Carbon Nanotube Thermal Transport: Ballistic to Diffusive
  We propose to use l_0/(l_0+L) for the energy transmission covering both ballistic and diffusive regimes, where l_0 is mean free path and L is system length. This formula is applied to heat conduction in carbon nanotubes (CNTs). Calculations of thermal conduction show: (1) Thermal conductance at room temperature is proportional to the diameter of CNTs for single-walled CNTs (SWCNTs) and to the square of diameter for multi-walled CNTs (MWCNTs). (2) Interfaces play an important role in thermal conduction in CNTs due to the symmetry of CNTs vibrational modes. (3) When the phonon mean free path is comparable with the length L of CNTs in ballistic-diffusive regime, thermal conductivity \kappa goes as L^{\alpha} . The effective exponent \alpha is numerically found to decrease with increasing temperature and is insensitive to the diameter of SWCNTs for Umklapp scattering process. For short SWCNTs (<0.1 \mu m) we find \alpha \approx 0.8 at room temperature. These results are consistent with recent experimental findings. 
Jian Wang and Jian-Sheng Wang