Wetting on rough surfaces and contact angle hysteresis : numerical experiments based on a phase field model
ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 6, pp. 1027-1044.

We present a phase field approach to wetting problems, related to the minimization of capillary energy. We discuss in detail both the Γ-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 : 10.1051/m2an/2009016
Classification : 76D45, 74N30, 49S05
Mots-clés : wetting, contact angle hysteresis, super-hydrophobic surfaces 
@article{M2AN_2009__43_6_1027_0,
     author = {Turco, Alessandro and Alouges, Fran\c{c}ois and DeSimone, Antonio},
     title = {Wetting on rough surfaces and contact angle hysteresis : numerical experiments based on a phase field model},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {1027--1044},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {6},
     year = {2009},
     doi = {10.1051/m2an/2009016},
     mrnumber = {2588431},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2009016/}
}
TY  - JOUR
AU  - Turco, Alessandro
AU  - Alouges, François
AU  - DeSimone, Antonio
TI  - Wetting on rough surfaces and contact angle hysteresis : numerical experiments based on a phase field model
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2009
SP  - 1027
EP  - 1044
VL  - 43
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2009016/
DO  - 10.1051/m2an/2009016
LA  - en
ID  - M2AN_2009__43_6_1027_0
ER  - 
%0 Journal Article
%A Turco, Alessandro
%A Alouges, François
%A DeSimone, Antonio
%T Wetting on rough surfaces and contact angle hysteresis : numerical experiments based on a phase field model
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2009
%P 1027-1044
%V 43
%N 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2009016/
%R 10.1051/m2an/2009016
%G en
%F 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: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 6, pp. 1027-1044. doi : 10.1051/m2an/2009016. http://www.numdam.org/articles/10.1051/m2an/2009016/

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

[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 | Zbl

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

[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 | Zbl

[6] A. Braides, Γ-convergence for beginners. Oxford University Press (2002). | MR

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

[8] G. Dal Maso, An introduction to Γ-convergence. Birkhaüser (1993). | MR | Zbl

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

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

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

[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 | Zbl

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

[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 | Zbl

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

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

Cité par Sources :