Breathing Mode Frequency Research Articles

Electronic structure and radial breathing mode for carbon nanotubes with ultra‐high curvature

AbstractUltra‐high curvature single‐walled carbon nanotubes (SWCNTs) with diameters down to 0.37 nm were prepared by transformation of FeCp2 peapods to double‐walled CNTs (DWCNTs). Results from resonance Raman scattering and high resolution transmission electron microscopy (TEM) were compared to calculations on the molecular dynamical, many electron corrected extended tight binding, and density functional theory (DFT) level. The growth process was found to be catalytic from Fe3C particles inside the tubes with dimensions of a crystallographic unit cell. The electronic structure showed dramatic deviations from tight binding results. The family behavior leads to level crossing already for $E_{11} $ and $E_{22} $ transitions. Experimental results fitted well into a Kataura–Popov plot and allowed assignment for the observed Raman lines of the inner tubes. Experimental and calculated radial breathing mode (RBM) frequencies showed a systematic difference indicating a radial expansion of the smallest inner tubes and a radial compression for the larger tubes.

Host–guest interaction between single‐wall carbon nanotubes and encapsulated C60 probed by resonance Raman spectroscopy

AbstractSingle‐wall carbon nanotubes (SWCNTs) that are encapsulating C60 [nanopeapods (NPDs)] can be considered one of the best model systems for investigating host–guest interactions in π‐conjugated molecular systems. We thus have investigated the effects of C60 encapsulation on the radial breathing mode (RBM) frequencies of semiconducting SWCNTs in a wide range of diameters (dt = ∼1.23 − 1.5 nm). The observed frequency shifts show a characteristic behavior depending on the intermolecular spacing between encapsulated C60 and the host SWCNTs. The present findings clearly indicate the van der Waals nature of the SWCNT‐C60 interaction.

Effects of dimensional parameters and various boundary conditions on axisymmetric vibrations of multi-walled carbon nanotubes using a continuum model

Axisymmetric vibrations of multi-walled carbon nanotubes (MWCNTs) with finite length are investigated in this paper. A multi-walled carbon nanotube is modeled with multiple elastic isotropic shells. Based on a continuum model and considering van der Waals forces between tubes, a two-dimensional finite element model is developed to obtain axisymmetric natural frequencies and mode shapes of MWCNTs. First, the axisymmetric vibrational characteristics of single-walled carbon nanotubes are investigated, and then, they are compared with those of MWCNTs. The effects of van der Waals forces on radial vibration of MWCNTs are also explained. Moreover, influences of end conditions on radial and axial natural frequencies of carbon nanotubes (CNTs) with wide range of length-to-thickness ratio are studied by considering the free-free (F-F) and simply supported (S-S) end conditions. Besides, it is focused on dependence of axisymmetric mode frequencies on dimensional parameters such as length, diameter, as well as number of layers of MWCNTs. Through this, explicit expressions are found for calculating the radial breathing mode and longitudinal mode frequencies of MWCNTs.

Vibrational mode assignments for bundled single-wall carbon nanotubes using Raman spectroscopy at different excitation energies

This study establishes a generic fitting approach to assignment of nanotube chiralities based on radial breathing mode frequencies (ωRBM) of SWCNTs in as-produced bundles. Four laser lines with energies of 2.62 eV, 2.33 eV, 1.88 eV and 1.58 eV were employed. The observed RBM frequencies, ωRBM, were plotted as a function of the possible diameters, d, as identified from the so-called Kataura plot and reported values of the parameters A and B, where ωRBM=A/d+B, assuming that SWCNTs resonant at the respective laser frequencies dominate the spectrum. The refined values of A and B, obtained by the best fit of a linear regression between ωRBM and 1/d, were found to vary significantly for different laser frequencies. This variation is interpreted in terms of the differences in electronic properties of SWCNTs resonant at different frequencies. The assigned nanotubes match well with those identified in the Kataura plot, falling within a resonant line width of ±0.2 eV of the respective laser lines.

Electronic Structure of Carbon Nanotubes with Ultrahigh Curvature

The electronic and the vibrational structure of carbon nanotubes with ultrahigh curvature was systematically studied by resonance Raman scattering, high-resolution transmission electron microscopy (HRTEM), molecular dynamics, and ab initio DFT calculations. The ultrahigh curvature tubes were grown inside commercial HiPco tubes after filling the latter with the small but carbon-rich molecule ferrocene. TEM showed partial filling of the outer tubes with inner tubes and mobility of the latter in the electron beam. The smallest analyzed tube was of (5,0) chirality and had a DFT determined diameter of 0.406 nm and a radial breathing mode frequency of 570 cm(-1). For all inner tubes which had transitions in the visible spectral range, transition energies and RBM frequencies were determined with a resonance width of only 45 meV. Experimentally determined transition energies revealed dramatic deviations up to several electronvolts compared to tight-binding calculations and a significant family spread of more than 2 eV but were in agreement with many electron contribution corrected extended tight-binding results and with results from DFT calculations.

Classical phase-space approach for coherent matter waves

We investigate a classical phase-space approach of matter-wave propagation based on the truncated Wigner equation (TWE). We show that such a description is suitable for ideal matter waves in quadratic time-dependent confinement as well as for harmonically trapped Bose-Einstein condensates in the Thomas-Fermi regime. In arbitrary interacting regimes, the TWE combined with the moment method yields the low-energy spectrum of a condensate as predicted by independent variational methods. TWE also gives the right breathing-mode frequency for long-ranged interactions decaying as $1/{r}^{2}$ in three dimensions and for a contact potential in two dimensions. Quantum signatures, beyond the TWE, may only be found in the condensate dynamics beyond the regimes of classical phase-space propagation identified here.

Horizontal single-walled carbon nanotubes on MgO(1 1 0) and MgO(0 0 1) substrates

Horizontal single-walled carbon nanotubes (SWNT) on MgO(110) and MgO(001) single-crystal substrates were synthesized in a temperature range of 830–910°C by chemical vapor deposition using Co catalyst particles. SWNTs grow preferentially along the 〈110〉 directions of the two MgO surface planes. For synthesis at 830°C, the distribution of the radial breathing mode (RBM) frequencies of SWNTs grown on MgO(110) is centered at 142 and 168cm−1, attributed to semi-conducting SWNTs. For the same synthesis time, the RBM distribution of SWNTs grown on MgO(001) is significantly broader. This difference would be due to the larger surface energy of MgO(110) which reduces the probability of coalescence between Co particles.

Radial breathing mode in silicon nanowires: Anab initiostudy

We study by means of first-principles calculations the phonon spectrum of silicon nanowires oriented along the $⟨110⟩$ direction with a diameter ranging from 1.0 to 2.4 nm. We analyze in particular the evolution of the radial breathing mode frequency as a mean to calibrate experimentally the diameter through Raman analysis. Remarkably, the results of elastic theory analysis are very reliable even in the limit of ultrathin wires with significant deviations from the cylindrical shape.

A molecular based anisotropic shell model for single-walled carbon nanotubes

Single-walled carbon nanotubes (SWCNTs) are frequently modeled as isotropic elastic shells. However, there are obvious evidences showing that SWCNTs exhibit remarkable chirality induced anisotropy that should not be neglected in some cases. In this paper, we derive the closed-form expressions for the anisotropic elastic properties of SWCNTs using a molecular mechanics model. Based on these anisotropic elastic properties, we develop a molecular based anisotropic shell model (MBASM) for predicting the mechanical behavior of SWCNTs. The explicit expressions for the coupling of axial, circumferential, and torsional strains, the radial breathing mode frequency, and the longitudinal and torsional wave speeds are obtained. We show that the MBASM is capable of predicting the effects of size and chirality on these quantities. The efficiency and accuracy of the MBASM are validated by comparisons of the present results with the existing results.

Chiral interaction in double-wall carbon nanotubes: Simple rules deduced from a large sampling of tubes

We study a large sampling of chiral double-wall carbon nanotubes to propose simple formula describing the dependence of the interwall energy, the chiral discrimination energy, and the radial breathing mode frequencies as a function of the main characteristics of the tubes, i.e., their radius, length and chiral angle. It is shown that tube pairs with the same handedness are more stable than enantiomeric pairs, and this discrimination, though small, mainly occurs in the first step of the growth of an inner tube inside an outer one. Chiral splittings of the breathing mode frequencies for the two DWCNT diastereoisomers (n(i),m(i))@(n(o),m(o)) and (m(i),n(i))@(n(o),m(o)) can reach a few wave numbers.

Wall thickness and elastic moduli of single-walled carbon nanotubes from frequencies of axial, torsional and inextensional modes of vibration

Abstract Free vibrations of thirty-three armchair, zigzag and chiral single-walled carbon nanotubes (SWCNTs) of aspect ratios (length/diameter) between ∼3 and ∼15 and having free ends have been studied using the MM3 potential. It is found that these tubes exhibit Rayleigh and Love inextensional modes of vibration. The lowest natural frequency of a mode of vibration corresponds to a circumferential wave number greater than one. Recall that a cylindrical shell of small aspect ratio and comprised of a linear elastic and isotropic material also exhibits the Rayleigh and the Love inextensional modes of vibration. In order to quantitatively compare frequencies of a shell with those of a SWCNT, we find geometric and material parameters for the shell in two ways. In the first approach, we require that the lowest Rayleigh and the radial breathing mode frequencies and the lowest frequencies of the axial and the torsional modes of vibration of a SWCNT match with the corresponding ones of the shell having length and mean diameter equal to those of the SWCNT. In the second technique, we account for the transverse inertia effects, and equate frequencies of the lowest Love, axial and torsional modes of vibration of a SWCNT to that of a shell. Each one of these two methods determines Young’s modulus and Poisson’s ratio of the material of the shell and its thickness, and enables us to explore similarities and differences between vibrations of a shell and of a SWCNT. It is found that the two techniques give very close results for the material and the geometric parameters of the shell, and hence of the SWCNT. The SWCNT thickness increases from ∼0.88 A to 1.37 A when the tube radius is increased from ∼3.6 A to 15 A and stays at 1.37 A for further increases in the tube radius. The wall thickness is essentially independent of the tube chirality. We use these results to provide an expression for the wall thickness in terms of the tube radius and the bond length in the initial relaxed configuration of a SWCNT. We also compare higher vibrational modes of shells and hollow cylinders with those of the corresponding SWCNTs. For a shell we use a first-order shear deformable shell theory (FSDST). Frequencies of a shell using the FSDST and of the hollow cylinder using the three-dimensional linear elasticity theory are computed with the finite element method. It is found that for low to moderate circumferential and axial wave numbers frequencies and mode shapes of the shell and of the hollow cylinder agree well with those of the corresponding SWCNT computed with the MM simulations.

Interaction between single-wall carbon nanotubes and encapsulated C60 probed by resonance Raman spectroscopy

The effects of C(60) encapsulation on the radial breathing mode (RBM) frequencies of single-wall carbon nanotubes (SWCNTs) are investigated over a wide range of diameters (d(t) approximately 1.25-1.5 nm). The observed frequency shifts show a characteristic behavior depending on the inter-spacing between C(60) and SWCNTs. The present findings clearly indicate the van der Waals nature of the SWCNT-C(60) interaction and an importance of hybridization between the electronic states of C(60) and SWCNTs.

The effect of environment on the radial breathing mode of supergrowth single wall carbon nanotubes

It has been shown that “supergrowth” single wall carbon nanotubes (SWNTs) exhibit a radial breathing mode frequency ωRBM dependence on tube diameter dt given by ωRBM=227/dt. This result gave rise to two distinct scenarios for SWNTs: one for the supergrowth radial breathing mode and another for all the other samples reported in the literature. Here we show that, by dispersing the supergrowth SWNTs in surfactant or bringing them into interacting bundles, it is possible to merge these two scenarios, where now the supergrowth SWNT properties are similar to all SWNT properties reported so far in the literature.

Interactive mechanisms between the internal blast loading and the dynamic elastic response of spherical containment vessels

The interactive mechanisms between internal blast loading and dynamic elastic response of spherical containment vessels are studied in this paper. The blast loading history in containment vessels can be divided into three periods, i.e. the primary-shock period, the shock-reflection period and the pressure-oscillation period. It is shown that the initial response of the containment vessel depends on both the impulse and the shape of the primary-shock depending on the ratio of the loading period to the breathing mode period. However, during the shock-reflection period, the response of the containment vessel can be coupled with the reflected shock waves in the vessel, especially when the dominant frequency of reflected shock waves is close to the breathing mode frequency of the vessel. During the pressure-oscillation period, the dynamic loading is mainly the oscillation of the internal pressure due to the oscillatory volume change of the vessel, which couples dissipatedly with the vibration of the vessel leading to reduced vibration amplitudes. The effects of the influential non-dimensional parameters on the resonant interaction in shock-reflection period are discussed, based on which guidelines are recommended for avoiding the strain growth in the shock-reflection period in the design of spherical containment vessels.

Ab initio study on the electronic structure and vibration modes of alkali and alkaline-earthamides and alanates

We study the electronic structure and vibrational modes of several amidesM(NH2)n andalanates M(AlH4)n (M = K, Na, Li, Ca and Mg), focusing on the role of cation states. Calculated breathingstretching vibration modes for these compounds are compared with measuredinfrared and Raman spectra. In the amides, we find a significant tendencysuch that the breathing mode frequencies and the structural parameters ofNH2 vary in accordance with the ionization energy of cation. The tendency may be explained bythe strength in hybridization between cation orbitals and molecular orbitals of(NH2)−. The microscopic mechanism of correlations between the vibration frequencies andstructural parameters is elucidated in relation to the electronic structure. A possible similartendency in the alanates is also discussed.

Anisotropy effects on the time-resolved spectroscopy of the acoustic vibrations of nanoobjects

The impact of spherical symmetry breaking and crystallinity on the response of the vibrational acoustic modes of a nanoobject in frequency- and time-domain experiments is investigated using the finite-element analysis method. The results show that introduction of shape anisotropy, i.e., evolution from a nanosphere to a nanorod, modifies the periods of the fundamental radial and quadrupolar modes and leads to activation of a quadrupolar-like mode in time-domain studies. In contrast, crystallinity is shown to have a negligible impact on the breathing mode frequency of a nanosphere formed by a cubic crystal and does not activate any quadrupolar mode. The effect of an anisotropic excitation in time-resolved measurements of a large nanosphere is also discussed.

N-Octyl-O-sulfate chitosan stabilises single wall carbon nanotubes in aqueous media and bestows biocompatibility

A non-covalent approach to debundle single wall carbon nanotubes using a biocompatible chitosan-derivative, namely N-octyl-O-sulfate chitosan (NOSC), was investigated. The resulting stable dispersions were characterised by Raman spectroscopy, UV-Vis spectroscopy, atomic force microscopy (AFM), transmission electron microscopy (TEM) and zeta-potential measurements. Both AFM and TEM studies revealed the presence of individual carbon nanotubes wrapped with the polymer (diameters up to 7 nm). Raman spectra showed radial breathing mode frequency shifts, after the addition of NOSC, due to the wrapping of the biomolecules onto the graphitic sidewalls. Molecular modelling studies were employed to investigate the mode of binding of the NOSC chains to the surface of the nanotubes. In agreement with the experiments, modelling studies predicted that the wrapped tube has a maximum thickness of approximately 7 nm. Studies on the anticoagulant properties of these complexes revealed that NOSC coated SWCNTs exhibit similar activity to the polymer alone, this property would eliminate the risk for SWCNTs to induce coagulation as a host reaction process when used in vivo.

Structural support of the external tubes in double-wall carbon nanotubes

High-pressure Raman studies of double-wall carbon nanotubes (DWCNTs) and their parent single-wall carbon nanotubes provide experimental evidence of the structural support of carbon nanotubes due to the encapsulation of smaller diameter tubes in their interior. The structural support is manifested by the reduced slope of the radial breathing mode frequency of the external tubes in DWCNTs, which further decreases at higher pressures as the inner–outer tube (intratube) interaction increases.

The Boron Buckyball and Its Precursors: An Electronic Structure Study

Using ab initio calculations, we analyze electronic structure and vibrational modes of the boron fullerene B(80), a stable, spherical cage similar in shape to the well-known C(60). There exist several isomers, lying close in structure and energy, with total energy difference within approximately 30 meV. We present detailed analysis of their electronic structure and geometry. Calculated radial breathing mode frequency turns out to be 474 cm(-1), which can be a characteristic of B(80) in Raman spectroscopy. Since the B(80) structure is made of interwoven double-ring clusters, we also investigate double-rings with various diameters. We present their structure and HOMO-LUMO dependence on the diameter, and find out that the gap alternates for different sizes and closes its value for infinite double-ring.

The degenerate Fermi gas with density-dependent interactions in the large-N limit under the K-harmonic approximation

We present a simple implementation of a density-dependent, zero-range interaction in a degenerate Fermi gas using hyperspherical coordinates in the large particle number limit. The method produces a 1D effective potential that accurately describes the ground-state energy as a function of the hyperradius of the two spin component gas throughout the vicinity of the unitarity regime. In the unitarity regime the breathing mode frequency is found to limit to the non-interacting value. A dynamical instability, similar to the Bosenova, is predicted to be possible for a gas containing more than three spin components, for large, negative, two-body scattering lengths. While the situation is less clear for the gas with three components, collapse might also be possible.