October  2016, 21(8): 2905-2926. doi: 10.3934/dcdsb.2016079

A numerical study of three-dimensional droplets spreading on chemically patterned surfaces

1. 

School of Mathematics and Statistics, Guangdong University of Finance and Economics, China

2. 

Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong, China

3. 

Computational Transport Phenomena Laboratory, King Abdullah University of Science and Technology, Saudi Arabia

Received  August 2015 Revised  March 2016 Published  September 2016

We study numerically the three-dimensional droplets spreading on physically flat chemically patterned surfaces with periodic squares separated by channels. Our model consists of the Navier-Stokes-Cahn-Hilliard equations with the generalized Navier boundary conditions. Stick-slip behavior and contact angle hysteresis are observed. Moreover, we also study the relationship between the effective advancing/receding angle and the two intrinsic angles of the surface patterns. By increasing the volume of droplet gradually, we find that the advancing contact line tends gradually to an equiangular octagon with the length ratio of the two adjacent sides equal to a fixed value that depends on the geometry of the pattern.
Citation: Hua Zhong, Xiao-Ping Wang, Shuyu Sun. A numerical study of three-dimensional droplets spreading on chemically patterned surfaces. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2905-2926. doi: 10.3934/dcdsb.2016079
References:
[1]

S. Brandon, N. Haimovich, E. Yeger and A. Marmur, Partial wetting of chemically patterned surfaces: The effect of drop size,, J. Colloid Interf. Sci., 263 (2003), 237.  doi: 10.1016/S0021-9797(03)00285-6.  Google Scholar

[2]

S. Brandon and A. Marmur, Simulation of contact angle hysteresis on chemically heterogeneous surfaces,, J. Colloid Interf. Sci., 183 (1996), 351.  doi: 10.1006/jcis.1996.0556.  Google Scholar

[3]

S. Brandon, A. Wachs and A. Marmur, Simulated contact angle hysteresis of a three-dimensional drop on a chemically heterogeneous surface: a numerical example,, J. Colloid Interf. Sci., 191 (1997), 110.  doi: 10.1006/jcis.1997.4912.  Google Scholar

[4]

A. B. D. Cassie and S. Baxter, Wettability of porous surfaces,, Trans. Faraday Soc., 40 (1944), 546.  doi: 10.1039/tf9444000546.  Google Scholar

[5]

D. Chatain, D. Lewis, J. Baland and W. Carter, Numerical analysis of the shapes and energies of droplets on micropatterned substrates,, Langmuir, 22 (2006), 4237.  doi: 10.1021/la053146q.  Google Scholar

[6]

M. Gao and X. P. Wang, A gradient stable scheme for a phase field model for the moving contact line problem,, J. Comput. Phys., 231 (2012), 1372.  doi: 10.1016/j.jcp.2011.10.015.  Google Scholar

[7]

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

[8]

M. Iwamatsu, Contact angle hysteresis of cylindrical drops on chemically heterogeneous striped surfaces,, J. Colloid Interf. Sci., 297 (2006), 772.  doi: 10.1016/j.jcis.2005.11.032.  Google Scholar

[9]

J. F. Joanny and P. G. de Gennes, A model for contact angle hysteresis,, Simple Views on Condensed Matter, 12 (2003), 457.  doi: 10.1142/9789812564849_0048.  Google Scholar

[10]

R. Johnson and R. Dettre, Contact-angle hysteresis. 3. study of an idealized heterogeneous surface,, J. Phys. Chem., 68 (1964), 1744.  doi: 10.1021/j100789a012.  Google Scholar

[11]

H. Kusumaatmaja and J. M. Yeomans, Modeling contact angle hysteresis on chemically patterned and superhydrophobic surfaces,, Langmuir, 23 (2007), 6019.  doi: 10.1021/la063218t.  Google Scholar

[12]

S. T. Larsen and R. Taboryski, A Cassie-like law using triple phase boundary line fractions for faceted droplets on chemically heterogeneous surfaces,, Langmuir, 25 (2009), 1282.  doi: 10.1021/la8030045.  Google Scholar

[13]

A. Marmur, Contact-angle hysteresis on heterogeneous smooth surfaces,, J. Colloid Interf. Sci., 168 (1994), 40.  doi: 10.1006/jcis.1994.1391.  Google Scholar

[14]

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

[15]

W. Q. Ren, Wetting transition on patterned surfaces: Transition states and energy barriers,, Langmuir, 30 (2014), 2879.  doi: 10.1021/la404518q.  Google Scholar

[16]

L. Schwartz and S. Garoff, Contact angle hysteresis on heterogeneous surfaces,, Langmuir, 1 (1985), 219.  doi: 10.1021/la00062a007.  Google Scholar

[17]

L. Schwartz and S. Garoff, Contact angle hysteresis and the shape of the 3-phase line,, J. Colloid Interf. Sci., 106 (1985), 422.  doi: 10.1016/S0021-9797(85)80016-3.  Google Scholar

[18]

P. S. Swain and R. Lipowsky, Contact angles on heterogeneous surfaces: A new look at Cassie's and Wenzel's laws,, Langmuir, 14 (1998), 6772.  doi: 10.1021/la980602k.  Google Scholar

[19]

X. P. Wang, T. Z. Qian and P. Sheng, Moving contact line on chemically patterned surfaces,, J. Fluid Mech., 605 (2008), 59.  doi: 10.1017/S0022112008001456.  Google Scholar

[20]

R. N. Wenzel, Resistance of solid surfaces to wetting by water,, Ind. Eng. Chem., 28 (1936), 988.  doi: 10.1021/ie50320a024.  Google Scholar

[21]

G. Wolansky and A. Marmur, The actual contact angle on a heterogeneous rough surface in three dimensions,, Langmuir, 14 (1998), 5292.  doi: 10.1021/la960723p.  Google Scholar

[22]

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

[23]

T. Young, An essay on the cohesion of fluids,, Philos. Trans. R. Soc. London, 95 (1805), 65.  doi: 10.1098/rstl.1805.0005.  Google Scholar

[24]

H. Zhong, X. P. Wang, A. Salama and S. Sun, Quasistatic analysis on configuration of two-phase flow in Y-shaped tubes,, Comput. Math. Appl., 68 (2014), 1905.  doi: 10.1016/j.camwa.2014.10.004.  Google Scholar

show all references

References:
[1]

S. Brandon, N. Haimovich, E. Yeger and A. Marmur, Partial wetting of chemically patterned surfaces: The effect of drop size,, J. Colloid Interf. Sci., 263 (2003), 237.  doi: 10.1016/S0021-9797(03)00285-6.  Google Scholar

[2]

S. Brandon and A. Marmur, Simulation of contact angle hysteresis on chemically heterogeneous surfaces,, J. Colloid Interf. Sci., 183 (1996), 351.  doi: 10.1006/jcis.1996.0556.  Google Scholar

[3]

S. Brandon, A. Wachs and A. Marmur, Simulated contact angle hysteresis of a three-dimensional drop on a chemically heterogeneous surface: a numerical example,, J. Colloid Interf. Sci., 191 (1997), 110.  doi: 10.1006/jcis.1997.4912.  Google Scholar

[4]

A. B. D. Cassie and S. Baxter, Wettability of porous surfaces,, Trans. Faraday Soc., 40 (1944), 546.  doi: 10.1039/tf9444000546.  Google Scholar

[5]

D. Chatain, D. Lewis, J. Baland and W. Carter, Numerical analysis of the shapes and energies of droplets on micropatterned substrates,, Langmuir, 22 (2006), 4237.  doi: 10.1021/la053146q.  Google Scholar

[6]

M. Gao and X. P. Wang, A gradient stable scheme for a phase field model for the moving contact line problem,, J. Comput. Phys., 231 (2012), 1372.  doi: 10.1016/j.jcp.2011.10.015.  Google Scholar

[7]

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

[8]

M. Iwamatsu, Contact angle hysteresis of cylindrical drops on chemically heterogeneous striped surfaces,, J. Colloid Interf. Sci., 297 (2006), 772.  doi: 10.1016/j.jcis.2005.11.032.  Google Scholar

[9]

J. F. Joanny and P. G. de Gennes, A model for contact angle hysteresis,, Simple Views on Condensed Matter, 12 (2003), 457.  doi: 10.1142/9789812564849_0048.  Google Scholar

[10]

R. Johnson and R. Dettre, Contact-angle hysteresis. 3. study of an idealized heterogeneous surface,, J. Phys. Chem., 68 (1964), 1744.  doi: 10.1021/j100789a012.  Google Scholar

[11]

H. Kusumaatmaja and J. M. Yeomans, Modeling contact angle hysteresis on chemically patterned and superhydrophobic surfaces,, Langmuir, 23 (2007), 6019.  doi: 10.1021/la063218t.  Google Scholar

[12]

S. T. Larsen and R. Taboryski, A Cassie-like law using triple phase boundary line fractions for faceted droplets on chemically heterogeneous surfaces,, Langmuir, 25 (2009), 1282.  doi: 10.1021/la8030045.  Google Scholar

[13]

A. Marmur, Contact-angle hysteresis on heterogeneous smooth surfaces,, J. Colloid Interf. Sci., 168 (1994), 40.  doi: 10.1006/jcis.1994.1391.  Google Scholar

[14]

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

[15]

W. Q. Ren, Wetting transition on patterned surfaces: Transition states and energy barriers,, Langmuir, 30 (2014), 2879.  doi: 10.1021/la404518q.  Google Scholar

[16]

L. Schwartz and S. Garoff, Contact angle hysteresis on heterogeneous surfaces,, Langmuir, 1 (1985), 219.  doi: 10.1021/la00062a007.  Google Scholar

[17]

L. Schwartz and S. Garoff, Contact angle hysteresis and the shape of the 3-phase line,, J. Colloid Interf. Sci., 106 (1985), 422.  doi: 10.1016/S0021-9797(85)80016-3.  Google Scholar

[18]

P. S. Swain and R. Lipowsky, Contact angles on heterogeneous surfaces: A new look at Cassie's and Wenzel's laws,, Langmuir, 14 (1998), 6772.  doi: 10.1021/la980602k.  Google Scholar

[19]

X. P. Wang, T. Z. Qian and P. Sheng, Moving contact line on chemically patterned surfaces,, J. Fluid Mech., 605 (2008), 59.  doi: 10.1017/S0022112008001456.  Google Scholar

[20]

R. N. Wenzel, Resistance of solid surfaces to wetting by water,, Ind. Eng. Chem., 28 (1936), 988.  doi: 10.1021/ie50320a024.  Google Scholar

[21]

G. Wolansky and A. Marmur, The actual contact angle on a heterogeneous rough surface in three dimensions,, Langmuir, 14 (1998), 5292.  doi: 10.1021/la960723p.  Google Scholar

[22]

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

[23]

T. Young, An essay on the cohesion of fluids,, Philos. Trans. R. Soc. London, 95 (1805), 65.  doi: 10.1098/rstl.1805.0005.  Google Scholar

[24]

H. Zhong, X. P. Wang, A. Salama and S. Sun, Quasistatic analysis on configuration of two-phase flow in Y-shaped tubes,, Comput. Math. Appl., 68 (2014), 1905.  doi: 10.1016/j.camwa.2014.10.004.  Google Scholar

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