Frictional Properties of Nanojunctions Including Atomically Thin Sheets

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Letter pubs.acs.org/NanoLett

Frictional Properties of Nanojunctions Including Atomically Thin Sheets Wengen Ouyang,†,‡ Ming Ma,§,∥ Quanshui Zheng,†,‡ and Michael Urbakh*,§,∥ †

Applied Mechanics Laboratory, Department of Engineering Mechanics and ‡Center for Nano and Micro Mechanics, Tsinghua University, Beijing 100084, People’s Republic of China § School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel ∥ The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978, Israel S Supporting Information *

ABSTRACT: Using nonequilibrium molecular dynamics simulations and a coarse-grained description of a system, we have investigated frictional properties of nanojunctions including atomically thin sheets embedded between metal surfaces. We found that the frictional properties of the junctions are determined by the interplay between the lattice mismatch of the contacting surfaces and out-of-plane displacements of the sheet. The simulations provide insight into how and why the frictional characteristics of the nanojunctions are affected by the commensurate−incommensurate transition. We demonstrated that in order to achieve a superlow friction, the graphene sheet should be grown on or transferred to the surface with morphology, which is close to that of the graphene (for instance, Cu), while the second confining surface should be incommensurate with the graphene (e.g., Au). Our results suggest an avenue for controlling nanoscale friction in layered materials and provide insights in the design of heterojunctions for nanomechanical applications. KEYWORDS: Nanoscale friction, graphene, commensurability, lattice mismatch, bending rigidity, out-of-plane deformation

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between moiré patterns and out-of-plane deformations of the graphene film may significantly influence tribological properties of the junctions including the films. Despite the numerous studies of the effect of nanoscale landscape on electronic and mechanical properties of graphene films, understanding a relation between the film morphology and its frictional response remains challenging. Current qualitative understanding of friction on the nanoscale is based mostly on two models: the Prandtl−Tomlinson (PT) and Frenkel−Kontorova (FK) models.23,24 These models do not account for the effect of film bending rigidity and out-of-plane deformations of the film on friction, which could be important for metal−graphene junctions. In this Letter, we propose a minimal two-dimensional (2D) model for the description of these effects. We demonstrate that the interplay between incommensurability of contacting materials and the out-ofplane deformations of the film determines the frictional dissipation in such layered systems. We establish conditions needed to achieve a superlow friction at metal surfaces coated by graphene films. Figure 1a displays schematically the basic tribological setup, where a monolayer graphene sheet is embedded between two

raphene exhibits extraordinary mechanical properties, such as the extremely high strength and stiffness,1 ultralow interlayer friction,2 and even superlubricity.3−7 Because of the unique combination of high in-plane stiffness and low bending rigidity, this one-atom-thick membrane is always wrinkled to a certain degree.8−12 Out-of-plane deformations of graphene may strongly modify its electronic and mechanical properties, and recently much effort has been done to control them.8,13 Friction force microscope measurements (FFM)2,14,15 and simulations16−18 demonstrated that friction dissipation in graphene films, which are freely suspended or weakly adhered to substrates, is determined by formation of out-of-plane deformations of the film in front of the nanoscale sliding tip. In the majority of tribological applications, contacts between metallic substrates are employed, therefore it is especially important to investigate the frictional properties of graphene in contact with metal surfaces. Recent FFM measurements and simulations2,19 showed that graphene can be an excellent coating for low friction and wear under the conditions that it is not damaged during sliding. Graphene monolayers supported on hexagonal metal substrates are found to exhibit, apart from their intrinsic periodicity, an additional long-range order characterized by two-dimensional moiré patterns20−22 that result from the mismatch and/or the relative rotation angle between the lattices of graphene and supporting substrates. The interplay © XXXX American Chemical Society

Received: December 8, 2015 Revised: January 31, 2016

A

DOI: 10.1021/acs.nanolett.5b05004 Nano Lett. XXXX, XXX, XXX−XXX

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model, as illustrated in Figure 1b. The use of this approximation reduces significantly the simulation cost that allows the detailed study of the problem. The proposed 2D model reproduces well the results of 3D simulations,27 (see also Supporting Information (SI)). Here, the graphene sheet is modeled as a chain of particles interacting with nearest neighbors through linear springs of stiffness kint and angular springs of stiffness kθ. The values of kint and kθ have been estimated from the Young’s modulus and bending rigidity of graphene28 (see SI). The van der Waals interactions between the graphene and the two confining surfaces have been described using the Steele’s29 method (see SI) that gives the following equations for the potentials of interaction between the particles in the chain and the substrate and slider

Figure 1. (a) Schematic sketch of experimental setup. (b) The simplified 2D model. (c,d) Friction force ⟨FL⟩ and RMS height of film corrugation as functions of normal force ⟨FN⟩ calculated for graphene embedded between different metal surfaces. The FCC(111)-(1 × 1)R0° direction of surfaces and zigzag direction of graphene lay along the pulling axis x. For comparison, the dashed lines in panels c and d show a friction force and RMS height calculated for the confining surfaces, which have the same periodicity in x-direction as the graphene, and the same interaction parameters as Cu surfaces. In order to be on the same scale, the force for the commensurate contact is reduced by five times. Panel a was plotted with VMD.34

∞ 0 Usub(xi , zi) = Usub (z i ) +

n (zi)cos ∑ Usub n=1

2πn xi asub

(1)

∞ 0 Usld(xi , zi) = Usld (Zsld − zi) +

∑ Usldn (Zsld − zi) n=1

2πn (xi − Xsld) × cos asld

(2)

where (xi, zi) and (Xsld, Zsld) are the coordinates of the ith particle in the chain, and of the center of mass (COM) of the slider, respectively, and asub(sld) are the lattice constants of the substrate (slider). The expressions for the parameters U0sub(sld), Unsub(sld) in terms of the strength, ϵgs, and diameter, σgs, of Lenard-Jones interactions between the chain particles and the plates are presented in SI. The values of ϵgs and σgs have been fitted to give correct equilibrium distances and binding energies for graphene on the slider and substrate surfaces, as provided by the density functional theory calculations30 (see also SI). Both the slider and substrate are treated as rigid bodies. The equations of motion for the slider and particles in the chain that mimics the graphene sheet can be written in the form

solid plates. The top plate (slider) is coupled by a spring of spring constant Kdr to a stage that moves at constant velocity Vdr in x-direction, while the bottom plate (substrate) is kept at rest. The plates are held together by a normal load applied in the z-direction through the spring of stiffness KZ that is connected to the fixed normal stage. Because of its extremely high in-plane stretching and shear stiffnesses and a low bending rigidity, 1D ripples are the lowest energy deformation modes of the graphene.8 Recently, the 1D ripples of graphene have been observed experimentally11,13,25,26 and studied in simulations.11,25 Thereby, considering the effect of the graphene deformations on friction the 3D frictional junction in Figure 1a can be approximately described by a 2D

N N ⎧ ∂U (x , z ) ⎪ MẌsld + ∑ η sld(zi)m(Ẋsld − xi̇ ) + ∑ sld i i + Kdr(Xsld − Vdrt ) = 0 x ∂Xsld ⎪ ⎪ i=1 i=1 ⎨ N N ⎪ ∂U (x , z ) sld ̈ ̇ ̇ MZ Z ( z ) m ( Z z ) η + Γ + − + ̇ ⎪ sld ∑ z i ∑ sld i i + KZ(Zsld − Zdr0) = 0 i sld sld ⎪ ∂Zsld ⎩ i=1 i=1

(3)

⎧ ∂Usub(xi , zi) ∂Usld(xi , zi) + = ξxi(t ) ⎪ mxï + ηxsub(zi)mxi̇ + ηxsld(zi)m(xi̇ − Ẋsld ) + ∂xi ∂xi ⎪ ⎨ ⎪ ∂Usub(xi , zi) ∂Usld(xi , zi) sub sld ̇ )+ + = ξzi(t ) ⎪ mzï + ηz (zi)mzi̇ + ηz (zi)m(zi̇ − Zsld ∂zi ∂zi ⎩ (i = 1, ⋯ , N ).

(4)

Here, N is the number of particles in the chain, M and m are the mass of the slider and a particle in the chain. In order to ensure the right transformation from the 3D to the 2D model, the mass of chain particles is calculated as m = nmCLy/aygr, where mC is the mass of carbon atom, Ly and aygr are the width

of the sheet and the size of the graphene unit cell in y-direction, respectively, and n is number of carbon atoms in the graphene sub sld sld unit cell (see SI). The parameters ηsub x , ηz and ηx , ηz are the damping coefficients responsible for the dissipation of the kinetic energy due to the motion in the lateral and normal B

DOI: 10.1021/acs.nanolett.5b05004 Nano Lett. XXXX, XXX, XXX−XXX

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Nano Letters directions, and ξix and ξiz are the random forces acting on the ith particle in x- and z-direction, which satisfy the fluctuation− dissipation theorem. We assume that the damping coefficients decrease exponentially with the increase of the distance between the particles and the substrate and the slider,31,32 as sub sld sld ηsub α (z) = ηα0 exp(1 − z/σsub), ηα (z) = ηα0 exp[1 − (Zsld − z)/ σsld], α = x, z. The position of the normal stage, Zdr0, defines the applied normal load. The potential energy of the embedded graphene sheet includes contributions of interactions between the graphene and confining plates and the stretching and bending energies of the sheet. It can be written as

substrates at an average velocity equal to half of the pulling velocity. However, in the heterojunction, Cu/gr/Au, the sheet is trapped by the Cu plate, which interacts more strongly with graphene. The presented results show an unexpected variation of the friction force with the lattice mismatch. Predictably, the friction force for the ideally commensurate junction (the dashed line in Figure 1c) is much larger than in other cases. However, surprisingly the friction force found for the nearly commensurable junction Cu/gr/Cu (δ = −0.04) is significantly lower than for the incommensurate Au/gr/Au junction (δ = −0.147), where due to the structural incompatibility of contacting lattices the lateral forces between them are expected to cancel out systematically. Furthermore, the friction force for the heterojunction, Au/gr/Cu, where the significant lattice mismatch occurs only at one of the frictional interfaces, is lower than for both Cu/gr/Cu and Au/gr/Au junctions. Given that the difference between the strengths (ϵgs) of graphene−Au and graphene−Cu interactions is less than 5%, these findings indicate that considering the in-plane commensurability alone does not allow to reveal laws of friction in the layered systems. Comparing the results presented in Figure 1c,d, one can see a pronounced correlation between the friction forces and the root-mean-square (RMS) height of the out-of-plane corrugation, ⟨hrms⟩, of the embedded film. The RMS height has been calculated as, ⟨hrms⟩ = ⟨[N−1∑i N= 1(zi − zcm)2]1/2⟩, where zcm = N−1∑i N= 1zi is the normal coordinate of the COM of the film. For the Cu/gr/Cu, Au/gr/Au, and Au/gr/Cu junctions, where the lattice mismatch δ ≠ 0, the friction force increases with increasing of the RMS height of corrugation; however, for the ideally commensurate junction (δ = 0) exhibiting the smallest ⟨hrms⟩ the friction force is very high. The presented results demonstrate that in order to determine the microscopic mechanism of friction in junctions including atomically thin films, one has to consider the interplay between incommensurability of contacting materials and out-of-plane deformations of the film. This is especially important for graphene, which is intrinsically nonflat and tends to be corrugated due to the instability of two-dimensional crystals.8 Figure 2a shows momentary distances of the chain particles from the slider, zgr−sld = Zsld − zi, calculated during sliding. The upper and lower curves have been calculated for the Au/gr/Au and Cu/gr/Cu junctions at ⟨FN⟩ = 300 nN, respectively, and the corresponding friction forces are marked by the circles in Figure 2b,c. Our simulations show a number of important features. First, for the incommensurate contact Au/gr/Au (δ = −0.147) the amplitude of the film corrugations is larger than that for Cu/gr/Cu, where the lattice mismatch, δ = −0.04, is very small (see Figure 1d). The reason is that the out-of-plane displacements of the film involve the lateral displacements of its particles, which for the commensurate contacts are energetically unfavorable. Second, the film embedded between Au surfaces exhibits, apart from its intrinsic periodicity, additional longrange modulations (moiré patterns20−22,27) resulted from the mismatch between the contacting lattices. The period of these roughly sinusoidal modulations, λ, can be calculated as21,22 λ = asub(1 + δ)/|δ| that for the Au/gr/Au junction gives λ = 1.67 nm. The latter value agrees well with the average period of the graphene out-of-plane corrugation shown in Figure 2a. At low temperature, we find small-amplitude periodic modulations also for the Cu/gr/Cu junctions, however at room temperature, which is the case studied here, the patterns become highly irregular. It should be noted that the morphology of supported

N

Utot =

∑ [Usub(xi , zi) + Usld(xi − Xsld , Zsld − zi)] i=1

+

1 2

N−1

∑ k int(ri ,i+ 1 − agr)2 + i=1

1 2

N−1

∑ kθ(θi − π )2 i=2

(5) th

th

where ri,i+1 is the distance between the i and (i + 1) particles, agr is the equilibrium distance between the particles in the chain, and θi is the angle between the neighboring C−C bonds. In the simulations, the lattice spacing in the chain is chosen as agr = 2.46 or 4.26 Å that correspond to the lattice constants for the graphene along zigzag and armchair directions, respectively. The time-averaged friction force has been calculated as ⟨FL⟩ = ⟨Kdr(Vdrt − Xsld)⟩. The presented results have been obtained for the pulling velocity Vdr = 10 m/s, which is sufficiently low for the system to exhibit stick−slip behavior. We found that for 0.1 m/s < Vdr < 10 m/s the friction force changes logarithmically with the velocity. All the simulations have been performed at room temperature (300 K). We used velocity−Verlet algorithm to solve the Langevin eqs 3 and 4. We employed open boundary conditions for the graphene sheet. The results of simulations presented below have been −1 sub sld sld obtained for33 ηsub x0 = ηz0 = ηx0 = ηz0 = 4 ps , and for the critical value of the damping coefficient of the slider in zdirection, Γ = 2(KzM)1/2. The width of the sheet is Ly = 1 nm, and the number of particles in the chain is N = 100. We have verified that with further increase of N the friction force grows proportionally to N. The stiffnesses of the external springs are chosen as Kdr = 5 N/m and KZ = 20 N/m that are close to the values used in the experiments.3 We present in Figure 1c the friction force as a function of normal load calculated for tribological junctions including the graphene embedded between two Cu(111) surfaces (denoted as Cu/gr/Cu), two Au(111) surfaces (Au/gr/Au), and Au(111)/Cu(111) surfaces (Au/gr/Cu). The contacting surfaces are aligned along their FCC(111)-(1 × 1)R0° direction that is chosen as a sliding direction, x. The lattice spacing for Cu(111) and Au(111) in this direction are 2.56 and 2.88 Å, respectively. In the simulations presented in Figure 1c, the zigzag direction of graphene lays along the axis x and the lattice spacing in the chain, agr = 2.46 Å. For comparison, the dashed line shows a friction force calculated for the confining surfaces, which are commensurate with the graphene, and have the same interaction parameters as Cu surfaces. The lattice mismatches, δ = (agr − asub(sld))/asub(sld), between the considered confining surfaces and the graphene are 0.0 for the commensurate junction, −0.04 for Cu and −0.147 for Au. It should be noted that in symmetric junctions, Cu/gr/Cu and Au/gr/Au, the graphene sheet slides with respect to both C

DOI: 10.1021/acs.nanolett.5b05004 Nano Lett. XXXX, XXX, XXX−XXX

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The results obtained allow to conclude that (1) the out-ofplane deformations of graphene give the major contribution to the friction in the nearly incommensurate junctions (e.g., Au/ gr/Au), whereas they do not contribute to ⟨FL⟩ in the commensurate junctions (Cu/gr/Cu); (2) a superlow friction can be achieved in the incommensurate junctions if one finds a way to suppress the out-of-plane deformations of the film. To confirm these conclusions, we carried out simulations for the system mimicking the Cu/gr/Cu contact, where the armchair direction of graphene lays along the x-axis coinciding with the FCC(111)-(1 × 1)R0° direction of the confining surfaces. In this case, the period of the graphene sheet in the xdirection is agr = 4.26 Å, and we are dealing with the incommensurate Cu/gr/Cu junction with the lattice mismatch δ = 0.66. This new junction can be obtained from the one considered in Figure 2c by rotating the graphene sheet by 90° with respect to the Cu plates. As shown in Figure 2d, the frictional properties of the incommensurate Cu/gr/Cu junction are similar to those outlined in Figure 2b for Au/gr/Au. In this case, the out-of-plane displacements of the film play the major role and the friction force strongly depends on the bending rigidity of the film. Another interesting result emerging from the simulations presented in Figures 1c and 2b−d is a remarkable difference in the dependence of the friction force on the normal load for the nearly commensurate and incommensurate junctions. The incommensurate Au/gr/Au and Cu/gr-armchair/Cu junctions exhibit a linear dependence of ⟨FL⟩ on ⟨FN⟩ (the Amontons’ law), whereas the almost commensurate Cu/gr-zigzag/Cu junction shows a superlinear dependence of ⟨FL⟩ on ⟨FN⟩. This effect can be understood by calculating the load dependence of the potential energy landscape for the sliding of a single graphene atom confined between two plates. For the commensurate Cu/gr/Cu junction, the heights of the slidingenergy barriers increase superlinearly with the normal force, whereas for the incommensurate contacts the barriers heights grow approximately linearly with ⟨FN⟩. To further study the effects of interplay between the incommensurability of contacting surfaces and out-of-plane deformations of the atomically thin film on friction, we consider a model system in which the lattice mismatch δ = (af − asub)/ asub varies changing the lattice spacing in the film, af. The other parameters are kept the same as for Cu/gr/Cu junction. Experimentally, such study can be carried out by rotating the graphene sheet with respect to the confining plates. Figure 3 shows a commensurate−incommensurate transition20,21 in the mechanical characteristics of the junction, which occurs with the variation of the mismatch, δ. The simulations have been carried out for a fixed value of normal load, ⟨FN⟩ = 300 nN, and for the two values of the bending rigidity, D = 1.6 and 160 eV. Figure 3a exhibits two distinct regimes in the frictional response: (i) For very small values of the mismatch, |δ| < 0.05, for which the film and the substrates lattices are nearly commensurate, the friction force decreases rapidly with increasing |δ|, and it is almost independent of the bending rigidity. (ii) For larger lattice mismatches, |δ| > 0.05, the bending rigidity greatly affects the friction force, which can be reduced by more than an order of magnitude with an increase in D (see inset to Figure 3a). The variation of ⟨FL⟩ with δ correlates well with the dependencies of the RMS height of the film corrugation, ⟨hrms⟩, and the average distance between the film and the slider, ⟨Zsld − zcm f ⟩, on δ that are shown in Figure 3b,c. In the region |δ| < 0.05

Figure 2. Simulation results for graphene embedded between different metal surfaces. (a) The momentary distances of the chain particles from the slider zgr−sld as a function of the particle index in the chain. Upper and lower curves in (a) correspond to Au/gr/Au and Cu/gr/ Cu junctions, respectively. The corresponding values of friction forces are marked by circles in panels b and c. (b,c) The friction force ⟨FL⟩ as a function of normal force ⟨FN⟩ for Au/gr/Au and Cu/gr/Cu junctions. The zigzag direction of graphene lays along the x-axes, and agr = 2.46 Å. (d) The friction force for the graphene sheet embedded between two Cu(111) surfaces. Here the armchair direction of graphene is along the x-axes, and agr = 4.26 Å. The insets to panels b− d show the time averaged values of RMS heights of film corrugation in the sliding state. The black and red curves are calculated for D = 1.6 eV that is the bending rigidity of the graphene film and for the much larger value of D = 160 eV, respectively. The lattice mismatches for the systems in panels b−d are −0.147, −0.04 and 0.66, respectively.

graphene films given by the reduced model considered here agrees well with the predictions of 3D simulations27 (see also SI for details). To identify the effect of out-of-plane corrugation on friction, we present in Figure 2b−d the results of simulations performed for two values of bending rigidity, D = 1.6 eV, that corresponds to the graphene (black curves), and for an artificially high value of D = 160 eV for which the film corrugations are significantly reduced (red curves). All other parameters are kept the same. The insets to Figure 2b−d show the time-averaged RMS height of the film corrugation, ⟨hrms⟩, as a function of the normal load that is calculated for both values of D during sliding. For a freestanding graphene, ⟨hrms⟩ is inverse proportional8 to D−1/2. For the graphene sheet embedded between two metal surfaces, we find that ⟨hrms⟩ depends not only on D but also on the normal load and the mismatch between the film and surface lattices. As discussed above, for D = 1.6 eV, the RMS height is larger for the incommensurate junction Au/gr/Au (δ = −0.147) than that for Cu/gr/Cu, where the lattice mismatch, δ = −0.04, is very small. Figure 2b,c demonstrates that the friction force in the Au/gr/ Au junction strongly decreases with increase of bending rigidity that reduces the out-of-plane corrugation of the film, whereas for the Cu/gr/Cu and for the ideally commensurate junctions the friction force is insensitive to the variation of D. It is important to note that the simulations performed for the highbending rigidity predict significantly lower friction force for the incommensurate Au/gr/Au junction than for the almost commensurate junction, Cu/gr/Cu, although for D = 1.6 eV the situation is the opposite. D

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⟨Ustretch⟩ is varying rapidly with |δ| for |δ| < 0.05. Above the critical value, |δ| ∼ 0.05, the decrease in the film−substrates interaction energy can no longer compensate for the increase of the stretching energy, and the junction forms an incommensurate state characterized by a low, almost uniformly distributed strain and relatively high out-of-plane deformations of the film (see Figure S8 in SI). Hence, for |δ| > 0.05 ⟨Ustretch⟩ saturates with increasing |δ|. Similar variations of the strain profiles at the commensurate−incommensurate transition have been observed experimentally and in simulations for the graphene films deposited on h-BN substrate.20,21 Our simulations predict that the junctions, which parameters correspond to a boundary between the commensurate and incommensurate regimes (δ ≈ 0.05 in Figure 3), should exhibit a superlow friction. In such junctions the lattice mismatch is already large enough to provide a structural incompatibility of contacting surfaces, whereas the out-of-plane corrugation of the film is relatively small. Comparing the Cu/gr/Cu and Au/gr/ Au junctions, one can see that the first one with δ = −0.04 satisfies these conditions, whereas the second one with δ = −0.147 is far away from the boundary region, and therefore the friction force for Cu/gr/Cu is smaller than for Au/gr/Au. In the low-load regime, ⟨FN⟩ < 100 nN, a fine interplay between the effect of ⟨FN⟩ on the out-of-plane deformations of the film and interfacial commensurability leads to an unusual load dependence of friction force in the Cu/gr/Cu junctions, showing a plateau and even a small negative slope (negative friction coefficient15) as a function of ⟨FN⟩ (see Figure 2c). However, it should be noted that such behavior is sensitive to the interaction potentials and dimensionality of the model. With the increase of the normal load, the boundary between the two regimes is shifted to larger values of the mismatch and, as shown in SI, for ⟨FN⟩ = 3000 nN the boundary is situated at |δ| ∼ 0.1. In this case, the frictional properties of the Cu/gr/Cu junction are mainly determined by the commensurability of the contacting lattices, and the friction force for Cu/gr/Cu is significantly higher than for Au/gr/Au. As shown above, the suppression of the film corrugation reduces significantly the friction force in the incommensurate junctions. In the simulations presented in Figures 2 and 3, this was demonstrated by increasing the bending rigidity of the film that is, of course, impractical. However, this effect can be achieved in heterogeneous junctions where the film is embedded between two different metals. For this purpose, one of the confining surfaces should be (close to) commensurate with the graphene film, thus suppressing the film corrugation, whereas the second surface should be incommensurate to the graphene providing frictionless motion across the interface between these materials. Figure 4 shows that such superlow friction junction can be produced using, for instance, the graphene film embedded between Cu(111) and Au(111) surfaces. The zigzag direction of the film should be oriented along the FCC(111)-(1 × 1)R0° direction of the confining surfaces. The Cu/gr/Au junction provides a small RMS height of corrugation and superlow friction coefficient of ∼3.6 × 10−3. In contrast, the graphene film embedded between Au(111) and Ag(111) surfaces, which are both incommensurate with the graphene, exhibits higher film corrugation and considerably larger friction coefficient, ∼1.4 × 10−2. It should be noted that the strengths of interaction between the graphene and Au, Ag, and Cu differ from one another by no more than 12%. The results of 3D simulations presented in SI support our

Figure 3. Effects of lattice mismatch on friction for graphene between two identical (111) surfaces. The black and red curves correspond to D = 1.6 eV and D = 160 eV, respectively. (a) The friction force as a function of the lattice mismatch, δ. Inset shows a magnification of the region that corresponds to incommensurate junctions and marked by the blue rectangular. (b) The average value of RMS height (⟨hrms⟩) of graphene corrugation as a function of δ. (c) The time averaged distance between the film center of mass (⟨zcm f ⟩) and the slider (⟨Zsld⟩) as a function of δ. (d) The average value of stretching energy (⟨Ustretch⟩) of graphene as a function of δ. The curves are calculated during sliding. In the simulations, the parameters of interactions between the graphene and metals are the same as for graphene between Cu surfaces.

the film corrugation sharply grows with increasing δ, and for |δ| > 0.05 the RMS height becomes highly sensitive to the value of the bending rigidity with ⟨hrms⟩ being much larger for D = 1.6 eV than for D = 160 eV. As shown in SI, this behavior becomes even more pronounced at low temperatures. For |δ| < 0.05, the low film corrugation results in a small distance between the film and plates (Figure 3c), which leads to high barriers in the sliding energy landscape, and correspondingly to large friction. With the increase in ⟨hrms⟩ that occurs at |δ| ∼ 0.05, the distance ⟨Zsld − zcm f ⟩ also increases. The variation of ⟨hrms⟩ and ⟨Zsld − zcm f ⟩ with δ presented in Figure 3b,c are similar to those found in 3D simulations,27 and in SI we present an analytical model explaining these behaviors. Summing up, we found that for |δ| > 0.05 the out-of-plane corrugations of the film give the major contribution to the friction force, while they do not affect the friction for |δ| < 0.05. This conclusion is in line with the results presented in Figures 1c and 2 for the Au/gr/Au and Cu/ gr/Cu junctions. In order to characterize the effect of the lattice mismatch on the state of the film during the sliding, it is instructive to look at the time averaged stretching energy of the film, 1 N−1 Ustretch = 2 k int ∑i = 1 [ri , i + 1(t ) − a f ]2 , where ri,i+1 is the

distance between the ith and (i + 1)th particles in the film. As shown in Figure 3d, we again find two distinct regimes in the variation of ⟨Ustretch⟩ with δ. For |δ| < 0.05, it is energetically favorable to adjust the film lattice to the lattices of the plates to become commensurate, which leads to an increase in the stretching energy of the graphene but reduces the film− substrates interaction energy. The film stretches locally to achieve the energetically favorable state, which results in relatively large areas of commensurate stacking and deformations concentrated in narrow strained regions. Accordingly, E

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conclusion that the superlow friction can be achieved in the Cu/gr/Au heterojunction. In summary, we suggest that the frictional properties of junctions including atomically thin sheets embedded between metal surfaces are determined by the interplay between the lattice mismatches of contacting surfaces and out-of-plane displacements of the sheet. The simulations reveal a direct connection between the height of the sheet corrugations and the friction forces. The results presented here provide insight into how and why the frictional characteristics of the heterojunctions are affected by the commensurate−incommensurate transition. We establish conditions needed to achieve a superlow friction in the heterojunctions.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.5b05004. Results of simulations for 3D model, the calibration of analytical potentials, the equilibrium configuration of graphene and the distribution of strain on surfaces with different lattice mismatches, the results of simulations for high normal load, and a comparison between theoretical model and simulations. (PDF)

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Figure 4. Friction properties of heterogeneous junctions. (a) The friction force as a function of normal force for graphene embedded between (111) surfaces of Au and Cu and Au and Ag. (b) The RMS height of graphene corrugation (⟨hrms⟩) as a function of normal force for graphene embedded between Au and Cu and Au and Ag surfaces. The results for graphene embedded between Au(111) and Cu(111) surfaces are shown by black squares, and for graphene between Au(111) and Ag(111) surfaces are shown by red circles. The zigzag direction of graphene is along the x-axis (agr = 2.46 Å).

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Letter

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS W.O. acknowledges the financial support of the National Key Basic Research Program of China (Grant 2013CB934201) and a fellowship program for short-term study abroad for doctoral students from Tsinghua University. M.M. acknowledges the financial support from a fellowship program for outstanding postdoctoral researchers from China and India in Israeli Universities. M.U. acknowledges the financial support of the Israel Science Foundation, Grant 1316/13 and of COST Action MP1303. Q.Z. and M.U. acknowledge the support of XIN center, Tel Aviv−Tsinghua Universities. F

DOI: 10.1021/acs.nanolett.5b05004 Nano Lett. XXXX, XXX, XXX−XXX