Intertube Van Der Waals Forces Research Articles

Dynamic analysis of multiple-nanobeam-systems acted upon by multiple moving nanoparticles accounting for nonlocality, lag, and lateral inertia

This paper deals with nonlocal transverse vibrations of vertically aligned single-layered membranes from single-walled carbon nanotubes (SWCNTs) for transporting of nanoparticles. The consisting nanotubes are modeled via classical and shear deformable beams, while the moving nanoparticles is assumed as rigid-in contact-particles with dissimilar velocities accounting for their weight force, inertia, and lags. The nonlocal equations of motion are derived by considering the intertube van der Waals (vdW) forces and then solved for elasto-dynamic fields methodically. For two decorations of the nanosystem, the effects of inertia, shear deformation, nonlocality, velocities and lags of moving nanoparticles on the dynamic response of the nanosystem are explained in some detail. Among various scenarios of loadings, lags, and velocities of moving nanoparticles, the most critical case is introduced and discussed. By increasing the slenderness ratio of nanotubes, the role of shear deformation diminishes and the results of various models approach each other. Further, the influence of vdW forces on the dynamic deflections reduces and the discrepancies between the results of two understudy nanosystem’s patterns would reduce. The newly developed models in the present study can be employed for modeling of more complex nanosystems like as vertically aligned forests of SWCNTs for delivery of nanoparticles with different lags and velocities.

Voltage effect on amplitude–frequency response of parametric resonance of electrostatically actuated double-walled carbon nanotube resonators

This work deals with the amplitude–frequency response of coaxial parametric resonance of electrostatically actuated double-walled carbon nanotubes (DWCNTs). Nonlinear forces acting on the DWCNT are intertube van der Waals and electrostatic forces. Soft alternating current (AC) excitation and small viscous damping are assumed. In coaxial vibration, the outer and inner carbon nanotubes move synchronously (in-phase). Euler–Bernoulli beam model is used for DWCNTs of high length-to-diameter ratio. Modal coordinates are used for decoupling the linearized differential equations of motion without damping. The reduced-order model (ROM) method is used in this investigation. All ROMs using one through five modes of vibration (terms) are developed in terms of modal coordinates. ROM using one term is solved and frequency–amplitude response predicted by using the method of multiple scales (MMS). All other ROMs using two through five terms are numerically integrated using MATLAB in order to simulate time responses of the structure and also solved using AUTO-07P, a software package of continuation and bifurcation, in order to predict the frequency–amplitude response. All models and methods are in agreement at lower amplitudes, while in higher amplitudes only ROM with five terms provides reliable results. The effects of voltage and damping on the amplitude–frequency response of electrostatically actuated DWCNTs are reported. It is shown that increasing voltage and/or decreasing damping results in a larger range of frequencies for which pull-in occurs.

Investigation on mechanical vibration of double-walled carbon nanotubes with inter-tube Van der waals forces

This work represents the study of the vibration response of the double walled carbon nanotubes (DWCNT) for various boundary conditions. The inner and outer carbon nanotubes are modeled as two individual Euler-Bernoulli's elastic beams interacting each other by Van der waals force. Differential transform method (DTM) is used as a numerical method to solve the governing differential equations and associated boundary conditions. The influence of Winkler elastic medium on vibration frequency is also examined and results are interpreted. MATLAB is used as a tool for solving the governing differential equations. The fundamental natural frequencies are validating with those available in literature and observed a good agreement between them.

Nonlocal free dynamic analysis of periodic arrays of single-walled carbon nanotubes in the presence of longitudinal thermal and magnetic fields

This paper deals with transverse vibrations of three-dimensional vertically aligned periodic arrays of single-walled carbon nanotubes acted upon by both longitudinal magnetic and thermal fields. For this purpose, a nonlocal higher-order beam theory is employed for modeling of the nanosystem. Accounting for the intertube van der Waals forces, a discrete-based model as well as a continuous-based model is established. For various geometries of the nanosystem and under various applied thermal and magnetic fields, the results of the continuous model are successfully checked with those of the discrete model. Through a comprehensive parametric study, the effects of the nanotube’s radius, slenderness ratio, nonlocality, ensemble’s population, temperature change, and strength of magnetic field on fundamental frequencies of the nanosystem are addressed and displayed.

Effect of Longitudinal Magnetic Field on Vibration Response of Double-Walled Carbon Nanotubes Embedded in Viscoelastic Medium

This paper aims to study the effect of externally applied longitudinal magnetic field on the transverse vibration of viscoelastic double-walled carbon nanotubes (visco-DWCNTs) embedded in a viscoelastic medium. The analyses are carried out based on the nonlocal viscoelastic model and Euler–Bernoulli beam theory. Governing equations are derived for the vibration of the embedded visco-DWCNT subjected to a magnetic field, where the Lorentz magnetic force, the surrounding viscoelastic medium, the intertube van der Waals forces and viscoelasticity of the DWCNT are taken into consideration. In this study, the transfer function method is employed to solve the governing equations, which enables one to obtain the natural frequencies and the corresponding mode shapes in closed form for the DWCNTs with arbitrary boundary conditions. Here the developed mechanics model is first compared with the existing techniques available in the literature in a few particular cases, where excellent agreement is achieved. The validation of the model is followed by a detailed parametric study of the effects of longitudinal magnetic field, nonlocal parameter, boundary conditions, structural damping coefficient and aspect ratio of the DWCNTs on their vibration. The study demonstrates the efficiency of the present technique designed for vibration analysis of a complicated multi-physics system comprising DWCNTs, the viscoelastic medium and a magnetic field in longitudinal direction.

Theoretical Study of Sp2-sp3 Hybrid Carbon Networks for Energy Storage

Carbon nanostructures have attracted a lot of attentions in hydrogen storage for its light weight and high surfaces area. However, carbon nanotubes, graphene expanded structures suffer from the weak H2 physisorptions for practical use.[1] Light metal doping is thought to be a feasiable way to improve the H2 binding and those structures are predicted to have ultra-high hydrogen storage capacity.[2] Nevertheless, no experiment has yielded comparable capacity to the theoretical expectations. Sigal et al. [3] doubt about the practicability of hydrogen storage on metal-doped two-dimensional (2D) systems since even slight oxygen content in H2 atmosphere can block the metal sites which suppresses the H2 adsorption. The metal-decorated porous structures could be more practical by hindering the oxygen access to the metal sites. Three-dimensional (3D) porous structures such as carbide-derived carbons,[4] metal organic framework (MOF)[5] and covalent organic framework (COF) [6] are designed and exhibit promising capacity close to the DOE target. However, none of nanoporous structures could meet the requirement of hydrogen storage in both weight percentage and volumetric density up to now. We present here that the bundles of carbon nanotubes could undergo phase transition to a type of 3D sp2-sp3 hybrid carbon networks (HCN) by changing intertube van der Waals forces to strong sp3 covalent bonding. With well-aligned and size-tunable 1-D nanopores, we show high Li capacity, stable Li-ion intercalation voltage profile, low Li diffusion barriers, slight volumetric change upon Li-doping in HCN, making it a potential Li-ion battery anode material. [7] Moreover, the unique nanoporous strcuture is beneficial for the achoring Li atoms and induces improved H2 binding energy due to the polarized interaction between H2 and Li ion. A maximum hydrogen capacity of 6.28 wt % is reached with average binding energy of -0.19 eV in Li-doped HCN. Combined with an ultrahigh volumetric density of hydrogen (102 g/L), it is a promising candidate for the hydrogen storage medium.[8]

Nonlocal axial load-bearing capacity of two neighboring perpendicular single-walled carbon nanotubes accounting for shear deformation

This study is devoted to examine load-bearing capacity of a nanosystem composed of two adjacent perpendicular single-walled carbon nanotubes (SWCNTs) which are embedded in an elastic matrix. Accounting for the nonlocality and the intertube van der Waals forces, the governing equations are established based on the nonlocal Euler–Bernoulli, Timoshenko, and higher-order beam theories. These are sets of coupled integro-ordinary differential equations whose analytical solutions are unavailable. Hence, an efficient meshless methodology is proposed and the discrete governing equations are obtained via Galerkin approach. By solving the resulting set of eigenvalue equations, the axial buckling load of the elastically embedded nanosystem is evaluated. The roles of the radius and slenderness ratio of the constitutive SWCNTs, free distance between two tubes, small-scale parameter, aspect ratio, transverse and rotational stiffness of the surrounding matrix on the axial buckling load of the nanosystem are comprehensively addressed. The obtained results can be regarded as a pivotal step for better understanding the mechanism of elastic buckling of more complex systems such as elastically embedded-orthogonal membranes or even forests of SWCNTs.

Vibrations of double-nanotube systems with mislocation via a newly developed van der Waals model

This study deals with transverse vibrations of two adjacent-parallel-mislocated single-walled carbon nanotubes (SWCNTs) under various end conditions. These tubes interact with each other and their surrounding medium through the intertube van der Waals (vdW) forces, and existing bonds between their atoms and those of the elastic medium. The elastic energy of such forces due to the deflections of nanotubes is appropriately modeled by defining a vdW force density function. In the previous works, vdW forces between two identical tubes were idealized by a uniform form of this function. The newly introduced function enables us to investigate the influences of both intertube free distance and longitudinal mislocation on the natural transverse frequencies of the nanosystem which consists of two dissimilar tubes. Such crucial issues have not been addressed yet, even for simply supported tubes. Using nonlocal Timoshenko and higher-order beam theories as well as Hamilton's principle, the strong form of the equations of motion is established. Seeking for an explicit solution to these integro-partial differential equations is a very problematic task. Thereby, an energy-based method in conjunction with an efficient meshfree method is proposed and the nonlocal frequencies of the elastically embedded nanosystem are determined. For simply supported nanosystems, the predicted first five frequencies of the proposed model are checked with those of assumed mode method, and a reasonably good agreement is achieved. Through various studies, the roles of the tube's length ratio, intertube free space, mislocation, small-scale effect, slenderness ratio, radius of SWCNTs, and elastic constants of the elastic matrix on the natural frequencies of the nanosystem with various end conditions are explained. The limitations of the nonlocal Timoshenko beam theory are also addressed. This work can be considered as a vital step towards better realizing of a more complex system that consists of vertically aligned SWCNTs of various lengths.

Vibration analysis of multi-walled carbon nanotubes embedded in elastic medium

We propose a method to estimate the natural frequencies of the multi-walled carbon nanotubes (MWCNTs) embedded in an elastic medium. Each of the nested tubes is treated as an individual bar interacting with the adjacent nanotubes through the inter-tube Van der Waals forces. The effect of the elastic medium is introduced through an elastic model. The mathematical model is finally reduced to an eigen value problem and the eigen value problem is solved to arrive at the inter-tube resonances of the MWCNTs. Variation of the natural frequencies with different parameters are studied. The estimated results from the present method are compared with the literature and results are observed to be in close agreement.

Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field

Based on nonlocal theory of thermal elasticity mechanics, an elastic multiple column model is developed for column buckling of MWNTs with large aspect ratios under axial compression coupling with temperature change. The present model treats each of the nested tubes as an individual column interacting with adjacent nanotubes through the intertube van der Waals forces. The thermal effect is incorporated in the formulation. In particular, an explicit expression is derived for the critical axial strain of a double-walled carbon nanotube which clearly demonstrates that small scale effects contribute significantly to the thermo-mechanical behavior of multiwalled carbon nanotubes and cannot be ignored.

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.

Non-Twisted and Twisted CNT Bundles under Axial Tensile and Compressive Loads

Compressive and tensile properties of non-twisted and twisted carbon nanotube (CNT) bundles are studied using molecular dynamics (MD) simulations. The results reveal that non-twisted CNT bundles exhibit better compressive and tensile properties than twisted CNT bundles. When the twist angle of a CNT bundle is greater than 60°, its buckling load dropped considerably due to the significant curve geometries of the surrounding single-walled carbon nanotubes (SWCNTs). Twisted CNT bundles also do not demonstrate good tensile properties. During tensile loading, the intertube distance in CNT bundle with twisting angle greater than 75° will decrease as the strain increases. This gives rise to increasing intertube van der Waals forces until the intertube distance falls below 2.0Å. At this time, the repulsive force between the SWCNTs suddenly increases causing the SWCNTs to repel one another at junction areas where they overlap. As a result, twisted CNT bundles tend to fail at these junction areas with lower failure load and strain.

Vibration of a double-walled carbon nanotube aroused by nonlinear intertube van der Waals forces

Vibration of a double-walled carbon nanotube aroused by nonlinear interlayer van der Waals (vdW) forces is studied. The interlayer vdW forces as a nonlinear function are described by the interlayer spacing. The inner and outer carbon nanotubes are modeled as two individual elastic beams. Detailed results are demonstrated for double-walled carbon nanotubes (DWCNTs) with an aspect ratios of 10 and 20, based on the simply supported, fixed, or free end conditions, respectively. Harmonic balance method is used to analyze the relation between the amplitudes of deflection and the frequencies of coaxial and noncoaxial free vibrations. Our results indicate that the nonlinear factors of vdW forces have little effect on the coaxial free vibration, and that the deflection amplitudes increase rapidly with the increasing frequency, which are almost the same with those of the linear free vibration. On the other hand, the nonlinear factors of vdW forces have a great effect on noncoaxial free vibration. The relation between the deflection amplitudes and the frequencies shows nonlinear trend, which indicates that the aspect ratio and end condition almost have no affect on the noncoaxial amplitudes of DWCNTs.

Investigation of the dynamic buckling of doublewalled carbon nanotube subjected to axial periodic disturbing forces

The dynamic response of a double-walled carbon nanotube embedded in elastic medium subjected to periodic disturbing forces is investigated. Investigation of the dynamic buckling of a double-walled carbon nanotube develops continuum model. The effect of the van der Waals forces between two tubes and the surronding elastic medium for axial dynamic buckling are considered. The buckling model subjected to periodic disturbing forces and the critical axial strain and the critical frequencies are given. It is found that the critical axial strain of the embedded multi-walled carbon nanotube due to the intertube van der Waals forces is lower than that of an embedded single-walled carbon nanotube. The van der Waals forces and the surrounding elastic medium affect region of dynamic instability. The van der Waals forces increase the critical frequencies of a double-walled carbon nanotube. The effect of the surrounding elastic medium for the critical frequencies is small.

The van der Waals interactions of concentric nanotubes

Concentric fullerene tubes (nanotubes) are stabilized by a competition between the short-range kinetic-energy repulsion and the long-range van der Waals binding. At the relevant binding distances (3–3.5 Å or 6–7 Bohr radii) the traditional first-principle density-functional theory calculations cannot account for these combined interactions. An accurate quantum-physics account of the multi-walled nanotube structure and dynamics must therefore also include a determination of the intertube van der Waals forces. We use a successful model of the electrodynamical response of the nanotube electron gas to provide such a combined description in which we reflect the important self-consistent screening effects arising within the nanotube electron gases. Our description differs significantly from traditional asymptotic calculations of the van der Waals interactions and offers an approximate determination at the relevant intertube binding distances.

Axially compressed buckling of a doublewalled carbon nanotube embedded in an elastic medium

An elastic double-shell model is presented for infinitesimal buckling of a double-walled carbon nanotube embedded in an elastic matrix under axial compression. The analysis is based on a Winkler model for the surrounding elastic medium and a simplified model for the van der Waals interaction between the inner and outer nanotubes. An explicit formula is derived for the critical axial strain, which indicates the effects of the surrounding elastic matrix combined with the intertube van der Waals forces. In particular, the present model predicts that the critical axial strain of the embedded double-walled nanotube is lower than that of an embedded single-walled nanotube under otherwise identical conditions. This implies that inserting an inner tube lowers the critical axial strain of an embedded single-walled carbon nanotube, although the total critical compressive force could be increased due to the increase in the cross-sectional area of the nanotube. The reduced critical axial strain is attributed to the intertube slips between the inner and outer tubes. This result indicates that embedded multi-walled carbon nanotubes could be even more susceptible to infinitesimal axial buckling than embedded single-walled carbon nanotubes.

Column buckling of multiwalled carbon nanotubes with interlayer radial displacements

An elastic model is presented for column buckling of a multiwalled carbon nanotube embedded within an elastic medium. The emphasis is placed on the role of interlayer radial displacements between adjacent nanotubes. In contrast to an existing model which treats the entire multiwalled nanotube as a single column, the present model treats each of the nested tubes as an individual column interacting with adjacent nanotubes through the intertube van der Waals forces. Based on this model, a condition is derived in terms of the parameters describing the van der Waals interaction, under which the effect of the noncoincidence of all deflected column axes is so small that it does not virtually affect the critical axial strain. In particular, this condition is met for carbon multiwalled nanotubes provided that the half-wavelength of the buckling mode is much larger than the outermost diameter. In this case, the critical axial strain can be predicted correctly by the existing single-column model. On the other hand, the existing model could overestimate the critical axial strain when the half-wavelength of the buckling mode is close to or smaller than the outermost radius.

Elastic properties of crystals of single-walled carbon nanotubes

The elastic constants, Young's and bulk moduli, and Poisson ratio of triangular close-packed crystal lattices of single-walled carbon nanotubes are calculated for various tube types using analytical expressions. Some of these quantities exhibit up to three different regimes of behavior which are due to the interplay of the intertube van der Waals forces and the elastic forces in the tubes. These regimes are most prominent in the case of the bulk modulus which has a maximal value of 38 GPa at tube radius R≈6 A ̊ and decreases for larger radii.