Abstract
In this study, we considered the wetting phenomenon on a general substrate from a new viewpoint of continuum mechanics. The analyses first show how the Wenzel and the Cassie models deviate the practical results in some special substrates, and then elucidate the mechanism of the triple contact line (TCL) moving. Based upon variational theory of the total free functional dealing with the movable boundary condition, we show that the macroscopic contact angle (MCA) expression is the corresponding transversality condition. It manifests that the MCA depends only on the chemical and geometric property at the TCL, and is not affected by the gravity of the droplet and the contact area beneath the liquid. Our continuum model also shows the exploration of the pinning effect on a sharp wedge or the interface between two different phases. This investigation will help designing super-hydrophobic materials for novel micro-fluidic devices.
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References
Young T. An essay on the cohesion of fluids. Trans R Soc London, 1805, 95:65–87
de Gennes P G. Wetting: statics and dynamics. Rev Mod Phys, 1985, 57:827–863
Quéré D. Rough ideas on wetting. Physica A, 2002, 313:32–46
Yuan Q Z, Zhao Y P. Topology-dominated dynamic wetting of the precursor chain in a hydrophilic interior corner. Proc R Soc A, 2012, 468:310–322
Ceihuis C, Barthlott W. Characterization and distribution of water-repellent, self-cleaning plant surfaces. Ann Bot, 1997, 79:667–677
Hu D L, Chan B, Bush J W. The hydrodynamics of water strider locomotion. Nature, 2004, 424:663–666
Feng X Q, Gao X F, Wu Z N, et al. Superior water repellency of water strider legs with hierarchical structures: experiments and analysis. Langmuir, 2007, 23:4892–4896
Liu J L, Feng X Q. Buoyant force and sinking conditions of a hydrophobic thin rod floating on water. Phys Rev E, 2007, 76:066103
Shi F, Niu J, Liu J L, et al. Towards understanding why superhydrophobic coating is needed by water striders. Adv Mater, 2007, 19:2257–2261
Wu C W, Kong X Q, Wu D. Micronanostructures of the scales on a mosquito’s legs and their role in support. Phys Rev E, 2007, 76: 017301
Parker A R, Lawrence C R. Water capture by a desert beetle. Nature, 2001, 414: 33–34
Zhai L, Cebeci F C, Cohen R E, et al. Stable superhydrophobic coatings from polyelectrolyte multilayers. Nano Lett, 2004, 4:1349–1353
Lau K K S, Bico J, Teo K B K, et al. Superhydrophobic carbon nanotube forests. Nano Lett, 2003, 3:1701–1705
Hosono E, Fujihara S, Honma I, et al. Superhydrophobic perpendicular nano-pin film by the bottom-up process. J Am Chem Soc, 2005, 127:13458–13459
Yoshimitsu Z, Nakajima A, Watanabe T, et al. Effects of surface structure on the hydrophobicity and sliding behavior of water droplets. Langmuir, 2002, 18:5818–5822
Onda T, Shibuichi S, Satoh N, et al. Super-water-repellent fractal surfaces. Langmuir, 1996, 12:2125–2127
Bico J, Thiele U, Quéré D. Wetting of textured surfaces. Colloid Surf A-Physiochem Eng Asp, 2002, 206:41–46
Patankar N A. On the modeling of hydrophobic contact angles on rough surfaces. Langmuir, 2003, 19:1249–1253
Lee J, He B, Patankar N A. A roughness-based wettability switching membrane device for hydrophobic surfaces. J Micromech Microeng, 2005, 15:591–600
Liu J L, Xia R, Li B W, et al. Directional motion of droplets in a conical tube or on a conical fibre. Chin Phys Lett, 2007, 24:3210–3213
Blossey R. Self-cleaning surface: Virtual realities. Nat Mater, 2003, 2:301–306
Dai W, Zhao Y P. An electrowetting model for rough surfaces under low voltage. J Adhes Sci Tech, 2008, 22:217–229
Wenzel R N. Resistance of solid surfaces to wetting by water. Ind Eng Chem, 1936, 28:988–994
Cassie A B D, Baxter S T. Wettability of porous surfaces. Faraday Soc, 1944, 40:546–551
Pease D C. The significance of the contact angle in relation to the solid surface. J Phys Chem, 1945, 49:107–110
Bartell F E, Shepard J W. Surface roughness as related to hysteresis of contact angles. II. The systems paraffin-3 molar calcium chloride solution-air and paraffin-glycerol-air. J Phys Chem, 1953, 57:455–458
Extrand C W. Contact angles and hysteresis on surfaces with chemically heterogeneous islands. Langmuir, 2003, 19:3793–3796
Gao L, McCarthy T J. How Wenzel and Cassie were wrong. Langmuir, 2007, 23:3762–3765
Gao L, McCarthy T J. An attempt to correct the faulty intuition. Langmuir, 2009, 25:7249–7255
Liu J L, Mei Y, Xia R. A new wetting mechanism based upon TCL pinning. Langmuir, 2011, 27:196–200
McHale G. Cassie and Wenzel: were they really so wrong? Langmuir, 2007, 23:8200–8205
Nosonovsky M. On the range of applicability of the Wenzel and Cassie equations. Langmuir, 2007, 23:9919–9920
Panchagnula M V, Vedantam S. Comment on how Wenzel and Cassie were wrong by Gao and McCarthy. Langmuir, 2007, 23:13242
Marmur A, Bittoun E. When Wenzel and Cassie are right: Reconciling local and global considerations. Langmuir, 2009, 25:1277–1281
Swain P S, Lipowsky R. Contact angles on heterogeneous surfaces: a new look at Cassie’s and Wenzel’s laws. Langmuir, 1998, 14:6772–6780
Bormashenko E. A variational approach to wetting of composite surfaces: is wetting of composite surfaces a one-dimensional or two-dimensional phenomenon? Langmuir, 2009, 25:10451–10454
Bormashenko E. Young, Boruvka-Neumann, Wenzel and Cassie-Baxter equations as the transversality conditions for the variational problem of wetting. Colloid Surf A-Physiochem Eng Asp, 2009, 345:163–165
Yu Y S, Zhao Y P. Deformation of PDMS membrane and microcantilever by a water droplet: comparison between Mooney-Rivlin and linear elastic constitutive models. J Colloid Inter face Sci, 332:467–476
Koch K, Bohn H F, Barthlott W. Hierarchically sculptured plant surfaces and superhydrophobicity. Langmuir, 2009, 25:14116–14120
Quéré D. Surface wetting: Model droplets. Nat Mater, 2004, 3: 79–80
Yuan Q Z, Zhao Y P. Precursor film in dynamic wetting, electrowetting, and electro-elasto-capillarity. Phys Rev Lett, 2010, 104: 246101
Calabri L, Pugno N, Menozzi C, Valeri S. AFM nanoindentation: Tip shape and tip radius of curvature effect on the hardness measurement. J Phys-Condens Matter, 2008, 20: 474208
Wolansky G, Marmur A. Apparent contact angles on rough surfaces: The Wenzel equation revisited. Colloid Surf A-Physiochem Eng Asp, 1999, 156:381–388
Brandon S, Haimovich N, Yeger E, et al. Partial wetting of chemically patterned surfaces: the effect of drop size. J Colloid Interface Sci, 2003, 263:237–243
Yu X, Wang Z, Jiang Y, et al. Surface gradient material: from superhydrophobicity to superhydrophilicity. Langmuir, 2006, 22:4483–4486
Lipowsky R, Lenz P, Swain P S. Wetting and dewetting of structured and imprinted surfaces. Colloid Surf A-Physiochem Eng Asp, 2000, 161:3–22
Li Y, Wang J, Yin Y, et al. Division and microstructure feature in the interface transition zone of Fe3Al/Q235 diffusion bonding. J Colloid Interface Sci, 2005, 288:521–525
Yu Y, Wu Q, Zhang K, et al. Effect of triple-phase contact line on contact angle hysteresis. Sci China-Phys Mech Astron, 2012, 55:1045–1050
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Liu, J., Xia, R. & Zhou, X. A new look on wetting models: continuum analysis. Sci. China Phys. Mech. Astron. 55, 2158–2166 (2012). https://doi.org/10.1007/s11433-012-4895-2
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DOI: https://doi.org/10.1007/s11433-012-4895-2