  We present a phenomenological force-constant model developed for the
description of lattice dynamics of sp2 hybridized carbon networks. Within this
model approach, we introduce a new set of parameters to calculate the phonon
dispersion of graphene by fitting the ab initio dispersion. Vibrational modes
of carbon nanotubes are obtained by folding the 2D dispersion of graphene and
applying special corrections for the low-frequency modes. Particular attention
is paid to the exact dispersion law of the acoustic modes, which determine the
low-frequency thermal properties and reveal quantum size effects in carbon
nanotubes. On the basis of the resulting phonon spectra, we calculate the
specific heat and the thermal conductance for several achiral nanotubes of
different diameter. Through the temperature dependence of the specific heat we
demonstrate that phonon spectra of carbon nanotubes show one-dimensional
behavior and that the phonon subbands are quantized at low temperatures.
Consequently, we prove the quantization of the phonon thermal conductance by
means of an analysis based on the Landauer theory of heat transport.
