# American Institute of Mathematical Sciences

October  2016, 21(8): 2839-2850. doi: 10.3934/dcdsb.2016075

## Analysis for wetting on rough surfaces by a three-dimensional phase field model

 1 The State Key Laboratory of Scienti c and Engineering Computing, Institute of Computational Mathematics and Scienti c/Engineering Computing, Chinese Academy of Sciences, Beijing 100190, China

Received  October 2015 Revised  March 2016 Published  September 2016

In this paper, we consider the derivation of the modified Wenzel's and Cassie's equations for wetting phenomena on rough surfaces from a three-dimensional phase field model. We derive an effective boundary condition by asymptotic two-scale homogenization technique when the size of the roughness is small. The modified Wenzel's and Cassie's equations for the apparent contact angles on the rough surfaces are then derived from the effective boundary condition. The homogenization results are proved rigorously by the $\Gamma$-convergence theory.
Citation: Xianmin Xu. Analysis for wetting on rough surfaces by a three-dimensional phase field model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2839-2850. doi: 10.3934/dcdsb.2016075
##### References:
 [1] Y. Achdou, P. Le Tallec, F Valentin and O. Pironneau, Constructing wall laws with domain decomposition or asymptotic expansion techniques, Compt. Methods Appl. Mech. Engrg, 151 (1998), 215-232. doi: 10.1016/S0045-7825(97)00118-7.  Google Scholar [2] R. A. Adams, Sobolev Spaces, Academic Press INC., 1978. Google Scholar [3] G. Alberti and A. DeSimone, Wetting of rough surfaces: A homogenization approach, Proc. R. Soc. A, 461 (2005), 79-97. doi: 10.1098/rspa.2004.1364.  Google Scholar [4] D. Bonn, J. Eggers, J. Indekeu, J. Meunier and E. Rolley, Wetting and spreading, Rev. Mod. Phys. 81 (2009), 739-805. doi: 10.1103/RevModPhys.81.739.  Google Scholar [5] D. Bonn and D. Ross, Wetting transitions, Rep. Prog. Phys. 64 (2001), 1085-1163. Google Scholar [6] A. Braids, $\Gamma$-convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications, Vol. 22, Oxford University Press, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar [7] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.  Google Scholar [8] A. Cassie and S. Baxter, Wettability of porous surfaces, Trans. Faraday Soc., 40 (1944), 546-551. doi: 10.1039/tf9444000546.  Google Scholar [9] J. Cahn, Critical point wetting, J. Chem. Phys., 66 (1977), 3667-3672. Google Scholar [10] W. Choi, A. Tuteja, J. M. Mabry, R. E. Cohen and G. H. McKinley, A modified Cassie-Baxter relationship to explain contact angle hysteresis and anisotropy on non-wetting textured surfaces, J. Colloid Interface Sci., 339 (2009), 208-216. doi: 10.1016/j.jcis.2009.07.027.  Google Scholar [11] G. Dal Maso, Introduction to $\Gamma$-convergence, Birkhauser, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar [12] H. Y. Erbil, The debate on the dependence of apparent contact angles on drop contact area or three-phase contact line: A review, Surface Science Reports, 69 (2014), 325-365. doi: 10.1016/j.surfrep.2014.09.001.  Google Scholar [13] L. Gao and T. J. McCarthy, How Wenzel and Cassie were wrong, Langmuir, 23 (2007), 3762-3765. doi: 10.1021/la062634a.  Google Scholar [14] P. G. de Gennes, Wetting: Statics and dynamics, Rev. Modern Phy., 57 (1985), 827-863. Google Scholar [15] P. G. de Gennes, F. Brochard-Wyart and D. Quere, Capillarity and Wetting Phenomena, Springer, Berlin, 2004. doi: 10.1007/978-0-387-21656-0.  Google Scholar [16] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003. doi: 10.1142/9789812795557.  Google Scholar [17] A. L. Madureira and F. Valentin, Asymptotics of the Poisson problem in domains with curved rough boundaries, SIAM J. Math. Anal., 38 (2007), 1450-1473. doi: 10.1137/050633895.  Google Scholar [18] M. De Menech, Modeling of droplet breakup in a microfluidic t-shaped junction with a phase-field model, Phys. Rev. E, 73 (2006), 031505. Google Scholar [19] G. Mchale, Cassie and Wenzel: Were they really so wrong, Langmuir, 23 (2007), 8200-8205. doi: 10.1021/la7011167.  Google Scholar [20] A. Marmur and E. Bittoun, When Wenzel and Cassie Are Right: Reconciling Local and Global Considerations, Langmuir, 25 (2009), 1277-1281. doi: 10.1021/la802667b.  Google Scholar [21] J. Nevard and J. B. Keller, Homogenization of rough boundaries and interfaces, SIAM J. Appl. Math. 57 (1997), 1660-1686. doi: 10.1137/S0036139995291088.  Google Scholar [22] M. V. Panchagnula and S. Vedantam, Comment on how wenzel and cassie were wrong by gao and McCarthy, Langmuir, 23 (2007), 13242-13242. doi: 10.1021/la7022117.  Google Scholar [23] N. A. Pantanka, On the modeling of hydrophyobic angles on rough surfaces, Langmuir, 19 (2003), 1249-1253. Google Scholar [24] D. Quere, Wetting and roughness, Annu. Rev. Mater. Res., 38 (2008), 71-99. Google Scholar [25] R. N. Wenzel, Resistance of solid surfaces to wetting by water, Ind. Eng. Chem., 28 (1936), 988-994. doi: 10.1021/ie50320a024.  Google Scholar [26] G. Whyman, E. Bormashenko and T. Stein, The rigorous derivative of Young, Cassie-Baxter and Wenzel equations and the analysis of the contact angle hysteresis phenomenon, Chem. Phy. Letters, 450 (2008), 355-359. Google Scholar [27] G. Wolansky and A. Marmur, Apparent contact angles on rough surfaces: The Wenzel equation revisited, Colloids and Surfaces A, 156 (1999), 381-388. doi: 10.1016/S0927-7757(99)00098-9.  Google Scholar [28] X. Xu and X.-P. Wang, Derivation of Wenzel's and Cassie's equations from a phase field model for two phase flow on rough surface, SIAM J. Appl. Math., 70 (2010), 2929-2941. doi: 10.1137/090775828.  Google Scholar [29] X. Xu and X.-P. Wang, A modified Cassie's equation and contact angle hysteresis, Colloid Polymer Sci., 291 (2013), 299-306. doi: 10.1007/s00396-012-2748-1.  Google Scholar [30] X. Xu, Modified Wenzel and Cassie equations for wetting on rough surfaces, preprint, 2015. Google Scholar [31] T. Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805), 65-87. Google Scholar

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##### References:
 [1] Y. Achdou, P. Le Tallec, F Valentin and O. Pironneau, Constructing wall laws with domain decomposition or asymptotic expansion techniques, Compt. Methods Appl. Mech. Engrg, 151 (1998), 215-232. doi: 10.1016/S0045-7825(97)00118-7.  Google Scholar [2] R. A. Adams, Sobolev Spaces, Academic Press INC., 1978. Google Scholar [3] G. Alberti and A. DeSimone, Wetting of rough surfaces: A homogenization approach, Proc. R. Soc. A, 461 (2005), 79-97. doi: 10.1098/rspa.2004.1364.  Google Scholar [4] D. Bonn, J. Eggers, J. Indekeu, J. Meunier and E. Rolley, Wetting and spreading, Rev. Mod. Phys. 81 (2009), 739-805. doi: 10.1103/RevModPhys.81.739.  Google Scholar [5] D. Bonn and D. Ross, Wetting transitions, Rep. Prog. Phys. 64 (2001), 1085-1163. Google Scholar [6] A. Braids, $\Gamma$-convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications, Vol. 22, Oxford University Press, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar [7] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. doi: 10.1007/BF00254827.  Google Scholar [8] A. Cassie and S. Baxter, Wettability of porous surfaces, Trans. Faraday Soc., 40 (1944), 546-551. doi: 10.1039/tf9444000546.  Google Scholar [9] J. Cahn, Critical point wetting, J. Chem. Phys., 66 (1977), 3667-3672. Google Scholar [10] W. Choi, A. Tuteja, J. M. Mabry, R. E. Cohen and G. H. McKinley, A modified Cassie-Baxter relationship to explain contact angle hysteresis and anisotropy on non-wetting textured surfaces, J. Colloid Interface Sci., 339 (2009), 208-216. doi: 10.1016/j.jcis.2009.07.027.  Google Scholar [11] G. Dal Maso, Introduction to $\Gamma$-convergence, Birkhauser, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar [12] H. Y. Erbil, The debate on the dependence of apparent contact angles on drop contact area or three-phase contact line: A review, Surface Science Reports, 69 (2014), 325-365. doi: 10.1016/j.surfrep.2014.09.001.  Google Scholar [13] L. Gao and T. J. McCarthy, How Wenzel and Cassie were wrong, Langmuir, 23 (2007), 3762-3765. doi: 10.1021/la062634a.  Google Scholar [14] P. G. de Gennes, Wetting: Statics and dynamics, Rev. Modern Phy., 57 (1985), 827-863. Google Scholar [15] P. G. de Gennes, F. Brochard-Wyart and D. Quere, Capillarity and Wetting Phenomena, Springer, Berlin, 2004. doi: 10.1007/978-0-387-21656-0.  Google Scholar [16] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003. doi: 10.1142/9789812795557.  Google Scholar [17] A. L. Madureira and F. Valentin, Asymptotics of the Poisson problem in domains with curved rough boundaries, SIAM J. Math. Anal., 38 (2007), 1450-1473. doi: 10.1137/050633895.  Google Scholar [18] M. De Menech, Modeling of droplet breakup in a microfluidic t-shaped junction with a phase-field model, Phys. Rev. E, 73 (2006), 031505. Google Scholar [19] G. Mchale, Cassie and Wenzel: Were they really so wrong, Langmuir, 23 (2007), 8200-8205. doi: 10.1021/la7011167.  Google Scholar [20] A. Marmur and E. Bittoun, When Wenzel and Cassie Are Right: Reconciling Local and Global Considerations, Langmuir, 25 (2009), 1277-1281. doi: 10.1021/la802667b.  Google Scholar [21] J. Nevard and J. B. Keller, Homogenization of rough boundaries and interfaces, SIAM J. Appl. Math. 57 (1997), 1660-1686. doi: 10.1137/S0036139995291088.  Google Scholar [22] M. V. Panchagnula and S. Vedantam, Comment on how wenzel and cassie were wrong by gao and McCarthy, Langmuir, 23 (2007), 13242-13242. doi: 10.1021/la7022117.  Google Scholar [23] N. A. Pantanka, On the modeling of hydrophyobic angles on rough surfaces, Langmuir, 19 (2003), 1249-1253. Google Scholar [24] D. Quere, Wetting and roughness, Annu. Rev. Mater. Res., 38 (2008), 71-99. Google Scholar [25] R. N. Wenzel, Resistance of solid surfaces to wetting by water, Ind. Eng. Chem., 28 (1936), 988-994. doi: 10.1021/ie50320a024.  Google Scholar [26] G. Whyman, E. Bormashenko and T. Stein, The rigorous derivative of Young, Cassie-Baxter and Wenzel equations and the analysis of the contact angle hysteresis phenomenon, Chem. Phy. Letters, 450 (2008), 355-359. Google Scholar [27] G. Wolansky and A. Marmur, Apparent contact angles on rough surfaces: The Wenzel equation revisited, Colloids and Surfaces A, 156 (1999), 381-388. doi: 10.1016/S0927-7757(99)00098-9.  Google Scholar [28] X. Xu and X.-P. Wang, Derivation of Wenzel's and Cassie's equations from a phase field model for two phase flow on rough surface, SIAM J. Appl. Math., 70 (2010), 2929-2941. doi: 10.1137/090775828.  Google Scholar [29] X. Xu and X.-P. Wang, A modified Cassie's equation and contact angle hysteresis, Colloid Polymer Sci., 291 (2013), 299-306. doi: 10.1007/s00396-012-2748-1.  Google Scholar [30] X. Xu, Modified Wenzel and Cassie equations for wetting on rough surfaces, preprint, 2015. Google Scholar [31] T. Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805), 65-87. Google Scholar
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