Article pubs.acs.org/Langmuir
The Role of Multiscale Roughness in the Lotus Effect: Is It Essential for Super-Hydrophobicity? Eyal Bittoun and Abraham Marmur* The Wolfson Department of Chemical Engineering Technion, Israel Institute of Technology, 32000 Haifa, Israel ABSTRACT: The role of multiscale (hierarchical) roughness in optimizing the structure of nonwettable (superhydrophobic) solid surfaces was theoretically studied for 2D systems of a drop on three different types of surface topographies with up to four roughness scales. The surface models considered here were sinusoidal, flat-top pillars, and triadic Koch curves. Three criteria were used to compare between the various topographies and roughness scales. The first is the transition contact angle (CA) between the Wenzel (W) and Cassie−Baxter (CB) wetting states, above which the CB state is the thermodynamically stable one. The second is the solid−liquid (wetted) interfacial area, as an indicator for the ease of roll-off of a drop from the superhydrophobic surfaces. The third is the protrusion height that reflects the mechanical stability of the surface against breakage. The results indicate that multiscale roughness per se is not essential for superhydrophobicity; however, it mainly decreases the necessary protrusion height. Thus, multiscale roughness is beneficial for the Lotus effect mostly with regard to mechanical stability. The sinusoidal topography with three levels of roughness scales is best for nonwettability out of the topographies studied here. This observation may partially explain why Nature chose rounded-top protrusions, such as those on the Lotus leaf. The least useful topography is the flat-top pillars with three roughness scales. In the case of the triadic Koch topography, four roughness scales are required to have nonwettable surface.
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INTRODUCTION Superhydrophobic (nonwettable) solid surfaces are abundant in Nature.e.g.1−15 Out of these surfaces, the Lotus leaf has gained special attention, to the extent that nonwettability is frequently referred to as the “Lotus effect”. However, nonwettable surfaces are also prevalent among animals; for example, wings of flying insects,4,16,17 shark skin,18 and water striders.3,19−21 Due to the wide variety of nonbiological applications that nonwettable surfaces may have (such as self-cleaning windows,22,23 waterproofing of cloths and textiles,24 and antibiofouling surfaces25−27), there has been much interest in understanding the Lotus effect and mimicking Nature in producing such surfaces in the lab and industry. Pictures of the Lotus leaf10,15 show protrusions that somewhat resemble paraboloids,28 which are covered by hairy structures smaller by at least 1 order of magnitude than the main protrusions. It also appears that most, if not all, natural nonwettable surfaces involve multiscale roughness. However, successful, nonwettable surfaces have been produced, for example, by photolithography, without necessarily employing multiscale roughness.29−31 Therefore, a key question that needs to be answered is whether multiscale roughness is an essential ingredient in preparing high-quality, useful, nonwettable surfaces. This question was partially answered by various thermodynamics analyses that have been published during the past few years.32−37 The effects of the hierarchical, geometrical features of roughness on the nonwettability of surfaces made of 2D pillars, up to three levels of roughness scales, were theoretically studied for different scenarios of condensed36 and sessile drops.32−35,37 The main focus in the existing papers was on achieving the highest possible, stable CA, as an indication for optimality of a nonwettable surface. As an example, a © 2012 American Chemical Society
wetting map for two levels of roughness scales of star-shaped pillars was constructed, describing all possible wetting states and CAs as a function of the distance between the pillars.35 It was shown, using various simple model surfaces,34,38−40 that multiplicity of roughness scales increases the resistance to water penetration into the roughness grooves and reduces the solid− liquid contact area,34,41 thus improving nonwettability. It was also suggested that a combination of mechanical stability and nonwettability properties can be achieved by at least two levels of roughness.33,39 The objective of the present paper is thus to further the theoretical understanding of the effect of multiscale roughness on nonwettability of solid surfaces. This is done based on studies of the Gibbs energy maps of wetting systems, representing three very different surface-roughness geometries, and multiscale levels. These include sinusoidal surfaces with up to three scales of roughness, flat-top pillars with up to three scales of roughness, and surfaces made of self-similar, triadic Koch curve with up to four scales of roughness (see Figure 1). The various situations are then compared in terms of (a) the transition contact angle (CA) between the Wenzel42 (W) and Cassie−Baxter43 (CB) states; (b) the wetted area at the transition CA; and (c) the height of the main protrusion at this state (the W state refers to complete penetration of the liquid into the roughness grooves, while in the CB state, the drop sits, at least partially, on a layer of air). The first comparison criterion was suggested41 since it represents the lowest CA for which the CB state becomes stable; thus, it is a measure of the Received: July 21, 2012 Revised: September 3, 2012 Published: September 4, 2012 13933
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Figure 1. Unit cells of solid two-dimensional surface topographies. Three levels of roughness scales of a sinusoidal surface topography are shown in (a−c) (in (c) the picture is magnified compared with the others). Three levels of roughness scales of flat-top pillars are shown in (d−f). Four levels of roughness scales for surfaces made of triadic Koch curves are presented in (g−j).
system stability. The second criterion was proposed28,41 because it must be related to the ease of removing the drop off the surface; hence, it measures, to some extent, the nonwettability of the surface. The third criterion is proposed here as an indication of the mechanical stability.
order to check whether the fractal nature leads to unique properties, as has been claimed.49,50 It is assumed that each of the topographies shown in Figure 1 is centered in a “unit cell” that is characteristic of the whole surface. The width of the unit cell, which is actually the longest wavelength of the roughness, λ, is defined as unity. Thus, all geometric parameters of the topographies are, in fact, normalized with respect to λ. The z-coordinate points in the direction of liquid penetration, from the top of the roughness peaks to the bottom of the grooves. The specific surface model with three roughness scales of sinusoidal topography is given by
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THEORETICAL MODELING Three types of periodic surface topographies were studied to explore the effect of additional roughness scales on the transition CA and the wetted area. This was done by calculating the Gibbs energy of a drop, as it depends on the position of the liquid−air interface inside the grooves of the surface and the geometric CA (the CA that serves as the independent variable for the Gibbs energy). The wetting systems in this study are two-dimensional (2D) and consist of a cylindrical drop on a 2D rough surface that is, however, chemically uniform. The reason for choosing 2D systems is, obviously, their simplicity. However, 2D studies are justified, because, so far, the qualitative results of 2D calculations were found to fit the 3D reality.44,45 The three roughness topographies are (a) sinusoidal protrusions with up to three scales of roughness; (b) flat-top pillars with up to three scales of roughness; and (c) self-similar, triadic Koch curve with up to four scales of roughness. The geometric details are shown in Figure 1. These topographies were chosen because they somewhat mimic well-known natural as well as artificial surfaces. Sinusoidal topography resembles the shape of protrusions on the Lotus leaf and was also prepared using microfabrication techniques at different roughness scales.46 Flat-top-pillar topography has been frequently used and optimized in terms of high apparent CA or low CA hysteresis on various arrays.33,35−37,47,48 The triadic Koch curve is a simple example of a fractal that was analyzed in this study in
z(x) = φ1[1 − cos(2πx)] + φ2[1 − cos(2πn2x)] + φ3[1 − cos(2πn3x)]
(1)
where φ is the normalized amplitude of oscillations, x is the normalized horizontal distance from the center of the unit cell, and n is an integer that determines the wavelength of each roughness scale. The subscript numbers indicate the consecutive number of the roughness scale. In this study, the amplitudes were reduced by an order of magnitude from each roughness scale to the next φi = 10−1φi − 1
i = 2, 3
(2)
Obviously, φi = 0 for a roughness scale that does not exist. n2 and n3 were taken to be 10 and 100, respectively. The flat-top pillars (Figure 1d−f) are constructed by carving inside each roughness scale to attain finer pillars in such a way that the geometric dimensions of the pillars are reduced by an order of magnitude. The normalized width of the coarse pillars, a1, was taken as 0.5, and their normalized height, h1, was varied between approximately 0 and 0.65, in order to be comparable 13934
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Figure 2. Illustration of different intersection points between the solid surface (solid lines) and the liquid−air interface (dashed line). The numbers stand for the number of intersection points.
A sl = 2fsl R sin θ
with the sinusoidal topography. The width and height of the pillars were reduced from each roughness scale to the next ai = 10−1ai − 1
i = 2, 3
(3a)
hi = 10−1hi − 1
i = 2, 3
(3b)
where fsl is the 2D wetted area per unit base area of the drop and its range is 0 ≤ fs1 ≤ r where r is the roughness ratio of the surface. The 2D solid−air interfacial area is the difference between the total solid surface area (which is constant), Atotal, and the wetted solid area
The numbers of pillars in the second and third roughness scale were taken to be 5 and 10, respectively. The triadic Koch curve is made of equilateral triangles that are reduced by a factor of 3 and duplicated three times from one roughness scale to the next. This can be clearly seen in Figure 1g−j. For this topography, the geometric parameters are self-determined: b1 = 1/3 (by definition), where b is the length of the triangle side and the subscript indicates the roughness scale, b2 = 1/9, b3 = 1/27, and b4 = 1/81. During the calculations, it was assumed, for simplicity, that the effect of gravity is negligible. Consequently, the shape of the drop cross section is circular. Furthermore, the air pressure inside the roughness grooves is equal to the atmospheric pressure, since the roughness grooves are interconnected among themselves and to the outside atmosphere. This implies that the radius of curvature of the liquid−air interface inside the roughness grooves is equal to the radius of the drop. It is also assumed that the size of the drop is much larger than the roughness scales. This assumption has two important implications: (a) combined with the previous assumption, it indicates that the liquid−air interface inside the roughness grooves is approximately planar, since its radius of curvature is much larger than the roughness scale (typically by about 3 orders of magnitude); and (b) the volume of the liquid inside the roughness grooves is negligible compared with drop volume. In addition, line tension effects were also assumed negligible.51 In general, the Gibbs energy of a solid−liquid−air wetting system is given by G = Al σl + A sl σsl + A sσs
A s = A total − A sl = A total − 2fsl R sin θ
(7)
The “radius” of the drop base is related to the drop radius and its drop volume by r 2 = (R sin θ )2 = V
2 sin 2 θ 2θ − sin 2θ
(8)
Introducing eqs 5−8 into eq 4, the Gibbs energy can be written in the following dimensionless form G* ≡
G = F1/2(θ )(θ − Ω sin θ ) 2 2V σl
(9)
where F(θ ) ≡
1 2θ − sin 2θ
(10)
and Ω ≡ fsl cos θY − fl
(11)
It should be noted that Atotal is constant and does not affect the minimization; therefore, it is taken as zero. It is also important to notice that fsl and f l both depend on the depth of liquid penetration into the roughness grooves, z; therefore, fsl = fsl(z) and f l = f l(z). The details of the penetration process of the liquid drop into the roughness grooves on the solid surface are very complicated and insufficiently understood.52,53 However, as explained above, it is assumed that the liquid−air interface inside the roughness grooves is, on average, planar and horizontal. For each given penetration depth of the liquid, z, it is known that the minimal G is associated with the CB CA if the drop is sufficiently large:54
(4)
where σ is the surface/interfacial tension, A is the interfacial area, and the subscripts l and s stand for liquid and solid, respectively. The 2D liquid−air interfacial area consists of two parts, the outside interface of the drop and the liquid−air interfaces within the roughness grooves Al = 2Rθ + 2fl R sin θ
(6)
cos θCB = fsl cos θY − fl
(12) 42
The CB equation becomes the Wenzel equation when the liquid drop completely penetrates in between the roughness grooves. Then, f l = 0, fsl = r, and
(5)
R is the radius of the cylindrical drop, θ is the geometric CA, and f l is the fraction of the base area of the drop which is exposed to the air inside the roughness grooves. Therefore, it is limited to be in the range of 0 ≤ f1 ≤ 1. The 2D solid−liquid interfacial area is given by
cos θ W = r cos θY
(13)
However, the dependence of G on z may lead to a saddle point,55 which implies an unstable equilibrium. Therefore, the dependence of G(θCB, z) on z was followed. To this end, the 13935
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Figure 3. Dimensionless Gibbs energy curves vs the dimensionless penetration depth of the drop base into the roughness grooves, for sinusoidal topography: (a) 2φ1 = 0.4; (b) 2φ1 = 0.6. Dotted, dashed, and solid lines stand for a single (RS1), two (RS2), and three (RS3) roughness scales, respectively. The thin lines point at the minima points in the curves that stand for metastable or stable equilibrium states.
⎧ ⎪ ⎪ ⎪ ⎪ fsl (z) = ⎨ ⎪ ⎪ ⎪ ⎪ ⎩
terms fsl and f l in eq 11 were numerically calculated for the above-described surface topographies as follows. For each fixed value of z, the intersection points between the solid surface profile and the liquid−air (assumed) plane were identified, as qualitatively demonstrated for various situations in Figure 2. These points were used to compute the wetted area of the solid by the liquid drop and the liquid−air interface between the protrusions. f l was calculated from the accumulated length of horizontal lines that do not intersect the solid surface. fsl was calculated by summation of the lengths of the wetted arcs or straight lines. For the flat-top pillars (Figure 1d−f)
(14)
(15)
fsl (z) =
where a, h, and N are the width, height, and the number of pillars, respectively, and the subscript numbers indicate the roughness scale. For the triadic Koch curve, the equations for only the first two roughness scales are presented below as examples, since the equation for the higher roughness scales are too tedious to present here. For the first roughness scale (Figure 1g)
fl (z) = 1 −
0≤z
