Wetting and Roughness

ANRV347-MR38-04 ARI 28 May 2008 6:32

Wetting and RoughnessDavid QuereLaboratoire de Physique et Mecanique des Milieux Heterogenes, ESPCI, 75005 Paris,France; email: [email protected]

Annu. Rev. Mater. Res. 2008. 38:7199

First published online as a Review in Advance onApril 7, 2008

The Annual Review of Materials Research is online atmatsci.annualreviews.org

This articles doi:10.1146/annurev.matsci.38.060407.132434

Copyright c 2008 by Annual Reviews.All rights reserved

1531-7331/08/0804-0071$20.00

Key Words

microtextures, superhydrophobicity, wicking, slip

AbstractWe 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 an-gle hysteresis can be dramatically affected by the presence of roughness. Ow-ing to the development of refined methods for setting very well-controlledmicro- or nanotextures on a solid, these effects are being exploited to inducenovel wetting properties, such as spontaneous filmification, superhydropho-bicity, superoleophobicity, and interfacial slip, that could not be achievedwithout roughness.

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1. WETTING WITHOUT ROUGHNESS

1.1. Ideal Wetting

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, inwhich a liquid brought into contact with a solid spontaneously makes a film. In the case of awindshield, for example, this film maintains the transparency of the glass; in addition, the filmflows in the gravity field (if the car is stopped) or due to air friction (when it moves), taking dustparticles with it. The second limit is complete drying: Liquid drops remain spherical withoutdeveloping any contact with the substrate. They are thus readily evacuated, which prevents liquidcontamination of the solid surface.

It is of obvious interest to determine which parameters favor both these situations. The basiclaws were first established for ideal solids, which are both flat and chemically homogeneous. Asunderstood 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 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 forceapplies along the IJ surface to minimize the corresponding (positive) surface energy. We denotethe liquid/air surface energy simply as .

Hence, we arrive at a construction first imagined by Marangoni: A film spreads from a reservoirof liquid (a drop or a bath) onto a solid, as sketched in Figure 1a, provided that the solid/airsurface tension SA (which entrains this film) is larger than SL + , the sum of the solid/liquidand liquid/air surface tensions (which both resist the spreading because complete wetting expandsthe two corresponding surface areas). The sign of the spreading parameter S = SA SL will thus determine the behavior of a drop on a solid: For S > 0, a drop spreads, whereas it formsa small lens in the opposite case. This lens meets the solid with a well-defined contact angle ,whose value is similarly given by a force balance (Figure 1b). Projecting on the solid plane thedifferent surface tensions acting on the contact line provides the equilibrium condition of the drop(1). The balance at equilibrium can be written as

SA = SL + cos . 1.The contact angle is thus fixed univocally by the chemical nature of the different phases. Herewe show that this statement can be dramatically affected if the solid is rough. We refer below tothe angle as the chemical or Young angle. In many common situations, this angle lies between0 and 90 (i.e., the hydrophilic case). Very qualitatively, a solid/liquid surface tension (between

SL

SA

SL

SA

a b

Figure 1Two classical wetting situations for an ideal material. (a) A liquid film spreads, drawn by the solid/air surfacetension, despite the action of the liquid/air and solid/liquid tensions. (b) Wetting is only partial, and thebalance of surface tensions determines the contact angle .

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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 1results, implying an acute angle .

Conversely, we could define a drying parameter D = SL SA . If SL is larger than SA + , the contact line will be withdrawn by surface forces until a film of air comes between the solidand the liquid: D > 0 is the criterion for complete drying. (This criterion is also simply derivedby making cos < 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 thesame 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 boilingtemperature of the liquid, a vapor film forms between the solid and the liquid, which sits on itsown vapor. Considering in Equation 1 that the solid role is played by the vapor, we determine thatD = 0. But the situation is quite different at standard temperatures. Although complete wetting canbe 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 contactangle never exceeds approximately 120 (3), to which corresponds a negative parameter D (ofapproximately /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 forfilling the (large) gap existing between 120 and (nearly) 180, thus generating ultrahydrophobicbehaviors 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) caseof cylindrical pores of constant diameter and consider one of these pores. A liquid will penetratesuch a tube if the surface energy of the solid is lower wet than dry (Figure 2a).

We can introduce a wicking parameter W = SA SL, whose sign indicates if liquid penetratesthe tube. Wicking will occur if W > 0, that is, if the contact angle is smaller than 90 (as we see fromEquation 1). Then, the meniscus formed by the liquid inside the tube must be curved in such a waythat the Laplace pressure (associated with curved surfaces) is negative below the surfaceanotherway 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 thetube is lower wet than dry. This means that, as deduced from Equation 1, the contact angle of the liquid onthe tube walls must be acute. (b) Conversely, for an obtuse contact angle, the tube tends to remain dry. Ifgravity is present, both the rise and descent are limited.

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the tube. As stressed above, contact angles are generally acute, which means that most spongesabsorb 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 themechanism of invasion in this case. Therefore, as shown in Figure 1a, a liquid film (typically ofa molecular thickness) progresses along the tube walls (4) such that the meniscus behind the filmadvances 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 = , which isindeed 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 cav-ities can remain filled with air, even if the solid is exposed to a liquid, and thus approach theLeidenfrost limit. As we see, not only can roughness modify the wettability of a solid but alsoperhaps the main message of this reviewroughness can result in new and specific propertiessuch as water repellency. We first show that the natural roughness of most solids is likely to inducepinning of the contact line and thus variability of the contact angle (apparently contrasting withwhat 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 andpinning 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 aslamination) may generate striations or microgrooves. Materials resulting from the compaction ofgrains exhibit roughness at the scale of the grains. Coating can also induce roughness, in particularwhen the coating film dewets, thus producing microdrops at the surface. Conversely, very few solidsare molecularly flat. Most often, molecularly flat solids result from solidifying a liquid film, eitherfree or suspended on another liquid; in such cases, the roughness can correspond to the thermalroughness 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, dropletson 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 thefront (of high curvature) and the rear (of smaller curvature) and, thus, a force able to resist gravityprovided that the drop is small enough (6). Both chemical heterogeneities and roughness can actas 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 )makes the contact angle flexible at this place. We measure a (Young) angle before the edge anda (Young) angle + after the edge, considering the horizontal as the reference (we ignorewith our naked eye the existence of the defect). Hence it is possible to have any angle between and + at the edge (7). A groove can thus stop the front of a liquid drop (as if it werenonwetting), and a tip will act in the opposite way so that a solid decorated with both kinds ofdefects 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 itsconfiguration is that of a drop pinned on wetting defects. The way to quantify this contact angle

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Figure 3Apparent pinning of a contact line on an edge. The Young condition stipulates that the liquid meets the solidwith a contact angle . Hence the contact angle at the edge can take any value (if the horizontal direction isconsidered as the reference one) between and + , as illustrated by the colored region.

hysteresis consists of slowly increasing the volume of a drop: The contact line first remains stuckbefore it suddenly jumps above a critical volume (for which the line suddenly depins and movestoward a next series of pinning defects). The maximum observed angle is the so-called advancingcontact angle a. Conversely, sucking the liquid from the drop flattens it until it depins and retractsto the next wetting series of events, which stop and pin the line; the minimum corresponding angleis the receding contact angle r.

Contact angle hysteresis can be seen as beneficial (e.g., when it is exploited for guiding aflow along a line of defects, following a predefined route) or detrimental (e.g., water drops stuckon window panes distort their transparency and contribute to degradation of the glass). It is thuscrucial 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, whichaverages on many defects). We give further an example of such a calculation. More generally, wesee that the contact angle hysteresis = a r varies dramatically on a rough solid, fromnearly zero to a giant value, of the order of a itself (8).

2.2. The Wenzel Model

We see above that roughness impacts the contact angle hysteresis. But it also affects the typical orapparent angle, which is (often very) different from the one expected from Equation 1. This wasfirst 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 configu-ration, characterized by a contact angle (possibly different from the Young angle ). The key

Vapor

Solid

dx

Liquid

Figure 4The Wenzel picture. One can obtain the apparent contact angle by considering a small apparentdisplacement of the contact line and looking at the corresponding variation in surface energy, assuming thatthe liquid follows the accidents of the solid surface.


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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/airinterface gets replaced by a solid/liquid interface of the same surface area. The surface energyvariation dE arising from an apparent displacement dx of the line can be written, per unit lengthof the contact line, as

dE = r(SL SA) dx + dx cos , 2.where the second term on the right corresponds to the change of liquid/vapor surface area as thedrop spreads. The roughness increases both the solid energies, enhanced geometrically by a factorr. The minimum of E (dE = 0) yields Equation 1 if the solid is flat (r = 1); if not, we find (9)

cos = r cos , 3.where is the chemical angle given by Equation 1.

The Wenzel relation (Equation 3) predicts that roughness enhances wettability. If the factorr is larger than 1, a hydrophilic solid ( < 90) becomes more hydrophilic when rough ( < ).Conversely, a hydrophobic solid ( > 90) shows increased hydrophobicity ( > ). Althoughthese tendencies are generally (but not always) observed, agreement with Equation 3 is far fromquantitative (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 arbitrarilylarge, which seems to imply that complete wetting (cos > 1) or complete drying (cos < 1)should be induced by large roughness (r 1). We show that such behavior is not observed becauseWenzel assumptions often are not satisfied.

Even when Wenzel relation is likely to be obeyed, it is difficult to check directly whether therelation is being followed. Because liquid conforms to the roughness, pinning of the contact lineis particularly strong in this state, both on the edges of and along the defects. Besides, pushinga Wenzel drop leaves behind cavities filled with liquid such that the drop can also be pinned bythe liquid itself. As a consequence, a Wenzel state is generally characterized by very low recedingangles and thus giant hysteresis ( a). In such conditions, it is very difficult to extract thesole angle or to check Equation 3. Modern and more detailed discussions on the validity of theWenzel model can be found in References 1013, which stress in particular that drops should bemuch larger than the defects to use such an averaged model. In the converse limit, liquid rearrangessuch 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 Youngequation (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 thesurface properties of a solid and, even better, to induce properties that could not be generatedotherwise, 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 the1990s, researchers from the Kao Corporation in Japan showed that extremely large angles could beobtained 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 sametime, Neinhuis and Barthlott in Germany systematically analyzed the structures on the surfacesof 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 toanimals, and new fascinating designs were (re)discovered and discussed (2124). There have since

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been many attempts to mimic these natural patterns so as to understand their efficiency. (c) Therecent development of microfabrication techniques allows us today to construct very well-definedmicro- or nanostructures, which has pushed researchers, e.g., to imagine new designs and tooptimize given designs. We now summarize different findings related to these three factors (ac).

3.1. The Kao Experiment

Kao researchers constructed different well-characterized fluorinated substrates, either rough orflat (15, 16). They compared the contact angles on the rough samples with the contact angleson flat materials, which should be close to the Young angle . This comparison was performedthrough the use of several liquids to vary . A typical result is displayed in Figure 5, in which cos

is plotted as a function of cos , showing the modification of the wetting properties generated bysurface roughness. This plot provides only one angle (which seems to be close to the advancingangle, according to References 15 and 16), so we ignore the hysteresis associated with each datapoint.

We first notice that the abscissa is far from exploring the complete interval [1, 1]: cos isnever smaller than 0.3, corresponding to an angle of approximately 110. This data point wasobtained by the use of water as a liquid; as emphasized above, contact angles on flat solids are neverlarger 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 < 0), wesee a spectacular effect. As soon as we enter this domain, the apparent contact angle jumps andreaches a value of 170 (much larger than the chemical angle); roughness here induces a wettingbehavior that could not be achieved otherwise. This state is often referred to as superhydrophobic.

In the hydrophilic domain (cos > 0), cos first increases linearly with cos ; the slope islarger 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 agreementwith 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 be-havior is not obeyed when the contact angle becomes smaller than some critical value c. Instead,we observe a second linear regime (with a slope smaller than unity), which tends toward = 0 as = 0 (quite trivially, a wettable solid remains wettable if rough). We see in Section 4.2 that this

c

cos

*

cos 1 10

1

0

Figure 5Cosine of theapparent contactangle on atextured surface, as afunction of thecosine of the Youngangle measured onthe same surface, yetflat (16). The linesshow the behaviorexpected fromEquations 3, 10,and 14.


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regime results from the penetration of the liquid inside the microtextures; this liquid surrounds thedrop on which the contact angle is measured. Then, the lens of liquid sits upon a mixture of solid andliquid, at odds with the Wenzel hypothesis, which assumes a dry solid beyond the drop, as shown inFigure 4. We call this second regime superhydrophilic, and we describe this regime in Section 4.

3.2. Natural Microtextures

Microtextures are also found on the surfaces of many plants and animals (Figure 6). In his NaturalHistory, Pliny the Elder noticed that water on a leaf forms perfect spheres, provided that the leafsurface is woolly (25, p. 32). The old literature (and poetry) sporadically reports the special wettingbehavior of plants and animals, such as a review by Dufour in 1833 (26, pp. 6874), a paper byFogg in 1944 (27), and a comment on Foggs paper by Cassie & Baxter (28). More systematicstudies 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 1050 m (Figure 6a,b). For the mostpopular of these hydrophobic plants, namely the lotus, and many other ones, these papillae arecovered by fine nanostructures at the scale of 100 nm. The coexistence of two scales of roughnesscontributes significantly to the quality of the superhydrophobicity (2936). However, despite many

a b

c d

500 nm50 m

50 m50 m

Figure 6A few examples of natural superhydrophobic materials, as revealed by SEM. (a) Leaf of the so-calledelephants 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 (courtesyof Lei Jiang). (d ) Surface of a mosquito (Culex pipiens) eye. From Reference 40 (courtesy of Lei Jiang). Notethe difference in scale between panels ac and panel d.

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stimulating hypotheses, there is no real understanding for this hierarchical structure, which is notnecessary for reaching very high degrees of hydrophobocity (37). Actually, such structures oftenprovide both a high contact angle and a low contact angle hysteresis. For lotus, for example, theadvancing angle is approximately 160, and the receding angle is larger than 150, which confersto water drops a high mobility on these leaves. In some cases, such as the rice leaf, the arrangementof 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/orsuperhydrophobic, 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 protectsthe 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 thebody or on the legs (Figure 6c), allowing some animals to float on water or even to live underwaterowing to the air spread on their body (4348); see details in the recent and comprehensive reviewby Bush et al. (24). The setae often consist of tapered hairs with a length of 30 m, a diameter of110 m, and an angle of inclination of typically 30 (Figure 6c). As for plants, there is a secondarytexture, namely nanogrooves, whose exact role is still questioned (23). Other structures can bevery different: Figure 6d shows the pattern that decorates the eye of Culex pipiens, the classicalmosquito. It is very simple and well-ordered, at an impressively small scale (posts of size and heightof approximately 100 nm) (40). We show further that some applications indeed require reductionof the pattern size.

3.3. Synthetic Microtextures

As is pointed out above, many recent papers are devoted to the creation of microtextured surfaceswith particular wetting properties (most often, superhydrophobic ones). Many techniques existfor producing such materials, even primitive ones: Approaching a piece of glass from a flamegenerates soot, which quickly darkens the glass. If you put water on this substrate, you will see itbehaving as if it were a soft solid, rolling and bouncing off the surface! More generally, templatesynthesis, phase separation, all kinds of etching, crystallization, and electrospinning of microfiberswere proposed to construct more elaborate materials (4950). As a result, many different textures,from highly disordered or fractal to ordered and well-defined, were obtained (Figure 7). Most ofthese surfaces provide specific wetting properties, and we still have to understand which surfacesare 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 gravitydominates surface tension effects) can induce superhydrophilicity. We discuss above the Wenzeleffect, 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, thestructures 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 progressesin the microstructures, it develops an interface with air, leaving (possibly) a few dry islands behindit. 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 considerationsof the dynamics of these films.


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a b

c d

5 m

500 nm

20 m

10 m

Figure 7Different examples of synthetic microtextured surfaces. (a) The simplest possible surface, with regularmicropillars. (Courtesy of M. Reyssat.) (b) A surface decorated with nanofibers. From Reference 51 (courtesyof L. Gao and T.J. McCarthy). (c) A surface planted with carbon nanotubes. From Reference 52 (courtesy ofJ. 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 forspecific purposes, such as directional wetting. We consider, for example, a rectangular groove ofwidth w and depth , 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 beingwet ( SL < SA). But this is not enough because a liquid/vapor interface also develops at the top.

dx

w

Figure 8A liquid (in blue) invading a rectangular groove on a solid. Here, we ignore the menisci (at the liquid frontand 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 correspondingsurface area. Hence, for a liquid progression in the groove by a quantity dx, the surface energieschange by an amount (ignoring gravity effects)

dE = (SL SA)(2 + w) dx + w dx. 4.Using the Young equation, we find that the liquid progression is favorable (dE < 0) if we have

< c, 5.

with

cos c = w/(2 + w). 6.Whatever the values of w and , the latter quantity (which depends only on the aspect ratio /w)defines a number between 0 and 1, from which the angle c can be made explicit. If the grooveis narrow and/or deep (w is small and/or is large), we recover the criterion (discussed above, inthe context of Figure 2) of spontaneous penetration in a classical porous medium: c = 90. Ina general case ( w), c is somewhere between 0 and 90: It is more demanding to impregnatea groove than a 3-D porous medium. This is all the more true because the groove is shallow: As/w tends toward 0, so does 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). Wecharacterize such a surface by its pillar density S and roughness r. We show in Figure 9 the topview of an ethanol drop on/in a forest of pillars (with 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 surroundedby a film of ethanol; in this situation hemiwicking takes place. In some cases, the film conformsto the micropattern, so the film can take a square shape on a square array of microposts (55).

Figure 9Top view of anethanol drop (with adiameter of a fewmillimeters) on asilicon waferdecorated by siliconmicroposts. Here,hemiwicking takesplace, as deducedfrom the observationof a thick film aheadof the drop. FromReference 54(courtesy of ChiekoIshino and MathildeReyssat).


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Liquid

Air dx p b

h

Solid

Figure 10Liquid film (in blue) propagating on a solid, within a forest of microposts of height h, mutual distance p, andradius 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 theliquid on a surface area proportional to r S, whereas the (flat) liquid/vapor interface developson a surface area proportional to 1 S.

For a film progressing by a distance dx (larger than the scale of the defects), the variation insurface energy per unit length perpendicular to the figure can be written as

dE = (SL SA)(r s)dx + LV(1 s) dx. 7.The progression is favorable provided (once again) that the Young angle is smaller than a criticalvalue c, which depends only on the design of the solid (56):

cos c = (1 s)/(r s). 8.Liquid invasion on a microstructured solid can thus be tuned by the geometry of the structures.For dilute defects (small S), we have cos c 1/r: The rougher the substrate, the larger c, i.e.,the more likely that hemiwicking occurs. For a substrate composed of disconnected defects (suchas 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, al-lowing the liquid to reach the next row. In other cases, the contact line can remain pinned in ametastable Wenzel state, and an external source of energy (such as vibrations) must be employed tonucleate a contact with the next rows of pillars. We can even imagine equilibrium situations inwhich the drop coexists with a wet ring of finite extension (looking a bit like a fried egg). If, forexample, the energy barrier is passed owing to the action of the Laplace pressure, the progressioncan stop once the drop spreads enough to make its Laplace pressure too small for inducing afurther motion.

We can interpret the second regime (for < 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 thevalue of the parameter S, we expect it to be smaller than 1, so cos c should be of the order of1/r. The second regime indeed starts close to the abscissa where the Wenzel regime (of slope r)intercepts the line cos = 1, that is, for cos 1/r. The apparent angle then hardly depends on , which can be understood qualitatively: The drop sits on a composite surface consisting mainlyof liquid, apart from a few solid islands. The angle should be very close to 0 (the value it wouldtake if there were only liquid), but it cannot reach this value owing to the islands on which theangle is > 0. The number of islands should be a function of , which makes it difficult to producea general theory. The value expected for the angle is based on Figure 11, in which we sketchthe drop coexisting with the impregnating film.

We consider a displacement of the contact line by a quantity dx. The solid becomes wet ona fraction of surface S, and liquid interfaces are eliminated on a fraction 1 S. Because thedisplacement also implies an increase of the liquid/vapor interface of the drop, the total change ofsurface energies eventually becomes (per unit length of the line)

dE = s(SL SA) dx (1 s) dx + dx cos . 9.

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Liquid

dx

Solid

Figure 11Drop coexisting with a film that self-propagated within the material textures. The apparent angle is obtainedby considering a displacement of the contact line and computing the corresponding variation of surfaceenergy.

The minimum of E yields the apparent angle , as first shown by Cassie (57, 58):

cos = 1 (1 s) cos . 10.

For small S, the angle hardly varies with , as observed in Figure 5 for < c. This variationis linear when the cosines of both angles are plotted, and the slope provides S. We would deduce,for example, from Figure 5 that S 0.15. However, the actual behavior should not be linear fora disordered surface, for which the proportion of emerged islands should itself be a function of (the smaller the , the smaller the S), making the actual variation ( ) 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 = (r S) (cos cos c), which is fixed by the quality of wetting and by thedesign of the surface. If wetting is complete ( = 0), a molecular layer propagates ahead of theimpregnated film, which lowers the surface energy via the suppression of liquid/vapor interfacesonly, on a surface area proportional to r 1. Hence, the force then is (r 1). It only dependson 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 balancedby viscous force. This resistance to the flow should scale as Vx, denoting x as the impregnateddistance and as the liquid viscosity. Balancing both forces, we find that the film should progressas the square root of time, with dynamics similar to the wicking Washburn law inside a porousmedium (59). For rough substrates as for grooves, the liquid films indeed progress in a Washburnfashion (6062). This is also true within forests of microposts (56). In this case, we can calculatethe coefficient characterizing the dynamics (if we note x2 = Dt, D is this coefficient), allowing usto 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 paragraphspredict a wicking force scaling as bh/p2. The exact form of the viscous resistance depends on thepillar height. (a) For short pillars (h p), the dissipation is fixed by the depth h of the flow (thevelocity gradients take place between the bottom solid and the free liquid interface). We deducea viscous force (per unit length) Vx/h and thus a dynamic coefficient D ( /) (bh2/p2), whichis efficiently tuned by the pillar height. (b) For tall pillars (h p), dissipation takes place mainlyaround the pillars. The viscous force per pillar is Vh (omitting an Oseen logarithmic factor),with x/p2 pillars per unit length. Hence, there is a viscous force Vhx/p2, which eventually yieldsa very simple dynamic coefficient: D b/. We thus see how the design can be optimized: If fora given application a film of height h must propagate, we select pillars of height h (hemiwicking


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is a good way for setting films of a desired thickness). Then, p is chosen to maximize the speedof propagation; it will be taken of order h because there is no dynamic benefit for having morecompact networks.

5. SUPERHYDROPHOBICITY

5.1. Air Trapping

On hydrophobic solids, the situation is of course different from that for hydrophilic solids. Ifthe solid is rough enough, we do not expect that the liquid will conform to the solid surface, asassumed in the Wenzel model (Figure 4). Rather, air pockets should form below the liquid [this isthe so-called Cassie or fakir state (4849)], provided that the energetic cost associated with all thecorresponding 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. Ifthe liquid/air interfaces are assumed to be flat (which can be justified by a condition of constantLaplace pressure in the liquid, which, for defects much smaller than the drop size, can be takenas null), the wet and liquid surface areas are proportional to (r S) and (1 S), respectively.Hence, air pockets are favored, provided that (6365)

(r s) (SL SA) > (1 s), 11.which (through the use of the Young formula) can be reformulated as > c, with

cos c = (1 s)/(r s). 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 minus thecritical angle below which hemiwicking takes place (Equation 8). For very rough solids (r 1),this criterion is always satisfied. Then, c tends toward 90, and is indeed larger than thisvalue, because we assumed chemical hydrophobicity. For materials decorated with long hairs, forexample, the roughness factor r 2bh/p2 can typically be 5 to 10 (as deduced from Figures 6cor 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).

1 cm

a b

Figure 12(a) A Cassie state in action: Microvelia walking on water (scale bar, 1 cm). (b) Thin (hydrophobic) hairs allowthe bug to be repelled by and to skate on water. From Reference 24 (courtesy of D. Hu and J.W.M. Bush).

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Liquid

dx

Solid

Air

Figure 13Displacing the contact line in the Cassie regime: The energy balance must include the creation of liquid/airinterfaces below the drop, as indicated by the dotted lines.

The situation is more ambiguous for modest roughness factors, such as those provided bysmall pillar density S. The chemical contact angle is typically 100 to 110, and its cosine isslightly negative. Thus, the criterion for air trapping is not obeyed. However, Cassie states areoften observed in spite of a higher interfacial energy (64, 66, 67). The air present before we placea drop can remain trapped in a metastable state, as long as the drop does not nucleate a contactwith 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 asmall contact angle hysteresis, owing to the presence of the air cushion. As a consequence, we termsuperhydrophobic the only Cassie regime, which generates amazing properties such as reducedadhesion, 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 . Ifthere is only air, Young (Equation 1, where we replace the index S by A) predicts a contact angleof 180 (i.e., no contact). Any deviation from this value tells us the proportion of solid actuallycontacting the liquid.

The variation of interfacial energy arising from a displacement of the contact line by a quantitydx (as sketched in Figure 13) is related to the creation of new wet solid surface and liquid/vaporinterfaces. The final balance can be written as

dE = s(SL SA) dx + (1 s) dx + dx cos . 13.It is crucial to assume flat liquid/vapor interfaces. Consideration of some curvature would modifythe result (liquid/vapor interfaces would then have a larger surface area), suggesting that theangle 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 precisemeasurement of this curvature.

At equilibrium, E is the minimum, which yields the apparent angle (69):

cos = 1 + (1 s) cos . 14.This description must be complemented by a (local) Young condition at each contact line (atthe edge of the drop and for each liquid/vapor interface below). This condition is satisfied bythe presence of edges on the posts (or more generally of large slopes on the rough material). Asdiscussed above in the context of Figure 3 and because we have > 90, sharp angles permitthis condition. We expect stronger pinning if these edge angles are smaller: Re-entrant designswill make more robust Cassie states, and we see below (Section 5.3) that they can even induce airtrapping in a hydrophilic situation (29).

Equation 14 usually predicts large angles. For 110120 and S between 5% and 10%,we get apparent angles of 160170. In Figure 14, we see a millimetric water drop on a silicon


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Figure 14Millimetric water dropon a hydrophobicsurface textured withregularly spacedmicropillars. Thetexture acts as adiffraction grating,which inducesstructural colors. FromReference 70 (courtesyof M. Reyssat).

substrate where silicon micropillars (similar to Figure 7a, with S = 5%) were etched and coatedwith a fluoropolymer (70). The drop is like a pearl (69, 71), sitting on a solid whose iridescencesreveal the regular array of defects, which diffracts light (structural color) (66, 72, 73). Conversely,structures much smaller than the wavelength of light yield transparency (7476). This raises theinteresting question of the smallest size generating water repellency, which remains to be solved.

The smaller is S, the larger is . But for a more complex topology, S should be a function of (69). Conversely, the measurement of in a Cassie situation should provide the solid fractions contacting the liquid, a quantity of interest for characterizing not only wetting but also hydro-dynamic slip (see Section 6.3) or any properties related to a solid/liquid contact (e.g., electrical orchemical). In the limit of small S, we note that = (with 1), and Equation 14 rewritesto

2 cos (/2)1/2s , 15.whose critical behavior in S emphasizes the difficulty for achieving a strict nonwetting situation( 0). However, Gao & McCarthy (51) approached, and perhaps reached, this limit (withinthe uncertainty of the measurements) by using nonwoven assemblies of nanofibers (Figure 7b) onwhich both the advancing and receding angles were 180. The same authors reported similar resultsfor pulverulent hydrophobic solids obtained by compressing commercially available lubricant (37).In both cases, the texture is submicrometric and quite regular, with smooth rounded defects, whichshould induce a very low hysteresisthe 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 con-sequence, drops are unusually mobile, which generates novel properties, such as bouncing, thatare not observed on conventional materials (77). Liquids behave to a large extent as Leidenfrostdrops, for which the underlying vapor minimizes the friction. However, one generally observes onsuperhydrophobic materials a residual hysteresis (and thus adhesion), whose value is still debatedtoday (7883). A key factor in this discussion is the possibility of pinning the contact line on the

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y

p

2b

Figure 15In a Cassie state, a drop is likely to pin on the edges of the defects as we displace it. The drop becomesdistorted, 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 evidencedin a classical experiment by Oner & McCarthy (71) in which they varied the post shape.

Here we restrict the discussion to the case of strong pinning on dilute defects, elucidatedin 1984 by Joanny & de Gennes (84) and Pomeau & Vannimenus (85). The model is based onFigure 15, in which we see a few defects on which the line pins as we move the liquid, using aforce F. The line meets S defects per unit area and thus, for a displacement dx, Sdx defects perunit length. Passing each of them, an energy is stored and then released in the liquid (where itgets dissipated by viscosity) as the line depins. Hence, we have

F dx = S dx. 16.

We guess that will depend on the shape of the defects (for example, complex contours willgenerate a higher , and thus a larger hysteresis, unlike small and rounded defects). For the sakeof 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 S b2/p2.

As the line pins on a defect of size b, the drop gets distorted, as shown in Figure 15. Its tailsform surfaces of zero curvature: r = b cosh (x/b); that is, for x b, r 1/2 b exp(x/b). Thedeformation is maximum (x = u) for the largest lateral deformation r, i.e., for the typical distancep 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 b , which yields a relationship between force and deformation,

f u/ log (p/b). 18.

Equation 18 defines a linear spring of stiffness K = /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, perunit length, as

F = (cos r cos a). 19.


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Putting together these equations, we finally get

(cos r cos a) S log(1/S). 20.The contact angle hysteresis vanishes with the density of defects, but the presence of a logarithm inEquation 20 makes this behavior quite pathological: At small S, the hysteresis decreases becausethere are fewer defects (the linear term), but the logarithm term (slowly) diverges, making theresidual 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 morecomplex patterns.

Hysteresis makes drops stick on solids, despite gravity field or air flow. A general calculationof the sticking force is difficult, but a simplified argument allows us to evaluate how the hysteresisenters this quantity (86). We assume that the rear half of the drop joins the solid with an angle r,whereas the front half meets it with an angle a. The capillary sticking force can be written as (cos r cos a), denoting l as the radius of the solid/liquid contact (quasi-circular for 1).Assuming a geometric contact l R (where is the difference between the mean angle and ,and R is the drop radius) and using Equations 15 and 20, we find that a drop will move in thegravity field (on a vertical window pane) provided that

3/2s log(1/S) < R22, 21.

in which we introduce the capillary length 1 = ( /g)1/2 (2.7 mm for water) and ignore allthe numerical coefficients. Once again, the density of defects is crucial for driving the wettingproperties (here the degree of adhesion of a drop on a solid). As we could guess, small densities arerequired for suppressing adhesion. However, such a limit also weakens the stability of the Cassiestate, 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 millimetricwater drops on a microtextured substrate (with a pillar density S of 1% and pillar height h of12 m). The first drop is placed without any impact, and the second drop is released from a fewcentimeters such that it meets the solid with a velocity of a few tens of centimeters per second.

Figure 16Millimetric water drops of the same volume on a superhydrophobic substrate covered with dilute pillars(S = 0.01, h = 2 m). The drop on the right was thrown on the substrate, whereas the one on the left wascarefully deposited. As observed, Cassie and Wenzel states (left and right, respectively) can coexist. FromReference 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 isin a Wenzel state: On this substrate of small roughness (r 1.1), the contact angle is close to theYoung angle, slightly larger than 90 in this example.

When the Wenzel state is the less energetic one, any perturbation of a Cassie drop can provokeits transition to this state. Conversely, a Wenzel drop is firmly bound to its stable configuration. Itis of practical importance to quantify the robustness of metastable Cassie states and to understandthe conditions provoking impalement. Obviously, an energy barrier must be overcome to find theground state (8790). One can evaluate this barrier by considering the penetration of a Cassiedrop. Assuming unchanged liquid/vapor interfaces as the drop sinks, the only change in surfaceenergy corresponds to the (unfavorable) wetting of the posts walls. This implies a (positive) energyper unit area E = ( SL SA)(r 1) = (r 1) cos (this quantity becomes negative in ahydrophilic situation). We thus find an energy barrier E 2bh/p2 cos . It is proportionalto the pillar height h, which appears as a natural parameter for tuning this quantity. The energybarrier E is generally too large to be overcome by thermal energy (we need defects of moleculardimensions to get E of the order of kT ). However, the energy can be supplied by pressing onthe drop (66, 90), by vibrating the substrate (91), or by an impact (92, 93). Indeed, the higher theposts, the larger is the resistance to impalement. Once the liquid penetrates the texture, it remainsstrongly 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 andfollowing a zipping mechanism: One row of cavities gets filled (in a time of approximately 10 son a length of 100 m!) before jumping to the next row. This process is somehow reminiscent ofthe progression in a groove sketched in Figure 8. The surface force here involves the creationof wet surfaces and the suppression of suspended liquid/air interfaces, whereas the resisting forceshould be viscous. We thus expect a Washburn law for the progression (see Section 4.3), which canbe extremely quick at the small scale of the phenomenon. For a pattern composed of posts whoseheight h is comparable to the pitch p (of approximately 10 m), the typical time for invading a rowof length x scales as x2/ h, of the order of 10 s for water and x = 100 m.

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). Ifthe drop becomes small enough, the liquid can reach the underlying solid and then propagate. Thesize of a drop should thus impact the drops wetting state. Indeed, small droplets are more likelyto be in a Wenzel state than are large droplets (66), which can also be evidenced by observing theevaporation of a drop. Then, the drops size varies continuously, and investigators have reportedthe 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 /p2 (for h < p). Equatingit with the drop curvature 1/R yields the depth of penetration of the interface inside the texture: p2/R; the smaller the drop, the larger is. When it becomes of the order of the pillar height

h

p

Figure 17The liquid/vapor interface is curved, owing to the curvature of the drop, but the interface also can be curvedif we apply pressure to the drop. denotes the lowering of the interface below the top of the posts.


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h, a solid/liquid contact can nucleate on the bare substrate and propagate if the Cassie state ismetastable. This implies a critical radius for a Cassie drop scaling as (96)

R p2/h. 22.

Note that, as deduced from Figure 3, depinning from the edge will also occur if the drop radius issmaller than p/|cos |, a limiting condition for modest chemical hydrophobicity. Here we assumethat this second condition is screened by the first one, i.e., p > h/|cos |.

The radius R can be much larger than p if h < p. The Cassie state will be all the more robustbecause this critical radius is small (no drops fall in the Wenzel drops, except invisible ones). Onecan achieve this in two ways: either by making h large, using micro- or nanofibers for decoratingthe solids (see Figures 6c and 7c) (96), or by reducing both p and h. Miniaturizing the pattern sizeenhances the resistance of the Cassie state, which may explain the existence of such small scalesin many natural materials.

Jiang and coauthors (40) recently reported that the eye of C. pipiens apparently remains dryeven if exposed to tiny drops, as encountered in the foggy and moist environments where thesemosquitoes usually circle. Figure 18 is a close-up of C. pipiens after the mosquito passed through anaerosol of water droplets. Water condenses on most of the animal, but the eyes remain dry, whichof course preserves its vision (renowned as excellent). Figure 6d displays a microphotographof 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 alsoevaporates quasi-instantaneously. In a cloud, drops are quite polydisperse, with a typical radiusof 10 msuch small drops would impale on most microtextured surfaces but might resist theWenzel transition for the nanopattern worn by the mosquitos eye.

Other promising metastable Cassie states are those obtained on hydrophilic materials witha particular design. Oils having contact angles of the order of 40 can be suspended on specialtextures, producing an increase of the angle by approximately 100 (the superoleophobic effect)(53, 97102). As shown by Herminghaus, fakir wetting drops are possible on overhangs or re-entrant angles, that is, sites where a hydrophilic Young condition can be satisfied (29). Fibersand defects with overhangs do provide quite robust oleophobicity and are able to resist a Wenzeltransition by pressing on the liquid with the Laplace pressure related to the size of the holes

Figure 18Close-up of Culexpipiens after exposureto water aerosol.Droplets condense onthe antennas, but theblack eyes remaindrya condition forpreserving the eyesightof the insect. FromReference 40 (courtesyof L. Jiang).

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at the surface. With structures with pronounced re-entrant profiles, such as those displayed inFigure 7d (where the pattern evokes a mushroom, or a hoodoo), Tuteja et al. (53) spectacularlyreported that even a liquid as wetting as octane can be suspended in a Cassie state with advancingand 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 fewspecial 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 mi-crowrinkles, that consequently generate anisotropic wetting, in particular, in the Cassie regime(69, 103105). Owing to a differential pinning of contact lines, the contact angles (and the hys-teresis) are quite different along and perpendicular to the grooves. Axial motion is preferred, andsuch 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 thescales covering the wings of the butterfly Papilio ulysses. Both the arrangement and microtexture ofthe tiles contribute to the directionality of this material. Another kind of anisotropy is exploitedby water striders (see its inclined hairs in Figure 6c): Striders strike the surface perpendicular tothe grooves, which generates a large contact force, before swinging the legs by 90 to align themin the direction of the motion for skating. Motion will arise from alternating pinning and glidingevents (24).

100 m

Figure 19The wings of Papilioulysses. The way thetiles are displayedtogether with thedetail of the textureconfer anisotropy tothe texture. FromReference 106(courtesy of S.Berthier).


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6.2. Wettability Switches

Textured surfaces undergo a brutal change of wettability as the contact angle exceeds 90

(Figure 5). This behavior can be exploited to achieve superhydrophobic/superhydrophilicswitches. Different physicochemical effects affecting the solid wettability, such as light on pho-tocatalytic textures (107, 108) or heat for temperature-sensitive coatings (109), can be used fortriggering the transition. The comprehensive review by Feng & Jiang (50) provides details.

Electric field, as we have learned from Lippman, also affects wettability. Applying a voltageacross a drop lowers its contact angle, and this effect is amplified on a textured surface. A dropwith an angle of approximately 160 can nearly spread under the action of modest voltages (ap-proximately 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 expectationsfor a switch. Krupenkin et al. (110) proposed to use a short and intense pulse of current (througha thin conductive layer on the sample), which evaporates the liquid close to the surface, hencerestoring a Cassie-suspended state. More generally, there is today no clear example of a Wenzelstate (even potentially metastable) spontaneously transforming into a Cassie state. This situationis detrimental as a vapor condenses: This naturally forces a Wenzel situation, which often evolvestoward mixed and ambiguous Cassie/Wenzel situations (111115). Much remains to be done toachieve genuine antidew materials.

6.3. Giant Slip

Experiments confirm that superhydrophobic materials can provide slippage as water flows onthem, provided that these materials are in the Cassie state. The amplitude of the phenomenon iscaptured by the so-called slip length, which is the extrapolated distance on which the liquid velocityvanishes (Figure 20). First introduced by Navier, this length is generally molecular. However, itcan become of the order of 10 nm on flat hydrophobic solids, as experiments using a surfaceforce apparatus show (116). Similar to what we saw for wetting, this hydrophobic behavior can bedramatically enhanced if the solid is rough (117).

Ou et al. (118) reported micrometric slip lengths from measurements of the pressure necessaryto drive a given flux of water along a square channel striated with microgrooves. For a Poiseuilleflow, the flux varies as W 4p/, denoting p as the pressure gradient along the flow and W as thewidth and depth of the channel. For a large slip ( > W ), the flux instead scales as W 3p/. Slipat the wall reduces the pressure gradient by a factor /W. Ou et al. found pressure reduction byapproximately 40%, suggesting slip lengths of the order of 10 m (see also References 119121).

Figure 20The slip length is the distance inside the solid for which the velocity profile of a flowing liquid vanishes.

92 Quere

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0

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Figure 21Slip length onsuperhydrophobicpatterns ofnanotubes ofconstant density, butwith a varying pitch.In the Cassie state(circles), the sliplength ismicrometric andincreases with thepitch. In the Wenzelstate (squares), thereis no measurable slip.

Using microparticle imaging velocimetry, Joseph et al. (122) directly measured slip lengths for wa-ter flowing on hydrophobic carbon nanotubes. As shown in Figure 21, the slip length dramaticallydepends 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 thatroughness kills any (potential) slippage if the liquid conforms to it. Conversely, micrometric sliplengths are observed in the Cassie state, and increase linearly with the post distance p, in theseexperiments performed at constant S. This slip can be dramatically damped if liquid/air interfacesare 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 largeviscosity ratio between water and air (typically a factor of 100). However, part of the liquid contactsthe top of the posts, which limits the total slip on the surface. We can make this argument morequantitative, following a recent analysis by Ybert et al. (125). For a flow of typical velocity V, thesize affected by the presence of a post should scale as b, the post radius (b p, the pitch of thepost array). The friction force per pillar, and thus per surface area p2, should scale as Vb, denoting as the liquid viscosity. This yields a viscous stress Vb/p2. This stress dominates the onearising from the underlying air flow, provided that Vb/p2 > aV/h, in which we introduced theair viscosity a. Hence there is a geometric requirement, bh/p2 > a/, which can be achieved byadjusting the post height h. With water, a/ is of the order of 102, and the latter criterion willbe satisfied with posts of characteristics b = 1 m, h = 10 m, and p = 10 m, for which thefactor bh/p2 is 101. As seen in Figure 20, the stress can also be written as V/, from which wededuce an effective slip length :

p2/b p/1/2S . 23.For a constant pillar density S, is linear with the pillar spacing p, as seen in Figure 21. Forb p (S 1), we expect very large slip lengths, compared with the values found on flat solids. will typically be between a few p, i.e., 110 m (as reported experimentally), 1000 times largerthan the slip length on a flat hydrophobic solid! If air friction dominates the pillar friction (h

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Wetting and Roughness