Role of Statistical Properties of Randomly Rough Surfaces in

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Role of Statistical Properties of Randomly Rough Surfaces in Controlling Superhydrophobicity F. Bottiglione and G. Carbone* Tribology LAB, Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, v. le Japigia 182, 70126 Bari, Italy ABSTRACT: We investigate the effect of statistical properties of the surface roughness on its superhydrophobicity. In particular, we focus on the liquid−solid interfacial structure and its dependence on the coupled effect of surface statistical properties and drop pressure. We find that, for self-affine fractal surfaces with Hurst exponent H > 0.5, the transition to the Wenzel state first involves the short wavelengths of the roughness and, then, gradually moves to larger and larger scales. However, as the drop pressure is increased, at a certain point of the loading history, an abrupt transition to the Wenzel state occurs. This sudden transition identifies the critical drop pressure pW, which destabilizes the composite interface. We find that pW can be strongly enhanced by increasing the mean square slope of the surface, or equivalently the Wenzel roughness parameter rW. Our investigation shows that, even in the case of randomly rough surface, rW is still the most crucial parameter in determining the superhydrophobicity of the surface. An analytical approach is, then, proposed to show that, for any given value of Young’s contact angle θY, a threshold value (rW)th = 1/(−cos θY) exists, above which the composite interface is strongly stabilized and the surface presents robust superhydrophobic properties. Interestingly, this threshold value is identical to the one that would be obtained in pure Wenzel regime to guarantee perfect superhydrophobicity.

1. INTRODUCTION Controlling surface superhydrophobicity is of utmost importance in a countless number of applications.1 Examples are antiicing coatings,2−4 friction reduction,5−7 antifogging properties,8 antireflective coatings,9−11 solar cells,12−14 stabilization of Leidenfrost vapor layers and reduction of the risk of explosion in power plants,15 chemical microreactors and microfluidic microchips,16,17 self-cleaning paints, optically transparent surfaces.18−23 Water droplets on superhydrophobic surfaces usually present very low contact angle hysteresis, i.e., very low rolling and sliding friction values, which make them able to move very easily and quickly on the surface, capturing, in the meanwhile, external impurities. There is concrete evidence for surfaces with such properties in many biological systems, as the sacred lotus leaves,24 water striders,25,26 or mosquito eyes,8 where the presence of surface asperities causes the liquid to rest on top of the surface summits and air to remain entrapped between the drop and the substrate. This type of “fakir-carpet” configuration is referred to as the Cassie−Baxter state.27 However, depending on the surface chemical properties and its microgeometry, a drop in a Cassie−Baxter state may become unstable when the liquid pressure p increases above a threshold value pW.28−37 The increase of liquid pressure may occur during drop impacts at high velocities.29,38,39 When this happens, the liquid undergoes a transition to the Wenzel state,40 which makes the droplet rest in full contact with the substrate.32,41 This transition to the Wenzel state is usually irreversible because of the high energy barrier the drop should overcome to return to the Cassie−Baxter state.28,30,41,42 However, the © 2012 American Chemical Society

presence of multiscale or hierarchical microstructures may favor the transition back to the Cassie−Baxter state,43−47 or the stabilization of the “fakir-carpet” configuration,28,48 thus explaining why many biological systems present such multiscale geometry.49−51 Although many artificial microstructured surfaces have been shown to possess superhydrophobic properties, their design has not yet been driven by a clear understanding of the underlying physical mechanisms. In particular, it has been shown that, in the case of surfaces made of a regular array of micropillars, the shape of the pillar has a considerable influence on the superhydrophobic properties of the surface.52,53 However, only a few works have tried to provide suggestions on what should be the optimal geometry of the micropillars.31,54 In particular, ref 31 shows that conical-topped cylindrical pillars may be a very promising solution since this kind of geometry stabilizes the Cassie−Baxter state by strongly increasing the critical drop pressure threshold pW, and, at the same time, in contrast with flat-topped cylindrical pillars, also guarantees very low pull-off (menisci) forces during detachment. Noteworthy examples of this kind of conical-topped pillars can be found in water striders25 and flower leaves.55 By moving from regular microstructured surfaces to disordered randomly rough surfaces, some studies56,57 have shown that many man-made fractal surfaces possess super Received: October 15, 2012 Revised: December 4, 2012 Published: December 5, 2012 599

dx.doi.org/10.1021/la304072p | Langmuir 2013, 29, 599−609

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Figure 1. Schematic picture of a drop deposited onto a rigid substrate. The contact patch has an apparent length equal to D. A magnified view of the contact patch is shown in the lower part of the figure. At this scale, the surface appears rough. We assume a spatial periodicity L.

When S > 0, the surface presents robust superhydrophobic properties (higher values being better than smaller ones), while when S < 0, the surface is weakly superhydrophobic, despite the fact that a composite interface can also be formed in this case, as a consequence of the presence of local energy minima. The limiting condition S(rW, θY) = 0 allows us to determine, for any given value of Young’s equilibrium contact angle θY, the threshold value (rW)th of the Wenzel roughness above which the surface possesses robust superhydrophobic properties. We propose a very simple analytical model to estimate the parameter S(rW, θY) and the Wenzel roughness rW in terms of the mean square slope ⟨∇h2⟩ of the roughness (the symbol ⟨⟩ is the ensemble average operator) both for 1D roughness and 2D isotropic rough surfaces. This allows to identify a threshold value ⟨∇h2⟩th and propose the simple relation ⟨∇h2⟩ > ⟨∇h2⟩th as a criterion to design robust superhydrophobic randomly rough surfaces.

water-repellent properties with contact angles up to 174°. Such hierarchical structures have been also exploited by biological systems to enhance their hydrorepellent properties. In fact, a closer analysis of the micro- and nano-geometry of many leaf surfaces shows the presence of random roughness at the smallest scales.58 The statistical properties of such random structures are usually characterized through the surface power spectral density (PSD), which should have a strong influence on the superhydrophobic properties of the surface. Unfortunately, only a few studies have been presented in the literature focusing on this aspect of the problem.58−63 In particular, molecular dynamics simulations59,60 show that, due to strong thermal fluctuations, contact angle hysteresis is almost completely suppressed, and suggest that the fractal dimension of the surface has only a weak influence on the apparent contact angle values. However, some experimental investigations61 show, instead, that the fractal dimension may have an important effect on the superhydrophobic properties of the surface, since the fractal dimension indirectly affects the mean square slope of the surface and, hence, the so-called Wenzel roughness parameter rW,62 which, as shown by a very recent numerical study,63 has a significant influence on superhydrophobicity. However, how large rW should be to make the randomly rough substrate strongly superhydrophobic is still an open question. Moreover, almost nothing is known about how surface statistical properties affect the threshold pressure pW, which causes the transition to the Wenzel state. This parameter is, indeed, a good estimator of the robustness of the superhydrophobicity of randomly rough surfaces, with higher values being highly desirable. In this respect, we present a numerical methodology to study the wet contact between a drop and a self-affine randomly rough surface with roughness in only one direction, and investigate how its statistical properties affect the quantity pW. However, besides pW we also propose another parameter to measure and assess the degree of superhydrophobicity. This parameter is shown to depend only on the Wenzel roughness parameter rW and Young’s contact angle θY, and is defined as the change of interfacial surface energy which occurs as the liquid−solid real contact area increases of a unit area. We call it the energy−slope parameter S(rW, θY).

2. FORMULATION AND NUMERICAL PROCEDURE We consider a liquid drop in contact with a rigid randomly 1D rough periodic surface, described by a Fourier series M

h(x) =

∑ hk cos(kqLx + ϕk) k=1

(1)

where qL = 2π/L is the long-distance cutoff wave vector, with L being the spatial periodicity of the rough profile, and M is the number of wave components. The profile has been numerically generated as described in Appendix A. We have opted for a fractal self-affine rough profile. In this case, only three parameters are enough to completely characterize the statistical properties of the profile. We choose the following ones: (i) the mean square roughness ⟨h2⟩ of the profile, i.e., the zero-th order moment ⟨h2⟩ = m0 = ∫ C(q) dq of the profile power spectral density C(q) = (2π)−1∫ ⟨h(x′)h(x′ + x)⟩e−iqx dx; (ii) the mean square slope ⟨h′2⟩ of the profile (observe that in the case of 1D rough surfaces ∇h2 = h′2), i.e., the second moment ⟨h′2⟩ = m2 = ∫ q2C(q) dq of the PSD; and (iii) the Hurst exponent H, i.e., the fractal dimension Df = 2 − H. We assume that the spectral content lies in a range of wavelengths much smaller than the 600


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Figure 2. Geometrical quantities needed to define the geometry of the composite interface and the state of the system.

linear size D of the apparent liquid−solid contact area, i.e., L/D ≪ 1 (see Figure 1). As a consequence of this very strong separation of length scales, when the interface is observed at the roughness microscale (local observer in Figure 1), the drop will appear as a semi-infinite space, and the influence of Laplace pressure (due to the curvature of the liquid−air free interface far from the rough substrate) and gravity will simply result in a drop pressure p at the liquid−solid interface. The drop pressure must be taken into account by the local observer to find the liquid−substrate configuration at equilibrium. In Figure 2, the general case of partial contact between liquid and solid is shown, and the principal geometrical quantities are defined. We take as reference frame an observer fixed to the mean plane of the liquid profile. In this reference frame, the height distribution of the 1D rough substrate is

xi = −

uL(x) = yi − sgn p R2 − (x − xi)2

(2)

N

∑ (∫ i=1

ai

bi

uS(x)dx +

∫b

ai + 1

uL(x , bi , ai + 1)dx)

i

= f (a1 , b1 , ..., ai , bi , ..., aN , bN , p , Δ) (5)

=0

Equation 5 can be equivalently written as Δ(a1 , b1 , ..., aN , bN , p) = hmax −

1 L

N

∑ [∫ i=1

ai

bi

h(x)dx +

∫b

ai + 1

uL(x , bi , ai + 1)dx]

i

(6)

Assuming isothermal conditions, liquid incompressibility, and constant drop pressure, the total interfacial energy (per unit t h ic k n es s ) o f t h e s y s t e m i s t h e G i bb s e n e r g y .(a1 , b1 , ..., ai , bi , ..., aN , bN , p). .(a1 , b1 , ..., ai , bi , ..., aN , bN , p) is the sum of the pressure potential −pLΔ plus the interfacial Helmholtz free surface energy per unit thickness - = γLA(lLA − lLS cos θY ), where lLS and lLA are the liquid−solid and liquid−air interfacial length, respectively. Moreover, −γLA cos θY = γLS − γSA, and θY is Young’s contact angle. The Gibbs energy becomes

x∈Ω x∉Ω

(4)

The equation of state of the system, which links the penetration Δ to the quantities a1, b1, ..., ai, bi, ..., aN, bN, and p, is calculated by enforcing the condition that the y-coordinate of the mean plane of the liquid profile is constantly equal to zero. This yields

where h(x) is the profile height above the mean plane of the roughness, hmax is the highest peak of the profile above the mean plane of the rough substrate, s is the separation between the mean plane of surface roughness and the mean plane of the liquid profile, Δ = hmax − s is the penetration of the solid substrate into the liquid. The quantities ai and bi are the abscissae of the ith liquid−substrate contact spot, with ai < bi and i = 1, 2, ..., N, where N is the unknown number of contacts. The contact domain is Ω = ∪Ni = 1 [ai, bi]. The liquid height spatial distribution is uL(x). Out of the liquid−solid contact area, this quantity is unknown and must be determined. However, in the noncontact regions the free liquid surface obeys the Laplace equation and, therefore, must take the shape of a circular arc with radius R = γLA/p, where γLA is the surface tension at liquid−air interface. Because the liquid pressure at the interface is uniformly distributed, the radius of curvature R of the free liquid−air interface is also constant. Therefore, one can write uL(x) = uS(x) = h(x) − hmax + Δ

1

⎧ 4R2 − (a − b )2 − [h(a ) − h(b )]2 ⎫ 2 i+1 i i+1 i ⎬ ×⎨ (ai + 1 − bi)2 + [h(ai + 1) − h(bi)]2 ⎭ ⎩ ai + 1 + bi + 2 1 yi = sgn p(ai + 1 − bi) 2 1 ⎧ 4R2 − (a − b )2 − [h(a ) − h(b )]2 ⎫ 2 i+1 i i+1 i ⎬ ×⎨ (ai + 1 − bi)2 + [h(ai + 1) − h(bi)]2 ⎭ ⎩ h(ai + 1) + h(bi) + − hmax + Δ 2

uS(x) = h(x) − s = h(x) − hmax + Δ(a1 , b1 , ..., aN , bN , p)

1 sgn p[h(ai + 1) − h(bi)] 2

(3)

The quantities xi and yi represent, respectively, the abscissa and the ordinate of the center of the circular free liquid surface at the ith noncontact spot, and both depend on the quantities ai, bi, and p = γLA/R through the relations 601


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ately large variations of pressure, we observed that the most reliable approach involves many small increments of pressure starting from zero up to the desired value, while optimizing the solution at every intermediate step. The solution of the problem is indeed not unique: for the same pressure, different configurations (local minima) may be found depending on the loading history. Nonetheless, the contact pattern occurring with increasing pressure is uniquely identified and it is the most suitable solution to represent the real behavior of a drop gently posed on the rough rigid solid. The incremental process ends when the drop undergoes a transition to the Wenzel state, i.e., when the drop pressure reaches the critical pressure pW, and the liquid jumps into full contact with the substrate.

.(a1 , b1 , ..., ai , bi , ..., aN , bN , p) = γLA(lLA − lLS cos θY ) − pLΔ

(7)

The quantities lLS and lLA depend on a1, b1, ..., ai, bi, ..., aN, bN through the following relations N

lLS =

bi

∑∫ i=1 N

lLA =

∑∫ i=1

1 + [h′(x)]2 dx

ai

bi

ai + 1

1 + [uL′ (x)]2 dx (8)

For any given pressure p, the configuration at equilibrium is numerically determined by minimizing the total interfacial energy . , i.e., by minimizing eq 7. This allows us to calculate the coordinates ai and bi of each contact patch, the liquid configuration, and all the other thermodynamical quantities. From the point of view of the implementation of the numerical procedure, we have followed an approach similar to the one presented by one of the authors to investigate a differet problem related to the elastic contact mechanics of rough solids.64,65 Here, we briefly summarize the methodology. At a low level in our numerical implementation of the algorithm there is the Solver,64,65 a software code that, given the drop pressure and the contact regions, calculates everything else. On top of it, we built another piece of software to adjust the position of the contacts boundaries a1, b1, ..., aN, bN to minimize the interfacial energy. We employed a conjugate gradient method in the version given by Polak and Ribiére.66 Unfortunately, the problem is more complex than a minimization in a 2N-dimensional space. Starting from an arbitrary configuration, some of the contact boundaries can acquire the same value, meaning that either a contact is detaching or two contacts are coalescing together into a single, bigger one. If this happens, the minimization has to be restarted in a different number of dimensions. Furthermore, a constraint must be taken into account: in principle, we can determine the configuration corresponding to any given pressure and contacts, but we must impose that in the noncontact regions the liquid profile never intersects the substrate. For instance, if we consider a physical configuration minimizing the interfacial energy, and then we increase the drop pressure pushing the substrate against the liquid drop, the starting point for the new conjugate gradient minimization may show an intersection between the liquid profile and some peaks of the substrate in the noncontact regions. This indicates that a new contact has to be added before starting the minimization. A specific procedure inside our software is designed to detect all the intersections between the liquid and the substrate, to enforce the physical constraints of the problem. Given the drop pressure p, the search of the contact domain Ω = ∪Ni = 1 [ai, bi] that minimizes the interfacial energy is challenging: not only are the 2N variables a1, b1, ... aN, bN unknown, but the number N of contact regions is also unknown. The solution of the problem resorts to conjugate gradient minimization alternated to searches for intersections between the elastic layer and the block. The minimization procedure stops when the conjugate gradient ends successfully, i.e., it is not interrupted by a coalescence or detachment of contacts, and the successive search for intersections confirms that there are no intersections in the noncontact regions. Although we took special care to guarantee that the minimization procedure would converge for moder-

3. NUMERICAL RESULTS Given the statistical properties of the rough substrate, represented by the PSD of its roughness, we have generated twenty statistically equivalent rough profiles, whose spectral content comprises 3 orders of magnitude of length scales, i.e., M = 1000. To guarantee good resolution even at the smallest length scales, the discretization grid is composed of 104 sampling points. For each of these twenty profiles, we have carried out the numerical simulations by following the aforementioned incremental approach, i.e., by increasing the drop pressures p from zero until the liquid jumps into full contact with the solid substrate. The ensemble average of the numerical results has then been calculated and considered representative of the wetting properties of the randomly rough substrate. Figure 3 shows

Figure 3. Different snapshots of the drop configuration at increasing liquid pressure, for two different rigid rough substrates, (a) and (b). The two profiles present the same values of m2 = 50 and hrms = 10 μm, but different fractals dimensions: Df = 1.2 (a); Df = 1.3 (b). The dashed line is the shape of the liquid profile at the previous pressure step.

the liquid profiles obtained as a result of simulations carried out for two different rough surfaces. Both surfaces are characterized by the same surface root-mean-square roughness hrms = m1/2 0 = ⟨h2⟩1/2 = 10 μm, and the same value of the mean square slope m2 = ⟨h′2⟩ = 50. However, they present different values of the fractal dimension: Df = 1.2 (i.e., H = 0.8) in Figure 3a, and Df = 1.3 (i.e., H = 0.7) in Figure 3b. In both cases, Young’s contact 602


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angle is θY = 108°. In Figure 3, the cyan color identifies the liquid domain, whereas the blue color identifies the rigid rough substrate. For each of the two rough surfaces, four different snapshots are shown at increasing drop pressure values until the critical value pW is reached. In each snapshot, a dashed red line is visible. It represents the liquid configuration at the previous pressure step, and serves to better clarify the evolution of the contact configuration as the pressure is increased. We observe that the drop rests in partial contact with the substrate. However, a closer look at the figure shows that the drop does not simply touch the top of the asperities, but rather, at contact spots, it adheres in full contact with the finest structures of the rough surfaces. Therefore, the real liquid−solid interfacial configuration is a mix between the Cassie−Baxter state, which occurs at the larger scales, and the Wenzel state at the smallest scales. However, since an increase of the fractal dimension Df (i.e., a reduction of H), at fixed hrms = 10 μm and m2 = 50, determines a significant increase of the slope qkhk of the highest frequency Fourier components, one should expect that increasing Df must lead to: (i) a reduction of the liquid−solid contact area fraction ϕ = |Ω|/L, where |Ω| = ΣNi = 1 (bi − ai) is the liquid−solid contact area projected on the mean plane of the rough substrate, (ii) an increase of the number N of liquid− solid contact spots, and (iii) a reduction of the average liquid− solid contact area per contact spot ϕ/N, which is accompanied by a reduction of the size of the smallest liquid−solid contact spot (see Appendix B for additional details). This behavior is confirmed in Figures 4 and 5, which show that the liquid−solid contact fraction ϕ is an increasing

Figure 5. Number of liquid−solid contact spots N, (a), and the mean dimensionless size ϕ/N of the contact spots, (b), as a function of the liquid pressure p, for hrms = 10 μm, m2 = 50, and different fractal dimensions.

jumps to unit, and only one single contact is then formed (i.e., N = 1). The pressure at which the jump to the Wenzel state occurs is identified as the threshold value pW. Interestingly, Figures 4 and 5 show that the critical pressure value pW decreases as the fractal dimension Df is increased. This result, that may appear rather counterintuitive, has an easy explanation if one considers that the Wenzel transition does not occur contemporaneously on all length scales. In fact, let us observe that for Hurst exponents H > 0.5 (i.e., the one considered in our investigation and most commonly found in practice) the maximum slope qkhk of each profile spectral component decreases as the wavenumber k is increased. Therefore, as the liquid pressure is increased, the Wenzel transition will first involve the short wavelengths and gradually the larger scales. Within this picture, one necessarily concludes that the final transition to the Wenzel state is dominated by the largest wavelength, i.e., the critical pressure pW should be identified with the value of pressure which causes a transition to the Wenzel state on the largest length scale. For the case of a single sinusoidal profile with large aspect ratio, the pressure pW can be estimated as28

Figure 4. The fraction ϕ = |Ω|/L of liquid−solid contact as a function of the liquid pressure p, for Df = 1.1, 1.2, 1.3, 1.4; m2 = 50; and hrms = 10 μm. The abrupt transition to the Wenzel state is shown by the vertical dashed lines, which also identify the critical pressure pW at which the transition occurs. The simulation at Df = 1.4 has been stopped before the occurrence of the transition to Wenzel state.

function of the liquid pressure p. At small pressures, the increase is slow, since the liquid only touches the top of a few asperities. As the pressure is further increased, an increasing number of bigger asperities will touch the liquid drop. However, since the top of each bigger asperity is covered by a large number of small asperities, the liquid−solid contact will actually be established with these smaller asperities on top of bigger ones. This will cause a fast increase of the number of contact spots (see Figure 5a) and, hence, of liquid−solid contact area. At large drop pressures, almost all big asperities have gone in contact with their tops, so that the liquid−solid contact will now increase slowly again. An additional increase of pressure may cause the transition to a complete contact condition (a perfect Wenzel state): The area fraction ϕ, indeed,

pW = c(θY )qL 2h1

(9)

where c(θY) is a constant depending only on Young’s contact angle θY. Thus, observing that eqs 26, 28, and 29 yield 2

qL h1 =

2

m2 m01/2

M

∑k = 1 k−1 − 2H M

∑k = 1 k1 − 2H

(10)

one concludes that increasing the fractal dimension Df (i.e., reducing H), at constant m0 and m2, necessarily causes a reduction of the quantity qL2h1, and, in turn, of the critical pressure pW. Of course, the above argument is only in qualitative agreement with Figure 4, since eq 9 holds true 603


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only for the case of a perfectly sinusoidal substrate, i.e., it does not take into account all the additional effects due to the presence of random roughness. Figure 6 reveals that the

Figure 6. Fraction ϕ = |Ω|/L of liquid−solid contact as a function of the liquid pressure p, for hrms = 10 μm, Df = 1.3, and two values of m2 = 10 and 50.

parameter m2 = ⟨h′2⟩ has a huge influence in determining the hydrophobic properties of the substrate. In fact, given the value of m0 and the fractal dimensions Df = 2 − H, pW is much larger for m2 = 50 than m2 = 10, and the liquid−solid contact area fraction ϕ strongly decreases as m2 is increased. This is a consequence of the strong influence of m2 on the maximum slope qkhk of each single Fourier component of the rough surfaces.

Figure 7. Dimensionless interfacial Helmholtz free energy - /(γLAL) as a function of the dimensionless liquid−solid contact area ϕ, for different values of the fractal dimension Df = 1.1, 1.2, 1.3, and 1.4; three values of mean square slope m2 = 5, 10, and 50; and hrms = 10 μm (a); and for Df = 1.3, m2 = 10, and two values of hrms = 10 and 50 μm (b).

4. SIMPLE CRITERION FOR ROBUST SUPERHYDROPHOBICITY The effect of m2 is even more clear if one plots the dimensionless Helmholtz free interfacial energy - /(γLAL) as a function of the liquid−solid contact area fraction ϕ (see Figure 7a). Results are presented for hrms = (m0)1/2 = 10 μm, for three values of the mean square slope of the rough substrate m2 = 5, 10, and 50, and different fractal dimensions ranging from Df = 1.1 to 1.4. A linear dependence of the surface energy as a function of the liquid−solid contact area is observed in all cases. More importantly, the only parameter which strongly affects the slope of the linear relation is m2, while both the fractal dimension Df and the root-mean-square roughness hrms = (m0)1/2 do not have any appreciable influence (see Figure 7b). We can easily explain this observed trend of the interfacial free energy by recalling that - = γLA(lLA − lLS cos θY ) and rephrasing this equation as = (1 − ϕ)β − ϕα cos θY γLAL

and h(x) are independent statistical properties (i.e., that the joint probability density distribution P(h,h′) = P(h′)P(h)), one concludes that the distribution P(h′) cannot depend on the height of the asperity of the rough surface in contact with the liquid. In this case, one may approximate α ≈ rW 1 = L

∞

=

lLA 1 = L L

N

∑∫ i=1

(11)

P(h′) =

∑∫

bi

rW = ai + 1

1 + h′2 P(h′) dh′

(13)

⎛ h′2 ⎞ 1 exp⎜ − ⎟ 2πm2 ⎝ 2m2 ⎠

(14)

and from eq 13

1 + [h′(x)]2 dx = αϕ

ai

N i=1

bi

∫−∞

1 + [h′(x)]2 dx

where we have assumed that the subset of values of h′(x), with x belonging to the liquid-solid contact domain is statistically representative of the entire distribution of slope values. For a Gaussian surface, one writes

where the parameters α ≥ 1 and β ≥ 1 have been defined through the following relations lLS 1 = L L

∫L

1 + [uL′ (x)]2 dx = β(1 − ϕ)

⎛ 1 1 ⎞ 2m2 U ⎜ − , 0, ⎟ 2m2 ⎠ ⎝ 2

(15)

where U(a, b, z) is the confluent hypergeometric function of the second kind, also known as Tricomi’s function. Equation 15 shows that rW(m2) is an increasing function of m2, i.e., there is a one-to-one relation between rW and m2, so that, in this respect, the two quantities may be considered equivalent from a statistical point of view. Using eq 15, we can rephrase eq 11 as

(12)

Now, let us observe that, for any given drop pressure p, the local slope u′L (x) of the liquid−air interface is u′L (x) ≪ 1. Thus, we may assume with a negligible error β ≈ 1, i.e., lLA/L ≈ 1 − ϕ. Moreover, considering that for a Gaussian surface h′(x) 604


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from eq 15 one obtains m2 = ⟨h′2⟩ > 13.75 ≫ 1. Moreover, recalling that U[−1/2, 0, 1/(2m2)] < π−1/2 and observing that U[−1/2, 0, 1/(2m2)] rapidly converge to π−1/2, i.e., that rW = (2m2/π)1/2, for m2 = ⟨h′2⟩ ≫ 1, a more strict but very simple criterion for robust superhydrophobicity of 1D rough surfaces is π m2 = ⟨h′2 ⟩ > ⟨h′2 ⟩th = 2 cos2 θY (18)

(16)

which shows that the dimensionless interfacial energy - /(γLAL) is always a linear function of the liquid−solid contact area fraction ϕ, and its slope is solely affected by the mean square slope of the underlying rough solid surface, in perfect agreement with the presented numerical results. The dashed straight lines in Figure 7 represent the plots of eq 16 for the same values of m2 utilized to carry out the simulations. The agreement with numerical result is amazingly perfect, thus confirming the validity of the proposed simple theoretical interpretation. Moreover, we also observe that the values of m2, which lead to an increasing trend of - /(γLAL) as a function of ϕ, will be certainly beneficial in terms of superhydrophobicity, since the increase of - will tend to: (i) balance the decrease of pressure potential −pLΔ, which occurs as the drop penetrates into the substrate, (ii) hamper the spread of the drop on the substrate, and (iii) stabilize the Cassie−Baxter state. In other words, since larger contact areas lead to higher interfacial energy drop configurations, a large amount of work must be provided by the external source of energy (e.g., the mechanical energy associated with the drop pressure) to move the drop in intimate contact with the substrate and destabilize the Cassie− Baxter state. On the other hand, values of m2 which lead to a negative slope of the linear relation 16 should be considered deleterious, since this time, apart from local energy minima, that may weakly hamper the evolution of the contact interface, an increase of contact area will lead, on the average, to a reduction of the free energy - . In this case, under the action of the pressure p, the liquid−solid contact will easily spread toward an energetically more favorable state of the system. These arguments suggest a criterion to distinguish between robust (or strongly) superhydrophobic rough surfaces and weakly superhydrophobic ones: A robust superhydrophobic surface is obtained when the energy-slope parameter S(rW , θY ) = ∂- /∂ϕ = ( −cos θY )rW − 1 is S(rW, θY) > 0. This criterion leads to the following condition for robust superhydrophobicity rW > (rW )th = 1/( −cos θY )

For the specific value θY = 108°, eq 18 gives m2 > 16.44 (slightly larger but more conservative than the previous calculated value). The same criterion eq 17 can be extended to 2D isotropic surfaces (see Appendix C). In this case, rW = A−1 ∫ d2 x (1 + [∇h(x)]2)1/2 differs from the value rW found in the 1D case. In the 2D case, the Wenzel parameter rW becomes (see Appendix C) rW = 1 +

π ⟨(∇h)2 ⟩1/2 exp[⟨(∇h)2 ⟩−1] 2

erfc[⟨(∇h)2 ⟩−1/2 ]

(19)

For the specific value θY = 108°, one obtains ⟨(∇h) ⟩ > 11.65 ≫ 1. However, observing that rW → (π⟨(∇h)2⟩)1/2/2 for ⟨(∇h)2⟩ ≫ 1, the condition rW > (rW)th = 1/(−cos θY) simply becomes 2

⟨(∇h)2 ⟩ > ⟨(∇h)2 ⟩th =

4 π cos2 θY

(20)

Notice that eq 18 is more conservative than eq 20; thus one may always use the condition eq 18 to design robust superhydrorepellent surfaces.

5. CONCLUSIONS We present a numerical methodology to study the wet contact between a drop and a self-affine randomly rough surface h(x), with roughness in only one direction (1D rough surfaces), and propose a simple theoretical approach to assess the superhydrophobic properties of both 1D rough and 2D isotropic rough surfaces. The numerical methodology has been exploited to investigate the effect of statistical properties of the surface on the ability of the surface to stabilize the composite interface, i.e., the fakir-carpet state of the liquid drop. In particular, we focus on the evolution of the liquid−solid contact area, which occurs as the drop pressure is increased, and investigate how this evolution change depending on the statistical parameters of the rough surface. In particular, we find that, for Hurst exponent H > 0.5, the transition to the Wenzel state will first involve the short wavelengths and gradually the larger scales, although an abrupt transition occurs a certain point of the loading history. The critical drop pressure pW, which destabilize the system causing the abrupt transition to the Wenzel state, is strongly increased as the mean square slope ⟨∇h2⟩ of the surface, is increased. Indeed, both the numerical approach and the theoretical calculation shows that ⟨∇h2⟩ is the most crucial parameter affecting the superhydrophobicity of the surface. The fractal dimensions and the root-mean-square roughness have, instead, only a limited influence on superhydrophobicity. By exploiting the proposed analytical model, given the Young contact angle θY, we show that, if the mean square slope of the surface ⟨∇h2⟩ exceeds the threshold value ⟨∇h2⟩th ≈ π/(2 cos2 θY), the surface energy at the interface increases as the liquid− solid contact area is increased. When this happens, the fakir-

(17)

Interestingly, this threshold value is identical to the limiting value obtained in pure Wenzel regime for perfect superhydrophobicity. In the Wenzel state,40 the apparent contact angle θa is related to Young’s contact angle θY through the simple relation cos θa = rW cos θY; hence, by requiring that θa = 180° (perfect superhydrophobicity), one obtains the same threshold value (rW)th = 1/(−cos θY). Interestingly, eq 17 appears to agree well with some existing experimental results. In fact, in the case of surfaces made of perfluoroalkanes (θY = 120°) or alkanes (θY = 110°), as those investigated in ref 62, values of rW = 1.5 and 2.5 are respectively needed to stabilize the fakir carpet state. The corresponding values of (rW)th calculated from eq 17 are instead 2.0 and 2.92, respectively. We observe that these values are slightly larger than 1.5 and 2.5 reported in ref 62. However, this is expected since eq 17 is a criterion for robust superhydrophobicity and not simply a criterion for the occurrence of the Cassie−Baxter state. Now, assuming θY = 108° as in our case, the criterion for robust superhydrophobicity requires rW > (rW)th = 3.24. Then, 605


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carpet state is strongly stabilized and the drop cannot spread on the substrate, unless a significant amount of work is provided by an external force: a robust superhydrophobic surface is obtained. Interestingly, we show that ⟨∇h2⟩ is related by a one-to-one relation to the Wenzel roughness parameter rW, so that from a geometric point of view, both ⟨∇h2⟩ and rW measure the same statistical property of the surface. In terms of rW, the criterion for robust superhydrophobicity becomes rW > (rW)th = 1/(−cos θY). Interestingly, this threshold value is identical to the one that would be obtained in pure Wenzel regime to guarantee perfect superhydrophobicity (contact angle θa = 180°). We show that this criterion can be exploited even in the case of randomly rough surface when the system is in a mixed Cassie−Wenzel state. The criterion allows us to tune the statistical properties of the roughness and design very robust superhydrophobic randomly rough surfaces.

C( −kqL) = C(kqL) = Ck =

Using eq 22 and observing that C0 = ⟨h21⟩ δ(0)/4, one obtains ⟨hk2⟩ = ⟨h12⟩k−(2H + 1)

m0 = ⟨h(x)2 ⟩ =

(21)

(22)

where qL = 2π/L, the symbol ⟨⟩ stands for the ensemble average linear operator, and H is the Hurst exponent. It is related to the fractal dimension Df = 2 − H. We observe that, because of translational invariance, the autocorrelation function ⟨h(x′)h(x′ + x)⟩ satisfies the relation ⟨h(x′)h(x′ + x)⟩ = ⟨h(0) h(x)⟩. Equation 22 shows that only three parameters are needed to fully characterize the PSD of the surface. We have utilized a periodic profile with Fourier components up to the value qM = MqL. The amplitudes hk and the phases ϕk of the harmonic terms (see eq 1) need to be determined in such a way that the resulting profile is Gaussian with the PSD given in eq 22. First, observe that, in order to satisfy the translational invariance of the profile statistical properties, it is enough to assume that the random phases ϕk are uniformly distributed on the interval [−π, π]. In such a case, the autocorrelation function takes the form

k=1

⟨hk2⟩ cos(kqLx) 2

M

m0 = ⟨h2(x)⟩ = 2qLC0 ∑ k−1 − 2H k=1

M

∑ ∫ dx k=1

m2 = ⟨h′2 (x)⟩ = ⟨(∇h)12D ⟩ =

k=1

k=1

4

(29)

Interestingly, we observe that since 0 < H < 1 the sum ΣM k=1 k1−2H does not converge as M is increased, i.e., as more and more spectral components are included in the rough profile. However, we also observe that, when H > 0.5, the slope qkhk = 2(q3LC0)1/2 k(1−2H)/2 of each single Fourier component of the rough profile decreases as k is increased, with a faster decrease for higher values of H. Since real surfaces most of the time present Hurst exponents larger than 0.7, we may say that for real surfaces the shortest-wavelength components of the roughness are smoother than long-wavelength components.

(23)

■

APPENDIX B: AUTOCORRELATION FUNCTION FOR LIQUID−SOLID CONTACT AREAS To investigate the statistical structure of the liquid−solid composite interface, we make use of the autocorrelation function of the liquid−solid contact areas. This quantity is defined in terms of contact characteristic function χ(x)

⟨hk2⟩ cos(kqLx) e−iqx 2

∑ 1 [⟨hk2⟩δ(q − kqL) + ⟨hk2⟩δ(q + kqL)]

∫ q2C(q) dq = 2qL3

M

C0 ∑ k1 − 2H

M

=

(28)

and the second moment of the PSD

Now, we need to calculate the quantities ⟨h2k ⟩. To this purpose, let us calculate the PSD of the periodic profile given in eq 1. By using the definition, we get 1 C(q) = 2π

(27)

i.e., we assume that hk = 2(qLCk) . In the above relations, we 67 have used δ(q = 0) ≈ q−1 L = L/(2π). It can be shown that eq 27 also guarantees that the random profile h(x) is Gaussian. The three parameters employed to characterize the statistical properties of the surface are the Hurst exponent H, the zero-th order moment of the roughness PSD

⎛ |q| ⎞ ⎟⎟ ⎝ L⎠

M

(26)

k=1

1/2

∫ ⟨h(x′)h(x′ + x)⟩ e−iqx = C0⎜⎜ q

∑

M

∑ k−(2H + 1)

p(hk ) = δ(hk − 2 qLCk )

−(2H + 1)

⟨h(x′)h(x′ + x)⟩ =

⟨h12⟩ 2

Therefore, if one knows the zero-th order moment m0 = ∫ C(q) −(2H+1) dq = 1/2 ⟨h21⟩ ΣM of the power spectral density, i.e., k=1 k the mean square value of the profile heights m0 = h2rms = ⟨h(x)2⟩, one can calculate ⟨h21⟩ and therefore all other quantities ⟨h2k⟩. However, to completely characterize the rough profile we still need the probability distribution of the amplitudes hk. There are several choices; however, the simplest assumption, as suggested by Persson et al. in ref 67, is that the probability density function of hk is just a Dirac delta function centered at [4Ck/δ(0)]1/2 ≈ 2(qLCk)1/2

In such a case, it can be shown that the power spectral density (PSD) is 1 2π

⟨h21⟩

Hence, the quantities can be determined once and the Hurst exponent of the surface are known. Now, observe that 2 from eq 23 ⟨h(x)2⟩ = ΣM k = 1 ⟨hk ⟩/2, and using eq 25 yields

APPENDIX A: ROUGH PROFILE GENERATION In order to carry out the simulations, we need to numerically generate a rough profile. We have opted for a fractal self affine geometry. For any self-affine profile h(x), the statistical properties are invariant under the transformation

C(q) =

(25)

⟨h2k⟩

■

x → tx ; h → t Hh

⟨hk2⟩ δ(0) 4

χ (x) = 1; x ∈ Ω

(24)

χ (x) = 0; x ∉ Ω

from which it follows that 606


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Observe that the average value of χ(x) is ⟨χ(x)⟩ = ϕ. The liquid−solid contact autocorrelation function is then R χ (x1 , x 2) = ⟨χ (x1)χ (x 2)⟩

(30)

Since the statistical properties of the rough surface are translational, invariant one obtains Rχ(x1, x2) = Rχ(x2 − x1) = Rχ(z), i.e., the liquid−solid contact autocorrelation function Rχ only depends on the coordinate difference z = x2 − x1. Moreover, one also has Rχ(z) = Rχ(−z). Notice that 1 L 1 = L

∫L χ(x1)χ(x1 + z) dx1

R χ (z ) =

=ϕ

∫Ω χ(x1 + z) dx1 1 |Ω|

∫ χ(x1 + z) dx1

= ϕ⟨χ (x1 + z)|x1 ∈ Ω⟩

(31)

where |Ω| is the projected liquid−solid contact area and the quantity ⟨χ(x1 + z)|x1∈Ω⟩ is the conditional expectation value of χ at given distance z from x1, if it is known that x1∈Ω (i.e., if it is known that that χ(x1) = 1). Interestingly, we observe that, for z = 0, Rχ(0) = ϕ, whereas for z → ∞, the quantities χ(x1) and χ(x1 + z) become uncorrelated and we obtain Rχ(z→∞) = ⟨χ(x1)χ(x1 + z)⟩ = ⟨χ(x1)⟩⟨χ(x1 + z)⟩ = ϕ2, i.e., ⟨χ(x1 + z)| x1∈Ω⟩ = ⟨χ(x1 + z)⟩ = ϕ. In the context of surface wetting, one of the most interesting properties of the liquid−solid contact autocorrelation function is that Rχ can be used to statistically estimate the size of the smallest liquid−solid contact spot. In fact, the correlation function Rχ(z) very weakly decays on length scales comparable to the size d of the smallest contact spot, whilst a rapid decay must be observed over distance |z| > d. Figure 8 shows the reduced correlation function Rχ/ϕ as a function of qLz in a semi-logarithmic diagram. The results are presented for different values of liquid pressure p, for m2 = 50 and Df = 1.2 (see Figure 8a); and for different values of fractal dimension Df, given m2 = 50 and p/(qLγLA) = 0.16 (see Figure 8b). The estimation of the size d of the smallest contact spot is calculated assuming that at z = d the quantity Rχ/ϕ has been subjected to a decrease of 10%. Interestingly, Figure 8a shows that the drop pressure does not affect significantly the quantity d, which therefore seems to be an intrinsic properties of both the substrate roughness and Young’s contact angle. Figure 8b shows, indeed, that, increasing the fractal dimension, nonnegligibly reduces the size d of the smallest contact spot. This is a consequence of the presence of very sharp surface structures at higher fractal dimensions.

Figure 8. Quantity Rχ/ϕ as a function of the dimensionless distance qLz in a log−linear diagram. Calculations have been carried out at different drop pressures and Df = 1.2 (a); at different fractal dimensions and fixed dimensionless pressure p/(qLγLA) = 0.16 (b). In all cases hrms = 10 μm and m2 = 50. The horizontal line is placed at Rχ/ϕ = 0.9 and its intersection with the curves allows to estimate the size of the smallest liquid−solid contact spot. Observe the strong influence of the fractal dimension: increasing Df determines a significant reduction of the smallest liquid−solid contact spot.

where A is the nominal (or apparent) contact area, x = (x,y) is the position vector in the mean plane of the rough surface, and ξ1 = ∂h/∂x, ξ2 = ∂h/∂y, and P(ξ1,ξ2) is the joint density probability distribution of ξ1 and ξ2. We observe that ξ1 and ξ2 are in general statistically correlated,68 i.e., ⟨ξ1,ξ2⟩ = m11 ≠ 0. However, for a isotropic surface the surface statistical properties are direction independent and m11 = 0. In this case, ξ1 and ξ2 are noncorrelated and ⟨ξ21⟩ = ⟨ξ22⟩ = 1/2 ⟨(∇h)2⟩ = m2 where m2 is the mean square slope of the profile obtained by cutting the surface along any direction, and therefore P(ξ1) = P(ξ2). As a consequence, the joint density probability distribution becomes

■

APPENDIX C: WENZEL ROUGHNESS PARAMETER FOR 2D ISOTROPIC ROUGH SURFACES In the case of 2D isotropic rough surfaces, the coefficient rW is rW =

=

1 A

∫

1 A

∫A

P(ξ1 , ξ2) = P(ξ1)P(ξ2) =

∫A 1+

1 + [∇h(x)]2 dx dy 2 ⎛ ∂h ⎞2 ⎛ ∂h ⎞ ⎜ ⎟ + ⎜ ⎟ dx dy ⎝ ∂x ⎠ ⎝ ∂y ⎠

= P(ξ1 , ξ2) 1 +

ξ12

+

ξ22

dξ1 dξ2

(32)

1 2πm2

⎛ ξ2 ⎞ ⎛ ξ2 ⎞ 1 exp⎜ − 2 ⎟ exp⎜ − 1 ⎟ 2πm2 ⎝ 2m2 ⎠ ⎝ 2m2 ⎠

⎛ ξ 2 + ξ2 ⎞ 1 1 ⎟ exp⎜ − 2 2πm2 2 m ⎝ 2 ⎠ ⎡ (∇h)2 ⎤ 1 = exp ⎢− ⎥ ⎣ ⟨(∇h)2 ⟩ ⎦ π ⟨(∇h)2 ⟩ =

(33)

(34)

(35)

then one obtains 607


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Article

∫ P(ξ1, ξ2)

2 ⟨(∇h)2 ⟩

=1 +

∫0

+∞

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1 + ξ12 + ξ22 dξ1 dξ2 ⎛ ξ2 ⎞ ξ 1 + ξ 2 dξ exp⎜ − 2 ⎟ ⎝ ⟨(∇h) ⟩ ⎠

π ⟨(∇h)2 ⟩1/2 exp(⟨(∇h)2 ⟩−1) erfc(⟨(∇h)2 ⟩−1/2 ) 2 (36)

where ξ = + = |∇h|. The value calculated in eq 36 coincides with the one found (by following a complete different approach) by Persson.59 Now, observe that for robust superhydrorepellence one needs ⟨(∇h)2⟩1/2 ≫ 1, hence the above expression simplifies as (ξ21

rW ≈

ξ22)1/2

π ⟨(∇h)2 ⟩1/2 2

(37)

Equation 37 allows a very simple calculation of the Wenzel roughness parameter rW.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

REFERENCES

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Role of Statistical Properties of Randomly Rough Surfaces in

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