Wetting of Structured Hydrophobic Surfaces by Water Droplets

Wetting of Structured Hydrophobic Surfaces by WaterDroplets

Oskar Werner,*,† Lars Wågberg,† and Tom Lindstrom‡

KTH, Fibre and Polymer Technology, Div. Fibre Technology, Drottning Kristinas Vag 53,10044 Stockholm, Sweden, and STFI, Box 5604, 10044 Stockholm, Sweden

Received September 5, 2005

Super-hydrophobic surfaces may arise due to an interplay between the intrinsic, relatively high, contactangle of the more or less hydrophobic solid surface employed and the geometric features of the solid surface.In the present work, this relationship was investigated for a range of different surface geometries, makinguse of surface free energy minimization. As a rule, the free energy minima (and maxima) occur when theLaplace and Young conditions are simultaneously fulfilled. Special effort has been devoted to investigatingthe free energy barriers present between the Cassie-Baxter (heterogeneous wetting) and Wenzel(homogeneous wetting) modes. The predictions made on the basis of the model calculations compare favorablywith experimental results presented in the literature.

BackgroundIn recent years, there has been growing interest in

super-hydrophobic surfaces. The list of possible applica-tions of such surfaces is long, including both fundamentalresearch and commercial applications. Many differenttypes of water-repellent and self-cleaning surfaces havebeen developed.1-9 A few years ago, Nakajima et al.presented a review concerning this topic.10

The wetting of a heterogeneous solid surface is a rathercomplex matter, and a number of previous works dealwith its theory.11-18,24,26,28,29 Even though it can in principlebe defined for equilibrium cases at every point by theLaplace and Young conditions, the number of differentstates that need to be examined is often vast. To generatea model that is practical and useful, it is often better toconsider the entire system as a whole. Such an approachusually requires the use of certain approximations.Moreover, any such model should be as simple as possible,

yet physically correct, and should enable good enoughpredictions.

The best known theoretical approach to heterogeneouswetting is presumably that of Cassie and Baxter.12 In thisapproach, the difference in free energy per projected unitarea beneath and beside a resting sessile drop is used toaccount for the location of the droplet base perimeter,whereas the penetration of the pores in the base structureis governed by the fulfilment of the Young condition.25

The two major assumptions made are that (a) the structureof the solid surface can be treated in an average fashionand (b) the effects of the hydrostatic and droplet curvaturepressures are negligible.

The main purpose of this work was to develop a usefultool to support the development of super-hydrophobiccellulose surfaces. The aim was that the model shouldindicate the apparent contact angle and whether a certaindrop-on-surface system will would be in heterogeneous(Cassie-Baxter) or homogeneous (Wenzel) wetting mode.These modes correspond to free energy minima. Toinvestigate not just their existence but also their stability,the free energy barriers between the heterogeneous andhomogeneous wetting modes have to be studied. To thisend, for the special case of square pillars, the energybarriers were studied by Patankar.19

Most of the droplet-on-surface systems of interest residein the earth’s gravitational field. For small droplets,however, the effect of gravity is usually minor, comparedwith that of the interfacial free energy changes, and canthus be disregarded, though there are some exceptions.By ignoring gravity, one may predict contact angles equalto 180°,17 both in the heterogeneous and homogeneousmodes. The former case implies droplets resting on aninfinitesimally small solid surface area; if that view ischosen, the system is regarded as kept in the heteroge-neous mode by infinitesimally short three-phase contactlines. It may also be the case that gravity-less models failto predict interesting wetting behaviors, especially forintrinsic contact angles, θintr, approaching 90°. One of theoriginal objectives of this work was to present a modelthat takes gravity into account in an appropriate fashion.In the end, however, it turned out that such gravity effectsare often negligible. Instead, the focus has been shifted

* Corresponding author. E-mail: [email protected].† KTH.‡ STFI.(1) Shibuichi, S.; Onda, T.; Satoh, N.; Tsujii, K. J. Phys. Chem 1996,

100, 19512-19517.(2) Onda, T.; Shibuichi, S.; Satoh, N.; Tsujii, K. Langmuir 1996, 12,

2125-2127.(3) Bico, J.; Marzolin, C.; Quere, D. Europhys. Lett. 1999, 47, 220-

226.(4) Youngblood, J.; McCarthy, T. Macromolecules 1999, 32, 6800-

6806.(5) Oner, D.; McCarthy, T. Langmuir 2000, 16, 7777-7782.(6) Nakajima, A.; Hashimoto, K.; Watanabe, T. Thin Solid Films

2000, 140-143.(7) Miwa, M.; Nakajima, A.; Fujishima, A.; Hashimoto, K.; Watanabe,

T. Langmuir 2000, 16, 5754-5760.(8) Yoshimitsu, Z.; Nakajima, A.; Watanabe, T.; Hashimoto, K.

Langmuir 2002, 18, 5818-5822.(9) Feng, L.; Song, Y.; Zhai, J.; Liu, B.; Xu, J.; Jiang, L.; Zhu, D.

Angew. Chem. 2003, 42, 800-802.(10) Nakajima, A.; Hashimoto, K.; Watanabe, T. Monatshefte Chem.

2001, 132, 31-41.(11) Wenzel, R. Ind. Eng. Chem 1936, 28, 988-994.(12) Cassie, A.; Baxter, S. Trans Faraday Soc 1944, 40, 546-551.(13) Johnson, R.; Dettre, R. In Surface and Colloid Science; Science,

E. M. C., Ed.; John Wiley and Sons, Inc.: New York, 1969; Vol. 2, ChapterWettability and Contact Angles, pages 85-153.

(14) Oliver, J.; Huh, C.; Mason, S. Colloids Surf. 1980, 1, 70-104.(15) Kijlstra, J.; Reihs, K.; Klamt, A. Colloids Surf. 2002, 206, 521-

529.(16) Patankar, N. Langmuir 2003, 19, 1249-1253.(17) Marmur, A. Langmuir 2003, 19, 8343-8348.(18) Otten, A.; Herminghaus, S. Langmuir 2004, 20, 2405-2408. (19) Patankar, N. A. Langmuir 2004, 20, 7097-7102.

12235Langmuir 2005, 21, 12235-12243

10.1021/la052415+ CCC: $30.25 © 2005 American Chemical SocietyPublished on Web 11/19/2005

toward estimating the free energy minima correspondingto homogeneous wetting and quantifying the barriersbetween them.

General Features of the Present Model

A global approach is taken by considering the overallexcess free energy of the whole droplet resting on the solidsurface. Surface free energies are invoked as well as thepotential energy of the water in the droplet in thegravitational field. In this way, every point in the state-space, in which each dimension represents one of thesystem variables, corresponds to a certain free energyvalue. This energy surface cannot only be used todetermine the thermodynamically stable and metastablestates but also for determining the depth of the corre-sponding minima and the height of the barriers betweenthem.

The model uses two variables: the apparent contactangle, θ, and the penetration of the liquid into the basestructure, z. Hence, the state-space is one of two dimen-sions. The geometrical surface parameters, f (i.e., thewetted fraction of the projected area) and r (i.e., real wettedarea per projected wetted area), are considered to befunctions of z (Figure 1). Thus f (z) and r(z), together withthe interfacial free energies of the solid-liquid and solid-vapor interfaces, γsl and γsv, are the sole characteristicsof the solid surface. For reasons of convention, the notationrf and r1-f will be used as roughness factors for the wettedand unwetted parts of the solid surface for the reason ofclarification in some cases.

The thermodynamic scheme of the Young as well as theCassie-Baxter equations describing the droplet-on-surface problem can be backtracked to a minimization ofthe free energy (hereafter denoted E) with respect to θ.In the Cassie-Baxter approach,12 the average free energyper unit area beneath the droplet is considered whencalculating the apparent contact angle, θ, the result beingthe Cassie-Baxter equation, viz.

where the intrinsic contact angle, θintr, is to be distin-guished from the apparent contact angle, θ. Along the zaxis, the fulfilled Young condition (the zero hydrostaticpressure approximation, resulting in a planar liquid-vapor interface beneath the droplet) and knowledge ofthe geometry of the surface (parallel cylinders in theoriginal work)12 are used to calculate the penetration ofthe liquid into the structure. In the case of rf ) 1, eq 1becomes cos θ ) f cos θintr + f - 1, and in the case of

complete wetting, it reduces to the Wenzel equation11

Concerning the hydrostatic and capillary pressures, onemight invoke a pressure-induced curvature of the liquid-vapor interface beneath the droplet. For a global model,this is, however, not very convenient, since the curvaturewill be a quite complex function of the spatial coordinates,for even a rather simple surface topography. In the presentmodel, the flat, liquid-vapor interface assumption isretained but merely in the sense that the extension of theliquid-vapor interface is estimated as if it were actuallyplanar. [The planar approximation is good for pore sizesup to approximately one tenth of the droplet radius; thiscan be checked using the chord theorem.] Instead, thechanges of the droplet gravitational energy are explicitlyconsidered. Hence, when the interface propagates in thez direction, the gravitational energy lowers. To determinethe equilibrium penetration, z, this lowering is balancedagainst the net interfacial free energy change due to thechange in interfacial area. The situation encountered hereis in principle the same as the one we face when treatingcapillary rise. The effect of gravity can be accounted foreither by considering the Laplace pressure drop acrossthe liquid-vapor interface or in terms of the potential ofthe rising liquid in the gravitational field.20

The applicability of a strictly thermodynamic treatmentof heterogeneous wetting has been questioned.4,5,21 Hys-teresis effects may cause the droplet system to be unableto reach its true free energy minimum, meaning that E(θ)has local minima corresponding to metastable states. Thisdoes, however, mean that smoothing E(θ), i.e., makingthe local energy minima in E(θ) less deep,4 is moreimportant than having the global minimum at a high θvalue in those cases.

A droplet resting on a solid surface is treated as a systemlocated somewhere in a 2-dimensional state-space speci-fied by the two variables θ and z. For each system, everystate (θ, z) corresponds to a certain E value. It is thenpossible to determine the path along which the systemhas to move in order to reach a state of lower E. A plotof such a state-space is shown in Figure 2. The separatecontributions to E are shown in eqs 4-7 below. The firstterm, describing the gravity contribution (4), displays howthe free energy will decrease when the droplet’s center ofmass declines with increasing values of z. For example,if f(z) is 0.5 for all values of z, the center of mass will

(20) Henriksson, U.; Eriksson, J. C. J. Chem. Educ. 2004, 81, 150-154.

(21) Chen, W.; Fadeev, A.; Hsieh, M.; Oner, D.; Yongblood, J.;McCarthy, T. Langmuir 1999, 15, 3395-3399.

Figure 1. Schematic representation of the z-dependent version of eq 1 used in the model with a 2D analogy. In this figure, where∑ici is the real wetted sample distance and ∑iai is the projected wetted sample distance, f2D ) ∑iai/L, r2D ) ∑ici/∑iai. Both ai andci, and hence r2D and f2D, are dependent on z. The liquid-vapor interface beneath the droplet can be approximated as being flatif the structure is small in scale compared to the droplet radius.

cos θ ) r cos θintr (2)

cos θ )frf(γsv - γsl)

γlv+ f - 1 ) frf cos θintr + f - 1 (1)

12236 Langmuir, Vol. 21, No. 26, 2005 Werner et al.

decline 1 when z increases 2 µm. Equations 5-7 implythat the interfacial free energy contributions are directlyproportional to the area of each kind of interface. Theconstant in eq 7 represents the total interfacial energy ofthe solid-vapor interface of the nonwetted sample. It hasbeen set to zero in this model; however, this is not arestriction, since only the area change is of importance.Summing up we have:

variables: θ, zconstants: γlv,γsv,γsl, V, g, Fknown functions: R(θ), r(z), f (z)

where θ is the equilibrium apparent contact angle(degrees), z is the liquid’s penetration depth into thesurface structure (m), γij is the interfacial energy (J m-2),V is the volume (m3), F is the mass density (kg m-3), g isthe gravitational acceleration (9.82 ms-2), R is the dropradius (m), f(z) is the wetted fraction of the projected area(dimensionless), r(z) is the real wetted area per projectedwetted area (dimensionless), and E is the free energy (J)

that is

where the functions r(z) and f (z) are specific for eachsurface topography. A schematic explanation is found inFigure 1.

If the influence of gravity is neglected, i.e., Egravity ) 0,z is set to a constant value and θi ) (γsv - γsl)/γlv, thenexpression 8 reduces to E(θ) ) πR2(θ)(sin2 θ(γsl - γsv) +2γlv(1 - cos θ)) + constant, and δE/δθ ) 0 will give eq 1.

Implicit in eq 8 is the assumption that the amount ofliquid penetrating the surface structure is small comparedwith the droplet volume. [Taking account of the decreasein volume above the droplet base and the ensuing decreasein R and Elv corresponds to considering the Laplacepressure of the spherical droplet surface in the Young-Laplace approach.] The present model accounts for gravityonly to a limited extent. The pressure differences insidethe droplet and the nonspherical droplet shape they causeare not considered in this model.

It should be mentioned that during the early develop-ment of the present theoretical model describing wettingof super-hydrophobic surfaces,22 a similar but still sig-nificantly different model was simultaneously developedby Jopp et al.23

HysteresisAlthough propagation of a droplet in the z direction

through a rough structure typically is in the micrometerrange, propagation of the droplet base perimeter in thex-y direction may very well occur in the millimeter range.For this reason, the lateral and normal directions aretreated differently in the model. For example, energybarriers originating in the roughness are accounted foralong the z axes, where they are most critical, but not inthe x-y plane, i.e., the θ-dependent part of the model.However, if there is a large difference between theadvancing and the equilibrium apparent contact angles,it may be of interest to investigate the z directionindependently. If the model is restricted in this way, itcan be determined at what contact angles a droplet of aparticular volume can no longer stay in heterogeneouswetting mode on a particular surface. This angle is denotedby θfallthrough in this work. If this angle is compared withthe equilibrium contact angle, the degree of stability of asuper-hydrophobic surface in Cassie mode can be esti-mated.

Development of a Computer ToolA computer tool was coded in MATLAB in order to test

the applicability of the main expression presented in eq8. For a structure of given parameters and surfacefunctions, the program calculates a matrix consisting of

(22) Werner, O. Computer Modelling of the Influence of SufaceTopography on Water Repellency and a Study on Hydrophobic PaperSurfaces with Partly Controlled Roughness. Master’s thesis, LinkopingUniversity, 2003 http://www.ep.liu.se/exjobb/ifm/tff/2003/1168/exjobb.p-df.

(23) Jopp, J.; Grull, H.; Yerushalmi-Rosen, R. Langmuir 2004, 20,10015-10019.

(24) Roura, P.; Fort, J. Langmuir 2002, 18, 566-569.(25) Young, T. Philos. Trans. R. Soc. London 1804, 1, 65.(26) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.;

Academic Press: New York, 1992.(27) Reference deleted in proof.(28) Wolansky, G.; Marmur, A. Langmuir 1998, 14, 5292-5297.(29) de Gennes, P. Rev. Mod. Phys. 1985, 57, 827-863.

Figure 2. 3D plot of the free energy E(θ, z) of a 4-nL dropleton an idealized surface partly covered with hemispheres withradii of 2 µm. The step at z ) 2 µm corresponds to transitionto completewetting,whereby theareaof thehigh-energy liquid-vapor interface decreases. A spherical droplet placed on top ofthe surface is assumed to be in the initial state, (180°, 0 µm).Since this is not an equilibrium state, the system will changetoward a lower free energy and settle in the free energyminimum (150°, 1 µm) (see also Figure 3). If the droplet isdisturbed, for example, by being mechanically pressed down,it might be forced past the free energy threshold and settle inthe global minimum at (135°, 2 µm), corresponding to homo-geneous wetting for the system in question.

E ) Elv + Esl + Esv - Egravity (3)

Egravity ) VFgz(1 -∫0

zf(x) dx

z ) (4)

Elv ) γlvSlv ) 2πR2γlv(1 - cos θ) +

(1 - f (z))γlvπR2 sin2 θ (5)

Esl ) γslSsl ) f(z)r(z)πR2γsl sin2 θ (6)

Esv ) γsvSsv ) const. - f(z)r(z)πR2γsvsin2 θ (7)

E(z,θ) ) πR2(2γlv(1 - cos θ) + sin 2θ((1 - f (z))γlv +

f )(z)r(z)(γsv - γsl))) - VFgz(1 -∫0

zf(x) dx

z ) + const.

(8)

Wetting of Structured Hydrophobic Surfaces Langmuir, Vol. 21, No. 26, 2005 12237

E(θ, z) values, according to eq 8, for all relevant values of(θ, z). This matrix, the state-space, maps the possible statesof the system. The global minimum represents a stablestate, whereas the local minima represent metastablestates.

The model is controlled by a MATLAB graphical userinterface (GUI). The input parameters are the interfacialenergy values, droplet volume, and density of the liquid.Here, the geometry of the modeled solid surface is alsocontrolled. New geometry types can easily be added to theprogram, since the geometry is only represented by f(z)and r(z). The geometries currently implemented in themodel, apart from a flat solid surface, are hemispheres,infinite posts, finite posts, cones, and holes. For eachgeometry, except the flat surface, parameters such asheight, base area, and surface coverage can be varied. Anumber of display options are available for investigatingdifferent aspects of a system’s behavior.

The program-generated matrix containing the energyvalues of all (θ, z) values is displayed as a free-energysurface (Figure 2). Bearing in mind that the system willgo from higher to lower free energy states, this plot givesa general idea as to how the system will behave. A droplet-on-surface system will move from its initial state towardstates of lower free energy until it comes to rest in an freeenergy minimum, which might be either local or global.For a system at rest to start moving toward higher freeenergy states, external disturbances such as vibrationsare needed.

To more accurately describe where these minima arelocated, the lowest free energy, and the corresponding θ,are found for every value of z (Figure 3). These results areobtained by minimizing E(θ, z) for every z with respect toθ (eq 8). The plot of this path in the state-space is ofparticular interest, as it shows how the free energy of thesystem will vary as z increases. Furthermore, it isinstructive to plot the surface-specific functions, f(z) andr(z) (Figure 4). For all simulations presented in this study,it holds that only one local or border minimum exists forE(θ)z, which is also the global minimum of E(θ)z.

If one wishes to investigate a system in which theapparent contact angle is fixed for reasons of hysteresis,

a graph showing E(z) for a specific θ can be generated. Bymeans of this function, a test can be carried out, hereafterdenoted the fixed-θ test, determining at what contactangles a droplet of a particular volume can no longer stayin heterogeneous wetting mode on a particular surface.This angle will be called θfallthrough in this work.

Comparison with Experimental Results

A new model should, of course, be tested againstexperimental results. Since there are numerous relevantand well-documented experiments in the literature,simulations using the available data were conducted, andthe results were compared with the experimental values.The geometry was obtained from the tables and, in somecases, from scanning electron microscopy images includedin the original works.3,5 For the liquid-vapor interfacialenergy, i.e., the surface tension of water, the value at 291K, 73 mJ/m2, was used. The droplet volumes were takendirectly from the reports or calculated from the dropletradii. The work presented by Oner et al.5 used aninstrument with variable droplet size. However, a dropletvolume of 4 m was used in these simulations. Theinterfacial energy difference, γsv - γsl, was obtained usingthe Young equation with the tabulated value of the surfacetension of water, together with cos θ for the flat referencesurface of the used material. If both the advancing contactangle, θa, and the receding contact angle, θr, weremeasured in the original experiment, θa was used in thesimulation. For all tables appearing in this work, mea-sured values are taken from the documentation of theoriginal experiments. The simulated values are all pre-pared using the model presented in the present work.

System with an Observed Metastable State. Theexperimental values according to Bico et al.3 are shownin Table 1, together with the calculated and simulatedvalues. For the simulated hole-patterned surface, cylin-drical cavities with radii of 1 µm and depths of 0.5 µmwere used. The pillars were simulated as standingcylinders with radii of 0.7 µm and heights of 2.2 µm. Thedroplet volume used was 4.2 nL. The locations of theminima are, not surprisingly, very close to those calculatedusing the Wenzel and Cassie-Baxter equations. In theoriginal experiment, it was observed that when a dropletplaced on a pillar surface was depressed, the contact angledecreased to 130°; this is close to the Wenzel value,

Figure 3. Graph showing the E(θeq, z) function for thehemisphere-covered surface described in Figure 2. For everyvalue of z, the minimal free energy and corresponding value ofθ is shown. For this surface, there is a local minimum at (142°,1 µm) and a global minimum at (130°, 2 µm) for both the freeenergy and contact angle. The straight horizontal line to thefar right is included simply to make the value at z ) zmax easierto read.

Figure 4. Surface characteristic functions f(z) and r(z) for thesame hemisphere-covered surface depicted in Figures 2 and 3.


indicating that when the droplet is pressed down it isforced over the threshold, as seen in Figure 5. For the holesurface, the energy threshold is much more significant(see Figure 6).

Scaling of Surface Features. In the work presentedby Oner et al.,5 three different kinds of chemicals wereused to modify the surfaces to enhance their hydropho-bicity. The simulations made here are all based on thesurfaces modified with dimethyldichlorosilane (DMDCS).In one experiment, θa and θr were measured for sampleswith 40-µm-high posts of different side lengths andspacings, such that 25% of the sample area was coveredwith posts in all samples. A droplet volume of 4 µL wasused in the simulation. Table 2 displays the measuredand simulated values. A large hysteresis is observed, andthe simulated angles are closer to the receding than to theadvancing contact angles.

Since the advancing contact angles are much greaterthan the equilibrium ones, and the apparent contact angleis most likely determined by hysteresis effects due to localeffects,4 the simulations of this experiment have focusedon the balance between gravitational and capillary forcesalong the z direction. Hysteresis in the x-y plane can beseen as a restriction on the system, so that it can no longer

move freely along the θ axis in the state-space (Figure 2).Thus, the only degree of freedom left will be z. The fixed-θtest was used to examine whether the droplet would reston the pillars for a particular value of θ (i.e., the dropletbase area described solely by the droplet volume and θ)other than the equilibrium θ. The lowest θ for which nofree energy minimum existed at z ) 0 is designated asθfallthrough (Table 2). As this is an investigation of E(z)θ ratherthan of E(θ, z), the present model can be used, even thoughthe difference between θadvancing and θequilibrium is significant.

Another experiment in the same work5 studied surfacesincorporating square-shaped posts of a certain side lengthbut spaced at various distances, prepared in the sameway as in the experiment referred to above. Measuredand simulated values are shown in Table 3. In general,the θa values were stable at ∼173° as the spacingincreased. However, as the spacing was increased from32 µm to 56 µm, both θa and θr decreased dramatically,most likely due to a transition from the Cassie-Baxterto the Wenzel mode. At the same spacing, the simulatedvalue of θfallthrough decreased to below 175°. A possible courseof events might be for θ to increase until it reaches eithera particular θa, which is dependent on hysteresis, orθfallthrough.

Table 1. Measured and Simulated Values for theSurfaces with Different Topographies Presented by Bico

et al.3a

measured calculated simulated

pattern φs θa θr θCassie θWenzel θlocal θglobal

flat 1 118 100 118 118holes 0.64 138 75 131 130 131 130pillars 0.05 170 155 167 128 167 128

a Φs represents the total combined area of all pillar tops as afraction of the total projected sample area (equivalent to f(0) in thepresent model). θlocal corresponds to a local energy minimum wherethe droplet is resting on the structure. θglobal corresponds to theglobal energy minimum where the droplet has fully wetted thesurface area under the droplet base. There exists a local energyminimum corresponding to the cassie mode for both the hole andthe pillar surface. This is also shown in Figures 5 and 6.

Figure 5. Minimum free energy for every value of z as obtainedfrom simulations of the experiment described by Bico et al.3 inwhich a 4.2-nL droplet is resting on a pillar surface. There isa local free energy minimum at (167°, 0 µm) and a global oneat (128°, 2.2 µm). The height of the free energy barrier is 2 pJwith the peak situated at 0.4 µm. The scale of the y axis in theenergy graph has been chosen so as to show the features of theenergy barrier, leaving the Wenzel minimum of 8.9 nJ outsidethe figure.

Figure 6. Minimum free energy for every value of z in thesimulation of the hole surface.3 There is a local minimum at(131°, 0 µm) and a global minimum at (130°, 0.5 µm). The heightof the free energy barrier is 175 pJ, and the peak is situatedat 5 µm. Even though the contact angles, corresponding to theminima, are so similar, the high and wide threshold, andcomparison with the situation depicted in Figure 5, indicatesthat the droplet is in the Cassie-Baxter mode.

Table 2. Measured and Simulated Data for SurfacesConstructed by O2 ner et al.5 with Posts of Different Side

Lengthsa

measured simulated

post side length θa θr θz)0µm θz)40µm θfallthrough

flat 107° 102° 107°b

2 µm 176° 141° 145°b 180° 179°8 µm 173° 134° 145°b 180° 17716 µm 171° 144° 145°b 180° 176°32 µm 168° 142° 145°b 180° 174°64 µm 139° 81° 145° 118°b 171°128 µm 116° 80° 145° 112°b 167°

a The posts, of whatever side length, together cover 25% of thesurface. θfallthrough is the lowest value of θ for which the gravitationalforce exceeds the capillary force. b These contact angles correspondto global energy minima.


DiscussionPost and needle structures are classic examples of water-

repellent surfaces. The following discussion concernssimulations of the equilibrium wetting of such structures.Two simulated surfaces, each having an intrinsic contactangle of 118°, were covered with cylinders and cones,respectively. Both pillars (cylinders) and needles (cones)had heights of 10 µm, radii of 0.5 µm and covered 10% ofthe area; 4-nL droplets were used in this simulation.Figures 7 and 8 show that the energy functions of thesesurfaces are quite different. Although a droplet placed on

the pillar surface will stay on top of the pillars, a dropplaced on the needle surface will fall through the wholestructure way and completely wet the area. The contactangle of the pillar surface might also be obtained from theCassie-Baxter equation, but the situation is more complexfor the needle structure depicted in Figure 8. The modelestimates a contact angle of 180°. This corresponds to thedegenerated Wenzel equation when r cos θintr > 1;17 thatis, the free energy per wetted projected sample area isgreater than γlv, resulting in both the model and theWenzel equation predicting a completely spherical droplet.Though this is not the case in reality when gravity ispresent, such a surface may have a very high contact angle.

Figure 9 shows the situation when no gravity is present.Here nothing indicates that the droplet will stick to thesurface (note the scale on the ordinate). A fixed-θ testexecuted for 179°, shown in Figure 10, displayed an energyminimum at z ) 3 µm, indicating that the droplet will bepartly pinned on the 10-µm-long needles. These twoexamples illustrate how sensitive the system is in theexceptionally high contact angle region.

In this work, the hypothesis that the free energyperspective can be used to predict not only contact angles

Table 3. Measured and Simulated Data for Square Postsof Side Length 8 µm, All Covering Various Fractions (Os)

of the Surface5a

measured simulated

spacing φs θa θr θz)0µm θz)40µm θfallthrough

16 µm 0.25 173° 134° 145°b 180° 177°23 µm 0.12 175° 146° 156°b 171° 175°32 µm 0.06 173° 154° 163° 130°b 173°56 µm 0.02 121° 67° 170° 114°b 167°

a The simulated contact angles are the angles that for a givenz value correspond to the lowest value of E. The measured valuesare from the original work by Oner and McCarthy.5 b These contactangles correspond to global energy minima.

Figure 7. Minimum free energy for each z value in thesimulation of a pillar surface. Note the minimum at z ) 0.

Figure 8. Minimum free energy for each value of z in thesimulation of a needle surface. This function has no minimumexcept when the area is completely wetted.

Figure 9. If gravity is not present, and hence does not influenceconditions on the needle surface, min(E) will be at (θ ) 180, z) 0). The energy scale of the upper diagram has been adjustedso that variations in the 10-15 J region can be distinguished.

Figure 10. Investigation of E(z) when θ ) 179° for the needlesurface. At this still very high contact angle, ∂E/∂z ) 0 for z )3 µm.


but also transitions between the Wenzel and Cassie modesis argued. The model was built around the same math-ematics used to derive the Young, Wenzel, and Cassieequations but with the addition of gravity contributionsto balance the capillary forces, or, more descriptively, thechange in free energy as the wetted area changes whenthe three phase contact line moves along the z axis. Also,the lowering of the center of mass as the contact angleincreases is taken account of.

Influence of Gravity. Most models of droplet-on-surface systems account neither for gravity nor dropletsize. This is typically done because the model deals withdroplets sufficiently small that gravity does not affectthem. However, gravity does influence the system if oneconsiders the droplet-on-surface system from an energypoint of view. It does so mainly in two ways. First, gravityacts as a force balancing the capillary forces. Thisdetermines how close together pillars have to be placedon a surface in order to keep the droplet in the Cassie-Baxter regime. This contribution is dependent on thedroplet size, since gravitational forces are proportional tothe volume of the droplet, V, although the capillary forcesare proportional to V2/3. This contribution is included inthe present model.

A surface like that of the pillar surface presented byBico et al.3 was simulated to depict the importance of thiscontribution. To this end, the spacing between pillars(expressed as the total combined area of all pillar tops asa fraction of total projected sample area) needed to keepa droplet of a particular size in heterogeneous wettingmode was investigated. If gravity was not accounted forand the intrinsic contact angle was greater than 90°, aninfinitesimal fraction was enough. In this case, thestatistical treatment naturally becomes misleading sincea drop cannot rest on less than one pillar. If thegravitational contribution mentioned above is taken intoaccount, a particular capillary force per area unit, andthus a particular pillar density, is needed to balance thegravitational force. For a 4-µL droplet, 2% of the surfacehad to be covered by pillars to keep the system in Cassie-Baxter mode. For a 4-nL droplet, 0.4% coverage wassufficient. Documented experiments8 also indicate thatlarger droplets can end up in Wenzel mode when placedon a surface on which smaller droplets remain in Cassiemode.

Flattening of the Droplet. The next influence ofgravity is due to the fact that it flattens the droplets, suchthat the curvature is lower at the apex of the droplet thanat the contact line. Taking account of this contributionprevents predictions of infinitesimal droplet base areasin both the Wenzel and Cassie-Baxter modes. Anotheraspect of flattening is that it actually counteracts thepreviously mentioned contribution in Cassie-Baxtermode, as a greater base area leads to a larger balancingcapillary force. This contribution has not been includedin the present model which only was designed to copewith the main features. Because of the compromisesinvolved in satisfying the demands mentioned in theBackground section, one effect and not another was takeninto account in the model. If gravity had not beenconsidered at all, the importance of the relationshipbetween pillar top area and pillar circumference wouldhave been missed. However, properly taking account ofthe flattening would mean involving the Adams-Bash-forth technique, or similar, which would complicate themodel considerably.

Why Droplets Stay in Heterogeneous Mode: Ex-istence of Minima. When a structure is scaled in sucha way that the proportions of the features in the x-y plane

are preserved5 (as in Table 2, where parameter f in theCassie equation is kept constant), the contact angle isalso predicted to stay the same. However, the reason thedroplet remains in the Cassie-Baxter mode and does notdecline is that it would wet more surface area if it did.Since the pillars have vertical sides in this case, this areachange is directly proportional to the added circumferenceof the pillars per sample area unit, a quantity whichdecreases as the structure is scaled up. This is the reasonfor the decreasing θfallthrough in Table 2. In the same work,5posts with cross-sections shaped like stars and rhombiwere investigated. It was concluded that these shapeswould lead to a more contorted contact line and lesshysteresis. According to the present model, pillars of theseshapes would also have the benefit of a larger circumfer-ence and thus not need to be spaced as closely as wouldsquare pillars of the same top area in order to keep adroplet from falling down. Experiments involving varyingthe distance between nonsquare pillars have, to theknowledge of the authors, not been conducted.

When used to predict contact angles for droplets restingon pillars or completely wetting the base area, the presentmodel is built on almost the same theory as are the Wenzeland Cassie equations, and hence, it should give the sameresults. Geometries in which energy minima exist betweenz ) 0 and zmax will give results similar to those of eq 1.

Model-based predictions of transition from the Cassie-Baxter to the Wenzel mode agree well with the experi-mental results of Oner et al.5 For both referred-toexperiments, the super-water-repellent effect ceased atthe spacing at which θfallthrough was close to or smaller thanθa. A plausible course of events is that when the dropletis placed on a pillar surface its base perimeter willpropagate until θa is reached. However, if θa > ) θfallthrough,the capillary forces will not be able to mach the gravi-tational forces, and the droplet will enter the homogeneouswetting mode. In the homogeneous mode, however, thedroplet will come to rest in the Wenzel minimum, whichcorresponds to a much lower contact angle.

Stability of Minima and Height of Barriers. Aninvestigation of a droplet on a solid surface is not completewhen the minima is located, as it is also important tounderstand the barriers between the thermodynamicminima. If a droplet is in a metastable state and the barrieris low, even slight vibration might suffice to push thedroplet into the stable state. In this work “barrier height”by itself should be understood as the relative free energydifference between the local free energy minimum inquestion and the point with the highest free energy(usually a saddle point in the state-space) the system hasto pass to get to the stable minimum. The “barrier heightfrom X” should be understood as the barrier height whichmust be crossed in order to leave state X.

The above section discussed a case in which hysteresisin the θ direction restricts the projected area beneath thedroplet. If only surface energy is taken into account, therelationship between the free energy of the base areas inthe Wenzel versus the Cassie modes would be the same.What differs is the height of the free energy barrier. If thebarrier is low enough, it might not be able to match thegravitational forces, and the droplet would fall down tothe Wenzel mode. This effect is accounted for by addingthe gravitational term in eq 3.

The situations depicted in Figures 5 and 6 are goodexamples of two distinctly different barriers. In Figure 5,the barrier is rather low and situated close to the Cassie-Baxter minimum. It has also been shown that it could bebypassed by applying a slight pressure.3 Describing thesituation from left to right: At the far left the droplet is


resting on top of the pillars in a metastable Cassie-Wenzelminimum. When the droplet is pressed down, the freeenergy per projected sample area increases until it exceedsγlv, at which point the θ will reach 180°. This state isunstable, and the droplet will continue downward untilthe dramatic decrease in free energy as the liquid-vaporinterface below the droplet is destroyed at completewetting. In Figure 6, the peak of the barrier is situatedimmediately before the Wenzel minimum. The barrier ishigh and wide making the Cassie-Baxter minimum muchmore stable.

A way to explicitly describe the height of the barrierfrom both directions is depicted in Figure 11, in which thefree energy for complete wetting, for the Cassie-Baxterminimum (if existing) and for the threshold, is plottedagainst θintr for a particular surface geometry and dropletvolume. The maximum should be understood as a maxi-mum in E(z), as in Figure 3, which represents the thresholdbetween two (meta)stable states, if such exist. For thegiven geometry, the barrier from Cassie-Baxter can beobtained from the diagram by reading the vertical distancefrom the Cassie-Baxter curve to the maximum curve atthe intrinsic contact angle of interest.

As seen in Figure 11, the barrier from Wenzel to Cassie-Baxter, read as the vertical distance from the Wenzel curveto the maximum curve, is not surprisingly several ordersof magnitude higher, even for very large intrinsic contactangles.

However, this is not the case for all geometries. Theheight of the barrier from Cassie-Baxter is stronglydependent on the vertical or slanting area of the solidsurface, since this has to be wetted before the liquid-vaporinterface can be destroyed. A larger f leads to a lower θand thus a larger base area, which also leads to a higherbarrier. The barrier from Wenzel to Cassie-Baxter

depends largely on how much liquid-vapor interface thathas to be created when the drop is leaving completewetting.

As seen in Figure 12, for a surface with small, closelyspaced cavities, both the Wenzel and the Cassie-Baxterminima might be of the same depth; for high enough valuesof θintr, the Wenzel minimum will even cease to exist. Thisresults in not only transitions from the Cassie-Baxter tothe Wenzel mode being of interest but also from the Wenzelto the Cassie-Baxter mode. When investigating suchtransitions, it will likely be very important to study localeffects, such as the spontaneous formation of vapor bubblesand whether they will have the ability to spread, merge,and form a continuous liquid-vapor interface.

Conclusions

The essentially thermodynamic model resulting in theCassie-Baxter equation has been generalized into a modelwhich additionally covers the balance between the freeenergy change due to gravity (potential energy) and thefree energy change due to area changes in the liquid-vapor, solid-vapor, and solid-liquid interfaces. For agiven surface topography and a given intrinsic contactangle, free energy as a function of the contact angle andthe liquid’s penetration of the surface structure, E(θ, z),gave more information regarding the surface’s water-repellent properties than did formulas such as Wenzel’sor Cassie’s, because it also showed which modes werestable. In the special case of constant apparent contactangle in the heterogeneous mode when f (0) is changed,a restricted version of the model, in which θ in E(θ, z) isconsidered to be a parameter rather than a variable, canbe used to predict how the geometry can be altered andstill retain a droplet in the heterogeneous mode.

Also important is the concept of free energy barriers.It is not sufficient to know where the minima are locatedand which is the lowest; it is also of interest to identifythe energy barriers between the minima themselves and

Figure 11. Free energies for complete wetting, for the Cassie-Baxter minimum (if existing), and for the threshold are plottedagainst θintr for a particular surface geometry (here the pillarsurface presented by Bico et al.3 and droplet volume V ) 4.2nL). For each θintr the height of the thresholds can be read fromthe diagram, as can the number of minima and their positions.At point A in the figure, the maximum energy no longercorresponds to that of the droplet resting on top of the pillars.At this point, the metastable Cassie-Baxter minimum is found.This minimum is nonexistent below this point. The thresholdthe system must pass to get from the Cassie-Baxter minimumto the much deeper Wenzel minimum should be read as thevertical distance between the respective curves. In comparison,the potential energy of a 4.2-nL droplet elevated 1 dm is 4.1 nJ.

Figure 12. Free energies for complete wetting, for the Cassie-Baxter minimum (if existing), and for the threshold are plottedagainst θintr for a certain surface geometry and droplet volume(here a simulated surface covered with holes and V ) 4.2 nL).As can be seen, the Wenzel minimum is the only one existinguntil A. Between A and B, the Cassie-Baxter minimum is lessdeep than the Wenzel minimum is. At B, the two minima havethe same free energy, and for even higher contact angles, thethreshold from Wenzel to Cassie-Baxter shrinks until it ceasesto exist at C.


alsobetweentheminimaandthedroplet’s startingpositionin the state-space.

Simulations show that droplet on surface systems thatexperiments,3 showed to be highly instable indeed had toovercome lower energy barriers compared to those seenin a systems in which no transitions from Cassie-Baxtermode to Wenzel mode were experimentally observed.Simulations, together with experimental data,5 supportthat gravitational effects play a role in transitions fromthe Cassie to the Wenzel mode.

Water repellency is produced by an interplay betweenthe intrinsic contact angle of the material and its geometricstructure. A high equilibrium apparent contact angle isthe result of a high free energy cost per projected wettedsurface area compared to the surface tension of water.One way to achieve this is through the heterogeneouswetting with a low rff factor of a surface, for example apillar surface with an intrinsic contact angle greater than90°. The stability of such a state can be estimated by theheight and width of the free energy barrier in the state-space which has to be passed in order to reach completewetting. A large overall pillar circumference per projectedsample area and high intrinsic contact angles producesteep barriers, whereas a large unwetted vertical areaper projected sample area and high intrinsic contact anglesproduce high barriers. In special cases in which thesequantities are sufficiently small, gravitational effects are

significant. If the r factor is large and θintr > 90, the freeenergy cost per projected wetted surface area is high evenin the homogeneous wetting mode. In some cases, no freeenergy minimum existed in the homogeneous mode.

The present model could be used for the design andevaluation of water-repellent geometries at the microscale.It could also be used to give a general idea of how sucha surface should be constructed and how its features shouldbe dimensioned; for example, high intrinsic contact angle,large added circumference, steep features, and a largedroplet base area promote the existence of a minimum inthe Cassie-Baxter mode and a high energy barrier. Agreater barrier width is usually promoted by highgeometric features. Exactly how a barrier is passed is tobe investigated with experiments and models in whichkinetics are taken more into account, presumably withtheory of action.

Acknowledgment. The authors thank Professors JanChrister Eriksson and Abraham Marmur for valuablecomments on the manuscript, and M.S. Andrew Horwathand Dr. Shannon Notley for proofreading. Finally theForest Products Industrial Research College and Vinnovaare thanked for financing the work of O.W.

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