# American Institute of Mathematical Sciences

October  2021, 26(10): 5551-5566. doi: 10.3934/dcdsb.2020366

## The effect of surface pattern property on the advancing motion of three-dimensional droplets

 1 School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China 2 School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China 3 Computational Transport Phenomena Laboratory, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

* Corresponding author

Received  December 2017 Revised  October 2020 Published  October 2021 Early access  December 2020

We investigate numerically the advancing motion of 3D droplets spreading on physically flat chemically heterogeneous surfaces with periodic structures. We use the Navier-Stokes-Cahn-Hilliard equations with the generalized Navier boundary conditions to model the motion of droplets. Based on a convex splitting scheme, we have done numerical simulations and compared the results between different surface patterns quantitatively. We study the effect of pattern property on the advancing motion of three phase contact lines, the critical volume at the contact line jump and the effective advancing angles. By increasing the volume of droplet slowly on heterogeneous surfaces with different pattern property, we find that the advancing contact line approaches an equiangular octagon for the patterned surface with periodic squares separated by channels and approaches a regular hexagon for the patterned surface with periodic circles in regular hexagonal arrays. The shape of three-phase contact line is much more determined by the macro structure of the pattern than the micro structure of the pattern in each period.

Citation: Hua Zhong, Xiaolin Fan, Shuyu Sun. The effect of surface pattern property on the advancing motion of three-dimensional droplets. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5551-5566. doi: 10.3934/dcdsb.2020366
##### 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. [2] S. Brandon and A. Marmur, Simulation of contact angle hysteresis on chemically heterogeneous surfaces, J. Colloid Interf. Sci., 183 (1996), 351-355. [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. [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.-P. Baland and W. C. Carter, Numerical analysis of the shapes and energies of droplets on micropatterned substrates, Langmuir, 22 (2006), 4237-4243.  doi: 10.1021/la053146q. [6] P. G. de Gennes, Wetting: Statics and dynamics, Rev. Mod. Phys., 57 (1985), 827-863.  doi: 10.1103/RevModPhys.57.827. [7] 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. [8] H. Gouin, The wetting problem of fluids on solid surfaces. Ⅰ. The dynamics of contact lines, Contin. Mech. Thermodyn., 15 (2003), 581-596.  doi: 10.1007/s00161-003-0136-2. [9] H. Gouin, The wetting problem of fluids on solid surfaces. Ⅱ. The contact angle hysteresis, Contin. Mech. Thermodyn., 15 (2003), 597-611.  doi: 10.1007/s00161-003-0137-1. [10] 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. [11] J. F. Joanny and P. G. de Gennes, A model for contact angle hysteresis, J. Chem. Phys., 81 (1984), 552-562.  doi: 10.1142/9789812564849_0048. [12] R. E. Johnson and R. H. Dettre, Contact-angle hysteresis. 3. study of an idealized heterogeneous surface, J. Phys. Chem., 68 (1964), 1744-1749.  doi: 10.1021/j100789a012. [13] 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. [14] 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. [15] L. Luo, X.-P. Wang and X.-C. Cai, An efficient finite element method for simulation of droplet spreading on a topologically rough surface, J. Comput. Phys., 349 (2017), 233-252.  doi: 10.1016/j.jcp.2017.08.010. [16] A. Marmur, Contact-angle hysteresis on heterogeneous smooth surfaces, J. Colloid Interf. Sci., 168 (1994), 40-46.  doi: 10.1006/jcis.1994.1391. [17] T. 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. [18] W. Ren, Wetting transition on patterned surfaces: Transition states and energy barriers, Langmuir, 30 (2014), 2879-2885.  doi: 10.1021/la404518q. [19] L. W. Schwartz and S. Garoff, Contact angle hysteresis on heterogeneous surfaces, Langmuir, 1 (1985), 219-230.  doi: 10.1021/la00062a007. [20] L. W. 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. [21] 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. [22] X.-P. Wang, T. Qian and P. Sheng, Moving contact line on chemically patterned surfaces, J. Fluid Mech., 605 (2008), 59-78.  doi: 10.1017/S0022112008001456. [23] R. N. Wenzel, Resistance of solid surfaces to wetting by water, Ind. Eng. Chem., 28 (1936), 988-994.  doi: 10.1021/ie50320a024. [24] 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. [25] X. Xu, Analysis for wetting on rough surfaces by a three-dimensional phase field model, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2839-2850.  doi: 10.3934/dcdsb.2016075. [26] X. Xu and X. Wang, Analysis of wetting and contact angle hysteresis on chemically patterned surfaces, SIAM J. Appl. Math., 71 (2011), 1753-1779.  doi: 10.1137/110829593. [27] T. Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805), 65-87.  doi: 10.1098/rstl.1805.0005. [28] 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. [29] H. Zhong, X.-P. Wang and S. Sun, A numerical study of three-dimensional droplets spreading on chemically patterned surfaces, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2905-2926.  doi: 10.3934/dcdsb.2016079.

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), 237-243. [2] S. Brandon and A. Marmur, Simulation of contact angle hysteresis on chemically heterogeneous surfaces, J. Colloid Interf. Sci., 183 (1996), 351-355. [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. [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.-P. Baland and W. C. Carter, Numerical analysis of the shapes and energies of droplets on micropatterned substrates, Langmuir, 22 (2006), 4237-4243.  doi: 10.1021/la053146q. [6] P. G. de Gennes, Wetting: Statics and dynamics, Rev. Mod. Phys., 57 (1985), 827-863.  doi: 10.1103/RevModPhys.57.827. [7] 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. [8] H. Gouin, The wetting problem of fluids on solid surfaces. Ⅰ. The dynamics of contact lines, Contin. Mech. Thermodyn., 15 (2003), 581-596.  doi: 10.1007/s00161-003-0136-2. [9] H. Gouin, The wetting problem of fluids on solid surfaces. Ⅱ. The contact angle hysteresis, Contin. Mech. Thermodyn., 15 (2003), 597-611.  doi: 10.1007/s00161-003-0137-1. [10] 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. [11] J. F. Joanny and P. G. de Gennes, A model for contact angle hysteresis, J. Chem. Phys., 81 (1984), 552-562.  doi: 10.1142/9789812564849_0048. [12] R. E. Johnson and R. H. Dettre, Contact-angle hysteresis. 3. study of an idealized heterogeneous surface, J. Phys. Chem., 68 (1964), 1744-1749.  doi: 10.1021/j100789a012. [13] 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. [14] 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. [15] L. Luo, X.-P. Wang and X.-C. Cai, An efficient finite element method for simulation of droplet spreading on a topologically rough surface, J. Comput. Phys., 349 (2017), 233-252.  doi: 10.1016/j.jcp.2017.08.010. [16] A. Marmur, Contact-angle hysteresis on heterogeneous smooth surfaces, J. Colloid Interf. Sci., 168 (1994), 40-46.  doi: 10.1006/jcis.1994.1391. [17] T. 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. [18] W. Ren, Wetting transition on patterned surfaces: Transition states and energy barriers, Langmuir, 30 (2014), 2879-2885.  doi: 10.1021/la404518q. [19] L. W. Schwartz and S. Garoff, Contact angle hysteresis on heterogeneous surfaces, Langmuir, 1 (1985), 219-230.  doi: 10.1021/la00062a007. [20] L. W. 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. [21] 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. [22] X.-P. Wang, T. Qian and P. Sheng, Moving contact line on chemically patterned surfaces, J. Fluid Mech., 605 (2008), 59-78.  doi: 10.1017/S0022112008001456. [23] R. N. Wenzel, Resistance of solid surfaces to wetting by water, Ind. Eng. Chem., 28 (1936), 988-994.  doi: 10.1021/ie50320a024. [24] 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. [25] X. Xu, Analysis for wetting on rough surfaces by a three-dimensional phase field model, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2839-2850.  doi: 10.3934/dcdsb.2016075. [26] X. Xu and X. Wang, Analysis of wetting and contact angle hysteresis on chemically patterned surfaces, SIAM J. Appl. Math., 71 (2011), 1753-1779.  doi: 10.1137/110829593. [27] T. Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805), 65-87.  doi: 10.1098/rstl.1805.0005. [28] 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. [29] H. Zhong, X.-P. Wang and S. Sun, A numerical study of three-dimensional droplets spreading on chemically patterned surfaces, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2905-2926.  doi: 10.3934/dcdsb.2016079.
Schematic diagram of a flat surface with a ring pattern
A spherical cap
Schematic diagram of physically flat surfaces with square-channel like pattern (upper), circular patches in square arrays (middle) and in regular hexagonal arrays (lower)
Effective advancing angle versus Young's angle of channel with Young's angle of square $60^\circ$
Advancing contact line and cubic root of critical volume with $a:b = 12:5, \theta_a:\theta_b = 60^\circ:100^\circ$
Advancing contact line (part) and cubic root of critical volume with $a:b = 12:5, \theta_a:\theta_b = 60^\circ:110^\circ$
Advancing contact line and cubic root of critical volume with $a:b = 12:5, \theta_a:\theta_b = 60^\circ:120^\circ$
Advancing contact line and cubic root of critical volume with $a:b = 12:12, \theta_a:\theta_b = 60^\circ:110^\circ$
Circle pattern in square arrays and in regular hexagon arrays
 [1] Antonio DeSimone, Natalie Grunewald, Felix Otto. A new model for contact angle hysteresis. Networks and Heterogeneous Media, 2007, 2 (2) : 211-225. doi: 10.3934/nhm.2007.2.211 [2] 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 [3] Haiyan Yin, Changjiang Zhu. Convergence rate of solutions toward stationary solutions to a viscous liquid-gas two-phase flow model in a half line. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2021-2042. doi: 10.3934/cpaa.2015.14.2021 [4] Theodore Tachim Medjo. A two-phase flow model with delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3273-3294. doi: 10.3934/dcdsb.2017137 [5] Yi Shi, Kai Bao, Xiao-Ping Wang. 3D adaptive finite element method for a phase field model for the moving contact line problems. Inverse Problems and Imaging, 2013, 7 (3) : 947-959. doi: 10.3934/ipi.2013.7.947 [6] T. Tachim Medjo. Averaging of an homogeneous two-phase flow model with oscillating external forces. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3665-3690. doi: 10.3934/dcds.2012.32.3665 [7] Esther S. Daus, Josipa-Pina Milišić, Nicola Zamponi. Global existence for a two-phase flow model with cross-diffusion. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 957-979. doi: 10.3934/dcdsb.2019198 [8] Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control and Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006 [9] Changyan Li, Hui Li. Well-posedness of the two-phase flow problem in incompressible MHD. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5609-5632. doi: 10.3934/dcds.2021090 [10] Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541 [11] K. Domelevo. Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves. Discrete and Continuous Dynamical Systems - B, 2002, 2 (4) : 591-607. doi: 10.3934/dcdsb.2002.2.591 [12] Yangyang Qiao, Huanyao Wen, Steinar Evje. Compressible and viscous two-phase flow in porous media based on mixture theory formulation. Networks and Heterogeneous Media, 2019, 14 (3) : 489-536. doi: 10.3934/nhm.2019020 [13] Feimin Huang, Dehua Wang, Difan Yuan. Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3535-3575. doi: 10.3934/dcds.2019146 [14] Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157 [15] Helmut Abels, Harald Garcke, Josef Weber. Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants. Communications on Pure and Applied Analysis, 2019, 18 (1) : 195-225. doi: 10.3934/cpaa.2019011 [16] Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks and Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401 [17] Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. An improved homogenization result for immiscible compressible two-phase flow in porous media. Networks and Heterogeneous Media, 2017, 12 (1) : 147-171. doi: 10.3934/nhm.2017006 [18] Marie Henry, Danielle Hilhorst, Robert Eymard. Singular limit of a two-phase flow problem in porous medium as the air viscosity tends to zero. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 93-113. doi: 10.3934/dcdss.2012.5.93 [19] Theodore Tachim Medjo. On the convergence of a stochastic 3D globally modified two-phase flow model. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 395-430. doi: 10.3934/dcds.2019016 [20] G. Deugoué, B. Jidjou Moghomye, T. Tachim Medjo. Approximation of a stochastic two-phase flow model by a splitting-up method. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1135-1170. doi: 10.3934/cpaa.2021010

2021 Impact Factor: 1.497