Nonlinear vibration of a double-walled carbon nanotube embedded in a polymer matrix

In the current work, the nonlinear vibration of an embedded double-walled carbon nanotube (DWCNT) aroused by nonlinear van der Waals (vdW) interaction forces from both surrounding medium and adjacent tubes is studied. Using both Euler–Bernoulli and Timoshenko beam models, the relation between deflection amplitudes and resonant frequencies of the DWCNT is derived through harmonic balance method. It is found that the nonlinear vdW forces from the surrounding medium result in noncoaxial vibration of the embedded DWCNT. The noncoaxial vibration includes both uni-directional and bi-directional vibration modes. It is found that the surrounding matrix has more prominent effect on the uni-directional vibration in comparison to the bi-directional vibration. The axial load effect on the vibrational behavior of the embedded DWCNT is also discussed. Due to the influence of the surrounding polymer, the prediction on the resonant frequencies of embedded CNTs is quite different from that for free-standing CNTs. A softening behavior for the deflection amplitude-resonant frequency relation is observed for the first time in the bi-directional vibration of the embedded DWCNT, which can only be obtained using the Timoshenko beam theory.     

Nonlinear vibration and stability analysis of a double-walled carbon nanotube under electrostatic actuation

In this paper, forced vibrations of a double-walled clamped–clamped carbon nanotube (DWNT) are studied. Two Euler–Bernoulli beams are used to model the inner and outer layers of the DWNT. An electrostatic actuation, which is comprised of DC and AC voltages is applied between the nanotubes and the electrode. In the system model, the nonlinear form of the interlayer van der Waals (vdW) force, and also, the mid-plane stretching are considered. The obtained equations are solved through Galerkin and multiple scales methods for primary and secondary resonances. The frequency response of the system is obtained as a function of some of the system parameters. A stability analysis of the response is conducted and bifurcation points are determined. The results demonstrate that the DWNT shows different behavior by changing the value of DC voltage. It is also observed that both layers of the DWNT vibrate with the same frequency under the primary and secondary resonance conditions.     

Sound wave propagation in zigzag double-walled carbon nanotubes embedded in an elastic medium using nonlocal elasticity theory

In the present work, nonlocal Euler–Bernoulli beam theory is used to investigate the wave propagation in zigzag double-walled carbon nanotube (DWCNT) embedded in an elastic medium. Winkler-type foundation model is employed to simulate the interaction of the DWCNT with the surrounding elastic medium. The DWCNTs are considered as two nanotube shells coupled through the van der Waals interaction between them. It is noticed in the presented study that the equivalent Young’s modulus for zigzag DWCNT is derived using an energy-equivalent model. Influences of nonlocal effects, the chirality of zigzag DWCNT, Winkler modulus parameter, and aspect ratio on the frequency of DWCNT are analyzed and discussed. The new features of the vibration behavior of zigzag DWCNTs embedded in an elastic medium and some meaningful results in this paper are helpful for the application and the design of nanostructures in which zigzag DWCNTs act as basic elements.     

Graphene-nanotube structures: Constitution and formation energy

A new class of carbon nanostructures is considered: covalently (or molecularly) bound graphene and carbon nanotube fragments. As found using semiempirical computations and molecular dynamics methods, the structures formed by bonding nanotube edge atoms to a planar segment of a graphene nanofragment or its edge atoms to nanotube atoms arranged at the element of cylinder and graphene nanoribbons attached to a nanotube by van der Waals forces along the nanotube are the most energetically stable systems.     

Nonlinear vibration of viscoelastic embedded-DWCNTs integrated with piezoelectric layers-conveying viscous fluid considering surface effects

This article studies the nonlinear vibration of viscoelastic embedded nano-sandwich structures containing of a double walled carbon nanotube (DWCNT) integrated with two piezoelectric Zinc oxide (ZnO) layers. DWCNT and ZnO layers are subjected to magnetic and electric fields, respectively. This system is conveying viscous fluid and the related force is calculated by modified Navier–Stokes relation considering slip boundary condition and Knudsen number. Visco–Pasternak model with three parameters of the Winkler modulus, shear modulus, and damp coefficient is used for simulation of viscoelastic medium. The nano-structure is simulated as an orthotropic Timoshenko beam (TB) and the effects of small scale, structural damping and surface stress are considered based on Eringen's, Kelvin-voigt and Gurtin–Murdoch theories. Energy method and Hamilton's principle are employed to derive motion equations which are then solved using differential quadrature method (DQM). The detailed parametric study is conducted, focusing on the combined effects of small scale effect, fluid velocity, thickness of piezoelectric layer, boundary condition, surface effects, van der Waals (vdW) force on the frequency and critical velocity of nano-structure. Results indicate that the frequency and critical velocity increases with assume of surface effects.     

Nonlinear free vibrations of curved double walled carbon nanotubes using differential quadrature method

Nonlinear free vibration analysis of curved double-walled carbon nanotubes (DWNTs) embedded in an elastic medium is studied in this study. Nonlinearities considered are due to large deflection of carbon nanotubes (geometric nonlinearity) and nonlinear interlayer van der Waals forces between inner and outer tubes. The differential quadrature method (DQM) is utilized to discretize the partial differential equations of motion in spatial domain, which resulted in a nonlinear set of algebraic equations of motion. The effect of nonlinearities, different end conditions, initial curvature, and stiffness of the surrounding elastic medium, and vibrational modes on the nonlinear free vibration of DWCNTs is studied. Results show that it is possible to detect different vibration modes occurring at a single vibration frequency when CNTs vibrate in the out-of-phase vibration mode. Moreover, it is observed that boundary conditions have significant effect on the nonlinear natural frequencies of the DWCNT including multiple solutions.     

Buckling analysis of carbon nanotube bundles under axial compressive,bending and torsional loadings via a structural mechanics model

A structural mechanics model is employed for the investigation of the buckling behavior of carbon nanotube bundles of three single-walled carbon nanotubes (SWCNTs) under axial compressive, bending and torsional loadings. The effects of van der Waals (vdW) forces are further modeled using a nonlinear spring element.The effects of different types of boundary conditions are studied for nanotubes with various aspect ratios. The results reveal that bundles comprising longer SWCNTs exhibit lower critical buckling load. Moreover, for the fixed-free boundary condition the rate of critical buckling load reduction is highest, while the lowest critical buckling load occurs. Simulations show good agreement between our model and molecular dynamics results.     

Molecular dynamic investigation of mechanical properties of armchair and zigzag double-walled carbon nanotubes under various loading conditions

Using molecular dynamic simulation (MDS), effects of chirality and Van der Waals interaction on Young's modulus, elastic compressive modulus, bending, tensile, and compressive stiffness, and critical axial force of double-walled carbon nanotube (DWCNT) and its inner and outer tubes are considered. Achieving the highest safety factor, mechanical properties have been investigated under applied load on both inner and outer tubes simultaneously and on each one of them separately. Results indicate that as a compressive element, DWCNT is more beneficial than single-walled carbon nanotube (SWCNT) since it carries two times higher compression before buckling. Except critical axial pressure and tensile stiffness, in other parameters zigzag DWCNT shows higher amounts than armchair type. Outer tube has lower strength than inner tube; therefore, most reliable design of nanostructures can be attained if the mechanical properties of outer tube taken as the properties of DWCNT.     

Modelling the nonlinear behaviour of double walled carbon nanotube based resonator with curvature factors

This paper deals with the nonlinear vibration analysis of a double walled carbon nanotube based mass sensor with curvature factor or waviness, which is doubly clamped at a source and a drain. Nonlinear vibrational behaviour of a double-walled carbon nanotube excited harmonically near its primary resonance is considered. The double walled carbon nanotube is harmonically excited by the addition of an excitation force. The modelling involves stretching of the mid plane and damping as per phenomenon. The equation of motion involves four nonlinear terms for inner and outer tubes of DWCNT due to the curved geometry and the stretching of the central plane due to the boundary conditions. The vibrational behaviour of the double walled carbon nanotube with different surface deviations along its axis is analyzed in the context of the time response, Poincaré maps and Fast Fourier Transformation diagrams. The appearance of instability and chaos in the dynamic response is observed as the curvature factor on double walled carbon nanotube is changed. The phenomenon of Periodic doubling and intermittency are observed as the pathway to chaos. The regions of periodic, sub-harmonic and chaotic behaviour are clearly seen to be dependent on added mass and the curvature factors in the double walled carbon nanotube. Poincaré maps and frequency spectra are used to explicate and to demonstrate the miscellany of the system behaviour. With the increase in the curvature factor system excitations increases and results in an increase of the vibration amplitude with reduction in excitation frequency.     

Effect of temperature and geometrical nonlinearity on wave propagation characteristics of carbon nanotubes

Considering the effect of temperature and geometrical nonlinearity in the constitutive relation, the equation of motion for a carbon nanotube is obtained based on the Euler–Bernouli beam model. Also, the effect of van der Waals forces is taken into account in the formulation. The carbon nanotube is assumed to be under the application of a constant distributed external load. At any temperature, the equilibrium solutions of the governing equations for a single-walled carbon nanotube (SWCNT) and a double-walled carbon nanotube (DWCNT) are obtained. A small perturbation is assumed around the equilibrium solution. Using this perturbation, the nonlinear equations of motion are linearized. Using the linearized form of the equations of motion, the characteristic equations and dispersion relations are obtained. It is shown that in the linear case and for the case of high temperature there exists a temperature beyond which the phase velocity does not exist. It is shown that in the case of room or low temperature there is no critical value for temperature. Based on the dispersion equation, a relation for the critical value of temperature is obtained. It is found that when the large deformation effect is taken into account, the critical value for temperature does not exist. Also, the effect of large deformations on phase velocities and lateral deformations of single-walled and double-walled carbon nanotube beams are studied. It is found that unlike the linear theory, the nonlinear theory predicts a non-zero phase velocity at the temperature corresponding to linear critical temperature.     

Double-walled carbon nanotube with surrounding elastic medium under axial pressure

In this paper, the buckling behavior and critical axial pressure of double-walled carbon nanotubes (DWCNTs) with surrounding elastic medium are investigated. A double-shell (circular cylindrical shell) model is presented and the effects of surrounding elastic medium on the outer tube and the van der Waals forces between two adjacent tubes are taken into account. The analysis and the numerical solution method are based on the classical theory of plates and shells and the Galerkin method. Equations are derived for the critical axial forces and pressures of DWCNTs; the critical axial forces and pressures are calculated for different axial half sine wavenumbers and circumferential sine wavenumbers and compared with those for single-walled carbon nanotubes (SWCNTs).Results indicate that the critical axial force of a DWCNT is higher than that of an SWCNT, but the critical axial pressure of a DWCNT is lower than the critical axial pressure of a SWCNT. Although the critical axial force of a DWCNT decreases as the axial half sine wavenumbers increase, it rises as the circumferential sine wavenumbers increase.     

Vibration response of double-walled carbon nanotubes subjected to an externally applied longitudinal magnetic field: A nonlocal elasticity approach

The magnetic properties of carbon nanotubes and their mechanical behaviour in a magnetic field have attracted considerable attention among the scientific and engineering communities. This paper reports an analytical approach to study the effect of a longitudinal magnetic field on the transverse vibration of a magnetically sensitive double-walled carbon nanotube (DWCNT). The study is based on nonlocal elasticity theory. Equivalent analytical nonlocal double-beam theory is utilised. Governing equations for nonlocal transverse vibration of the DWCNT under a longitudinal magnetic field are derived considering the Lorentz magnetic force obtained from Maxwell's relation. Numerical results from the model show that the longitudinal magnetic field increases the natural frequencies of the DWCNT. Both synchronous and asynchronous vibration phases of the tubes are studied in detail. Synchronous vibration phases of DWCNT are more affected by nonlocal effects than asynchronous vibration phases. The effects of a longitudinal magnetic field on higher natural frequencies are also presented. Vibration response of DWCNT with outer-wall stationary and single-walled carbon nanotube under the effect of longitudinal magnetic field are also discussed in the paper.     

Nanotube bundle oscillators: Carbon and boron nitride nanostructures

In this paper, we investigate the oscillation of a fullerene that is moving within the centre of a bundle of nanotubes. In particular, certain fullerene–nanotube bundle oscillators, namely C₆₀-carbon nanotube bundle, C₆₀-boron nitride nanotube bundle, B₃₆N₃₆-carbon nanotube bundle and B₃₆N₃₆-boron nitride nanotube bundle are studied using the Lennard–Jones potential and the continuum approach which assumes a uniform distribution of atoms on the surface of each molecule. We address issues regarding the maximal suction energies of the fullerenes which lead to the generation of the maximum oscillation frequency. Since bundles are also found to comprise double-walled nanotubes, this paper also examines the oscillation of a fullerene inside a double-walled nanotube bundle. Our results show that the frequencies obtained for the oscillation within double-walled nanotube bundles are slightly higher compared to those of single-walled nanotube bundle oscillators. Our primary purpose here is to extend a number of established results for carbon to the boron nitride nanostructures.     

Nonlinear resonances of a single-wall carbon nanotube cantilever

The dynamics of an electrostatically actuated carbon nanotube (CNT) cantilever are discussed by theoretical and numerical approaches. Electrostatic and intermolecular forces between the single-walled CNT and a graphene electrode are considered. The CNT cantilever is analyzed by the Euler–Bernoulli beam theory, including its geometric and inertial nonlinearities, and a one-mode projection based on the Galerkin approximation and numerical integration. Static pull-in and pull-out behaviors are adequately represented by an asymmetric two-well potential with the total potential energy consisting of the CNT elastic energy, electrostatic energy, and the Lennard-Jones potential energy. Nonlinear dynamics of the cantilever are simulated under DC and AC voltage excitations and examined in the frequency and time domains. Under AC-only excitation, a superharmonic resonance of order 2 occurs near half of the primary frequency. Under both DC and AC loads, the cantilever exhibits linear and nonlinear primary and secondary resonances depending on the strength of the excitation voltages. In addition, the cantilever has dynamic instabilities such as periodic or chaotic tapping motions, with a variation of excitation frequency at the resonance branches. High electrostatic excitation leads to complex nonlinear responses such as softening, multiple stability changes at saddle nodes, or period-doubling bifurcation points in the primary and secondary resonance branches.     

碳纳米管/聚苯胺纳米复合管的制备及其微波介电特性研究

用原位乳液聚合法在碳纳米管表面包覆聚苯胺，制备出了碳纳米管/聚苯胺一维纳米复合管.复合管的直径为50—80 nm，聚苯胺包覆层的厚度为20—30 nm，聚苯胺在碳纳米管表面以层状和枝晶状两种形态生长.研究了碳纳米管/聚苯胺复合管在2—18 GHz的微波介电特性.与纯碳纳米管相比，碳纳米管/聚苯胺复合管的介电常数的实部ε′和虚部ε″在2—18 GHz随频率变化较小，在低频波段介电常数值较小，作为微波吸收剂容易实现与自由空间的阻抗匹配，而且它的介电损耗角正切(tanδ=ε″/ε′)较高，是一种很好的微波吸收剂. 关键词： 碳纳米管 聚苯胺 微波介电特性 微波吸收剂     

Structure of graphene nanotube hybrid materials produced via single-stage CVD

The structures of graphene layer-carbon nanotube hybrid films produced via CVD with a single-stage flow of acetylene into a chamber containing a prepared substrate are studied. It is shown that such films have a hybrid double-layer structure consisting of a graphene layer and a dense continuous network of nanotubes. The graphene layer contains continuous extended areas 10–50 μm in size and island areas ～0.1 μm in size. TEM images lead to the conclusion that the graphene layer and carbon nanotubes are bound by covalent bonds.     

Preparation of large-area double-walled carbon nanotube films and application as film heater

Large-area (larger than 30×30 cm²) double-walled carbon nanotube (DWCNT) films are prepared and application as a heating element for film heaters is demonstrated. A high heating efficiency is observed. Measurements indicate that the use of the DWCNT film heater would save energy consumption up to 20–30% when compared with a commercial film-like metal-based heater. Morphological analysis reveals that the special surface structure, appropriate electric and high thermal conductivities of the film formed by the network of entangled nanotube bundles may lead to the high heating performance. Considering large-area, shape flexibility, negligible weight and easy manipulation, the film exhibits promising potential applications as a film heater for thermal control in aircrafts, medical equipment, home appliances and other industrial fields at low temperature (below 400 °C).     

Transforming carbon nanotubes by silylation: an ab initio study

We use ab initio density functional calculations to study the chemical functionalization of single-wall carbon nanotubes and graphene monolayers by silyl (SiH(3)) radicals and hydrogen. We find that silyl radicals form strong covalent bonds with graphene and nanotube walls, causing local structural relaxations that enhance the s p(3) character of these graphitic nanostructures. Silylation transforms all carbon nanotubes into semiconductors, independent of their chirality. Calculated vibrational spectra suggest that specific frequency shifts can be used as a signature of successful silylation.     

Molecular dynamics investigation into the electric charge effect on the operation of ion-based carbon nanotube oscillators

The fabrication of nanoscale oscillators working in the gigahertz (GHz) range and beyond has now become the focal center of interest to many researchers. Motivated by this issue, this paper proposes a new type of nano-oscillators with enhanced operating frequency in which both the inner core and outer shell are electrically charged. To this end, molecular dynamics (MD) simulations are performed to investigate the mechanical oscillatory behavior of ions, and in particular chloride ion, tunneling through electrically charged carbon nanotubes (CNTs). It is assumed that the electric charges with similar sign and magnitude are evenly distributed on two ends of nanotube. The interatomic interactions between carbon atoms and van der Waals (vdW) interactions between ion and nanotube are respectively modeled by Tersoff-Brenner and Lennard-Jones (LJ) potential functions, whereas the electrostatic interactions between ion and electric charges are modeled by Coulomb potential function. A comprehensive study is conducted to get an insight into the effects of different parameters such as sign and magnitude of electric charges, nanotube radius, nanotube length and initial conditions (initial separation distance and velocity) on the oscillatory behavior of chloride ion-charged CNT oscillators. It is shown that, the chloride ion frequency inside negatively charged CNTs is lower than that inside positively charged ones with the same magnitude of electric charge, while it is higher than that inside uncharged CNTs. It is further observed that, higher frequencies are generated at higher magnitudes of electric charges distributed on the nanotube.     

The importance of solvent properties for optical limiting of carbon nanotube dispersions

The optical limiting performances of single-walled carbon nanotube dispersions in N-methyl-2-pyrrolidone were enhanced significantly by blending a range of organic solvents or by increasing the temperature of the dispersions up to 100 °C. While both nanotube bundle size and various solvent parameters have an influence on the optical limiting responses, we verified experimentally that the surface tension of the solvent plays a more important role than the viscosity or boiling point; the appropriate solvent properties contribute to the nonlinear scattering dominated optical limiting phenomenon more than the bundle size.