A carbon nanotube is an ideal object for understanding the atomic scale
aspects of interface interaction and friction. Using molecular statics and
dynamics methods different types of motion of nanotubes on a graphite surface
are investigated. We found that each nanotube has unique equilibrium
orientations with sharp potential energy minima. This leads to atomic scale
locking of the nanotube.
  The effective contact area and the total interaction energy scale with the
square root of the radius. Sliding and rolling of nanotubes have different
characters. The potential energy barriers for sliding nanotubes are higher than
that for perfect rolling. When the nanotube is pushed, we observe a combination
of atomic scale spinning and sliding motion. The result is rolling with the
friction force comparable to sliding.Comment: 4 pages (two column) 6 figures - one ep

A. Buldum

B. Bhushan

B. N. J. Persson

C. M. Mate

G. Binnig

Jian Ping Lu

L. A. Girifalco

M. R. Falvo

R. Saito

R. W. Pastor

S. Ijima

T. Hertel

English

arXiv

Using molecular statics and dynamics methods we investigate the motion of nanotubes on a graphite surface. Each nanotube has unique equilibrium orientations with sharp potential energy minima which lead to atomic scale locking of the nanotube. The effective contact area and the total interaction energy scale with the square root of the radius. Sliding and rolling nanotubes have different characters. The potential energy barriers for sliding nanotubes are higher than that for perfect rolling. When the nanotube is pushed, we observe a combination of atomic scale spinning and sliding motion

Buldum, A.

Lu, J.P.

Carolina Digital Repository

VOLUME 83, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 13 DECEMBER199950Atomic Scale Sliding and Rolling of Carbon NanotubesA. Buldum* and Jian Ping Lu†Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599(Received 4 June 1999)Using molecular statics and dynamics methods we investigate the motion of nanotubes on a graphitesurface. Each nanotube has unique equilibrium orientations with sharp potential energy minima whichlead to atomic scale locking of the nanotube. The effective contact area and the total interactionenergy scale with the square root of the radius. Sliding and rolling nanotubes have different characters.The potential energy barriers for sliding nanotubes are higher than that for perfect rolling. When thenanotube is pushed, we observe a combination of atomic scale spinning and sliding motion.PACS numbers: 62.20.Qp, 61.16.Ch, 61.48.+c, 68.35.–peneuraectennogs-naunaeaautheac-ce.-besion-n-ged.ndontheinchine.rpck-eygleueAlthough the fundamental aspects of friction have bestudied for more than centuries, our knowledge aboutmicroscopic aspects is very limited [1]. The invention oatomic force microscope (AFM) [2] and its application imeasurements of atomic scale friction [3] [friction forcmicroscope (FFM)] have made a great impact on the sties of friction. A carbon nanotube is a stable nano-objehaving cylindrical shape [4], thus ideal for understandinatomic scale friction. Falvoet al. showed that it is possibleto slide, rotate, and roll carbon nanotubes on a graphite sface [5]. They demonstrated that a nanotube has prefeorientations on the graphite surface and prefer rolling thsliding when it is in atomic scale registry with the surfacIn this study, we carried out molecular statics, dynamicalculations, and studies of stick-slip motion for a varieof nanotubes. We found the following: (i) A nanotubhas sharp potential energy minima leading to orientatiolocking. The locking angles are directly related to the chral angle. (ii) Sliding and rolling nanotubes have differecharacters. The energy barriers for sliding are higher ththe barriers for perfect rolling. (iii) The effective contacarea and total interaction energy scale with the square rof the radius. (iv) A combination of sliding and spinninmotion is observed when the tube is pushed. The net reis rolling with the friction force comparable to the corresponding force for sliding.The character of interaction between the moving obje(atom, molecule, or any nanoparticle) and the underlyisurface defines the motion [6]. The interaction energy mconsist of short-range, attractive interaction energy dto chemical bonding; short-range repulsive energy along-range, attractive van der Waals energy. The intertion between a carbon nanotube and a graphite surfacsimilar to that between two graphite planes which is weand van der Waals in origin. To investigate the overbehavior of the motion of a carbon nanotube on a graphsurface, we represent the interaction between the tand the graphite surface atoms by an empirical potenof Lennard-Jones type [7] which was used extensiveto study solid C60 and nanotube [8]. Recent theoreticacalculations [9] showed that multiwall nanotubes on50 0031-90079983(24)5050(4)$15.00nitsfd-ctgur-redn.syali-tantotultctgyedc-iskllitebeiallylagraphite surface are not deformed significantly. Tatomic scale motion is determined mostly by the intertion of the outmost layer of the nanotube with the surfaIn this work we studied rigid single wall nanotubes with different chiralities and radii.The energy barriers related to the motion of nanotucan be conveniently analyzed by calculating the variatof the potential energy,EP , and corresponding force during the motion. Four different types of motion are cosidered: spinning, rotating, sliding, and rolling. Durineach step of motion the height of the tube is optimizWe first spin and rotate the nanotubes in order to fithe equilibrium positions. Figure 1 shows the interactienergy as a function of the rotation angle betweentube axis and the graphite lattice (all the data giventhis study is for per Å length of the nanotubes). Eananotube has unique equilibrium orientations repeatingevery 60±, reflecting the lattice symmetry of the graphitThe variation of energy near the minimum is very shawhich causes atomic scale locking of the nanotube. Loing angles are different for different nanotubes, and th-0.161-0.160-0.209-0.208-0.207 EP  ( eV )-60 -30 0 30 60 90-0.197-0.196(10,10)(30,0)(20,10)FIG. 1. The interaction energy as a function of rotation anbetween the nanotube axis and the graphite lattice for10, 10,30, 0, and 20, 10 nanotubes. Each nanotube has uniqminimum energy orientations repeating in every 60±.© 1999 The American Physical SocietyVOLUME 83, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 13 DECEMBER 1999are the direct measure of the chiral angle. This providesa novel method for measuring the chiralities of carbonnanotubes. Another important point in Fig. 1 is that theenergy variation between two consecutive energy minimais very small (except near the minima). Thus the forceneeded to rotate a nanotube is very small when the tube isout-of-registry. These results are in good agreement withthe recent experiments by Falvo et al. [5].Next, we studied sliding and rolling of carbon nano-tubes. In Fig. 2(a) the variation of the total interactionenergy EPs of a 20, 10 nanotube as a function ofsliding distance s is shown. The energy variation is verysmall for rolling since perfect atomic scale registry isalways maintained in the contact region. On the otherhand, in sliding motion, all the atoms in the contact regionmove simultaneously out-of-registry to higher energypositions and contribute to the energy barrier of motion.When the nanotube is attached to an AFM tip, stick-slip motion occurs. The tube first sticks and then slipssuddenly (slides or rolls or both) when the force exertedby the tip is sufficiently large [10]. The friction force instick-slip motion of the nanotube when it is attached toan AFM tip (with string constant 8.0 3 1023 Nm) isshown in Fig. 2(b). The area in the hysteresis curve givesus the amount of energy dissipated during the stick-slipmotion.0 5 10 15 20 s ( A )-0.194-0.193-0.192-0.191-0.19EP ( eV )0 10 20 30 40 s ( A )-0.01-0.00500.0050.01Ff ( nN )(a)(b)slidingrollingFIG. 2. (a) The variation of the interaction energy EP of a20, 10 nanotube as a function of sliding and rolling distances.(b) The friction force in stick-slip motion of the 20, 10nanotube when attached to an AFM tip with spring constant8.0 3 1023 Nm.To investigate the tube dependence, we performedcalculations with tubes in different chiralities or radii.Figure 3(a) shows the interaction energy as a functionof the nanotube radius, R. We found that the effectivecontact area and the interaction energy scale with thesquare root of the radius of the nanotube. The interactionenergy is independent of chirality. In-registry slidingforce (when the tube is in a minimum energy orientation)is also scaled with the square root of the radius. However,the sliding force is different for nanotubes with differentchiralities and the same radii. Figure 3(b) shows theforce for in-registry sliding and rolling. For a typicalnanotube (radius  13 nm, length  600 nm), the slidingforce value is estimated as 87 nN for an armchair tubeand 43 nN for a zigzag tube, in good agreement withthe friction force values measured in the experiments [5](50 nN).The static calculations we discussed give insight ofideal sliding and rolling. However, the dynamical be-havior is crucial for the competition between sliding androlling in the course of motion. In our model, the graphitesubstrate and the nanotube are considered to be rigid, butthe nanotube as a whole is able to move having transla-tional and spinning degrees of freedom. Constant lateralforce was applied to the nanotube for a short period oftime (50 ps) and then the motion of the nanotube wasanalyzed. In the same way, we applied constant torque orcombinations of torque and lateral force to the nanotube.The total energy of the system was kept constant.0 2 4 6 8 10-0.50-0.40-0.30-0.20-0.10 EP ( eV )0 2 4 6 8 10R1/2 ( A1/2 )024681012Fs ( pN )slidingarmchairsliding zigzagrolling(a)(b)FIG. 3. (a) The interaction energy as a function of square rootof the nanotube radius,pR. The filled circles correspond toarmchair tubes, and the hollow circles correspond to zigzagtubes with different radii. (b) The corresponding force forin-registry sliding and rolling of the nanotubes as a functionofpR.5051VOLUME 83, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 13 DECEMBER 1999After constant lateral force is applied on the nanotube,slide-spin motions in the atomic scale are observed [11].When the atoms in the contact region are in atomic scaleregistry it is easier for the nanotube to slide. Then thenanotube atoms move from in-registry to out-of-registrypositions and it is easier for the nanotube to spin. Byspinning the nanotube decreases its potential energy andthe atoms recover the in-registry positions. Switchingof the tube motion between spinning and sliding can beclearly seen in the sliding and the spinning components ofthe kinetic energy shown in Fig. 4(a). The switching isin the atomic scale and directly related to the corrugationof interaction energy. The total force acting on the tubein the direction of motion is shown in Fig. 4(b). Themaximum value of the force is comparable to the forcerequired to slide the nanotube [see Fig. 3(b)]. Sliding andspinning distances (angle multiplied by R) as functionsof time are shown in Fig. 4(c). Ideal rolling wouldbe a perfect overlap of sliding and spinning distances;meanwhile slide-spin motion gives oscillations, and thenet result is equivalent to rolling.For a better understanding of the slide-spin motion,we plot the interaction energy as a function of sliding0123EK ( meV )-0.00500.005Fy ( nN )0 0.5 1 1.5 2 time ( ns )10203040 s ( A )spin slidespin slide(a)(b)(c)FIG. 4. (a) After the initial push, sliding and spinning compo-nents of the kinetic energy of the 10, 10 nanotube as functionsof time are represented by the solid and dashed lines, respec-tively. Notice the switching between spinning and sliding mo-tions in the atomic scale. (b) The total force acting on the tubein the direction of motion. (c) The sliding and spinning dis-tances (angle multiplied by R) as functions of time.5052distance and spinning angle in Fig. 5(a). The trajectoryfor ideal rolling is a line at the bottom of the valleylikeregions. However, a nanotube performing slide-spinmotion follows an oscillating path in these valleys [seeFig. 5(b)] with an amplitude of oscillation depending onthe initial kinetic energy.When the system is coupled to a heat bath or energydissipation due to friction is considered, there are changesin the slide-spin motion. We modeled the dissipation byadditional velocity dependent forces [12] on the atomsclose to the contact region. The results are presented inFig. 6. Without energy dissipation the tube oscillates inthe valleylike regions of potential energy surface [seeFig. 5(a)] in the slide-spin motion. When the energy dis-sipation is considered, the total kinetic energy decreasedand there is more mixing between sliding and spinning.Eventually, the nanotube performs ideal rolling. If thenanotube has very high kinetic energy, it slides over manysurface unit cells. But because of energy dissipation thetube’s total kinetic energy decreases and slide-spin motionstarts. Afterwards the motion is close to ideal rolling [asseen in Fig. 6(b)]. This atomic scale picture of rolling isvery similar to the rolling of macroscopic objects. Recentmolecular dynamics simulations [13] with a full relaxedsystem by Schall and Brenner find similar conclusions.0 50 100 150 200 spin ( deg. ) 10203040 slide ( A )High K.E.Low K.E. (b)FIG. 5. (a) The variation of the interaction energy as afunction of sliding distance and spinning angle. (b) Thetrajectories correspond to the slide-spin motion of a nanotubehaving lower and higher total kinetic energies on the planedefined in (a).VOLUME 83, NUMBER 24 P H Y S I C A L R E V I E W L E T T E R S 13 DECEMBER 19992 3 4 time ( ns )010203040EK ( meV )0 100 200 300 400 500 spin ( deg. )020406080100 slide ( A )slidespin(a)(b)5FIG. 6. (a) Sliding and spinning components of the kineticenergy of the 10, 10 nanotube as functions of time arerepresented by the solid and dashed lines, respectively. (b) Thenanotube’s trajectories on the surface defined in Fig. 5(a) fordifferent initial kinetic energies. The trajectories from higherto lower initial kinetic energies are represented by dot-dashed,solid, and dashed lines, respectively. The kinetic energy valuesin (a) correspond to the trajectory plotted by solid line.To conclude, we investigated different types of motionof carbon nanotubes on a graphite surface. Each nanotubehas unique minimum energy orientations with respect tothe surface structure. The variation of interaction energyis very sharp leading to orientational locking of the nano-tubes. The locking angles are a direct measure of thechiral angles, and this provides a novel method for mea-suring the nanotube chirality. The effective contact areaand the total interaction potential energy scale with thesquare root of the radius of the nanotube. A combinationof atomic scale spinning and sliding motion is observedwhen the nanotube is pushed.The authors are grateful to R. Superfine, M. R. Falvo,J. Steele, D W. Brenner, J. D. Schall, and S. Washburn formany stimulating discussions. This work is supported byU.S. Office of Naval Research (N00014-98-1-0593).*Email address: buldum@physics.unc.edu†Email address: jpl@physics.unc.edu[1] For a general review and references see B. Bhushan, J. N.Israelachvili, and U. Landman, Nature (London) 347, 607(1995); B. N. J. Persson, Sliding Friction, Springer Seriesin Nanoscience and Technology (Springer-Verlag, Berlin,1998), p. 1.[2] G. Binnig, C. F. Quate, and Ch. Gerber, Phys. Rev. Lett.56, 930 (1986).[3] C. M. Mate, G. M. McClelland, R. Erlandsson, and S.Chiang, Phys. Rev. Lett. 59, 1942 (1987).[4] S. Ijima, Nature (London) 354, 56 (1991); for a generalreview see R. Saito, G. Dresselhaus, and M. S. Dressel-haus, Physical Properties of Carbon Nanotubes (ImperialCollege Press, London, 1998); Carbon Nanotubes: Prepa-ration and Properties, edited by T. W. Ebbesen (CRC,Boca Raton, 1997).[5] M. R. Falvo, R. M. Taylor, A. Helser, V. Chi, F. P. Brooks,S. Washburn, and S. Superfine, Nature (London) 397, 236(1999).[6] A. Buldum and S. Ciraci, Phys. Rev. B 54, 2175 (1996);A. Buldum, S. Ciraci and Ş. Erkoç, J. Phys. Condens.Matter 7, 8487 (1995).[7] L. A. Girifalco and R. A. Lad, J. Chem. Phys. 25, 693(1956).[8] Jian Ping Lu, X. P. Li, and R. M. Martin, Phys. Rev. Lett.68, 1551 (1992); Jian Ping Lu, Phys. Rev. Lett. 79, 1297(1997).[9] T. Hertel, R. E. Walkup, and P. Avouris, Phys. Rev. B58, 13 870 (1998); A. Buldum and Jian Ping Lu (to bepublished).[10] A. Buldum and S. Ciraci, Phys. Rev. B 55, 2606 (1997).[11] Slide-spin motions can be clearly seen in the movieof the simulation at http://www.physics.unc.edu/jpl/friction/movie.[12] R. W. Pastor, B. R. Brooks, and A. Szabo, Mol. Phys. 65,1409 (1988).[13] J. D. Schall and D. W. Brenner (private communication).5053

Atomic Scale Sliding and Rolling of Carbon Nanotubes

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