Thermal Behavior of Double-Walled Carbon Nanotubes and Evidenceof Thermal RectificationKhoa Bui,†,‡ Huong Nguyen,†,‡ Cedric Cousin,§ Alberto Striolo,†,‡ and Dimitrios V. Papavassiliou*,†,‡

†School of Chemical, Biological and Materials Engineering, University of Oklahoma, Norman, Oklahoma 73019-0390, United States‡Carbon Nanotube Technology Center (CANTEC), University of Oklahoma, Norman, Oklahoma 73019-0390, United States§Clermont-Ferrand’s Superior National School of Chemistry, Aubier̀e, France

ABSTRACT: The thermal boundary resistance (TBR, also known as Kapitza resistance) between carbon nanotubes and asurrounding polymeric matrix (e.g., epoxy, polystyrene) is known to hinder the development of thermally conductive, nanotube-based composite materials. Because the carbon nanotube to carbon nanotube TBR can be even higher than that between ananotube and the polymer matrix, a fundamental question is whether thermal transfer along multiwalled carbon nanotubesoccurs solely via the outmost nanotube. To address this question, molecular dynamics simulations were conducted in double-walled carbon nanotubes. It is found that the resistance to heat transfer between the two nanotubes is higher than the TBRtypically reported between nanotubes and polymeric matrix materials, although heat transfer occurs more easily when the twonanotubes have different chirality. Further, and probably more importantly in terms of fundamental knowledge, the thermalboundary resistance is found to be up to 120% lower when thermal energy is transferred from the outer toward the innernanotube than vice versa, indicating thermal rectification in the radial direction of the carbon nanotube.

■ INTRODUCTIONRecent advances in synthesis and characterization of double-walled carbon nanotubes (DWCNTs) suggest that thesematerials possess enhanced properties compared to those ofsingle-walled carbon nanotubes (SWCNTs).1−4 Kim et al.3

concluded that DWCNTs are more thermally stable thanSWCNTs, as they can withstand temperatures as high as 2000°C without experiencing considerable morphological changes.Koziol et al.4 reported the synthesis of nanotube fibers bydirectly spinning an aerogel of DWCNTs. These fibers were upto 3 times stronger than commercial Kevlar fibers. Cumings etal.5 characterized multiwalled carbon nanotubes (MWCNTs)using TEM and found ultralow-friction and wear-free surfacesin these systems.In terms of thermal properties, the focus of the present work,

the exceptional thermal conductivity of carbon nanotubes(CNTs) and graphene sheets6 has generated promises towardmanufacturing nanocomposites with superior thermal manage-ment capabilities. Unfortunately, the thermal boundaryresistance (TBR, also known as Kapitza resistance)7,8 currentlyhinders such applications. The TBR is the major barrier to heat

transfer between the CNTs and the surrounding polymericmatrix.9−12 The TBR has been estimated, using moleculardynamics (MD) simulations, for several CNT−polymer andgraphene−polymer interfaces (epoxy, polystyrene, PMMA,etc.).10,13−15 Values on the order of 10−8 K m2 W−1 havebeen reported. Functional groups can significantly reduce thisresistance.16−19 Increasing the thermal transport of nano-composites by increasing CNT loading has been found not tobe applicable in general because of the tendency of nanotubesto form bundles and because the CNT−CNT TBR is expectedto be even higher than that between the CNT and thesurrounding matrix.20−23 Yang et al.24 concluded that theCNT−CNT TBR is strongly dependent on contact area. Theyreported a value of CNT−CNT TBR equal to 1.2 × 10−8 K m2

W−1 when two CNTs are aligned and partially overlap eachother. On the other hand, this value is 10 times smaller, 1.2 ×10−9 K m2 W−1, when two CNTs are crossing each other.

Received: November 9, 2011Revised: January 26, 2012Published: January 27, 2012

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© 2012 American Chemical Society 4449 dx.doi.org/10.1021/jp2107878 | J. Phys. Chem. C 2012, 116, 4449−4454

Prior MD simulations have been performed for DWCNTs.Rivera et al.25 suggested the possibility of creating nanosizedelements that oscillate along the axial direction at gigahertzfrequencies. Zhang et al.26 performed MD simulations ofDWCNT as rotational bearings and concluded that theinterlayer friction coefficient is extremely small, suggestingapplications as wear-less bearings. The present work stems fromincreasing interest focused on DWCNTs, with special attentionon thermal management opportunities. We seek to determine,via MD simulations, the Kapitza resistance between the twoconcentric nanotubes forming a DWCNT. Our results clarifywhether the thermal energy transfer along a DWCNT occursonly through the outer CNT or whether both CNTs contributeto heat transfer. Our results suggest that the TBR between thetwo nanotubes depends on the direction of heat flux (heattransfer occurs more easily from the outer to the inner CNTthan in the opposite direction), suggesting possible applicationsfor DWCNTs in thermal rectification in the radial direc-tion.27,28

Thermal rectification properties have been reported forsystems with defects in their structure or when impurities arepresent. Wu et al.29 reported thermal rectification at intra-moecular junctions formed by introducing pentagon−heptagonrings in SWCNTs. Chang et al.30 concluded that as carbon andboron nitride nanotubes were mass-loaded with heavymolecules, the resulting nanoscale system yielded asymmetricaxial thermal conductance with greater heat flow in thedirection of decreasing mass density. Graphene ribbons alsopossess thermal rectification properties when their structure isasymmetric.31 Through this work, we find that thermalrectification properties can be achieved in the radial directionfor a pure, symmetric (around its longitudinal axis), and defect-free DWCNT.

■ SIMULATION SETUPDWCNTs in vacuum were generated with different lengths,ranging from 2.8 to 30 nm. The chirality of the two nanotubesin a DWCNT was either the same [i.e., a (5,5) SWNT inside a(10,10) SWNT, indicated as (5,5)-(10,10) DWCNT], ordifferent [i.e., (6,6)-(19,0) DWCNT and (5,5)-(16,5)DWCNT]. In all cases the intertube distance was 0.34 nm.See Figure 1 for a simulation snapshot.The MD simulations were performed using the package

LAMMPS.32 The Tersoff potential33 was employed to describethe interactions among carbon atoms within one nanotube. Theinteraction between carbon atoms of different nanotubes wasdescribed using the 12−6 Lennard-Jones (LJ) potential.Details about the models and the parameters used and about

the implemented protocol have been described else-where.9,16,18,34 The methodology is based on nonequilibriumMD simulations. The DWCNT in vacuum was heated from 50to 300 K and then equilibrated at 300 K for 600 ps (NVTensemble). For the next 200 ps, the system relaxed under theNVE ensemble in order to record 15 different and independentconfigurations. Each of the 15 configurations was used toinitiate nonequilibrium simulations used to extract the thermalboundary resistance. One of the two SWNTs within theDWCNT was instantaneously heated to 500 K (“hot” SWNT).The hot SWNT was maintained at 500 K for 10 ps to relax thenewly excited vibration modes (NVT ensemble), while theother SWNT was maintained rigid. Then the simulationprogressed in the NVE ensemble with one SWNT at initial T of500 K and the other at 300 K. During the latter NVE portion of

the simulation, heat flowed from the hot SWNT to the coldone. The temperature difference between the two SWNTs wasrecorded as a function of time. The observed decayingtemperature difference, ΔT, was fitted with the exponentialfunction

Δ = Δ = − τT t T t( ) ( 0)e t/(1)

to obtain the time relaxation constant τ. The Kapitza resistance,Rk, is obtained from the time constant via the followingrelation,9,16,35,36 based on the lumped capacitance model:

=τ

RAC

T

Tk

(2)

In eq 2, AT is the interfacial area available for heat transfer,which is calculated as the logarithmic mean of the surface of thetwo nanotubes.37 CT is the heat capacity, calculated viaindependent simulations for one CNT in vacuum as16

= ⟨ ⟩ − ⟨ ⟩C

E E

k TT

2 2

b2

(3)

In eq 3, E is the total energy of the system, kb is the Boltzmannconstant, and T is the absolute temperature of the system. Toobtain CT, each nanotube was isolated and simulated within theNVT ensemble at 300, 400, or 500 K for 400 ps.Equation 2, based on the lumped capacitance model, is

expected to hold when the temperature of the surroundingsystem does not change significantly during the NVE stage ofthe simulation. Although in the system simulated here thetemperature of the cold SWNT changes with time, it isnecessary to assess the validity of the underlying assumptions.The Biot number for this system, defined as Bi = hL/k, where h

Figure 1. The DWCNT is composed of one armchair (6,6) SWNTinside a zigzag (19,0) CNT. The figure represents the lateral view ofone section of the DWCNT collected during the relaxation stage (a)and two cross sections obtained at the midpoint of the DWCNT (b).The two cross sections are obtained at a 1 ps interval to illustrate howthe SWNTs deform, relative to their original cylindrical shape, as thesimulation progresses.

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is thermal conductance (equal to inverse of the Kapitzaresistance in this case), L is the distance through which heat istransferred, and k is the thermal conductivity of the SWNT, canbe used to define a criterion for the applicability of the lumpedcapacitance model.36,20 When Bi < 0.1 the lumped capacitancemodel can be used, and eq 2 is expected to yield reliableestimates for the Kapitza resistance.36 In our case, assuming atypical value of k = 2900 W/(m K), L = 0.34 nm, and h = 0.1 ×108 W/(m2 K), the Biot number is found to be 1.17 × 10−4,well below the value of 0.1.

■ RESULTS AND DISCUSSION

Shenogin et al.9 investigated the Kapitza resistance of a CNT−octane system and found that the Kapitza resistance changes asa function of the CNT length. The results converged when thenanotube length approached 8.6 nm. The reason for this lengthdependence is the availability of more long-wavelengthphonons for heat transfer as the tube length increases. Ourresults show that the Kapitza resistance between inner andouter nanotube increases as the length of the nanotubeincreases from 2.8 to 20.0 nm, but then it reaches a plateau

when the DWCNT length is in the range of 20.0−30.0 nm (seeFigure 2). Unlike the CNT−octane case, in which the softphonons with frequencies less than 5 THz govern heat transfer,high-frequency optical phonons in the range 60−65 THzappear to govern the heat transfer for the CNT−CNT systemconsidered here (see Figure 3). Our results are consistent withresults from Shiomi et al.,38 who studied the heat conduction ina single-walled CNT using the heat pulse technique. Theyreported that at nonequilibrium most of the energy isdistributed to the bands higher than 50 THz, correspondingto optical phonons. These can play a major role in heat transfer.Because of the structural similarities, one could expect that

the Kapitza resistance is less when both SWNTs composingone DWCNT have the same chirality. Our results (see Figure2) indicate the contrary. The Kapitza resistance is smaller forthe armchair−zigzag DWCNTs and larger for armchair−armchair systems. Results for the DWCNT composed by one(6,6) SWNT in a (16,5) SWNT lie in between the other twocases. To explain this finding, the Fourier transform of thevelocity autocorrelation function for carbon atoms is plotted inFigure 3. The frequencies at which maxima and minima of the

Figure 2. Kapitza resistance between the two SWNTs composing DWCNTs of different lengths and chiralities. The notations “i” and “o” indicatethat either the inner or the outer nanotube was heated, respectively. The notations “acac”, “aczz”, and “acch” indicate (5,5)-(10,10), (6,6)-(19,0), and(6,6)-(16,5) DWCNTs, respectively. The error bars were calculated from 15 simulations with different initial configurations with Student t test (95%level of confidence).

Figure 3. Fourier transform of the velocity autocorrelation function for carbon atoms in (6,6)-(19,0) (a) and (5,5)-(10,10) DWCNTs (b). BothDWCNTs are of length 4.8 nm.

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vibration spectrum occur for the interior (6,6) SWNT matchthe frequencies of maxima and minima of the vibrationspectrum of the (19,0) SWNT in a (6,6)-(19,0) DWCNT. Incomparison, the frequencies at the maxima and minima of thespectrum of the (5,5) SWNT do not match as well the maximaand minima of the (10,10) SWNT in a (5,5)-(10,10) DWCNT.This means that heat can transfer more easily between anarmchair and a zigzag nanotube than in the other case. We alsocalculated the Kapitza resistance of a (6,6)-(16,5) DWCNT oflength 7.8 nm, and the result lies in between those obtained forarmchair−armchair and armchair−zigzag DWCNTs (seeFigure 2).Visual analysis of our simulation results (sequences of

snapshots such as those in Figure 1) showed that the twonanotubes in a DWCNT can rotate relative to each otherabehavior explored by others with focus on wear-less bearings.26

For the scope of this article, it is worth to point out that whenthe inner SWNT is heated the rotational period for theDWCNTs is about 7 ps for the armchair−armchair DWCNTand 40 ps for the armchair−zigzag DWCNT (see Figure 4).

Considering the period of rotation obtained from Figure 4 andthe Kapitza resistance of the corresponding DWCNTs, weobserve that higher rotation speeds lead to higher resistances toheat transfer, probably because of weaker coupling between themechanical vibrations of the carbon atoms belonging to the twointeracting nanotubes. On the basis of this observation, it islikely that those DWCNTs that provide poorer thermaltransport properties will be those more suitable for theproduction of wear-less bearings.Our estimates for the Kapitza resistance within concentric

DWCNTs are consistent with estimates reported by others forthat between CNTs touching via their externalwall.9,10,14,18,20,21 More importantly, the values of the Kapitzaresistance obtained in all cases considered in this work arealways larger than those typically observed at the CNT−polymer interface (e.g., 2.61 × 10−8 K m2 W−1 for CNT−epoxy,39 0.95 × 10−8 K m2 W−1 for CNT−PMMA,40 and 2.21× 10−8 K m2 W−1 for CNT−polystyrene22). Our results, thus,suggest that when DWCNTs are used in thermally conductivecomposites, the heat is likely to transfer mainly through theouter SWNT, while the inner SWNT does not contribute tothermal management. (Note also that values of the Kapitzaresistance in other solid−solid interfaces are also smaller thanthe CNT−CNT resistances calculated herein, e.g., 0.12 × 10−8

K m2 W−1 for GaN/SiC and 0.1 × 10−8 K m2 W−1 for GaN/sapphire.41,42)Another important observation can be obtained from the

results shown in Figure 2. Specifically, given one DWCNT, theKapitza resistance estimated when the outer SWNT is heated islower than that obtained when the inner SWNT is heated. Thissuggests that the heat flux preferably travels from the outer tothe inner SWNT than vice versa. This behavior is explained bymechanical observations. When a SWNT is heated, it expandsin the radial direction, and vice versa, when it is cooled itcontracts. When heat transfers from the outer to the innerSWNT within a DWCNT, the inner SWNT expands uponheating, while the outer SWNT contracts upon cooling,improving mechanical coupling between the two nanotubes.On the other hand, when heat transfers from the inner to theouter SWNT the mechanical expansion/contraction leads toless mechanical coupling as the heat-transfer process pro-gresses. This different behavior is quantified in Figure 5, inwhich the average distance between the two concentric SWNTs

Figure 4. Distance between two neighbor carbon atoms belonging toinner and outer nanotubes in a DWCNT during heat transfer. Thenotations “inn” and “out” indicate the nanotube that is heated. Thelength of the DWCNTs is 4.8 nm. The periodicity observed in the datacorresponds to the periodicity of the relative rotation of the nanotubescomposing the DWCNT.

Figure 5. Average distance between two SWNTs decreases during the relaxation step as the outer or inner one is heated for (5,5)-(10,10) DWCNT,as seen on the left panel. On the contrary, this average distance increases when the inner nanotube is heated due to the expansion/contractionbehavior, as seen on the right panel. A similar tendency is also observed for other DWCNTs.

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is shown when heat transfers from the outer to the inner orfrom the inner to the outer. The average distance between thetwo SWNTs decreases in the former case and increases in thelatter. A small change in the tube−tube distance causes strongeffects on the dispersion forces between the two nanotubes,thus affecting the Kapitza resistance.In order to assess whether the difference in Kapitza

resistance observed when heat transfers from the inner to theouter SWNT in DWCNTs is sufficient to suggest thermalrectification in the radial direction, we calculate the thermalrectification factor, RREC, defined as

=−

×+ −−

RK K

K100%REC (4)

In eq 4, K+ is the thermal conductance (i.e., the inverse of theKapitza resistance) as heat transfers from the outer to the innerSWNT and K− is thermal conductance in the oppositedirection. As can be seen from Table 1, RREC is slightly smaller

for the (6,6)-(19,0) compared to the (5,5)-(10,10) DWCNT.As the DWCNT length increases from 2.8 and 4.8 nm, RRECdecreases about 20%. It reaches a plateau for longer DWCNTs.This trend is similar to the one reported by Yang et al.31 forasymmetric graphene ribbons. The rectification ratio was ashigh as 240% for short ribbons. As the length of the ribbonsincreased from 3.4 to 5.1 nm, this ratio decreased from 240% to92% and remained constant for longer ribbons. The authorsreported that, for asymmetric graphene ribbons, the gradedgeometric asymmetry is the key for the increase of therectification ratio. On the other hand, evidence of thermalrectification in DWCNTs is attributed here to the expansion/contraction behavior of individual CNTs. Interestingly, ourresults suggest that symmetric (around their longitudinal axis)and defect-free DWCNTs exhibit a RREC in the range of 103−114% for DWCNTs of length exceeding 10 nm.

■ CONCLUSIONSTo summarize, we employed nonequilibrium moleculardynamics simulations to estimate, within the lumpedcapacitance model, the resistance to heat transfer (known asthermal boundary resistance or Kapitza resistance) betweentwo concentric single-walled carbon nanotubes forming adouble-walled carbon nanotube (DWCNT) in vacuum. Thecalculations were performed for DWCNTs of increasing length,and it was found that high-frequency optical phonons in therange 60−65 THz govern heat transfer in the systemconsidered here, unlike soft phonons of frequency less than 5THz that were found to govern heat transfer between carbonnanotubes and octane. In agreement with results obtained forcarbon nanotubes in configurations other than the oneconsidered here, our results show that the resistance to heattransfer between the concentric nanotubes is larger than thatbetween carbon nanotubes and organic matrixes, suggestingthat when DWCNTs are used in thermally conductingnanocomposites heat travels predominantly along the outer

nanotube, while the inner nanotube does not contribute tothermal management. DWCNTs of different chirality wereconsidered, and it was found that the Kapitza resistance issmaller in armchair−zigzag DWCNTs than in armchair−armchair DWCNTs. By examining the relative rotation of thetwo nanotubes, it was found that those carbon nanotubes thatseem to have frictionless movement (faster rotation) arecharacterized by higher thermal boundary resistance, suggestingthat mechanically coupling the vibrations of carbon atoms inthe two concentric carbon nanotubes facilitates heat transfer.Comparing the thermal boundary resistance for DWCNTs inwhich either the outer or the inner carbon nanotube washeated, it was found that, because of better mechanicalcoupling, heat transfer occurs much more easily from a hotouter carbon nanotube to a cold inner one than vice versa,yielding thermal rectification factors in the radial direction inthe range 103−114% for longitudinal symmetric, defect-freeDWCNTs of length exceeding 10 nm. Even though atemperature difference of about 200 K between two wallsmight not be practical, these results help understand heattransfer in carbon nanotubes, and they might be useful foropening up new opportunities for applications of double-walledcarbon nanotubes in thermal rectification applications.

■ AUTHOR INFORMATION

Corresponding Author*Phone (405) 3255811, fax (405) 3255813, e-mail dvpapava@ou.edu.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

This work was supported by the Center for Applications ofSingle-Walled Carbon Nanotubes, funded by the Departmentof Energy-Basic Energy Sciences (Grant DE-FG02-06ER64239). We also acknowledge support by DOD-EPSCOR: FA9550-10-1-0031. Computations were carried outat the University of Oklahoma Supercomputing Center forEducation & Research (OSCER).

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Table 1. Thermal Rectification Factor in the RadialDirection, RREC (%), for (6,6)-(19,0) and (5,5)-(10,10)DWCNTs of Different Lengths

length (nm) 2.8 4.8 10.0 20.0 30.0

(6,6)-(19,0) DWCNT 120.1 106.3 100.7 100.6 103.6(5,5)-(10,10) DWCNT 123.6 122.4 112.4 108.3 114.7

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