Theoretical investigation of nonlinear resonances in a carbon nanotube cantilever with a tip-mass under electrostatic excitation

With excellent mechanical and electrical properties, carbon nanotubes (CNTs) are ideal materials for nano-electromechanical systems. Nano-devices that take advantage of the low density, high aspect ratio, and high conductivity of a CNT include nano-resonators^1–4 or mass sensors.^5–7 Nano-resonators are based on electromechanical resonance responses of a cantilevered CNT to electrostatic excitation. Mass sensors measure shifts in resonance frequency due to an additional mass on the CNT tip, such as that reported in the experimental study⁸ of a CNT with an attached spheroidal carbon particle. In addition, the static analysis of nano-switches^9,10 was based on experiments of cantilevered CNT tweezers.^11,12 The dynamics of nano-resonators has been examined to predict the linear¹ or nonlinear responses² of a cantilevered CNT. Other studies include those on mass- or bio-sensors for detecting metal atoms, proteins, or bacteria.^5–7 

Ke¹ predicted the time responses of a CNT nano-resonator by using a linear cantilever beam model. Ouakad and Younis² studied the nonlinear behavior of a nano-resonator by incorporating the geometric nonlinearity of the beam model. A mass sensor for detecting a metal atom can be used in vacuum or in air, but a biosensor needs to be operated in an environment that preserves the shape, functions, and mechanical or chemical properties of biological entities. In this respect, Sawano et al.³ examined the frequency response of the CNT resonator in water or air. Adhikari and Chowdhury⁶ investigated the resonance frequency shift due to an attached point mass or a distributed load over the surface of CNT biosensors. Elishakoff et al.⁷ modeled a bacterium on the tip of CNT as a spherical body or as a vertical or horizontal cylinder.

The studies cited above are basically limited to small oscillating amplitudes and near-resonant motion. When a CNT has a large amplitude oscillation, however, the effects of geometric nonlinearities will be significant. Also, it will be shown that behaviors different from the linear model occur because of the inertial nonlinearity as the tip mass changes with attached metal particles or biomolecules. According to previous report,¹³ the geometric nonlinearity leads to stiffening, whereas the inertial nonlinearity causes softening of nano-cantilever beams. Therefore, we derived a general equation of motion for a simulated nano-resonator considering the geometric and inertial nonlinearities of cantilevered nano-devices. We used the cantilever model of Chaterjee and Pohit¹⁴ and considered external forces such as electrostatic and van der Waals interactions between the CNT and a surface plane. We derived the nonlinear responses of the nano-resonator driven by electrostatic excitation and various tip mass effects. The tip mass of the attached particle, such as a metal atom or a biomaterial, was assumed to be non-conducting and in the atto-gram range, i.e., lighter than the CNT itself. For increased detection sensitivity, the CNT is employed for mass sensor in nano-scale due to the high aspect ratio and low density of CNT. Also, because the sensitivity increases with varying positions of attached mass from fixed end to free end of cantilever system,¹⁵ the attached mass was located at the functionalized tip of the CNT cantilever. The axial force¹⁶ induced by the tip mass was ignored here. The dynamic stability of the nano-resonator was determined by frequency and phase responses; we verified the stability using transient and steady-state time responses with varying tip masses.

A schematic diagram (from Ref. 1) of an electrostatically actuated nanomechanical resonator, based on a double-wall CNT, with tip mass is shown in Fig. 1. The diagram depicts the equilibrium force incorporating the elastic force, electrostatic forces, and the intermolecular interactions between the CNT and a graphene ground plane. In Fig. 1, $qelec$ denotes electrostatic forces and $qLJ$ denotes intermolecular forces based on a Lennard-Jones potential.¹⁷ For a non-conductive tip mass, we assume no interactions between it and the ground plane, including non-axial forces. Given these assumptions, we derived the equation of motion for a cantilevered nano-resonator with tip mass from the dynamic force equilibrium, and then non-dimensionalized the equation.

For the configuration in Fig. 1, we obtained the strain and kinetic energy of the CNT including axial displacement $u(x,t)$ and transverse displacement $w(x,t)$⁠, as in Ref. 14. Using Hamilton's principle and the inextensibility condition between $u(x,t)$ and $w(x,t)$⁠, we derived the nonlinear equation for the bending motion of a CNT cantilever with tip mass with a single degree of freedom. We took into account the cubic nonlinear terms, as in Ref. 14. By applying the viscous damping term to the derivation, the damped version of the equation of motion can be written as

with the boundary conditions:

The external forces $qex$ across instantaneous gap D consist of electrostatic and intermolecular forces (e.g., attractive van der Waals forces) and repulsive Pauli exclusion forces.^10,17

Using the non-dimensional variables,

a non-dimensionalized version Eq. (1a) is

and the boundary conditions are

where

The first, second, and third terms on the right side of Eq. (3a) are electrostatic forces, van der Waals attractive interactions, and Pauli repulsive forces, respectively. The superscript (*) of Eqs. (3) and (4) is eliminated for convenience.

To simulate the cantilevered nano-resonator, we discretized Eq. (3) using Galerkin's approximation. The approximate deflection of the CNT can then be expressed as

where $yi(t)$ is the temporal function and $ϕi(x)$ is the normalized eigenfunction of the i-th mode of the cantilever beam.¹⁸ The response of the nano-resonator is assumed to be single mode; therefore, for additional tip mass changes, it is sufficient to measure the frequency and phase shifts. The eigenfunction Eq. (5b) is normalized such that $∫01ϕ1(x)2dx=1$⁠. Then, using a one-mode approximation of the nano-resonator, Eq. (3) can be written as

where $ηG$ and $ηI$ denote the coefficients of the CNT geometric and inertial nonlinearities, respectively. Table I summarizes parameters and properties of the CNT nano-resonator. The length L = 500 nm, the outer radius $RO$ = 3.075 nm, the inner radius $RI$ = 1.4 nm, and the initial gap $Dinit$ between the CNT and ground surface is 100 nm. Considering the gas damping effect¹⁹ in Ref. 1, the quality factor Q = 1000. With these values, the coefficients of Eq. (6) are calculated with various mass ratios $μ=M/ρAL$⁠, as shown in Table II.

It is difficult to evaluate directly the Galerkin integrals in terms of the external excitations of Eq. (3), because the external forces impose implicit temporal functions for the spatial integrals in Eqs. (3)–(6). Considerable computation time is required to integrate numerically at every time step in the time solution of Eq. (6). Hence, for faster integration, we used curve fitting to approximate each external force with a fractional function form. For example, the maximum non-dimensional deflection of the CNT is 1.0. The range of the time solution $y1(t)$ has a lower value than 1/$ϕmax$⁠. Finally, if the Galerkin integration of the external forces can be considered to be over the distribution of external forces with $y1(t)$⁠, then the external excitation terms in Eq. (6) are written, respectively, as

where the coefficients of Eqs. (7) and (8) are equal to the values in Table III with various mass ratio $μ$⁠.

After the above procedures, the fitted electrostatic force function has a 7.3% error relative to the original force. The fitted functions for van der Waals attractive and Pauli repulsive interactions have 4.3% errors relative to their original interactions. The fitted fractional functions like Eqs. (7) and (8) matched well with the original functions of the external forces within the effective range of the initial gap between the static position and the electrode (Fig. 2). Because these fitted functions are expressed as an explicit form of $y1(t)$⁠, the numerical calculation of the nano-resonator equation is much faster than the case for the original external forces as an implicit form. Therefore, we used the fitted external forces in Eqs. (7) and (8), instead of the original forces, to analyze the electrostatically actuated nano-resonator. In particular, for the fitted functions of the van der Waals attraction and the Pauli repulsion in Eq. (8), each order of the denominator differs by the 6th order. This result indicates an equivalent relation such as the difference in order between the attractive and repulsive intermolecular forces in a Lennard-Jones potential.¹⁷ 

We analyzed the different responses of the linear and nonlinear models for the electrostatically actuated nano-resonator with an attached mass at the tip. For cantilevered CNT nano-resonators, we used AUTO²⁰ to compute the frequency responses and MATLAB to simulate the time responses. The frequency responses of a nano-resonator were computed with various AC and DC harmonic excitations. Because the nano-resonator was harmonically actuated, it had complex nonlinear responses with applied voltages and tip mass ratios μ. While the geometric nonlinearity affected stiffening of the nano-cantilever, the nonlinear inertia and the electrostatic force that is proportional to $V2$ as $γ2$ in Eq. (4) softened the nano-cantilever without tip mass.¹³ 

The nonlinear geometry and inertia effects can be analyzed by calculating the non-dimensional coefficients of the geometric and inertial nonlinear terms in Eq. (6) with tip mass ratio variations. As shown in Table II and Fig. 3, the geometric nonlinear coefficients significantly changed with increasing tip mass, whereas the inertial nonlinear coefficients changed only slightly. In Fig. 3, each nonlinear coefficient for a specific tip mass was normalized by the corresponding nonlinear coefficient without tip mass. While the inertial nonlinearity had not significantly changed with tip mass, the geometric nonlinearity led to stiffening or softening of the system depending on whether the tip mass ratio was higher or lower, respectively, than $μ=0.2$ (Fig. 3). The nonlinear inertia had a major effect near $μ=0.2$ because the geometric nonlinear coefficient was close to zero. Consequently, the nano-resonator softened because of the inertial nonlinearity and nonlinear electrostatic forces, regardless of tip mass effects. The geometric nonlinear effect depended on the tip mass ratio. Therefore, the geometric and inertial nonlinearities should be taken into account to understand the nonlinear behaviors of the nano-resonator when detecting mass.

When the CNT nano-resonator was driven by AC voltage only, the excitation force was proportional to $VAC2(1+cos2Ωt)/2$⁠. Therefore, the excitation frequency was twice the AC harmonic frequency $Ω$⁠. We can then predict the superharmonic²¹ response because the frequency of the response from the electrostatic excitation with half the primary harmonic frequency was doubled by the excitation frequency. In Fig. 4, the nano-resonator showed the superharmonic responses of the linear (Fig. 4(a)), the geometric nonlinear (Fig. 4(b)), the inertial nonlinear (Fig. 4(c)), and both the geometric and inertial nonlinear (Fig. 4(d)) models with different tip mass ratios at the same AC voltage excitation (⁠$VAC$ = 0.15 V). For convenient comparison of the nonlinearities of CNT and tip mass effects, the excitation frequencies and amplitudes in Fig. 4 were normalized by each fundamental frequency according to its attached tip mass-induced frequency shift (see Table II) and the initial gap $Dinit$⁠, respectively. With increasing tip mass ratio, both the fundamental frequency and the geometric nonlinearity decreased, while the inertial nonlinearity slightly increased, as shown in Table II and Fig. 3. The additional tip mass led to an increase in the response amplitude, as shown in Fig. 4. The linear model had a linear resonance peak, regardless of tip mass effects (Fig. 4(a)). If geometric nonlinearity alone was effective (⁠$ηG≠0$⁠, $ηI=0$ in Eq. (6)), the nano-resonator stiffened in the frequency response amplitude because the geometric nonlinear coefficient $ηG$ was greater than zero (⁠$μ<0.2$⁠), as shown in Fig. 4(b). If inertial nonlinearity alone was effective (⁠$ηG=0$⁠, $ηI≠0$ in Eq. (6)), the nano-resonator always softened, as shown in Fig. 4(c). When both geometric and inertial nonlinearities were effective, the nonlinear model (Fig. 4(d)) indicated that the resonant response was similar to that of the linear model at $μ=0$⁠. For an increasing tip mass ratio, the resonance branch softened more than the case of only geometric nonlinearity. The softening effects²¹ at the resonance branch occurred because the cubic nonlinearity of the equation of motion became strongly negative with increasing applied voltage⁴ or inertial nonlinearity. The stiffening effects at the resonance branch were due to the positive cubic term.²¹ In the nonlinear model, these results compare favorably with a previous experimental study²² and the reduced-order model analysis²³ of a resonating microcantilever beam. The nonlinear behavior of a nano-resonator with tip mass exhibits jump-to-contact that indicates stability changes between the saddle and the center on the phase plane at a saddle node (SN) bifurcation with softening effects. In Fig. 4, the solid and dashed lines denote a stable and unstable branch, respectively.

With DC and AC voltages, the nano-resonator with or without tip mass had the primary and secondary harmonic response in the frequency domain as shown in Figs. 5 and 6. These responses were mainly due to parametric excitations²¹ such as the $cos2Ωt$ and $cosΩt$ terms in V² of γ₂ (Eq. (4)). Figure 5 plots the frequency and phase response of the nonlinear model for the nano-resonator without tip mass for the fundamental and superharmonic response branches under DC voltage variations (⁠$VDC$ = 0.1, 0.5, and 1.0 V) and a fixed AC voltage $VAC$ = 0.05 V. Under a low DC bias of $VDC$ = 0.1 V (Fig. 5(a)), the nano-resonator exhibited a very weak superharmonic resonance peak and no phase difference at half the fundamental harmonic frequency because of the low ratio of AC voltage between DC and AC harmonic excitation, unlike in our previous work.⁴ However, the primary resonance peak occurred at the fundamental frequency with a phase difference of $π$⁠. Figures 5(b) and 5(c) show that the nano-resonator softened and the SN bifurcation occurred at the fundamental resonance branch under higher DC voltage. At increasing excitation voltages, the natural frequency decreased,²⁴ but the unstable region increased. The stability change at the SN bifurcation compared favorably with previous work,^2,4 but the phase response, which had a stability change at the bifurcation point, did not.² 

Figure 6 shows the frequency and phase response of the nonlinear model of the nano-resonator with tip mass (⁠$μ=0.2$⁠) for the primary and secondary response branches under DC voltage variations (⁠$VDC$ = 0.1, 0.5, and 1.0 V) and a fixed AC voltage $VAC$ = 0.05 V. This nano-resonator had a slightly larger superharmonic and fundamental resonance peak and more softened responses than those of the nano-resonator without tip mass. Under a low DC bias $VDC$ = 0.1 V, as shown in Fig. 6(a), the SN bifurcation with a weak softening effect occurred at the fundamental resonance peak, unlike the case with no tip mass effects, as described in Fig. 5(a). For increasing DC voltage bias, the natural frequency decreased, but the unstable region increased, as illustrated in Figs. 6(a)–6(c). Under a higher DC voltage of $VDC$ = 1.0 V, the nano-resonator softened at the fundamental and subharmonic resonance branches. In the subharmonic behavior, there was a period doubling (PD) bifurcation at twice the primary resonance frequency due to the electrostatic periodic excitation.²¹ This was due to a parametric resonance that occurs in a nonlinear vibration system when the nano-resonator is periodically actuated at its harmonics. In addition, the subharmonic resonance branch indicated a softening with increasing DC voltage, as observed for the fundamental resonance branch. Moreover, the phase response had the same change in stability at the PD bifurcation point.

To understand the characteristics of the time response at the superharmonic, primary, or subharmonic resonances, it was compared with the time response for a similar period at a different excitation frequency. The transient and steady-state time response of the nano-resonator with or without tip mass was performed with a MATLAB simulation, and the time responses with the superharmonic, fundamental, and subharmonic resonances in Figs. 5(c) and 6(c) are shown in Figs. 7 and 8, respectively (⁠$VDC$ = 1.0 V, $VAC$ = 0.05 V, and $Q$ = 1000). The time responses of the nano-resonator with and without tip mass at the superharmonic excitation (⁠$Ωμ=0=0.4954$ and $Ωμ=0.2=0.4910$⁠) in Figs. 5(c) and 6(c) are displayed in Figs. 7(a) and 8(a). The steady-state amplitude in the time response (Figs. 7(a) and 8(a)) matched the amplitude of the superharmonic frequency response in Figs. 5(c) and 6(c). As explained earlier, the frequency of response was twice of that of the applied voltage (Fig. 9(a)).

For the primary resonance behavior, Figs. 7(b) and 8(b) are the time responses from Figs. 5(c) and 6(c). Because there is no escape band,^2,4 the nano-resonator with or without tip mass showed a stable response for the primary resonance excitation (⁠$Ωμ=0=0.9867$⁠, $Ωμ=0.2=0.9713$⁠), as shown in Fig. 9(b). The frequency of the time response was equivalent to the excitation frequency as illustrated in Fig. 9(b). Conversely, in previous work,⁴ we monitored the pull-in behavior when it was difficult to use a resonator driven by a frequency within the escape band. Furthermore, when the Pauli repulsion force was considered, there was an unstable response like bouncing or tapping as the CNT approached the ground plane.

Regarding subharmonic behavior, Figs. 7(c) and 8(c) are the time responses from Figs. 5(c) and 6(c), and the steady-state amplitude in the time response correlated with the subharmonic resonance amplitude. If there was no tip mass, the nano-resonator did not have a subharmonic resonance response (Fig. 7(c)). However, the nano-resonator with tip mass had the PD bifurcation at the subharmonic resonance branch (Fig. 8(c)). The time response within the PD bifurcation (Fig. 9(c)) was similar to the response of the fundamental frequency excitation (Fig. 9(b)) but had the time response at half-frequency (doubling the period) compared to the excitation frequency $Ωμ=0.2=1.964$⁠. Therefore, mass detection based on a nano-resonator can be implemented by monitoring the subharmonic resonance through PD bifurcation as well as the fundamental frequency shift. However, except for tip mass effects, the nano-resonator had the PD bifurcation at subharmonic resonance branch only the under conditions of a high electrostatic field.^2,4 In this study, the tip mass effects led to the PD bifurcation of the system. These results show that the PD bifurcation occurred not only under high electrostatic forces but also with a tip mass effect, as illustrated in Fig. 10. The onset of the period doubling was $VDC$ = 1.11 V (⁠$Ωμ=0=1.977$⁠) for nano-resonator with no tip mass, while the period doubling started at $VDC$ = 0.61 V (⁠$Ωμ=0.2=1.987$⁠) for nano-resonator with tip mass ratio $μ=0.2$⁠. Generally, the high electrostatic force-induced parametric excitation caused the PD bifurcation. However, the tip mass effect facilitated the PD bifurcation under a lower DC voltage than the case of the nano-resonator without tip mass.

We analyzed the nonlinear responses of a nano-resonator in the frequency and time domains with tip mass effects and various excitation conditions. The nonlinear model of a nano-resonator had a softer response in contrast to the linear model. The nano-resonator exhibited nonlinear frequency responses in amplitude and in phase such as softening, and SN or PD bifurcations, with increasing applied voltage and tip mass. We identified the time responses of the nano-resonator under different applied excitation frequencies as super- or subharmonics. The superharmonic response was dominant with AC voltage and AC-only excitation analysis, while the subharmonic response occurred with DC and AC voltages as parametric excitations, with tip mass effects. The primary and secondary resonances were affected by increasing DC voltages and an AC voltage. When the tip mass ratio was increased, the fundamental frequency decreased, but its amplitude increased. In particular, the geometric nonlinearity had significant changes, whereas the inertial nonlinearity changed slightly with the tip mass ratio. Regardless of the mass effects, the inertial nonlinearity that caused softening of the nano-cantilever had a major effect on the nano-resonator. In addition, under lower DC voltages than the case of the nano-resonator without tip mass, tip mass effects can lead to or facilitate a complex motion of the electromechanical response because of the PD bifurcation at the subharmonic resonance branch. Thus, for nano-resonator mass detection at the cantilever tip, we can monitor either the subharmonic resonance or changes in the resonance amplitude, as well as measuring the fundamental frequency shift. The sensitivity of mass detection can be increased by not only controlling the AC excitation but also the AC and DC voltages simultaneously.

This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (MEST) (NRF-2012-0002982) and by the Human Resources Development program (No. 20124010203260) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Trade, Industry and Energy.