ELASTIC PROPERTIES OF SINGLE-WALLED CARBON NANOTUBES

Analytical expressions for the velocities of the longitudinal and the torsional sound waves in single-walled carbon nanotubes are derived using Born's perturbation technique within a lattice-dynamical model. These expressions are compared to the formulas for the velocities of the sound waves in an elastic hollow cylinder from the theory of elasticity to obtain analytical expressions for the Young's and shear moduli of nanotubes. The calculated elastic moduli for different chiral and achiral ~armchair and zigzag! nanotubes using force constants of the valence force field type are compared to the existing experimental and theoretical data. vibration amplitude of several isolated MWNT's was ana- lyzed in a transmission electron microscope to eventually obtain 1.8 TPa for the average Young's modulus. Later on, this technique was applied to measure Young's modulus of isolated SWNT's in the diameter range 1.0 21.5 nm and an average value ^Y&51.2520. 35/10.45 TPa was derived. 7 In another experimental approach 8 the MWNT's were pinned to a substrate by conventional lithography and the force was measured at different distances from the pinned point by atomic force microscope ~ATM! The average Young's modulus for different MWNT's with diameters from 26 to 76 nm was found to be 1.2860. 59 TPa. Recently, Young's and shear moduli of ropes of SWNT's were measured by sus- pending the ropes over the pores of a membrane and using ATM to determine directly the resulting deflection of the rope. 9 The theoretical estimation of the elastic moduli was accomplished exclusively by numerical second derivatives of the energy of the strained nanotubes. In the calculation of the elastic moduli of various SWNT's within a simple force- constant model 10 it was found that the moduli were insensi- tive to tube size and helicity and had the average values of ^Y&50.97 TPa and ^G&50.45 TPa. In several works, molecular-dynamics ~MD! simulation algorithms using the Tersoff-Brenner potential for the carbon-carbon interactions were implemented to relax the strained nanotubes and calcu- late their energy. 11-13 For tubes of diameter of 1 nm values for Y of 5.5 TPa ~Ref. 12! and 0.8 TPa ~Ref. 13! were ob- tained. A non-orthogonal tight-binding ~TB! scheme was ap- plied to calculate Young's modulus of several chiral and achiral SWNT's yielding an average value of 1.24 TPa. 14 Recently, the second derivative of the strain energy with re- spect to the axial strain, calculated with a pseudopotential density-functional-theory ~DFT! model for a number of SWNT's, 15 was found to vary slightly with the tube type and to have the average value of 56 eV. In this paper, we choose a different approach to the cal- culation of the elastic properties of SWNT's. Namely, we derive analytical expressions for the elastic ~Young's and shear! moduli of SWNT's using a perturbation technique due to Born 16 within a lattice-dynamical model for nanotubes. 17 This scheme has the advantage that the elastic moduli are consistent with the lattice dynamics of the nanotubes and that each of these moduli is obtained in one calculational step only. The essential features of a model of the lattice dynamics of SWNT's based on the explicit accounting for the helical symmetry of the tubes 17 are summarized in Sec. II A. This model is applied to study of the long-wavelength vibrations in nanotubes using Born's perturbation technique 16 and to obtain analytical expressions for the velocities of the longi- tudinal and the torsional sound waves in SWNT's ~see Sec. II B! The comparison of these expressions with the formulas from the theory of elasticity for the velocities of these waves in an elastic hollow cylinder allows one to determine the Young's and shear moduli of the nanotubes. The calculated phonon dispersion of a (10,10) nanotube and elastic moduli for various chiral and achiral ~armchair and zigzag! nano- tubes using force constants of the valence force field ~VFF! type 18 are presented in Sec. III and discussed in comparison with the existing experimental and theoretical data.