parameter-free dissipation in simulated sliding friction

Parameter‐freedissipationinsimulatedslidingfriction

A.Benassi1,2G.E.Santoro1,2,3A.Vanossi1,2andE.Tosatti1,2,3

Abstract

Non‐equilibrium molecular dynamics simulations,of crucial importance in sliding friction, arehamperedbyarbitrarinessanduncertaintiesintheway Joule heat is removed. We implement in arealisticfrictionalsimulationaparameter‐free,non‐markovian,stochasticdynamics,which,asexpectedfromtheory,absorbsJouleheatpreciselyasasemi‐infiniteharmonicsubstratewould.Simulatingstick‐slip friction of a slider over a 2D Lennard‐Jonessolid, we compare our virtually exact frictionalresults with approximate ones from commonlyadoptedempirical dissipationschemes.Whilethelatteraregenerallyinseriouserror,weshowthatthe exact results can be closely reproducedby aviscousLangevindissipationattheboundarylayer,once the back‐reflected frictional energy isvariationallyoptimized.

1.  SISSAScuolaInternazionaleSuperioseStudiAvanzati,Trieste(Italy)2.  DEMOCRITOSNationalSimulationCenter,Trieste(Italy)3.  InternationalCenterforTheoreticalphysics(ICTP),Trieste(Italy)

Simulatinga2Dsemi‐infiniteLenard‐Jonessubstrate

i)  FollowingAdelmanwewritetheHamilton’s equationsforall theatoms,distinguishingbetween3differentregions(fig.2a)

ii)  Underthehypothesisofanharmonicheatbathwithdynamicaltensor φ weavoidtosimulateexplicitlytheheatbath(fig.2b)accountingforitspresencethrougheffectiveequationsofmotionfortheatomsinthedissipationlayer

iii)  Theseequationsallowustodissipatetheenergyinjectedinthesubstratehasifthesubstratewasreallyasemi‐infiniteobject

iv)  Theeffectiveequationsarenon‐markovianLangevinequationswithmanymemorykernelsKandstochasticforcesR.

fig.2(b)

…z

x

fig.2(a)

1 2 3Dissipation

layer

Explicitlysimulatedatoms

Infiniteheatbath

(notsimulated)

i, j = 1, 2, 3, ... µ, ! = x, z

mq̈iµ = +

!

j,!

qj!(t)

"Ki,j

µ,!(0)! !i,jµ,!

#!m

!

j,!

$ t

0q̇iµ(s)Ki,j

µ,!(t! s)ds + Riµ(t)

Directinteraction

Indirectinteraction+selfinteraction

Heatbathcontribution

!

Comparisonwithotherdissipationschemes

Wecomparedtheresultsforthesemi‐infinitesubstratewithothertwodissipationschemesbasedonmarkovianLangevinequations:

Thememorykernelsandthestochasticnoise

Thekernels are not chosenapriori, they come fromthe microscopictheorytoo:

λiandωi2beingtheeigenvectorsandeigenvaluesofthedynamicmatrix

φoftheheatbath.Allthiskernelsareoscillatinganddecayingfunctions,anexampleisgiveninfig.3.

Kk,mµ,! =

!

i

(!"i · !#kµ,µ)(!"i · !#m

!,!)$2

i

cos($it)

!R(t)iµ" = 0 !R(t)i

µR(t!)j!" = mKBTKi,j

µ,!(t # t!)

fig.3Accordingly to the fluctuation dissipation theory, memory kernels arealso needed to correlate the stochastic noise that arise at finitetemperature:

k v0

fig.4

Incommensuratedryfriction

Puttingaslideronthefreesurfaceofoursemi‐infinitesubstrate(fig.4)enableustostudy friction phenomena without any a priori assumption on the shape of thedissipativeforce.

Theenergyofthesliderisdissipatedexcitingthephononicmodesofthesubstrate,oncethatthephononsreachthedissipationlayertheyareabsorbedasiftheywherecontinuingtopropagateinthenonsimulatedpartofthesubstrate.

• Thesliderisdriventhroughaspringconnectedtotheslidercenterofmass.

•  The slider is slightly incommensurate with respect to the substrate, an anti‐kinkappearsmovingbackwardwithjumpsof5‐7atomsatonce

• Periodicboundaryconditionsareappliedalongtheslidingdirection

Atipicalstick‐slipprofileisshowninfig.5(a)where the friction force is plotted againsttime.

9atomssliderovera10x20Lennard‐JonessubstrateKBT=0.035fewKelvindegrees,v0=0.01,k=5.0,verticalload=10.0(LJunits)

fig.5

BibliographyandAcknowledgmentsConclusions

i) Throughanon‐markovianLangevindissipationschemewecansimulatethedissipationof semi‐infiniteharmonicsubstratesinarathersmallsimulationcell

ii)Frictionrelatedphenomenacanbeexactlysimulatedwithinthisframework,withnoneedforempiricalparameters

iii)Acomparisonwithviscousdampingdissipationschemesshowsastrongdependenceofthefrictionforce,andrelatedquantities,ontheempiricalparameters

iv)Using theexactresultsasareference,wedemonstratedthatevenaviscousdampingdissipationschemecanbetailoredinsuchawaytoreproducethecorrectfrictionforce,oncethatthedampingparameterischosenaccordingtoasimpleandselfstandingprocedure.

ThisactivityhasbeenfundedbyESFEurocoreFANAS‐AFRI

[1]S.AdelmanandJ.Doll,J.Chem.Phys.642375(1976)R.J.Rubin,J.Math.Phys.1309(1960)[2]X.LiandW.E,Phys.Rev.B76104107(2007)[3]L.kantorovich,Phys.Rev.B78094304(2008)L.kantorovichandN.Rompotis,Phys.Rev.B78094305(2008)

Thanksto:AlexanderFilippov‐‐DonetskInstituteforPhysicsandEngineeringofNASU(Ukraine)RosarioCapozza‐‐Universita’deglistudidiModenaeReggioEmilia(Italy)GiovanniBussi‐‐SISSAScuolaInternazionaleSuperioseStudiAvanzati(Italy)forinterestingandhelpfuldiscussions.

k v0

k v0

(b)Viscousdampingappliedtotheslideratomswhilethesubstrateatomsarefrozen

(equivalenttoaFrenkel‐Kontorovamodel)

(c)Viscousdampingappliedtothesubstrateatomsonly

!!(vi ! vCM )

ii

!!vi

Thefrictionforcenowdependsonthechoiceofthedampingparameterγ. Fig.6showsthisdependencefortheaveragefrictionforceandforitsvariance:dashedlineforcase(b)anddottedlineforcase(c),thebluestripesindicatetheexactvaluesobtainedwiththenon‐markovianaproach.

fig.6numbersrefertofig.5wheresomeselected

stik‐slipprofilesareshownincomparisonwiththeexactresult(a)

Whenweplaceatoohighviscousdampingonthemovingsliderortooclosetotheslider‐substrateinterface,wepreventthesliderfromexchangingtherightquantityofenergywiththesubstrate.Thisresultsinatoolargefrictionforce.

Ifweplaceatoosmallviscousdampingonthesubstrateatoms,wearenotremovingtheenergyefficiently.Thesubstrateheatsupandthefrictionforceusuallyresultstobesmallerthantheexactvalue.

The viscous damping must be switched on far fromthe slidinginterface:

top:layerresolvedkineticenergyforaslipeventona semi‐infinitesubstrate(up)andonafinitesubstrate(down)

Bottom:phononsexcitedbyaslipeventina2Dsubstrate

Averaging over many long simulations, theaveragefrictionforceis1.17(LJunits),wecannow compare this exact result (a) with theones obtained employing other dissipationschemes.

(d)Viscousdampingappliedtothelastsubstrateatomsonly

k v0

i

Inthelattercase(d),itexistsarangeofγvalues(between2and20)inwhichthefrictionforceanditsvarianceareindependentofγandarereallyclosetotheexactresults.

MoreinterestinglytheexactresultisreproducedbythoseγvalueswhichminimizethesubstrateaverageinternalenergyW(seefig6(c)):

Nowevenwithouttheexactresultasareference,theoptimalγ valueofcanbevariationallyobtained.

W = !E(T, !, v0)" # !E(T, !, 0)"

!!vi

fig.1