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Surface Science 601 (2007) 3676–3681

Nonlinear mobility of a driven system: Temperature and disorder effects

Roberto Guerra, Andrea Vanossi *, Mauro Ferrario

CNR-INFM National Research Center S3, Department of Physics, University of Modena and Reggio Emilia, Via Campi 213/A, 41100 Modena, Italy

Available online 21 July 2007

Abstract

We consider the dissipative nonlinear dynamics of a model of interacting atoms driven over a substrate potential. The substrateparameters can be suitably tuned in order to introduce disorder effects starting from two geometrically opposed ideal cases: commensu-rate and incommensurate interfaces. The role of temperature is also investigated through the inclusion of a stochastic force via a Lange-vin molecular dynamics approach. Here, we focus on the most interesting tribological case of underdamped sliding dynamics. Fordifferent values of the chain stiffness, we evaluate the static friction threshold and consider the depinning transition mechanisms as afunction of the applied driving force. As experimentally observed in QCM frictional measurements of adsorbed layers, we find that dis-order operates differently depending on the starting geometrical configuration. For commensurate interfaces, randomness lowers consid-erably the chain depinning threshold. On the contrary, for incommensurate mating contacts, disorder favors static pinning destroying thepossible frictionless (superlubric) sliding states. Interestingly, thermal and disorder effects strongly influence also the occurrence of para-metric resonances inside the chain, capable of converting the kinetic energy of the center-of-mass motion into internal vibrational exci-tations. We comment on the nature of the different dynamical states and hysteresis (due to system bi-stability) observed at differentincreasing and decreasing strengths of the external force.� 2007 Elsevier B.V. All rights reserved.

Keywords: Atomic scale friction; Nanotribology; Models of non-linear phenomena; Molecular dynamics; Computer simulations

1. Introduction

During the last decade, due to its practical importanceand the relevance to basic scientific questions, there hasbeen a major increase in the activity of investigating inter-facial dynamics in nanoscale systems. In particular, thestudy of surface interactions has revealed numerous inter-esting aspects at the basis of friction. It has recently beenshown [1] that simple (low dimensional) phenomenologicalmodels of friction gave reasonable agreement with experi-mental results and more complex simulation data on nano-scale tribology. With respect to this, the application ofdriven and generalized Frenkel–Kontorova (FK) like mod-els (see, e.g. [2] and references therein), describing the dissi-pative motion of a chain or layer of interacting particlesthat slides over a rigid substrate potential due to the appli-

0039-6028/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.susc.2007.07.014

* Corresponding author. Tel.: +39 059 2055685; fax: +39 059 374794.E-mail addresses: robguerra@unimore.it (R. Guerra), vanossi.an-

drea@unimore.it (A. Vanossi).

cation of an external driving force, has found an increasinginterest as a possible interpretative key to understand theatomic processes occurring at the interface of two materialsin relative motion.

The minimal threshold force Fs required to induce mo-tion and the ratio B = VCM/F of the time-averaged cen-ter-of-mass (CM) velocity to the external applied force(i.e. the chain mobility) turn out to be strongly dependenton the geometrical features of the sliding interface. Obser-vation of finite static friction implies that the contactingsolids have locked into a local energy minimum, and Fs

represents the force needed to lift them out of it. AboveFs, the layer mobility is usually a highly nonlinear functionof the applied force. Its value is intrinsically determined bythe kinetic frictional force, which arises from some dissipa-tive mechanism converting the energy of translational mo-tion into excitation of various degrees of freedom of thesystem and, eventually, into heat.

Commensurate (CC) and incommensurate (IC) slidinginterfaces represents two opposite and ideal cases. While

R. Guerra et al. / Surface Science 601 (2007) 3676–3681 3677

perfectly aligned commensurate surfaces are always pinned(Fs 5 0), when two crystalline workpieces with incommen-surate lattices are brought into contact, then the static fric-tional force may vanish, provided the two substrates arestiff enough. Based on the pioneering work of Aubry [3],this remarkable conclusion can be drawn in the contextof the standard FK model. In the case of incommensuratelength scales (i.e. a/b irrational, where a and b define thechain natural lattice constant and the substrate spatial per-iod, respectively) and for fixed substrate potential ampli-tude, the FK ground state undergoes a transition bybreaking of analyticity (or Aubry transition) if the chainstiffness K drops below a critical value Kc. From a physicalpoint of view, this means that for K > Kc there exists a con-tinuum of ground states that can be reached by the chainthrough nonrigid displacements of its atoms with no energycost. This sliding mode corresponds to the regime of zerostatic friction. On the contrary, below Kc, the chain andsubstrate become pinned with nonzero static friction, irre-spective of incommensurability.

Relevant generalizations of the FK model have beenproposed so far in the literature to cover a large class ofphysically interesting phenomena (e.g. see Ref. [2]). Fordescribing realistic physical systems, it is sometimes helpfulto consider substrate potentials, which may vary substan-tially from the simple sinusoidal shape assumed in the stan-dard FK model, with a possible consequent drastic changein the statics and dynamics occurring between the surfaces.For example, for atoms adsorbed on metal surfaces, theon-site potential is usually characterized by sharp bottomsand flat barriers [4,5]. Moreover, if the substrate is charac-terized by a complex unit cell, the substrate potential hasmore complicated shape with several minima and/or max-ima [6,7]. In this situation, from a tribological point ofview, different types of sliding behavior may be expected.Another possibility is the inclusion of quasiperiodicity [8]or disorder [9–11] in the substrate.

In this work, we focus on the effects of temperatureand surface disorder in the underdamped dynamics of aone-dimensional (1D) chain of interacting atoms slidingover a substrate potential. The model parameters can betuned in order to introduce disorder effects starting fromthe two geometrically opposite ideal cases of commensu-rate and incommensurate interfaces. The Langevinmolecular dynamics approach allows us to introduce tem-perature via the inclusion of a stochastic force. Workingin the low-dissipative regime, where inertia effects cannotbe neglected, we analyze, due to system bi-stability, thedisplay of hysteretic behavior in the B(F) characteristicsfor adiabatic increase and decrease of the external drivingforce.

2. Model and computational details

We investigate the dynamics of a driven 1D model,whose N particle positions xi satisfy the following equa-tions of motion

m€xi ¼ F þ F intðjxi � xi�1jÞ þ F subðxiÞ þ CðtÞ � mc _xi; ð1Þ

where m is their mass. The dots denote time derivatives andF represents the external driving force applied to all atomsof the chain. The force Fint arises from the nearest-neigh-bors interparticle interaction Vint. The numerical simula-tions are carried out by choosing a Morse-type potential

V intðrÞ ¼K2½1� eða�rÞ�2; ð2Þ

with strength K and natural equilibrium spacing a. Thechain is subject to the force Fsub due to a substrate poten-tial corrugation Vsub defined as sum of identical Gaussianfunctions of amplitude U and width l:

V subðxÞ ¼XM

i¼0

2Ulb exp� x� ibþ qRi

lb

� �2

: ð3Þ

The set {Ri,i = 0,M} of M random numbers uniformly dis-tributed in the interval (0,1), weighted by the factor q, en-sure a random distribution in space of the Gaussianfunctions around the mean value b of the substrate latticeperiodicity.

As long as the thermal energy is not negligible (com-pared to the amplitude of the external substrate potential),the qualitative dynamical behavior of the system canchange significantly and an analysis in the absence of aan external noise, such as thermal fluctuations, is not rea-sonable anymore. The last two terms in Eq. (1) take intoaccount thermal effects in the framework of the Langevindynamics approach. The random force C(t) and the viscousdamping term mc _xi are related by the fluctuation–dissipa-tion theorem:

hCðtÞCðt0Þi ¼ 2cmKBTdjt � t0j; ð4Þ

with KB and T denoting the Boltzmann constant and tem-perature, respectively. In describing the microscopic dissi-pative dynamics of nonlinear models, the inclusion of aphenomenological viscous damping c has become a wellestablished procedure. This coefficient represents degreesof freedom inherent in the real physical system, which arenot explicitly included in the model (e.g. substrate pho-nons, electronic excitations, etc.). Simulations [12] haveprovided indirect evidence that such phenomenological vis-cosity terms serves well this purpose.

The numerical integration of Eq. (1) is carried out usingthe velocity Verlet algorithm, suitably modified to handlethe contribution of the stochastic force. It has the advan-tage, compared to the standard Verlet scheme, of using(as basic variables) positions and velocities at the same timeinstant t. The system is initialized with the chain particlesplaced at rest at the uniform separation a. After relaxation,the chain is driven by the external force F, which is in-creased (or decreased) adiabatically. For every value of F

and after reaching the dynamical steady state, the systemcharacteristics of relevant physical interest are measured.If not stated differently, the values m = 1, U = 1, b = 1,

1T=0.0T=0.3

3678 R. Guerra et al. / Surface Science 601 (2007) 3676–3681

and c = 0.7 define our system units. Bf = (mc)�1 denotesthe maximum asymptotic value of the chain mobility.

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

Mobility (B/Bf)

Force (F)

T=1.0T=5.0

Fig. 1. Static friction reduction due to finite temperature effects on theB(F) characteristics for the perfectly CC case (b/a = 1). Hysteresis tends toclose at increasing values of T. Here, as in all subsequent figures, trianglesand circles denote, respectively, the increasing and decreasing process of F.The model parameters are N = 233, K = 5, and q = 0 (no disorder).

3. Results and discussion

Previous works [9–11] report of very interesting resultsconcerning the sliding motion of simple 1D models overdisordered substrates; however, these studies mainly focuson the strongly overdamped regime, where inertia effectsbecome negligible compared to dissipative forces and, con-sequently, hysteresis cannot be observed in the B(F) char-acteristics. For different values of the model parameters q

and T, we explore here the static frictional properties withthe related depinning transition mechanisms and thebehavior of the chain mobility as a function of adiabaticvariations of F.

As experimentally observed in Quartz-Crystal-Micro-balance (QCM) frictional measurements of adsorbed lay-ers, we find that disorder operates differently dependingon the starting geometrical configuration. For commensu-rate interfaces, randomness lowers considerably the staticfriction threshold of the chain. On the contrary, for incom-mensurate mating contacts, disorder favors static pinningdestroying the possible frictionless (superlubric) slidingstates, occurring above the Aubry transition. Indepen-dently of interface geometry and substrate corrugation,the presence of nonzero temperatures can, in turn, inducesliding at very low values of the external driving (thermolu-bricity) [13]. Interestingly, thermal and disorder effectsstrongly influence also the occurrence of parametric reso-nances [14] inside the chain. We note that so far, we havetaken into account thermal (T 5 0, q = 0) and disorder(T = 0, q 5 0) effects separately.

3.1. Temperature effects

Fig. 1 shows, for four different values of temperature T

and no disorder (q = 0), the behavior of the chain mobilityB as a function of increase and decrease of the driving force

Fig. 2. Detailed depinning transition mechanism (atomic trajectories)

F. This figure corresponds to the perfectly commensuratecase (when the number of chain atoms N coincides withthe number M of minima of the substrate potential). Whilefor T = 0, a large hysteresis is clear, at finite temperaturevalues the hysteretic loop tends to close. Recent studiesof the one-dimensional low-dissipative FK model [15] hasshown the occurrence of hysteresis in the depinning transi-tion at sufficiently low temperature. Disturbing the in-phase particles-substrate commensurability and inducingthe formation of chain defects, thermal fluctuations reduceconsiderably, and even suppress, the static friction thresh-old at which the depinning transition takes place. Fig. 2displays the characteristic detailed behavior (atomic trajec-tories versus time) at the depinning transition at a smallnonzero temperature. As shown, the scenario starts withthe thermally induced creation of localized defects (kinks

at a small, but finite, temperature for the perfectly CC interface.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Mobility (B/Bf)

Force (F)

T=0.0T=0.3T=1.0T=5.0

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Mobility (B/Bf)

Force (F)

T=0.0T=0.3T=1.0T=5.0

Fig. 3. Temperature effects on static friction, hysteresis and parametric resonances of the B(F) characteristics for the golden mean IC case, below (K = 2,left) and above (K = 5, right) the Aubry transition. Note that, for K = 5, static friction vanishes event at T = 0. Remaining parameters are N = 233,b=a ¼ 233=144 �

ffiffiffi5pþ 1Þ=2, and q = 0 (no disorder).

0.2

0.4

0.6

0.8

1

Mobility (B/Bf)

q=0.0q=0.1

R. Guerra et al. / Surface Science 601 (2007) 3676–3681 3679

and antikinks), which initiates an avalanche-like growth ofa domain of running atoms, bringing finally the system tothe running state. The reason why defects are so importantdynamically, is that they can move along the chain mucheasily than the atoms themselves. The activation energyfor kink or antikink motion (the so called Peierls–Nabarrobarrier) is always smaller or much smaller than the ampli-tude of the substrate potential. Because the kinks (anti-kinks) correspond to local compressions (extensions) ofthe chain, their motion provides a mechanism for masstransport along the system. Therefore, namely kinks areresponsible for mobility, conductivity, diffusivity, etc. insuch nonlinear models. The higher is the concentration ofkinks, the higher will be the system mobility.

Fig. 3 shows the temperature effects on the mobility-driving characteristics for an IC interface correspondingto the irrational golden mean b=a ¼ ð

ffiffiffi5pþ 1Þ=2.1 For this

incommensurate case, following the FK theory briefly out-lined above, we have to distinguish between the tribologicalbehavior of the system for low and high values of the chainstiffness K, that is, below and above the Aubry transition,respectively. Left panel of Fig. 3 highlights the capacityof thermal fluctuations to suppress the static frictionalforce, which is finite at T = 0 below the Aubry transition.Above Aubry (right panel), the existence of regimes withvanishing depinning threshold is not hampered by thermaleffects. It should be noted that in both these plots, if, onone hand, temperature tends to reduce significantly thehysteretic loops clearly visible at T = 0, on the other its

1 Theoretically, when simulating an infinite system, we are forced by theperiodic boundary constraints to approximate the desired irrationalnumber b/a by a ratio of integers. The continued fraction expansiontechnique [16] allows us to check quantitatively that a given rationalapproximately (e.g. N/M = 233/144 for the golden mean case) is ofsufficiently high order to model the desired incommensurability.

fluctuations destroy the occurrence of parametric reso-nances inside the chain, as possible channels for energy dis-sipation, and favor sliding with higher mobility values.

3.2. Substrate disorder effects

In order to introduce substrate randomness effectivelyand to avoid possible artificial effects due to periodicboundary conditions in mimicking an infinite system, weneed to increase the system size considerably; this shouldguarantee an efficient (reasonable exhaustive) sampling ofthe Gaussian center distribution.

As shown in Fig. 4, we find that substrate disorder low-ers considerably the static friction threshold starting from a

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Force (F)

q=0.5q=0.9

Fig. 4. Static friction reduction due to different degrees q of substratedisorder on the B � F characteristics for the perfectly CC case b/a = 1 atzero temperature, T = 0. Hysteresis remains clearly visible even for highlydisordered substrates. Simulation parameters are K = 5, N = 1597.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Mobility (B/Bf)

Force (F)

q=0.0q=0.1q=0.5q=0.9

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Mobility (B/Bf)

Force (F)

q=0.0q=0.1q=0.5q=0.9

Fig. 5. Disorder effects on the mobility-force curves for the golden mean IC case, below (K = 2, left) and above (K = 5, right) Aubry. Deviating from theideal incommensurate geometry, even a tiny amount of substrate disorder leads to static pinning. Parameters are N = 1597, b=a ¼ 1597=987 �

ffiffiffi5pþ 1Þ=2,

and T = 0.

3680 R. Guerra et al. / Surface Science 601 (2007) 3676–3681

perfectly commensurate configuration (N = M). Random-ness induces chain spatial defects having a much lower acti-vation energy than the particles themselves. Similarly towhat happens in the presence of thermal fluctuations, thesharp feature of the depinning transition is smeared outfor increasing values of the disorder parameter q; however,the hysteretic loops still remain well visible here. We pointout that for large enough q, there exists a finite probabilityfor two spatially contiguous Gaussians to overlap in such away to get rid of the minimum in between. This fact altersthe commensurability ratio N/M at the interface, with pos-sible drastic changes in the static and dynamic frictionalproperties.

Fig. 5 displays the disorder effects on the B(F) charac-teristics for the golden mean IC interface below (left panel)and above (right panel) the Aubry transition, respectively.Substrate randomness destroys the ideal sliding configura-tion due to the incommensurate mating contact favoringpinning. Below the transition (left panel), the static frictiontends to increase with increasing values of disorder and thedepinning transition becomes smooth. Interestingly, forhigh enough values of q, we observe a crossing betweenthe curves obtained for adiabatic increase and decreaseof F. The system seems to lock again (B = 0) at a largervalue of the external applied force as compared to the cor-responding depinning threshold. Above the Aubry transi-tion (right panel) and any q > 0 the frictionless regimedisappears and the static friction becomes finite. Due tothe relevant chain stiffness, the depinning transition re-mains here quite sharp. At some high applied forces F, dis-order (even if less efficient than temperature) can assistsliding by hampering the occurrence of parametric reso-nances inside the chain, capable of converting the kineticenergy of the CM motion into internal vibrationalexcitations.

4. Conclusions

In this work, we consider the effects of temperature andsubstrate disorder on the underdamped frictional dynamicsof a one-dimensional chain of interacting atoms drivenby an external dc force. Starting from the two geometri-cally opposite ideal cases of commensurate and incommen-surate interface, the model parameters can be tuned inorder to introduce randomness effects. Via a Langevinmolecular dynamics approach, the role of temperature isinvestigated through the inclusion in the equations of mo-tion of a stochastic force coupled to a viscous dampingterm. While the temperature always favors the depinningtransition, we find that disorder operates differentlydepending on the starting geometrical configuration. Forcommensurate interfaces, randomness lowers significantlythe static friction threshold. On the contrary, for incom-mensurate mating contacts, disorder favors pinningdestroying the possible frictionless (superlubric) slidingstates. Interestingly, temperature and disorder also affectsignificantly the occurrence of parametric resonances insidethe chain, as possible channels for energy dissipation.

Acknowledgements

This research was partially supported by PRRIITT(Regione Emilia Romagna), Net-Lab ‘‘Surfaces & Coat-ings for Advanced Mechanics and Nanomechanics’’(SUP&RMAN) and by MIUR Cofin 2004023199.

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