Radial breathing mode of carbon nanotubes subjected to axial pressure
© Lei et al; licensee Springer. 2011
Received: 14 May 2011
Accepted: 11 August 2011
Published: 11 August 2011
In this paper, a theoretical analysis of the radial breathing mode (RBM) of carbon nanotubes (CNTs) subjected to axial pressure is presented based on an elastic continuum model. Single-walled carbon nanotubes (SWCNTs) are described as an individual elastic shell and double-walled carbon nanotubes (DWCNTs) are considered to be two shells coupled through the van der Waals force. The effects of axial pressure, wave numbers and nanotube diameter on the RBM frequency are investigated in detail. The validity of these theoretical results is confirmed through the comparison of the experiment, calculation and simulation. Our results show that the RBM frequency is linearly dependent on the axial pressure and is affected by the wave numbers. We concluded that RBM frequency can be used to characterize the axial pressure acting on both ends of a CNT.
Radial breathing mode (RBM) of carbon nanotubes (CNTs) is a low frequency mode, but accounts for the strongest feature observed in the CNT Raman spectrum. For the RBM, all of the carbon atoms in a CNT move in the radial direction synchronously, which generates an effect similar to "breathing" [1, 2]. This mode is unique to CNTs, and is not observed in other carbon systems . Resonant Raman measurement of the RBM in CNTs is a standard, straightforward method for precisely determining the diameter of a CNT, distinguishing the CNT chiral-index assignments, or characterizing CNT conglomerates [4–7]. For CNTs, pressure studies are motivated by the need to investigate mechanical stability, pressure-induced phase transitions (such as vibrational characteristics), and the effects of intertube interactions. In this letter, the RBM frequency of CNTs subjected to axial pressure is studied using an elastic continuum mechanics model. Single-walled carbon nanotubes (SWCNTs) are described as an individual elastic shell and double-walled carbon nanotubes (DWCNTs) are considered as two shells coupled through the van der Waals (vdW) force interaction between them. The interaction of the vdW force between the inner and outer tubes and the effect of axial pressure are incorporated into the formulation. We consider the effects of the influences of the axial half-wave number m, circumference wave number n, nanotube diameter, and the aspect ratio L/D of the nanotubes on the RBM frequency for SWCNTs and DWCNTs exposed to varying axial pressures. Through comparison with previous results obtained from experiments and simulations, it can be seen that the continuum shell model can be used to predict the RBM frequency of CNTs exposed to various axial pressures.
2. Theoretical approach
2.1 Governing equations of SWCNTs under axial pressure
2.2 van der Waals interaction forces
where a is the carbon-carbon bond length (0.142 nm); R i and R j are the inner and outer radii of the DWCNTs; and σ and ε are the vdW radius and the well depth of the Lennard-Jones potential, respectively. The vdW parameters of the DWCNTs in the Lennard-Jones potential are ε = 2.967 meV and σ = 0.34 nm (from Saito et al. ).
2.3 RBM frequency of DWCNTs
To model the vdW force, we substitute Eqs. (4) and (5) into Eqs. (1)-(3). The governing equations for the RBM frequency of inner and outer tubes of DWCNTs subjected to an axial pressure can be expressed as:
where A k , B k and C k are the longitudinal, circumferential and radial amplitudes of the displacements in the inner (k = 1) and outer tubes (k = 2), respectively. L is the length of CNT which is shown in Figure 1. The wave numbers m and n are the axial half-wave and circumferential numbers, respectively.
3. Numerical results and discussion
For the present analysis, an individual SWCNT was assumed to be a graphene sheet rolled into a cylinder and the DWCNT is considered to be two layered nanotube shells coupled by vdW interactions. The value of the thickness of sheet is 0.34 nm; the elastic modulus is 1.0 TPa; the Poisson's ratio is 0.27; the mass density of the CNTs is 2.3 g/cm3; and the inner and outer diameters of the DWCNTs are 2.2 nm and 3.0 nm, respectively .
Based on our proposed theoretical approach, we first calculate the RBM frequency of an isolated SWCNT subjected to zero pressure varying with radius. Note that the commonly used unit for the frequency of the RBM f is in cm-1 for Raman spectroscopy experiments. However, the unit Hertz (Hz) for ω has been adopted for convenience in this study. Our results show that when the diameter is increased from 1.43 to 1.59 nm, the RBM frequency of a SWCNT monotonically decreases from 33.03 to 26.76 THz. To extract the correct parameters for the elastic continuum model, we compared the present work with other approaches. According to Raman scattering technique by Jorio et al. , the frequency of a RBM ranges from 33.18 to 28.46 THz for SWCNTs with diameters of 1.43 to 1.65 nm when structural factors are considered. Furthermore, Peica et al.  used tip-enhanced Raman spectroscopy (TERS) to show that with a change in diameter from 1.35 to 1.63 nm, the RBM frequency changes from 33.44 to 28.29 THz for SWCNTs. Kurti et al.  used first-principles calculations to show that the frequency of the RBMs were between 32.72 and 28.14 THz for SWCNTs with diameters ranging from 1.35 to 1.56 nm. Batra et al.  also investigated the RBM frequency of CNTs using the molecular dynamics (MD) method and finite element method (FEM). They found that the RBM frequency ranged from 31.55 (31.22 for FEM) to 27.40 (27.26) THz for SWCNTs with diameters between 1.35 to 1.56 nm. Our calculated frequencies of the RBM of SWCNTs with varying diameters not exposed to axial pressure agree closely with these previously reported values. This verifies that the continuum elastic shell model can accurately describe the RBM frequency of CNTs.
where f p is the frequency of the RBM with exposed to axial pressure, and f 0 is the frequency of RBM not exposed to axial pressure. Note that the RBM frequency ratio η is sublinear with respect to increasing axial pressure. We also found that the axial half-wave number m plays a critical role in this increasing frequency ratio. When the axial half-wave number increases, the increment of change in the speed of the frequency ratio becomes much smaller. The results from nonlinear stick-spiral model, presented by Chang et al. , are also shown for comparison. The models show good agreement when the axial half-wave number m is 1. These results confirm that the largest contribution to the RBM for a SWCNT comes when m = 1. By contrast, m ≥ 2 wave numbers are in the minority.
Based on elastic continuum mechanics, we studied the RBM frequency of simply-supported CNTs exposed to axial pressure. The SWCNTs were modeled as individual elastic shells, and the DWCNTs were considered to be two layered nanotube shells coupled by vdW interactions. The effects of the wave numbers, aspect ratio and axial pressure are discussed in detail. It can be seen through comparison with previous experimental and simulation investigations on the RBM frequency of isolated SWCNTs with increasing radius and the RBM frequency ratio with increasing pressure, the continuum shell model can be used to predict the RBM frequency of CNTs subject to an axial pressure. The results of the CNTs exposed to an axial pressure show that the RBM vibration frequency is sensitive to both the vibrational mode and axial pressure, while the frequency of the RBM is hardly affected by the aspect ratio. We are now processing the theoretical analysis on vibrational properties of SWCNTs and DWCNTs subjected to axial pressure in order to provide further quantitative and qualitative experiments and simulations on RBM of CNTs.
radial breathing mode
double-walled carbon nanotube
van der Waals
single-walled carbon nanotube
tip-enhanced Raman spectroscopy
finite element method.
This work was supported by a Grant-in-Aid from the Global COE Program from the Ministry of Education, Culture, Sports, Science and Technology and by CLUSTER (second stage) from the Ministry of Education, Culture, Sports, Science and Technology (Japan).
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