Derivation of the Wenzel and Cassie equations from a phase field model for two phase flow on rough surface

SIAM Journal on Applied Mathematics

Volumn 70, Issue 8, 2010, Pages 2929-2941

  Xu, Xianmin ^a,b   Wang, Xiaoping ^a  

^a INSTITUTE OF COMPUTATIONAL MATHEMATICS AND SCIENTIFIC ENGINEERING COMPUTING   (China)

^b HONG KONG UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Hong Kong)

Author keywords

Cassie equation; Homogenization; Wenzel equation

Indexed keywords

APPARENT CONTACT ANGLE; CASSIE EQUATION; CONVERGENCE THEORY; EQUILIBRIUM BEHAVIOR; HOMOGENIZATION; HOMOGENIZATION TECHNIQUES; MULTIPLE-SCALE EXPANSION; PHASE FIELD EQUATION; PHASE FIELD MODELS; ROUGH SURFACES; TOTAL FREE ENERGY; TWO PHASE FLUID; WENZEL EQUATION;

BOUNDARY CONDITIONS; CONTACT ANGLE; HOMOGENIZATION METHOD; SURFACE MEASUREMENT;

TWO PHASE FLOW;

EID: 78751550704     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/090775828     Document Type: Article

References (29)

1. Y. ACHDOU, P. LE TALLEC, F. VALENTIN, AND O. PIRONNEAU, Constructing wall laws with domain decomposition or asymptotic expansion techniques, Comput. Methods Appl. Mech. Engrg., 151 (1998), pp. 215-232.
2. R. A. ADAMS, Sobolev Spaces, Academic Press, New York, 1978.
3. G. ALBERTI AND A. DESIMONE, Wetting of rough surfaces: A homogenization approach, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 451 (2005), pp. 79-97.
4. D. BONN AND D. ROSS, Wetting transitions, Rep. Progr. Phys., 64 (2001), pp. 1085-1163. (Pubitemid 32908326)
5. A. BRAIDS, F-Convergence for Beginners, Oxford Lecture Ser. Math. Appl. 22, Oxford University Press, Oxford, UK, 2002.
6. G. CAGINALP, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), pp. 205-245.
7. G. CAGINALP AND P. C. FIFE, Dynamics of layered interfaces arising from phase boundaries, SIAM J. Appl. Math., 48 (1988), pp. 506-518.
8. A. CASSIE AND S. BAXTER, Wettability of porous surfaces, Trans. Faraday Soc., 40 (1944), pp. 546-551.
9. J. CAHN, Critical point wetting, J. Chem. Phys., 66 (1977), pp. 3667-3672.
10. G. Dal Maso, Introduction to F-Convergence, Birkhäuser Boston, Boston, 1993.
11. L. GAO AND T. J. MCCARTHY, How Wenzel and Cassie were wrong, Langmuir, 23 (2007), pp. 3762-3765.
12. L. GAO AND T. J. MCCARTHY, Reply to "Comment on How Wenzel and Cassie Were Wrong by Gao and McCarthy," Langmuir, 23 (2007), pp. 13243-13243.
13. P. G. DE GENNES, Wetting: Statics and dynamics, Rev. Modern Phys., 57 (1985), pp. 827-863.
14. P. G. DE GENNES, F. BROCHARD-WYART, AND D. QUERE, Capillarity and Wetting Phenomena, Springer, Berlin, 2003.
15. E. GIUSTI, Direct Methods in the Calculus of Variations, World Scientific, River Edge, NJ, 2003.
16. A. L. MADUREIRA AND F. VALENTIN, Asymptotics of the Poisson problem in domains with curved rough boundaries, SIAM J. Math. Anal., 38 (2007), pp. 1450-1473.
17. M. DE MENECH, Modeling of droplet breakup in a microfluidic t-shaped junction with a phasefield model, Phys. Rev. E, 73 (2006), article 031505.
18. G. MCHALE, Cassie and Wenzel: Were they really so wrong, Langmuir, 23 (2007), pp. 8200-8205.
19. A. MARMUR AND E. BITTOUN, When Wenzel and Cassie are right: Reconciling local and global considerations, Langmuir, 25 (2009), pp. 1277-1281.
20. J. NEVARD AND J. B. KELLER, Homogenization of rough boundaries and interfaces, SIAM J. Appl. Math., 57 (1997), pp. 1660-1686.
21. M. V. PANCHAGNULA AND S. VEDANTAM, Comment on How Wenzel and Cassie Were Wrong by Gao and McCarthy, Langmuir, 23 (2007), pp. 13242-13242.
22. N. A. PANTANKA, On the modeling of hydrophyobic angles on rough surfaces, Langmuir, 19 (2003), pp. 1249-1253.
23. T. QIAN, X.-P. WANG, AND P. SHENG, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), article 016306.
24. D. QUERE, Wetting and roughness, Annu. Rev. Mater. Res., 38 (2008), pp. 71-99.
25. A. TURCO, F. ALOUGES, AND A. DESIMONE, Wetting on rough surfaces and contact angle hysteresis: Numerical experiments based on a phase field model, M2AN Math. Model. Numer. Anal., 43 (2009), pp. 1027-1044.
26. R. N. WENZEL, Resistance of solid surfaces to wetting by water, Ind. Eng. Chem., 28 (1936), pp. 988-994.
27. 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. Phys. Lett., 450 (2008), pp. 355-359.
28. G. WOLANSKY AND A. MARMUR, Apparent contact angles on rough surfaces: The Wenzel equation revisited, Colloids and Surfaces A, 156 (1999), pp. 381-388.
29. T. YOUNG, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805), pp. 65-87.