Heat Pulse Propagation in SWNT

MD simulations in [16] showed higher thermal conductivity for the (10,0) zigzag SWNT compared to the (5,5) armchair nanotube especially as can be seen in Figure 7.7b. Further MD simulations in [48] also showed that zigzag (20,0) SWNT had higher thermal conductivity compared to armchair (11,11) and chiral (10,13) nanotubes. Also calculations using lattice dynamics in [58] confirmed the same trend. The diameter of the (10,0) nanotube is only 16% larger than that of the (5,5) nanotube. This raises a question about the nature of energy transfer mechanisms in each nanotube and whether the mechanism is affected by chirality or the diameter of the nanotube. For example, excitation of different phonon modes may be responsible for the earlier reported difference in the thermal conductivities of zigzag and armchair nanotubes of same diameter [9,48,58]. The Fourier approximation used in steady-state thermal conductivity calculations, however, does not provide any information about the participating phonon modes and the energy carried by each phonon mode.

The application of strong heat pulses, on the other hand, generates several waves propagating at different speeds corresponding to different phonon modes, and can provide information about individual modes and their contribution towards the overall heat transport [59-61]. Heat pulse measurements in NaF at low temperatures revealed ballistic LA and TA phonon mode propagation as well as second sound waves at temperatures below 14 K [60]. Therefore heat pulse experiments provide quantitative information on transport by diffusion, ballistic phonons, and second sound waves. The second sound waves can be observed, as in the experiments reported in [60], when the momentum-conserving normal phonon scattering processes are dominant compared to the momentum-randomizing Umklapp phonon scattering processes. Both experimental measurements on thermal conductivity and some theoretical models as well point to the long phonon mean free path length as being responsible for the higher thermal conductivity and its increase up to room temperature. As Umklapp processes mainly determine the mean free path length, one can also raise the question about whether this implies that normal phonon-phonon scattering processes (N-processes) dominate up to room temperature or at least over a wider range of temperatures compared to that in NaF. If this is true, then one can expect second sound waves to be observed in nanotubes. The speed of the leading edges of pulses arriving at the detector is determined from the arrival time and the sample length and used to determine the ballistic phonon mode (LA or TA) [59,60]. This approach assumes accurate knowledge of the speed of each phonon mode in the material under investigation. Consequently, heat pulse simulations can provide insight into how heat flow occurs in nanotubes, the important contributing phonon modes, and their dependence on the nanotube chirality and diameter.

Earlier molecular dynamics studies on pulsed heat propagation in alpha iron demonstrated energy flow by LA and TA modes and second sound waves [59]. Low temperatures and pure crystalline materials are required to observe second sound waves that are not attenuated by dissipative scattering processes. The molecular dynamics simulations thus provide an ideal platform for investigating these properties by controlling the temperature of the CNT, shape, and duration of heat pulses assuming the perfect crystalline structure. Thermal energy transport in single-wall carbon nanotubes subjected to intense heat in (7,0), (10,0), and (5,5) single-wall carbon nanotubes with particular emphasis on the role of nanotube chirality and diameter was examined in [62]. The heat pulse was found to generate wave packets that move at the speed of sound corresponding to different phonon modes, second sound waves, and diffusive components. The waves corresponding to ballistic LA, TW phonon modes, and second sound in zigzag nanotubes carried more heat energy than in armchair nanotubes of similar diameter [62]. The thermal conductivity of zigzag nanotubes due to the ballistic phonon contribution therefore is expected to be larger as compared to the armchair nanotubes as reported recently [16,48].

For MD simulations, each nanotube was divided into 250 slabs. The number of atoms per slab was 40 atoms in a (10,0) nanotube resulting in 10,000 atoms for the whole nanotube. The number of atoms per slab increases as the diameter of the nanotube increases. The axis of the tube was aligned along the z-axis with the free end of the nanotube at z = 0, and the far end of the nanotube was held rigid. In order to apply heat pulse and ensure smooth temperature transition from the boundary to the region of interest on the nanotube, the boundary region was chosen to extend over five slabs and the temperature was adjusted equally during the rise and fall times of the pulse. Additionally, the temperature at each slab was spatially averaged over 5 slabs centered at the slab of interest. The resulting temperature was also time-averaged over 200 time steps. The boundary at the far end of the nanotube consisted of several slabs with the outermost slab rigidly held in place. The remaining slabs were held in equilibrium at the simulation temperature by applying a Gaussian random force, satisfying the fluctuation-dissipation theorem, which ensured that no reflection from the boundary to the oncoming propagating waves occurred. The one-dimensional nature of the nanotube simplifies the boundary conditions, as one does not have to worry about the scattering in the transverse directions. The atom's motion and configurations in the transverse or radial directions are governed by the intrinsic forces that maintain the shape of the nanotube.

Initially, MD simulations of single-wall carbon nanotubes were run to achieve thermal equilibrium at low temperatures close to 0.1 K. This was followed by the application of a strong heat pulse of finite duration, and rise and fall times. The simulation was stopped before the leading waves reached the right boundary, even though the boundary slabs and conditions would have ensured no reflection back into the system. The instantaneous temperature distribution along the nanotube was recorded every 200 time steps in addition to the time and spatially averaged temperatures. The minimum time scale Tm for time-averaging is taken to be the time required for a ballistic LA mode to traverse a single slab which corresponds to about 25 fs for a (5,5) nanotube which is significantly larger than the time step of 0.5 fs used in the simulations. In order to determine the propagation speeds of the leading stress waves, the slab numbers corresponding to the peak locations at given time intervals are recorded. The speed is then determined from the spatial distance traversed by the particular wave during a given time interval. The average speed for each peak is determined from linear fit of the peak locations versus time.

An example of the temporal and spatial variation of the kinetic temperature in a (10,0) nanotube obtained from the MD simulations is shown in Figure 7.14. The

(10,0) nanotube

Figure 7.14. Spatial and temporal distribution of kinetic temperature along a (10,0) SWNT [62].

(10,0) nanotube

Figure 7.14. Spatial and temporal distribution of kinetic temperature along a (10,0) SWNT [62].

Figure 7.15. Spatial distribution of kinetic temperature along (10,0) nanotube at time 3 and 4 ps after the application of the heat pulse [62].

Figure 7.15. Spatial distribution of kinetic temperature along (10,0) nanotube at time 3 and 4 ps after the application of the heat pulse [62].

heat pulse induces clearly defined wave packets that propagate on the nanotube. The leading wave packets move at higher speeds as compared to the diffusive background. The shapes of wave packets P2 and P3, as shown in Figure 7.15, change and the peak intensity of P2 decreases by 15% between 3 ps and 5 ps whereas that of P3 stays constant during the same period. On the other hand, the peak intensity of the weak leading wave packet P1 stays constant and the shape does not change with time. The leading wave packets are followed by a dual-peak wave packet that undergoes very minor changes in its shape and the peak intensities slowly decay to a final kinetic temperature around 40 K at 5 ps. The time axis for this analysis was started at 1.5 ps to avoid the initial transient large kinetic temperature values that make it difficult to observe and analyze the low-intensity leading propagating waves from left to right. By shifting the time axis to 1.5 ps, the plotted maximum temperature has been reduced from 400 K to less than 100 K, which allows for better resolution of wave packets with smaller peaks. The distance along the nanotube axis has been normalized by the slab width and is plotted as slab numbers.

The three-dimensional view in Figure 7.14 provides a clear picture of how waves develop and propagate as separate wave packets on the nanotube. The details of the low-amplitude higher-speed wave packets are not clearly seen in the figure because of the slow-moving large diffusive components. Figure 7.15 shows the details of the frontal leading waves after leaving out the diffusive part of temperature distributions at times 4 and 5 ps. The average speed of peaks P1

Figure 7.16. Shape of LA leading mode peak in (10,0), (7,0), and (5,5) SWNT at time = 4 ps. The line through the data points represents the best Guassian fit [62].

Figure 7.16. Shape of LA leading mode peak in (10,0), (7,0), and (5,5) SWNT at time = 4 ps. The line through the data points represents the best Guassian fit [62].

and P2 were found to be 20.6 km/s and 16.6 km/s, respectively. These speeds are very close to the speed of sound associated with longitudinal acoustic and twisted phonon modes, respectively [22]. The values obtained from the MD simulations are similar to theoretically calculated sound velocities of LA and TW phonons, in (10,10) armchair nanotubes, which were reported to be 20.35 km/s and 15 km/s, respectively [22]. The shape of the leading wave packet (identified as arising from the LA mode) did not change and had a peak kinetic temperature of 0.6 K and a full width at half maximum of 2.4 nm as shown in Figure 7.16. However, the wave packet P2, corresponding to the TW mode, has higher peak energy and has a fast decay from an initial value of 17 K to 7.0 K at 5 ps with a decay time constant of 1.06 ps as shown in Figure 7.17a. For comparison, the peak temperature for P1 is shown in Figure 7.17b. At 5 ps after the onset of the pulse, wave packet P2 had a much higher peak kinetic temperature of 7 K and a width at half maximum of 4.3 nm. Therefore the atomic motions within the TW wave packet P2 carry about 20 times higher overall kinetic energy as compared to the atomic motions within the P1 wave packet

The strongest wave packets in Figure 7.15 are P4 and P5 and they propagate together at a speed of 12.2 Km/s. The peak energies for P4 and P5 in a (10,0) nanotube decay with time constants of 2.1 and 1.0 ps, respectively, as shown in Figure 7.17a. At 5 ps after the onset of the pulse, wave packets P4 and P5 had peak temperatures 39.0 K and 44.0 K and full width at half maximum of 1.9 nm and 2.3 nm, respectively. Note that these two wave packets also remain spatially confined as compared to the leading wave packets P1, P2, and P3. Furthermore,

Figure 7.17. Temporal change of peak temperature of (a) TW mode wave packets; (b) second sound wave packet. The line through the data points represents the best exponential decay fit [62].

the total energy in the wave packets P4 and P5, as computed from the area of the curve under the Gaussian fit, is three times larger than the combined P2 and P3 wave packets. Consequently, one can conclude that the leading wave packets in zigzag (10,0) nanotubes propagate at the sound velocities of LA and TW modes. However, the largest amount of energy is carried by wave packets propagating at 12.2 Km/s. Additional simulations of (7,0) and (5,5) SWNT also revealed a similar trend. The results for the sound speed of wave packets P1 through P5 in (10,0), (5,5), and (7,0) nanotubes are summarized in Table 7.1.

Table 7.2 Speeds in km/sec of the leading propagating wave packets in (10,0), (7,0), and (5,5) single-wall carbon nanotubes (from heat pulse simulation).

Peak

(10,0)

(7,0)

(5,5)

P1

20.6 (LA)

21.3 (LA)

20.3 (LA)

P2

16.4 (TW)

18.1 (TW)

16.2 (TW)

P3

15.8 (TW)

16.2 (TW)

17.0 (TW)

P4

12.2

12.7

12.9

P5

12.2

12.6

12.3

The wave packets P4 and P5 propagate at speed approximately equal to 12 Km/sec in all the nanotubes as can be seen from Table 7.2. These wave packets can also be analyzed in comparison with the propagation speed data of known phonon modes. The propagation speed data, as seen in Table 7.3, shows that for TW modes propagation speeds vary from 12.0 km/s to 15.0 km/s whereas for the transverse breathing (TB) phonon modes vary from 12.3 km/s to 42.0 km/s [22,63-67]. The TB modes are also considered optical modes because their frequency is nonzero at q = 0. This large variation, however, arises from the differences in values of force constants and Young's modulus values used in different methods. To examine the possibility of the occurrence of the second sound wave, the ratio R of the speed of wave packet P1 (VLA mode) to the average speed of the wave packets P4 and P5 (VS2), was computed for all three nanotubes. The ratio R was found to be 1.69 in (10,0), 1.68 in (7,0), and 1.62 in (5,5) nanotubes. These ratios deviate fromV3, by 2%, 3%, and 6% in (10,0), (7,0), and (5,5) nanotubes, respectively. This ratio was used to predict the presence of second sound waves at low temperatures in a-iron and NaF [59-61], and it is expected to be V3 for second sound waves in good crystalline materials. The second sound waves arise when momentum-conserving phonon-scattering events (N-processes) are far more frequent as compared to momentum-destroying collisions (U-processes) as temperature increases in the material. Under such conditions, the energy propagates as a collective temperature pulse characterized by velocity (second sound wave velocity) that is determined by the interaction of different phonon modes in the system [61].

Table 7.3 Speeds of sound waves associated with longitudinal (VL), transverse (VT), twisted (VTW), and breathing (VB) phonon modes in carbon nanotubes.

Vl (km/s)

vt (km/s)

VTw(km/s)

Vb (km/s)

References

20.35

9.43

15.0

1

21.7

14.0

42.8 Km/s

2

20.35

24.0, 9.0

15.0

3

20.7

14.1

14.1

4

19.9

12.3

5,6

19.9

12.3

12.3

6

15.0(13.0)

7

In carbon nanotubes, the momentum-destroying Umklapp scattering processes are far less frequent at low temperatures as compared to the N-process and only begin to play a major role above room temperature which is responsible for the monotonic increase in the thermal conductivity of carbon nanotubes up to room temperature [10-13]. The local kinetic temperature within P4 and P5 wave packets is well below room temperature and the nanotube temperature is below 0.1 K which makes N-processes dominant. Furthermore, as can be seen from the three dimensional plot in Figure 7.14, other modes (TW) are generated as the peaks of modes P4 and P5 decay; this supports the selection that the peaks P4 and P5 represent second sound waves. Further dynamic investigations with the Tersoff-Brenner potential to analyze the dynamics of individual phonon modes are required to confirm the peak assignments. We note, however, that the above assignment of P4 and P5 as the second sound wave does not affect in any way the main results and conclusions of this work; that is, in general the nondiffusive energy carried by the modes P1-P5 for zigzag type nanotubes (7,0) and (10,0) are larger than the energy carried by similar modes in the armchair type (5,5) nanotube.

Assuming that similar propagating modes carry the bulk of thermal energy transport under equilibrium conditions as well, the above comments support the observation that for the same temperature gradient, one can expect higher heat flux in (10,0) zigzag nanotubes as compared to the (5,5) armchair nanotubes [16,58]. Furthermore, even the energy carried by second sound waves is smaller in (5,5) compared to (10,0) and (7,0) nanotubes. This translates to higher thermal conductivity for the zigzag nanotubes compared to armchair nanotubes. As the nanotube diameter increases, more modes are excited which can also contribute to heat flow. A more detailed analysis of the propagation speeds of the individual phonon modes and simulation of larger-diameter nanotubes are required to obtain more quantitative dynamic results.

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