Free vibration characteristics of double-walled carbon nanotubes embedded in an elastic medium

In this Letter, a theoretical analysis of the resonant vibration of double-walled carbon nanotubes (DWCNTs) and the DWCNTs embedded in an elastic medium is presented based on Euler-Bernoulli beam model and Winkler spring model. The vibration modes of DWCNTs are quite different from those of single-walled carbon nanotubes (SWCNTs). The resonant vibrations of DWCNTs are found to have in-phase and anti-phase modes, in which the deflections of the inner and outer nanotubes occur in the same and opposite directions, respectively. For the vibration of DWCNTs with the same harmonic numbers, the resonant frequencies of anti-phase mode are larger than the ones of in-phase mode. Moreover, influence of the surrounding medium on the resonant vibrations is investigated using the Winkler spring model. The results show that surrounding medium makes a strong impact on the vibration frequencies of in-phase mode, but little on those of anti-phase mode.     

Nonlinear size-dependent longitudinal vibration of carbon nanotubes embedded in an elastic medium

In this paper, we study the longitudinal linear and nonlinear free vibration response of a single walled carbon nanotube (CNT) embedded in an elastic medium subjected to different boundary conditions. This formulation is based on a large deformation analysis in which the linear and nonlinear von Kármán strains and their gradient are included in the expression of the strain energy and the velocity and its gradient are taken into account in the expression of the kinetic energy. Therefore, static and kinetic length scales associated with both energies are introduced to model size effects. The governing motion equation along with the boundary conditions are derived using Hamilton's principle. Closed-form solutions for the linear free vibration problem of the embedded CNT rod are first obtained. Then, the nonlinear free vibration response is investigated for various values of length scales using the method of multiple scales.     

Nonlocal shear deformable shell model for bending buckling of microtubules embedded in an elastic medium

A nonlocal shear deformable shell model is developed for buckling of microtubules embedded in an elastic matrix of cytoplasm under bending in thermal environments. The results reveal that the lateral constraint has a significant effect on the buckling moments of a microtubule when the foundation stiffness is sufficiently large.     

Dynamic column buckling of multi-walled carbon nanotubes under axial impact load

This paper studies the dynamic column buckling of multi-walled carbon nanotubes (MWNTs) under axial impact load. The analysis is based on the continuum mechanics model and a simplified model for the van der Waals forces between adjacent layers. By introducing initial imperfections for MWNTs and applying the method of preferred mode, a buckling condition is derived for the buckling load and associated buckling mode. In particular, explicit expressions are obtained for double-walled carbon nanotubes (DWNTs). Finally, numerical calculations are worked out for a DWNT and a five-layer MWNT with different length-to-radius ratios.     

Vibration and buckling analyses of nanobeams embedded in an elastic medium

Boundary characteristic orthogonal polynomials are used as shape functions in the Rayleigh–Ritz method to investigate vibration and buckling of nanobeams embedded in an elastic medium. The present formulation is based on the nonlocal Euler–Bernoulli beam theory. The eigen value equation is developed for the buckling and vibration analyses. The orthogonal property of these polynomials makes the computation easier with less computational effort. It is observed that the frequency and critical buckling load parameters are dependent on the temperature, elastic medium, small scale coefficient,and length-to-diameter ratio. These observations are useful in the mechanical design of devices that use carbon nanotubes.     

Effect of small length scale on elastic buckling of multi-walled carbon nanotubes under radial pressure

A nonlocal multiple-shell model is developed for the elastic buckling of multi-walled carbon nanotubes under uniform external radial pressure on the basis of theory of nonlocal elasticity. The effect of small length scale is incorporated in the formulation. An explicit expression is derived for the critical buckling pressure for a double-walled carbon nanotube. The influence of the small length scale on the buckling pressure is examined. It is concluded that the critical buckling pressure for a carbon nanotube could be overestimated by the classic (local) shell model due to ignoring the effect of small length scale.     

Radial buckling of multi-walled carbon nanotubes under hydrostatic pressure

Radial buckling stresses of carbon nanotubes (CNTs) need to be studied in high-pressure resonance Raman scattering spectrum. In this work, the closed-form expression of the critical buckling stress of multi-walled carbon nanotubes (MWCNTs) under hydrostatic pressure is derived that can be conveniently employed. Using the derived formulae, the critical buckling stresses of single-walled carbon nanotubes and double-walled carbon nanotubes with different diameters are calculated. The results are in good agreement with other reported literatures. In addition, the critical buckling stresses of each layer of a quintuple-walled CNT in different buckling modes are predicted, showing that the buckling instability can occur not only in the outermost rolled layer, but also in other rolled layer of MWCNTs by considering different diameters and buckling modes.     

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.     

Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory

Variational principles are derived for multi-walled carbon nanotubes undergoing buckling using the semi-inverse method. Derivations are based on the continuum modelling of nanotubes taking into account small scale effects via the nonlocal theory of elasticity. Natural and geometric boundary conditions for multi-walled nanotubes are derived which leads to a set of coupled boundary conditions.     

Transverse elasticity of multi-walled carbon nanotubes

A discrete shell model is proposed to describe the radial deformation of carbon nanotubes under a hydrostatic pressure and the radial Young's modulus of (single- or multi-walled) nanotubes is obtained. It is found that the radial modulus decreases with increasing tube diameter while increases with increasing number of layers. The computational results agree well with the previous results of SWNTs and indicate that the radial modulus of carbon nanotubes is independent of the Poisson's ratio.     

Effect of oxyfluorination on gas sensing behavior of polyaniline-coated multi-walled carbon nanotubes

Polyaniline-coated multi-walled carbon nanotubes were prepared by in situ chemical polymerization method for the novel sensing materials of ammonia gas. The thickness of the polyaniline coatings was controlled by the oxyfluorination treatment on the multi-walled carbon nanotubes. The oxyfluorination with higher oxygen content produced the more hydrophilic functional groups on the surface of multi-walled carbon nanotubes. Both the resistivity change and the response time were significantly improved with high repeatability using the more hydrophilic multi-walled carbon nanotubes which were modified with oxyfluorination.     

Dynamic behavior of triple-walled carbon nanotubes conveying fluid

In this study, the instability of triple-walled carbon nanotubes (TWCNTs) conveying fluid is studied based on an Euler–Bernoulli beam model. The van der Waals (vdW) interactions between different carbon nanotubes (CNTs) are taken into account in the analysis, and the Galerkin discretization approach is used to solve the coupled equations of the motions. Numerical simulations show that the interlayer vdW interactions play a significant role in the natural frequencies and the stability of TWCNTs. The critical flow velocities—associated with divergence, restabilization and flutter—are determined. The effects of different inner radius and the value of mode N used in Galerkin discretization on the dynamical behaviors of the fluid-conveyed TWCNTs are also examined in detail. Results reveal that the internal moving fluid plays an important role in the instability of TWCNTs.     

In-plane vibration analysis of curved carbon nanotubes conveying fluid embedded in viscoelastic medium

The effect of the induced vibrations in the carbon nanotubes (CNTs) arising from the internal fluid flow is a critical issue in the design of CNT-based fluidic devices. In this study, in-plane vibration analysis of curved CNTs conveying fluid embedded in viscoelastic medium is investigated. The CNT is modeled as a linear elastic cylindrical tube where the internal moving fluid is characterized by steady flow velocity and mass density of fluid. A modified-inextensible theory is used in formulation and the steady-state initial forces due to the centrifugal and pressure forces of the internal fluid are also taken into account. The finite element method is used to discretize the equation of motion and the frequencies are obtained by solving a quadratic eigenvalue problem. The effects of CNT opening angle, the elastic modulus and the damping factor of the viscoelastic surrounded medium and fluid velocity on the resonance frequencies are elucidated. It is shown that curved CNTs are unconditionally stable even for a system with sufficiently high flow velocity. The most results presented in this investigation have been absent from the literature for fluid-induced vibration of curved CNTs embedded in viscoelastic foundations.     

On the buckling behavior of connected carbon nanotubes with parallel longitudinal axes

The application of hetero-junction carbon nanotubes (CNTs) is increasing continuously due to their outstanding properties in nano-mechanical systems. Several investigations have been conducted to study the behavior of CNTs. In this paper, straight hetero-junctions and their constituent CNTs (armchair and zigzag) were simulated by a commercial finite element package. Then, the buckling behavior of CNTs was evaluated by comparing the critical buckling load of each straight hetero-junction and its constituent CNTs. Both obtained, i.e. analytical calculations and computational, results were compared. The investigations showed that, first, the behavior of homogeneous CNTs under cantilevered boundary conditions follows the assumption of the classical Euler equation. Second, the analytical solutions are in good agreement with the finite element simulation results. In addition, it was shown that the first critical buckling load of hetero-junctions lies within the value of the fundamental homogeneous CNT range. It was also concluded that the buckling load of straight hetero-junctions and their fundamental CNTs increases by increasing the chiral number of both armchair and zigzag CNTs. The current study provides a better insight towards the prediction of straight hetero-junction CNTs behavior.     

Surface characteristics of Ni catalystic films on growth behavior of multi-walled carbon nanotubes

The effect of the surface characteristics of Ni catalyst films on the growth behavior of multi-walled carbon nanotubes (MWCNTs) were investigated using Ni catalyst films prepared by different physical vapor deposition methods, electron-beam evaporation and sputtering. The growth behavior of MWCNTs was dependent upon the surface roughness of the Ni films. After a pretreatment process with NH₃, the root mean squares of surface roughness of e-beam evaporated and sputtered Ni catalyst films increased to 16.6 and 3.2 nm, respectively. Curled-MWCNTs and carbon-encapsulated Ni nanoparticles were formed on the Ni film deposited by e-beam evaporation while vertically aligned-MWCNTs were grown on the sputter-deposited film. In addition, the surface roughness of the Ni films affected the field emission properties of the MWCNTs. This was considered to originate from the specific growth behavior of the MWCNTs which was primarily caused by the initial surface roughness of the Ni films.     

Dispersion of multi-walled carbon nanotubes in biocompatible dispersants

Owing to their phenomenal electrical and mechanical properties, carbon nanotubes (CNT) have been an area of intense research since their discovery in 1991. Different applications for these nanoparticles have been proposed, among others, in electronics and optics but also in the medical field. In parallel, emerging studies have suggested potential toxic effects of CNT while others did not, generating some conflicting outcomes. These discrepancies could be, in part, due to different suspension approaches used and to the agglomeration state of CNT in solution. In this study, we described a standardized protocol to obtain stable CNT suspensions, using two biocompatible dispersants (Pluronic F108 and hydroxypropylcellulose) and to estimate the concentration of CNT in solution. CNT appear to be greatly individualized in these two dispersants with no detection of remaining bundles or agglomerates after sonication and centrifugation. Moreover, CNT remained perfectly dispersed when added to culture medium used for in vitro cell experiments. We also showed that Pluronic F108 is a better dispersant than hydroxypropylcellulose. In conclusion, we have developed a standardized protocol using biocompatible surfactants to obtain reproducible and stable multi-walled carbon nanotubes suspensions which can be used for in vitro or in vivo toxicological studies.     

Solitary waves in an inhomogeneous cylindrical shell interacting with an elastic medium

A new generalized sixth-order nonintegrable equation is derived to model axisymmetric longitudinal wave propagation in an inhomogeneous cylindrical shell interacting with a nonlinear elastic medium. Exact soliton-like solutions to this equation are constructed with allowance for geometric and physical nonlinearities, both individually and in combination.     

Guided waves characteristics of multi-walled carbon nanotubes

The theoretical analysis of propagation of guided waves in the multi-walled carbon nanotubes is presented within the framework of the classical electrodynamics. Electronic excitations of each wall of the system are modeled as an infinitesimally thin cylindrical layer of the π-electrons, whose dynamics are described by means of the fluid theory. General expressions of dispersion relations are obtained for the electromagnetic wave with the transverse magnetic and transverse electric modes, respectively, by solving Maxwell and fluid equations with appropriate boundary conditions.     

XPS study of fluorinated carbon multi-walled nanotubes

Arc-produced carbon multi-walled nanotubes (MWNTs) were fluorinated at 420 °C in a flow of diluted F₂ gas containing small admixture of HF gas. Fluorinated materials (F-MWNTs) with 10–55 wt.% fluorine content were studied by XPS. It was shown that fluorination begins at the external layers of nanotubes and the reaction front propagates inside the multi-layer particles in concert with structural deterioration of graphene layers. The C₂F stoichiometry still allows MWNT wall integrity, similar to known for SWNTs. The fluorine contents in the product can noticeably exceed this higher fluorine limit for tube stability. The position of the F 1s line at 688.2 eV does not depend on the fluorine concentration. Nearly covalent C–F bonds dominate the F-MWNT samples, with a small quantity (2–9%) of ionic bonds also present. Fluorinated carbon tends to spatially separate from non-fluorinated carbon.