Smooth sliding and superlubricity in the nanofriction of collapsedcarbon nanotubes

Hao Xu a, Jihong Al-Ghalith b, Traian Dumitric�a a, b, *

a Department of Aerospace Engineering and Mechanics, University of Minnesota, Twin Cities, MN 55455, United Statesb Department of Mechanical Engineering, University of Minnesota, Twin Cities, MN 55455, United States

a r t i c l e i n f o

Article history:Received 22 February 2018Received in revised form29 March 2018Accepted 6 April 2018Available online 11 April 2018

a b s t r a c t

We present the characteristics of the sliding friction in large-diameter collapsed carbon nanotubes(CNTs) as emerged from molecular dynamics simulations. The friction force is found to depend stronglyon the CNT sliding velocity, the interface contact area, interface commensuration, and temperature. Thenon-classical smooth sliding and superlubric behaviors identified at the molecular level give a usefulstarting reference to the ongoing efforts aimed at engineering the mechanical load transfer in materialsystems comprising collapsed CNTs.

© 2018 Elsevier Ltd. All rights reserved.

1. Introduction

Advances in synthesis have brought industrial level productionof carbon nanotube (CNT) sheets suitable for developing super-lightweight structural materials [1]. In an attempt to translate theexceptional mechanical attributes of individual CNTs [2] to the CNTsheets, significant efforts are directed at manipulating by stretchingthe random CNT orientation, which is characteristic for the rawsheet [3]. Recent atomic-resolution transmission electron micro-scopy on the stretched sheets found the massive presence ofcollapsed CNTs, which were aligned, packed, and free of catalystparticles. The achieved high-concentration of aligned CNTs isinteresting for the development of superstrong composites, antic-ipated to competewith carbon fibers [1]. Nevertheless, conceiving agood load transfer between the CNT structural units will beessential in this endeavor. So far, the carbon nanotribologicalstudies have focused on C60 [3], concentric CNTs [4e7], graphene[4,8,9], and graphene nano scrolls [10], and the knowledge of thefriction at the collapsed CNT interfaces is still missing. To uncoverthe characteristics of the sliding friction, here we make recourse toclassical molecular dynamics (MD) simulations of collapsed CNTsstacked in bundle arrangements similar with the ones observed inexperiment [11].

2. Methods

Our MD simulations were carried out with LAMMPS [12] andrely on the standard empirical AIREBO potential among intra-tubeatoms [13] and the KolmogoroveCrespi (KC) (vdW) potentialamong inter-tube atoms [14]. The registry-dependent KC potentialwas selected since the Lennard-Jones potential of AIREBO cannotcapture the characteristic corrugation between commensurategraphene layers. The adequacy of the KC potential is demonstratedin Fig. 1. In Fig. 1a and b, we display two graphene layers in whichthe a lattice vectors of the two layers are parallel and perpendic-ularly oriented, respectively. Next, in Fig. 1c, it can be seen that theshifting of the top layer along the armchair and zigzag lattice di-rections leads to a periodic corrugation of the vdW KC energy forthe commensurate periodicities (periodicity ratio 1), but a vanish-ing corrugation when the periodicities are incommensurate (peri-odicity ratio 1/

ffiffiffi

3p

).To prepare the collapsed CNTs, a Nose-Hoover thermostat was

used initially to equilibrate the cylindrical CNTs at 5 K. After thesystem reached equilibrium, a 0.01eV/atom force was added in thetransverse direction for 15 ps, to squeeze the structure. The systemwas then freed in the transverse direction and equilibrated again inmicrocanonical ensemble for another 500 ps. Periodic boundaryconditions (PBC) were applied in the axial direction throughout thewhole process. In the end, we used conjugate gradient relaxation torelax the collapsed structures at 0 K.

To study friction, we employed MD setups consisting of three(Fig. 2a) and four (Fig. 2b) collapsed CNTs placed under PBC. Thesesetups allow to investigate the sliding of CNT1 in vdW contact with

* Corresponding author. Department of Mechanical Engineering, University ofMinnesota, Twin Cities, MN 55455, United States.

E-mail address: dtraian@umn.edu (T. Dumitric�a).

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Carbon 134 (2018) 531e535

CNT2 (Fig. 2a), and CNT2 and CNT2’ (Fig. 2b). After carrying out aninitial conjugate gradient relaxation of the composed CNTs system,a one-unit ring belonging to CNT3 is kept fixed to prevent themotion of its center of mass. CNT3 is linked to CNT2 and to CNT20 bytwenty sp-type covalent cross-links, which are created by addingten interstitial atoms. Note that in Fig. 2b, CNT3 and CNT20 arecross-linked by PBC.

Before the production runs, the system was equilibrated for aninitial 10 ps at the desired temperature imposed on the systemusing Nose-Hoover dynamics. Next, velocity v controlled sliding isrealized by imposing a desired v to a one-unit ring belonging toCNT1. A Nose-Hoover thermostat is applied during MD to the un-constrained atoms. The friction force F measurements are doneafter 100 ps MD time, in order to allow for equilibration after theinitial acceleration period. The reported F is the net vdW forceexerted onto CNT1 projected onto its axis, averaged over the sub-sequent 500 to 3,000 ps. Note that during our MD studies, the CNTstructures remained collapsed.

An alternative way to the direct measurement is to extract Ffrom the energy balance DQ¼ Fvt: Here, DQ is the kinetic energyextracted from the reservoirs and t is the sliding time. In Fig. 3 weplot DQ vs. vt computed from the MD simulations of (30,30)collapsed CNTs with L¼ 13.6 nm sliding with v ¼ 100m/s at roomtemperature. With the energy method, we obtain F¼ 7.58 eV/Å, ingood agreement with the direct forcemeasurement of 7.52 eV/Å (or0.89 nN/nm). We emphasize that the role of the 20 added cross-links in our MD setups is to constrain the sliding of CNT2 andCNT2’. The results reported next are converged with the number ofcross-links. For example, in the above example, with 20 cross-linkswe measured F ¼ 0:89 ±0:12 nN/nm, while with 16 cross-links isF ¼ 0:88 ±0:12 nN/nm.

Previous MD studies [5] on dynamical friction in oscillatingCNTs obtained relative v in the range of 120m/s to 790m/s,

depending on the magnitude of the oscillations. These simulationsgive us an indication of the v values that could be expected duringthe mesoscale CNT sliding processes occurring during the sponta-neous CNT film formation. In addition to considering such a high v

regime, we also lowered v up to 10m/s in order to approach theregime of the conventional pulling velocity exerted on materials.

The MD setups of Fig. 2 are selected because they are robust tothe thermostat location (see Fig. S1), and eliminate the non-representative large F peaks at selected v, observed in a two-CNTsetup (see Fig. S2). Note that friction enhancements at selected v

have been encountered earlier, in the sliding of coaxial cylindricalCNTs [7]. Additionally, any stretching or breaking of the interstitialbonds give an indication on the load transfer.

3. Results and discussions

We recently reported a low rate of thermal energy dissipation atthe vdW interfaces of collapsed CNTs [15]. Focusing here on relativemotion, we uncover a v dependent mechanical energy dissipationreflected by a complex dependence of F on v. This behavior differsfrom the classical v-independent F of the macroscopic scale. Fig. 4,which summarizes our MD results obtained in the three (35,35)collapsed CNT setup, reveals in fact three contiguous F regimes: (i)linear smooth sliding (v<80 m/s), (ii) peak (80< v<250 m/s), and(iii) weakening (250< v<500m/s). In Fig. 5a, a similar behavior canbe observed in the room temperature sliding of (50,0) collapsedCNTs. Both simulations are examples of commensurate configura-tions as the ratio of their unit cell sizes is 1. We relate next these Fregimes to the popular Prandtl-Tomlison (PT) model of the dampedmotion of an atom driven over the external periodic potential bymeans of a spring [16].

Despite the relative weakness of the vdW binding, we expectimportant energy corrugation U at the collapsed CNT interfaces.

Fig. 1. Two stacked graphene layers with the lattice vectors of the rectangular unit cells (shown by arrows) (a) parallel (AB stacking) and (b) 90� rotated. (c) The KC interlayer vdWenergy versus in-plane relative displacements of the two layers. The displacement directions for each layer are indicated by the arrows. The interlayer distance is kept at 0.335 nm.(A colour version of this figure can be viewed online.)

H. Xu et al. / Carbon 134 (2018) 531e535532

This is because layer stacking arrangements, where the carbonatoms do not eclipse the each other, Fig. 1a, are most favorable.Indeed, for the commensurate example of Fig. 4, we measured anenergy barrier of 1.5 eV/nm, Fig. S3, when sliding over a distance a,where a¼ 0.24 nm is the corrugation periodicity. In the low v

regime, the thermal motion of atoms plays an important role inhelping CNT creep out from these energy barriers. As in the PTmodel [16], the creep out motion results in release of the elasticenergy stored in the intratube sp2 bonds into thermal energy. Inagreement with our MD results for v<80 m/s, PT predicts smooth

sliding when there is a large disparity between the strengths of theatom-external potential (vdW bonding here) and the spring con-stant (intratube sp2 bonding here). Next, the peak F regime in-dicates that the timescale for the thermal creep becomes negligiblein comparison with the competing transport timescale a=v. Finally,the F weakening obtained as v is approaching the thermal speed ofcarbon atoms (~790m/s at room temperature) can be attributed tothe decrease in the energy dissipation when interfacial atoms haveless time to release their kinetic energies before the next barrierjump.

According to Krim [17], the phononic contribution to the smoothsliding F ¼ gv should be sensitive to the contact interface area Aand crystallographic commensuration, via g ¼ a$A$U2. Here, g isthe coefficient of friction while parameter a captures the temper-ature T dependence. Previous MD on concentric cylindrical CNTsfound F to be independent on A and commensuration [6]. On thecontrary, in collapsed CNTs we find that F is very sensitive to A; Uand T. First, (i) Fig. 5b shows that F=A vs. v data for (30,30), (35,35)and (50,50) collapsed CNTs falls onto a single curve, when A wasapproximated with A ¼ wL. Here, w is the width of the collapsedCNTs in vdW contact, and L¼ 13.6 nm the length of the trans-lational periodicity. For the linear regime, our MD givesg=A¼ 2.9 nN ps/nm3. Second, (ii) Fig. 5c considers the dynamicsliding at the (29,29)-(50,0) collapsed CNT incommensurate inter-face (periodicities ratio 1/

ffiffiffi

3p

). Here L¼ 6.4 nm in order to closelymatch the periodicities of 26 (29,29) CNT units and 15 (50,0) CNTunit cells. During MD, the collapsed (29,29)-(50,50) CNT interfacestays incommensurate, i.e., it doesn't develop local portions withmatching lattice constants. Since these two CNTs have similarwidths, the reduced friction relative to the results reported inFig. 5a can be attributed to the vanishing corrugation under crys-tallographic mismatch. The substantial reduction of friction relativeto the commensurate case, caused by incommensurability is calledsuperlubricity [4]. Third, (iii) Fig. 5d shows F decreasing as T in-creases, and that the three F regimes for commensurate sliding arepreserved at all T. In the PT model, the decrease of F with T is due tothe increased influence of thermal activation on the barrier jump-ing process.

Our MD showed that the bond length fluctuations of the CNT2-CNT3 cross-links are virtually independent from v. Such poor loadtransfer at the collapsed CNTs represents a potential culprit formaterial underperformance. Increasing load transfer, for exampleby polymer addition, will be critical for fulfilling the CNT's potentialas building blocks of ultrastrong materials. In this quest, MD could

Fig. 2. Schematics (side views) and atomistic models (axial views) for the MD setupswith (a) three and (b) four collapsed CNTs in the computational cell. Velocity isimposed to the one-unit cell “ring” (blue) belonging to CNT1. One-unit cell ring (red)from CNT3 is kept fixed. The vertical lines between CNT3 and CNT2 (and CNT20)indicated the introduced cross-links. The dashed lines indicate the (a) rectangular and(b) triclinic periodic boundary conditions. Note that in (a) the vertical periodicity islarge so CNT1 and CNT3 are virtually non-interacting by vdW. (A colour version of thisfigure can be viewed online.)

Fig. 3. The energy extracted by the thermostats DQ vs. sliding distance vt in the three-(30,30) collapsed CNT setup at room temperature. Here v ¼ 100m/s.

H. Xu et al. / Carbon 134 (2018) 531e535 533

be useful for making predictions about the frictional attributes ofdiverse polymer-coated CNTs. Since at this level of MD complexity,it is often more convenient to enforce full PBC, we have tested as apreliminary step, the transferability of our results to the four-CNTMD setup (Fig. 2b). In (35,35) collapsed CNTs, our computed F of

1.56± 0.07 nN/nm at v ¼ 50m/s is nearly twice as large than F of0.85± 0.05 nN/nm computed with setup of Fig. 1a. Furthermore,the four-CNT setup also allows to compute F under applied lateralpressure. Our MD simulations under 5.5% vertical compressionmeasured F ¼ 1.97± 0.12 nN/nm at v ¼ 50m/s. This change reflectsthe increased of corrugation, from 1.5 eV/nm to 2.9 eV/nm, as CNTfaces are getting in closer contact (Fig. S3). Finally, it is worth tostress out the important role of commensuration for the CNTmesoscale dynamics. With the setup of Fig. 2a, the simulated zero-input microcanonical evolution of CNT1 under an initial v¼ 50m/sgave a stopping time of only 3 ps for the commensurate (30,30)-(30,30) collapsed CNT and about 400 ps for the (29,29)-(50,0)collapsed CNT case.

4. Conclusion

In summary, because of the important role played by contactarea and commensuration, collapsed CNTs present frictional at-tributes more similar with graphene [9] than with cylindricalCNTs [6]. The commensuration dependence of F manifested insmooth sliding and superlubricity, points out a critical weaknessin these material systems, and gives a useful starting reference forthe ongoing efforts aimed at improving the load transfer. The MDderived F can be used to inform coarse-grained mesoscopicmodels that can explicitly incorporate the CNT tribology [18,19]and thus can be used to predict the spontaneous bundle forma-tion and its mechanics with accounting for the CNT sliding. Whilethe presented MD simulations considered vacuum betweencollapsed CNTs, the approach can be generalized to study thenanofriction at collapsed CNT interfaces in the presence of

Fig. 5. MD results with the three-CNT MD setup. (a) F (per unit length) vs. v for (50,0) collapsed CNTs. (b) F (per contact area) vs. v for collapsed armchair CNTs of different widths.The interface contact area A is 81.5 nm2, 96.3 nm2, and 138.4 nm2 for (30,30), (35,35), and (50,50) collapsed CNTs, respectively. (c) F (per unit length) vs. v at the interface of (29,29)and (50,0) collapsed CNTs. In (a)e(c) T¼300 K. (d) F (per unit length) vs. v for (30,30) collapsed CNTs at different T. (A colour version of this figure can be viewed online.)

Fig. 4. F (per unit length) vs. v for (35,35) collapsed CNTs. Dashed (red) lines are trendlines. Results were obtained with the three-CNT MD setup. (A colour version of thisfigure can be viewed online.)

H. Xu et al. / Carbon 134 (2018) 531e535534

polymeric layers [20].

Acknowledgments

We acknowledge useful discussions with G. Odegard and O.Hod. This work was supported by NASA's Space TechnologyResearch Grant NNX16AE03G and by the Institute for Ultra-StrongComposites by Computational Design, Grant NNX17AJ32G. Re-sources supporting this work were provided by the NASA High EndComputing Program through the NASA Advanced SupercomputingDivision at Ames Research Center.

Appendix A. Supplementary data

Supplementary data related to this article can be found athttps://doi.org/10.1016/j.carbon.2018.04.011.

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Supporting Information for

Smooth Sliding and Superlubricity in the Nanofriction of Collapsed Carbon Nanotubes

Hao Xu1, Jihong Al-Ghalith2, and Traian Dumitrică1,2,*

1Department of Aerospace Engineering and Mechanics, University of Minnesota, Twin Cities, MN 55455

2Department of Mechanical Engineering, University of Minnesota, Twin Cities, MN 55455

* Corresponding author. Fax: +1-612-6245230, E-mail: dtraian@umn.edu

1

Computational Methods

We give here further details about the computational method, including method verification.

In Figure S1 we demonstrate an alternative three-collapsed CNT setup in which the vdW

contacting orange (non-contacting gray) faces of CNT1 and CNT2 were evolved in the

microcanonical (canonical) ensemble, Figure S1a. As demonstrated in Figure S1b, the friction

forces computed this way have similar magnitudes with the MD set-up used in the main paper.

Figure S1. (a) Schematics (side view) for the MD setup containing three collapsed CNTs in the

computational cell. Unlike in the set-up described in Fig.2a of the main paper, the interface

atoms between CNT1 and CNT2 are evolved in the microcanonical ensemble (NVE). (b)

Comparison between the 𝐹 computed with the set-up used in the main paper (NVT) and the

combined NVT+NVE set-up described in (a).

Figure S2. (a) Schematics (side view) for the MD setup containing two collapsed CNTs in the

computational cell. (b) The computed 𝐹 vs. 𝑣 presents peaks at selected 𝑣.

2

In Figure S2a, we are showing a simpler two collapsed CNT setup, which was used in the early

stages of this work. As shown in Figure S2b, this setup gives large frictional peaks at selected

sliding speeds, such as 𝑣 = 350 m/s.

In the two-CNT setup, we observed that the corresponding velocities where peak F exists have a

linear dependence on the inverse of our selected tube length 1/L. Phonon density of state analysis

was carried out next. While Fourier transforming the velocity-velocity correlation function, by

merely considering longitudinal component and transversal component of velocities, respectively,

we obtained large longitudinal and transversal DOS at the previously identified peak velocities.

The frequencies of these excited modes also depend linearly on 1/L, thereby proving a correlation

between these excited vibration modes at the selected peak velocities. The provided movies clearly

show strong vibrational excitations at the peak velocity in comparison with sliding at non-peak

velocities. We have determined that the modes are standing waves created in CNT2 due to the

fixing of the ring of atoms (marked in red) to prevent the center of mass motion. In the three and

four CNT setups of Figure 2, this task is accomplished instead by cross-linking CNT2 and

CNT2’ with CNT3, whose center of mass motion is prevented. With these slightly more complex

setups (used in the main paper), we can successfully avoid excitation of the peak modes, and are

able to provide representative measurements of the friction force.

In Figure S3 we are showing the computed van der Waals corrugation during commensurate

sliding, strain free and under 5.5% lateral compression.

Figure S3. Variations of the static inter-tube van der Waals energy (measured from the most

favorable stacking) at the (35,35)-(35,35) collapsed CNT interface under relative axial sliding.

Both the relaxed and 5.5% compressed cases are shown.

3

Finally, in Table S1 we give the lengths (L), widths (𝑤), and contact areas (A) for the collapsed

CNTs used in the main paper.

Table S1. The characteristic dimensions of collapsed CNTs studied in this work.

CNT Length

(nm)

Width

(nm)

Area

(nm2)

(30,30) 13.6 6 81.6

(35,35) 13.6 7.1 96.6

(50,50) 13.6 10.2 138.7

(50,0) 14.1 5.6 79.0