An efficient numerical model for vibration analysis of single-walled carbon nanotubes

This paper presents a linear spring-based element formulation for computation of vibrational characteristics of single-walled carbon nanotubes (CNTs). Three-dimensional nanoscale elements and corresponding elemental equations are developed for the numerical treatment of the dynamic behaviour of single-walled CNTs, including appropriate stiffness and mass characteristics. The atomistic microstructure of nanotubes is used to assemble the elemental equations and construct the dynamic equilibrium equation. The developed elements simulate the relative translations and rotations between atoms as well as the mass of the atoms. In this way, molecular mechanics theory can be applied directly because the atomic bonds are modelled by using exclusively physical variables such as bond stretching. The modelling is regenerative and can provide simulations for different geometric characteristics of the nanotubes. Numerical results are presented that illustrates new natural frequencies and mode shapes, going beyond the usual ones for various nanotubes under different support conditions and defects. Comparisons with corresponding numerical predictions from the literature, where they are possible, show very good agreement.     

Nonlocal anisotropic elastic shell model for vibrations of single-walled carbon nanotubes with arbitrary chirality

A nonlocal anisotropic elastic shell model is developed to study the effect of small scale on shell-like vibration of single-walled carbon nanotubes (SWCNTs) with arbitrary chirality. Anisotropic elastic shell model is reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion are derived and analytical solution for the vibration of anisotropic SWCNTs is presented by using the Flügge shell theory and complex method. The suggested model is justified by a good agreement between the results given by the present model and available data in literature. Furthermore, the model is used to elucidate the effect of small scale on the vibration of zigzag, armchair and chiral SWCNTs. Our results show that small scale is essential for vibration of SWCNTs when the axial wave-length is not extremely long. Moreover, the results show that local model substantially overestimates vibrational frequencies of almost all modes.     

Thermal vibration of double-walled carbon nanotubes predicted via double-Euler-beam model and molecular dynamics

The paper presents how to study the thermal vibration of a double-walled carbon nanotube (DWCNT) by using a model of double-Euler beams, together with the law of energy equipartition, with the energy of van der Waals interaction between layers taken into consideration. The basic finding of the study is the relation, derived via the model of double-Euler beams and the law of energy equipartition, between the temperature and the root-of-mean-squared (RMS) amplitude of the thermal vibration at any cross section of the DWCNT. The molecular dynamics simulations of thermal vibration of the DWCNT in argon atmosphere show that the model of double-Euler beams can predict the RMS amplitude of the thermal vibration of the DWCNT reasonably well.     

Self-assembly of single-walled carbon nanotubes into multiwalled carbon nanotubes in water: molecular dynamics simulations

We report discoveries from a series of molecular dynamics simulations that single-walled carbon nanotubes, with different diameters, lengths, and chiralities, can coaxially self-assemble into multiwalled carbon nanotubes in water via spontaneous insertion of smaller tubes into larger ones. The assembly process is tube-size-dependent, and the driving force is primarily the intertube van der Waals interactions. The simulations also suggest that a multiwalled carbon nanotube may be separated into single-walled carbon nanotubes under appropriate solvent conditions. This study suggests possible bottom-up self-assembly routes for the fabrication of novel nanodevices and systems.     

Analytical and numerical modeling of higher order free vibration characteristics of single-walled carbon nanotubes

In this article, analytical and numerical investigations of small, higher, and asymptotic order vibration eigenmodes and natural eigenfrequencies of single-walled carbon nanotubes (CNT) are elaborated. The nonlocal elasticity theory is used and the numerical simulation is based on the differential quadrature method. Due to numerical instability, the common analytical forms of eigenmodes can be only used for the first 12 modes or so. New mathematical models for higher eigenmodes and associated eigenfrequencies of CNT are developed. The obtained eigenmodes are well conditioned and numerically stable and may be used as modal bases at any required frequency range.     

Generation of single-walled carbon nanotubes from alcohol and generation mechanism by molecular dynamics simulations

Recent advances in high-purity and high-yield catalytic chemical vapor deposition (CVD) generation of single-walled carbon nanotubes (SWNTs) from alcohol are comprehensively presented and discussed on the basis of results obtained from both experimental and numerical investigations. We have uniquely adopted alcohol as a carbon feedstock, and this has resulted in high-quality, low-temperature synthesis of SWNTs. This technique can produce SWNTs even at a very low temperature of 550 degrees C, which is about 300 degrees C lower than the conventional CVD methods in which methane or acetylene is typically used. We demonstrate the excellence of the proposed alcohol catalytic CVD method for high-yield production of SWNTs when Fe-Co on USY-zeolite powder was used as a catalyst. At optimum CVD conditions, a SWNT yield of more than 40 wt % was achieved over the weight of the catalytic powder within the reaction time of 120 min. In addition to the advantages for mass production, this method is also suitable for the direct synthesis of high-quality SWNTs on Si and quartz substrates when combined with the newly developed liquid-based "dip-coat" technique to mount catalytic metals on the surface of substrates. This method allows easy and costless loading of catalytic metals without the need for any support or underlayer materials that were usually required in previous studies for the generation of a sufficient quantity of SWNTs on an Si surface. Finally, the result of molecular dynamics simulation for the SWNT growth process is presented to obtain a fundamental insight into the initial growth mechanism on the catalytic particles.     

Nonlinear free vibration of non-prismatic single-walled carbon nanotubes by a non-local shear deformable beam p-element

The nonlinear free vibration of non-prismatic single-walled carbon nanotubes (SWNTs) is studied using a new non-local shear deformable beam p-element. The effects of the internal length scale parameter, transverse shear deformation, rotary inertia, and geometrical nonlinearity are taken into account. The principle of virtual displacements and the harmonic balance method are used to derive the nonlinear equations of motion, which are solved iteratively by the linearized updated mode method to obtain the fundamental nonlinear frequencies and mode shapes of H–H, C–H, and C–C SWNTs with uniform, linear, and quadratic radius variation. The convergence and accuracy of the non-local shear deformable beam p-element are demonstrated through comparison with other methods. It is shown that the non-uniformity parameters influence significantly the backbone curves and mode shapes of non-prismatic SWNTs.     

An elastic shell model for characterizing single-walled carbon nanotubes

This paper proposes a two-dimensional elastic shell model to characterize the deformation of single-walled carbon nanotubes using the in-plane rigidity, Poisson ratio, bending rigidity and off-plane torsion rigidity as independent elastic constants. It was found that the off-plane torsion rigidity of a single-walled carbon nanotube is not zero due to the off-plane change in the π-orbital electron density on both sides of the nanotube. It was concluded that a three-dimensional elastic shell model of single-walled carbon nanotubes can be established with well-defined effective thickness.     

Buckling of defective single-walled and double-walled carbon nanotubes under axial compression by molecular dynamics simulation

Buckling of defective single-walled and double-walled carbon nanotubes (SWCNTs and DWCNTs, respectively) due to axial compressive loads has been studied by molecular dynamics simulations, and results compared with those of the perfect structures. It is found that single vacancy defect greatly weakens the carrying capacity of SWCNTs and DWCNTs, though it does slight harm to the effective elastic modulus of the tubes. The influence of defects on the buckling properties of nanotubes is related to the density of the defects, and the relative position of defects also plays an important role in buckling of DWCNTs. The van der Waals force among atoms in the inner and the outer tubes of short defective DWCNTs makes the critical buckling strain of DWCNTs greater than that of the inner tube.     

The bending-kinking analysis of a single-walled carbon nanotube—a combined molecular dynamics and continuum mechanics technique

This paper investigates the deformation mechanism of a single-walled carbon nanotube in pure bending. The molecular dynamics analysis and continuum mechanics characterisation were used together to achieve a deeper understanding. It was found that at a bending angle of 24° the nanotube buckles locally, forming a kink in the middle of the nanotube. As the bending angle increases, the kink progresses along the nanotube and varies its shape in both longitudinal and circumferential directions. The kink formation can be considered as the result of rotations of planes/surfaces about the moving and stationary hinge lines. It was also found that the kink deformation influences the load bearing capacity of the nanotube.     

Numerical algorithms for prediction of mechanical properties of single-walled carbon nanotubes based on molecular mechanics model

The objective of this paper is to develop the numerical algorithms for the prediction of mechanical properties of single-walled carbon nanotubes (SWCNTs). By using the energy method, the analytical expressions are obtained and the five independent variables algorithm is developed for the prediction of the elastic properties of SWCNTs via a molecular mechanics model in which the geometrical relationship of carbon nanotube is introduced. It can be found that due to the introduction of the geometrical approximate conditions some errors may exist in the calculation of mechanical properties of SWCNTs in terms of the five independent variables algorithm. Therefore, two improved algorithms, i.e., eigenvalues modified method (EMM) and eigenvalues and eigenvectors modified method (EEMM) are proposed to analyze the possible errors in the numerical results. It is found that the results obtained by the three kinds of algorithms are almost consistent with one another, but EMM and EEMM are preferred to be used because they have properties similar to those of the finite element method, where the consistent equation works just as the constraint condition to void the singularity of the element stiffness matrix. The computational results also reveal that both the surface Young’s modulus and Poisson’s ratio depend on the diameter of carbon nanotubes, and finally converge to the values of the graphite sheet with an increase in the tube diameter in the inverse trends. For SWCNTs with approximately the same diameters, the surface Young’s modulus is in direct and Poisson’s ratio is in inverse proportion to chiral angles, respectively.     

Nonlocal and shear effects on column buckling of single-layered membranes from stocky single-walled carbon nanotubes

Axial buckling behavior of single-layered membranes from vertically aligned single-walled carbon nanotubes is studied in the context of the nonlocal continuum theory of Eringen. To this end, useful discrete models based on the nonlocal Rayleigh, Timoshenko, and higher-order beam theories are developed to evaluate critical buckling loads associated with both in-plane and out-of-plane buckling modes. In discrete models, the size of the eigenvalue equations to be solved drastically magnifies for highly populated membranes. Thereby, development of models whose computational efforts do not affected by the population of the membrane is of great advantageous. To bridge this scientific gap, appropriate nonlocal continuous models are established based on the developed discrete models. The accuracy of the proposed discrete and continuous models is checked and remarkable results are achieved. Subsequently, the roles of the influential factors on both in-plane and out-of-plane axial buckling loads are addressed. The obtained results can be regarded as a basic step in examining of axial buckling mechanisms of more complex systems consist of multi-layered membranes from parallel or even orthogonal single-walled carbon nanotubes.     

Cutting single-walled carbon nanotubes

A two-step process is utilized for cutting single-walled carbon nanotubes (SWNTs). The first step requires the breakage of carbon-carbon bonds in the lattice while the second step is aimed at etching at these damage sites to create short, cut nanotubes. To achieve monodisperse lengths from any cutting strategy requires control of both steps. Room-temperature piranha and ammonium persulfate solutions have shown the ability to exploit the damage sites and etch SWNTs in a controlled manner. Despite the aggressive nature of these oxidizing solutions, the etch rate for SWNTs is relatively slow and almost no new sidewall damage is introduced. Carbon-carbon bond breakage can be introduced through fluorination to ～C(2)F, and subsequent etching using piranha solutions has been shown to be very effective in cutting nanotubes. The final average length of the nanotubes is approximately?100?nm with carbon yields as high as 70-80%.     

单壁碳纳米管振动特性有限元分析

利用有限元方法对悬臂梁式单壁碳纳米管固有频率和手性角、长度、直径等几何参数之间的关系进行研究.C-C共价键作用通过等效梁单元来模拟,碳原子等效为质点,建立碳纳米管分子结构力学有限元模型,并引入了质量比例因子β来消除数值误差.数值结果表明,手性角对单壁碳纳米管的固有频率影响很小;固有频率随长度的增加而减小,但达到一定长度之后基频随长度的变化不再明显;前4阶固有频率随直径增大而增大,第5阶固有频率不随直径变化,而高阶固有频率随直径的变化不规律;在低阶模态时长度对固有频率的影响比直径的影响更加显著.     

Formylation of single-walled carbon nanotubes

We report an improved, elegant method for the covalent formylation of single-wall carbon nanotubes (SWNTs) via formyl transfer from N-formylpiperidine, which could potentially open the gateway for more versatile chemical modification of carbon nanotube (CNT) walls than is possible via other reported functionalisation methods. The formylation reaction does not inflict damage upon the pristine CNT structure, unlike the currently commonly used carboxylation route, and involves much fewer steps, and takes considerably less time, than most other reported routes. The modified SWNTs have been characterised by Raman spectroscopy, ultraviolet-visible-near infrared (UV-vis-NIR) spectroscopy and "covalent tagging" with derivatising groups followed by thermogravimetric analysis-mass spectroscopy (TGA-MS). UV-vis-NIR spectroscopy shows that there is only limited disruption of the intrinsic electronic structure of the SWNTs. This is confirmed from estimates of the extent of functionalisation from TGA-MS, which suggest that it may be as low as 2 atomic per cent.     

Nonlocal beam model for nonlinear analysis of carbon nanotubes on elastomeric substrates

Postbuckling, nonlinear bending and nonlinear vibration analyses are presented for single-wall carbon nanotubes (SWCNTs) resting on a two-parameter elastic foundation in thermal environments. The SWCNT is modeled as a nonlocal nanobeam which contains small scale effects. The elastomeric substrate with finite depth is modeled as a two-parameter elastic foundation. The thermal effects are included and the material properties of both SWCNTs and the substrate are assumed to be temperature-dependent. The governing equation that includes beam–foundation interaction is solved by a two-step perturbation technique. The numerical results reveal that the small scale parameter e₀a reduces the postbuckling equilibrium paths, the static large deflections and natural frequencies of SWCNTs resting on an elastic foundation. The results also reveal that the effect of the small scale parameter is significant for compressive buckling, but less pronounced for static bending and marginal for free vibration of SWCNTs resting on an elastic foundation.     

Investigating the effect of chirality on structural parameters of chiral single-walled carbon nanotubes by molecular dynamics simulation

The structural parameters of thin single-walled carbon nanotubes (SWCNTs) vs. chiral angle were investigated using molecular dynamics (MD) simulation. A comparison was made between nanotube radius obtained from MD simulation and that obtained from ideal rolling graphene model. Brenner empirical bond order potential was used to describe the interaction between carbon atoms. SWCNTs (n, m) with n + m = 6, 8, 10 and 12 were considered. It was observed that chiral nanotubes have three unequal bond lengths and three unequal bond angles, while for armchair and zigzag SWCNTs there are two unequal parameters.     

Purification of single-walled carbon nanotubes

The discovery of carbon nanotubes (CNTs) created much excitement and stimulated extensive research into the properties of nanometer-scale cylindrical networks. From then on, various methods for the synthesis and characterization of aligned CNTs-both single-walled (SWCNTs) and multi-walled (MWCNTs) by different methods have been hotly pursued. Unfortunately, most methods currently in use produce raw multi component solid products, only a small fraction of which contains carbon nanotubes. The balance of the material is composed of residual catalyst particles (some of which are encased in concentric graphitic shells), fullerenes, other graphitic materials and amorphous carbon. These impurities cause a serious impediment for their detailed characterization and applications. If the carbon nanotube is ever to fulfill its promise as an engineering material, large, high quality aliquots will be required. A number of purification methods involving elimination processes such as physical separation, gas phase and liquid phase oxidation in combination with chemical treatments have been developed for nanotube materials. Though the quantitative determination of purity remains controversial, reported yields are best regarded with an appropriate level of skepticism on the method of assay. In this article, a review is given on the past and recent advances in purification of SWCNTs.     

Co-synthesis of single-walled carbon nanotubes and carbon fibers

A new vertical floating catalytic technique is developed and used to prepare both single-walled carbon nanotubes (SWNTs) and carbon fibers (CFs). Scanning electron microscopy (SEM) observation shows a clear separation of these two materials. Thin films of SWNTs can be peeled easily from the CF substrate which just acts as a catalyst support for the SWNT growth. The production process is also semicontinuous, resulting in a yield of ∼1.0 g h⁻¹ of SWNTs film with high purity. Structure and vibrational properties of these materials are investigated by electron microscopy and Raman spectroscopy, respectively.     

Nonlocal effect on the free vibration of short nanotubes embedded in an elastic medium

In this paper, the small size effect on the free vibration behavior of finite length nanotubes embedded in an elastic medium is investigated. The problem is formulated based on the three-dimensional (3D) nonlocal elasticity theory. Since the 3D nonlocal constitutive relations in a cylindrical coordinate system are used, in addition to displacement components, the stress tensor components are chosen as degrees of freedom. The surrounding elastic medium is modeled as the Winkler’s elastic foundation. The differential quadrature method as an efficient and accurate numerical tool in conjunction with the series solution is used to discretize the governing equations. Very fast rate of convergence of the method is demonstrated. The effects of the nonlocal parameter together with the other geometrical parameters and also the stiffness parameter of the elastic medium on the natural frequencies are studied.