Application of elastic wave dispersion relations to estimate thermal properties of nanoscale wires and tubes of varying wall thickness and diameter.

This paper reports dependency of specific heat and ballistic thermal conductance on cross-sectional geometry (tube versus rod) and size (i.e., diameter and wall thickness), in free-standing isotropic non-metallic crystalline nanostructures. The analysis is performed using dispersion relations found by numerically solving the Pochhammer-Chree frequency equation for a tube. Estimates for the allowable phonon dispersion relations within the crystal lattice are obtained by modifying the elastic acoustic dispersion relations so as to account for the discrete nature of the material's crystal lattice. These phonon dispersion relations are then used to evaluate the specific heat and ballistic thermal conductance in the nanostructures as a function of the nanostructure geometry and size. Two major results are revealed in the analysis: increasing the outer diameter of a nanotube while keeping the ratio of the inner to outer tube radius (gamma) fixed increases the total number of available phonon modes capable of thermal population. Secondly, decreasing the wall thickness of a nanotube (i.e., increasing gamma) while keeping its outer diameter fixed, results in a drastic decrease in the available phonon mode density and a reduction in the frequency of the longitudinal and flexural acoustic phonon modes in the nanostructure. The dependency of the nanostructure's specific heat on temperature indicates 1D, 2D, and 3D geometric phonon confinement regimes. Transition temperatures for each phonon confinement regime are shown to depend on both the nanostructure's wall thickness and outer radius. Compared to nanowires (gamma = 0), the frequency reduction of acoustic phonon modes in thinner walled nanotubes (gamma = 0.96) is shown to elevate the ballistic thermal conductance of the thin-walled nanotube between 0.2 and 150 K. At 20 K, the ballistic thermal conductance of the thin-walled nanotube (gamma = 0.96) becomes 300% greater than that of a solid nanowire. For temperatures above 150 K, the trend in ballistic thermal conductance inverts. The greater number of phonon modes in nanostructures with increased outer diameter and wall thickness is shown to have a larger contribution to ballistic thermal conductance when compared to the increased contribution from the frequency reduction of acoustic phonon modes in thinner walled nanotubes.