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

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

Xianmin Xu 1,

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.
Keywords: the modified Cassie's equation, The modified Wenzel's equation, homogenization..
Mathematics Subject Classification: Primary: 41A60, 49J45; Secondary: 76T1.
Citation: Xianmin Xu. Analysis for wetting on rough surfaces by a three-dimensional phase field model. Discrete and 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.

[2]

R. A. Adams, Sobolev Spaces, Academic Press INC., 1978.

[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.

[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.

[5]

D. Bonn and D. Ross, Wetting transitions, Rep. Prog. Phys. 64 (2001), 1085-1163.

[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.

[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.

[8]

A. Cassie and S. Baxter, Wettability of porous surfaces, Trans. Faraday Soc., 40 (1944), 546-551. doi: 10.1039/tf9444000546.

[9]

J. Cahn, Critical point wetting, J. Chem. Phys., 66 (1977), 3667-3672.

[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.

[11]

G. Dal Maso, Introduction to $\Gamma$-convergence, Birkhauser, 1993. doi: 10.1007/978-1-4612-0327-8.

[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.

[13]

L. Gao and T. J. McCarthy, How Wenzel and Cassie were wrong, Langmuir, 23 (2007), 3762-3765. doi: 10.1021/la062634a.

[14]

P. G. de Gennes, Wetting: Statics and dynamics, Rev. Modern Phy., 57 (1985), 827-863.

[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.

[16]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003. doi: 10.1142/9789812795557.

[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.

[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.

[19]

G. Mchale, Cassie and Wenzel: Were they really so wrong, Langmuir, 23 (2007), 8200-8205. doi: 10.1021/la7011167.

[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.

[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.

[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.

[23]

N. A. Pantanka, On the modeling of hydrophyobic angles on rough surfaces, Langmuir, 19 (2003), 1249-1253.

[24]

D. Quere, Wetting and roughness, Annu. Rev. Mater. Res., 38 (2008), 71-99.

[25]

R. N. Wenzel, Resistance of solid surfaces to wetting by water, Ind. Eng. Chem., 28 (1936), 988-994. doi: 10.1021/ie50320a024.

[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.

[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.

[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.

[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.

[30]

X. Xu, Modified Wenzel and Cassie equations for wetting on rough surfaces, preprint, 2015.

[31]

T. Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805), 65-87.

show all references

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.

[2]

R. A. Adams, Sobolev Spaces, Academic Press INC., 1978.

[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.

[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.

[5]

D. Bonn and D. Ross, Wetting transitions, Rep. Prog. Phys. 64 (2001), 1085-1163.

[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.

[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.

[8]

A. Cassie and S. Baxter, Wettability of porous surfaces, Trans. Faraday Soc., 40 (1944), 546-551. doi: 10.1039/tf9444000546.

[9]

J. Cahn, Critical point wetting, J. Chem. Phys., 66 (1977), 3667-3672.

[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.

[11]

G. Dal Maso, Introduction to $\Gamma$-convergence, Birkhauser, 1993. doi: 10.1007/978-1-4612-0327-8.

[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.

[13]

L. Gao and T. J. McCarthy, How Wenzel and Cassie were wrong, Langmuir, 23 (2007), 3762-3765. doi: 10.1021/la062634a.

[14]

P. G. de Gennes, Wetting: Statics and dynamics, Rev. Modern Phy., 57 (1985), 827-863.

[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.

[16]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, 2003. doi: 10.1142/9789812795557.

[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.

[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.

[19]

G. Mchale, Cassie and Wenzel: Were they really so wrong, Langmuir, 23 (2007), 8200-8205. doi: 10.1021/la7011167.

[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.

[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.

[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.

[23]

N. A. Pantanka, On the modeling of hydrophyobic angles on rough surfaces, Langmuir, 19 (2003), 1249-1253.

[24]

D. Quere, Wetting and roughness, Annu. Rev. Mater. Res., 38 (2008), 71-99.

[25]

R. N. Wenzel, Resistance of solid surfaces to wetting by water, Ind. Eng. Chem., 28 (1936), 988-994. doi: 10.1021/ie50320a024.

[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.

[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.

[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.

[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.

[30]

X. Xu, Modified Wenzel and Cassie equations for wetting on rough surfaces, preprint, 2015.

[31]

T. Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805), 65-87.

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