Eur. Phys. J. B 6, 59 6 (008 DOI: 0.0/epj/e008-009-9 THE EUROPEAN PHYSICAL JOURNAL B Generlizion of nonliner fricion relion for dimer sliding on periodic susre M. Tiwri,,S.Gonçles, nd V.M. Kenkre Consorium of he Americs for Inerdisciplinry Science nd Deprmen of Physics nd Asronomy, Uniersiy of New Mexico, Aluquerque, New Mexico 873, USA Insiuo de Físic, Uniersidde Federl do Rio Grnde do Sul, Cix Posl 505, 950-970 Poro Alegre RS, Brzil Receied 9 Ferury 008 / Receied in finl form 6 April 008 Pulished online My 008 c EDP Sciences, Socieà Ilin di Fisic, Springer-Verlg 008 Asrc. An omic cluser moing long solid surfce cn undergo dissipion of is rnslionl energy hrough direc mode, inoling he coupling of he cener-of-mss moion o herml exciions of he susre, nd n indirec mode, due o dmping of he inernl moion of he cluser, o which he cener-of-mss moion cn e coupled s resul of surfce poenil. Focusg only on he less well undersood indirec mode, on he sis of numericl soluions, we presen, deprures from recenly repored simple relionship eween he force nd elociy of nonliner fricion. A generlizion of he nlyic considerions h erlier led o h relionship is crried ou nd shown o explin he deprures sisfcorily. Our generlizion res for he sysem considered (dimer sliding oer periodic susre he complee dependence on seerl of he key prmeers, specificlly inernl dissipion, nurl frequency, susre corrugion, nd lengh rio. Furher predicions from our generlizions re found o gree wih new simulions. The sysem nlyzed is relen o nnosrucures moing oer crysl surfces. PACS. 8.0.Pq Fricion, luricion, nd wer 6.55.+d Triology nd mechnicl concs Inroducion The fricion experienced y oms, smll molecules nd dlyers moing on susres is n cie opic of curren reserch [ 7,6] ecuse of is relence o he fundmenl undersnding of riey of processes such s hose inoled in omic force microscopy [8,9]. Tking he nurl poin of iew h horough undersnding of he moion of idelized omic srucures oer idelized susres should precede he sudy of he moion of relisic omic eniies oer relisic surfces, seerl inesigors he explored he moion of dimers (wo-om sysems moing oer fixed surfce represened y usoidl [0 5] poenil. I is perhps surprig u rue h, in spie of he fc h numer of horough sudies of more relisic (e.g., exended sysems he lredy ppered in he lierure, inereg resuls coninue o e found in he simple dimer sysem. If dimer chrcerized y he mss m of ech om conneced y spring of spring consn m 0 is hrown wih n iniil cener of mss elociy 0 oer -dimensionl susre represened y usoidl poenil of mpliude u 0 nd welengh, one of hese recen numericl inesigions [0] showed h, if he only dmping in he sysem cs on he inernl coordine of he dimer, he cener of mss moion of he dimer is effecielydmpedinremrkle e-mil: mukesh@unm.edu fshion represened y nonliner fricion relion. Th relion ses h he effecie fricion force experienced y he dimer cener of mss moion is inersely proporionl o he cue of he cener of mss elociy. The purpose of he presen pper is o inesige he rnge of lidiy of h fricion relion. We crry ou numericl simulions h go eyond he limiing se of condiions explored in reference [0], show h cler deprures re oined in he ehior of he dimer including in he dependence of is sopping ime on rious sysem prmeers, nd proide sisfcory heoreicl explnion for our findings. The pper is se ou s follows. We firs inesige numericlly he deprures from he simple nonliner fricion relion of reference [0] wih chnge in sysem prmeers (Sec. nd hen gie simple explnion in erms of n exended nonliner fricion relion o suppor our numericl findings (Sec. 3. In Secion we mke predicions on he sis of our nlysis nd find h hey re erified y new numericl soluions. Concluding remrks re presened in Secion 5. Deprures from he / 3 fricion relion The equions of moion goerning he sysem under sudy, dimer moing on -dimensionl periodic
60 The Europen Physicl Journl B susre, re mẍ = k(x x + ( πu 0 mγ (ẋ ẋ mẍ = k(x x + ( πu 0 ( πx ( πx mγ (ẋ ẋ ( where x, re he coordines of he wo dimer pricles of equl mss m, oerdos denoe ime deriies, γ is he dmping coefficien nd k,,, u 0 re, respeciely, he spring consn, equilirium lengh of he dimer, welengh of he susre poenil nd hlf he mpliude of he poenil. Noe h he dmping of ech mss is proporionl no o is solue elociy u o he elociy relie o he oher mss. This mens h dmping cs only on he inernl coordine. Such siuion represens dissipion of energy hrough chnnels inernl o he moing srucure. We consider he sysem emperures low enough h rndom (Brownin moion erms need no e dded in equion (. This we do for simpliciy s in reference [0]. I hs een poined ou in rious ppers such s reference [] how he Brownin erms my e incorpored. The uhors of reference [0] he shown, hrough simulions nd nlyic indicions sring from equion (, h he dimer dynmics cn resul in he remrkly simple nonliner fricion relion: d d = γ ( u0 m ( π 3 ( Equion ( predics h he sopping ime (defined s heimewhichheelociyofhecenerofmssdrops o zero would e gien s: s = γ ( u 0 m ( (3 π This mens h he sopping ime for he cener of mss moion should ry inersely s he coefficien γ which conrols he dmping in he inernl coordine of he dimer. The firs of he deprures from he fricion relion gien y equion ( h we repor in he presen pper is he iolion of he monoonic γ-dependence of he sopping ime. We plo in Figure he riion, wih chnge in he dmping coefficien γ, of he sopping ime oined y soling numericlly equion (. The plo clerly shows h deprure. The soluion we oin (solid cure coincides wih he predicion of equions (, 3iniilly, showing decrese in sop wih increg γ. Howeer, furher increse in γ resuls in n increse in sop,in conrs o he predicions of equions (, 3. In exmining he dependence of sop on γ, wehefoundsriking liner relionship eween γ nd he relie difference eween sop nd he sopping ime s prediced y he / 3 relion se ou in reference [0]. We exhii his liner relionship in Figure. The second deprure from predicions of he relion in equions (, 3 h we found inoles he frequencydependence of he sopping ime. While equion ( is 0 sop 300 00 00 0 0 0 0 60 80 00 0 γ presen work / 3 lw Fig.. Deprure from he fricion relion of reference [0]: comprison eween he non-monoonic sopping ime sop oined from numericl simulions (solid line wih he monoonic predicion (see Eq. (3 of he / 3 relion (dshed line of reference [0] for differen lues γ of dmping. Dmping is expressed in unis of (/ u 0/m nd sopping ime is in unis of m/u 0.Iniilelociy 0 is 5 u 0/m. Oher prmeers re / =0.5 ndk /u 0 =0.0. sop / s 0 0 numericl soluion liner fi 0 000 000 6000 8000 0000 000 γ Fig.. Liner relionship eween γ nd he relie difference eween he sopping ime nd he predicion of he fricion relion in reference [0]. The circles represens he numericl soluion while he solid line is n (excellen liner fi. Oher prmeers re s in Figure. independen of he nurl frequency 0 = k/m of he dimer, our numericl soluions show h sop does exhii dependence on 0, showing minimum nd increse on eiher side of he minimum. We disply his ehior in Figure 3, where we plo sop normlized o is limi 0 s 0 0. The mrked oscillory ehior o he lef of he minimum in Figure 3 is represenie of he resonnce s will e cler from Figure elow. The res of he jgged nure of he cure in Figure 3 hs no physicl significnce nd rises from numericl sources.
M. Tiwri e l.: Generlizion of nonliner fricion relion... 6 sop / 0. 0.9 0.8 0.7 presen work / 3 lw 0.6 0 00 00 300 0 Fig. 3. Deprure from he fricion relion of reference [0]s seen in he dependence of he sopping ime on he dimer nurl frequency 0. Shown is he sopping ime (normlized o is lue 0 for 0 0, see ex from our numericl soluions (solid line nd from he predicion of equions (, 3 (dshed line, i.e., of he erlier nlysis reference [0]. Frequency is expressed in unis of (/ u 0/m nd γ (m/u 0=.Oher prmeers re s in Figures,. 3 Anlyic sudies from n exended ersion of he fricion relion In order o undersnd he source of he deprures from he / 3 relion repored in Secion, weexminehe equions of moion ( y following he procedure se ou in reference [0]. By defining scled nd rnsled coordine ξ, hrough ξ = (x x, ( we oin he equion of moion gien in reference [0], d ξ d + γ dξ d + 0 ξ = πu { 0 π } m ( + ξ cos(ζ, (5 where ζ, he scled cener of mss coordine oeys d ζ d = [ (π (u0 ζ = π(x + x, (6 m ] { π } cos ( + ξ (ζ (7 I ws rgued in reference [0] h, lhough equion (7 predics n inoled dependence of ζ on ξ nd, herefore, eenully on, he simple liner pproximion ζ( where = π(/ is he so clled wshord frequency, ( eing he elociy of he cener of mss, leds, for smll ξ (ξ, o d ξ d + γ dξ d + 0ξ = πu ( 0 π m cos ( (8 Wih equion (8 s he sring poin, pproximing he ssumed wekly -dependen o e consn, one cn sole for he inernl coordine ξ( = πu ( 0 π m ( 0 + γ cos( δ (9 where δ is he phse ngle gien y n(δ =γ /(0. The uhors of reference [0] deried he nonliner fricion lw y ug power lnce condiion. To e le o ddress he deprures repored y us in Figures 3 oe, we inroduce n imporn modificion in h rgumen. Insed of neglecing 0 nd γ erms in d d = γ ( u0 m ( π [ ( 0 π ] ( + γ π, (0 we elec o use equion (0 s our generlized fricion relion. Clerly, under he pproximion h he wshord frequency is much lrger hn oh he nurl frequency 0 nd dmping γ, one ges he / 3 relion of reference [0]. Insed of simplifying equion (0 for 0 /π nd γ/π, wesole equion (0 excly: [{ ( } = 0 + η 0 { ( } 0 ( ( γ π 0 0 0 ( ( ] 0 ln ( π0 0 where, η = γ ( u0 m ( π. The definiion of he frequency =π 0 /, whichisheiniil lue of,nd corresponds o he iniil cener of mss elociy of he dimer, llows us o rewrie our soluion ( s [ ( ] ( [ ( ] = 0 + γ 0 0 s ( 0 ( ln 0 ( where s = 0 η.equion( is one of he principl nlyic resuls of he presen pper. 3. Anlysis of he γ dependence of he sopping ime for smll inernl frequency ( 0 The ide of puing = 0 in equion(, nd clculing he corresponding sopping ime sop, nurlly suggess iself. Howeer, i is no lid in generl: he exisence of he logrihmic erm ensures h he elociy hs longime il which mens h he sopping ime is, sricly, lwys infinie. On he oher hnd, he simulions do show sopping ime wihin which he dimer flls ino one of he susre wells, nd hen rles ck nd forh. A good esime of sop cn e oined from equion ( for lues of 0 / sufficienly smll h he logrihmic
6 The Europen Physicl Journl B erm cn e negleced. Then, if he inernl irion frequency is smll enough ( 0, u if specificlly we llow he dmping γ o e unresriced, we oin s = [ ( 0 ] + γ [ ( ] (3 0 0.9 0.8 0.7.5 c = 0.0 0.5 0 00 00 300 numericl nlyicl fi Puing =0inequion(3, we ge he sopping ime ( γ ] sop = s [+ ( sop 0.6 0.5 0. c = /5 Equion (, noher of he new resuls of he presen nlysis, displys he sopping ime sop s he produc of wo fcors. The firs fcor is he sopping ime s = 0 η gien y he / 3 relion of reference [0]. The second fcor is he correcion proided y our presen nlysis: +(γ/. If dmping is reliely smll, which ws he cse reed erlier, (γ/ my e negleced nd here is negligile correcion. If, howeer, dmping is susnil, sop s gien y equion (cnincrese wih γ. Indeed, ( ledso sop s = γ. (5 Our nlyic resul (5 proides he highly ccure fi o he simulions displyed in Figure. 3. Anlysis of he 0 dependence of he sopping ime for smll dmping (γ If, inequion(, we mke he pproximion h he dmping in inernl coordine is smll (γ, u llow he frequency 0 o e unresriced, we oin s = [ ( ] 0 ( 0 ( [ 0 ( ln 0 ( ] 0 (6 Unlike in equion ( we cnno oin sop in his cse y puing =0inequion(6, ecuse, s explined oe, he sopping ime is lwys infinie. To ge n esime of sop from (6, we pu / 0 equl o sufficienly smll non-zero lue c nd sole (6 numericlly. We cll he sopping ime hus oined sop rher hn sop (ecuse c 0, lhough smll nd disply is dependence on he oscillor frequency in Figure. Thelue of c we he ken is oined y equing he cener of mss kineic energy o he susre poenil energy. The significn feure in he oe cse is he deelopmen of long ils of ( h rise from he logrihmic erm in he righ hnd side of (6. In he low-elociy limi ( 0 /π he denominor of he righ hnd side of equion (0 ecomes independen of nd we recoer he liner fricion relion. d d = γ ( u0 m s ( π, (7 0.3 0. 0 00 00 300 00 500 600 700 0 Fig.. Dependence of he ppren sopping ime sop on he oscillor frequency 0 showing close greemen eween he numericl (solid line nd nlyic expression gien y equion (6 (dshed line s explined, for differen lues of he nurl frequency of he oscillor. The rio c (see ex is ken o e /5. Frequency is expressed in unis of (u0/m/ nd γ (m/u 0 =. Oher prmeers re he sme s in Figure. Inse shows he comprison for smller lue of c = 0.0 which shows poor greemen of simulion nd heory. where, s = 0 π is he sliding elociy corresponding o he nurl frequency of he oscillor. By fixing / 0 = c nd puing d/d 0 = 0 in our nlyic equion (6, we cn oin resonle lue of 0 which he minimum of sop occurs in Figure : [(c /lnc] /. This mnner of geing sopping ime would e compleely useless if he resuls we oin were srongly dependen on he chosen lue of c. Wehecrriedou numerous clculions for differen c s nd found h for c 0. we ge fine greemen eween simulion nd nlyic predicion. For smller lues, we sill find quliie greemen (e.g. in he exisence nd pproxime locion of he minimum which ecomes worse s c is mde smller (see inse of Fig.. The oscillions seen o he lef of he minimum occur when he driing frequency is close o resonnce. This is oiously no presen in our simple nlyic resuls. The resonnce is eween he nurl frequency of he dimer 0 nd he wshord frequency =π/ (see Eq. (9. Comprison of he predicions of he generlized fricion relion wih simulions Gien he sisfcory greemen of he nlyic expression, equion (, for he sopping ime deried here, we presen Figures 5, 6, where we compre he cener of mss elociy ccording o our nlyic expression ( wih numericl simulions. In Figure 5 we he kep he dmping coefficien γ unchnged nd he ried he nurl frequency of he dimer o produce cures,, c from our nlyic predicion for hree differen lues of 0 s
M. Tiwri e l.: Generlizion of nonliner fricion relion... 63 5 5 3 35 0 c,, c 3 c 0 c 0 0 50 00 50 00 50 300 350 00 50 Fig. 5. Comprison eween our presen nlyic predicion from he generlized fricion relion (dshed lines nd numericl simulions (solid lines wih oscillions for he cener of mss elociy (. Iniil elociy 0 is 5 u 0/m. Oherprmeers re / =0.5 ndγ m/u 0 =. The differen lues of frequency 0 re 3.6, 0 nd. expressed in unis of (u0/m/. They correspond o he spring consn k eing in he rio :0:0 respeciely. The lines, nd c re he predicions of he / 3 fricion relion of reference [0] nd coincide wih one noher ecuse of he lck of ppernce of he nurl frequency in h relion. Velociy is expressed in unis of u 0/m nd ime is in unis of m/u 0.Inseshows he resonnce srucure h is isile in he min figure round = 0. shown. Our prediced cures si righ ono he simulion resuls unil he ler rek ino oscillions which our nlyic predicions cnno descrie. Clerly, our presen nlysis does considerly eer is--is he simulions hn he / 3 relion which, eing insensiie o 0, produces he gle cure denoed,,c in Figure 5. The oher feure our presen simulions show is he resonnce phenomenon when he wshord frequency equls he nurl frequency. I is responsile for he srucure seen in Figure 5 round = 0 (mgnified in he inse nd is he sme s he one h produces he srucure o he lef of he minimum in Figure. In Figure 6 we he lef 0 consn nd ried he dmping. Once gin our nlyic predicions, cures,, c (for γ = 5, 50, 00 respeciely in unis (u 0 /m/ do ery well wih respec o he simulions. No only do hey si on op of he simulion cures (excep for he oscillions while cures,,c oined from he / 3 relion do no, u hey lso exhii non-monoonic ehior s γ is ried: cure is o he lef of u cure c is o he righ of. By conrs, cures,,c exhii monoonic endency in h hey moe lefwrd in he - -c progression. 5 Concluding remrks The simple sysem of dimer sliding on periodic susre offers eg ground for heoreicl pproches 0 0 0 30 0 50 60 70 Fig. 6. Comprison eween our presen nlyic predicion from he generlized fricion relion (dshed lines nd numericl simulions (solid lines wih oscillions for he cener of mss elociy ( s.ime. Iniilelociy 0 is 5 u 0/m. Oher prmeers re / =0.5 ndk /u 0 =0.0. The differen lues of dmping γ expressed in unis of (u 0/m/ re 5, 50 nd 00. The doed lines, nd c show he predicions of he / 3 fricion relion. Velociy is expressed in unis of u 0/m nd ime is in unis of m/u 0. imed undersnding he nure of omic fricion on meril susres. In exmining recenly repored nonliner fricion relion [0], we he found, i, our simulions, deprures from he predicion of h relion. The deprures re inereg, re displyed in Figures 3, nd cn e explined heoreiclly, s we he shown. Predicions mde on he sis of our nlysis re orne ou excellenly y simulions s shown in Figures 5, 6. Our new nlyic resuls re equion ( for he sopping ime of he dimer when hrown oer he susre wih n iniil elociy, is pproxime forms equions (3, 6, nd deried expressions such s equions (, 5. All hese resuls sem from he generlizion, equion (, of he fricion relion of reference [0]. Much of he ehior seen in he eoluion of he cener of mss elociy cn e undersood physiclly from he concep of resonnce. The ell shpe of he resonnce gien y equion (0 is conrolled y dmping nd he nurl frequency of he dimer. Mximum energy is los in he irionl moion nd close o resonnce. A smll lues of dmping he mximum lue of he ell shpe is lrge u he widh is nrrow so he inernl mode does no lose much energy nd hence he sopping ime is lrge. Increg he lue of dmping rings down he resonnce pek u he sme ime increses he widh of he ell shpe. Thus he dimer loses more energy s i spends more ime in he resonnce region. Lrge lues of dmping desroys he widh of he resonnce cure nd gin he inernl mode is no le o lose much energy; hence he increse in sopping ime. The nurl frequency deermines he posiion of he pek. Iniilly, he driing frequency is fr wy from resonnce nd no much
6 The Europen Physicl Journl B energy is eing rnsferred o he inernl coordine. As he cener of mss elociy drops down o close o resonnce, he rnsfer of energy reches locl mximum. Howeer, ecuse he dmping is no lrge he energy cnno e dissiped wy immediely nd hence is rnsferred ck ino rnslionl moion. This ck nd forh rnsfer of energy eween rnslionl nd irionl moion leds o srong periodic oscillions elociies close o resonnce. The nonliner fricion relion gien y equion (0 is no le o cpure hese oscillions. The coupling dries he dimer ou of resonnce window o where he driing frequency ecomes much smller hn he resonn frequency nd hence he elociy decys exponenilly wih ime (see Eq. (7 nd he oscillions ecome periodic. This srong oscillory ehior cn e wshed ou y increg he dmping of he sysem due o resons lredy menioned efore. Increse in he lue of nurl frequency shifs he resonnce pek o lrger lues nd hus he dimer goes ino resonnce region lrger driing frequencies. Our im in he presen pper hs een o ddress some feures of omisic fricion. Excep for he simplificion h he model is confined o gle spil dimension, i hs he necessry ingrediens o represen rel dimer or molecule in conrolled microscopic sliding experimen (e.g. [] low emperures, o he exen h exernl dmping my e considered negligile. Dissipion of rnslionl energy of n ojec long surfce cn occur in wo wys, direcly nd indirecly. Indeed, here hs een n ongoing dee in he lierure ou he relie impornce of he source of sliding fricion: wheher i is elecronic or phononic. As explined in deil in he conclusion secion of recen pper y some of he presen uhors [] i migh mke sense o idenify ckground liner fricion wih elecronic, nd resonnce fricion wih phononic, sources. For sufficienly lrge corrugions of he susre, he ler (he only chnnel considered in he presen pper cn domine in principle. Neerheless if he modulion rio / is smll, he chnnel we he considered cn e unimporn y comprison, nd dlyer elociies relie o he susre migh need o e s high s 30 300 m/s for he resonnce fricion o e pprecile. Experimens repored so fr do no inole such high elociies. Our sudy hs no ddressed ps oserions u rges relizle fuure scenrios. We hope h in he ligh of he cler nlysis presened in his pper nd elsewhere [0], he focus on he resonnce fricion chnnel used oe will e of help in fuure oserions where h chnnel my no e oerlooked. I is plesure o hnk Birk Reichench for numerous discussions. This work ws suppored in pr y he NSF under grn INT-033633. References. M. Urkh, J. Klfer, D. Gourdon, J. Isrelchili, Nure (London 30, 55 (00. B.N.J. Persson, Phys. Re. B 8, 80 (993 3. J. Krim, Surf. Sci 500, 7 (00. J.B. Sokoloff, Phys. Re. B, 760 (990 5. A.S. Kole, A.I. Lndu, Low Temp. Phys. 8, 3 (00 6. O.M. Brun, R. Ferrndo, D.E. Tommei, Phys. Re. E 68, 050 (003 7. L. Consoli, H.J.F. Knops, A. Fsolino, Phys. Re. Le. 85, 30 (000 8. C.M. Me, G.M. McClellnd, R. Erlndsson, S. Ching, Phys. Re. Le. 59, 9 (987 9. E. Gnecco, R. Bennewiz, T. Gylog, E. Meyer, J. Phys.: Condens. Mer 3, R69 (00 0. S. Gonçles, V.M. Kenkre, A.R. Bishop, Phys. Re. B 70, 955 (00. S. Gonçles, C. Fusco, A.R. Bishop, V.M. Kenkre, Phys. Re. B 7, 958 (005. C. Fusco, A. Fsolino, T. Jnssen, Eur. Phys J. B 3, 95 (003 3. C. Fusco, A. Fsolino, Thin Solid Films 8, 3 (003. A.H. Romero, A.M. Lcs, J.M. Sncho, Phys. Re. E 69, 0505 (00 5. O.M. Brun, Phys. Re. E 63, 00 (00 6. S.Yu. Krylo, K.B. Jinesh, H. Vlk, M. Dienwieel, J.W.M. Frenken, Phys. Re. E 7, 0650(R (005