THE SLIDING AND ROLLING OF A CYLINDER AT THE NANO-SCALE

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1 Proceeings of 4 ASME/STLE Interntionl Joint Tribolog Conference Long Bech, Cliforni USA, October 4-7, 4 TRIB THE SLIDING AND ROLLING OF A CYLINDER AT THE NANO-SCALE O. Tln Sri, George G. Ams, Sinn Müftü Deprtment of Mechnicl n Instril Engineering Northestern Universit, Boston, MA 5 Emil: ms@ne.e ABSTRACT The behvior of nno-scle clinricl bo (e.g. fiber, ling on sbstrte n cte pon b combintion of norml n tngentil forces, is the sbject of this investigtion. As the scle ecreses to the nno level, hesion becomes n importnt isse in this contct problem. Ths this investigtion trets the two-imensionl plne strin elstic eformtion of both the cliner n the sbstrte ring rolling/sliing motion, incling the effect of hesion sing the Mgis moel. For the initition of sliing, the Minlin pproch is se, wheres for rolling, the Crter pproch is tilize. Ech cse is moifie for nno-scle effects b incling the effect of hesion on the contct re n b sing the hesion theor of friction for the friction stress. Anlticl reslts re given for the norml n tngentil loing problems, incling the initition of sliing n rolling in terms of imensionless qntities representing hesion, cliner size, n pplie forces. Bo h Bo Sliing Direction T F Figre. Contct of cliner with hlf-spce ner norml n tngentil loing. e R c c h INTRODUCTION Ahesion of clinricl boies on sbstrte is encontere in nno-wires, crbon nno-tbes n nno-fibers, n in ifferent fiels sch s microbiolog, microelectronics, n MEMS/NEMS evices. Determintion of the forces necessr to roll or slie clinricl bo on the sbstrte re importnt qntities to know in these pplictions. In some cses it is importnt to mniplte these single fibers to form strctre wheres in other instnces the sliing n rolling motions re importnt in contmintion removl processes. In this pper the hesion of contcting cliners, or eqivlentl cliner in contct with hlf spce, t the nnoscle is consiere. If the cliner is sbjecte to combine tngentil n norml loing it m remin t rest, roll, slie or nergo comple motion epening on the mgnites n the ppliction points of the loing. The elstic behvior of the clinricl bo n hlf-spce with hesion re investigte sing the plne strin theor of elsticit. A similr problem ws trete in the thesis b Sri []. Nmeros sties hve been concte on the herence of sphericl boies. Brle [] fon the pll-off force reqire to seprte two rigi sphericl boies, of rii R n R, to be F wr where R = R R /( R + R is the eqivlent ris of crvtre, w is the work of hesion w= + -, with the srfce energies of the contcting boies n, n the interfce energ of the two srfces. Johnson, Kenll n Roberts (JKR presente theor on the herence of eformble elstic boies [3]. In the JKR pproimtion, the hesion otsie the contct region is ssme to be zero, n the contct re is lrger thn the Hertz contct re. The pll-off force ws fon to be F = (3/wR. Derjgin, Mller n Toporov (DMT presente nother theor where the hesion force is consiere otsie the contct re, bt the form of the contct stress istribtion is ssme to be nffecte [4]. The sme pll-off force s the Brle reltion ws fon. It shol be note tht F is inepenent of the elstic properties of the mterils, for both JKR n DMT theories. Copright 4 b ASME

2 Althogh the JKR n DMT theories seem to be competitive, Tbor [5] showe tht these two theories represent /3 the etremes of prmeter ( Rw / E z, where E is the composite Yong s mols n z is the eqilibrim spcing. Ths boies in which the elstic eformtion is lrge compre to the rnge of srfce forces re in the JKR regime, wheres the DMT regime correspons to elstic eformtions which re mch less thn the rnge of srfce forces. Greenwoo constrcte n hesion mp tht covers these regimes [6]. Mgis presente moel for the trnsition between the JKR n DMT theories [7]. Similr to the Dgle moel of crck, the Mgis moel ssmes constnt tensile srfce stress in regions where the srfces re seprte b istnce less thn h, where the hesion seprtion istnce h is obtine from the reltion w= h. When this work of hesion is set eql to tht for the Lennr-Jones potentil, h is fon to be.97 z, where is tken s the theoreticl strength. Bne n Hi [8] se the Mgis moel to investigte the twoimensionl problem of the hesion of two circlr cliners. Reslts re given for the contct n hesion regions s fnctions of the pplie norml force n lso for the pll-off force. The nno-scle sliing n rolling nlsis trete in this pper iffers from the corresponing mcro-scle problem in two importnt ws. First, e to the smll scle of the contct re, hesion becomes importnt. The Mgis moel is se to pproimte the hesive stress otsie the contct region in mnner similr to Bne n Hi. Secon, in the mcro scle Colomb friction, which sttes tht frictionl force is proportionl to the norml lo, is consiere vli. At the nno-scle, however, where the contct ris is on the orer of nm or less, single rel contct re n constnt sher stress re ssme for sliing. THEORY AND DISCUSSION OF THE RESULTS The contct of clinricl bo of ris R with flt srfce is investigte. The reslts re eqll vli for the contct of two cliners b sing the eqivlent ris of crvtre. Liner plne strin elsticit is se throghot, which implies tht the forces re given per nit length. Accoring to plne strin liner elsticit [9], the erivtive of the srfce norml isplcements cn be written in terms of norml n sher stresses s, ( ( ( A 4 c c p ( B p 4 (. ( Similrl the reltion between the erivtive of the reltive isplcements of the boies in the tngentil irection n the bonr stresses is given b ( ( ( A 4 p ( B p 4 ( In (-( p is the tngentil trction in the -irection n the contct pressre p is consiere positive in compression. The mteril prmeters A n B re given b 4( 4( A G G 8, E B 4 4, G G ( E, Ei G i i E E (3 where G, G re the sher moli,, re the Poisson's rtios n E, E re the moli of elsticit of boies n respectivel, n E is the composite mols. Eqtion ( cn be simplifie for either ienticl mterils (B= or for the frictionless cse (p (=. Even if the mterils re not ienticl, the effect of the constnt B is sll smll [] n is often neglecte. Ths norml/sher stresses o not proce reltive tngentil/norml isplcements. Norml Loing Consier norml loing in which norml lo F is pplie to cliner with the tngentil force T eql to zero. This problem hs been solve b Bne n Hi [8], bt their nlsis is smmrize here becse the reslts of the norml loing problem etermine the contct region se in the sliing n rolling nlsis. There eists centrl contct zone (-<< srrone b two hesion zones (<<c in which the seprte srfces re ner constnt tensile stress s escribe b the Mgis hesion moel [7]. This configrtion is shown in Fig. where, b smmetr, c =c =c, h =h =h, n e=. The tensile hesive stress is effective p to seprtion h, beon which it vnishes. The reltion for the eformtions in the norml irection t the contct interfce is ( (, -<< (4 R in the contct zone, where is the mimm cliner penetrtion which occrs t the center of the contct zone. B proceeing s Bne n Hi [8], Eqs. ( n (4 re combine sing the Mgis conition in the hesion zones p (, c, c (5 with B=. The soltion is given b the sperposition of the Hertz soltion, the soltion for n eterior crck [], n the homogeneos soltion of (. Tht sperposition of soltions is lso sbject to the conitions tht the stress is bone t both ens ( =. The reslt is [8], p E ( R tn c, (6 Becse the contct hlf-with ( n the hesion hlf-with (c re both nknown, two etr eqtions re necessr. These eqtions re obtine b sing the force eqilibrim in the -irection, n b sing the reltion for the reltive seprtion of the two boies t =c n =. These give the following nonimensionl eqtions, F m, (7 n m m ln( m m m ln( m m mln( m where the following nonimensionl vribles re se,. (8 Copright 4 b ASME

3 F F,, /3 /3 ( Ew R ( R w / E 4 c, m /3 ( Ew/ R It is note tht Eqs. (7 n (8 form pir of cople nonliner eqtions with F known n ā n m nknown. These eqtions re solve nmericll. The reslts for the imensionless contct hlf-with ( vs. the imensionless norml force ( F for vrios vles of re shown in Fig.. As iscsse in [8], lrge correspon to the JKR regime, smll pproch Hertz contct, n when is of orer nit the reslts cn be pproimte b the DMT theor. For nonzero there is pll-off force which, for sfficientl lrge vle of, occrs t nonzero contct ris. Initition of Sliing Tngentil forces cn be trnsmitte b friction in contcting boies. Consier cliner in contct with hlfspce, compresse b norml force F, n cte pon b tngentil force T (Fig.. With the norml lo constnt, the tngentil force is grll increse in orer to initite sliing. The problem is solve for the ncople cse, i.e. B= in Eq. (. Ths the contct re remins in stte of stick ring the ppliction of the norml lo n the contct re remins constnt ring the ppliction of the tngentil force. Minlin stie the initition of mcro-scle sliing of cliner sing Colomb friction withot hesion n showe tht slip will occr t the eges of the contct zone []. Accoring to Minlin s theor there eists centrl stick zone ( < srrone b slip zones smmetricll locte in both the leing n triling eges. In nno-scle contcts, hesion between the two boies will ffect the contct with. This effect in clinricl contcts is escribe in the previos section. With respect to tngentil loing, in the mcro-scle Colomb friction is se, wheres t the nno-scle the friction stress in the slip zone is ssme to be constnt, s in the cse of the hesion theor of friction. The reltion between the reltive tngentil eformtions of the boies to the bonr stresses is given b Eq. (. In the slip regions the following eqtion previls, p (, ( where is the sher strength. In the stick region, we hve, ( ( ( (9, -<<. ( Prior to the initition of globl sliing motion, the horizontl shift (i.e. ( - ( of the boies in the stick zone (-<< will be constnt. The soltion of Eq. ( sbject to (-( is fon b sperposition of the soltion for n eternl crck loe b smmetric constnt sher stress in finite region ner the crck tip [7, p. ], tn, - < < ( n the homogeneos soltion of Eq. ( [9], - < < ( D where D is etermine from the conition tht is bone t =. Ths the sher stress in the stick region becomes tn, - < <. (3 Force blnce in the horizontl irection cn be se to fin the reltion between the hlf-length of the stick zone ( n the pplie sher force (T. This procere gives. (4 T ( ( The integrl in (4 hs been evlte in [8]. As the tngentil force reches the criticl vle of, stte of complete slip (i.e. globl sliing occrs with =. Pre Sliing If the contcting cliner hs reltive sliing motion with respect to the plne, there nee not be smmetr e to the nonliner ntre of hesion. In this cse, the origin of the coorinte sstem will be chosen to be in the center of the contct region. The eccentricit e inictes the vle of corresponing to the pe of the neforme cliner. The leing hesion zone will be strip (<<c n the triling hesion zone will be nother strip (-c <<-, s shown in Fig.. The reltion between the eformtions of the boies in the -irection insie the contct is ( ( ( e. (5 R The sme elsticit formltion se for the smmetric norml contct cn be se here i.e. Eq. ( sbject to (5 in the contct region -<<. The soltion cn be fon b the sperposition of for problems. The first problem is the soltion for constnt tensile stress in the leing ege (<<c n the secon is for constnt tensile stress in the triling ege (-c <<-. These soltions, which correspon to n eternl crck, re in T et l. [, p. 7]. The Moe I stress intensit fctors t the leing n triling eges re K I K I ( ( m m ( m m ( ( m m ( m m (6 where m =c / n m =c /. The thir problem correspons to the soltion of ( n (5 withot hesion, i.e. Hertz tpe soltion with n eccentricit which cn be fon sing [3] E ( e R, -<< (7 p Y n the forth is the homogeneos soltion of (, i.e. p Y ( D /, -<< (8 3 Copright 4 b ASME

4 The sm of these for soltions mst be sch tht the norml stress is bone t the ens t =. Recll tht the Moe I stress intensit fctor t = is efine b K I ( lim (, (9 The bone norml stress conition t = gives D ( m m ( m m, ( wheres the reqirement tht the norml stress mst be bone t = -, becomes R DR e ( m m ( m m ( E E When the cliner is in stte of ste sliing, the hesion effect in the triling ege is ssme to be lrger thn in the leing ege. This ssmption is consiere vli becse the srfce will be prtill clene e to the sliing motion of the contcting srfces. This effect is cconte for b tking the hesion seprtion istnce in the triling ege (h lrger thn in the leing ege (h, wheres is ssme nchnge. De to the ifference between h n h, the hesion with vles will not be the sme in the leing n the triling eges. Since there re two more nknowns, the seprtion eqtions mst be written for both the leing n triling eges. At the leing ege this procere gives: ( c e ( e R ( ( c ( h ( n t the triling ege it iels: ( c e ( e R ( ( c ( h (3 Sperposition of the for soltions iels [ m m m m ( m m cosh Eh Eh m m ] [( / Rh m m m m e m D ( e/ ln( m m ln( m m h Eh ( 4 h from the leing ege conition n m m m m ( m m cosh Eh Eh m m ] [( / Rh D ( e/ ln( m m ln( m m [ m m m m e m ] ] Eh (5 for the triling ege. The pplie norml force cn be fon from force eqilibrim in the -irection. De to the smmetr, resltnt moment will ct on the pper bo e to the smmetric norml stress istribtion. If moment eqilibrim is written with respect to the center of contct, the resltnt moment (clockwise irection cting on the hlf-spce tken to be positive. The pplie norml force n the resltnt moment re given in nonimensionl form b, 3 F D e, M e, (6 where the following nonimensionl qntities re se, D D D,, D, Eh 4 Eh 4 e M, e, M Rh wr (7 in which w= h. Ths Eqs. (4-(5, long with the eplicit epressions in (-(, n the nonimensionl qntities in (7, represent pir of eqtions which, for specifie n h / h, which cn be solve for m n m. Finll the nonimensionl force n moment re fon from (6. The reslts for the imensionless contct hlf-with ( vs. the imensionless norml force ( F for vrios vles of re shown in Fig. 3 for h / h 5. When Figs. n 3 re compre, it is seen tht s h / h increses from to 5, the force reqire to proce given contct with lso increses. This effect is greter for lrge (JKR region, e to elstic eformtion thn it is for moerte (DMT regime, limite elstic eformtion or smll (Hertz regime, smll hesion. Figs. 4 shows the imensionless hesion hlf-with ifference (m -m vs. imensionless contct hlf-with ( for ifferent vles of with h / h 5. This mesre of the smmetr of the hesion zones becomes lrge for smll vles of the imensionless contct ris (. It is lso mch greter for smll thn for lrge. This reslt m pper conterintitive. However the contct hlf-with is normlize b qntit which incles the cbe-root of the work of hesion, wheres vries s the two-thirs power of w. Also lrge correspons to greter elstic eformtion which is better cpble of ccommoting the smmetr in the work of hesion. The reslts for the imensionless verge hesion length (m +m / vs. imensionless contct ris re shown for vrios vles of in Fig. 5 for h / h 5. Finll the imensionless moment ( M vs. the imensionless norml force ( F re shown for vrios vles of in Fig. 6 for h / h 5. Note tht s the norml force pproches the plloff force, the moment pproches finite vle, even in the smll regime where vnishes t pll-off. This reslt is the combine effect of the smmetr in the work of hesion long with the smll elstic eformtion. 4 Copright 4 b ASME

5 Rolling The problem of ste stte rolling of n elstic cliner on n elstic hlf-spce (or eqivlentl one cliner rolling on nother with Colomb friction ws solve b Crter [4]. Accoring to Crter s soltion the leing ege of the contct zone (<< is in stte of stick wheres the triling ege (-<< is in stte of slip. As in the ppliction to locomotive wheel [4], the pper bo is the riving cliner. Dring this rolling motion, the liner velocit of the center of the cliner is slightl less thn R, where is the nglr velocit. The creep velocit represents this velocit ifference. At the nno-scle the sher stress (friction stress in the slip zone is ssme to be constnt, s previosl iscsse. As with sliing, hesion ffects the reltion between the norml force n contct with. The tngentil reltive isplcement (shif between the boies is epresse s ( ( s(, (,, (,, C(, (8 where C represents the rigi bo motion of the pper bo reltive to the lower bo. In the stick zone the time erivtive of the shift in the moving coorinte sstem is zero. The stick conition cn be written s, s (, V ( C, << (9 where C ( is the constnt rigi bo slip (or creep velocit. Frthermore the sher stress is constnt in the slip region, i.e. p (, -<< (3 The soltion of Eq. (, sbject to (9-(3, cn be fon b sperposition of the soltion for crck eternl to the stick region n loe in sher on one sie, i.e. the slip zone (T et l. [, p. 7], the soltion of ( e to the constnt creep velocit, n the homogeneos soltion of (. The Moe II stress intensit fctors for the eternl crck problem t the ens of the stick zone re [] ( 3 KII ( cosh ( 3 KII ( cosh (3 wheres the soltion to the creep velocit n the homogeneos soltion re CE D, << (3 V ( ( p ( The reqirement tht the soltion be bone t = n = les to the following eqtions, 3 D cosh C 3 cosh V E (33. (34 If force eqilibrim is written in the -irection, the pplie sher force cn be relte to the contct with prmeter ( b T p ( ( (35 Eqtion (35 gives the vrition of the pplie tngentil force with the etent of the slip zone. The greter the trction force, the lrger is the slip zone. As the rolling motion pproches complete slip. Eqtion (34 gives the imensionless creep velocit, which is liner in /E n vries nonlinerl with the slip zone prmeter (/. As the trction force increses, increses n hence the mgnite of the creep velocit increses logrithmicll ccoring to (34. The mening of the negtive creep velocit is tht R for the riving wheel is greter thn the velocit of the contct zone. CONCLUSIONS This pper trets the two-imensionl elstic contct problem of cliner on sbstrte ring rolling/sliing motion n incles the effect of hesion sing the Bne n Hi version of the Mgis-Dgle moel. Dring initition of sliing, there is centrl stick zone srrone b slip regions in the leing n triling eges. As the tngentil force T increses, the lengths of the slip zones increse ntil complete slip occrs t certin vle of T. Dring ste sliing the brsive ction of the sher stress cn be epecte to prtill clen the srfce, reslting in ifferent leing n triling ege hesive properties. This effect is incle in the moel of ste nno-scle sliing. Vritions of the creep velocit n the length of the stick zone with T re etermine. As the trction force increses, the stick zone length ecreses n the creep velocit increses eventll leing to pre slip with rottion. REFERENCES. O. T. Sri, Agst 3, Mster of Science Thesis, Mechnicl Engineering Dept., Northestern Universit, Boston, MA.. R. S. Brle, 93, The cohesive force between soli srfces n the srfce energ of solis, Phil. Mg., 3, pp K. L. Johnson, K. Kenll n A. D. Roberts, 97, Srfce energ n the contct of elstic solis, Proc. Rol Soc. Lonon Ser. A, 34, pp B. V. Derjgin, V. M. Mller n Y. P. Toporov, 975, Effect of contct eformtions on the hesion of prticles, J. Coll. n Interfce Sci., 67, pp D. Tbor, 976, Srfce forces n srfce interctions, J. Coll. n Interfce Sci., 58, pp J. A. Greenwoo, 997, Ahesion of Elstic Spheres, Proc. R. Soc. Lon. A, 453, pp D. Mgis, 99, Ahesion of spheres: the JKR-DMT trnsition sing Dgle moel, J. Coll. n Interfce Sci., 5, pp J. M. Bne n C.-Y. Hi, 997, A Cohesive Zone Moel for the Ahesion of Cliners, J. Ahesion Sci. n Technolog,, pp J. R. Brber,, Elsticit, secon eition, Klwer Acemic Pblishers.. K. L. Johnson, 985, Contct Mechnics, Cmbrige Universit Press.. H. T, P. C. Pris, n G. R. Irwin,, The Stress Anlsis of Crcks Hnbook, thir eition, ASME Press. 5 Copright 4 b ASME

6 . R. D. Minlin, 949, Complince of elstic boies in contct, J. Appl. Mech., 7, pp A. Eréli, Tbles of Integrl Trnsforms, Btemn Mnscript Project, McGrw-Hill Book Compn, New York, ( F. W. Crter, 96, On the ction of locomotive riving wheel, Proc. Ro. Soc. (Lonon, A, pp Ahesion Hlf-With Difference: m -m =4 = = =. = Contct Hlf-With: Figre 4. The triling n leing ege hlf-with ifference (m -m vs. contct hlf-with (ā ring sliing with h /h =.5 Contct Hlf-With: Contct Hlf-With: =. - - Norml Force: F Figre. The vrition of the imensionless contct hlfwith (ā with the imensionless norml lo ( F for vrios vles of..5 = =4 = =4 =. Hertz = =. Hertz - - Norml Force: F Figre 3. The vrition of the imensionless contct hlfwith (ā with the imensionless norml lo ( F for vrios vles of ring sliing with h /h =5. Averge Ahesion Hlf-With: (m +m / Moment: M = =4 = Contct Hlf-With: = = =4 =. Figre 5. The triling n leing ege verge (m +m / vs. contct hlf-with (ā ring sliing with h /h =5. = =. - Norml Force: F Figre 6. The imensionless resltnt moment ( M vs. imensionless contct hlf-with (ā for vrios vles of ring sliing with h /h =5. 6 Copright 4 b ASME