PAPER

www.rsc.org/faraday_d | Faraday Discussions

Mechanism of ultra low friction of multilayer
graphene studied by coarse-grained molecularsimulation
Hitoshi Washizu,*a Seiji Kajita,a Mamoru Tohyama,aToshihide Ohmori,a Noriaki Nishino,b Hiroshi Teranishib
and Atsushi Suzukib
Received 25th November 2011, Accepted 9th December 2011
DOI: 10.1039/c2fd00119e

Coarse-grained Metropolis Monte Carlo Brownian Dynamics simulations areused to clarify the ultralow friction mechanism of a transfer film of multilayered
graphene sheets. Each circular graphene sheet consists of 400 to 1,000,000 atoms
confined between the upper and lower sliders and are allowed to move in 3
translational and 1 rotational directions due to thermal motion at 300 K. The
sheet–sheet interaction energy is calculated by the sum of the pair potential of thesp2 carbons. The sliding simulations are done by moving the upper slider at
a constant velocity. In the monolayer case, the friction force shows a stick-slip
like curve and the average of the force is high. In the multilayer case, the friction
force does not show any oscillation and the average of the force is very low. Thisis because the entire transfer film has an internal degree of freedom in themultilayer case and the lowest sheet of the layer is able to follow the equipotential
surface of the lower slider.

1 Introduction
Lamellar materials, such as graphite, often show low friction in air. Graphite andmolybdenum disulfide are typical low friction materials, while mica is not.1 Thesematerials are used as solid lubricants in an artificial satellite in which liquid lubricants, such as hydrocarbons, are not used because of the high vacuum environment.
Some additives are mixed into automotive lubricants in order to generate a solid
lubricant film in the sliding contact area due to the heat and shear which often
undergo tribochemical reactions.2
The mechanism of low friction in such lamellar materials, whether the interlayer
actions are due the van der Waals interactions or Coulomb ion pair interactions, iscommonly explained by the slip generated between the layers. When we put the pileof papers on the desk and slide the uppermost paper, it appears that the inner paper
slipping generates the low friction. However, this macroscopic mechanism is notconfirmed to occur in the nanoscopic range.
Assume that a steel ball with a radius of several millimetres is sliding on an infinitely long graphene3 sheet on the graphite. In this case, the graphene cannot slideas the pulling force will be infinitely high when the pulling force is finite per unit
length. The effect of bending should also be considered in the realistic case as theshearing between the layers is hard to put into practice for geometrical reasons.

aToyota Central R&D Labs., Inc., Nagakute, Aichi 480-1192, Japan. E-mail: h@washizu.org;
Fax: +81 561 63 6920; Tel: +81 561 71 7531b
Toyota Motor Corporation, 1 Toyota-cho, Toyota, Aichi 471-8572, Japan

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To consider the realistic phenomena, the low friction should be due to the sliding
motion between the small pieces of graphene layers which are transferred to the steel
ball and the lower substrate graphite. The surface has a roughness of a micron orderand the transfer film is assumed to be physically adsorbed. Some transfer films arevisualized by a Transmission Electron Microscope.4 Recently, the molecular
dynamics simulations concerning interactions between a graphite surface and therigid pyramidal nanoasperity have shown the cleavage and flake formation ofgraphite.5
The question is then the mechanism of friction between the transfer film and thegraphite surface. Does the friction occur inside the transfer film similar to the macroscopic friction of the pile of papers? If not, what is the difference in the friction mechanism between general solid materials and lamellar materials? In general crystal
materials, sliding between the same materials produces a high friction.1 The caseof low friction is attributed to the incommensurate systems in which the rotation(yaw) angles of two surfaces are twisted.6,7 The physically adsorbed transfer film
may rotate in a commensurate condition for a long time due to the stability of thethermal equilibrium structure. Therefore, the macroscopically observed low frictionof lamellar materials is not totally understood from the viewpoint of an incommensurate friction mechanism.
From a molecular point of view, the most underlying problem when investigatingthese phenomena is the size of the transfer film. The transfer film consists of a huge
number of atoms4 which is difficult to treat by an atom based simulation, such as
molecular dynamics (MD) or molecular mechanics (MM). Based on this difficulty,
MD and MM simulations are either performed for a small flake of graphene,8,9 orfor an infinitely large sheet by adopting periodical boundary conditions.10 Acoarse-grained molecular simulation is needed to overcome this difficulty.
The transfer film consists of graphene sheets. One concept is to model each graphene sheet as a rigid body. The covalent bond of the carbon atoms is on the orderof 100 kBT and the van der Waals interlayer interaction is on the order of kBT,11
where T is the absolute temperature and kB is the Boltzmann constant. This canbe one foundation to support such a rough molecular model. Each graphene sheet
has several internal vibration modes. The fluctuation of the sp2 electron cloud
due to this internal vibration can be the source of the thermal fluctuation of thevan der Waals force which affects the thermal motion of neighboring graphenesheets. The rigid body has 3 translational and 3 rotational degrees of freedom.
The important degrees of freedom are the 3 translational and yaw rotational angles
which are the motion around the axis in the direction of the film thickness. Based onthese assumptions, the sliding dynamics of a transfer film can be described as the
motion of a set of rigid graphene sheets which move by thermal brownian motion
in 4 degrees of freedom.
In this study, coarse-grained Metropolis Monte Carlo Brownian dynamics(MCBD)12–14 simulations are employed to investigate the friction dynamics of thetransfer film of multilayered graphene sheets. MCBD was first introduced by Kikuchi et al.12 in order to calculate the dynamics of colloid suspensions. This method ismathematically the same as Langevin dynamics, but has advantages in calculation
stability and speed. MCBD is suitable for a system for which the effect of the thermal
fluctuation and the inner sheet interactions are comparable in order.

2 Simulation method
2.1 Monte Carlo Brownian dynamics
The MCBD is the numerical solver of the Fokker–Plank diffusion equation andphysically equal to the Langevin dynamics.
The Metropolis Monte Carlo (MC) method,15 when applied to part of the degrees
of freedom of a system, e.g., the center of graphene sheets, turns into a simulation
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procedure for the Brownian motion of the sheets. The Metropolis MC was originallydeveloped as a method for calculating statistical mechanical configuration integrals.
Since the MC sampling is a Markovian process, if we introduce a ‘‘time’’ scale twhich actually labels the order of subsequent configurations X, the ‘‘dynamic’’evolution of the probability distribution function is governed by the masterequation16ðnovPðX ; tÞ¼
W ðX jX 0 X 0 ÞPðX 0 ; tÞ  W ðX 0 jX ÞPðX ; tÞ dX 0(1)
vt
where W(X|X0 ) is the transition probability per unit ‘‘time’’ for a transition from
configuration X0 to X. The master equation may be approximately solved by an
expansion method in powers of a parameter U, the size of the system. Given that
W(X|X0 ) has the canonical form
W(X|X0 ) ¼ F0(x0 ;r) + U1F1(x0 ;r) + U2F2(x0 ;r) + .

(2)

with x0 ¼ X0 /U and r ¼ X  X0 , the jump moments are defined byð
a n;l ðxÞ ¼ rn Fl ðx; rÞdr(3)

When a1,0 ¼ 0, the U expansion yields the lowest approximation of a nonlinear Fokker–Planck equation:vPðX ; sÞv
1 v2¼  a1;1 ðxÞP þ
a2;0 ðxÞP
vs
vx
2 vx2

(4)

with s ¼ U2t. In the Metropolis MC, the reciprocal of the maximum displacement
allowed for an MC move during a ‘‘time’’ interval Dt may be taken as the parameter
U. The transition probability then has the canonical form with1
for jrj#1
2Dt¼ 0 for jrj.1

F0 ðx; rÞ ¼
and

and, supposing, for example, the derivative of the potential energy of the system
U(x) with respect to x is negative, i.e., U(x) < 0, with
F1 ðx; rÞ ¼ 
and1
0
U ðxÞr for
2kB Dt¼ 0 forr\  1and 1#jrj\0
r$0

for a sufficiently large U. From these one obtains
a1;0 ¼ 0; a2;0 ¼

1
1
0; a1;1 ¼ 
U ðxÞ
3Dt
2Nf kB TDt(5)

where Nf is the number of the degree of freedom, so that eqn (4) reduces to the diffusion equationvPðX ; tÞ
D v 0
v2
¼
U ðxÞP þ D 2 Pvx
vt
kB T vx(6)

where the diffusion constant D is defined by
2NfDDtU2 ¼ 1(7)Thus, by making the size of the maximum allowed displacement U1 in the Metropolis MC very small the forces acting on a particle remain essentially constant during
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the Monte Carlo ‘‘time’’ step, the method turns into a Brownian dynamics simulation.
The merit of MCBD compared to a molecular dynamics based simulation is thatthe calculation of the forces acting on each sheet is not needed and only the calculation of the potential energy U(x) is needed so that the calculation remains stable.
Compared to the general Monte Carlo method, MCBD provides the time evolutionof a physical quantity, where averaging of multiple trajectories is needed.
As described above, the reality of the MCBD simulation relies on the reality of theinteracting energy U(x) and the diffusion coefficients D. In this paper, we constructed U(x)s from a widely used pairwise potential parameter proposed by Girifalco et al.17 The potential is used in sp2 carbons, such as graphites, carbon
nanotubes, and C60 and evaluated in elastic and thermal properties. In order toobtain the diffusion coefficients, D, ether experiments and atom-based simulations
are available. The parameters can be more precisely obtained from ab initio quantum
calculations.
2.2 Potential between graphene sheets
In the structure of graphites under a thermal equilibrium stable AB stuck structure, and the average distance between
the distance between carbon atoms la is 1.42 A,
 The interaction between graphene sheets calculated and stored
sheets lz is 3.35 A.
during the simulation are taken from the literature.17 This potential reproduces
the stable AB stuck structure and the stress during separation of the two graphenesheets. Using this pairwise potential for the interactions between rigid sheets, theanisotropy of potential landscape due to the motion of sheets is obtained.
The potential between the sp2 carbon atoms Uatom–atom(ra) is described by thefollowing Lennard–Jones type function
A
B
Uatomatom ðra Þ ¼  6 þ 12
ra ra

(8)

where ra is the distance between carbon atoms, A and B are constants. This function
can be written using the parameter which describes the depth of the potential 3 andthe equilibrium distance s.
"     #612ss
Uatomatom ðra Þ ¼ 43 
(9)
þ
ra
ra
3, and ra0 are rewritten as follows:
3¼

ra 0 ¼ 2

1=6

A2
;
4B

 1=6
2B
s¼
:
A

(10)

(11)

6,
We used the pairwise inter atom potential on the graphene sheets as A ¼ 15.2 eV A
12, r0 ¼ 3.83 A,
 3 ¼ 2.89 meV.
B ¼ 24.1  103 eV A
We then calculated the potential between a carbon atom and a graphene sheet by
the following equation using eqn (8).
Uatomsheet(x,y) ¼

PUatomatom(ra)(12)

We set x as the sliding direction in Cartesian coordinates, y as the perpendicular
direction to the sliding direction, and the summation at the rhs as the sum of theinteractions between a set of carbon atoms in a honeycomb structure of unit
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This journal is ª The Royal Society of Chemistry 2012length la in the z ¼ 0 plane and a carbon atom at z ¼ Dz. During thesimulation,
we employed the coordinate system based on the vectors
pﬃﬃﬃ
a ¼ ðð 3=2Þla ; ð1=2Þla Þ; b ¼ ð0; la Þ considering the symmetry and calculated
Uatom–sheet before the simulation on the lattice point divided by 100 in the a,b
directions, and returned to the Cartesian coordinates during the motion of the sheets
in order to calculate the potential energy between the sheets.
Each Uatom–sheet on the lattice points are calculated until the difference in theenergy converges to DUatom–sheet < 106 eV from the neighbor to farther atoms.
Fig. 1 shows the Uatom–sheet(x,y) in the Cartesian coordinates.
As two carbon atoms, which are facing each other on the honeycomb structures,
can be geometrically distinguished by the periodicity, the potential between the
sheets Usheet–sheet is calculated using Uatom–sheet(x, y).
Usheet–sheet ¼ (Uatom–sheet(Dx,Dy) + Uatom–sheet(Dx,Dy  (1/2)la))$S(Dx,Dy)(13)

where Dx and Dy are the distances between center of two circle shaped sheets in the x
and y directions respectively, and S(Dx,Dy) is the number of the carbon atoms in an
overlapped area of two circles in a position Dx and Dy. By eqn (13), the end effectwhich is the effect of the finite size of the graphene sheets is able to be included in
the simulation. The potential surface Usheet–sheet of the system is plotted in Fig. 2.
Dx ¼ 0 and Dy ¼ 0 correspond to the AB stuck structure and the interaction between
the sheets decreases when two sheets are in separated positions.

Fig. 1 Potential between a carbon atom and a graphene sheet DUatom–sheet (x,y plane).

Fig. 2 Potential between two graphene sheets DUsheet–sheet (x, y plane).
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The interaction between a graphene sheet and an infinitely large graphite plate
Usheet–plate is described as follows:
Usheet–plate ¼ (Uatom–sheet(x,y) + Uatom–sheet(x,y  (1/2)la))$Nxy/2(14)where x,y are the coordinates of the sheet, and Nxy is the number of atoms in the
sheet. During this interaction, the inclined baseline shown in Fig. 2 does not exist
and the potential becomes a periodical function.
The effect of the incommensurate surface interaction is included by adopting
a smooth energy surface when two sheets are twisted at the yaw angle q. The effectof the yaw angle is included as follows. The potential between graphene sheets
Usheet–sheet is set to an average value Usheet–sheet ¼ Uave when the difference between
the sheet rotation angles becomes 0.1 degree. In Fig. 3, this simulation condition isplotted using the direct calculation due to the interactions of the finite number ofcarbon atoms. Although the direct calculation shows cyclic functions, this effect isneglected for simplicity.
For the z direction, which is the direction of load and the film thickness, eachpotential energy is already calculated Usheet–sheet(x,y) as a summation of each
atom pair potential Uatom–sheet for the distance between both sheets Dz. During
the simulations, Usheet–sheet(x,y) is stored as a table calculated before the MC simulation run starts. The values of Usheet–sheet(x,y) is linearly complemented from themesh value.

Fig. 3 Potential between two graphene sheets DUsheet–sheet as a function of the yaw angle q.

Fig. 4 Potential between two graphene sheets Usheet–sheet as a function of z.
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Fig. 4 shows the sheet potential Usheet–sheet as a function of Dz under a commensurate condition. The averaged U under the incommensurate condition is also plotted
as a curve. Each bar shows the range of the Usheet–sheet distribution in the (x, y) plane.
A small Dz corresponds to a high pressure under a heavy load which shows a large
range of energy in the commensurate condition. As Dz increases, which corresponds
to a lower pressure, the range of Usheet–sheet decreases and when over the equilibrium
width of 0.335 nm, Usheet–sheet does not significantly change.
2.3 Simulation conditions
The MCBD simulation conditions are described as follows. Each circular graphenesheets consist of 400 (Nxy ¼ 20  20) to 1 000 000 (Nxy ¼ 1 000  1 000) atoms allowed to move in 3 translational and 1 rotational directions due to thermal motion
at 300 K. The transfer layer consists of Nz graphene sheets piled in the z direction.
The initial conditions of the graphene sheets are set for the AB stuck structure at
the origin and confined by the upper and lower sliders. The lower slider is set to an
infinitely wide plane. The interaction between the lowest confined graphene sheet
and the lower slider is set to the same interaction without the end effect. The upper
slider is fixed to the same size as the confined graphene sheets. The interaction
between the top confined graphene sheet and the upper slider is set to the same interaction including the end effect. The effect of the load in the z direction is given by
setting the total width of the upper and the lower sliders to a fixed value lz 
(Nz + 1) where lz ¼ 0.330 nm is the average of the interlayer distance.
The sliding simulation is done by moving the upper slider in the x direction 1.0 
108 nm MC1 step. In the heaviest case Nxy ¼ 1 000  1 000, Nz ¼ 1, the number ofatoms is 1 000 000 which may take a huge amount of time for an all atom molecular
dynamics simulation. In this method, the total simulation time depends on Nz andnot on Nxy due to the calculation method described in the former subsection i.e.,
we first calculate the potential table Uatom–sheet in the infinite system, then calculate
Usheet–sheet by multiplying the number of atoms Nxy including the end effect. The
radius of the transfer film on Nxy ¼ 100  100, Nz ¼ 10 is equivalent to the flake
diameter of 18.2 nm and thickness of 3.7 nm which is almost as the same scale as
the transfer film observed on the AFM tip.4
1

The maximum allowed displacement is set to U1x ¼ Uy ¼ 0.01 A for the x,y direc1
tion, U1
¼
0.001
A
for
the
z
direction
and
U
¼
0.01
for
the
q direction. EachzqMonte Carlo time step in the MCBD method and physical time Dt is related to using
the diffusion coefficient D in eqn (4). Although the correlations between the directions of fluctuations are assumed to be independent in this study, the correlation
between the x,y directions should be considered in the future.

3 Results and discussions
The sliding simulations are done by moving the upper slider at a constant velocity.
Fig. 5 shows the typical time developments of the x coordinates of the graphenesheets. As the upper slider moves in the x direction, the graphene sheets follows
the upper slider. The lower the sheet number Nz, the larger the delay in the motionof the sheets and the curve shows a wavy pattern.
The trajectory during the sliding motion of the lowermost sheet Nz ¼ 10 is plottedvs. the potential surface between the lower slider and the 10th sheet in Fig. 6. It isclearly shown that even the upper slider moves against the potential hill, and thelowest sheet slides toward the valley of the potential surface. More precisely,
a stay motion at the bottom of the potential surface is also shown. Apart fromthe stay, the trajectory shows that the motion of the lowermost sheet is almost on
the plane energy surface.
Time development of the friction forces Fx ¼ DU/Dx is plotted in Fig. 7 for Nz ¼ 1
and Nz ¼ 10, where DU is the difference in the total potential energy of transfer film
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Fig. 5 Motion of each graphene sheet in x direction under sliding condition. The lowest curveis the 10th layer. Nxy ¼ 10,000, Nz ¼ 10, Degrees of freedom: x,y,z,q, lz ¼ 0.330 nm.

Fig. 6 Trajectory of the center of the 10th sheet of the transfer layer and the potential surface
between the lower slider and the 10th sheet.

and Dx is the motion in the x direction. In the Nz ¼ 1 system, the stick-slip like oscillation of the friction force is found. On the other hand, in the Nz ¼ 10 system, thefriction force fluctuates around zero and no oscillation is found. This is due to the
motion shown in Fig. 6 that the lowest sheet moves the plane, which is almost equal
to DU ¼ 0, and the friction force, which is calculated from this quantity, becomesvery small. We call this motion of multilayer graphene sheets which generates
a low friction as the thermal escaping motion.
In order to quantitatively discuss the low friction force due to the thermal
escaping motion, the averaged friction forces Fx in several Nxy and Nz are plottedin Fig. 8. When the sheets are large enough (Nxy $ 602), the large friction force
Fx is found when Nz ¼ 1 and decreases when Nz > 1 which is due to the thermal
escaping motion. The friction force Fx increases as the number of Nxy increases
when Nz ¼ 1. The friction force Fx slightly increases as Nz increases in Nz $ 3and does not show any sheet size Nxy dependence. This may be due to the friction
between the inner sheets; in other words, the weakening of the spring constant of
a whole transfer film. When the sheets are small (Nxy # 402), the friction forcesmonotonously increase with the increase in Nz and the two curves are different in
Nz $ 3 which suggests the lack of any thermal escaping motion.
Considering the mechanism of the thermal escaping motion, a conceptual illustration is shown in Fig. 9. The confined graphene sheet interacts with both the upper
slider and the lower slider shown in Fig. 9(a). When the upper slider moves right,
the interaction region of the upper side decreases whereas the lower side does notchange. The graphene sheet moves in the right direction in order to compensate
for the lost interacting region and the stability recovers (Fig. 9(b)). This is the end
effect which is the basic reason that the graphene sheet follows the upper slider.
In other words, an asymmetric situation due to the existence of the ends of the upper
and the lower sliders makes the basic motion. In realistic sheets, the potential energy
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Fig. 7 Time development of the friction force for a number of sheets Nz ¼ 1 (a), Nz ¼ 10 (b).

Fig. 8 Dependence of the size of the sheets (Nxy), the number of sheets (Nz) on friction force
Fx. The friction force is averaged by the MCBD simulation of 2  108 MC step at 16 calculations under independent initial conditions. The error bars are almost within the size of themarkers.

function drawn in Fig. 9(b) has the landscape precisely shown as Usheet–sheet in Fig. 3
which has local minima (Fig. 9(c)). In order to move out from the local minima,
thermal fluctuation is necessary. This is strongly related to the size Nxy and numberof sheets Nz dependence on the friction force Fx found in Fig. 8.
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Fig. 9 A conceptual illustration of the end effect.

The trajectories on the xy plane under the three different sheet numbers Nz ¼ 1, 3,
10 are plotted in Fig. 10. In the Nz ¼ 1 system, because of the restricted free lateralmotion, the trajectory is narrow in the y direction and the graphene sheet climbs thepotential hill. In the Nz ¼ 3, 10 systems, the widths of the fluctuation range in the y
direction are almost the same and thermal escape is found. Note that due to thestochastic motion of the sheets, the sheets randomly move both positively and negatively in the y direction when Nz ¼ 3, 10.
From an energetic point of view, the time evolution of the total energy Ut of thethree different sheet numbers Nz ¼ 1, 3, 10 are plotted in Fig. 11. In the Nz ¼ 3, 10
systems, the fluctuation of Ut is very small and almost used as a single quantity. In
the Nz ¼ 1 system, the Ut separates into two quantities and the fluctuation of eachquantity is high. This means that when Nz ¼ 1, the confined graphene sheet mustmove to two stable points which are made by the upper and lower sliders. The binaryenergy difference DUB is about 20 meV and the transfer motion between the twostable points generates a large hysteresis which causes a high energy dissipation.
The increase in the friction force Fx on Nxy when Nz ¼ 1 shown in Fig. 8 can be explained from this point of view, i.e., the increase in DUB causes an increase in thefriction.
The friction behavior of a small flake of a sheet (Nxy ¼ 202, 402) shown in Fig. 8 isdifferent compared to that of a larger sheet. This can be understood by the thermal
fluctuation. Fig. 12 shows the fluctuation of the total energy Ut for different sheet
sizes Nxy. In the Nxy ¼ 202 system, it shows a very large fluctuation with the range
DUt ¼ 0.5 meV. For Nxy ¼ 402, it also show a large fluctuation which is smaller thanthe Nxy ¼ 202 system whereas the larger system shows almost the same range of fluctuation. In the Nxy ¼ 202 system, it does not show any oscillation of the energy which

Fig. 10 The trajectories on xy plane under three different sheet numbers Nz (Nxy ¼ 1002). Each
sheet moves from the origin at time t ¼ 0 to the positive direction of the x axis.
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Fig. 11 The time evolution of the total energy Ut of the three different sheet numbers Nz(Nxy ¼ 1002).

Fig. 12 The time evolution of the total energy Ut of the five different sheet sizes Nxy (Nz ¼ 10).

is found in the others. This means that in the Nxy ¼ 202 system, the thermal fluctuation exceeds the depth of the periodical potential well in the potential landscape ofthe lowermost sheet and the lower slider. A sufficient depth of the potential well isneeded to stick the graphene sheets together. For this reason, the small sheets cannot
exhibit the thermal escape motion found in the larger sheet systems.
In contrast, when the sheet size is very large, the depth of the potential well in eachgraphite interaction exceeds the thermal fluctuation. The friction behavior of a verylarge system Nxy ¼ 1 0002 is plotted in Fig. 13 with the potential surface of the potential between the sheet and the lower slider. When compared to the trajectory for
Nxy ¼ 1002 shown in Fig. 10, the motion in the y direction is completely suppressed
and the sheet climbs the potential hill. In this case, the friction may be due to theenergy dissipation of the atom vibration. This is same as in the case for phonon dissipation in general solid materials.18,19
Using the coarse-grained Metropolis Monte Carlo Brownian Dynamics simulations, we found a low friction behavior in multilayer graphene sheets which we
called the thermal escape motion. The low friction of lamellar materials has been
explained by the weak interaction between the layers. However, even if the interaction is weak in a commensurate system, the atoms must overcome the potential hillwhich produces a large energy dissipation.18,19 Note that the van der Waals interaction energy is on the same order for the graphite sp2 carbon and the otherelements. The weakness of the interlayer bond in the depth direction should causea weak spring constant, then the energy dissipation becomes high. Therefore, wecannot directly understand the low friction of lamellar materials from this pointof view.
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Fig. 13 Trajectory of the very large single sheet (Nxy ¼ 1 0002, Nz ¼ 1).

The largest difference between general solid materials and lamellar materials isthat, in the transfer film of lamellar materials, each sheet has high translationaland rotational degrees of freedom. Due to the flexibility of the whole transfer filmby this degree of freedom, the sheet can escape the potential hills. This is the only
situation for the multilayer case already shown in our simulation. This motion
also occurs only when the potential well depth and the degree of thermal fluctuation
are comparable. If the thermal fluctuation is high in a small flake, the upper slider
cannot hold the film. If the potential well depth is high in the very large sheet, thesheets cannot escape the potential hill by thermal fluctuation. This thermal escapemotion has not been reported as far as the authors know by a study using an allatom simulation such as MD or MM. This can be explained by the restriction of
the MD or MM the precise simulation can treat a very small flake,8,9 or infinitely
large sheet by adopting periodic boundary conditions.10
The total friction force due to the sliding is the sum of the energy dissipation due
to the atomistic lattice vibration18,19 and the energy dissipation due to the conformational change in the entire transfer film which is calculated in this paper. To compare
it with the experiments, the effect of the atomistic lattice vibration should be
included for a precise discussion. We have, however, already found the difference
between the single layer and the multilayer case for the amount of energy dissipation
due to the conformational change. By including the atomistic lattice vibration, thethermal escaping motion will decrease the impulse of atoms on the opponent surface
so that the difference between the single and the multilayer case will become larger.
Our simulation also shows that the very low friction of the lamellar materials isable to exist even in the commensurate system. We think that the low friction of
the lamellar materials obtained from macroscopic experiments can be explainedby this mechanism.
The effects of the chemical nature at the edge20 are not included in our rough simulation. Also, the effect of a vacancy and defect are not included.21 In another study,
these can be treated using a modified potential surface. The effect of the internal
atom vibration should also be studied by using direct all-atom simulations. In thesemeans, the all-atom simulations and our coarse-grain simulation are complementaryin each other. A comparison with the experiments is also desired. This can be realized by fixing the size and the number of graphene sheets on the atomistic friction
experiments such as by the atomistic force microscope. Recently, the friction of
a fixed number of graphene sheets was measured,22–24 whereas the size of the sheets
was obviously larger than the transfer film which is our concern. We would verify themechanism for the thermal escape motion by measuring the friction force of a transfer film for a fixed size and number of graphene sheets.
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4 Conclusions
The Monte Carlo Brownian Dynamics simulation was used to investigate the friction motion for a transfer layer of graphite. In the monolayer case, the friction force
showed a stick-slip like curve and the average of the force was high. In the multilayercase, the friction force did not show any oscillation and the average of the force wasvery low. This is because the entire transfer film obtained the internal degree of
freedom in the multilayer case and the lowest sheets of the layer were able to follow
the equipotential surface of the lower slider which is called the thermal escapemotion. The low friction of lamellar materials obtained in macroscopic experiments
is able to be explained by this mechanism.

Acknowledgements
This study was partially supported by the Next Generation Super Computing
Project, Nanoscience Program, MEXT, Japan. We also thank Dr Shoichi Shirai,
Prof. Shi-aki Hyodo, Prof. Kazuo Kikuchi and Prof. Mark Robbins for their useful
discussions.

References
1 F. Bowden and D. Tabor, in The friction and lubrication of solids, Oxford: Clarendon Press,
Oxford, 1964.
2 A. Erdemir and E. J.-M. Martin, in Superlubricity, Elsevier Science, Amsterdam, 2007.
3 A. Geim and K. Novoselov, Nat. Mater., 2007, 6, 183.
4 M. Dienwiebel, et al., Surf. Sci., 2005, 576, 197.
5 A. Khomenko and N. Prodanov, Carbon, 2010, 48, 1234.
6 M. Hirano and K. Shinjo, Phys. Rev. B: Condens. Matter, 1990, 41, 11837–11851.
7 K. Shinjo and M. Hirano, Surf. Sci., 1993, 283, 473–478.
8 N. Sasaki, K. Kobayashi and M. Tsukada, Phys. Rev. B: Condens. Matter, 1996, 54, 2138–
2149.
9 G. S. Verhoeven, M. Dienwiebel and J. W. M. Frenken, Phys. Rev. B: Condens. Matter
Mater. Phys., 2004, 70, 165418.
10 N. Itamura, K. Miura and N. Sasaki, Jpn. J. Appl. Phys., 2009, 48, 060207.
11 Israelachvili, in Intermolecular and Surface Forces, Third Edition, Academic Press,
Burlington, 2010.
12 K. Kikuchi, M. Yoshida, T. Maekawa and H. Watanabe, Chem. Phys. Lett., 1991, 185,
335–338.
13 H. Washizu and K. Kikuchi, J. Phys. Chem. B, 2002, 106(43), 11329.
14 H. Washizu and K. Kikuchi, J. Phys. Chem. B, 2006, 110(6), 2855–2861.
15 N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, J. Chem.
Phys., 1953, 21, 1087–1092.
16 N. G. van Kampen, in Stochastic processes in physics and chemistry, revised and enlarged
edition, North-Holland, Amsterdam, 1992, ch. 11.
17 L. Girifalco, M. Hodak and R. S. Lee, Phys. Rev. B: Condens. Matter, 2000, 62, 13104.
18 S. Kajita, H. Washizu and T. Ohmori, Europhys. Lett., 2009, 87(6), 66002.
19 S. Kajita, H. Washizu and T. Ohmori, Phys. Rev. B: Condens. Matter Mater. Phys., 2010,
82, 115424.
20 N. Sasaki, H. Okamoto, S. Masuda, K. Miura and N. Itamura, Journal of Nanomaterials,
2010, 742127.
21 Y. Guo, W. Guo and C. Chen, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 76,
155429.
22 C. Lee, Q. Li, W. Kalb, X.-Z. Liu, H. Berger, R. Carpick and J. Hone, Science, 2010, 328,
76–80.
23 C. Lee, X. Wei, Q. Li, R. Carpick, J. W. Kysar and J. Hone, Phys. Status Solidi B, 2009,
246, 2562–2567.
24 T. Filleter and R. Bennewitz, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 155412.

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