Mass Nature of Heat and Its Applications II: Non-Fourier Heat Conduction in Carbon Nanotubes

Carbon nanotubes (CNTs) have attracted much attention in nanotechnology fields because of their unique thermal properties. The thermal conductivity of CNTs was reported to be as high as several thousand W/mK. The heat flux in CNTs can reach 10⁹−10¹² W/m² under normal heat conduction conditions. In this paper we demonstrate that Fourier’s heat conduction law breaks down for so high heat flux. Based on a novel concept of thermomanss, which is defined as the equivalent mass of thermal energy according to Einstein’s mass-energy relation, heat conduction in CNTs can be regarded as the flow of a phonon gas governed by its mass and momentum conservation equations like in fluid mechanics. The momentum conservation equation, including driving force, inertial force and resistance terms, reduces to Fourier’s law as the heat flux is not very high and the inertial force of phonon gas is negligible with respect to the driving force. However, Fourier’s law of heat conduction no longer holds if the heat flux is very high such that the inertial force of the phonon gas is not negligible. The heat conduction behavior deviates from Fourier’s law even for steady state conditions so that the heat conduction is characterized by a non-linear relationship between the heat flux and the temperature gradient. In this case, the thermal conductivity of the CNTs can no longer be defined as the ratio of the heat flux to the temperature gradient in experiments or numerical computations. Furthermore, the ratio of the phonon gas velocity to the thermal sound speed can be defined as the thermal Mach number. Heat flow in CNTs will be choked, just like gas flows in a converging nozzle, and a temperature jump will be observed when the thermal Mach number equals or exceeds unity. In this case, the predicted temperature profile of the CNTs based on Fourier’s law is much lower than that based on the thermomass theory considering a CNT electrically heated by high-bias current flows. The intrinsic thermal conductivity can be only calculated by the present thermomass theory, rather than Fourier’s heat conduction law. The present study shows that the thermomass based theory should be applied for high flux heat conduction in CNTs where Fourier’s heat conduction law breaks down.