A strong correlation between the bending rigidity and the length of single-walled carbon nanotubes

Continuous efforts to discover the novel carbon nanotube ultrahigh frequency resonators or sensors have being made since past two decades. The bending rigidity plays a key role in determining the frequency magnitude. Although it is previously justified that the bending rigidity has the almost linear dependence on the cubic of tube diameter, its dependence on another characteristic scale, i.e., the tube length is missing. Considering that the direct experimental observation faces significant challenge due to the low measurement precision by the inevitable thermodynamic fluctuation, we theoretically explored such size effect by means of three approaches respectively at different scale levels including quantum mechanics lattice dynamics calculations, molecular mechanics simulations and nonlocal continuum modeling for single-walled carbon nanotubes. The results from the different approaches give the consistent conclusion that there exists a strong correlation between the tube length and the bending rigidity, i.e., the rigidity increases with the tube length and converges to the value predicted by continuum theories. Moreover, we also find the nonlocal parameter reflecting the microscopic lattice effect in present continuum modeling almost independent of the chirality and linearly increases with the tube diameter with a scale factor 1.5. The comprehensive study may not only guide the design of ultrahigh frequency carbon nanotube devices but also provide insight to the bending nanomechanics of other devices made from nanotubes, nanobeams and nanowires.