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

We propose a continuum model to predict long-wavelength vibrational modes of empty and liquid-filled tubules that are very hard to reproduce using the conventional force-constant matrix approach based on atomistic ab initio calculation. We derive simple quantitative expressions for long-wavelength longitudinal and torsional acoustic modes, flexural acoustic modes, as well as the radial breathing mode of empty or liquid-filled tubular structures that are based on continuum elasticity theory expressions for a thin elastic plate. We furthermore show that longitudinal and flexural acoustic modes of tubules are well described by those of an elastic beam resembling a nanowire. Our numerical results for biological microtubules and carbon nanotubes agree with available experimental data.

Figure 1

Schematic representation of important deformations of a tubular structure. (a) Longitudinal acoustic (LA, stretching), (b) torsional acoustic (TA, torsion), (c) flexural acoustic (doubly degenerate ZA, bending), and (d) the radial breathing mode (RBM). (e) Schematic dispersion relations of the corresponding long-wavelength phonon modes. The tilde denotes liquid-filled tubules.

Figure 2

(a) Frequency of vibrational modes depicted in Fig. 1 in empty and water-filled carbon nanotubes. (b) Dependence of the flexural coefficient $c_{ZA}\left(R\right)$, defined in Eq. (9), on the radius $R$ of empty and water-filled carbon nanotubes. The tilde denotes liquid-filled nanotubes.

Figure 3

(a) Frequency of vibrational modes depicted in Fig. 1 in empty and water-filled tubulin-based microtubules. (b) Dependence of the flexural coefficient $c_{ZA}\left(R\right)$, defined in Eq. (9), on the radius $R$ of empty and water-filled tubulin-based microtubules. The tilde denotes liquid-filled tubules.

Figure 4

Nature and coupling of vibration modes $\omega \left(k\right)$ of a carbon nanotube that may be filled with water. (a) Coupling between the dispersionless RBM of a CNT, shown by the black line, and the pressure wave of water enclosed in the CNT, shown by the green dotted line. (b) Coupling between the RBM of a CNT, shown by the black line, and the longitudinal acoustic mode of the CNT, shown by the green dotted line. Results are presented for a CNT with a radius of 1 nm.