Article pubs.acs.org/Langmuir
Wetting Transitions and Depinning of the Triple Line E. Bormashenko,*,†,‡ A. Musin,†,‡ G. Whyman,† and M. Zinigrad‡ †
Department of Physics, and ‡Department of Chemical Engineering and Biotechnology, Ariel University Center of Samaria, Post Office Box 3, Ariel 40700, Israel ABSTRACT: Physical mechanisms of Cassie−Wenzel wetting transitions are discussed. The origin of the potential barrier separating the Cassie and Wenzel wetting states is clarified. It may contain contributions originating from the filling of hydrophobic pores and displacement of the triple line along the smooth portions of the relief. One- and two-dimensional scenarios of wetting transitions are considered. We demonstrate that the contribution to the potential barrier because of the displacement of the triple line is not negligible in both cases.
1. INTRODUCTION Revealing the natural surfaces demonstrating pronounced water repellence (the so-called “lotus effect”) stimulated the extended theoretical and experimental research of wetting phenomena occurring on rough surfaces. One of the most studied phenomena occurring on rough surfaces are so-called wetting transitions.1−36 Understanding the physical mechanism of wetting transitions is crucial for the design of highly stable superhydrophobic materials.37 The physics of wetting transitions has been cleared up in a series of experimental and theoretical works.1−36 Various wetting states may coexist on rough surfaces because of multiple minima of the free energy of a droplet deposited on a rough surface.38 Main wetting regimes are described by the Cassie and Wenzel equations and their extensions.39−47 Our paper focuses on the Cassie−Wenzel transitions. Thus, it is suggested that the Cassie (“Fakir”) state is metastable and the Wenzel state is stable (in principle, the opposite experimental situation is also possible). The wetting transitions promoted by external stimuli, such as vibration, bouncing, or evaporation of droplets, are well-known to be irreversible.1−36 It has been demonstrated that the Cassie and Wenzel wetting states are separated by the energetic barrier.5,16,36 The irreversibility of wetting transitions is explained by the fact that the energy barrier from the side of the metastable (higher energy) wetting state is always much lower than that from the side of the stable wetting state.36 It is generally agreed that the potential barrier separating the Cassie and Wenzel states originates from filling hydrophobic grooves with liquid, which is energetically unfavorable.5,16,36 Our paper demonstrates that barriers may comprise additional contribution related to the pinning of the triple (three-phase) line by smooth elements of the relief. © 2012 American Chemical Society
2. EXPERIMENTAL SECTION The energy barrier related to the displacement of the triple line on smooth substrates has been studied as follows. A 10 μL water droplet was evaporated on the polytetrafluoroethylene (Teflon) film with the roughness (Ra) of 4.82 nm. The roughness was measured with Dimensions Icon atomic force microscopy (AFM, Bruker AXS). A three-dimensional (3D) AFM image of the substrate is given in Figure
Figure 1. Three-dimensional AFM image of the Teflon substrate. 1. The contact angle and the radius of a droplet were measured during evaporation by a Rame−Hart goniometer (model 500) with the accuracy of 0.1° and 10−6 m. The resistivity of bidistilled water measured with a LRC-meter Motech MT 4090 was 2 MΩ cm−1. All Received: November 10, 2011 Revised: January 2, 2012 Published: January 20, 2012 3460
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experiments were performed in standard laboratory conditions: temperature of 24 °C and relative humidity of about 50%.
han−Sefiane approach extremely suitable for the analysis of wetting transitions accompanied by such a displacement.48,49 The main ideas of this approach are as follows. A droplet on a flat substrate takes the shape of a spherical cap, with the contact radius R0 and the contact angle θ0 (Figure 3). When it
3. RESULTS AND DISCUSSION On the rough substrate manufactured from strongly hydrophobic material, such as Teflon, the Cassie wetting state corresponds to a higher free energy of the droplet/substrate system than the Wenzel wetting state. Generally, two main scenarios of wetting transitions are possible, as shown in Figure 2. The first scenario depicted in Figure 2a corresponds to a
Figure 3. Geometrical parameters of a droplet.
evaporates, we observe the stick−slip process typical for polymer surfaces with low pinning (Figure 4). During evaporation, the droplet may be in the state with a larger contact radius than the equilibrium state (for the same volume of the droplet), R = R0 + δR, and with a lower contact angle, θ = θ0 − δθ. The free surface energy G can be evaluated as48 ⎡ ⎤ 2 G(R , θ) = γπR2⎢ − cos θ0⎥ ⎦ ⎣ (1 + cos θ)
Figure 2. Scheme of two scenarios of wetting transitions.
wetting transition taking place under pinned triple (threephase) line. When the triple line is pinned, the Cassie and Wenzel wetting states are separated by the potential barrier.5,16,36 Indeed, to proceed to the Wenzel state, the liquid has to wet hydrophobic grooves constituting the relief. Such filling is energetically unfavorable; this gives rise to the energetic barrier separating the Cassie and Wenzel wetting states.5,16,36 Let us denote this energetic barrier U1. The second scenario occurs when the triple line is depinned, as shown in Figure 2b. Wetting transitions accompanied by the depinning of the triple line were observed under vibration and bouncing droplets.12−14,22−24 In this case, it is necessary not only to fill hydrophobic grooves of the surface but also to displace the triple line, as shown in Figure 2b. For the sake of simplicity, we suggest that this displacement occurs on the smooth horizontal portion of the relief. Obviously, the mechanical work should be performed for such a displacement, giving rise to the additional energetic barrier U2 (the units of both U1 and U2 are joules, and they are related to the entire droplet, when U1 = SŨ 1 and U2 = lŨ 2, where S and l are the area underneath the droplet filled by water and the perimeter of the triple line, respectively). Hence, the resulting energetic barrier to be surmounted for the Cassie−Wenzel transitions equals U = U1 + U2
(2)
where γ is the surface tension on the liquid−air interface. After a slip, the droplet is in a new equilibrium state, with the radius R1 and the contact angle θ1, which differs from the Young angle θE because the droplet is surrounded now with a wet area.48 In the pinned state, before the slip, a droplet with radius R and contact angle θ had the free-energy excess equal to the energy barrier to be surmounted for the slip motion U2 = 2πRŨ 2. ⎧ ⎡ ⎡ ⎤ 2 2 γπ⎨R2⎢ − cos θ0⎥ − R12⎢ ⎦ ⎣ (1 + cos θ1) ⎩ ⎣ (1 + cos θ) ⎪
⎪
⎤⎫ − cos θ0⎥⎬ = 2πRU2̃ ⎦⎭ ⎪
⎪
(3)
The changes in the free energy for seven pronounced “jumps” in the graph of Figure 4 calculated according to eq 3 are presented in Table 1, with the mean value of Ũ 2 being on the order of 10−6 J/m, which is close to the values of the potential barriers reported for various materials in refs 48 and 49. This value is also close to the upper limit of reported values of the linear tension; however, it remains disputable whether Ũ 2 could be identified with the linear tension.48 It should be mentioned that the reported method of establishment of Ũ 2 is possible only on the surfaces where a stick−slip motion is observed, and it could not be applied for the study of strongly pinning substrates (such as metallic substrates).49 Now compare U2 = lŨ 2 with U1, i.e., the energy barrier originating from the filling of hydrophobic pores (grooves). The simple model of posts with dimensions a × a × h separated with grooves with the width a is used (Figure 5). The energy barrier U1 equals the maximal change in surface energy when liquid wets the walls of posts but does not yet touch the substrate bottom (Figure 5). The change in the surface energy when one “cell” is filled with liquid is
(1)
It could be supposed that U2 ≪ U1, but we will demonstrate that the situation is more complicated and the interrelation between U1 and U2 depends upon the topography of the relief. The potential barrier U2 originates from the intermolecular interactions between molecules of the liquid and solid. Thus, the calculation of U2 is a challenging task. However, it can be measured using the technique developed recently by Shanahan and Sefiane for the study of stick−slip motion of evaporated droplets. The stick−slip motion of evaporated droplets and the Cassie−Wenzel wetting transitions depicted in Figure 2b involve the triple-line displacement. This makes the Shana-
ΔUcell = 4ah(γSL − γSA ) = −4ah γ cos θE 3461
(4)
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Figure 4. Dependence of the contact angle (●) and the contact radius (○) upon the time of evaporation.
Table 1. Change in the Free Energy Per Unit Length of Contact Line for Seven Slips during Evaporation of a 10 μL Water Droplet on the Smooth Teflon Surface (Figure 4) Ũ 2 (J/m) 1
2
3
4
5
6
7
average
3.6 × 10−7
1.1 × 10−6
8.5 × 10−7
4.4 × 10−7
9.8 × 10−7
1.1 × 10−6
1.2 × 10−6
8.7 × 10−7
Figure 5. Scheme illustrating the filling of grooves with liquid for the Cassie−Wenzel state transition.
Figure 6. Scheme illustrating 1D and 2D scenarios of wetting transitions.
where γSL and γSA are the surface tensions on the solid−liquid and solid−air interfaces, respectively. The number of such cells depends upon a wetting scenario.10,11,22,26 According to the first scenario, only the pores in the nearest vicinity of the triple line are filled in the course of wetting transition. This 1D mechanism is justified by the fact that the apparent contact angle is totally governed by the wetting situation in the vicinity of the triple line.26,50,51 The existence of the 1D mechanism of wetting transitions has been demonstrated experimentally in ref 11 (see also refs 22−24). The 2D scenario requires filling all of the pores underneath the droplet (see Figure 6). Both possibilities will be considered. When only pores adjacent to the triple line (circle in Figure 6) are filled (“1D” transition) with liquid, the number of “cells”
to be filled is N1 = 2πR/2a = πR/a. Therefore, the surfaceenergy change per a droplet is expressed as U11D = 4ah1D(γSL − γSA )N1 = −4πh1DR γ cos θE
(5)
If all of the surface beneath the droplet is filled (“2D” transition), the number of “cells” to be filled by water is N2 = πR2/4a2 and U12D = 4ah2D(γSL − γSA )N2 =
−πh2DR2 γ cos θE a
(6)
Let us calculate the height of posts for which the surface component of the barrier U11D is equal to the depinning energy of the triple line: U11D = −4πh1DRγ cos θE = πRŨ 2. Assuming Ũ 2 3462
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≈ 10−6 J/m, γ = 72 × 10−3 J/m2, and cos θE = −0.34, one obtains h1D =
U2̃ ≈ 10 μm −4γ cos θE
Similarly, for the 2D mechanism = −πh2DR γ cos θE/a = πRŨ 2, and assuming a ∼ 10−5 m and R ∼ 10−3 m, we obtain h2D =
U2̃ a ≈ 0.04 μm −R γ cos θE
2
(8)
It could be recognized that, when only the nearest to the triple line pores are filled in the course of the wetting transition, the energy of filling pores and the energy of the triple-line depinning are comparable at a quite reasonable posts’ height h1D ∼ 10 μm, which is typical for superhydrophobic surfaces.52 Thus, in this case, the energy necessary for depinning of the triple line on the smooth portion of the relief is at least not negligible. When h ≫ h1D, the energy barrier separating the Cassie and Wenzel state is determined by the energy of filling the pores (grooves); if h ≪ h1D, it is governed by the energy of depinning of the triple line. The low value (eq 8) of h2D obtained for the 2D scenario shows that, in this case, wetting transitions are governed by the filling of pores and not by the depinning of the triple line (if microstructured substrates are considered). However, the mentioned depinning may be important for wetting transitions on nanostructures typical for natural and artificial superhydrophobic surfaces.53−60 The pinning of the triple line is responsible on a variety of wetting phenomena, observed on rough surfaces. In particular, because of the effect of pinning, the Cassie apparent contact angle is not the maximal possible contact angle observed on a rough surface.61 A droplet that traps air in the Cassie wetting state could be inflated, and the advancing apparent contact angle becomes larger than the Cassie contact angle.61
4. CONCLUSION The Cassie−Wenzel wetting transitions may proceed under the pinned and depinned triple line. The latter is the case when the droplet is vibrated or bounced. We discuss the origin of the potential barrier separating wetting states when the triple line is depinned. It comprises the contributions related to filling of hydrophobic pores and to the displacement of the triple line. A stick−slip study of the droplets evaporated on nanometrically smooth Teflon substrates was studied. The value of the potential barrier for the displacement of the triple line was calculated. This value was established as 10−6 J/m per unit length of the triple line and is at least comparable to the barrier, which is due to filling microscopically scaled hydrophobic grooves. We conclude that, for wetting transitions accompanied by the motion of the triple line, the energy related to its displacement should be considered.
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AUTHOR INFORMATION
Corresponding Author
*Telephone: +972-3-9066134. Fax: +972-3-9066621. E-mail:
[email protected].
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ACKNOWLEDGMENTS A. Musin is grateful for the support of the Milken Family Foundation for her fellowship. 3463
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