# 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.
 [1] Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364 [2] Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119 [3] Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251 [4] Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218 [5] Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346 [6] Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457 [7] Vieri Benci, Sunra Mosconi, Marco Squassina. Preface: Applications of mathematical analysis to problems in theoretical physics. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020446 [8] Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323 [9] Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256 [10] Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219 [11] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [12] Yichen Zhang, Meiqiang Feng. A coupled $p$-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 [13] Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318 [14] Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341 [15] A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441 [16] Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136 [17] Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345 [18] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [19] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317 [20] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

2019 Impact Factor: 1.338

## Metrics

• PDF downloads (78)
• HTML views (64)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]