# American Institute of Mathematical Sciences

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 and 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), 237-243. doi: 10.1016/S0021-9797(03)00285-6. [2] S. Brandon and A. Marmur, Simulation of contact angle hysteresis on chemically heterogeneous surfaces, J. Colloid Interf. Sci., 183 (1996), 351-355. doi: 10.1006/jcis.1996.0556. [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), 110-116. doi: 10.1006/jcis.1997.4912. [4] A. B. D. Cassie and S. Baxter, Wettability of porous surfaces, Trans. Faraday Soc., 40 (1944), 546-551. doi: 10.1039/tf9444000546. [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), 4237-4243. doi: 10.1021/la053146q. [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), 1372-1386. doi: 10.1016/j.jcp.2011.10.015. [7] P. G. de Gennes, Wetting: Statics and dynamics, Simple Views on Condensed Matter, 12 (2003), 357-394. doi: 10.1142/9789812564849_0041. [8] M. Iwamatsu, Contact angle hysteresis of cylindrical drops on chemically heterogeneous striped surfaces, J. Colloid Interf. Sci., 297 (2006), 772-777. doi: 10.1016/j.jcis.2005.11.032. [9] J. F. Joanny and P. G. de Gennes, A model for contact angle hysteresis, Simple Views on Condensed Matter, 12 (2003), 457-467. doi: 10.1142/9789812564849_0048. [10] R. Johnson and R. Dettre, Contact-angle hysteresis. 3. study of an idealized heterogeneous surface, J. Phys. Chem., 68 (1964), 1744-1749. doi: 10.1021/j100789a012. [11] H. Kusumaatmaja and J. M. Yeomans, Modeling contact angle hysteresis on chemically patterned and superhydrophobic surfaces, Langmuir, 23 (2007), 6019-6032. doi: 10.1021/la063218t. [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), 1282-1284. doi: 10.1021/la8030045. [13] A. Marmur, Contact-angle hysteresis on heterogeneous smooth surfaces, J. Colloid Interf. Sci., 168 (1994), 40-46. doi: 10.1006/jcis.1994.1391. [14] T. Z. Qian, X. P. Wang and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306. doi: 10.1103/PhysRevE.68.016306. [15] W. Q. Ren, Wetting transition on patterned surfaces: Transition states and energy barriers, Langmuir, 30 (2014), 2879-2885. doi: 10.1021/la404518q. [16] L. Schwartz and S. Garoff, Contact angle hysteresis on heterogeneous surfaces, Langmuir, 1 (1985), 219-230. doi: 10.1021/la00062a007. [17] L. Schwartz and S. Garoff, Contact angle hysteresis and the shape of the 3-phase line, J. Colloid Interf. Sci., 106 (1985), 422-437. doi: 10.1016/S0021-9797(85)80016-3. [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), 6772-6780. doi: 10.1021/la980602k. [19] X. P. Wang, T. Z. Qian and P. Sheng, Moving contact line on chemically patterned surfaces, J. Fluid Mech., 605 (2008), 59-78. doi: 10.1017/S0022112008001456. [20] R. N. Wenzel, Resistance of solid surfaces to wetting by water, Ind. Eng. Chem., 28 (1936), 988-994. doi: 10.1021/ie50320a024. [21] G. Wolansky and A. Marmur, The actual contact angle on a heterogeneous rough surface in three dimensions, Langmuir, 14 (1998), 5292-5297. doi: 10.1021/la960723p. [22] X. Xu and X. P. Wang, Analysis of wetting and contact angle hysteresis on chemically patterned surfaces, SIAM J. Appl. Math., 71 (2011), 1753-1779. doi: 10.1137/110829593. [23] T. Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805), 65-87. doi: 10.1098/rstl.1805.0005. [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), 1905-1914. doi: 10.1016/j.camwa.2014.10.004.

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##### 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), 237-243. doi: 10.1016/S0021-9797(03)00285-6. [2] S. Brandon and A. Marmur, Simulation of contact angle hysteresis on chemically heterogeneous surfaces, J. Colloid Interf. Sci., 183 (1996), 351-355. doi: 10.1006/jcis.1996.0556. [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), 110-116. doi: 10.1006/jcis.1997.4912. [4] A. B. D. Cassie and S. Baxter, Wettability of porous surfaces, Trans. Faraday Soc., 40 (1944), 546-551. doi: 10.1039/tf9444000546. [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), 4237-4243. doi: 10.1021/la053146q. [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), 1372-1386. doi: 10.1016/j.jcp.2011.10.015. [7] P. G. de Gennes, Wetting: Statics and dynamics, Simple Views on Condensed Matter, 12 (2003), 357-394. doi: 10.1142/9789812564849_0041. [8] M. Iwamatsu, Contact angle hysteresis of cylindrical drops on chemically heterogeneous striped surfaces, J. Colloid Interf. Sci., 297 (2006), 772-777. doi: 10.1016/j.jcis.2005.11.032. [9] J. F. Joanny and P. G. de Gennes, A model for contact angle hysteresis, Simple Views on Condensed Matter, 12 (2003), 457-467. doi: 10.1142/9789812564849_0048. [10] R. Johnson and R. Dettre, Contact-angle hysteresis. 3. study of an idealized heterogeneous surface, J. Phys. Chem., 68 (1964), 1744-1749. doi: 10.1021/j100789a012. [11] H. Kusumaatmaja and J. M. Yeomans, Modeling contact angle hysteresis on chemically patterned and superhydrophobic surfaces, Langmuir, 23 (2007), 6019-6032. doi: 10.1021/la063218t. [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), 1282-1284. doi: 10.1021/la8030045. [13] A. Marmur, Contact-angle hysteresis on heterogeneous smooth surfaces, J. Colloid Interf. Sci., 168 (1994), 40-46. doi: 10.1006/jcis.1994.1391. [14] T. Z. Qian, X. P. Wang and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306. doi: 10.1103/PhysRevE.68.016306. [15] W. Q. Ren, Wetting transition on patterned surfaces: Transition states and energy barriers, Langmuir, 30 (2014), 2879-2885. doi: 10.1021/la404518q. [16] L. Schwartz and S. Garoff, Contact angle hysteresis on heterogeneous surfaces, Langmuir, 1 (1985), 219-230. doi: 10.1021/la00062a007. [17] L. Schwartz and S. Garoff, Contact angle hysteresis and the shape of the 3-phase line, J. Colloid Interf. Sci., 106 (1985), 422-437. doi: 10.1016/S0021-9797(85)80016-3. [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), 6772-6780. doi: 10.1021/la980602k. [19] X. P. Wang, T. Z. Qian and P. Sheng, Moving contact line on chemically patterned surfaces, J. Fluid Mech., 605 (2008), 59-78. doi: 10.1017/S0022112008001456. [20] R. N. Wenzel, Resistance of solid surfaces to wetting by water, Ind. Eng. Chem., 28 (1936), 988-994. doi: 10.1021/ie50320a024. [21] G. Wolansky and A. Marmur, The actual contact angle on a heterogeneous rough surface in three dimensions, Langmuir, 14 (1998), 5292-5297. doi: 10.1021/la960723p. [22] X. Xu and X. P. Wang, Analysis of wetting and contact angle hysteresis on chemically patterned surfaces, SIAM J. Appl. Math., 71 (2011), 1753-1779. doi: 10.1137/110829593. [23] T. Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805), 65-87. doi: 10.1098/rstl.1805.0005. [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), 1905-1914. doi: 10.1016/j.camwa.2014.10.004.
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