Friction Dynamics of Confined Weakly Adhering Boundary Layers

Mar 8, 2008 - of adhesive bonds between the two shearing surfaces with an additional viscous ... Thin, low-shear-strength liquid layers are often inte...
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Langmuir 2008, 24, 3857-3866

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Friction Dynamics of Confined Weakly Adhering Boundary Layers Denis Mazuyer,† Juliette Cayer-Barrioz,*,† Andre´ Tonck,† and Fre´de´ric Jarnias‡ Laboratoire de Tribologie et Dynamique des Syste` mes - UMR 5513 CNRS, Ecole Centrale de Lyon, 36 aVenue guy de collongue, 69134 Ecully Cedex, France, and TOTAL France Centre de Recherche de Solaize, BP 22, Chemin du Canal, 69360 Solaize, France ReceiVed October 11, 2007. In Final Form: January 18, 2008 The nanotribological behavior of self-assembled monolayers is investigated. The latter accommodate friction through transient relaxation and dilatancy effects whose kinetics depends on the structure of the confined layers. Thus, the molecular ordering onto the surfaces controls the level and the stability of the friction coefficient. Moreover, the behavior of these systems is theoretically accounted for using a model based on the kinetics of formation and rupture of adhesive bonds between the two shearing surfaces with an additional viscous term.

Introduction Thin, low-shear-strength liquid layers are often interposed between two surfaces in contact in order to facilitate the relative motion of the solids, to reduce friction, and to prevent damage. The shearing of the lubricant accommodates the sliding velocity, and the frictional dissipation is mainly dependent on viscosity.1 Thus, in order to increase the lifetime of contacts, tribologists need to quantify the basic properties of the lubricant and the surfaces that it is separating. Previously, the bulk properties of these materials (lubricant and solids) were sufficient. In most lubrication process (metal-forming, valve train, bearings, etc.), this is no longer true, and a knowledge of the mechanical properties of solids and lubricants is required on a scale that is small or comparable with the film thickness. Nowadays, this means a scale in the range of 10-9-10-7 m at which the surface phenomena and the interface confinement cannot be neglected anymore.1 On this scale, the interfacial behavior changes significantly: relaxation times become orders of magnitude higher than those of the bulk, and thin lubricant films may have solidlike properties.2 This is why the study of sliding friction has recently come into focus in tribology in order to understand its genesis on the molecular scale.2-4 However, in spite of its great practical importance,5,6 the microscopic origin of sliding friction is not well understood.7-10 Nevertheless, its physics has benefited from significant progress over the past 15 years with the extension of the SFA technique toward shear solicitation. This is the basis of * Corresponding author. Tel: +33 4 72 18 62 8. Fax: + 33 4 78 43 33 83. E-mail: [email protected]. † Laboratoire de Tribologie et Dynamique des Syste ` mes - Ecole Centrale de Lyon. ‡ TOTAL France - Centre de Recherche de Solaize. (1) Mazuyer, D.; Tonck, A.; Cayer-Barrioz, J. In Superlubricity; Erdemir, A., Martin, J.-M., Eds.; Elsevier: Amsterdam, 2007; p 397. (2) Granick, S. Science 1991, 253, 1374. (3) Singer, I. L., Pollock, H. M., Eds. Fundamentals of Friction: Macroscopic and Microscopic Processes; NATO ASI Series E : Applied Science; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1992; p 220. (4) Persson, B. N. J., Tosatti, E., Eds. Physics of Sliding Friction; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1996. (5) Robbins, M. O.; Mu¨ser, M. H. In Handbook of Modern Tribology; Bhushan, B., Ed.; CRC Press, 2000. (6) Bowden, F. P.; Tabor, D. Friction and Lubrication of Solids; Clarendon Press: Oxford, England, 1964. (7) Rabinowicz, E. Friction and Wear of Materials; Wiley: New York, 1965. (8) Landman, U.; Luedtke, W. D.; Burnham, N. A.; Colton, R. J. Science 1990, 248, 454. (9) Sokoloff, J. B. J. Appl. Phys. 1992, 72, 1262. (10) Shinjo, K.; Hirano, M. Surf. Sci. 1993, 283, 473.

the research line initiated by Israelachvili11 and Georges12 that deals with single microcontacts between smooth surfaces with a typical lateral extension in the range of 1-10 µm under normal stress from 10 to 100 MPa. This approach allows an accurate investigation of the rheology of individual asperities and the effects associated with the interfacial materials involved in boundary lubrication. In this regime, a thin film is confined between the surfaces and displays specific properties that are very different from the viscous behavior of the bulk. It is usually admitted that the change in material properties under confinement can be attributed to a liquid/solid transition.13,14 In some cases, layering occurs in the interfacial material, and the situation is similar to a liquid-crystal transition.13,15-17 Usually, the liquid/ solid transition occurs without any ordering and belongs to a large class of structural transitions.16,18,19 For some lubricants, either a glassy or layering transition can be observed depending on temperature,18 dwell time, surface roughness,20 and commensurability of the surface and the film.17 Generally, the solidlike ordering in the film leads to the development of a static yield stress and stick-slip instabilities. The existence of this selfsustained regime is related to different parameters such as the relaxation time, τmech (roughly the reciprocal of the eigenfrequency) of the measuring apparatus, the operating conditions (pressure, velocity, and temperature), the nature of the surfaces (dry or covered with adsorbed molecules, grafted, etc.), and their characteristics (mechanical properties, rough or corrugated surfaces, etc.), and the shape of the confined fluid molecules.21 Even though in practice the use of adequate lubricants may reduce or even cause these instabilities to vanish, transient effects and stick-slip oscillations reveal the history dependence of the frictional signature. Therefore, their investigation provides richer data concerning the microscopic mechanisms responsible for the dissipation than do stationary states. Indeed, if simple (11) Israelachvili, J. N.; Mc Guiggan, P.; Homola, A. M. Science 1988, 240, 189. (12) Georges, J.-M.; Tonck, A.; Mazuyer, D. Wear 1994, 175, 59. (13) Gee, M. L.; Mc Guiggan, P.; Israelachvili, J. N.; Homola, A. M. J. Chem. Phys. 1990, 93, 1895. (14) Thompson, P. A.; Robbins, M. O. Phys. ReV. 1990, A41, 6830. (15) Schoen, M.; Rhykerd, C. L.; Diesler, D. J.; Cushman, J. H. Science 1989, 254, 1223. (16) Thompson, P. A.; Grest, G. S.; Robbins, M. O. Phys. ReV. Lett. 1992, 68, 3448. (17) Gao, J.; Luedtke, W. D.; Landman, U. Phys. ReV. Lett. 1996, 79, 705. (18) Gourdon, D.; Israelachvili, J. N. Phys. ReV. 2003, E 68, 21602. (19) Demirel, A. L.; Granick, S. Phys. ReV. Lett. 1996, 77, 2261. (20) Gao, J.; Luedtke, W. D.; Landman, U. Tribol. Lett. 2000, 9, 3. (21) Drummond, C.; Israelachvili, J. N. Phys. ReV. 2001, E63, 41506.

10.1021/la703152q CCC: $40.75 © 2008 American Chemical Society Published on Web 03/08/2008

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lubricants consisting of small molecules have fast relaxation processes, then long memory effects are observed for complex fluids.22,23 In the case of polymer surfaces and self-assembled monolayers, nanoscale shear measurements also display long relaxation times and a characteristic “memory distance”.24-26 The measured times are much longer than τmech, showing that the dynamics is directly related to the boundary layer. The glassy state of the confined interface and the existence of memory effects indicate that the tribological behavior of these molecular systems is very close to that observed on a much more macroscopic scale with multicontact interfaces27-30 or in seismic phenomena.31 Even though the nature of the stick-slip oscillations and the transition from stable to unstable sliding are very different according to the molecular organization under confinement, they have common characteristics: Their dynamics is determined by intrinsic properties of the boundary layers. Most of the experiments reported in the literature are carried out within over-damped conditions in which the response time of the mechanical system is much shorter than characteristic slip time of the film itself. Whatever the shape of the fluid molecules (spherical, linear, branched, brush, etc.), their friction trace is governed by long memory distances. This first indicates the presence of slip domains or long-range cooperativity extended over lateral distances that are very large compared with molecular dimensions.22,32 Because the associated relaxation times are retarded by the confinement, a detailed description of the sliding history is required to obtain a complete picture of the physical shear processes and to be able to predict how the system will accommodate the change in tribological conditions. These memory effects that link the microscopic and the macroscopic scales in friction processes are the basis of the socalled phenomenological “state and rate” models used to describe the frictional response of dry/boundary lubricated single-asperity or multiasperity contacts as initiated by Ruina for the friction of rocks.33 The rate variable refers to the instantaneous sliding velocity, and the state variable is meant to capture all of the history-dependent effects. This approach assumes that the interfacial area is large enough to be self-averaging. Therefore, the mean-field state variable is sufficient to model the collective dependence of friction both on the internal degrees of freedom of the interfacial materials and on the dynamical variables characteristic of the shear motion. This is why by relating the state variable to the average lifetime of individual contacts Ruina’s constitutive equations have been successfully applied to dry friction between solids with micrometer-scale roughness.27,31,34,35 Carlson and Batista36 used this phenomenological approach to describe the temporal evolution of a single contact in boundary lubrication by associating the state variable with the degree to which the lubricant is melted. Even if they are effective in (22) Drummond, C.; Israelachvili, J. N. Macromolecules 2000, 33, 4910. (23) Dhinogwala, A.; Cai, L.; Granick, S. Langmuir 1996, 12, 4537. (24) Luengo, G.; Heuberger, M.; Israelachvili, J. N. J. Chem. Phys. 2000, 104, 7944. (25) Cayer-Barrioz, J.; Mazuyer, D.; Tonck, A.; Kapsa, Ph.; Chateauminois, A. Tribol. Int. 2006, 39, 62. (26) Georges, J.-M.; Tonck, A.; Loubet, J.-L.; Mazuyer, D.; Georges, E.; Sidoroff, F. J. Phys.1996, 6, 57. (27) Baumberger, T.; Berthoud, P. Phys. ReV. 1999, B60, 3928. (28) Berthoud, P.; Baumberger, T. Phys. ReV. 1999, B59, 14313. (29) Bureau, L.; Baumberger, T.; Caroli, C. Eur. Phys. J. 2002, 8, 331. (30) Bureau, L. Ph.D. Dissertation, University of Paris 7, 2002. (31) Dieterich, J. J. Geophys. Res. 1979, B 84, 2161. (32) Demirel, A. L.; Granick, S. Phys. ReV. Lett. 1996, 77, 4330. (33) Ruina, A. L. J. Geophys. Res. 1983, B 88, 10359. (34) Rice, J. R.; Ruina, A. L. J. Appl. Mech. 1983, 105, 343. (35) Baumberger, T.; Caroli, C.; Perrion, B.; Ronsin, O. Phys. ReV. 1995, E51, 4005. (36) Carlson, J. M.; Batista, A. A. Phys. ReV. 1996, E53, 4153.

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capturing steady-state and transient effects on a wide variety of materials, the friction laws that these mechanisms inspire are not based on the underlying microscopic physics of the interface. Indeed, either in dry friction or in boundary lubrication, the dissipative pinning/depinning of domains whose lateral dimension is between the size of a microcontact and the molecular size governs the interfacial rheology and the frictional response of contacts. These elementary units are the analogs of the elementary volumes of plastic deformation or the shear transformation zone (STZ)37 used to describe the plastic flow of amorphous materials. The latter are at the origin of the multistability of microcontacts in dry friction considering that their stress response involves nanometer-thick amorphous adhesive joints in which the shear is localized.27 Within this framework, Persson38 proposed a model inspired by the older spring-block theory of Burridge-Knopoff39 used to describe the shear of confined lubricants. As in the nanoblock model used in ref 27 to schematize a single contact, the boundary layer is made of a set of pinned, solidlike islands called blocks or domains immersed in a 2D confined fluid. Persson introduced thermal processes that activate the nucleation, growth, or death of these fluid and/or frozen domains during sliding and stopping.40 By simulating these islands, with coupled mechanical oscillators, the local liquid/solid transitions are correlated to the nature of sliding instabilities. Assuming a size distribution of the blocks, a broadband spectrum of relaxation times emerges that is often associated with chaotic instabilities as observed for branched molecules.21 The transition between stick-slip/steady sliding and the transient response of the friction force in stopstart tests have also been successfully modeled by an intermediate statistical theory that generalizes the STZ theory where the effects of glassy relaxation are treated via the introduction of a state variable related to the internal free volume.41 The comparison between experiment and theory clearly shows that the understanding of the tribological response of engineering-type contacts has to account for the coupling between the lifetime of the bearing asperities and the interfacial rheology that both contribute, through their own dynamics, to the level and stability of the friction force. However, to be related to the friction coefficient, this rheology cannot be restricted to only average mechanical properties such as viscosity or elastic modulii but should include the heterogeneity of the interface induced by both shearing and confinement. Considering the interface materials as a set of statistically treated interacting mesoscopic domains seems to be a promising way of modeling the friction and controlling its level in boundary lubrication.1 To reach this goal, long-chain, oil-soluble surfactants are often added to liquid lubricants and form an important class of lubricant additives termed organic friction modifiers. The traditional view concerning the mechanism by which these compounds control friction is that they form physically or chemically adsorbed monolayers on polar solid surfaces and that these monolayers reduce adhesion between contacting asperities and thus limit junction growth.6 This concept has been supported by studies that have shown that both deposited42 and self-assembled43 monolayers on solid surfaces can indeed reduce friction. Nevertheless, even if this assumption seems acceptable, then it is not sufficient to explain the microscopic physical processes that govern the level and stability of the friction coefficient. In (37) Falk, M. L.; Langer, J. S. Phys. ReV. 1998, E57, 7192. (38) Persson, B. N. J. Phys. ReV. 1994, B 50, 4771. (39) Burridge, R.; Knopoff, L. Bull. Seismol. Soc. Am. 1967, 57, 341. (40) Persson, B. N. J. Phys. ReV. 1995, B51, 13568. (41) Lemaıˆtre, A.; Carlson, J. Phys. ReV. 2004, E69, 61611. (42) Dominguez, D. D.; Mowery, R. L.; Turner, N. H. Tribol. Trans. 1994, 37, 59. (43) Spikes, H. A.; Cameron, A. Proc. R. Soc. London 1974.

Friction Dynamics of Boundary Layers

this article, we are dealing with the molecular mechanisms and the associated dynamics of friction between weakly adhering compressed brushes that are often used in boundary lubrication. Although the asperity interactions in a rubbing contact produce very high local stresses and surface deformations, which may lead to chemical processes, we show that the monolayer shearing behavior in idealized smooth contacts can bring about new insight into friction control on the macroscopic scale by playing with the shape and the organization of friction-modifier molecules. The extremely low friction that such layers promote under moderate pressure ( 100 nm), the surfaces can be considered to be rigid solids, and the contact pressure results from the Derjaguin approximation. The latter links the normal force FZ(D) between a sphere (radius of R) and a plane and the energy per unit surface W(D) between two planes at the same distance D as follows: FZ(D) ) 2πRW(D)

(2)

Therefore, at large distance when no elastic deformation appears, the mean contact pressure is the disjoining pressure and can be calculated from the relation Π(D) )

dW(D) 1 dFZ(D) ) dD 2πR dD

(3)

Procedures. A couple of solid samples (sphere and plane) are prepared and cleaned for each test. The sample is mounted on the molecular tribometer. As soon as the sphere/plane distance D reaches 10 µm, a droplet of liquid is carefully deposited between the two surfaces. The actual experiments are started after an adsorption time of 12 h at this distance. The sphere/plane separation is monitored with the contact capacitance. From an initial operating distance of 1 µm, the quasi-static force is measured as a function of D by making inward and outward

motions with a constant speed of 0.15 nm/s. From this quasi-static squeeze of the interface, the thickness of the surface layers within the contact can be evaluated as detailed in ref 1. The sphere/plane distance of 2L corresponding to the onset of normal force increase during the initial loading characterizes the first contact between the molecules layers adsorbed on both surfaces. When the sphere/plane distance is reduced, the thickness of the adsorbed layer decreases from L to LC as a result of confinement effects. The interaction is similar to “hard sphere” repulsion, and the layers behave as an elastic wall. Its thickness is referred to as LC. It depends on the normal force FZ. In the dynamic mode, the mechanical impedance of the interface that is measured by superimposing an oscillatory motion of a given amplitude and pulsation, ω, is divided into two additive components. The first one is the conservative part coming from the in-phase response of the interface, which gives its elastic stiffness KZ(ω). The second one is the dissipative part coming from the out-of-phase response of the interface, which gives its viscous damping AZ(ω). For large sphere/plane distances and with a homogeneous Newtonian liquid, the Stokes’ law describes the hydrodynamic flow, and the associated damping function AZ is given by AZ )

6πηR2 D

(4)

where D is the sphere/plane distance, η is the bulk viscosity of the liquid, and R is the sphere radius. The presence of an adsorbed layer drifts the wall where the flow velocity vanishes toward LH over the solid surface. This defines an infinite viscosity layer within which the molecules are not perturbed by the flow. Thus, accounting for this new boundary condition, the damping function becomes AZ )

6πηR2 D - 2LH

(5)

According to eq 5, the plot of 1/AZ as a function of the distance D allows us to determine the bulk viscosity of the lubricant and the hydrodynamic length LH. When lower than LC, LH results from the capability of the fluid to flow through the molecules that build up structures that are more or less ordered by their interactions with the surfaces. This local organization slightly roughens the surface, and the volume fraction of the adsorbed molecules can be deduced by an appropriate modeling of the hydrodynamic flow through the surface microgeometry. Using a multilevel method51 to solve the Reynolds equation numerically and to compute the viscous fluid damping as a function of the sphere/plane distance, it has been found that the ratio LH/LC was exactly equal to the surface coverage of molecules (particles) whatever their shape and height.52 Thus, in the case of brushlike layers, the ratio of LH to LC is related to the layer heterogeneity: if LH/LC is close to 1, then the surfaces are covered by a dense homogeneous monolayer, but if LH/LC < 1, then the monolayer is incomplete. The surface density of the layer before confinement is given by LH/L. Loading/unloading experiments also serve as a systematic control to examine the adsorbed layer integrity following shear measurements. (51) Venner, C. H.; Lubrecht, A. A. In MultileVel Methods in Lubrication; Tribology Series 37; Elsevier: Amsterdam, 2000. (52) Bec, S.; Tonck, A.; Georges, J.-M.; Yamaguchi, E. S.; Ryason, P. R. Tribol. Trans. 2003, 46, 522.

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Figure 3. Schematic evolution of the frictional force versus the sliding distance during an experiment at constant normal load (a) with no static force and (b) with static force. After a linear phase characterized by the elastic stiffness KX, the tangential force Fx reaches a limited value FXl. The transition between these two periods occurs at a sliding distance X* ) FXl/KX. The vibration superimposed on the tangential motion results in small reversible cycles with a slope KX, leading to a continuous measurement of the tangential stiffness during the friction process. Table 2. Experimental Conditions Used for the Interface Characterization in Dynamic Mode, Both in the Normal and Tangential Directions, during the Squeeze and Nanofriction Experiments experimental conditions

normal direction

tangential direction

velocity (nm/s) vibrational frequency Ω (Hz) vibrational amplitude ∆X (nm)

0.01 37 0.1

0.05-500 70 0.03

To characterize the tribological behavior of the friction-modifier layers, friction experiments have been performed at constant normal force and for various sliding velocities ranging from 0.05 to 500 nm/s. The relative tangential displacement of the plane and the sphere is defined as the sliding distance, X. The friction force is measured as a function of the sliding distance at constant normal load. The effects of sliding velocity (from 0.1 to 500 nm/s) on the accommodation of adsorbed layers to shear are more particularly investigated. As schematically shown in Figure 3a,b (in the case of the existence of static friction), the evolution of the tangential force FX follows two periods: - a linear reversible period described by tangential stiffness KX and - a nonlinear period where the tangential force increases (and then decreases for Figure 3b) until an equilibrium value of FXl is reached. The length noted X*, which represents the threshold beyond which the interface is no longer elastically deformed and starts sliding with energy loss, is defined in both cases as X* )

FXl KX

(6)

Moreover, oscillatory motions are superimposed on the normal and tangential displacements to measure the viscoelastic properties of the sliding interface in both the normal and tangential directions. For that purpose, appropriate amplitudes and frequencies are chosen

Figure 4. (a) Attractive and repulsive forces measured during the interface quasi-static squeeze of the two liquids as a function of the sphere-plane distance for (∆) the amine solution and (0) the phosphite solution. The inset shows that the adhesion force is well fitted by the van der Waals law, which is consistent with a Hamaker constant of 1.5 × 10-19 J. There is no (little) hysteresis for the amine solution (phosphite solution) between loading and unloading. (b) Plot of the damping function versus the sphere-plane distance D for (∆) the amine solution during the quasi-static squeeze of the interface. At D > 200 nm, the slope is constant and corresponds to a viscosity of 17.0 ( 0.2 mPa s. This value is coherent with the value obtained for the bulk solution at 23 °C. The hydrodynamic thickness LH is measured from the intercept of the distance axis with the fit of the curve at large distance. to avoid microslip that could be induced by the tangential sinusoidal displacement. As a consequence, the amplitude ∆X of the tangential vibration must be inferior to the critical distance X* associated with the adsorbed layers (Table 2). The experimental conditions that must also take into account the loss of resolution due to the nonlinearity of the friction process, are summarized in Table 2. Because the amplitude ∆X of the oscillatory tangential motion is sufficiently small to prevent additional slipping and any change in the friction measurement, the measured tangential force corresponds to the force T ) KX∆X that is required to shear the interface of thickness D elastically along a sliding distance ∆X. Assuming that the interface is homogeneously sheared over all of its thickness in its elastic domain, the associated mean shear stress τ is given by Hooke’s law τ)

T KX∆X ∆X ) )G h S S D

(7)

where S is the contact area. Because the mean contact pressure is P h ) FZ/S, eq 7 leads to the mean shear modulus G h normalized by pressure P h: G h K XD ) P h FZ

(8)

Results and Discussion Interface Characteristic Thicknesses. The evolutions of the normal static force according to the sphere/plane distance

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Table 3. Thickness of the Adsorbed Surface Layers and Surface Coverage Ratio for the Two Liquidsa

liquid amine/PAO phosphite/PAO a

2LC 2LC (nm at (nm at 2LH 2L LH/LC (nm) (nm) 0.1 mN) 1 mN) LH/L (at 0.1 mN) 4.25 4.50

6.30 8.40

4.50 5.80

4.25 4.50

0.67 0.53

0.94 0.78

The thickness of the confined interface at 1 mN is also reported.

measured for the amine and phosphite solutions show very little hysteresis between the loading and the unloading curves (Figure 4a). This indicates that the interface made of the contacting monolayers behaves as an elastic wall. Before the repulsive part of the curve, an attractive force is measured. In both liquids, it is well fitted by the van der Waals’ law with a Hamaker constant of 1.5 × 10-19 J (nonretarded effects over a distance range of 5-20 nm). Thus, this weak adhesion is taken into account in the JKR contact model to determine the parameters of the elastic contact versus normal FZ (i.e., contact radius, mean contact pressure, and elastic deformation of the solids. The variation of the damping function 1/(Azω) for the amine solution is plotted versus D in Figure 4b: it follows eq 5. At large distance (D > 200 nm), the slope of the curve gives the viscosity. The best fit of the curve at large distance intercepts the distance axis at 2 LH, which defines the hydrodynamic thickness. All of the thicknesses that characterize the interface are summarized in Table 3. The distance of the onset of repulsion, 2L, and the confinement distance, 2LC (measured at a normal load of 0.1 mN corresponding to a mean contact pressure of 10 MPa) divided by the thickness of the immobile layer, 2LH, is equal to the surface density before and during confinement, respectively. From the values of Table 3, it can be deduced that - The hydrodynamic distance LH and the thickness of the confined layer at 1 mN are close to the length of the individual molecule in both cases. This suggests that the plane and the sphere surfaces are covered with a nearly complete, homogeneous monolayer of friction modifier. The low thickness of these layers combined with their low compliance validates the use of JKR theory to estimate the contact pressure. - The distance 2L is much larger than the size of two monolayers in contact for both friction modifiers. It can be assumed that at low confinement PAO molecules remain within the contact, between the two monolayers. - As confinement increases, the distance L reaches the value of Lc at Fz ) 1 mN. Considering that 2LC is approximately the size of two monolayers in contact, one may suppose that PAO molecules are finally squeezed out at severe confinement (i.e., for Fz ) 1 mN). Therefore, friction experiments will be carried out at a normal force of 1 mN. - The softer increase in the static normal force observed for phosphite molecules compared to that for amine molecules, when the sphere/plane distance is close to the distance 2L, might be attributed to the interactions between PAO and the different monolayers. - The confinement increases the surface density of the layer adsorbed from both liquids. Moreover, before and under confinement, the surface density of the amine layer is higher than that of the phosphite layer. Friction and Elasticity of the Friction Modifier Layers: Pressure Effect. The elastic properties of the friction modifiers layers contribute to their frictional behavior and depend on their confinement state. According to ref 1 and eq 8, the compressive E h and shear elastic modulii G h have been calculated for the two friction modifiers under confinement during a squeeze experiment,

Figure 5. Evolution of the elastic properties of the adsorbed layers of amine and phosphite molecules under confinement during a squeezing experiment. G h is the shear elastic modulus, and E h is the compressive elastic modulus. For the amine solution, the distance D ranges from 4 to 6 nm: (∆) G h and (2) E h . For the phosphite solution (6 < D < 8 nm), (-) G h and (--) E h.

and their evolution is shown in Figure 5. The large variability in the measured compressive elastic modulus corresponds to the noise of the measurement. These curves clearly show that the elastic modulii of the interface increase with confinement. At low confinement, one may assume that PAO molecules remain within the contact whereas at high confinement the interface consists of two monolayers in contact. Therefore, the elastic properties of the monolayers can be determined at high confinement. As soon as the contact is loaded, the compressive elastic modulus of the amine layer immediately reaches a high value of about 2 GPa. G h reaches 1.2 MPa at high confinement. The phosphite monolayers are stiffer: the shear elastic modulus G h attains a stationary value of 25 MPa at high confinement. For both friction modifiers, the shear elastic modulus is much lower than the compressive elastic modulus: the ratio E h /G h is 2000 for the amine layer and 130 for the phosphite layer. This suggests that the confined layers exhibit a significant nonisotropic mechanical behavior. Sliding Velocity Accommodation. The frictional behavior of the monolayers has been tested at constant normal load Fz ) 1 mN (contact pressure of 30 MPa) and varying sliding velocities. The normal load has been chosen to ensure that the PAO molecules are squeezed out of the contact. Figure 6 shows the evolution of the friction force for the amine solution versus sliding distance X and the kinetics of its variations when a sequence of increasing and decreasing sliding velocities is applied. During the whole test, the normal force is maintained constant at 1 mN, and corresponding contact diameters are roughly 6 µm. The sliding speed is increased or diminished in four successive stages: 0.1, 0.6, 3, and 12 nm/s. The sliding distance is less than 0.5 µm (i.e., much less than a contact diameter). However, the main results obtained with this lubricant are detailed as follows: The amine layers exhibit an extremely low friction coefficient (only few thousandths), offering outstanding superlubricity properties in the studied range of sliding velocities. The molecular tribometer is able to measure accurately significant tangential forces for sliding speeds as small as 0.1 nm/s, which is sufficiently slow to assume that the resulting friction force is comparable to a static friction coefficient. This point is confirmed by the occurrence of a sharp peak at the beginning of the sliding (Figure 6). The amine lubricant exhibits liquidlike behavior because each increase (decrease) in the sliding velocity produces an instantaneous transient increment (decrement) of the friction force. This interface accommodation of the interface to speed variations

Friction Dynamics of Boundary Layers

Figure 6. Friction trace of an amine bilayer induced by a series of successive increasing/decreasing sliding velocities. The interface accommodates these shear solicitations by an immediate transient viscous response followed by a slow relaxation of the friction force along a memory length of about 20 nm. This memory length is independent of the velocity. Beyond this displacement, the interface forgets the history of previous sliding kinetics, which explains that these transient effects are completely reversible. The dilatancy of the confined film is induced by shear. Thickness variations of a few tenths of a nanometer are associated with the change in velocity: when the velocity is increased, the tangential force decreases, and the thickness of the interface increases.

suggests that a viscous component contributes to the friction coefficient. After three successive increases in sliding speed by a factor 4, the increments of the stabilized force preceding the speed shift are 13, 17, and 9%, respectively. As explained in the following part, the shear viscosity of the interface can be deduced by combining these experimental results with the estimation of the apparent contact area and the thickness of the shear plane. A lower steady-state frictional force characterizes the friction regime that follows this transient response. To be achieved, the latter requires an adaptation of the interface to the change in the contact kinematics over a sliding distance D0 of a few tens of nanometers. This distance is velocity-independent. A slight increase/decrease in the interface thickness (a few tenth of nanometers) is associated with this accommodation: for the amine solution, a 0.1 nm dilatancy of the confined film induced by shear is measured. This variation in film thickness is temporary: the film recovers its original thickness after a sliding distance of D0. This value of dilatancy seems to be independent of the velocity. The frictional response of the phosphite layers to similar increases/decreases in sliding velocities is presented in Figure 7 versus the sliding distance. It can be shown that - the phosphite layers exhibit a higher friction coefficient of about 0.055; - the velocity dependence on the tangential force is less important and the interface thickness remains constant; and - the viscous contribution to the frictional force (instantaneous effect) is lower in relative value but twice as high in absolute value. With this formulation, the ratio between the dissipative and the conservative parts (respectively ωAx and Kx) of the tangential mechanical impedance is lower than that of the amine formulation: ωAx/Kx is about 0.11 for the phosphite solution versus 0.40 for the amine solution. However, the dissipative part for the phosphite is higher than that of the amine solution, which leads to a shear viscosity of the interface of 3.2 × 103 Pa s, as explained in the following text. In the case of the phosphite monolayer, the velocity effect seems only partially reversible. However, this difference can be

Langmuir, Vol. 24, No. 8, 2008 3863

Figure 7. Evolution of the frictional force of a phosphite layer as a function of sliding distance. The sliding velocity is increased stepwise from 0.1 to 12 nm/s and is then decreased following the same procedure. No evolution of the interface thickness is observed.

attributed to the surface coverage. Indeed, in the case of the phosphite monolayer, LH/Lc is only 0.78, which reveals that the phosphite does not cover completely cover the surfaces. Therefore, the molecular structure of the layer along the sliding zone may be modified, and the friction force can then vary slightly. The loading/unloading experiment also serves as a systematic control to examine the adsorbed layer integrity following shear measurements. Irreversible modifications in the layer structure under these circumstances seem unlikely. Long-term drift in capacitance bridges is also unlikely. When the sliding speed is increased, the tangential force decreases (after a small transient viscous increase). This evolution is all the more rapid because the sliding speed is fast, suggesting that for both friction modifiers these phenomena are governed by a relaxation length rather than by a relaxation time. This length D0 (20 nm for the amine and 8 nm for the phosphite) appears to be a memory distance beyond which the interface has forgotten the history of the previous sliding kinematics. It is velocity-independent. This explains why these transient effects are completely reversible. Indeed, the interface recovers its initial state after a succession of identical steps of increasing/decreasing sliding velocities. According to eq 6, the sliding distance threshold X* is also calculated at high confinement for a normal load of 1 mN. It is longer for the amine molecule (0.72 nm) than for the phosphite molecule (0.24). This difference might be related to the higher compliance of the amine layer. Viscous Dissipation in Friction. Figures 6 and 7 have revealed that the kinetic friction is velocity-dependent: an increase ∆V (respectively decrease) in sliding velocity is accommodated by a transient increase ∆FX(respectively decrease) in the friction force. This clear evidence of a viscouslike contribution to the frictional behavior can be used to estimate an effective viscosity of the interface, thanks to the following relation

ηp )

∆FXd ∆VS

(9)

where S is the mean contact area and d is the thickness involved in viscous shearing. This latter cannot be easily determined because it depends on the properties of the flow in the interface during sliding: - If the shearing concerns the whole interface, then d should be the total thickness of the confined layers. - If the flow is not homogeneous, then a localization of the shear occurs, and d should be the thickness of the shear plane.

3864 Langmuir, Vol. 24, No. 8, 2008

Figure 8. Schematic picture of the shear processes at the molecular level in the kinetic friction regime of the amine layer. The shear is supposed to be localized within an interpenetration zone of thickness δ.

Nevertheless, for the amine layers, the very low dilatancy (less than 0.1 nm, which is much lower than the thickness of the layers themselves) associated with friction accommodation processes suggests that the tangential damping ωAX results from the viscous response of the interpenetration zone between the layers. Thus, the slip is located through the thickness of this zone, which acts as a shear plane (Figure 8), whereas the elastic shear of the interface concerns its whole thickness. In this zone, it can be assumed that the end of the molecules can diffuse through each other as previously mentioned for hydrogenated DLC53 films or for grafted polymer brushes.44,45 The mutual interaction of end-tethered friction modifier chains in the sheared interpenetration zone may explain the viscous contribution to the frictional force and the relaxation effects that have been observed even for purely elastic layers such as amineand phosphite-confined films (Figures 6 and 7). Using eq 9 and assuming that d has an order of magnitude of 0.1 nm for the amine layer, the three speed increments of 0.5, 2.4, and 9 nm/s (Figure 6) induce shear viscosities of 8 × 103, 103, and 102 Pa s, respectively, for a normal load of 1 mN (leading to a contact area of 28.3 µm2). This suggests that the interface has shearthinning behavior. At the same load, the sliding viscosity of the phosphite layer is 30 × 103 Pa s for a speed increment of 0.5 nm/s. Even if this viscosity is higher than that of the amine layer, the viscous contribution to friction is lower because the damping function, ωAX, is only 1/10 of the elastic tangential stiffness KX. These results tend to demonstrate that the amine molecules form liquidlike layers whereas phosphite molecules tend to form solidlike layers, which is consistent with the differences in their frictional properties. It is noteworthy that this behavior can be exhibited only by transient variations of the tangential force that steep changes of speed induce. Physical Interpretation. These experimental results cited in several papers in the literature show that the friction dynamics of the confined layer depends on the organization at the molecular level. The latter is governed by the local interactions of the molecules with the surface but is modified by the sliding process. This results in various times and memory lengths that characterize the relaxation and the accommodation processes involved in the frictional response of the boundary layers. One of the most important difficulties is the proposal of a realistic physical modeling of the shearing of these confined interfaces formed with complex fluids. Because of the extremely long times and length scales involved, the molecular dynamics simulation cannot give all of the answers to this problem. To achieve a satisfactory description of the phenomena, a different or complementary approach has to be taken.1 Persson38 has proposed the use of a thermally activated 2D model of pinned islands in the contact region to describe the (53) Harrison, J. A.; White, C. T.; Colton, R. J.; Brenner, D. W. J. Chem. Phys. 1993, 97, 6573.

Mazuyer et al.

Figure 9. Sketch of junction shearing in a bonded state during a friction test. (a) At rest, the domain is bonded through an interpenetration zone of thickness γ. At the onset of sliding (b), the junction is elastically stretched, and the cooperative motions of the molecules in the entanglement zone result in a viscous contribution to friction. When the junction breaks (c), heat dissipation induced by the vibration of molecules is responsible for the friction dissipation.1

behavior of confined lubricants under shear, with promising results. This theory uses a statistical treatment of the dynamics of the pinning/depinning sites that is responsible for the evolution of the friction coefficient. However, like the phenomenological “state-rate” approach, it does not account for the kinetics of the pinning/depinning activation. Moreover, the major drawback of the existing models is the lack of physical meaning for the variables used in the equations. It is usually unclear how to relate the parameters in the models to molecular properties or to experimentally measurable quantities. This is why we used a general model based on the kinetics of formation and rupture of adhesive bonds between the two shearing surfaces with an additional viscous term to understand the frictional behavior of the friction modifiers on the molecular level.1 This model is derived from a theory of adhesive friction, originally developed for “elastomer” surfaces.40,54 Drummond et al.47 have successfully applied this model to predict new instability regimes (inverted stick-slip) and friction/velocity traces of wetted surfactant monolayers. The confined interface is treated as a viscoelastic medium with a shear elastic modulus G h and a sliding viscosity η. At any shearing time, the total contact area A is assumed to consist of N independent bonds or adhesive nanodomains, called junctions, each of average area δA. During motion, the whole contact area A does not slide as a single unit: individual junctions are continually formed and broken incoherently. Each junction can be either in a bonded state or in a free state, and its activation involves two characteristic times: - τ0, the mean time to break a junction due to thermal fluctuations under zero shear force, is given by

( )

τ0 ) τ* exp -

U0 kT

where τ* is an elementary time and U0 is the energy barrier that has to be overcome to break an adhesive junction - τ, the mean time to activate or reactivate a junction thermally. During sliding, the junctions are elastically stretched until a yield point beyond which they are totally depinned (Figure 9). This results in reducing the energy barrier to transit from the bonded state to the free state. According to Schallamach, the reduction is proportional to the elementary elastic force Fel ) G h δA(Vt/d) applied to one junction of thickness d after a sliding time t. Thus, the energy barrier becomes U ) U0 - γFel, where γ is a constant length. In addition, a junction is assumed to be always depinned when it is stretched up to a yield point reached at a critical deformation (Figure 9b), where tb is the lifetime of the junction. Thus, the average lifetime 〈tb〉 of an adhesive junction can be deduced from the calculation of its survival probability. Following this model, during shear, a junction detaches either (54) Schallamach, A. Wear 1971, 17, 301.

Friction Dynamics of Boundary Layers

Langmuir, Vol. 24, No. 8, 2008 3865 Table 4. Input Parameters Used to Fit the Friction Dynamics of Both Solutions. These Input Parameters Are Issued from Experimentsa 6

G (10 Pa) l* ) 2X* (10-9 m) A (10-12 m2) η (Pa s)

amine

phosphite

3.9 1.6 28.3 900

35 0.5 28.3 1000

a The viscosity η is larger than the bulk viscosity as a result of the confinement effect.

Figure 10. Sliding-velocity dependence of the friction force between two adsorbed monolayers of (a) amine and (b) phosphite at room temperature under a normal load of 1 mN. The black line is the fit obtained with the adhesive elastic friction model. The red (blue) dotted line is the elastic (viscous) contribution calculated from the model. (d) Thickness of the confined interface.

spontaneously, by thermal excitation, or by the external shear force and reforms further after various thermoelastic relaxation processes have occurred. This elastic-adhesive model is not sufficient to capture the friction regimes at high velocity. As a consequence, it is necessary to introduce an additional viscous contribution that arises from the free junctions.47 Therefore, the friction force can be written as

∫1e

kT R

F ) φN

R

(

)

tb du A ln(u) exp u + (1 - φ) ηV (10) u Rτ0 d

where φ(V) ) 〈tb〉/(〈tb〉 + τ) is the fraction of bonded junctions and R ) γδAG h l*/kTd. In eq 10, the first term represents the elastic contribution, and the second one corresponds to the viscous contribution. According to Chernyak,55 eq 10 can be simplified and then takes the following expression

Vτ0 1 - (1 + (l*/Vτ0))e-1*/Vτ0 A + (1 - φ) ηV (11) -1*/Vτ0 l* d 1-e

F ) φF0

where F0 ) A/dGl* is the mean elastic shearing force. The sliding velocity dependence of the friction force obtained for both friction modifiers can be analyzed within the framework of this model. This allows us to give a physical meaning to the model parameters and relate them to experimental data. Figure 10a (Figure 10b) presents the experimental evolution of the friction force as a function of the sliding velocity for the amine (55) Chernyak, Y. B.; Leonov, A. I. Wear 1986, 108, 105.

solution (for the phosphite solution). The main trends of these friction force-sliding velocity curves can be accounted for by theoretical lines issued from the model and plotted in Figure 10a,b. The model predicts that the friction force at low velocity is h l*/2D. By comparison with eq 6, we deduce FT(V f 0) ) AG X* ) l*/2. Then, the threshold sliding distance X* measured by the ratio of the stabilized friction force to the tangential stiffness of the interface gives the critical deformation l* at the yield point beyond which all of the junctions are depinned. Table 4 summarizes the input parameters used for the fit and issued from the experiments. The superlubricity properties of the friction modifier layers are due to an important viscous component of the friction force in addition to the nonisotropic elastic behavior of the layers.1 The analysis of eq 11 shows that the elastic component of the friction force eventually vanishes at high velocities. However, it has been experimentally observed that the friction remains at a nonzero value independent of the velocity for V > 10 nm/s. From eq 11, it can be deduced that the friction force has a mainly viscous origin. As shown previously, the amine interface exhibits shear-thinning behavior. As a consequence, this result has been taken into account in the model in order to predict the increase in friction force with velocity for V > 10-7 m/s. Moreover, for sliding velocities varying from 10-10 to 10-5 m/s, the friction force displayed by both friction modifiers is in the range of few micronewtons under a 1 mN load (Figure 10a,b). Even though the measured viscosity of the confined layer is high, a viscous flow through the whole thickness of the interface (∼5 nm) is not sufficient to explain such high values of the shear force. Thus, the flow should be localized in a much thinner layer, such as the interpenetration zone and the viscous term in eq 11 becoming Aη(1 - φ)V/γ. The output data obtained by comparison between the experimental investigation and the viscous-elastic-adhesive model are detailed in Table 5. The model predicts the existence of various friction regimes, provided that τ , τ0. From Table 5, it can be seen that this condition is respected. Therefore, the evolution of the friction force-velocity curves can be explained for both friction modifiers. For the phosphite, two regimes are observed: in the first one, where l*/τ0 ≈ 10-12 < V < l*/τ ≈ 10-8 m/s, the junctions are mainly elastically depinned, reaching the critical friction force in an uncorrelated manner.47 The friction force is almost velocity-independent because the number of junctions in the bonded state is almost constant. Beyond this “plateau” regime, as V becomes comparable to or larger than l*/τ, the number of junctions in the bonded state N0 decreases because each junction now spends relatively more and more time in the unbonded state.47 For the amine solution, no “plateau” regime is observed, and the friction force decreases as the sliding velocity increases. Moreover, the superlubricity and the importance of the viscous behavior suggest that only a low fraction of junctions in a bonded state is involved in the shearing motion

3866 Langmuir, Vol. 24, No. 8, 2008

Mazuyer et al.

Table 5. Output Data Obtained from the Fit and Experimental Values of the Accommodation Length D0 for Both Friction Modifiersa amine N (junction/area, m-2) N0 (bonded junction/ area, m-2) τ (s) τ0 (s) 〈tb〉 (s) activation volume, γδA (10-27 m3) thickness, γ (10-9 m) δA (10-18 m2) D0 (10-9 m)

5.0 × 10

15

phosphite 1.6 × 1016

5.0 × 1013-2.5 × 1015 1.6 × 1015-1.6 × 1016 1.4 × 10-2-18 1000 1.6 × 10-3-16 20.6

4.5 × 10-4 100 5 × 10-5-5 6.5

0.1

0.1

200

65

20

8

a It can be seen that the time, τ , needed to break a junction is velocity0 independent for both friction modifiers. However, the time to (re)activate a junction for the amine solution is velocity-dependent.

and that the average lifetime of a bonded junction should be shorter than the time required to activate a free nanodomain.1 However, as the sliding velocity increases, the number of bonded junctions decreases: at high speed, most of the junctions are in the free state, and the main contribution to the dissipation comes from the viscous component. Consequently, the lifetime of a bonded junction is expected to be shorter when the sliding is fast. Moreover, once the monolayers have slipped against each other over a sliding distance δA1/2, the whole population of the junctions in the total contact area has been completely renewed. Similarly to what has been demonstrated for solid friction in multi-contact interface,56 δA1/2 defines a characteristic length of the sliding process that can be regarded as the memory length,

D0, governing the friction dynamics1 as illustrated in Figure 6 and Figure 7. In that purpose, experimental values of D0 are reported in Table 5 for comparison with δA. Therefore, the junctions of size δA could correspond to the elementary mesoscopic area around which local shear arrangements occur. Thus, the interpenetration zone behaves as a thin glassy interface whose deformation involves cooperative molecular motion, as modeled by STZ theory.41

Conclusions By using the SFA technique, the friction dynamics of confined, weakly adhering boundary layers has been studied. Two frictionmodifier lubricants have been investigated. The amine layer exhibits liquidlike behavior associated with a sliding-velocitydependent friction force. The phosphite interface behaves as a solid, and the friction force is weakly sliding-velocity-dependent in the studied range of sliding velocities. However, for both friction modifiers, an increase in sliding velocity produces an instantaneous and transient increment of the friction force. These phenomena seem to be governed by a relaxation distance rather than by a relaxation time. Physical modeling of the shearing of these confined layers based on the kinetics of formation and rupture of adhesive bonds between the two shearing surfaces has been used to interpret these transient effects and the associated memory distances from the dynamics of shear-activated pinned and free junctions. It is noteworthy that the model input parameters are issued from experiments. The accommodation distance can be seen as a memory distance beyond which the interface has forgotten the history of the previous sliding kinematics. Moreover, the properties of the entanglement area between the monolayers can be deduced. LA703152Q (56) Heslot, F.; Baumberger, T.; Perrin, B.; Caroli, B.; Caroli, C. Phys. ReV. 1994, E49, 4973.