Giant Thermal Rectification from Single-Carbon NanotubeâGraphene

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Giant Thermal Rectification from Single Carbon Nanotube-Graphene Junction Xueming Yang, Dapeng Yu, and Bingyang Cao ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.7b04464 • Publication Date (Web): 21 Jun 2017 Downloaded from http://pubs.acs.org on June 22, 2017

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ACS Applied Materials & Interfaces

Giant Thermal Rectification from Single Carbon Nanotube-Graphene Junction Xueming Yang,*, ‡ Dapeng Yu, † and Bingyang Cao,*, ‡ † ‡

Department of Power Engineering, North China Electric Power University, Baoding 071003, China Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of

Engineering Mechanics, Tsinghua University, Beijing 100084, China

KEYWORDS: thermal rectification, carbon nanotube-graphene junctions, phonon density of states, standing wave, molecular dynamics

Abstract We describe the influence of the geometry parameters on the thermal rectification of single carbon nanotube-graphene junction. The two-dimensional (2D) distribution of the thermal rectification with respect to the tube length and the side length of the graphene nanosheet are calculated and visualized. The maximum thermal rectification ratios of the designed single carbon nanotube-graphene junction can reach 1244.1% and 1681.6% at average temperatures of 300 K and 200 K, respectively. These values are much higher than those reported for singlematerial nanostructure-based thermal rectifiers. The thermal rectification ratios of the nanotubegraphene junction are fairly sensitive to geometry size, and are almost entirely dominated by the degree of overlap of the power spectra under negative thermal bias. These findings could offer useful guidelines for the design and performance improvement of thermal diodes.

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1. INTRODUCTION As most fundamental thermal controlling devices, thermal rectifiers have attracted considerable research attention due to their great potential in various applications, such as on-chip cooling, energy-saving buildings and phononics applications1-4. Numerous theoretical and experimental studies have predicted or demonstrated that thermal rectification (TR) exists in bulk or nanosized system, for example, shape asymmetric structures5-14, bi-segment or bi-material systems15-20, hybrid structures21,

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and systems based on phase change23-26. However, the maximum

rectification efficiency of these thermal rectifiers seems to be still much lower than that of their electronic counterparts27. Accordingly, the development of more efficient thermal rectifiers is becoming of primary importance for their future applications. In our recent study,28 we reported the considerable thermal rectification from the designed graded two-stage pillared graphene structures and preliminarily discussed the mechanism. Such kind of nanostructures can be considered to be constructed with building blocks of carbon nanotube-graphene junctions.29 However, the role played by the carbon nanotube-graphene junctions in the thermal rectification effect of the pillared graphene is not yet fully understood. Moreover, whether ultrahigh thermal rectification can be exhibited in a single carbon nanotubegraphene junction is still unknown. Furthermore, how the thermal rectification is affected by its geometry parameters remains unknown and should be investigated, as this knowledge is crucial for the successful design of thermal rectifiers with pillared graphene structures. It is mentioning that the thermal rectification of graded nanostructures is rather sensitive to their geometry size, but the optimization is challenging due to the huge computational cost. In this study, thermal rectification of two models of single-walled carbon nanotube (SWCNT)graphene junctions with different boundary thermal contacts are investigated using


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nonequilibrium molecular dynamics (NEMD) simulations. The two-dimensional (2D) distribution of the thermal rectification with respect to the tube length and side length of the graphene nanosheet are calculated and visualized. We find that the values of the maximum thermal rectification ratios can reach 1244.1% and 1681.6% at average temperatures of 300 K and 200 K, respectively. 2. MD SIMULATION METHOD NEMD simulations are conducted for SWCNT-graphene junctions using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) package.30 To facilitate the comparison with the literature data, the simulation details adopted are almost identical to those by Wang et al.,7 and were also described in detail in our previous study. 28 The schematics of the configuration of the two models of SWCNT-graphene junctions with different boundary thermal contacts6, Junction A and Junction B, are illustrated in Figure 1a, d, in which the green, red, and cyan areas denote the fixed part, thermostated part and free part of the system, respectively. The SWCNT-graphene junctions are considered to be constructed via an armchair SWCNT (6, 6), which is erected on and covalently joined to a graphene nanosheet with an appropriate hole. In our study, the graphene nanosheets within the SWCNT-graphene junctions have a square shape with the side length L2 6 as shown in Figure 1a. Following the idea of Lee, the free part of the system is considered as the

nanodevice, and the thermostated parts are considered as the thermal contacts. In all simulations, the optimized Tersoff potential31 is adopted, and a free boundary condition is used. The time step is set to 0.4 fs. Initially, the suspended junctions are relaxed in the NVT ensemble at T0 for 2 ns using the Nosé-Hoover thermostat. Then, to establish a temperature gradient along the longitudinal direction of the tube, NEMD is performed for another 4.8 ns; the constant temperatures in the two heating bath zones are controlled by a Berendsen thermostat. The temperatures in the hot and cold bath zones are set to be T0(1+ ∆) and T0(1- ∆), respectively, 3 ACS Paragon Plus Environment


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where, T0 and ∆ represent the average temperature and the normalized temperature difference, respectively. The TR ratio is defined as:

η=

(J+ − J− ) × 100% J−

(1)

where, J + is the heat current from the wide side to the narrow side, corresponding to ∆ > 0, and J − is the heat current from the narrow side to the wide side when ∆ < 0.

3. RESULTS AND DISCUSSION First, we investigate the dependence of thermal rectification on the geometry size of the SWCNT-graphene junctions, which are characterized by the tube length L1 and the graphene nanosheet L2 (depicted in Figure 1a). The 2D gridded data of the thermal rectification by L1 varying from 3.69 to 9.84 nm (15-40 unit cells) with a step of 5 unit cells, and L2 varying from 6 to 16 nm with a step of 2 nm are calculated by MD simulations. In all simulations, T0 is set as 300 K, and | ∆ | is set as 0.5. Surprisingly, ultrahigh thermal rectification ratios with maximum values of 1244.1% and 1063% are observed by Junction A and Junction B with geometry sizes L1 = 4.9 nm and L2 = 10 nm, respectively. These thermal rectification ratio values are much higher than the rectification value of 790.8% established for the designed graded two-stage pillared graphene structures we reported in our previous study28. Then, a cubic interpolation is performed on the gridded simulation results to obtain a smoother surface distribution of the thermal rectification ratio. The distribution of the thermal rectification with respect to L1 and L2 for Junction A and Junction B is illustrated in Figure 1b, c, e, and f, in which thermal rectification of the nanotube-graphene junctions that is fairly sensitive to geometry size can be observed. L1 varies from 3.69 to 9.84 nm (15-40 unit cells), and L2 from 6


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to 16 nm, which lead to changes in the values of the thermal rectification ratio of Junction A and Junction B from 69% to 1244.1%, and from 15% to 1063%, respectively. The characteristics of the spatial distribution of the thermal rectification with geometry size for Junction A and Junction B are very similar, except that the highest peak and the second peak for Junction A are higher than those for Junction B. These findings may indicate that, for SWCNT-graphene junctions, the difference between the thermal contacts influences the values more than the overall distribution characteristics of the thermal rectification with respect to geometry size. Specifically, for both Junction A and Junction B, the highest peak with the maximum value of the thermal rectification ratio appears at geometry sizes L1 = 4.9 nm and L2 = 10 nm, and the second peak is observed at L1 = 6.1 nm and L2 = 8 nm; as can be seen in Figure 1, where most of the regions with larger rectification values are located in the narrow areas of the connections of the two peaks. The causes for the ultrahigh thermal rectification in these junction nanostructures and for the sensitivity of thermal rectification to the geometry size also need to be elucidated. Thus, we analyze the dependence of the phonon spectra overlap of the thermal rectification on the geometry size of the junctions. It is worth noting that among the working principles of thermal rectification, phonon spectra overlap is the most commonly used in the analysis and explanation of TR in heterojunctions and graded systems. The phonon spectra are obtained via calculation of the vibration density of states (vDOS). In the calculations of the power spectra of the wide and narrow sides, we choose the atoms of the device in the first two unit cells away from each thermal contact. The overlaps (S) of the power spectra of the wide and narrow sides of the device are calculated as follows:



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S± =

∫ P (ω ) ⋅ P (ω )dω ( ∫ P (ω ) dω ) ⋅ ( ∫ P (ω) w

n

1/ 2

2

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w

2

n

dω

(2)

)

1/ 2

where S + and S− are the overlaps in the case of ∆ >0 and ∆ 0) and a negative thermal bias ( ∆ < 0). These results have important implications in the design of thermal diodes and suggest that thermal rectification ratio can be improved by increasing the value of S + or decreasing that of S− . The distribution of S+, 1/ S− , ( S + / S − )δ R − 1 with respect to the geometry sizes L1 and L2 for model Junction A are calculated (Figure 2a, b, and c). We find that S+ is insensitive to the geometry size (Figure 2a). Conversely, S− is rather sensitive to geometry size (Fig. 2b), and the variations of 1/ S− with geometry size are similar to those of the thermal rectification ratio (Figure 1c). Accordingly, our results indicate that the strong effect of the geometry size on the 6 ACS Paragon Plus Environment

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thermal rectification ratio is dominated by S− and has an insignificant relationship with S+. The distribution of ( S + / S − )δ R − 1 ( δ R = 1.8) is also calculated and shown in Figure 2c, which follows the equation TR = ( J + − J − ) / J − : ( S + / S − )δ R − 1 . Moreover, the distribution of J + and 1 / J − with respect to geometry size is also established in our study (Figure 2d, e). Although J + increases with increasing L2 and decreases with increasing L1, its variation is much smaller than that of 1 / J − . The latter is more sensitive to the geometry size, and its distribution with geometry size appears to be similar to that of the TR ration and ( S + / S − )δ R − 1 . Thus, we can conclude that the decrease of S− is more important for improving the TR ratio than increasing S + . If a device exhibiting large rectification is to be designed, the solutions attempted to decrease S− should be preferentially considered. To further understand the underlying mechanism of the strong geometry size effect on S− , we calculate the power spectra and the velocity autocorrelation function (VACF) for three Junctions A with different geometry sizes, which are presented in Figure 3a-l. In the calculations, the atoms groups used for the calculations of S + and S− described above are selected. In Figure 2c, the geometry size sets selected are labeled as follows: A: L1= 4.9 nm, L2= 10 nm; B: L1= 6.1 nm, L2= 8 nm; C: L1= 9.84 nm, L2= 16 nm. In addition, the sets A, B, and C refer to the first peak, the second peak, and the point with the maximum geometry size, respectively, in Figure 1b. Significant low-frequency peaks at 14.5 THz are observed in the power spectra of the narrow side for geometry size sets A, B, and C under negative thermal bias and the higher peak leads to larger TR ratio. These significant low-frequency peaks have been found to occur only when the narrow end of the graded nanostructure is subjected to a temperature that is higher than that of the wide end.6, 9, 10, 28 In this study, they are demonstrated to be of standing wave nature by 7 ACS Paragon Plus Environment


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VACFs as shown in Figure 3 c, g, and k. The standing wave greatly hinder the propagation of phonon waves and the transfer of thermal energy, which indicates that the formation of the standing wave is sensitive to the geometry size, and stronger standing wave leads to smaller S− . In general, the thermal rectification can also be explained by phonon localization, visualized by the spatial distribution of phonon modes equivalent to the weight distribution of the specific modes. The weight of a specific range (Λ) of phonon modes on the whole spectral range for the ith atom can be expressed as:32

wΛ ,i =

∫ Pi (ω )dω Λ

∝

∫0

Pi (ω )dω

(3)

The phonon spatial distribution of the atoms of the nanodevice part of the system are illustrated in Figure 4 a, b at ∆ = -0.5 and ∆ = 0.5 with Λ = 12.5-16.5 THz and Λ = 0-30 THz, respectively. As can be seen in the figure, the phonon spatial distribution is almost uniform in the case of ∆ = 0.5. However, the phonon localization is substantial when ∆ = -0.5, and atoms with higher wi are located in the region near the tube ends. The higher wi result from the extremely high peaks of vDOS at 14.5 THz which are of standing wave nature as discussed above. These findings indicate that the standing waves in the area near the tube end are vigorous and decay fast along the heat transfer direction. We also calculated the natural frequencies of Junction A with geometry size sets A, B, and C, which are depicted in Figure 4c. It is apparent from this figure that the natural frequencies of 14.5 THz coincides with the peaks of 14.5 THz in Figure 3a, e, and i for the vDOS and the frequencies of standing waves in Figure 3c, g, and k. This means that the standing wave is a resonance at the natural frequencies of the junction, and smaller peaks with a frequency of 29 THz may be harmonic with the doubled frequency also caused by the


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resonance at the natural frequency of 14.5 THz. To illustrate the standing wave more clearly that formed along the junction, the free part of the system of Junction A has been used, and four atom groups (each consisting of 48 atoms) are chosen and labeled 1-4 (see Fig. 5a). The VACF are calculated for atom groups 1-4, as shown in Fig. 5b, respectively. In the heat transport direction (from narrow side to wide side when ∆ = 0.5), the strength of the standing wave continuously decreases, and gradually disappears at the region of atom groups 3 and 4, thus suggesting that the standing waves decay fast along the heat transport direction of the free part. Fig. 5 c shows a plot of the temperature profile along the transport direction for junction A with geometry size set A. This indicates that the two curves of the temperature profile are asymmetric when ∆ = -0.5 and ∆ = 0.5. A temperature jump at the structure region between SWCNT and graphene for ∆ = 0.5 can also be seen, which is much larger than that in case of ∆ = -0.5. More importantly, for ∆ = -0.5, an undulation of temperature can be observed. Such a behavior of the temperature has been reported, and was also confirmed to be the results of the standing wave.9, 10 For junction A with geometry sizes L1 = 4.9 nm and L2 = 10 nm, the TR ratio versus | ∆ | and the average temperature are shown in Figure 6a and b, respectively. The TR ratio increased almost linearly with | ∆ | , which corresponded to the fast and low increase rate of J + and J − . Even under a small thermal bias | ∆ | = 0.1, a rectification ratio as high as 188.6% can be achieved. The rectification ratio of Junction A at | ∆ | = 0.1 is slightly higher than that of the designed graded two-stage pillared graphene structure PGN-B we have reported earlier;

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however, the rectification ratio of Junction A at | ∆ | = 0.5 is much higher than that of PGN-B. These results indicate that the rectification ratio of Junction A increases faster than that of PGNB with | ∆ | . It is noteworthy that the rectification ratio of Junction A decreased linearly with the 9 ACS Paragon Plus Environment


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elevation in the average temperature and reached 1681.6% and 912.7% at temperatures of 200 and 400 K, respectively. This can be explained as the weakening effect of Umklapp scattering. In a typical U-process, randomization of the heat flow direction occurs and the net heat flux along the axis of the tube is reduced. With increasing temperature, scattering of the phonons steadily increases and the geometric asymmetry effect of the thermal rectification will steadily decrease, thus resulting in decreasing TR with temperature. Before closing, we would like to emphasize that the geometry size of the designed graded twostage pillared graphene nanostructures (PGN) in our previous study28 was not optimized. Thus, it is not appropriate to make a direct comparison between the thermal rectification of the pillared graphene structures and that of the optimized single carbon nanotube-graphene junction studied in this work. However, the rectification effect exhibited by a single carbon nanotube-graphene junction should relate with the giant thermal rectification from the designed graded two-stage pillared graphene structures. It should also be noted that the obtained optimized geometry parameters for the single carbon nanotube-graphene junction cannot be directly used to obtain an optimized design of the pillared graphene-based thermal rectifier, because the coupling of the phonons of the junctions need to be considered and further studied. We have shown that both designed single carbon nanotube-graphene junction and graded two-stage pillared graphene nanostructures can exhibit giant thermal rectification, and would therefore be excellent thermal rectifiers. In reality, however, assembling such a small device is still quite challenging. Advanced technology, such as atomic force microscopy, can already manipulate single junctions or single molecules will be helpful for fabricating such a small nanodevice.

4. CONCLUSIONS


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In summary, the effect of the geometry size on the thermal rectification of single carbon nanotube-graphene junctions has been investigated. Additionally, the distribution of the thermal rectification with respect to the tube length and side length of the graphene nanosheet was calculated and visualized. The optimized carbon nanotube-graphene junction was found to have an ultrahigh thermal rectification ratio. In addition, the maximum thermal rectification ratios for Junction A under | ∆ | = 0.5 can reach 1244.1% and 1681.6% at average temperatures of 300 K and 200 K, respectively. These are much higher than those of other reported single-material nanostructure-based thermal rectifiers. The thermal rectification of the nanotube-graphene junctions is fairly sensitive

to the geometry size and almost completely dominated by the degree of overlap of the power spectra under negative thermal bias. Our findings could offer useful guidelines for the design and improvement of the performance of thermal diodes.

ASSOCIATED CONTENT Supporting Information Comparison and discussion on the different equations for the calculation of overlaps of the vDOS.

AUTHOR INFORMATION Corresponding Authors: *E-mail: [email protected] *E-mail: [email protected]

Notes The authors declare no competing financial interest. 11 ACS Paragon Plus Environment


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ACKNOWLEDGMENTS This research is supported by the National Natural Science Foundation of China (Grant Nos. 51576066, 51322603, 51136001 and 51356001), the Natural Science Foundation of Hebei Province of China (Grant No. E2014502042).

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Figure 1. Distribution of the thermal rectification with respect to a tube length L1 and a side length of graphene nanosheet L2 for model Junction A and Junction B of the SWCNT-graphene junctions: (a) model Junction A; (b) three-dimensional (3D) view of the distribution of the TR ratio η with respect to L1 and L2 for Junction A; (c) 2D view of the distribution of the TR ratio

η with respect to L1 and L2 for Junction A; (d) model Junction B; (e) 3D view of the distribution of the TR ratio η with respect to L1 and L2 for Junction B; (f) 2D view of the distribution of the TR ratio η with respect to L1 and L2 for Junction B.


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Figure 2. Distribution of S+, 1/S-, ( S + / S − )δ R − 1 , J+ and 1/J- with respect to geometry sizes L1 and L2 for model Junction A: (a) S+; (b) 1/S-; (c) ( S + / S − )δ R − 1 ; (d) J+; (e) 1/J-



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Figure 3. vDOS per atom and normalized velocity autocorrelation functions for Junction A with geometry size sets A, B, and C: (a), (e), and (i) vDOS when ∆ = -0.5; (b), (f), and (j) vDOS when ∆ = 0.5; (c), (g), and (k) VACF for narrow sides; (d), (h), and (l) VACF for wide sides.


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Figure 4. (a) phonon (12.5 to 16.5 THz) spatial distribution of the atoms of the free part of the system for junction A with geometry sizes set A when ∆ = -0.5 and ∆ = 0.5; (b) phonon (0 to 30 THz) spatial distribution of the atoms of the free part of the system for junction A with geometry sizes set A when ∆ = -0.5 and ∆ = 0.5; (c) calculated natural frequencies of junction A with geometry sizes sets A, B, and C.



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Figure 5 Illustrations for the standing wave formed and temperature profile along the junction A with geometry sizes set A (a) the free part of the system for Junction A and chosen atom groups 1-4; (b) VACF calculated for group 1-4 when ∆ = -0.5; (c) temperature profile of Junction A along the heat transport direction.


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Figure 6. TR ratio and heat flux for junction A with geometry sizes L1 = 4.9 nm and L2 = 10 nm: (a) TR ratio and corresponding heat flux vs | ∆ | ; (b) TR ratio and corresponding heat flux vs average temperature.



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Giant Thermal Rectification from Single-Carbon NanotubeâGraphene

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