Transition from Cassie to Wenzel state in patterned soft elastomer sliding contacts E´lise Degrandi-Contraires,1 Christophe Poulard,1 Fr´ed´eric Restagno,1 Rapha¨el Weil,1 and Liliane L´eger1 1Laboratoire de physique des solides, CNRS & Universit´e Paris-Sud, 91405 Orsay cedex (Dated: January 10, 2012) Inthispaper,wepresentedanexperimentalandtheoreticalanalysisoftheformationofthecontact betweenasmoothelastomerlensandanelastomersubstratemicropatternedwithhexagonalarrays ofcylindricalpillars. WeshowusingaJKRmodelcoupledwithafulldescriptionofthedeformation ofthesubstratebetweenthepillarsthatthetransitionbetweenthetoptothefullcontactisobtain whenthenormalloadisincreasedaboveawellpredictedthreshold. Wehavealsoshownthatabove the onset of full contact, the evolution of the area of full contact was obeying a simple scaling. 2 Roughness is known to deeply influence the contact vent the two surfaces to establish a full contact, due to 1 mechanicsofelasticbodiesandtodrasticallyaffectprop- adhesion forces. The same kind of jump to full contact 0 erties such as adhesion and friction. Due to roughness, duetoadhesionforcesonroughmodelelasticsurfaceshas 2 only partial contact can usually be established between also been studied even more recently on rippled surfaces n two solids. The real area of contact is then smaller than [? ]. a theapparentone,anddependsonbothexperimentaland J In the present letter, we present what we think to be material parameters. As a result, molecular forces are 9 the first systematic analysis of the nature of the con- usuallyunabletoproduceanoticeableadhesionbetween tact between a smooth elastomeric lens (R = 1.2 mm ] solids but it has also been shown that dividing a surface t and El = 2.1 MPa) and series of elastomeric substrates f in a set of parallel soft asperities can increase adhesion o (Es = 1.8 MPa) patterned with regular hexagonal ar- in the case of biomimetical surfaces [? ]. Concerning s rays of cylindrical micropillars having a fixed height h . friction, roughness is thought to be at the origin of the t and diameters (h = 2.2 µm and d = 4 µm) and various a classical Amonton’s law which which predicts that the spacing. Thisallowedustoputaspecialemphasisonthe m friction coefficient between two surfaces is independent incidenceoftheappliedloadandofthepatterngeometry - of the apparent contact area between the surfaces. In d on the evolution of the nature of the contact. Surfaces their pionneer work, Fuller and Tabor [1] gave the first n were made by moulding and reticulation of PDMS (Syl- o microscopical understanding of Amonton’s law based on gard 184) using a know well established technique [6]. c the idea that due to plastic deformations of the asperi- The homemade JKR apparatus that have been used to [ ties,theirshouldbealinearrelationshipbetweenthereal investigate the contact have yet been described in [8]. 1 contact area between two surfaces and the normal load. It is devoted to the characterization of the size of the v More recently, statistical distributions of elastic asperi- contact versus applied normal load. An essential point 9 tieshavebeenstudiedbyGreenwood[? ] orPersson[? ]. is the possibility of continuously monitoring the contact 1 Molecular friction is usually studied using SFA ou AFM 7 through an optical microscope equipped with a video and is related to monocontact friction but real friction 1 camera, when progressively pushing to or pulling off the . deals with multicontacts. There is however, at present, lens from the substrate. All data reported below corre- 1 no real understanding on how one can go from the mono 0 spond to final static state occurring after a micrometric to the multicontact behaviors when changing the rough- 2 displacement step and a waiting for relaxation. 1 nessortheload,despitetheobviouspracticalimportance : ofbeingabletoadjustandcontrolfriction. Howdoesthe Asshowninfigure1,twoquitedifferentsituationscan v applied normal load affect the nature of the contact be- be identified, depending on the applied load. For low i X tweentwosolids? Therecentdevelopmentofmicrofabri- enough loads, the lens only touches the top of the pillars r cation techniques [4] allows a relatively easy preparation over the whole contact zone (figure 1-a). Air is trapped a ofsurfaceswithwellcontrolledmicropatternshavingspe- between the lens and the substrate around the pillars, cific geometrical characteristics providing a unique tool whichthenremainwellvisibleduetotheindexofrefrac- to experimentally try to answer to the above questions. tion mismatch between air and elastomer. For the same A first exploration of the incidence of micropatterning pattern geometry, when the normal load is increased, a on sliding friction for elastomeric contacts has been re- centralzoneappearsinthecontact(figure1-b), withthe ported recently [5] and has pointed out the influence of lens touching the substrate in between the pillars which thenatureofthecontactonthefrictionbetweensurfaces then become hardly visible. This zone of full contact re- with controlled asperities made of regular pillars. More mains surrounded by a corona in which the contact is precisely, they observed that tall pillars were leading to only established on the top of the pillars (contrast simi- partial contact, with the contact only established on the lar to that in 1-a). Such a contact will be called a mixed topofthepillars,whileshortpillarswerenotabletopre- contact. Thistransitionbetweentopandfullcontactap- pears quite reminiscent of the well known Cassie-Wenzel 2 (a) (b) 1.0 1.0 0.8 0.8 A0.6 A0.6 A/ f 0.4 A/ f 0.4 φ= 0.227 φ= 0.179 φ= 0.145 0.2 0.2 φ= 0.120 φ= 0.101 0.0 0.0 0 10 20 30 40 50 0 5 10 15 20 Force (mN) F / Fc (a) (b) FIG. 3. Evolution of the full contact area A normalized by f theapparentcontactareaAversusthenormalload(left)and the normal load normalized by the critical one (right). the range over which the lens needs be deformed to ac- FIG.1. Visualizationofthecontactintopviewandschematic commodate the shape of the patterned substrate. representationoftheinterfacefortop(a)andmixed(b)con- In figure 3-a, the evolution of the full contact area A f tact. normalized by the apparent contact area A with the ap- pliednormalloadisreportedfordifferentsurfacedensity of pillars. All curves appear rather similar, except for 100 the value of the critical load Fc which depends on φ. It is then tempting to scale all data of the figure 3-a nor- malizing the applied load by the critical load. Such a 10 N) scaling is shown in figure 3-b. A single master curve is m approximately obtained with an evolution of the scaled c ( 1 F area of full contact with the scaled load highly non lin- Exp ear: a rapid increase in the very vicinity of the threshold Hertz 0.1 JKR (F/F (cid:39)1) is followed by a slower long evolution to sat- c JKR + C uration. A small deviation is visible to the scaled curve 0.015 6 7 8 9 2 3 4 5 forhighsurfacedensityofpillars. Theapproximatescal- 10 f (%) ing shown in figure 3-b is a remarkable result. We have developed a mechanical description of the contact based FIG.2. log-logrepresentationofthecriticalnormalloadver- on the classical JKR calculation [12? ] which allows one sus the fraction of surface occupied by pillars with different to understand the origin of this simple scaling. It point spacing. out possible reasons for the departures from scaling well visible in figure 3-b. We present below the main steps of that description. transition observed when a liquid drop is deposited on a A first important point is to check for the validity of rough non wetting substrate [10]. the JKR approach (well known to correctly describe the We have first analyzed in details how the critical load, adhesive smooth contact) for describing either the pure F , for the transition between top and mixed contact c top or the mixed full and top contacts formed in exper- was affected by the geometrical parameters of the pat- iments. In figure 4, the relation between the measured tern. Theresultsarereportedinfigure2, intermsofthe normalload(afterrelaxationateachstep)F andthera- evolution of the critical load as a function of the fraction dius of the apparent contact a is reported in the scaled of surface of the substrate occupied by the pillars φ, for units of the linear form of the JKR equation: pillarsallhavingthesamediameter. Withthehexagonal array of cylindrical pillars, φ= 2√π3(d/i)2. The range of F (cid:18) a3/2 (cid:19) (cid:112) investigated spacing varying from 5 µm to 12 µm, φ was √ =K √ − WeffK (1) 6πa3 R 6π varied from 58% to 10% (since d = 4 µm). It appears clear in figure 2 that the critical force increases with the where R is the radius of curvature of the lens and W eff pillar density φ. Since an increase of φ means decreas- theeffectiveworkofadhesion. Alldatacanclearlybede- ing the pillar spacing i, decreasing the distance between scribed by a unique linear dependence whatever the na- pillars makes more and more difficult the formation of tureofthecontact(topormixed)expectforsmallradius. a zone of full contact in the center of the contact zone. TheslopeK =0.86±0.05MPaisexpectedtobethestiff- This is at least qualitatively easy to understand, as, for ness of the contact. Knowing this value, the intercept a given height of the pillars, their relative distance fixes at origin allows to define an effective work of adhesion, 3 1400 ξ 1200 K = 0.85 ± 0.05 MPa h−δ l 3/2m ) 1000 W= 6 ± 3 mJ/m² ξS 31/2a) ( N/ 860000 f f f ===000...121720971 FIG. 5. Schematic representation of the result of a finite 400 f =0.145 p6 F/( 200 f =0.296 element simulation for the different deformation δ, ξl and ξs f =0.120 with periodic boundary condition for d=4 µm and i=12 µm 0 f =0.057 (φ=10%). -200 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.61.8x10-3 a3/2/R(6p )1/2 (m1/2) with δ the deformation of the pillar located closest to the center of contact, ξ and ξ the deformation respec- s l tively of the substrate and of the lens, and h the initial FIG. 4. Evolution of the rescaled normal load as a function height of the pillars. The evaluation of F thus amounts of the rescaled radius of the contact area for series of pat- c terned surfaces with various patterns characteristics and one to an evaluation of all deformations δ, ξs and ξl. Con- elastomer lens with a radius R=1.2 mm. sidering δ (cid:28) h, ξ (cid:28) R and ξ (cid:28) H, the total height of l s thesubstrate,alldeformationsaresmallandtheHooke’s law can be used. One has to notice that, because forces W =10±5mJ/m2 havingacorrectorderofmagnitude eff are transmitted through the interfaces, the local stress to be identified to φW (with W the work of adhesion on exerted on one pillar is related to the local stress σ(r) smoothPDMS elastomer, W =43mJ/m2) forthe range given by the JKR stress profile (eq. 2) by: ofφvaluesspanbytheexperiments(0.1<φ<0.3). One cannoticeinfigure4aslighttendencytodepartfromthe σ(r) σ = (4) linear dependence at small effective areas of contact, an p φ effect which can be attributed to finite size effects when, at very small areas of contact, the lens senses essentially When the distance between pillars is large enough, both thelayerofpillars: indeed, intheframeworkoftheover- deformations of lens and substrate due to the compres- all JKR analysis of the contact, this layer of pillars can sion of the more central pillar can be described by the be viewed as having an average elastic modulus smaller deformationofanincompressiblesemi-infinitemediasub- than a dense smooth elastomer substrate [11]. Two im- mitted to a uniform pressure through a flat cylindri- portantconclusionscanbedrawnfromthedatareported cal punch, and are given by classical mechanics: ξs,l = in figure 4: first, the lens - patterned substrate contact 2(1−νs2,l) dσp, with ν respectively the Poisson ratio and globally obeys at the JKR’s law, and second, this is true E πrespEecs,tlivelythese,llasticmodulusforthesubstrateand s,l whatever the nature (top or mixed top and full) of the the lens. The displacement δ due to the deformation of contact. Globally,thecontactcanbedescribedthrougha a cylindrical pillar under the local stress σ is: δ = σph. rigidity modulus, given by those of the PDMS elastomer In fact, as it has been discussed recently, pfor small iE/sd, and lens, and an effective work of adhesion renormalized thedeformationofthesubstrate(andlens)cannolonger by the fraction of contact. relaxtotheunperturbedpositionbetweentwocloseback As the JKR’s law is observed to correctly account for pillars (coupled behavior of all pillars through the defor- the global mechanics of the contact, the local stress pro- mationoftheunderlyingsubstrate[6]). Theevaluationof file inside the contact zone can be given by JKR contact the deformations to be plugged in each deformation can mechanics, i.e. [12]: then no longer be conducted analytically, but, as shown in[6],bothdeformationandthestoredelasticenergycan (cid:114) (cid:112) 3f r2 6πa3φWK 1 σ = H 1− − (2) be estimated numerically. The two deformations of lens 2πa2 a2 2πa2 (cid:113)1− r2 and substrate need then to be corrected by a numeri- a2 cally determined correcting function, which is the same where r is the distance from the center of the contact. on both sides of the interface and only depends on the Themaximumlocalstressisatthecenterofthecontact, geometry of the pattern. In figure 2, the experimental which will thus be the location of the maximum defor- dataforthecriticalforceFc arefirstestimatedusingthe mation of both the lens and the substrate. F , which is JKRstressdistributioninsidethecontact,theestimation c the applied normal force corresponding to the first ap- of the critical force assuming independent pillars (dash pearance of a full contact, can thus be obtained by the line)andcoupledpillars(fullline)havebeenplotted. For condition: comparison, the case of independent pillars with a stress distribution given by a Hertz analysis of the contact is ξ +ξ =h−δ (3) also reported as the dotted line. It’s important to notice s l 4 theslopeofthelinearvariationoflog(1−(a /a)2)versus f 1 log(F/F ). It also appears clear in figure 6 that if the Hertz c 7 JKR + C JKRpluscoupledpillarsapproachpredictstheexponent 6 5 of the power law dependence (slope of the dotted lines 4 a/ a)²f 3 iancctohuenltogfo-lrogthpeloptroeffaficgtourr.e 6W),eithiasvneortigahbtlentoowconrorecretlayl 1-( 2 φ =0.227 explanationforthatfactwhichmaybeduetoanincrease φ =0.179 difficulty in increasing the radius of the full contact (a φ =0.145 0.1 kind of hysteresis of the contact) when the density of φ =0.120 7 φ =0.101 pillars is increased. 6 5 As a conclusion, we have presented a detailed analysis 5 6 78 2 3 4 5 6 78 2 3 4 5 1 10 of the formation of the contact between a smooth elas- tomer lens and an elastomer substrate micropatterned F/Fc with hexagonal arrays of cylindrical pillars. We have FIG. 6. Log-log variation for experimental and theoretical shown that the transition between top and full contact values of (1−(a /a)2) versus the normalized force F/F . previouslyobservedinslidingcontactswhenchangingthe f c height of the pillars, could indeed be induced, without sliding motion, when changing the normal load applied thattherearenoadjustableparametersinthesecompar- in this JKR type contact and studied in a more detail isons. One can see that the Hertz approach is only able the evolution of the contact area. These findings rep- to predict qualitatively the observed critical force while, resent the first step of an extensive investigation of the as expected, the JKR mechanics does far better. It is incidence of patterning in sliding contacts. also clear in figure 2 that one needs to consider the cou- pling of the deformations when φ increases above 0.2 to correctly account for the data. Itisnowpossibletopushfurtherthemechanicalanal- [1] K. Fuller and D. 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Brochard-Wyart, and A. Buguin, Physical Review lead to a simple scaling defined by: E 82, 031806 (2010). [6] C. Poulard, F. Restagno, R. Weil, and L. Leger, Soft (cid:16)a (cid:17)2 (cid:18)F (cid:19)−2/3 Matter 7, 2543 (2011). 1− f = (5) a F [7] C. Cohen, F. Restagno, C. Poulard, and L. Leger, Soft c Matter 7, 8535 (2011). This is represented as the dashed line in figure 6. The [8] M.Deruelle,H.Hervet,G.Jandeau, andL.Leger,Jour- nalOfAdhesionScienceAndTechnology12,225(1998). full line represents the result of a similar analysis but [9] L. Bureau and L. Leger, Langmuir 20, 4523 (2004). using the JKR stress distributions and the numerically [10] A.LafumaandD.Quere,NatureMaterials2,457(2003). estimatedcoupledpillarsdeformations(JKR+C).Again, [11] E. Barthel, A. Perriot, A. Chateauminois, and one can clearly see that if the Hertz approach captures C. Fretigny, Philosophical Magazine 86, 5359 (2006). the essential features of the problem, it cannot account [12] D. 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