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We propose a continuum model to predict long-wavelength vibrational modes ofempty and liquid-filled tubules that are very hard to reproduce using theconventional force-constant matrix approach based on atomistic ab initiocalculation. We derive simple quantitative expressions for long-wavelengthlongitudinal and torsional acoustic modes, flexural acoustic modes, as well asthe radial breathing mode of empty or liquid-filled tubular structures that arebased on continuum elasticity theory expressions for a thin elastic plate. Wefurthermore show that longitudinal and flexural acoustic modes of tubules arewell described by those of an elastic beam resembling a nanowire. Our numericalresults for biological microtubules and carbon nanotubes agree with availableexperimental data.

Long-wavelength deformations and vibrational modes in empty and liquid-filled microtubules and nanotubes: A theoretical study