Model Experimental Study of Scale Invariant Wetting Behaviors inCassie−Baxter and Wenzel RegimesValentin Hisler, Laurent Vonna,* Vincent Le Houerou, Stephan Knopf, Christian Gauthier,Michel Nardin, and Hamidou Haidara

Institut de Science des Materiaux de Mulhouse (IS2M) CNRS - UMR 7361, Universite de Haute Alsace, 15 rue Jean Starcky BP2488,68057 Mulhouse Cedex, France

ABSTRACT: In this work, we discuss quantitatively two basicrelations describing the wetting behavior of microtopo-graphically patterned substrates. Each of them contains scaleinvariant topographical parameters that can be easily expressedonto substrates decorated with specifically designed micro-pillars. The first relation discussed in this paper describes thecontact angle hysteresis of water droplets in the Cassie−Baxterregime. It is shown that the energy at the origin of thehysteresis, that has to be overcome for moving the triple line,can be invariantly expressed for hexagonal pillars by varyingthe pillars width and interpillar distance. Identical contactangle hystereses are thus measured on substrates expressingthis scale invariance for pillar widths and interpillar distancesranging from 4 to 128 μm. The second relation we discuss concerns the faceting of droplets spreading on microtopographicallypatterned substrates. It is shown in this case that the condition for pinning of the triple line can be fulfilled by simultaneouslyvarying the height of the pillars and the interpillar distance, leading to faceted droplets of similar morphologies. The invariance ofthese two wetting phenomena resulting from the simultaneous and homothetic variation of topographical parameters isdemonstrated for a wide range of pattern dimensions. Our results show that either of those two wetting behaviors can be simplyachieved by the proper choice of a dimensionless ratio of topographical length scales.

■ INTRODUCTION

Roughness is a parameter that is known for long to stronglyaffect the wetting of a surface.1 For droplets on a smoothsubstrate, characterized by a water contact angle <90°, addingroughness on that substrate is expected to increase thespreading of the droplet, leading to lower apparent contactangles. In the wetting regime characterized by a water contactangle of >90°, on the contrary, increasing the roughness of asubstrate will result in an increase of its hydrophobicity and thusto higher apparent contact angles. This enhancement ofhydrophobicity that may lead to superhydrophobic behavior(contact angle > 140°) can be described by two models. In theWenzel model, the liquid droplet totally wets the solid surfacebelow it, but the triple line becomes pinned.2 In the Cassie−Baxter model, the droplet rests on the peaks of the surfaceasperities and entraps air pockets.3 These two models wereextensively studied since the work of Neinhuis and Barthlottthat linked the water-repellent properties of the lotus leaf to itssurface roughness.4

Motivated by the development of microstructuring techni-ques and the discovery of a wide range of natural systemsshowing remarkable wetting properties, micropatterned surfaceshave been extensively studied as model rough surfaces tounderstand the role of topography on wetting phenomena. Inthe case of the Cassie−Baxter regime, many efforts have been

made to understand the causes of the extremely low contactangle hysteresis that characterizes superhydrophobic surfaces.Several authors have thus studied the pinning/depinning of thecontact line at the origin of the contact angle hysteresis, as afunction of the morphology and surface density of posts, holesor stripes pinning the contact line.5−9 To predict the contactangle hysteresis of droplet resting on micropatterned substrates,Reyssat and Quere10 as well as Dubov et al.11 proposed amechanical model in which the pinned liquid tail that locallyforms on a micropost during the receding of the contact line isconsidered as a spring. More recently Paxson and Varanasi12

examined the morphology of the contact line pinned at the topof microposts, and suggested a self-similar depinning mecha-nism to explain the observed contact angle hysteresis onmicropatterned surfaces. Fewer studies have considered thewetting of a microtopographically patterned substrate for liquidsand solids showing high affinity (equilibrium contact angle <90°). In this framework, Bico et al.13 defined a critical contactangle depending on the roughness of the substrate, which allowsprediction of the spontaneous filling of roughness by a liquid.The dynamics of spreading in the microtopography as well as

Received: March 31, 2014Revised: July 15, 2014Published: July 15, 2014

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the final shape of the drop was studied as a function of themicroposts’ morphology and surface density.14−17 Recently,several authors reported asymmetric wetting induced byasymmetric post arrays.18,19 A large number of applicationsare concerned with the wetting of such micropatterned surfaces,including the design of water-repellent surfaces or microfluidicsystems, for example. The now available micropatterning toolsallowing the fabrication of different topography length scales,shapes, and chemistries offer multiple ways to establish basicrelations driving the wetting of such substrates, and moregenerally of rough substrates.In this study, we explore two of these relations on the basis of

model experiments. The first describes the contact anglehysteresis of droplets in the Cassie−Baxter regime, and thesecond describes the topographic conditions for dropletfaceting, the liquid being imbibed into the micropattern. Wehere examine more precisely the scale invariant wettingparameters found in these two relations that can be expressedby a series of specifically designed self-similar surface patterns.Although explicitly expressed in these two basic relationsdescribing the wetting of microtopographically patternedsubstrates, this invariance is rarely discussed in the literature,especially its range and conditions of validity. In the first part ofthis paper, we will thus discuss the contact angle hysteresis ofwater droplets suspended on micropillars (Cassie−Baxterregime), which is expected to be constant for a given ratio l/d, with d being the width of the pillars and l being the interpillardistance. In the second part, we will discuss the morphologies ofthe more wetting ethanol droplets that are formed on a series ofsubstrates decorated with pillars of identical aspect ratio d/H,with d and H being the width and the height of the pillars,respectively. This aspect ratio d/H was shown indeed to rule thepinning of the triple line and thus the faceting and morphologyof the droplet. In the following, the invariance of the wettingparameters that characterize these two laws is examined onsurfaces decorated with micropillar arrays defined by identicalratio l/d or aspect ratio d/H, which are invariantly expressed fordifferent widths d, interpillar distances l, and heights H, asdescribed in the Materials and Methods section.

■ MATERIALS AND METHODSFabrication of the Micropatterned Surfaces. A polydimethylsi-

loxane (PDMS) elastomer (Sylgard-184, Dow Corning) was used tofabricate the micropatterned surfaces. The Sylgard 184 prepolymer andcross-linker was mixed at 10:1 ratio and poured on silanized siliconmolds. The PDMS was cured at 80 °C during 4 h and then peeled fromthe silicon molds. The smooth PDMS samples used as control surfaceswere obtained from a PDMS molded on a flat silicon wafer using asimilar procedure.The silicon molds were micropatterned to obtain PDMS substrates

decorated with hexagonal micropillars for which width d and interpillardistance l are varied in a homothetic manner over the samples (with d= l) as shown in Figure 1A and B. The micropillars are distributed on ahexagonal lattice and thus display a constant surface fraction ϕ (ϕ =25%). We used four series of surface pattern as a microtopographiestoolbox (Figure 1C), to address the question of the scale invariantwetting parameters characterizing the two relations discussed in theIntroduction. The two first series (series 1 and 2) each have constantpillar height, H = 16 and H = 4 μm, respectively. The width d andinterpillar distance l are varied from 4 to 128 μm (with d = l). The twolast series (series 3 and 4) are each characterized by pillars of constantaspect ratio with d/H = 2/1 and d/H = 4/1, respectively. The width dis varied for these two series from 8 to 32 μm and 16 to 128 μm,respectively. For these surfaces, the roughness factor r defined as r =(total surface area/projected area) is here r = 1 + H/d, which is the

same for a given aspect ratio d/H. The root mean squared roughness ofthe smooth PDMS (measured with a Nanoscope IIIa from DigitalInstruments, in the tapping mode; Veeco, Santa Barbara, CA) is ∼0.8nm on a 400 μm2 large area.

Surface Characterization and Wetting Experiments. Theprofiles of the patterned surfaces were characterized by scanningelectron microscopy (SEM; FEI Quanta 400 ESEM operating atelectrical potentials of between 15 and 20 kV). The droplet shapes (topview) were characterized by optical microscopy (Olympus BX60) andrecorded with a COHU camera coupled with a computer. Freshdeionized water (Elix from Merck, Millipore, Germany) and analyticalgrade ethanol (Carlo Erba, Italy) were used as wetting liquids. Thecontact angles were measured on a manual goniometer. In the case ofthe advancing and receding contact angles measurements, a droplet ofwater (∼8 μL) was first formed at the tip of a syringe. The drop wasthen slowly moved down until it touched the substrate. This procedureensures an exclusive contact between the liquid and the top of thepillars, especially in the case of large interpillar distances for which theCassie−Baxter state (suspended droplet) is particularly unstable. Theadvancing contact angle was then measured by growing the size of thedroplet, and the receding contact angle by decreasing the size of thedroplet. A droplet volume of ∼8 μL leads to a droplet contact basediameter of ∼4 mm which is large compared to the maximal interpillardistance of 128 μm used in this work.

■ RESULTS AND DISCUSSIONContact Angle Hysteresis of Water Droplets in the

Cassie−Baxter Regime. The contact angle θCB of a droplet inthe Cassie−Baxter regime for flat-topped pillars is given by thefollowing relation:

θ ϕ θ= + −cos (1 cos ) 1CB eq (1)

with ϕ being the wetted surface fraction and θeq being theequilibrium contact angle measured on a flat surface of the samematerial. Even though the equilibrium wetting contact angle isoften considered, it is more likely the contact angle hysteresisthat unambiguously describes the superhydrophobic or water-repellence feature of a surface. Indeed, even if exhibiting a highcontact angle, a droplet may be in the Wenzel state which isassociated with a large wetting contact area. This results indroplets characterized by high wetting contact angle hysteresisand that, nonetheless, adhere to the rough substrate. In theCassie−Baxter state on the other hand, the droplet mainly

Figure 1. (A) Schematic of the surface pattern with the interpillardistance l and the width of the pillar d. (B) Scanning electronmicrograph of a micropillar array with the height of the pillar H. (C)Microtopographies tool box describing the four series of surfacepatterns used in this work.

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interacts with air, which reduces its pinning. The associatedwetting contact angle hysteresis is thus remarkably reduced, andthe droplet easily rolls over the superhydrophobic surface whichcan be defined this time as water-repellent. The Cassie−Baxterrelation (eq 1) fairly predicts the equilibrium contact angle of asuspended droplet. It was shown however that it does notcapture the contact angle hysteresis.11,20,21

A good prediction of both the contact angle hysteresis andadvancing/receding contact angles was proposed on the basis ofthe pinning/depinning model of the triple line proposed byJoanny and de Gennes.10,11,22 In this model applied tosuspended droplets, each top of the asperities acts like apinning site that deforms the contact line. Rupture of thepinning is considered to occur when the adhesive wettingenergy per defect equals the elastic energy εel stored in theliquid tail that forms at each pinning site, with22

ε = ku12el

2(2)

In this equation, k is the elastic stiffness of the liquid tail whichwas shown to scale as

ω∝kl dln(2 / ) (3)

with ω being the adhesive energy on the pillar, d being thewidth of the pillars, and l being the interpillar distance. u in eq 2is the maximal elongation length of the liquid tail which wasshown to scale as

∝u d l d[ln(2 / )] (4)

The energy εw per unit area arising from the adhesion on thepillars, that has to be overcome for moving a line, that meets anumber density of pillars n = 1/(l + d)2 can thus be written as

ε ω= ∝+

nkul d

l d12

12

ln(2 / )(1 / )w

22

(5)

This energy which defines the amplitude of the contact anglehysteresis clearly shows a scale invariant behavior according toits dependence on the ratio l/d. The same energy should thusbe required for a line moving on substrates of the sametopographical ratio l/d, leading to similar contact anglehysteresis.Following the pinning/depinning model, Reyssat and

Quere10 proposed an expression for the contact angle hysteresisΔ cos θ, with εw = γ Δ cos θ and ϕ = 1/(1 + l/d)2 where γ is theliquid−vapor surface tension:

θ αϕ πϕ

Δ ≈⎛⎝⎜

⎞⎠⎟cos

14

ln(6)

with α being a dimensionless geometrical parameter describingthe true contact line around a pinning site. Following the sameenergetic arguments, and considering this time εw = γ (1 + cosθ), Dubov et al.11 proposed an expression for the advancing andreceding contact angles, θr and θa:

θγ

ω=+

−k l d

cos1

2 (1 / )1r

2

2(7)

θγ

λ=+

−k l dl d

cos2

( / )(1 / )

1a

2 2

2(8)

with λ being a dimensionless geometrical parameter describingthe deformation of the pinned contact line. Finally, in these twoapproaches leading respectively to eqs 6−8, the geometry of theliquid tail formed at the pinning point is adjusted by the authors(through α in eq 6, and k or λ in eqs 7 and 8) to fit theirexperimental data. It is indeed this geometry of the liquid tailwhich defines the elastic energy counterbalancing the pinningenergy, and which determines thus the amplitude of the contactangle hysteresis.Even though explicitly expressed in the former equations,

none of these authors explored the scale invariant feature of thecontact angle hysteresis. They considered indeed varying theratio l/d or, equivalently, varying the pillar surface fractions, ϕ =1/(1 + l/d)2. As suggested earlier it is possible, however, toconsider a micropattern morphology that keeps the ratio l/dconstant, or equivalently the surface fraction of pillars ϕ, butwith varying interpillar distance l and pillar width d. Equation 5that expresses this scale invariant wetting behavior through theratio l/d should lead to similar contact angle hysteresis(according to eq 6), or similar advancing/receding contactangles (according to eqs 7 and 8), for a given ratio l/d. It is thisapproach which is considered in the following, to assess for thescale invariant feature of eq 5 over a wide range of interpillardistances l and pillar widths d, or otherwise stated, over a widerange of micropillars with number densities at fixed surfacefraction.To address this question, we considered a series of substrates

with the microtopography depicted in Figure 1C (series 1),which consists of hexagonal micropillars (of width d)distributed on a hexagonal lattice (of interpillar distance l). Asshown in Figure 1, the width of the pillars d and the interpillardistance l are varied in a homothetic manner from one sampleto another, with d = l. The pillar widths d (or interpillar distancel) were 4, 8, 16, 32, 64, and 128 μm, whereas the height H of thepillars was 16 μm in all cases. We benefited here from theopportunity of invariantly formulating the ratio l/d, leading tothe same micropillar surface density for all samples, ϕ = 25%,but different micropillar number densities. By gentle manipu-lation of the droplet (see the Materials and Methods section), itwas possible to measure the equilibrium, advancing, andreceding contact angles of a water droplet in the Cassie−Baxterstate (droplet suspended on the top of the pillars) over thewhole series of substrates. All contact angles are presented inFigure 2.The equilibrium contact angle of water θeq is similar for all

pillar widths d and interpillar distances l, with θeq = 137° ± 2°.This result agrees fairly well with the contact angle of a dropletin the Cassie−Baxter regime predicted by eq 1, θCB = 146° ± 2°,considering an equilibrium contact angle measured on a smoothPDMS substrate, θeq

smooth = 110° ± 2° (with θa = 116° ± 2°, θr= 80° ± 2°, and Δθ = θa − θr = 36° ± 4°). Both the advancingand receding contact angles of a water droplet, respectively θaand θr, are similar for all micropillars number densities, with θa≈ 157° and θr ≈ 118°. The high advancing and receding contactangles both denote here a superhydrophobic behavior, with thedroplets being in the Cassie−Baxter regime. As expected fromeq 6, the contact angle hysteresis Δ cos θ is quite constant forthe surface fraction ϕ considered here (ϕ = 25%). This contactangle hysteresis of Δθ = θr − θa ≈ 40° is also well predicted byeq 6 in our case. With α = 2, this equation leads to a contactangle hysteresis Δ cos θ = cos θr − cos θa = 0.31, correspondingto a receding contact angle of 127° (at given advancing contactangle of 157°), to be compared to the measured receding

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contact angle of 118°. On the other hand the advancing andreceding contact angles expected from eqs 7 and 8 are,respectively, θr = 122° and θa = 165° in the case of the systemsconsidered here (for k/γ = 0.114 and λ = 1.5). Although slightlysmaller, the advancing and receding contact angles we measuredover the samples of varying micropillar number density welldemonstrate the scale invariance of the wetting behaviorexpected from eq 5.Our results finally show that the scale invariant wetting

behavior expressed in eq 5 through the ratio l/d is valid over awide range of pillar number densities (18 042 to 17 mm−2) or,equivalently, over a wide range of pillar widths d and interpillardistances l, with d = l (d = 4 μm to d = 128 μm), leading to thesame hysteresis or advancing/receding contact angles at a givenl/d ratio. In other words, a self-similar variation of the size anddistribution of pinning sites (micropillars) that keeps constantthe surface fraction ϕ of those pinning sites does not affect thecontact angle hysteresis or advancing/receding contact angles ofa droplet in the Cassie−Baxter state. From a phenomenologicalpoint of view, this result arises from the antagonist variation ofthe square of the elongation length u2, which scales as d 2, andthe number density of pillars n, which scales as 1/d 2 (eq 5).Otherwise stated, there is an invariance of the elastic energy/unit area εw stored in the liquid tails, that is ensured through theconstancy of the term (nu2) in eq 5. This scale invariance alsosuggests that the pinning sites (top of the pillars) can be hereconsidered as diluted and not interacting with each other.Indeed, such cross-interaction between defects would haveimpacted the shape of the tail (through the maximal elongationlength u) and thus the elastic energy. A breakdown of theobserved scale-invariance should be expected at interpillardistances smaller than those considered in this work (higherpillar number densities), for which pinning sites no longer actindividually on the triple line. Instead, the liquid tails from thedifferent pillars overlap, leading to a rather smoothened contactline and collective behavior of the pillars (depinning).5,10,23,24

Droplet Morphologies in the Wenzel-like Regime. Weconsider in the following the case where the microtopo-graphically patterned substrates are totally wetted by the liquid.For this case that we referred to as the Wenzel-like wettingregime, the equilibrium contact angle of a liquid on the smooth

substrate of the material is θeq < 90°. Microtopography may leadin this case to a lower equilibrium contact angle or to aspontaneous imbibition within the roughness. On the basis ofenergetic arguments, Bico et al.13 defined the critical contactangle θc below which a liquid is spontaneously imbibed in theroughness voids. This critical contact angle is obtained byconsidering the interfacial energy variation dE corresponding toa displacement dx of a liquid of surface tension γ invading theroughness:

γ γ ϕ γ ϕ= − − + −E r x xd ( )( )d (1 )dSL SV s s (9)

with γ, γSL, and γSV, respectively, being the surface tension of theliquid, the solid/liquid interfacial tension, and the solid/vaporinterfacial tension. r is the roughness factor (ratio of the realsurface area to the projected one) and ϕs is the surface fractionof the pillars. Considering that the liquid wets the substrate if dEis negative and introducing Young’s relation (γcos θeq = γSV -γSL) in eq 9 allows defining a condition for the spontaneous 2Dimbibition to occur:

θ θ θ ϕ ϕ< = − −rwith cos (1 )/( )eq c c s s (10)

with θeq being the equilibrium contact angle of the liquid on aideal flat surface. A droplet with a wetting contact angle θeq < θcon such a substrate should spontaneously fill the roughness, toform at equilibrium a film or a film connected to a droplet(droplet sitting on a mixture of solid and liquid).13,25,26 In thecase of topographically patterned substrates, however, the tripleline can be pinned by a row of micropillars. Instead of forming afilm, the progression of the liquid stops and faceted droplets areformed.13 Simple geometric arguments were proposed topredict the critical distance lc between the nearest neighborrow of micropillars, beyond which the fluid does notspontaneously imbibe the roughness but is pinned by a rowof micropillars.13 This critical length lc can indeed be defined asthe length of the meniscus joining the substrate and the top of a

micropillar as shown in Figure 3. To a first approximation, thelength lc can be written as

θ θ= =l H H l/tan or tan /c eq eq c (11)

with H being the height of the micropillars and θeq being theequilibrium contact angle on the smooth substrate. Thecondition for the pinning to occur is that the nearest neighborrow of micropillars stands at a distance l > lc. As otherwisestated by Bico et al.,13 this approach is equivalent to defining aminimum contact angle θmin given by tan θ min = H/lc abovewhich the contact line is pinned by a pillar row (Figure 3). Forθmin < θeq < θc, the liquid will not spread into the roughness butwill be stopped by a pillar row and form faceted droplets withshapes reflecting the symmetry of the micropillars array.13,18,26

If θmin > θeq, the liquid will spread into the roughness to fullyform a film or a film connected to a droplet.

Figure 2. Advancing contact angle (△), receding contact angle (○),and equilibrium contact angle (□) of a water droplet, measured on thesmooth substrate and on the substrates micropatterned with pillars ofdimensions d/H.

Figure 3. Schematic of the meniscus at the front of the fluid imbibingthe surface micropattern.

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Figure 4 shows the morphologies of ethanol dropletsdeposited on substrates micropatterned with hexagonal pillars

of identical height (H = 4 μm), but with increasing widths d andinterpillar distances l (homothetic variation of d and l, with d =l) as depicted in Figure 1C (series 2). SEM images of the surfacemicropattern as well as the corresponding θc retrieved from eq10 are shown in Figure 4. The imbibition criterion defined in eq10 is fulfilled for the three first surface microtopographies (4/4,8/4, and 16/4), based on a equilibrium contact angle of ethanolmeasured on the smooth PDMS surface θeq = 37°, a surfacefraction ϕs = 0.25, and a roughness factor r = 1 + H/d with r = 2,r = 3, and r = 5 for the pillar dimensions 4/4, 8/4, and 16/4,respectively. The three droplets of Figure 4A′−C′ are thus inthe situation of hemiwicking described by Bico et al.13 Thedroplet morphology of Figure 4A′ is that corresponding to adroplet sitting on a mixture of solid and liquid, while the dropletof Figure 4C′ clearly shows a faceted droplet morphology. Thedroplet in Figure 4B′ rather corresponds to a transition stateshowing at some edges very narrow wetting films, and completepinning at others. The conditions for the pinning of the tripleline expected from eq 11 can be checked for the pillar aspectratios and pillar surface distribution considered here. Thecritical length lc of the meniscus formed at the edge of thedroplet and bounding the substrate to the top of a pillar (Figure3) is lc = 5.3 ± 0.4 μm according to eq 11. This length has to becompared with the distances l* between the nearest pillar rowsfor the two patterns leading to a faceting of the droplet (Figure4B′ and C′), l* = 4.6 μm (for pillar dimensions 8/4) and l* =9.2 μm (for pillar dimensions 16/4). As discussed earlier, asimilar comparison can be made on the basis of the minimum

contact angle θmin defined by the surface topography, with,respectively, θmin = 41° (for pillar dimensions 8/4) and θmin =23° (for pillar dimensions 16/4). It appears finally that thecondition for faceting is fulfilled for the droplet of Figure 4C′(pillar dimensions 16/4), with l* > lc (l* = 9.2 μm and lc = 5.3μm) or θmin < θeq < θc (with θmin = 23°, θeq = 37°, and θc = 41°).This is not the case however for the droplet of Figure 4B′ (pillardimensions 8/4), for which l* < lc (l* = 4.6 μm and lc = 5.3 μm)or θeq < θmin < θc (with θmin = 41°, θeq = 37°, and θc = 53°). Inthe case of the 32/4, 64/4, and 128/4 pillar dimensions, nospontaneous 2D imbibition is expected (θc < θeq in all cases),and thus also no faceting of the droplets. These droplet profilesare indeed circular as observed in Figure 4.On the basis of eq 11, a scale invariant behavior is expected

while varying in a homothetic manner the height H and length lof the surface pattern (leading to a constant aspect ratio H/l).Pinning conditions of the triple line should indeed bereproduced for a constant aspect ratio H/l, or equivalently H/d in our case, leading to a similar macroscopic droplet shape.We experimentally checked for this assertion by considering themorphologies of ethanol droplets deposited on two series ofsubstrates decorated each with micropillars of identical aspectratio, respectively, d/H = 2/1 and d/H = 4/1 as depicted inFigure 1C (series 3 and 4). Both series should thus lead todroplet morphologies similar to the hexagonal shape observedin Figure 4B′ (aspect ratio 2/1) and Figure 4C′ (aspect ratio 4/1). Figure 5 shows the morphologies of ethanol dropletsdeposited on these six different substrates. The condition forspontaneous imbibition (θeq < θc) is fulfilled for the two sets ofsamples since each of them is characterized by the same aspectratio H/d, with cos θc = 0.75/(0.75 + H/d) according to eq 10.The hexagonal shape of the droplet observed on the substratedecorated with pillars of dimensions 8/4 (Figure 5A) is indeed

Figure 4. Optical microscope images of ethanol droplets deposited onthe micropatterned substrates and corresponding scanning electronmicroscope images of the surface pattern, for pillars of constant heightH = 4 μm and widths of (A, A′) 4 μm, (B, B′) 8 μm, (C, C′) 16 μm,(D, D′) 32 μm, (E, E′) 64 μm, and (F, F′) 128 μm.

Figure 5. Optical microscope images of ethanol droplets deposited onmicropatterned substrates and corresponding scanning electronmicroscope images of the surface pattern, for micropatterns of aspectratio 2/1 built with pillars of (A, A′) 8/4 μm, (B, B′) 16/8 μm, and (C,C′) 32/16 μm; and aspect ratio 4/1 built with pillars of (D, D′) 16/4μm, (E, E′) 32/8 μm, and (F, F′) 64/16 μm.

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also observed with pillars of dimensions 16/8 and 32/16(Figure 5B and C). The droplet morphology is remarkably wellreproduced if the height of the pillars is increased together withtheir width and spacing to keep a constant aspect ratio.Constant pillar height with increasing pillar width and interpillardistance led on the contrary to droplets that rounded up asshown in Figure 4. The same remark stands for the aspect ratioof 4/1 corresponding to pillar dimensions of 16/4, 32/8, and64/16 (Figure 5D−F). The observation of similar dropletmorphologies for a given aspect ratio H/d confirms the scaleinvariance expressed in eq 11 which defines the condition forpinning of the triple line at the origin of the droplet faceting,over the range of pillar dimensions considered in this work.

■ CONCLUSIONWe give experimental evidence for the scale invariance thatcharacterizes the basic relations predicting the contact anglehysteresis of suspended droplets and the faceting of dropletswhile being imbibed into hexagonally packed micropatternedsubstrates consisting of hexagonal cross-section micropillars.This scale invariance is here demonstrated for a wide range ofpattern dimensions, although critical pattern length scalesshould exist which bound the domain of validity of thisinvariance. Invariant contact angle hysteresis of suspendeddroplets was observed for micropillar dimensions (l = d)ranging from 4 to 128 μm, whereas invariant faceting of dropletswas observed for micropillars ranging from 8 to 64 μm. From atechnological point of view, our results demonstrate thatspecific wetting behaviors such as the adhesion of suspendeddroplets or spreading of droplets can be equally achieved andcontrolled with either small or large microstructures, as long asthe dimensionless pattern ratio that drives the wetting behavioris maintained constant. Otherwise stated, we demonstrate that,in the frame of the design of functional surfaces and for thewetting phenomena considered here, small microstructures (∼1 μm) can efficiently be replaced by larger microstructures(∼100 μm) if this condition of constant pattern ratio is fulfilled.These phenomena observed here for hexagonal pillars mayeventually differ over the same length scales in the case of morecomplex pillar shapes. Pillars with sharp edges (star-shaped, forexample) or with an overprinted topography, for example, willindeed not necessarily reproduce the same anchoring andrelated self-similar variation of the triple line at the basis of thescale invariance.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: laurent.vonna@uha.fr.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThe authors would like top thanks Boris Lakard for technicalsupport. This work was supported by the MICA CarnotInstitute, the French RENATECH network, and its FEMTO-ST technological facility.

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Langmuir Article

dx.doi.org/10.1021/la501225m | Langmuir 2014, 30, 9378−93839383