Effects of Hierarchical Surface Roughness on Droplet Contact Angle

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Effects of Hierarchical Surface Roughness on Droplet Contact Angle Michael S. Bell,† Azar Shahraz,‡ Kristen A. Fichthorn,*,†,‡ and Ali Borhan‡ †

Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, United States Department of Chemical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, United States

‡

S Supporting Information *

ABSTRACT: Superhydrophobic surfaces often incorporate roughness on both micron and nanometer length scales, although a satisfactory understanding of the role of this hierarchical roughness in causing superhydrophobicity remains elusive. We present a two-dimensional thermodynamic model to describe wetting on hierarchically grooved surfaces by droplets for which the influence of gravity is negligible. By creating wetting phase diagrams for droplets on surfaces with both single-scale and hierarchical roughness, we find that hierarchical roughness leads to greatly expanded superhydrophobic domains in phase space over those for a single scale of roughness. Our results indicate that an important role of the nanoscale roughness is to increase the effective Young’s angle of the microscale features, leading to smaller required aspect ratios (height to width) for the surface structures. We then show how this idea may be used to design a hierarchically rough surface with optimally high contact angles.

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INTRODUCTION Control over wetting of surfaces has many potential applications, such as self-cleaning windows and antifouling surfaces,1−3 low-drag surfaces,2−4 and water-repellent and stainresistant clothing.3,4 These applications involve superhydrophobic surfaces, for which the static equilibrium contact angle (CA) is greater than 150° and the contact angle hysteresis (CAH) is smaller than 10°.2,5,6 Studies have shown that both the chemistry and the roughness of a surface are important in determining its wetting properties.7 The surface chemistry affects the intrinsic, or Young’s, CA8 of the surface, referred to here as θe, which is the CA attained by a droplet on a flat smooth surface of a given material. However, the largest θe observed on any surface to date, whether natural or artificial, is only approximately 120°,9−12 which alone cannot confer superhydrophobicity. Thus, superhydrophobic surfaces must also employ roughness to their advantage. Some of the most superhydrophobic surfaces possess roughness on multiple length scales, which is commonly referred to as hierarchical roughness.5 In fact, hierarchical roughness is a common motif in naturally occurring superhydrophobic surfaces: the lotus leaf, one of the most well-known natural surfaces with hierarchical roughness,13 demonstrates a water CA of about 160° and a CAH on the order of a few degrees;14 some flower petals are hierarchically rough, resulting in a water CA of about 152°;15 hierarchical roughness on bird feathers helps to prevent the penetration of water through the feathers;16 and hierarchical roughness on the legs of water striders allows these insects to stand and even jump on water.4 Advances in nanotechnology have allowed for the design of biologically inspired superhydrophobic surfaces,17 many of which have utilized hierarchical roughness to achieve water CAs greater than 160°.2,18−21 It has also been demonstrated that superhydrophobicity can be achieved on artificial surfaces with © XXXX American Chemical Society

a single length scale of roughness. Feng et al. created a surface using polyacrylonitrile nanostructures for which the CA for water was 173.8°,14 while Lau et al. were able to achieve water CAs of 160−170° using carbon nanotube forests.22 A droplet on a rough surface may be found in one of two wetting modes: in the Cassie mode,23 the droplet remains suspended above the asperities in the surface, and in the Wenzel mode,24 the droplet penetrates the asperities in the surface. While there is a general consensus that surfaces which allow more air to be trapped under a droplet are more hydrophobic,19,22,25 the precise role of hierarchical roughness in causing superhydrophobicity is not completely understood.3,26−28 Some have argued that, while hierarchical structure is not necessary for superhydrophobicity, it does allow for a smaller aspect ratio (defined as the ratio of height to width) of the macroscale structures, which increases the durability of the surface.2,29,30 Recent studies on the lotus leaf31 and on bird feathers16 show that hierarchical structures help make the Cassie wetting mode more stable under higher pressures than would be possible with only a single scale of roughness. Still others have argued that hierarchical roughness is important for both the durability and the superhydrophobic stability of surfaces.32,33 A number of simple models, typically relying on the Cassie and Wenzel equations (included in the Supporting Information), which are applicable only in the limit of surface features much smaller than the size of the drop,34−38 have been used in attempts to understand the role of hierarchical roughness in causing superhydrophobicity. One such model is that proposed by Herminghaus,39 in which he adds multiple generations of Received: March 20, 2015 Revised: May 27, 2015

A

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Figure 1. Model surface consisting of parallel, infinitely long rectangular grooves. (A) The macroscale grooves are defined by groove width G, step width W, and step height H. (B) The microscale grooves are cut into the macroscale steps and are defined by groove width G′, step width W′, and groove depth H′.

Figure 2. Four wetting states considered in the model. All droplets shown have n = 2; n′ = 5, where n and n′ denote the number of macroscale and microscale grooves, respectively, covered by the droplet. (A) The Cassie−Cassie (Cc) state (C2c5); (B) the Wenzel−Cassie (Wc) state (W2c5); (C) the Cassie−Wenzel (Cw) state (C2w5); (D) the Wenzel−Wenzel (Ww) state (W2w5).

roughness to a surface with macroscale roughness greatly increases the range of surface-feature sizes or, equivalently, increases the range of droplet sizes on a fixed surface, for which high CAs can be attained. Our results indicate that an important effect of the microscale roughness may be to increase the effective Young’s angle of the macroscale roughness. Using this idea, we are able to identify optimal ranges of roughness parameters for enhancing the hydrophobicity of the surface using multiple scales of roughness.

hierarchical roughness to a surface and, by considering the Cassie model, he shows that such structures can exhibit water CAs approaching 180° even for substrates with θe < 90°. Related experiments, however, showed that the model only worked for droplets that began in the Cassie wetting mode, which was unstable for such surfaces.12 Another simple model was provided by Patankar,30 who showed for a threedimensional surface consisting of square pillars that a hierarchically rough surface allows high CAs to be achieved for lower aspect-ratio nanostructures (compared to surfaces with only one scale of roughness), presumably resulting in a more durable surface.30,40 In addition to these simple models based on the Cassie and Wenzel equations, many other thermodynamic models of wetting have been proposed. Nosonovsky and Bhushan developed a model which showed that the presence of hierarchical roughness on the inside edges of the macroscale grooves makes the Cassie mode more stable than the Wenzel mode.41,42 Liu et al. considered the wetting of hierarchically rough surfaces comprised of self-similar square pillars.26 A similar three-dimesional model based on the Cassie and Wenzel equations was presented by Zhang et al.6 Sajadinia and Sharif presented a detailed thermodynamic model for droplets on a surface with hierarchical roughness, and showed that the addition of nanostructures caused changes in the energy barriers between the Cassie and Wenzel wetting modes.43 They also showed that the addition of hierarchical roughness leads to increased CA for fixed-width pillars. However, their model again required that the droplets be sufficiently larger than the surface features. In this paper, we use a two-dimensional thermodynamic model to demonstrate the effects of hierarchical structure on the CA of droplets on a physically patterned surface. By constructing wetting phase diagrams of CAs for a variety of surface parameters, we show that the addition of microscale

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THERMODYNAMIC MODEL We consider the wetting of a cylindrical drop on a surface consisting of parallel, rectangular grooves with two different length scales, as shown in Figure 1. Cylindrical drops with their (infinite) axes parallel to the grooves are considered to simplify the problem, effectively reducing it to a two-dimensional one. The macroscale grooves are defined by a groove width of G, a step width of W, and a step height of H, as shown in Figure 1A, while the microscale grooves are cut into the macroscale steps (cf. Figure 1B) and are defined by width G′, spacing W′, and depth H′. The hierarchical geometry consists of N′ + 1 microscale steps and N′ microscale grooves above each macroscale step. In this study, we do not explicitly account for microscale roughness on the side walls of the grooves, as we are primarily interested in wetting configurations with the highest CA, for which the Cassie mode is most stable. However, we do consider the Wenzel mode and, as we will discuss below, we developed an implicit expression for side-wall roughness. We consider four combinations of wetting modes for a droplet on the surface, as shown in Figure 2, with each state being designated by its macroscale wetting mode (C or W) followed by its microscale wetting mode (c or w), as has been done elsewhere in the literature.21,44 The four states considered here are Cassie−Cassie (Cc) (Figure 2A), Wenzel−Cassie (Wc) (Figure 2B), Cassie−Wenzel (Cw) (Figure 2C), and B

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specify the surface have been made dimensionless with R0 (e.g., G = G/R0). The Wc State.

Wenzel−Wenzel (Ww) (Figure 2D). The equilibrium configuration of a droplet is determined by both the wetting state and the number of grooves covered. Thus, two additional parameters n and n′, defined as the number of macroscale and microscale grooves covered by the droplet, respectively, are required to specify the droplet’s equilibrium configuration. For all droplets shown in Figure 2, n = 2 and n′ = 5. The configuration of the droplet in Figure 2B, for instance, is designated as W2c5 for Wenzel on the macroscale with n = 2 and Cassie on the microscale with n′ = 5. To investigate the wetting of the hierarchical surfaces, we compute the free energy difference per unit length parallel to the grooves, ΔE, between a surface with a droplet and the same surface without a droplet, given by the expression45,46 ΔE = Alv σlv + A sl (σsl − σsv) + Ug

ΔE ̅ Wncn′ = 2R̅ Wc(θ − cos θe sin θ ) + n′G′(1 + cos θe) ⎡2 − 2nH̅ cos θe + Bo⎢ R̅ Wc 3 sin 3 θ ⎣3 + (π − nGH )(H̅ − R̅ Wc cos θ ) +

(6)

where R̅ Wc =

(1)

⎡2 − 2n′H′ cos θe + Bo⎢ R̅ Cw 3 sin 3 θ ⎣3

(2)

R̅ Cw =

2(π − n′G′H′) 2θ − sin 2θ

(9)

ΔE ̅ Wnwn′ = 2R̅ Ww(θ − cos θe sin θ ) − 2(nH̅ + n′H′) cos θe

(3)

⎡ + Bo⎢(π − nGH − n′G′H′)(H̅ − R̅ Ww cos θ ) ⎢⎣ ⎛ 2 H′ ⎞ R̅ Ww 3 sin 3 θ + n′G′H′⎜H̅ − ⎟ ⎝ 3 2 ⎠ ⎤ 1 + nGH 2⎥ 2 ⎦

(10)

2π − 2(nGH + n′G′H′) 2θ − sin 2θ

(11)

+

where R̅ Ww =

We note here that Shahraz et al. considered an additional mode, called the “mixed mode”, in which the droplet was allowed to partially penetrate into the grooves.45 However, the mixed mode was never a stable wetting mode in the absence of gravity. Here, we consider only small droplets for which the effects of gravity are negligible, i.e., in the limit Bo → 0. Hence, we will not consider a mixed mode in the current model. Our expressions for the free energies as functions of n, n′, and θ and the surface geometry parameters were used to determine the wetting configuration associated with a global minimum in the free energy for a particular surface. The general procedure was to scan over n and n′ to find the minimumenergy configuration for each of the four states, with the lowest energy configuration among the four minima then being designated as the global minimum. A more detailed description

(4)

where θ is the CA (in radians) and 2π 2θ − sin 2θ

(8)

The Ww State.

ΔEC̅ ncn′ = 2R̅ Cc(θ − cos θe sin θ )

R̅ Cc =

+ (π − n′G′H′)(H̅ − R̅ Cw cos θ ) ⎛ H ′ ⎞⎤ + n′G′H′⎜H̅ − ⎟⎥ ⎝ 2 ⎠⎦

where

where the overbars denote the dimensionless counterparts of the quantities appearing in eq 2. We consider the droplet to have a constant radius of curvature above the asperities, an assumption which has been shown to be a reasonable approximation for sufficiently small droplets.47 We also neglect the curvature of the menisci formed over grooves. Using these approximations, the free energy for each wetting state can be determined as described by Shahraz et al.45,46 for a droplet on a surface with a single scale of roughness. Here, we use the same strategy to calculate ΔE for a droplet in each of the four wetting states depicted in Figure 2. The Cc State.

+ (nG̅ + n′G′)(1 + cos θe) ⎡2 + Bo⎢ R̅ Cc 3 sin 3 θ ⎣3 ⎤ + π (H̅ − R̅ Cc cos θ )⎥ ⎦

(7)

ΔEC̅ nwn′ = 2R̅ Cw(θ − cos θe sin θ ) + nG̅ (1 + cos θe)

As a final step, eq 2 is made dimensionless using the characteristic scale σlvR0, where R0 is the cylindrical equivalent radius of the droplet, defined as the radius of a cylindrical drop of equal volume not in contact with the surface.45,46 This leads to an expression for the dimensionless energy difference that is given by ΔE ̅ = A̅ lv − A̅ sl cos θe + Ug̅

2(π − nGH ) 2θ − sin 2θ

The Cw State.

where Ug is the gravitational potential energy per unit length of the droplet in the z direction (parallel to the grooves) and Ai and σi represent the area per unit length in the z direction and the interfacial energy per unit area, respectively, for the various interfaces which are specified by the subscripts lv for liquid− vapor, sl for solid−liquid, and sv for solid−vapor. Equation 1 can be simplified using Young’s equation for the same drop on a flat substrate to give ΔE = (Alv − A sl cos θe)σlv + Ug

⎤ 1 nGH 2 ⎥ ⎦ 2

(5)

is the radius of the drop’s circular cap. Bo is the Bond number, defined as Bo = ρgR02/σlv, with ρ denoting the droplet’s density and g the gravitational acceleration. All parameters used to C

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Figure 3. (A) Two-dimensional scan of the wetting phase diagram for a surface with a single scale of roughness, with H = 0.5. The contact angle is indicated by the color scale. The dashed lines are used to delineate boundaries between several adjacent equilibrium configurations and wetting modes. The points labeled A−D are further discussed and elaborated in Figures S3A−D (Supporting Information). (B) The phase diagram from part A, displayed on a log−log scale.

Interestingly, the linear isocontours for θ for G, W ≲ 0.01 in Figure 3B suggest a power law relationship between G and W for a given value of θ. Inspection of the model for the region of phase space in which G, W ≪ 1 is helpful in understanding this observed relationship. For a droplet in the Cassie mode, the free energy is given by45

of the minimization procedure is included in the Supporting Information.

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RESULTS AND DISCUSSION We begin by considering a surface with a single scale of roughness, as was done by Shahraz et al.45,46 We choose θe = 115°, representing a surface made of a hydrophobic material. Figure 3 shows the equilibrium CA as a function of W and G for a fixed value of H = 0.5. The nearly horizontal dashed line in Figure 3A near G = 0.4 delineates the boundary between the Cassie and Wenzel wetting modes, with the Cassie mode being preferred for G ≲ 0.4. The fact that the line separating the two modes is approximately horizontal for fixed H indicates that the boundary between the Wenzel and Cassie modes is relatively insensitive to the step width. This is an effect which we have also observed for other values of H, and it is discussed in more detail in the Supporting Information. From Figure 3A, we note two general trends for the largest CAs. One is that for smaller drops (i.e., G, W ∼ 1) large CAs are observed near the boundaries between adjacent equilibrium configurations (e.g., the boundary between C1 and C2). In these regions, it is the pinning effects that are most important in affecting the CA, as has been observed elsewhere.45 Specifically, within a given configuration, the CAs increase with decreasing G and W due to pinning effects (decreasing G and W tends to squeeze the base of the drop, resulting in a larger CA). However, we observe a much different trend for the large CAs in the region where W ≪ 1, which is shown in more detail in Figure 3B. In this region, though the CAs still increase with decreasing W, they now display the opposite trend in G, actually increasing as G increases (until G reaches a critical value, above which the droplet enters the Wenzel mode). In this small-W region, the pinning effects are less important than the amount of air under the droplet. Here, increasing G introduces even more air under the drop and causes it to prefer a higher CA, with CAs approaching 180° for the smallest W and the largest allowed G in the Cassie mode (for a fixed H = 0.5). It should be noted, however, that, for the larger drops, the increasing CAs are associated with a change in configuration (a decrease in the number of grooves spanned), unlike in the small drop limit, where the changes in CAs resulted from squeezing the base of the drops while the drops remained in a single configuration.

ΔEC̅ n = 2R̅ C(θ − sin θ cos θe) + nG̅ (1 + cos θe)

(12)

where

R̅ C =

2π 2θ − sin 2θ

(13)

For drops much larger than the size of the surface features, the number of grooves spanned by the base of the droplet, n, becomes approximately a continuous function of θ (i.e., small changes in θ lead to changes in n) according to n≃

2R̅ C sin θ G̅ + W̅

(14)

Substituting this expression into eq 12 and recognizing that the solid fraction of the surface (defined as the fraction of area occupied by pillar tops in the plane containing them) is given by fs = 1 − [G/(W + G)] leads to ΔEC̅ = 2R̅ C[θ − sin θ(fs cos θe − (1 − fs ))]

(15)

which produces a minimum in the free energy when the contact angle satisfies cos θ = fs cos θe − (1 − fs )

(16)

Substitution for fs in terms of G and W in this expression leads to a linear relationship between G and W for a fixed value of θ, which is the basis for the power-law relationship inferred earlier from Figure 3B. Equation 16 is identical to the Cassie−Baxter equation23 (see the Supporting Information), indicating that our general 2D model reduces to the Cassie model in the limit where G and W ≪ 1, while, in the smaller drop limit corresponding to larger G and W, our model predicts regions of high CA related to pinning effects near the boundaries of adjacent equilibrium configurations, which are not predicted by the Cassie or Wenzel models. In the top left corner of Figure 3A, we note a region of CAs less than θe. This effect has been observed before,45,48 and stems from the fact that, in this region of the phase diagram, the D

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Langmuir groove widths are on the order of the size of the drop. This makes the Wenzel mode the preferred wetting mode. Because the steps are relatively narrow in this region (W ≲ 0.4), the drops prefer the W2 configuration rather than W1, in which the contact lines would have to be pinned to the outside of the bounding steps with high CAs. By entering the W 2 configuration instead of W1, the increase in solid−liquid interfacial energy resulting from filling an additional groove is more than compensated by the reduction in liquid−vapor interfacial energy of the droplet cap. Because the grooves in this region are similar in size to the drop, the volume of liquid stored in the filled grooves reduces the amount left in the cap of the drop by enough that θ < θe. Finally, we note that the region of phase space from our 2D model in which the largest CAs are attained is in agreement with the observations made by Lau et al.22 of 3D superhydrophobic carbon nanotube forest surfaces. The carbon nanotubes had an average diameter of 50 nm, had an average areal density of 10 tubes per μm2 corresponding to an average linear density of 3.16 tubes per μm, and were coated with a material for which θe = 108°. The authors tested tube heights of 2, 1.1, 0.9, and 0.2 μm. The smallest droplets for which they reported observing superhydrophicity (with CAs between about 150 and 170°) had diameters of 3 μm, leading to the following upper limits on the corresponding dimensionless macroscale parameters in our model: G = 0.177, W = 0.0333, and H = 1.33, 0.733, 0.633, or 0.133 for the four heights listed above, respectively. With these parameters, our model predicts a global minimum configuration in the Cassie mode with θ = 154° for all but the shortest nanotubes, for which our model predicts a global minimum configuration in the Wenzel mode with θ = 125°. It is interesting that Lau et al. observed droplets in the Cassie mode for even the shortest nanotubes. In comparing the dimensionless energies of the Wenzel and the lowest-energy Cassie mode for nanotubes with H = 0.133, we find that they are similar, with values of 5.80 and 6.20, respectively. Though the authors did not observe a transition from Wenzel to Cassie after “prolonged periods of time”, it is still conceivable that the Wenzel mode was actually the global minimum on their surface, with Cassie being only metastable and a free-energy barrier separating the two modes. Our recent molecular-dynamics simulation studies employing forward flux sampling to obtain transition rates between the Cassie and Wenzel states indicate that transition times can be on the order of years,49 so this seems possible. The discrepancy in observations may also be related to a possible inherent difference between the 2D and 3D problems, though whether such a difference actually exists has not yet been examined. To understand the transition from the small-droplet regime, where pinning effects are important, to the large-droplet regime, where the pinning effects are unimportant, we plot the free energy versus CA in Figure 4 for droplets in the Cassie mode on surfaces with varying feature sizes from G + W = 0.75 down to G + W ≃ 0.05 (all with G = 2W giving a solid fraction of fs = 1/3, with θe = 115°). The prediction of the Cassie model is shown by the solid black (smooth) curve for a surface with a solid fraction of 1/3, for which θc = 144°, where θc is the Cassie CA. The many local minima correspond to the existence of multiple metastable wetting configurations for droplets on rough surfaces, and these have been discussed elsewhere.1,50,51 Here, we note that, as the surface features become smaller with respect to the drop size, it becomes apparent that the CAs corresponding to the global minima of the free energy curves

Figure 4. Dimensionless free energy vs CA for droplets in the Cassie mode on surfaces with varying feature sizes. For each surface, θe = 115°, and G = 2W giving fs = 1/3. The curves were made continuous by constraining one edge of the contact line to remain pinned to the outside of a step at all times, while allowing the other contact line to move with increasing θ. As the features are made smaller, the pinning effects become less important, and the global minimum approaches that predicted by the Cassie model.

converge to that predicted by the Cassie model, indicating that, for smaller feature sizes, the effects of pinning become less important than the solid fraction in determining the equilibrium CA, which helps to explain our observations in Figure 3. Figure 5 shows the effect of a typical microscale pattern while scanning over G and W for a fixed value of H = 0.5 (corresponding to the macroscale roughness considered in Figure 3). Two additional parameters characterize the microscale roughness, namely, α and N′. To consider constant α and N′, we require α = G′/W′ and W = (N′ + 1)W′ + N′G′. Here, we consider α = 10.0 and N′ = 10 for several different values of H′, ranging from 0.01H up to 0.25H. The values of α and N′ were selected as an example, after investigating the effect of microscale groove spacing and density on CA (Figures S3A−D, Supporting Information), to simulate surfaces on which the droplets may achieve large CAs and for which the droplets can be found in either the Cassie mode or the Wenzel mode on the microscale, depending on the width of the underlying macroscale step. Comparing the phase diagrams in Figure 5 with Figure 3, we see that the addition of the microscale structure, particularly for the larger H′ values, greatly expands the region of phase space in which large CAs may be attained. Figure 5A shows that the addition of shallow microgrooves leads to larger CAs than those attained in Figure 3 for very small W and G ≲ 0.4, while Figure 5B−D shows that, with increasing depth of the microgrooves, the region of phase space characterized by the highest CAs expands significantly. Figure 5C shows that CAs of at least 140° can be attained over most of the phase space, while, in Figure 5D, a large region of phase space exhibits CAs of greater than 160°. Thus, hierarchical roughness greatly increases the hydrophobicity of the surface, particularly for droplets in the Cassie mode on the microscale. A few remarks about Figure 5 are in order. First, we note the nearly horizontal dashed line at G ≃ 0.4, below which the largest CAs are generally found. This line divides the phase space into Cassie-dominated and Wenzel-dominated regions on the macroscale, as was observed in Figure 3A, and its location is determined by the macroscale surface parameters, specifically G E

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Figure 5. Effect of adding microscale roughness with fixed α = 10.0 and N′ = 10 to a surface with macroscale roughness with H = 0.5. The CA is shown over a range of W and G. (A) H′ = 0.01H. (B) H′ = 0.05H. (C) H′ = 0.10H. (D) H′ = 0.25H. The dashed lines delineate the divisions between the labeled wetting states.

Figure 6. CA versus G and W for self-similar surfaces. (A) A surface with a single scale of roughness with H/W = 1.5. (B) A surface with hierarchical roughness with H/W = 1.5, H′/W′ = 10, α = G′/W′ = 1.24, and N′ = 83/(1 + G/W) − 1/(1 + α) to simulate a lotus-like surface. The dashed curves delineate the macroscale Cassie−Wenzel boundary, while the solid white lines G/W = 13.6/5 and G/W = 7.6/11 bound the range of G/W reported for the lotus leaf, as discussed in the Supporting Information. The dotted black line G/W ≃ 5.0 represents surfaces with a single scale of roughness that produce the same solid fraction (and hence the same CA for large droplets in the Cassie mode) as the surfaces with hierarchical roughness represented by the black line in Figure 6B.

Finally, we note the region of CAs smaller than θe = 115° in the upper left corner of Figure 5, corresponding to large G. Though this region is also present with only single-scale roughness (cf. Figure 3), it becomes more pronounced with the addition of microscale roughness. Thus, while the addition of microscale roughness can lead to much higher CAs over a wide range of macroscale parameter space, it simultaneously expands a small region of phase space in which the roughness actually lowers the CAs. We observed this effect in previous work45 and it was also observed by Jopp et al.,48 although they consider the effect to be unphysical and attribute it to inadequate

and H. The line can be shifted up or down in G by making H larger or smaller, respectively. We also note the vertical dashed line between high and low CA regions for G < 0.4, corresponding to the division between the Cassie and Wenzel modes on the microscale. Larger CAs are generally observed on the left side of this vertical line, where the droplet is in the Cassie mode on the microscale. For larger H′, this line shifts farther to the right (to larger W), for the same reasons that the horizontal line dividing the Cassie and Wenzel modes on the macroscale shifts up (to larger G) with increasing H. F

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altering the lotus-like parameters changes the wetting phase diagram, but such an investigation is beyond the scope of this paper. We have seen above (cf. Figure 3) that large CAs may be obtained even for surfaces with a single scale of roughness, and it is natural to consider what type of single-scale roughness would be required to produce the same CAs as those obtained for large drops (for which drop size becomes irrelevant to CA) on an average lotus-like surface (corresponding to the solid black line in Figure 6B, with θ ≃ 155°). Such lotus-like surfaces have air fractions of 1 − fs ≃ 0.83 (as derived in the Supporting Information and given in eq S7), leading to a value of G/W = 5.0 (for θe = 115°), as illustrated by the black dotted line in Figure 6A. We note that, for this particular choice of H/W = 1.5, surfaces with G/W = 5.0 fall well within the Wenzel regime (see Figure 6A). In fact, to make the Cassie mode the most stable wetting mode for large drops on surfaces with G/W = 5.0 requires H/W ≳ 3.4. (A derivation is provided in the Supporting Information.) Figure 7 shows a to-scale comparison between the average lotus-like surface and its single-scale equivalent surface. This

assumptions made in the Wenzel relation for the structures they used. In all studies, this effect occurs for droplets small enough that the volume stored in the surface asperities is a significant fraction of the total droplet volume, and we believe that it is a physical effect, since we have not made any assumptions about the Wenzel relation. In Figure 6, we investigate the effects of hierarchical roughness for surfaces with macroscale and microscale patterns of constant aspect ratios (H/W and H′/W′). Such figures are useful in determining the effect of drop size on the CA, which may be observed by monitoring the CA while moving along any line passing through the origin, with the slope of the line fixing the value of G/W. Moving away from the origin, toward larger G and W, corresponds to decreasing drop size. The geometrical parameters (described below) were chosen in such a way as to mimic the surface of a lotus leaf,28,31,52 while leaving θe = 115°. This value of θe falls within the wide range of reported values θe ∈ [75°, 119°] for the wax material making up the nanostructures on the lotus leaf.25,53 Similarly, one can find a range of values reported for the sizes of the surface features on the lotus leaf. The slopes of the white lines in Figure 6 (m = 2.7 and m = 0.7) demonstrate a range of G/W values reported in the literature, with the “actual” values of G/W for the lotus leaf likely lying somewhere between the two extremes. A more detailed explanation of the particular choices of lotus-like parameter values is included in the Supporting Information. Figure 6A shows the CAs for a surface with a single scale of roughness with H/W = 1.5, while Figure 6B shows the corresponding CAs for the same surface with added microscale roughness for which H′/W′ = 10, α = 1.24, and N′ = 83/(1 + G/W) − 1/(1 + α) (rounded to the nearest integer) is varied with G/W to maintain a constant density of microscale features on the macroscale steps, where the microscale density is defined as (G + W)/(G′ + W′) = 83 for our choice of lotus-like parameter values. From the figures, we see again that hierarchical roughness is much more effective than a single scale of roughness in increasing the hydrophobicity of the surface. In particular, we note the regions of the phase diagrams lying between the white lines, which represent a set of lotus-like surfaces. It is interesting that, in the region near the origin (corresponding to drops much larger than the surface features), most of the lotus-like surfaces (all but those with the highest values of G/W) are predicted to yield droplets in the Cassie mode, which is the wetting mode observed experimentally on the lotus leaf.54 For those lotus-like surfaces for which our model predicts the Wenzel mode on the macroscale in the region of small G and W, we note that the predicted CAs are still quite high. We also expect a barrier to transition from Cassie to Wenzel49 so that droplets that are initially in the Cassie mode may remain in the Cassie mode for long periods of timean observation similar to that for droplets on carbon nanotube forest-like surfaces in the discussion of Figure 3. We also note in Figure 6 that the surfaces predicted by our model to produce the largest CAs in the phase diagrams are largely contained in the set of lotus-like surfaces. In other words, our two-dimensional model seems to validate nature’s choice of the topography for the surface of the lotus leaf, at least as far as large CAs are concerned. We note, however, this does not prove that the lotus-like hierarchical roughness is the best type for superhydrophobicity. The purpose of the current study is to understand the role of hierarchical roughness on superhydrophobic surfaces, and here we have used the lotus leaf as a prototype. It would be interesting to look more deeply at how

Figure 7. Comparison of an average lotus-like surface (top, with H/W = 1.5, G/W = 1.71, α = 1.24, N′ = 30, H′/W′ = 10) with the singlescale (macroscale) roughness (bottom, H/W = 3.4, G/W = 5.0) required to produce droplets in the Cassie mode with the same CAs. The red marks at the top of the small pillars (top) represent the lotuslike nanoscale roughness, which is too fine to be depicted clearly here. The drawings are to scale.

particular figure was created by assuming the single-scale equivalent surface was designed using macroscale roughness, rather than microscale roughness. However, there is no reason, a priori, to prefer the macroscale roughness over the microscale roughness in designing such an equivalent surface. In fact, the previous argument for designing an equivalent single-scale surface also holds for the nanoscale roughness, though the lotus-like nanohairs already have an aspect ratio of 10,52 which is sufficient for stability of the Cassie mode for moderately sized drops. In both cases, the relatively large aspect ratios required for the single-scale roughness, combined with their wide spacing, would likely result in a relatively fragile surface structure. Thus, we see that hierarchical roughness can result in drops with large CAs for surface structures with lower aspect ratios, at least on the macroscale. A similar observation related to the aspect ratios was discussed by Patankar, as well as by Li and Amirfazli.30,40 It is interesting, however, that, although an argument has been made for hierarchical roughness allowing for smaller aspect ratios, the nanohairs on the lotus leaf seem to have intrinsically high aspect ratios. Perhaps it is the high density of the nanohairs (in the case of multiscale roughness) which helps to relieve the pressure on individual hairs (when compared to a single-scale equivalent surface consisting of only nanohairs, which must be spaced farther apart), thereby reducing the risk of damage due to the Euler instability associated with their high aspect ratio.32 G

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Figure 8. (A) CA vs log G and log W in the large-drop limit corresponding to the lower-left corner of Figure 6A, with θe = 115° and H/W = 1.5. (B) The difference in CA predicted by our model and that predicted by the Wenzel model or the Cassie model for drops in the Wenzel or Cassie mode, respectively, for the same region of phase space as is shown in part A. In both figures, the dashed line delineates the Cassie−Wenzel boundary.

Figure 9. (A) CA for a single-scale self-similar surface for which θe* = 138°, corresponding to the CA predicted by the Cassie model for a droplet on the lotus-like microscale roughness alone. (B) The difference in CA of the surface represented in part A from the corresponding value for surfaces in Figure 6B, for which the microscale pillars were considered individually. The solid white lines bound the region of lotus-like G/W values, while the dashed black curve delineates the division between the Wenzel and Cassie modes. The white dashed curve in part B represents the macroscale Cassie−Wenzel boundary for surfaces in Figure 6B, which did not account for microscale roughness in the macroscale grooves.

We have also seen in Figures 3 and 4 that, for a fixed step height, the CAs predicted by our model agree with those predicted by the Cassie model in the large-drop regime in which the pinning effects become unimportant. In Figure 8, we investigate the CA in more detail over a small region of G-W phase space of a self-similar (fixed aspect ratio) surface, corresponding to the lower-left corner of Figure 6A. Figure 8A shows the CA versus log G and log W. Here we see that the largest CAs are attained at the boundary between the Cassie and Wenzel modes, as has been noted elsewhere.20,30 This effect occurs because the pinning effects are unimportant for large drops and so the CA for droplets in the Cassie mode is affected mainly by the amount of air under the drop. Increasing G causes the CA to increase until G becomes so large that the drop prefers the Wenzel mode, after which increasing G causes CAs to decrease (due to a decreasing roughness factor, r, in the Wenzel equation, as shown in the Supporting Information, eq S4). In Figure 8B, we show the difference between the CAs predicted by our model and those predicted by either the Cassie or the Wenzel model, depending on whether our model predicts a drop in the Cassie or the Wenzel mode, respectively. Here we see that both the Cassie and Wenzel model predictions agree very well with those of our model, especially for G, W < 0.01a scale which is characteristic of the microscale roughness on a surface with hierarchical roughness

even for smaller drops with sizes on the order of those of the macroscale features. This suggests that perhaps a modification to the CA, rather than any pinning effects, is the main duty of the microscale roughness. In other words, perhaps the main role of the microscale roughness is simply to create an effective Young’s angle, θ*e , which is greater than the intrinsic θe of the material from which the surface is constructed, similar to the idea discussed by Herminghaus39 and others.20,30 Indeed, as mentioned earlier, the largest θe observed to date is only approximately 120°. By covering the macroscale features with microscale roughness, one can effectively overcome this upper limit on the intrinsic θe, thereby potentially allowing larger CAs to be attained. On the basis of the large number of natural examples of hydrophobic surfaces with hierarchical roughness, it appears that nature has already adopted this trick. In Figure 9, we use the idea of representing hierarchical roughness as a single scale of roughness with a modified θe to reexamine the effect of lotus-like microscale roughness on the CAs of droplets on self-similar surfaces of macroscale roughness. The microscale features of the lotus leaf are characterized by a solid fraction of fs = W′/(G′ + W′) = 1/(1 + α) = 0.45, corresponding to a Cassie CA of θC = 138° when θe = 115°. By using θC as the effective θe (or θ*e ) for the macroscale roughness, we are effectively modeling a hierarchically rough surface with the microscale roughness covering not only the tops of the large steps (as we have been considering H

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Figure 10. Guide for desiging an optimally hydrophobic surface for large drops (G + W = 0.01): (A) θe* = 110°; (B) θe* = 130°; (C) θe* = 140°; (D) θ*e = 150°. In each of the figures, the dashed line represents the Cassie−Wenzel boundary.

drop in the Cassie mode, which is represented by the regions below the dashed lines. To maximize the CA, one should increase G/W as much as possible while remaining below the Cassie−Wenzel boundary, which is described by

until now) but also the sides and bottoms of the large grooves, similar to an actual lotus leaf. The results of a scan over G and W for fixed H/W = 1.5 (as in Figure 6) are shown in Figure 9A. The solid white lines represent a range of lotus-like surfaces as before, and we note that all such surfaces now give droplets in the Cassie mode, unlike in Figure 6. In Figure 9B, we directly compare the CAs of droplets on the surfaces represented by Figure 9A with their corresponding values on the hierarchically rough surfaces shown in Figure 6B, by plotting the difference between the two. For droplets that were originally in the Cassie mode on the macroscale in Figure 6B, we find no significant change in CA, except near the boundaries between different Cassie configurations. For droplets that were originally in the Wenzel mode on the macroscale in Figure 6B, however, we observe larger changes in the CA, mainly due to the droplets now preferring the Cassie mode instead of the Wenzel mode. This change stems from the fact that the original model did not include microscale roughness within the macroscale grooves, while using a modified θe does inherently include the effects of the microscale roughness within the large grooves. The most important aspect of this figure, however, is the large region of agreement (below the white dashed curve in Figure 9B) between the two methods of modeling hierarchical roughness. This suggests that the detailed structure of the microscale roughness may be safely ignored by using the Cassie or Wenzel model to compute an equilibrium CA for the microscale roughness alone, and then applying that as the effective θe for the macroscale roughness. In Figure 10, we provide a guide for optimizing the hydrophobicity of a surface for a given θe (or θe*) and a given aspect ratio, H/W. For a fixed aspect ratio, to obtain the highest CAs (and presumably the lowest CAH), we desire a

⎛ 2 cos θ* ⎞⎛ H̅ ⎞ ⎛ G̅ ⎞ e ⎜ ⎟ = −⎜ ⎟⎜ ⎟ ⎝ W̅ ⎠ ⎝ 1 + cos θe* ⎠⎝ W̅ ⎠

(17)

as derived in the Supporting Information. From Figure 10A, it is clear that, for some (hydrophobic) θe, such as θe = 110° represented here, by adjusting G/W and H/W, one may arbitrarily increase the CA for a surface. With a material for which θe = 110°, we can design a surface that gives lotus-like CAs of ∼160° for H/W > 10. In fact, one may achieve CAs of nearly 180° for H/W ≳ 100, though in practice such large aspect ratios may be unfeasible. In that case, to attain the desired high CAs with a single scale of roughness, one must change the chemistry of the surface to cause a larger θe, which will allow for smaller aspect ratios. However, this approach is again difficult in practice, as the largest intrinsic CAs observed so far on any natural or artificial surface have been limited to θe ≤ 120°. Faced with such limitations in θe and aspect ratios, nature has adopted a trick for creating superhydrophobic surfaces, which relies on the use of multiple scales of roughness. On such hierarchically rough surfaces, the microscale roughness may serve to increase the effective θ*e of the surface to a value greater than 120°. The nanohairs on the lotus leaf, for example, appear to produce θe* ≈ 140° (as mentioned earlier), which corresponds to the θ*e in Figure 10C. For such surfaces, we see that, to attain CAs of ∼160°, the macroscale aspect ratios need not be much more than 1, as seen on the lotus leaf. On the basis of Figure 10, one potential algorithm for desigining a hydrophobic, hierarchically rough surface is to first I

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determine θ*e for the macroscale roughness, based on θe for the microscale material and the maximum feasible microscale aspect ratio (which will allow for the largest θe*). Once θe* has been set by the microscale, the macroscale G/W and H/W may be adjusted accordingly to maximize the CA. For general values of θe (other than those shown in Figure 10), eq 17 may be used to determine the maximum allowed G′/W′ for the Cassie mode, and then, assuming pinning effects are unimportant on the microscale, the Cassie equation may be used to compute the corresponding CA for the microscale roughness, with this value then being used as θe* for the macroscale roughness.

CONCLUSION We presented a two-dimensional thermodynamic model to describe the wetting of hierarchically patterned surfaces by small droplets for which the effects of gravity may be neglected. The results of our analysis show that hierarchical roughness is effective in producing large CAs for droplets on solid surfaces. Although a surface with a single scale of roughness is capable of producing large CAs, we find that this is possible only in two small regions of phase spacenamely, in pinning-dominated regions near the boundaries between adjacent configurations for smaller drops and in liquid−vapor interface-dominated regions where W ≪ 1 and G > W (for large drops). Further, we find that such surfaces require a high aspect ratio to prevent droplets from entering the Wenzel mode. However, with the addition of microscale roughness to create a hierarchically rough surface, we find that the region of G-W phase space in which large CAs are achieved is greatly expanded as compared with the case of a single scale of roughness. Thus, at least one important aspect of hierarchical roughness is to increase the robustness of the hydrophobicity of a surface. This robustness probably stems from the fact that the microscale roughness modifies the effective θe for the macroscale roughness. Finally, we showed how this idea of the microscale roughness causing a modified effective θe* can be used to design optimal hydrophobic surfaces based on limits to the feasible aspect ratios of the structures on multiple scales of roughness. ASSOCIATED CONTENT

S Supporting Information *

Cassie and Wenzel model descriptions, energy minimization scheme, pinning criteria, effects of microscale feature density, choice of lotus-like parameters, effects of changing drop size, and derivation of single-scale equivalent of hierarchical roughness. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ acs.langmuir.5b01051.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: fi[email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by Grant CBET 0730987 from the National Science Foundation. J

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