Thermal energy exchange between carbon nanotube and airMing Hu, Sergei Shenogin, Pawel Keblinski, and Nachiket Raravikar

Citation: Applied Physics Letters 90, 231905 (2007); doi: 10.1063/1.2746954 View online: http://dx.doi.org/10.1063/1.2746954 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/90/23?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Thermal resistance between crossed carbon nanotubes: Molecular dynamics simulations and analyticalmodeling J. Appl. Phys. 114, 224308 (2013); 10.1063/1.4842896 Interfacial thermal resistance between metallic carbon nanotube and Cu substrate J. Appl. Phys. 110, 124314 (2011); 10.1063/1.3670011 Contact thermal resistance between individual multiwall carbon nanotubes Appl. Phys. Lett. 96, 023109 (2010); 10.1063/1.3292203 The interfacial thermal conductance between a vertical single-wall carbon nanotube and a silicon substrate J. Appl. Phys. 106, 034307 (2009); 10.1063/1.3191673 Interfacial thermal conductance between silicon and a vertical carbon nanotube J. Appl. Phys. 104, 083503 (2008); 10.1063/1.3000441

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 80.197.85.17

On: Thu, 08 May 2014 09:16:17

Thermal energy exchange between carbon nanotube and airMing Hu,a� Sergei Shenogin, and Pawel Keblinskib�

Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, Troy, New York 12180and Rensselaer Nanotechnology Center, Rensselaer Polytechnic Institute, Troy, New York 12180

Nachiket RaravikarIntel Corporation, 5000 W. Chandler Blvd. Chandler, Arizona 85226

�Received 4 April 2007; accepted 14 May 2007; published online 5 June 2007�

Using molecular dynamics simulations the authors impose a heat flux between single-walled carbonnanotubes and air to study thermal interfacial conductance. They estimate that the nanotube-airinterfacial thermal conductance is about 0.1 MW/m2 K at room temperature and atmosphericpressure. The associated interfacial thermal resistance is equivalent to the resistance of 250 nm thicklayer of air. They also show that the interfacial resistance is a strong function of the interactionparameters between air atoms and carbon nanotubes. © 2007 American Institute of Physics.�DOI: 10.1063/1.2746954�

Efficient heat dissipation is one of the crucial challengesthat limits the development of disruptive microelectronic de-vice technologies. The International Technology Roadmapfor Semiconductors �ITRS� 2004 �Refs. 1 and 2� estimatessuggest that an 8 nm feature-size device may generate localheat fluxes as high as 100 000 W/cm2 that would need to bedissipated efficiently to preserve device integrity, reliability,and performance. The anticipated heat fluxes at the die levelare �1000 W/cm2, which is tenfold higher than presentcomplementary metal oxide semiconductor �CMOS�devices,3 making power dissipation perhaps the most impor-tant factor that eventually limits the device density on eachchip.

Carbon nanotubes �CNs� due to their very high intrinsicthermal conductivity4 are considered for improvement ofheat dissipation in microelectronic devices. However, thethermal performance of materials incorporating carbon nano-tubes is so far well below expectations mainly because of theinterfacial thermal resistance or contact resistance that limitsthe heat flow between CNs and surroundings.5 While theinterfacial energy transfer between nanotubes and fluids wasalready studied experimentally5 and theoretically,6 under-standing of thermal transfer between nanotubes and gasesreceived less attention, despite significant research effort onadsorption of gases on carbon nanotubes.7,8

From the thermal application perspective nanotube-airinterface is the most relevant. In fact, a recent experiment9

demonstrated that vertical arrays of CNs on silicon improvedthermal energy transfer only moderately, despite a very largesurface area involved. A number of factors might be respon-sible for this behavior, including air flow through the tubearray, air-tube interfacial resistance, or tube-surface contactresistance. In this letter, using molecular dynamics �MD�simulations, we will focus on the understanding of oneof these factors: interfacial thermal resistance at air-tubeinterface.

In our studies we performed classical MD simulations ofa single-walled carbon nanotube surrounded by air �22% ofoxygen and 78% of nitrogen molecules� with interatomic

interactions described by the polymer consistent force field.10

In this force field the nonbonded van der Waals interactionsbetween the air �O2 and N2� and carbon atoms is describedby the 9-6 Lennard-Jones interatomic potential, ��2�� /r�9

−3�� /r�6� where r is the interatomic spacing, and the param-eters � and � are energy and length scales, respectively. Thestandard value of the � parameter for oxygen-carbon andnitrogen-carbon interactions in the force field is 2.51�10−3

and 3.06�10−3 eV respectively, which gives the adsorptionenergy �noted by �ad thereafter� for O2 molecule on the wallof �10, 10� carbon nanotube of about 0.068 eV correspond-ing to 2.64kT, where k is the Boltzmann constant and T is theroom temperature �300 K� that we used throughout our stud-ies. Considering the significant spread of the values of theoxygen adsorption energy as obtained by ab initiocalculations,11 instead of selecting just one, we study thethermal exchange as a function of � with bonding energyranging from less than kT to over 10kT.

The simulation box consists of one �10, 10� �1.34 nmdiameter� 12.3 nm long CN. The tube is surrounded by amixture of 2000 oxygen and nitrogen molecules filling thesimulation box with dimensions 24.7�24.7�12.3 nm3. Pe-riodic boundary conditions are used in all directions, suchthat the tube has no open ends. In the first stage of simula-tions we equilibrate the system at T=300 K and at fixedvolume for several nanoseconds using integration time stepof 0.5 fs. For weak adsorption parameters the pressure of oursystems is about 10 atm. For stronger gas-nanotube attrac-tion, because of the adsorption of gas molecules on the nano-tube, the pressure in the simulation box gradually decreasesduring equilibration and stabilizes at the level of 7–10 atmdepending on the strength of the attraction. For these stron-ger absorption cases, we correct our results for the pressurechange.

Following equilibration, we applied a constant rate heatsource at a carbon nanotube and removed the heat at thesame rate in a cylindrical shell of 5.83 Å thickness, concen-tric with the nanotube axis, at the distance 9.6 nm from thecenter of the tube. The heat source and sink were realizedusing the energy and momentum conserving velocity rescal-ing algorithm.12 After a transient period ranging from severalhundred picaseconds to a few nanoseconds, depending on the

a�Electronic mail: hum2@rpi.edub�Electronic mail: keblip@rpi.edu

APPLIED PHYSICS LETTERS 90, 231905 �2007�

0003-6951/2007/90�23�/231905/3/$23.00 © 2007 American Institute of Physics90, 231905-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 80.197.85.17

On: Thu, 08 May 2014 09:16:17

strength of the air-carbon interactions, the system reaches asteady state. Once the steady state is reached we analyzeradial temperature profiles �separate for gas and solid�, whichare obtained as time averages for cylindrical shells of thick-ness of 5.83 Å.

Figure 1 shows a typical steady-state temperature profileobtained with the simulation setup described above. In theair the temperature profile fits very well to the solution of thediffusive heat flow equation. However, as clearly seen in Fig.1, the temperature profile is dominated by a large tempera-ture drop at the carbon nanotube–air interface. From thesedata we can calculate the thermal conductance of the systemG from

J = G�T , �1�

where J and �T are the heat flux and temperature jump at theinterface, respectively. We calculated the heat flux asdQ /dt /A, where dQ /dt is the rate of the heat supplied to thetube �equal and opposite to the heat rate removed from thegas� and A is the area of the tube-air interface which weassumed is the area of the tube cylinder with a radius of6.7 Å.

For O2 adsorption energy of 2.64kT, which is the stan-dard value for the force fields used in simulations, we obtainthe value of G of 0.90 MW/m2 K. This result can be com-pared with interfacial conductance associated with fully dif-fusive collisions.13 During such collisions it is assumed thatmolecules are scattered back into the gas with the tempera-ture of the solid. In consequence, the accommodation coef-ficient, which measures the efficiency of the energy transfer,is equal to unity and the interfacial conductance has a maxi-mum theoretical value, GM �4.5 MW/m2 K. Consequently,G=0.90 MW/m2 K corresponds to the accommodation coef-ficient of 20%.

In addition to simulations at 10 atm, by reducing thenumber of gas molecules by half at the beginning of thesimulation run, we performed simulation at 5 atm pressurewith �=2.64kT and obtained G=0.44 MW/m2 K, which ishalf of the value at p=10 atm. These expected results arisefrom the fact that at a given temperature the conductance is

proportional to the number of collision per unit time, whichis simply proportional to pressure. This allows us to estimateG at 1 atm and room temperature to be 0.09 MW/m2 K. Thesignificance of this value can be assessed via the so-calledequivalent air thickness, i.e., the thickness of the air slabover which the temperature drop in planar heat flow geom-etry is the same as that at the interface. This thickness, alsoknown as the Kapitza length lK, is given by

lK = �/G , �2�

where � is the thermal conductivity of air. UsingG=0.09 MW/m2 K and �=0.024 W/mK, we obtainlK=270 nm.

Another way of assessing the importance of the interfa-cial resistance is the value the “critical” nanotube length Lcover which the heat flown into the tube at one end is dissi-pated into the gas, which is given by14

Lc = �rtube�tube/G . �3�

Using tube conductivity �tube=3000 W/mK and tuberadius rtube=0.7 nm, and G=0.09 MW/m2 K, one obtainsLc�5 �m. Both values lK and Lc are quite significant indi-cating that the interfacial resistance is an important limitingfactor for the heat flow between nanotubes and gases.

In the top panel of Fig. 2 we demonstrate the sensitivityof the interfacial conductance to the adsorption energy pa-rameter. It is interesting to note that at low adsorption energythe interfacial conductance is extremely low. The quadraticextrapolation of the results to the zero bonding energy givesG=0.15±0.04 MW/m2 K at 10 atm and room temperature.This limit corresponds to a very low accommodation coeffi-cient of about 3.3%. Increasing the adsorption energy steeplyincreases the conductance up to the value of 2.5 MW/m2 Kat bonding energy of 5.33kT. Further increase of the surfaceattraction leads to only minor increase of the interfacial con-ductance.

To better understand the dependence of the conductanceon the adsorption energy, in the middle panel of Fig. 2 weshow the surface coverage. As expected, the coverage in-creases with increasing attraction. At the strongest attractionthe coverage is about 10 molecules/nm2, which is essentially100% coverage. Comparison with the conductance datashows that 50% coverage corresponds to a point at which theconductance ceases to increase steeply. This can be under-stood in terms of a large fraction of tube area being coveredby air molecules thus preventing direct collisions of the gasmolecules with the tube and adsorption on the surface.

Further insight into the collision process is provided bythe distribution of the residence times of air molecules attube surface, which are shown in Fig. 3. At low adsorptionenergies the residence times are very short and correspond to“ballistic flight” collisions with the tube. At larger adsorptionenergies, an increasing number of molecules resides on thetube for a period of time before it is desorbed into the gasphase. This adsorption-desorption process leads to diffusivescattering and a more efficient energy transfer.

The above analysis of the interfacial conductance andunderlying scattering process suggests the following sce-nario. At low surface attraction the scattering corresponds tomore or less instantaneous collision with the tube surface.This collision is almost elastic leading to accommodationcoefficient as little as 3%. With increasing attraction, empiri-cal observation from Fig. 2 suggests that the conductance

FIG. 1. Typical temperature profile for the system of �ad=0.068 eV. Thedotted line assuming bulk air thermal conductivity represents the logarith-mic decay of the temperature of air according to the solution of the diffusiveheat flow equation in cylindrical geometry.

231905-2 Hu et al. Appl. Phys. Lett. 90, 231905 �2007�

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 80.197.85.17

On: Thu, 08 May 2014 09:16:17

increase proportionally to the bonding energy. However, theresidence time and the coverage remain low. In this case, theincreased attraction strength effectively increases the rangeof the interaction potential, thus leading to increased timeover which the air molecules interact with the tube. Finally,when the bonding energy becomes strong enough to adsorb�at least temporarily� the air molecules, the scattering be-comes diffuse and relies on adsorption/desorption processesinvolving large residence times.

In summary, using MD simulations we have studied thethermal interfacial conductance between single-walled car-bon nanotubes and air by imposing a heat flux between them.We estimate that the nanotube-air interfacial thermal conduc-tance is about 0.1 MW/m2 K at room temperature and atmo-spheric pressure. The associated interfacial thermal resis-tance is equivalent to the resistance of about 250 nm thick

layer of air. We also show that the interfacial resistancestrongly depends on the interaction parameters between airatoms and carbon nanotube. Therefore, further refinement ofthe predictions on the interfacial thermal resistance requiresan accurate description of the interactions between nanotubeand oxygen and nitrogen molecules.

This work was supported by the gift from the IntelCorporation and by the New York State Interconnect FocusCenter at RPI.

1Emerging Research Devices, International Technology Roadmap for Semi-conductors 2004 update �http://www.itrs.net/Links/2004Update/2004_05_ERD.pdf�.

2R. S. Prasher, J. Y. Chang, I. Sauciuc, S. Narasimhan, D. Chau, G.Chrysler, A. Myers, S. Prstic, and C. Hu, Intel Technol. J. 9, 285 �2005�.

3R. C. Chu, Proceedings of the Rohsenow Symposium on Future Trends inHeat Transfer, Cambridge, MA, 16 May 2003 �unpublished�.

4P. Kim, L. Shi, A. Majumdar, and P. L. McEuen, Phys. Rev. Lett. 87,215502 �2001�.

5S. T. Huxtable, D. G. Cahill, S. Shenogin, L. P. Xue, R. Ozisik, P. Barone,M. Usrey, M. S. Strano, G. Siddons, M. Shim, and P. Keblinski, Nat.Mater. 2, 731 �2003�.

6S. Shenogin, L. P. Xue, R. Ozisik, P. Keblinski, and D. G. Cahill, J. Appl.Phys. 95, 8136 �2004�.

7H. Ulbricht, G. Moos, and T. Hertel, Surf. Sci. 532-535, 852 �2003�.8H. Ulbricht, R. Zacharia, N. Cindir, and T. Hertel, Carbon 44, 2931�2006�.

9K. Kordás, G. Tóth, P. Moilanen, M. Kumpumäki, J. Vähäkangas, A.Uusimäki, R. Vajtai, and P. M. Ajayan, Appl. Phys. Lett. 90, 123105�2007�.

10H. Sun, S. J. Mumby, J. R. Maple, and A. T. Hagler, J. Am. Chem. Soc.116, 2978 �1994�.

11T. Savage, S. Bhattacharya, B. Sadanadan, J. Gaillard, T. M. Tritt, Y. P.Sun, Y. Wu, S. Nayak, R. Car, N. Marzari, P. M. Ajayan, and A. M. Rao,J. Phys.: Condens. Matter 15, 5915 �2003�.

12P. Jund and R. Julien, Phys. Rev. B 59, 13707 �1999�.13V. V. Goldman, Phys. Rev. Lett. 56, 612 �1986�.14S. T. Huxtable, D. G. Cahill, and L. M. Phinney, J. Appl. Phys. 95, 2102

�2004�.

FIG. 2. Thermal conductance G �top panel�, surface coverage �c �middlepanel�, and residence time �R �bottom panel� as a function of the adsorptionenergy.

FIG. 3. Probability of the residence time of air molecules residing at thecarbon nanotube surface to form a range of interaction strengths.

231905-3 Hu et al. Appl. Phys. Lett. 90, 231905 �2007�

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 80.197.85.17

On: Thu, 08 May 2014 09:16:17