Surface fractal dimension of single-walled carbon nanotubes

Isolated single-walled carbon nanotubes (SWNTs), SWNT bundles, and ropes (or strands) show a structural self-similar characteristic. By calculating the Hausdorff dimension, it was found that their self-similar organization leads to surface fractality and the value of the surface dimension ${(D}_s)$ depends on self-similar patterns. Experimentally, $D_s$ obtained by nitrogen adsorption measurements at 77.3 K and by the small-angle x-ray scattering technique in our study proved our calculation that the surface dimension of SWNTs is nonintegral, $2<\mathrm{}{D}_s<3,$ which indicates that the surface of SWNTs is fractal. According to our calculation and analysis, the fractality is determined by the self-similar arrangement of SWNTs, but the value of $D_s$ is also influenced by the effect of finite length and irregular aggregation of real SWNT samples.