Analysis of Wetting and Contact Angle Hysteresis on Chemically Patterned Surfaces

Wetting and contact angle hysteresis on chemically patterned surfaces in two dimensions are analyzed from a stationary phase-field model for immiscible two phase fluids. We first study the sharp-interface limit of the model by the method of matched asymptotic expansions. We then justify the results rigorously by the $\Gamma$-convergence theory for the related variational problem and study the properties of the limiting minimizers. The results also provide a clear geometric picture of the equilibrium configuration of the interface. This enables us to explicitly calculate the total surface energy for the two phase systems on chemically patterned surfaces with simple geometries, namely the two phase flow in a channel and the drop spreading. By considering the quasi-static motion of the interface described by the change of volume (or volume fraction), we can follow the change-of-energy landscape which also reveals the mechanism for the stick-slip motion of the interface and contact angle hysteresis on the che...

[1]  Robert V. Kohn,et al.  Local minimisers and singular perturbations , 1989, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[2]  S. Herminghaus,et al.  Wetting: Statics and dynamics , 1997 .

[3]  E. B. Dussan,et al.  LIQUIDS ON SOLID SURFACES: STATIC AND DYNAMIC CONTACT LINES , 1979 .

[4]  R. N. Wenzel.  RESISTANCE OF SOLID SURFACES TO WETTING BY WATER , 1936 .

[5]  S. C. Malik,et al.  Introduction to convergence , 1984 .

[6]  David Jacqmin,et al.  Contact-line dynamics of a diffuse fluid interface , 2000, Journal of Fluid Mechanics.

[7]  E. Giusti.  Direct methods in the calculus of variations , 2003 .

[8]  Thomas Young,et al.  An Essay on the Cohesion of Fluids , 1800 .

[9]  D. Bonn,et al.  Wetting and Spreading , 2009 .

[10]  David Qu,et al.  Wetting and Roughness , 2008 .

[11]  Ping Sheng,et al.  Molecular scale contact line hydrodynamics of immiscible flows. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Ping Sheng,et al.  Moving contact line on chemically patterned surfaces , 2008, Journal of Fluid Mechanics.

[13]  DERIVATION OF WENZEL’S AND CASSIE’S EQUATIONS FROM A PHASE FIELD MODEL FOR TWO PHASE FLOW ON ROUGH SURFACE∗ , 2010 .

[14]  Ya-Guang Wang,et al.  The Sharp Interface Limit of a Phase Field Model for Moving Contact Line Problem , 2007 .

[15]  G. Caginalp.  An analysis of a phase field model of a free boundary , 1986 .

[16]  Abraham Marmur,et al.  When Wenzel and Cassie are right: reconciling local and global considerations. , 2009, Langmuir : the ACS journal of surfaces and colloids.

[17]  L. Modica,et al.  Gradient theory of phase transitions with boundary contact energy , 1987 .

[18]  B. Dacorogna.  Direct methods in the calculus of variations , 1989 .

[19]  A. Cassie,et al.  Wettability of porous surfaces , 1944 .

[20]  S. Vedantam,et al.  Constitutive modeling of contact angle hysteresis. , 2008, Journal of colloid and interface science.

[21]  Xianmin Xu,et al.  Derivation of the Wenzel and Cassie Equations from a Phase Field Model for Two Phase Flow on Rough Surface , 2010, SIAM J. Appl. Math..

[22]  Robert L. Pego,et al.  Front migration in the nonlinear Cahn-Hilliard equation , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[23]  G. D. Maso,et al.  An Introduction to-convergence , 1993 .

[24]  Antonio DeSimone,et al.  A new model for contact angle hysteresis , 2007, Networks Heterog. Media.

[25]  J. NEVARD,et al.  Homogenization of Rough Boundaries and Interfaces , 1997, SIAM J. Appl. Math..

[26]  P. Gennes.  Wetting: statics and dynamics , 1985 .

[27]  A. DeSimone,et al.  Wetting of rough surfaces: a homogenization approach , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[28]  Paul C. Fife,et al.  Dynamics of Layered Interfaces Arising from Phase Boundaries , 1988 .

[29]  François Alouges,et al.  Wetting on rough surfaces and contact angle hysteresis: numerical experiments based on a phase field model , 2009 .

[30]  Chunfeng Zhou,et al.  Sharp-interface limit of the Cahn–Hilliard model for moving contact lines , 2010, Journal of Fluid Mechanics.

[31]  Edward Bormashenko,et al.  The rigorous derivation of Young, Cassie–Baxter and Wenzel equations and the analysis of the contact angle hysteresis phenomenon , 2008 .

[32]  Andrea Braides.  Γ-convergence for beginners , 2002 .

[33]  J. Yeomans,et al.  Modeling contact angle hysteresis on chemically patterned and superhydrophobic surfaces. , 2007, Langmuir : the ACS journal of surfaces and colloids.

[34]  Xinfu Chen,et al.  Exixtance of equilibria for the chn-hilliard equation via local minimizers of the perimeter , 1996 .