Axisymmetric compressive buckling of multi-walled carbon nanotubes under different boundary conditions

The paper studies the axisymmetric compressive buckling behavior of multi-walled carbon nanotubes （MWNTs） under different boundary conditions based on continuum mechanics model. A buckling condition is derived for determining the critical buckling load and associated buckling mode of MWNTs, and numerical results are worked out for MWNTs with different aspect ratios under fixed and simply supported boundary conditions. It is shown that the critical buckling load of MWNTs is insensitive to boundary conditions, except for nanotubes with smaller radii and very small aspect ratio. The associated buckling modes for different layers of MWNTs are in-phase, and the buckling displacement ratios for different layers are independent of the boundary conditions and the length of MWNTs. Moreover, for simply supported boundary conditions, the critical buckling load is compared with the corresponding one for axial compressive buckling, which indicates that the critical buckling load for axial compressive buckling can be well approximated by the corresponding one for axisymmetric compressive buckling. In particular, for axial compressive buckling of double-walled carbon nanotubes, an analytical expression is given for approximating the critical bucklingload. The present investigation may be of some help in fur- ther understanding the mechanical properties of MWNTs.

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 surrounding 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.

Dynamic torsional buckling of multi-walled carbon nanotubes embedded in an elastic medium

In this paper the dynamic torsional buckling of multi-walled carbon nanotubes （MWNTs） embedded in an elastic medium is studied by using a continuum mechanics model. By introducing initial imperfections for MWNTs and applying the preferred mode analytical method, a buckling condition is derived for the buckling load and associated buckling mode. In particular, explicit expressions are obtained for embedded double-walled carbon nanotubes （DWNTs）. Numerical results show that, for both the DWNTs and embedded DWNTs, the buckling form shifts from the lower buckling mode to the higher buckling mode with increasing the buckling load, but the buckling mode is invari- able for a certain domain of the buckling load. It is also indicated that, the surrounding elastic medium generally has effect on the lower buckling mode of DWNTs only when compared with the corresponding one for individual DWNTs.

A Continuum Shell Model Including van derWaals Interaction for Free Vibrations of Double-Walled Carbon Nanotubes

This paper proposes the free vibration analysis of Double-Walled Carbon NanoTubes(DWCNTs).A continuum elastic three-dimensional shell model is used for natural frequency investigation of simply supported DWCNTs.The 3D shell method is compared with beam analyses to show the applicability limits of 1D beam models.The effect of van der Waals interaction between the two cylinders is shown for different Carbon NanoTube(CNT)lengths and vibration modes.Results give the van der Waals interaction effect in terms of frequency values.In order to apply the 3D shell continuum model,DWCNTs are defined as two concentric isotropic cylinders(with an equivalent thickness and Young modulus)which can be linked by means of the interlaminar continuity conditions or by means of an infinitesimal fictitious layer which represents the van der Waals interaction.

A Revisited Definition of the Three Solute Descriptors Related to the Van der Waals Forces in Solutions

It is currently admitted that the intermolecular forces implicated in Gas Liquid Chromatography (GLC) can be expressed as a product of parameters (or descriptors) of solutes and of parameters of solvents. The present study is limited to those of solutes, and among them the three ones are involved in the Van der Waals forces, whereas the two ones involved in the hydrogen bonding are left aside at this stage. These three studied parameters, which we call δ, ω and ε, respectively reflect the three types of Van der Waals forces: dispersion, orientation or polarity strictly speaking, and induction-polarizability. These parameters have been experimentally obtained in previous studies for 121 Volatile Organic Compounds (VOC) via an original Multiplicative Matrix Analysis (MMA) applied to a superabundant and accurate GLC data set. Then, also in previous studies, attempts have been made to predict these parameters via a Simplified Molecular Topology procedure (SMT). Because these last published results have been somewhat disappointing, a promising new strategy of prediction is developed and detailed in the present article.

Updated Definition of the Three Solvent Descriptors Related to the Van der Waals Forces in Solutions

Innovative viewpoint on the older topic of the van der Waals forces, is of interesting and significant issue to be concerned in both the fields related to the fundamental investigation and thus valuable in guiding the new physiochemical phenomena and processes for both academic research and practical applications. The intermolecular Van der Waals forces involved in solutions have been recently deeply reconsidered as far as the solute side is concerned. More precisely, the solute descriptors (or parameters) experimentally established, have been accurately related to molecular features of a Simplified Molecular Topology. In the present study, an equivalent result is reached on the solvent side. Both experimental parameters have been obtained simultaneously in previous Gas Liquid Chromatographic studies for 121 Volatile Organic Compounds and 11 liquid stationary phases, via an original Multiplicative Matrix Analysis. In that experimental step, five groups of forces were identified, two of hydrogen bonding and three of Van der Waals: 1) dispersion (London), 2) orientation or polarity strictly speaking (Keesom), and 3) induction-polarizability (Debye). At this stage, an attempt of characterization the solvent parameters via the SMT procedure has been limited to those related to the Van der Waals forces, those related to the hydrogen bonding being for now left aside.

NONLINEAR DYNAMIC INSTABILITY OF DOUBLE-WALLED CARBON NANOTUBES UNDER PERIODIC EXCITATION

A multiple-elastic beam model based on Euler-Bernoulli-beam theory is presented to investigate the nonlinear dynamic instability of double-walled nanotubes. Taking the geometric nonlinearity of structure deformation, the effects of van der Waals forces as well as the non- coaxial curvature of each nested tube into account, the nonlinear parametric vibration governing equations are derived. Numerical results indicate that the double-walled nanotube （DWNT） can be considered as a single column when the van der Waals forces are sufficiently strong. The stiffness of medium could substantially reduce the area of the nonlinear dynamic instability region, in particular, the geometric nonlinearity can be out of account when the stiffness is large enough. The area of the principal nonlinear instability region and its shifting distance aroused by the nonlinearity both decrease with the increment of the aspect ratio of the nanotubes.

Hard-to-Soft Transition in Radial Buckling of Multi-Concentric Nanocylinders

We investigate the cross-sectional buckling of multi-concentric tubular nanomaterials, which are called multiwalled carbon nanotubes (MWNTs), using an analysis based on thin-shell theory. MWNTs under hydrostatic pressure experience radial buckling. As a result of this, different buckling modes are obtained depending on the inter-tube separation d as well as the number of constituent tubes N and the innermost tube diameter. All of the buckling modes are classified into two deformation phases. In the first phase, which corresponds to an elliptic deformation, the radial stiffness increases rapidly with increasing N. In contrast, the second phase yields wavy, corrugated structures along the circumference for which the radial stiffness declines with increasing N. The hard-to-soft phase transition in radial buckling is a direct consequence of the core-shell structure of MWNTs. Special attention is devoted to how the variation in d affects the critical tube number Nc, which separates the two deformation phases observed in N -walled nanotubes, i.e., the elliptic phase for N Nc. We demonstrate that a larger d tends to result in a smaller Nc, which is attributed to the primary role of the interatomic forces between concentric tubes in the hard-to-soft transition during the radial buckling of MWNTs.

Dynamic shell buckling behavior of multi-walled carbon nanotubes embedded in an elastic medium

This paper studies the dynamic shell buckling behavior of multi-walled carbon nanotubes(MWNTs) embedded in an elastic medium under step axial load based on continuum mechanics model.It is shown that,for occurrence of dynamic shell buckling of MWNTs or MWNTs embedded in an elastic medium,the buckling stress is higher than the critical buckling stress of the corresponding static shell buckling under otherwise identical conditions.Detailed results are demonstrated for dynamic shell buckling of individual double-walled carbon nanotubes(DWNTs) or DWNTs embedded in an elastic medium.A phenomenon is shown that DWNTs or embedded DWNTs in dynamic shell buckling are prone to axisymmetric buckling rather than non-axisymmetric buckling.Numerical results also indicate that the axial buckling form shifts from the lower buckling mode to the higher buckling mode with increasing buckling stress,but the buckling mode is invariable for a certain domain of buckling stress.Further,an approximate analytic formula is presented for the buckling stress and the associated buckling wavelength for dynamic axisymmetric buckling of embedded DWNTs.The effect of radii is also examined.

DYNAMIC BUCKLING BEHAVIOR OF MULTI-WALLED CARBON NANOTUBES SUBJECTED TO STEP AXIAL LOADING

This paper studies the dynamic buckling behavior of multi-walled carbon nanotubes （MWNTs） subjected to step axial loading. A buckling condition is derived, and numerical results are presented for MWNTs under fixed boundary conditions. It is shown that the critical buckling load of MWNTs is of multi-branches and decreases as the time elongates. The associated buckling modes for different layers of MWNTs can be either in-phase or out of phase, which is related to the branch that the critical buckling load belongs to. For MWNTs with the same innermost tube radius, the critical buckling load is decreased when increasing the layers.

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

In this paper the effects of temperature on the radial breathing modes （RBMs） and radial wave propaga- tion in multiwall carbon nanotubes （MWCNTs） are inves- tigated using a continuum model of multiple elastic isotropic shells. The van der Waals forces between tubes are simulated as a nonlinear function of interlayer spacing of MWCNTs. The governing equations are solved using a finite element method. A wide range of innermost radius-to-thickness ratio of MWCNTs is considered to enhance the investigation. The presented solution is verified by comparing the results with those reported in the literature. The effects of temperature on the van der Waals interaction coefficient between layers of MWCNTs are examined. It is found that the variation of the van der Waals interaction coefficient at high temperature is sensible. Subsequently, variations of RBM frequencies and radial wave propagation in MWCNTs with temperatures up to 1 600 K are illustrated. It is shown that the thick MWC- NTs are more sensible to temperature than the thin ones.

GUIDED CIRCUMFERENTIAL WAVES IN DOUBLE-WALLED CARBON NANOTUBES

A model of guided circumferential waves propagating in double-walled carbon nanotubes is built by the theory of wave propagation in continuum mechanics, while the van der Waals force between the inner and outer nanotube has been taken into account in the model. The dispersion curves of the guided circumferential wave propagation are studied, and some dispersion characteristics are illustrated by comparing with those of single-walled carbon nanotubes. It is found that in double-walled carbon nanotubes, the guided circumferential waves will propagate in more dispersive ways. More interactions between neighboring wave modes may take place. In particular, it has been found that a couple of wave modes may disappear at a certain frequency and that, while a couple of wave modes disappear, another new couple of wave modes are excited at the same wave number.

Nanotube-based heterostructures for electrochemistry: A mini-review on lithium storage, hydrogen evolution and beyond

Nanotube-based mixed-dimensional or one-dimensional heterostructures have attracted great attention recently because of their unique physical properties and therefore potential for novel devices. Their chemical properties, however, were less explored but can be utilized for energy storage and conversion.In this review, we summarize the recent progress of nanotube-based low dimensional materials for electrochemistry, in particular, lithium storage and hydrogen evolution. First, we describe the atomic structure of low-dimensional heterostructures and briefly touch previous work on planar van der Waals heterostructures(2D+2D) in electrochemistry applications. Then we focus this review on the more recently developed nanotube-based, i.e., 1D+2D and 1D + 1D heterostructures, and discuss their various preparation approaches and electrochemical performances. Finally, we outline the challenges and opportunities in this direction and particularly emphasize the possibility of building high-performance electrodes using a single-walled carbon nanotube-based ultra-thin 1D heterostructure, and the importance of understanding the fundamental mechanism at atomic precision.

Theoretical calculations of thermophysical properties of single-wall carbon nanotube bundles

Carbon nanotube bundles are promising thermal interfacial materials due to their excellent thermal and mechanical characteristics. In this study, the phonon dispersion relations and density of states of the single-wall carbon nanotube bundles are calculated by using the force constant model. The calculation results show that the inter-tube interaction leads to a significant frequency raise of the low frequency modes. To verify the applied calculation method, the specific heat of a single single-wall carbon nanotube is calculated first based on the obtained phonon dispersion relations and the results coincide well with the experimental data. Moreover, the specific heat of the bundles is calculated and exhibits a slight reduction at low temperatures in comparison with that of the single tube. The thermal conductivity of the bundles at low temperatures is calculated by using the ballistic transport model. The calculation results indicate that the inter-tube interaction, i.e. van der Waals interaction, hinders heat transfer and cannot be neglected at extremely low temperatures. For （5, 5） bundles, the relative difference of the thermal conductivity caused by ignoring inter-tube effect reaches the maximum value of 26% around 17 K, which indicates the significant inter-tube interaction effect on the thermal conductivity at low temperatures.

Modelling of the Usefulness of Carbon Nanotubes as Antiviral Compounds for Treating Alzheimer Disease

The new generations of nano-devices successfully apply with great promise as drug carriers in the treatment of different diseases. The proposed model aims to determine the pharmacological targets and evaluate the bio-safety of usefulness of carbon nanotube conjugated with two different antiviral compounds, Acetylcholine and Ravastigmine, for treating Alzheimer disease. We also obtain the medicinal model mathematically to evaluate the interaction energy arising from encapsulation of each antiviral compound inside the single-walled carbon nanotube. Acetylcholine is modelled as two-connected spheres, while Ravastigmine has two possible structures which are an ellipsoid and cylinder, all interacting with the interior wall of single-walled carbon nanotubes with variant radii rc . Our calculations show that the single-walled carbon nanotube of radius rc greater than 3.391 Åthat will accept both drugs which are quite closer to the recent findings.

Encapsulation of a Chloroform Molecule in a Peptide Nanotube

We determine the encapsulation of a chloroform molecule into a D,L-Ala cyclopeptide nanotube by investigating the interaction energy between the two molecular structures. We employ the Lennard-Jones potential and a continuum approach which assumes that the atoms are evenly distributed over the molecules providing average atomic densities. Our result demonstrates that the encapsulation depends on the size of the molecule and the internal diameter of the peptide nantube. In particular, the on-axis chloroform molecule is only accepted into a peptide nanotube whose internal radius is greater than 5 ?. If located near the edge of the nanotube, then it is unlikely that the chloroform molecule will enter the nanotube. This is due to the energy valley that the molecule will need to overcome to move past the edge into the open end of the nanotube.

Critical Length of Double-Walled Carbon Nanotubes Based Oscillators

The critical lengths of an oscillator based on double-walled carbon nanotubes(DWCNTs)are studied by energy minimization and molecular dynamics simulation.Van der Waals(vdW)potential energy in DWCNTs is shown to be changed periodically with the lattice matching of the inner and outer tubes by using atomistic models with energy minimization method.If the coincidence length between the inner and outer tubes is long enough,the restoring force cannot drive the DWCNT to slide over the vdW potential barrier to assure the DWCNT acts as an oscillator.The critical coincidence lengths of the oscillators are predicted by a very simple equation and then confirmed with energy minimization method for both the zigzag/zigzag system and the armchair/armchair system.The critical length of the armchair/armchair system is much larger than that of the zigzag/zigzag system.The vdW potential energy fluctuation of the armchair/armchair system is weaker than that of the zigzag/zigzag system.So it is easier to slide over the barrier for the armchair/armchair system.The critical lengths of zigzag/zigzag DWCNTbased oscillator are found increasing along with temperature,by molecular dynamics simulations.

Detecting van der Waals forces between a single polymer repeating unit and a solid surface in high vacuum

Ubiquitous van der Waals(vdW)forces are very importa nt for nano structures.Although the vdW forces between two surfaces(or two layers)have been measured for several decades,a direct detection at the single-molecule level is still difficult.Herein,we report a novel method to solve this problem in high vacuum by means of AFM-based sin gle-molecule force spectroscopy(SMFS).Solve nt molecules and surface adsorbed water are removed thoroughly under high vacuum so that the situation is greatly simplified.A constant force plateau can be observed when a polymer chain is peeled off from a substrate in high vacuum.Accordingly,the vdW forces between one polymer repeating unit and the substrates can be obtained.The experimental results show that the vdW forces(typical range:21-54 pN)are dependent on the species of substrates and the size of polymer repeating unit,which is in good accordance with the theoretical results.It is expected that this novel method can be applied to detect other non-covalent interactions(such as hydrogen bond andπ-πstacking)at the single-molecule level in the future.

Effects of combination modes of favorable growth unit of Al(OH)_3 crystals precipitating on Van der Waals and chemical bond force

The dipole moment, total energy, atomic charge, orbital population and orbital energy of four representative combination models of the favorable growth unit Al6（OH）18（H2O）6 of Al（OH）3 crystals precipitating are calculated by ab initio at RHF/STO-3G, RHF/3-21G, RHF/6-31G levels and DFT at RB3LYP/STO-3G, RB3LYP/3-21G, RB3LYP/6-31G levels with Dipole & Sphere solvent model. The effect of various combination models on Van der Waals force is analyzed using dipole moment and molecular radius, and that on chemical bond force is analyzed using total energy, orbital population and orbital energy.

A kinematic model describing particle movement near a surface as effected by Brownian motion and electrostatic and Van der Waals forces

Nanoparticle movement near a surface is greatly influenced by electrostatic and Van der Waals forces between the particle and the surface,as well as by Brownian motion.In this paper,several precise equations are derived to describe the Van der Waals and electrostatic forces between a particle and a surface when the particle is removed from the surface.These include an equation for particle displacement under the electrostatic force,and a numerical calculation for particle displacement under the Van der Waals force.Finally,a kinematic model is constructed to describe the particle distribution under the effects of the electrostatic and Van der Waals forces,as well as the particle’s Brownian motion.The results show that increasing the multiply of the particle and surface zeta potential values and decreasing the ionic strength of the detergent can prevent a particle from redepositing onto a surface.