Nose–Hoover thermostat length effect on thermal conductivity of single wall carbon nanotubes

International Journal of Heat and Mass Transfer 53 (2010) 5884–5887

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Nose–Hoover thermostat length effect on thermal conductivity of single wall carbon nanotubes Robert A. Shelly, Kasim Toprak, Yildiz Bayazitoglu ⇑ Department of Mechanical Engineering and Materials Science, Rice University, 6100 Main Houston, TX 77005-1892, USA

a r t i c l e

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Article history: Received 16 April 2010 Received in revised form 18 June 2010 Accepted 18 June 2010

Keywords: Molecular dynamics Conduction Carbon nanotube Thermal conductivity

a b s t r a c t Non-equilibrium molecular dynamics simulations are used to determine the thermal conductivities of single wall carbon nanotubes. By fixing opposing ends of an armchair single wall carbon nanotube with a Nose–Hoover thermostat, the length dependence of thermal conductivities of single wall carbon nanotubes were studied in a vacuum. Specifically, single wall carbon nanotubes of 12.3 nm, 24.6 nm, and 36.9 nm lengths with varying fixed end temperatures were analyzed to determine thermal conductivities. In addition, the fixed end temperature lengths of single wall carbon nanotubes were varied to see convergence of the temperature profiles. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction The potential applications for carbon nanotubes (CNTs) have been growing since Iijima first discovered CNTs in 1991 [1]. Experimental measurements and theoretical predictions have shown that single wall carbon nanotubes (SWNT) have high thermal conductivities even at short lengths and small chiralities [2]. Their high thermal properties make them good candidates in improving NEMS and MEMS devices. In addition, current computer processors are getting smaller and require more efficient heat dissipation to increase chip speed, making CNTs a logical choice to aid in heat removal. Therefore, the study of SWNTs is a high interest research area for scientists. SWNTs have been used in polymer composites to effectively increase heat transfer [3]. Keblinski and co-workers [4] studied the radial heat flux with the heat source and sink in the radial direction. Creating a heat sink with SWNTs in an array has the potential to efficiently increase heat flux axially. Che et al. [5] has shown that experimental measurements of thermal conductivity for an individual CNT are inadequate. Technological improvements are necessary to produce a higher quality and well ordered CNTs. The purpose of this study is to investigate the length dependence of thermal conductivity in a SWNT. Recent advances in computer power bring increased popularity and power to the molecular dynamics (MD) simulation techniques. The MD simulation method is a computational method to predict behavior of each molecule and following their motion in time based on the

⇑ Corresponding author. E-mail address: [email protected] (Y. Bayazitoglu). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.06.054

Newton’s second law of motion [6,7]. MD methods are well-known methods for predicting the thermal properties of the molecules in statistical mechanics and chemistry [8]. Two main types of MD techniques have been used: equilibrium molecular dynamics (EMD) and non-equilibrium molecular dynamics (NEMD) simulations [9]. NEMD was used to study the axial temperature profile of a SWNT. The MD simulation package GROningen MAchine for Chemical Simulations (GROMACS) is used to implement Nose– Hoover and Berendsen thermostats with harmonic potential [10]. Several cases were examined using a (5, 5) SWNT of 12.3 nm, 24.6 nm, and 36.9 nm lengths at a temperature of 305 K. 2. Analysis An outline of the process implemented, the governing equations, parameters, and software used to determine the overall heat transfer of a SWNT in a pin fin environment is given in the following sections. Using the axial temperature profiles created by the thermostats, the heat flux across the tube was calculated and used to determine the overall thermal conductivity. This study presents a parametric study of the effects on thermal conductivity when varying lengths and end temperatures are applied. 2.1. Simulation parameters In the presented MD simulations, harmonic potentials are used to model the Carbon–Carbon (C–C) bonded interaction within the SWNT using values from Guo et al. [11]

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Nomenclature area, m2 equilibrium bond length, nm diameter, nm applied force, kJ/mol nm Boltzmann’s constant, J/K bonded force constant, kJ/mol nm nanotube length, nm the length of the temperature-controlled length, nm simulation box length, nm number of degrees of freedom magnitude of momentum of atom, kg m/s

A bij d Fi kb b kij L Lc l Ndf pi

Uðrij Þ ¼

1 b k ðrij  bij Þ2 2 ij

ð1Þ

b

Here, kij describes the bonded force constant and rij is the bond length. In addition to the harmonic potential, which bond the nearest neighbors, all C–C pair interactions are calculated using a standard Lennard-Jones potential:

Uðrij Þ ¼ 4e

"  12

r

r ij

 6 # 

r

r ij

ð2Þ

where for the intermolecular interactions, the values of e = 0.0023 kJ/mol and r = 0.3374  105 nm are used [12]. A simulation box with dimensions l  l  l nm3 containing one SWNT, where l is the total length of the nanotube including the temperature-controlled lengths multiplied by 1.4, was simulated using GROMACS. The simulation box shape is cubic which saves CPU time rather when compared to other shapes during the simulations. Periodic boundary conditions are applied to minimize the edge effects in the x, y, and z directions of the system. After energy minimization, equilibrium is reached at a system temperature of 250 K by implementing a Berendsen thermostat [13] alone for 1 ns. Next, using Nose–Hoover thermostat for temperature coupling, the simulation was continued with a different end temperature with a time step of 0.5 ps for 4 ns to 6 ns depending on the tube length. For the simulations, a constant temperature of 330 K is placed at one tip of the SWNT while the other tip has a temperature of 280 K. Fig. 1 shows a basic model of the nanotube divided into temperature control sections and the body tube section. 2.2. Calculation of thermal conductivity The thermal conductivity of an individual SWNT is calculated using non-equilibrium MD using a similar procedure as Shiomi and Maruyama [14]. Tubegen, a nanotube structure generator was used to generate the SWNT over various lengths [15]. The SWNT is placed in a vacuum, and a Nose–Hoover thermostat is implemented to set the end temperatures and create a heat flux along the SWNT [2]. The Nose–Hoover thermostat is used because

Q rij T T0 U(rij)

mass parameter bond length, nm instantaneous temperature, K reference temperature, K potential energy, kJ/mol

Greek symbols e depth of the potential well, kJ/mol n heat bath parameter r finite distance, nm sT relaxation time, ns

it is more efficient for extended systems and relaxing time [16]. The magnitude of the equation of motion for a particle under the influence of a Nose–Hoover thermostat can be represented as

p_ i ¼ F_ i  npi

ð3Þ

and

1 n_ ¼ ðT  T 0 Þ Q

ð4Þ

where p is the magnitude of the momentum of atom i, F is the applied force on atom i, T0 is the reference temperature, and T is the instantaneous temperature. The term Q is a constant which is defined as

Q¼

s2T T 0 4p2

ð5Þ

where sT is the relaxation time. After we get the temperature distribution by applying Nose– Hoover thermostat, the heat transfer per unit area would then be

q ¼ n

Ndf kb T A

ð6Þ

In the heat flux (q), the cross sectional area, A, is defined as A ¼ pbd where b = 0.34 nm is the Van der Waals thickness. After the system has reached steady state, using the slope of the linear section of the SWNT in Fig. 2 and the heat flux from Eq. (6), the thermal conductivity, k, can be calculated from Fourier’s Law. This method is named NEMD and also can be called direct method [17]. Here, several lengths of the temperature-controlled sections are used to determine the overall changes in thermal conductivities. Various lengths of the temperature-controlled sections are applied to the SWNT, Fig. 2 shows how the axial temperature profiles change with nanotube length for the temperature-controlled length temperatures of 330 K and 280 K. Fig. 2 also presents the expected temperature jump at the interface of thermostats and nanotubes.

Fig. 1. A single wall carbon nanotube model.

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Fig. 2. Axial temperature distribution of (5, 5) SWNTs with its temperature-controlled lengths.

Fig. 4. Comparisons of calculated thermal conductivity of SWNTs [2,19–21]. Fig. 3. Effect of temperature-controlled lengths on the thermal conductivity of SWNTs.

Using the procedure described above, the thermal conductivity values calculated vary with the heat flux into the SWNT. Fig. 3 shows the linear dependence on the length of the temperaturecontrolled length. In this paper, the calculation for the values of thermal conductivity are similar to Shiomi and Maruyama’s [2,14] and Lukes and Zhong’ [18] when the overall length of the SWNT is included and similar boundary conditions are used. Here the thermal conductivity is calculated by measuring the temperature profile of the nanotubes while excluding the temperature-controlled lengths.

Fig. 4 shows the comparison between this work and the theoretical studies [2,19–21]. The current MD study with a harmonic potential gives good results when it is compared to the previous theoretical models. SWNTs with lengths from 12.3 nm to 98.4 nm are simulated with 36.9-nm-long temperature-controlled sections and converged thermal conductivity values.

3. Conclusion As the lengths of the temperature-controlled sections increase, the apparent thermal conductivity values increase and they converge to constant values. As illustrated in Figs. 3 and 4, there is a

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length effect for SWNTs and thermal conductivity increases with tube length. Thermal conductivity rise is coherent with the tube length but not fully linear, although, overall values agree well with previous studies as seen in Fig. 4. Acknowledgements This work is supported by the Lockheed Martin Corporation, within the LANCER project and by the Shared University Grid at Rice funded by NSF under Grant EIA-0216467, and a partnership between Rice University, Sun Microsystems, and Sigma Solutions, Inc. The authors also acknowledge Professor D. Tang for providing the numerical values [19] to produce Fig. 4. References [1] S. Iijima, Helical microtubules of graphitic carbon, Nature 354 (6348) (1991) 56–58. [2] J. Shiomi, S. Maruyama, Molecular dynamics simulations of diffusive-ballistic heat conduction in carbon nanotubes, Mater. Res. Soc. Symp. Proc. 1022 (2007) 32–37. [3] Y. Wu, C.H. Liu, H. Huang, S.S. Fan, Effects of surface metal layer on the thermal contact resistance of carbon nanotube arrays, Appl. Phys. Lett. 87 (21) (2005) 213108-1–213108-3. [4] M. Hu, S. Shenogin, P. Keblinski, N. Raravikar, Thermal energy exchange between carbon nanotube and air, Appl. Phys. Lett. 90 (23) (2007) 231905-1– 231905-3. [5] J. Che, T. Cagin, W.A. Goddard III, Thermal conductivity of carbon nanotubes, Nanotechnology 11 (2) (2000) 65–69. [6] A.J.H. McGaughey, M. Kaviany, Thermal conductivity decomposition and analysis using molecular dynamics simulations. Part I. Lennard-Jones argon, Int. J. Heat Mass Transfer 47 (2004) 1783–1798. [7] L. Consolini, S.K. Aggarwal, S. Murad, A molecular dynamics simulation of droplet evaporation, Int. J. Heat Mass Transfer 46 (2003) 3179–3188.

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[8] C.H. Cheng, H.W. Chang, Simulation of phase states for water in nanoscale systems by molecular dynamics method, Int. J. Heat Mass Transfer 47 (2004) 5011–5020. [9] S. Srinivasan, R.S. Miller, On parallel nonequilibrium molecular dynamics simulations of heat conduction in heterogeneous materials with three-body potentials: Si/Ge superlattice, Numer. Heat Transfer B 52 (2007) 297–321. [10] E. Lindahl, B. Hess, D.V.D. Spoel, GROMACS 3.0: a package for molecular simulation and trajectory analysis, J. Mol. Model. 7 (8) (2001) 306–317. [11] Y. Guo, N. Karasawa, W. Goddard, Prediction of fullerene packing in C60 and C70 crystals, Nature 351 (6326) (1991) 464–467. [12] V.R. Cervellera, M. Alberti, F. Huarte-Larranaga, A molecular dynamics simulation of air adsorption in single-walled carbon nanotube bundles, Int. J. Quantum Chem. 108 (10) (2008) 1714–1720. [13] H.J.C. Berendsen, J.P.M. Postman, W.F. van Gunsteren, A. DiNola, J.R. Haak, Molecular dynamics with coupling to an external bath, J. Chem. Phys. 81 (8) (1984) 3684–3690. [14] J. Shiomi, S. Maruyama, Molecular dynamics of diffusive-ballistic heat conduction in single-walled carbon nanotubes, Jpn. J. Appl. Phys. 47 (4) (2008) 2005–2009. [15] J.T. Frey, D.J. Doren, TubeGen 3.3, ,University of Delaware, Newark, DE, 2005, . [16] P.H. Hunenberger, Advances in Polymer Science, Springer, Berlin/Heidelberg, 2005. pp. 105–149. [17] R.J. Stevens, L.V. Zhigilei, P.M. Norris, Effect of temperature and disorder on thermal boundary conductance at solid–solid interfaces: nonequilibrium molecular dynamics simulations, Int. J. Heat Mass Transfer 50 (2007) 3977– 3989. [18] J. Lukes, H. Zhong, Thermal conductivity of individual single-wall carbon nanotubes, J. Heat Transfer 129 (6) (2007) 705–716. [19] Z. Wang, D. Tang, X. Zheng, W. Zhang, Y. Zhu, Length-dependent thermal conductivity of single-wall carbon nanotubes: prediction and measurements, Nanotechnology 18 (47) (2007) 475714-1–475714-4. [20] M. Alaghemandi, E. Algaer, M.C. Bohm, F. Muller-Plathe, The thermal conductivity and thermal rectification of carbon nanotubes studied using reverse non-equilibrium molecular dynamics simulations, Nanotechnology 20 (11) (2009) 115704-1–115704-8. [21] P. Rui-Qin, X. Zi-Jian, Z. Zhi-Yuan, Length dependence of thermal conductivity of single-walled carbon nanotubes, Chin. Phys. Lett. 24 (5) (2007) 1321–1323.