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

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Abstract
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).