A measurement of K(T) yields the combined contribution of the electrons and phonons. However, a simultaneous measurement of the electrical conductivity s provides a measure of the electron thermal conductivity Ke from the Wiedemann-Franz law[1]
In this way, the phonon contribution can be deduced by subtracting the electronic contribution from the total measured thermal conductivity.
Fig. 6 Calculated thermal conductivity of an isolated SWNT, as a function of temperature.
Fig. 7 Calculated nanotube thermal conductivity (solid line) compared to the thermal conductivity of a 2-D graphene sheet (dot-dashed line) and 3-D graphite (dotted line).
Thermal Conductivity: Theory
Berber et al.[13] have calculated the phonon thermal conductivity of isolated nanotubes. Fig. 6 shows the results of theoretical calculations of the phonon thermal conductivity of an isolated SWNT. K(T) peaks near 100 K, and then decreases with increasing temperature. The value of K at the peak (37,000 W/m K) is comparable to the highest thermal conductivity ever measured (41,000 W/m K for an isotopically pure diamond sample at 104 K). Even at room temperature, the thermal conductivity is quite high (6600 W/m K), exceeding the reported room-temperature thermal conductivity of isotopically pure diamond by almost a factor of 2.
Fig. 7 shows the calculated nanotube thermal conductivity compared to the calculated thermal conductivity of a single plane of graphene and that of 3-D graphite. In graphite, the interlayer interactions quench the thermal conductivity by nearly 1 order of magnitude. It is likely that the same process occurs in nanotube bundles. Thus it is significant that the coupling between tubes in bundles is weaker than expected. It may be that this weak coupling, which is problematic for mechanical applications of nanotubes, is an advantage for thermal applications.
Measured K(T) of SWNTs
Fig. 8 shows the measured K(T) of a bulk sample of SWNTs.[14-16] K(T) increases with increasing temperature to 300 K; the decreasing slope at high temperature may indicate the onset of Umklapp scattering. It is difficult to ascertain the intrinsic thermal conductivity of an individual tube from these measurements, although they point strongly to a very high value. In disordered ”mat” samples, the room-temperature thermal conductivity is ~35 W/m K.[15] However, the nanotubes in such a sample are highly tangled, and the thermal path is considerably longer than the direct distance between points. This effect can be reduced by aligning the nanotubes; in samples where the nanotubes have been aligned by filtration in a magnetic field, the thermal conductivity is significantly higher, above 200 W/m K,[14] which is comparable to that of a good metal. Even in these samples, the thermal conductivity is likely to be limited by tube-tube junctions, so that the intrinsic single-tube thermal conductivity is certainly higher. Significantly, the temperature dependence of the thermal conductivity is roughly the same for both types of samples, suggesting that the measured K(T) reflects the intrinsic temperature dependence of the single-tube K(T).
Fig. 8 Temperature-dependent thermal conductivity of a bulk sample of SWNTs which have been aligned by filtration in a high magnetic field.
Using Eq. 7, it is possible to calculate the electronic contribution to the thermal conductivity. In all samples, simultaneous measurement of the electrical and thermal conductivity shows that the electronic contribution to the thermal conductivity is only ~ 1% of the total, so that phonons dominate K(T) at all temperatures.
At low temperature, SWNT samples exhibit a linear K(T), strongly suggesting quantum effects. Because of the large number of nanotubes in a bulk sample, it is not possible to directly observe the thermal conductivity quantum measured by Schwab et al.[10] However, it is possible to measure the K(T) of SWNT samples with varying diameters: the phonon subband splitting is higher in smaller-diameter tubes, so that the linear K(T) behavior should extend to higher temperature. Fig. 9 shows the thermal conductivity divided by temperature, K/T, of two nanotube samples, one with average diameter 1.2 nm and the other with average diameter 1.4 nm.[16] In both samples, K/T approaches a constant value at low T, just as is expected for 1-D channels. At higher temperatures, K/T increases as more phonon modes contribute. In the 1.2-nm diameter sample, the upturn in K/T occurs ~ 5 K higher than in the 1.4-nm diameter sample. This shift provides additional evidence that the low-T linear behavior is true 1-D thermal conductivity. However, one unresolved issue is the different temperature ranges of the 1-D regime in heat capacity vs. thermal conductivity. For constant scattering time, the temperature ranges should be approximately identical. One possible explanation is that the phonons in the optical bands are much more strongly scattered, and so do not begin to contribute to the thermal conductivity until higher temperatures.
Measured K(T) of MWNTs
Because of the large diameter of MWNTs, the temperature scale for quantum effects should be quite small, and their thermal conductivity should be that of a 2-D system with linear acoustic phonons. The K(T) of such a 2-D sheet should follow a T2 temperature dependence. Graphite shows a temperature dependence closer to T2.3, because of the effect of the quadratically dispersing out-of-plane mode.[17] As was discussed above, interlayer effects can be ignored when considering the thermal conductivity.
Yi et al.[18] have measured K(T) for bulk samples of MWNTs. They found a roughly T2 temperature dependence up to 100 K, as expected. The room-temperature thermal conductivity of these samples is only ~ 25 W/m K, possibly as a result of the effects of tube-tube contacts, or also of the incomplete graphitization in their samples.
Fig. 9 Thermal conductivity divided by temperature for SWNT samples with different average diameters. The smaller-diameter tubes exhibit linear K(T) up to higher temperature, consistent with quantization effects.
Fig. 10 Measured thermal conductivity of a single MWNT.
Recently, Kim et al.[19] have used a microfabricated structure (inset to Fig. 10) to directly measure the thermal conductivity of individual MWNTs. The data in Fig. 10 show the measured thermal conductivity of one MWNT. K(T) increases as T2 up to ~ 100 K, peaks near 300 K, and decreases above this temperature. Again, the quadratic temperature dependence is exactly what would be expected for large-diameter nanotubes that essentially act as 2-D sheets. The room-temperature value of K(T) is over 3000 W/m K.
Applications
The high thermal conductivity of nanotubes may be useful for a number of thermal management applications, such as heat sinking of silicon processors, or to increase the thermal conductivity of plastics in such areas as housing for electric motors. Although many groups have studied nanotube-epoxy composite materials for their mechanical properties, their possible thermal properties have only recently attracted attention.
Biercuk et al.[20] have measured the thermal conductivity of epoxy resin loaded with SWNTs. Fig. 11 shows the enhancement in the thermal conductivity for loadings up to 1% SWNTs, and the enhancement for identical loadings of graphitic carbon fibers. Addition of 1% SWNTs doubles the thermal conductivity of the epoxy, while the same loading of carbon fibers provides only a ~ 40% increase. This initial result is quite promising for the development of composites for thermal management.
Fig. 11 Enhancement of the thermal conductivity of epoxy resin as a function of loading by SWNTs and by carbon fibers.
CONCLUSION
The thermal properties of carbon nanotubes are dominated by phonons. The measured specific heat of SWNTs closely matches calculations based on the phonon band-structure of isolated nanotubes, and shows direct evidence of 1-D quantization of the phonon bandstructure. This shows that coupling between nanotubes in a bundle is relatively weak; detailed modeling permits direct measurement of the tube-tube coupling strength, as well as the low-energy phonon structure. Theoretical work predicts a room-temperature thermal conductivity of 6600 W/m K for individual nanotubes. Measurements show a room-temperature thermal conductivity over 200 W/m K for bulk samples of single-walled nanotubes, and over 3000 W/m K for individual multiwalled nanotubes. Addition of nanotubes to epoxy resin can double the thermal conductivity for a loading of only 1%, showing that nanotube composite materials may be useful for thermal management applications.
