Zipping, entanglement, and the elastic modulus ofaligned single-walled carbon nanotube filmsYoonjin Wona, Yuan Gaoa, Matthew A. Panzera, Rong Xiangb, Shigeo Maruyamac, Thomas W. Kennya, Wei Caia,and Kenneth E. Goodsona,1

aDepartment of Mechanical Engineering, Stanford University, Stanford, CA 94305; bSchool of Physics and Engineering, Sun Yat-sen University,Guangzhou 510275, China; and cDepartment of Mechanical Engineering, The University of Tokyo, Tokyo 113-8656, Japan

Edited by Mark C. Hersam, Northwestern University, Evanston, IL, and accepted by the Editorial Board November 4, 2013 (received for review July 9, 2013)

Reliably routing heat to and from conversion materials is a dauntingchallenge for a variety of innovative energy technologies––fromthermal solar to automotive waste heat recovery systems––whoseefficiencies degrade due to massive thermomechanical stresses atinterfaces. This problemmay soon be addressed by adhesives basedon vertically aligned carbon nanotubes, which promise the revolu-tionary combination of high through-plane thermal conductivityand vanishing in-plane mechanical stiffness. Here, we report thedata for the in-plane modulus of aligned single-walled carbonnanotube films using a microfabricated resonator method. Molecularsimulations and electron microscopy identify the nanoscale mecha-nisms responsible for this property. The zipping and unzipping ofadjacent nanotubes and the degree of alignment and entanglementare shown to govern the spatially varying local modulus, therebyproviding the route to engineered materials with outstandingcombinations of mechanical and thermal properties.

nanostructured materials | van der Waals | thermal interface materials |thermoelectrics | energy conversion

Nanostructured materials provide unique combinations ofproperties that promise performance breakthroughs for

applications ranging from energy conversion to data storage andcomputation (1–4). In many cases it is the very unusual combi-nation of two properties (2), neither of which is an extreme valuewhen considered alone, that leads to adoption and major perfor-mance benefits. An example is the search for a mechanically com-pliant thermal conductor that can, for example, link semiconductingmaterials with the metals used for heat spreading and exchange. Aparticularly compelling case is thermoelectric waste heat recoverysystems (3), for which the interfaces between the thermoelectricmaterials, electrodes, insulators, and heat exchangers must accom-modate enormous, repetitive thermomechanical stress. However,nature does not provide a material combining the necessary highthermal conductivity with the required low elastic modulus (5, 6).Vertically aligned carbon nanotube (CNT) films may combine

mechanical compliance with high thermal conductivity (7–18), butthere have been few reports of the in-plane modulus of these filmsand little physical explanation for the wide range for the data (2–300 MPa) (19–25). Our previous data for the thickness-dependentin-plane modulus of multiwalled CNT films indicated a strongdependence on the nanotube density and alignment (21), which arelinked to the detailed growth details (26–28). Other approaches,such as mesoscopic simulations or atomistic models, found thatCNT networks exhibit unique self-organization, including bendingand bundling (29–31). Therefore, relating the nanoscale morpholo-gical details to the mechanical properties is critical.Combined experimental, theoretical, and computational tech-

niques applied to the more complex and fundamentally chal-lenging single-walled CNT system are presented in this paper,along with the in-plane data for the modulus for single-walledCNT films. Because single-walled CNT films have higher densitiesand smaller tube–tube distances, these films exhibit more complexdynamics and tube–tube interactions than multiwalled CNTfilms. We present coarse-grained molecular simulations, and this

approach is consistent with a much simpler cellular model treatingthe films as foam. The coarse-grained simulations not only predictthe modulus, but also describe the manner in which the nanotubesself-organize into bundles and highly entangled regions due to vander Waals forces. The computational results reveal the signifi-cance of deformation mechanisms in determining the mechanicalresponse of the single-walled CNT films.The microfabricated resonator depicted in Fig. 1A measures

the effective in-plane modulus of single-walled CNT films (E1,exp).The measurement method (21) and nanotube growth process(33) are described in SI Text. Fig. 1A shows that E1,exp decreaseswith increasing nanotube film thickness, which we attribute toinhomogeneities (with respect to the film-normal coordinate) inthe density and alignment. The measured, single-walled CNTfilms have a disordered and dense crust layer on top of an alignedmiddle region. This is similar to observations of multiwalled CNTfilms in past work (21, 27, 34). Our single-walled CNTs have beenobserved to be denser and stiffer due to a smaller diameter aswell as better quality. The theoretical curves, shown in Fig. 1B,capture the effect that as the film grows taller, the modulus of themiddle layer becomes more dominant, and the overall modulus ofthe film decreases. The measurements and three-layer model (21)suggest that large differences between the modulus of the crustand middle layer are due to their morphological variations.To understand the mechanisms governing the mechanical re-

sponse of the single-walled CNT films, we use two separate sim-ulation methods. The first is a simple model for cellular solids.Because carbon nanotubes are low-density solids, previous studieshave shown that the mechanical response and nanostructure offilms closely resemble those of a foam or cellular solid (10, 14).Gibson and Ashby introduced a rectangular cellular model that has

Significance

Aligned carbon nanotube films promise the unusual combinationof high thermal conductivity and mechanical compliance. Here,the mechanical compliance of single-walled nanotube films hasbeen measured and linked to their morphology and microscopicmotions, including zipping, unzipping, and entanglement. Thephysical mechanisms governing the mechanical response includebending forces or van der Waals interactions, with the dominantmechanism depending on the nanotube density and alignment.The dependence of film morphology on mechanical modulusexplored here provides the foundation for modeling of a varietyof other properties including thermal and electrical conductivity.

Author contributions: Y.W., M.A.P., S.M., T.W.K., and K.E.G. designed research; Y.W., Y.G.,and W.C. performed research; Y.W. and Y.G. analyzed data; X.R. and S.M. preparedsamples; and Y.W., Y.G., M.A.P., X.R., W.C., and K.E.G. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. M.C.H. is a guest editor invited by the Editorial Board.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. E-mail: goodson@stanford.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1312253110/-/DCSupplemental.

20426–20430 | PNAS | December 17, 2013 | vol. 110 | no. 51 www.pnas.org/cgi/doi/10.1073/pnas.1312253110

been widely used to estimate the modulus of foams (35). Becausethe film structure consists of angled tube segments rather thanperfectly horizontal and vertical beams, we present a modifiedcellular model in which trusses connect the corners and the centerof each unit cell, as shown in Fig. 2A (SI Text) (7). The aspect ratio(AR) of a unit cell is related to the angle made by the trusses withrespect to the horizontal plane, where AR = H/L =

ffiffiffi2

ptan θ. By

considering the bending of the trusses in response to horizontal andvertical forces applied to the film, we can predict the in-planemodulus (E1,cell) and out-of-plane modulus (E2,cell) of the film as

E1;cell =C sin θ cos4 θ ·�2− cos2 θ

�−1; [1]

E2;cell =C sin3 θ; [2]

with C = α  Etruss Itruss   f 2truss Atruss�2, where Etruss, Itruss, and Atruss

are the Young’s modulus, moment of inertia, and cross-section

area of each truss, respectively, and α is a numerical constant.The volume fraction occupied by the nanotubes is ftruss = Atrussl/HL2, where l is the length of the truss. Because nanotubes tendto bundle together, the trusses in our cellular model do notnecessarily represent individual nanotubes. Hence, Itruss andAtruss in our model are unknown, and constant C in Eqs. 1 and2 is treated as a fitting parameter. Fig. 2B plots the predicteddependence of the moduli on the AR at different volume frac-tions (f) of nanotubes. Our model predicts the ftruss

2 dependenceof the film moduli as the Gibson and Ashby model, but showsa stronger dependence of the film moduli on the AR. The pre-diction is largely consistent with our experimental data on thefilm moduli as well as SEM estimates of the AR and f. The ratiobetween the predicted E1,cell of the middle and crust layer isconsistent with the measurements, which is in the range of 30–60. The cellular model provides insight into the overall mechan-ical behavior of the CNT array. This model assumes that theelastic deformation of the film is entirely caused by truss bend-ing, which is not precisely correct in accounting for the complexmorphology in the CNT array. Nonetheless, we apply the cellularmodel here owing to its simplicity and the possibility, when com-bined with the coarse-grained approach also developed in thispaper, to provide insight into nanotube alignment by means ofthe AR parameter. In reality, the nanotubes are held together byweak van der Waals forces, and zipping of nanotube bundles mayplay a role in the elastic compliance of the film. To remove theempirical fitting parameter C and to gain a deeper understandingon the realistic structure and dynamics of the nanotubes in thefilms, we use a coarse-grained molecular simulation.In the coarse-grained molecular simulation, each nanotube is

modeled as a fiber discretized into a chain of nodes. Neighboringnodes on the same fiber interact to give the bending andstretching stiffness of the nanotube, whereas nodes on differentfibers interact through van der Waals forces (31) (SI Text). In thismodel, the range of AR (1.4–4.2) and f (2–12%) of interest aredetermined from the characterization results including imageprocessing (7, 9, 11, 13). The moduli calculated using the simu-lation E1,sim and E2,sim are shown in Fig. 2B. Fig. 3 shows that themolecular simulation is qualitatively similar to the SEMs, wherethe middle layer is more aligned compared with the crust. Theaverage tube orientation θ in the relaxed structure is determinedfor comparison with the cellular model. A consistency is ob-served between the molecular simulations, the cellular model(with a fitting parameter C = 60,000), and our experimentalmeasurements, as shown in Fig. 2B. Better agreement is ob-served particularly at AR < 3.0, where bending is dominantrather than van der Waals forces. However, due to the cancel-lation effect of the van der Waals interaction and bending energy,the overall trend of the curve (Fig. 2B) remains qualitatively thesame. The simulation predicts that the moduli in both directionsscale with f 1.5–2.0, depending on the AR, consistent with the f 2

dependence predicted by the cellular model. A better-alignedfilm with greater AR has not only lower in-plane modulus, whichis preferable for the interface applications, but also higher out-of-plane modulus (SI Text). This is because longer beams areweaker in bending, so as the AR increases, the lengthened beamsbend more easily in the direction perpendicular to their axis (Fig.2A). If the beams are aligned better, they are more difficult tostretch in the out-of-plane direction.The molecular simulations also provide more details about the

mechanisms of nanotube deformation when the film is subjectedto an elastic strain. There are many complex deformation modesnot considered in the cellular model. For example, bundles ofnanotubes may move together, a bundle formed by several nano-tubes may partially zip (Fig. 4A) or unzip, two nanotubes or bundlesmay rotate around a contact point (Fig. 4B), and nanotubes withina bundle (oriented perpendicular to the straining direction) mayslide relative to each other.

0 5 10 15 20 25 300

50

100

150

200

250

300

350

400

450

Single-Walled CNT Film Thickness (μm)

In-P

lan

e M

od

ulu

s (M

Pa

)

A

B

Dim

en

sio

nle

ss M

od

ulu

s

LDV

CNT Film

Crust

Cantilever Bottom

Middle

100

101

102

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Theoretical fitting curves

for multi-walled CNT films

Theoretical fitting curves

for single-walled CNT films

CNT Film Thickness (μm)

single-walled CNT multi-walled CNT

Crust Region

Crust+Middle Region

Top

1 μm Film

Fig. 1. (A) Effective in-plane modulus (E1,exp) of vertically aligned single-walled CNT samples measured using a resonator technique. The thicknessvariations and the resulting modulus variations due to the nonuniformitiesin nanotube growth are illustrated by horizontal and vertical error bars,respectively. The samples are from several deposition batches. This can resultin groups of samples that deviate from the trend. (Inset) Schematic ofa resonator with a vertically aligned CNT film and a schematic showing thedifferent layers in a film are overlaid on the plot. (B) Dimensionless modulus,Ep = ECNT

ErefρrefρCNT

, including our past data on multiwalled CNT films (21, 32) andbest-fit lines calculated using a three-layer analysis. Dashed lines indicate theeffect of varying modulus of the crust (600 MPa) and middle (10 MPa) layersby ±10% on the effective modulus.

Won et al. PNAS | December 17, 2013 | vol. 110 | no. 51 | 20427

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To probe the importance of the different deformation mech-anisms, we quantify the relative contributions of bending, stretch-ing, and van der Waals energies to the total energy and predictmodulus by measuring the average and curvature of these energycontributions as a function of applied strain, respectively. Formost films, the van der Waals energy is the dominant contribu-tion to the total energy of the film, meaning that the energygained in bundling together nanotubes and forming the filmstructure during the relaxation steps greatly exceeds the energycost due to nanotube bending and stretching. However, once thefilm structure has been formed, either bending energy or van derWaals interaction dominates the effective modulus under ap-plied strain, depending on the CNT morphological details (Fig.4 C and D). For example, for a film of AR = 1.4 and f = 4%,representing the crust layer, the relative contributions of bend-ing, stretching, and van der Waals energies to the total modulusare 72%, 5%, and 23%, respectively. In this case, bending is the

dominant contribution in determining the total modulus. How-ever, for a film of AR = 3.5 and f = 2%, representing the middlelayer, the contribution of the van der Waals energy to themodulus rises up to 60%. Therefore, the contribution of van derWaals interactions gains importance as the nanotubes becomemore aligned or as the volume fraction increases, because botheffects promote the nanotubes to zip into bundles (Fig. 4E).These results also suggest that the cellular model, which onlyincludes bending forces, is most accurate for structures with lowAR and low f (which is the case for many multiwalled CNTsamples), although the good agreement between the modelssuggests that, when fitting parameters are allowed, it is still ap-plicable to higher AR films.The combination of data and two approaches advances the

understanding of the nanostructural effect on the mechanicalproperties of single-walled CNT films. This understanding isessential for tuning the properties of nanotube films by engineering

1 1.5 2 2.5 3 3.5 4 4.5

10

10

10

10

10

−1

0

1

2

3

E1

E2

f=2%

f=6%

f=12%

f=2%

f=12%

f=6%

Aspect Ratio

Mo

du

lus

(MP

a)

E1,cell=E2,cell at H=L

E1,exp,crust

E1,exp,middle

In−Plane (E1), Out−of−Plane (E2) Modulus vs Aspect RatioA BCellular Model

2H

2L

θ

Cellular Model, EcellMolecular Simulation, EsimExperimental Data, Eexp

E2,exp,Gao

Fig. 2. (A) Schematic of a cellular model unit cell comprising eight trusses. The gray region depicts the elastic bending of one truss at an angle θ between thevector along the truss and the plane of the horizontal. (B) Predicted in-plane (E1) and out-of-plane modulus (E2) using the cellular models (lines) and coarse-grained molecular simulations (squares) with varied AR and f. The error bars of the simulation results represent variations in the modulus over repeatedcalculations of the same initial conditions. The line colors indicate the different f (2%, 6%, and 12%). The circled regions include the experimental data ofsingle-walled and multiwalled CNT films (7, 21).

Fig. 3. Examples of SEM images (A–C) and simulation snapshots (800 Å × 800 Å × 400 Å) after the relaxation (D–F) of the crust and middle. (A) Cross-sectionalSEM images of a single-walled CNT film showing aligned nanotubes in the middle region and (B) randomly oriented nanotubes in the crust region, re-spectively. (C) Top view of the crust layer. The crust has a value of AR of ∼1.4 and f of 4–8% whereas the middle layer has AR of 3–5 and f of 2–4%. In thesimulations, the nanotubes are constructed using a specific density and orientation to represent the different film morphologies. (D) Simulated alignednanotubes (AR = 4.2, f = 2%) and (E) the entangled nanotubes (AR = 2.8, f = 4%). (F) Top view of the entangled nanotubes (AR = 1.4, f = 4%). SEM images ofeach region of the film are analyzed using an image analysis procedure. The selected SEM images are representative of those obtained for each region.

20428 | www.pnas.org/cgi/doi/10.1073/pnas.1312253110 Won et al.

their nanostructure, which can be achieved by altering the syn-thesis conditions or by adding surfactant molecules. The de-velopment of self-consistent simulations and the experimentaldata can provide guidance on detailed simulation of other ma-terial properties including the effective thermal and electricalconductivities. The approach presented here is also applicable toa wide range of films with a fibrous nanostructure, such as filmsmade of nanowires and microwhiskers.

MethodsCoarse-Grained Nanotube Simulation. We adopt a coarse-grained molecularmodel in which each nanotube is represented by a set of nodes. The in-teraction between neighboring nodes on the same chain is designed to re-produce the bending and stretching response of an elastic tube with Young’smodulus Etube, inner radius rin, and outer radius rout. A Lennard-Jones (LJ)-type interaction is introduced between nonneighboring nodes to account forthe van der Waals attraction and steric repulsion between nanotubes. Thepotential energy as a function of nodal positions {ri, rj} can be written as (36)

V��

ri , rj��

=Xi

12ksðjri + 1 − ri j− l0Þ2 +

Xi

12kB

ðri−1 − riÞ · ðri+1 − riÞjri−1 − ri j · jri+1 − ri j + 1

!

+Xi,j

C12��ri − rj��12 − C6��ri − rj

��6!: [3]

The first summation is over all neighboring nodal pairs on the same nano-tube; the second summation is in which i is the neighbor of nodes i−1 and i+1on the same nanotube; the third summation is over all nodal pairs that donot belong to the same nanotube (where j ≠ i±1). kS = EtubeAtube/l0 and kB =EtubeItube/l0, where Itube = π(rout4 −rin4 )/4 is the moment of inertia of the tubecross-section, and l0 is the discretization length of the nanotubes. Here weadopt rout = 5 Å, rin = 4.4 Å by using the effective tube thickness (37). The LJcoefficients between the nodes are proportional to the LJ coefficients de-scribing the interaction between individual carbon atoms. Specifically,

C12 = Ncc12 and C6 = Ncc6, where Nc is the number of carbon atoms representedby each node (36), c12 = 2,516,582.4 kcal·mol−1·Å12, and c6 = 1,228.8kcal·mol−1·Å6 (38). For single-walled CNTs, we estimate Nc to be 2π rout l0/Ac,where Ac is the average area covered by each carbon atom. The nanotubes areinitialized as straight lines randomly positioned in the simulation cell. Thenumber of nanotubes and the length of each nanotube are selected toachieve the desired density. The initial nanotube orientations are parallel tovector t = (x1, x2+p, x3), where x1, x2, and x3 are random numbers uniformlydistributed in [−0.5, 0.5]. A larger p value corresponds to a stronger alignmentto the y axis. The structure is then allowed to relax by minimizing the potentialenergy using the conjugate gradient algorithm. During the relaxation, thenanotubes bend and bundle together. The relaxation algorithm stops whenthe potential energy change in one iteration is smaller than 10−4 eV.

After the structure has been relaxed, the components of the elasticstiffness tensor (Cij) connecting normal stress (σi), and normal strain («j)(where σi = Cij «j) are computed by stretching the simulation cell along the x,y, and z axes, respectively, relaxing the structure again, and computing thestress change. The maximum strain applied to the simulation cell is ±0.1%,which is also the maximum applied strain in our experiments. The effectivemodulus of E1,sim and E2,sim is computed from the components of the elasticstiffness tensor. For example, E1,sim corresponding to the equivalent value ofE1,exp in the current work is

E1,sim =C11 − ðC11C12 −C12C13Þ�C11C22 −C2

12

�−1C12

−�C212 −C13C22

��C212 −C11C22

�−1C13, [4]

where σy and σz are assumed to be zero because of the free surfaces of thesingle-walled CNT films. Because vertically aligned CNT films have entangledmorphologies showing foam-like behavior (10, 15, 17, 18), we assume thatthe film has a Poisson’s ratio of zero. Thus, the E2,sim is assumed to be C22.

The model accounts for the nanotube morphology by incorporatingvarious density and alignment parameters (i.e., AR and f). The coarse-grainedmolecular simulation predicts the mechanical response of single-walled CNTfilms depending on the film nanostructures and the intrinsic mechanicalbehavior of nanotubes.

Co

ntr

ibu

tio

n (%

)

0 2 4 6 8 10 12 14 16 180

20

40

60

80

100

Volume Fraction (%)

Co

ntr

ibu

tio

n (%

)StretchingBendingvan der Waals

C D

Alignment

Den

sity

Bendingdominant

van der Waals dominant

MiddleLayer

CrustLayer

alignedrandomlyoriented

low f

high f

E2 E1

E

A B

Zipping EntanglementZipping

-0.1% Strain -0.1% Strain

Bundled Nanotubes

1 1.5 2 2.5 3 3.5 4 4.50

20

40

60

80

100StretchingBendingvan der Waals

Aspect Ratio

Volu

me

Frac

tion

Fig. 4. Nanotube interactions when a compressive 0.1% strain is applied to the film (f = 2%, AR = 3.5). Snapshots from the molecular simulation illustrate thedifferent types of nanotube displacements under strain, including (A) zipping and unzipping (governed by van der Waals forces) and (B) crossed nanotubes.The blue lines show the nanotubes in the initial relaxed structure. The gray lines represent the new positions of the nanotubes when the cell is compressed.The contribution of bending, stretching, and van der Waals energy in the calculation of elastic modulus for varying (C) AR and (D) volume fraction. (E) Phasediagram to show the governing physics to decide the modulus of nanotubes with varying nanostructures.

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ACKNOWLEDGMENTS. This work was sponsored by the Office of NavalResearch (N00014-09-1-0296-P00004), the National Science Foundation(NSF), the Air Force Office of Scientific Research (AFOSR) through FA9550-12-1-0195, and the Semiconductor Research Corporation. W.C. acknowl-edges support from the NSF Grant CMS-0547681. S.M. acknowledgesfinancial support from Grant-in-Aid for Scientific Research (22226006 and

25630063), Indium replacement by single-walled carbon nanotube thinfilms (IRENA) project by Japan Science and Technology Agency, andStrategic International Collaborative Research Program. We performedthis work in part at the Stanford Nanofabrication Facility, which issupported by the NSF through the National Nanotechnology Infrastruc-ture Network.

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20430 | www.pnas.org/cgi/doi/10.1073/pnas.1312253110 Won et al.