Superlubricity between incommensurate surfaces provides a desired low-friction state essential for the function  of  small-scale  machines.  Here  we  demonstrate  experimentally  and  theoretically  that  superlubricity in contacts lubricated by lamellar solids might be eliminated due to torque-induced reorientation coupled to lateral motion. We find that the possibility of reorientation always leads to stabilization of a high frictional state which corresponds to a commensurate configuration

Dienwiebel, Martin

Filippov, Alexander E.

Frenken, Joost W.M.

Klafter, Joseph

Urbakh, Michael

English

Frenken, Joost W. M.

KITopen

Torque and Twist against SuperlubricityAlexander E. Filippov,1 Martin Dienwiebel,2,3 Joost W. M. Frenken,2 Joseph Klafter,4 and Michael Urbakh41Donetsk Institute for Physics and Engineering of NASU, 83144, Donetsk, Ukraine2Kamerlingh Onnes Laboratory, Leiden University, 2300 RA Leiden, The Netherlands3Fraunhofer Institute for Mechanics of Materials, 79108 Freiburg, Germany4School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel(Received 30 August 2007; published 29 January 2008)Superlubricity between incommensurate surfaces provides a desired low-friction state essential for thefunction of small-scale machines. Here we demonstrate experimentally and theoretically that super-lubricity in contacts lubricated by lamellar solids might be eliminated due to torque-induced reorientationcoupled to lateral motion. We find that the possibility of reorientation always leads to stabilization of ahigh frictional state which corresponds to a commensurate configuration.DOI: 10.1103/PhysRevLett.100.046102 PACS numbers: 68.35.Af, 07.79.Sp, 46.55.+d, 68.37.PsThe practical importance and the relevance to basicscientific questions have motivated studies towards under-standing the conditions which lead to superlow friction. Amechanism for superlow dry friction, which arises from thestructural incompatibility of two contacting solids, wasfirst suggested by Hirano and Shinjo [1,2]. This phenome-non is also referred to as superlubricity. However, incom-mensurability of the interfaces results in a cancellation ofonly one of the channels of energy dissipation, whichoriginates from a stick-slip instability. Other dissipativeprocesses, such as electronic and phononic friction, etc.,still persist, and therefore even in the case of completeincommensurability the net friction force will not be iden-tical to zero. Nonetheless, typical to the superlubricity stateis a reduction of the friction force by orders of magnitude.Detailed experimental studies of superlubricity havebeen performed recently by Dienwiebel et al. [3–5], whomeasured friction between a graphite flake attached to thetip of a frictional force microscope (FFM) and an atomi-cally flat graphite surface. Superlow friction forces(<50 pN) have been found for most relative orientationsof the flake and the substrate, for which the contactingsurfaces find themselves in incommensurate states. Fornarrow ranges of orientations corresponding to commen-surate contacts, stick-slip motion was observed and frictionwas high (typically 250 pN). A few earlier experiments[6,7] also provided indications of the superlubricity phe-nomenon in dry friction.Measuring friction between graphite flakes and anunderlying graphite surface we found that the ‘‘lifetime‘‘of a superlubric state can be finite and therefore, superlowfriction does not persist. We demonstrate that the dominat-ing contribution to such residual friction in contacts lubri-cated by lamellar solids may stem from torque-inducedreorientation of the flakes attached to the tip. Dynamics ofthe flake reorientation result in irregular sharp peaks of thelateral force shown in Fig. 1, which are observed during therestoration of the high friction state. The lifetime maydepend on the structure of the tip and the density of defectson the graphite surface. Our calculations show that thepossibility of flake rotation always stabilizes the high fric-tional state which corresponds to the commensurate con-figuration of the contacting surfaces in relative motion.Reorganization of a sliding layer leading to a locking ina ‘‘commensurate’’ state has been previously observed inmolecular dynamics simulations [8]. A superlow frictionFIG. 1 (color). Measurements of the lateral force between atungsten tip with a graphite nanoflake attached [3–5] and ahighly oriented, pyrolytic graphite substrate. Results were ob-tained with high-resolution FFM [28] at a scan speed of 30 nm=sunder a constant positive normal force of 24.3 nN. (a) Frictionforce (average lateral force) as function of time (scan linenumber); the arrows indicate the locations of the three individualfriction loops in (b)–(d); (b) scan line 1 (green arrow); (c) scanline 45 (red arrow); (d) scan line 195 (blue arrow). One scan lineequals to 6 nm (3 nm forward and backward) or 0.2 s. Thedistance between subsequent scan lines equals 0.024 nm.PRL 100, 046102 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending1 FEBRUARY 20080031-9007=08=100(4)=046102(4) 046102-1 © 2008 The American Physical Societybetween two incommensurate surfaces may be also elim-inated due to the presence of adsorbed molecules betweenthe surfaces [9–12], or the elasticity of the contactingsolids [13].Beyond explaining the results of FFM experiments ongraphite, this consideration has important implications forunderstanding the macroscopic properties of graphite andother solid lamellar lubricants [14,15]. Transmission elec-tron microscopy observations [14] on MoS2 indicated thatalso in a macroscopic sliding contact of lamellar solidsrotated flakes are created.In order to demonstrate the effect of temporal reorienta-tion of the flake on friction, we propose a generalization ofthe Tomlinson model to include the in-plane rotations ofthe rigid flake pulled along the surface. The Tomlinsonmodel has proven powerful in describing the dynamics offriction [16,17]. Previous generalizations of the original,one-dimensional Tomlinson model, which included effectsof two-dimensional structure of surfaces, lateral-normalcoupling, and thermal fluctuations [18–24], marked newsteps toward understanding and implementation of fric-tional phenomena. In particular they led to the introductionof frictional imaging of interfaces [18], mechanical controlof friction [25,26], and thermolubricity [24].Introducing coupling between the lateral motion and in-plane rotation of the flake which is attached to the FFM tipand driven along the surface, leads to the followingcoupled Langevin equations: m rc  rrcUflakerc;    _rc  Krc  Vt  frct;(1) I rUflakerc;    _ F0sgn _  ft: (2)Here the first equation describes the translational motion ofthe center of mass of the rigid flake of an effective massm,whose position is given by the two-dimensional coordinaterc  xc; yc, and the second one describes the rotation ofthe flake having an effective moment of inertia I in theplane (x, y) with respect to the point of its attachment to thetip. We assume that the flake is attached to the tip in itscenter of mass, and  defines the time-dependent rota-tional (misfit) angle [see Fig. 2(a)]. The tip (and thereby theflake center) is coupled to the FFM support by springsKx  Ky  K in the x and y directions, and the supportmoves with a constant velocity at an angle with respect tothe x axis, V  V cos; V sin. In our experiments andsimulations we kept   70. The parameters  and  areresponsible for the dissipation of the flake’s kinetic energydue to the translational and rotational motion, respectively.The graphite flake is treated as a rigid body, and theN-atom flake layer contacting with the substrate is mod-eled as a finite lattice, composed of hexagonal carbon rings[see Fig. 2(a)]. The interaction between a single carbonFIG. 2 (color). The 150-atom symmetric graphite flake on the graphite surface. (a) Schematic view of the system. (b) Time-averagedfriction force as a function of the misfit angle  for nonrotating flakes of different sizes. (c),(d) Frictional force in the direction ofpulling and the misfit angle as functions of the support displacement, X, for the rotating flake. Dash-dotted lines indicate a fewinstances where the flake enters or leaves commensurate orientations. (e),(f ). 2D maps for the potential and torque experienced by theflake as functions of  and the flake displacement in the pulling direction, Lk:f. Distribution function for the misfit angle  calculatedfor hundred trajectories, one of which is shown in Figs. 2(c) and 2(d). Parameter values: FK  K1, NU0=FK1  2:5, V=FK 0:33, V=FK21  0:13, F0=FK1  1:25. Under the experimental conditions, where K  1:8 N=m, 1  0:246 nm, V 30 nm=s, we have FK  0:44 nN, NV0  0:34 eV,   4:8 103 kg=s,   1:2 104 kg nm2=s, F0  0:14 nN nm. Note thatthe typical magnitudes of the friction forces obtained in simulations are in agreement with the experimental data presented in Fig. 1.Both experiments and calculations have been performed under overdamped conditions.PRL 100, 046102 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending1 FEBRUARY 2008046102-2atom in the flake and the graphite surface is approximatedby the potential used in Ref. [4], Ux; y  U02 cos2x=1 cos2y=2 cos4y=2	 (3)with 1  0:246 nm and 2  0:426 nm are periods of thegraphite surface, and U0 is a constant. Then the rigid flake-surface interaction potential Uflakexc; yc;  is simply ob-tained by summing Ux; y over N atomic contributions. Inorder to include the effects of energy barriers which resistthe rotation of the flake with respect to the tip, we intro-duced in Eq. (2), in addition to a ‘‘viscous’’ term,  _, afrequency-independent, internal friction torque,F0sgn _, analogous to the known static friction. Theeffect of thermal fluctuations on the translational and rota-tional motion of the flake is given by a random force andtorque, f rct and ft, which are -function correlated,hfrc;itfrc;jt0i  2kBTt t0i;j and hftft0i 2kBTt t0, where i; j  x; y.First, we show in Fig. 2(b) the time-averaged frictionforce as a function of the misfit angle , calculated usingLangevin Eq. (1) for rotationally locked symmetric flakesof different sizes, one of which corresponds to Fig. 2(a).The flakes exhibit 60 rotational symmetry, and thereforethe results are presented over a 60 range of the anglesonly. In order to compare different flakes the forces arenormalized to the maximal value for each flake. The fric-tion force is maximal for a misfit angle of   0, forwhich the lattices of flake and substrate form a commen-surate structure, and it is much lower for most otherorientations. The angular width of the friction maximadecreases with the size of the flake. These results are inagreement with the experimental data [3–5] and previousquasistatic calculations [4] which provided firm evidencefor superlubricity.However, the flake driven along the surface experiencesthe influence of the torque and may change its orientation.Including the flake rotation which is governed by Eq. (2),changes qualitatively the dynamics of friction, as shown inFig. 2(c). Preparing the contact in the superlubric state(  10), we found that initially the flake exhibits asuperlubric sliding motion with low friction, but after somelag time it performs an erratic stick-slip motion character-ized by higher friction. The origin of this phenomenonbecomes obvious when one looks at the variation of themisfit angle  during the lateral displacement of the flakeshown in Fig. 2(d). One can see that after a lag time, whichdepends on the value of the internal friction F0, the flakestarts changing its orientation. It should be noted that forF0 > 5K21 the flake remains trapped in the initial state onthe time scale of our simulations. In the experiment, suchtrapping has been the exception, rather than the rule.Nonrotating flakes have been observed only on ‘‘high-grade’’ graphite substrates.The dynamics of the flake rotation can be represented asa set of alternating segments of rotational stick and slipmotion. During the stick intervals the flake is locked in oneof the commensurate configurations, with misfit anglescom  60  n, where n  0;1;2; . . . . The slip in-tervals correspond to the escape from a trapped state andthe return to either the same or to an adjacent trapped state.The slip events are caused by strong fluctuations of thetorque which arise from the stick-slip motion in the lateraldirection and from the thermal fluctuations. Thus, the flakeperforms a correlated stick-slip motion along both transla-tional and rotational coordinates.Figure 2(e) shows the two-dimensional (2D) map of thepotential, which describes the interaction between the flakewith a given orientation and the substrate, as a function ofthe flake displacement along the pulling direction Ljj andthe misfit angle . Regions corresponding to high and lowvalues of the potential are displayed by red and blue colors,respectively, with the color scale running from light greenfor zero potential to full red (blue) for maximum (mini-mum) values of the potential. One can see that the trapped,commensurate configurations correspond to the cases ofthe most corrugated potential in the pulling direction. As aresult we observe a stick-slip motion for these configura-tions, as shown in Fig. 2(c). Outside the stripes, which arelocated around   com, the potential is almost flat, andthe flake performs a ‘‘smooth’’ sliding in incommensuratecontact with the substrate.In order to clarify the nature of the rotational motion wepresent in Fig. 2(f) the 2D map of the torque experiencedby the flake with a given orientation in the (Ljj, ) space.Here regions of anticlockwise and clockwise torque aredisplayed by red and blue colors, respectively, and thecolor scale is the same as in Fig. 2(e). Regions betweenthe stripes, which are located around   com, are char-acterized by low values of the torque, and in these regionsthe rotation motion has a diffusive nature, as one can seefrom Fig. 2(d). However, when the flake configuration isbrought close to the commensurate one (falls into thestripes) the torque prevents the escape from the commen-surate state and stabilizes the stick-slip motion [seeFig. 2(f)]. Thus, when starting in experiments and insimulations from an incommensurate configuration, theflake approaches one of the stable, commensurate configu-rations diffusively, and stays there until strong enoughfluctuations kick it out of the stripes and bring it backinto a diffusion region. Our simulations demonstrate thatsuperlubric states of flakes that are not rotationally lockedto the tip are in a sense ‘‘free’’ with respect to the action ofthe torque while, in contrast, the states with a high frictionare the ‘‘stable’’ ones.The probability distribution function (PDF) for the mis-fit angles calculated for hundred time series of , whichare similar to that shown in Fig. 2(d), provides quantitativeevidence that the flake spends most time in the locked,commensurate configurations. The PDF was normalized toPRL 100, 046102 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending1 FEBRUARY 2008046102-3its maximal value, and differences in heights and widths ofthe maxima are the result of the limited statistics.We found that the above conclusions obtained for sym-metric graphite flakes are robust and hold for a broad rangeof systems. We show in Fig. 3 the results obtained for anasymmetric graphite flake which is driven along a foreignsurface with a square symmetry. As in the previous case,we find that starting from a fully incommensurate configu-ration with low friction the flake, after some lag time, findsitself in a more commensurate state with higher friction.Unlike the symmetric flake, here for the flake pulled at theangle   70 we find four different quasicommensurateconfigurations which are shown in Figs. 3(a)–3(d). Itshould be noted that for the flake pulled along one of thesubstrate axes of symmetry the four different configura-tions merge into two degenerate ones. The PDF for themisfit angle in Fig. 3(f) shows that the flake spends most ofits time in the quasicommensurate states performing acorrelated stick-slip motion in lateral and rotationaldimensions.Our calculations demonstrate that even in the case of adisordered flake the main conclusions of this Letter hold[27]. Namely, the flake performs reorientations during thetranslational motion and finds ‘‘optimal’’ configurationswhich exhibit a stick-slip motion with high friction.Based on our observations, we propose to overcome thisenhancement in friction by fixing the flake orientations toensure persistent incommensurability.M. D. and J. W. M. F. thank the ‘‘NederlandseOrganisatie voor Wetenschappelijk Onderzoek’’ for finan-cial support. J. K. and M. U. acknowledge the support bythe German Research Foundation via Grant Ha No. 1517/26-1-3. M. U. acknowledges the support by the IsraelScience Foundation (Grant No. 1116/05).[1] M. Hirano and K. Shinjo, Phys. Rev. B 41, 11 837(1990).[2] M. Hirano and K. Shinjo, Surf. Sci. 283, 473 (1993).[3] M. Dienwiebel et al., Phys. Rev. Lett. 92, 126101 (2004).[4] G. S. Verhoeven, M. Dienwiebel, and J. W. M. Frenken,Phys. Rev. B 70, 165418 (2004).[5] M. Dienwiebel et al., Surf. Sci. 576, 197 (2005).[6] M. Hirano et al., Phys. Rev. Lett. 78, 1448 (1997).[7] P. E. Sheehan and C. M. Lieber, Science 272, 1158 (1996).[8] Ph. Depondt, A. Ghazali, and J.-C. S. Lévy, Surf. Sci. 419,29 (1998).[9] G. He, M. H. Müser, and M. O. Robbins, Science 284,1650 (1999).[10] G. He and M. O. Robbins, Phys. Rev. B 64, 035413(2001).[11] M. H. Müser, L. Wenning, and M. O. Robbins, Phys. Rev.Lett. 86, 1295 (2001).[12] C. Daly, J. Zhang, and J. B. Sokoloff, Phys. Rev. Lett. 90,246101 (2003).[13] M. H. Müser, Europhys. Lett. 66, 97 (2004).[14] J. M. Martin et al., Phys. Rev. B 48, 10 583 (1993).[15] J. A. Heimberg et al., Appl. Phys. Lett. 78, 2449 (2001).[16] M. H. Müser, M. Urbakh, and M. O. Robbins, Adv. Chem.Phys. 126, 187 (2003).[17] M. Urbakh et al., Nature (London) 430, 525 (2004).[18] T. Gyalog et al., Europhys. Lett. 31, 269 (1995).[19] V. Zaloj, M. Urbakh, and J. Klafter, Phys. Rev. Lett. 82,4823 (1999).[20] A. Socoliuc et al., Phys. Rev. Lett. 92, 134301 (2004).[21] E. Gnecco et al., Phys. Rev. Lett. 84, 1172 (2000).[22] Y. Sang, M. Dube, and M. Grant, Phys. Rev. Lett. 87,174301 (2001).[23] O. Dudko et al., Chem. Phys. Lett. 352, 499 (2002).[24] S. Yu. Krylov et al., Phys. Rev. E 71, 065101 (2005).[25] M. G. Rozman, M. Urbakh, and J. Klafter, Phys. Rev. E57, 7340 (1998).[26] A. Socoliuc et al., Science 313, 207 (2006).[27] See EPAPS Document No. E-PRLTAO-100-081803 forresults obtained for a disorder flake. For more informationon EPAPS, see http://www.aip.org/pubservs/epaps.html.[28] M. Dienwiebel et al., Rev. Sci. Instrum. 76, 043704(2005).FIG. 3 (color). An asymmetric graphite flake on a surface ofsquare symmetry. (a)–(d) Schematic view of four configurationscorresponding to the most commensurate states of the flake.(e) Frictional force in the direction of pulling as a function of thesupport displacement, X, for the rotating flake. (f) Distributionfunction for the misfit angle  calculated for hundred trajecto-ries, one of which is shown in Fig. 3(e). Parameter values as inFig. 2, and the period of the surface lattice   1.PRL 100, 046102 (2008) P H Y S I C A L R E V I E W L E T T E R Sweek ending1 FEBRUARY 2008046102-4

Torque and twist against superlubricity

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Torque and twist against superlubricity

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