Computation of constant mean curvature surfaces - NCSU COE People

Journal of Colloid and Interface Science 314 (2007) 184–198 www.elsevier.com/locate/jcis

Computation of constant mean curvature surfaces: Application to the gas–liquid interface of a pressurized fluid on a superhydrophobic surface E.J. Lobaton 1,2 , T.R. Salamon ∗ Mathematical and Algorithmic Sciences Research Center, Bell Laboratories, Alcatel-Lucent, 600 Mountain Avenue, Murray Hill, NJ 07974, USA Received 4 January 2007; accepted 4 May 2007 Available online 25 May 2007

Abstract The interface shape separating a gas layer within a superhydrophobic surface consisting of a square lattice of posts from a pressurized liquid above the surface is computed numerically. The interface shape is described by a constant mean curvature surface that satisfies the Young–Laplace equation with the three-phase gas–liquid–solid contact line assumed pinned at the post outer edge. The numerical method predicts the existence of constant mean curvature solutions from the planar, zero curvature solution up to a maximum curvature that is dependent on the post shape, size and pitch. An overall force balance between surface tension and pressure forces acting on the interface yields predictions for the maximum curvature that agree with the numerical simulations to within one percent for convex shapes such as circular and square posts, but significantly over predicts the maximum curvature for non-convex shapes such as a circular post with a sinusoidal surface perturbation. Changing the post shape to increase the contact line length, while maintaining constant post area, results in increases of 2 to 12% in the maximum computable curvature for contact line length increases of 11 to 77%. Comparisons are made to several experimental studies for interface shape and pressure stability. © 2007 Elsevier Inc. All rights reserved. Keywords: Superhydrophobic surface; Pressure stability; Interfacial shape; Constant mean curvature surface; Lagrangian evolution equation; Static contact line; Overall force balance

1. Introduction There has been substantial recent interest in so-called superhydrophobic surfaces, which utilize a combination of chemical treatment and local roughness to increase the hydrophobicity, i.e., decrease the wettability, of a surface. An example of a naturally occurring superhydrophobic surface is the lotus leaf, which is highly water repellent and is revered for its ability to remain pristine even after immersion in muddy water [1]. Recent advances in microfabrication techniques have allowed the creation of a variety of man-made superhydrophobic surfaces with precisely controlled local roughness and which exhibit * Corresponding author.

E-mail address: [email protected] (T.R. Salamon). 1 Present address: Department of Electrical Engineering and Computer Sci-

ence, University of California, Berkeley, CA 94720, USA. 2 Bell Labs Graduate Research Fellowship Program award recipient. 0021-9797/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2007.05.059

such interesting properties as decreased wettability and reduced friction in laminar flows. These characteristics make superhydrophobic surfaces a potential enabling technology for a variety of applications including microfluidics, lab-on-a-chip devices, self-cleaning surfaces and drag-reduction. Superhydrophobic surfaces achieve most of their hydrophobicity not through surface chemistry but rather via microscale surface roughness. Cassie and Baxter [2] and Wenzel [3] were the first to provide a theoretical explanation for the wettability properties of rough and porous surfaces based on modifications to Young’s equation [4], which describes the equilibrium contact angle θ0 of a liquid in contact with a smooth, solid surface cos θ0 =

γsg − γsl , γlg

(1)

where γsg , γsl and γlg , respectively, are the surface tensions, or more appropriately, interfacial energies, of the solid–gas, solid–liquid and liquid–gas interfaces. Cassie and Baxter’s [2]

E.J. Lobaton, T.R. Salamon / Journal of Colloid and Interface Science 314 (2007) 184–198

equation cos θCB = f (1 + cos θ0 ) − 1

(2)

describes the contact angle θCB of a liquid in contact with the top of a composite surface consisting of a porous or rough solid with air contained within the roughness features present in the surface, where f is the area fraction of fluid in contact with the solid portion of the composite surface and θ0 is the contact angle appearing in Young’s equation. Wenzel’s [3] equation describes the contact angle θW of a liquid that completely fills the surface features present on a rough solid cos θW = r cos θ0 ,

(3)

where r is a roughness factor, often taken to be the total area of a rough surface relative to its projected area. Equations (2) and (3) predict that for a hydrophobic surface (θ0 > 90◦ ), increasing the roughness factor r or decreasing the area fraction f increases the magnitude of the contact angle. More detailed theoretical studies [5–8] based on minimization of the total energy of a droplet in the shape of a spherical cap on a superhydrophobic surface include such additional effects as droplet size, droplet shape and partial wetting of the fluid into the porous surface. Experimental work examining liquid droplets on superhydrophobic surfaces are numerous [8–18] and include: contact angle dependence on the method of droplet deposition on a superhydrophobic surface [8]; effect of droplet pressure on macroscopic contact angle measurements for a fluid in contact with a superhydrophobic surface consisting of carbon nanotubes [18]; effect of post shape and pitch on advancing contact angle behavior [9]; use of electric fields for dynamic tuning of wettability [10]. We highlight details of a few studies that are relevant to the work presented here. Journet et al. [18] examined the effect of pressure on the receding contact angle for an evaporating water droplet placed between a silanized glass plate and a superhydrophobic surface consisting of carbon nanotubes, where the pressure differential across the air/water interface is due to surface tension forces arising from the droplet curvature. Journet et al. [18] observed that the receding contact angle for a water droplet in contact with a carbon nanotube surface with typical tube diameters ranging from 50 to 100 nm and tube-to-tube spacings of 100 to 250 nm was relatively insensitive to droplet pressures ranging up to 10 kPa, implying that the composite interface formed at the water/nanotube surface is maintained in the Cassie state. This is in contrast to experimental results of Quéré [14], who observed a decrease of the contact angle and significant contact angle hysteresis for pressures above 0.2 kPa for textured surfaces with 2 micron heights and spacings, thereby implying wetting of the fluid into the structured surface. Journet et al.’s [18] and Quéré’s [14] results demonstrate that small surface feature sizes are necessary to withstand large pressure differentials across the gas–liquid interface. Öner and McCarthy [9] examined the effect of post shape on the advancing and receding contact angle behavior on silicon surfaces prepared by photolithography and hydrophobized

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using silanization reagents. They observed that changing the post shape from square to indented square, star or staggered rhombus shape did not affect the advancing contact angle, but resulted in increases of up to 22◦ in receding contact angle. Öner and McCarthy [9] attributed these changes to more contorted and longer contact lines. Öner and McCarthy’s results suggest that post shape plays an important role in determining certain aspects of the wettability properties of superhydrophobic surfaces, in their case the receding contact angle behavior. In Section 4 we explore the effect of post shape on the pressure stability of superhydrophobic surfaces. Application of superhydrophobic surfaces to confined and unconfined flows is of significant interest due to the potential for drag reduction relative to a conventional smooth surface, particularly in the laminar flow regime. Several studies [19–24] have demonstrated drag reduction in the range of 10–40% for the confined laminar flow of water in pipes and microchannels with superhydrophobic walls. These superhydrophobic surfaces consist of arrays of posts [20] or grooves [19–24] that are constructed using microfabrication [20–24] or chemical coating [19] techniques and with typical roughness dimensions ranging from fractions of a micron [23] to tens of microns [20–22]. Gogte et al. [25] observed drag reduction of 10% and larger for unconfined flow past a hydrofoil in a water tunnel where the hydrofoil has roughness feature sizes on the order of 10 µm and a superhydrophobic coating. The authors of these studies attribute the drag reducing properties of these surfaces to air that is trapped within the superhydrophobic surface and which decreases the effective contact area of the channel wall with the fluid. The experiments of Ou et al. [20] and Choi et al. [23] are of particular relevance to the work presented here. Ou et al. [20] measured pressure drop reductions of up to 40 percent for the flow of water in a 127 mm high × 2.54 mm wide × 50 mm long microchannel with a superhydrophobic surface consisting of a square array of 30 × 30 µm square posts with post-to-post pitches ranging from 45 to 180 µm that are etched into silicon and coated with silane on the bottom channel wall and a glass slide on the top channel wall. Measurements of the shape of the air/water interface using a confocal surface metrology system showed increasing deflection of the interface into the intervening space between posts with increasing flow rate, while the contact line remained pinned at the edge of the top surface of the post in the Cassie state. Choi et al. [23] experimentally showed that it was necessary to use sub-micron roughness feature sizes to maintain the meniscus in the superhydrophobic state for hydrodynamic pressures approaching one atmosphere. Choi et al.’s [23] work is important because it is the first to show drag reduction in confined flows using nanometer size surface roughness features. Semi-analytical [20,26–28] and numerical [21,24,29] studies have focused on quantifying the magnitude of drag reduction for flow in pipes and channels with superhydrophobic walls. These studies assume the flow to be laminar and steady while the superhydrophobic walls are typically modeled as containing no-slip regions corresponding to the solid roughness features of the surface and shear-free regions that correspond

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to a low-viscosity air layer that is trapped within the surface. Estimates for the magnitude of drag reduction are in reasonable agreement with experimental values for no-slip and shear-free region sizes that are consistent with experiment, and support the hypothesis that the low-viscosity layer is responsible for the drag-reduction properties of these surfaces. One drawback to all of the aforementioned simulation work is that the air–liquid interface is idealized as a flat surface that is in the plane corresponding to the top of the superhydrophobic surface features. Such an approximation is expected to be valid for large surface tension fluids at low hydrodynamic pressures within the system, e.g., at low values of the capillary number Ca ≡ ηU/γ , where η is the fluid viscosity, U is a characteristic velocity of the flow, and γ is the gas–liquid surface tension. The experimental results of Ou et al. [20] coupled with the simulation results presented in this paper and the recent study of Zheng et al. [30], described below, clearly suggest that the interface deformation can become appreciable, and that including such effects in hydrodynamic models may be necessary for a quantitative comparison of modeling with experiment. Zheng et al. [30] have addressed the important issue of interface deformation by using a finite element method to compute the shape of the gas–liquid interface for a statically pressurized fluid in contact with a superhydrophobic surface consisting of a square array of square posts. The interface shape is governed by the Young–Laplace equation [4,31], while the gas–liquid– solid contact line is assumed to satisfy either a local pinning condition or contact angle condition depending on the magnitude of the local contact angle relative to the side surface of the post. For the case where the local contact angle is smaller than an assumed advancing contact angle associated with the fluid, the contact line remains pinned along the top, outer edge of the square post. For the case where the computed local contact angle exceeds the advancing contact angle, the pinning condition is relaxed and the interface is assumed to intersect the post side with an angle equal to the advancing contact angle. Zheng et al. [30] also applied a force balance analysis to obtain an analytical expression for the maximum pressure sustainable by a square array of square posts. Zheng et al. [30] used the finite element and force balance results to estimate the maximum sustainable pressure that a fluid on a superhydrophobic surface with particular geometric properties is able to maintain. They also introduce the notion of the pillar slenderness ratio and use this ratio to characterize the stability and transition from the Cassie–Baxter to the Wenzel wetting state. Although Zheng et al.’s [30] work is extremely important as it is the first paper to address the role of interface shape and local wetting behavior on the pressure stability of superhydrophobic surfaces, several assumptions used in their analysis deserve mention. First, although the notion that there exists a range of admissible static contact angles θstatic , and hence equilibrium surface shapes, bounded by the receding θrec and advancing θadv contact angles, e.g., θrec < θstatic < θadv , has been well established [32] for fluids on smooth surfaces, it is not clear that Zheng et al.’s [30] prescription for the con-

Fig. 1. Schematic of a liquid on a superhydrophobic surface.

tact line behavior applies locally at surfaces with roughness features such as sharp corners. The work of Patankar [7], He et al. [8] and Bico et al. [33] examining metastability of fluids on superhydrophobic surfaces suggests that the energy barrier separating equilibrium states and mechanisms for state transition also play a critical role in describing such wetting phenomena. Second, Zheng et al.’s [30] force balance analysis relies on an implicit assumption that the local contact angle is uniform around the circumference of the post. The simulation results presented in this paper show that this is a reasonable assumption for circular posts, but not the square posts studied in their work. Third, Zheng et al.’s [30] finite element formulation restricts their analysis to interfacial shapes that can only be represented as a single-valued function W (x, y) of the physical coordinates (x, y) associated with the plane of the superhydrophobic surface. The numerical method presented in this paper is more general and allows, for example, the calculation of re-entrant shapes, where the surface can no longer be described as a single-valued function of the underlying physical coordinates. The outline of this paper is as follows. In Section 2 the equations and boundary conditions governing the static shape of the gas–liquid interface for a fluid in contact with a superhydrophobic surface consisting of a square array of posts are presented. The numerical method used to solve for constant mean curvature surfaces is presented in Section 3. In Section 4 results are presented examining the effects of post size, post shape and post-to-post pitch on the interface shape and local contact angle at the post edge. The validity of a simple relation for the maximum admissible curvature/pressure difference across the interface based on a balance of surface tension and pressure forces is presented, along with application of the numerical and analytical calculations to several experimental studies of fluids on superhydrophobic surfaces. 2. Problem formulation A schematic of the interfacial shape problem is shown in Fig. 1. A liquid at pressure pliq is in contact with a superhydrophobic surface consisting of a square array of circular posts of diameter D and post-to-post pitch L. A gas layer at pressure pgas occupies the intervening spaces between the posts and below the liquid. The analysis is restricted to a computational unit cell consistent with the symmetry properties present in the square array of circular posts (see Fig. 2). The shape of the gas– liquid interface satisfies the Young–Laplace equation [4,31] 2γ κmean = p,

(4)


Fig. 2. Array of circular posts and domain of computation Ω (- - -).

where γ is the liquid–gas surface tension, κmean is the mean curvature, and p ≡ (pliq − pgas ) is the pressure difference across the interface. The interfacial shape problem is specified by Eq. (4) and appropriate boundary conditions along the edges of the computational unit cell shown in Fig. 2. Along the dashed lines denoted by Γsym the interface shape is assumed to be symmetric. The liquid is assumed to completely wet the top surface of the posts, such that the intersection of the interface with the post, or equivalently the location of the gas–liquid–solid contact line Γcl , occurs at the outer edge of the top surface of the post. This description of the contact line position, often referred to as the Cassie state [2], is consistent with experimental results [20] and with static equilibrium analysis [8] based on interfacial energies that show the pinning of interfaces at sharp corners often give rise to local energy minima. Note that for some of the computational results presented in the following the local contact angle exceeds 90◦ relative to the horizontal, even though the average contact angle around the post remains less than 90◦ . Such a configuration is physically realizable for the case where the post resembles a nail-head shape (see Fig. 4b), where the contact line is pinned at the lower, outer boundary of the nail-head. Nail-head and mushroom cap shapes have been proposed by several authors [33–35] as potential post geometries allowing the creation of superhydrophobic surfaces out of hydrophilic materials. Such nail-head geometries can be created using standard silicon processing techniques. For example, Krupenkin et al. [36] used 248 nm photolithography and dry reactive ion etching to create regular arrays of oxide dots on a silicon wafer with an initial 200 nm oxide layer. Post structures were then etched into the Si using the Bosch process, which uses two separate steps to create a vertical or anisotropic etch. The first step uses SF6 to etch the Si and the second uses C4 F8 to deposit a protective layer of flouropolymer. This process results in a nearly vertical sidewall, but with a certain amount of undercut relative to the mask. Modification of the etch recipe allows tailoring the degree of the undercut. Since a high degree of selectivity exists between the Si and oxide the silicon is undercut with little loss of the oxide layer, which corresponds to the top of the nailhead.

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Fig. 3. Force diagram for circular post case. Hollow arrows with solid stems denote surface tension forces and hollow arrows with dashed stems denote pressure forces.

Equation (4) can be recast into the form κmean =

p , 2γ

which for a constant pressure difference across the interface yields a constant curvature for the surface. Solutions of the interfacial shape problem are thus equivalent to finding surfaces with constant mean curvature that satisfy the aforementioned symmetry and pinning conditions of the interface. max ) and pressure difference 2.1. Maximum curvature (κmean max (p )

In this section a total force balance is used to bound the maximum curvature and pressure difference that a particular nanopost configuration is able to support. Similar analyses have been used to obtain estimates for meniscus configurations and curvatures for wetting in porous media [37–42]. Consider the section of interface associated with the computational domain Ω shown in Fig. 3. External forces acting on this section of surface include surface tension forces (hollow arrows with solid stems) that act along the interface boundary Γ and the force associated with the pressure difference (hollow arrows with dashed stems) that act normal to the interface. The equilibrium condition of no net force acting on this section of interface in the z-direction is expressed as (5) γ sin θcl ds + γ sin θcl ds = p cos θp dA, Γcl

Γsym

Ω

where θcl is the angle that the vector tangent to the interface and normal to the boundary forms with the xy-plane and θp is the angle formed by a vector normal to the interface and the z-direction. Noting that the symmetry condition along the boundary Γsym implies θcl = 0 and that γ and p are constant, Eq. (5) simplifies to γ sin θcl ds = pAp , (6) Γcl

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Fig. 4. Advancing contact angles θ adv for posts with different shapes: (a) circular post; (b) circular post and nail-head top; and (c) tapered circular post with taper angle α. Note that the contact angle is defined relative to the dashed line, which is parallel to the surface over which the fluid would advance if it were to wet, and hence represents the appropriate reference frame for defining the advancing contact angle.

where Ap ≡ Ω cos θp dA is the projected area of the surface onto the xy-plane. The maximum contribution to the surface tension forces acting along the contact line in the z-direction occurs when the interface intersects the contact line vertically (θcl = 90◦ ). This results in the following upper bound on the maximum pressure difference that the interface can maintain p max = γ

S (Arc-length) ≡γ , Ap (Projected interface area)

(7)

where S is the arc-length of the section of the boundary associated with the contact line. Substitution of Eq. (7) into the Young–Laplace equation (4) results in the following upper bound for the maximum mean curvature of the interface max κmean =

S (Arc-length) ≡ . 2Ap 2(Projected interface area)

(8)

Note that Eq. (8) contains purely geometric properties of the nanostructured surface and is independent of the gas–liquid surface tension γ . Equation (7) yields the following estimates for the maximum pressure difference for a surface consisting of a square array of circular posts of diameter D and post-to-post pitch L p max = γ

πD , L2 − π4 D 2

(9)

a surface consisting of a square array of square posts of width W and post-to-post pitch L 4W p =γ 2 , L − W2 and a surface consisting of grooves of width w max

(10)

2 (11) . w Equations (8)–(11) will be used in Section 4 as a check on the numerical simulations and to compare with experimental studies of superhydrophobic surfaces. Knowledge of the advancing contact angle θ adv for a particular gas–liquid–solid system can be used in conjunction with Eq. (6) to obtain a tighter bound for the maximum pressure sustainable for a particular superhydrophobic surface, although this requires making several assumptions: (i) the local contact angle is fairly uniform around the post periphery. The results

p max = γ

presented in the this paper suggest that this approximation is only applicable to cylindrically-symmetric posts, and we thus restrict our analysis to this case; and (ii) the local wetting behavior near the sharp corner associated with the post is welldescribed by the macroscopically observed advancing contact angle and issues of metastability and transition [7,8,33] do not play a substantial role. As mentioned earlier, the notion that the advancing contact angle provides an upper bound for static contact angles on smooth surfaces is well accepted [32], although it is not clear that this behavior applies locally at surfaces with roughness features such as sharp corners. With these assumptions one can arrive at the following forms for the maximum sustainable pressure for the various cylindrically symmetric geometries depicted in Fig. 4 in terms of the advancing contact angle on a flat substrate θ adv p max = γ p

max

=

S sin θ adv − π/2 , Ap

circular post,

γ ASp sin(θ adv ),

for θ adv π2 ,

γ ASp ,

for θ adv > π2 ,

nail head shaped post, p max = γ

(12)

(13)

S sin θ adv − π/2 − α , Ap

post with taper angle α and with α > 0,

(14)

where S = πD, Ap = L2 − π4 D 2 and D is the diameter of the post at the gas–liquid–solid contact line. Several points are worth noting about Eqs. (12)–(14). First, for the case of a hydrophilic surface where θ adv < π/2, the pressure stability analysis suggests that the cylindrical and tapered posts offer no pressure stability. Only the nail-head configuration is expected to be able to support any static pressure load. Second, for the case of hydrophobic liquids where θ adv > π/2, the nail-head configuration offers the greatest pressure stability relative to the cylindrical and tapered posts. Furthermore, for the nail head case the pressure stability is expected to be independent of the degree of hydrophobicity, e.g., the magnitude of θ adv plays no role as the maximum sustainable pressure corresponds to the interface intersecting the bottom of the nail head surface vertically.


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Details of the calculation of the normal vector and mean curvature used in Eq. (16), along with results demonstrating the convergence and accuracy of the numerical method are presented in the supplementary material. 4. Results

Fig. 5. 3D view of the initial conditions for the domain Ω.

3. Numerical method 3.1. Lagrangian evolution process The solution method presented in this section is motivated by models of viscous sintering [43,44]. In the sintering process a material such as a ceramic powder is heated to elevated temperatures which allows the material to flow. The ceramic powder then attempts to lower its surface energy through the coalescense of particles to form larger entities with reduced surface area. This process is driven by surface tension forces, which act through the mean curvature of the particle-gas interface and cause adjacent particles to coalesce into a single larger particle with smaller surface area, and hence lower surface energy. Surfaces with constant mean curvature are obtained by solving the following Lagrangian evolution problem for the location of points x along the interface x ) , in Ω and Γsym , x˙ · n = κ0 − κmean ( (15) where x˙ is the time derivative of the position vector, n is the unit normal vector to the interface, κ0 is the desired mean curvax ) is the mean curvature of the current interface ture, and κmean ( shape at position x . The initial condition for Eq. (15) is taken to be a flat interface that is in the plane of the top surface of the posts and which corresponds to the zero mean curvature solution (see Fig. 5). At steady state the surface points are no longer changing, e.g., x˙ = 0. This implies that the interface satisfies the constant curvature condition x ) = κ0 , κmean ( at all points x along the interface. It is important to reiterate that Eq. (15) is used solely for the computation of constant mean curvature surfaces, and that it has no physical basis in describing the dynamics of interface evolution for liquids in contact with superhydrophobic surfaces. The discrete analog to Eq. (15) is obtained by choosing an appropriate set of points { xi }, hereafter referred to as tracking points, that follow the evolution of the surface, i.e., xi ) , x˙i · ni = κ0 − κmean ( (16) where ni and κmean ( xi ) are the unit normal vector and mean curvature at the ith node. These points also serve as the nodes of a surface triangulation that is used for computing the normal vector and mean curvature appearing in Eq. (16). A sample discretization of the initial planar surface is shown in Fig. 5.

The outline of this section is as follows. The characteristic features of constant mean curvature surfaces describing the gas–liquid interface for a pressurized fluid in contact with a superhydrophobic surface consisting of a square array of circular posts are presented in Section 4.1. The effect of post shape on the maximum admissible curvature is explored in Section 4.2. Application of the numerical and analytical calculations to experimental studies of fluids on superhydrophobic surfaces is presented in Section 4.3. 4.1. Circular posts In this section results are presented for a superhydrophobic surface with feature sizes characteristic of the surfaces studied by Krupenkin et al. [10]. In Fig. 6a the interface shape for a superhydrophobic surface consisting of circular posts with D = 0.35 µm, L = 2 µm and a value of κ0 = 0.141 µm−1 is shown. Fig. 6b shows the shape of the surface spanning the square array associated with four adjacent posts while Fig. 6c shows the surface shape around an individual post. Figs. 6b and 6c are obtained by replicating the solution on the computational domain Ω shown in Fig. 6a based on the symmetry properties of the superhydrophobic surface. Note that the surface exhibits its maximal deflection at the center of the square array formed by four adjacent posts (see Fig. 6b), while the surface exhibits a rapid decrease in height adjacent to the post with a more gradual change away from the post surface. The contact angle θcl (φ), defined as the angle that the vector that is tangent to the surface and normal to the contact line forms with the horizontal (xy) plane (see Fig. 3), is plotted in Fig. 7 as a function of the angle around the post for the solution shown in Fig. 6. Note that the contact angle is fairly constant around the post, exhibiting a variation within ±2◦ of its average value. This is in contrast to the results of Section 4.2, where significant variation of the local contact angle is observed for non-circular posts. The variation in contact angle for the circular post case is due to long-range lattice effects in the superhydrophobic surface and not the post shape, which is symmetric with respect to angular position around the post. In Fig. 8a the mean contact angle θ¯cl , defined as Γ θcl (s) ds θ¯cl = cl (17) , Γcl ds where s is the arc-length around the post boundary along the contact line, is plotted as a function of the mean curvature for circular posts with D = 0.35 µm and three different values of the post-to-post pitch L. The mean contact angle monotonically increases with increasing mean curvature, approaching a maximum value close to 90◦ for a post-to-post pitch of L = 0.9 µm

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(a)

(b)

(c)

Fig. 6. (a) Interface shape in the computational domain Ω for a superhydrophobic surface with circular posts with D = 0.35 µm, L = 2 µm and κ0 = 0.141 µm−1 , (b) interface shape between four adjacent posts, and (c) interface shape around a single post.

most cusp-like, increase in slope as it approaches the maximum curvature for which solutions exist. In Fig. 8b the normalized surface area An , defined as An (κ0 ) =

A(κ0 ) , A(0)

(18)

where A(κ0 ) is the area of the surface at a mean curvature value of κ0 and A(0) is the area of the undeflected (planar) surface, is plotted as a function of the mean curvature. As the post-topost pitch increases the range of normalized surface area values decreases from 1 An 1.183 for L = 0.9 µm to 1 An 1.02 for L = 4 µm. This is consistent with the observation that the primary surface deformation is localized near the post and that for larger post-to-post pitches the majority of the surface is becoming more planar due to the decrease in the maximum curvature sustainable by the geometry. In Fig. 9a the maximum surface deflection δz, defined as Fig. 7. Contact angle θcl (φ) as measured from the horizontal as a function of angle φ around the post for the solution shown in Fig. 6. Note that φ = 0◦ corresponds to orientation along the +x-axis while φ = 90◦ corresponds to orientation along the +y-axis.

with slightly lower maximum values for L = 2 and 4 µm. For each of the curves depicted in Fig. 8a no steady solutions to Eq. (16) were found to exist above a critical mean curvature value κ0max . This maximum mean curvature agrees to within max in one percent with the value predicted by Eq. (9) for κmean Section 2.1. This suggests that the simple force balance is a reliable predictor of the maximum curvature for regular arrays of circular posts. As expected, decreasing the post-to-post pitch allows computation to higher values of mean curvature. For all three values of L the mean contact angle exhibits a sharp, al-

δz =

max

i=1,...,Nnodes

zi ,

(19)

is plotted as a function of the mean curvature. The maximum interface deflection ranges from approximately 0.5 µm for L = 4 µm, decreasing to 0.25 µm for L = 0.9 µm. The deflection at the maximum curvature κ0max appears to obey the following scaling with respect to the post-to-post pitch L δz κ0max ∼ O L+1/2 . (20) In Fig. 9b the decrease in the gas-phase volume δV , defined as δV =

Vs , V0

(21)


(a)

191

(a)

(b) Fig. 8. (a) Mean contact angle θ¯cl as a function of mean curvature κ0 , and (b) normalized surface area An as a function of mean curvature κ0 for circular posts with D = 0.35 µm and L = 0.9, 2 and 4 µm.

where Vs is the volume occupied by the surface below the xyplane, V0 ≡ H A(0) is the gas-phase volume below the computational domain Ω for the case of an undeflected surface and H is the height of the posts, is plotted as a function of the mean curvature for circular posts with height H = 7 µm, a representative value for the study of Krupenkin et al. [10]. The maximum gas-phase volume decrease ranges from approximately 6.2% for L = 4 µm to about 2.6% for L = 0.9 µm. This suggests that for a superhydrophobic surface consisting of circular posts of diameter D = 0.35 µm, pitch ranging from L = 0.9 to 4 µm, and containing a confined gas that is initially at 1 atm pressure when the gas–liquid interface is in the undeflected (planar) configuration, surface deformation will result in pressure increases within the gas-phase of up to 6200 Pa. Note that this does not account for any re-equilibration of the gas-phase due to dissolution of the gas into the liquid. Such an analysis is beyond the scope of this paper.

(b) Fig. 9. (a) Depth of surface δz as a function of mean curvature κ0 , and (b) decrease in gas-phase volume δV as a function of mean curvature κ0 with D = 0.35 µm, L = 0.9, 2 and 4 µm and a post height of H = 7 µm.

4.2. Effect of post shape The force balance analysis presented in Section 2.1 suggests max that one method for increasing the maximum curvature κmean that a superhydrophobic surface is able to support is to increase the arc-length of the contact line where the fluid wets the post. In this section the effect of increasing the contact line length is explored by varying the shape of the post while maintaining constant post area. The particular post shapes considered are shown in Fig. 10. The parameterization for the shapes are as follows R(φ) = R0 + R cos(nφ), 0 φ 2π, for shapes A, C, D and E

(22)

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Fig. 10. Shapes considered for the posts and their normalized arc-length as defined by Eq. (24).

Table 1 Values for R0 , R, n and sratio for formulas (22) and (23) for the shapes shown in Fig. 10 Shape

R0 (µm)

R (µm)

n

sratio

A B C D E

0.175 0.1935 0.1746 0.1733 0.1611

0 – 0.01746 0.03466 0.0967

0 – 8 8 4

1 1.11 1.14 1.47 1.77

and

π φ R(φ) = R0 cos φ − 1 + 2 × int , 4 π/2 0 φ 2π, for shape B,

Fig. 11. Average contact angle from horizontal θ¯cl as a function of mean curvature κ0 for shapes A, B, C, D and E with L = 2 µm and all post areas equal to post A. Table 2 Calculated maximum mean curvature κ0max and predicted maximum mean curmax as given by Eq. (9) for the results shown in Fig. 11 vature κmean

(23)

where R(φ) is the radius of the post as a function of the angular position φ around the post. In Table 1 values for R0 , R and n are given for the shapes shown in Fig. 10. Note that these values result in an area equivalent to that of the circular post with D = 0.35 µm. The quantity sratio , defined as Shape ds Γ sratio = cl (24) , Shape A ds Γ cl

which is the ratio of the arc-length along the boundary of the shape relative to the arc-length of the corresponding circular post (shape A), is also shown. In Fig. 11 the average contact angle θ¯cl is plotted as a function of mean curvature κ0 for the shapes A through E in Fig. 10 for L = 2 µm. As the value of sratio increases from 1 (shape A) to 1.77 (shape E) the average contact angle decreases for a fixed value of κ0 . The maximum curvature for which solutions exist increases with increasing sratio , from approximately a 2% increase in κ0max for shape A to a 12% increase in κ0max for shape E. In Table 2 the calculated values for κ0max are commax based on Eq. (8) pared with the theoretical predictions for κmean in Section 2.1. Note that the theoretical predictions significantly overestimate the maximum curvature for shapes B–E, which, in contrast to the circular shape A, are non-convex, i.e., it is possible to choose two points within the shape such that a straight line connecting the two points passes outside the shape [45]. In Fig. 12 the contact angle θcl (φ) is plotted as a function of angular position φ around the post for shapes D and E for the results shown in Fig. 11 for selected values of the mean curvature. Shapes D and E exhibit local maxima and minima in the contact angle that correspond to angular positions where the shape

Shape

Simulation, κ0max (µm−1 )

max Force balance, κmean (µm−1 )

A B C D E

0.141 0.144 0.144 0.148 0.158

0.141 0.157 0.161 0.207 0.250

parameterization function R(φ) exhibits local maxima and minima, respectively, e.g., local maxima occur when R (φ) = 0 and R (φ) < 0 and local minima occur when R (φ) = 0 and R (φ) > 0. Another important feature is that, in contrast to the circular post case, the local contact angle exceeds 90◦ in certain regions of the post. Note that in the problem formulation and numerical method outlined in Sections 2 and 3, respectively, the only constraint that is placed on the surface shape is that it is pinned along the contact line Γcl , and, as such, the numerical method admits solutions with contact angles larger than 90◦ . These results suggest two things. First, for non-axisymmetric shapes the fluid may begin to wet the post at a mean curvature value lower than that of the corresponding circular post with the same area. And second, a configuration where the post shape contains an undercut, similar to that of a nail-head or mushroom, as suggested by several authors [33–35], where the interface is pinned at the corner formed by the side and undercut of the nail head, may allow local contact angles that are larger than 90◦ . In Fig. 13 the contact angle θcl (φ) is plotted as a function of angular position φ around the post for shapes A, D and E for the results shown in Fig. 11 for κ0 = 0.14 µm−1 . Note that the circular post exhibits the smallest variation in contact angle, on the order of ±2◦ around its average value. In contrast shapes D and E exhibit contact angles that are significantly lower than


(a)

193

Fig. 14. Interface displacement as a function of distance y along the line x = 0 and for a square post of width W = 30 µm, L = 60 µm and κ0 = 0.00837 and 0.01193 µm−1 . Experimental data reproduced from Fig. 9 in Ou et al. [20].

shape A with the exception of positions near the local maxima in the shape parameterization function R(φ), as discussed earlier. 4.3. Comparison with experiments

(b) Fig. 12. Contact angle θcl (φ) as a function of angle around the post φ for varying values of κ0 for (a) shape D and (b) shape E for the results shown in Fig. 11.

Fig. 13. Contact angle θcl (φ) as a function of angle around the post φ for shapes A, D and E with L = 2 µm and κ0 = 0.14 µm−1 .

In this section we compare directly with experiments involving the flow of liquids in channels and pipes with superhydrophobic walls. We note that our approach neglects hydrodynamic stresses at the gas–liquid interface, which may play an important role in determining the shape and stability of the gas–liquid interface. Although such an effort is not covered by the scope of the present work, a more detailed analysis coupling fluid flow and interface deformation would shed important insight into the nature of these combined effects. 4.3.1. Experimental results of Ou et al. [20] In this section the numerical simulations are compared to Ou et al.’s [20] experimental measurements of the air–water interface shape using a confocal surface metrology system for flow in a microchannel that is 5 cm in length, 127 µm in height and with an upper wall consisting of a glass slide and a lower wall consisting of a superhydrophobic surface made of 30 µm square posts with 60 µm post-to-post pitch. In Fig. 14 the interface location z is plotted as a function of position y along the line x = 0 for square posts with W = 30 µm, L = 60 µm, and two values of κ0 . The values of κ0 are chosen such that the maximum computed displacement corresponds to the experimentally observed values for the experimental flow conditions, e.g., κ0 = 0.00837 µm−1 for Q = 300 ml/h and Pexp = 3200 Pa, and κ0 = 0.01193 µm−1 for Q = 420 ml/h and Pexp = 4500 Pa, where Q is the flow rate through the microchannel and Pexp is the pressure drop across the microchannel. Note that choosing the κ0 value to match the maximum displacement is necessary as the experimental

194


measurements of the gas-phase pressure are not reported. The computed shape of the interface is in good agreement with the experimental results of Ou et al. [20] for both sets of flow conditions. The simulations provide additional useful information as they can be used to infer the pressure difference across the air– water interface via Eq. (4), the Young–Laplace equation. For an air–water surface tension of γ = 72 dynes/cm this corresponds to a pressure difference across the meniscus of p = 1205 and 1718 Pa, respectively, for κ0 = 0.00837 and 0.01193 µm−1 . Assuming that the pressure gradient in the microchannel is linear with respect to axial position, and viscous stress contributions to the normal stress balance at the air–liquid interface are negligible, the pressure in the liquid phase at the air–liquid interface mid-way along the channel is given by 1 pliq = pexit + Pexp , (25) 2 where pexit is the pressure at the exit of the microchannel. Substitution of Eq. (25) into Eq. (4) results in the following form for the gas-phase pressure relative to the microchannel exit pressure 1 (pgas − pexit ) = (26) Pexp − 2γ κ0 . 2 Substitution of the experimentally reported values for Pexp and the simulation values of κ0 that match the experimentally observed maximum interface displacement into Eq. (26) results in a gas-phase pressure that is 395 and 532 Pa, respectively, above pexit for the flow conditions Q = 300 and 420 ml/h. Noting that the pressure at the microchannel exit is at least the ambient pressure of 1 atm, this suggests that, if the gas-phase volume is enclosed, pressurization of the gas-phase is on the order of 0.4–0.5% of an atmosphere. In Fig. 15a the maximum surface deflection is plotted as a function of mean curvature for square posts with W = 30 µm, L = 60 and 180 µm. The maximum surface deflection is in the range 0 δz 15 µm for L = 60 µm and 0 δz 37 µm for L = 180 µm. Note that this is two orders of magnitude larger than the interface deflections observed for the superhydrophobic surfaces studied in Section 4.1. The maximum curvature κ0max is also in excellent agreement with the estimates provided by Eq. (10) in Section 2.1. The agreement for the maximum computed curvature with the analytical estimates for the circular and square posts suggests that for convex post shapes, where a straight line connecting any two points within the shape always remains within the shape [45], the calculated maximum admissible curvature κ0max is well approximated by the algemax presented in Section 2.1. The maxibraic formulae for κmean mum curvature for the non-convex shapes studied in Section 4.2 does not appear to obey this analysis. In Fig. 15b the decrease in the gas-phase volume is plotted as a function of curvature for the results of Fig. 15a with posts of height H = 30 µm. The gas-phase volume decrease is in the range 0 δV 37% for L = 60 µm and 0 δV 75% for L = 180 µm. This suggests that for a superhydrophobic surface with appropriately chosen post size, pitch and height,

(a)

(b) Fig. 15. (a) Depth of surface δz and (b) gas-phase volume decrease δV plotted as a function of mean curvature κ0 for square posts with W = 30 µm, L = 60 and 180 µm and assuming a post height of H = 30 µm. Note that surface deflections larger than 30 µm have not been included in (b).

appreciable compression of an enclosed gas-phase is possible. For example, a 50% reduction of the gas-phase volume would result in a two-fold increase in the gas pressure relative to the undeflected state. If the initial gas-phase pressure is 1 atm, the compression effect due to deformation of the interface will result in a gas-phase pressure of 2 atm. Again, note that this analysis does not account for any potential reequilibration of the gas phase due to dissolution of the gas into the liquid. In Fig. 16 the contact angle θcl (φ) is plotted as a function of angular position φ around the post for W = 30 µm, L = 180 µm and various values of the curvature. The contact angle exhibits a maximum at the corner of the square post (φ = 45◦ ), consistent with the observations of Section 4.2 for non-circular posts. For curvature values larger than κ0 = 0.0016 µm−1 the contact angle near the post corner exceeds 90◦ .


4.3.2. Estimates for maximum pressure difference p max for various superhydrophobic surfaces In this section the analytical formulae for the maximum pressure difference p max for the analysis presented in Section 2.1 are used to provide estimates for the pressure stability of water in contact with various superhydrophobic surfaces used in the study of drag reduction and controlled wetting.

195

In Table 3 values of p max as predicted by Eqs. (9)–(11) for water on various superhydrophobic surfaces for the drag reduction experiments of Ou et al. [20], Watanabe et al. [19], Woolford et al. [21], Choi et al. [23], Davies et al. [24] and the controlled wetting experiments of Krupenkin et al. [10] are shown. max , which is defined as the maximum hydroThe quantity Pexp dynamic pressure that the superhydrophobic surface in the drag reduction studies is exposed to inlet max = pamb + Pexp , pliq

Fig. 16. Contact angle θcl (φ) as a function of angle around the post φ for a square post with W = 30 µm, L = 180 µm and varying values of κ0 .

(27)

inlet is the pressure at the inlet to the superhydrophowhere pliq bic section of the flow apparatus, is also shown. The values max for the experiments of Ou et al. [20] and Choi for Pexp et al. [23] correspond to data directly measured on their flow system and assuming that the exit of the microchannel is at ambient pressure, while the experiments of Watanabe et al. [19], Woolford et al. [21] and Davies et al. [24] are inferred from fully-developed laminar flow theory for the flow conditions max neglects visof the experiment. Note that although Pexp cous stress contributions to the normal stress balance at the air–liquid interface, it is expected to provide a reasonable order of magnitude estimate for the fluid stresses at this interface. Several points are worth noting about the results in Table 3. First, with the exception of the results of Choi et al. [23], the maximum pressure difference pmax for a working fluid such as water is limited to fractions of an atmosphere. To achieve

Table 3 Maximum pressure difference p max predicted by Eqs. (9)–(11) for various superhydrophobic surfaces used in studies of laminar drag reduction (a, b, c, d, e) and max corresponds to the maximum hydrodynamic pressure that the superhydrophobic surface is exposed wettability (f) and using a value of γ = 72 dynes/cm. Pexp to, as reported experimentally or inferred from fully-developed laminar flow theory for the reported flow conditions Experiment

Formula for pmax

Formula parameter values (µm)

(a) Ou et al. [20]—water flowing in a microchannel of length 5 cm and height H = 127 µm with a smooth upper wall and a lower wall patterned with a square array of square posts with size W and pitch L

Eq. (10)

W W W W

(b) Woolford et al. [21]—water flowing in a microchannel of length 7 cm and height H = 99 µm with upper and lower walls patterned with parallel ridges

Eq. (11)

(c) Watanabe et al. [19]—water flowing in a circular pipe of length 475 mm and a diameter of 6∗ or 12∗∗ mm with pipe walls coated with an acrylic resin containing grooves of size 10 µm

= 30, L = 45 = 30, L = 60 = 30, L = 90 = 30, L = 180

pmax (Pa)

max Pexp (Pa)

7680 3200 1200 274

200–1600 100–1400 100–1200 100–1200

w = 20 w = 30

7200 4800

1680–2120 1660–2760

Eq. (11)

w = 10 w = 10

14,400 14,400

(d) Choi et al. [23]—water flowing in a microchannel of length 4 mm and height ranging from 3 to 11 µm with upper and lower walls patterned with parallel ridges

Eq. (11)

w = 0.18

(e) Davies et al. [24]—water flowing in a microchannel of length 7 cm and height H = 80 µm with upper and lower walls patterned with parallel ridges

Eq. (11)

w = 20 w = 30 w = 35 w = 38 w = 39

(f) Krupenkin et al. [10]—water droplets on a superhydrophobic surface consisting of a square array of circular posts

Eq. (9)

D = 0.35, L = 0.9 D = 0.35, L = 2 D = 0.35, L = 4

800,000

7200 4800 4110 3790 3690 111,000 20,300 4980

58–389* 7–47**

10,100–101,000

5490–7500 4960–6840 4440–6320 3890–5980 3250–3930 Not applicable

196


pressure stability larger than an atmosphere Choi et al. [23] required sub-micron scale surface roughness features. Second, for nanostructured surfaces where the gas phase is in equilibrium with the ambient the interface experiences a pressure differenmax . This is clear from substitution of tial equal to precisely Pexp Eq. (27) into Eq. (4) max − (pgas − pamb ), 2γ κmean = Pexp

(28)

and noting that for the case pgas = pamb Eq. (28) reduces to max . 2γ κmean = Pexp

(29)

This indicates that surface tension alone is responsible for maintaining the gas–liquid interface in the superhydrophobic configuration. Third, the form of Eq. (28) suggests that to achieve max greater than p max pressurhydrodynamic pressures Pexp ization of the gas phase may be required, although the practicality of such an approach remains in question due to issues related to the solubility of air in water and associated degassing phenomena. Finally, for two of the experiments in Table 3 the predictions of the force balance analysis for p max suggest that surface tension alone is not able to support the gas–liquid interface: (i) for the case W = 30 µm and L = 180 µm in the study of Ou et al. [20] this corresponds to conditions where max > 274 Pa. The results of Section 4.3.1 indicate that dePexp formation of the interface causes a significant decrease in the gas phase volume that may pressurize the gas phase and provide such a restoring force. However, recent numerical simulations by one of the authors for the associated three-dimensional hydrodynamic problem [29] show that an interface configuration where the fluid wets into the post structure results in better quantitative agreement with the experimentally observed drag reduction than for a non-wetting configuration. This suggests that for the largest pitch case of L = 180 µm wetting of the superhydrophobic surface has occurred; and (ii) for all of the groove widths w in the study of Davies et al. [24] the pressure drop estimated from laminar flow theory is marginally in excess of that predicted by the force balance analysis. Davies et al. [24] noted that their experimentally observed drag reduction is consistently less than theoretical predictions, where the gas–liquid interface is assumed to be flat and in the plane of the top of the ridges. They suggest meniscus deformation and possible penetration into the cavity between intervening ridges may account for the dicrepancy in predicted and observed drag reduction. This is consistent with the aforementioned hypothesis that meniscus deformation and penetration into the grooves will result in a decrease in the gas-phase volume and hence may contribute the restorative force necessary to maintain the gas– liquid interface. 5. Conclusions A numerical method based on motion under mean curvature is presented for computing constant mean curvature surfaces. The method is applied to studying the shape of the gas–liquid interface separating a pressurized fluid from a gas layer contained within a superhydrophobic surface consisting of a square array of posts with constant post-to-post pitch. The interface

shape satisfies the Young–Laplace equation, which relates a balance of surface tension forces acting through the mean curvature of the interface with the pressure difference between the gas and liquid, and is assumed to be pinned at the gas– liquid–solid contact line located at the outer edge of the top surface of the post. Application of the method to the known analytical solution of a section of a circular arc connecting two infinite parallel ridges (see the supplementary material) demonstrates that the method is convergent and accurate. The method also accurately predicts the existence of a maximum curvature, which corresponds to the minimum radius cylinder that is able to span the gap between the parallel ridges, and above which no constant mean curvature solutions are computable (see the supplementary material). Application of the numerical method to superhydrophobic surfaces consisting of square arrays of posts also predicts the existence of constant mean curvature solutions starting from the planar, zero curvature solution up to a maximum curvature value κ0max that is dependent on the post shape, size and pitch. An overall force balance between surface tension acting along the contact line and the pressure differential across the interface yields an analytical formula for the maximum curvature that is dependent on the ratio of the contact line length to twice the interfacial area in the undeflected, planar state. The overall force balance analysis agrees to within one percent with the numerical simulations for convex shapes such as circular and square posts, but significantly overpredicts the maximum curvature for non-convex shapes such as a circular post with a sinusoidal perturbation added to the surface. The numerical simulations presented in this paper have also provided useful information about such interfacial properties as the average and local contact angle around the post, normalized surface area, maximum interface deflection and decrease in gas-phase volume as a function of mean curvature for a variety of post shapes, sizes and pitches. All of the aforementioned properties are monotonically increasing functions of the mean curvature, and exhibit a rapid, cusp-like behavior as the limiting curvature value κ0max is approached for a particular superhydrophobic surface. For all of the superhydrophobic surfaces studied here, the mean contact angle ranges from 0◦ for the planar, undeflected state, to values approaching 90◦ at the limiting curvature κ0max . Of particular importance are estimates for the interface deflection and the decrease in the gas-phase volume, as these have implications in terms of the stability of the interface. Large deflections may place the interface close to the bottom of the post and hence make the superhydrophobic surface more susceptible to wetting due to, for example, external mechanical vibrations. In contrast, large gas-phase volume decreases may pressurize the gas phase and provide a restoring force to counteract the pressure in the liquid phase. The overall force balance analysis and simple physical considerations suggest that larger maximum curvature values are attainable by increasing the length of the contact line while maintaining constant post area and pitch. Calculations with several different non-convex post shapes that increase the contact line length between 11 and 77% result in increases of 2–12%


in the maximum computable curvature. The properties of the interface for the non-convex shapes used to study this effect differ from the circular post case primarily in terms of the contact angle behavior. The mean contact angle tends to decrease as the arc length of the contact line is increased for a fixed mean curvature value, while the local contact angle exhibits significantly larger variation with angular position around the post, with maxima and minima occurring where the post shape parameterization function has local maxima and minima, and often exceeds 90◦ . The presence of local contact angles larger than 90◦ suggests that, unless a nail-head post design is used, where the top of the post has an undercut and the interface finds it energetically favorable to be pinned along the outer edge of the undercut, the liquid may begin to wet the post sides sooner for the non-convex post shapes than for the convex circular post. The marginal increase in maximum curvature indicates that varying the post shape may not be an effective method for increasing the pressure stability of a superhydrophobic surface. The good agreement between the simulations and the overall force balance analysis for the maximum computable curvature for surfaces composed of square arrays of convex post shapes suggests the force balance analysis is an adequate predictor of the maximum pressure differential supportable by superhydrophobic surfaces composed of circular and square posts. For the case of parallel grooves the force balance analysis is exact. Application of this analysis to the superhydrophobic surface geometries used in the study of drag reduction [19–23] and controlled wetting [10] suggests that surface tension forces are able to support air/water interface pressure differentials ranging from 0.003 to 1.1 atmospheres, with surfaces with smaller relative post size and pitches tending to support larger pressure differentials. This potentially places considerable constraints, for example, on the range of flow rates for which a superhydrophobic surface will maintain its drag reducing properties without wetting of the liquid into the superhydrophobic surface. This is of particular concern for flow in microchannels, where the small hydraulic diameter of the channel results in large pressure drops. The analysis suggests that active or passive pressurization of the gas phase within the superhydrophobic surface may be one method for achieving larger hydrodynamic pressures. Finally, two comments about the limitations of the analysis presented in this paper deserve mention. First, the effect of hydrodynamic stresses at the gas–liquid interface have been neglected. These stresses may play an important role in determining the shape and stability of the gas–liquid interface for fluid flow past superhydrophobic surfaces. And second, while theories [7,8] have been developed that appear to explain wetting to non-wetting transitions for static drops on superhydrophobic surfaces, no theory exists for predicting such a transition in the presence of flow. Developing a combined theoretical and experimental framework for determining parameters governing such a transition would be extremely beneficial for understanding the pressure stability of superhydrophobic surfaces in the presence of flow, and would also shed fundamental insight into the understanding of wetting behavior.

197

Acknowledgments The authors thank Tom Krupenkin, Marc Hodes, Paul Kolodner, Ryan Enright, Alan Lyons, John Mullins and Ashley Taylor for many valuable discussions and suggestions. Supplementary material The online version of this article contains additional supplementary material. Please visit DOI: 10.1016/j.jcis.2007.05.059. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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