-radius ratio, and Eringen's nonlocal parameter) (Asghar et al, 2020;Hussain et al, 2019Hussain et al, , 2020Singh and Dutt, 2020). The authors' previous article discusses the state-of-the-art research and derives a reduced form of fundamental transverse modes of SWCNTs (Singh and Dutt, 2020).…”