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Wetting and Roughness
David Quéré
Laboratoire de Physique et Mécanique des Milieux Hétérogènes, ESPCI, 75005 Paris,
France; email: [email protected]
Annu. Rev. Mater. Res. 2008. 38:71–99
Key Words
First published online as a Review in Advance on
April 7, 2008
microtextures, superhydrophobicity, wicking, slip
The Annual Review of Materials Research is online at
matsci.annualreviews.org
Abstract
This article’s doi:
10.1146/annurev.matsci.38.060407.132434
c 2008 by Annual Reviews.
Copyright All rights reserved
1531-7331/08/0804-0071$20.00
We discuss in this review how the roughness of a solid impacts its wettability.
We see in particular that both the apparent contact angle and the contact angle hysteresis can be dramatically affected by the presence of roughness. Owing to the development of refined methods for setting very well-controlled
micro- or nanotextures on a solid, these effects are being exploited to induce
novel wetting properties, such as spontaneous filmification, superhydrophobicity, superoleophobicity, and interfacial slip, that could not be achieved
without roughness.
71
1. WETTING WITHOUT ROUGHNESS
1.1. Ideal Wetting
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Controlling the wettability of solid materials is a classical and key issue in surface engineering.
Roughly speaking, two extreme limits are often desired. The first limit is complete wetting, in
which a liquid brought into contact with a solid spontaneously makes a film. In the case of a
windshield, for example, this film maintains the transparency of the glass; in addition, the film
flows in the gravity field (if the car is stopped) or due to air friction (when it moves), taking dust
particles with it. The second limit is complete drying: Liquid drops remain spherical without
developing any contact with the substrate. They are thus readily evacuated, which prevents liquid
contamination of the solid surface.
It is of obvious interest to determine which parameters favor both these situations. The basic
laws were first established for ideal solids, which are both flat and chemically homogeneous. As
understood by Young and Laplace, surfaces carry a specific energy, the so-called surface tension,
that reflects the cohesion of the underlying condensed phase (either solid or liquid). This quantity,
denoted as γ IJ for an interface between phases I and J (below the indices are S, L, A for solid,
liquid, and air, respectively), is an energy per unit area and thus a force per unit length: This force
applies along the IJ surface to minimize the corresponding (positive) surface energy. We denote
the liquid/air surface energy simply as γ .
Hence, we arrive at a construction first imagined by Marangoni: A film spreads from a reservoir
of liquid (a drop or a bath) onto a solid, as sketched in Figure 1a, provided that the solid/air
surface tension γ SA (which entrains this film) is larger than γ SL + γ , the sum of the solid/liquid
and liquid/air surface tensions (which both resist the spreading because complete wetting expands
the two corresponding surface areas). The sign of the spreading parameter S = γ SA − γ SL − γ
will thus determine the behavior of a drop on a solid: For S > 0, a drop spreads, whereas it forms
a small lens in the opposite case. This lens meets the solid with a well-defined contact angle θ,
whose value is similarly given by a force balance (Figure 1b). Projecting on the solid plane the
different surface tensions acting on the contact line provides the equilibrium condition of the drop
(1). The balance at equilibrium can be written as
γSA = γSL + γ cos θ.
1.
The contact angle is thus fixed univocally by the chemical nature of the different phases. Here
we show that this statement can be dramatically affected if the solid is rough. We refer below to
the angle θ as the chemical or Young angle. In many common situations, this angle lies between
0◦ and 90◦ (i.e., the hydrophilic case). Very qualitatively, a solid/liquid surface tension (between
a
b
γ
γSL
γSA
θ
γ
γSA
γSL
Figure 1
Two classical wetting situations for an ideal material. (a) A liquid film spreads, drawn by the solid/air surface
tension, despite the action of the liquid/air and solid/liquid tensions. (b) Wetting is only partial, and the
balance of surface tensions determines the contact angle θ.
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Quéré
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two condensed phases) tends to be smaller than a solid/air one (with only one condensed phase)
because the phases are less contrasted in the first case. Hence, a positive cosine in Equation 1
results, implying an acute angle θ .
Conversely, we could define a drying parameter D = γ SL − γ SA − γ . If γ SL is larger than γ SA +
γ , the contact line will be withdrawn by surface forces until a film of air comes between the solid
and the liquid: D > 0 is the criterion for complete drying. (This criterion is also simply derived
by making cos θ < −1 in Equation 1.) There is a first case in which this criterion is fulfilled:
For a system in which complete wetting is achieved (for example, water on freshly cleaned glass),
inverting the liquid and the air immediately provides D > 0; a bubble of air at the bottom of the
same glass filled with water will completely dewet glass.
In a particular circumstance, the so-called Leidenfrost effect (2), D is forced to vanish: If water
(or any volatile oil) is deposited on a solid whose temperature is much larger than the boiling
temperature of the liquid, a vapor film forms between the solid and the liquid, which sits on its
own vapor. Considering in Equation 1 that the solid role is played by the vapor, we determine that
D = 0. But the situation is quite different at standard temperatures. Although complete wetting can
be achieved (for example, with most light oils on most solids), complete drying of water (or any oil)
on a flat solid is never observed. On the most hydrophobic solids (fluorinated materials), the contact
angle never exceeds approximately 120◦ (3), to which corresponds a negative parameter D (of
approximately −γ /2). We term hydrophobic these situations in which obtuse angles are observed.
Another aim of this review is to show how one can take advantage of the surface roughness for
filling the (large) gap existing between 120◦ and (nearly) 180◦ , thus generating ultrahydrophobic
behaviors of obvious practical interest (water repellency).
1.2. Ideal Wicking
Materials are often not fully solid, yet are porous. We restrict our discussion to the (ideal) case
of cylindrical pores of constant diameter and consider one of these pores. A liquid will penetrate
such a tube if the surface energy of the solid is lower wet than dry (Figure 2a).
We can introduce a wicking parameter W = γ SA − γ SL , whose sign indicates if liquid penetrates
the tube. Wicking will occur if W > 0, that is, if the contact angle is smaller than 90◦ (as we see from
Equation 1). Then, the meniscus formed by the liquid inside the tube must be curved in such a way
that the Laplace pressure (associated with curved surfaces) is negative below the surface—another
way to understand liquid penetration as resulting from this depression that sucks the liquid inside
a
b
Figure 2
(a) A liquid brought into contact with a tube or a slot will penetrate it provided that the surface energy of the
tube is lower wet than dry. This means that, as deduced from Equation 1, the contact angle of the liquid on
the tube walls must be acute. (b) Conversely, for an obtuse contact angle, the tube tends to remain dry. If
gravity is present, both the rise and descent are limited.
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73
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the tube. As stressed above, contact angles are generally acute, which means that most sponges
absorb most liquids. A system for which we have S > 0 will necessarily satisfy the condition W >
0 (then, the contact angle is zero, indeed smaller than 90◦ ), and it is worth discussing carefully the
mechanism of invasion in this case. Therefore, as shown in Figure 1a, a liquid film (typically of
a molecular thickness) progresses along the tube walls (4) such that the meniscus behind the film
advances on a prewet tube. Because this meniscus suppresses the liquid/air interface as it moves,
the penetration is always favorable. The wicking parameter can be written as W = γ , which is
indeed the limit of W as the contact angle vanishes, as seen in Equation 1.
Conversely, wicking is not favorable for W < 0, and liquid is expelled from the pore
(Figure 2b)—hence the development of the idea that a solid decorated with hydrophobic cavities can remain filled with air, even if the solid is exposed to a liquid, and thus approach the
Leidenfrost limit. As we see, not only can roughness modify the wettability of a solid but also—
perhaps the main message of this review—roughness can result in new and specific properties
such as water repellency. We first show that the natural roughness of most solids is likely to induce
pinning of the contact line and thus variability of the contact angle (apparently contrasting with
what can be expected from Equation 1). Subsequently, we discuss how special kinds of roughness
(well-designed microstructures) can be created at the solid surface to control both wettability and
pinning and, beyond, special hydrodynamic properties such as slip.
2. ROUGH SOLIDS
2.1. Contact Angle Hysteresis
Most solids are naturally rough, often at a micrometric scale. Processes of fabrication (such as
lamination) may generate striations or microgrooves. Materials resulting from the compaction of
grains exhibit roughness at the scale of the grains. Coating can also induce roughness, in particular
when the coating film dewets, thus producing microdrops at the surface. Conversely, very few solids
are molecularly flat. Most often, molecularly flat solids result from solidifying a liquid film, either
free or suspended on another liquid; in such cases, the roughness can correspond to the thermal
roughness of a liquid interface, generally of the order of a few angstroms. This is the case of glass,
solidified from its molten state after deposition onto a bath of molten tin.
Gibbs pointed out that defects on a solid can pin a contact line. As a consequence, droplets
on an incline stay at rest; the front and rear contact nonwetting and wetting defects, respectively
(5). The resulting asymmetry in contact angles creates a Laplace pressure difference between the
front (of high curvature) and the rear (of smaller curvature) and, thus, a force able to resist gravity
provided that the drop is small enough (6). Both chemical heterogeneities and roughness can act
as pinning sites. It is useful to think of a single defect such as is sketched in Figure 3.
Even on a chemically homogeneous surface, the edge of the defect (of characteristic angle φ)
makes the contact angle flexible at this place. We measure a (Young) angle θ before the edge and
a (Young) angle π − φ + θ after the edge, considering the horizontal as the reference (we ignore
with our naked eye the existence of the defect). Hence it is possible to have any angle between
θ and π − φ + θ at the edge (7). A groove can thus stop the front of a liquid drop (as if it were
nonwetting), and a tip will act in the opposite way so that a solid decorated with both kinds of
defects yields both small and large apparent angles.
Contact angles therefore generally depend on the history of the process of liquid deposition.
A drop gently deposited spreads and stops when it is surrounded by primarily nonwetting defects,
which prevent it from exploring the solid further. After a while, the drop evaporates, and thus its
configuration is that of a drop pinned on wetting defects. The way to quantify this contact angle
74
Quéré
θ
Φ
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θ
Figure 3
Apparent pinning of a contact line on an edge. The Young condition stipulates that the liquid meets the solid
with a contact angle θ . Hence the contact angle at the edge can take any value (if the horizontal direction is
considered as the reference one) between θ and π − φ + θ , as illustrated by the colored region.
hysteresis consists of slowly increasing the volume of a drop: The contact line first remains stuck
before it suddenly jumps above a critical volume (for which the line suddenly depins and moves
toward a next series of pinning defects). The maximum observed angle is the so-called advancing
contact angle θ a . Conversely, sucking the liquid from the drop flattens it until it depins and retracts
to the next wetting series of events, which stop and pin the line; the minimum corresponding angle
is the receding contact angle θ r .
Contact angle hysteresis can be seen as beneficial (e.g., when it is exploited for guiding a
flow along a line of defects, following a predefined route) or detrimental (e.g., water drops stuck
on window panes distort their transparency and contribute to degradation of the glass). It is thus
crucial to understand it, but there is still a debate about the laws that relate the microscopic picture
(pinning on a single defect) to the macroscopic observations (measurement of the hysteresis, which
averages on many defects). We give further an example of such a calculation. More generally, we
see that the contact angle hysteresis θ = θ a − θ r varies dramatically on a rough solid, from
nearly zero to a giant value, of the order of θ a itself (8).
2.2. The Wenzel Model
We see above that roughness impacts the contact angle hysteresis. But it also affects the typical or
apparent angle, which is (often very) different from the one expected from Equation 1. This was
first appreciated by Wenzel (9), using a geometrical argument based on the roughness factor, r,
the ratio between the actual surface area and the apparent surface area of a rough surface.
A drop placed on a rough surface (Figure 4) will spread until it finds its equilibrium configuration, characterized by a contact angle θ ∗ (possibly different from the Young angle θ). The key
dx
Liquid
Vapor
θ
Solid
Figure 4
The Wenzel picture. One can obtain the apparent contact angle θ ∗ by considering a small apparent
displacement of the contact line and looking at the corresponding variation in surface energy, assuming that
the liquid follows the accidents of the solid surface.
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75
assumption of the model is sketched in Figure 4: As the contact line progresses on the dry solid,
it is assumed to follow all the topological variations of the material so that each piece of liquid/air
interface gets replaced by a solid/liquid interface of the same surface area. The surface energy
variation dE arising from an apparent displacement dx of the line can be written, per unit length
of the contact line, as
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dE = r(γSL − γSA ) dx + γ dx cos θ ∗ ,
2.
where the second term on the right corresponds to the change of liquid/vapor surface area as the
drop spreads. The roughness increases both the solid energies, enhanced geometrically by a factor
r. The minimum of E (dE = 0) yields Equation 1 if the solid is flat (r = 1); if not, we find (9)
cos θ ∗ = r cos θ,
3.
where θ is the chemical angle given by Equation 1.
The Wenzel relation (Equation 3) predicts that roughness enhances wettability. If the factor
r is larger than 1, a hydrophilic solid (θ < 90◦ ) becomes more hydrophilic when rough (θ ∗ < θ).
Conversely, a hydrophobic solid (θ > 90◦ ) shows increased hydrophobicity (θ ∗ > θ). Although
these tendencies are generally (but not always) observed, agreement with Equation 3 is far from
quantitative (see next section). We can guess that the Wenzel relation implies strange features.
For example, there is no limitation for the effect: The roughness factor can be made arbitrarily
large, which seems to imply that complete wetting (cos θ ∗ > 1) or complete drying (cos θ ∗ < −1)
should be induced by large roughness (r 1). We show that such behavior is not observed because
Wenzel assumptions often are not satisfied.
Even when Wenzel relation is likely to be obeyed, it is difficult to check directly whether the
relation is being followed. Because liquid conforms to the roughness, pinning of the contact line
is particularly strong in this state, both on the edges of and along the defects. Besides, pushing
a Wenzel drop leaves behind cavities filled with liquid such that the drop can also be pinned by
the liquid itself. As a consequence, a Wenzel state is generally characterized by very low receding
angles and thus giant hysteresis (θ ∼ θ a ). In such conditions, it is very difficult to extract the
sole angle θ ∗ or to check Equation 3. Modern and more detailed discussions on the validity of the
Wenzel model can be found in References 10–13, which stress in particular that drops should be
much larger than the defects to use such an averaged model. In the converse limit, liquid rearranges
such that the contact angle depends on the drop size (10, 11, 14).
3. MICROTEXTURED SOLIDS
We show in the previous section that roughness modifies both the ideal character of the Young
equation (the angle is not unique) and the value of the apparent observed angle. Therefore,
wettability can be tuned by roughness. We can take advantage of roughness to modulate the
surface properties of a solid and, even better, to induce properties that could not be generated
otherwise, a theme that has been extremely popular during the past decade.
Let us quote here three factors that contributed to the burst of this domain. (a) At the end of the
1990s, researchers from the Kao Corporation in Japan showed that extremely large angles could be
obtained by the use of fluorinated rough (fractal) surfaces (15, 16). This result was not fully novel;
similar results had been obtained in the 1940s (17) but somehow forgotten (18). (b) At the same
time, Neinhuis and Barthlott in Germany systematically analyzed the structures on the surfaces
of hydrophobic plants. These researchers showed the remarkable variety of the surface designs,
suggesting that nature had optimized the patterns (19, 20). This kind of study was extended to
animals, and new fascinating designs were (re)discovered and discussed (21–24). There have since
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3.1. The Kao Experiment
Kao researchers constructed different well-characterized fluorinated substrates, either rough or
flat (15, 16). They compared the contact angles θ ∗ on the rough samples with the contact angles
on flat materials, which should be close to the Young angle θ. This comparison was performed
through the use of several liquids to vary θ . A typical result is displayed in Figure 5, in which cos θ ∗
is plotted as a function of cos θ , showing the modification of the wetting properties generated by
surface roughness. This plot provides only one angle θ ∗ (which seems to be close to the advancing
angle, according to References 15 and 16), so we ignore the hysteresis associated with each data
point.
We first notice that the abscissa is far from exploring the complete interval [−1, 1]: cos θ is
never smaller than −0.3, corresponding to an angle θ of approximately 110◦ . This data point was
obtained by the use of water as a liquid; as emphasized above, contact angles on flat solids are never
larger than approximately 120◦ , corresponding to the maximum existing chemical hydrophobicity
(7). However, even if there are only a few data points on the hydrophobic side (cos θ < 0), we
see a spectacular effect. As soon as we enter this domain, the apparent contact angle θ ∗ jumps and
reaches a value of 170◦ (much larger than the chemical angle); roughness here induces a wetting
behavior that could not be achieved otherwise. This state is often referred to as superhydrophobic.
In the hydrophilic domain (cos θ > 0), cos θ ∗ first increases linearly with cos θ; the slope is
larger than unity (approximately 3). It is tempting to interpret this behavior as a Wenzel regime
(Equation 3). The material roughness deduced from micropictures is indeed in good agreement
with this slope (16). It is amazing to deduce this complex (and invisible) quantity from such a simple
(and cheap) experiment, in which just a few drops are used to probe the surface. However, this behavior is not obeyed when the contact angle θ becomes smaller than some critical value θ c . Instead,
we observe a second linear regime (with a slope smaller than unity), which tends toward θ ∗ = 0 as
θ = 0 (quite trivially, a wettable solid remains wettable if rough). We see in Section 4.2 that this
Figure 5
1
cos θ *
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been many attempts to mimic these natural patterns so as to understand their efficiency. (c) The
recent development of microfabrication techniques allows us today to construct very well-defined
micro- or nanostructures, which has pushed researchers, e.g., to imagine new designs and to
optimize given designs. We now summarize different findings related to these three factors (a–c).
Cosine of the
apparent contact
angle θ ∗ on a
textured surface, as a
function of the
cosine of the Young
angle θ measured on
the same surface, yet
flat (16). The lines
show the behavior
expected from
Equations 3, 10,
and 14.
0
θc
–1
0
1
cos θ
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77
regime results from the penetration of the liquid inside the microtextures; this liquid surrounds the
drop on which the contact angle is measured. Then, the lens of liquid sits upon a mixture of solid and
liquid, at odds with the Wenzel hypothesis, which assumes a dry solid beyond the drop, as shown in
Figure 4. We call this second regime superhydrophilic, and we describe this regime in Section 4.
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3.2. Natural Microtextures
Microtextures are also found on the surfaces of many plants and animals (Figure 6). In his Natural
History, Pliny the Elder noticed that water on a leaf forms perfect spheres, provided that the leaf
surface is woolly (25, p. 32). The old literature (and poetry) sporadically reports the special wetting
behavior of plants and animals, such as a review by Dufour in 1833 (26, pp. 68–74), a paper by
Fogg in 1944 (27), and a comment on Fogg’s paper by Cassie & Baxter (28). More systematic
studies performed (only) in the past decade generated a remarkable collection of microtextures,
of which Figure 6a displays a few examples.
On plant leaves, we often see bumps at the scale of 10–50 μm (Figure 6a,b). For the most
popular of these hydrophobic plants, namely the lotus, and many other ones, these papillae are
covered by fine nanostructures at the scale of 100 nm. The coexistence of two scales of roughness
contributes significantly to the quality of the superhydrophobicity (29–36). However, despite many
a
b
50 µm
c
50 µm
d
50 µm
500 nm
Figure 6
A few examples of natural superhydrophobic materials, as revealed by SEM. (a) Leaf of the so-called
elephant’s ear (Colocasia esculenta). From Reference 39 (courtesy of Peter Wagner and Christoph Neinhuis).
(b) Lotus leaf. Courtesy of Barthlott & Neinhuis (20). (c) Leg of a water strider. From Reference 23 (courtesy
of Lei Jiang). (d ) Surface of a mosquito (Culex pipiens) eye. From Reference 40 (courtesy of Lei Jiang). Note
the difference in scale between panels a–c and panel d.
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stimulating hypotheses, there is no real understanding for this hierarchical structure, which is not
necessary for reaching very high degrees of hydrophobocity (37). Actually, such structures often
provide both a high contact angle and a low contact angle hysteresis. For lotus, for example, the
advancing angle is approximately 160◦ , and the receding angle is larger than 150◦ , which confers
to water drops a high mobility on these leaves. In some cases, such as the rice leaf, the arrangement
of the papillae at the surface can be anisotropic, and thus wetting and adhesion are also anisotropic
(38). On such materials, water will flow preferentially along certain directions.
Feathers of many birds [such as those of pigeons (41) and ducks] are hydrophobic and/or
superhydrophobic, as are insects such as cicada, butterflies, and of course water striders (24, 42).
Insects’ cuticles are covered with a layer of epicuticular wax (of typical thickness of 250 nm),
which prevents the intrusion of water into the body (a serious threat for the insect) and protects
the animals from dessication. Without this protection, the insect rapidly dies if exposed to dry air.
But the most impressive superhydrophobic properties are related to the presence of setae on the
body or on the legs (Figure 6c), allowing some animals to float on water or even to live underwater
owing to the air spread on their body (43–48); see details in the recent and comprehensive review
by Bush et al. (24). The setae often consist of tapered hairs with a length of 30 μm, a diameter of
1–10 μm, and an angle of inclination of typically 30◦ (Figure 6c). As for plants, there is a secondary
texture, namely nanogrooves, whose exact role is still questioned (23). Other structures can be
very different: Figure 6d shows the pattern that decorates the eye of Culex pipiens, the classical
mosquito. It is very simple and well-ordered, at an impressively small scale (posts of size and height
of approximately 100 nm) (40). We show further that some applications indeed require reduction
of the pattern size.
3.3. Synthetic Microtextures
As is pointed out above, many recent papers are devoted to the creation of microtextured surfaces
with particular wetting properties (most often, superhydrophobic ones). Many techniques exist
for producing such materials, even primitive ones: Approaching a piece of glass from a flame
generates soot, which quickly darkens the glass. If you put water on this substrate, you will see it
behaving as if it were a soft solid, rolling and bouncing off the surface! More generally, template
synthesis, phase separation, all kinds of etching, crystallization, and electrospinning of microfibers
were proposed to construct more elaborate materials (49–50). As a result, many different textures,
from highly disordered or fractal to ordered and well-defined, were obtained (Figure 7). Most of
these surfaces provide specific wetting properties, and we still have to understand which surfaces
are the “best.” The answer depends on the required properties, which we now discuss.
4. HEMIWICKING
Patterns on a hydrophilic solid at a scale much smaller than the capillary length (above which gravity
dominates surface tension effects) can induce superhydrophilicity. We discuss above the Wenzel
effect, in which the roughness enhances hydrophilicity, provided that liquid fits in the pattern
(Figure 4), leaving dry the rest of the solid as in usual partial wetting (Figure 1b). However, the
structures may also guide the liquid within the array they form, in a manner similar to wicking.
The phenomenon that occurs here is not classical wicking but hemiwicking: As the film progresses
in the microstructures, it develops an interface with air, leaving (possibly) a few dry islands behind
it. We examine the conditions for observing hemiwicking, starting with the case of a single groove.
We discuss how this phenomenon impacts the wetting laws and conclude with a few considerations
of the dynamics of these films.
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a
b
10 µm
c
500 nm
d
5 µm
20 µm
Figure 7
Different examples of synthetic microtextured surfaces. (a) The simplest possible surface, with regular
micropillars. (Courtesy of M. Reyssat.) (b) A surface decorated with nanofibers. From Reference 51 (courtesy
of L. Gao and T.J. McCarthy). (c) A surface planted with carbon nanotubes. From Reference 52 (courtesy of
J. Bico). (d ) Mushroom pattern (with a flat hat). From Reference 53 (courtesy of G. McKinley).
4.1. Grooves
As stressed above, many solids are naturally striated by grooves. Such defects can also be etched for
specific purposes, such as directional wetting. We consider, for example, a rectangular groove of
width w and depth δ, as sketched in Figure 8, in which we ignore the detail of the different menisci.
For observing a spontaneous invasion of the groove, the solid must lower its energy by being
wet (γ SL < γ SA ). But this is not enough because a liquid/vapor interface also develops at the top.
dx
δ
w
Figure 8
A liquid (in blue) invading a rectangular groove on a solid. Here, we ignore the menisci (at the liquid front
and along the corners, ahead of it) and consider a progression of the liquid by a quantity dx.
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We expect the interface to be flat (as represented in Figure 8), which minimizes the corresponding
surface area. Hence, for a liquid progression in the groove by a quantity dx, the surface energies
change by an amount (ignoring gravity effects)
dE = (γSL − γSA )(2δ + w) dx + γ w dx.
4.
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Using the Young equation, we find that the liquid progression is favorable (dE < 0) if we have
θ < θc ,
5.
with
cos θc = w/(2δ + w).
6.
Whatever the values of w and δ, the latter quantity (which depends only on the aspect ratio δ/w)
defines a number between 0 and 1, from which the angle θ c can be made explicit. If the groove
is narrow and/or deep (w is small and/or δ is large), we recover the criterion (discussed above, in
the context of Figure 2) of spontaneous penetration in a classical porous medium: θ c = 90◦ . In
a general case (δ ∼ w), θ c is somewhere between 0◦ and 90◦ : It is more demanding to impregnate
a groove than a 3-D porous medium. This is all the more true because the groove is shallow: As
δ/w tends toward 0, so does θ c , meeting the criterion of complete wetting on a flat solid.
4.2. Assembly of Pillars
We can have similar arguments for a solid decorated with microposts (such as in Figure 7a). We
characterize such a surface by its pillar density φ S and roughness r. We show in Figure 9 the top
view of an ethanol drop on/in a forest of pillars (with φ S = 0.05 and r = 2) (54). The drop is a lens,
which deforms the colors generated by the regular array of microposts, and it is clearly surrounded
by a film of ethanol; in this situation hemiwicking takes place. In some cases, the film conforms
to the micropattern, so the film can take a square shape on a square array of microposts (55).
Figure 9
Top view of an
ethanol drop (with a
diameter of a few
millimeters) on a
silicon wafer
decorated by silicon
microposts. Here,
hemiwicking takes
place, as deduced
from the observation
of a thick film ahead
of the drop. From
Reference 54
(courtesy of Chieko
Ishino and Mathilde
Reyssat).
www.annualreviews.org • Wetting and Roughness
81
Air
dx
p
b
h
Liquid
Solid
Figure 10
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Liquid film (in blue) propagating on a solid, within a forest of microposts of height h, mutual distance p, and
radius b. The condition for the progression is deduced from the variation of surface energy associated with it.
The impregnating front should propagate as pictured in Figure 10: The solid is coated by the
liquid on a surface area proportional to r − φS , whereas the (flat) liquid/vapor interface develops
on a surface area proportional to 1 − φS .
For a film progressing by a distance dx (larger than the scale of the defects), the variation in
surface energy per unit length perpendicular to the figure can be written as
dE = (γSL − γSA )(r − φs )dx + γLV (1 − φs ) dx.
7.
The progression is favorable provided (once again) that the Young angle is smaller than a critical
value θ c , which depends only on the design of the solid (56):
cos θc = (1 − φs )/(r − φs ).
8.
Liquid invasion on a microstructured solid can thus be tuned by the geometry of the structures.
For dilute defects (small φ S ), we have cos θ c ≈ 1/r: The rougher the substrate, the larger θ c , i.e.,
the more likely that hemiwicking occurs. For a substrate composed of disconnected defects (such
as posts), the liquid front must somehow be activated to achieve the jumps sketched in Figure 10.
For wetting liquids, this is made possible via the menisci, which form around each post, allowing the liquid to reach the next row. In other cases, the contact line can remain pinned in a
metastable Wenzel state, and an external source of energy (such as vibrations) must be employed to
nucleate a contact with the next rows of pillars. We can even imagine equilibrium situations in
which the drop coexists with a wet ring of finite extension (looking a bit like a fried egg). If, for
example, the energy barrier is passed owing to the action of the Laplace pressure, the progression
can stop once the drop spreads enough to make its Laplace pressure too small for inducing a
further motion.
We can interpret the second regime (for θ < θ c ) in Figure 5 as resulting from hemiwicking.
In this experiment, the solid is very rough, with a fractal structure. Even if we do not know the
value of the parameter φS , we expect it to be smaller than 1, so cos θ c should be of the order of
1/r. The second regime indeed starts close to the abscissa where the Wenzel regime (of slope r)
intercepts the line cos θ ∗ = 1, that is, for cos θ ≈ 1/r. The apparent angle θ ∗ then hardly depends on
θ , which can be understood qualitatively: The drop sits on a composite surface consisting mainly
of liquid, apart from a few solid islands. The angle θ ∗ should be very close to 0 (the value it would
take if there were only liquid), but it cannot reach this value owing to the islands on which the
angle is θ > 0. The number of islands should be a function of θ, which makes it difficult to produce
a general theory. The value expected for the angle θ ∗ is based on Figure 11, in which we sketch
the drop coexisting with the impregnating film.
We consider a displacement of the contact line by a quantity dx. The solid becomes wet on
a fraction of surface φS , and liquid interfaces are eliminated on a fraction 1 − φS . Because the
displacement also implies an increase of the liquid/vapor interface of the drop, the total change of
surface energies eventually becomes (per unit length of the line)
dE = φs (γSL − γSA ) dx − (1 − φs )γ dx + γ dx cos θ ∗ .
82
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9.
dx
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Liquid
θ
Solid
Figure 11
Drop coexisting with a film that self-propagated within the material textures. The apparent angle is obtained
by considering a displacement of the contact line and computing the corresponding variation of surface
energy.
The minimum of E yields the apparent angle θ ∗ , as first shown by Cassie (57, 58):
cos θ ∗ = 1 − (1 − φs ) cos θ.
10.
For small φS , the angle θ ∗ hardly varies with θ, as observed in Figure 5 for θ < θ c . This variation
is linear when the cosines of both angles are plotted, and the slope provides φS . We would deduce,
for example, from Figure 5 that φS ≈ 0.15. However, the actual behavior should not be linear for
a disordered surface, for which the proportion of emerged islands should itself be a function of θ
(the smaller the θ, the smaller the φS ), making the actual variation θ ∗ (θ) less simple.
4.3. Dynamics of Hemiwicking
The force that drives hemiwicking can be derived from Equation 7. We get, per unit length,
F = −dE/dx = γ (r − φS ) (cos θ − cos θ c ), which is fixed by the quality of wetting and by the
design of the surface. If wetting is complete (θ = 0), a molecular layer propagates ahead of the
impregnated film, which lowers the surface energy via the suppression of liquid/vapor interfaces
only, on a surface area proportional to r − 1. Hence, the force then is γ (r − 1). It only depends
on the roughness and logically vanishes if the material is flat (r = 1).
Owing to the small scale of the textures, this (constant) driving force is generally balanced
by viscous force. This resistance to the flow should scale as ηVx, denoting x as the impregnated
distance and η as the liquid viscosity. Balancing both forces, we find that the film should progress
as the square root of time, with dynamics similar to the wicking Washburn law inside a porous
medium (59). For rough substrates as for grooves, the liquid films indeed progress in a Washburn
fashion (60–62). This is also true within forests of microposts (56). In this case, we can calculate
the coefficient characterizing the dynamics (if we note x2 = Dt, D is this coefficient), allowing us
to be more precise about the way the dynamics can be tuned by the design of the posts (54).
For wetting liquids and posts of height h, distance p, and radius b (b p), the last paragraphs
predict a wicking force scaling as γ bh/p2 . The exact form of the viscous resistance depends on the
pillar height. (a) For short pillars (h p), the dissipation is fixed by the depth h of the flow (the
velocity gradients take place between the bottom solid and the free liquid interface). We deduce
a viscous force (per unit length) ηVx/h and thus a dynamic coefficient D ∼ (γ /η) (bh2 /p2 ), which
is efficiently tuned by the pillar height. (b) For tall pillars (h p), dissipation takes place mainly
around the pillars. The viscous force per pillar is ηVh (omitting an Oseen logarithmic factor),
with x/p2 pillars per unit length. Hence, there is a viscous force ηVhx/p2 , which eventually yields
a very simple dynamic coefficient: D ∼ γ b/η. We thus see how the design can be optimized: If for
a given application a film of height h must propagate, we select pillars of height h (hemiwicking
www.annualreviews.org • Wetting and Roughness
83
is a good way for setting films of a desired thickness). Then, p is chosen to maximize the speed
of propagation; it will be taken of order h because there is no dynamic benefit for having more
compact networks.
5. SUPERHYDROPHOBICITY
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5.1. Air Trapping
On hydrophobic solids, the situation is of course different from that for hydrophilic solids. If
the solid is rough enough, we do not expect that the liquid will conform to the solid surface, as
assumed in the Wenzel model (Figure 4). Rather, air pockets should form below the liquid [this is
the so-called Cassie or fakir state (48–49)], provided that the energetic cost associated with all the
corresponding liquid/vapor interfaces is smaller than the energy gained not to follow the solid.
This criterion can be made more quantitative by consideration of, again, pillar-like textures. If
the liquid/air interfaces are assumed to be flat (which can be justified by a condition of constant
Laplace pressure in the liquid, which, for defects much smaller than the drop size, can be taken
as null), the wet and liquid surface areas are proportional to (r − φS ) and (1 − φS ), respectively.
Hence, air pockets are favored, provided that (63–65)
(r − φs ) (γSL − γSA ) > (1 − φs )γ ,
11.
which (through the use of the Young formula) can be reformulated as θ > θ c , with
cos θc = −(1 − φs )/(r − φs ).
12.
This criterion is similar to the one established for propagating a film of liquid inside the texture.
Air here replaces liquid, so the critical angle expected from Equation 12 is just π minus the
critical angle below which hemiwicking takes place (Equation 8). For very rough solids (r 1),
this criterion is always satisfied. Then, θ c tends toward 90◦ , and θ is indeed larger than this
value, because we assumed chemical hydrophobicity. For materials decorated with long hairs, for
example, the roughness factor r ≈ 2πbh/p2 can typically be 5 to 10 (as deduced from Figures 6c
or 7c, for example). Figure 12 confirms that the leg of Microvelia, a small bug walking on water,
does not contact the liquid, as evidenced by the distance visible between the leg and its reflection
(24).
a
b
1 cm
Figure 12
(a) A Cassie state in action: Microvelia walking on water (scale bar, 1 cm). (b) Thin (hydrophobic) hairs allow
the bug to be repelled by and to skate on water. From Reference 24 (courtesy of D. Hu and J.W.M. Bush).
84
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Liquid
Air
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θ
Solid
Figure 13
Displacing the contact line in the Cassie regime: The energy balance must include the creation of liquid/air
interfaces below the drop, as indicated by the dotted lines.
The situation is more ambiguous for modest roughness factors, such as those provided by
small pillar density φS . The chemical contact angle θ is typically 100◦ to 110◦ , and its cosine is
slightly negative. Thus, the criterion for air trapping is not obeyed. However, Cassie states are
often observed in spite of a higher interfacial energy (64, 66, 67). The air present before we place
a drop can remain trapped in a metastable state, as long as the drop does not nucleate a contact
with the ground surface of the solid. We discuss in Section 5.3 the metastability of Cassie states.
This Cassie regime is the one of interest because, in addition to a large contact angle, it provides a
small contact angle hysteresis, owing to the presence of the air cushion. As a consequence, we term
superhydrophobic the only Cassie regime, which generates amazing properties such as reduced
adhesion, water repellency, and slip (partially discussed in Sections 5.2 and 6.3).
Because the drop sits on a mixture of solid and air, we expect a large apparent angle θ ∗ . If
there is only air, Young (Equation 1, where we replace the index S by A) predicts a “contact” angle
of 180◦ (i.e., no contact). Any deviation from this value tells us the proportion of solid actually
contacting the liquid.
The variation of interfacial energy arising from a displacement of the contact line by a quantity
dx (as sketched in Figure 13) is related to the creation of new wet solid surface and liquid/vapor
interfaces. The final balance can be written as
dE = φs (γSL − γSA ) dx + (1 − φs )γ dx + γ dx cos θ ∗ .
13.
It is crucial to assume flat liquid/vapor interfaces. Consideration of some curvature would modify
the result (liquid/vapor interfaces would then have a larger surface area), suggesting that the
angle should (slightly) increase as the drop gets smaller. In recent experiments, Rathgen et al.
(68) analyzed the light diffracted through (transparent) Cassie materials, providing a very precise
measurement of this curvature.
At equilibrium, E is the minimum, which yields the apparent angle θ ∗ (69):
cos θ ∗ = −1 + (1 − φs ) cos θ.
14.
This description must be complemented by a (local) Young condition at each contact line (at
the edge of the drop and for each liquid/vapor interface below). This condition is satisfied by
the presence of edges on the posts (or more generally of large slopes on the rough material). As
discussed above in the context of Figure 3 and because we have θ > 90◦ , sharp angles permit
this condition. We expect stronger pinning if these edge angles are smaller: Re-entrant designs
will make more robust Cassie states, and we see below (Section 5.3) that they can even induce air
trapping in a hydrophilic situation (29).
Equation 14 usually predicts large angles. For θ ≈ 110–120◦ and φS between 5% and 10%,
we get apparent angles of 160–170◦ . In Figure 14, we see a millimetric water drop on a silicon
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85
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Figure 14
Millimetric water drop
on a hydrophobic
surface textured with
regularly spaced
micropillars. The
texture acts as a
diffraction grating,
which induces
structural colors. From
Reference 70 (courtesy
of M. Reyssat).
substrate where silicon micropillars (similar to Figure 7a, with φS = 5%) were etched and coated
with a fluoropolymer (70). The drop is like a pearl (69, 71), sitting on a solid whose iridescences
reveal the regular array of “defects,” which diffracts light (structural color) (66, 72, 73). Conversely,
structures much smaller than the wavelength of light yield transparency (74–76). This raises the
interesting question of the smallest size generating water repellency, which remains to be solved.
The smaller is φS , the larger is θ ∗ . But for a more complex topology, φS should be a function of
θ (69). Conversely, the measurement of θ ∗ in a Cassie situation should provide the solid fraction
φs contacting the liquid, a quantity of interest for characterizing not only wetting but also hydrodynamic slip (see Section 6.3) or any properties related to a solid/liquid contact (e.g., electrical or
chemical). In the limit of small φS , we note that θ ∗ = π − ε (with ε 1), and Equation 14 rewrites
to
ε ≈ 2 cos (θ /2)φs1/2 ,
15.
whose critical behavior in φS emphasizes the difficulty for achieving a strict nonwetting situation
(ε ≈ 0). However, Gao & McCarthy (51) approached, and perhaps reached, this limit (within
the uncertainty of the measurements) by using nonwoven assemblies of nanofibers (Figure 7b) on
which both the advancing and receding angles were 180◦ . The same authors reported similar results
for pulverulent hydrophobic solids obtained by compressing commercially available lubricant (37).
In both cases, the texture is submicrometric and quite regular, with smooth rounded defects, which
should induce a very low hysteresis—the quantity we now discuss.
5.2. Toward Nonsticking Water
The most important property of a Cassie surface is its small contact angle hysteresis. As a consequence, drops are unusually mobile, which generates novel properties, such as bouncing, that
are not observed on conventional materials (77). Liquids behave to a large extent as Leidenfrost
drops, for which the underlying vapor minimizes the friction. However, one generally observes on
superhydrophobic materials a residual hysteresis (and thus adhesion), whose value is still debated
today (78–83). A key factor in this discussion is the possibility of pinning the contact line on the
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y
p
2b
Figure 15
In a Cassie state, a drop is likely to pin on the edges of the defects as we displace it. The drop becomes
distorted, and the energy stored in this deformation fixes the amplitude of the hysteresis.
defects. Therefore, the shape of the defects or the sharpness of the edges is significant, as evidenced
in a classical experiment by Öner & McCarthy (71) in which they varied the post shape.
Here we restrict the discussion to the case of strong pinning on dilute defects, elucidated
in 1984 by Joanny & de Gennes (84) and Pomeau & Vannimenus (85). The model is based on
Figure 15, in which we see a few defects on which the line pins as we move the liquid, using a
force F. The line meets φ S defects per unit area and thus, for a displacement dx, φ S dx defects per
unit length. Passing each of them, an energy is stored and then released in the liquid (where it
gets dissipated by viscosity) as the line depins. Hence, we have
F dx = φS dx.
16.
We guess that will depend on the shape of the defects (for example, complex contours will
generate a higher , and thus a larger hysteresis, unlike small and rounded defects). For the sake
of simplicity, we assume an equilibrium (Young) angle of 90◦ ; in addition, our defects are pillars
(or disks) of radius b and mutual distance p (with b p). We thus have φ S ∼ b2 /p2 .
As the line pins on a defect of size b, the drop gets distorted, as shown in Figure 15. Its tails
form surfaces of zero curvature: r = b cosh (x/b); that is, for x b, r ≈ 1/2 b exp(x/b). The
deformation is maximum (x = u) for the largest lateral deformation r, i.e., for the typical distance
p between two defects. Hence we get
u ≈ b log ( p/b).
17.
The pinning force on a defect f is related to b (the line pins on the contour of each defect): f ≈
πbγ , which yields a relationship between force and deformation,
f ≈ π γ u/ log ( p/b).
18.
Equation 18 defines a linear spring of stiffness K = π γ /log(p/b) (84). Hence, there is an energy
= 1/2Ku2 stored in the deformation. The force necessary to move the line can be written, per
unit length, as
F = γ (cos θr − cos θa ).
19.
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87
Putting together these equations, we finally get
(cos θr − cos θa ) ∼ φS log(1/φS ).
20.
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The contact angle hysteresis vanishes with the density of defects, but the presence of a logarithm in
Equation 20 makes this behavior quite pathological: At small φ S , the hysteresis decreases because
there are fewer defects (the linear term), but the logarithm term (slowly) diverges, making the
residual hysteresis appreciable. This result seems to be in agreement with many existing data,
but more remains to be done to check these models quantitatively and to extend them to more
complex patterns.
Hysteresis makes drops stick on solids, despite gravity field or air flow. A general calculation
of the sticking force is difficult, but a simplified argument allows us to evaluate how the hysteresis
enters this quantity (86). We assume that the rear half of the drop joins the solid with an angle θ r ,
whereas the front half meets it with an angle θ a . The capillary sticking force can be written as π γ
(cos θ r – cos θ a ), denoting l as the radius of the solid/liquid contact (quasi-circular for θ 1).
Assuming a geometric contact l ≈ Rε (where ε is the difference between the mean angle and π ,
and R is the drop radius) and using Equations 15 and 20, we find that a drop will move in the
gravity field (on a vertical window pane) provided that
φs3/2 log(1/φS ) < R2 κ 2 ,
21.
in which we introduce the capillary length κ −1 = (γ /ρg)1/2 (2.7 mm for water) and ignore all
the numerical coefficients. Once again, the density of defects is crucial for driving the wetting
properties (here the degree of adhesion of a drop on a solid). As we could guess, small densities are
required for suppressing adhesion. However, such a limit also weakens the stability of the Cassie
state, which we now discuss.
5.3. Metastable Cassie States
As stressed above, drops are often observed in a Cassie state in spite of a smaller Wenzel energy.
As a consequence, these metastable Cassie states are fragile (70). Figure 16 shows two millimetric
water drops on a microtextured substrate (with a pillar density φ S of 1% and pillar height h of
12 μm). The first drop is placed without any impact, and the second drop is released from a few
centimeters such that it meets the solid with a velocity of a few tens of centimeters per second.
Figure 16
Millimetric water drops of the same volume on a superhydrophobic substrate covered with dilute pillars
(φS = 0.01, h = 2 μm). The drop on the right was thrown on the substrate, whereas the one on the left was
carefully deposited. As observed, Cassie and Wenzel states (left and right, respectively) can coexist. From
Reference 70 (courtesy of M. Callies).
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The first globule is a Cassie drop (we even see light passing below it), whereas the second one is
in a Wenzel state: On this substrate of small roughness (r ≈ 1.1), the contact angle is close to the
Young angle, slightly larger than 90◦ in this example.
When the Wenzel state is the less energetic one, any perturbation of a Cassie drop can provoke
its transition to this state. Conversely, a Wenzel drop is firmly bound to its stable configuration. It
is of practical importance to quantify the robustness of metastable Cassie states and to understand
the conditions provoking impalement. Obviously, an energy barrier must be overcome to find the
ground state (87–90). One can evaluate this barrier by considering the penetration of a Cassie
drop. Assuming unchanged liquid/vapor interfaces as the drop sinks, the only change in surface
energy corresponds to the (unfavorable) wetting of the posts’ walls. This implies a (positive) energy
per unit area E = (γ SL − γ SA )(r − 1) = −γ (r − 1) cos θ (this quantity becomes negative in a
hydrophilic situation). We thus find an energy barrier E ≈ 2πbh/p2 γ cos θ. It is proportional
to the pillar height h, which appears as a natural parameter for tuning this quantity. The energy
barrier E is generally too large to be overcome by thermal energy (we need defects of molecular
dimensions to get E of the order of kT ). However, the energy can be supplied by pressing on
the drop (66, 90), by vibrating the substrate (91), or by an impact (92, 93). Indeed, the higher the
posts, the larger is the resistance to impalement. Once the liquid penetrates the texture, it remains
strongly pinned, and the “printed” drop is even able to conform to the network of microposts (92).
Sbragaglia et al. (94) described the dynamics of the Cassie/Wenzel transition as very quick and
following a zipping mechanism: One row of cavities gets filled (in a time of approximately 10 μs
on a length of 100 μm!) before jumping to the next row. This process is somehow reminiscent of
the progression in a groove sketched in Figure 8. The surface force here involves the creation
of wet surfaces and the suppression of suspended liquid/air interfaces, whereas the resisting force
should be viscous. We thus expect a Washburn law for the progression (see Section 4.3), which can
be extremely quick at the small scale of the phenomenon. For a pattern composed of posts whose
height h is comparable to the pitch p (of approximately 10 μm), the typical time for invading a row
of length x scales as ηx2 /γ h, of the order of 10 μs for water and x = 100 μm.
The way in which solid/liquid contacts nucleate for triggering the transition is interesting.
Interfaces above the air pockets are curved, fitting the global curvature of the drop (Figure 17). If
the drop becomes small enough, the liquid can reach the underlying solid and then propagate. The
size of a drop should thus impact the drop’s wetting state. Indeed, small droplets are more likely
to be in a Wenzel state than are large droplets (66), which can also be evidenced by observing the
evaporation of a drop. Then, the drop’s size varies continuously, and investigators have reported
the existence of a critical radius below which the drop suddenly falls into the Wenzel regime
(95, 96).
Following the notations in Figure 17, the interface curvature scales as δ/p2 (for h < p). Equating
it with the drop curvature 1/R yields the depth of penetration δ of the interface inside the texture:
δ ∼ p2 /R; the smaller the drop, the larger δ is. When it becomes of the order of the pillar height
δ
h
p
Figure 17
The liquid/vapor interface is curved, owing to the curvature of the drop, but the interface also can be curved
if we apply pressure to the drop. δ denotes the lowering of the interface below the top of the posts.
www.annualreviews.org • Wetting and Roughness
89
h, a solid/liquid contact can nucleate on the bare substrate and propagate if the Cassie state is
metastable. This implies a critical radius for a Cassie drop scaling as (96)
R∗ ∼ p 2 / h.
22.
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Note that, as deduced from Figure 3, depinning from the edge will also occur if the drop radius is
smaller than p/|cos θ|, a limiting condition for modest chemical hydrophobicity. Here we assume
that this second condition is screened by the first one, i.e., p > h/|cos θ|.
The radius R ∗ can be much larger than p if h < p. The Cassie state will be all the more robust
because this critical radius is small (no drops fall in the Wenzel drops, except invisible ones). One
can achieve this in two ways: either by making h large, using micro- or nanofibers for decorating
the solids (see Figures 6c and 7c) (96), or by reducing both p and h. Miniaturizing the pattern size
enhances the resistance of the Cassie state, which may explain the existence of such small scales
in many natural materials.
Jiang and coauthors (40) recently reported that the eye of C. pipiens apparently remains dry
even if exposed to tiny drops, as encountered in the foggy and moist environments where these
mosquitoes usually circle. Figure 18 is a close-up of C. pipiens after the mosquito passed through an
aerosol of water droplets. Water condenses on most of the animal, but the eyes remain dry, which
of course preserves its vision (renowned as excellent). Figure 6d displays a microphotograph
of the textures observed on the surface of the eye; these are remarkably small, with p ∼ h ∼
100 nm. With these values, we get R ∗ ∼ 100 nm: A drop at this scale not only is invisible but also
evaporates quasi-instantaneously. In a cloud, drops are quite polydisperse, with a typical radius
of 10 μm—such small drops would impale on most microtextured surfaces but might resist the
Wenzel transition for the nanopattern worn by the mosquito’s eye.
Other promising metastable Cassie states are those obtained on hydrophilic materials with
a particular design. Oils having contact angles of the order of 40◦ can be suspended on special
textures, producing an increase of the angle by approximately 100◦ (the superoleophobic effect)
(53, 97–102). As shown by Herminghaus, fakir wetting drops are possible on overhangs or reentrant angles, that is, sites where a hydrophilic Young condition can be satisfied (29). Fibers
and defects with overhangs do provide quite robust oleophobicity and are able to resist a Wenzel
transition by pressing on the liquid with the Laplace pressure related to the size of the “holes”
Figure 18
Close-up of Culex
pipiens after exposure
to water aerosol.
Droplets condense on
the antennas, but the
black eyes remain
dry—a condition for
preserving the eyesight
of the insect. From
Reference 40 (courtesy
of L. Jiang).
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at the surface. With structures with pronounced re-entrant profiles, such as those displayed in
Figure 7d (where the pattern evokes a mushroom, or a hoodoo), Tuteja et al. (53) spectacularly
reported that even a liquid as wetting as octane can be suspended in a Cassie state with advancing
and receding contact angles as high as 160◦ and 140◦ , respectively.
6. SPECIAL PROPERTIES
As we see above, hydrophobic Cassie materials generate high contact angles and small hysteresis,
ideal conditions for making water drops very mobile. We conclude this article by discussing a few
special properties potentially generated by these surfaces, such as anisotropy, wettability switches,
and slip.
6.1. Anisotropy
Many natural (Figure 6a,b,d ) and synthetic (Figure 7a,b,c,d ) textures are isotropic. However,
it can be interesting to design directional structures, such as arrays of parallel grooves or microwrinkles, that consequently generate anisotropic wetting, in particular, in the Cassie regime
(69, 103–105). Owing to a differential pinning of contact lines, the contact angles (and the hysteresis) are quite different along and perpendicular to the grooves. Axial motion is preferred, and
such designs are appropriate when liquid must be guided.
There are examples of such patterns in nature (38, 41, 106), as in Figure 19, which shows the
scales covering the wings of the butterfly Papilio ulysses. Both the arrangement and microtexture of
the tiles contribute to the directionality of this material. Another kind of anisotropy is exploited
by water striders (see its inclined hairs in Figure 6c): Striders strike the surface perpendicular to
the grooves, which generates a large contact force, before swinging the legs by 90◦ to align them
in the direction of the motion for skating. Motion will arise from alternating pinning and gliding
events (24).
Figure 19
The wings of Papilio
ulysses. The way the
tiles are displayed
together with the
detail of the texture
confer anisotropy to
the texture. From
Reference 106
(courtesy of S.
Berthier).
100 µm
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91
6.2. Wettability Switches
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Textured surfaces undergo a brutal change of wettability as the contact angle exceeds 90◦
(Figure 5). This behavior can be exploited to achieve superhydrophobic/superhydrophilic
switches. Different physicochemical effects affecting the solid wettability, such as light on photocatalytic textures (107, 108) or heat for temperature-sensitive coatings (109), can be used for
triggering the transition. The comprehensive review by Feng & Jiang (50) provides details.
Electric field, as we have learned from Lippman, also affects wettability. Applying a voltage
across a drop lowers its contact angle, and this effect is amplified on a textured surface. A drop
with an angle of approximately 160◦ can nearly spread under the action of modest voltages (approximately 10 V) (73). However, the liquid gets irreversibly pinned in the superhydrophilic state
(or in any state in which it intimately contacts the rough solid), contrasting with our expectations
for a switch. Krupenkin et al. (110) proposed to use a short and intense pulse of current (through
a thin conductive layer on the sample), which evaporates the liquid close to the surface, hence
restoring a Cassie-suspended state. More generally, there is today no clear example of a Wenzel
state (even potentially metastable) spontaneously transforming into a Cassie state. This situation
is detrimental as a vapor condenses: This naturally forces a Wenzel situation, which often evolves
toward mixed and ambiguous Cassie/Wenzel situations (111–115). Much remains to be done to
achieve genuine antidew materials.
6.3. Giant Slip
Experiments confirm that superhydrophobic materials can provide slippage as water flows on
them, provided that these materials are in the Cassie state. The amplitude of the phenomenon is
captured by the so-called slip length, which is the extrapolated distance on which the liquid velocity
vanishes (Figure 20). First introduced by Navier, this length is generally molecular. However, it
can become of the order of 10 nm on flat hydrophobic solids, as experiments using a surface
force apparatus show (116). Similar to what we saw for wetting, this hydrophobic behavior can be
dramatically enhanced if the solid is rough (117).
Ou et al. (118) reported micrometric slip lengths from measurements of the pressure necessary
to drive a given flux of water along a square channel striated with microgrooves. For a Poiseuille
flow, the flux varies as W 4 ∇p/η, denoting ∇p as the pressure gradient along the flow and W as the
width and depth of the channel. For a large slip (λ > W ), the flux instead scales as λW 3 ∇p/η. Slip
at the wall reduces the pressure gradient by a factor λ/W. Ou et al. found pressure reduction by
approximately 40%, suggesting slip lengths of the order of 10 μm (see also References 119–121).
λ
Figure 20
The slip length λ is the distance inside the solid for which the velocity profile of a flowing liquid vanishes.
92
Quéré
Figure 21
2
Slip length on
superhydrophobic
patterns of
nanotubes of
constant density, but
with a varying pitch.
In the Cassie state
(circles), the slip
length is
micrometric and
increases with the
pitch. In the Wenzel
state (squares), there
is no measurable slip.
(µm)
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1.5
1
0.5
0
0
2
4
p (µm)
6
Using microparticle imaging velocimetry, Joseph et al. (122) directly measured slip lengths for water flowing on hydrophobic carbon nanotubes. As shown in Figure 21, the slip length dramatically
depends on the nature of hydrophobicity. In a Wenzel state (induced by pressing on the liquid),
there is no measurable slip, in agreement with the paper of Richardson (123), who stipulated that
roughness kills any (potential) slippage if the liquid conforms to it. Conversely, micrometric slip
lengths are observed in the Cassie state, and increase linearly with the post distance p, in these
experiments performed at constant φ S . This slip can be dramatically damped if liquid/air interfaces
are curved (as in Figure 17), owing, for example, to a pressure exerted on the liquid (124).
It is natural to expect a large slip in a Cassie situation: Liquid glides on air, owing to the large
viscosity ratio between water and air (typically a factor of 100). However, part of the liquid contacts
the top of the posts, which limits the total slip on the surface. We can make this argument more
quantitative, following a recent analysis by Ybert et al. (125). For a flow of typical velocity V, the
size “affected” by the presence of a post should scale as b, the post radius (b p, the pitch of the
post array). The friction force per pillar, and thus per surface area p2 , should scale as ηVb, denoting
η as the liquid viscosity. This yields a viscous stress σ ∼ ηVb/p2 . This stress dominates the one
arising from the underlying air flow, provided that ηVb/p2 > ηa V/h, in which we introduced the
air viscosity ηa . Hence there is a geometric requirement, bh/p2 > ηa /η, which can be achieved by
adjusting the post height h. With water, ηa /η is of the order of 10−2 , and the latter criterion will
be satisfied with posts of characteristics b = 1 μm, h = 10 μm, and p = 10 μm, for which the
factor bh/p2 is 10−1 . As seen in Figure 20, the stress σ can also be written as ηV/λ, from which we
deduce an effective slip length λ:
1/2
λ ∼ p 2 /b ∼ p/φS .
23.
For a constant pillar density φ S , λ is linear with the pillar spacing p, as seen in Figure 21. For
b p (φ S 1), we expect very large slip lengths, compared with the values found on flat solids.
λ will typically be between a few p, i.e., 1–10 μm (as reported experimentally), 1000 times larger
than the slip length on a flat hydrophobic solid! If air friction dominates the pillar friction (h <
ηa p2 /ηb), the slip length becomes λ ∼ ηh/ηa , which can be very large as well.
The contact angle θ ∗ is also determined by the density φ S (Equation 14), so θ ∗ and λ should
be correlated. At small φ S , we note θ ∗ = π − ε. Introducing Equation 22 in Equation 15 [where
www.annualreviews.org • Wetting and Roughness
93
2 cos (θ /2) is taken of order unity] yields
λ ∼ p/ε,
24.
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which directly connects the quality of the nonwetting with the amplitude of the slip.
It is of obvious practical interest to maximize the slip and thus to work with low post densities.
However, we see that the liquid in this limit is likely to sink inside the texture, provoking a
complete failure of the slip properties. There again, there is an optimum to find, and the design
to be given to the microstructures might itself be questioned. The design can also induce slip
anisotropy. If grooves are considered, for example, slip is expected to be larger along the grooves
than perpendicular to them. One can prove that the slip length (which is again of the order of p,
the distance between grooves) differs by a factor of two in both directions (126–128), reflecting
the factor-of-two difference of the viscous force on a slender object (such as a cylinder) in both
directions.
7. CONCLUSION
Textured surfaces, which provide superwetting, superslip, and superhydrophobicity, are supersurfaces. But so what? After a decade of intense research, we have hundreds of materials for which
drops behave in the eccentric ways we described, but there is no real large-scale application (contrasting with many other smart surfaces, such as self-cleaning ones, for example). The fragility
of these materials (in both a mechanical sense and a thermodynamic sense) seems to limit industrialization. Conversely, the textures in the natural world can be repaired and protected from
contamination (insects spend an appreciable part of their existence grooming their legs). However, there may be short-term applications for synthetic materials, such as disposable devices (for
microfluidics) or temporary surface treatments with microbeads (sticking on the material and
forming there the desired microstructures)—both cases in which aging is not a major obstacle.
On a more fundamental point of view, many interesting questions remain unsolved, among
which we select (a) the question of the reduction of size—it would be useful to quantify how
the wetting properties vary as the size of microtextures vanishes; (b) the question of optimization:
What is the “best” of the microstructures, according to the searched application? We need to define
a battery of tests to build a classification of the existing textures, before going further to more
detailed models; (c) the search for new properties, such as antidew, for which the use of flexible
microfibers seems promising; and (d ) a quantitative understanding of contact angle hysteresis,
which remains to be fully done. As we see, wetting and roughness should continue to play their
entertaining game with each other in the coming years!
DISCLOSURE STATEMENT
The author is not aware of any biases that might be perceived as affecting the objectivity of this
review.
ACKNOWLEDGMENTS
It is a real pleasure to thank Anne-Laure Biance, José Bico, Aurélie Lafuma, Mathilde Reyssat,
and Denis Richard for the many exchanges on pearl drops. I also thank Yong Chen, Chieko
Ishino, Anne Pépin, and Ko Okumura for stimulating collaborations. I am very grateful to Hervé
Arribart, Serge Berthier, Lydéric Bocquet, John Bush, Christophe Clanet, Lichao Gao, Stephan
Herminghaus, L. Mahadevan, Thomas McCarthy, Glen McHale, Gareth McKinley, Lei Jiang,
94
Quéré
Christoph Neinhuis, and Julia Yeomans for precious discussions and providing documents for this
report. Finally, the help of A. Dechy and V. Dolies regarding the pictures was greatly appreciated.
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