Wetting of Structured Hydrophobic Surfaces by Water Droplets

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Wetting of Structured Hydrophobic Surfaces by Water Droplets Oskar Werner,*,† Lars Wågberg,† and Tom Lindstro¨m‡ KTH, Fibre and Polymer Technology, Div. Fibre Technology, Drottning Kristinas Va¨ g 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, contact angle 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, making use of surface free energy minimization. As a rule, the free energy minima (and maxima) occur when the Laplace and Young conditions are simultaneously fulfilled. Special effort has been devoted to investigating the 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 favorably with experimental results presented in the literature.

Background In recent years, there has been growing interest in super-hydrophobic surfaces. The list of possible applications of such surfaces is long, including both fundamental research and commercial applications. Many different types of water-repellent and self-cleaning surfaces have been 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 rather complex matter, and a number of previous works deal with its theory.11-18,24,26,28,29 Even though it can in principle be defined for equilibrium cases at every point by the Laplace and Young conditions, the number of different states that need to be examined is often vast. To generate a model that is practical and useful, it is often better to consider the entire system as a whole. Such an approach usually requires the use of certain approximations. Moreover, any such model should be as simple as possible, * 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.; Que´re´, D. Europhys. Lett. 1999, 47, 220226. (4) Youngblood, J.; McCarthy, T. Macromolecules 1999, 32, 68006806. (5) O ¨ ner, 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, Chapter Wettability 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, 521529. (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.

yet physically correct, and should enable good enough predictions. The best known theoretical approach to heterogeneous wetting is presumably that of Cassie and Baxter.12 In this approach, the difference in free energy per projected unit area beneath and beside a resting sessile drop is used to account for the location of the droplet base perimeter, whereas the penetration of the pores in the base structure is governed by the fulfilment of the Young condition.25 The two major assumptions made are that (a) the structure of the solid surface can be treated in an average fashion and (b) the effects of the hydrostatic and droplet curvature pressures are negligible. The main purpose of this work was to develop a useful tool to support the development of super-hydrophobic cellulose surfaces. The aim was that the model should indicate the apparent contact angle and whether a certain drop-on-surface system will would be in heterogeneous (Cassie-Baxter) or homogeneous (Wenzel) wetting mode. These modes correspond to free energy minima. To investigate not just their existence but also their stability, the free energy barriers between the heterogeneous and homogeneous wetting modes have to be studied. To this end, for the special case of square pillars, the energy barriers were studied by Patankar.19 Most of the droplet-on-surface systems of interest reside in the earth’s gravitational field. For small droplets, however, the effect of gravity is usually minor, compared with that of the interfacial free energy changes, and can thus be disregarded, though there are some exceptions. By ignoring gravity, one may predict contact angles equal to 180°,17 both in the heterogeneous and homogeneous modes. The former case implies droplets resting on an infinitesimally small solid surface area; if that view is chosen, the system is regarded as kept in the heterogeneous mode by infinitesimally short three-phase contact lines. It may also be the case that gravity-less models fail to predict interesting wetting behaviors, especially for intrinsic contact angles, θintr, approaching 90°. One of the original objectives of this work was to present a model that takes gravity into account in an appropriate fashion. In the end, however, it turned out that such gravity effects are often negligible. Instead, the focus has been shifted (19) Patankar, N. A. Langmuir 2004, 20, 7097-7102.

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

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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 and ci, and hence r2D and f2D, are dependent on z. The liquid-vapor interface beneath the droplet can be approximated as being flat if the structure is small in scale compared to the droplet radius.

toward estimating the free energy minima corresponding to homogeneous wetting and quantifying the barriers between them. General Features of the Present Model A global approach is taken by considering the overall excess free energy of the whole droplet resting on the solid surface. Surface free energies are invoked as well as the potential energy of the water in the droplet in the gravitational field. In this way, every point in the statespace, in which each dimension represents one of the system variables, corresponds to a certain free energy value. This energy surface cannot only be used to determine the thermodynamically stable and metastable states but also for determining the depth of the corresponding minima and the height of the barriers between them. The model uses two variables: the apparent contact angle, θ, and the penetration of the liquid into the base structure, z. Hence, the state-space is one of two dimensions. The geometrical surface parameters, f (i.e., the wetted fraction of the projected area) and r (i.e., real wetted area per projected wetted area), are considered to be functions of z (Figure 1). Thus f (z) and r(z), together with the interfacial free energies of the solid-liquid and solidvapor interfaces, γsl and γsv, are the sole characteristics of the solid surface. For reasons of convention, the notation rf and r1-f will be used as roughness factors for the wetted and unwetted parts of the solid surface for the reason of clarification in some cases. The thermodynamic scheme of the Young as well as the Cassie-Baxter equations describing the droplet-onsurface problem can be backtracked to a minimization of the free energy (hereafter denoted E) with respect to θ. In the Cassie-Baxter approach,12 the average free energy per unit area beneath the droplet is considered when calculating the apparent contact angle, θ, the result being the Cassie-Baxter equation, viz.

cos θ )

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

where the intrinsic contact angle, θintr, is to be distinguished from the apparent contact angle, θ. Along the z axis, the fulfilled Young condition (the zero hydrostatic pressure approximation, resulting in a planar liquidvapor interface beneath the droplet) and knowledge of the geometry of the surface (parallel cylinders in the original work)12 are used to calculate the penetration of the liquid into the structure. In the case of rf ) 1, eq 1 becomes cos θ ) f cos θintr + f - 1, and in the case of

complete wetting, it reduces to the Wenzel equation11

cos θ ) r cos θintr

(2)

Concerning the hydrostatic and capillary pressures, one might invoke a pressure-induced curvature of the liquidvapor interface beneath the droplet. For a global model, this is, however, not very convenient, since the curvature will be a quite complex function of the spatial coordinates, for even a rather simple surface topography. In the present model, the flat, liquid-vapor interface assumption is retained but merely in the sense that the extension of the liquid-vapor interface is estimated as if it were actually planar. [The planar approximation is good for pore sizes up to approximately one tenth of the droplet radius; this can be checked using the chord theorem.] Instead, the changes of the droplet gravitational energy are explicitly considered. Hence, when the interface propagates in the z direction, the gravitational energy lowers. To determine the equilibrium penetration, z, this lowering is balanced against the net interfacial free energy change due to the change in interfacial area. The situation encountered here is in principle the same as the one we face when treating capillary rise. The effect of gravity can be accounted for either by considering the Laplace pressure drop across the liquid-vapor interface or in terms of the potential of the rising liquid in the gravitational field.20 The applicability of a strictly thermodynamic treatment of heterogeneous wetting has been questioned.4,5,21 Hysteresis effects may cause the droplet system to be unable to reach its true free energy minimum, meaning that E(θ) has local minima corresponding to metastable states. This does, however, mean that smoothing E(θ), i.e., making the local energy minima in E(θ) less deep,4 is more important than having the global minimum at a high θ value in those cases. A droplet resting on a solid surface is treated as a system located somewhere in a 2-dimensional state-space specified by the two variables θ and z. For each system, every state (θ, z) corresponds to a certain E value. It is then possible to determine the path along which the system has to move in order to reach a state of lower E. A plot of such a state-space is shown in Figure 2. The separate contributions to E are shown in eqs 4-7 below. The first term, describing the gravity contribution (4), displays how the free energy will decrease when the droplet’s center of mass 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, 150154. (21) Chen, W.; Fadeev, A.; Hsieh, M.; O ¨ ner, D.; Yongblood, J.; McCarthy, T. Langmuir 1999, 15, 3395-3399.

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

(

f )(z)r(z)(γsv - γsl))) - VFgz 1 -

Figure 2. 3D plot of the free energy E(θ, z) of a 4-nL droplet on an idealized surface partly covered with hemispheres with radii of 2 µm. The step at z ) 2 µm corresponds to transition to complete wetting, whereby the area of the high-energy liquidvapor interface decreases. A spherical droplet placed on top of the surface is assumed to be in the initial state, (180°, 0 µm). Since this is not an equilibrium state, the system will change toward a lower free energy and settle in the free energy minimum (150°, 1 µm) (see also Figure 3). If the droplet is disturbed, for example, by being mechanically pressed down, it might be forced past the free energy threshold and settle in the global minimum at (135°, 2 µm), corresponding to homogeneous wetting for the system in question.

)

∫0zf(x) dx z

+ const. (8)

where the functions r(z) and f (z) are specific for each surface topography. A schematic explanation is found in Figure 1. If the influence of gravity is neglected, i.e., Egravity ) 0, z is set to a constant value and θi ) (γsv - γsl)/γlv, then expression 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 of liquid penetrating the surface structure is small compared with the droplet volume. [Taking account of the decrease in volume above the droplet base and the ensuing decrease in R and Elv corresponds to considering the Laplace pressure of the spherical droplet surface in the YoungLaplace approach.] The present model accounts for gravity only to a limited extent. The pressure differences inside the droplet and the nonspherical droplet shape they cause are not considered in this model. It should be mentioned that during the early development of the present theoretical model describing wetting of super-hydrophobic surfaces,22 a similar but still significantly different model was simultaneously developed by Jopp et al.23 Hysteresis

decline 1 when z increases 2 µm. Equations 5-7 imply that the interfacial free energy contributions are directly proportional to the area of each kind of interface. The constant in eq 7 represents the total interfacial energy of the solid-vapor interface of the nonwetted sample. It has been set to zero in this model; however, this is not a restriction, since only the area change is of importance. Summing up we have: variables: θ, z constants: γlv,γsv,γsl, V, g, F known functions: R(θ), r(z), f (z) where θ is the equilibrium apparent contact angle (degrees), z is the liquid’s penetration depth into the surface structure (m), γij is the interfacial energy (J m-2), V is the volume (m3), F is the mass density (kg m-3), g is the gravitational acceleration (9.82 ms-2), R is the drop radius (m), f(z) is the wetted fraction of the projected area (dimensionless), r(z) is the real wetted area per projected wetted area (dimensionless), and E is the free energy (J)

E ) Elv + Esl + Esv - Egravity

(

Egravity ) VFgz 1 -

)

∫0z f(x) dx z

(3)

(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)

that is

Although propagation of a droplet in the z direction through a rough structure typically is in the micrometer range, propagation of the droplet base perimeter in the x-y direction may very well occur in the millimeter range. For this reason, the lateral and normal directions are treated differently in the model. For example, energy barriers originating in the roughness are accounted for along the z axes, where they are most critical, but not in the x-y plane, i.e., the θ-dependent part of the model. However, if there is a large difference between the advancing and the equilibrium apparent contact angles, it may be of interest to investigate the z direction independently. If the model is restricted in this way, it can be determined at what contact angles a droplet of a particular volume can no longer stay in heterogeneous wetting mode on a particular surface. This angle is denoted by θfallthrough in this work. If this angle is compared with the equilibrium contact angle, the degree of stability of a super-hydrophobic surface in Cassie mode can be estimated. Development of a Computer Tool A computer tool was coded in MATLAB in order to test the applicability of the main expression presented in eq 8. For a structure of given parameters and surface functions, the program calculates a matrix consisting of (22) Werner, O. Computer Modelling of the Influence of Suface Topography on Water Repellency and a Study on Hydrophobic Paper Surfaces with Partly Controlled Roughness. Master’s thesis, Linko¨ping University, 2003 http://www.ep.liu.se/exjobb/ifm/tff/2003/1168/exjobb.pdf. (23) Jopp, J.; Gru¨ll, 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.

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Figure 3. Graph showing the E(θeq, z) function for the hemisphere-covered surface described in Figure 2. For every value of z, the minimal free energy and corresponding value of θ 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 free energy and contact angle. The straight horizontal line to the far right is included simply to make the value at z ) zmax easier to read.

E(θ, z) values, according to eq 8, for all relevant values of (θ, z). This matrix, the state-space, maps the possible states of the system. The global minimum represents a stable state, whereas the local minima represent metastable states. The model is controlled by a MATLAB graphical user interface (GUI). The input parameters are the interfacial energy values, droplet volume, and density of the liquid. Here, the geometry of the modeled solid surface is also controlled. New geometry types can easily be added to the program, since the geometry is only represented by f(z) and r(z). The geometries currently implemented in the model, apart from a flat solid surface, are hemispheres, infinite posts, finite posts, cones, and holes. For each geometry, except the flat surface, parameters such as height, base area, and surface coverage can be varied. A number of display options are available for investigating different aspects of a system’s behavior. The program-generated matrix containing the energy values of all (θ, z) values is displayed as a free-energy surface (Figure 2). Bearing in mind that the system will go from higher to lower free energy states, this plot gives a general idea as to how the system will behave. A dropleton-surface system will move from its initial state toward states of lower free energy until it comes to rest in an free energy minimum, which might be either local or global. For a system at rest to start moving toward higher free energy states, external disturbances such as vibrations are needed. To more accurately describe where these minima are located, the lowest free energy, and the corresponding θ, are found for every value of z (Figure 3). These results are obtained by minimizing E(θ, z) for every z with respect to θ (eq 8). The plot of this path in the state-space is of particular interest, as it shows how the free energy of the system will vary as z increases. Furthermore, it is instructive to plot the surface-specific functions, f(z) and r(z) (Figure 4). For all simulations presented in this study, it holds that only one local or border minimum exists for E(θ)z, which is also the global minimum of E(θ)z. If one wishes to investigate a system in which the apparent contact angle is fixed for reasons of hysteresis,

Werner et al.

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

a graph showing E(z) for a specific θ can be generated. By means of this function, a test can be carried out, hereafter denoted the fixed-θ test, determining at what contact angles a droplet of a particular volume can no longer stay in 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 against experimental results. Since there are numerous relevant and well-documented experiments in the literature, simulations using the available data were conducted, and the results were compared with the experimental values. The geometry was obtained from the tables and, in some cases, from scanning electron microscopy images included in the original works.3,5 For the liquid-vapor interfacial energy, i.e., the surface tension of water, the value at 291 K, 73 mJ/m2, was used. The droplet volumes were taken directly from the reports or calculated from the droplet radii. The work presented by O ¨ ner et al.5 used an instrument with variable droplet size. However, a droplet volume of 4 m was used in these simulations. The interfacial energy difference, γsv - γsl, was obtained using the Young equation with the tabulated value of the surface tension of water, together with cos θ for the flat reference surface of the used material. If both the advancing contact angle, θa, and the receding contact angle, θr, were measured in the original experiment, θa was used in the simulation. For all tables appearing in this work, measured values are taken from the documentation of the original experiments. The simulated values are all prepared using the model presented in the present work. System with an Observed Metastable State. The experimental values according to Bico et al.3 are shown in Table 1, together with the calculated and simulated values. For the simulated hole-patterned surface, cylindrical cavities with radii of 1 µm and depths of 0.5 µm were used. The pillars were simulated as standing cylinders with radii of 0.7 µm and heights of 2.2 µm. The droplet volume used was 4.2 nL. The locations of the minima are, not surprisingly, very close to those calculated using the Wenzel and Cassie-Baxter equations. In the original experiment, it was observed that when a droplet placed on a pillar surface was depressed, the contact angle decreased to 130°; this is close to the Wenzel value,

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Table 1. Measured and Simulated Values for the Surfaces with Different Topographies Presented by Bico et al.3a measured

calculated

simulated

pattern

φs

θa

θr

θCassie

θWenzel

θlocal

θglobal

flat holes pillars

1 0.64 0.05

118 138 170

100 75 155

131 167

118 130 128

131 167

118 130 128

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

Figure 6. Minimum free energy for every value of z in the simulation of the hole surface.3 There is a local minimum at (131°, 0 µm) and a global minimum at (130°, 0.5 µm). The height of the free energy barrier is 175 pJ, and the peak is situated at 5 µm. Even though the contact angles, corresponding to the minima, are so similar, the high and wide threshold, and comparison with the situation depicted in Figure 5, indicates that the droplet is in the Cassie-Baxter mode. Table 2. Measured and Simulated Data for Surfaces Constructed by O 2 ner et al.5 with Posts of Different Side Lengthsa measured post side length

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

indicating that when the droplet is pressed down it is forced over the threshold, as seen in Figure 5. For the hole surface, the energy threshold is much more significant (see Figure 6). Scaling of Surface Features. In the work presented by O ¨ ner et al.,5 three different kinds of chemicals were used to modify the surfaces to enhance their hydrophobicity. The simulations made here are all based on the surfaces modified with dimethyldichlorosilane (DMDCS). In one experiment, θa and θr were measured for samples with 40-µm-high posts of different side lengths and spacings, such that 25% of the sample area was covered with posts in all samples. A droplet volume of 4 µL was used in the simulation. Table 2 displays the measured and simulated values. A large hysteresis is observed, and the simulated angles are closer to the receding than to the advancing contact angles. Since the advancing contact angles are much greater than the equilibrium ones, and the apparent contact angle is most likely determined by hysteresis effects due to local effects,4 the simulations of this experiment have focused on the balance between gravitational and capillary forces along the z direction. Hysteresis in the x-y plane can be seen as a restriction on the system, so that it can no longer

flat 2 µm 8 µm 16 µm 32 µm 64 µm 128 µm

θa 107° 176° 173° 171° 168° 139° 116°

simulated

θr

θz)0µm

102° 141° 134° 144° 142° 81° 80°

107°b 145°b 145°b 145°b 145°b 145° 145°

θz)40µm

θfallthrough

180° 180° 180° 180° 118°b 112°b

179° 177 176° 174° 171° 167°

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

move freely along the θ axis in the state-space (Figure 2). Thus, the only degree of freedom left will be z. The fixed-θ test was used to examine whether the droplet would rest on the pillars for a particular value of θ (i.e., the droplet base area described solely by the droplet volume and θ) other than the equilibrium θ. The lowest θ for which no free energy minimum existed at z ) 0 is designated as θfallthrough (Table 2). As this is an investigation of E(z)θ rather than of E(θ, z), the present model can be used, even though the difference between θadvancing and θequilibrium is significant. Another experiment in the same work5 studied surfaces incorporating square-shaped posts of a certain side length but spaced at various distances, prepared in the same way as in the experiment referred to above. Measured and simulated values are shown in Table 3. In general, the θa values were stable at ∼173° as the spacing increased. However, as the spacing was increased from 32 µm to 56 µm, both θa and θr decreased dramatically, most likely due to a transition from the Cassie-Baxter to the Wenzel mode. At the same spacing, the simulated value of θfallthrough decreased to below 175°. A possible course of events might be for θ to increase until it reaches either a particular θa, which is dependent on hysteresis, or θfallthrough.

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Table 3. Measured and Simulated Data for Square Posts of Side Length 8 µm, All Covering Various Fractions (Os) of the Surface5a measured spacing 16 µm 23 µm 32 µm 56 µm

φs 0.25 0.12 0.06 0.02

θa 173° 175° 173° 121°

simulated

θr

θz)0µm

θz)40µm

θfallthrough

134° 146° 154° 67°

145°b

180° 171° 130°b 114°b

177° 175° 173° 167°

156°b 163° 170°

a The simulated contact angles are the angles that for a given z value correspond to the lowest value of E. The measured values are from the original work by O ¨ ner and McCarthy.5 b These contact angles correspond to global energy minima.

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

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

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

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

Discussion Post and needle structures are classic examples of waterrepellent surfaces. The following discussion concerns simulations of the equilibrium wetting of such structures. Two simulated surfaces, each having an intrinsic contact angle 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% of the area; 4-nL droplets were used in this simulation. Figures 7 and 8 show that the energy functions of these surfaces are quite different. Although a droplet placed on

the pillar surface will stay on top of the pillars, a drop placed on the needle surface will fall through the whole structure way and completely wet the area. The contact angle of the pillar surface might also be obtained from the Cassie-Baxter equation, but the situation is more complex for the needle structure depicted in Figure 8. The model estimates a contact angle of 180°. This corresponds to the degenerated Wenzel equation when r cos θintr > 1;17 that is, the free energy per wetted projected sample area is greater than γlv, resulting in both the model and the Wenzel equation predicting a completely spherical droplet. Though this is not the case in reality when gravity is present, 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 the surface (note the scale on the ordinate). A fixed-θ test executed for 179°, shown in Figure 10, displayed an energy minimum at z ) 3 µm, indicating that the droplet will be partly pinned on the 10-µm-long needles. These two examples illustrate how sensitive the system is in the exceptionally high contact angle region. In this work, the hypothesis that the free energy perspective can be used to predict not only contact angles

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but also transitions between the Wenzel and Cassie modes is argued. The model was built around the same mathematics used to derive the Young, Wenzel, and Cassie equations but with the addition of gravity contributions to balance the capillary forces, or, more descriptively, the change in free energy as the wetted area changes when the three phase contact line moves along the z axis. Also, the lowering of the center of mass as the contact angle increases is taken account of. Influence of Gravity. Most models of droplet-onsurface systems account neither for gravity nor droplet size. This is typically done because the model deals with droplets sufficiently small that gravity does not affect them. However, gravity does influence the system if one considers the droplet-on-surface system from an energy point of view. It does so mainly in two ways. First, gravity acts as a force balancing the capillary forces. This determines how close together pillars have to be placed on a surface in order to keep the droplet in the CassieBaxter regime. This contribution is dependent on the droplet size, since gravitational forces are proportional to the volume of the droplet, V, although the capillary forces are proportional to V2/3. This contribution is included in the present model. A surface like that of the pillar surface presented by Bico et al.3 was simulated to depict the importance of this contribution. To this end, the spacing between pillars (expressed as the total combined area of all pillar tops as a fraction of total projected sample area) needed to keep a droplet of a particular size in heterogeneous wetting mode was investigated. If gravity was not accounted for and the intrinsic contact angle was greater than 90°, an infinitesimal fraction was enough. In this case, the statistical treatment naturally becomes misleading since a drop cannot rest on less than one pillar. If the gravitational contribution mentioned above is taken into account, a particular capillary force per area unit, and thus a particular pillar density, is needed to balance the gravitational force. For a 4-µL droplet, 2% of the surface had to be covered by pillars to keep the system in CassieBaxter mode. For a 4-nL droplet, 0.4% coverage was sufficient. Documented experiments8 also indicate that larger droplets can end up in Wenzel mode when placed on a surface on which smaller droplets remain in Cassie mode. Flattening of the Droplet. The next influence of gravity is due to the fact that it flattens the droplets, such that the curvature is lower at the apex of the droplet than at the contact line. Taking account of this contribution prevents predictions of infinitesimal droplet base areas in both the Wenzel and Cassie-Baxter modes. Another aspect of flattening is that it actually counteracts the previously mentioned contribution in Cassie-Baxter mode, as a greater base area leads to a larger balancing capillary force. This contribution has not been included in the present model which only was designed to cope with the main features. Because of the compromises involved in satisfying the demands mentioned in the Background section, one effect and not another was taken into account in the model. If gravity had not been considered at all, the importance of the relationship between pillar top area and pillar circumference would have been missed. However, properly taking account of the flattening would mean involving the Adams-Bashforth technique, or similar, which would complicate the model considerably. Why Droplets Stay in Heterogeneous Mode: Existence of Minima. When a structure is scaled in such a way that the proportions of the features in the x-y plane

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are preserved5 (as in Table 2, where parameter f in the Cassie equation is kept constant), the contact angle is also predicted to stay the same. However, the reason the droplet remains in the Cassie-Baxter mode and does not decline is that it would wet more surface area if it did. Since the pillars have vertical sides in this case, this area change is directly proportional to the added circumference of the pillars per sample area unit, a quantity which decreases as the structure is scaled up. This is the reason for the decreasing θfallthrough in Table 2. In the same work,5 posts with cross-sections shaped like stars and rhombi were investigated. It was concluded that these shapes would lead to a more contorted contact line and less hysteresis. According to the present model, pillars of these shapes would also have the benefit of a larger circumference and thus not need to be spaced as closely as would square pillars of the same top area in order to keep a droplet from falling down. Experiments involving varying the distance between nonsquare pillars have, to the knowledge of the authors, not been conducted. When used to predict contact angles for droplets resting on pillars or completely wetting the base area, the present model is built on almost the same theory as are the Wenzel and Cassie equations, and hence, it should give the same results. Geometries in which energy minima exist between z ) 0 and zmax will give results similar to those of eq 1. Model-based predictions of transition from the CassieBaxter to the Wenzel mode agree well with the experimental results of O ¨ ner et al.5 For both referred-to experiments, the super-water-repellent effect ceased at the spacing at which θfallthrough was close to or smaller than θa. A plausible course of events is that when the droplet is placed on a pillar surface its base perimeter will propagate until θa is reached. However, if θa > ) θfallthrough, the capillary forces will not be able to mach the gravitational forces, and the droplet will enter the homogeneous wetting mode. In the homogeneous mode, however, the droplet will come to rest in the Wenzel minimum, which corresponds to a much lower contact angle. Stability of Minima and Height of Barriers. An investigation of a droplet on a solid surface is not complete when the minima is located, as it is also important to understand the barriers between the thermodynamic minima. If a droplet is in a metastable state and the barrier is low, even slight vibration might suffice to push the droplet into the stable state. In this work “barrier height” by itself should be understood as the relative free energy difference between the local free energy minimum in question and the point with the highest free energy (usually a saddle point in the state-space) the system has to pass to get to the stable minimum. The “barrier height from X” should be understood as the barrier height which must be crossed in order to leave state X. The above section discussed a case in which hysteresis in the θ direction restricts the projected area beneath the droplet. If only surface energy is taken into account, the relationship between the free energy of the base areas in the Wenzel versus the Cassie modes would be the same. What differs is the height of the free energy barrier. If the barrier is low enough, it might not be able to match the gravitational forces, and the droplet would fall down to the Wenzel mode. This effect is accounted for by adding the gravitational term in eq 3. The situations depicted in Figures 5 and 6 are good examples of two distinctly different barriers. In Figure 5, the barrier is rather low and situated close to the CassieBaxter minimum. It has also been shown that it could be bypassed by applying a slight pressure.3 Describing the situation from left to right: At the far left the droplet is

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Figure 11. Free energies for complete wetting, for the CassieBaxter minimum (if existing), and for the threshold are plotted against θintr for a particular surface geometry (here the pillar surface presented by Bico et al.3 and droplet volume V ) 4.2 nL). For each θintr the height of the thresholds can be read from the diagram, as can the number of minima and their positions. At point A in the figure, the maximum energy no longer corresponds 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 threshold the system must pass to get from the Cassie-Baxter minimum to the much deeper Wenzel minimum should be read as the vertical distance between the respective curves. In comparison, the potential energy of a 4.2-nL droplet elevated 1 dm is 4.1 nJ.

resting on top of the pillars in a metastable Cassie-Wenzel minimum. When the droplet is pressed down, the free energy per projected sample area increases until it exceeds γlv, at which point the θ will reach 180°. This state is unstable, and the droplet will continue downward until the dramatic decrease in free energy as the liquid-vapor interface below the droplet is destroyed at complete wetting. In Figure 6, the peak of the barrier is situated immediately before the Wenzel minimum. The barrier is high and wide making the Cassie-Baxter minimum much more stable. A way to explicitly describe the height of the barrier from both directions is depicted in Figure 11, in which the free energy for complete wetting, for the Cassie-Baxter minimum (if existing) and for the threshold, is plotted against θintr for a particular surface geometry and droplet volume. The maximum should be understood as a maximum in E(z), as in Figure 3, which represents the threshold between two (meta)stable states, if such exist. For the given geometry, the barrier from Cassie-Baxter can be obtained from the diagram by reading the vertical distance from the Cassie-Baxter curve to the maximum curve at the intrinsic contact angle of interest. As seen in Figure 11, the barrier from Wenzel to CassieBaxter, read as the vertical distance from the Wenzel curve to the maximum curve, is not surprisingly several orders of magnitude higher, even for very large intrinsic contact angles. However, this is not the case for all geometries. The height of the barrier from Cassie-Baxter is strongly dependent on the vertical or slanting area of the solid surface, since this has to be wetted before the liquid-vapor interface can be destroyed. A larger f leads to a lower θ and thus a larger base area, which also leads to a higher barrier. The barrier from Wenzel to Cassie-Baxter

Werner et al.

Figure 12. Free energies for complete wetting, for the CassieBaxter minimum (if existing), and for the threshold are plotted against 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 existing until A. Between A and B, the Cassie-Baxter minimum is less deep than the Wenzel minimum is. At B, the two minima have the same free energy, and for even higher contact angles, the threshold from Wenzel to Cassie-Baxter shrinks until it ceases to exist at C.

depends largely on how much liquid-vapor interface that has to be created when the drop is leaving complete wetting. As seen in Figure 12, for a surface with small, closely spaced cavities, both the Wenzel and the Cassie-Baxter minima might be of the same depth; for high enough values of θintr, the Wenzel minimum will even cease to exist. This results in not only transitions from the Cassie-Baxter to the Wenzel mode being of interest but also from the Wenzel to the Cassie-Baxter mode. When investigating such transitions, it will likely be very important to study local effects, such as the spontaneous formation of vapor bubbles and whether they will have the ability to spread, merge, and form a continuous liquid-vapor interface. Conclusions The essentially thermodynamic model resulting in the Cassie-Baxter equation has been generalized into a model which additionally covers the balance between the free energy change due to gravity (potential energy) and the free energy change due to area changes in the liquidvapor, solid-vapor, and solid-liquid interfaces. For a given surface topography and a given intrinsic contact angle, free energy as a function of the contact angle and the liquid’s penetration of the surface structure, E(θ, z), gave more information regarding the surface’s waterrepellent properties than did formulas such as Wenzel’s or Cassie’s, because it also showed which modes were stable. In the special case of constant apparent contact angle in the heterogeneous mode when f (0) is changed, a restricted version of the model, in which θ in E(θ, z) is considered to be a parameter rather than a variable, can be used to predict how the geometry can be altered and still 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 located and which is the lowest; it is also of interest to identify the energy barriers between the minima themselves and

Wetting of Structured Hydrophobic Surfaces

also between the minima and the droplet’s starting position in the state-space. Simulations show that droplet on surface systems that experiments,3 showed to be highly instable indeed had to overcome lower energy barriers compared to those seen in a systems in which no transitions from Cassie-Baxter mode to Wenzel mode were experimentally observed. Simulations, together with experimental data,5 support that gravitational effects play a role in transitions from the Cassie to the Wenzel mode. Water repellency is produced by an interplay between the intrinsic contact angle of the material and its geometric structure. A high equilibrium apparent contact angle is the result of a high free energy cost per projected wetted surface area compared to the surface tension of water. One way to achieve this is through the heterogeneous wetting with a low rff factor of a surface, for example a pillar surface with an intrinsic contact angle greater than 90°. The stability of such a state can be estimated by the height and width of the free energy barrier in the statespace which has to be passed in order to reach complete wetting. A large overall pillar circumference per projected sample area and high intrinsic contact angles produce steep barriers, whereas a large unwetted vertical area per projected sample area and high intrinsic contact angles produce high barriers. In special cases in which these quantities are sufficiently small, gravitational effects are

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significant. If the r factor is large and θintr > 90, the free energy cost per projected wetted surface area is high even in the homogeneous wetting mode. In some cases, no free energy minimum existed in the homogeneous mode. The present model could be used for the design and evaluation of water-repellent geometries at the microscale. It could also be used to give a general idea of how such a surface should be constructed and how its features should be dimensioned; for example, high intrinsic contact angle, large added circumference, steep features, and a large droplet base area promote the existence of a minimum in the Cassie-Baxter mode and a high energy barrier. A greater barrier width is usually promoted by high geometric features. Exactly how a barrier is passed is to be investigated with experiments and models in which kinetics are taken more into account, presumably with theory of action. Acknowledgment. The authors thank Professors Jan Christer Eriksson and Abraham Marmur for valuable comments on the manuscript, and M.S. Andrew Horwath and Dr. Shannon Notley for proofreading. Finally the Forest Products Industrial Research College and Vinnova are thanked for financing the work of O.W. LA052415+