Statistical mechanics of nanotubes*

We investigate how thermal fluctuations affect mechanical properties of nanotubes by employing renormalization group procedure. For 2D sheets it was previously shown that thermal fluctuations effectively renormalize elastic constants beyond a characteristic thermal length scale (a few nanometers for graphene at room temperature), where the bending rigidity increases, while the in-plane elastic constants reduce with universal power law exponents. However, the curvature of nanotubes produces new phenomena. Specifically, we find that in the axial direction the in-plane elastic constants stop renormalizing at the elastic length scale (proportional to geometric mean of the radius and the effective thickness of the shell), while in the circumferential direction they continue to renormalize albeit with different universal exponents. On the other hand, the bending rigidity stops renormalizing in the circumferential direction at the elastic length scale. These results were verified with molecular dynamics simulations by measuring the mechanical response to axial loads and external pressure. We also comment on how these temperature dependent properties affect the critical buckling loads for nanotubes.