parameter-free dissipation in simulated sliding friction
TRANSCRIPT
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