How to Make the Cassie Wetting State Stable? - American Chemical

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How to Make the Cassie Wetting State Stable? Gene Whyman and Edward Bormashenko* Applied Physics Department and Department of Chemistry and Biotechnology Engineering, The Research Institute, Ariel University Center of Samaria, Post Office Box 3, Ariel 40700, Israel ABSTRACT: Wetting of rough hydrophilic and hydrophobic surfaces is discussed. The stability of the Cassie state, with air trapped in relief details under the droplet, is necessary for the design of true superhydrophobic surfaces. The potential barrier separating the Cassie state and the Wenzel state, for which the substrate is completely wetted, is calculated for both hydrophobic and hydrophilic surfaces. When the surface is hydrophobic, the multiscaled roughness of pillars constituting the surface increases the potential barrier separating the Cassie and Wenzel states. When water fills the hydrophilic pore, the energy gain due to the wetting of the pore hydrophilic wall is overcompensated by the energy increase because of the growth of the high-energetic liquid
air interface. The potential barrier separating the Cassie and Wenzel states is calculated for various topographies of surfaces. Structural features of reliefs favoring enhanced hydrophobicity are elucidated.

1. INTRODUCTION Wetting of rough surfaces attracted significant attention of investigators in the last decades.1
25 Interest in the problem was stimulated by revealing the so-called “lotus effect” giving rise to surfaces demonstrating pronounced water repellency.2,3 The paradigm of the wetting of rough surfaces supplied by the Cassie and Wenzel models is already well-grounded and wellunderstood.1,14,15,18,26
30 It is generally agreed that the Cassie (“fakir”) wetting is due to a high amount of air trapped by a droplet located on a superhydrophobic surface.1,15,22
25 Various experimental techniques applied to manufacturing lotuslike surfaces allowed fabrication of surfaces with high contact angles and low contact angle hysteresis.4,5,19,22
25 However, one superhydrophobicity-related problem remains actual: that is, increasing the stability of the Cassie wetting state. Wetting transitions, which could be spontaneous or induced by external stimuli such as pressure, vibrations, or droplets’ impact, prevent a broad technological penetration of superhydrophobic surfaces. The critical pressure necessary for Cassie
Wenzel wetting transitions on micrometrically scaled lotuslike surfaces was established by different groups as approximately 300 Pa. Recalling that the dynamic pressure of rain droplets may be as high as 104
105 Pa, which is much larger than 300 Pa (see ref 31), we conclude that creating highly stable superhydrophobic reliefs is of practical importance in a view of outdoor applications such as covering elements of photovoltaic devices, windows, etc. 2. RESULTS AND DISCUSSION Wetting transitions govern the life and death of fakir droplets.31
49 When the droplet is in the Cassie (“fakir”) state it sits mainly on air pockets, as shown in Figure 1a. The Cassie wetting is characterized by low contact angle hysteresis and low sliding r 2011 American Chemical Society

angles.13 Stability of the Cassie wetting, which is desirable for various technological applications, was studied in refs 50 and 51. In the course of the Cassie
Wenzel transition, liquid fills grooves, forming the relief (see Figure 1b), and the contact-angle hysteresis increases dramatically. What is the physical mechanism of the Cassie
Wenzel wetting transition? One of the possible mechanisms of a wetting transition, called recently by Patankar a “sag” transition, is shown in Figure 2. Consider a droplet deposited on the pillar-based relief. It was supposed that when external pressure applied to the droplet increased, the meniscus will move toward the relief bottom, as shown in Figure 2. The meniscus will eventually touch the relief bottom, leading to the collapse of the Cassie wetting and to the Cassie
Wenzel transition.41
49 Thus, it is clear that to increase the stability of the “fakir” state, the pillars forming the relief should be sufficiently long.48 This suggestion has been validated experimentally in ref 48. Obviously, for sufficiently long pillars the “sag” transition becomes impossible.41 It was supposed that in this case the pressure increase inside a droplet causes depinning of the threephase (triple) line.31 Depinning of the triple line occurs when a certain value of critical pressure in a droplet is surmounted.31 This pressure grows with decreasing the width of pillars forming the relief.7,31,38 It should be stressed that surmounting the critical pressure is a necessary but not sufficient condition of the wetting transition. Indeed, when the critical pressure is surmounted, the three-phase line may spontaneously slide downward along the side pillars’ surface only if this surface will be wetted. For this reason, in what follows, we consider different transition mechanisms for hydrophobic and hydrophilic surfaces. Received: March 31, 2011 Revised: May 4, 2011 Published: June 06, 2011 8171

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Figure 3. Composite wetting state.

Figure 1. (a) Cassie and (b) Wenzel wetting states.

Figure 2. “Sag” wetting transitions.

Figure 4. (a) Geometric parameters of the model relief used for calculation of Cassie
Wenzel transition energetic barrier. (b) Pillars possessing rough side facets.

2.1. Hydrophobic Surfaces. If pillars are built of a hydrophobic material, the wetting of their side surface is thermodynamically unfavorable. Thus, for wetting transition to proceed, the energy barrier should be surmounted.11,32,36 It was supposed that this energy barrier corresponds to the surface energy variation between the Cassie state and the hypothetical composite state with the almost complete filling of surface asperities by water, keeping the liquid
air interface under the droplet and the contact angle constant, as shown in Figure 3. In terms of physicalchemistry notions, such a state with the maximal energy is called the transition state. For a simple topography depicted in Figure 4a, the energy barrier W for the droplet could be calculated as follows:

with the parameters p = h = 20 μm, R = 1 mm, θY = 114° [corresponding to poly(tetrafluoroethylene)], and γ = 72 mJ/m2 gives a value of W = 180 nJ. It could be seen from eq 1 that high and narrow pillars make the Cassie state more stable. Note that the barrier value in eq 1 is positive since cos θY 0 is supposed to be satisfied below. In the opposite case, only one state exists and no transitions can happen. Now the question is how the value of the energetic barrier could be significantly increased. It could be increased if the side surface of the pillars is rough, as shown in Figure 4b. If the additional-scale roughness of the side surface equals ζ, the energetic barrier to be surpassed by the droplet will be given by

W ¼ 2πR 2 hðγSL
γSA Þ=p ¼
2πR 2 hγ cos θY =p

W ¼
2ζπR 2 γh cos θY =p

ð1Þ

where h and p are the geometrical parameters of the relief, shown in Figures 1a, 4a; R is the radius of the droplet contact area; and γSL, γSA, and γ are the surface tensions at the solid
liquid, solid
air, and liquid
air interfaces, respectively.52 It is supposed that the characteristic length scales of the relief are much less than the capillary length; hence, the effects due to gravity are neglected. The Young formula, γ cos θY = γSA
γSL, for the droplet contact angle θY on flat surfaces was used in eq 1. Numerical estimation of the energetic barrier according to eq 1

ð2Þ

It should be stressed that eq 2 implies the Wenzel wetting of side surfaces of pillars when the roughness ζ (defined as the ratio of the wetted surface to the projection area of a substrate) satisfies to the inequality 1 < ζ <
1/cos θY. As a result, the theoretical limit in the above example is Wmax = 2ζπR2γh/p that increases the barrier estimation about 2.5 times for the abovementioned angle θY = 114°. The Cassie wetting of the side surfaces of the hydrophobic pillars also increases the energetic barrier to be surpassed. In this 8172

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Figure 5. Geometrical air trapping on a hydrophilic relief.

case, the existence of two scales of the substrate texture is implied. In addition to the “large” scale shown in Figure 4, air may be trapped in cavities of some “small” second scale characterized by the fraction 1
fS of the small-scale cavities in the whole substrate area. It is not difficult to understand that the generalization of eq 1 to this case is W ¼
2R 2 γhðfS cos θY þ fS
1Þ=p

ð3Þ

with the same theoretical limit as in the case of the Wenzel wetting. It is well-known that the double roughness is important for constituting superhydrophobic properties of the surfaces. Gao and McCarthy6 supposed that two lengths of topography inherent to natural superhydrophobic objects (lotus leaves, birds wings, etc.) diminish the contact angle hysteresis, thus providing easy sliding of drops. It has been also demonstrated that the double roughness makes Cassie wetting energetically favorable.53 It turns out from our analysis that the additional small-scale roughness at the side surfaces of hydrophobic pillars increases the potential barrier to be surpassed for the Cassie
Wenzel transition, thus making the “fakir” wetting state more stable for both Cassie and Wenzel wetting of side surfaces. 2.2. Hydrophilic Surfaces. The situation on hydrophilic surfaces is more complicated. It is noteworthy that the lotus leaf, which gave rise to the intensive research on superhydrophobicity, is inherently hydrophilic.54 The equilibrium contact angle θY of the carnauba wax constituting the surface of lotus leaves is about 74°.54 Artificial superhydrophobic surfaces based on hydrophilic (metallic) material also have been reported.22
25,55,56 Wetting of inherently hydrophilic surfaces is always energetically favorable, but actually the Cassie wetting of hydrophilic surfaces is possible due to the “geometrical air trapping effect” discussed in ref 57 and illustrated in Figure 5. Consider a hydrophilic surface (θY < 90°) comprising pores as depicted in Figure 5. It is seen that air trapping is possible only if θY > j0, where j0 is the angle between the tangent in the highest point of the pattern and the horizontal symmetry axis O1O. Indeed, when the liquid level is descending, the angle j is growing (see Figure 5), and if the condition θY > j0

is violated, the equilibrium θY = j will be impossible. Geometrical air trapping gives rise to an energetic barrier to be surmounted for the total filling of a pore. When water fills the hydrophilic pore described in Figure 5, the energy gain due to the wetting of the pore hydrophilic wall is overcompensated by the energy increase at the expense of the growth of the highenergetic liquid
air interface. To perform the quantitative analysis, consider a spherical model of the cavity drawn in Figure 6. The cavity surface energy E is expressed as E ¼ 2πr 2 γ cos θY ðcos j
cos j0 Þ þ γπr 2 sin2 j

ð4Þ

where the first and second terms are the energies of the liquid
solid and liquid
air interfaces, respectively, and r is the cavity radius. The energy maximum corresponds to j = θY. Note that a central angle j, which defines the liquid level, is simultaneously a current contact angle. So, the energetic barrier per cavity w from the side of the Cassie state (j = j0) is w ¼ πr 2 γðcos j0
cos θY Þ2

on the condition θY > j0 ð5Þ

The counterpart of w in eq 5 per one droplet can roughly be evaluated as W ∼ πr2γN, where N ∼ S/4r2 is the number of unit cells in the liquid
solid interface area S for a plane quadratic close-packed lattice with the lattice constant 2r. Thus, for the droplet with a contact radius R ∼ 1 mm the upper limit W ∼ πSγ/4 ∼ 102 nJ, which is of the same order of magnitude as the barrier inherent to microscopically scaled hydrophobic surfaces. The energy difference between the Cassie state and the Wenzel one (j = π) EC
EW ¼ γπr 2 ð1 þ cos j0 Þð1
cos j0 þ 2 cos θY Þ ð6Þ is always positive for hydrophilic material. Consequently, the Cassie state is energetically unfavorable in this case but may be stabilized by a high energy barrier separating the Cassie and Wenzel wetting states. Thus, the Cassie
Wenzel transition proceeds here in the following way. In the initial Cassie state, the cavity is completely filled by air. Spontaneous liquid penetration into the cavity does 8173

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Figure 6. Formation of a transition state in a spherical cavity.

not take place because of the energy increase (on the condition θY > j0). When some external factor (pressure, etc.) promotes liquid penetration into the cavity, the energy attains its maximum when the contact angle reaches the Young value j = θY. After that, liquid spontaneously fills the cavity with a large energy gain. Another hydrophilic relief stabilizing the Cassie state, which presents a system of overturned truncated cones, is shown in Figure 7. The energy of varying interfaces is expressed as (per unit lattice cell) E ¼ γðab
πx2 Þ


πðr 2max
x2 Þ γ cos θY sin ðR=2Þ

ð7Þ

where x is the current cone radius at the liquid level, rmax is the radius of the upper base of the cone, and R is the opening angle of the cone. The first and second terms correspond to the liquid
air and liquid
solid interfaces, respectively. The energy increases monotonically with decreasing x and reaches its maximum at the minimal cone radius x = rmin. From eq 7, the barrier of the Cassie
Wenzel transition is 2 2
rmin Þγ½1
cos θY =sin ðR=2Þ w ¼ πðrmax

ð8Þ

while the energy difference of the equilibrium Cassie and Wenzel states is determined independently as
2 EC
EW ¼ γ ðab
πrmax Þð1 þ cos θY Þ   1 2 2 þ πðrmax þ1
rmin Þcos θY sin ðR=2Þ ð9Þ The barrier exists, that is, w > 0, if the opening angle R of cones is sufficiently large: π R
< θY ð10Þ 2 2 This simply means that the actual local contact angle, which equals π/2
R/2, must be less than the Young angle (see Figure 7). This is possible due to the phenomenon of contact-angle hysteresis. The minimal value of eq 9 corresponds to the minimal product ab, which is equal to 4rmax2 from geometrical reasons (a = b = 2rmax, Figure 7). Since 4 > π, it follows from eq 9 that the Cassie state is metastable (EC > EW) but the maximal possible energy barrier from the Cassie state side is higher than in the case of spherical cavities: wmax ¼

2 πrmax γð1
cos

θY Þ

ð11Þ

Figure 7. Transition state for the hydrophilic relief of overturned truncated cones. a and b are the parameters of a rectangular twodimensional lattice. The radius of the upper cone base is rmax and that of the bottom base is rmin, and the opening angle of the cone is R.

energy maximum is reached just before liquid touches bottoms of the relief, not at an intermediate liquid level, as for the spherical cavities relief. It should be mentioned that the mechanism of wetting transitions discussed in this section remains valid for hydrophobic surfaces (section 2.1). It means that the energy increase due to the growth of the liquid
air interface in the course of liquid penetration into pores or grooves constituting the relief enhances the total energy barrier separating the Cassie and Wenzel states. 2.3. Surfaces Built of Ensembles of Balls. An interesting example is presented by a system of spheres resembling the lotus surface.23,25 The surface energy of the model system shown in Figure 8 (for designation see Figure 8) E ¼ γab
πr 2 γ½sin2 j þ 2ð1
cos jÞcos θY 

ð12Þ

has a local minimum at j = π
θY. This means that the Cassie state is characterized here by the liquid penetration below spheres’ equators (and, for moderate hydrophilicity, close to them).58 The maximal value of the energy in eq 12 is attained for j = π, that is, when liquid completely fills the cavities (but does not touch relief bottoms). Thus the barrier is w ¼ πr 2 γð1
cos θY Þ2

ð13Þ

The difference between the energies of the Cassie and Wenzel states is EC
EW ¼ γabð1 þ cos θY Þ
πr 2 γð1
cos θY Þ2

ð14Þ

The Cassie state is thermodynamically favorable when EC < EW or in other words

(compare eq 11 to eq 5). The Cassie
Wenzel transition for the present relief proceeds in the similar way as in the preceding case with one difference: the

θY ab > 1þ tan 2 4πr 2 2

8174

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 8πr 2 1þ ab

ð15Þ

ð16Þ

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11, and 13 differ for various topographies and depend on the characteristic lengths. It was supposed that experimentally observed wetting transitions are of a local character, whether they start from the triple line or from the center of the contact area.32,35,45
47 In this case w will define the stability of the Cassie wetting. If wetting transitions imply simultaneous filling of all pores, W will describe adequately the stability of the “fakir” wetting. It also should be mentioned that not only thermodynamic parameters govern wetting transitions but also the critical pressure necessary for depinning of the triple line and the transition time are important.31

Figure 8. Transition state for the relief of spheres. a and b are the parameters of a rectangular two-dimensional lattice.

For the close-packed quadratic lattice a = b = 2r o

θY > 94:7

ð17Þ

This is in accordance with the general conclusion that on the hydrophilic surfaces the Cassie state is thermodynamically unfavorable. For the discussed relief, the stability of this state may be attained due to weak hydrophobicity according to eq 17 or due to the energy barrier (eq 13) separating the Cassie and the Wenzel states. In nature, lotus leaves have similar relief but on two scales, which are micro- and nanoscales. The nanoscale may be modeled in the present example by an effective angle θY > 95° (see ref 53). The barrier given by eq 13 is maximal for θY = π; however, the largest known Young angle is 114°, registered for poly(tetrafluoroethylene). It is also seen that the barrier becomes considerable starting from angles close to π/2. This explains the pronounced superhydrophobicity of reliefs based on nanoscaled poly(vinylidene fluoride) spheres (θY = 85°).23 The order of magnitude of the barrier W for the droplet with a contact area S may be evaluated as W = γπr2N, where N = S/(ab) ∼S/4r2 is the number of unit cells in the area S. Thus, for the droplet of a 1 mm contact radius the barrier W ∼ Sγπ/4 ≈ 200 nJ is close to the barriers calculated in sections 2.1 and 2.2. The question is what is the quantitative parameter describing the stability of the “fakir” state? Is it w, related to filling a single pore, or perhaps it is W, related to the entire droplet? It could be seen that the energy barrier W is on the order of magnitude of γS, and it is independent of the microtopography of the hydrophilic relief. However, the final results for the unit-cell barrier w in eqs 5,

3. CONCLUSIONS Various reliefs are analyzed from the viewpoint of the stability of the Cassie (fakir) state. For hydrophobic materials, the multiscaled roughness of posts constituting the surface increases the potential barrier separating the Cassie and Wenzel states. For hydrophilic materials the Cassie state always corresponds to the higher energy state compared to the Wenzel one but is stabilized by the energy barrier. The condition for the existence of such a barrier is a geometrical property of the relief that provides a sufficient increase of the liquid
air interface in the course of the liquid penetration into details of this relief. In this way the apparent hydrophobicity of the reliefs based on inherently hydrophilic materials can be understood. Some examples of such reliefs are given, including one resembling the lotus leaf structure. The increase of liquid
air interface in the course of the liquid penetration into details of this relief also enhances the barrier separating the Cassie and Wenzel states on the inherently hydrophobic surfaces. ’ ACKNOWLEDGMENT We are thankful for Yelena Bormashenko for her help in preparing the manuscript. ’ REFERENCES (1) de Gennes, P. G.; Brochard-Wyart, F.; Qu
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(57) Bormashenko, E.; Bormashenko, Ye.; Whyman, G.; Pogreb, R.; Stanevsky, O. J. Colloid Interface Sci. 2006, 302, 308–311. (58) Sometimes such a state is called the composite state (see Figure 3), keeping in mind that some part of the cavity is in contact with liquid, as in the Wenzel state, while the other part is in contact with air, as in the Cassie state. However, then the “pure” Cassie state would correspond to the unphysical situation with liquid touching each ball at only one point.

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