Divergent and Ultrahigh Thermal Conductivity in Millimeter-Long Nanotubes

Low-dimensional materials could display anomalous thermal conduction that the thermal conductivity ($\kappa$) diverges with increasing lengths, in ways inconceivable in any bulk materials. However, previous theoretical or experimental investigations were plagued with many finite-size effects, rendering the results either indirect or inconclusive. Indeed, investigations on the anomalous thermal conduction must demand the sample length to be sufficiently long so that the phenomena could emerge from unwanted finite-size effects. Here we report experimental observations that the $\kappa$’s of single-wall carbon nanotubes continuously increase with their lengths over 1 mm, reaching at least $8640\mathrm{W}/\mathrm{mK}$ at room temperature. Remarkably, the anomalous thermal conduction persists even with the presence of defects, isotopic disorders, impurities, and surface absorbates. Thus, we demonstrate that the anomalous thermal conduction in real materials can persist over much longer distances than previously thought. The finding would open new regimes for wave engineering of heat as well as manipulating phonons at macroscopic scales.

Figure 1

(a) SEM image of a CNT anchored on a test fixture consisting of parallel resistive thermometers (${\mathrm{RT}}_i$’s) made by Pt films on ${\mathrm{SiN}}_x$ beams. (b) The corresponding thermal circuits when ${\mathrm{RT}}_1$ is used as a heater. (c) Measured background thermal conductance due to radiation heat transfer (from heater to sensor) for various heater-to-sensor distances. The measured values for forward and reversed biases (i.e., exchanging the role of the heater and the sensor) are shown, demonstrating $f_{ij}=1\pm 0.04$. We have noticed that the background thermal conductance is sensitive to the environment (such as whether the Si substrate is fully etched through or partially etched), so that the measured values are different even if the heater-to-sensor distances are similar. (d) Measure $\mathrm{\Delta }{T}_s$ vs $P$ for driving frequency at 2 Hz, which gives a noise equivalent thermal conductance of $4.7\times {10}^{-12}\mathrm{W}{\mathrm{K}}^{-1}{\mathrm{Hz}}^{-1/2}$ at room temperature.

Figure 2

SEM panorama of sample 9 (divided into three parts), where a CNT is suspended across a heater and a sensor (the horizontal beams in the top right and the bottom left images). The total suspended length of sample 9 is 1.039 mm. The arrows in the figures denote the CNT.

Figure 3

$\kappa$ vs $L$ relations for nine different CNTs. Both measured ${\kappa }_m$’s (open symbols) and corrected $\kappa$’s (solid symbols, after incorporating radiation heat loss from the surface of CNTs) are shown for each sample. The measured ${\kappa }_m$’s and corrected $\kappa$’s are almost identical for $L<100\mu \mathrm{m}$. For the longest CNT investigated ($L=1.039\mathrm{mm}$), the measured ${\kappa }_m$ and the corrected $\kappa$ reach $8640$ and $13300\mathrm{W}/\mathrm{mK}$, respectively. The fits (by parametrizing $\kappa \sim L^{\alpha }$) to the corrected $\kappa$’s and measured ${\kappa }_m$’s are shown by solid curves and dashed curves, respectively.

Figure 4

Normalized $\kappa$ vs $L$ for the investigated samples. Here the corrected $\kappa$’s (solid symbols) and measured ${\kappa }_m$’s (open symbols) are normalized, respectively, by those of each sample’s shortest $L$. The effects of contact thermal resistance from small ($K_s/K_c=0.2$) to large ($K_s/K_c=5$) are calculated using Eq. (4) (with $\alpha =0$), demonstrating that the observed divergent of $\kappa$ or ${\kappa }_m$ cannot be attributed to contact thermal resistances adding to a diffusive thermal conductor. A controlled experiment on a ${\mathrm{SiN}}_x$ beam shows the expected normal thermal conduction.