Thermal boundary resistance and thermal conductivity of multiwalled carbon nanotubes

Thermal boundary resistance (Kapitza resistance) and thermal conductivity of multiwalled carbon nanotubes (MWCNTs) are calculated assuming the properties of graphite in the temperature range of $10–100\mathrm{K}$. By including the thermal boundary resistance between the MWCNT and the measuring device, calculated thermal conductivity of MWCNT is in very good agreement with the experimental data of Kim et al. [Phys. Rev. Lett. 87, 215502 (2001)], showing that the thermal behavior of MWCNTs is similar to graphite. Thermal conductivity of MWCNT as measured by Kim et al. is smaller than the thermal conductivity of graphite fibers and pyrolytic graphite at low temperatures due to thermal boundary resistance. The intrinsic mean free path of phonons in MWCNTs in the temperature range of $10–100\mathrm{K}$ is found to be similar to the mean free path in graphite fibers. Finally, it is shown that the thermal boundary resistance could be one of the main reasons that the thermal conductivity of MWCNT bundles as measured by Kim et al. is lower than the thermal conductivity of a single MWCNT.

Figure 1

Comparison between the thermal conductivity of MWCNTs and graphite. Kim et al. (Ref. 1) reported an error of $2\mathrm{nm}$ in MWCNT diameter with a nominal diameter of $14\mathrm{nm}$. Thermal conductivity from their data is shown by assuming diameters of $14\mathrm{nm}$ and $16\mathrm{nm}$. Larger diameter gives smaller thermal conductivity, making it closer to the graphite thermal conductivity at higher temperatures (after the maxima).

Figure 2

Theoretical specific heat of graphite, graphene, and MWCNTs assuming the properties of graphite. Curves for 3.4 and $2\mathrm{nm}$ MWCNTs taken from Ref. 43. The figure clearly shows that deviation from the bulk behavior due to phonon confinement takes place at very low temperatures. For $14\mathrm{nm}$ MWCNT used by Kim et al. (Ref. 1), significant deviation will take place at $T\sim 2.5\mathrm{K}$ (see text for details).

Figure 3

(Color online) Calculated specific heat of MWCNT in comparison with experimental data. $c_{44}=0.7\mathrm{GPa}$ and $c_{33}=20\mathrm{GPa}$ reflect the reduced coupling between the graphene layers in MWCNTs (see text for details).

Figure 4

Comparison between prediction and experimental data for MWCNT without including the contact resistance.

Figure 5

Schematic of the device used by Kim et al. (Ref. 1) for the measurement of thermal conductivity of MWCNT. The heat enters the MWCNT through a very narrow constriction formed due to van der Waals forces between the MWCNT and the substrate.

Figure 6

(Color online) Schematic showing the different orientations of graphite (MWCNT) [(a) horizontal contact and (b) vertical contact] on a substrate for which thermal boundary resistance has been calculated.

Figure 7

Thermal contact conductance between graphite and platinum for the basal plane (vertical contact) and $c$-axis (horizontal contact) directions as a function of temperature. Similar to the thermal conductivity, the anisotropy in contact conductance decreases with decreasing temperature.

Figure 8

(Color online) Comparison between the predicted and experimental thermal conductivities by including the thermal boundary resistance and using the recommended properties of graphite (Table ).

Figure 9

(Color online) Comparison between the predicted and experimental thermal conductivities by including the thermal boundary resistance and assuming $c_{44}=0.7\mathrm{GPa}$ and $c_{33}=20\mathrm{GPa}$. Reduced values of $c_{44}$ and $c_{33}$ are used to reflect reduction in intertube coupling in MWCNTs, as compared to graphite (see text for details).

Figure 10

Comparison between the prediction and data for bundles assuming the bundles to be solid cylinders. The contact width is calculated using Eq. (20). Trends obtained from the prediction by including the contact resistance are consistent with the experimental trends, i.e., thermal conductivity of single MWCNT (not shown here) is higher than the thermal conductivity of the $80\mathrm{nm}$ bundle which in turn is higher than the thermal conductivity of the $200\mathrm{nm}$ bundle.

Figure 11

Comparison between the prediction and data for bundles assuming that only a few nanotubes in the bundles touch the substrate. Effective contact width of the $80\mathrm{nm}$ bundle is assumed to be $2\mathrm{nm}$ and for the $200\mathrm{nm}$ bundle is assumed to be $3\mathrm{nm}$.