Elastic axial buckling of carbon nanotubes via a molecular mechanics model

Based on a molecular mechanics model, analytical solutions for the critical buckling strain of single-walled achiral carbon nanotubes under axial compression are obtained. The results show that zigzag tubes are more stable than armchair tubes with the same diameter. Comparison with the results given by continuum mechanics model shows that the continuum mechanics model underestimates the critical buckling strain for smaller tubes if a Young’s modulus for larger tubes (or for graphene sheets) is adopted. The effect of intertube van der Waals interaction from the inner tube of multi-walled carbon nanotubes on the buckling of the outermost tube is also qualitatively discussed and it is found that the van der Waals interaction has little effect on the critical buckling strain for double-walled carbon nanotubes.     

The effect of Stone–Thrower–Wales defects on mechanical properties of graphene sheets – A molecular dynamics study

The mechanical properties of graphene may be strongly affected by defects. In our work, the effects of the orientation and tilting angles of Stone–Thrower–Wales (STW, 5-7-7-5) defects on the mechanical properties of graphene have been investigated based on molecular dynamics simulations. When there is one centred STW defect including STW-1 defect and STW-2 defect, our study reveals that the orientation with respect to the chirality governs the mechanical properties of graphene. For STW-1 defective graphene, the strength of the armchair direction is much lower than that of zigzag direction. While the circumstance for STW-2 defective graphene is opposite, the strength of the armchair direction is much higher than the strength of the zigzag direction. Furthermore, when there is more than one STW defect, the mechanical properties of graphene depends on the tilting angle of STW defects. The breaking strength of graphene decreases with the increasing tilting angle. Our present work could provide significant insights into the effect of STW defects on the mechanical properties of graphene.     

On the prediction of graphene’s elastic properties with reactive empirical bond order potentials

The elastic properties of graphene as described by the reactive empirical bond order potential are studied through uniaxial tensile tests calculations at both zero temperature, with a conjugate gradient approach, and room temperature, with molecular dynamics simulations. A perfect linear elastic behavior is observed at 0 K up to ≈0.1% strain. The Young’s modulus and Poisson’s ratio obtained with this potential are of ≈730 GPa and 0.39, respectively, with little chirality effects. These values differ significantly from former estimations, much closer to experimental values. We show that these former values have certainly been obtained by neglecting the effect of atomic relaxation, leading to a severe inaccuracy. At larger strains, an extended apparent linear domain is observed in the stress–strain curves, which is relevant to Young’s modulus calculations at finite temperature. Our molecular dynamics simulations at 300 K have allowed obtaining the following, chirality dependent, apparent Young’s moduli, 860 and 761 GPa, and Poisson’s ratios, 0.12 and 0.23, for armchair and zigzag loadings, respectively.     

Two least squares analyses of bond lengths and bond angles for the joining of carbon nanotubes to graphenes

In order to transmit signals from future nanoelectromechanical graphene sheets to other materials, connections with carbon nanotubes need to be effected. Here, we examine three particular perpendicular connections of carbon nanotubes employing two simple distinct least squares approaches and using Euler’s theorem. Firstly, for (8, 0) and (4, 4) carbon nanotubes, we apply a least squares approach to the bond lengths. Sixteen distinct defects and two possible orientations for the armchair tube (4, 4) are identified. Assuming that only pentagons, hexagons, heptagons and occasionally octagons are accepted, the number of possibilities are greatly reduced. By excluding octagonal rings, the number of possible configurations may be further reduced to only one and two most likely configurations for the zigzag (8, 0) and the armchair (4, 4) tubes, respectively. Secondly, for (6, 0) and (8, 0) carbon nanotubes, we apply a least squares approach to bond angles, and for one particular (8, 0) junction, we show that the two least squares approaches produce similar structures in terms of atom locations.These purely geometric approaches can be formally related directly to certain numerical energy minimisation methods used by a number of authors.     

Structural stability of carbon nanosprings

A carbon nanospring is formed by coiling a single-walled carbon nanotube (n, m) surrounding a cylinder surface with a uniform pitch length and spring rising angle. Both armchair and zigzag carbon nanosprings are studied. Due to the limitations of hexagonal carbon-atom rings and the bond length of carbon–carbon atoms, a maximum rising angle exists for each type of nanospring (n, m), and zigzag nanosprings have greater maximum rising angles than armchair nanosprings with the same spring radius. In this study, a molecular mechanics (MMs) calculation is performed to investigate the structural properties of carbon nanosprings. Flat carbon nanosprings are obtained with small spring radii and rising angles, and round carbon nanosprings are found to have a larger spring radius than flat ones. The results show that the structural stability of a nanospring mainly depends on the parameters of its tube radius, spring radius, and rising angle, whereas its chiral type has little effect.     

Edge effect on electronic transport properties of graphene nanoribbons and presence of perfectly conducting channel

Numerical calculations have been performed to elucidate unconventional electronic transport properties in disordered nanographene ribbons with zigzag edges (zigzag ribbons). The energy band structure of zigzag ribbons has two valleys that are well separated in momentum space, related to the two Dirac points of the graphene spectrum. The partial flat bands due to edge states make the imbalance between left- and right-going modes in each valley, i.e. appearance of a single chiral mode. This feature gives rise to a perfectly conducting channel in the disordered system, i.e. the average of conductance 〈g〉 converges exponentially to 1 conductance quantum per spin with increasing system length, provided impurity scattering does not connect the two valleys, as is the case for long-range impurity potentials. Ribbons with short-range impurity potentials, however, through inter-valley scattering, display ordinary localization behavior. Symmetry considerations lead to the classification of disordered zigzag ribbons into the unitary class for long-range impurities, and the orthogonal class for short-range impurities. The electronic states of graphene nanoribbons with general edge structures are also discussed, and it is demonstrated that chiral channels due to the edge states are realized even in more general edge structures except for armchair edges.     

Ab initio density functional studies of the restructuring of graphene nanoribbons to form tailored single walled carbon nanotubes

A method based on density functional theory calculations is proposed for the preparation of chiral controlled single walled carbon nanotubes (SWCNTs) by tailoring the edges of bi-layered graphene nanoribbons (GNRs). We find that armchair edged bi-layered GNRs are highly stable and need to be compressed to overcome the energy barrier to form zigzag SWCNTs, while the zigzag edged bi-layered GNRs are intrinsically highly unstable and immediately form armchair SWCNTs. We have investigated the rehybridization of orbitals of carbon atoms in the process of nanotube formation. Nanotube formation is found to be assisted by the edge ripples along with the intrinsic edge reactivity of different types of bi-layered GNRs. Utilizing these results we show that it may be possible to produce high specificity chiral controlled SWCNTs and pattern them for nanoscale device applications.     

Multi-wall carbon nanotubes with uniform chirality: evidence for scroll structures

Multi-wall carbon nanotubes (MWCNT) obtained by catalytic decomposition of iron phthalocyanine are investigated by high resolution electron microscopy and electron diffraction (ED). The evaluation of the ED patterns shows that all MWCNTs studied have a uniform chirality, i.e. all tubes of a given MWCNT show the same chiral angle. The conditions for the formation of nested tubes are discussed. A comparison of the values of the chiral angles with those of the corresponding interwall spacings, both obtained from the ED patterns, leads to the conclusion that these MWCNTs have a scroll structure, very probably consisting of one single rolled-up graphene sheet.     

Knitted graphene-nanoribbon sheet: a mechanically robust structure

In this paper, a new nanostructure is proposed, namely, the knitted graphene-nanoribbon sheet (KGS), which consists of zigzag and/or armchair graphene nanoribbons. The knitting technology is introduced to graphene nanotechnology to produce large area graphene sheets. Compared with pristine graphene, the chirality of a knitted graphene-nanoribbon sheet is much more flexible and can be designed on demand. The mechanical properties of KGSs are investigated by molecular dynamics simulations, including the effect of vacancies. With hydrogen atoms saturating the ribbon edges, the structure (KGS + H) is found to be of significant mechanical robustness, whose fracture does not rely on the critical bonds. The fracture strain of KGS + H remains nearly unchanged as long as there remains a single defect-free graphene nanoribbon in the tensile direction. This graphene nano knitting technique is experimentally feasible, inspired by a recent demonstration by Fournier et al. [Phys. Rev. B, 2011, 84, 035435] of lifting a single molecular wire using a combined frequency-modulated atomic force and tunnelling microscope.     

Buckling of functionalized single-walled nanotubes under axial compression

Buckling characteristics of functionalized single-walled carbon nanotubes under axial compression are investigated by molecular mechanics simulation. The influences of the content, the distribution density and the location of the sp³-hybridized carbon atoms as well as the chirality on the critical buckling strains of functionalized single-walled carbon nanotubes are carefully studied. The results indicate that the chirality and the distribution density have dominant effect on the critical buckling strains. The critical buckling strains of present armchair (5, 5) and zigzag (10, 0) carbon nanotube are degraded by about 43% and 70%, respectively, due to the dense distribution of the sp³-hybridized carbon atoms. The reduction amplitude of the critical strain increases with increasing the tubule radius of an armchair or zigzag single-wall carbon nanotube. The dramatic reduction of the critical strain could cause a great loss of reinforcing role of carbon nanotubes in composites.     

Beryllium doping graphene,graphene-nanoribbons,C60-fullerene,and carbon nanotubes

Beryllium substitutional doping within graphene, graphene nanoribbons, and carbon nanotubes are investigated using first-principles density functional theory calculations. Nanoribbons with armchair and zigzag edges, semiconducting (10,0) and metallic (6,6) carbon nanotubes, and C₆₀ fullerene structures are analyzed. Binding energy, doping energy, band structure, electronic density of states (DOS), and magnetic ordering are calculated. Our results demonstrate that conversely to perfect graphene, Be-doped graphene reveals a semiconducting behavior with an indirect band gap of 0.298 eV. Formation energy analysis reveals that Be into graphene and ribbons is more energetically favorable, but the energies involved are larger than those obtained for B- and N-doped nanocarbons. For nanoribbons, two different ways of incorporating the Be atom are explored (dopant placed in the center or edge), demonstrating that armchair nanoribbons preserve the semiconducting behavior with a reduced band-gap whereas that zigzag nanoribbons exhibit a half-metallic behavior with magnetic order along the edges. Results on Be-doping zigzag (10,0) semiconducting and armchair (6,6) metallic nanotubes and C₆₀ fullerene reveal the appearance of additional electronic states around the Fermi level. We envisage that the present investigation could motivate the realization of future experiments to introduce Be into sp² graphite-like lattice using high temperature chemical vapor deposition method.     

Stability comparison of simulated double-walled carbon nanotube structures

The focus of this research is to systematically study and classify electronic energy trends in different double-walled carbon nanotube (DWCNT) structures through ab initio simulations. Simulations comparing the stability of DWCNTs with different interwall spacings, tube types (armchair or zigzag), lengths, diameters, and endcaps were performed at a variety of computational levels. These simulations showed that DWCNTs nucleate from end caps and become energetically more stable as length and diameter increase. Another finding of this research was that the interwall spacing is dependent on which type of tube is in the outer position of the DWCNT. High stability configurations occurred when the interwall spacing was approximately 3.3 Å and a zigzag tube was in the outer position or when the interwall spacing was approximately 3.5 Å and an armchair tube was in the outer position. It was also seen that endcaps affected which tube combinations were more stable; the armchair@armchair DWCNT was the most energetically stable combination for capped tubes, while the armchair@zigzag DWCNT had the highest stability of uncapped tubes. Understanding if there is a preferred structural motif for DWCNTs and clarifying which nucleation and growth paths are favored by nanotubes will elucidate if controlled fabrication can be achieved.     

Stress simulation of carbon nanotubes in tension and compression

Based on a continuum shell model, a structural mechanics approach is presented to simulate stress-strain behavior of carbon nanotubes (CNTs). The nanoscale continuum theory is established to directly incorporate the Morse potential function into the constitutive model of CNTs. According to the present model, the mechanical properties of both zigzag and armchair tubes are investigated. The result shows that the atomic structures of CNTs have a significant influence on the stress-strain behavior. The armchair zigzag tube exhibits larger stress-strain response than the zigzag tube under tensile loading, but its relationship turns over between the tension and compression deformations. The theoretical approach supplies a set of very simple formulas and able be serve as a good approximation on the mechanical properties for CNTs.     

Energetics and structures of carbon nanorings

Molecular dynamics (MD) simulation is adopted to study the stability of carbon nanorings, where the force-field function describes the interactions of the carbon atoms. A nanoring is formed by bending a straight nanotube (n, m) and connecting its two ends together. Both armchair and zigzag nanorings are investigated, and the critical diameters for stable nanorings are obtained. It is found that zigzag nanorings have a larger critical diameter than armchair nanorings.     

Spectral phonon thermal properties in graphene nanoribbons

This work provides a comprehensive investigation on the spectral phonon properties in graphene nanoribbons (GNRs) by the normal mode decomposition (NMD) method, considering the effects of edge chirality, width, and temperature. We find that the edge chirality has no significant effect on the phonon relaxation time but has a large influence to the phonon group velocity. As a result, the thermal conductivity of around 707 W/(m K) in the 4.26 nm-wide zigzag GNR at room temperature is higher than that of 467 W/(m K) in the armchair GNR with the same width. As the width decreases or the temperature increases, the thermal conductivity reduces significantly due to the decreasing relaxation times. Good agreement is achieved between the thermal conductivities predicted from the Green–Kubo method and the NMD method. We find that optical phonons dominate the thermal transport in the GNRs while the relative contribution of acoustic phonons to the thermal conductivity is only 10.1% and 13% in the zigzag GNR and the armchair GNR, respectively. Interestingly, the ZA mode is found to contribute only 1–5% to the total thermal transport in GNRs, being much lower than that of 30–70% in single layer graphene.     

A molecular dynamics study of the thermal conductivity of graphene nanoribbons containing dispersed Stone–Thrower–Wales defects

Classical molecular dynamics with the AIREBO potential is used to investigate the thermal conductivity of both zigzag and armchair graphene nanoribbons possessing different densities of Stone–Thrower–Wales (STW) defects. Our results indicate that the presence of the defects can decrease thermal conductivity by more than 50%. The larger the defect density, the lower the conductivity, with the decrease significantly higher in zigzag than in armchair nanoribbons for all defect densities. The effect of STW defects in the temperature range 100–600 K was also determined. Our results showed the same trends in thermal conductivity decreases at all temperatures. However, for higher defect densities there was less variation in thermal conductivity at different temperatures.     

An atomistic simulation study of the mechanisms and kinetics of surface bond strengthening in thermally-treated cone-stacked carbon nanofibers

Bonding mechanisms and rates between the active edges of a cone-stacked CNF are examined by molecular dynamics simulations at temperatures up to 2273 K. Thermally treated nanofibers subjected to tensile deformation show a substantial increase in the elastic strain limit, albeit no change in elastic modulus, due to the resistance of surface bonds to crack propagation. Two bonding mechanisms; i.e., the formation of energetically stable loops from single dangling atoms and the folding of zigzag and armchair graphene bilayer edges, are shown to display predominant, yet distinct kinetics. This study reveals a critical transition temperature at 1000 K beyond which bilayer edge folding dominates over the formation of single atom loops in strengthening the surface of CNFs. This study also underscores the critical roles played by surface bond types, numbers, and distributions on the large failure strength dispersion observed experimentally in CNFs.     

Persistent currents in finite zigzag carbon nanotubes

Magnetic properties of finite zigzag carbon nanotubes are studied within the tight-binding model. The spin-B interaction (Zeeman splitting) causes the metal-semiconductor transition and thus produces a large persistent current (J) with special jump structures. This effect makes all zigzag carbon nanotubes exhibit a gigantic paramagnetism. It also destroys the periodicity of magnetic properties. The dependence on the magnetic flux, the length (w), the radius (r), the temperature (T), and the chirality (zigzag or armchair) is strong. The amplitude of J quickly decreases with increasing of (w, r, T). Zigzag carbon nanotubes differ significantly from armchair carbon nanotubes (or infinite zigzag carbon nanotubes) in features such as magnetic susceptibility and in special structures in J.     

Correlation between atomistic morphology and electron transport properties in defect-free and defected graphene nanoribbons: An interpretation through Clar sextet theory

Structural, electronic and transport properties of defect-free, defected and functionalized armchair and zig-zag graphene nanoribbons (GNRs) are investigated with density functional theory and non-equilibrium Green’s function calculations and rationalized in terms of Clar’s theory of the aromatic sextet. Calculations suggest a tight relationship between the transport properties of nanoribbons and the underlying bond patterning as described by valence bond and Clar sextet theory. Namely, armchair GNRs exhibit a strong dependence of the transport properties on the ribbon width, as a consequence of different valence bond representations. The occurrence of localized defects involving electron pairs does not significantly alter this behavior. Conversely, transport properties of zigzag GNRs are less affected by morphological details, such as width and occurrence of defects, as expected from the application of Clar’s theory. However, controlled edge functionalization and morphology modifications in zigzag GNRs can potentially lead to localization of aromatic sextets and, consequently, to strong changes in the transport properties. Our work indicates Clar sextet theory as a powerful and accurate tool to rationalize and predict the electronic and transport properties of complex carbon nanostructures based on GNRs. These principles can be extended to the design of novel systems with potential applications in nanoelectronics.     

Energetic stability of graphene nanoflakes and nanocones

We investigate the relative energetic stability of a variety of nanographene structures such as graphene nanoflakes, nanoribbons, nanodisks, and nanocones. We calculate the cohesive energies with respect to hydrogen passivation, edge nature (zigzag versus armchair) and shape (triangular, rectangular, hexagonal). The cohesive energy is confirmed to increase with size for all these structures. We pay particular attention to optimally-compact circular flakes and compare our theoretical results with round disks produced in a plasma torch atmosphere. We find in the calculations that round shape does not have preferred relative stability. This suggests that the observed disks are grown under conditions where carbon atoms are highly mobile. For graphene nanocones we obtain a similar result. Experimentally, the open base of a 19-degree-cone is observed perpendicular to the cone axis, but this does not correspond to the most stable configuration as obtained by the calculations. Instead, we find that both, disks and cones, prefer minimal length of the edge termination rather than a maximum in the cohesive energy. With respect to our results we discuss for polycyclic aromatic hydrocarbons (PAH) and atomic clusters, as models for graphene flakes, the significance of the cohesive energy for the observed abundances.