Deformation Of Single-walled Carbon Nanotubes Research Articles

Electromechanical Property Calculation of Carbon Nanotubes Using Linear Augmented Cylindrical Wave Method

Deformations of single-walled carbon nanotubes (SWNTs) change their band structure in the nanoelectromechanical systems. In this study, we investigated the response of the electronic structure of chiral and nonchiral SWNTs (8,7), (9,6), (10,5), (7,7), (11,0), (12,0), and (13,0) to twisting and axial tension modes by using the symmetrized linear augmented cylindrical wave technique. We showed that perturbations of the band structures depend on a “family” index p = (n1 − n2)mod 3 (where p = −1, 0 or 1). Twisting the semiconducting (8,7) tubule with p = 1 in the direction of the screw axis is accompanied by the large broadening of minimum gap E11 and narrowing of the second gap E22, while these gaps drastically change in the tubule (10,5) with p = −1. In these tubules, changing the direction of twisting leads to the reversal in direction of the gap shifts. Regardless of the twisting direction, in metallic (7,7) and quasi-metallic (9,6) SWNTs with p = 0, the E11 gap rapidly increases from 0.0 and 0.035 eV to about 1 eV. When twisting the zigzag tubules (13,0) p = 1 and (11,0) p = −1, the gaps E11 equal to about 0.8 eV increase and decrease by several hundredths of eV. On the contrary, the compression and extension of these tubules cause a sharp change in their band structure with approximately a twofold change in the gaps E11 and E22 and inversion in the sequence of the boundary bands. The similar deformation of the armchair nanotube (7,7) has practically no effect on its electronic levels. In the case of zigzag (12,0) p = 0 SWNT, all deformation modes transform the quasi-metallic tubule into the semiconductor.

Radial deformation of single-walled carbon nanotubes adhered to solid substrates and variations of energy: Atomistic simulations and continuum analysis

Radial deformation of carbon nanotubes (CNTs) adhered to substrates can strongly influence their physical properties and the performance of CNT-based nanodevices. Here we explore the radial deformation and related energy variations of single-walled carbon nanotubes (SWCNTs) attached to solid substrates by adopting the classical molecular dynamics (MD) simulations and continuum analysis. The radial deformation of a SWCNT can be divided into three stages: the drop-shaped stage, the half-hourglass-shaped stage and the half-dumbbell-shaped stage. Three continuum models are established to mimic the radially deformed configurations. Minimization of the total free energy provides the equilibrium profiles of the SWCNT during the radial deformation. There is a good agreement between the theoretical predictions and the results of MD simulations. By analyzing the variations of free energy, we find that the favorable configuration of a substrate-supported SWCNT can be characterized by two threshold diameters. For SWCNTs with diameter below the first threshold diameter, the circular state is stable. For SWCNTs with diameter exceeding the second threshold diameter, the collapsed half-dumbbell-shaped state is energetically most favorable. The energy barrier, for obtaining a fully collapsed SWCNT, decreases with the increasing diameter, which approximately keeps a negative exponential relationship. The adhesion work is demonstrated to play an important role in the radial deformation and morphology stability of substrate-supported SWCNTs.

Two-Dimensional Elastica Model for Describing the Flexural Behavior of Single-Walled Carbon Nanotubes

Two-dimensional elastica is defined as the shape of a surface into which an elastic film is deformed. In the present analysis, graphene is considered to be a plate of non-vanishing thickness and, moreover, a single-walled carbon nanotube (SWCNT) is viewed as the result of rolling up a sheet of graphene into a cylindrical tube. Therefore, when the obtained tube is subjected to bending, it can be treated as a two-dimensional elastica. In order to take into consideration the geometrical and material nonlinearities, the Lagrangian description of deformation is adopted and, in addition, a nonlinear anisotropic constitutive relation is employed that can also capture appropriately the effect of chirality of SWCNTs. By formulating the equilibrium of the corresponding two-dimensional elastica, the exact solution to the bending-induced deformation field is determined, and the exact closed-form expression for the flexural rigidity of SWCNTs is obtained. It is shown that deformation of SWCNTs in pure bending is accompanied by the warping and radial components. In addition to the exact solution, a perturbation analysis is also performed to obtain simplified approximate expressions.

Radial deformation of single-walled carbon nanotubes on quartz substrates and the resultant anomalous diameter-dependent reaction selectivity

Owing to the unique conjugated structure, the chemical-reaction selectivity of single-walled carbon nanotubes (SWNTs) has attracted great attention. By utilizing the radial deformation of SWNTs caused by the strong interactions with the quartz lattice, we achieve an anomalous diameter-dependent reaction selectivity of quartz lattice-oriented SWNTs in treatment with iodine vapor; this is distinctly different from the widely reported and well accepted higher reaction activity in small-diameter tubes compared to large-diameter tubes. The radial deformation of SWNTs on quartz substrate is verified by detailed Raman spectroscopy and mappings in both G-band and radial breathing mode. Due to the strong interaction between SWNTs and the quartz lattice, large-diameter tubes present a larger degree of radial deformation and more delocalized partial electrons are distributed at certain sidewall sites with high local curvature. It is thus easier for the carbon–carbon bonds at these high-curvature sites on large-diameter tubes to break down during reaction. This anomalous reaction activity offers a novel approach for selective removal of small-bandgap large-diameter tubes.

Generation of Temperature Dependent Transversely Isotropic Properties for Zigzag and Armchair Single-Walled Carbon Nanotubes

Finite element analysis has been carried out to obtain temperature dependent transversely isotropic properties of the single-walled carbon nanotubes (SWCNTs). Finite element models of SWCNTs are generated by specifying the C-C bond rigidities. The five independent transversely isotropic properties for different chiralities are evaluated using the stress fields of thick-walled cylinders and the elastic deformations of SWCNTs subjected to pure extension, internal pressure and pure torsion loads. Empirical relations are provided for the five independent elastic constants useful to armchair and zigzag SWCNTs.

Quasi-Continuum Contact Model for the Simulation of Severe Deformation of Single-Walled Carbon Nanotubes at Finite Temperature

Quasi-Continuum Contact Model for the Simulation of Severe Deformation of Single-Walled Carbon Nanotubes at Finite Temperature

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.

Nonlinear finite-element modeling of graphene and single- and multi-walled carbon nanotubes under axial tension

An effective finite-element (FE) approach for modeling the structure and the deformation of single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) is presented. An individual tube was modeled using a frame-like structure with beam elements. The effect of van der Waals forces, crucial in MWCNTs, was modeled by spring elements. The success of this new carbon nanotube (CNT) modeling approach was verified by comparing the simulation results for single- and multi-walled nanotubes and graphene with other experimental and computational results available in the literature. Simulations of final deformed configurations were in excellent agreement with the atomistic models for various deformations. The proposed approach successfully predicts the experimentally observed values for mechanical behavior of SWCNTs and MWCNTs. The results demonstrated that the proposed FE technique could provide a valuable tool for studying the mechanical behavior of different types of nanotubes, as well as their effectiveness as load-bearing entities in nanocomposite materials.

A rod model for three dimensional deformations of single-walled carbon nanotubes

A continuum model for single-walled carbon nanotubes (SWCNT) is presented which is based on an extension to the special Cosserat theory of rods ( Kumar and Mukherjee, 2011). The model allows deformation of a nanotube’s lateral surface in a one dimensional framework and hence is an efficient substitute to the commonly used two dimensional shell models for nanotubes. The model predicts a new coupling mode in chiral nanotubes – coupling between twist and cross-sectional shrinkage implying that the three deformation modes (extension, twist and cross-sectional shrinkage) are all coupled to each other. Atomistic simulations based on the density functional based tight binding method (DFTB) are performed on a (9, 6) SWCNT and the simulation data is used to estimate material parameters of this rod model. A peculiar behavior of the nanotube is observed when it is axially stretched – induced rotation of each cross-section is equal in magnitude but opposite to that of its two neighboring cross-sections. This is shown to be an effect of relative shift/inner-displacement between the two SWCNT sub-lattices.

On the coupling of axial and shear deformations of single-walled carbon nanotubes and graphene: a numerical study

This paper presents a numerical investigation for the evaluation of elastic coupling of extension and twist in single-walled carbon nanotubes. The carbon nanotubes are modelled according to their atomistic structure. The harmonic approximation is utilized for describing the interaction potential energies. The force field is simulated via suitable straight and torsional springs that express the interatomic interactions and interconnect nodes placed on the atomic positions. By using appropriate boundary conditions, the nanotubes are loaded axially and the resulted twist is numerically predicted using finite element procedures. The numerical outcomes reveal that the stretching deformation of single-walled carbon nanotubes leads to their torsion only for the chiral case. A complete parametric study with respect to geometric characteristics of nanotubes shows that the coupling is strongly dependent on the diameter, chiral angle, and nanotube length. To explain these observations, the inherent graphene sheet geometries used for the generation of single-walled carbon nanotubes are also studied under extension. The results prove similar coupling phenomena occur between extension and in-plane shear. Comparisons with results provided in the open literature are also performed, where possible.

Mechanical Properties of Single-Walled Carbon Nanotubes under Large Axial Deformation

The deformation of single-walled carbon nanotubes (SWCNTs) under large axial strain is studied by a geometrical mapping method. The interactions between atoms in carbon nanotubes (CNTs) are described by Tersoff-Brenner potential. Results show the strain energy depends on chirality but hardly on tubes’ radii. For graphitic sheet under large axial deformation, the elastic moduli decrease with the increase of engineering strain under tension. The modulus reaches the peak value as the axial engineering strain reaches -0.08 for armchair pattern and -0.15 for zigzag pattern under compression.

Effects of hydrogen chemisorption on the structure and deformation of single-walled carbon nanotubes

We report results of molecular-dynamics simulations for the effects of atomic hydrogen chemisorption on the structure and deformation of single-walled carbon nanotubes. Upon hydrogenation, the nanotubes expand and the degree of expansion depends on the hydrogen coverage. There is a critical hydrogen coverage, ∼25%–30%, that marks the onset of a structural transition associated with the sp2-to-sp3 bonding transition: at lower-than-critical coverage, sp2 C–C bonding dominates and nanotube swelling is negligible; at higher-than-critical coverage, however, sp3 C–C bonding dominates and radial and axial strains increase monotonically with coverage. This behavior is independent of nanotube chirality and diameter and of temperature.

Bending Deformation of Single-Walled Carbon Nanotubes Caused by 5–7 Pair Couple Defect

The characteristics of 5–7 pair couple defects formed in single-walled carbon nanotubes were investigated. For armchair nanotubes with diameters from 6.8 to 13.6 Å, such defects were formed by removing atoms. Structural optimization, which was carried out using tight-binding molecular dynamics, resulted in the bending deformation of the nanotubes due to the defects. The resulting stable structure was found to change with the number of atoms removed, showing a bending angle of 16.6° for a diameter of 10.7 Å. The dependence of bending angle on the diameter of nanotubes was analysed using a simple model. As results, several sets of 5–7 pair couple defects were found to be necessary in order to induce plastic deformation that was experimentally observed in nanotubes.

Development of an equation to predict radial modulus of elasticity for single-walled carbon nanotubes

Finite element (FE) method is used to model radial deformation of single-walled carbon nanotube (SWCNT) under hydrostatic pressure. Elastic deformation of the nanostructure is simulated via elastic beams. Properties of the beam element are calculated by considering the stiffness of the covalent bonds between the carbon atoms in the hexagonal lattice. By applying the beam elements in a three-dimensional space, elastic properties of the SWCNT in transverse direction are obtained. In this regard, influences of diameter and tube wall thickness on the radial and circumferential elastic moduli of zigzag and armchair SWCNTs are considered. It is observed that there is a good agreement between the FE results and data available in the literatures. The FE data are employed to develop a predictive equation through a statistical non-linear regression model.

Deformation of single-walled carbon nanotubes under large axial strains

The deformations of single-walled carbon nanotubes (SWCNTs) under large axial strains are investigated. The Tersoff–Brenner potential is adopted to describe the interactions between carbon atoms. Results show that the changes of strain energies are dependent on chirality but independent of the tubes' radii. Under large axial tension, abrupt changes of the configurations of SWCNTs may occur when the strains exceed 0.382 for armchair patterns and 0.43 for zigzag patterns. The reason is that the changes of bond lengths and angles lead to corresponding changes of many-body coupling interactions between atoms. Then the tubes reach new equilibrium states. Such abrupt changes of configurations must trigger fracture of CNTs in a dynamic deformation.

Measurement of radial deformation of single-wall carbon nanotubes induced by intertube van der Waals forces

A cylindrical single-wall carbon nanotube (SWCNT) deforms in a nanotube bundle by van der Waals forces. The deformation is hard to measure, but is required in order to understand the properties of bundles. Here, we show that such deformations can be measured from changes in electron diffraction intensities. We demonstrate this for a bundle of two SWCNTs ($d=2.05$ and $1.56\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$). Deformation model predicted by atomistic simulations is scaled to fit the experiment. We show that the best fit gives flattening values of 0.7 and $0.35\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ at the middle of the binding surface for the two SWCNTs.

Buckling behaviors of single-walled carbon nanotubes filled with metal atoms

Molecular dynamics method is employed to investigate the buckling deformations of single-walled carbon nanotubes (SWCNTs) filled with nickel (Ni), copper (Cu), and platinum (Pt) atoms under axial compression. The critical buckling strains of filled tubes decrease linearly before a critical number of metal atoms and then increase linearly when more atoms are encapsulated. For SWCNT completely filled with metals, its critical strain is larger than that of the hollow tube. Furthermore, the critical strain of SWCNT completely filled with Ni atoms is larger than that of the tube fully filled with Cu or Pt atoms.

Numerical study of equilibrium shapes and deformation of single-wall carbon nanotubes

A method is proposed for constructing equilibrium shapes and studying the energetics of perfect single-wall carbon nanotubes in terms of five geometrical parameters including the twist, radial expansion, axial stretch, and two components of the inner displacement of the interwoven Bravais lattices defining the unrolled, planar, hexagonal graphene sheet. Values of these quintuplets at equilibrium are computed over an extensive range of chiralities by solving an unconstrained energy minimization problem using Brenner’s interatomic potential. An analogous formulation for computing the parameters defining the deformed tube shape under the auspices of molecular membrane mechanics is discussed. Three fundamental modes of deformation that preserve the cylindrical shape are identified describing extension that allows for expansion and twist, and their two cyclic permutations. The tube response is studied in each case by solving constrained energy minimization problems in four scalar variables. The results demonstrate the coupling between the elementary modes of deformation, and reveal topological transitions when the deformation exceeds certain thresholds. Under these critical conditions, a change in neighbors occurs signaling reconstruction and plastic deformation. Graphene rolling and tube deformation are shown to have a significant effect on the electronic band gap.

Theoretical studies on the charge-induced failure of single-walled carbon nanotubes

Electromechanical coupling in single-walled carbon nanotubes has been studied theoretically. The charge distribution on single-walled carbon nanotubes in an electric field is obtained by an atomistic moment method based on classical electrostatics theory. The electrostatic interactions between charged carbon atoms are calculated using the Coulomb law. The charge-induced deformations of single-walled carbon nanotubes in axial and radial directions are obtained by using the molecular structural mechanics method and considering the electrostatic interactions as external loads acting on carbon atoms. The electrical failure of charged carbon nanotubes is found to be controlled by the charge level and also affected by the caps on the nanotube ends. The results indicate that the bond breaking first appears at the tube ends and the end-caps can enhance the stability of the nanotubes.

Evaluation of Nonuniform Strain in Carbon Nanotube with Bend Junction

Carbon nanotubes (CNTs) have been attracting attention because of their prominent mechanical and electronic properties. In this study, we investigate the deformation of single-walled carbon nanotubes (SWCNTs) with a bend junction using molecular dynamics in order to analyze strain concentration due to the tube shape (macroscopic effect) and the membered ring shape (microscopic effect). At first, we propose a method for evaluating the local strain at each memberded ring. Then, we analyze the strain concentration at the bend junction due to the both of macroscopic and microscopic effects. The strain concentration caused by the microscopic effect is separated by a rough analysis; and it is about 1.5 times higher than that caused by the macroscopic effect.