American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020366

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

Hua Zhong 1, , Xiaolin Fan 2, and  Shuyu Sun 3,,

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  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.

Keywords: Advancing motion, contact angle hysteresis, patterned surfaces, contact line, two-phase flow.
Mathematics Subject Classification: Primary: 76T10, 76D05, 76D45; Secondary: 97M10.
Citation: Hua Zhong, Xiaolin Fan, Shuyu Sun. The effect of surface pattern property on the advancing motion of three-dimensional droplets. Discrete & Continuous Dynamical Systems - B, 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.   Google Scholar

[2]

S. Brandon and A. Marmur, Simulation of contact angle hysteresis on chemically heterogeneous surfaces, J. Colloid Interf. Sci., 183 (1996), 351-355.   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), 110-116.   Google Scholar

[4]

A. B. D. Cassie and S. Baxter, Wettability of porous surfaces, Trans. Faraday Soc., 40 (1944), 546-551.  doi: 10.1039/tf9444000546.  Google Scholar

[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.  Google Scholar

[6]

P. G. de Gennes, Wetting: Statics and dynamics, Rev. Mod. Phys., 57 (1985), 827-863.  doi: 10.1103/RevModPhys.57.827.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

A. Marmur, Contact-angle hysteresis on heterogeneous smooth surfaces, J. Colloid Interf. Sci., 168 (1994), 40-46.  doi: 10.1006/jcis.1994.1391.  Google Scholar

[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.  Google Scholar

[18]

W. Ren, Wetting transition on patterned surfaces: Transition states and energy barriers, Langmuir, 30 (2014), 2879-2885.  doi: 10.1021/la404518q.  Google Scholar

[19]

L. W. Schwartz and S. Garoff, Contact angle hysteresis on heterogeneous surfaces, Langmuir, 1 (1985), 219-230.  doi: 10.1021/la00062a007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

R. N. Wenzel, Resistance of solid surfaces to wetting by water, Ind. Eng. Chem., 28 (1936), 988-994.  doi: 10.1021/ie50320a024.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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), 237-243.   Google Scholar

[2]

S. Brandon and A. Marmur, Simulation of contact angle hysteresis on chemically heterogeneous surfaces, J. Colloid Interf. Sci., 183 (1996), 351-355.   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), 110-116.   Google Scholar

[4]

A. B. D. Cassie and S. Baxter, Wettability of porous surfaces, Trans. Faraday Soc., 40 (1944), 546-551.  doi: 10.1039/tf9444000546.  Google Scholar

[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.  Google Scholar

[6]

P. G. de Gennes, Wetting: Statics and dynamics, Rev. Mod. Phys., 57 (1985), 827-863.  doi: 10.1103/RevModPhys.57.827.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

A. Marmur, Contact-angle hysteresis on heterogeneous smooth surfaces, J. Colloid Interf. Sci., 168 (1994), 40-46.  doi: 10.1006/jcis.1994.1391.  Google Scholar

[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.  Google Scholar

[18]

W. Ren, Wetting transition on patterned surfaces: Transition states and energy barriers, Langmuir, 30 (2014), 2879-2885.  doi: 10.1021/la404518q.  Google Scholar

[19]

L. W. Schwartz and S. Garoff, Contact angle hysteresis on heterogeneous surfaces, Langmuir, 1 (1985), 219-230.  doi: 10.1021/la00062a007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

R. N. Wenzel, Resistance of solid surfaces to wetting by water, Ind. Eng. Chem., 28 (1936), 988-994.  doi: 10.1021/ie50320a024.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Schematic diagram of a flat surface with a ring pattern
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  A spherical cap
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Schematic diagram of physically flat surfaces with square-channel like pattern (upper), circular patches in square arrays (middle) and in regular hexagonal arrays (lower)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Effective advancing angle versus Young's angle of channel with Young's angle of square $ 60^\circ $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Advancing contact line and cubic root of critical volume with $ a:b = 12:5, \theta_a:\theta_b = 60^\circ:100^\circ $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Advancing contact line (part) and cubic root of critical volume with $ a:b = 12:5, \theta_a:\theta_b = 60^\circ:110^\circ $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Advancing contact line and cubic root of critical volume with $ a:b = 12:5, \theta_a:\theta_b = 60^\circ:120^\circ $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Advancing contact line and cubic root of critical volume with $ a:b = 12:12, \theta_a:\theta_b = 60^\circ:110^\circ $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Circle pattern in square arrays and in regular hexagon arrays
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Antonio DeSimone, Natalie Grunewald, Felix Otto. A new model for contact angle hysteresis. Networks & 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 & Continuous Dynamical Systems, 2017, 37 (2) : 1061-1073. doi: 10.3934/dcds.2017044
[3]

Theodore Tachim Medjo. A two-phase flow model with delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3273-3294. doi: 10.3934/dcdsb.2017137
[4]

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 & Applied Analysis, 2015, 14 (5) : 2021-2042. doi: 10.3934/cpaa.2015.14.2021
[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 & 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 & 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 & 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 & Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006
[9]

Barbara Lee Keyfitz, Richard Sanders, Michael Sever. Lack of hyperbolicity in the two-fluid model for two-phase incompressible flow. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 541-563. doi: 10.3934/dcdsb.2003.3.541
[10]

K. Domelevo. Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 591-607. doi: 10.3934/dcdsb.2002.2.591
[11]

Yangyang Qiao, Huanyao Wen, Steinar Evje. Compressible and viscous two-phase flow in porous media based on mixture theory formulation. Networks & Heterogeneous Media, 2019, 14 (3) : 489-536. doi: 10.3934/nhm.2019020
[12]

Feimin Huang, Dehua Wang, Difan Yuan. Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3535-3575. doi: 10.3934/dcds.2019146
[13]

Guochun Wu, Yinghui Zhang. Global analysis of strong solutions for the viscous liquid-gas two-phase flow model in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1411-1429. doi: 10.3934/dcdsb.2018157
[14]

Helmut Abels, Harald Garcke, Josef Weber. Existence of weak solutions for a diffuse interface model for two-phase flow with surfactants. Communications on Pure & Applied Analysis, 2019, 18 (1) : 195-225. doi: 10.3934/cpaa.2019011
[15]

Stefan Berres, Ricardo Ruiz-Baier, Hartmut Schwandt, Elmer M. Tory. An adaptive finite-volume method for a model of two-phase pedestrian flow. Networks & Heterogeneous Media, 2011, 6 (3) : 401-423. doi: 10.3934/nhm.2011.6.401
[16]

Brahim Amaziane, Leonid Pankratov, Andrey Piatnitski. An improved homogenization result for immiscible compressible two-phase flow in porous media. Networks & Heterogeneous Media, 2017, 12 (1) : 147-171. doi: 10.3934/nhm.2017006
[17]

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 & Continuous Dynamical Systems - S, 2012, 5 (1) : 93-113. doi: 10.3934/dcdss.2012.5.93
[18]

Theodore Tachim Medjo. On the convergence of a stochastic 3D globally modified two-phase flow model. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 395-430. doi: 10.3934/dcds.2019016
[19]

G. Deugoué, B. Jidjou Moghomye, T. Tachim Medjo. Approximation of a stochastic two-phase flow model by a splitting-up method. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1135-1170. doi: 10.3934/cpaa.2021010
[20]

Brahim Amaziane, Mladen Jurak, Leonid Pankratov, Anja Vrbaški. Some remarks on the homogenization of immiscible incompressible two-phase flow in double porosity media. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 629-665. doi: 10.3934/dcdsb.2018037

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (32)
  • HTML views (174)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences