American Institute of Mathematical Sciences

Advanced
February  2017, 37(2): 1061-1073. doi: 10.3934/dcds.2017044

A dynamic theory for contact angle hysteresis on chemically rough boundary

Xiao-Ping Wang 1,, and  Xianmin Xu 2,

1. 

Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
2. 

LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author: Xiao-Ping Wang

Received  August 2015 Revised  April 2016 Published  November 2016

Fund Project: This publication was supported in part by the Hong Kong RGC-GRF grants 605513 and 605311, Hong Kong RGC-CRF grant C6004-14G and NSFC grant 11571354

We study the interface dynamics and contact angle hysteresis in a two dimensional, chemically patterned channel described by the Cahn-Hilliard equation with a relaxation boundary condition. A system for the dynamics of the contact angle and contact point is derived in the sharp interface limit. We then analyze the behavior of the solution using the phase plane analysis. We observe the stick-slip of the contact point and the contact angle hysteresis. As the size of the pattern decreases to zero, the stick-slip becomes weaker but the hysteresis becomes stronger in the sense that one observes either the advancing contact angle or the receding contact angle without any switching in between. Numerical examples are presented to verify our analysis.

Keywords: Contact angle hysteresis, Cahn-Hilliard equation, asymptotic analysis.
Mathematics Subject Classification: Primary:41A60, 34A36;Secondary:76T1.
Citation: Xiao-Ping Wang, Xianmin Xu. A dynamic theory for contact angle hysteresis on chemically rough boundary. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1061-1073. doi: 10.3934/dcds.2017044
References:
[1]

X. Chen, X.-P. Wang and X. Xu, Analysis of the cahn-hilliard equation with a relaxation boundary condition modeling the contact angle dynamics, Arch. Rational Mech. Anal., 213 (2014), 1-24.  doi: 10.1007/s00205-013-0713-x.  Google Scholar

[2]

P. de Gennes, Wetting: Statics and dynamics, Simple Views on Condensed Matter, 12 (2003), 357-394.  doi: 10.1142/9789812564849_0041.  Google Scholar

[3]

A. DeSimone, N. Gruenewald and F. Otto, A new model for contact angle hysteresis, Networks and Heterogeneous Media, 2 (2007), 211-225.  doi: 10.3934/nhm.2007.2.211.  Google Scholar

[4]

R. Pego, Front migration in the nonlinear cahn-hilliard equation, Proc. R. Soc. Lond., A, 422 (1989), 261-278.  doi: 10.1098/rspa.1989.0027.  Google Scholar

[5]

T. Qian, X. Wang and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306.  doi: 10.1103/PhysRevE.68.016306.  Google Scholar

[6]

T. Qian, X. Wang and P. Sheng, Power-law slip profile of the moving contact line in two-phase immiscible flows, Phys. Rev. Lett., 93 (2004), 094501.  doi: 10.1103/PhysRevLett.93.094501.  Google Scholar

[7]

A. Turco, F. Alouges and A. DeSimone, Wetting on rough surfaces and contact angle hysteresis: Numerical experiments based on a phase field model, ESAIM: Mathematical Modelling and Numerical Analysis, 43 (2009), 1027-1044, http://www.esaim-m2an.org/article_S0764583X09000168. Google Scholar

[8]

X. Wang, T. Qian and P. Sheng, Moving contact line on chemically patterned surfaces, Journal of Fluid Mechanics, 605 (2008), 59-78.  doi: 10.1017/S0022112008001456.  Google Scholar

[9]

X. Xu and X. P. Wang, Analysis of wetting and contact angle hysteresis on chemically patterned surfaces, SIAM J. Appl. Math., 71 (2011), 1753-1779.  doi: 10.1137/110829593.  Google Scholar

[10]

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

show all references

References:
[1]

X. Chen, X.-P. Wang and X. Xu, Analysis of the cahn-hilliard equation with a relaxation boundary condition modeling the contact angle dynamics, Arch. Rational Mech. Anal., 213 (2014), 1-24.  doi: 10.1007/s00205-013-0713-x.  Google Scholar

[2]

P. de Gennes, Wetting: Statics and dynamics, Simple Views on Condensed Matter, 12 (2003), 357-394.  doi: 10.1142/9789812564849_0041.  Google Scholar

[3]

A. DeSimone, N. Gruenewald and F. Otto, A new model for contact angle hysteresis, Networks and Heterogeneous Media, 2 (2007), 211-225.  doi: 10.3934/nhm.2007.2.211.  Google Scholar

[4]

R. Pego, Front migration in the nonlinear cahn-hilliard equation, Proc. R. Soc. Lond., A, 422 (1989), 261-278.  doi: 10.1098/rspa.1989.0027.  Google Scholar

[5]

T. Qian, X. Wang and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306.  doi: 10.1103/PhysRevE.68.016306.  Google Scholar

[6]

T. Qian, X. Wang and P. Sheng, Power-law slip profile of the moving contact line in two-phase immiscible flows, Phys. Rev. Lett., 93 (2004), 094501.  doi: 10.1103/PhysRevLett.93.094501.  Google Scholar

[7]

A. Turco, F. Alouges and A. DeSimone, Wetting on rough surfaces and contact angle hysteresis: Numerical experiments based on a phase field model, ESAIM: Mathematical Modelling and Numerical Analysis, 43 (2009), 1027-1044, http://www.esaim-m2an.org/article_S0764583X09000168. Google Scholar

[8]

X. Wang, T. Qian and P. Sheng, Moving contact line on chemically patterned surfaces, Journal of Fluid Mechanics, 605 (2008), 59-78.  doi: 10.1017/S0022112008001456.  Google Scholar

[9]

X. Xu and X. P. Wang, Analysis of wetting and contact angle hysteresis on chemically patterned surfaces, SIAM J. Appl. Math., 71 (2011), 1753-1779.  doi: 10.1137/110829593.  Google Scholar

[10]

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

Figure 1.  Contact angle formed by the liquid-air interface with the solid boundary
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Interface motion in a channel
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The phase plane for the homogeneous boundary
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The phase plane and the attracting region bounded by the nullclines.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The advancing contact angle trajectory for large (left) and small (right) pattern period $T$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Advancing and receding contact angles for large ($T=2\pi/5$) and small ($T=2\pi/40$) pattern period. For large $T$, we observe stick-slip of the contact point (see Fig. 7). For small $T$, we observe contact angle hysteresis.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  The contact point stick-slip behaviour for pattern surface with period $T=2\pi/5$. Top: advancing trajectory of contact angle $\theta$. Middle: contact point velocity $x_t$. Below: $x_t$ in log scale
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Advancing and receding contact angles for inhomogeneous surfaces with smooth $\theta_Y$ with large and small patterned period.
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Pierluigi Colli, Gianni Gilardi, Paolo Podio-Guidugli, Jürgen Sprekels. An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 353-368. doi: 10.3934/dcdss.2013.6.353
[2]

Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511
[3]

Antonio DeSimone, Natalie Grunewald, Felix Otto. A new model for contact angle hysteresis. Networks & Heterogeneous Media, 2007, 2 (2) : 211-225. doi: 10.3934/nhm.2007.2.211
[4]

Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207
[5]

Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013
[6]

Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033
[7]

Alain Miranville, Ramon Quintanilla, Wafa Saoud. Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2257-2288. doi: 10.3934/cpaa.2020099
[8]

Ciprian G. Gal, Hao Wu. Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1041-1063. doi: 10.3934/dcds.2008.22.1041
[9]

Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
[10]

Peter Howard, Bongsuk Kwon. Spectral analysis for transition front solutions in Cahn-Hilliard systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 125-166. doi: 10.3934/dcds.2012.32.125
[11]

Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163
[12]

Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037
[13]

Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31
[14]

Alain Miranville, Sergey Zelik. The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 275-310. doi: 10.3934/dcds.2010.28.275
[15]

S. Maier-Paape, Ulrich Miller. Connecting continua and curves of equilibria of the Cahn-Hilliard equation on the square. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1137-1153. doi: 10.3934/dcds.2006.15.1137
[16]

Amy Novick-Cohen, Andrey Shishkov. Upper bounds for coarsening for the degenerate Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 251-272. doi: 10.3934/dcds.2009.25.251
[17]

Gianni Gilardi, A. Miranville, Giulio Schimperna. On the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (3) : 881-912. doi: 10.3934/cpaa.2009.8.881
[18]

Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012
[19]

Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517
[20]

Pierluigi Colli, Gianni Gilardi, Elisabetta Rocca, Jürgen Sprekels. Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 37-54. doi: 10.3934/dcdss.2017002

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (33)
  • HTML views (16)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences