Curvature effects on pressure-induced buckling of empty or filled double-walled carbon nanotubes

Summary The curvature effects of interlayer van der Waals (vdW) forces on pressure-induced buckling of empty or filled double-walled carbon nanotubes (DWNTs) are studied for various radii, length-to-radius ratios, end conditions and internal-to-external pressure ratios. The analysis is based on a double-elastic shell model and assumes that the interlayer vdW pressure at a point between the inner and outer tubes depends not only on the change of the interlayer spacing, but also on the change of the curvatures of the inner and outer tubes at that point. Here the role of filling substances inside DWNTs is modeled by a uniformly distributed internal pressure. The present work aims to study the curvature effects on critical radial pressure. An explicit formula is obtained for the external buckling pressure of empty or filled DWNTs. The critical value of external pressure is estimated with various internal-to-external pressure ratios. It is shown that the curvature effects play a more significant role in buckling problems under radial pressure for small radii DWNTs than under pure axial stress. Our results show that loading transfer through vdW forces prior to buckling is important for the pressure-induced buckling of DWNTs rather than axially compressed buckling.     

Analysis of wave propagation in carbon nanotubes via elastic shell theories

This paper investigates wave propagation in both single-walled carbon nanotubes (SWNTs) and double-walled carbon nanotubes (DWNTs) via two developed elastic shell theories: Love’s thin cylindrical shell theory and the Cooper–Naghdi thick cylindrical shell theory. In studying DWNTs, the van der Waals effect is accounted for and modeled with the two theories. The elastic thick shell theory, in which the shear and inertia effects are taken into account, is developed first to investigate the wave propagations of CNTs to provide more accurate wave dispersions for higher modes. The material properties of the CNTs that are used in the two shell theories are proposed, and the expression of the inertia moment of the cross area in the thick shell theory is recommended. The dispersion results that are derived via the two theories are compared to show the feasibility of those theories in studying CNTs. Radius-dependent wave propagation results in SWNTs and DWNTs are also studied via the two theories.     

Thermal-gradient-induced interaction energy ramp and actuation of relative axial motion in short-sleeved double-walled carbon nanotubes

We investigate the phenomenon of actuation of relative linear motion in double-walled carbon nanotubes (DWNTs) resulting from a temperature gradient. Molecular dynamics simulations of DWNTs with short outer tube reveal that the outer tube is driven towards the cold end of the long inner tube. It is also found that the terminal velocity of the sleeve roughly depends linearly on the applied thermal gradient. We calculate the inter-tube interaction energy surface which is revealed to have a gradient depending upon the applied thermal gradient. Consequently, it is proposed that the origin of the thermophoretic motion of the outer tube may be attributed partially to the existence of such an energy gradient. A simple analytical model is presented accounting for the gradient in energy profile as well as the effect of biased thermal noise. It is shown that the proposed model predicts the dynamical behaviour of the long-time performance reasonably well.     

Selective tuning of the electronic properties of coaxial nanocables through exohedral doping

The electronic properties of exohedrally doped double-walled carbon nanotubes (DWNTs) have been investigated using density functional theory and resonance Raman spectroscopy (RRS) measurements. First-principles calculations elucidate the effects of exohedral doping on the M@S and S@M systems, where a metallic (M) tube is either inside or outside a semiconducting (S) one. The results demonstrate that metallic nanotubes are extremely sensitive to doping even when they are inner tubes, in sharp contrast to semiconducting nanotubes, which are not affected by doping when the outer shell is a metallic nanotube (screening effects). The theoretical predictions are in agreement with RRS data on Br2- and H2SO4-doped DWNTs. These results pave the way to novel nanoscale electronics via exohedral doping.     

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.     

High-temperature thermal stability and axial compressive properties of a coaxial carbon nanotube inside a boron nitride nanotube

The structural performance of double-walled C(5, 5)@BN(10, 10) and C(5, 5)@C(10, 10) nanotubes subject to high temperatures is investigated through molecular dynamics simulations. It is found that the inner tube C(5, 5) in the C(5, 5)@BN(10, 10) exhibits less distortion than that in the C(5, 5)@C(10, 10) at annealing temperatures of 3500 and 4000 K. The C(5, 5)@BN(10, 10) and C(5, 5)@C(10, 10) models with different axial compressive strains are optimized using the universal force field (UFF) method. It is found that the critical buckling strains of the inner tubes in the C(5, 5)@BN(10, 10) and C(5, 5)@C(10, 10) are 12.74% and 9.1%, respectively. The critical buckling strain of the former is larger than that of the latter; although the former exhibits greater deformation and energy loss after buckling than does the latter. These phenomena are also analyzed on the basis of the radial distribution function (RDF) and system energy. The results of this study indicate that the outer tube boron nitride nanotube (BNNT) has a better protective effect on the inner tube than does the outer tube carbon nanotube (CNT) under both high-temperature and lower compressive strain conditions. In these cases, the thermal stability and compressive resistance properties of the C(5, 5)@BN(10, 10) are superior to those of the C(5, 5)@C(10, 10).     

Buckling of circular/annular Mindlin nanoplates via nonlocal elasticity

The present study proposes an analytical solution for the axisymmetric/asymmetric buckling analysis of moderately thick circular/annular Mindlin nanoplates under uniform radial compressive in-plane load. In order to consider small-scale effects, nonlocal elasticity theory of Eringen is employed. To ensure the efficiency and stability of the present methodology, the results are compared with other ones presented in the literature. Further the exact closed-form solution is obtained using three potential functions. In addition, the effect of small scales on buckling loads for different parameters such as geometry of the nanoplate, boundary conditions, and axisymmetric/asymmetric mode numbers, is investigated. It is observed that the buckling mode shape for annular nanoplates, which corresponds to the lowest critical buckling load, may be axisymmetric or asymmetric depending on boundary conditions, inner to outer radius ratios, and thickness of the nanoplate. In other words, for stiffer boundary conditions and smaller inner to outer radius ratios, the mode shape corresponding to the lowest critical buckling load is an asymmetric mode. Also, the difference between axisymmetric and asymmetric buckling loads for higher mode numbers, greater thickness to outer radius ratios and smaller outer radii decreases by increasing the nonlocal parameter.     

Eccentric compression stability of multi-walled carbon nanotubes embedded in an elastic matrix

This paper reports the results of an investigation on the eccentric compression stability of multi-walled carbon nanotubes embedded in an elastic matrix. Based on continuum modeling, a multilayer shell model is presented for the eccentric compression buckling of multi-walled carbon nanotubes embedded in an elastic matrix, in which the effect of van der Waals forces between two adjacent tubes is taken into account. The critical bending moment and the eccentric compression mode for three types of multi-walled carbon nanotubes with different layer numbers and ratios of radius to thickness are calculated. Results obtained show that the eccentric compression buckling mode corresponding the critical bending moment is unique, and is different from the purely axial compression buckling of an individual multi-walled carbon nanotube. For different types of multi-walled carbon nanotubes, the effect of matrix stiffness on the critical bending moment of multi-walled carbon nanotubes under eccentric compression loading is obviously different, and is dependent on the innermost radius and layer numbers of the multi-walled carbon nanotubes. The critical bending stress exerted on the center tubes of nearly solid multi-walled carbon nanotubes does not change as the ratio of the axial compression loading to the bending membrane force increases. The new features and meaningful numerical results in this paper are helpful for the application and the design of nanostructures in which multi-walled carbon nanotubes act as basic elements.     

Vibration and buckling analysis of two-layered functionally graded cylindrical shell,considering the effects of transverse shear and rotary inertia

This research investigates the free vibration and buckling of a two-layered cylindrical shell made of inner functionally graded (FG) and outer isotropic elastic layer, subjected to combined static and periodic axial forces. Material properties of functionally graded cylindrical shell are considered as temperature dependent and graded in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. Theoretical formulations are presented based on two different methods of first-order shear deformation theory (FSDT) considering the transverse shear strains and the rotary inertias and the classical shell theory (CST). The results obtained show that the transverse shear and rotary inertias have considerable effect on the fundamental frequency of the FG cylindrical shell. The results for nondimensional natural frequency are in a close agreement with those in literature. It is inferred from the results that the geometry parameters and material composition of the shell have significant effect on the critical axial force, so that the minimum critical load is obtained for fully metal shell. Good agreement between theoretical and finite element results validates the approach. It is concluded that the presence of an additional elastic layer significantly increases the nondimensional natural frequency, the buckling resistance and hence the elastic stability in axial compression with respect to a FG hollow cylinder.     

In situ Raman study on single- and double-walled carbon nanotubes as a function of lithium insertion

We investigated the electrochemical lithium ion (Li(+)) insertion/desertion behavior on highly pure and bundled single- and double-walled carbon nanotubes (SWNTs and DWNTs) using an in situ Raman technique. In general, two storage sites could host Li(+) in SWNT and DWNT bundles when varying an external potential: a) the outer surface sites, and b) the interstitial spaces within the bundles. The most sensitive changes in the tangential mode (TM) of the Raman spectra upon doping with Li(+) can be divided into two regions. The first region was found from 2.8 to 1.0 V (the coverage of Li(+) on the outer surface of a bundled nanotube) and was characterized by the loss of resonant conditions via partial charge transfer, where the G(+) line of the SWNT and the TM of the outer tube of DWNTs experienced a highly depressed intensity, but remained almost constant in frequency. The appearance of a Breit-Wigner-Fano (BWF) profile provided strong evidence of metallic inner tubes within DWNTs. The second region was observed when the applied potentials ranged from 0.9 to 0 V and was characterized by Li(+) diffusion into the interstitial sites of the bundled nanotube material. This phenomenon invoked a large downshift of the G(-) band in SWNTs, and a small downshift of the TM of the inner tube of DWNTs caused by expansion of the C--C bonds due to the charge transferred to the nanotubes, and the disappearance of the BWF profile through the screening effect of the interstitial Li(+) layers.     

Controllable preparation and properties of single-/double-walled carbon nanotubes

In this paper, we discussed recent studies done in our laboratories with a floating catalyst chemical vapor deposition (CVD) method. We can grow single- or double-walled carbon nanotubes (SWNTs/DWNTs) with different kinds of catalysts. Single-walled carbon nanotubes without amorphous carbon coating were prepared by thermally decomposing acetylene (C₂H₂) at the temperature range of 750–1200 °C with ferrocene as catalyst. While with sulfur promoted ferrocene catalyst, double-walled carbon nanotubes were mass-produced by pyrolizing C₂H₂ at the temperature range of 900–1100 °C. Furthermore, tunable growth of DWNTs with different diameter was achieved in our experiment. It is found that DWNTs produced at lower carbon partial pressure have much smaller inner tubes, even DWNTs with the smallest inner diameter of 0.4 nm was found in here. As convenient and effective tool, radial breathing mode (RBM) of Raman scattering technique can be used to distinguish SWNTs from DWNTs. In further studies of Raman scattering with DWNTs, the possible match of the inner tubes and the outer tubes according to the RBM bands was assigned, and different chirality types were discussed according to the diameter and chirality dependence of resonant Raman vibration. We also investigated the temperature-dependent frequency shift of resonant Raman spectra of DWNTs in the range of 78–650 K. We found that different RBM peaks, which are relative to different tube diameters, have different temperature coefficient of frequency shift, and the larger diameter carbon nanotubes have more RBM frequency downshift with increasing temperature. It is ascribed to the RBM frequency variation to the temperature dependence of the stretching force constant of C–C bond. Besides, Polarized Raman spectra were preformed on well-aligned SWNTs structure fabricated through post-growth method and found that the angular dependence of Raman intensity is consistent well with the predictions of the resonance Raman theory.     

Experimental-computational study of shear interactions within double-walled carbon nanotube bundles

The mechanical behavior of carbon nanotube (CNT)-based fibers and nanocomposites depends intimately on the shear interactions between adjacent tubes. We have applied an experimental-computational approach to investigate the shear interactions between adjacent CNTs within individual double-walled nanotube (DWNT) bundles. The force required to pull out an inner bundle of DWNTs from an outer shell of DWNTs was measured using in situ scanning electron microscopy methods. The normalized force per CNT-CNT interaction (1.7 ± 1.0 nN) was found to be considerably higher than molecular mechanics (MM)-based predictions for bare CNTs (0.3 nN). This MM result is similar to the force that results from exposure of newly formed CNT surfaces, indicating that the observed pullout force arises from factors beyond what arise from potential energy effects associated with bare CNTs. Through further theoretical considerations we show that the experimentally measured pullout force may include small contributions from carbonyl functional groups terminating the free ends of the CNTs, corrugation of the CNT-CNT interactions, and polygonization of the nanotubes due to their mutual interactions. In addition, surface functional groups, such as hydroxyl groups, that may exist between the nanotubes are found to play an unimportant role. All of these potential energy effects account for less than half of the ~1.7 nN force. However, partially pulled-out inner bundles are found not to pull back into the outer shell after the outer shell is broken, suggesting that dissipation is responsible for more than half of the pullout force. The sum of force contributions from potential energy and dissipation effects are found to agree with the experimental pullout force within the experimental error.     

Performance of a fiber reinforced polymer web core skew bridge superstructure. Part I: field testing and finite element simulations

An analytical method is presented to investigate hydrothermal effects on locally buckling for an elliptical delamination near the surface of cylindrical laminated shells. The critical strain of non-linear buckling for a locally elliptic delamination of cylindrical laminated shells is obtained by considering transverse displacements of the elliptically sub-laminated shells. The stacking sequence of sub-laminated shells may be multilayer and asymmetric. The geometrical axis of sub-laminated shells may be arbitrary. The Young's modulus and the thermal and humidity expansion coefficients of the material are considered as functions of temperature change in base-laminated shells. The critical strains of locally buckling for cylindrical laminated shells subjected to hydrothermal effects are extracted for different stacking sequences, and different radii of base-laminated shells, by applying Rayleigh–Ritz method based on second variation of potential energy. From results carried out, it is found that the critical strain from the non-linear buckling for a locally elliptic delamination near the surface of a cylindrical laminated shell presents a lower value than that from linear considerations.     

Raman spectroscopy study of isolated double-walled carbon nanotubes with different metallic and semiconducting configurations

A double-walled carbon nanotube (DWNT) provides the simplest system to study the interaction between concentric layers in carbon nanotubes. The inner and outer walls of a DWNT can be metallic (M) or semiconducting (S), and each of the four possible configurations (M@M, M@S, S@S, S@M) has different electronic properties. Here we report, for the first time, detailed Raman spectroscopy experiments carried out on individual DWNTs, where both concentric tubes are measured under resonance conditions, in order to understand the dependence of their electronic and optical properties according to their configuration. Interestingly, for the three DWNTs that were studied, the inner-outer tube distance (e.g., 0.31-0.33 nm) was less than the interlayer spacing in graphite. We believe these results have important implications in the fabrication of electronic devices using different types of S and M tubular interconnects.     

On the buckling properties of concentric carbon/boron-nitride multi-walled nanotubes: A finite element investigation

Finite element modeling approach is used here to investigate the behavior of concentric multi-walled boron-nitride and carbon nanotubes under the compressive loadings. Double-walled and triple-walled concentric boron-nitride and carbon nanotubes with different arrangements and geometries are considered. It is shown that multi-walled boron-nitride nanotubes lose their stabilities at larger compressive forces than other arrangements in which at least one carbon wall exists. Comparing the armchair and zig-zag multi-walled nanotubes, the latter one has larger buckling force than the former one. It is also shown that the nanotubes with smaller radii have larger critical compressive forces than those with larger radii.     

Bolt-and-Nut Pairs Made from Carbon Nanotubes with Artificial Defects

Structures of double-wall carbon nanotubes (DWNTs) with atomic scale defects that can operate as bolt-and-nut pair are analyzed. The relative thread depth for such bolt-and-nut pair is calculated for various types of defects in inner and outer walls of (4,6)@(12,8) and (8,2)@(16,4) DWNTs. It is found that a type of defect determines only the absolute thread depth but weakly influences on the relative thread depth. Possibility of producing of the DWNT (which can operate as bolt-and-nut pair) by self-organization is proposed.     

The eigenvalue problem for hollow circular cylinders

In three-dimensional elasticity the solution of the biharmonic equation for a hollow circular cylinder can be presented in terms of Bessel functions. If there are no surface tractions on either of the radial faces and no thermal effects, an eigenvalue problem arises. A method of establishing these eigenvalues and tables of them for various types of hollow cylinders are presented. Two special cases are investigated, namely, as the ratio of radii tends to unity, that is, a ‘thin shell’, and as the ratio tends to infinity, which can either be regarded as the inner radius tending to zero for a fixed outer radius or as the outer radius tending to infinity for a fixed inner.The eigenvalues are subsequently used for the calculation of the effect of end loading on a semi-infinite length cylinder. From this a quantitative comparison can be made with thin shell theory in the transition region between thin and thick shells of this type.     

Postbuckling of nanotube-reinforced composite cylindrical shells in thermal environments,Part II: Pressure-loaded shells

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to lateral or hydrostatic pressure in thermal environments. The multi-scale model for functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shells under external pressure is proposed and a singular perturbation technique is employed to determine the buckling pressure and postbuckling equilibrium path. Numerical results for pressure-loaded, perfect and imperfect, FG-CNTRC cylindrical shells are obtained under different sets of thermal environmental conditions. The results for uniformly distributed CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell. The results show that the linear functionally graded reinforcements can increase the buckling pressure as well as postbuckling strength of the shell under external pressure. The results reveal that the carbon nanotube volume fraction has a significant effect on the buckling pressure and postbuckling behavior of CNTRC shells.     

A molecular dynamics study on the thickness and post-critical strength of carbon nanotubes

This paper presents a molecular dynamics (MDs) study on the linear, buckling and post-buckling behaviour of carbon nanotubes (CNTs) under pure shortening and pure twisting. Its objectives are (i) to clarify the issue about the most correct thickness value to adopt in the simulation of CNTs using shell models and (ii) to evaluate their post-critical strength. Three CNTs with similar length-to-diameter ratio but different atomic structures (zig-zag, armchair and chiral) are selected for this study. Then, MD simulations are performed to investigate the pre-critical, critical buckling and post-critical behaviour of CNTs under pure shortening and pure twisting. Using available analytical formulae derived from shell models, the influence of CNT thickness on their critical strain and critical angle of twist is investigated. Some conclusions are drawn regarding (i) the most appropriate choice of the thickness value to use in shell models and (ii) the effectiveness of post-critical stiffness and strength of CNTs.     

Structure determinations of double-wall carbon nanotubes grown by catalytic chemical vapor deposition

We report an application of nanoarea electron diffraction for structure determination of double-wall carbon nanotubes (DWNT) grown by catalytic chemical vapor deposition. The structures of 30 tubes were determined from experimental diffraction patterns. Among these tubes, the inner and outer wall structure of 18 tubes was precisely determined by comparison with kinematic electron diffraction simulations. For the structure of the DWNTs, our experiment revealed a mixture of semiconducting-metallic (S-M), S-S and M-M tubes. The spacing between the two walls ranges from 0.335 nm to 0.384 nm. Most DWNTs are incommensurate and chiral.