Contact-Angle Hysteresis on Super-Hydrophobic Surfaces G. McHale,* N. J. Shirtcliffe, and M. I. Newton School of Biomedical and Natural Sciences, The Nottingham Trent University, Clifton Lane, Nottingham NG11 8NS, United Kingdom Received June 1, 2004. In Final Form: August 10, 2004 The relationship between perturbations to contact angles on a rough or textured surface and the super- hydrophobic enhancement of the equilibrium contact angle is discussed theoretically. Two models are considered. In the first (Wenzel) case, the super-hydrophobic surface has a very high contact angle and the droplet completely contacts the surface upon which it rests. In the second (Cassie-Baxter) case, the super-hydrophobic surface has a very high contact angle, but the droplet bridges across surface protrusions. The theoretical treatment emphasizes the concept of contact-angle amplification or attenuation and distinguishes between the increases in contact angles due to roughening or texturing surfaces and perturbations to the resulting contact angles. The theory is applied to predicting contact-angle hysteresis on rough surfaces from the hysteresis observable on smooth surfaces and is therefore relevant to predicting roll-off angles for droplets on tilted surfaces. The theory quantitatively predicts a “sticky” surface for Wenzel-type surfaces and a “slippy” surface for Cassie-Baxter-type surfaces. Introduction How a droplet of a liquid sits and rolls on a surface is determined by both the surface chemistry and the surface roughness or topography. 1-3 A flat and smooth hydro- phobic surface exhibiting an equilibrium contact angle to water of, say, 115° can be converted into a super- hydrophobic surface exhibiting a contact angle of greater than 150° simply by roughening it, even without altering any surface chemistry. Two models of contact-angle enhancement exist. In the Cassie-Baxter 4,5 model, a drop rests on the peaks of surface protrusions and bridges the air gaps in between. In the Wenzel 6,7 model, the liquid droplet retains contact at all points with the solid surface below it. Experimentally, when the response of a drop to a tilt of the surface is investigated, it is found that all super-hydrophobic surfaces are not equivalent. In some cases, the drop rolls off the surface easily, while for others the drop clings to the surface, even for high tilt angles. 8-10 Que ´re ´ refers to these two contrasting cases as “slippy” and “sticky” super-hydrophobic surfaces. 11 The ability of a drop to move on a surface is determined by the contact- angle hysteresis. 12,13 When a drop resting on a horizontal surface has liquid steadily injected by a syringe until the contact line advances, the contact angle of the drop observed when it is just set in motion is defined as the advancing angle, θ A (Figure 1a). Similarly, when liquid is steadily retracted, the contact angle observed when the contact line is just set in motion is the receding angle, θ R (Figure 1b). If a surface on which a drop rests is tilted slightly, the drop remains at rest but with differing contact angles at each side of the drop (Figure 1c). The difference in forces per unit length at the two sides of the drop is proportional to γ LV (cos θ L - cos θ U ) where γ LV is the liquid-vapor surface tension and θ L and θ U are the contact angles at the lower and upper sides of the drop. When the upper angle reaches the receding angle and the lower angle reaches the advancing angle, the drop just begins to move. The range of angles, Δθ H ) (θ A - θ R ), is the contact-angle hysteresis and is an important parameter in understanding drop motion on a surface. Que ´re ´ has discussed how a Cassie- Baxter surface with a high contact angle and low hysteresis can result in a reduction in the force required to set a drop into motion. 12 Few quantitative attempts have been reported on modeling how contact-angle hysteresis might be influenced in the transition of a surface from hydrophobic to super- hydrophobic. Two lines of thought exist in the literature on super-hydrophobicity. In the first case, contact-angle hysteresis is viewed as a consequence of solid surface area, while in the second, it is the contact perimeter of the solid surface area which is viewed as important. That these views are not necessarily equivalent is evident by con- sidering a surface of square pillars across which a drop sits in contact with the tops of the pillars but bridging the gaps between (i.e., the Cassie-Baxter case). For pillars having sides of length D and arranged in a square array of sides L (Figure 2a), the solid contact area is D 2 per L 2 of area and the perimeter is 4D per L 2 of area; the solid * Corresponding author: E-mail: glen.mchale@ntu.ac.uk. Tel: +44 (0)115 8483383. (1) Onda, T.; Shibuichi, S.; Satoh, N.; Tsujii, K. Langmuir 1996, 12, 2125-2127. (2) Neinhuis, C.; Barthlott, W. Ann. Bot. 1997, 79, 667-677. (3) Blossey, R. Nat. Mater. 2003, 2, 301-306. (4) Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546- 551. (5) Johnson, R. E.; Dettre, R. H. Contact angle, Wettability and Adhesion; Advances in Chemistry Series 43; American Chemical Society: Washington, DC, 1964; p 112. (6) Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988-994. (7) Wenzel, R. N. J. Phys. Colloid Chem. 1949, 53, 1466-1467. (8) Miwa, M.; Nakajima, A.; Fujishima, A.; Hashimoto, K.; Watanabe, T. Langmuir 2000, 16, 5754-5760. (9) O ¨ ner, D.; McCarthy, T. J. Langmuir 2000, 16, 7777-7782. (10) Feng, L.; Li, S.; Li, Y.; Li, H.; Zhang, L.; Zhai, J.; Song, Y.; Liu, B.; Jiang, L.; Zhu, D. Adv. Mater. 2002, 14, 1857-1860. (11) Que ´re ´ , D.; Lafuma, A.; Bico, J. Nanotechnology 2003, 14, 1109- 1112. (12) Que ´re ´, D. Physica A 2002, 313, 32-46. (13) Marmur, E. Langmuir 2004, 20, 3517-3519. Figure 1. (a) Advancing contact angle, (b) receding contact angle, and (c) advancing and receding contact angles determined by tilting experiment. 10146 Langmuir 2004, 20, 10146-10149 10.1021/la0486584 CCC: $27.50 © 2004 American Chemical Society Published on Web 10/02/2004