Models

The Wenzel Equation

The first attempt to understand the effect of roughness on wettability is that of Wenzel (1936). Wenzel was interested in ways of improving the water-proofing of fabrics, which are naturally textured materials (at the scale of the monofilaments which make the yarns). [81] He had noticed that the natural tendency of a material (hydrophilic or hydrophobic) is enhanced by the presence of textures. Wenzel’s interpretation of these facts is based on the increasing of the surface area of a material because of its roughness: a liquid will tend to spread more on a rough hydrophilic substrate, since it allows it to develop more solid/liquid contact (which is favorable in a hydrophilic situation). Conversely, a rough hydrophobic material appears (apparently) more hydrophobic, because the liquid would have to develop a much larger (unfavorable) contact with the solid if the contact angle is kept unchanged.

The key parameter is thus the roughness factor, defined by Wenzel as the ratio of the true surface area A (taking into account the peaks and valleys on the surface) to the apparent surface area A’ is defined as the roughness factor r = A/A’. It is thus a dimensionless number, larger than unity, and all the larger since the surface is rough.

The main assumption of Wenzel is that the liquid follows the defects of the solid surface, as it is deposited on it. The apparent contact angle is the one which minimizes the (surface) energy of the drop as shown in Figure 2-24.

Figure 2-24. The Wenzel state: the liquid follows the solid surface.

θ

For r = 1 (flat solid), we get back Young’s law (equation (2.5)). For a rough surface (r > 1), we derive Wenzel’s relation: [81, 82]

cosθ_w =rcosθ (2.36)

In Eq. (2.36), θ is the intrinsic CA on a smooth surface, θ_w is that on a rough surface made of the same material, and r is the roughness factor. The Wenzel relation qualitatively agrees with the main observations: both hydrophobicity and hydrophilicity are enhanced by roughness, since we deduce from equation (2.36) that increasing surface roughness results in actual CA decrease for hydrophilic materials (θ < 90°) and increase for hydrophobic materials (θ > 90°).. This looks like a simple and attractive solution for inducing superhydrophobicity: the rougher the material, the higher the contact angle. However, this is not that simple, for two reasons: firstly, contact angles generally spread in quite a large interval, contrasting with equation (2.36) which predicts a unique angle. This interval, often referred to as the contact angle hysteresis, is responsible for the sticking of drops, an effect in contradiction with water repellency. In a Wenzel state, the contact angle hysteresis will be very large: trying to remove a liquid makes it contact itself (owing to the fraction left in the textures), which yields a low ‘‘receding’’ contact angle—values as low as 40° were reported, making this state hydrophilic-like in the receding stage. [83]

The second reason which makes it impossible to reach high values of θ_w, as expected from equation (2.36) for r large and θ > 90°, can be guessed quite easily: for very rough hydrophobic materials, the energy stored for following the solid surface is much larger than the energy associated with the air pockets sketched in Fig. 2-25.

[83-86]

Figure 2-25. The Cassie state, the liquid only contacts the top of the asperities, leaving air below.

The Cassie and Baxter Equation

In Cassie and Baxter state (Figure 2-25), the liquid only contacts the solid through the top of the asperities, on a fraction that we denote as f1. [80] If only air were present between the solid and the liquid (as for a water drop on a very hot plate), the ‘‘contact angle’’ would be 180°: the smaller f1, the closer to this extreme situation, and thus the higher the hydrophobicity. More precisely, the contact angle θ_c of such a ‘‘fakir’’ drop (Figure 2-24) is an average between the angles on the solid (of cosine

cosθ ), and on the air (of cosine-1), respectively weighed by the fractions f1 and 1 - f1, which yields:

cosθ_c = f1(cosθ + − (2.37) 1) 1

For θ = 110° and f1 = 10%, we find that θ_c is about 160°. In this case, 90% of the drop base contacts air! This makes it understandable that the corresponding hysteresis is observed to be very low (typically around 5 to 10°), as first reported by Johnson and Dettre: [88] the liquid has very little interactions with its substrate. Hence, this state will be the (only) repellent one, since it achieves both a large contact angle and a small hysteresis (this can be observed further, in Figure 2-26).

θ

Figure 2-26. Millimetric water drops (of the same volume) deposited on a superhydrophobic substrate consisting of dilute pillars (f1 = 0.01). (a) The right drop has been pressed, which induced a Wenzel state, characterized by a smaller angle (the roughness is very low, and equal to 1.1). The light passes below the left drop, indicating a Cassie state. (b) Ten minutes later, the drop volumes have decreased, owing to evaporation, and angles became receding ones. The difference of hysteresis between both states is clearly visible: the Wenzel drop even became hydrophilic

θc monotonously increases as f1 decreases, suggesting that f1 should be made as small as possible. But reducing f1 also makes the roughness decrease, so that we reach the critical roughness γc below which the Wenzel state is favoured. The quantity rc is easily deduced from the intersection of equation (2.36) and (2.37), and is found to be (f1 - 1)/cosθ + f1, which is generally close to -1/cosθ (since we will often have: f1 = 1). For θ = 120° (a high value for the Young angle, obtained on fluorinated substrates), the fakir state will thus be favored for roughness factors larger than 2. Conversely, Öner and McCarthy experimentally observed that below a critical

density of defects (i.e. below a critical roughness), there is indeed a serious deterioration of the water-repellent properties. [89]