Radial breathing mode frequencies of carbon nanotubes for determination of their diameters

In this paper, exact formulas are obtained for the radial breathing mode (RBM) frequencies of triple-walled carbon nanotubes (TWCNTs) using symbolic package in MAPLE software. For this purpose, TWCNT is considered as triple concentric elastic thin cylindrical shells, which are coupled through van der Waals (vdW) forces between two adjacent tubes. Lennard–Jones potential is used to calculate the vdW forces between adjacent tubes. Then, explicit formulas for RBM frequencies of single-walled (SW), and double-walled (DW) CNTs have been deduced from TWCNT formulas that show an excellent agreement with the available experimental results and the other theoretical model results. The advantage of this analytical approach is that the elastic shell model considers all degrees of freedom in the vibrational analysis of CNTs. To demonstrate the accuracy of this work, the RBM frequencies of different multi-walled carbon nanotubes (MWCNTs) are compared with the available experimental or atomistic results with relative errors of less than 1.5%. To illustrate the application of this approach, the diameters of DWCNTs are obtained from their RBM frequencies which show an excellent agreement with the available experimental results. Also, this approach can be used to determine the diameters of the TWCNTs and MWCNTs. The influence of changing the geometrical and mechanical parameters of a TWCNT on its RBM frequencies has been investigated, too.     

多壁碳纳米管储氢的物理吸附特性

采用巨正则蒙特卡罗方法 ,模拟常温、1 0MPa下氢在扶手椅型多壁壁碳纳米管中的物理吸附过程 .氢分子之间、氢分子与碳原子之间的相互作用采用Lennard Jones势能模型 .研究了双壁碳纳米管外 (内 )径固定而内 (外 )径改变时的物理吸附储氢情况 ,发现氢分子主要储存在双壁碳纳米管的管壁附近 ,当双壁碳纳米管的内外管壁间距由 0 .34nm增大到 0 .6 1或 0 .88nm时可有效增加物理吸附储氢量 ,并给出了相应的理论解释 .在此基础上 ,计算了管壁间距为 0 .34、0 .6 1和 0 .88nm时的三壁碳纳米管的物理吸附储氢量 ,并与相同条件下单壁和双壁碳纳米管的物理吸附储氢量作了比较 ,发现多壁碳纳米管的物理吸附储氢量随碳管层数的增加而减小 .     

Nonlinear dynamics of bi-layered graphene sheet,double-walled carbon nanotube and nanotube bundle

Due to strong van der Waals (vdW) interactions, the graphene sheets and nanotubes stick to each other and form clusters of these corresponding nanostructures, viz. bi-layered graphene sheet (BLGS), double-walled carbon nanotube (DWCNT) and nanotube bundle (NB) or ropes. This research work is concerned with the study of nonlinear dynamics of BLGS, DWCNT and NB due to nonlinear interlayer vdW forces using multiscale atomistic finite element method. The energy between two adjacent carbon atoms is represented by the multibody interatomic Tersoff–Brenner potential, whereas the nonlinear interlayer vdW forces are represented by Lennard-Jones 6–12 potential function. The equivalent nonlinear material model of carbon–carbon bond is used to model it based on its force–deflection relation. Newmark’s algorithm is used to solve the nonlinear matrix equation governing the motion of the BLGS, DWCNT and NB. An impulse and harmonic excitations are used to excite these nanostructures under cantilevered, bridged and clamped boundary conditions. The frequency responses of these nanostructures are computed, and the dominant resonant frequencies are identified. Along with the forced vibration of these structures, the eigenvalue extraction problem of armchair and zigzag NB is also considered. The natural frequencies and corresponding mode shapes are extracted for the different length and boundary conditions of the nanotube bundle.     

Thermodynamic Curvature of the Binary van der Waals Fluid

The thermodynamic Ricci curvature scalar R has been applied in a number of contexts, mostly for systems characterized by 2D thermodynamic geometries. Calculations of R in thermodynamic geometries of dimension three or greater have been very few, especially in the fluid regime. In this paper, we calculate R for two examples involving binary fluid mixtures: a binary mixture of a van der Waals (vdW) fluid with only repulsive interactions, and a binary vdW mixture with attractive interactions added. In both of these examples, we evaluate R for full 3D thermodynamic geometries. Our finding is that basic physical patterns found for R in the pure fluid are reproduced to a large extent for the binary fluid.     

ZnO-CNT composite nanotubes as nanoresonators

This Letter reports the very first vibration analysis of the novel composite nanotubes (NTs) synthesized by coating carbon nanotubes (CNTs) with piezoelectric zinc oxide (ZnO). Timoshenko beam theory was used and modified to account for the interlayer van der Waals (vdW) interaction in the inner CNT and hybrid structures of the NTs. The distinctive vibration behaviours of the NTs were captured and the physics behind these unique features was investigated in terms of the critical role of the vdW interaction and the effect of the ZnO coating layer on the structural rigidity of the NTs. The composite NTs are found to be promising for gigahertz/terahertz electromechanical nanoresonators whose frequency can be even higher than that of the core CNTs.     

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.     

Wave characteristics of single-walled fluid-conveying carbon nanotubes subjected to multi-physical fields

Wave propagation in single-walled carbon nanotubes (SWCNTs) conveying fluids and placed in multi-physical fields (including magnetic and temperature fields) is studied in this paper. The nanotubes are modelled as Timoshenko beams. Based on the nonlocal beam theory, the governing equations of motion are derived using Hamilton's principle, and then solved by Galerkin approach, leading to two second-order ordinary differential equations (ODEs). Numerical simulations are carried out to verify the analytical model proposed in the present study, and determine the influences of the nonlocal parameter, the fluid velocity and flow density, the temperature and magnetic field flux change, and the surrounding elastic medium on the wave behaviour of SWCNTs. The results show that the nonlocal parameter has a considerable influence on dynamic behaviour of the nanotube and the fluid flow inside it. The results also show that the magnetic and temperature fields play an important role on the wave propagation characteristics of SWCNTs.     

Instability of single-walled carbon nanotubes conveying Jeffrey fluid

We report instability of the single-walled carbon nanotubes(SWCNT) filled with non-Newtonian Jeffrey fluid.Our objective is to get the influences of relaxation time and retardation time of the Jeffrey fluid on the vibration frequency and the decaying rate of the amplitude of carbon nanotubes.An elastic Euler-Bernoulli beam model is used to describe vibrations and structural instability of the carbon nanotubes.A new vibration equation of an SWCNT conveying Jeffrey fluid is first derived by employing Euler-Bernoulli beam equation and Cauchy momentum equation taking constitutive relation of Jeffrey fluid into account.The complex vibrating frequencies of the SWCNT are computed by solving a cubic eigenvalue problem based upon differential quadrature method(DQM).It is interesting to find from computational results that retardation time has significant influences on the vibration frequency and the decaying rate of the amplitude.Especially,the vibration frequency decreases and critical velocity increases with the retardation time.That is to say,longer retardation time makes the SWCNT more stable.     

Error estimation and anisotropic mesh refinement for 3d laminar aerodynamic flow simulations

This article considers a posteriori error estimation and anisotropic mesh refinement for three-dimensional laminar aerodynamic flow simulations. The optimal order symmetric interior penalty discontinuous Galerkin discretization which has previously been developed for the compressible Navier–Stokes equations in two dimensions is extended to three dimensions. Symmetry boundary conditions are given which allow to discretize and compute symmetric flows on the half model resulting in exactly the same flow solutions as if computed on the full model. Using duality arguments, an error estimation is derived for estimating the discretization error with respect to the aerodynamic force coefficients. Furthermore, residual-based indicators as well as adjoint-based indicators for goal-oriented refinement are derived. These refinement indicators are combined with anisotropy indicators which are particularly suited to the discontinuous Galerkin (DG) discretization. Two different approaches based on either a heuristic criterion or an anisotropic extension of the adjoint-based error estimation are presented. The performance of the proposed discretization, error estimation and adaptive mesh refinement algorithms is demonstrated for 3d aerodynamic flows.     

Potential energy,force distribution and oscillatory motion of chloride ion inside electrically charged carbon nanotubes

In this research, a continuum-based model is presented to explore potential energy, force distribution and oscillatory motion of ions, and in particular chloride ion, inside carbon nanotubes (CNTs) decorated by functional groups at two ends. To perform this, van der Waals (vdW) interactions between ion and nanotube are modeled by the 6-12 Lennard-Jones (LJ) potential, whereas the electrostatic interactions between ion and functional groups are modeled by the Coulomb potential and the total interactions are analytically derived by summing the vdW and electrostatic interactions. Making the assumption that carbon atoms and charge of functional groups are all uniformly distributed over the nanotube surface and the two ends of nanotube, respectively, a continuum approach is utilized to evaluate the related interactions. Based on the actual force distribution, the equation of motion is also solved numerically to arrive at the time history of displacement and velocity of inner core. With respect to the proposed formulations, comprehensive studies on the variations of potential energy and force distribution are carried out by varying functional group charge and nanotube length. Moreover, the effects of these parameters together with initial conditions on the oscillatory behavior of system are studied and discussed in detail. It is found out that chloride ion escapes more easily from negatively charged CNTs which is followed by uncharged and positively charged ones. It is further shown that the presence of functional groups leads to enhancing the operating frequency of such oscillatory systems especially when the electric charges of ion and functional groups have different signs.     

Nucleation and growth of SWNT: TEM studies of the role of the catalyst

This paper reviews transmission electron microscopy studies, combining high resolution imaging and electron energy loss spectroscopy, of the nucleation and growth of carbon single wall nanotubes with a particular emphasis on the nanotubes obtained from the evaporation-based elaboration techniques. Inspection of samples obtained from different synthesis routes shows that in all cases nanotubes are found to emerge from catalyst particles and that they have grown perpendicular or parallel to the surface according to whether they have been synthesized via evaporation-based methods or CCVD methods. Whereas the latter case corresponds to the well-known situation of carbon filaments growth, the former case strongly suggests another formation and growth process, which is described and its different steps discussed in detail. In this model, formation of the nanotubes proceeds via solvation of carbon into liquid metal droplets, followed by precipitation, at the surface of the particles, of excess carbon in the form of nanotubes through a nucleation and root growth process. It is argued that the nucleation of the nanotubes, which compete with the formation of graphene sheets wrapping the surface of the particle, necessarily results from a surface instability induced by the conditions of segregation. The nature and the origin of this instability was studied in the case of the class of catalyst Ni–R.E. (R.E.=Y, La, Ce, …) in order to identify the influence of the nature of the catalyst. The respective roles played by Ni and R.E. have been identified. It is shown that carbon and rear-earth co-segregate and self-assemble at the surface of the particle in order to form a surface layer destabilizing the formation of graphene sheets and providing nucleation sites for nanotubes growing perpendicular to the surface. To cite this article: A. Loiseau et al., C. R. Physique 4 (2003).     

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.     

基于碳纳米薄膜/砷化镓范德华异质结的高性能自驱动光电探测器研究

基于碳纳米材料/体半导体范德华(vdW)异质结的光电器件可以同时实现碳纳米材料的超高载流子迁移率以及体半导体的优异光电性能,且具有结构简单、工艺简便、易于调控界面等优点.尤其是通过调控单壁碳纳米管(SWCNT)的直径/手性、费米能级等可以与体半导体形成能带匹配、具有原子级界面的新型混合维度vdW异质结.本文报道了一种基...     

Longitudinal vibration and instabilities of carbon nanotubes conveying fluid considering size effects of nanoflow and nanostructure

In this study, the effects of small-scale of the both nanoflow and nanostructure on the vibrational response of fluid flowing single-walled carbon nanotubes are investigated. To this purpose, two various flowing fluids, the air-nano-flow and the water nano-flow using Knudsen number, and two different continuum theories, the nonlocal theory and the strain-inertia gradient theory are studied. Nano-rod model is used to model the fluid-structure interaction, and Galerkin method of weighted residual is utilizing to solve and discretize the governing obtained equations. It is found that the critical flow velocity decreases as the wave number increases, excluding the first mode divergence that it has the least value among of the other instabilities if the strain-inertia gradient theory is employed. Moreover, it is observed that Kn effect has considerable impact on the reduction of critical velocities especially for the air-flow flowing through the CNT. In addition, by increasing a nonlocal parameter and Knudsen number the critical flow velocity decreases but it increases as the characteristic length related to the strain-inertia gradient theory increases.     

The thermal effect on vibration and instability of carbon nanotubes conveying fluid

Based on the theory of thermal elasticity mechanics, an elastic Bernoulli–Euler beam model is developed for vibration and instability analysis of fluid-conveying single-walled carbon nanotubes (SWNTs) considering the thermal effect. Results are demonstrated for the dependence of natural frequencies on the flow velocity as well as temperature change. The influence of temperature change on the critical flow velocity at which buckling instability occurs is investigated. It is concluded that the effect of temperature change on the instability of SWNTs conveying fluid is significant.     

On the compressive response of carbon nanotube tangles

The nonlinear bulk compressibility of entangled multiwalled carbon nanotubes is studied. The analogy with textile fibre assemblies is explored by means of the well established van Wyk model. In view of the small diameter of the nanotubes, the possible effect of adhesive van der Waals interactions at tube-tube contacts is analysed. It is found, however, that the contribution of adhesive contacts to the bulk stress should be negligible. Compression experiments are performed on multi-walled carbon nanotubes and show that van Wyk's model is able to describe the response, although the values of the dimensionless parameter k of van Wyk's model were lower than expected. There is indeed no indication that van der Waals interactions play any significant role.     

Longitudinal vibration and stability analysis of carbon nanotubes conveying viscous fluid

Nowadays, carbon nanotubes (CNT) play an important role in practical applications in fluidic devices. To this end, researchers have studied various aspects of vibration analysis of a behavior of CNT conveying fluid. In this paper, based on nonlocal elasticity theory, single-walled carbon nanotube (SWCNT) is simulated. To investigate and analyze the effect of internal fluid flow on the longitudinal vibration and stability of SWCNT, the equation of motion for longitudinal vibration is obtained by using Navier-Stokes equations. In the governing equation of motion, the interaction of fluid-structure, dynamic and fluid flow velocity along the axial coordinate of the nanotube and the nano-scale effect of the structure are considered. To solve the nonlocal longitudinal vibration equation, the approximate Galerkin method is employed and appropriate simply supported boundary conditions are applied. The results show that the axial vibrations of the nanotubesstrongly depend on the small-size effect. In addition, the fluid flowing in nanotube causes a decrease in the natural frequency of the system. It is obvious that the system natural frequencies reach zero at lower critical flow velocities as the wave number increases. Moreover, the critical flow velocity decreases as the nonlocal parameter increases.     

Gas-phase synthesis of nitrogen-containing carbon nanotubes and their electronic properties

Nitrogen-containing carbon nanotubes are synthesized using a gas-phase reaction. The synthesis of nitrogen-doped carbon nanotubes from 100 to 500 Å in diameter is accomplished through pyrolysis of acetonitrile (CH₃CN) at a temperature of 800°C. Cobalt and nickel metallic particles formed upon thermal decomposition of a mixture of maleate salts are used as catalysts. The materials synthesized are investigated by scanning and transmission electron microscopy. Analysis of the x-ray photoelectron spectra demonstrates that the content of nitrogen atoms in three nonequivalent charge states is approximately equal to 3%. A comparison of the CK _α x-ray fluorescence spectrum of the carbon nanotubes synthesized through electric-arc evaporation of graphite and the x-ray fluorescence spectrum of the nitrogen-containing carbon nanotubes prepared by catalytic decomposition of acetonitrile indicates that, in the latter case, the spectrum contains a certain contribution from the sp ³ states of carbon atoms. The temperature dependences of the electrical conductivity for different types of multi-walled carbon nanotubes are compared. The difference observed in the temperature dependences of the electrical conductivity is associated with the presence of additional scattering centers in nitrogen-containing carbon nanotubes.     

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.     

Radial deformation and adhesion of carbon nanotube

The van der Waals (vdW) interactions of carbon nanotube (CNT)-substrate and CNT-CNT can cause strong adhesion. The adhesion can lead to radial deformation of CNTs, which is shown in both experiments and theoretical analysis. A scaling approach is used to predict the mechanical properties, vdW adhesion, and the elastic deformation of CNTs. It is found that the indentation of CNT is proportional to R _7/4 and h ^?3/2 in nanotube-substrate system and two same CNT system. Here, R and h are the radius and the wall thickness of CNT, respectively. The indentation ratio H ₁/H ₂ for CNT-CNT is proportional to (R ₁/R ₂)^3/2 and (h ₂/h ₁)^3/2.