Anisotropy of Atomic-Scale Peeling of Graphene

Anisotropy of atomic-scale peeling of the monolayer graphene sheet adsorbed onto the graphite substrate surface is numerically studied by molecular mechanics simulation. During the peeling process the surface contact area of the graphene sheet takes atomic-scale sliding behavior, which strongly depends on the initial contact orientation angle θin between the graphene sheet and the graphite surface within the lateral plane. When the initial contact is commensurate AB stacking orientation (θin = 0 ◦), the mean lateral force during the peeling process takes a maximum peak value. However, as the initial orientation angle θin increases (0 ◦ < θin ≤ 30◦), the effect of the incommensurate contact is further enhanced, and the mean lateral force decreases toward near zero value. At an intermediate incommensurate angle (θin = 9 ◦), the peeled area of the graphene sheet twists around the perpendicular axis during the peeling process since the surface contact area discretely slips toward metastable AB stacking orientation. The above anisotropic sliding mechanics of the graphene sheet appeared during the peeling can be applied to the mechanical control of the material properties of the π-conjugated sheet as a novel device. [DOI: 10.1380/ejssnt.2016.204]


I. INTRODUCTION
The graphene is a highly promising material which has attracted great interests of many researchers in its application to electronic, magnetic, spintronic and optical devices [1][2][3].Among various material properties the mechanics of the graphene is essential because elastic deformation changes atomic arrangements with lattice periodicity, which leads to control of all the material properties of graphene.In this paper, as one of examples of graphene mechanics, the sliding and deformation of the graphene sheet during the peeling process is discussed.
In our previous works, both numerical and experimental studies of the atomic-scale peeling and adhesion mechanics of the graphene adsorbed onto the graphite surface were performed.Numerically peeling of the monolayer graphene sheet was studied based on the molecular mechanics simulation [4][5][6].Experimentally peeling of the monolayer graphene sheet and the multilayer graphene plate with a thickness of several µm was studied by using atomic-force microscopy [7].Our simulation succeeded in the interpretation of the following experimental features: 1) The transition from the surface to the line contact of the graphene sheet adsorbed onto the graphite substrate surface occurs as the peeling proceeds.2) Atomic-scale stick-slip sliding of the graphene sheet occurs during the surface contact of the peeling process.
We have so far discussed mainly the peeling process along the specific crystal orientation.On the other hand, atomic-force microscopy measurement showed the marked anisotropy of the atomic-scale friction of the graphene sheets adsorbed onto the graphite substrate surface, where the incommensurate contact between the graphene sheets and the graphite surface induces ultralow frictional state due to superlubricity [8][9][10][11].The anisotropy derived from the superlubricity [12] can be expected to give influences on the peeling process of graphene.Therefore effect of the initial lattice orientation of the graphene sheet adsorbed onto the graphite substrate surface on the atomicscale peeling is studied using the molecular mechanics simulation.The close relation among the anisotropy of the mean lateral force, the lateral force curve, and the sliding mechanics is shown.

II. MODEL AND METHOD OF SIMULATION
As shown in Fig. 1(a), we used a simulation model of the monolayer graphene sheet physically adsorbed onto the rigid graphite substrate surface.The rigid graphite surface is modeled by a hexagonal graphene sheet with each side of 66 Å comprised of 4056 atoms.Then a monolayer rectangular graphene sheet with each side of 38 Å×20 Å comprised of 298 carbon atoms is used as illustrated in Fig. 1(b).
First initial stable structures of both the hexagonal and rectangular graphene sheets are separately calculated by energy minimization using the conjugate gradient (CG) method [13].Here the covalent bonding energy V cov described by Tersoff potential energy function [14] is minimized.The convergence criterion is set so that the maximum of absolute value of all the forces acting on the movable atoms becomes lower than 1.6×10 −5 nN.
Next, as shown in Fig. 1(b), a monolayer graphene sheet is put onto the rigid graphite surface with an initial orientation angle, θ in .Here it is noted that θ in = 0 • is set so that the AB stacking registry between the graphene sheet and the graphite surface is satisfied.The rotation center of the graphene sheet is set as a carbon atom located above the hollow site of the graphite surface.Furthermore the region of 0 The peeling process is now simulated as illustrated in Fig. 1(a).The outermost array of atoms of the left edge of the graphene sheet [Fig.1(b)] is gradually moved upward along the z direction, parallel to the [0001] axis of the graphite surface, by 0.1 Å.For each peeling height z, V total = V cov +V vdW is minimized using the CG method, where V vdW is nonbonding van der Waals interaction energy described by modified Lennard-Jones potential energy function [15,16].Here, if z 0 is denoted as an initial stable vertical height of the left edge of the graphene sheet, ∆z = z−z 0 can be regarded a peeling distance.Thus the metastable deformed shape of the graphene sheet and the forces acting on the lifting edge are obtained as a function of ∆z.Here we discuss the lateral force F l just opposite to the sliding direction along the initial orientation angle θ in .In the present simulation, only the region 0 ≤ ∆z ≤ ∆z s is discussed, where the surface contact between the graphene sheet and the graphite substrate is conserved.Here the maximum peeling height ∆z s for the surface contact is defined so as to satisfy the criterion that only the outermost array of the free right edge of the graphene sheet [Fig.1(b)] receives the repulsive interaction force per atom from the graphite surface.In the present simulation for 0 • ≤ θ in ≤ 30 • , ∆z s is obtained to range the region of 29.8 Å ≤ ∆z s ≤ 30.2 Å.
Figures 3(a)-3(c) show the lateral forces F l , trajectories of the two carbon atoms of the free graphene edge within the lateral plane, and orientation angles of the free edge θ f , respectively.Here it is noted that θ f = θ in is satisfied before the peeling.It is clarified that the peak to peak amplitudes of the lateral force curves rapidly decrease as The transition of the shape of the lateral force curves mentioned above [Figs.3(a-1)-3(a-5)] can be explained by that of the atomic-scale sliding motion of the graphene sheet [Figs.3(b-1)-3(b-5)].For θ in = 0 • , surface contact area of the graphene sheet takes nearly one-dimensional straight stick-slip motion along the commensurate stacking direction [Fig.3(b-1)], which appears as repeated sawtooth behavior in the lateral force curve [Fig.3(a-1)].Therefore orientation angle of the free edge is nearly con-stant, θ f ≃ θ in = 0 • during all the surface contact region [Fig.3(c-1)].However, for θ in = 1 • , since the commensurate stacking contact slightly breaks, the graphene sheet takes two-dimensional zigzag stick-slip motion [Fig.3(b-2)], which appears as step-like discrete change of θ f , between θ in = 1 • and minus several degrees [Fig.3(c-2)].
For θ in = 9 • , due to the incommensurate stacking contact between the graphene sheet and graphite substrate, the lateral force F l markedly decreases, and the graphene sheet takes nearly one-dimensional continuous motion for the first half of the surface contact region A as shown in Fig. 3  However, for the last half of the surface contact region B, as the peeled area of the graphene sheet increases and the surface contact area decreases, surface contact area discretely slips toward the stable AB stacking orientation (θ f ≃ 0 • ) several times [Fig.3(b-3)B].Thus repeated discrete change of θ f toward nearly 0 • occurs [Fig.3(c-3)B].As a result the peeled area twists around the perpendicular axis.For θ in = 20 • and 30 • , since the incommensurate stacking feature is further enhanced, the graphene sheet takes nearly one-dimensional continuous motion [Figs.3(b-4) and 3(b-5)] and therefore θ f ≃ θ in holds [Figs.3(c-4) and 3(c-5)] for all the surface contact regions.

IV. DISCUSSIONS AND CONCLUSIONS
In this paper we simulated anisotropy of the sliding mechanics of the monolayer graphene sheet adsorbed onto the graphite surface during its peeling process.It is found that the mean lateral force ⟨F l ⟩ takes a maximum peak value within a narrow region around the commensurate stacking orientation angle θ in = 0 • .For other incommensurate region, 0 • < θ in ≤ 30 • , ⟨F l ⟩ decreases toward 0 pN.The above behavior of ⟨F l ⟩ shows the transition of the sliding mechanics of the graphene sheet from the commensurate-to the incommensurate-sliding as follows: For the commensurate direction (θ in = 0 • ), the regular stick-slip sliding occurs.However, as the effect of the incommensurate stacking is enhanced, that's to say, θ in increases, transition from the stick-slip sliding to the continuous sliding occurs.At an intermediate incommensurate direction (θ in = 9 • ), the surface contact area discretely slips toward the metastable AB stacking orientation, which induces the twist of the peeled area of the graphene sheet.
Here we mention the anisotropy, ⟨F l ⟩−θ in curve.Our preliminary simulation shows that the graphene sheet with a larger width becomes harder and cannot bend easily within a lateral plane.Therefore, unlike the case of θ in = 9 • (Figs.3(b-3) and 3(c-3)), it seems more difficult for the wider graphene sheet to take a stick-slip motion to bend toward θ f = 0 • .As a result intermediate region for 1 • ≤ θ in ≤ 10 • in the ⟨F l ⟩−θ in curve tends to vanish.Thus simulated anisotropy obtained in this work strongly depends on the width or the ratio of the width and length of the graphene sheet, which will be discussed in our future work.
Then the atomic-scale behavior depending on the peeling direction is mentioned.As shown in our previous work [5], the surface contact area of the graphene sheet with a sufficient large width, continuously slides at the beginning of the peeling process, and then it takes stick-slip sliding motion.Thus the small jump of θ f seems to appear at the transient region from the continuous sliding to the stick-slip sliding.Since the surface contact area can move only continuously at the beginning of the peeling process, it slides only between the nearest neighboring AB stacking sites around the orientation of θ f ≃ θ in .This is a reason that, for θ in = 1 • , θ f continuously changes for the positive direction, θ f ≥ θ in = 1 • , at 6.0 Å ≤ ∆z ≤ 8.3 Å as shown in Fig. 3(c-2).However, after the stick-slip sliding starts, the surface contact area can rotate discretely to-ward θ f = 0 • , nearly AB stacking orientation of the whole surface contact area, which results in the negative jump of θ f .
On the other hand, for θ in = 0 • , since graphene/graphene interface has initially a structural symmetry with respect to the orientation of θ f = θ in (= 0 • ), events of the sliding of the surface contact area toward θ f > 0 • and θ f < 0 • are stochastically equivalent to each other.Therefore the positive jump of θ f at 7.4 Å ≤ ∆z ≤ 8.2 Å as shown in Fig. 3(c-1) is due to the numerical reason such as the convergence criterion of the structural optimization.Similarly for θ in = 30 • , there initially exists structural symmetry with respect to the orientation of θ f = θ in (= 30 • ).Therefore the positive jump of θ f at 27.3 Å ≤ ∆z ≤ 27.8 Å as shown in Fig. 3(c-5) is also due to the numerical reason.
Furthermore the relation between our simulation and Amonton-Coulomb's law is discussed.In the present work based on molecular mechanics simulation, atomic-scale dynamic friction under the quasi-static limit, v → 0, is discussed, where the mechanical stable equilibrium state is achieved for each peeling position.Frictional-force maps observed by frictional-force microscopy with the scanning velocity of v ≤ 100 nm/s can be reproduced by our previous molecular mechanics simulation [17][18][19].Therefore our simulation holds for this quasi-static limit.Here, if energy dissipation due to the stick-slip sliding of the graphene sheet is assumed to be proportional to (dynamic friction force)×(sliding velocity v), our simulation satisfies Amonton-Coulomb's law saying, 'dynamic friction force does not depend on the sliding velocity.'Therefore, in the present simulation on the peeling of the graphene sheet from the graphite substrate, it can be said that Amonton-Coulomb's law holds for the quasi-static limit, v ≤ 100 nm/s, for which molecular mechanics method also holds.
In our preliminary experiment, the similar anisotropy was obtained by atomic-force microscopy measurement.However, in order to compare directly the present simulated results with observed data, effect of the cantilever has to be included in our simulation model, which is our future work.Kawai et al. [11] has recently measured atomic-scale friction of the graphene ribbon by controlling the distance between the atomic-force microscopy tip and the substrate surface.Based on such an AFM technique, the scale-up of the atomic-scale superlubricity will become possible.If the well-defined large flat area on the order of more than micrometer is prepared, anisotropy of the sliding friction during the peeling process studied in the present work can be used to control the macroscopic sliding friction.If such mechanics is combined with its characteristic electronics, magnetics, spintronics and optics of the graphene sheet, novel device of π -conjugated sheet will be developed.

FIG. 1 .
FIG. 1.(a) The schematic illustration of the simulation model of the rectangular monolayer graphene sheet peeled from the rigid hexagonal graphite substrate surface.The outermost array of the carbon atoms on the left edge of the monolayer graphene sheet is gradually lifted along the z or [0001] direction by 0.1 Å.(b) For each initial orientation angle, θin, the graphene sheet is structurally optimized by the CG method to obtain the initial adsorbed structure.

FIG. 2 .
FIG.2.The mean lateral force ⟨F l ⟩ plotted as a function of the initial orientation angle of the monolayer graphene sheet θin while the graphene sheet takes a surface contact with the graphite surface for the peeling distance of 0 ≤ ∆z ≤ ∆zs.Anisotropy of ⟨F l ⟩ clearly appears as a function of θin.(1)-(5) correspond to those in Fig.3, respectively.

4 FIG. 3 .
FIG. 3. Nanomechanical behaviors of the graphene sheet during the surface contact region of the peeling process of 0 ≤ ∆z ≤ ∆zs for θin = (1) 0 • , (2) 1 • , (3) 9 • , (4) 20 • , and (5) 30 • .(a) The lateral force F l plotted as a function of the peeling distance of the lifting edge, ∆z.(b) The trajectory of the two carbon atoms on the free right edge of the graphene sheet.(c) The orientation angle of the free edge, θ f , plotted as a function of the peeling distance ∆z.The boundary between A and B for (3) θin = 9 • is 14.5 Å.