Transient processes in “soft” tribology: ageing, jamming, healing, …

Transient Processes in Tribology G. Dalmaz et al. (Editors) 9 2004 Elsevier B.V. All fights reserved

Transient processes in "soft" tribology: ageing, jamming, healing, ... Tristan Baumberger Groupe de Physique des Solides, Universit~s Paris 6 & 7, CNRS Campus Boucicaut, 140 rue de Lourmel, 75015 Paris, France I present some selected examples of transient processes at work in model systems under "soft" tribological conditions where wear effects are minimized and true steady frictionnal states can be achieved. I show that, however, structural ageing and shear-induced rejuvenation of the interface induce rich and complex dynamical behaviours. 1. H Y S T E R E T I C

FRICTION

Many frictional interfaces are incompletely characterized by their steady sliding characteristics. When a time-varying sliding velocity is forced, the friction response is hysteretic [1-3]. This also emerges from the detailed analysis of stick-slip oscillations, when these occur, especially close to the bifurcation between steady sliding and stick-slip. Figure 1 shows two examples of such hysteretic force responses for paper sliding on paper and for a rosined contact which mimics the bow/string contact in a violin [2]. One is therefore led to conclude that, for these systems, besides the instantaneous sliding rate, the friction coefficient depends at least on one dynamical "state" variable. This is obvious when the steady-state characteristic exhibits a negative slope. Were the sliding rate the only relevant variable, steady sliding would be a l w a y s unstable with respect to stick-slip, whatever the values of e.g. the stiffness of the driving stage. This is generally not the case since one can get rid of stick-slip by using a stiff enough tribometer. Although the state-dependent part of the friction coefficient does have important dynamical consequences, e.g. as regards the sliding stability, it is generally relatively small. It may therefore be masked by other sources of drift or fluctuations due, e.g. to wear. This is probably why most of the systems where transient friction has been studied belong to what can be termed "soft" tribology, involving soft materials (polymers, paper .... ) and/or soft tribological conditions (low velocities and loads, pairs of materials of comparable

hardnesses, . . . ) i n order to minimize wear effects.

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Figure 1. Hysteretic friction forces under nonsteady motion. (a) Paper/paper friction, from Ref. [1] (b) Wood/rosin/PMMA, adapted from Ref. [2]. The dashed lines are steady-state characteristics. The loops in (b) around the rest point is an artifact. Even in the favorable cases where hysteretic friction is clearly observable,

establishing a closed set of dynamical equations for the friction coefficient, valid over a clearly identified range of dynamical variables, is a very difficult task for which there is no systematic phenomenological approach. One challenging issue of such modelling is to identify the physical nature of the underlying state variable. Only a few systems have been brought so far to such a level of understanding. 2. THE P A R A D I G M STATE-DEPENDENT

OF RATEFRICTION

AND

The most extensively studied model of

rate-and state-dependent friction has been established with remarkable insight by Rice & Ruina [4] after the experimental work of Dieterich [5] on granite/granite friction.

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I Figure 2. Multicontact interface between rough surfaces. It has been found that the natural framework to which it applies if that of multicontact interfaces "& la Greenwood" between nominally flat surfaces, rough on the micrometer scale [6]. It is now well established that the state of such an interface is characterized by its real area of contact T-.r, a fruitful concept introduced by Bowden & Tabor [7]. The friction force reads F = ~sT-,r where ~s is the shear strength of the interface. For not too brittle materials, T-.r depends not only on the normal load, but also on the contact time, due to plastic creep of the loadbearing asperities [8]. The area T-.r increases with time at rest: the interface ages and the static friction force increases with the contact duration. As sliding proceeds, the microcontacts get destroyed and replaced by new ones, formed at random between fresh asperities. The interface is progressively rejuvenated and Y-.rtends to decrease. The full dynamics is ruled by the competition between plastic ageing and rejuvenation by sliding. This

can be formulated by introducing the "age" ~(t) of the interface, which evolves according to a simple dynamics [4, 9]: t ~(t)= f dt' exp-(X(t)-x(t')~ Do / It involves a characteristic memory length DO which is the length to be sled for the contact population to be renewed. Hence it is of the order the average diameter of the microcontacts, typically l~m. At rest, ~(t) is the time elapsed since the creation of the interface. On steady sliding at velocity V the age reads r = D0/V, i.e. the faster the slip, the younger the interface. For a wide range of materials (paper, PMMA, PS, granite .... ) the real area of contact T-,ris found related to r through: Y-.r(r = Y-,0[1 + m In(1 +~/'c)] with m and ~ constants of typical order, respectively,10 - 2 and 10-2 s, and T_.0 the short time area, proportional to the normal load. We have checked that such a logarithmic dependence is fully compatible with the bulk creep behaviour of polymer glasses PMMA and PS, at temperatures up to their glass transition [10]. The friction model is said to be state- and rate-dependent because the interfacial strength ~s depends on the instantaneous sliding rate v. More precisely, it is found that within a low velocity range, typically 10-2 < v < 10 2 l~m/s, and again for an amazing variety of materials, the interfacial strength ~s increases quasi-logarithmically with v. This behaviour has been successfully related to thermally activated flow processes occuring within the nanometer-thin layer at the junction between load bearing asperities, where shear is accomodated [11-12]. It is remarkable that such a simple model where the rate and state variables affect independently both components of the friction force, namely F(v, r = ~s(V) T-.r(~), describes quantitatively most of the transient dynamical behaviour of multicontact interfaces.

2.1.

Stick-slip bifurcation [9,13]

2.2. Transients between dynamic friction.

The state-dependence of the real area of contact results in a negative contribution to the slope of the friction force vs. steady state velocity. The rate-dependent contribution is positive. In many cases the low velocity characteristics is actually negatively slopped, therefore promoting stick-slip. The stability of the system, however, depends on the values of the control parameters, driving velocity, stiffness of the tribometer, normal load (Fig. 3). With soft tribological systems (paper/paper, PMMA/PMMA, ...) the bifurcation between stick-slip and steady sliding has been studied in great detail, definitely validating the model of Rice & Ruina. It is important to notice that the behaviour of the system in the vicinity of the bifurcation is critically sensitive not only to the values of the parameters but also to the details of the dynamical equations. This has been turned to profit for refining the phenomenological model [14].

static

and

This is a long-standing issue, first addressed by Rabinowicz [15], which can be re-formulated in terms of the competition between the geometrical ageing a n d rejuvenation processes at the interface. In the simplest tribometric arrangement, a slider standing on a tilted track, macroscopic sliding begins as the tilt angle reaches the static repose angle which increases with the time elapsed at rest. As sliding proceeds over the first micrometers the interface is rejuvenated and the friction force drops accordingly, resulting in a self-accelerated motion. The high-school physics account for this behaviour is that "the dynamic friction coefficient is lower than the static one". Introducing the real area of contact of the interface provides some physical support to this oversimplified, though useful, empirical property. 0.35 0.3 0.25 0.2 "~0.15

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Figure 3. Stick slip oscillations for a spring-block system (see inset) with a paper/paper interface. The remote loading point is driven at V = 1 pm/s. Different traces corresponding to different normal loads W have been shifted vertically for the sake of clarity. The bifurcation from stick-slip to steady sliding is continuous - of the direct Hopf type.

Figure 4. Transient slip to rest after cessation of drive. At t<0, the slider was in steady sliding at V = 10pm/s. A more versatile arrangement involves a driving stage with some necessarily finite compliance (see inset of Fugure 3.). Starting from steady sliding at velocity V, when the remote drive is stopped abruptly, the slider creeps its way to rest over a finite slip length, the scale of which is given by DO. Here, ageing wins and results in a self decelerating slip

motion (Fig. 4). The slip velocity goes asymptotically to zero in a way which is quantitatively accounted for by the model [16].

rejuvenation wins. In between these trivial regimes, as ageing during the rest fraction of the fast oscillating motion overcomes rejuvenation during the sliding one, the slider is observed to start creeping slowly. Although the shear excitation is steadily maintained, this motion develerates until arrest: this creep gets jammed after sliding a distance of order DO. The amplitude of the AC component controls the bifurcation between self-accelerated unbounded slip and what has been referred to as a jamming creep regime (Fig. 5.b.). 3. T R I B O L O G I C A L INVESTIGATION OF THE S T R U C T U R A L A G E I N G OF ADHESIVE J U N C T I O N S

Figure 5. (a) Oscillating tilted plane arrangement which results in a DC + AC force loading of the slider. The angle 0 is chosen close below the static friction angle. (b) Transition from finite "jamming" creep to unbounded slip downwards when increasing slightly the AC acceleration amplitute Ymax, in g units. Bottom curve is the quasistatic slip distance Xdc, upper curve is the excitation Ymax. Coming back to the tilted plane arrangement, it is possible to disentangle the subtle interplay between geometric ageing at rest and slip-induced rejuvenation by adding to a constant (DC) force just below the static threshold an oscillating (AC) component which may bring the slider into the sliding state during some adjustable fraction of its period (Fig. 5.a.) [17-18]. If the amplitude of its AC component is low enough, the slider remains in the pinned state, the interface ages and responds elastically. On the contrary, if the AC component is large, the slider performs an unbounded slip downwards; here again

The real area of contact concept is relevant for many systems. However, it is possible to achieve homogeneous contacts with soft and/or smooth enough surfaces. For these systems, the real area of contact is independent of the sliding history. The sliding dynamics is then fully describable in terms of the shear strength ~s of the interface. If care is taken in order to avoid ploughing or wear, and in the absence of lubrication, shear is ultimately accomodated in a nanometer thin layer involving the near surface atoms or molecules (Fig. 6). This layer we refer to in the following as the "adhesive junction". So far, ~s has been decribed as depending only on the instantaneous sliding velocity v, hence as an essentially rate-dependent stress. This is not the full story and we describe in the following some examples of state-dependent interfacial strengths of adhesive junctions.

Adhesive junction Figure 6. Schematic representation of an homogeneous contact with a flat and smooth body. When shear is accomodated within the nanometer thin layer involving, e.g. the near surface molecules, we refer to it as the adhesive junction.

3.1. The tribology, rheology, plasticity of rosin [2] I first consider for heuristic purpose of an interesting case where the adhesive junction is quasi-macroscopic. Friction of the bow on the string of, say a violin, is deeply influenced by the use of an anti-lubricant, rosin, which is rubbed before playing against the hair of the bow where it forms a micrometer thick layer. Rosin is an amorphous, glassy material, which shows thermal softening around 60~ As shown on Figure l.b, the steady state friction characteristic of a rosined contact exhibits a negative slope, responsible for the stick-slip excitation of the string. It is amazing that, despite the amount of work devoted to the physics of the bowed spring, it is only recently that the hysteretic nature of the friction of a rosined bow against a string has been studied. Smith & Woodhouse [2] have shown that the state of the rosin layer is mainly controlled by the "flash" temperature at the interface. The real area of contact remains a constant but the yield stress of the interface reads ~s(V, T), the temperature T being coupled to the sliding velocity through the heat flux generated by frictional dissipation. The authors propose two limiting phenomenological expressions for the rate dependence of ~s. The first one amounts to describing the rosin layer as a viscous fluid while the other treats it as a plastic material. It is not the place here to discuss about these models further but I find it essential to notice that tribology, plasticity of amorphous materials and rheology of complex fluids naturally gather here. I will argue in the following that the links between these different fields are also relevant in the case of dry contacts where the adhesive junctions are molecular-thin.

3.2. Adhesive junctions as soft glassy systems In the case of low velocity (< 100 #m/s) sliding at a multicontact interface, self heating is negligible and cannot influence the rheology of the interface. However, it has

recently been suggested that shear may play a role similar to temperature for a wide class of bulk systems, referred to in the litterature [19] as soft glassy materials, e.g. colloidal glasses, pastes, foams, which exhibit a so called "jammed" state in which they behave as elastic amorphous solids. Such systems never truly reach their thermodynamic equilibrium and their properties, e.g. mechanical, evolve slowly with time. They are said to age. When a shear stress is applied above a characteristic threshold, or yield stress, they flow and generally exhibit non-newtonian rheologies. Note that most of the above-mentioned systems are athermal, i.e. cannot be brought out of the jammed state by heating, by contrast to e.g. glass forming liquids. Shear plays therefore a major role in the structure of these systems, sinec they flow, or "melt" under high enough shear stress [20]. Such "shear-melting" has been also studied on quasi-2D systems consisting of molecular thin layers of lubricants, confined between atomically smooth surfaces in the Surface Force Apparatus [21-22]. Due to confinement, these liquids jam (become solid) and exhibit a yield stress. Ageing of the yield stress with time has been also clearly evidenced [23]. Another example pertains to nominally dry friction. We have recently studied friction between a rough surface of glassy PMMA and a flat, smooth plate of silica glass [17]. Such a configuration makes it possible to get rid of the geometrical state variable since the loadbearing asperities are not destroyed when sliding. Hence, the real area of contact reaches a quasi-constant value at long contact times. It is then no longer a dynamical variable. However, we have clearly evidenced that the static friction coefficient nevertheless exhibits a quasi-logarithmic increase with the time elapsed at rest [24]. Although its area is here time-independent, the junction still ages (Fig. 7), Le. its internal structure slowly relaxes and strengthens.

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Figure 7. Stop and go test of the interface (population of adhesive junctions) between a rough PMMA surface and a smooth glass plate. After cessation of drive at t = 0, the slider creeps until the interface jams. On resuming tangential loading, the friction coefficient exhibits a stiction peak at Ps before reaching its steady state value Pd< Pd. This transient dynamics reveals that the junctions have aged meanwhile and are then rejuvenated by sliding. In all these cases, it is clear that the structural state of the confined layer must be specified in order to account for the transient dynamics of the sliding contact. The nature of the state variables of a soft glassy material and the way they couple dynamically to the shear strain rate (and/or temperature) has recently motivated different theoretical as well as experimental and numerical studies in the frame of rheology of complex fluids [19] and plasticity of amorphous materials [25-26]. The problem is far from being solved and a lot can be gained from the exchange of experimental procedures and concepts between different fields. As such links may lead to cross-fertilization, it has been shown simultaneously in two different systems, namely on the one hand the adhesive joint between PMMA and silica glass [17, 24], and on the other hand a colloidal glass [27], that whereas a large enough shear strain rejuvenates the system, a small shear stress, close below the static threshold, is able to induce overageing ! Though non-standard in either of both fields, such experiments are

quite valuable for the understanding of the underlying physical mechanisms. In the tribological case, it has been clearly shown that the role of the shear stress is to bias the energy lanscape of the glassy joint, i.e. to lower the energy barriers to be overcome through thermal activation in order to trigger the molecular rearrangments responsible for the plastic deformation of the junction. 4. I N H O M O G E N E O U S HEALING PULSES.

SLIDING"

SELF-

Up to now we have described the rate- and state-dependent processes occuring at a sliding interface as if a single macroscopic degree of freedom, the shear rate v , was enough. Soft tribological systems have provided clear evidence that macroscopic sliding may occur inhomogeneously. The most studied case are Schallamach waves occuring at rubber-glass contacts [28]. These are detachment waves and the apparent frictional energy dissipation originates from the adhesion and/or viscoelastic hysteresis. It has recently been suggested by Marder [29] that frictional dissipation between atomically smooth surfaces of cristalline hard materials could originate from the phonon radiation associated with the propagation of selfhealing, sonic mode I crack. The size of these pulses and their short travel duration through a typical homogeneous hard contact make them probably undetectable, at least directly. Indirect, remote detection of an interfacial slip field is however within seismologists' grasp. It is now taken for likely that some earthquakes are associated with the propagation of selfhealing slip pulses, i.e. involving essentially mode II cracks [30].The existence of such pulses is, however, still an open problem in fracture mechanics. A key point is that a simple Coulomb "local" friction law relating the shear stress to the normal one in the sliding part of the pulse, and especially close to the crack tip, cannot account at all for the fracture dynamics: a transient friction law is required [30-31].

4.1. Gelatin/glass

friction

[32-33]

We have recently evidenced directly the existence of self-healing slip pulses in soft plane-plane contacts between a gelatin hydrogel block and a silica glass plate (Fig. 8). The slip field is measured by analyzing the motion of near-interface optical defects of the transparent gel block. Slip is initiated as the rear edge of the block. It propagates into the contact as a crack-like field at a markedly subsonic velocity of the order mm/s. So to say, it unzips the adhering contact, leaving it in a sliding state.

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Figure 8. Schematic drawing of a gel/glass contact in the presence of a self-healing slip pulse. The upper part of the gel block is driven at constant velocity. The contact is partly sliding (arrows), partly sticking (thick lines). The slipping part is moving at a velocity larger than the driving one. The absence of optical contrast when looking through the contact during slip propagation suggests that no macroscopic detachment occurs. This is confirmed by studying the rheology of the adhesive joint in steady sliding, only compatible with the shearing of a layer of water/gelatin solution of thickness comparable to the nanometric meshsize of the gel.

Steady, homogeneous sliding cannot be established, however, if the driving velocity is smaller than a critical value Vc, typically of order 100 lam/s. More precisely, when the local slip velocity reaches Vc, the contact suddenly heals and goes back to the pinned, sticking state (Fig. 9). The interface is then the sit of a travelling slip pulse since only a fraction of the contact is slipping. Pulses sweep periodically the contact and result in the macroscopic sliding motion of the block.

4.2. Healing as a rheologial instability As far as steady sliding is concerned, we have shown that whenever it is possible to achieve steady sliding, the shear strength ~s of the interface is only rate-dependent, i.e. it is fully described by the instantaneous sliding velocity v. More precisely, % increases with v according to a power law.

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Figure 10. Measured steady state characteristic of the gel/glass friction (dots). The solid curve is a guide to the eyes? The velocity weakening part is tentative, as explained in the text. However, the experimental fact that gelatin/glass contacts heal at a finite velocity Vc suggests that this simple rheology cannot remain valid when v reaches down to Vc. We have proposed that the steady state ~s(V) characteristics could exhibit a branch with a negative slope, clearly unstable (Fig. 10). Such a velocity-weakening characteristics would necessarily result from an underlying structural dynamics. This is confirmed by the fact that at rest the interface ages, as revealed by the increase of the stress required for nucleating slip at the rear edge of the block. The state of the interface is presumably related to the number of bonds between gelatin molecules and adhesive sites on the glass plate. Sliding necessarily results in limiting bond lifetime, which is then ruled by the balance between advection and rebonding dynamics. Several authors [34] have modelled this situation in the case of elastomer or gel friction and show that it results in a velocityweakening contribution to the steady-sliding stress which should vanish at high enough v, where advection becomes too fast for bonds to get formed. We believe that this is precisely what happens in the gelatin-glass system.

CONCLUSION

Transient processes in soft tribology, where it is rather easy to find model systems, can in many cases be described in terms of state-dependent friction. The state of the interface refers to its structure. The first gross structural characteristic of an interface is its geometry, more precisely its real area of contact. The way it couples to the sliding dynamics in the case of multicontact interfaces is now well understood. It provides us with the simplest solved case of ageing/rejuvenation competition. In the case of bulk soft glasses, the structure refers to the microscopic configurations the system adopts within its highly multistable energy landscape. Nanoconfined lubricant layers as well as adhesive joints made of the near surface molecules of the bodies in dry contact behave as quasibidimensional soft glassy materials. The corresponding ageing/rejuvenation dynamics is however often hidden by the real area of contact effect. The role of shear is ambivalent: depending on its amplitude, it can either promote structural ageing or induce rejuvenation. The modelling of such dynamical processes, though of fundamental importance, is still in its infancy. Extended contacts, e.g. involving a very compliant gel or elastomer, may exhibit inhomogeneous sliding which can occur as periodic self-healing slip pulses. The transient friction characteristic controls the healing process hence affects the existence and dynamics of the pulses. In the case of gel/glass friction, the structure of the interface is presumably controlled by the number of adhesive bonds between the glass and the polymer network. A quantitative theory is however still lacking. REFERENCES

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