Wetting on rough surfaces and contact angle hysteresis : numerical experiments based on a phase field model
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 6, p. 1027-1044

We present a phase field approach to wetting problems, related to the minimization of capillary energy. We discuss in detail both the $\Gamma$-convergence results on which our numerical algorithm are based, and numerical implementation. Two possible choices of boundary conditions, needed to recover Young’s law for the contact angle, are presented. We also consider an extension of the classical theory of capillarity, in which the introduction of a dissipation mechanism can explain and predict the hysteresis of the contact angle. We illustrate the performance of the model by reproducing numerically a broad spectrum of experimental results: advancing and receding drops, drops on inclined planes and superhydrophobic surfaces.

DOI : https://doi.org/10.1051/m2an/2009016
Classification:  76D45,  74N30,  49S05
Keywords: wetting, contact angle hysteresis, super-hydrophobic surfaces
@article{M2AN_2009__43_6_1027_0,
author = {Turco, Alessandro and Alouges, Fran\c cois and DeSimone, Antonio},
title = {Wetting on rough surfaces and contact angle hysteresis : numerical experiments based on a phase field model},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {43},
number = {6},
year = {2009},
pages = {1027-1044},
doi = {10.1051/m2an/2009016},
zbl = {pre05636845},
mrnumber = {2588431},
language = {en},
url = {http://www.numdam.org/item/M2AN_2009__43_6_1027_0}
}

Turco, Alessandro; Alouges, François; DeSimone, Antonio. Wetting on rough surfaces and contact angle hysteresis : numerical experiments based on a phase field model. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 6, pp. 1027-1044. doi : 10.1051/m2an/2009016. http://www.numdam.org/item/M2AN_2009__43_6_1027_0/

[1] G. Alberti and A. De Simone, Wetting of rough surfaces: a homogenization approach. Proc. R. Soc. A 461 (2005) 79-97. | MR 2124194 | Zbl 1145.82321

[2] G. Alberti and A. Desimone, Quasistatic evolution of sessile drops and contact angle hysteresis. In preparation (2009).

[3] G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with line-tension effect. Arch. Rat. Mech. Anal. 144 (1998) 1-46. | MR 1657316 | Zbl 0915.76093

[4] S. Baldo and G. Bellettini, $\Gamma$-convergence and numerical analysis: an application to the minimal partition problem. Ricerche di Matematica 1 (1991) 33-64. | MR 1191885 | Zbl 0755.65064

[5] W. Bao and Q. Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow. SIAM J. Sci. Comp. 25 (2004) 1674. | MR 2087331 | Zbl 1061.82025

[6] A. Braides, $\Gamma$-convergence for beginners. Oxford University Press (2002). | MR 1968440 | Zbl pre01865939

[7] M. Callies and D. Quéré, On water repellency. Soft Matter 1 (2005) 55-61.

[8] G. Dal Maso, An introduction to $\Gamma$-convergence. Birkhaüser (1993). | MR 1201152 | Zbl 0816.49001

[9] P.-G. De Gennes, F. Brochard-Wyart and D. Quéré, Capillarity and Wetting Phenomena. Springer (2004). | Zbl 1139.76004

[10] A. Desimone, N. Grunewald and F. Otto, A new model for contact angle hysteresis. Networks and Heterogeneous Media 2 (2007) 211-225 | MR 2291819 | Zbl 1125.76011

[11] R. Finn, Equilibrium Capillary Surfaces. Springer (1986). | MR 816345 | Zbl 0583.35002

[12] A. Lafuma and D. Quéré, Superhydrophobic states. Nature Materials 2 (2003) 457-460.

[13] L. Modica, Gradient theory of phase transitions with boundary contact energy. Ann. Inst. H. Poincaré Anal. Non Linéaire 5 (1987) 497. | Numdam | MR 921549 | Zbl 0642.49009

[14] L. Modica and S. Mortola, Un esempio di $\Gamma$-convergenza. Boll. Un. Mat. It. B 14 (1977) 285-299. | MR 445362 | Zbl 0356.49008

[15] N.A. Patankar, On the modeling of hydrophobic contact angles on rough surfaces. Langmuir 19 (2003) 1249-1253.

[16] S.J. Polak, An increased accuracy scheme for diffusion equations in cylindrical coordinates. J. Inst. Math. Appl. 14 (1974) 197-201. | MR 421097 | Zbl 0286.65041

[17] P. Seppecher, Moving contact lines in the Cahn-Hilliard theory. Int. J. Engng. Sci. 34 (1996) 977-992. | Zbl 0899.76042

[18] J.C. Strikwerda, Finite Difference Schemes and PDE. SIAM (2004). | MR 2124284