Computation of constant mean curvature surfaces: Application to the gas-liquid interface of a pressurized fluid on a superhydrophobic surface.

cien
m
in
op
, T
orie
07;
ne 2Abstract
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 super-
hydrophobic 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]. Re-
cent 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: tsalamon@alcatel-lucent.com (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.
such interesting properties as decreased wettability and reduced
friction in laminar flows. These characteristics make superhy-
drophobic 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 hydropho-
bicity 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 con-
tact angle θ0 of a liquid in contact with a smooth, solid surface
(1)cos θ0 =
γsg − γsl
γlg
,
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]Journal of Colloid and Interface S
Computation of constant
Application to the gas–liquid
on a superhydr
E.J. Lobaton 1,2
Mathematical and Algorithmic Sciences Research Center, Bell Laborat
Received 4 January 20
Available onli0021-9797/$ – see front matter © 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.jcis.2007.05.059ce 314 (2007) 184–198
www.elsevier.com/locate/jcis
ean curvature surfaces:
terface of a pressurized fluid
hobic surface
.R. Salamon ∗
s, Alcatel-Lucent, 600 Mountain Avenue, Murray Hill, NJ 07974, USA
accepted 4 May 2007
5 May 2007

loidE.J. Lobaton, T.R. Salamon / Journal of Col
equation
(2)cos θCB = f (1 + cos θ0) − 1
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
(3)cos θW = r cos θ0,
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◦), in-
creasing 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 super-
hydrophobic 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 superhy-
drophobic 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 con-
tact with a superhydrophobic surface consisting of carbon nan-
otubes [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 re-
ceding contact angle for an evaporating water droplet placed
between a silanized glass plate and a superhydrophobic sur-
face consisting of carbon nanotubes, where the pressure dif-
ferential across the air/water interface is due to surface tension
forces arising from the droplet curvature. Journet et al. [18] ob-
served that the receding contact angle for a water droplet in
contact with a carbon nanotube surface with typical tube di-
ameters ranging from 50 to 100 nm and tube-to-tube spacings
of 100 to 250 nm was relatively insensitive to droplet pres-
sures ranging up to 10 kPa, implying that the composite inter-
face 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 signif-
icant 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. Jour-
net 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 sili-
con surfaces prepared by photolithography and hydrophobizedand Interface Science 314 (2007) 184–198 185
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 con-
torted 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 superhydropho-
bic 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 con-
structed using microfabrication [20–24] or chemical coating
[19] techniques and with typical roughness dimensions rang-
ing 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 hy-
drofoil has roughness feature sizes on the order of 10 µm and
a superhydrophobic coating. The authors of these studies at-
tribute the drag reducing properties of these surfaces to air that
is trapped within the superhydrophobic surface and which de-
creases 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 interven-
ing 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 fea-
ture 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] stud-
ies have focused on quantifying the magnitude of drag re-
duction 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 con-
taining no-slip regions corresponding to the solid roughness
features of the surface and shear-free regions that correspond

loid186 E.J. Lobaton, T.R. Salamon / Journal of Col
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 characteris-
tic velocity of the flow, and γ is the gas–liquid surface tension.
The experimental results of Ou et al. [20] coupled with the sim-
ulation results presented in this paper and the recent study of
Zheng et al. [30], described below, clearly suggest that the in-
terface 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 inter-
face 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 magni-
tude 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 condi-
tion 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 an-
alytical expression for the maximum pressure sustainable by
a square array of square posts. Zheng et al. [30] used the fi-
nite 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 lo-
cal wetting behavior on the pressure stability of superhydropho-
bic 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-and Interface Science 314 (2007) 184–198
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 transi-
tion also play a critical role in describing such wetting phenom-
ena. 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 sur-
face. 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 equa-
tions and boundary conditions governing the static shape of the
gas–liquid interface for a fluid in contact with a superhydropho-
bic surface consisting of a square array of posts are presented.
The numerical method used to solve for constant mean cur-
vature 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 an-
gle 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 superhy-
drophobic surface consisting of a square array of circular posts
of diameter D and post-to-post pitch L. A gas layer at pres-
sure 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]
(4)2γ κmean = �p,

loidE.J. Lobaton, T.R. Salamon / Journal of Col
Fig. 2. Array of circular posts and domain of computation Ω (- - -).
where γ is the liquid–gas surface tension, κmean is the mean cur-
vature, 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 com-
putational unit cell shown in Fig. 2. Along the dashed lines
denoted by Γsym the interface shape is assumed to be symmet-
ric. 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 en-
ergies 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 mush-
room cap shapes have been proposed by several authors [33–35]
as potential post geometries allowing the creation of superhy-
drophobic 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 reg-
ular 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 C4F8 to deposit a protective
layer of flouropolymer. This process results in a nearly verti-
cal 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 ex-
ists between the Si and oxide the silicon is undercut with little
loss of the oxide layer, which corresponds to the top of the nail-
head.and Interface Science 314 (2007) 184–198 187
Fig. 3. Force diagram for circular post case. Hollow arrows with solid stems
denote surface tension forces and hollow arrows with dashed stems denote pres-
sure 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 in-
terfacial shape problem are thus equivalent to finding surfaces
with constant mean curvature that satisfy the aforementioned
symmetry and pinning conditions of the interface.
2.1. Maximum curvature (κmaxmean) and pressure difference
(�pmax)
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 compu-
tational 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)
∫
Γcl
γ sin θcl ds +
∫
Γsym
γ sin θcl ds =
∫
Ω
�p cos θp dA,
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
∫(6)γ
Γcl
sin θcl ds = �pAp,

loid
ar p
is p
tact
4W where S = πD, Ap = L − 4 D and D is the diameter of the(10)�pmax = γ
L2 − W 2
,
and a surface consisting of grooves of width w
(11)�pmax = γ 2
w
.
Equations (8)–(11) will be used in Section 4 as a check on the
numerical simulations and to compare with experimental stud-
ies of superhydrophobic surfaces.
Knowledge of the advancing contact angle θadv for a partic-
ular gas–liquid–solid system can be used in conjunction with
Eq. (6) to obtain a tighter bound for the maximum pressure
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 θadv < π/2, the
pressure stability analysis suggests that the cylindrical and ta-
pered posts offer no pressure stability. Only the nail-head con-
figuration is expected to be able to support any static pressure
load. Second, for the case of hydrophobic liquids where θadv >
π/2, the nail-head configuration offers the greatest pressure sta-
bility 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 magni-
tude of θadv plays no role as the maximum sustainable pressure188 E.J. Lobaton, T.R. Salamon / Journal of Col
Fig. 4. Advancing contact angles θadv for posts with different shapes: (a) circul
angle α. Note that the contact angle is defined relative to the dashed line, which
hence represents the appropriate reference frame for defining the advancing con
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 act-
ing 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
(7)�pmax = γ S
Ap
≡ γ
(Arc-length)
(Projected interface area) ,
where S is the arc-length of the section of the boundary associ-
ated 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
(8)κmaxmean =
S
2Ap
≡
(Arc-length)
2(Projected interface area) .
Note that Eq. (8) contains purely geometric properties of the
nanostructured surface and is independent of the gas–liquid sur-
face 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
(9)�pmax = γ πD
L2 − π4 D
2 ,
a surface consisting of a square array of square posts of width
W and post-to-post pitch Lsustainable for a particular superhydrophobic surface, although
this requires making several assumptions: (i) the local contact
angle is fairly uniform around the post periphery. The resultsand Interface Science 314 (2007) 184–198
ost; (b) circular post and nail-head top; and (c) tapered circular post with taper
arallel to the surface over which the fluid would advance if it were to wet, and
angle.
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 be-
havior near the sharp corner associated with the post is well-
described 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 sta-
tic contact angles on smooth surfaces is well accepted [32],
although it is not clear that this behavior applies locally at sur-
faces with roughness features such as sharp corners. With these
assumptions one can arrive at the following forms for the max-
imum sustainable pressure for the various cylindrically sym-
metric geometries depicted in Fig. 4 in terms of the advancing
contact angle on a flat substrate θadv
(12)�pmax = γ S
Ap
sin
(
θadv − π/2
)
, circular post,
�pmax =
{
γ SAp sin(θ
adv), for θadv � π2 ,
γ SAp , for θ
adv > π2 ,
(13)nail head shaped post,
�pmax = γ
S
Ap
sin
(
θadv − π/2 − α
)
,
(14)post with taper angle α and with α > 0,
2 π 2corresponds to the interface intersecting the bottom of the nail
head surface vertically.

loidE.J. Lobaton, T.R. Salamon / Journal of Col
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 inter-
face and cause adjacent particles to coalesce into a single larger
particle with smaller surface area, and hence lower surface en-
ergy.
Surfaces with constant mean curvature are obtained by solv-
ing the following Lagrangian evolution problem for the location
of points �x along the interface
(15)˙�x · �n = (κ0 − κmean(�x)
)
, in Ω and Γsym,
where ˙�x is the time derivative of the position vector, �n is the
unit normal vector to the interface, κ0 is the desired mean curva-
ture, and κmean(�x) is the mean curvature of the current interface
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 solu-
tion (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
κmean(�x) = κ0,
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 describ-
ing 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.,
(16)˙�xi · �ni =
(
κ0 − κmean(�xi)
)
,
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 nor-
mal vector and mean curvature appearing in Eq. (16). A sample
discretization of the initial planar surface is shown in Fig. 5.and Interface Science 314 (2007) 184–198 189
Details of the calculation of the normal vector and mean cur-
vature used in Eq. (16), along with results demonstrating the
convergence and accuracy of the numerical method are pre-
sented in the supplementary material.
4. Results
The outline of this section is as follows. The characteris-
tic features of constant mean curvature surfaces describing the
gas–liquid interface for a pressurized fluid in contact with a su-
perhydrophobic 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 ex-
perimental 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 sur-
face. 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 vec-
tor 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 aver-
age 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 circu-
lar post case is due to long-range lattice effects in the superhy-
drophobic surface and not the post shape, which is symmetric
with respect to angular position around the post.
In Fig. 8a the mean contact angle ¯θcl, defined as
(17)¯θcl =
∫
Γcl
θcl(s)ds
∫
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 maxi-
mum value close to 90◦ for a post-to-post pitch of L = 0.9 µm

loid
(a)
ic s
sin
In Fig. 8b the normalized surface area An, defined asFig. 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 κmax0 . This maximum mean curvature agrees to within
one percent with the value predicted by Eq. (9) for κmaxmean in
Section 2.1. This suggests that the simple force balance is a re-
liable predictor of the maximum curvature for regular arrays of
(18)An(κ0) = A(κ0)
A(0)
,
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-to-
post 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
(19)δz = max
i=1,...,Nnodes
‖zi‖,
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 κmax0 appears to obey the following
scaling with respect to the post-to-post pitch L
(20)δz(κmax0
)
∼ O
(
L+1/2
)
.
In Fig. 9b the decrease in the gas-phase volume δV , defined
as190 E.J. Lobaton, T.R. Salamon / Journal of Col
(b)
Fig. 6. (a) Interface shape in the computational domain Ω for a superhydrophob
(b) interface shape between four adjacent posts, and (c) interface shape around acircular 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-and Interface Science 314 (2007) 184–198
(c)
urface with circular posts with D = 0.35 µm, L = 2 µm and κ0 = 0.141 µm−1,
gle post.
most cusp-like, increase in slope as it approaches the maximum
curvature for which solutions exist.(21)δV = Vs
V0
,

loidE.J. Lobaton, T.R. Salamon / Journal of Col
(a)
(b)
Fig. 8. (a) Mean contact angle ¯θcl 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 xy-
plane, V0 ≡ HA(0) is the gas-phase volume below the compu-
tational 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 representa-
tive 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) con-
figuration, 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 disso-
lution of the gas into the liquid. Such an analysis is beyond the
scope of this paper.and Interface Science 314 (2007) 184–198 191
(a)
(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
that one method for increasing the maximum curvature κmaxmean
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
followsR(φ) = R0 + �R cos(nφ),
(22)0� φ � 2π, for shapes A, C, D and E

loid192 E.J. Lobaton, T.R. Salamon / Journal of Col
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 0.175 0 0 1
B 0.1935 – – 1.11
C 0.1746 0.01746 8 1.14
D 0.1733 0.03466 8 1.47
E 0.1611 0.0967 4 1.77
and
R(φ) = R0 cos
(
φ −
π
4
[
1 + 2 × int
(
φ
π/2
)])
,
(23)0� φ � 2π, for shape B,
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 equiv-
alent to that of the circular post with D = 0.35 µm. The quantity
sratio, defined as
(24)sratio =
∫
Γ Shapecl
ds
∫
Γ Shape Acl
ds
,
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 ¯θcl is plotted as a func-
tion 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 ex-
ist increases with increasing sratio, from approximately a 2%
increase in κmax0 for shape A to a 12% increase in κ
max
0 for
shape E. In Table 2 the calculated values for κmax0 are com-
pared with the theoretical predictions for κmaxmean based on Eq. (8)
in Section 2.1. Note that the theoretical predictions signifi-
cantly 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 re-
sults shown in Fig. 11 for selected values of the mean curvature.
Shapes D and E exhibit local maxima and minima in the con-
tact angle that correspond to angular positions where the shapeand Interface Science 314 (2007) 184–198
Fig. 11. Average contact angle from horizontal ¯θcl as a function of mean curva-
ture 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 κmax0 and predicted maximum mean cur-
vature κmaxmean as given by Eq. (9) for the results shown in Fig. 11
Shape Simulation, κmax0
(µm−1)
Force balance, κmaxmean
(µm−1)
A 0.141 0.141
B 0.144 0.157
C 0.144 0.161
D 0.148 0.207
E 0.158 0.250
parameterization function R(φ) exhibits local maxima and min-
ima, 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 curva-
ture 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

E.J. Lobaton, T.R. Salamon / Journal of Colloid and Interface Science 314 (2007) 184–198 193
(a)
(b)
Fig. 12. Contact angle θcl(φ) as a function of angle around the post φ for vary-
ing values of κ0 for (a) shape D and (b) shape E for the results shown in Fig. 11.
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 ear-
lier.
4.3. Comparison with experiments
In this section we compare directly with experiments in-
volving the flow of liquids in channels and pipes with super-
hydrophobic walls. We note that our approach neglects hydro-
dynamic 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 in-
sight 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 inter-
face 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 func-
tion 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 ex-
perimental 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 rateFig. 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.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

loid194 E.J. Lobaton, T.R. Salamon / Journal of Col
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 con-
ditions.
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. As-
suming 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 negli-
gible, the pressure in the liquid phase at the air–liquid interface
mid-way along the channel is given by
(25)pliq = pexit + 12�Pexp,
where pexit is the pressure at the exit of the microchannel. Sub-
stitution of Eq. (25) into Eq. (4) results in the following form for
the gas-phase pressure relative to the microchannel exit pres-
sure
(26)(pgas − pexit) =
(
1
2
�Pexp − 2γ κ0
)
.
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 superhydropho-
bic surfaces studied in Section 4.1. The maximum curvature
κmax0 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 circu-
lar and square posts suggests that for convex post shapes, where
a straight line connecting any two points within the shape al-
ways remains within the shape [45], the calculated maximum
admissible curvature κmax0 is well approximated by the alge-
braic formulae for κmaxmean presented in Section 2.1. The maxi-
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 plot-
ted 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,and Interface Science 314 (2007) 184–198
(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 de-
flections larger than 30 µm have not been included in (b).
appreciable compression of an enclosed gas-phase is possi-
ble. For example, a 50% reduction of the gas-phase volume
would result in a two-fold increase in the gas pressure rela-
tive to the undeflected state. If the initial gas-phase pressure
is 1 atm, the compression effect due to deformation of the in-
terface will result in a gas-phase pressure of 2 atm. Again,
note that this analysis does not account for any potential re-
equilibration 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◦.

erh
e m
oryFig. 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.
Table 3
Maximum pressure difference �pmax predicted by Eqs. (9)–(11) for various sup
wettability (f) and using a value of γ = 72 dynes/cm. �P maxexp corresponds to th
to, as reported experimentally or inferred from fully-developed laminar flow the
Experiment Formula for
�pmax
(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)
(b) Woolford et al. [21]—water flowing in a micro-
channel 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
Eq. (11)
(d) Choi et al. [23]—water flowing in a micro-
channel of length 4 mm and height ranging from 3 to
11 µm with upper and lower walls patterned with
parallel ridges
Eq. (11)
(e) Davies et al. [24]—water flowing in a micro-
channel of length 7 cm and height H = 80 µm with
upper and lower walls patterned with parallel ridges
Eq. (11)
(f) Krupenkin et al. [10]—water droplets on a Eq. (9)
superhydrophobic surface consisting of a square
array of circular postsmaximum pressure difference �pmax for a working fluid such
as water is limited to fractions of an atmosphere. To achieve
ydrophobic surfaces used in studies of laminar drag reduction (a, b, c, d, e) and
aximum hydrodynamic pressure that the superhydrophobic surface is exposed
for the reported flow conditions
Formula parameter
values (µm)
�pmax
(Pa)
�P maxexp
(Pa)
W = 30, L = 45 7680 200–1600
W = 30, L = 60 3200 100–1400
W = 30, L = 90 1200 100–1200
W = 30, L = 180 274 100–1200
w = 20 7200 1680–2120
w = 30 4800 1660–2760
w = 10 14,400 58–389*
w = 10 14,400 7–47**
w = 0.18 800,000 10,100–101,000
w = 20 7200 5490–7500
w = 30 4800 4960–6840
w = 35 4110 4440–6320
w = 38 3790 3890–5980
w = 39 3690 3250–3930
D = 0.35, L = 0.9 111,000 Not applicableE.J. Lobaton, T.R. Salamon / Journal of Colloid and Interface Science 314 (2007) 184–198 195
4.3.2. Estimates for maximum pressure difference �pmax for
various superhydrophobic surfaces
In this section the analytical formulae for the maximum pres-
sure difference �pmax 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.
In Table 3 values of �pmax as predicted by Eqs. (9)–(11) for
water on various superhydrophobic surfaces for the drag reduc-
tion experiments of Ou et al. [20], Watanabe et al. [19], Wool-
ford et al. [21], Choi et al. [23], Davies et al. [24] and the con-
trolled wetting experiments of Krupenkin et al. [10] are shown.
The quantity �P maxexp , which is defined as the maximum hydro-
dynamic pressure that the superhydrophobic surface in the drag
reduction studies is exposed to
(27)pinletliq = pamb + �P maxexp ,
where pinletliq is the pressure at the inlet to the superhydropho-
bic section of the flow apparatus, is also shown. The values
for �P maxexp for the experiments of Ou et al. [20] and Choi
et al. [23] correspond to data directly measured on their flow
system and assuming that the exit of the microchannel is at am-
bient 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
of the experiment. Note that although �P maxexp neglects vis-
cous stress contributions to the normal stress balance at the
air–liquid interface, it is expected to provide a reasonable or-
der of magnitude estimate for the fluid stresses at this inter-
face.
Several points are worth noting about the results in Table 3.
First, with the exception of the results of Choi et al. [23], theD = 0.35, L = 2 20,300
D = 0.35, L = 4 4980

loid196 E.J. Lobaton, T.R. Salamon / Journal of Col
pressure stability larger than an atmosphere Choi et al. [23] re-
quired 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 differen-
tial equal to precisely �P maxexp . This is clear from substitution of
Eq. (27) into Eq. (4)
(28)2γ κmean = �P maxexp − (pgas − pamb),
and noting that for the case pgas = pamb Eq. (28) reduces to
(29)2γ κmean = �P maxexp .
This indicates that surface tension alone is responsible for main-
taining the gas–liquid interface in the superhydrophobic con-
figuration. Third, the form of Eq. (28) suggests that to achieve
hydrodynamic pressures �P maxexp greater than �pmax pressur-
ization of the gas phase may be required, although the practi-
cality 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 �pmax suggest that
surface tension alone is not able to support the gas–liquid in-
terface: (i) for the case W = 30 µm and L = 180 µm in the
study of Ou et al. [20] this corresponds to conditions where
�P maxexp > 274 Pa. The results of Section 4.3.1 indicate that de-
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 hy-
drodynamic 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 pres-
sure 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 reduc-
tion 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 hypothe-
sis 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 con-
tained within a superhydrophobic surface consisting of a square
array of posts with constant post-to-post pitch. The interfaceand Interface Science 314 (2007) 184–198
shape satisfies the Young–Laplace equation, which relates a
balance of surface tension forces acting through the mean cur-
vature 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) demon-
strates 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 κmax0 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 in-
terfacial area in the undeflected, planar state. The overall force
balance analysis agrees to within one percent with the numer-
ical 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 per-
turbation 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, nor-
malized surface area, maximum interface deflection and de-
crease in gas-phase volume as a function of mean curvature for
a variety of post shapes, sizes and pitches. All of the afore-
mentioned properties are monotonically increasing functions
of the mean curvature, and exhibit a rapid, cusp-like behav-
ior as the limiting curvature value κmax0 is approached for a
particular superhydrophobic surface. For all of the superhy-
drophobic surfaces studied here, the mean contact angle ranges
from 0◦ for the planar, undeflected state, to values approach-
ing 90◦ at the limiting curvature κmax0 . 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 con-
siderations 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 sev-
eral different non-convex post shapes that increase the contact
line length between 11 and 77% result in increases of 2–12%

loidE.J. Lobaton, T.R. Salamon / Journal of Col
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 con-
tact 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 signif-
icantly larger variation with angular position around the post,
with maxima and minima occurring where the post shape para-
meterization 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 en-
ergetically 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 sur-
face.
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 superhy-
drophobic 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 rang-
ing 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 superhy-
drophobic 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 pres-
sure drops. The analysis suggests that active or passive pressur-
ization of the gas phase within the superhydrophobic surface
may be one method for achieving larger hydrodynamic pres-
sures.
Finally, two comments about the limitations of the analy-
sis presented in this paper deserve mention. First, the effect of
hydrodynamic stresses at the gas–liquid interface have been ne-
glected. These stresses may play an important role in determin-
ing the shape and stability of the gas–liquid interface for fluid
flow past superhydrophobic surfaces. And second, while theo-
ries [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 exper-
imental 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 un-
derstanding of wetting behavior.and Interface Science 314 (2007) 184–198 197
Acknowledgments
The authors thank Tom Krupenkin, Marc Hodes, Paul
Kolodner, Ryan Enright, Alan Lyons, John Mullins and Ash-
ley Taylor for many valuable discussions and suggestions.
Supplementary material
The online version of this article contains additional supple-
mentary material.
Please visit DOI: 10.1016/j.jcis.2007.05.059.
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