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Contact angle formed by the liquid-air interface with the solid boundary
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Analysis of an irregular boundary layer behavior for the steady state flow of a Boussinesq fluid
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 |
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.
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 and Continuous Dynamical Systems,
2017, 37
(2)
: 1061-1073.
doi: 10.3934/dcds.2017044
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References:
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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. |
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P. de Gennes,
Wetting: Statics and dynamics, Simple Views on Condensed Matter, 12 (2003), 357-394.
doi: 10.1142/9789812564849_0041. |
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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. |
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [10] |
T. Young,
An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805), 65-87.
|
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. |
| [2] |
P. de Gennes,
Wetting: Statics and dynamics, Simple Views on Condensed Matter, 12 (2003), 357-394.
doi: 10.1142/9789812564849_0041. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [10] |
T. Young,
An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805), 65-87.
|
Figure 2.
Interface motion in a channel
Figure 3.
The phase plane for the homogeneous boundary
Figure 4.
The phase plane and the attracting region bounded by the nullclines.
Figure 5.
The advancing contact angle trajectory for large (left) and small (right) pattern period $T$ .
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 8.
Advancing and receding contact angles for inhomogeneous surfaces with smooth $\theta_Y$ with large and small patterned period.
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