Behavior Of Single-walled Carbon Nanotubes Research Articles

Nanoscale finite element models for vibrations of single-walled carbon nanotubes: atomistic versus continuum

By the atomistic and continuum finite element models, the free vibration behavior of single-walled carbon nanotubes (SWCNTs) is studied. In the atomistic finite element model, the bonds and atoms are modeled by the beam and point mass elements, respectively. The molecular mechanics is linked to structural mechanics to determine the elastic properties of the mentioned beam elements. In the continuum finite element approach, by neglecting the discrete nature of the atomic structure of the nanotubes, they are modeled with shell elements. By both models, the natural frequencies of SWCNTs are computed, and the effects of the geometrical parameters, the atomic structure, and the boundary conditions are investigated. The accuracy of the utilized methods is verified in comparison with molecular dynamic simulations. The molecular structural model leads to more reliable results, especially for lower aspect ratios. The present analysis provides valuable information about application of continuum models in the investigation of the mechanical behaviors of nanotubes.

An enhanced continuum modeling of the ideal strength and the angle of twist in tensile behavior of single-walled carbon nanotubes

By utilizing the fourth-, sixth-, eighth-, and tenth-order elastic moduli tensors of graphene a highly nonlinear constitutive model for it is proposed. Subsequently, an accurate analytical formulation, describing the entire tensile behavior of single-walled carbon nanotubes (SWCNTs) from their initial unloaded states through their ideal strengths, is made possible. The angle of twist which is a critical parameter that varies with the tensile loading is also calculated within the current framework. The estimated value of the theoretical strength of SWCNTs with different chiralities and radii as well as that of graphene ranges from 0.39 to 0.44 TPa. Some peculiarities associated with chirality of 15° are observed.

Temperature and Chirality Dependent Mechanical Properties of Single-Walled Carbon Nanotube

Presented herein is an investigation into the stretching and fracture behavior of single-walled carbon nanotubes (SWCNT) subjected to axial tensile load at various temperatures. This study is carried out using molecular dynamics (MD) simulation at temperatures ranging from 200 K to 2200 K. The authors find that single-walled carbon nanotubes exhibit obvious temperature and chirality dependent properties. The values of tensile strain and tensile strength of zigzag (12, 0) and armchair (7, 7) SWCNTs with similar radii decrease with increase in temperature, but decline slightly on different slopes. The stress-strain curves of zigzag and armchair SWCNTs vary significantly when the temperature is more than 1200 K. Our studies suggest temperature and chirality play an important role in determining the tensile strength and tensile strain of SWCNT.

Energy absorption characteristics of single-walled carbon nanotubes

The excellent mechanical properties of carbon nanotubes make them potential candidates for engineering application. In this paper, the impact and failure behaviors of single-walled carbon nanotubes (SWCNTs) are investigated. The effects of diameter, length, and chirality on their energy absorption characteristics under lateral impact and axial crush are studied. By integrating the principle of molecular structural mechanics (MSM) into finite element method (FEM), the locations and directions of fracture process can be predicted. It is shown that the specific energy absorption (SEA) of SWCNTs is 1–2 order of magnitude higher than that of the ordinary metallic materials and composites in axial impact, indicating that carbon nanotubes are promising energy absorption materials for engineering applications.

Allometric Pressure versus Volume Behavior of Single-Walled Carbon Nanotubes Under High Pressure

Single-walled carbon nanotubes (SWCNT) under high pressure have been studied by Raman spectroscopy. Considering cylindrical nanotubes with circular cross sections at all pressures the experimental data show an allometric pressure versus volume (PV) behavior under non-hydrostatic and quasi-hydrostatic conditions. We show that there is no need to consider a structural phase transition to explain the dependence of the radial breathing mode (RBM) phonon frequency with pressure. The bulk modulus (Bo) of the bundled nanotubes obtained from the PV data increases with pressure following a fit given by Bo = 2.38+10.77P GPa.

Free vibration analysis of carbon nanotubes by using three-dimensional theory of elasticity

This paper studies vibration behavior of single-walled carbon nanotubes based on three-dimensional theory of elasticity. To accounting for the size effect of carbon nanotubes, nonlocal theory is adopted to the shell model. The nonlocal parameter is incorporated into all constitutive equations in three dimensions. Governing differential equations of motion are reduced to the ordinary differential equations in thickness direction by using Fourier series expansion in axial and circumferential direction. The state equations obtained from constitutive relations and governing equations are solved analytically by making use of the state space method. A detailed parametric study is carried out to show the influences of the nonlocal parameter, thickness-to-radius ratio and length-to-radius ratio. Results reveal that excluding small-scale effects caused decreasing accuracy of natural frequencies. Furthermore, the obtained closed form solution can be used to assess the accuracy of conventional two-dimensional theories.

A molecular structural mechanics model applied to the static behavior of single-walled carbon nanotubes: New general formulation

A new general formulation for the mechanical behavior of Single-Walled Carbon Nanotubes is presented. Carbon atoms are located at the nodes of an hexagonal honeycomb lattice wrapped into a cylinder. They are linked by covalent C–C bonds represented by a truss or spring element, and the three-body interaction among two neighboring covalent bonds is reproduced by a rotational spring. The main advantage of our approach is to allow general load conditions (and any chirality) with no need of specific formulation for each load case, in contrast with previous works [26,27,31]. Four load configurations are adopted: tension, compression, bending and torsion of cantivelered SWCNTs. Calculations with our own codes for both AMBER and Morse potential functions have been carried out, aimed to compare their final results. Initial positions of the atoms (nodes) into nanotube cylindrical geometry has been reproduced in great detail by means of a conformal mapping from the planar graphene sheet. Therefore, the effect of initial SWCNTs curvature has been introduced explicitly through a system of initial stresses (prestressed state) which contribute to maintain their circular cross-section. Numerical results and deformed shapes for nanotubes with several diameters and chiralities under each load case are used to obtain their mechanical parameters with the only objective of checking the present formulation with previous works [28,30,20,24]. Also, the significance of the atomistic discrete simulations at the nano-scale size against other continuum models is underlined.

Prediction of compressive post-buckling behavior of single-walled carbon nanotubes in thermal environments

In the present investigation, the axial buckling and post-buckling configurations of single-walled carbon nanotubes (SWCNTs) are studied including the thermal environment effect. For this purpose, Eringen’s nonlocal elasticity continuum theory is implemented into the classical Euler–Bernoulli beam theory to represent the SWCNTs as a nonlocal elastic beam model. A closed-form analytical solution is carried out to analyze the static response of SWCNTs in their post-buckling state in which the axial buckling load is assumed to be beyond the critical axial buckling load. Common sets of boundary conditions, named simply supported–simply supported (SS–SS), clamped–clamped (C–C), and clamped–simply supported (C–SS), are considered in the investigation. Selected numerical results are given to represent the variation of the carbon nanotube’s mid-span deflection with the applied axial load corresponding to various nonlocal parameters, length-to-diameter aspect ratios, temperature changes, and end supports. Moreover, a comparison between the post-buckling behaviors of SWCNTs at low- and high-temperature environments is presented. It is found that the size effect leads to a decrease of the axial buckling load especially for SWCNTs with C–C boundary conditions. Also, it is revealed that the value of the temperature change plays different roles in the post-buckling response of SWCNTs at low- and high-temperature environments.

A modal analysis of carbon nanotube using elastic network model

Although it is widely known that both size and chirality play significant roles in vibration behaviors of single-walled carbon nanotubes (SWCNTs), there haven’t been yet enough studies specifying the relationship between structure and vibration mode shape of SWCNTs. We have analyzed the chirality and length dependence of SWCNT by using normal mode analysis based elastic network model in which all interatomic interactions of the given SWCNTs structure are represented by a network of linear spring connections. As this method requires relatively short computation time compared to molecular dynamics simulation, we can efficiently analyze vibration behavior of SWCNTs. To ensure the relationship between SWCNT structure and its vibration mode shapes, we simulated more than one hundred SWCNTs having different types of chirality and length. Results indicated that the first two major mode shapes are bending and breathing. The minimum length of nanotube for maintaining the bending mode does not depend on chirality but on its diameter. Our simulations pointed out that there is a critical aspect ratio between diameter and length to determine vibration mode shapes, and it can be empirically formulated as a function of nanotube length and diameter. Therefore, uniformity control is the most important premise in order to utilize vibration features of SWCNTs. It is also expected that the obtained vibration aspect will play an important role in designing nanotube based devices such as resonators and sensors more accurately.

Probe diffusion of single-walled carbon nanotubes in semidilute solutions of polyacrylonitrile homo- and copolymers: Effects of topological constraints and polymer/Nanorod interactions

With varying the molecular weight, copolymer chemical structure and concentration, the probe diffusion behavior of single-walled carbon nanotubes (SWCNTs) in N, N-dimethylacetamide (DMAc) solutions of polyacrylonitrile (PAN) was studied by dynamic light scattering (DLS) technique. In combination with the systematic viscosity property investigation for PAN/DMAc system, the probe diffusion results provided insight to understand the interactions between PAN and SWCNTs. It was found that, due to depletion interaction, the SWCNT diffusion positively deviates from the Stokes–Einstein (SE) relation in PAN homopolymer solutions. However, in PAN copolymer solution, the SWCNT diffusion mostly obeys the SE relation, which was speculated as a result of hydrogen-bonding formation between the carboxylic acid functional groups in PAN copolymers and on purified SWCNTs. Moreover, due to the unique features of SWCNT probes – very long, extremely rigid, and adjustable bundle diameter, the SWCNT probe diffusion unambiguously revealed the topological constraints of polymer network responsible for the retardation of thin rod diffusion. Quantitative analysis indicated that the correlation length rather than the “tube” diameter of polymer liquid is the critical length scale to dictate the diffusion retardation of SWCNTs in polymer solutions.

Various gradient elasticity theories in predicting vibrational response of single-walled carbon nanotubes with arbitrary boundary conditions

This article deals with the development and use of different gradient beam theories in order to predict the vibrational behavior of single-walled carbon nanotubes (SWCNTs). To address the problem of free vibration, the Euler–Bernoulli and Timoshenko beam theories in conjunction with the gradient elasticity theories including stress, strain and combined strain/inertia are implemented. The generalized differential quadrature method is employed to numerically solve the problem which can treat various boundary conditions. The results generated from the present gradient models are compared with those from molecular dynamics simulations as a benchmark of good accuracy and the proper values of small length scales used in the gradient models are proposed. This study shows prominent differences between various gradient models when the nanotube becomes very short (for aspect ratios of approximately lower than six). It is indicated that applying the strain gradient elasticity by incorporation of inertia gradients yields more reliable results especially for shorter length SWCNTs on account of two small-scale factors related to the inertia and strain gradients. Moreover, since with the reduction in the aspect ratio of nanotubes the effects of boundary conditions become dominant, a discussion is given to investigate the influence of end conditions on the vibrational characteristics of SWCNTs.

Dependence of mechanical characteristics and the fracture and buckling behavior of single-walled carbon nanotubes on their geometry

Dependence of mechanical characteristics and the fracture and buckling behavior of single-walled carbon nanotubes on their geometry

Theoretical Prediction of Tensile Behavior of Single-Walled Carbon Nanotubes

Based on molecular mechanics, a finite element model is proposed to evaluate the tensile behaviour of single-walled carbon nanotubes (SWCNT). The SWCNT is modelled as a space-frame structure. The bonds between the carbon atoms are simulated as beam members to carry the loads, while the carbon atoms are the joints of the members. The modified Morse potential is adopted to characterize the non-linear behaviour of C-C bonds. The simulation are carried out for two well-known SWCNTs, armchair and zigzag with different diameters to investigate the influence of chirality and diameter on the mechanical properties including the Youngs modulus, tensile strength and fracture strain of SWCNTs. It is found that the Youngs moduli of both armchair and zigzag nanotubes increase monotonically with the increase of diameter. The armchair nanotube exhibits larger stress-strain response than the zigzag nanotube. Numerical simulations reveal that the Youngs modulus and tensile strength of armchair nanotubes are larger than that of zigzag nanotubes. These results are in good agreement with the existing numerical and experimental results.

Conformational–Induced Doping Effect of Sodium Dodecyl Benzene Sulfonate on Single Walled Carbon Nanotubes

The doping behavior of single-walled carbon nanotubes (SWCNTs) was investigated with an emphasis on the control of the conformation of sodium dodecylbenzene sulfonate (NaDDBS) with sulfonate groups acting as an electro-withdrawing group. The conformation of adsorbed NaDDBS on SWCNTs was controlled as a function of the amount of NaDDBS. The doping behavior of SWCNTs was significantly affected by the dosing amount of NaDDBS due to the conformational change of NaDDBS adsorbed on the SWCNT surface, which affected the spatial distance between the SWCNT surface and the sulfonate groups in NaDDBS. At a higher concentration, the spatial distance between the sulfonate group in NaDDBS and SWCNT was not sufficiently close enough to dope SWCNT due to the repulsive forces between the sulfonate groups in NaDDBS. Alternatively, at a lower concentration, NaDDBS acted as a p-type dopant for SWCNTs. To this end, this paper demonstrates a new tendency of doping that is related to the adsorbed behavior of a dispersant.

Numerical simulation for finite deformation of single-walled carbon nanotubes at finite temperature using temperature-related higher order Cauchy-Born rule based quasi-continuum model

In the present study, a meshless computational framework for simulating the large deformation behaviors of single-walled carbon nanotubes (SWCNTs) at finite non-zero temperatures is established based on the so-called temperature-related higher order Cauchy-Born rule (THCB rule) where the second-order deformation gradient is involved in the kinematic description of the deformation of SWCNTs. The Helmholtz free energy is used as the thermodynamic potential and the local harmonic approximation (LHA) is adopted to construct the nanoscale quasi-continuum constitutive model. The proposed numerical approach is then applied to four numerical examples under different loading conditions. It is found that the results obtained by the proposed numerical framework agree well with those from molecular dynamics simulations.

Buckling Behaviour and Natural Frequency of Zigzag and Armchair Single-Walled Carbon Nanotubes

In this study, single-walled carbon nanotubes were generated in their perfect state as finite element models in the MSC.Marc software. The buckling behaviour and resonant frequency modes of the two limiting cases of carbon nanotubes, i.e. the armchair and zigzag models, were studied. The obtained results were compared with the classical analytical solutions related to a similar continuum structure of a hollow cylinder. The buckling behaviour of single-walled carbon nanotubes under cantilever boundary conditions proved to be almost identical to the prediction of the classical Euler equation. Furthermore, there was very good agreement between the analytical and finite element results of the studied single-walled carbon nanotubes; though the achieved value of the first mode of frequency, obtained from the finite element results, was more accurate than the higher modes.

Dependence of mechanical characteristics and the fracture and buckling behavior of single-walled carbon nanotubes on their geometry

The dependence of the mechanical characteristics of single-walled carbon nanotubes on their length, radius, and chirality is investigated by molecular dynamics simulation. The observed changes in the mechanical properties with radii are compared with the results reported by other authors under both tensile and compressive loading. The tensile properties of zigzag and armchair CNTs are not dependent on their lengths. Slight changes in the mechanical properties with chirality are also reported. However, their compressive properties, primarily their stiffness and critical compressive stress, vary with length for all types of CNTs. The fracture patterns show different modes in each case. Buckling produces kinks, wavy structures, shell-wall buckling, columnar buckling, bending, and twisting.

Thermal Buckling Analysis of Embedded Single-Walled Carbon Nanotubes with Arbitrary Boundary Conditions Using the Nonlocal Timoshenko Beam Theory

In this paper, the thermal buckling behavior of single-walled carbon nanotubes (SWCNTs) embedded in an elastic medium is studied. To this end, the SWCNTs are modeled based on the nonlocal Timoshenko beam theory into which the effect of the elastic medium is incorporated The generalized differential quadrature (GDQ) method is employed to discretize the governing differential equations and to consider different commonly used boundary conditions (BCs). For simply supported BCs, the results obtained from the present analysis are compared with the ones from the exact solution and an excellent agreement has been achieved. The effects of the aspect ratio, nonlocal parameter and the Winkler parameter on the dimensionless critical buckling temperature are carefully investigated.

Controlling the Electrical Behavior of Semiconducting Carbon Nanotubes via Tube Contact

The electromechanical behavior of single-walled carbon nanotubes (SWNTs) in contact with different materials is investigated by scanning probe microscopy. An anomalous diamond/semiconducting nanotube behavior is observed, which is consistent with ab initio calculations: the formation of a broken-gap heterojunction between semiconducting SWNTs and a hydrogenated diamond surface results in a metallic response for such SWNTs.

A theoretical investigation of thermal effects on vibrational behaviors of single-walled carbon nanotubes

Thermal effects on the vibrational behaviors and dynamic Young’s modulus of single-walled carbon nanotubes (SWCNTs) are investigated through both constant temperature molecular dynamics (MD) simulation and modified molecular structural mechanics (MMSM) modeling. The MD simulation incorporates a modified Nosé-Hoover thermostat model to control the system temperature. In the MMSM modeling, the covalent and nonbonded interactions between carbon atom pairs are modeled with the second generation force field and the Lennard–Jones potential, respectively, where the covalent bonds are treated as Euler–Bernoulli beam and the temperature-dependent bond length and angle are determined through the Badger’s rule and MD simulation. The results derived from these two approaches are compared with each other and the published theoretical and experimental data. Results show that the dynamic Young’s modulus of the SWCNTs tends to be smaller than the published static one obtained from uniaxial tensile tests, and their natural frequency and dynamic Young’s modulus would decrease with increasing temperature. Moreover, a comparable frequency ratio of the first two flexural modes is achieved by these two approaches. The frequency ratio is highly dependent on their aspect ratio but independent of temperature, and would converge to the literature experimental data (about 6.1–6.2) as the aspect ratio becomes very large.