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Thermal Conductivity of Single-Walled Carbon Nanotubes under Axial Stress Cuilan Ren,†,‡ Wei Zhang,*,† Zijian Xu,† Zhiyuan Zhu,† and Ping Huai† Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, P.R. China, and Graduate UniVersity of the Chinese Academy of Sciences, Beijing 100049, P.R. China ReceiVed: October 29, 2009; ReVised Manuscript ReceiVed: January 21, 2010
The thermal conductivity of single-walled carbon nanotubes (SWCNTs) under axial stress is studied by nonequilibrium molecular dynamics simulation. The thermal conductivity is found to increase and then decrease with the tube elongation changing from an axially compressed state to a stretched state. The phonon density of states of the systems is analyzed to elucidate the variation of heat conduction with respect to the stress in CNTs. The primary peak of the phonon spectrum shows a blue shift or red shift as the SWCNT is compressed or stretched. These shifts correspond to the change of the elasticity coefficient of the CNTs. The variation trend of primary peak height of radial phonon spectra with axial strain is similar to that of the thermal conductivity, which indicates that the radial phonon modes, especially the high-frequency modes, play a dominant role in the heat conduction mechanism of CNTs. Introduction Carbon nanotubes (CNTs) have noble mechanical, thermal, and electronic properties, making them one of the most promising candidates for building nanometer-scale electronic devices.1-7 In the past decade, an increasing amount of studies in heat conduction of nanoscale materials has been evoked both experimentally and theoretically.8-11 These results show that CNTs have a high thermal conductivity, which can be compared to that of the diamond.12-14 The thermal conductivity of CNTs has also been found to have a dependence on the temperature, tube length, defects, and impurities.8,9,11,14-16 However, there are noteworthy differences in the theoretical results due to the theoretical methods.13,15-18 The remarkable dependence on tube length has made it difficult to directly compare the values of thermal conductivity.18-21 The thermal conductivity also depends on the temperature gradient to some extent when the nonequilibrium molecular dynamics (NEMD) simulation method is used.15 As is well-known, phonons play a dominant role in the heat conduction of carbon nanostructures as compared to electrons. The phonon contribution to the heat capacity is ∼102 times larger than that of electrons for CNTs at room temperature, while it is ∼104 times for graphite,22 which makes it reasonable to neglect the role of electron contribution. There are varieties of phonon modes in CNTs, such as the transverse acoustic modes (TA, doubly degenerate), the longitudinal acoustic mode (LA), the twist mode (TW), as well as the radial breathing mode (RB).23 For the CNT thermal conduction simulation, the heat transportation could be assumed as a long steady process, and various phonon modes contribute to the heat conductivity.24 The phonon mean free path of pristine SWCNTs is estimated to be on the order of micrometers at room temperature.14,23 However, it is limited by the tube length when the simulation is conducted in a small box because of the anharmonic phonon-phonon scattering (Umklapp scattering process) by the boundary. This should be the main reason for the significant differences between * To whom correspondence should be addressed. Phone: (86)2139194793. Fax: (86)21-39194793. E-mail:
[email protected]. † Shanghai Institute of Applied Physics, Chinese Academy of Sciences. ‡ Graduate University of the Chinese Academy of Sciences.
the reported thermal conductivities obtained from molecular dynamics simulations.18-21 The effective approach to understanding the mechanism of heat conduction in CNTs is to investigate the difference between the varieties of phonon modes and furthermore to know which kinds of phonon modes play the dominant role. The study of the propagation of heat pulses in CNTs by MD simulation could provide some information on propagation of the specific phonon modes and their contributions to the instantaneous heat conduction in CNTs.25-27 Osman et al. have studied the intense heat pulse in CNTs (both SWCNTs and MWCNTs).25,26 Their work revealed that the intense heat pulse split into several heat wave packets moving at different speeds which correspond to different phonon modes. The leading heat wave packets are LA and TW modes. However, the energy carried by the second sound wave packets is larger than those carried by the leading heat wave packets. Shiomi et al. have shown that acoustic phonons behave ballistically and optical phonons play a major role when heat pulse is applied on SWCNTs.27 However, the heat wave packet with a specified speed may contain phonon modes of different frequencies. Studying propagation of heat pulse could not reveal how the phonon modes at different frequencies contribute to the heat conduction. The phonon modes may change as the tube is compressed or stretched. Therefore, the study of thermal conductivity as well as the phonon spectra of the CNTs with regular change of geometrical form (under axial strain) could help us understand the contributions of individual phonon modes to the thermal conduction of CNTs. In the present work, the thermal conductivity of (10, 10) SWCNTs under axial strain is studied using the NEMD method. The length effect can be of important influence on thermal conductivity of SWCNTs. The thermal conductivity λ usually becomes larger with the tube increasing, and has a relationship with tube length L as Lβ in a range of L up to micrometers.19,20 However, herein we focus on a series of SWCNTs with the same length of about 60 Å, so that the comparison between them makes sense and the length effects need not be involved. It is found that the radial phonon modes, especially the high-frequency radial
10.1021/jp910339h 2010 American Chemical Society Published on Web 03/04/2010
Thermal Conductivity of SWCNTs under Axial Stress
J. Phys. Chem. C, Vol. 114, No. 13, 2010 5787 nsteps
vibrational modes, play a dominant role in the long-term steadystate heat conduction process of CNTs.27
1 J) A
Computational Details Thermal conductivity of SWCNTs under axial stress is studied using the NEMD method with empirical potentials.15,28 The interaction between C atoms in SWCNTs is modeled by Tersoff-Brenner bond-order potential with the cutoff distance of 2.0 Å.29 This potential has been widely used for simulating carbon systems.21,28 The long-range van der Waals (vdW) interaction modeled with Lennard-Jones potential is also taken into account. The L-J parameters for carbon are ε ) 0.002 84 eV, σ ) 3.4 Å, respectively.30 The classical equations of motion are integrated by the velocity Verlet algorithm with a fixed time step of 0.5 fs for the simulation. The thermal conductivity λ is defined by Fourier’s law
J)
dT 1 dQ ) -λ A dt dl
( )
∑
|∆E1(j) + ∆E2(j)|
j)1
(4)
nsteps
∑ ∆t j)1
where A is the cross-sectional area which is a ring of van der Waals width 3.4 Å. ∆E1(j) and ∆E2(j) correspond to the additional kinetic energy added to the hot slab and removed from the cold slab, respectively, at each time step to keep the thermostatted slabs at specified temperatures. Besides the heat flux density, the temperature profile is obtained by the averaging instantaneous temperature of each slab during the statistical process. The temperature gradient is calculated by a linear fit of the central 15 slabs’ temperature. The temperature calculation is based on the theorem of energy equipartition
(1) nk
where J is the steady-state heat flux density which represents the energy Q transmitted across the area A in a time interval dt, while dT/dl is the temperature gradient along the tube axis of the system. As shown in Figure 1, the SWCNT is divided into 25 equal slabs. After evolving at the desired temperature T0 for 5 ps to reach a steady state, a gradient is introduced by keeping the two slabs at the ends of the tube at the temperature of T1 ) T0 - ∆T and T2 ) T0 - ∆T with ∆T ) 20 K, respectively. Then a run of 1.5 ns (3 × 106 time steps) is taken for simulating the heat transport process. The last 1.25 ns is used to gather statistics in order to get the heat flux density and the temperature gradient. In the simulation, the velocity scaling method is used to control the temperature of the cold or hot slab as follows
υi′ ) υi
Tdef Tk
(2)
where Tdef represents T1 or T2, and Tk is the instantaneous temperature of the kth slab. The additional kinetic energy added to the hot slab or removed from the cold slab at each time step is described as nk
∆E )
∑
m (υ2 - υi2) 2 i)1 i
(3)
where nk is the number of atoms in the kth slab. The heat flux density is defined by
∑
m Tk ) 〈υ2〉 3nkkB i)1 i
(5)
〈υ2i 〉 is the mean-square speed of carbon atoms in the kth slab and kB is Boltzmann’s constant. In this work, the phonon spectrum of CNTs has also been calculated using the same potentials as the calculation of thermal conductivity. The correlation between two different quantities is measured in the usual statistical sense, via the correlation coefficient. The non-normalized time correlation function of a variable B is defined as
lim C(t) ) 〈B(t) · B(0)〉 ) τf∞
∫-ττ B(t' + t)B(t') dt' ∫-ττ dt'
(6)
The phonon spectrum of CNTs is calculated by Fourier transformation of the normalized atomic velocity autocorrelation function (VACF) N
Cυυ(ω) )
k k ∞ iωt 〈υj (t) · υj (0)〉 e 〈υjk(0) · υjk(0)〉
∑ ∑ ∫0
k)x,y,z j)1
dt
(7)
where υkj is the component of velocity of the jth atom in the k direction (k ) x, y, z denotes the Cartesian axes), dt is the time step, and N is the total number of atoms in the system. The method to calculate the phonon spectrum has been used to obtain the phonon spectra of carbon clusters, silicon
Figure 1. Schematic simulation setup. The black parts of the system are fixed rigidly during the simulation, the red and blue parts are thermostated for relative high and cool slabs to get the temperature gradient, the gray parts between fixed and temperature-controlled slabs are in an attempt to reduce reflecting of heat from the edge, and the yellow parts are free during simulation.
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Ren et al.
Figure 2. Parameters of the system: (a) bond length, (b) bond angle, (c) average radius, and (d) axial stress of a SWCNT under axial strain as a function of tube strained length L (L0 is the length of pristine CNT).
thin films, as well as other metal systems.31-33 The phonon spectrum was used to investigate the thermal stability of these systems. Results and Discussion In this work, the mechanical effects and geometrical structures of SWCNTs under axial strain (compressing and stretching) are studied by the method above. Initially, a (10, 10) SWCNT with 1000 atoms is compressed or stretched to a specific length by changing the axial coordinates proportionally. Then, with two ends of the strained SWCNT being fixed, the rest of the strained SWCNT is relaxed using the conjugate gradient method. Since the C-C bond in graphite is the strongest bond in nature, CNTs are widely regarded as the ultimate fiber with regard to its strength in the direction of the nanotube axis.34 Sears and Batra have summarized that the value of Young’s modulus of CNTs varied in the range of 320-2400 GPa in experimental measurements and 940-1240 GPa in theoretical simulations due to the difference between potential models.35 For the Brenner model, extremely high strain (L/L0 < 0.96, or >1.20) causes the (10, 10) SWCNT to collapse into a random network or break into fragments. In order to study the thermal conductivity of the tube under axial stress, the geometrical shape of the CNT should be well preserved during the long-time simulation process. The structural information of a series of (10, 10) SWCNTs under axial strain is summarized in Figure 2. It explicitly displays the bond length, bond angle, average radius, and axial stress. Figure 2 shows that the bond length and bond angle changed under the stain. The tube radius increases with the tube compressed and decreases with the tube stretched. Stress of the tube is calculated by the formula σ ) F/A, where F is the force exerted on the two ends of the tube. In this work, we use a constant radius of pristine SWCNT of 6.78 Å. Although the radius of the SWCNT changes a little after being strained, this small change of radius affects the result of the stress or thermal conductivity very weakly. For example, the maximum deviation in stress or thermal conductivity is (6.78 - 6.70)/6.78 ) 1.1% at L/L0 ) 1.05. The cylindrical form of the CNT preserves well
during the simulation process within a compression stress of 23.5 GPa and tensile stress of 37.1 GPa. The elasticity coefficient of the SWCNT as a function of tube strained length (0.975 e L/L0 e 1.05) is shown in Figure 3. It is estimated by the formula
Kb ) -
FL+∆L - FL ∆L
(8)
where FL is the force exerted on the two ends of the tube when the tube length is L and ∆L ) 0.001L0. The elasticity coefficient of the SWCNT decreases with the strained tube length. There is a change of slope at L/L0 ) 1.0. In the Brenner potential, the elasticity coefficient of the bond angle with respect to the change of the bond angle increases quickly when the bond angle is less than 120°, while it decreases slowly when the bond angle is larger than 120°. In the sp2 structure, the profile of elasticity coefficient is the collective result of the changes of three bond angles.
Figure 3. Elasticity coefficients of SWCNTs under axial strain as a function of tube strained length L.
Thermal Conductivity of SWCNTs under Axial Stress
Figure 4. Thermal conductivity of SWCNTs under axial strain as function of tube strained length at different temperatures.
J. Phys. Chem. C, Vol. 114, No. 13, 2010 5789 It could be seen from Figure 4 that the thermal conductivity of SWCNTs under axial stress is sensitive to the axial strain of the nanotube. In the temperature range of 200-400 K, the thermal conductivity increases with the tube length first and then decreases after reaching its maximum at about L/L0 ) 1.02. The value of thermal conductivity of pristine (10, 10) SWCNT is about 1430 W m-1 K-1 at room temperature, which is consistent with that from other literature.16,18,36 To understand the role of phonons in the heat conduction of CNTs, the normalized total phonon spectrum at 300 K is calculated by Fourier transformation of the velocity autocorrelation functions of the simulated systems with a correlation time length of 5 ps. Figure 5a-c shows the radial, longitudinal, and total phonon spectrum of the system, respectively. It can be seen from Figure 5a that the shift of lower-frequency phonon modes is small while the higher-frequency optical phonon modes (the primary peak of the phonon spectrum) are more sensitive to the axial strain than the lower-frequency modes. The higher-
Figure 5. Phonon spectra of SWCNT under axial strain: (a) radial phonon spectra, (b) axial phonon spectra, (c) total phonon spectra, and (d) height of the primary peak of radial phonon spectra as a function of tube strained length.
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frequency modes, which mainly characterize the C-C bond stretching, show a blue shift when the tube is compressed and a red shift when the tube is stretched. The primary peak of the longitudinal and total phonon spectrum has the same shift trend as that of the radial phonon spectrum, as shown in Figure 5b and c. The shifts of the primary peaks in frequency are partially due to the change of the elasticity coefficients. The shift of the primary peak of the SWCNT phonon spectrum under axial stress shows a monotonic relation, which has also been reported in early literature.37 By using tight binding molecular dynamics method with Xu’s orthogonal model, Yu et al. have studied the phonon dispersion curves and phonon spectra of SWCNTs under axial stress via diagonalization of a dynamical matrix (DM).37 Gao et al. have studied the Raman frequency variation of stretched SWCNTs experimentally.38 These studies have shown the same shifting trend of the primary peak as in our results. As reported by Osman et al., heat is carried mainly through the radial phonons during the heat conduction in SWCNTs.16 We also notice that the height of the primary peak in radial phonon spectra is changed when the stress is exerted on the tube. The relative change of this peak means the role of the dominant vibrational mode is weakened or strengthened. If the phonon modes of this peak give the most important contribution to the heat transport, the variation of the thermal conductivity could be understood. The primary peak of the phonon spectrum reflects the carbon bonding when the geometry of the carbon lattices is changed. In previous works,28,39,40 introducing point defects into the nanotubes or functionalizing CNTs with hydrogen atoms or phenyl rings also changed the relative height of the primary peak, which causes the decrease in the thermal conductivity. The height of the primary peak in the radial phonon spectrum (Figure 5a) as a function of the tube strained length has the same trend as the thermal conductivity, as shown in Figure 5d. This indicates that the radial phonon modes closely related to the carbon bonding are more important in heat conduction. Figure 5b shows that the primary peak height of longitudinal phonon spectra ascends with the tube strained length. It is known that there are 3N normal phonon vibrational modes in a CNT system of N atoms; 2N of them are radial vibrational modes while N of them are longitudinal ones. So the radial phonon modes are the majority among all the phonon modes of CNTs. Figure 5c exhibits the total phonon spectra of the CNTs which are equivalent to the sum of the 2N radial modes and the N longitudinal modes. As there are more radial phonon modes in the CNTs than longitudinal ones, the total phonon spectra (Figure 5c) show that the primary peak height has the same varying trend with the tube strain as that shown in Figure 5a. It is indicated that the heat conduction is dominated by radial phonon modes, especially the high-frequency modes (C-C characteristic peak), rather than the longitudinal modes. While decreasing of the height of the peak, the peak is broadened, even split into two peaks, as Figure 5a-c shows. The increased nonmonochromaticity may enhance the anharmonic effect in the heat conduction; that is, the Umklapp scattering of phonons is increased. The increase of Umklapp processes should reduce the thermal conductivity. The anharmonicity is inherently taken into consideration when the phonon spectrum is calculated by the Fourier transformation of the velocity autocorrelation function because no harmonic approximation is made in this method and the anharmonicity is also embodied in the bond-order potential function. Thus, the widening of the primary peaks results from the anharmonic
Ren et al. effects of interatomic interactions in the CNTs. In the pristine (10, 10) SWCNT, the C-C bond lengths are mostly 1.42 Å. The corresponding elasticity coefficients of C-C bonds are thus uniform, so the phonon modes they correspond to are distributed in a concentrated manner. However, the C-C bond lengths deviate from the normal value as the SWCNT is compressed or stretched, as shown in Figure 2a. As a result, the primary peak broadens because the elasticity coefficients of C-C bonds are distributed dispersedly under axial CNT strain. Conclusions Thermal conductivity of (10, 10) SWCNTs under axial strain is studied using NEMD simulation with empirical potentials. We found that the thermal conductivity increases and then decreases with the tube elongation from an axially compressed state to a stretched state. The normalized phonon density of states calculated by Fourier transformation of the velocity autocorrelation functions of the simulated systems is analyzed to elucidate the physical mechanism of heat conduction in SWCNTs. The phonon spectra show that the primary peaks of compressed SWCNTs have blue shift while the primary peaks of stretched SWCNTs have red shift. These shifts correspond to the change of the elasticity coefficient of the CNTs. The variation trend of the primary peak height with respect to the axial strain of the CNT is similar to that of the thermal conductivity. The results also show that the radial phonon modes, especially the high-frequency modes, contribute more to the heat conduction in CNTs than the longitudinal modes. In addition, the primary peak widens under axial strain, which is due to the anharmonic effects of the nanotube lattices. Acknowledgment. This work is partly supported by the Key Project of the Knowledge Innovation Program of the Chinese Academy of Sciences (KJCX3-SYW-N10), the CAS Hundred Talents Program, the National Natural Science Foundation of China (10874197), the Scientific Research Starting Foundation of the Ministry of Human Resources and Social Security of China for Returned Overseas Chinese Scholars, and the Shanghai Municipal Science and Technology Commission (09ZR1438300). We thank the Shanghai Supercomputer Centre for the use of the Dawning 5000A supercomputer. We thank Prof. Zhenxia Wang for helpful discussions. References and Notes (1) McEuen, P. L. Nature 1998, 393, 15. (2) Huang, Y.; Duan, X.; Cui, Y.; Lauhon, L. J.; Kim, K.; Lieber, C. M. Science 2001, 294, 1313. (3) Craighead, H. G. Science 2000, 290, 1532. (4) Chico, L.; Crespi, V. H.; Benedict, L. X.; Louie, S. G.; Cohen, M. L. Phys. ReV. Lett. 1996, 76, 971. (5) Bachtold, A.; Hadley, P.; Nakanishi, T.; Dekker, C. Science 2001, 294, 1317. (6) Postma, H. W. C.; Teepen, T.; Yao, Z.; Grifoni, M.; Dekker, C. Science 2001, 293, 76. (7) Romo-Herrera, J. M.; Terrones, M.; Terrones, H.; Dag, S.; Meunier, V. Nano Lett. 2007, 7, 570. (8) Nan, C. W.; Liu, G.; Lin, Y.; Li, M. Appl. Phys. Lett. 2004, 85, 3549. (9) Yu, C.; Shi, L.; Yao, Z.; Li, D.; Majumdar, A. Nano Lett. 2005, 5, 1842. (10) Chol, S.; Maruyama, S. J. Korean Phys. Soc. 2004, 45, 897. (11) Wang, J.; Wang, J. Appl. Phys. Lett. 2006, 88, 111909. (12) Kim, P.; Shi, L.; Majumdar, A.; McEuen, P. L. Phys. ReV. Lett. 2001, 87, 215502. (13) Berber, S.; Kwon, Y.; Tomanek, D. Phys. ReV. Lett. 2000, 84, 4613. (14) Hone, J.; Whitney, M.; Piskoti, C.; Zettl, A. Phys. ReV. B 1999, 59, R2514. (15) Kondo, N.; Yamamoto, T.; Watanabe, K. J. Surf. Sci. Nanotech. 2006, 4, 239. (16) Osman, M. A.; Srivastava, D. Nanotechnology 2001, 12, 21.
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