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parameter-free dissipation in simulated sliding friction

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Parameter‐free dissipation in simulated sliding friction A. Benassi 1,2 G.E. Santoro 1,2,3 A. Vanossi 1,2 and E. Tosatti 1,2,3 Abstract Non‐equilibrium molecular dynamics simulations, of crucial importance in sliding friction, are hampered by arbitrariness and uncertainties in the way Joule heat is removed. We implement in a realistic frictional simulation a parameter‐free, non‐ markovian, stochastic dynamics, which, as expected from theory, absorbs Joule heat precisely as a semi‐ infinite harmonic substrate would. Simulating stick‐ slip friction of a slider over a 2D Lennard‐Jones solid, we compare our virtually exact frictional results with approximate ones from commonly adopted empirical dissipation schemes. While the latter are generally in serious error, we show that the exact results can be closely reproduced by a viscous Langevin dissipation at the boundary layer, once the back‐reflected frictional energy is variationally optimized. 1. SISSA Scuola Internazionale Superiose Studi Avanzati, Trieste (Italy) 2. DEMOCRITOS National Simulation Center, Trieste (Italy) 3. International Center for Theoretical physics (ICTP), Trieste (Italy) Simulating a 2D semi‐infinite Lenard‐Jones substrate i) Following Adelman we write the Hamilton’s equations for all the atoms, distinguishing between 3 different regions (fig. 2a) ii) Under the hypothesis of an harmonic heat bath with dynamical tensor φ we avoid to simulate explicitly the heat bath (fig. 2b) accounting for its presence through effective equations of motion for the atoms in the dissipation layer iii) These equations allow us to dissipate the energy injected in the substrate has if the substrate was really a semi‐infinite object iv) The effective equations are non‐markovian Langevin equations with many memory kernels K and stochastic forces R. fig. 2 (b) … z x fig. 2 (a) 1 2 3 Dissipation layer Explicitly simulated atoms Infinite heat bath (not simulated) i, j =1, 2, 3, ... μ, ν = x, z m¨ q i μ = + j,ν q j ν (t) K i,j μ,ν (0) - φ i,j μ,ν - m j,ν t 0 ˙ q i μ (s)K i,j μ,ν (t - s)ds + R i μ (t) Direct interaction Indirect interaction + self interaction Heat bath contribution ∈ Comparison with other dissipation schemes We compared the results for the semi‐infinite substrate with other two dissipation schemes based on markovian Langevin equations: The memory kernels and the stochastic noise The kernels are not chosen a priori, they come from the microscopic theory too: λ i and ω i 2 being the eigenvectors and eigenvalues of the dynamic matrix φ of the heat bath. All this kernels are oscillating and decaying functions, an example is given in fig. 3. K k,m μ,ν = i ( λ i · φ k μ,μ )( λ i · φ m ν,ν ) ω 2 i cos(ω i t) R(t) i μ =0 R(t) i μ R(t ) j ν = mK B TK i,j μ,ν (t - t ) fig. 3 Accordingly to the fluctuation dissipation theory, memory kernels are also needed to correlate the stochastic noise that arise at finite temperature: k v0 fig. 4 Incommensurate dry friction Putting a slider on the free surface of our semi‐infinite substrate (fig. 4) enable us to study friction phenomena without any a priori assumption on the shape of the dissipative force. The energy of the slider is dissipated exciting the phononic modes of the substrate, once that the phonons reach the dissipation layer they are absorbed as if they where continuing to propagate in the non simulated part of the substrate. • The slider is driven through a spring connected to the slider center of mass. • The slider is slightly incommensurate with respect to the substrate, an anti‐kink appears moving backward with jumps of 5‐7 atoms at once • Periodic boundary conditions are applied along the sliding direction A tipical stick‐slip profile is shown in fig.5 (a) where the friction force is plotted against time. 9 atoms slider over a 10 x 20 Lennard‐Jones substrate K B T = 0.035 few Kelvin degrees, v 0 =0.01, k=5.0, vertical load=10.0 (LJ units) fig. 5 Bibliography and Acknowledgments Conclusions i) Through a non‐markovian Langevin dissipation scheme we can simulate the dissipation of semi‐infinite harmonic substrates in a rather small simulation cell ii) Friction related phenomena can be exactly simulated within this framework, with no need for empirical parameters iii) A comparison with viscous damping dissipation schemes shows a strong dependence of the friction force, and related quantities, on the empirical parameters iv) Using the exact results as a reference, we demonstrated that even a viscous damping dissipation scheme can be tailored in such a way to reproduce the correct friction force, once that the damping parameter is chosen according to a simple and self standing procedure. This activity has been funded by ESF Eurocore FANAS‐AFRI [1] S. Adelman and J. Doll, J.Chem.Phys. 64 2375 (1976) R. J. Rubin, J. Math. Phys. 1 309 (1960) [2] X. Li and W. E, Phys. Rev. B 76 104107 (2007) [3] L. kantorovich, Phys.Rev.B 78 094304 (2008) L. kantorovich and N. Rompotis, Phys.Rev.B 78 094305 (2008) Thanks to: Alexander Filippov ‐‐ Donetsk Institute for Physics and Engineering of NASU (Ukraine) Rosario Capozza ‐‐ Universita’ degli studi di Modena e Reggio Emilia (Italy) Giovanni Bussi ‐‐ SISSA Scuola Internazionale Superiose Studi Avanzati (Italy) for interesting and helpful discussions. k v0 k v0 (b) Viscous damping applied to the slider atoms while the substrate atoms are frozen (equivalent to a Frenkel‐Kontorova model) (c) Viscous damping applied to the substrate atoms only -γ(vi - vCM) i i -γvi The friction force now depends on the choice of the damping parameter γ. Fig.6 shows this dependence for the average friction force and for its variance: dashed line for case (b) and dotted line for case (c), the blue stripes indicate the exact values obtained with the non‐markovian aproach. fig. 6 numbers refer to fig.5 where some selected stik‐slip profiles are shown in comparison with the exact result (a) When we place a too high viscous damping on the moving slider or too close to the slider‐substrate interface, we prevent the slider from exchanging the right quantity of energy with the substrate. This results in a too large friction force. If we place a too small viscous damping on the substrate atoms, we are not removing the energy efficiently. The substrate heats up and the friction force usually results to be smaller than the exact value. The viscous damping must be switched on far from the sliding interface: top: layer resolved kinetic energy for a slip event on a semi‐infinite substrate (up) and on a finite substrate (down) Bottom: phonons excited by a slip event in a 2D substrate Averaging over many long simulations, the average friction force is 1.17 (LJ units), we can now compare this exact result (a) with the ones obtained employing other dissipation schemes. (d) Viscous damping applied to the last substrate atoms only k v0 i In the latter case (d), it exists a range of γ values (between 2 and 20) in which the friction force and its variance are independent of γ and are really close to the exact results. More interestingly the exact result is reproduced by those γ values which minimize the substrate average internal energy W (see fig 6 (c)): Now even without the exact result as a reference, the optimal γ value of can be variationally obtained. W = E(T, γ,v0)-E(T, γ, 0) -γvi fig. 1

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Page 1: 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

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