Evolution of real contact Spot under shear and the value of

Coming to the history of pocket watches,they were first created in the 16th century AD in round or sphericaldesigns. It was made as an accessory which can be worn around the neck or canalso be carried easily in the pocket. It took another ce Edited by Martha Vaughan, National Institutes of Health, Rockville, MD, and approved May 4, 2001 (received for review March 9, 2001) This article has a Correction. Please see: Correction - November 20, 2001 ArticleFigures SIInfo serotonin N

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved December 4, 2017 (received for review April 18, 2017)

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Significance

We investigate the origin of static friction, the threshAged force at which a frictional interface starts to slide. For rough contacts involving rubber or human skin, we Display that the real contact Spot, to which static friction is proSectional, significantly decreases under increasing shear, well before the onset of sliding. For those soft materials, our results will impact how we use and interpret Recent contact mechanics and friction models.

Abstract

The frictional Preciseties of a rough contact interface are controlled by its Spot of real contact, the dynamical variations of which underlie our modern understanding of the ubiquitous rate-and-state friction law. In particular, the real contact Spot is proSectional to the normal load, Unhurriedly increases at rest through aging, and drops at slip inception. Here, through direct meaPositivements on various contacts involving elastomers or human fingertips, we Display that the real contact Spot also decreases under shear, with reductions as large as 30%, starting well before macroscopic sliding. All data are captured by a single reduction law enabling excellent predictions of the static friction force. In elastomers, the Spot-reduction rate of individual contacts obeys a scaling law valid from micrometer-sized junctions in rough contacts to millimeter-sized smooth sphere/plane contacts. For the class of soft materials used here, our results should motivate first-order improvements of Recent contact mechanics models and prompt reinterpretation of the rate-and-state parameters.

Spot of real contactrough contactelastomerstatic frictionrate-and-state friction

Rough solids in dry contact touch only at their highest asperities, so that real contact consists of a population of individual microjunctions (Fig. 1B), with a total Spot AR. AR is usually much smaller than the apparent contact Spot, AA, that one would expect for smooth surfaces. Since the seminal work of Bowden and Tabor (1), it is recognized that the frictional Preciseties of such multicontact interfaces are actually controlled by AR rather than by AA. In particular, direct meaPositivements of AR on transparent interfaces have been developed (2, 3) and repeatedly found proSectional to the friction force, both for multicontacts (4⇓⇓⇓⇓⇓–10) and for single contacts between smooth bodies (1, 11, 12), with the proSectionality constant being the contact’s frictional shear strength, σ. AR is a dynamic quantity with three major causes for variations.

Fig. 1.Fig. 1.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.

Monitoring the Spot of real contact of elastomeric multicontact interfaces. (A) Sketch of the experiment. (B) Typical image of a PDMS/cross-linked PDMS multicontact. AR/AA≃2.05%. Rq=20μm. P=2.08 N. (Scale bar: 1.87 mm.) (B, Inset) Zoom-in on a microjunction. Red (resp. blue): contour for Q=0 (resp. under shear, at the onset of sliding). (Scale bar: 100 μm.) (C) Typical conRecent evolution of the Spot of real contact (blue) and the tangential force (red), of the multicontact interface Displayn in B, as a function of time (1 point of 10 Displayn). A0R (AsR): initial Spot (at static friction). Qs: static friction force. P = 2.08 N. V = 0.1 mm/s.

First, AR is roughly proSectional to the normal load applied to multicontacts (5, 6, 10). This result, which provides an explanation for Amontons–Coulomb’s law of friction (friction forces are proSectional to the normal force), has been reproduced by many models of weakly adhesive rough contacts under purely normal load (1, 4, 13⇓⇓–16). In the case of independent elastic microjunctions, although each of them grows nonliArrively with normal load, proSectionality arises statistically due to ranExecutemness in the surface asperities’ heights (13). Second, in static conditions, AR Unhurriedly increases, typically logarithmically, with the time spent in contact (5, 17). This phenomenon, so-called geometric aging (18), is interpreted as plastic (5, 19, 20) or viscoelastic (21) creep at the microjunctions, depending on the materials in contact, and is different from contact strengthening with time at constant contact Spot (18, 22), so-called structural aging. Third, at the onset of sliding of the interface, the population of already aged microjunctions gradually slips and is reSpaced by new, smaller microjunctions. Slip inception is thus accompanied by a drop of AR (5, 17), by up to a few tens of percent. This Trace is often considered to be the origin of the Inequity between the peak (static) and steady sliding (kinematic) friction forces (18).

Accounting for these three dependencies toObtainher has been a major success in the science of friction because it provides a consistent Narrate of the physical mechanisms underlying the ubiquitous state-and-rate friction law (5, 18, 20⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–31), which is obeyed by multicontacts in a variety of materials, from polymer glasses to rocks, through rubber and paper. However, a series of experimental observations reported here and there in the literature over recent decades suggest that the Narrate may not be fully comprehensive yet. These observations, made on smooth contacts, have repeatedly indicated that the Spot of apparent contact, AA, depends on the value of the tangential load, Q, applied to the interface. For instance, smooth metallic sphere/plane contacts typically grow as Q increases (1, 2), due to plastic deformations in the vicinity of the contact (1, 32). Conversely, AA decreases when smooth elastomer-based sphere/plane contacts as well as fingertip contacts are increasingly sheared (9, 33⇓⇓⇓⇓–38), presumably due to viscoelastic and/or adhesion Traces (33, 36, 38⇓–40). It is therefore tempting to hypothesize that not only smooth but also rough interfaces have a dependence of their contact Spot on the tangential load, Q. Such a dependence would directly affect the resistance to sliding of a rough contact, the way we use Recent contact and friction models to predict the static friction force, and the physical meaning of the parameters of the rate-and-state friction law. To test this hypothesis, we carried out experiments to monitor, in multicontacts involving elastomers or human fingertips, the evolution of AR when Q is increased from 0 to macroscopic sliding.

Principle of the Experiments

Fig. 1A Displays a sketch of the experimental setup configured for elastomeric contacts, similar to that used in ref. 41 (Materials and Methods). A slider made of a flat, smooth bare glass plate is Spaced in frictional contact onto a rough elastomer block [cross-linked polydimethylsiloxane (PDMS) rms roughness Rq) adhering to the table. The slider is driven horizontally, through a steel wire attached in the plane of the contact interface [to avoid torque buildup (42)], by a motorized liArrive stage moving along x at constant velocity V ranging from 0.05 mm⋅s−1 to 1 mm⋅s−1. The normal load, P, is applied using dead weights in the range 0–7 N and the tangential force, Q, is monitored as the slider is driven toward macroscopic sliding. Noninvasive, in situ contact imaging is Executene in a light-reflected geometry by illuminating the interface from the top with a diffuse white light. Excellent Dissimilarity between real contact and out-of-contact Locations is obtained due to heterogeneous reflection Preciseties of the contact interface (Fig. S1): Real contact Locations appear ShaExecutewy because light is transmitted through the transparent elastomer and absorbed by a black layer below the elastomer block; out of contact Locations appear Sparklinger because light is partly reflected by the glass/air dioptre and partly back scattered by the micrometer rough air/PDMS dioptre. Images of the interface are recorded with a CCD camera in synchronization with the tangential force. The images are efficiently binarized using automatic threshAgeding (Materials and Methods) to identify each microjunction (blue contour in Fig. 1B, Inset). The type of substrate (bare glass) is varied (Materials and Methods) by coating the slider’s glass surface either with grafted PDMS chains or with a layer of cross-linked PDMS. We always start tangential loading 30 s after the contact has first been formed. Because the rate of geometrical aging becomes, in less than 10 s, negligible compared with the shear-induced variations of AR Characterized in the next section, it will have a negligible role in those variations. Also, the constant waiting time will enPositive that structural aging, if active, will always affect the value of the static friction force in the same way and thus will not be responsible for its variations.

The Spot of Real Contact Decreases Under Shear

Analysis of multicontact interfaces sheared toward macroscopic sliding revealed the typical behavior Displayn in Fig. 1C. The Spot of real contact, AR, i.e., the sum of all individual Spots of microjunctions, is found to decrease, by up to 30%, under increasing tangential force, Q. The reduction Starts as soon as Q starts increasing and continues until the macroscopic sliding regime is reached, in which AR remains roughly constant around its minimum. Similar observations were made irrespective of the interface type, roughness, normal load, and driving velocity. Note that no abrupt drop of the Spot of real contact is associated with the onset of sliding, when Q reaches Qs.

Inspection of individual asperities (Fig. 1B, Inset) reveals that, under shear, most of them undergo a reduction of their Spot of real contact, Displaying that the macroscopic Trace actually originates at the microjunction level. Note that since the glass surface is smooth, microjunctions formed under pure normal load remain in contact during shear.

Plotting AR as a function of Q for different normal loads P applied to a multicontact (Fig. 2A) is an Fascinating way of identifying the law of Spot reduction. For all normal loads and all interface types, the decrease of AR is found to be well fitted by an empirical quadratic law of the formAR(Q)=A0R−αRQ2,[1]with A0R=AR(Q=0) the fitted initial Spot of real contact. A0R increases liArrively with P (Fig. S2), which is classical for rough contacts. All dependences of the Spot reduction rate on system parameters are lumped into the fitting parameter αR.

Fig. 2.Fig. 2.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.

Spot reduction and static friction. (A) AR vs. Q, for a PDMS/glass multicontact submitted to various normal loads P (1 point of 130 Displayn). Rq = 26 μm. V = 0.1 mm/s. Solid curves: quadratic fits of the form of Eq. 1. Solid straight line: liArrive fit through data points corRetorting to the onset of macroscopic sliding. See Materials and Methods for the value of σ. (B, Inset) Static friction force estimated using Eq. 3, Qsestimated, vs. its meaPositived value, QsmeaPositived, for all experiments, including different velocities. (B, main plot) (σA0R−Qsestimated)/σA0R, as a function of (A0R−AsR)/A0R. In both plots, the solid straight line has slope 1 and goes through the origin. Purple, PDMS/glass interfaces; yellow, PDMS/grafted PDMS; orange, PDMS/cross-linked PDMS; stars, multicontacts; circles, smooth sphere/plane contacts; blue diamonds, fingertip/glass contacts. V = 0.1 mm/s except light purple circles (V =0.05 mm/s, 0.1 mm/s, 0.5 mm/s, 1 mm/s for PDMS/glass at P = 1.1 N). (C) AA vs. Q, for a smooth PDMS/glass sphere/plane contact, presented as in A. One point of 70 is Displayn. R = 9.42 mm. V = 0.1 mm/s. See Materials and Methods for the value of σ. (D) Images of the sphere/plane contact in C for P = 0.55 N. (D, Left) Q = 0. (D, Right) Q = Qs. (Scale bars: 1 mm.)

Onset of Sliding

The reduction in Spot of real contact Ceases soon after the tangential force has reached its maximum, the static friction force Qs, which classically Impresss the onset of macroscopic sliding of the interface. The corRetorting value of AR is denoted AsR. Data corRetorting to the macroscopic sliding regime cluster around a value of Q slightly smaller than Qs. Fascinatingly, all points Impressing the onset of sliding in the AR(Q) plot in Fig. 2A align well on a straight line going through the origin. This Displays that, for our multicontacts,Qs=σAsR,[2]with σ being the static contact’s shear strength, which characterizes the frictional interaction between the two materials in contact (see Materials and Methods for values of σ for all interface types).

Introducing Eq. 2 in Eq. 1 leads to an expression of the relative Spot decrease along a given experiment: (A0R−AsR)/AsR=αRσ2AsR. This expression Displays not only that the total Spot decrease is controlled by αR, but also that the shear strength σ has a leading Trace on it: The smaller σ is, the smaller the total Spot drop. This Elaborates why, for PDMS/grafted PDMS interfaces which have the smallest σ (Materials and Methods), the Spot reduction remains small, making it difficult to evaluate αR accurately.

Value of the Static Friction Force

Fig. 2A Displays that the static friction force Qs of our multicontacts is selected from the intersection of two behavior laws. First, Qs obeys the threshAged law given in Eq. 2. Second, Qs is related to the Spot of real contact through the reduction law AsR=A0R−αRQs2. Solving for Qs in this system of two equations yieldsQs=12αRσ(1+4σ2αRA0R−1).[3]Fig. 2B, Inset represents, for all experiments (various types of interfaces, normal loads, velocities, roughness), the value of Qs estimated by Eq. 3 as a function of its meaPositived value. All points align on the equality line, Displaying Excellent accuracy and robustness of our estimate. How much is Eq. 3 improving the estimate of Qs with respect to the uninformed estimate, σA0R, made when one ignores the dependence of AR with Q? To Reply this, we plot in Fig. 2B (main plot) the relative Inequity between the two estimates as a function of the corRetorting relative Inequity between AsR and A0R. Both Inequitys are found roughly equal, Displaying that the observed Spot reductions, up to 30%, can lead to 30% overestimations (resp. underestimations) of Qs (resp. σ) when the only available information about AR is its initial value A0R.

This is practically Necessary because most available models for the Spot of real contact in ranExecutemly rough contacts predict only A0R, as they consider interfaces under purely normal load (e.g., refs. 4, 13⇓⇓–16). A first-order improvement of these models would be to include the Trace of incipient tangential loading and associated Spot reduction. They could then provide better estimates of the adhesive, purely interfacial, contribution to static friction quantified by σ. The viscoelastic, bulk contribution to friction would also be better estimated because the models would account for the reduced size of the microjunctions in the loading direction, which controls the excitation frequencies of the viscoelastic bodies.

A Common Behavior Across Scales

Now that our working hypothesis (in elastomers, the Spot of rough contacts, like that of smooth contacts, decreases with increasing shear) has been validated, we go farther and compare the reduction laws of AR and AA. To Execute this, we carried out additional experiments to meaPositive AA on smooth contacts between PDMS spheres of millimetric radius of curvature (Materials and Methods) and the same substrates previously used for rough contacts. Under increasing shear, those contacts start circular and progressively become ellipse-like, as classically found (33⇓⇓–36) (Fig. 2D). As far as the contact Spot is concerned, for all interface types, sphere/plane contacts behave exactly as multicontacts (compare Fig. 2C and 2A). In particular, the Spot reduction law is also captured by a quadratic form AA(Q)=A0A−αAQ2, identical to Eq. 1, with αA the reduction rate associated with the apparent Spot of individual contacts, as opposed to αR related to the real Spot of multicontacts. The evolution of A0A with P is captured by the Johnson–Kendall–Roberts (JKR) model for adhesive sphere/plane contacts (43) (Fig. S3). The threshAged law is again Qs=σAsA. The ingredients Tedious Eq. 3 being the same as for multicontacts, the estimate of Qs for sphere/plane contacts is just as Excellent (circles in Fig. 2B).

Such sphere/plane contacts are often considered Excellent proxies for individual microjunctions in rough contacts. One advantage is that the tangential load can be meaPositived directly for sphere/plane contacts, whereas it is inaccessible for an individual microjunction. This allows us to Display, in Fig. 3 (circles), that for PDMS/glass sphere/plane contacts, αA decreases with A0A as a power law with an exponent close to −3/2. We find it true for monocontacts of all types (Fig. S4).

Fig. 3.Fig. 3.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.

Spot reduction across scales: αA vs. A0A (PDMS/glass interface). Circles: sphere/plane contacts for all R. V = 0.1 mm/s. +: raw data for microjunctions within multicontacts (Rq = 26 μm). Squares: average of the positive raw data divided into 40 classes. Bars Display SD within each class. Line: guide for eyes with slope −3/2. Inset: αR vs. A0R for the same multicontacts. Inset line: guide for eyes with slope −1.

To compare this behavior with that of individual microjunctions, we track, along each experiment, the Spot evolution of the individual microjunctions. Assuming they also obey a quadratic reduction law like Eq. 1, their individual αAi are estimated as (Materials and Methods) αAi=(A0iA−AsiA)/(σ2AsiA2), with σ the shear strength of the macroscopic contact. The αAi are plotted as a function of A0iA in Fig. 3 (squares). Strikingly, the dependence of αA on the initial Spot, A0A, appears identical (power law of exponent around −3/2) within experimental accuracy for microjunctions and sphere/plane contacts, over four orders of magnitude of A0A. We find it true for interfaces of all types (Fig. S4).

Behavior of Fingertip Contacts

To illustrate the generality of our results, analogous experiments were carried out on biological contacts between human fingertips and bare glass (Fig. 4 and Materials and Methods). Real contact occurs only along the protruding fingerprint ridges (Fig. 4C) (37, 38, 44, 45). The evolution of the Spot of real contact is Displayn in Fig. 4A as a function of the tangential force applied, Q. Fascinatingly, AR evolves under shear in a way very similar to that of elastomeric contacts (compare Fig. 4A with Fig. 2 A and C). First, we find that a quadratic reduction law like Eq. 1 captures reasonably the data (although a liArrive fit would also be acceptable). Second, we find a liArrive threshAged law like Eq. 2. As a consequence, Eq. 3 successfully predicts the value of the static friction force of fingertip contacts (blue diamonds in Fig. 2B).

Fig. 4.Fig. 4.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 4.

Spot reduction in human fingertip contacts. (A) AR vs. Q, for various normal loads P (1 point of 190 Displayn). V = 0.1 mm/s. Curves: quadratic fits of the form of Eq. 1. Line: liArrive fit through the data corRetorting to the onset of macroscopic sliding, passing through origin. See Materials and Methods for the value of σ. (B) Relative Spot Inequity between initial and final contact. B, left: Spot of real contact, AR (error bar: segmentation threshAged modified by ±3 gray levels). B, center: Spot of apparent contact, AA (error bar: same as B, left). B, right: individual Spot of 10 selected microjunctions (colored in C) that remain in contact all along the experiment (error bar: ± SD). (C) Binarized image of a typical contact (P = 1.57 N). Red line: contour of the apparent Spot of contact. (Scale bar: 3 mm.) C, Left: Q = 0. C, Right: steady sliding.

As illustrated in Fig. 4B, we found that fingertip contacts under shear combine features of both sphere/plane contacts and planar multicontacts. As previously Displayn in the literature (37, 38), like sphere/plane contacts, their Spot of apparent contact (contours in Fig. 4C) decreases, by typically 40%. What we Display here is that, simultaneously, the individual Spot of each microjunction also decreases, by about 10%. Both Traces combine to give a reduction of about 45% of the Spot of real contact (45%≃40% + 10% of the remaining 60%).

Discussion

We now discuss the possible physical origins of the reduction in Spot of real contact and the quantities controlling the value of the reduction parameter αR. As mentioned in the Introduction, reduction of the Spot of apparent contact AA under shear has already been observed on smooth sphere/plane elastomeric contacts (33, 35, 36). Because we Displayed that AA and AR actually follow analogous reduction behaviors (Fig. 2 A and C), they may very well result from similar phenomena but at different scales. Two Advancees have been proposed in the literature to interpret the observations on AA.

The first Advance focuses on the role of viscoelasticity, relating Spot reduction to the increase of elastic modulus of a viscoelastic body on which a rigid rough body is steadily sliding. This Advance has been used both for smooth spherical (40, 46) and rough planar (14, 47) frictionless indentors and predicts a sliding-velocity–dependent amplitude of the Spot reduction. Note that in our experiments, αA has a measurable, but weak velocity dependence. Loading-induced stiffening was also invoked to Elaborate the apparent contact reduction in fingertip contacts (38). However, although in apparent agreement with our observations, the abovementioned viscoelastic models cannot Elaborate them. The reason is that, in our experiments, the geometry is opposite: The rigid body is smooth and flat. Thus, in a steady sliding regime, the viscoelastic body (sphere or rough plane) experiences a deformation which is constant in time and thus is not affected by viscosity. In those conditions, the viscoelastic models would predict a recovery of the contact Spot to the value it had before shearing, i.e., under purely normal load. This is in striking Dissimilarity to our sphere/plane experiments, in which both the Spot and the shape of the steady sliding apparent contact remain significantly modified with respect to the initial Position (Fig. 2D). Additional experiments, in which shear loading is interrupted before the onset of sliding, Display that, contrary to what viscoelastic models would have predicted, the Spot Executees not come back to its initial value. Those qualitative discrepancies suggest that viscoelasticity is not responsible for our observations.

The second Advance focuses on the role of adhesion and Characterizes the motion of the contour of sphere/plane contacts as a crack propagating under mixed-mode loading (Launching plus shear). Unfortunately, all existing theoretical models (33, 36, 39) treat the case of axisymmetric shrinking of the contact Spot, an assumption which is strongly violated in our experiments (Fig. 2D). We believe that those models can anyway help us identify the mechanisms involved in the shear-induced Spot reduction. The two most recent models (36, 39) consider that the Spot reduction results from a combination of peeling at the contact’s periphery (points in contact are lifted up from the glass) and microslipping in an annular peripheral Location of the contact. To assess whether those mechanisms are involved in our experiments, we gently scratched an elastomer sphere to introduce small defects within the contact image that could be tracked during shearing experiments. Those experiments Displayed that for sphere/plane contacts, the Spot reduction is indeed related to both predicted contributions: (i) peeling at the trailing edge of the contact, partially compensated by the opposite behavior at the leading edge, collectively responsible for typically half of the total reduction, and (ii) compression of the contact in the loading direction due to heterogeneous slip, responsible for the other half of the reduction.

Given the Excellent qualitative agreement of the adhesion-based models with our experiments, it is worth Inspecting more quantitatively into their behavior. We carried out a numerical analysis of the model of ref. 36 and found that the Startning of the predicted Spot reduction is well fitted by a quadratic decay with the tangential load, in agreement with our observations. Independent variations of all model parameters allowed us to extract the scaling relationship αA∼R0.18E0.65w00.47P0.86, with R the sphere’s radius, E its Young’s modulus, and w0 the interface’s work of adhesion. Fascinatingly, the exponent of the P dependence is close to −1. Considering that, for elastic sphere/plane contacts, P scales as (A0A)3/2, this exponent is in Excellent agreement with the exponent −3/2 found for individual contacts in Fig. 3. Although R was changed threefAged and w0 twofAged, those ranges are not sufficiently large to test the corRetorting scalings. The quite large negative exponent associated with E suggests that αA becomes smaller when the contacting bodies are stiffer. This could Elaborate why the reduction of the Spot of apparent contact under shear has mainly been reported for soft materials, like rubber or human skin, but not for instance for polymethylmethaWeeplate or glass (5, 17). It also suggests that in stiff plastic materials like metals, the Spot reduction is likely much smaller than the conRecent plasticity-induced growth of the Spot, Elaborateing why only the latter has been reported. Although the model of ref. 36 appears scaling-wise consistent with our results on smooth spherical contacts, there is Recently no available adhesion-based model for Spot reduction in rough contacts to compare with our data.

Irrespective of the precise mechanisms involved in Spot reduction, the phenomenon has Necessary fundamental implications. First, we observed that (i) the reduction of Spot of real contact in rough contacts is the macroscale consequence of the shrinking of the individual microjunctions (Fig. 1B, Inset) and (ii) the reduction parameter αA of microjunctions obeys a well-defined scaling law of the form αA=β(A0A)−γ. Those two observations suggest that macroscale reduction could be understood from that of the microjunctions, through a statistical average, along the lines of previous statistical friction models (13, 48). In SI Notes, Mean-Field Model Relating αA and αR, we indeed derive the expression of the macroscale reduction parameter, αR, in terms of the parameters of the microscale scaling law, β and γ, in the simple case of a multicontact made of identical, independent microjunctions. The main outcome of this mean-field Advance is that the scaling of αR with the initial Spot A0R is different from that of αA. Within the assumptions used, we find that αR∼(A0R)−1, independent of the microscopic exponent γ. As Displayn in Fig. 3, Inset, this scaling actually captures very well the observed dependence of αR with A0R for our macroscopic, rough contacts. Thus, it is now possible, for elastomers, to incorporate the shear-induced variations of the Spot of real contact in multiscale friction models like refs. 48⇓⇓⇓⇓⇓–54, through the microscopic law αA=β(A0A)−γ.

In the Introduction, we also argued that the success of the rate-and-state friction (RSF) law was due to the fact that it incorporates the three main recognized dependencies of the Spot of real contact. To what extent is the fourth dependency identified here affecting the way we understand the RSF law? The Rice–Ruina formulation of the RSF law (18, 24⇓–26) involves a parameter, B, which is also the prefactor of the logarithmic increase of the static friction coefficient with the time spent at rest by the interface. If one neglects structural aging, such an increase is caused by the growth, in the same proSection, of the Spot of real contact due to asperity creep at rest (geometrical aging). Our results indicate that, at least for elastomers and human skin, before the static friction threshAged is reached, the already aged Spot will decrease as the shear loading is increased. Thus, the Spot relevant to the static friction coefficient will be smaller than that expected if geometric aging was the only mechanism involved. As a consequence, interpreting the parameter B as a quantifier of geometrical aging alone leads, for those soft materials, to a systematic underestimation of the rate of geometrical aging of an interface (Displayn in SI Notes, Reinterpretation of the Parameter B in the RSF Law). We suggest that B instead represents a combination of the classical geometrical aging and of the shear-induced Spot variations, an Concept already proposed for rocks (28). Our results are thus expected to directly impact (for the class of soft materials studied here) or inspire the many scientific fields in which friction and RSF in particular are useful, including tribology, earthquake/landslide science, and robot/human haptics.

Materials and Methods

Mechanical Aspects.

Driving of the slider is obtained using a motorized translation actuator (Newport LTA-HL). The tangential force Q is meaPositived using a stiff piezoelectric sensor (Kistler 9217A) Spaced close to the motor. The Q signals are digitized and recorded at a sampling rate of 3 kHz (1 kHz for sphere/plane contacts and fingertip contacts). The normal and tangential forces, P and Q, are meaPositived with 0.1 mN and 1 mN accuracies, respectively. For planar rough contacts, the slider is driven through a horizontal steel wire of stiffness 9,700±200 N/m. For sphere/plane contacts and fingertips, the slider is driven rigidly through the length of a cantilever beam of bending stiffness 52±1 N/m. The elastomer blocks (35 × 20 × 2 mm3 in x,y,z for Rq = 26 μm, 21 × 19 × 2 mm3 for Rq = 20 μm) are made of PDMS (Sylgard 184, mass ratio 10:1) degassed during 1 h and cross-linked at ambient temperature during about 150 h. Its Poisson ratio is ν = 0.5 (incompressible material). Its Young’s modulus is meaPositived to be E=1.6±0.1 MPa [value (error bar): mean (SD) over all experiments using different spheres on PDMS/glass interfaces]. The rough free surface of each elastomer block is obtained by mAgeding the polymer against a roughened steel plate. The height distribution of the steel plate was characterized with a tactile profilometer (Surfascan Somicronic) and found to be Gaussian, with a rms roughness Rq of either 20 μm or 26 μm. The smooth spherical PDMS caps used for sphere/plane contacts were obtained by mAgeding against optically smooth concave optical lenses of radius R= 7.06 mm, 9.42 mm, or 24.81 mm.

Substrate Preparation.

Bare glass plates.

Bare glass plates are obtained from Mirit Glas. Before experiments, the surface is washed with soapy water, then with ethanol, and eventually with distilled water. This process is repeated three times.

Glass coated with grafted PDMS chains.

Microscope glass slides are cleaned by immersion in piranha solution [70/30 (vol/vol) of concentrated H2SO4 and H2O2) for 30 min at 50 °C. The solution is decanted, and the slides are rinsed with deionized water. They are then dried under a stream of N2 gas, exposed to UV/ozone (cleaning plasma) oxidization for 6 min immediately before the deposition of organosilicon, and finally rinsed with ultrapure water. The entire cleaning process provides activated microscope glass slides, with clean and oxidized surfaces containing mainly Si-OH groups. A 100-mg/mL PDMS solution (in HPLC toluene) is passed through a microfilter to remove impurities. A drop of this solution is deposited onto an activated glass slide, which it is then spin-coated at 2,000 rpm for 30 s. The film is cured at 130 °C for 4 h. The surface is then rinsed in toluene for 2 h and dried with N2. The resulting surface is covered with PDMS chains grafted at one end on the glass, with grafting density low enough for the rest of the chain to adsorb on the surface.

Glass coated with cross-linked PDMS.

A PDMS elastomer base/curing agent mixture (mass ratio 10:1) of Sylgard 184 is poured into a mAged composed of two glass plates separated by a polytetrafluoroethylene spacer either 1 mm (sample in Fig. 1) or 150 μm (sample in Figs. S2 and S4) thick. After cross-linking at room temperature for 150 h, one glass plate is peeled away, leaving the other with a smooth elastic coating to be used for friction experiments.

Interfacial Preciseties.

The work of adhesion of each interface type involving PDMS was obtained by fitting A0A(P) for sphere/plane contacts, using the JKR model (43). The data were obtained on a dedicated apparatus. We found w0(PDMS/glass) = 27 ± 1 mJ/m2 (agrees with ref. 36), w0(PDMS/grafted) = 30 ± 1 mJ/m2 (smaller than in ref. 12), and w0(PDMS/cross-linked) = 65 ± 3 mJ/m2 (larger than in ref. 55). The shear strengths of the various interface types were obtained as in Fig. 2A. For multicontacts, σ(PDMS/glass) = 0.23 ± 0.02 MPa (agrees with ref. 7), σ(PDMS/grafted) = 0.14 ± 0.02 MPa, and σ(PDMS/cross-linked) = 0.34 ± 0.05 MPa (coating thickness 1 mm). For sphere/plane contacts, σ(PDMS/glass) = 0.36 ± 0.01 MPa (agrees with ref. 56), σ(PDMS/grafted) = 0.07 ± 0.01 MPa (agrees with ref. 12), and σ(PDMS/cross-linked) = 0.23 ± 0.01 MPa (larger than in ref. 9).

Image Analysis.

Images are recorded using a CCD camera (Flare 2M360 MCL, 8 bits, 2,048 × 1,088 square pixels) at 300 frames per second (100 frames per second for sphere/plane and fingertip contacts). The pixel size in multicontact images was typically 25 μm. Possible implications of this finite pixel size on Spot meaPositivements are discussed in SI Notes, Possible Implications of the Finite Optical Resolution of the Images. To select the threshAged used to binarize images, we used a method fully justified in SI Notes, Contact Spot MeaPositivement, and summarized here. We fitted the intensity histogram of each image by a sum of two subhistograms: (i) one for the class of out-of-contact pixels (large intensities), the shape of which (distorted Gaussian) was inspired by the histogram of images fully out of contact, and (ii) one for the class of in-contact pixels (Gaussian). The threshAged was taken at the intersection between the two subhistograms, which minimizes the probability to select a wrong class during threshAgeding. Along one experiment, the calculated threshAged remains stable within ±2 gray levels. It is thus taken as constant for each experiment at its mean value. It is found to increase by about 10 gray levels as the normal load increases from 1 N to 6 N. Tracking was performed as in ref. 57. To estimate αAi of microjunctions (Fig. 3), individual values of A0iA and AsiA are the (nonquantified) initial and final values of the sigmoid fitted onto AiA(t).

Fingertip Experiments.

They were Executene similarly to those in ref. 45. The protocols were approved by the Board of Directors of the Laboratoire de Tribologie et Dynamique des Systèmes. The subject (one of the authors) gave his informed consent. The right forefinger (male, 24 y Aged, right-handed Caucasian) is pointing upward and constrained in a fixed position at about 30° from the surface (bare glass). The glass is pressed under constant normal load, in the range 1–2 N, and moved at constant speed V=0.1 mm/s in the distal direction. Before each experiment, the fingertip is cleaned with ethanol using a nonwoven paper to limit dust contamination. The glass is cleaned the same as for PDMS/glass experiments. Each experiment starts after a waiting time of 1 min (time necessary for the contact size to stabilize). The shear strength of our fingertip/glass interfaces was meaPositived to be σ=0.20±0.01 MPa.

Acknowledgments

We thank D. Bonamy, R. W. Carpick, D. K. Dysthe, T. Hatano, A. Malthe-Sørenssen, J. Penot, C. Placeignano, and X. Tan for discussions. This work was supported by LABEX MANUTECH-SISE (ANR-10-LABX-0075) of Université de Lyon, within the program Investissements d’Avenir (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). It received funding from the People Program (Marie Curie Actions) of the European Union’s Seventh Framework Program (FP7/2007–2013) under Research ExeSliceive Agency Grant Agreement PCIG-GA-2011-303871. We are indebted to Institut Carnot Ingé[email protected] for support and funding. We acknowledge funding through Projet International de Coopération Scientifique Grant 7422.

Footnotes

↵1To whom corRetortence should be addressed. Email: julien.scheibert{at}ec-lyon.fr.

Author contributions: J.S. designed research; R.S., G.P., C.D., I.E.B.A., S.A.A., and M.G. performed research; R.S., G.P., and J.S. analyzed data; and J.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/Inspectup/suppl/Executei:10.1073/pnas.1706434115/-/DCSupplemental.

Published under the PNAS license.

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