Vibration analysis of multi-walled carbon nanotubes embedded in elastic medium

Abstract

We propose a method to estimate the natural frequencies of the multi-walled carbon nanotubes (MWCNTs) embedded in an elastic medium. Each of the nested tubes is treated as an individual bar interacting with the adjacent nanotubes through the inter-tube Van der Waals forces. The effect of the elastic medium is introduced through an elastic model. The mathematical model is finally reduced to an eigen value problem and the eigen value problem is solved to arrive at the inter-tube resonances of the MWCNTs. Variation of the natural frequencies with different parameters are studied. The estimated results from the present method are compared with the literature and results are observed to be in close agreement.

Keywords

natural frequencies / multi-walled carbon nanotubes (MWCNTS) / elastic medium

Introduction

The structural perfection, small size, low density, high stiffness, high strength and excellent electronic properties has rendered the application of Carbon Nanotubes (CNTs) in many fields like: material reinforcement [1,2], field emission pane display, chemical sensing [3], drug delivery [4,5], nano electronics [6], bio-engineering [7,8], to name a few. The unusually superior mechanical properties of CNTs over the other known materials hold substantial promise as constituents [9], in the form of super fibers, in composite materials of high strength/weight ratios, as components of nano scale instruments, including probes of high-resolution scanning force microscopes. Hence, the mechanical behavior of CNTs has been a vibrant subject of research ever since their discovery [9]. The most promising mechanical applications of carbon nano tubes is to take advantage of their exceptionally high stiffness combined with their excellent resilience. Owing to their large aspect ratio and hallow geometry, various buckling behaviors of CNTs, as a shell or a column and under bending or axial compression, have been subjects of numerous experimental and molecular dynamics simulations [10].

Multiscale methods [11-14] are computationally efficient techniques to simulate the nanoscale phenomena. However, the molecular dynamics (MD) simulations will play a key role in arriving at the properties of CNTs. The extreme Young’s modulus values CNTs helps in stiffening the polymers in which they are distributed. Lot of structural applications are found for the stiffened polymer structures [15], for example, as a stiffening layer on aircraft wing surface [16]. Zhang et al. [2] have investigated the influence of the polymer wrapped on two neighboring single-walled carbon nanotubes on their load transfer by MD simulations. They developed an analytical technique for the interface debonding [17], for large diameter carbon nanotube-reinforced composite with functionally graded variation inter-phase. CNTs are produced by rolling the graphene sheet into a cylinder. Due to the extremely small out-of-plane stiffness of graphene, the differential thermal expansion causes the formation of the wrinkles during the chemical vapor deposition [18]. The collapse of the standing wrinkles leads to folds in graphene [19]. A quasi-analytical solution based on the minimum length has been proposed in Ref [20], to estimate the system energy of the stable multi-layer folded graphene wrinkles. The effects of the dispersion of polymer wrapped two neighboring single walled carbon nanotubes (swnts) on nano-engineering load transfer are studied in [21]. Jiang et al. [22] have investigated the interlayer energy of the twisting bi-layer graphene by the molecular mechanics method using both the registry-dependent potential and the Lennard-Jones potential. The wave propagation in the armchair and zigzag single-walled CNTs based on the nonlocal elasticity and the molecular structural mechanics principles is studied in Ref. [23]. The high stiffness and large surface area of graphene combined with low mass of carbon atoms, makes it an excellent mass sensor to detect individual atoms or molecules [24]. The enhancement of the mass sensitivity and resonant frequency of graphene nano-mechanical resonators are investigated through MD simulation in Ref. [25]. by driving the nanoresonators into the nonlinear oscillation regime. The mechanisms underpinning the unresolved, experimentally observed temperature-dependent scaling transition in the quality factors of graphene nano-mechanical resonators is investigated in Ref. [26].

Zhao et al. [27] performed the MD simulations to estimate the mechanical properties of three types of carbon allotropes namely, supergraphene, cyclicgraphene and graphyne. They studied the chirality dependence of the mechanical properties including Young’s moduli, shear moduli, Poisson’s ratio, ultimate strength and ultimate strains, based on the adaptive intermolecular reactive empirical bond order (AIREBO) potential. The cohesive energy between the carbon nanotubes and the graphene and its substrates has been theoretically estimated in [28], by modeling the Van der Waals interactions based on the continuum principles. The extreme reduction of the thermal conductivity due to the presence of defects and folds in carbon nanocoils is studied in [29], based on the non-equilibrium MD simulations. The stick-spiral and beam models are developed in [30] to model the graphene based on the principles of continuum mechanics. The analytical solutions helps in arriving at quick estimates for understanding the interaction between the nanostructures and substrates, and designing composites and nano-electromechanical systems.

Several techniques have been developed in the last decade to perform the vibration analysis of CNTs. Xu et al. [31] and Elishakoff et al. [32] have studied the free vibrations of a double walled carbon nanotubes (DWCNTs), composed of two coaxial single-walled carbon nanotubes (SWCNTs) interacting with each other by the inter-layer Van der Waals forces. The inner and outer CNTs are modeled as two individual elastic beams. Ru [33] and Lanir et al. [34] modeled the CNTs as columns to study the buckling phenomena. Elastic model [35], nonlinear Multi-wall carbon nanotubes (MWCNTs) model [36], finite element model [37], three dimensional elasticity model [38], Timeshenkhov beam model [39] are few other techniques to estimate the vibration characteristics of the CNTs. The uncertainties propagation and their effects on the reliability of polymeric nanocomposite (PNC) continuum structures, in the framework of the combined geometry and material optimization are explained in Ref [40]. Nam et al. [41] have proposed the stochastic predictions of the interfacial characteristic of carbon nanotube polyethylene composites.

The vibration characteristics of the CNTs will be significantly improved when they are embedded in an elastic medium. Therefore, several researchers have studied the vibration characteristics of the CNTs embedded in an elastic medium. Horng [42] studied the transverse vibration analysis of SWCNTs embedded in an elastic medium using the Bernoulli-Fourier method. Li et al. [43] and Yoon et al. [44] have analyzed the vibration characteristics of MWCNTs embedded in elastic media by a non-local elastic shell model. Lourie et al. [45] have studied the buckling and collapse of thick and thin walled embedded CNTs through experiment. Natarajan et al. [46] have estimated the natural frequencies of cracked functionally graded material plates by the extended finite element method. Navid et al. [47] have studied the static and dynamic characteristics of functionally graded material plates using a non-uniform rational B-spline based on iso-geometric finite element method. The size-dependent free flexural vibration behavior of functionally graded nanoplates, using the iso-geometric based finite element method is investigated in Ref. [48].

In the present work, we made an attempt to estimate the natural frequencies of the MWCNTs embedded in the elastic medium. The arrangement of the article is as follows: the vibration analysis of the MWCNTs embedded in the elastic medium is introduced in Section 1. Section 2 explains the mathematical model and the results are discussed in Section 3. Vibration characteristics of several MWCNTs are discussed. Section 4 concludes the article.

Vibration analysis

In this section we developed the methodology to extract the natural frequencies of the MWCNTs. The mathematical model is developed based on the theory of beam bending.

The elastic beam model

Consider a bar subjected to transverse vibrations as shown in Fig. 1(a). The elastic properties of the bar and the external forces are indicated in Fig. 1(a). The free body diagram of an element of length dx is shown in Fig. 1(b). Based on the simple beam theory, the rotation of the element is insignificant as compared to the vertical translation, and the shear deformation is small compared to the vertical translation. Therefore, Fig. 1(b) shows the forces in the vertical direction and the moments about the axis normal to the x and the y directions and passing through the center of cross-sectional area. Neglecting the higher order terms of dx, the equations of motion of the flexural vibrations of the bar can be written as

(1)

where E is the Young’s modulus, I is the moment of inertia, w is the displacement in the transverse direction, f are the external forces, ρ is the mass density and A is the cross sectional area, of the bar. Considering a set of n individual concentric nano-tubes stacked one on the other as shown in Fig. 2, the co-planar transverse vibrations of n nested tubes are described by the following n coupled Eqs [44].

(2)\begin{array}{l}E{I}_{\mathrm{1}}{\displaystyle \frac{{\mathrm{d}}^{\mathrm{4}}{w}_{\mathrm{1}}}{\mathrm{d}{x}^{\mathrm{4}}}}+\rho A_{\mathrm{1}}{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}{w}_{\mathrm{1}}}{\mathrm{d}{t}^{\mathrm{2}}}}=c_{\mathrm{1}}(w_{\mathrm{2}}-w_{\mathrm{1}}),\hfill \\ E{I}_{\mathrm{2}}{\displaystyle \frac{{\mathrm{d}}^{\mathrm{4}}{w}_{\mathrm{2}}}{\mathrm{d}{x}^{\mathrm{4}}}}+\rho A_{\mathrm{2}}{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}{w}_{\mathrm{2}}}{\mathrm{d}{t}^{\mathrm{2}}}}=c_{\mathrm{2}}(w_{\mathrm{3}}-w_{\mathrm{2}})-c_{\mathrm{1}}(w_{\mathrm{2}}-w_{\mathrm{1}}),\hfill \\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \hfill \\ E{I}_n{\displaystyle \frac{{\mathrm{d}}^{\mathrm{4}}{w}_n}{\mathrm{d}{x}^{\mathrm{4}}}}+\rho A_n{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}{w}_n}{\mathrm{d}{t}^{\mathrm{2}}}}=p-c_{n-\mathrm{1}}(w_n-w_{n-\mathrm{1}}),\hfill \end{array}

where x is the axial coordinate, t is the time variable, w_j , I_j and A_j are the transverse deflection, moment of inertia and area of cross section of the j^th tube, respectively. The subscripts j = 1,2,...,n indicate the locations of the nano-tubes, first nanotube being at the innermost location and the n^th nanotube being at the outermost location. p is the interaction pressure per unit axial length between the outer most tube and the surrounding elastic medium. The Van der walls interaction coefficients (c_j) can be estimated as [33]

(3)c_j={\displaystyle \frac{\mathrm{200}(\mathrm{2}{r}_j)}{\mathrm{0.16}{d}^{\mathrm{2}}}},

where d is the bond length between the carbon atoms which is equal to 0.142 nm, r is the inner radius of each pair of nanotube and j = 1,2,...,(n-1). Note the units of the quantity in the numerator of Eq. (3) as erg/cm². In this work, the elastic medium is characterized by a spring constant k. Therefore, we adopt the Winkler model [34,35] to represent the pressure p per unit axial length acting on the outermost tube due to the surrounding elastic medium. Based on the Winkler model, p is given by

(4)p(x)=-k{w}_n(x),

where the negative sign indicates that the pressure p is opposite to the deflection of the outermost tube. The constant k is estimated by the material constants of the elastic medium, the outermost diameter of the embedded MWCNTs and the wavelength of the vibrational modes [34]. Figure 3 shows the variation of the spring constant k with the parametern\mathrm{\pi }d/\mathrm{2}L, with E = 2 GPa, ν = 0.35.

Extraction of the natural frequencies

Consider a MWCNTs of length L, embedded in an elastic medium, refer to Fig. 2. The equations of motion of such a system are derived in Eq. (2). Since the simultaneous equations in Eq. (2) are functions of both space and time, we consider a harmonic solution as given below:

(5)w_n(x,t)=a_n{e}^{\lambda x}{e}^{jwt},

where a_n is the amplitude of vibration and ω is the frequency of excitation. Substituting the solution in Eq. (5) into Eq. (2) and collecting the coefficients of like terms, system of equations in Eq. (2) can be reduced to

(6)\begin{array}{r}\hfill (E{I}_{\mathrm{1}}{\lambda }^{\mathrm{4}}-\rho A_{\mathrm{1}}{\omega }^{\mathrm{2}}+c_{\mathrm{1}})a_{\mathrm{1}}-c_{\mathrm{1}}{a}_{\mathrm{2}}=\mathrm{0},\\ \hfill -c_{\mathrm{1}}{a}_{\mathrm{1}}+(E{I}_{\mathrm{2}}{\lambda }^{\mathrm{4}}-\rho A_{\mathrm{2}}{\omega }^{\mathrm{2}}+c_{\mathrm{2}}+c_{\mathrm{1}})a_{\mathrm{2}}-c_{\mathrm{2}}{a}_{\mathrm{3}}=\mathrm{0},\\ \hfill \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ \hfill -c_{n-\mathrm{2}}{a}_{n-\mathrm{2}}+(E{I}_{n-\mathrm{1}}{\lambda }^{\mathrm{4}}-\rho A_{n-\mathrm{1}}{\omega }^{\mathrm{2}}+c_{n-\mathrm{1}}+c_{n-\mathrm{2}})a_{n-\mathrm{1}}-c_{n-\mathrm{1}}{a}_n=\mathrm{0},\\ \hfill -c_{n-\mathrm{1}}{a}_{n-\mathrm{1}}+(E{I}_n{\lambda }^{\mathrm{4}}-\rho A_n{\omega }^{\mathrm{2}}+k+c_{n-\mathrm{1}})a_n=\mathrm{0.}\end{array}

Further, Eq. (6) can be written in the matrix format as shown below:

(7)\left[\begin{array}{l}\begin{array}{llllll}{T}_{\mathrm{1}}& -c_{\mathrm{1}}& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}\\ -c_{\mathrm{1}}& T_{\mathrm{2}}& -c_{\mathrm{2}}& \mathrm{0}& \cdots & \mathrm{0}\\ \mathrm{0}& -c_{\mathrm{2}}& T_{\mathrm{3}}& -c_{\mathrm{3}}& \cdots & \mathrm{0}\end{array}\\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\ \begin{array}{llllll}\mathrm{0}& \cdots & \mathrm{0}& -c_{n-\mathrm{2}}& T_{n-\mathrm{1}}& -c_{n-\mathrm{1}}\\ \mathrm{0}& \cdots & \mathrm{0}& \mathrm{0}& -c_{n-\mathrm{1}}& T_n\end{array}\end{array}\right]\left\{\begin{array}{l}{a}_{\mathrm{1}}\hfill \\ a_{\mathrm{2}}\hfill \\ a_{\mathrm{3}}\hfill \\ \cdots \hfill \\ a_{n-\mathrm{1}}\hfill \\ a_n\hfill \end{array}\right\}=\mathrm{0},

where

(8)$$\begin{gathered} T_{\mathrm{1}}=E{I}_{\mathrm{1}}{\lambda }^{\mathrm{4}}-\rho A_{\mathrm{1}}{\omega }^{\mathrm{2}}+c_{\mathrm{1}}, \\ {T}_{\mathrm{2}}=E{I}_{\mathrm{2}}{\lambda }^{\mathrm{4}}-\rho A_{\mathrm{2}}{\omega }^{\mathrm{2}}+c_{\mathrm{2}}+c_{\mathrm{1}}, \\ {T}_{\mathrm{3}}=E{I}_{\mathrm{3}}{\lambda }^{\mathrm{4}}-\rho A_{\mathrm{3}}{\omega }^{\mathrm{2}}+c_{\mathrm{3}}+c_{\mathrm{2}}, \\ {T}_{n-\mathrm{1}}=E{I}_{n-\mathrm{1}}{\lambda }^{\mathrm{4}}-\rho A_{n-\mathrm{1}}{\omega }^{\mathrm{2}}+c_{n-\mathrm{1}}+c_{n-\mathrm{2}}, \\ {T}_n=E{I}_n{\lambda }^{\mathrm{4}}-\rho A_n{\omega }^{\mathrm{2}}+k+c_{n-\mathrm{1}}. \end{gathered}$$

Let

(9)P_n=E{I}_n{\lambda }^{\mathrm{4}}+c_{n-\mathrm{1}}+c_n

and

(10)Q_n=\rho A_n.

Therefore, after substituting Eqs. (8)-(10) into Eq. (7) and noting that c₀ = 0 and c_n = k, the coefficient matrix in Eq. (7) can be written as

(11)\left[\begin{array}{llllll}{P}_{\mathrm{1}}-{\omega }^{\mathrm{2}}{Q}_{\mathrm{1}}\hfill & -c_{\mathrm{1}}\hfill & \mathrm{0}\hfill & \mathrm{0}\hfill & \cdots \hfill & \mathrm{0}\hfill \\ -c_{\mathrm{1}}\hfill & P_{\mathrm{2}}-{\omega }^{\mathrm{2}}{Q}_{\mathrm{2}}\hfill & -c_{\mathrm{2}}\hfill & \mathrm{0}\hfill & \cdots \hfill & \mathrm{0}\hfill \\ \mathrm{0}\hfill & -c_{\mathrm{2}}\hfill & P_{\mathrm{3}}-{\omega }^{\mathrm{2}}{Q}_{\mathrm{3}}\hfill & -c_{\mathrm{3}}\hfill & \cdots \hfill & \mathrm{0}\hfill \\ \cdots \hfill & \cdots \hfill & \cdots \hfill & \cdots \hfill & \cdots \hfill & \cdots \hfill \\ \mathrm{0}\hfill & \cdots \hfill & \mathrm{0}\hfill & -c_{n-\mathrm{2}}\hfill & P_{n-\mathrm{1}}-{\omega }^{\mathrm{2}}{Q}_{n-\mathrm{1}}\hfill & -c_{n-\mathrm{1}}\hfill \\ \mathrm{0}\hfill & \cdots \hfill & \mathrm{0}\hfill & \mathrm{0}\hfill & -c_{n-\mathrm{1}}\hfill & P_n-{\omega }^{\mathrm{2}}{Q}_n\hfill \end{array}\right].

The coefficient matrix in Eq. (11) is a sparse, symmetric and positive definite matrix. For a nontrivial system, the determinant of the coefficient matrix in Eq. (11), should go to zero. Therefore, in simple form

(12)\left|P-{\omega }^{\mathrm{2}}Q\right|=\mathrm{0.}

The eigen value problem in Eq. (12) is solved for the natural frequencies of n walled carbon nanotubes embedded in the elastic medium.

Results and discussion

In the present work, unless and otherwise stated all the nano tubes are assumed to have the same Youngs Modulus E = 1 GPa, with the effective thickness of SWCNTs equal to 0.34 nm and the mass density ρ = 1.3 g/cm³. The camped-clamped boundary conditions are assumed for the CNTs.

Consider a SWCNTs with a constant inner diameter of 0.7 nm, embedded in a polyurethane elastic medium with the Youngs modulus E = 1 GPa and ν = 0.38. The variation of natural frequencies estimated from Eq. (12), with the length of the embedded SWCNTs for the first three modes is plotted in Fig. 4(a). From Fig. 4(a), it can be observed that the natural frequencies are converging to 0.2 THz, when the length of the SWCNTs is more than 30 nm. The difference of the natural frequencies between the higher order modes is significant at smaller lengths. Therefore, long tubes are poor in strengthening the elastic medium. In other words, when clusters of short nanotubes are embedded in the elastic medium the stiffness of the coupled system enhances significantly.

The variation of the resonances with the diameter, for a constant length of 20 nm, for the first three modes is plotted in Fig. 4(b). For diameters less than 1 nm the natural frequencies are less than 0.5 THz and diverges with the increase in the diameter. The divergence is observed to be larger for the higher order modes. Hence, embedded large diameter nanotubes are efficient in improving stiffness. Variation of the first natural frequency of the SWCNTs with the length for various inner diameters and with the diameter for various lengths, is plotted in Fig. 5(a). Similar trend with the length and the diameter as explained above, is observed for the variation of the first natural frequency. Therefore, embedded chunks of large diameter, small length nanotubes are the most efficient to enhance the stiffness and hence the natural frequencies of the system. To estimate the effect of the elastic medium on the stiffness of the SWCNTs, the variation of the natural frequencies with the Young’s modulus of the medium is plotted in Fig. 5(b). A smooth increase in natural frequencies is observed with the increase in the Young’s modulus of the medium. Note that the maximum value of the Young’s modulus considered in Fig. 5 is 10 GPa.

In the next study, we estimate the natural frequencies of an embedded DWCNTs. The same polyurethene medium considered in the analysis of SWCNTs, is adopted in this study. Figure 6(a) compares the estimated first mode natural frequencies of the DWCNTs without the elastic medium, with the values from Ref. [32]. The results are closely agreeing with each other. The effect of the elastic medium is plotted in Fig. 6(b). Figure 6(b) shows the variation of the first natural frequency of the first tube with the L/d ratio. From Fig. 6(b), the natural frequencies are significantly raised when the nanotubes are embedded in the elastic medium. Therefore, the elastic medium helps to significantly improve the stiffness of the nanotubes. Variations of the first and second mode frequencies of the first and second nanotubes with the length for the inner diameters of 1, 1.5 and 2 nm are plotted in Fig. 7. The notation adopted to denote the natural frequencies is as follows: ω_ij, the subscript i indicate the tube number and the j indicate the mode number. Therefore, ω₁₂ will indicate the first tube’s second natural frequency. As in the case of SWCNTs, the natural frequencies are decreasing and converging to 0.2 THz when the length is more than 30 nm. Variation of the first five natural frequencies of the DWCNTs with the length for a constant inner diameter of the first nanotube equal to 0.7 nm are plotted in Fig. 8(a) and the variation with the diameter for a constant length of 20 nm is plotted in Fig. 8(b). From Figs. 7 and 8, the natural frequencies of the inner most tube are observed to be the lowest compared to the rest of the frequencies. Also, the second mode frequency is observed to be always greater than the first mode frequency.

Finally, we studied the vibration characteristics of the five walled CNTs embedded in the elastic medium. The polyurethene medium considered in the analysis of SWCNTs, is adopted in the current study. The effect of the Young’s modulus on the stiffness of the CNTs is plotted in Fig. 9(a). When the Young’s modulus is more than 200 GPa, there is no effect on the first four CNTs. However, since the outermost nanotube is in direct contact with the elastic medium, the stiffness of the outermost nanotubes increases with the Young’s modulus. Variation of the first mode frequency of the five nanotubes with the L/d ratio is plotted in Fig. 9(b). The effect of the L/d ratio is negligible on the outer four nanotubes. The first natural frequency is decreasing with increase of the L/d ratio.

Conclusions

The natural frequencies of the MWCNTs embedded in an elastic medium are studied. A mathematical model based on the simple beam theory, to estimate the natural frequencies of MWCNTs has been developed. The effect of the elastic medium is included through a spring constant k. The methodology is applied to calculate the natural frequencies of the SWCNTs, the DWCNTs and the 5 wall CNTs. The natural frequencies are observed to decrease with the increase of the length and increase with the increase of, the diameter and the Young’s modulus of the elastic medium.

A platform has been developed to estimate the natural frequencies of MWCNTs embedded in an elastic medium. The method helps to quickly estimate of the impact of the elastic medium and the CNTs, on the stiffness of the structures embedded with the CNTs. A potential application of the CNTs embedded in the elastic medium is, to stiffen the large thin walled structures in the automobile and the aircraft industry, espcially the aircraft wing structure.

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