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Phil. Trans. R. Soc. A (2008) 366, 1539–1556
doi:10.1098/rsta.2007.2171
Published online 11 January 2008
Non-adhesive lotus and other
hydrophobic materials
B Y D AVID Q UÉRÉ *
AND
M ATHILDE R EYSSAT
Laboratoire de Physique et Mécanique des Milieux Hétérogènes, ESPCI, 75005
Paris, France
Superhydrophobic materials recently attracted a lot of attention, owing to the potential
practical applications of such surfaces—they literally repel water, which hardly sticks to
them, bounces off after an impact and slips on them. In this short review, we describe
how water repellency arises from the presence of hydrophobic microstructures at the
solid surface. A drop deposited on such a substrate can float above the textures,
mimicking at room temperature what happens on very hot plates; then, a vapour layer
comes between the solid and the volatile liquid, as described long ago by Leidenfrost. We
present several examples of superhydrophobic materials (either natural or synthetic),
and stress more particularly the stability of the air cushion—the liquid could also
penetrate the textures, inducing a very different wetting state, much more sticky, due to
the possibility of pinning on the numerous defects. This description allows us to discuss
(in quite a preliminary way) the optimal design to be given to a solid surface to make it
robustly water repellent.
Keywords: wetting; roughness; superhydrophobicity
1. Drop on flat solids
When depositing a small drop on a solid, we commonly observe that the liquid
makes contact with its substrate, whose size is most often of the order of the drop
diameter (figure 1). Owing to the existence of surface tension (denoted as g), the
liquid–vapour interface is spherical (this minimizes the corresponding surface
area), so that the contact size is geometrically set by the contact angle q with
which the liquid meets the solid
ð1:1Þ
[ Z R sin q;
where R is the radius of curvature of the spherical cap.
The contact angle itself is obtained by balancing the three surface tensions
acting on the contact line. Each interface draws on the line for reducing the
surface area of the corresponding interface, and the balance can be written on
the horizontal axis
g cos q C gSL Z gSV ;
ð1:2Þ
* Author for correspondence ([email protected]).
One contribution of 7 to a Theme Issue ‘Nanotribology, nanomechanics and applications to
nanotechnology II’.
1539
This journal is q 2008 The Royal Society
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D. Quéré and M. Reyssat
2
R
Figure 1. A liquid drop on a solid forms a spherical cap, which meets the solid with an angle q
(situation of partial wetting). We denote [ as the radius of the solid–liquid contact, and R as the
radius of curvature of the drop.
Figure 2. Provided it is small enough, a liquid drop generally remains stuck on an incline, which is
due to the difference of angles at the front (on the left) and at the rear (on the right). (Courtesy of
Marc Fermigier and Christophe Clanet.)
where gSV and gSL are the solid–vapour and solid–liquid surface tensions,
respectively. Equation (1.2), the so-called Young’s relation, thus stipulates that
the contact angle is uniquely defined by the nature of the solid and liquid (Young
1805). If this property were practically obeyed, a drop would necessarily slide
when tilting the substrate: displacing the liquid along the plane does not modify
any surface energy (proportional to the corresponding surface area), yet lowers
gravitational energy. However, drops generally remain stuck on inclines
(figure 2)—multiple angles are observed for a liquid sitting on a solid. As a
result, the apparent angle at the ‘front’ is larger than the angle at the ‘rear’ (this
can also be observed when trying to displace a liquid on a solid, using an airflow,
for example), which generates a capillary force opposing the motion (Furmidge
1962). This phenomenon is due to the presence of defects on the solid, on which
the contact line can pin, hence the existence of larger (or smaller) apparent
angles, although some Young condition can be satisfied locally, at the scale of the
(microscopic) defects.
The magnitude of the sticking force can easily be estimated. We first assume
that the angles at the rear and at the front are, respectively, the receding and
advancing angles qr and qa (minimum and maximum possible apparent contact
angles). This condition, even if not necessarily verified experimentally, yields a
maximum value for the sticking force (Krasovitski & Marmur 2005). Then, the
force per unit length acting at the rear of the drop is gSVKgSLKg cos qr (which
tends to retain it), while the force acting at the front is gSVCg cos qaKgSL
(which also tends to retain it). Hence, the maximum sticking force can be written
(in absolute value) g(cos qrKcos qa). If we naively integrate this force along the
contact line, we find a sticking force FZpgR sin q(cos qrKcos qa), which scales
linearly with the drop size and denoting q as an average contact angle (Dussan &
Chow 1983; Quéré et al. 1998).
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Non-adhesive lotus
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Since the sticking force is of the order of gR and the weight of the order of
rgR3 (here we assume a tilting angle of 908 and denote r as the liquid density),
only drops larger than the capillary length kK1Z(g/rg)1/2 will slide. This indeed
corresponds to the very common observation we can make, looking at window
panes on a rainy day—drops larger than a few millimetres slowly creep down,
while smaller ones remain immobile. If we wish to make water drops mobile on a
solid (to avoid optical distortions or corrosion), the strategy seems simple—
we must either minimize the contact angle hysteresis Dq (DqZqaKqr) and/or
maximize the (average) contact angle q. Here we show how both these objectives
can be achieved using textured hydrophobic materials, restricting our discussion
to the (important practically) case of water, as a liquid.
We could reach the same conclusion from a slightly different point of view. In
many cases, it is desired to detach from their substrate drops such as the one
sketched in figure 1 (think about detergency, for example). It is instructive to
calculate the difference of surface energies for the drop in both situations
(contacting its substrate or detached from it). If we denote R 0 as the drop
size once detached, this energy difference is simply found to be
DE Z 4pgR20 ð1K R0 =RÞ, where R is the radius of the spherical cap formed by
the drop when it meets the solid with the angle q (the smaller q, the larger R).
This difference is positive—the surface energy of a drop is lowered as it contacts a
solid and reaches its equilibrium position sketched in figure 1. It is commonly of
the order of 1 mJ, a very large quantity compared with energies ‘available’ for the
system (such as thermal energy). As we can see, DE can be made much smaller
by decreasing dramatically the surface tension (which can be done using special
surfactants if the interface is a water–oil one) or by approaching (here again)
the limit of zero-wetting (R/R 0), the most efficient way for getting rid of
adhesive drops.
2. The Leidenfrost effect
As we have understood from the first paragraph, the mobility (and detachment)
of a drop should be maximized when making the wetting as low as possible.
Mercury is called quicksilver, because it realizes this condition on most solids,
which indeed makes it non-sticking. As for water, there is a situation where it is
as quick as mercury—the so-called Leidenfrost effect. This happens, as
discovered in 1756 by the German physician Leidenfrost (1996), when depositing
water on very hot solids (typically at a temperature higher than 2008C, i.e. much
higher than the boiling point of water). Then, the water drops are observed to
run everywhere, as we also see when pouring liquid nitrogen on the floor.
Leidenfrost correctly interpreted the effect—the drop never contacts the solid,
because a film of vapour comes in between the substrate and the liquid, which
levitates on this vapour cushion (figure 3).
Young’s relation can be used to calculate the contact angle. We replace in
equation (1.2) the solid by the vapour, which yields g cos qCgZ0—a Leidenfrost
drop makes a ‘contact’ angle of 1808 on its substrate, which corresponds to a
situation of complete drying, for which the solid–liquid contact vanishes
(as observed in figure 3). Hence, the liquid cannot pin anymore on the solid
defects, which also yields a zero contact angle hysteresis. The Leidenfrost
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D. Quéré and M. Reyssat
Figure 3. A water drop on a very hot solid (here steel at 3008C) levitates, owing to the presence of a
vapour cushion below. The bar indicates 1 mm, from which we deduce that the vapour thickness is
of the order of 100 mm. (Courtesy of Anne-Laure Biance.)
situation thus achieves the ultimate non-wetting state (qZ1808, and DqZ08) and
provides the highest conceivable mobility for a drop—photos such as figure 3 are
extremely difficult to take (tiny slopes make the liquid immediately run away),
and Leidenfrost himself had to trap his drops in hot spoons in order to observe
them conveniently.
In spite of a zero wetting, the drop in figure 3 is observed to be nonspherical—the bottom is flattened, so that, if we cannot see the film, we may
wrongly measure a large (yet not maximum) contact angle. This flattening plays
a key role in the life of a Leidenfrost drop—in this region, the distance between
the plate and the liquid is minimal, so that evaporation is enhanced. The
existence and size of this zone affect the thickness of the vapour film, which
results from a compromise between evaporation (which tends to feed the film)
and gravity (the drop presses on the film; Biance et al. 2003). The weight is
precisely the reason for this pseudo-contact, as shown by Mahadevan & Pomeau
(1999)—it tends to lower the centre of the mass of the liquid, which surface
tension opposes. The pressure in the film can be seen as resulting from the
action of the weight (then, it is 4/3prgR3/p[ 2), or as the Laplace pressure inside
the drop 2g/R (crossing the liquid–vapour interface below the drop does not
induce a pressure jump, since the interface there is flat). Hence the law for the
‘contact’
[ wR2 k;
ð2:1Þ
which is valid as long as the drop is smaller than the capillary length kK1, so
that it is quasi-spherical ([!R). The law for this contact is very different from
the one found in wetting (equation (1.1))—the contact angle naturally does not
matter anymore, and the variation with the drop size is much quicker (R 2
instead of R). This remark is important in the limit (studied hereafter) of nearly
zero wetting. Since adhesion (and friction, if the drop moves) are imposed by the
size of the contact, this quickly vanishing law (as a function of the drop size)
makes us expect a remarkable mobility for small drops, in this limit (provided
that sticking forces arising from the contact hysteresis do not overcome driving
forces; Mahadevan & Pomeau 1999).
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Non-adhesive lotus
(a)
(b)
10 µm
50 µm
(c)
(d)
10 µm
500 nm
Figure 4. Different superhydrophobic surfaces, as revealed by electron microscope photographs. In
each case, micro- or nanotextures are present, such as (a) bumps (on a magnolia leaf ), (b) hairs (on
a water strider leg), (c) regular microposts (synthetic surface obtained after etching) and
(d ) fibrous or spongy design (another synthetic material). (Courtesy of (a) C. Neinhuis,
(b) L. Jiang and (d ) L. Gao.)
3. The lotus effect
Several natural materials approach the Leidenfrost limit, at room temperature.
Duck feathers or the legs of water striders repel water, and contact angles
measured on such substrates are indeed remarkably high, in the range of
150–1708 (Gao & Jiang 2004; Feng et al. 2007; Bush et al. in press). In addition,
the contact angle hysteresis on these substrates is also very low (smaller than
208), which also contributes to make water mobile. More than 200 plants are
similarly superhydrophobic (Neinhuis & Barthlott 1997; Bhushan & Jung 2006),
the most famous one being the lotus, whose leaf does not retain raindrops—the
typical contact angle on such a leaf is 1608, with a hysteresis of approximately
108. Since these mobile drops take dust with them, as they roll, it is often said
that such materials are self-cleaning, which is referred to as the lotus effect. Here
we extend the definition of this ‘effect’ to the more basic property of material
having both a high contact angle and a low hysteresis.
Using an electron microscope, one can see on these different materials the
presence of microtextures, at the scale of approximately 10 mm (figure 4).
Different designs are observed: bumps (for lotus, and for many other plants, see
figure 4a), spikes for water striders (figure 4b) or microfibres (for other plants,
such as Drosera, the little carnivorous plant). In addition, it is most often
reported that a second scale of texture is present (it is generally submicrometric), but the question remains open (Neinhuis & Barthlott 1997;
Bhushan & Jung 2006; Gao & McCarthy 2006a; Feng et al. 2007). It is also worth
mentioning that sandy soils may become superhydrophobic, in particular if
organic compounds coat them (Doerr et al. 2000)—the texture is given by the
sand grains themselves, and the organic coating provides hydrophobicity. This
coating can result from a forest fire or from microorganisms. Such soils are
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D. Quéré and M. Reyssat
particularly fragile—rains cannot be absorbed, so they flow, taking with them
the first few centimetres of soil, where most nutrients are present. Superhydrophobic soils are thus often precursors of desertification.
All these natural materials are coated with waxy substances, which follow the
microtextures. This provides a chemical hydrophobicity that is enhanced by
the presence of microstructures. On a flat hydrophobic substrate, the contact
angle of water is typically 100–1108, and the use of fluorinated substances
increases this angle up to approximately 1208. Microtextures do not only increase
significantly (by approx. 508) the chemical hydrophobicity, but they also
generate a degree of water repellency that the surface chemistry alone would not
permit (Shafrin & Zisman 1964). Owing to the large variety of designs at the
solid surface (as seen in figure 4, for example), it is often asked which is the most
efficient one. This question is not really solved today, maybe because it generally
does not make sense—the answer depends on the required function, and we do
not expect similar microstructures for anti-rain purposes, for anti-dew properties,
for gliding (and potentially stopping) on water surface, etc.
There is, however, a particular design that recently attracted a lot of
attention, not only because it might be the simplest one but also because it allows
one to establish quantitatively the laws of water repellency. This design consists
of regularly spaced microposts, of section either square or circular (Bico et al.
1999; Öner & McCarthy 2000). In principle, three parameters are necessary to
define such a topography, namely the post radius b, the post height h and the
pitch p of the array (supposed to be square; in addition we shall here only
consider the limit b/p). Such structures can be made by etching, or by casting,
allowing us to set independently these three parameters, typically in the range of
1–100 mm. We show in figure 4c an example of such a surface, obtained after
etching a silicon wafer decorated by a square array of circular spots of
chromium—after lifting off these masks, and coating the whole material with a
fluorinated substance, a regular array of hydrophobic posts is obtained, on which
water drops are observed to make a contact angle of the order of 160–1708,
depending on the characteristics of the posts.
4. The Wenzel and Cassie states of wetting
The first natural and simple effect arising from the presence of microstructures is
an increase of the surface area of the material. We define the roughness r of a
material as the ratio of its actual surface area to its projected (or apparent) one.
For regular arrays of long and dilute, cylindrical posts (b/h and b/p), the
roughness simply is
r Z 1 C 2pbh=p2 :
ð4:1Þ
On these substrates, we can define two levels; namely, the top of the posts and
the ground on which they sit. Hence, a second natural parameter is the ratio fS
between the top and bottom surface areas, which is also the post density. It can
be written as
fS Z pb2 =p2 :
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Non-adhesive lotus
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(a)
(b)
Figure 5. Two possible states for a drop on a microtextured surface: (a) the drop either follows the
structures on the solid (Wenzel situation) or (b) sits at the top on the posts (Cassie or fakir state).
The observed behaviour depends on the chemical properties of the solid (expressed by its Young
contact angle) and on the characteristics of the design. However, different states can coexist, as
described further in the text.
For the example displayed in figure 4c, we have by1 mm, hy10 mm and
py10 mm, which yields ry1.6 (moderate roughness) and fSy0.03 (low density).
Note that such structures conveniently allow us to vary independently both
parameters r and fS—for example, increasing the height of the pillars without
modifying their radii and distance increases r keeping fS constant. When
depositing a water drop on such a solid, it can adopt (at least) two types of
configuration—conforming to the solid surface (figure 5a) or sitting on the top of
the posts (figure 5b).
The first and simplest picture is the so-called Wenzel state (figure 5a)—the
drop just follows the structures on the material (Wenzel 1936). This will affect
both the wetting (the value of the equilibrium angle, that is the one which
minimizes the surface energy of the drop) and the adhesion (characterized here
by the value of the contact angle hysteresis). As for the wetting, it is as if the
solid surface energies in equation (1.2) are multiplied by the same factor r, which
immediately yields an equilibrium contact angle qW
cos qW Z r cos q:
ð4:3Þ
A Wenzel state will thus enhance the chemical affinity of a drop towards a
solid—a hydrophilic situation (q!p/2) becomes more hydrophilic (q!q), while
a hydrophobic one (qOp/2) tends to be superhydrophobic (qOq). This effect (in
principle) can be amplified by increasing the roughness (that is, in our case, by
making the posts either higher or denser).
As for the hysteresis, it will generally be large in this situation (Johnson &
Dettre 1964; Lafuma & Quéré 2003). If we try to remove a drop, it pins very
strongly on the sides and at the edges of the microstructures, allowing the
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receding angle to reach values as small as 0 (even on hydrophobic solids).
Therefore the Wenzel regime will be used to make a liquid stick on its substrate,
in contrast to what we expect from a lotus-type material.
We guess that the Wenzel configuration should not be obeyed if the solid
roughness is (too) large. Then, it is energetically unfavourable for the liquid to
follow the solid asperities, owing to the large solid–liquid energy stored in such a
configuration (Bico et al. 2002; Patankar 2003; Marmur 2004). Rather, the liquid
should sit on the top of the posts (Cassie & Baxter 1944), contacting the solid
surface on a surface area reduced by a factor r/fS, compared with the Wenzel
state (this factor can typically be 10–100 on surfaces decorated with pillars).
Conversely, this Cassie state (sometimes referred to as the fakir state, for an
obvious reason—see figure 5b) implies the creation of a liquid–vapour interface
below the drop, on a fraction of surface of the order of 1KfS, considering these
interfaces as flat (the drop is large compared with the pillars, implying a negligible
curvature of the liquid–vapour interface at the scale of pillars). The cosine of the
apparent contact angle is then an average between the cosine of the Young angle
and the cosine of 1808, the angle of a water drop on air, where the weights for the
average are, respectively, fS and 1KfS. Hence an angle qC
cos qC ZK1 C fS ð1 C cos qÞ:
ð4:4Þ
It is worth noticing that the Leidenfrost limit can be approached if the post
density fS vanishes. But it cannot be reached: the pillars must sustain the liquid,
which implies fS greater than 0. For a density of posts of the order of 5–10% and a
Young contact angle of 110–1208, the Cassie angle qC is expected from equation
(4.4) to be of the order of 1608, in agreement with many observations on such
surfaces. More complex designs such as fractal ones (for which the effective fS is not
obvious to predict) were observed to lead to still higher values of contact angles, up
to 1758 (Onda et al. 1996). For the loose network of thin fibres seen in figure 4d
(where the elasticity of the network might play a role in the value taken by the
parameter fS), Gao & McCarthy could not even distinguish the value of the contact
angle from 1808 (note, however, that, for such high values, the experimental
determination of the angles is difficult; Gao & McCarthy 2006b, 2007).
We guess that the contact angle hysteresis will be generally very small in the
Cassie state, since the liquid mainly sits on air, on which it cannot stick (Lafuma &
Quéré 2003). It contacts only the solid at the top of the posts (or more generally of
the structures present on the solid), on which it can pin—the smaller fS, the less
efficient the pinning, and thus the smaller the hysteresis: in the limit of dilute
defects, each one contributes to the pinning of the line, so that we (roughly) expect a
hysteresis proportional to the defect concentration fS (Joanny & de Gennes 1984).
Measurements indeed confirm a very low hysteresis in the Cassie state, as has been
emphasized by Johnson & Dettre (1964), of typically 5–208 instead of more than
1008 in a Wenzel situation. Öner & McCarthy (2000) interestingly reported that the
strength of each defect is related to the form of the pillar (pillars with a star
cross-section will pin more efficiently to the contact lines than to those with a
circular cross-section). When approaching the ultimate value of 1808, as observed
with the texture displayed in figure 4c, the value of the hysteresis is so low that
it becomes basically non-measurable (Gao & McCarthy 2006b, 2007): as it reaches
the maximum possible angle, a drop logically loses its adhesive properties, as if it
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Figure 6. Microvelia walking on water (the bar indicates 1 cm). A close-up on the leg shows that this
bug is in a Cassie state—the leg does not contact the water, owing to thin (hydrophobic) hairs, which
allows Microvelia to be repelled by water and to skate on it. (Courtesy of D. Hu and J. W. M. Bush.)
were of a Leidenfrost type. These examples stress that the Cassie situation, which is
characterized by both a large contact angle and a low hysteresis, is the one which
must be practically achieved for generating non-stick behaviours for drops. We now
discuss the condition to be given to the solid design for favouring a Cassie state
rather than a Wenzel one.
5. Stable and metastable states
In order to predict the most favoured state for a drop, we can recall that the
energy gained by a spherical drop when it is placed on a substrate is DE Z
4pR20 gðR0 =R K 1Þ. This energy gain will be (generally) different in a Wenzel or in
a Cassie state—the angles given by equations (4.3) and (4.4) being (generally)
not the same, we can define for each state a Cassie radius or a Wenzel radius.
From the relationship for DE, we learn that the state of lower energy is the one of
larger radius, that is, of smaller contact angle.
A comparison between the cosines given by equations (4.3) and (4.4) thus
immediately tells us the state of lower energy. For example, a Cassie state will be
favoured provided that cos qCOcos qW, that is
cos q! ðfS K1Þ=ðr K fS Þ:
ð5:1Þ
In the limit fS/r, this condition can be rewritten as rO1/jcos qj, which
implies roughness factors larger than 2–3 (the Young contact angle is commonly
in the range of 110–1208, for hydrophobic compounds). Using equation (4.1), we
thus find that the pillar height must exceed the ratio p2/2pb, which implies a
large aspect ratio h/b (larger than 1/2fS). This might make the structures very
fragile, explaining why in many cases fractal-like surfaces will be preferred (they
are very rough, yet more compact). However, long hairs such as those observed
on the legs of water striders fulfil this criterion, and their flexibility makes them
able to resist shocks and deformation. In figure 6, you can check that the leg of
Microvelia, a small bug walking on water, does not contact its reflection in the
water mirror, but is instead separated from the surface by thin hairs, clearly
indicating a Cassie state (Bush et al. in press).
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Figure 7. Millimetric drop (of radius RZ400 mm) gently deposited on a solid decorated with dilute
microposts (fSZ0.01), and of moderate hydrophobicity (qZ1008). However, the drop is clearly
observed to be in a Cassie state—the interval between the bottom of the drop and its reflection is
24 mm, i.e. twice the post height hZ12 mm, indicating that the drop sits on the top of the bed of
micronails (fakir effect).
However, it is often observed that a drop can sit at the top of dilute posts, in a
Cassie state instead of a Wenzel one of smaller surface energy (Yoshimitsu et al.
2002; Lafuma & Quéré 2003; Barbieri et al. 2004; Patankar 2004). Figure 7 shows
such a drop, deposited on a solid with fSZ0.01. For this particular surface, we
have bZ1.5 mm, pZ25 mm and hZ12 mm, from which we deduce rZ1.2. Since we
also have on this surface a Young angle qZ1008, the criterion rO1/jcos qjw6 is
far from being satisfied. However, we clearly see in figure 7 light passing below
the drop (behind which a lamp is placed), which implies a Cassie state. This can
be proved even more firmly by measuring the distance between the drop and its
reflection, which is found to be 24 mm, twice the pillar height.
How can we understand the existence of these metastable states? It is quite
obvious that a drop gently brought into contact with a microstructured solid will
be (at least transiently) in a Cassie state—the contact first occurs at the top of
the defects. Assuming a zero velocity at the contact (the meaning of the word
‘gently’ in the previous sentence), we can evaluate the energy gain passing from
this Cassie state to the Wenzel one. Looking at figure 5, we understand that this
transition implies the creation of a solid–liquid interface on a surface proportional to rKfS, the suppression of a solid–air interface on the same surface
area, together with the suppression of a liquid–vapour interface on an area
proportional to 1KfS. The change of surface energy per unit area is thus
DEZ(gSLKgSV) (rKfS)Kg(1KfS)ZKg[(rKfS)cos qC(1KfS)], a quantity
that is indeed negative in the case we are considering here (condition (5.1)).
However, there is an energy barrier to overcome, in order to find this
groundstate (Barbieri et al. 2004; Ishino et al. 2004; Patankar 2004). The height
of this barrier defines the robustness of a metastable Cassie state, and it can
be evaluated by considering the motion of the liquid–vapour interface along
the walls of the posts keeping this interface flat. While this interface sinks
along the post, its surface area remains unchanged, and so does the solid–liquid
interface at the top of each post—the only changed surface energy corresponds
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Figure 8. Evaporation of a water drop on a hydrophobic surface decorated with pillars of diameter
3 mm, height 5.5 mm and distance 20 mm. The time interval between two successive photos is 9 s. As
the drop radius reaches 160 mm (fifth photo), a sudden Cassie–Wenzel transition is observed. Note
the difference in receding angle (observed in these evaporation sequences) between both states.
to the (unfavourable) wetting of the posts walls, implying a (positive) energy
per unit area DEZ(gSLKgSV)(r K1)ZKg (r K1) cos q. Using equation (4.1),
we thus find for the energy barrier between the Cassie and the Wenzel states
DEZ2pbh/p2gjcos qj. The barrier is proportional to the height of the pillar,
which appears as a natural parameter (yet not the only one) for tuning
this quantity.
As emphasized earlier, this energy barrier is generally too large to be supplied
by thermal energy, or even by small vibrations of the substrate. However, it can be
brought about by an external source of energy—pressing on the drop (Lafuma &
Quéré 2003), an impact (Bartolo et al. 2006; Reyssat et al. 2006) or vibrating the
substrate (Bormashenko et al. 2007) can all provoke a transition to the Wenzel
state. As expected from our expression for the energy barrier, it is indeed found
that the higher the post, the better the drop resists impalement in the structure
(Reyssat et al. 2008). When the liquid penetrates the texture to the bottom, it
remains strongly pinned inside it, the ‘printed’ drop even being able to conform its
shape to the network of microposts (Reyssat et al. 2006). But the scenario for the
nucleation of the solid–liquid contact can be different from the one suggested here,
as proved by looking at the evaporation of a drop in a metastable Cassie state. It is
observed that the drop impales when its size is smaller than a critical size, which is
significantly larger than the inter-pillar distance p (McHale et al. 2005; Reyssat
et al. 2008), as seen in figure 8.
The same result is also observed by gently depositing small and large drops on
a given substrate—if condition (5.1) is not satisfied, the little drops are of a
Wenzel type, while the large ones are in a Cassie state (Lafuma & Quéré 2003).
In this quasi-static limit, we can think of two conditions for establishing a solid–
liquid contact—or expressed conversely, for conserving a drop in the non-stick
Cassie state, in spite of a higher surface energy (Ishino et al. 2004; Patankar
2004; Bartolo et al. 2006; Reyssat et al. 2006, 2008). The sketch in figure 9, which
shows the liquid–vapour interfaces below a Cassie drop state, helps us to
precisely know these conditions.
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D. Quéré and M. Reyssat
Figure 9. Liquid–vapour interfaces below a fakir drop are curved, owing to the curvature of the
drop (or to the pressure applied to the drop, if there is any). We characterize the lowering of the
interface below the top of the posts by a length d.
Firstly, all the different contact lines must be at equilibrium below the drop.
Here the presence of ‘favourable’ designs, for the structures, helps—if the solid
is decorated with microspheres, for example, there is always a position above
the equator of the sphere (supposed to be hydrophobic) where the Young
condition can be satisfied. For pillars, the equilibrium arises from the presence
of sharp edges at their tops—as first noted by Gibbs, there is a place on the
edge (at a nanoscopic scale) where the Young condition is also fulfilled. This
possibility of having an equilibrium of the contact line on the solid suggests that
it should be possible to make a superhydrophobic solid with a hydrophilic
surface, thus generating (metastable) super-oleophobicity (Morra et al. 1989;
Herminghaus 2000; Woodward et al. 2003; Cao et al. 2007; Tuteja et al. 2007).
For example, having overhangs on the posts (Herminghaus 2000) and/
or multiscale roughness (Nosonovsky & Bhushan 2007a,b), or using microspheres or microfibres, should generally allow such states (the problem being
the robustness of these metastable states, compared with the more stable
impregnated ones).
Secondly, the liquid–vapour interface itself is curved, either because of a
pressure applied on the drop (a dynamical one, for example, in the case of an
impact), or simply owing to the shape of the drop, which is curved at a large
scale. The interface curvature, as defined in figure 9, should scale as d/p2, while
the drop curvature increases as 1/R. Hence, the depth of penetration d of the
interface inside the texture should obey the relationship—dwp2/R: the smaller
the drop (as it becomes if it evaporates), the larger d. If this quantity becomes of
the order of the pillar height h, a solid–liquid contact should nucleate on the
bottom of the substrate and propagate if the Cassie state is of lower energy than
the Wenzel one (see the corresponding discussion around equation (5.1)). The
critical radius for a Cassie drop is thus of the order of p2/h (much larger than p,
if h!p). This quantity can be made very small (robust Cassie state) either by
making h large or by reducing the different sizes p and h—a reduction in size
affects the resistance of the Cassie state, which might explain the existence of
such very small scales in many natural materials. It would be useful to
understand how these simple ideas apply (or not) for multiple-scale textures, for
which we could imagine successive similar transitions, and thus mixed
Wenzel–Cassie states. The understanding of these kinds of substrates would
certainly help to design optimized superhydrophobic materials, able in particular
to resist high water pressure (for underwater applications, for example) or
other kinds of aggression (Marmur 2006; Shirtcliffe et al. 2006; Nosonovsky &
Bhushan 2007c).
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Note finally that the other types of ‘transitions’ were observed between the
two states. For example, condensing water vapour on a cold superhydrophobic
material should favour a Wenzel state, even on lotus leaves (Cheng &
Rodak 2005). The observed intermediate states can be Wenzel/Cassie mixtures
(Wier & McCarthy 2006), and the figures of condensation are the object of active
research (Dorrer & Rühe 2007), which might help to design anti-dew materials.
It would also be interesting to see if a material for which condition 8 is satisfied
(stable Cassie state), and on which water would be condensed (favouring a
Wenzel state), is subjected to a reverse Wenzel–Cassie transition, which should
be much more difficult to initiate owing to the large pinning in the Wenzel state.
Coming back to the more classical case of a Cassie–Wenzel transition, it was
shown that external fields can be used for triggering such a transition (and thus
for getting a material of tunable wettability)—let us quote here the possibility of
using an electrical field (Krupenkin et al. 2004), light for a photocatalytic texture
(Feng et al. 2004; Zhang et al. 2004), or heat for temperature-sensitive coatings
(Sun et al. 2004).
6. Slippery behaviour
A remarkable property of the lotus-like surfaces is their ability to (possibly)
generate slip at their surface. The classical hydrodynamic boundary condition
at a solid–liquid interface is a no-slip condition, which was shown to be verified
at a microscopic scale for hydrophilic solids. For hydrophobic materials,
however, a nanoscopic slip has often been reported (Cottin-Bizonne et al.
2005). A convenient way to quantify the amplitude of this slip consists of
evaluating, for a given flow, the distance inside the solid for which the
extrapolated velocity of the flow vanishes. For hydrophobic solids, this length
was observed to be of the order of 20 nm, which implies that significant
effect on the flow (for example, reduction of the viscous drag associated with it)
will be observed if the thickness of the flowing film is of the same order—most
of the usual flows (even in micrometric channels) will not be affected by such
a modest slip.
However, the slip can be amplified in a superhydrophobic case, as is the
hydrophobicity. There again, Wenzel and Cassie situations are very different.
Richardson showed that the roughness on a solid surface dramatically damps any
(potential) slippage, provided that the liquid follows the solid surface structure
(figure 5a; Richardson 1973). Conversely, we can guess that a flow in a Cassie
state should enhance the slip—liquid literally glides on air (as we can see with a
Leidenfrost drop), owing to the large viscosity ratio between water and air
(typically a factor of 100). The associated slip length can be taken as infinite,
while part of the drop (on a surface area fS) flows on the posts, so that the
smaller fS, the better the slippage.
This naive argument is correct, and it can be made more quantitative in order
to be precise about the amplitude of the slip (Ybert et al. 2007). For a flow of
typical velocity V, the spatial zone ‘affected’ by the presence of a post should
scale as b, the post radius (taken again as smaller than p, the pitch of the array of
posts). The total friction force on such a surface should thus scale as NhVb,
denoting N as the total number of posts. Since the total surface area of the
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D. Quéré and M. Reyssat
material can be written as Np2, we get a viscous stress on the surface scaling as
hVb/p2. By definition of the slip length, this stress can also be written as hV/l,
from which we deduce an effective slip length l
1=2
l wp2 =b wp=fS :
ð6:1Þ
This formula would be slightly different for an anisotropic texture such as a
groove. Then, denoting b as the width of the wall, and p the distance between
grooves (b/p), we expect the friction on each groove to be given by an Oseentype formula, a groove being highly analogous to a cylinder—for a groove of
length L, this friction should scale as hVL/ln( p/b). The logarithmic factor is a
characteristic of this friction on a slender body, and the cut-off lengths are taken
as the lateral distance on which the friction takes place (i.e. b) and the shortest
distance where another similar object is found (i.e. p). Again, we have a total
force scaling as NhVL/ln( p/b), with N the number of the grooves. Since we
also have a surface area scaling as NpL, the force per unit surface is now
hV/p ln( p/b), from which we deduce a slip length l
l wp lnðp=bÞ wp lnð1=fS Þ:
ð6:2Þ
Since we know that the friction along a cylinder is twice as small as that
perpendicular to it, we immediately deduce that the slip length should be twice
as small, if the flow is perpendicular to the grooves rather than parallel. The flow
is anisotropic, with an easier motion (larger slip) in the direction parallel to the
striations. This property was indeed demonstrated by Lauga & Stone (2003).
In the limit we consider here (b/p, that is, fS/1), these estimations imply
very large slip lengths, compared with values on flat solids—l is of the order of p,
the typical width of the groove, and thus can be approximately 10 mm, 1000
times larger than on a hydrophobic solid! This effect is even more spectacular
with posts, on which the slip length is ‘amplified’, compared with grooves, by
a factor p/b, which can typically be 10 (corresponding to a factor fS of approx.
1%). Note finally that this anisotropy of slip, for a solid decorated with grooves,
also exists as for the wetting properties—a liquid will glide much more easily
along the grooves rather than perpendicular to them, owing to a smaller contact
angle hysteresis in this direction (Bico et al. 1999; Yoshimitsu et al. 2002; Chen
et al. 2005; Zheng et al. 2007).
There is experimental evidence of very large slip in a Cassie state. Ou et al.
(2004) were the first ones to deduce micrometric slip lengths from the
measurements of the pressure needed to drive a given flux of water inside a
channel of typical size (width and depth) W. Poiseuille flow makes this flux to be
W 4Vp/h, denoting Vp as the pressure gradient along the flow. For a flow with a
large slip (lOW ), a similar calculation yields a flux of lW 3Vp/h. The reduction
of pressure needed for driving this flux is thus of the order of l/W, and Ou et al.
(2004) indeed reported significant pressure reduction (by approx. 40%),
suggesting slip lengths of the order of 10 mm (see also Truesdell et al. 2006).
More recently, Roach et al. (2007) found unusual responses of superhydrophobic
microbalance, which could be interpreted by considering an interfacial slip.
Joseph et al. (2006) were able to deduce slip lengths from microparticle imaging
velocimetry for water flowing on hydrophobized pyramids of carbon nanotubes.
These authors confirmed that the slip length dramatically depends on the state
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of hydrophobicity—in a Wenzel state (induced by the application of pressure on
the liquid), there is basically no slip; conversely, in the Cassie state, there is a slip
that increases linearly with the post distance p, in agreement with equation (6.1)
(these experiments are performed at a constant fS). Similar experiments are
currently done by the same group, with the hope of reaching even larger values
of l. Note however a natural difficulty. Decreasing the value of fS makes the
Cassie state fragile (as seen earlier), so that an irreversible transition towards the
sticky Wenzel state might be expected—all the more since some pressure is
required to drive the liquid in the channels, encouraging this transition.
7. Conclusion
As we saw, hydrophobic microstructures on solids allow one to generate waterrepellent materials, on which water does not stick, easily runs off and even slips.
The condition for achieving such a situation consists of placing a cushion of air
below the drop (Cassie state), which can be made with very rough substrates.
Highly robust metastable Cassie states are also possible, on substrates of smaller
roughness, and the optimization of the design remains to be fully understood. On
the other hand, many questions still remain to be solved, the most important one
being the applicability of these materials. For synthetic materials, water
repellency is practically a non-permanent property, owing to pollution or
fragility of the materials. As a consequence, very few products based on the
principles described here exist today on the market. Despite the numerous
potential applications, the domain of application of superhydrophobic materials
remains to be invented.
It is a pleasure to thank Anne-Laure Biance, José Bico, Aurélie Lafuma and Ko Okumura for
stimulating exchanges. In addition, I do thank Hervé Arribart, Catherine Barentin, Bharat
Bhushan, Lydéric Bocquet, John Bush, Stefan Doerr, L. Mahadevan, Thomas McCarthy, Glen
McHale, Lei Jiang, Christian Marzolin and Christoph Neinhuis for precious discussions and
providing pictures for this article.
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