The onset of instability in nanostructures: The role of nonlinear resonance

Classical trajectory methods are used to examine the vibrational dynamics of carbon nanotubes. The results clearly demonstrate an integral relationship between the diameter and length of a nanotube and its positional stability: tubes having diameters smaller than 0.7 nm undergo large-amplitude motion. The origin of this motion is due to strong coupling(s) between the longitudinal (vibration along the length) and a ring breathing mode (vibration about the axis of the cylinder). It is shown that the vibrational frequency of these modes follow a simple scaling law: ωc∝1/C, ωL∝1/L, where C is the contour length around the end of the tube and L is the length of the tube along its axis. This law should be applicable to any isotropic material with a cylindrical shape and provides an analytical equation for predicting mechanical stability: When the frequencies have small integer ratios with one another, in particular a 1:2 ratio, instability will occur on a short time scale (this phenomena represents a nonlinear resonance controlled by the geometry of the system).