Hostname: page-component-5c6d5d7d68-qks25 Total loading time: 0 Render date: 2024-08-18T12:23:54.626Z Has data issue: false hasContentIssue false
  • English
  • Français

Macroscopic contact angle and liquid drops on rough solid surfaces via homogenization and numerical simulations

Published online by Cambridge University Press:  10 March 2013

S. Cacace ,
A. Chambolle ,
A. DeSimone  and
L. Fedeli
S. Cacace
Affiliation:
Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185, Rome, Italy.. cacace@mat.uniroma1.it
A. Chambolle
Affiliation:
CMAP, Ecole Polytechnique, CNRS 91128, Palaiseau, France.; antonin.chambolle@cmap.polytechnique.fr
A. DeSimone
Affiliation:
SISSA–International School for Advanced Studies, Via Bonomea 265, 34136, Trieste, Italy.; desimone@sissa.it
L. Fedeli
Affiliation:
California Institute of Technology, 1200, E. California Blvd, 91125, Pasadena, CA .; fedeli@caltech.edu
Get access

Abstract

We discuss a numerical formulation for the cell problem related to a homogenization approach for the study of wetting on micro rough surfaces. Regularity properties of the solution are described in details and it is shown that the problem is a convex one. Stability of the solution with respect to small changes of the cell bottom surface allows for an estimate of the numerical error, at least in two dimensions. Several benchmark experiments are presented and the reliability of the numerical solution is assessed, whenever possible, by comparison with analytical one. Realistic three dimensional simulations confirm several interesting features of the solution, improving the classical models of study of wetting on roughness.

Keywords

Wettingsuper-hydrophobic surfacescontact-angle hysteresishomogenizationtotal variationnon-smooth optimizationaugmented Lagrangian
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 3 , May 2013 , pp. 837 - 858
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alberti, G. and DeSimone, A., Wetting of rough surfaces: a homogenization approach. Proc. R. Soc. A 461 (2005) 79–97. Google Scholar
Alberti, G. and DeSimone, A., Quasistatic evolution of sessile drops and contact angle hysteresis. Arch. Rat. Mech. Anal. 202 (2011) 295–348. Google Scholar
L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York (2000)
Caffarelli, L.A., The obstacle problem revisited. J. Fourier Anal. Appl. 4 (1998) 383–402. Google Scholar
Caffarelli, L.A. and Mellet, A., Capillary drops on an inhomogeneous surface, Perspectives in nonlinear partial differential equations. Contemp. Math. 446 (2007) 175–201. Google Scholar
DeSimone, A., Grunewald, N. and Otto, F., A new model for contact angle hysteresis. Netw. Heterog. Media 2 (2007) 211–225. Google Scholar
J. Eckstein and D.P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55 (1992)
E. Esser, Applications of Lagrangian-Based Alternating Direction Methods and Connections to Split Bregman (2009)
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Boca Raton, FL (1992).
Fedeli, L., Turco, A. and DeSimone, A., Metastable equilibria of capillary drops on solid surfaces: a phase field approach. Contin. Mech. Thermodyn. 23 (2011) 453–471. Google Scholar
H. Federer, Geometric measure theory. Springer-Verlag New York Inc., New York (1969).
Finn, R., Some properties of capillary surfaces. Milan J. Math. 70 (2002) 1–23. Google Scholar
M.J. Lai, B. Lucier and J. Wang, The convergence of a central-difference discretization of Rudin-Osher-Fatemi model for image denoising, Scale Space and Variational Methods in Computer Vision. Springer (2009) 514–526.
Lions, P.L. and Mercier, B., Splitting Algorithms for the Sum of Two Nonlinear Operators. SIAM J. Numer. Anal. 16 (1979) 964–979. Google Scholar
Quéré, D., Wetting and Roughness. Annu. Rev. Mater. Res. 38 (2008) 71–99. Google Scholar
Taylor, J.E., Boundary regularity for solutions to various capillarity and free boundary problems. Commun. Partial Differ. Equ. 2 (1977) 323–357. Google Scholar
Yang, H., Buguin, A., Taulemesse, J.M., Kaneko, K., Mery, S., Bergeret, A. and Keller, P., Micron-Sized Main-Chain Liquid Crystalline Elastomer Actuators with Ultralarge Amplitude Contractions. J. Amer. Chem. Soc. 131 (2009) 15000–15004. Google ScholarPubMed
Wang, J. and Lucier, B.J., Error Bounds for Finite-Difference Methods for Rudin-Osher-Fatemi Image Smoothing. SIAM J. Numer. Anal. 49 (2011) 845–868. Google Scholar