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External loads typically have a indirect influence on phonon curves, i.e.,they influence the phonon curves by changing the state about whichlinearization is performed. In this paper, we show that in nanotubes, the axialload has a direct first-order influence on the long-wavelength behavior of thetransverse acoustic (TA) mode. In particular, when the tube is force-free theTA mode frequencies vary quadratically with wave number and have curvature(second derivative) proportional to the square-root of the nanotube's bendingstiffness. When the tube has non-zero external force, the TA mode frequenciesvary linearly with wave number and have slope proportional to the square-rootof the axial force. Therefore, the TA phonon curves -- and associated transportproperties -- are not material properties but rather can be directly tuned byexternal loads. In addition, we show that the out-of-plane shear deformationdoes {\em not} contribute to this mode and the unusual properties of the TAmode are exclusively due to bending. Our calculations consist of 3 parts:first, we use a linear chain of atoms as an illustrative example that can besolved in close-form; second, we use our recently-developed symmetry-adaptedphonon analysis method to present direct numerical evidence; and finally, wepresent a simple mechanical model that captures the essential physics of thegeometric nonlinearity in slender nanotubes that couples the axial loaddirectly to the phonon curves. We also compute the Density of States and showthe significant effect of the external load.

Anomalous phonon behavior of carbon nanotubes: First-order influence of external load