Model Experimental Study of Scale Invariant Wetting Behaviors in

Jul 15, 2014 - contact angle hysteresis of water droplets in the Cassie−Baxter regime. It is shown that the energy at the origin of the hysteresis, ...
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Model Experimental Study of Scale Invariant Wetting Behaviors in Cassie−Baxter and Wenzel Regimes Valentin Hisler, Laurent Vonna,* Vincent Le Houerou, Stephan Knopf, Christian Gauthier, Michel Nardin, and Hamidou Haidara Institut de Science des Matériaux de Mulhouse (IS2M) CNRS - UMR 7361, Université de Haute Alsace, 15 rue Jean Starcky BP2488, 68057 Mulhouse Cedex, France ABSTRACT: In this work, we discuss quantitatively two basic relations describing the wetting behavior of microtopographically patterned substrates. Each of them contains scale invariant topographical parameters that can be easily expressed onto substrates decorated with specifically designed micropillars. The first relation discussed in this paper describes the contact angle hysteresis of water droplets in the Cassie−Baxter regime. It is shown that the energy at the origin of the hysteresis, that has to be overcome for moving the triple line, can be invariantly expressed for hexagonal pillars by varying the pillars width and interpillar distance. Identical contact angle hystereses are thus measured on substrates expressing this scale invariance for pillar widths and interpillar distances ranging from 4 to 128 μm. The second relation we discuss concerns the faceting of droplets spreading on microtopographically patterned substrates. It is shown in this case that the condition for pinning of the triple line can be fulfilled by simultaneously varying the height of the pillars and the interpillar distance, leading to faceted droplets of similar morphologies. The invariance of these two wetting phenomena resulting from the simultaneous and homothetic variation of topographical parameters is demonstrated for a wide range of pattern dimensions. Our results show that either of those two wetting behaviors can be simply achieved by the proper choice of a dimensionless ratio of topographical length scales.

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INTRODUCTION Roughness is a parameter that is known for long to strongly affect the wetting of a surface.1 For droplets on a smooth substrate, characterized by a water contact angle 90°, on the contrary, increasing the roughness of a substrate will result in an increase of its hydrophobicity and thus to higher apparent contact angles. This enhancement of hydrophobicity that may lead to superhydrophobic behavior (contact angle > 140°) can be described by two models. In the Wenzel model, the liquid droplet totally wets the solid surface below it, but the triple line becomes pinned.2 In the Cassie− Baxter model, the droplet rests on the peaks of the surface asperities and entraps air pockets.3 These two models were extensively studied since the work of Neinhuis and Barthlott that linked the water-repellent properties of the lotus leaf to its surface roughness.4 Motivated by the development of microstructuring techniques and the discovery of a wide range of natural systems showing remarkable wetting properties, micropatterned surfaces have been extensively studied as model rough surfaces to understand the role of topography on wetting phenomena. In the case of the Cassie−Baxter regime, many efforts have been © 2014 American Chemical Society

made to understand the causes of the extremely low contact angle hysteresis that characterizes superhydrophobic surfaces. Several authors have thus studied the pinning/depinning of the contact line at the origin of the contact angle hysteresis, as a function of the morphology and surface density of posts, holes or stripes pinning the contact line.5−9 To predict the contact angle hysteresis of droplet resting on micropatterned substrates, Reyssat and Quere10 as well as Dubov et al.11 proposed a mechanical model in which the pinned liquid tail that locally forms on a micropost during the receding of the contact line is considered as a spring. More recently Paxson and Varanasi12 examined the morphology of the contact line pinned at the top of microposts, and suggested a self-similar depinning mechanism to explain the observed contact angle hysteresis on micropatterned surfaces. Fewer studies have considered the wetting of a microtopographically patterned substrate for liquids and solids showing high affinity (equilibrium contact angle < 90°). In this framework, Bico et al.13 defined a critical contact angle depending on the roughness of the substrate, which allows prediction of the spontaneous filling of roughness by a liquid. The dynamics of spreading in the microtopography as well as Received: March 31, 2014 Revised: July 15, 2014 Published: July 15, 2014 9378

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the final shape of the drop was studied as a function of the microposts’ morphology and surface density.14−17 Recently, several authors reported asymmetric wetting induced by asymmetric post arrays.18,19 A large number of applications are concerned with the wetting of such micropatterned surfaces, including the design of water-repellent surfaces or microfluidic systems, for example. The now available micropatterning tools allowing the fabrication of different topography length scales, shapes, and chemistries offer multiple ways to establish basic relations driving the wetting of such substrates, and more generally of rough substrates. In this study, we explore two of these relations on the basis of model experiments. The first describes the contact angle hysteresis of droplets in the Cassie−Baxter regime, and the second describes the topographic conditions for droplet faceting, the liquid being imbibed into the micropattern. We here examine more precisely the scale invariant wetting parameters found in these two relations that can be expressed by a series of specifically designed self-similar surface patterns. Although explicitly expressed in these two basic relations describing the wetting of microtopographically patterned substrates, this invariance is rarely discussed in the literature, especially its range and conditions of validity. In the first part of this paper, we will thus discuss the contact angle hysteresis of water droplets suspended on micropillars (Cassie−Baxter regime), which is expected to be constant for a given ratio l/ d, with d being the width of the pillars and l being the interpillar distance. In the second part, we will discuss the morphologies of the more wetting ethanol droplets that are formed on a series of substrates 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 the pinning of the triple line and thus the faceting and morphology of the droplet. In the following, the invariance of the wetting parameters that characterize these two laws is examined on surfaces decorated with micropillar arrays defined by identical ratio l/d or aspect ratio d/H, which are invariantly expressed for different widths d, interpillar distances l, and heights H, as described in the Materials and Methods section.

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Figure 1. (A) Schematic of the surface pattern with the interpillar distance l and the width of the pillar d. (B) Scanning electron micrograph of a micropillar array with the height of the pillar H. (C) Microtopographies tool box describing the four series of surface patterns used in this work. same for a given aspect ratio d/H. The root mean squared roughness of the smooth PDMS (measured with a Nanoscope IIIa from Digital Instruments, in the tapping mode; Veeco, Santa Barbara, CA) is ∼0.8 nm on a 400 μm2 large area. Surface Characterization and Wetting Experiments. The profiles of the patterned surfaces were characterized by scanning electron microscopy (SEM; FEI Quanta 400 ESEM operating at electrical potentials of between 15 and 20 kV). The droplet shapes (top view) were characterized by optical microscopy (Olympus BX60) and recorded with a COHU camera coupled with a computer. Fresh deionized water (Elix from Merck, Millipore, Germany) and analytical grade ethanol (Carlo Erba, Italy) were used as wetting liquids. The contact angles were measured on a manual goniometer. In the case of the advancing and receding contact angles measurements, a droplet of water (∼8 μL) was first formed at the tip of a syringe. The drop was then slowly moved down until it touched the substrate. This procedure ensures an exclusive contact between the liquid and the top of the pillars, especially in the case of large interpillar distances for which the Cassie−Baxter state (suspended droplet) is particularly unstable. The advancing contact angle was then measured by growing the size of the droplet, and the receding contact angle by decreasing the size of the droplet. A droplet volume of ∼8 μL leads to a droplet contact base diameter of ∼4 mm which is large compared to the maximal interpillar distance of 128 μm used in this work.

MATERIALS AND METHODS

Fabrication of the Micropatterned Surfaces. A polydimethylsiloxane (PDMS) elastomer (Sylgard-184, Dow Corning) was used to fabricate the micropatterned surfaces. The Sylgard 184 prepolymer and cross-linker was mixed at 10:1 ratio and poured on silanized silicon molds. The PDMS was cured at 80 °C during 4 h and then peeled from the silicon molds. The smooth PDMS samples used as control surfaces were obtained from a PDMS molded on a flat silicon wafer using a similar procedure. The silicon molds were micropatterned to obtain PDMS substrates decorated with hexagonal micropillars for which width d and interpillar distance 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 a hexagonal lattice and thus display a constant surface fraction ϕ (ϕ = 25%). We used four series of surface pattern as a microtopographies toolbox (Figure 1C), to address the question of the scale invariant wetting parameters characterizing the two relations discussed in the Introduction. The two first series (series 1 and 2) each have constant pillar height, H = 16 and H = 4 μm, respectively. The width d and interpillar distance l are varied from 4 to 128 μm (with d = l). The two last series (series 3 and 4) are each characterized by pillars of constant aspect ratio with d/H = 2/1 and d/H = 4/1, respectively. The width d is 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

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RESULTS AND DISCUSSION Contact Angle Hysteresis of Water Droplets in the Cassie−Baxter Regime. The contact angle θCB of a droplet in the Cassie−Baxter regime for flat-topped pillars is given by the following relation: cos θCB = ϕ(1 + cos θeq) − 1

(1)

with ϕ being the wetted surface fraction and θeq being the equilibrium contact angle measured on a flat surface of the same material. Even though the equilibrium wetting contact angle is often considered, it is more likely the contact angle hysteresis that unambiguously describes the superhydrophobic or waterrepellence feature of a surface. Indeed, even if exhibiting a high contact angle, a droplet may be in the Wenzel state which is associated with a large wetting contact area. This results in droplets characterized by high wetting contact angle hysteresis and that, nonetheless, adhere to the rough substrate. In the Cassie−Baxter state on the other hand, the droplet mainly 9379

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with λ being a dimensionless geometrical parameter describing the deformation of the pinned contact line. Finally, in these two approaches leading respectively to eqs 6−8, the geometry of the liquid 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 their experimental data. It is indeed this geometry of the liquid tail which defines the elastic energy counterbalancing the pinning energy, and which determines thus the amplitude of the contact angle hysteresis. Even though explicitly expressed in the former equations, none of these authors explored the scale invariant feature of the contact angle hysteresis. They considered indeed varying the ratio l/d or, equivalently, varying the pillar surface fractions, ϕ = 1/(1 + l/d)2. As suggested earlier it is possible, however, to consider a micropattern morphology that keeps the ratio l/d constant, or equivalently the surface fraction of pillars ϕ, but with varying interpillar distance l and pillar width d. Equation 5 that expresses this scale invariant wetting behavior through the ratio l/d should lead to similar contact angle hysteresis (according to eq 6), or similar advancing/receding contact angles (according to eqs 7 and 8), for a given ratio l/d. It is this approach which is considered in the following, to assess for the scale invariant feature of eq 5 over a wide range of interpillar distances l and pillar widths d, or otherwise stated, over a wide range of micropillars with number densities at fixed surface fraction. 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). As shown in Figure 1, the width of the pillars d and the interpillar distance l are varied in a homothetic manner from one sample to another, with d = l. The pillar widths d (or interpillar distance l) were 4, 8, 16, 32, 64, and 128 μm, whereas the height H of the pillars was 16 μm in all cases. We benefited here from the opportunity of invariantly formulating the ratio l/d, leading to the same micropillar surface density for all samples, ϕ = 25%, but different micropillar number densities. By gentle manipulation of the droplet (see the Materials and Methods section), it was possible to measure the equilibrium, advancing, and receding contact angles of a water droplet in the Cassie−Baxter state (droplet suspended on the top of the pillars) over the whole series of substrates. All contact angles are presented in Figure 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 droplet in the Cassie−Baxter regime predicted by eq 1, θCB = 146° ± 2°, considering an equilibrium contact angle measured on a smooth PDMS substrate, θeqsmooth = 110° ± 2° (with θa = 116° ± 2°, θr = 80° ± 2°, and Δθ = θa − θr = 36° ± 4°). Both the advancing and receding contact angles of a water droplet, respectively θa and θr, are similar for all micropillars number densities, with θa ≈ 157° and θr ≈ 118°. The high advancing and receding contact angles both denote here a superhydrophobic behavior, with the droplets being in the Cassie−Baxter regime. As expected from eq 6, the contact angle hysteresis Δ cos θ is quite constant for the surface fraction ϕ considered here (ϕ = 25%). This contact angle hysteresis of Δθ = θr − θa ≈ 40° is also well predicted by eq 6 in our case. With α = 2, this equation leads to a contact angle hysteresis Δ cos θ = cos θr − cos θa = 0.31, corresponding to a receding contact angle of 127° (at given advancing contact angle of 157°), to be compared to the measured receding

interacts with air, which reduces its pinning. The associated wetting contact angle hysteresis is thus remarkably reduced, and the droplet easily rolls over the superhydrophobic surface which can be defined this time as water-repellent. The Cassie−Baxter relation (eq 1) fairly predicts the equilibrium contact angle of a suspended droplet. It was shown however that it does not capture the contact angle hysteresis.11,20,21 A good prediction of both the contact angle hysteresis and advancing/receding contact angles was proposed on the basis of the pinning/depinning model of the triple line proposed by Joanny and de Gennes.10,11,22 In this model applied to suspended droplets, each top of the asperities acts like a pinning site that deforms the contact line. Rupture of the pinning is considered to occur when the adhesive wetting energy per defect equals the elastic energy εel stored in the liquid tail that forms at each pinning site, with22

εel =

1 2 ku 2

(2)

In this equation, k is the elastic stiffness of the liquid tail which was shown to scale as ω k∝ ln(2l /d) (3) with ω being the adhesive energy on the pillar, d being the width of the pillars, and l being the interpillar distance. u in eq 2 is the maximal elongation length of the liquid tail which was shown to scale as u ∝ d[ln(2l /d)]

(4)

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

1 1 ω ln(2l /d) nku 2 ∝ 2 2 (1 + l /d)2

(5)

This energy which defines the amplitude of the contact angle hysteresis clearly shows a scale invariant behavior according to its dependence on the ratio l/d. The same energy should thus be required for a line moving on substrates of the same topographical ratio l/d, leading to similar contact angle hysteresis. Following the pinning/depinning model, Reyssat and Quere10 proposed an expression for the contact angle hysteresis Δ cos θ, with εw = γ Δ cos θ and ϕ = 1/(1 + l/d)2 where γ is the liquid−vapor surface tension: Δ cos θ ≈

⎛π⎞ 1 αϕ ln⎜ ⎟ 4 ⎝ϕ⎠

(6)

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

1 ω2 −1 2γk (1 + l /d)2

(7)

cos θa =

k λ 2(l /d)2 −1 2γ (1 + l /d)2

(8) 9380

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substrate of the material is θeq < 90°. Microtopography may lead in this case to a lower equilibrium contact angle or to a spontaneous imbibition within the roughness. On the basis of energetic arguments, Bico et al.13 defined the critical contact angle θc below which a liquid is spontaneously imbibed in the roughness voids. This critical contact angle is obtained by considering the interfacial energy variation dE corresponding to a displacement dx of a liquid of surface tension γ invading the roughness: dE = (γSL − γSV )(r − ϕs)dx + γ(1 − ϕs)dx

(9)

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

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

θeq < θc

with

cos θc = (1 − ϕs)/(r − ϕs)

(10)

with θeq being the equilibrium contact angle of the liquid on a ideal flat surface. A droplet with a wetting contact angle θeq < θc on such a substrate should spontaneously fill the roughness, to form at equilibrium a film or a film connected to a droplet (droplet sitting on a mixture of solid and liquid).13,25,26 In the case of topographically patterned substrates, however, the triple line can be pinned by a row of micropillars. Instead of forming a film, the progression of the liquid stops and faceted droplets are formed.13 Simple geometric arguments were proposed to predict the critical distance lc between the nearest neighbor row of micropillars, beyond which the fluid does not spontaneously imbibe the roughness but is pinned by a row of micropillars.13 This critical length lc can indeed be defined as the length of the meniscus joining the substrate and the top of a

contact angle of 118°. On the other hand the advancing and receding contact angles expected from eqs 7 and 8 are, respectively, θr = 122° and θa = 165° in the case of the systems considered here (for k/γ = 0.114 and λ = 1.5). Although slightly smaller, the advancing and receding contact angles we measured over the samples of varying micropillar number density well demonstrate the scale invariance of the wetting behavior expected 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 a wide range of pillar number densities (18 042 to 17 mm−2) or, equivalently, over a wide range of pillar widths d and interpillar distances l, with d = l (d = 4 μm to d = 128 μm), leading to the same hysteresis or advancing/receding contact angles at a given l/d ratio. In other words, a self-similar variation of the size and distribution of pinning sites (micropillars) that keeps constant the surface fraction ϕ of those pinning sites does not affect the contact angle hysteresis or advancing/receding contact angles of a droplet in the Cassie−Baxter state. From a phenomenological point of view, this result arises from the antagonist variation of the square of the elongation length u2, which scales as d 2, and the 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 the constancy of the term (nu2) in eq 5. This scale invariance also suggests that the pinning sites (top of the pillars) can be here considered as diluted and not interacting with each other. Indeed, such cross-interaction between defects would have impacted the shape of the tail (through the maximal elongation length u) and thus the elastic energy. A breakdown of the observed scale-invariance should be expected at interpillar distances smaller than those considered in this work (higher pillar number densities), for which pinning sites no longer act individually on the triple line. Instead, the liquid tails from the different pillars overlap, leading to a rather smoothened contact line and collective behavior of the pillars (depinning).5,10,23,24 Droplet Morphologies in the Wenzel-like Regime. We consider in the following the case where the microtopographically patterned substrates are totally wetted by the liquid. For this case that we referred to as the Wenzel-like wetting regime, the equilibrium contact angle of a liquid on the smooth

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

micropillar as shown in Figure 3. To a first approximation, the length lc can be written as lc = H /tan θeq

or

tan θeq = H /lc

(11)

with H being the height of the micropillars and θeq being the equilibrium contact angle on the smooth substrate. The condition for the pinning to occur is that the nearest neighbor row of micropillars stands at a distance l > lc. As otherwise stated by Bico et al.,13 this approach is equivalent to defining a minimum contact angle θmin given by tan θ min = H/lc above which the contact line is pinned by a pillar row (Figure 3). For θmin < θeq < θc, the liquid will not spread into the roughness but will be stopped by a pillar row and form faceted droplets with shapes reflecting the symmetry of the micropillars array.13,18,26 If θmin > θeq, the liquid will spread into the roughness to fully form a film or a film connected to a droplet. 9381

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Figure 4. Optical microscope images of ethanol droplets deposited on the micropatterned substrates and corresponding scanning electron microscope images of the surface pattern, for pillars of constant height H = 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.

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 the condition for faceting is fulfilled for the droplet of Figure 4C′ (pillar dimensions 16/4), with l* > lc (l* = 9.2 μm and lc = 5.3 μm) or θmin < θeq < θc (with θmin = 23°, θeq = 37°, and θc = 41°). This is not the case however for the droplet of Figure 4B′ (pillar dimensions 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°). In the case of the 32/4, 64/4, and 128/4 pillar dimensions, no spontaneous 2D imbibition is expected (θc < θeq in all cases), and thus also no faceting of the droplets. These droplet profiles are 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 l of the surface pattern (leading to a constant aspect ratio H/l). Pinning conditions of the triple line should indeed be reproduced 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 the morphologies of ethanol droplets deposited on two series of substrates decorated each with micropillars of identical aspect ratio, respectively, d/H = 2/1 and d/H = 4/1 as depicted in Figure 1C (series 3 and 4). Both series should thus lead to droplet morphologies similar to the hexagonal shape observed in Figure 4B′ (aspect ratio 2/1) and Figure 4C′ (aspect ratio 4/ 1). Figure 5 shows the morphologies of ethanol droplets deposited on these six different substrates. The condition for spontaneous imbibition (θeq < θc) is fulfilled for the two sets of samples since each of them is characterized by the same aspect ratio H/d, with cos θc = 0.75/(0.75 + H/d) according to eq 10. The hexagonal shape of the droplet observed on the substrate decorated with pillars of dimensions 8/4 (Figure 5A) is indeed

of identical height (H = 4 μm), but with increasing widths d and interpillar distances l (homothetic variation of d and l, with d = l) as depicted in Figure 1C (series 2). SEM images of the surface micropattern as well as the corresponding θc retrieved from eq 10 are shown in Figure 4. The imbibition criterion defined in eq 10 is fulfilled for the three first surface microtopographies (4/4, 8/4, and 16/4), based on a equilibrium contact angle of ethanol measured on the smooth PDMS surface θeq = 37°, a surface fraction 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 in the situation of hemiwicking described by Bico et al.13 The droplet morphology of Figure 4A′ is that corresponding to a droplet sitting on a mixture of solid and liquid, while the droplet of Figure 4C′ clearly shows a faceted droplet morphology. The droplet in Figure 4B′ rather corresponds to a transition state showing at some edges very narrow wetting films, and complete pinning at others. The conditions for the pinning of the triple line expected from eq 11 can be checked for the pillar aspect ratios and pillar surface distribution considered here. The critical length lc of the meniscus formed at the edge of the droplet and bounding the substrate to the top of a pillar (Figure 3) is lc = 5.3 ± 0.4 μm according to eq 11. This length has to be compared with the distances l* between the nearest pillar rows for the two patterns leading to a faceting of the droplet (Figure 4B′ and C′), l* = 4.6 μm (for pillar dimensions 8/4) and l* = 9.2 μm (for pillar dimensions 16/4). As discussed earlier, a similar comparison can be made on the basis of the minimum

Figure 5. Optical microscope images of ethanol droplets deposited on micropatterned substrates and corresponding scanning electron microscope images of the surface pattern, for micropatterns of aspect ratio 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′) 16/4 μm, (E, E′) 32/8 μm, and (F, F′) 64/16 μm.

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

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also observed with pillars of dimensions 16/8 and 32/16 (Figure 5B and C). The droplet morphology is remarkably well reproduced if the height of the pillars is increased together with their width and spacing to keep a constant aspect ratio. Constant pillar height with increasing pillar width and interpillar distance led on the contrary to droplets that rounded up as shown in Figure 4. The same remark stands for the aspect ratio of 4/1 corresponding to pillar dimensions of 16/4, 32/8, and 64/16 (Figure 5D−F). The observation of similar droplet morphologies for a given aspect ratio H/d confirms the scale invariance expressed in eq 11 which defines the condition for pinning of the triple line at the origin of the droplet faceting, over the range of pillar dimensions considered in this work.

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CONCLUSION We give experimental evidence for the scale invariance that characterizes the basic relations predicting the contact angle hysteresis of suspended droplets and the faceting of droplets while being imbibed into hexagonally packed micropatterned substrates consisting of hexagonal cross-section micropillars. This scale invariance is here demonstrated for a wide range of pattern dimensions, although critical pattern length scales should exist which bound the domain of validity of this invariance. Invariant contact angle hysteresis of suspended droplets was observed for micropillar dimensions (l = d) ranging from 4 to 128 μm, whereas invariant faceting of droplets was observed for micropillars ranging from 8 to 64 μm. From a technological point of view, our results demonstrate that specific wetting behaviors such as the adhesion of suspended droplets or spreading of droplets can be equally achieved and controlled with either small or large microstructures, as long as the dimensionless pattern ratio that drives the wetting behavior is maintained constant. Otherwise stated, we demonstrate that, in the frame of the design of functional surfaces and for the wetting 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 may eventually differ over the same length scales in the case of more complex pillar shapes. Pillars with sharp edges (star-shaped, for example) or with an overprinted topography, for example, will indeed not necessarily reproduce the same anchoring and related self-similar variation of the triple line at the basis of the scale invariance.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like top thanks Boris Lakard for technical support. This work was supported by the MICA Carnot Institute, the French RENATECH network, and its FEMTOST technological facility.

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REFERENCES

(1) Dettre, R.; Johnson, R. Contact Angle Hysteresis. IV. Contact Angle Measurements on Heterogeneous Surfaces. J. Phys. Chem. 1965, 69, 1507−1515. (2) Wenzel, R. N. Resistance of Solid Surfaces to Wetting by Water. Ind. Eng. Chem. 1936, 28, 988−994. 9383

dx.doi.org/10.1021/la501225m | Langmuir 2014, 30, 9378−9383