Air flow through carbon nanotube arraysMing Hu, Sergei Shenogin, Pawel Keblinski, and Nachiket Raravikar Citation: Applied Physics Letters 91, 131905 (2007); doi: 10.1063/1.2793174 View online: http://dx.doi.org/10.1063/1.2793174 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/91/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Controlling water flow inside carbon nanotube with lipid membranes J. Chem. Phys. 141, 094901 (2014); 10.1063/1.4893964 How fast does water flow in carbon nanotubes? J. Chem. Phys. 138, 094701 (2013); 10.1063/1.4793396 The role of activation energy and reduced viscosity on the enhancement of water flow through carbon nanotubes J. Chem. Phys. 134, 194509 (2011); 10.1063/1.3592532 Role of thermal boundary resistance on the heat flow in carbon-nanotube composites J. Appl. Phys. 95, 8136 (2004); 10.1063/1.1736328 Fluid flow in nanopores: Accurate boundary conditions for carbon nanotubes J. Chem. Phys. 117, 8531 (2002); 10.1063/1.1512643

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Air flow through carbon nanotube arraysMing Hu,a� Sergei Shenogin, and Pawel Keblinskib�

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

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

�Received 30 May 2007; accepted 11 September 2007; published online 26 September 2007�

Using molecular dynamics simulations, we studied the air flow through carbon nanotube arrays. Wefound that for 1.4 nm diameter tubes separated by 25 nm, the air flow can be well described by thefree molecular flow theory. We estimate that for such array, the pressure gradient is about0.1 atm/�m at 1 atm air pressure and 5 m/s flow velocity, indicating that the flowing air can onlypass through an array of no more than about 400 carbon nanotubes in series. Our findings providedesign rules for arrays of nanotubes for thermal energy exchange with air. © 2007 AmericanInstitute of Physics. �DOI: 10.1063/1.2793174�

Efficient heat dissipation is one of the crucial challengesthat limits the development of disruptive microelectronic de-vice technologies. The International Technology Roadmapfor Semiconductors 2004 �Refs. 1 and 2� estimates suggestthat an 8 nm feature-size device may generate local heatfluxes as high as 100 000 W/cm2 that would need to be dis-sipated efficiently to preserve device integrity, reliability, andperformance. The anticipated heat fluxes at the die level are�1000 W/cm2, which is tenfold higher than the presentcomplementary metal oxide semiconductor devices,3 makingpower dissipation perhaps the most important factor thateventually limits the device density on each chip.

Carbon nanotubes �CNTs� due to their very high intrinsicthermal conductivity4,5 are considered for improvement ofheat dissipation in microelectronic devices. However, thethermal performance of materials incorporating carbon nano-tubes, such as liquid suspensions,6 polymer composites,7 orcarbon nanotube arrays–air interfaces,8 is far below expecta-tions. One factor credited for these disappointing results isthe interfacial thermal resistance that limits the heat flowbetween CNTs and surroundings.6,9 In fact, our recent mod-eling studies10 demonstrated that at normal conditions, theinterface between air and a single-walled carbon nanotubehas interfacial resistance equivalent to resistance of 250 nmlayer of air, which is indeed significant.

Another important factor controlling gas-solid thermalenergy exchange is the ability to flow cold air to the tubeinterfaces. Considering that the majority of thermal solutionspropose arrays of carbon nanotubes to be used, the key ques-tion is what the nature is, and key parameters, such as pres-sure drop, associated with air �gas� flow through such arrays.

To address these questions, without resorting to any as-sumptions underlying theoretical analysis, we performed mo-lecular dynamics �MD� simulations. In our studies, one �10,10� �1.34 nm diameter� 12.3 nm long single-walled carbonnanotube �total of 2000 carbon atoms� is surrounded by 2000oxygen and nitrogen molecules �at air composition� fillingthe simulation box with dimensions 24.7�24.7�12.3 nm3.

The interatomic interactions are described by the polymerconsistent force field.11 Periodic boundary conditions areused in all directions, such that the tube has no open ends.Such boundary conditions mimic a square array of infinitelylong tubes separated by 24.7 nm.

In the first stage of simulations, we equilibrated the sys-tem at T=300 K and at fixed volume for several nanosecondusing integration time step of 0.5 fs and observed that systemequilibrium pressure was at designated value of 10 atm. Fol-lowing the equilibration period, we applied an external driv-ing force Fext to the nanotube in the x direction �distributeduniformly over all carbon atoms� and applied equal and op-posite forces to the air atoms to keep the total momentum ofthe system at zero, as shown in Fig. 1. Once the steady stateis reached, we calculate the average velocities of carbonnanotube and air atoms. We verified that in the steady state,the average drag force acting on the nanotube is equal andopposite to Fext.

Figure 2 presents the total drag force FD acting on thenanotube as a function of relative velocity between thecarbon nanotube and air. First, we note that the drag force isproportional to the velocity. This behavior is characteristic oflaminar flow. However, the analysis of the magnitude of this

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

FIG. 1. Schematic of carbon nanotube-air system. The bar indicates meanfree path of air molecules between collisions.

APPLIED PHYSICS LETTERS 91, 131905 �2007�

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force reveals large discrepancy between the predictions ofcontinuum hydrodynamics. In Fig. 2, we show the solutionof the Navier-Stokes equations for the fluid flow around anarray of cylinders,12,13 which was obtained with the use ofthe Oseen approximation14 �solid line in Fig. 2�. As shown inFig. 2, the actual drag force is nearly two orders ofmagnitude smaller than that predicted by continuumhydrodynamics.

The above results are not surprising considering that themean free path of air molecules in our system is about10 nm; i.e., it is much larger than the nanotube diameter �seeFig. 1�. Therefore, we also compare our results with theforce-velocity relationship derived under assumption of freemolecular flow and completely diffusive scattering.15 The re-sulting relationship is given by

FDfree =

�U2

2d���e−S2/2

S�S2 +

3

2I0S2

2

+ S2 +1

2I1S2

2� +

�3/2

4S�TR� . �1�

In Eq. �1�, U is the relative velocity between CNT and air, dis the diameter of the CNT, � is the density of the air,S=�� /2M is the molecular speed ratio, � is the specific heatratio of the air, M =U /c is the Mach number, c is the speedof sound in air at room temperature, TR is the temperatureratio between the CNT and air, and I0 and I1 are the modifiedBessel functions of zeroth and first orders, respectively. In allof our calculations, we take �=1.4, c=346 m/s, and TR=1.

As shown in Fig. 2, the results of our MD simulation arein very good agreement with the prediction of free molecularflow theory, indicating that free molecular flow is the domi-nant mechanism of gas flow through carbon nanotube arrays.This can also be understood considering that the Knudsennumber, Kn=� /Lc, which determines the degree of rarefac-tion of a gas and gauges the validity of the Navier-Stokesequations, reaches the value of the order of 10 with meanfree path of the air �10 nm and characteristic dimension ofthe flow geometry Lc taken as tube diameter. This renderscontinuum flow analysis inapplicable.

To provide a direct evidence for the free molecular flowbehavior, in Fig. 3, we show the steady state velocity �xcomponent� profile of air in cylindrical shells concentric withthe tube. We also plot the solution of the Oseen approxima-tion in Fig. 3 for comparison. We obtained the radial velocityprofiles as time averages x component of air atoms residingin cylindrical shells of thickness of 5.83 Å. As clearly seenin Fig. 3, the Oseen approximation under the framework ofcontinuum theory predicts a logarithmic decay of the veloc-ity of air near the carbon nanotube, which is quite differentfrom the MD results. Velocity profile obtained by MD isdominated by a large velocity drop at the carbon nanotube-air interface and almost constant and uniform velocity in therest of the simulation cell volume. This is consistent withfree molecular flow regime where the large mean free path�on the scale of the microstructural system length� preventsdevelopment of significant gradients in the velocity field,other than that present at the interfaces.

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 driving force of 15 pN and obtained U=146.3 m/s,which is exactly two times the value obtained at p=10 atmand the same driving force. This also implies that at a giventemperature, the drag force is proportional to the air pressure,which is an expected result from the free molecular flowtheory.15 This allows us to extrapolate our results to 1 atm.

To put our results in the context of practical consider-ations, we can estimate how much pressure gradient isneeded to keep the air flow at given velocity through thenanotube array. From our analysis, the pressure gradientalong the flow direction can be estimated asdP /dx=�P /Lx=FD/ALx, where A=LyLz is the cross-sectional area perpendicular to the flow. At 1 atm air pressureand U�5 m/s �which is typical for an electrical fan16� forthe tube array considered in the simulations, the pressuregradient is dP /dx�0.1 atm/�m. This pressure drop corre-sponds to 1.4 nm diameter tubes separated by 25 nm. For30 nm tubes separated by 100 nm, which is typical for mul-tiwalled carbon nanotube array,17–21 the pressure drop will beabout 0.13 atm/�m. Despite relatively small drag forces ex-

FIG. 2. Drag forces on carbon nanotube �per unit length� as a function ofrelative tube-air velocities. The dashed and solid lines represent the solutionof free molecular flow and Oseen viscous flow around cylinder arrays,respectively.

FIG. 3. Typical velocity profile of the CNT-air system with driving force of15 pN. The dashed line shows the comparison with the solution of cylinderarray moving in fluid using Oseen approximation.

131905-2 Hu et al. Appl. Phys. Lett. 91, 131905 �2007�

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hibited on carbon nanotubes in free molecular flow regime,the associated pressure gradients are very large. This origi-nates from very small spacing between carbon nanotubes.From these estimations, the size of the nanotube array whichcan be effectively cooled by air flow is limited to approxi-mately 400 nanotubes in a row. Beyond these dimensions,only diffusive air molecular motion is operational with nohydrodynamic flow.

Our findings suggest design rules for arrays of nanotubesfor thermal energy exchange with air. As shown in Fig. 4, thearrays could be in islands containing �100�100 tube arraysseparated by larger spaces �see dI in Fig. 4� with bare sur-faces �with no tubes�. This will allow for the driven air jet topenetrate into the majority of the tube interfaces and thusmaximize thermal energy exchange. The optimization ofspacing between carbon nanotube islands should be based oncontinuous flow analysis, since the distances involved aremuch larger than air molecule mean free path.

This work was supported by the gift from the Intel Cor-poration and by the New York State Interconnect Focus Cen-ter at RPI.

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FIG. 4. Schematic of suggested design for nanotube arrays.

131905-3 Hu et al. Appl. Phys. Lett. 91, 131905 �2007�

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