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Normal contact and friction of rubber with model randomly rough surfaces S. Yashimaa,d,‡, V. Romerob,c,‡, E. Wandersmanb,c, C. Frétignya, M.K. Chaudhurye, A. Chateauminoisa, and A. M. Prevostb,c,∗∗ a Soft Matter Science and Engineering Laboratory (SIMM), CNRS / UPMC Univ Paris 6, UMR 7615, ESPCI, F-75005 Paris, France b CNRS, UMR 8237, Laboratoire Jean Perrin (LJP), F-75005, Paris, France. E-mail: [email protected] c Sorbonne Universités, UPMC Univ Paris 06, UMR 8237, Laboratoire Jean Perrin, F-75005, Paris, France d Laboratory of Soft and Wet Matter, Graduate School of Life Science, Hokkaido Univ, Sapporo, Japan and 7 e Department of Chemical Engineering, Lehigh University, Bethlehem PA 18015, USA 1 0 Wereport onnormal contact andfriction measurementsof modelmulticontact interfaces formed 2 between smooth surfaces and substrates textured with a statistical distribution of spherical micro- asperities. Contacts are eitherformed between arigid textured lensand a smooth rubber,or a flat n texturedrubberandasmoothrigid lens. MeasurementsoftherealareaofcontactAversusnormal a J load P are performed by imaging the light transmitted at the microcontacts. For both interfaces, A(P) is found to be sub-linear with a power law behavior. Comparison to two multi-asperity 5 contact models, which extend Greenwood-Williamson (J. Greenwood, J. Williamson, Proc. Royal Soc. London Ser. A 295, 300 (1966)) model by takingintoaccount theelastic interaction between ] t asperities at different length scales, is performed, and allows their validation for the first time. We f o find that long range elastic interactions arising from the curvature of the nominal surfaces are the s main source of the non-linearity of A(P). At a shorter range, and except for very low pressures, . the pressure dependence of both density and area of micro-contacts remains well described by t a Greenwood-Williamson’s model, which neglects any interaction between asperities. In addition, in m steadysliding,friction measurementsrevealthatthemeanshearstressatthescaleoftheasperities - is systematically larger than that found for a macroscopic contact between a smooth lens and a d rubber. This suggests that frictional stresses measured at macroscopic length scales may not be n simply transposed tomicroscopic multicontact interfaces. o c PACSnumbers: [ Keywords: Friction,roughsurfaces,Contact, Rubber,Elastomer,Torsion 1 v 8 Introduction and Manners [6, 7], amongst others, in order to solve 6 the problem of elastic contacts between one dimen- 2 sional periodic wavy surfaces. Most of the subsequent 1 Surface roughness has long been recognized as a generalisations of elastic contact theories to randomly 0 key issue in understanding solid friction between rough surfaces are more or less based on a spectral . macroscopic bodies. As pointed out by the pioneering 1 description of surface topography [8–11]. Within the work of Bowden and Tabor [1], friction between rough 0 framework of linear (visco)elasticity or elasto-plastic 7 surfaces involves shearing of myriads of micro-asperity behavior, these theories allow estimation of the pressure 1 contacts of characteristic length scales distributed over dependence of the distribution of microcontactssize and : orders of magnitude. The statistical averaging of the v pressure at various length scales. From an experimental i contributions of individual micro-asperity contacts to perspective, elucidation and validation of these models X frictionremainsanopenissue whichlargelyreliesonthe using microscopic randomly rough surfaces such as r contactmechanicsdescriptionofmulticontactinterfaces. a abraded or bead blasted surfaces is compromised by the In early multi-asperities contact models such as the difficulties in the measurementofthe actualdistribution seminal Greenwood-Williamson’s model (GW) [2], ofmicrocontactareasatthe micrometerscale. Although randomly rough surfaces are often assimilated to a early attempts were made by Dieterich and Kilgore [12] heightdistributionofnoninteractingsphericalasperities with roughened surfaces of transparent materials using which obey locally Hertzian contact behavior. Along contact imaging techniques, direct comparison of the these guidelines, some early models also attempted experimental data with contact mechanics models lacks to describe the fractal nature of surface roughness by clarity. considering hierarchical distributions of asperities [3]. In this study, we take advantage of recent advances More refined exact elastic contact mechanics theories in sol-gel and micro-milling techniques to engineer were also developped by Westergard [4], Johnson [5] two types of model randomly rough and transparent surfaces with topographical characteristics compatible with GW’s model of rough surfaces [2]. They both consist of statistical distributions of spherical asperities ∗[email protected] 2 whose sizes ( 20 µm up to 200 µm) allow for an ∼ optical measurement of the spatial distributions of the microcontacts areas. In their spirit, these experiments are along the line of Archard’s previous investigations [3], which used model perspex surfaces consisting of millimeter sized spherical asperities of equal height. However, in Archard’s investigations, a small number of asperities were used. Furthermore, technical limitations in the estimation of variation of heights of asperities did not allow for a statistical analysis of the load dependence of the distributions of microcontact areas. Here, using a sphere-on-plane contact geometry with different statistical distributions of micro-asperities,we probe the elastic interactions between asperities (see e.g. [13–16]) by directly comparing the measured distributions of the real area of contact to the predictions of two different multi-asperity contact models. We show how the use of textured surfaces allows an accurate validation of these models that permits an investigation of the statistical distribution of contact pressure, number of microcon- FIG. 1: (a) SEM image topography of a RA+ sol-gel replica tacts and microcontact radii distributions. In the last (φ = 0.41). (b) Same with an SA PDMS replica of a micro- partofthe paper,wepresentthe resultsofapreliminary milled mold (φ=0.4). (c) microcontacts spatial distribution study that illustrates how such model systems can be with RA+ (P = 22 mN). (d) Same with the SA of (b) and used to investigate the relationship between frictional a lens of radius of curvature 128.8 mm (P = 20 mN). (c- properties and real contact areas. d) are image differences with a reference non-contact image. Notethesizedifferenceintheapparentcontactrelatedtothe differencein curvatureof both indenters. Materials and Techniques Two types of randomly rough surfaces covered with glass lenses using a sol-gel imprinting process fully de- spherical caps were designed using two different tech- scribed elsewhere [17]. An example of the resulting pat- niques as described below. The first surface (RA for ternwithsmoothsphericalcapsofvarioussizesis shown Rigid Asperities) consists of glass lenses (BK7, Melles- in Fig. 1a. By changing the time of exposure t of the exp Griot, radius of curvature 13 mm) covered with a distri- HMDS treated glass to water vapor, different surfaces bution of micrometer sized rigid asperities with varying with different asperity sizes and densities are obtained heights and radii of curvature. The second surface (SA as a result of droplet coalescence during the water con- for Soft Asperities) is made of a nominally flat silicone densationprocess. Twopatternswithsmall(resp. large) slabdecoratedwith a randomspatialdistribution ofsoft asperities were made with t =15 s (resp. 60 s). They exp spherical micro-asperities with equal radius of curvature are respectively referred to as RA− and RA+. Their to- and varying heights. pography at the apex was characterized with an optical profilometer (Microsurf 3D, Fogale Nanotech) to extract themeansurfacefractionφcoveredbytheasperities(Ta- RA lenses ble1)andthedistributionsoftheirheightshandradiiof curvatureR. BothdistributionsarefoundtobeGaussian RA’s topography was obtained by replicating con- (not shown) with means h¯, R¯ and standard deviations densed liquid droplets on a hydrophobic surface. Wa- giveninTable1. ForRA+, hisfoundtobe proportional ter evaporating from a bath heated at 70◦C was first al- to R (Fig. 2). This suggests that the spherical shape of lowed to condense on a HexaMethylDiSilazane (HMDS) the asperities is uniquely controlledby the contactangle treated hydrophobic glass slide kept at room temper- θ of water droplets on the HMDS treated surface prior ature, resulting in a surface with myriads of droplets. to molding. In this case, one expects, indeed, the re- This surface was then covered with a degassed mixture lationship h = R(1 cosθ). Fitting the data of Fig. 2 of a PolyDiMethylSiloxane cross-linkable liquid silicone yieldsθ 57◦,very−closeto55◦ whichisthevalueofthe (PDMS, Sylgard 184, Dow Corning) cured at 70◦C for 2 advancin∼g contact angle we measured for water droplets hours. One is left, upon demolding, with a PDMS sur- on HMDS treated glass. For RA− however, no evident facewithconcavedepressions,whicharenegativeimages correlationhasbeenobserved,forwhichwehavenoclear of the condensed water droplets. These PDMS samples explanation (Fig. 2, inset). then serve as molds to replicate rigid equivalents on the 3 TABLE I:RA’s mean topographical characteristics As detailed above, RA samples display spatial and texp(s) φ h¯(µm) R¯(µm) height distributions of asperities set by both the evapo- 15 0.34 ± 0.02 9.0 ± 2.4 49.6 ± 12.8 a rationand the sol-gelprocesses,which can only be char- 60 0.41 ± 0.05 29.6 ± 10.1 64.4 ± 19.6 b acterized a posteriori. SA samples however, have a sta- a from 293 asperities. tistical roughness which can be finely tuned with any b from 119 asperities. desired pattern, both in height and spacing. As a result, SA flat surfaces are very appropriate for the sta- 80 20 ) tistical investigation of contact pressure distribution as m they can be produced at centimeter scales thus allowing 70 15 h (µ for several realizations of the contact at different posi- tionsonthe patternedsurface. Nevertheless,contraryto 60 10 SAasperitieswhichalwayspresentamicroscopicsurface roughness inherent to the milling procedure, RA micro- 5 50 R (µm) asperitiesareverysmooth. Itthusmakesthemespecially m) 0 suitable for the investigation of frictional properties, as µ40 0 20 40 60 80 h ( microcontacts obtained with a smooth rubber substrate can be assimilated to single asperity contacts. 30 20 Experimental setups 10 For RA lenses, normal contact experiments were per- 0 0 20 40 60 80 100 120 formed by pressing the lenses against a thick flat PDMS R (µm) slab under a constant normal load P. Its thickness ( ∼ 15 mm) was chosen to ensure semi-infinite contact con- FIG. 2: (Color online) Height h of the spherical micro- ditions (i.e. the ratio of the contact radius to the speci- asperities as a function of their radius of curvature R for menthickness wasmorethan ten[19]). ForSA flatsam- the RA+ lens (φ = 0.41). Inset: Same for the RA− lens ples, sphere-on-plane contacts were obtained by press- (φ=0.34). The solid line is a linear fit of thedata. ing them against a clean BK7 spherical lens (LA1301, Thorlabs Inc.) with a radius of curvature of 128.8 mm, 10 times larger than the radius of curvature of the ∼ SA samples patternedRAlenses. Toensurecomparablesemi-infinite contactconditions,SAsamplesremainedinadhesivecon- SA samples were obtained by cross-linking PDMS in tact against a 15 mm thick PDMS slab. The experi- ∼ molds (2.5 mm deep) fabricated with a desktop CNC mentswereperformedwithahomemadesetupdescribed Mini-Mill machine (Minitech Machinary Corp., USA) in [20, 21]. Using a combination of cantilevers and ca- usingballendmillsofradius100µm,allowingtodesign, pacitive displacement sensors, both the normal (P) and with 1 µm resolution, patterns with controlled surface interfacial lateral (Q) forces are monitored in the range densities and height distributions (Fig. 1b). Spherical [0–2.5] N with a resolution of 10−3 N. This setup also cavities were randomly distributed over 1 cm2 with a providessimultaneousimagingofthemicrocontactswith non overlapping constraint with two different surface the combination of a high resolutionCCD camera (Red- densities φ = 0.1 and 0.4. Their heights as obtained lake ES2020M, 1600 1200 pixels2, 8 bits) and a long– × from a uniform random distribution were in the range working distance Navitar objective. Once illuminated in [30–60] µm. SA samples with φ = 0.1 are thus referred transmission with a white LED diffusive panel, micro- to asSA− further down,andthosewith φ=0.4as SA+. contacts appear as bright disks. Measuring their areas Half of the bottom of the mold was kept smooth so that using standardimage thresholdingtechniques providesa SA samples had both a patterned part and a smooth direct measure of their entire spatial distribution. The one. The smooth part was used in a JKR contact totaltrueareaofcontactAisthenobtainedbysumming configuration [18], which allowed measurement of each all microcontact areas. In addition, assuming the valid- sampleYoung’smodulusE. Secondly,itprovidedmeans ityofHertziancontacttheoryatthescaleoftheasperity to locate accurately the center of the apparent contacts andknowingE,radiiofcurvatureRofeachasperityand formed on the patterned part. Since contacts with the ν =0.5 the Poisson’sratio [20, 21], the disks radii ai are patterned part were obtained by a simple translation of a direct measure of the local normal forces pi since the sample, the center within the contact images was takenasthecenteroftheJKRcircularcontact,obtained 4Ea3 p = i (1) using standard image analysis. i 3(1 ν2)R − 4 As described previously [21], a linear relationship be- InGT’smodel,Hertztheoryofelasticcontactbetween tween the total normal load P = p and the mea- a smooth sphere and a smooth plane is extended by c i i sured P is systematically found for aPll SA samples, thus adding roughness to the plane. As a starting point, the validating Hertz assumption. However, the slope of P relationshipbetweenthelocalpressureandthe localreal c versus P depends slightly on the optical threshold used contact area within an elementary portion of the rough to detect a . To recover a unit slope, we thus calibrated contact is assumed to obey GW’s theory. Accordingly, i the optical threshold with a reference sample of known micro-asperitycontacts aresupposedto be Hertzianand Young’s modulus. For all other samples, we then kept to be independent, that is, the elastic displacements due the same optical threshold and tuned E for each sample to the normal force exerted on one asperity has negligi- withinitsmeasureduncertaintiestorecoveraunitslope. ble effect on any other asperity. However, use of GW’s Note that Hertz contact theory assumes that a /R 1 relationshiprequiresthatthe separationofbothsurfaces i ≪ in order to stay in the linear elastic range. In our exper- atanylocationwithinthemacroscopiccontactisknown, iments, we find that, at the highest normal load, a /R i.e. that the shape of nominal surfaces under deforma- i is at maximum of the order of 0.3. Investigations by tion is determined. This requirement is deduced from Liu and coworkers [22] using micro-elastomeric spheres linearelasticitytheory(Green’s tensor,seereference[24] in contact with a plane (contact radius a) have shown for instance) that introduces long range elastic interac- howeverthatHertztheoryremainsaccurateforvaluesof tions at the scale of the apparent Hertzian contact. As a/R up to 0.33. opposed to GW’s model, which can be derived analyti- ∼ For RA samples,sucha calibrationmethod could notbe cally, in GT’s model, calculation of the real contact area appliedasitrequiresknowingtheradiiofcurvatureofall and pressure distribution can only be done with an iter- asperities to evaluate p . Because of this limitation[28], ativenumericalintegrationofasetofcoupledequations, i we chose the threshold arbitrarily from the contact im- as described in [13]. ages between their two extremal values for which the In Ciaravella et al.’s model, the approach includes in changeintotalareawasfoundtovarymarginally. Conse- thefirstorder-senseelasticinteractionsbetweenHertzian quently, it was not possible to measure any local normal micro-asperitycontacts,i.e. foreveryasperityadisplace- force distribution for RA samples. ment is imposed which is sensitive to the effect of the Friction experiments with RA patterned lenses were spatial distribution of Hertzian pressures in the neigh- performed with another experimental setup described boringasperities. Foreachmicro-asperitycontact,ashift earlier [23]. RA lenses were rubbed against a smooth of the position of the deformable surface is introduced, PDMSslab(E = 3 0.1MPa)keepingbothP andthe which results from the vertical displacement caused by ± driving velocity v constant. The setup allowed variation theneighboringones. Accordingly,theindentationdepth of v from a few µms−1 up to 5 mms−1 thus allowing si- δ of the ith micro-asperity contact is i multaneous measurements of P and Q with a resolution of 10−2 N. N δ =δ0+ α δ3/2, (2) i i ij j Xj6=i Multi-asperity contact models where δ0 > 0 is the indentation depth in the absence i of any elastic coupling between microcontacts, and α ij To investigate quantitatively the effects of elastic in- are the elements of the interaction matrix. As shown in teractions between micro-asperity contacts on the real Fig.3,δ0isapurelygeometricaltermsimplygivenbythe i contact area and related pressure distribution, two dif- difference between the positions of the two undeformed ferent multi-asperity contact models were considered, surfaces for the prescribed indentation depth ∆. The both of which include elastic interactions at different sum in the rhs of eqn (2) representsthe interactionterm length scales. The first one was derived by Greenwood derived from Hertz contact theory. Our study slightly and Tripp (GT) [13] as an extension to the case of differs from Ciavarella et al.’s model as we take for α ij rough spheres of the seminal model of Greenwood and an asymptotic expansion of the Hertz solution for the Williamson(GW)forthe contactbetweennominallyflat vertical displacement of the surface, instead of its exact surfaces. The second one was developed more recently expression. Elements α of the interaction matrix thus ij by Ciavarella et al. [14, 15]. It consists in a modified read form of GW’s model, with elastic interactions between microcontacts incorporated in a first order-sense. Both 4 Rj 1 [α ]= ,i=j, (3) models describe the contact mechanics of rough surfaces ij − p3π r 6 ij with random distributions of spherical asperities, which is what we investigate here experimentally. As a conse- where r is the distance between asperities i and j ij quence of this simplified form of surface topography, it and R is the radius of curvature of the jth asperity. j wasnot necessaryto considermore refinedcontactmod- This approximation avoids evaluating at each step of elsbasedonaspectraldescriptionofthesurfacessuchas the calculation the interaction matrix [α ], which con- ij Persson’s model [8]. sequently depends only on the surface topography. Such 5 a a Δ RA R lens PDMS glass lens 10−7 2) m A ( b δ0 i 2m)10−7 SA A ( P (N) b 10−8 RA 0.01 0.1 0.5 10−8 0.01 0.1 0.2 P (N) FIG.4: Log-logplotoftherealareaofcontactAversusP for δi0 bothRA−(a)andRA+(b)lenses. Theupperandlowerlimits PDMS oftheerrorbarscorrespondtothetotalareasmeasuredwith thearbitrarilychosenextremalvaluesoftheopticalthreshold FIG. 3: Sketch of the geometric configuration for the inden- (seetext). Redshadedareascorrespondtothepredictionsof tation of (a) SA and (b) RA surface topography. For both Ciavarellaetal.’smodel[14,15]bysettingαij to0ineqn(2). configurations, ∆ is the prescribed indentation depth taking Greenareascorrespondtoαij 6=0. Areasextentcharacterizes asareferencefortheverticalpositionoftheindentingsphere thescatterinthesimulations,arisingfromuncertaintiesinthe the altitude at which the smooth surface is touching the up- experimental determination of the topography parameters. permost asperity. calculations were carried out using simulated lens to- anapproximationisvalidaslongastheaveragedistance pographies generated from Gaussian sets of asperity between asperities L is much larger than the average heightscalculatedusing the experimentalparametersre- asperity microcontact radius a. For RA samples, optical portedinTableI.Theradiiofcurvatureoftheasperities measurementsrevealthatthiscriterionissatisfiedasthe werevariedasafunctionoftheirheightsusingtheexper- ratio L/a, which is a decreasing function of P, remains imentally measured R(h) relationship. Asperities were between 6 and 8. For SA samples, one also measures spatially distributed according to a uniform distribution that L/a 16 32 for SA− and L/a 9 15 for SA+. with a non-overlapconstraint. In order to minimize bias ≈ − ≈ − The above detailed models are obviously valid as long intheirspatialdistribution,asperitieswerepositionedby as no contact occurs in regions between the top parts of decreasing size order. the spherical caps. Figure 4 shows the results of such simulations using Ciavarella’s model. Uncertainties in the experimental determination of surface parameters (mainly the R(h) relationship) were found to result in some scatter in the Normal contacts simulated A(P) response. In order to account for this scatter, the simulated curves are represented as colored areas in Fig. 4. A good agreement is observed between RA measurements theoryandexperimentsonlywhenelasticinteractionsare accounted for. Without such interactions (i.e. when the In order to stay consistent with the hypothesis of the termα ineqn(2)issettozero),theactualcontactarea ij contact models, true contact area measurements for RA at a given P is clearly underestimated. lenses were performed for normalloads P for which only topsofthemicro-asperitiesmakecontactwiththePDMS slab. While for RA+ lenses, this is observed for the entire range (up to 0.6 N) of P, for RA− lenses this occurs SA measurements as long as P 0.2 N. Figure 4 shows the total contact ≤ areaA versus P for both RA lenses contactinga smooth PDMS substrate. A(P) exhibits a non-linear power law ForSAsamplesincontactwiththeglasslensofradius behavior with the following exponents: 0.812 0.009 for of curvature 128.8 mm, microcontacts always occur at RA− and 0.737 0.042 for RA+. ± the top of the asperities for the whole investigated P ± To compare these results with Ciaravella et al.’s model, range up to 0.6 N. For each P, the real area of contact 6 A was averaged over N = 24 different locations on the 10−6 sample. This allowed us to probe statistically different 892m) cboynatafcatctcoorn√figNur.atFioignusrew5hislehorwesduthciengrestuhletinegrroArvoenrsuAs 567−7A (10 P for both SA− and SA+ samples. As found with RA 4 lenses,A(P)curvesarealsosub-linearandarewellfitted 3 by power laws. For both tested surface densities, power P (N) law exponents are found to be density independent, 2) 0.2 0.3 0.4 0.5 0.6 m with 0.945 0.014 for SA− and 0.941 0.005 for SA+. A ( ± ± Changing φ from 0.1 to 0.4 mainly results in an increase of A(P) at all P (Fig. 5). As previously done with RA 10−7 samples, both SA data sets are compared to Ciaravella glass et al.’s model [14, 15] predictions, with both α = 0 ij and αij = 0. Calculations were performed using the SA 6 exact topography used to make SA samples, and A versus P curves were obtained with the exact same 0.02 0.1 0.6 24 contact configurations. Errors on the calculated A P (N) valueswereobtainedbyvaryingYoung’smoduluswithin its experimental uncertainties, yielding the shaded areas FIG.5: Log-logplotoftherealareaofcontactAversusP for of Fig. 5. Red shaded areas correspond to αij to 0 in both SA− (φ=0.1, blue diamonds) and SA+ (φ=0.4, blue eqn (2), while green areas correspond to α = 0. At ij circles)samples. Theinsetisacloseupfor0.2 ≤ P ≤ 0.6N. 6 low normal loads (P 0.1 N), the effect of the elastic ErrorbarsaregivenbythestandarddeviationofAon24dif- ≤ interaction on A is almost negligible, but it becomes ferentcontactconfigurations. Redshadedareascorrespondto more pronounced at higher ones (P > 0.1 N), resulting thepredictions ofCiavarella et al.’smodel[14, 15] bysetting in a larger true contact A. As shown on Fig. 5, our αij to 0 in eqn (2). Green areas correspond to αij 6= 0. Ar- data at P > 0.1 N is clearly better captured by the easextentcharacterizesthescatterinthesimulations,arising interacting model rather than the non-interacting one from uncertainties in theexperimental determination of E. for both surface densities. These A(P) measurements, together with those ob- For all investigated SA topographies, a nearly linear re- tained with RA lenses, indicate that including an elastic lationship is found for A(P), which is consistent with interaction is thus essential to have a complete descrip- theconclusionsofthe paperofGreenwoodandTripp[13] tion of the contact mechanics of such systems. Yet, it that states that A(P) is ”approximately” linear. More remainsunclearwhichoftheshortrange(interactionbe- generally, our findings for SA surfaces do not depart tween neighboring asperities) and/or long range (deter- from most of asymptotic development at low P of most minedbythegeometryofthemacroscopiccontact)parts current rough contact models for nominally flat surfaces of the elastic interaction predominate. We now address [10]. Such models, indeed, also predict a linear A(P) precisely this question in the following. relationship. Conversely, for RA topographies, a non- linear power law like A(P) relationship is found. Such deviations from linearity was actually pointed out in recent theoretical works by Carbone and Bottiglione [25] Role of elastic interactions fornominallyplane–planeroughcontacts. Theseauthors pointed out indeed that asperity contact models deviate True contact area load dependence veryrapidly fromthe asymptotic linear relationeven for Using contact imaging techniques, we were able to very small, and in many cases, unrealistic vanishing ap- probe how the total true contact area varies with the plied loads. For our present sphere–on–plane contact, it applied load for contacts between a smooth surface and islegitimatetowonderifthemagnitudeofthedeviations the different model roughsurfacesdecoratedwith spher- arises either from the differences in the asperities height ical caps. For all sizes and spatial distributions of the andsizedistributionsand/orthemacroscopiccurvatures micro-asperities tested here, we found that A(P) curves of the spherical indenter. To provide an answer to this couldbesatisfactorilydescribedwithinthe frameworkof question, simulations using Ciaravella’s et al.’s model, asimpleroughcontactmodelwithaclassicalassumption with the exact same asperities distribution (height, ra- that Hertzian contact occurs at the scale of the micro- dius of curvature and lateral distribution) but different asperities. As opposed to both GW’s and GT’s models, radii of curvature R of the macroscopic lens indenter l ourapproachtakesintoaccountinanapproximateman- (R = 13 mm and R = 128.8 mm, as in the experi- l l ner the elasticcoupling betweenasperities whichis often ments) were performed. In both cases, A(P) curves are neglectedtofullydescribethecontactmechanicsofrough found to follow asymptotically (for 0.005 P 1N) a ≤ ≤ interfaces. power law, whose exponent is 0.86 with R = 13 mm l ∼ 7 FIG. 6: (Color online) (a), (b), (c) Images of the interface at P =0.02,0.2,0.5 N with the φ=0.4 SA sample. microcontacts appear as the white disks. Green (resp. red) circles indicate Ciaravella et al.’s model predicted microcontacts with αij 6= 0 (resp. αij = 0). On all images, the white dashed line circles delimit Hertz contacts for the corresponding P. (d),(e),(f) Angularly averaged pressure p distribution as a function of the distance to the center r on a SA sample with φ = 0.4 at increasingnormalloadsP. Bothpandrarenormalizedbyrespectively,Hertz’maximumpressurep0 andHertzcontactradius aH. The black dashed line corresponds to Hertz prediction. Blue solid lines are fits using Greenwood-Tripp model (GT) with a uniform asperity height density and same surface density φ. The red dot-dashed lines are predictions of Ciaravella et al.’s model [14, 15] setting the interaction term αij = 0, while the green dashed lines correspond to the full model with αij 6= 0. Both latter predictions are statistical averages over 1000 independent pattern realizations with φ= 0.4 and a uniform height distribution. and 0.93 with R = 128.8 mm. Decreasing R thus ing the Hertziancontact. This lengthis found to varyas l l enha∼nces the nonlinearity of the A(P) relationship. It is ∆ R5/9P−1/9. This confirms that for a given applied likely that such effects simply result from the fact that loa∝d, thle extension of the contact area from its Hertzian the increase in the gapbetween both the PDMS and the value, as resulting from microasperities contacts, should lensfromtheedgesofthecontactislargerforalenswith be enhanced when R increases. l a small radius of curvature. For a load increase δP, the Of course, it is expected that the non-linearity of the increaseinthenumberofmicrocontactsattheperiphery A(P) relationship could also depend on the statistical ofthe apparentcontactareais thus expectedto be more properties of the asperity distributions. This is indeed pronouncedwithalargeRl. Thisshouldtranslateinto a suggested by eqn. (A.9) which predicts that ∆ scales as more linear A(P) dependence for large Rl. This hypoth- σ2/3,whereσ isthe standarddeviationofthe heightdis- esis is further supported by a simple calculationdetailed tribution of asperities. One can also mention the early in Appendix A. Assuming that the rough contact obeys theoreticalworkofArchard[3],basedonhierarchicaldis- Hertz law at the macroscopic length scale, one can ex- tribution of spherical asperities on a spherical indenter. press the gap height between surfaces at the periphery ThismodelpredictsthatA(P)followsapowerlawwhose of the contact as a function of the Hertzian radius and exponentvariesbetween2/3(i.e. thelimitofthesmooth the radius of curvature of the indenting lens. Equating Hertziancontact)andunity(whenthenumberofhierar- this gap height to the standard deviation of the height chical levels of asperities is increased). distribution yields a characteristic length scale ∆ which Before addressing further the issue of the elastic inter- corresponds to the size of the annular region surround- actions between microcontacts, some preliminary com- 8 ments are warranted, regarding the sensitivity of the Totesttheeffectofincludinganelasticinteractionatdif- A(P) relationship to the details of the spatial distribu- ferent length scales, we also computed p(r) as predicted tion of microasperities. For that purpose, one can con- by GT’s model. As discussed earlier, this model indeed sider a comparison between experimental and theoreti- constitutes in some sense a ’zeroth order approximation’ cal results for RA patterns. While the micro-asperities of Ciaravellaet al.’s model, as it only takes into account weredistributedspatiallyaccordingtoauniformrandom long range elastic interactions whose extent is set by the distribution in the simulations,sucha distributionprob- size of the apparent contact. GT’s calculation was im- ably does not reproduce very accurately the features of plemented with Mathematica 9 (Wolfram Research Inc., the droplet pattern. As a result of droplet coalescence USA),usingarandomasperitiesheightdistributionwith during condensation, some short distance order is prob- heights chosen uniformly between 30 and 60 µm. ably achieved between asperities as suggested by a close Figures6d–e–fshowtheresultsontheexampleofSA+ examination of Fig. 1a. However, the good agreement for the three increasing loads P of Figs. 6a–b–c. As al- between the experiments and the simulations in Fig. 4a ready anticipated from Figs. 6a–b–c, Ciaravella et al.’s showsthattheloaddependenceoftheactualcontactarea model with α = 0 gives a reasonably good fit of the ij 6 is notverysensitivetothe details inthe spatialdistribu- measureddata. Taking α =0 yields largerdiscrepancy ij tion of asperities. As far as the normal load dependence withthe experimentalpoints,revealingthat, onaverage, of the real contact area is considered, the relevant fea- the effect of the elastic interaction is to increase signifi- tures of surface topography are thus likely to be mainly cantlytheapparentradiusofcontact,thehigherthenor- thesurfacedensityofmicro-asperities,andtheirsizeand mal load P. As pointed out by Greenwoood and Tripp height distributions. in their original paper, the effect of roughness is to add Microcontacts and pressure spatial distributions a small tail to the Hertzian pressure distribution which Sofar,weonlyconsideredtheeffectoftheelasticinter- corresponds to the annular region around the Hertzian action on the load dependence of A, and thus neglected contact in which the separation is comparable with the any spatial dependence of the microcontacts distribu- surface roughness. Indeed, as already mentioned earlier, tion. Direct comparison of such data with Ciaravella et an order of magnitude of this tail is provided by the al.’s model calculations is not easily accessible for RA characteristiclength∆whichscalesasR5/9σ2/3 (seeAp- samples since it would require a knowledge of all asper- pendix A). It can be noted that this scaling is very close ities positions and respective radii of curvature. With to that deduced fromdifferent arguments by Greenwood SA samples however, this can be easily done, as posi- and Tripp (i.e. ∆ √Rσ). ∝ tions and radii of curvature of asperities are known by Giventheexperimentalerrorbars,itisdifficulttoclearly design of the micromilled pattern. Figures 6a-b-c show delineatewhichofCiaravellaet al.’sinteractingmodelor such direct comparison at three increasing normal loads GT’smodelfitsbestthemeasureddata. Actually,tofirst P (P =0.02,0.2,0.5 N) for the case of the SA+ sample. order, both models fit equally well the experiments, and As expected, predicted microcontacts with α = 0 al- constitute, to our knowledge, the first direct experimen- ij 6 most always match the measured microcontacts (see the talvalidationofbothmodels. Thissuggestsinparticular, green circles on the figure). For comparison, red circles that if one needs to measure the spatial distribution of at the predicted positions of the model without elastic pressurep(r), GT’s modelis averygoodapproximation. interaction have been overlapped on the contact images. Second,itindicatesthatshortrangelocalelasticinterac- Clearly, the non-interacting model predicts contacts at tions effects cannot easily be caught when analyzing the locationswithintheapparentcontactwhicharenotseen radial pressure distribution, or that these effects are of in the experiment. second order. To perform a more quantitative comparison with the- The fact that p(r) distributions are very similar for oretical predictions, we computed for both the experi- both models motivates a closer examination of the dis- mental and calculated points, the local radial pressure tributions of quantities from which p(r) derives. For profiles p(r). The latter, which is expected to be ra- that purpose, the pressure dependence of surface den- dially symmetric for a sphere–on–plane normal contact, sity η and mean radius a of microcontacts was consid- was obtained by summing up local forces p exerted on ered (where η is defined as the number of microcontacts i all microcontacts located within an annulus of width per unit area). In Fig 7, theoretical (as calculated from dr =0.25mmandradiusrcenteredontheapparentcon- Ciavarella’smodelwithα =0)andexperimentalvalues ij 6 tactcenter(obtainedfromJKRexperiments). Toreduce of η and a are reported in a log-log plot as function of the statisticalerror,averagingof p(r) for severalcontact thecontactpressurep. Twodifferentdomainsareclearly configurations was then performed. For the experiment, evidenced. When the pressure is greater than a critical 24contactconfigurations(compatiblewiththesizeofthe value p∗, which is here of the orderof 50 Pa, η and a ex- SA pattern) at different locations on the same SA pat- hibit with p a power law behavior whose exponents are tern were used. For the calculated data (Ciaravella et found to be equal to 0.4 and 0.2, respectively, from the al.’smodel),1000statisticallydifferentSApatternswere simulated data. As detailed in Appendix B, these expo- usedandnormalloadingwasdoneatthecenteroftheSA nents are identical to that predicted by the GW model pattern. Both α =0 and α =0 data were computed. for nominally flat surfaces in the case of a uniform dis- ij ij 6 9 a 100 shortrangeeffectsofthepaircorrelationfunctionassoci- ated with asperity distribution. However, the important point is that p∗ always corresponds to very low contact −1 10 pressures. Fromanextendedsetofnumericalsimulations where parameters such as asperities density, radius of curvatureand heightdistribution werevariedby at least −2 10 one order of magnitude, p∗ was systematically found to η0 be in the range 101 103Pa. For the considered contact η/ − −3 conditions, such a pressure range corresponds to a very 10 narrow domain at the tail of the pressure distribution whosephysicalrelevanceisquestionable. Inotherwords, 10−4 both the simulations and the experimentaldata indicate that the GW theory is able to describe accurately the microcontacts distribution over most of the investigated −5 10 pressure range without a need to incorporate the effects −3 −2 −1 0 1 2 3 4 5 10 10 10 10 10 10 10 10 10 of short range elastic interactions in the rough contact b 102 p (Pa) description. Frictional properties 1 10 ) We now turn ontothe frictionalbehaviorof RA lenses m) µm against a smooth PDMS slab. As mentioned above, RA µ( a a ( asperities are very smooth which allows us to consider theassociatedmicro-asperitiescontactsassingle-asperity 0 10 contacts. RAsurfacesthusprovidesystemswithasingle roughnessscale as opposedto SA surfaces which present an additional microscopic roughness. In what follows, we address from preliminary results the issue ofthe con- −1 tribution of individual micro-asperities contact to the 10 10−3 10−2 10−1 100 101 102 103 104 105 macroscopic friction force. For P within [0.01–0.6] N p (Pa) and driving velocities v up to 5 mms−1, both RA+ and RA− lensessystematicallyexhibitedsmoothsteadystate FIG. 7: (Color online) (a) Microcontacts density η, normal- friction with no evidence of contact instabilities such as ized by the mean number of micro-asperities per unit area stick-slip,norstrongchangesintheirfrictionalbehavior. η0, versus local pressure p for the SA sample with φ = 0.4. Thus, only results obtained at the intermediate velocity (b) Mean microcontacts area a¯ versus local pressure p for of v =0.5 mms−1 are reported here. Figure 8 shows the the same sample. On both graphs, black disks are the re- resulting lateral force Q versus normal force P curves sults of GT’s model predictions, the green disks are predic- for both RA− (Fig. 8a) and RA+ (Fig. 8b) samples, as tions of Ciaravella et al.’s model with αij 6= 0 and crosses well as for a reference glass lens with the same radius of correspond to the experimental data at three different loads curvature and covered with a thin smooth layer of the P = 0.02,0.2,0.5 N. Thick black lines are power law fits of same sol-gelmaterialusedfor RA lenses (Fig. 8b, inset). GT’s model predicted data, while green solid lines are power lawfitsofCiaravella et al.’smodelpredicteddataforp<p∗, In all cases, Q is found to vary non-linearly with P. In with p∗ ≈50 Pa. the simplest description, the total friction force Q is expected to be the sum of local friction forces q acting i onallcontactingmicro-asperities. Accordingtoprevious tribution of asperities heights (η p2/5 and a p1/5). studies using glass/PDMS elastomer contacts [26, 27], a This means that as long as p > p∝∗, the pressure∝depen- constant, pressure independent, shear stress τ0 can be assumed to prevail at the intimate contact interface be- dence of η and a is insensitive to both the effects of the tween the asperities and the PDMS elastomer, yielding elasticcouplingbetweenmicro-asperitiescontactsandto the curvature of the nominal surfaces. Below the crit- qi = τ0(πa2i). Within this framework, Q should thus write as ical pressure p∗, a power law dependence of η and a is stillobservedbutwithexponents,respectively0.78±0.11 Q=τ0A (4) and 0.37 0.02, which depart from the GW predictions (Fig. 7). ±We do not yet have a definite explanation for with A = (πa2) the real area of contact. In the cal- i i these deviations which are systematically observed, irre- culation, wPe take for A the experimental values mea- spectiveofthenumberofsurfacerealizations(upto8000) sured under normal indentation after verifying from op- considered. Theycouldtentativelybeattributedtosome tical contact observations that the microcontacts areas 10 a 0.20 asperities respectively. There is thus some evidence of a dependence of the frictional shear stress on the contact length scale, the shear stress at the microcontacts scale being larger than that at the scale of a millimeter sized 0.15 contact ( 18% and 44% increase for RA- and RA+, ∼ ∼ respectively). Curvatures of the micro-asperity contacts ) being larger than that of the smooth contact with the N Q (0.10 glass lens, the increase in τ0 at small length scales could be attributed to bulk viscoelastic dissipation as a result of the ploughing of the PDMS substrate by the micro- asperities. However, the fact that Q does not vary sig- 0.05 nificantly when the sliding velocity is changed by nearly three orders of magnitude (from 0.01 to 5 mms−1) does not support this assumption. This weak contribution of viscoelastic dissipation to friction can be related to the 0.00 low glass temperature T = 120◦C of the PDMS elas- g − 0.00 0.05 0.10 0.15 0.20 0.25 tomer. Indeed, for the considered micro-asperities size P (N) b distributions, the characteristic strain frequency associ- 0.6 atedwiththe microcontactsdeformationisv/a 10Hz, ≈ 1.4 i.e. well below the glass transition frequency at room 1.2 N) temperature (more than 108 Hz). Other effects, arising 0.5 1.0 Q ( for example from non linearities in the highly strained smooth 0.8 lens microcontacts could be at play, which will be the scope 0.4 0.6 of further investigations. However, these experimental 0.4 N) 0.2 P (N) results show that frictional stresses measured at macro- Q ( 0.3 0.0 scopiclengthscalesmaynotbesimplytransposedtomi- 0.0 0.2 0.4 0.6 croscopic multicontact interfaces. 0.2 Conclusion 0.1 Inthiswork,wehavestudiedbothnormalcontactand 0.0 friction measurements of model multicontact interfaces 0.0 0.1 0.2 0.3 0.4 0.5 0.6 formedbetween smoothsurfaces androughsurfacestex- P (N) tured with a statistical distribution of spherical micro- asperities. Two complementary interfacial contacts were FIG. 8: Q versus P in steady sliding (v = 0.5 mms−1) for studied, namely a rigid sphere covered with rigid asper- contactsbetweenasmoothPDMSsubstrateandRA−(a)and ities against a smooth elastomer, and a smooth rigid RA+ (b) lenses. On both graphs, dashed lines are the theo- sphere againsta flat patterned elastomer. In both cases, retical Q given by eqn (4), taking for A its measured values experimental A(P) relationships were found to be non- andforτ0 =0.34MPatheaverageshearstressobtainedwith the smooth lens. Solid lines are fits of the experimental data linear and well fitted by Ciaravella et al.’s model taking with eqn (4), yielding τ0 =0.40 MPa for RA− and 0.49 MPa into account elastic interaction between asperities. Ad- for RA+. Inset: Q versus P for the smooth lens, in steady ditional information regarding the nature of the elastic sliding. Thesolidlineisafitofthedatausingeqn(4),taking couplingbetweenasperitieswasprovidedfromtheexam- for A its measured valuein steady sliding. inationoftheprofilesofcontactpressure,contactdensity and average radius of asperity contacts. While the long range elastic coupling arising from the curved profile of during sliding are not significantly different from that the indenter was found to be an essential ingredient in achieved under static loading [29]. As a first attempt, the description of the rough contacts, both experimen- thefrictionalshearstressτ0 wastakenasthe experimental andsimulation results demonstrate that, for the con- tal value calculated from the ratio of the friction force sideredtopographies,shortrangeelastic interactionsbe- to the actual contact area measured during steady state tweenneighboringasperitiesdoesnotplayanydetectable friction with the smooth lens. As shown by the dotted role. As a consequence, the pressure dependence of both lines in Figs. 8a-b, choosing this shear stress value un- the density and the radius of asperity contacts within derestimates the experimental data for both small and the macroscopic contact is very accurately described us- large size asperities RA samples. Fitting the experimen- ing GW model which neglects asperity interactions. To tal data with eqn (4) using a least square method yields our best knowledge, these results constitute the first di- however τ0 = 0.4 and 0.49 MPa for small and large size rectexperimentalvalidationofGWandGTmodels. The