4984 DOI: 10.1021/la9036452 Langmuir 2010, 26(7), 4984–4989Published on Web 01/21/2010

pubs.acs.org/Langmuir

© 2010 American Chemical Society

Nano toMicro Structural Hierarchy Is Crucial for Stable Superhydrophobic

and Water-Repellent Surfaces

Yewang Su,† Baohua Ji,*,†,‡ Kai Zhang,† Huajian Gao,§ Yonggang Huang,*,^ and Kehchih Hwang†

†Department of Engineering Mechanics, Tsinghua Universtiy, Beijing 100084, P. R. China, §Division ofEngineering, BrownUniversity, Provindence, Rhode Island 02912, and ^Department of Civil andEnvironmentalEngineering and Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208.‡Present address: Biomechanics and Biomaterials Laboratory, Department of Applied Mechanics, Beijing

Institute of Technology, Beijing 100081, P.R. China

Received September 27, 2009. Revised Manuscript Received November 19, 2009

Water-repellent biological systems such as lotus leaves and water strider’s legs exhibit two-level hierarchical surfacestructures with the smallest characteristic size on the order of a few hundreds nanometers. Here we show that such nanoto micro structural hierarchy is crucial for a superhydrophobic and water-repellent surface. The first level structure atthe scale of a few hundred nanometers allows the surface to sustain the highest pressure found in the naturalenvironment of plants and insects in order to maintain a stable Cassie state. The second level structure leads to dramaticreduction in contact area, hence minimizing adhesion between water and the solid surface. The two level hierarchyfurther stabilizes the superhydrophobic state by enlarging the energy difference between the Cassie and the Wenzelstates. The stability of Cassie state at the nanostructural scale also allows the higher level structures to restoresuperhydrophobicity easily after the impact of a rainfall.

Introduction

Nature has evolved hierarchical surface-structures to achievesuperior water-repellent capabilities in some plants and insects,1-6

allowing rain drops to roll freely on Lotus (Nelumbo nucifera)leaves1,2 andwater striders (Gerris remigis) to stand effortlessly andmove quickly on water.3,4 Lotus leaves exhibit two levels ofhierarchy,1,2,5 the first level having characteristic dimension onthe order of 100-500 nm and the second level on the order of20-100 μm (Figure 1a). Water strider’s legs also show a two-levelhierarchy with the smallest structure size around 400 nm6

(Figure 1b). The present paper aims to address why thesebiological surfaces are hierarchically structured from nanoscaleup and what are the underlying mechanisms that determine thelength scales and the number of hierarchies involved.

During the past decade,many experimental1-12 and theoreticalstudies13-22 have been dedicated to understanding the so-called“Lotus effect”. The connections between surface roughness andhydrophobicitywere first worked out byWenzel13 andCassie andBaxter.1,2,14 In the so-calledWenzel state,13 water forms seamlesscontact with a surface and the contact angle changes with thesurface roughness. In contrast, the Cassie state14 refers to waterforming incomplete contact with a rough surface with air trappedbetween the liquid and solid. Although both Wenzel and Cassiestates can lead to a high contact angle ofwater droplets ona roughsurface, there are large differences in their adhesion propertieswhich, together with their hydrophobic properties, determine thesuperhydrophobic properties of the surface. [Normally, a surfaceis called superhydrophobic if the contact angle of a water dropleton the surface exceeds 150�.6-8 More rigorously, however, super-hydrophobicity should be defined by both a large contact angleand a low roll-off angle (or low adhesion).16,18,19] Experimentaland theoretical studies1,2,7,16 have shown that the Cassie state ismore reasonable for the Lotus effect in view of its incompletecontact and relatively weak adhesion with water droplets. Fromthis point of view, it is critically important for water-repellentbiological surfaces to maintain a stable Cassie state even underharsh environmental conditions.

A number of previous studies7,15-21,23-31 have attempted toinvestigate the stability of the Cassie state and the transitionbetweenCassie toWenzel states. Johnson andDettre15 compared

*Corresponding authors. E-mail: (B.J.) bhji@bit.edu.cn; (Y.H.) y-huang@northwestern.edu(1) Barthlott, W.; Neinhuis, C. Planta 1997, 202, 1–8.(2) Neinhuis, C.; Barthlott, W. Ann. Bot. 1997, 79, 667–677.(3) Hu, D. L.; Chan, B.; Bush, J. W. M. Nature 2003, 424, 663–666.(4) Suter, R. B.; Rosenberg, O.; Loeb, S.; Wildman, H.; Long, J. H. J. Exp. Biol.

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2002, 18, 5818–5822.(11) Wang, Z.; Koratkar, N.; Ci, L.; Ajayan, P. M. Appl. Phys. Lett. 2007, 90,

143117.(12) Gao, L. C.; McCarthy, T. J. Langmuir 2006, 22, 5998–6000.(13) Wenzel, R. N. Ind. Eng. Chem 1936, 28, 988–994.(14) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546–551.(15) Johnson, R. E.; Dettre, R. H. In Contact angle, wettability, and adhesion;

Gould, R. F., Ed.; American Chemical Society: Washington, D. C., 1964; pp 112-135.(16) Marmur, A. Langmuir 2004, 20, 3517–3519.(17) Patankar, N. A. Langmuir 2004, 20, 8209–8213.(18) Zheng, Q. S.; Yu, Y.; Zhao, Z. H. Langmuir 2005, 21, 12207–12212.(19) Carbone, G.; Mangialardi, L. Eur. Phys. J. E 2005, 16, 67–76.(20) Herminghaus, S. Europhys. Lett. 2000, 52, 165–170.

(21) Liu, B.; Lange, F. F. J. Colloid Interface Sci. 2006, 298, 899–909.(22) Otten, A.; Herminghaus, S. Langmuir 2004, 20, 2405–2408.(23) Barbieri, L.; Wagner, E.; Hoffmann, P. Langmuir 2007, 23, 1723–1734.(24) Callies, M.; Quere, D. Soft Matter 2005, 1, 55–61.(25) Cao, L. L.; Hu, H. H.; Gao, D. Langmuir 2007, 23, 4310–4314.(26) He, B.; Patankar, N. A.; Lee, J. Langmuir 2003, 19, 4999–5003.(27) Nosonovsky, M. Langmuir 2007, 23, 3157–3161.(28) Patankar, N. A. Langmuir 2003, 19, 1249–1253.(29) Quere, D. Rep. Prog. Phys. 2005, 68, 2495–2532.(30) Marmur, A. Langmuir 2003, 19, 8343–8348.(31) Yang, C.; Tartaglino, U.; Persson, B. N. J. Phys. Rev. Lett. 2006, 97,

116103.

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the free energies associated with the Cassie andWenzel states of asinusoidal surface and found that the Cassie state is preferred atlarge wave heights. The criterion of the stability of a Cassie statewas discussed from the point of view of design of hydrophobicsurfaces.25-28 Qu�er�e and co-workers7,24,29 studied the transitionbetween the Cassie and Wenzel states by applying a pressure on aliquid drop on amicrotextured surface and observed a transition inwetting state as the pressure is increased. They found that there is acritical pressure that induces an irreversible transition from Cassieto Wenzel states. Extending the work of Johnson and Dettre,15

Carbone and Mangialardi19 found a critical height of a sinusoidalsurface structure beyond which the Cassie state becomes morestable than the Wenzel state, and determined the critical pres-sure for the Cassie-to-Wenzel (CW) transition. Zheng et al.18

performed more rigorous studies on the stability, metastabilityand instability of the Cassie and Wenzel states, as well as theirtransitions under pressure acting on the surface of a periodicarray of micropillars.

Recently, the effect of the characteristic size of the microstruc-ture on the stability of the Cassie state has been explored.21,32,33

The roles of the hierarchy of surface structures were studied by Liand Amirfazli34 and Yu et al.35 However, few studies have beendedicated to the quantitative understanding of the questions whythe biological surfaces are hierarchically structured from nano-scale up and what are the underlying mechanisms that determinethe length scales and the number of hierarchies involved. Thepoint of view adopted in the present study is that the nanometerlength scale should be vital to the water-repellent properties ofbiological surfaces and this size selection might stem from natural

evolution toward a stable Cassie state under harsh environmentalconditions. We will also show that the higher level structures inthe hierarchy will further stabilize the Cassie state and also lead todramatic reduction in contact area, thereby largely removingadhesion between water and the surface.

Results and Discussion

Nanostructure is Crucial for a Stable Cassie State. Let usfirst address the effect of structure size on the wetting state of asurface. Consider a two-dimensional model of water droplet incontact with a surface profile with periodic sinusoidal protru-sions,15,19

hsðxÞ ¼-h cosðkðxþ dÞÞ -d -λ=2e x < -d

-h -d e x < d-h cosðkðx-dÞÞ d e xe d þ λ=2

8<: ð1Þ

where h is the height of the surface protrusions, λ is the surfacewavelength, k = 2π/λ is the wavenumber, d is the half-distancebetween the bases of two adjacent protrusions (Figure 1c), andF=2h/λ is the slenderness ratio of the structure. If the surface is inthe Cassie state, the water will rest on top of the surface structurewith air trapped between the liquid and solid under the support ofsurface tension. As long as the contact angle is smaller than 180�,there will be a finite liquid-solid contact area near the top of thesurface, as shown inFigure 1c. The liquid imposes a pressureP onthe substrate as a result of an externally applied load or impact ofa droplet on the surface (as in rainfall).

Consideration of equilibrium at the triple-junction yields thefollowing normalized relationship between thewater pressure andproperties (geometrical and chemical) of the sinusoidal protru-sion (see the Supporting Information),

P ¼ -h sin a cos θ0 -sin θ0

β

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ h

2sin2 a

q ð2Þ

Here, P = P/(kγLA) is the normalized pressure, γLA is theliquid-air interfacial energy (similarly, γLS and γSA denote theliquid-solid and solid-air interfacial energies, respectively); β =a þ d, β= kβ, a= ka, h= kh, and θ0 is the contact angle of thewater droplet. This solution is valid only for the Cassie state withair trapped between the liquid and solid as shown inFigure 1c. Tofocus on the effect of structure size λ on the Cassie-to-Wenzel(CTW) transition, we set d = 0 in the following discussionswithout loss of generality. The projected fraction of contact area isthen defined as

f ¼ 1-2a=λ ¼ 1-a=π ð3ÞFrom eq 2, we can determine the evolution of the normalizedpressureP as a function of the projected fraction of contact area f.Figure 2a shows five curves of such evolution for the sinusoidal-like surface with different values of the slenderness ratio F. In eachcase,P increases as f increases at the initial stage, as expected. At acritical value fc, the pressure reaches a maximum value defined asthe critical nondimensional pressure Pc. Further increase of pres-sure beyond this critical value induces an instability that causesthe liquid bridge to collapse onto the solid surface, resulting incomplete contact with the solid with fc equal to 1. This is theessence of the CTW transition. Note that we have defined thesurface to be in the Cassie state when f< fc. TheWenzel state cor-responds to f=1.There exists an intermediate state for fc<f<1,which is not stable according to our analysis.

Figure 1. Biological surfaces and theoretical model of the Cassiewetting state of an N-level hierarchical surface structure. (a) Two-level hierarchical structure of Lotus leaves (N. nucifera)1,2 consist-ing of epidermal cells at microscale (papillae; bar = 20 μm) and asuperimposed layer of hydrophobic wax crystals at nanoscale (seeinset; bar = 1 μm);5 (b) Two-level hierarchical structure of waterstrider’s leg (G. remigis)6 exhibitingmicrosetae (bars=20μm)withfine nanoscale grooves (see inset; bar= 200 nm). (c) A liquid dropin contact with an N-level hierarchical wavy surface, where air istrapped between the liquid and structure at each level. The lowestpanel is for an illustration of the contact angle at the triple-line andthe physical quantities involved in the problem.

(32) Reyssat, M.; Pepin, A.; Marty, F.; Chen, Y.; Quere, D. Europhys. Lett.2006, 74, 306–312.(33) Tuteja, A.; Choi, W.; Ma, M. L.; Mabry, J. M.; Mazzella, S. A.; Rutledge,

G. C.; McKinley, G. H.; Cohen, R. E. Science 2007, 318, 1618–1622.(34) Li, W.; Amirfazli, A. Soft Matter 2008, 4, 462–466.(35) Yu, Y.; Zhao, Z. H.; Zheng, Q. S. Langmuir 2007, 23, 8212–8216.

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According to eq 2, the normalized pressure depends on thecontact angle θ0, as well as the normalized parameters h (or theslenderness ratio F) and a (or f). However, the critical nondimen-sional pressure Pc only depends on θ0 and F (as shown inFigure 2a). According to the normalization P = P/(kγLA), thecritical pressure can be expressed as

Pc ¼ PckγLA ¼ 2πPcðθ0, FÞγLAλ

ð4Þ

which is inversely proportional to the surface wavelength. There-fore, the critical pressure for the CTW transition is inverselyproportional to the characteristic size of the surface roughnesswhen all other chemical (e.g., contact angle) and geometrical (e.g.,the slenderness ratio) parameters of the surface are fixed. This sizeeffect is consistentwith the experimental studies byReyssat et al.,32

and eq 4 considers more chemical and geometrical parametersthanprevious studies. For a given contact angleθ0 (e.g.,θ0=106�on a smooth wax surface) and slenderness ratio F, the smaller thestructure size, the higher the critical pressure for the CTWtransition and the more stable the Cassie state is, as shown inFigure 2b. This result suggests that reduction in size of the

microstructure can significantly increase the critical pressure forthe CTW transition, hence stabilizing the Cassie state.

For Lotus leaves, the highest pressure found in nature comesfrom the impact by water droplets in a rainfall, which can be aslarge as 0.1 MPa.36 The critical nondimensional pressure isdefined once the chemical and normalized geometrical para-meters of the surface, e.g., the contact angle and the slendernessratio F, are given. The slenderness ratio of the nanostructure onLotus leaves is generally 3-5 (Figure 1a), while that on waterstrider’s legs is 1-3 (Figure 1b). For F = 3, the critical non-dimensional pressure is found to bePc=0.16. According to eq 4,under the highest impact pressureP=0.1MPa of rain drops, thecritical structure size is calculated to be 740 nm (taking γLA =0.073J/m2). Therefore, our analysis shows that the structure size λmust be smaller than λc=740 nm in order to maintain the Cassiestate under the highest impact pressure from rain drops.

To validate our theoretical model, we have numerically calcu-lated the relationship between the critical size of the surfacestructure and pressure, with or without the effects of gravity,for both 2D (with 1Ddistribution ofwavy protrusions, Figure 1c)and 3D surface configurations (with 2D distribution of revolvingsinusoidal protrusions, as shown by the inset in Figure 2b) usingthe software package Surface Evolver.37 The numerical results arecompared with the theoretical predictions in Figure 2b. It is seenthat the critical size calculated based on the 2Dmodel is the sameas our analytical result. The corresponding result based on the 3Dmodel is smaller than the 2Dvalue (the difference between 2Dand3D models can be rigorously obtained for a prismatic pillarstructure). At the raindrop pressure of 0.1 MPa, the 3D modelpredicts a critical structure size of 400 nm. This result is inagreement with the observed smallest structure on Lotus leavesand suggests that the nanometer length scale may have indeedbeen evolved to maintain a stable Cassie state under the mostsevere pressure from raindrops. Similarly, nanostructures allowwater striders to maintain their legs in a stable water-repellentstate to be able to stand and maneuver on water.The Roles of Hierarchical Surface Structures. Recently, a

number of studies11,12,38,39 have shown that surfaces with a singlelevel structure do not necessarily guarantee low adhesion even inthe Cassie state, and some systems exhibit strongly adhesivesurfaces no matter what the size of the surface structure is.However, the adhesion force dramatically decreases as soon asthe second level structure is introduced. What are the underlyingmechanisms of adhesion reduction via structural hierarchy?

To answer this question, let us consider a self-similar hierarch-ical structure in the Cassie state shown in Figure 1c. The contactarea fraction (the ratio of the contact area to the total surface areaof the hierarchical structure) of a self-similar N-level structure is(for derivation see the Supporting Information)

SN ¼YNn¼1

sn ð5Þ

where

sn ¼ φðan,πÞ,φðx1, x2Þ ¼Z x2

x1

ð1þh2sin2 xÞ1=2 dx=π ð6Þ

and an denotes the position of the triple line at the nth level struc-ture. Figure 3a shows that the contact area fraction exponentially

Figure 2. Critical pressure for water penetration into a sinusoidalsurface as a function of the characteristic structural size. (a)Normalized pressure versus the contact area fraction as liquidbridges penetrate into the surface at different slenderness ratios ofthe surface protrusions. (b) Scaling lawsof the penetrating pressureversus the structural size predicted by 2D theoreticalmodel and 2Dand 3D numerical simulations. The prediction of 2D numericalsimulationswith andwithout the gravity effects agreeswellwith thetheoretical results. The penetrating pressure predicted from the 3Dmodel is lower than that from the 2D model.

(36) Erpul, G.; Norton, L. D.; Gabriels, D. Catena 2002, 47, 227–243.(37) Brakke, K. Surface Evolver.(38) Jin,M.H.; Feng, X. J.; Feng, L.; Sun, T. L.; Zhai, J.; Li, T. J.; Jiang, L.Adv.

Mater. 2005, 17, 1977–1981.(39) Yeh, K. Y.; Chen, L. J.; Chang, J. Y. Langmuir 2008, 24, 245–251.

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decreases as the number of hierarchical levels increases (We choseP=0and 146 Pa in the calculation, where 146 Pa is the static pres-sure induced by the surface tension of a liquid drop of 2 mmin diameter). For example, the contact area fraction is S1 = 0.255for the single level structure but becomes S2 = 0.003 for the two-level structure. Since a small contact area between liquid and solid isan important factor for adhesion reduction,8,39 the structuralhierarchy keeps the adhesion force sufficiently low for free rollingof water droplets on lotus leaves and walking of water striders onwater.

In addition, the hierarchical structure can significantly enhancethe stability of the superhydrophobic state of a surface. Consider-ing different wetting states of the self-similar N-level hierarchicalstructure shown in Figure 1c, we can calculate the difference inHelmholtz free energy per unit area (unit length in the 2Dmodel)between the Cassie and the Wenzel states of the surface as (forderivation see the Supporting Information),

ΔFN ¼ FW

N -FCB

N ð7ÞIn this equation, FN

W and FNCB are determined by the recursive

equations FNW = FN-1

Wφ(0, π) and FN

CB = FN-1CB

φ (aN, π)þγLAψ(aN, 1/PN), whereF0

W= F0CB= γLS-γSA,ψ(x1, x2)= x2 arcsin

(x1/x2)/π, aN denotes the position of the triple line at theNth levelstructure, and PN denotes the normalized pressure of the Nthlevel. Figure 3b shows the relationship between this energydifference and the number of hierarchies in the cases of P = 0and 146 Pa. It is seen that the energy difference between theCassieand the Wenzel states increases exponentially as the number ofhierarchies increases. This suggests that the structural hierarchyalso tends to enlarge the energy difference between the Wenzeland the Cassie states of the surface, thereby stabilizing the Cassiestate.

Why do biological surfaces only have two levels of hierarchies?To address this question, we assume that the higher level surfacestructures can sustain at least the static pressure of liquid dropletor the pressure (on the order of 100 Pa) induced by the weight ofa water strider (10 dyn)3 (otherwise they will be in the Wenzel

state and useless for the hydrophobicity of biological surfaces).Generalizing eq 4, we have the critical size for higher levelstructure,

λN ¼ 2πPcðθN-1, FÞγLAPc

ð8Þ

According to eqs s10 and s11 (Supporting Information), θN-1 canbe calculated as

cos θN-1 ¼ γSA,N-1 -γLS,N-1

γLA¼ -

FCB

N-1

γLAð9Þ

For the static pressure of 146 Pa, the critical size of the thirdlevel structure is calculated to be 2.6 mm. This size is larger thanthe size of a typical rain droplet of intermediate size (e.g., 2 mm),and therefore useless for enhancing superhydrophobicity. For thewater strider, this critical size ismuch larger than the dimension ofits leg (on the order of 100 μm). If one reduces the size of the thirdlevel structure by 1 order of magnitude, i.e., 260 μm, it will beclose to the upper limit of the second level structure, which is20-100 μm, aswell as the dimension of thewater strider’s leg. Thethird level structure has no physicalmeaning either. Thismight bethe reason why the surface structures in nature generally has onlytwo levels of hierarchy. We therefore suggest that the staticpressure of liquid determines themaximumnumber of hierarchiesof the biological surfaces.The Reversibility of the Cassie State of Higher Level

Structures. How do the higher-level structures, e.g., the secondlevel structure, survive the impact from rain drops and maintainthe Cassie state? This question can be understood by studying theGibbs free energy of the system as the wetting state of the surfaceevolves under external pressure. The Gibbs free energy of thesystem loaded by a pressure P is written as G = F - PV, whereV= hλ-V0 is the volume of liquid penetrating into the structure,andV0 is the volume of air trapped under the liquid (see Figure 1cand the Supporting Information). In the calculation of freeenergy, we choose P = 146 Pa and 0.1 MPa, i.e., the staticpressure of a 2 mm water droplet and the raindrop pressure,respectively. Two different structure sizes λ=200 nm and 20 μmare selected in the calculation.

The upper panel inFigure 4 plots theGibbs free energy per unitarea of surfaceG=G/λ of a single level structure as a function ofthe area fraction of liquid-solid contact for the two differentstructure sizes. At the structure size of λ = 200 nm, the wettingstate stays as a stable Cassie state with minimum free energy atf = fc under both values of the external pressure. This suggeststhat the external load is not able to overcome the forward energybarrier for the CTW transition at this sufficiently small structuresize. However, for the large structure size of, λ=20 μm, althoughit canmaintain a stable Cassie state at the static pressure P=146Pa (due to the forward energy barrier ΔG1

f ), the wetting statewould evolve from the Cassie state at f= fc to theWenzel state atf = 1 under the raindrop pressure.

Let us examine the wetting behavior of the surface with largerstructure size λ = 20 μm when the surface is released from thepreviously loaded raindrop pressure back to the static pressure ofthe liquid. As the pressure is reduced from 0.1MPa to 146 Pa, thewetting state immediately transfers from state A to state B asshown in the upper panel of Figure 4, which is still aWenzel state.In order for the surface to restore its original Cassie state, i.e.,from B to C, a reverse energy barrier from the Wenzel to theCassie states, ΔG1

r, must be overcome.

Figure 3. Effects of structural hierarchy on superhydrophobicity.(a) Contact area fraction as a function of the number of hierarchi-cal levels under liquid pressures P = 0 and 146 Pa. (b) EnergydifferenceΔFN between theWenzel and Cassie states as a functionof the number of hierarchical levels under liquid pressures atP=0and146Pa.Theopen squares inblack stand forP=0and the solidtriangles in red are stand for P= 146 Pa. θ0 = 106�, and F= 3.

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The reverse energy barrier ΔGr for the Nth level structure canbe calculated as,

ΔGr

N ¼ -FCB

N-1φð0, acNÞþγLAψðacN , 1=PNÞþγLAPNVc

N=ð2πÞð10Þ

where aNc is the critical position of the triple line of the Nth

level at which the structure reaches the maximum value of theGibbs free energy, and VN

c 0 is the corresponding value ofV 0 ata = aN

c . According to eq 10, we find that the reverse energybarrier decreases exponentially with increasing number ofhierarchical levels (see Figure 5), a behavior similar to that ofthe contact area fraction. For the single-level structure of size20 μm, the reverse energy barrier ΔG1

r is about 1/20 of surfaceenergy (see Figure 5). However, for the second level structureof hierarchical surface of the same size, the reverse energybarrier ΔG2

r is 3 orders of magnitude smaller than ΔG1r (at the

same time the forward energy barrierΔG2f is much larger than

ΔG1f ) (see the lower panel in Figure 4 and Figure 5). For

example, the energy input needed for the transition at thesecond level structure is only 8 � 10-12J. In nature, adisturbance by a wind of velocity 0.1 m/s is on the order of4 � 10-8J, which is big enough for inducing the transition.Therefore, the structural hierarchy allows the Cassie state tobe restored after the impact of a rainfall, i.e., from B0 to C0.A very recent study by Boreyko and Chen40 has demonstratedhow a disturbance can restore superhydrophobicity from theWenzel state.

Conclusions

In summary, the present study indicates that the nano- tomicrostructural hierarchy is essential for the superhydrophobicsurfaces seen in water-repellent plants and insects. The nano-structure allows the surface to sustain the highest pressure innature in order to maintain a secure Cassie state, while themicrostructure significantly reduce the contact area, therebylargely removing adhesion between solid and fluid at the macro-scopic level. The two level hierarchy further stabilizes the super-hydrophobic state by enlarging the energy difference between theCassie and the Wenzel states. The stability of Cassie state at thenanostructural scale also allows the higher level structures torestore the Cassie state easily after the impact of a rainfall. Thisstudy shows that the characteristic size of the smallest structure ofbiological surfaces is determined by dynamic loading (rainfallpressure), while the number of hierarchical levels seems to bedetermined by the static loading (static pressure of the liquid) inthe natural environment of relevant plants and insects. Theprinciple of such superhydrophobic surfaces can provide usefulguidelines for engineering new materials for industrial applica-tions.

Compared to previous energetic studies of fractal sur-faces,15-20,23-28,30,31 the main contribution of our present studyis that we have considered the roles of an external force (fromeither the impact pressure of rain drops or static pressure of awater droplet due to surface tension) in the stability of the Cassiestate and the critical size of each level structure of the hierarchy.Our model provides feasible explanations for why the nanostruc-ture of the biological surface can sustain the impact by rain dropsand remains in a stable Cassie state. Our method has also beenused to study the energy barrier between the Cassie state and theWenzel state under the external force, and the effects of the higherlevel structure on energy barrier for the inverse state transitionfrom the Wenzel to the Cassie, and to discuss the reversibility ofthe Cassie state (from the Wenzel state) of the surface at higherlevel structures after impact by the rain drops.

Previous studies41-44 have shown that the nanometer structuresize and structural hierarchy play essential roles in mechanicalproperties of biological systems. There are interesting analogueies

Figure 4. Gibbs free energy per unit area (length in the 2Dmodel)of the liquid-air-solid system (θ0 = 106� and F = 3) as liquidpenetrates into the surface. The Cassie state, Wenzel state andenergy barrier between them can be clearly identified. The upperpanel for a single level structure shows that the energy barrier cannot be overcome when the structure size λ is 200 nm at the externalpressure conditions P=146 Pa and 0.1MPa. However, the CTWtransition spontaneously occurs at P = 0.1 MPa when the struc-ture size is λ = 20 μm. The reverse energy barrier for Wenzel-to-Cassie transition G1

r is about 1/20 of the surface tension of water;The lower panel is for a two-level hierarchical surface structurewith λ1=200 nmand λ2=20μm.The reverse energy barrierG2

r ofthe second level structure is about 3 orders of magnitude smallerthan G1

r due to the presence of the first level nanostructure, whichsignificantly enhances the reversibility of the Cassie state of thesurface.

Figure 5. Relationship between the reverse energy barrierGNr and

the structure level N in a self-similar hierarchical surface at P= 0and 146 Pa. The open squares in black are results for P=0, whilethe solid triangles in red are for P= 146 Pa.

(40) Boreyko, J. B.; Chen, C.-H. Phys. Rev. Lett. 2009, 103, 174502.

(41) Gao, H.; Ji, B.; Jager, I. L.; Arzt, E.; Fratzl, P.Proc. Natl. Acad. Sci. U.S.A.2003, 100, 5597–5600.

(42) Ji, B.; Gao, H. J. Mech. Phys. Solids 2004, 52, 1963–1990.(43) Gao, H.; Yao, H. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 7851–7856.(44) Yao, H.; Gao, H. J. Mech. Phys. Solids 2006, 54, 1120–1146.

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between the roles of the hierarchical structures in wetting behaviorsand those in mechanical properties of bone and shells as well as inadhesion ability of Gecko. For example, the nanometer scalerenders the critical structure components in bone and shellsinsensitive to crack-like flaws, thereby allowing these materials tofail near their theoretical strength,41,42 and hierarchical structuresfurther enhance the fracture toughness of bone and shells. Also, thenanoscale fibrillar structures on the feet of Gecko can achievetheoretical strengthof adhesionand structure hierarchy enlarges theadhesion energy so that Gecko can adhere and move along verticalwalls and ceilings with strong adhesion ability.43,44 In the super-hydrophobicity of surface of plants and insects, nanostructureensures the stability of theCassie state and the hierarchical structurereduces the contact area and at the same time provides the“robustness” of Cassie state at higher scales. Therefore, it seems

that nature uses similar strategies, i.e., with hierarchical design fromnanoscale, to optimize or control different material properties.

Acknowledgment. This research was supported from theNational Natural Science Foundation of China through GrantNo. 10502031, 10628205, 10732050, 10872115, and NationalBasic Research Program of China through Grant No.2004CB619304 and 2007CB936803. The work of HG is partiallysupported by the A*Star Visiting Investigator Program “SizeEffects in Small Scale Materials” hosted at the Institute of HighPerformance Computing in Singapore.

Supporting Information Available: Text giving the mathe-matical derivations. This material is available free of chargevia the Internet at http://pubs.acs.org.

Supporting Information for:

Nano to Micro Structural Hierarchy is Crucial for Stable

Superhydrophobic and Water-Repellent Surfaces

Yewang Su, Baohua Ji, Kai Zhang, Huajian Gao, Yonggang Huang, Kehchih Hwang

For the derivation of equation 2:

Consider the liquid-solid system shown in Figure 1c. We assume that a) the substrate can be considered

rigid in comparison with water and b) the effect of gravity is negligible. The latter assumption is justified

as the structure size of concern here is much smaller than the capillary length. Furthermore, since

typical dimensions of the surface structure, h and λ, are much smaller than the diameter of a water

droplet, we simplify the liquid geometry as a semi-infinite space.

Young’s equation ( ) LALSSA γγγθ −=0cos is assumed to hold at the triple-junction of liquid, air

and solid, where 0θ is the contact angle, and SAγ , LSγ , LAγ are the solid-air, liquid-solid and liquid-air

interfacial energies, respectively.

The contact angle is equal to the angle between the tangent vectors l and m (see Figure 1c) as

2/122

2/122

0 )](sin)(1[)/1(/)sin(cos

kakhrrkakh

+−+

−=ββθ (s1)

where da +=β . The relationship between the pressure P and the radius of curvature of the liquid-air

interface is defined by the Young-Laplace equation (see Figure 1c),

rP LAγ= (s2)

By substituting equation (s2), equation (s1) can be rewritten in a normalized form as

0cos)sin1()1(sin 02/1222/122

=++−+ θββ ahPahP (s3)

where )/( LAkPP γ= , kaa = , khh = , kdd = , and ββ k= . Solving Equation (s3), we can get the

analytical solution of nondimensional pressure P , i.e., equation 2 in the main text.

For the derivation of equation 5:

Consider the theoretical model for an N-level hierarchical structure as shown in Figure 1c. The contact

area fraction of the first level structure is (see the lowest panel in Figure 1c),

∫∫ +=+==πλ

πλλ

1

1

1

2/1222/

2/122

1111 )sin1(1)sin1(2

aa

xdxhdxxhSS (s4)

where 1S and 1λ are the contact area and the structure size of the first level structure, respectively; 1a

is the position of the triple line at the first level structure, and 1a is the nondimensional value of 1a .

The contact area fraction of the second level structure is,

ππ

∫ +=2

2/12212 )sin1(

a

xdxhSS (s5)

Defining ππ

∫ +=1

2/1221 )sin1(

a

xdxhs , and ππ

∫ +=2

2/1222 )sin1(

a

xdxhs , we have

11 sS = , 212 ssS = (s6)

Therefore, for the Nth level structure, equation (s6) can be generalized to be,

∏=

=N

nnN sS

1

(5)

where ),( πφ nn as = , ( ) πφ xdxhxxx

x∫ += 2

1

2/12221 )sin1(, , and na denotes the position of the triple

line at the nth level structure.

For the derivation of equation 7:

We begin with the first-level structure from which we can then consider the higher-level structures level

by level. Considering the N-level hierarchical structure as shown in Figure 1c, the apparent energy per

unit area (unit length in the 2D model) of the first-level structure can be described by the Helmholtz free

energy as,

)( 1,1,111 SACBLS

CBCB FF γγλ −== . (s7)

Here, CBLS 1,γ is the apparent solid-liquid interface energy of a typical Cassie wetting state, consisting of

interface energy of the solid-liquid interface, liquid-air interface, and solid-air interface as (see the

lowest panel in Figure 1c),

( ) ( ) ( )11111, ,0,, araa SALALSCBLS φγψγπφγγ ++= (s8)

where we define ( ) πφ xdxhxxx

x∫ += 2

1

2/12221 )sin1(, , and ( ) πψ )/arcsin(, 21221 xxxxx = . 1a

denotes the position of the triple line at the first-level structure, and 1λ denotes the size of the first-

level structure; 1,SAγ is the apparent solid-air interface energy,

( )πφγγ ,01, SASA = (s9)

It is noted that 1,SAγ is a constant which does not change with the wetting state. Substituting equation

(s8) and (s9) into equation (s7), we have ( ) ( )11101 1,, PaaFF LACBCB ψγπφ += , where

SALSCBF γγ −=0 and rkPP LA 1)/( == γ . Generalizing this to the Nth level of hierarchy in a self-

similar hierarchical surface, we obtain the recursive equation for the apparent surface energy of a

hierarchical surface in Cassie state,

( )πφγγ ,01,, −= NSANSA (s10)

( ) ( ) ( )NNLANNLSNNSANLS Paaa /1,,,0 1,1,, ψλπφγφγγ ++= −− (s11)

The Helmholtz free energy for each level can be calculated from,

( )NLSNSACB

NF ,, γγ −−= (s12)

in combination with the recursive equation

( ) ( )NNLANCB

NCB

N PaaFF 1,,1 ψγπφ += − (s13)

where Na denotes the position of the triple line at the Nth level structure, and NP denotes the

normalized pressure of the Nth level structure.

In a similar manner, the Helmholtz free energy of the first level structure in Wenzel state is

)( 1,111 SAWLS

WW FF γγλ −== (s14)

where ( )πφγγ ,0LSWLS = is the apparent solid-liquid interface energy of the Wenzel state. Therefore

( )πφ ,001WW FF = , where SALS

WF γγ −=0 . Thus we have the recursive equation of Helmholtz free

energy for the N-level hierarchical surface in the Wenzel state,

( )πφ ,01W

NW

N FF −= (s15)

The energy difference between the Cassie and Wenzel states of the surface with N levels of hierarchy

can be obtained based on equations (s13) and (s15),

CBN

WNN FFF −=∆ (7)

For the derivation of 'V :

The Gibbs free energy of the system loaded by a pressure P is written as, PVFG −= , where

'VhV −= λ is the volume of liquid penetrating into the structure, and

( )

+−+−−= ahaP

PaaP

PaahV sin1

2)arcsin(

2

1cosk

'2/122

2πλ

. (s16)

The free energy per unit area of the surface is πγλ 2VPFGG LA−== , where VkV 2 = . Then,

we can calculate the change of free energy as the liquid penetrates into the surface under fixed

pressure.