A semi-analytical approach for calculating the equilibrium structure and radial breathing mode frequency of single-walled carbon nanotubes

A semi-analytical model for determining the equi-librium configuration and the radial breathing mode (RBM) frequency of single-wall carbon nanotubes (CNTs) is pre-sented. By taking advantage of the symmetry characteristics, a CNT structure is represented by five independent vari-ables. A line search optimization procedure is employed to determine the equilibrium values of these variables by minimizing the potential energy. With the equilibrium con-figuration obtained, the semi-analytical model enables an efficient calculation of the RBM frequency of the CNTs. The radius and radial breathing mode frequency results obtained from the semi-analytical approach are compared with those from molecular dynamics (MD) and ab initio calculations. The results demonstrate that the semi-analytical approach offers an efficient and accurate way to determine the equilib-rium structure and radial breathing mode frequency of CNTs.     

Nonlinear optical vibrations of single-walled carbon nanotubes

We demonstrate the new specific phenomenon of the long-time resonant energy exchange in the carbon nanotubes (CNTs) in the two optical branches - the Circumferential Flexure Mode (CFM) and Radial Btreathing Mode (RBM). It is shown that the modified nonlinear Schrödinger equation, obtained in the framework of nonlinear elastic thin shell theory, allows to describe the CNT nonlinear dynamics connected with considered frequency bands. Comparative analysis of the oscillations of the CFM and RBM branches shows the principal difference between nonlinearity effects. If the nonlinear resonant interaction of the low-frequency modes in the CFM branch leads to the energy capture in the some domain of the CNT, the same interaction in the RBM branch does not appear any tendency to the energy localization. The reason of such a distinction is the difference of the non-linear terms in the equations of motion. If the CFMs are specified by the soft power nonlinearity, the RBM dynamics is determined by the hard gradient nonlinearity. Moreover, in contrast to CFM the importance of nonlinearity in the case of RBM oscillations decreases with increasing of the length to radius ratio. The numerical integration of the thin shell theory equations confirms the results of the analytical study.     

Coarse-grained potentials of single-walled carbon nanotubes

We develop the coarse-grained (CG) potentials of single-walled carbon nanotubes (SWCNTs) in CNT bundles and buckypaper for the study of the static and dynamic behaviors. The explicit expressions of the CG stretching, bending and torsion potentials for the nanotubes are obtained by the stick-spiral and the beam models, respectively. The non-bonded CG potentials between two different CG beads are derived from analytical results based on the cohesive energy between two parallel and crossing SWCNTs from the van der Waals interactions. We show that the CG model is applicable to large deformations of complex CNT systems by combining the bonded potentials with non-bonded potentials. Checking against full atom molecular dynamics calculations and our analytical results shows that the present CG potentials have high accuracy. The established CG potentials are used to study the mechanical properties of the CNT bundles and buckypaper efficiently at minor computational cost, which shows great potential for the design of micro- and nanomechanical devices and systems.     

A molecular based anisotropic shell model for single-walled carbon nanotubes

Single-walled carbon nanotubes (SWCNTs) are frequently modeled as isotropic elastic shells. However, there are obvious evidences showing that SWCNTs exhibit remarkable chirality induced anisotropy that should not be neglected in some cases. In this paper, we derive the closed-form expressions for the anisotropic elastic properties of SWCNTs using a molecular mechanics model. Based on these anisotropic elastic properties, we develop a molecular based anisotropic shell model (MBASM) for predicting the mechanical behavior of SWCNTs. The explicit expressions for the coupling of axial, circumferential, and torsional strains, the radial breathing mode frequency, and the longitudinal and torsional wave speeds are obtained. We show that the MBASM is capable of predicting the effects of size and chirality on these quantities. The efficiency and accuracy of the MBASM are validated by comparisons of the present results with the existing results.     

Measuring elastic property of single-walled carbon nanotubes by nanoindentation: A theoretical framework

Nanoindentation is an effective technique for deducing the elastic property of single-walled carbon nanotubes (SWCNTs). Following an atomistic study of the nanoindentation mechanism, reverse analysis algorithms are proposed by utilizing the indentation force-depth data measured from the initial uniaxial compression and post-buckling regimes, respectively, which lead to stretching stiffness of 382 Pa m and 429 Pa m, that are very close to those in the literature. Parallel finite element simulations incorporating atomic interactions are also carried out, which closely duplicates the indentation response of SWCNTs in atomistic simulations. The numerical studies carried out in this paper may be used to guide the nanoindentation experiments, explain and extract useful data from the test, as well as stimulate new experiments.     

A molecular mechanics study on size-dependent elastic properties of single-walled boron nitride nanotubes

We present an analytical study for the elastic properties of single-walled boron nitride nanotubes via a molecular mechanics model. Closed-form expressions for Young's modulus, Poisson's ratio and surface shear modulus are derived as functions of the nanotube diameter. The results are helix angle sensitive and comparable to those from ab initio calculations. This work is a first effort to establish analytical model of molecular mechanics for composite nanotubes and reveals the dissimilarities between size-dependent elastic properties of carbon and boron nitride nanotubes.     

A molecular mechanics approach for analyzing tensile nonlinear deformation behavior of single-walled carbon nanotubes

In this paper, by capturing the atomic information and reflecting the behaviour governed by the nonlinear potential function, an analytical molecular mechanics approach is proposed. A constitutive relation for single-walled carbon nanotubes (SWCNT’s) is established to describe the nonlinear stress-strain curve of SWCNT’s and to predict both the elastic properties and breaking strain of SWCNT’s during tensile deformation. An analysis based on the virtual internal bond (VIB) model proposed by P. Zhang et al. is also presented for comparison. The results indicate that the proposed molecular mechanics approach is indeed an acceptable analytical method for analyzing the mechanical behavior of SWCNT’s. The project supported by the National Natural Science Foundation of China (10121202, 90305015 and 10328203), the Key Grant Project of Chinese Ministry of Education (0306) and the Research Grants Council of the Hong Kong Special Administrative Region, China (HKU 7195/04E).     

径向压缩单壁碳纳米管的力学行为

寻求具有卓越性能而又易于制造的纳米开关是纳米科技界的一个研究热点. 首先基于碳纳米管的结构双稳性提出了一种纳米开关的原型设计, 其由关到开的转换可通过径向压缩实现; 然后利用基于REBO-II作用势的分子动力学模拟, 系统分析了径向压缩单壁碳纳米管的力学行为和不同纳米管的双稳性特征, 给出了一些纳米管双稳性的特征参数, 对于工程设计具有一定的指导意义.     

Finite deformation continuum model for single-walled carbon nanotubes

A continuum-based model for computing strain energies and estimating Young’s modulus of single-walled carbon nanotubes (SWCNTs) is developed by using an energy equivalence-based multi-scale approach. A SWCNT is viewed as a continuum hollow cylinder formed by rolling up a flat graphite sheet that is treated as an isotropic continuum plate. Kinematic analysis is performed on the continuum level, with the Hencky (true) strain and the Cauchy (true) stress being employed to account for finite deformations. Based on the equivalence of the strain energy and the molecular potential energy, a formula for calculating Young’s modulus of SWCNTs is derived. This formula, containing both the molecular and continuum scale parameters, directly links macroscopic responses of nanotubes to their molecular structures. Sample numerical results show that the predictions by the new model compare favorably with those by several existing continuum and molecular dynamics models.     

Predicting the elastic properties of single-walled carbon nanotubes

Advances in the prediction of the mechanical properties of single-walled carbon nanotubes (SWNTs) are reviewed in this paper. Based on the classical Cauchy-Born rule, a new computational method for the prediction of Young's modulus of SWNTs is investigated. Compared with the existing approaches, the developed method circumvents the difficulties of high computational efforts by taking into consideration of the microstructure of nanotube and the atomic potential of hydrocarbons. Numerical results of Young's modulus and its variation with respect to the deformation gradient tensor are given and discussed. The results obtained are in good agreement with those obtained by laboratory experiments and other numerical methods.     

Material and structural instabilities of single-wall carbon nanotubes

The nonlinear atomistic interactions usually involve softening behavior. Instability resulting directly from this softening are called the material instability, while those unrelated to this softening are called the structural instability. We use the finite-deformation shell theory based on the interatomic potential to show that the tension instability of single-wall carbon nanotubes is the material instability, while the compression and torsion instabilities are structural instability.     

A new method for characterizing the interphase regions of carbon nanotube composites

The elastic properties of a carbon nanotube (CNT) reinforced composite are affected by many factors such as the CNT–matrix interphase. As such, mechanical analysis without sufficient consideration of these factors can give rise to incorrect predictions. Using single-walled carbon nanotube (SWCNT) reinforced Polyvinylchloride (PVC) as an example, this paper presents a new technique to characterize interphase regions. The representative volume element (RVE) of the SWCNT–PVC system is modeled as an assemblage of three phases, the equivalent solid fiber (ESF) mimicking the SWCNT under the van der Waals (vdW) forces, the dense interphase PVC of appropriate thickness and density, and the bulk PVC matrix. Two methods are proposed to extract the elastic properties of the ESF from the atomistic RVE and the CNT-cluster. Using atomistic simulations, the thickness and the average density of interphase matrix are determined and the elastic properties of amorphous interphase matrix are characterized as a function of density. The method is examined in a continuum-based three-phase model developed with the aid of molecular mechanics (MM) and the finite element (FE) method. The predictions of the continuum-based model show a good agreement with the atomistic results verifies that the interphase properties of amorphous matrix in CNT-composites could be approximated as a function of density. The results show that ignoring either the vdW interaction region or the interphase matrix layer can bring about misleading results, and that the effect of internal walls of multi-walled carbon nanotubes (MWCNTs) on the density and thickness of the dense interphase is negligible.     

Thermal vibration and apparent thermal contraction of single-walled carbon nanotubes

It is of fundamental value to understand the thermo-mechanical properties of carbon nanotubes. In this paper, by using molecular dynamics simulation, a systematic numerical investigation is carried out to explore the natural thermal vibration behaviors of single-walled carbon nanotubes and their quantitative contributions to the apparent thermal contraction behaviors. It is found that the thermo-mechanical behavior of single-walled carbon nanotubes is exhibited through the competition between quasi-static thermal expansion and dynamic thermal vibration, while the vibration effect is more prominent and induces apparent contraction in both radial and axial directions. With increasing temperature, the anharmonic interatomic potential helps to increase the bond length, which leads to thermally induced expansion. On the other hand, the higher structural entropy and vibrational entropy of the system cause the carbon nanotube to vibrate, and the apparent length of nanotube decreases due to various vibration modes. Parallel analytical and finite element analyses are used to validate the vibration frequencies and provide helpful insights. The unified multi-scale study has successfully decoupled and systematically analyzed both thermal expansion and contraction behaviors of single-walled carbon nanotube from 100 to 800 K, and obtained detailed information on various vibration modes as well as their quantitative contributions to the coefficient of thermal expansion in axial and radial directions. The results of this paper may provide useful information on the thermo-mechanical integrity of single-walled carbon nanotubes, and become important in practical applications involving finite temperature.     

Ionic liquid coated single-walled carbon nanotube buckypaper as supercapacitor electrode

Effect of stacking structure of single-walled carbon nanotubes (SWCNTs) on its performance as electrode of supercapacitor was investigated in the present work. Considering SWCNTs easily formed bundles due to strong van de Waals attraction between tubes, we proposed first dispersion of SWCNTs by ionic liquids (ILs) of 1-ethyl-3-methyl imidazolium tetrafluoroborate (EMIMBF₄), followed by fabrication of buckypaper by compression. The debundling effect of ILs on SWCNTs increased the interface between electrode and electrolyte, decreased electrical resistance, and, consequently, increased performance of the supercapacitor. Since ILs, used to disperse SWCNTs, also functioned as electrolyte in supercapacitor, our method is a simple way to prepare buckypaper electrode with high performance.     

金沙江虎跳峡河段岸坡变形破坏的相关动力因子研究

石墨层和单臂碳纳米管都是以C---C共价键结合的. 在小变形条件下C---C键的势能可用谐和函 数来描述，这与梁单元的变形能具有相同的形式，因此可以用梁单元等效C---C键的作用. 提出了一种C---C键的等效梁单元有限元模型，该模型能够完备地替代谐和势描述C---C键的 伸长、面内键角变化、离面键角变化和扭转. 通过分析石墨层的典型受载情况得到了等效梁 单元的参数，以及等效梁单元参数与谐和势参数的关系，并用该模型计算了单臂碳纳米管的 杨氏模量和泊松比，计算结果为相关文献所验证.     

Modulational instability of gap solitons in single-walled carbon nanotube lattices

Modulational instability and nonlinear localized excitations are addressed, in the framework of a one-dimensional diatomic carbon nanotube (CNT) model, using the quasi-discrete approximation. Gap soliton solutions, based on the modulational instability criterion, are studied, where one considers the solutions arising in the upper and lower optical frequency cutoff regimes, and in the upper acoustic frequency cutoff mode. Solutions are found as breathers and double breathers, and their response to interatomic interaction parameters is discussed. Vibrations of the CNTs from the two modes are compared based on their capability of carrying the amount of energy required for specific purposes, either in Microelectronics or in Nano-devices.     

Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model

An analytical model based on a molecular mechanics approach is presented to relate the elastic properties of a single-walled carbon nanotube to its atomic structure. We derive closed-form expressions for elastic modulus and Poisson's ratio as a function of the nanotube diameter. Properties at different length scales are directly connected via these expressions. The analytically calculated elastic properties for achiral nanotubes using force constants obtained from experimental data of graphite are compared to those based on tight binding numerical calculations. This study represents a preliminary effort to develop analytical methods of molecular mechanics for applications in nanostructure modeling.     

A glance on the effects of temperature on axisymmetric dynamic behavior of multiwall carbon nanotubes

In this paper the effects of temperature on theradial breathing modes(RBMs) and radial wave propagation in multiwall carbon nanotubes(MWCNTs) are investigated using a continuum model of multiple elastic isotropicshells.The van der Waals forces between tubes are simulatedas a nonlinear function of interlayer spacing of MWCNTs.The governing equations are solved using a finite elementmethod.A wide range of innermost radius-to-thickness ratioof MWCNTs is considered to enhance the investigation.Thepresented solution is verified by comparing the results withthose reported in the literature.The effects of temperature onthe van der Waals interaction coefficient between layers ofMWCNTs are examined.It is found that the variation of thevan der Waals interaction coefficient at high temperature issensible.Subsequently,variations of RBM frequencies andradial wave propagation in MWCNTs with temperatures upto 1 600 K are illustrated.It is shown that the thick MWCNTs are more sensible to temperature than the thin ones.     

Deformation of metallic single-walled carbon nanotubes in electric field based on elastic theory

The electromechanical properties of metallic single-walled carbon nanotubes (SWCNTs) in the electric field are demonstrated with a column shell model in this paper.A hemisphere model is introduced to determine the charge distribution and the local electric field in SWCNTs.By treating the SWCNT as an elastic column shell,the analytical solutions of the charged SWCNT’s axial strain and the radial strain are obtained.SWCNTs with a larger aspect ratio show greater deformation.The greatest radial deformation appears at the end of the tube.The significant axial strain can be induced in CNTs with a large length (around 100 nm) even though the applied electric field is not strong enough.When SWCNTs are fixed at both ends,the radius of SWCNTs becomes small along the axial position.     

Effect of process conditions on the synthesis of carbon nanotubes by catalytic decomposition of methane

A new dual-composition catalyst based on Ni-Mo/MgO with high efficiency of producing carbon nanotubes （CNTs） from methane was reported recently. In the present article, with this type of catalyst, the impact of such experimental parameters as reaction temperature, reaction time, concentration of H2, flow rate ratio of CH4 to H2 on yield and graphitization were investigated, leading to the following optimal growth conditions： reaction time 60min, reaction temperature 900℃, CH4：H2 about 100：20mL/min, under which high-yield multi-walled CNTs bundles were synthesized. Raman measurement indicated that the as-synthesized product was well-graphitized, and the purity was estimated over 95% by TG-DSC analysis. In terms of the above results, an explanation of high-efficiency formation of CNTs bundles and the co-catalysis mechanism of Ni-Mo/MgO were suggested. 2007 Chinese Societv of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V.