# American Institute of Mathematical Sciences

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

## A dynamic theory for contact angle hysteresis on chemically rough boundary

 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.

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
Contact angle formed by the liquid-air interface with the solid boundary
Interface motion in a channel
The phase plane for the homogeneous boundary
The phase plane and the attracting region bounded by the nullclines.
The advancing contact angle trajectory for large (left) and small (right) pattern period $T$.
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.
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
Advancing and receding contact angles for inhomogeneous surfaces with smooth $\theta_Y$ with large and small patterned period.
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