Discontinuous Tangential Stress in Double Wall Carbon Nanotubes

We have examined the stability of double wall carbon nanotubes under hydrostatic pressures up to 10 GPa. The tangential optical phonon mode observed by inelastic light scattering is sensitive to the in-plane stress and splits into a contribution associated with the external and internal tube. While the pressure coefficient from the external tube is the same as in single wall carbon nanotubes, the pressure coefficient from the internal tube is found to be 45% smaller. The phonon band from the external tube broadens considerably with applied pressure in contrast with the phonon band of the internal tube which stays constant. These pressure dependent phonon shifts of the external and internal tubes and the contrasting phonon line broadening are explained by the elastic continuum shell model which takes into account both the continuous radial and discontinuous tangential stress components.

Figure 1

Raman spectra of double wall CNT in the region of optical mode at different hydrostatic pressure.

Figure 2

Frequencies deduced from the fitting of the spectra reported in Fig. 1.

Figure 3

Linewidth of the Lorentzians used for the fitting of the spectra reported in Fig. 1. Dashed lines are guide-eyes. Graphite value is taken from Ref. [24]

Figure 4

Tangential stress (${\sigma }_{\theta \theta }/-p$) at the surface of both external and internal tube for an intermediate radius ranging from 0.34 (no vacuum inside, $R_1=0\mathrm{n}\mathrm{m}$ and $R_2=0.68\mathrm{n}\mathrm{m}$) to 1.2 nm ($R_1=0.86\mathrm{n}\mathrm{m}$ and $R_2=1.54\mathrm{n}\mathrm{m}$).