Jun Fng，Bin Chen，?，Hui Pn?

aDepartment of Engineering Mechanics，Zhejiang University，Hangzhou 310027，China

bInstitute of Applied Physics and Materials Engineering，F(xiàn)aculty of Science and Technology，University of Macau，Macao SAR，China

Anomalous friction of graphene nanoribbons on waved graphenes

Jun Fanga，Bin Chena，?，Hui Panb，?

aDepartment of Engineering Mechanics，Zhejiang University，Hangzhou 310027，China

bInstitute of Applied Physics and Materials Engineering，F(xiàn)aculty of Science and Technology，University of Macau，Macao SAR，China

H I G H L I G H T S

·The friction increases with the amplitude at a fixed period.

·The friction anomalously increases and then decreases with the period.

·A theory is proposed to explain the simulation results.

A R T I C L EI N F O

Article history：

6 August 2015

Accepted 9 September 2015

Available online 25 September 2015

Friction

Small scale

Graphene

Molecular dynamics

Friction plays a critical role in the function and maintenance of small-scale structures，where the conventional Coulomb friction law often fails.To probe the friction at small scales，here we present a molecular dynamics study on the process of dragging graphene nanoribbons on waved graphene substrates.The simulation shows that the induced friction on graphene with zero waviness is ultra-low and closely related to the surface energy barrier.On waved graphenes，the friction generally increases with the amplitude of the wave at a fixed period，but anomalously increases and then decreases with the period at a fixed amplitude.These findings provide insights into the ultra-low friction at small scales，as well as some guidelines into the fabrication of graphene-based nano-composites with high performance.

?2015 The Authors.Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics.This is an open access article under the CC BY-NC-ND license（http：// creativecommons.org/licenses/by-nc-nd/4.0/）.

Friction is very important to the function and maintenance of small scale structures due to their large specific area ratios. Conventionally，the friction is described by the Coulomb law and inlinearproportiontothenormalcompressiveforce.However，the Coulomb law often fails at small scales because the friction can be coupled with adhesion or be significant even under a normal tensile force［1］.Recently，significant scientific efforts have been devoted to uncover the hidden principles for friction at small scales［2］，which，nevertheless，may have already been employed in nature.For example，a large pretension in the spatula pad under gecko toe can be induced through sliding movement，which can lead to highly reversible adhesion of gecko adhesion［3］.

Due to the remarkable properties［4-7］and probably also the simplicity，the friction of graphene-based small scale structures has attracted extensive interests［8-12］.With atomic force microscopy（AFM），the friction on SiO2covered with a layer of graphenewasfoundtobemuchlowerthanthatwithoutgraphene， while the adhesion of the former was much larger than that of the latter［13］.The friction between an AFM tip and a graphene substrate was showed to be mainly due to van der Waals force［13］.It was reported that the friction of a supported graphene depends on the supported height with a higher friction at a larger height［14］. Monte Carlo simulation indicated that the friction of a graphene layer manifested with a stick-slip pattern［15］.For multiple layers，the friction was found to be relatively stable and its average value was relatively small［15］.

A graphene layer is extremely flexible and can develop waviness on the surface due to thermal fluctuation［16-18］.The waviness can also be introduced in a graphene layer through mechanical compression or through cyclic heating and cooling［19］.Such waviness can dramatically change the properties of graphenes.For example，the electric conductivity of waved graphenes decreased due to the separation of electrons while the ferroelectricity could be highly improved［19］.The local chemistry of a waved graphene can even be different from a flat graphene［19］.Waved graphene layers were also observed in graphene based nano-composites［20］.

Here，we employ the molecular dynamics method to simulate theprocessofdragginggraphenenanoribbons（GNRs）ongraphenesubstrates with or without waviness.We find that the friction of GNRsonasuspendedgraphenelayerwithoutwavinessisultra-low and proportional to the width of a GNR.We show that，on waved graphenes，the friction generally increases with the amplitude of the waves at a fixed period，but anomalously increases and then intriguingly decreases with the period at a fixed amplitude.We further develop a theory to explain the simulation results.

Fig.1.Variation of the friction with the width of the GNR at a fixed length，L=12·7 nm（a），with the length of the GNR at a fixed width，W=1·23 nm（b），with the perimeter of the GNR（c）and with the area of the GNR（d）.Solid line in plot（a）is a linear fit to the simulation results.Inset in plot（a）illustrates the dragging of a GNR on a flat graphene substrate.Inset in plot（b）displays the stick-slip phenomenon observed in the simulation.

The molecular dynamics simulations are carried out with LAMMPS.In all simulations，the reactive empirical bond order（REBO）potential is adopted to describe the carbon-carbon interaction within each layer［21］，while the Lennard-Jones potential withσcc=3·4 ? andεcc=0·00284 eV［22］is used to describe the carbon-carbon interaction between neighboring layers.In the simulation，the temperature of the system is maintained at～300 K.

We firstly investigate the dragging of a GNR on a flat graphene substrate.The GNR has a width of W and a length of L.As shown in the inset of Fig.1（a），the right edge of the GNR layer is horizontally pulled at a constant velocity of 0.1 ?/ps.The calculated total resisting force on the GNR along the pulling direction（inset of Fig.1（b））has a period close to the size of the crystal lattice of a graphene.Though local out-of-plane bending of nanoribbons may be very small，local non-uniform in-plane shearing deformation of nanoribbonsisclearlyobservedinthesimulation.Intheslipphase，local interaction energy between the nanoribbon and graphene substrate is considered to be downhill，which leads us to believe thattheforcetraceshownintheinsetofFig.1（b）involvesunstable slip motion.The friction calculated from the average value of the negative peak of this resisting force along the pulling direction within multiple periods is very small.Figure 1（a）shows that the friction increases almost linearly with the width of the GNR at a slope of～0·15 nN/nm.Such a linear relation is robust.For example，when W=20 nm，the simulated friction is 3.11 nN.As shown in Fig.1（b），the friction initially increases with the length of theGNRuntilitsaturatesforlongGNRs.Thedependenceoffriction on perimeter or area will be affected by how these two parameters change in the simulation.As indicated in Figs.1（c）and（d），the friction significantly scatters around a fixed perimeter or around a fixed area，which leads us to conclude that the friction does not correlate with either the perimeter or the area of GNRs.

WetheninvestigatethedraggingofaGNRonawavedgraphene substrate，as illustrated in the inset of Fig.2（a）.The waviness is described by where A is the amplitude andλis the period.We vary A andλ in the simulation.The total resisting force on the GNR along the pulling direction is obtained in the simulation，which also exhibits a stick-slip phenomenon（Fig.2（b））.When A is relatively small，the period of the stick-slip is close to the size of the crystal lattice of a graphene.For a relatively large A，the period of the stick-slip is close to the period of the wave.The friction is subsequently calculated from the average value of the negative peak of the resisting force along the pulling direction within multiple periods. Similarly，we find that the friction is also very low.As shown in Fig.2（b），the friction generally increases with the amplitude of the waves at a fixed period.As shown in Fig.2（c），the friction increases and then decreases with the period at a fixed amplitude.Figures 2（b）and（c）indicate that the friction of GNRs on waved graphenes can be several times higher than that on flat graphene.

Recently，the peeling of a thin film adhering on a corrugated substrate was investigated［23］.For the special case of a thin filmadhering on a flat substrate，the theoretical prediction［23］was the same as that of the Kendall's peeling model［24］，which is different from the current model.The Kendall's peeling model［24］predictsthattherequiredpullingforceforaGNRonaflatgraphene substrate，F(xiàn)K，is given by

where E is the Young's modulus of a GNR，～1 TPa［4］，γis the surface energy density，～0·16 N/m［25］，and t is its thickness，～0·334 nm［23］.With Eq.（2），we find the Kendall's prediction would be at least one order of magnitude higher than our simulation results in Fig.1（a）.

Fig.2.Dragging of GNRs on waved graphene substrates.（a）Schematics of dragging of a GNR on a waved substrate with 1 eV/? being 1.6 nN；（b）time evolution of the total resisting force onthe GNRalong thepulling direction withλ=1·5nm；（c）variation of the frictionwith A；（d）variation of thefriction withλ.In the simulation，W=1·23 nm and L=12·7 nm are defaulted values.

Fig.3.Theoretical model for the friction of GNRs when being dragged along a waved graphene substrate.

However，the ultra-low value of friction between GNRs and the graphene substrate found in our simulation is consistent with previous report of the inter-shell friction of double-walled carbon nanotubes（DWCNT）［25］.In the process of pulling the inner tube out of the DWCNT，it was suggested that the shear stress vanished within the overlapped region and only the shear stress near the edge of the tube was responsible for the intershell friction.

To understand our simulation results，a simple model is proposed.As schematically shown in Fig.3，a dragging force，P，which is parallel to the pulling direction，is required to move a GNR along a curved surface.As indicated in Fig.1（b），the friction force is saturated when the ribbon is longer than～6 nm，which leads us to adopt a local criteria by assuming that the local friction，f，is equal to the local energy barrier along the moving path，Δγ.Such a local energy theory to evaluate the friction force can be similar to Griffith's approach for a crack problem.Due to the geometrical constraint，there may exist local normal force，F(xiàn)N，along the interface.Based on the force equilibrium，we can approximately get

whereθis the angle between the local tangential direction of the interface and the pulling direction andαreflects the edge effect，which is～1.Note that P may vary as the GNR is dragged along the substrate and the friction is then taken to be

It should be pointed out that local normal force can affect local Δγalong the interface，which may also depend on the pulling direction.For simplicity，letΔγbe a constant andαbe 1，and we get

Suppose that a GNR can be in perfect contact with the underlying graphene substrate.With Eqs.（2），（5），we find that

According to Eq.（6），the friction is independent of the length and is close toΔγ.For a flat graphene substrate，A=0 and F=ΔγW，which is consistent with our simulation results shown in Fig.1（a）.According to Fig.1（a），Δγ=0·15 N/m.It is interesting to note that the intershell cohesive energy density of DWCNT is～0·16 N/m［25］.We attribute the variation of friction with the length of the GNR at small lengths in the simulation（Fig.1（b））to the size effect.

According to Eq.（6），the friction on a waved graphene substrate increases with the amplitude，which is consistent with the results given in Fig.2（c）.However，the friction should decease with the period according to Eq.（6），which is not consistent with the results shown in Fig.2（d）.When quantifying surface roughness with the average absolute slope of the surface profile within a sampling length，the roughness of the waved graphene substrate would increase with the amplitude at a fixed period and decrease with the period at a fixed amplitude.Thus，F(xiàn)ig.2（d）indicates that the friction anomalously increases and then decreases with the roughness.As observed in simulations，the GNR can only attach to the top part of the waved graphene when the period of the waved graphene substrate is too small.In fact，the friction on such a graphene substrate approaches to that on a flat graphene substrate，as seen in Fig.2（d）.Thus，the inconsistency between the simulation and the theory is because that the GNR cannot make perfect contact with the substrate when the period of the waved graphene substrate is very small.

In conclusion，the process of dragging a GNR on graphenes with or without waviness is investigated with molecular dynamics simulation.It is found that the induced friction per unit width on GNRs in the current analysis is approximately the surface energy on graphenes without waviness.On waved graphenes，the friction on GNRs generally increases with the amplitude of the waves at a fixed period，but anomalously increases and then decreases with the period at a fixed amplitude.These results cannot be explained by the conventional Coulomb friction law or the Kendall's model［24］.This work should help understand the friction at small scales，as well as provide some guidance in fabricating graphenebased nano-composites with high performance.

Acknowledgments

Hui Pan thanks the support of the Science and Technology Development Fund from Macau SAR（FDCT-068/2014/A2 and FDCT-132/2014/A3）and Multi-Year Research Grants（MYRG2014-00159-FST and MYRG2015-00017-FST）from Research&Development Office at University of Macau.Bin Chen thanks the support of the National Natural Science Foundation of China（Grant No. 11372279）.Bin Chen also thanks Dr.Qunyang Li for helpful discussions.

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3 July 2015

s.

E-mail addresses：chenb6@zju.edu.cn（B.Chen），huipan@umac.mo（H.Pan）.

http：//dx.doi.org/10.1016/j.taml.2015.09.001

2095-0349/?2015 The Authors.Published by Elsevier Ltd on behalf of The Chinese Society of Theoretical and Applied Mechanics.This is an open access article under the CC BY-NC-ND license（http：//creativecommons.org/licenses/by-nc-nd/4.0/）.

in revised form

*This article belongs to the Biomechanics and interdiscipline