Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-05T11:16:55.013Z Has data issue: false hasContentIssue false
  • English
  • Français

Effective boundary conditions for dynamic contact angle hysteresis on chemically inhomogeneous surfaces

Published online by Cambridge University Press:  03 February 2022

Zhen Zhang Open the ORCID record for Zhen Zhang [Opens in a new window]  and
Xianmin Xu Open the ORCID record for Xianmin Xu [Opens in a new window]
Zhen Zhang
Affiliation:
Department of Mathematics, Guangdong Provincial Key Laboratory of Computational Science and Material Design, International Center for Mathematics, National Center for Applied Mathematics (Shenzhen), Southern University of Science and Technology (SUSTech), Shenzhen 518055, PR China
Xianmin Xu*
Affiliation:
NCMIS & LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Beijing 100190, PR China School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, PR China
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Recent experiments (Guan et al., Phys. Rev. Lett., vol. 116, issue 6, 2016a, p. 066102; Guan et al., Phys. Rev. E, vol. 94, issue 4, 2016b, p. 042802) showed many interesting phenomena on dynamic contact angle hysteresis while there is still a lack of complete theoretical interpretation. In this work, we study the time averaging of the apparent advancing and receding contact angles on surfaces with periodic chemical patterns. We first derive a new Cox-type boundary condition for the apparent dynamic contact angle on homogeneous surfaces using the Onsager variational principle. Based on this condition, we propose a reduced model for some typical moving contact line problems on chemically inhomogeneous surfaces in two dimensions. Multiscale expansion and averaging techniques are employed to approximate the model for asymptotically small chemical patterns. We obtain a quantitative formula for the averaged dynamic contact angles. It describes explicitly how the advancing and receding contact angles depend on the velocity and the chemical inhomogeneity of the substrate. The formula is a coarse-graining version of the Cox-type boundary condition on inhomogeneous surfaces. Numerical simulations are presented to validate the analytical results. The numerical results also show that the formula characterizes well the complicated behaviour of dynamic contact angle hysteresis observed in the experiments.

JFM classification

Interfacial Flows (free surface): Contact lines
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 935 , 25 March 2022 , A34
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Blake, T.D. 2006 The physics of moving wetting lines. J. Colloid Interface Sci. 299 (1), 1–13.CrossRefGoogle ScholarPubMed
Blake, T.D., Bracke, M. & Shikhmurzaev, Y.D. 1999 Experimental evidence of nonlocal hydrodynamic influence on the dynamic contact angle. Phys. Fluids 11 (8), 1995–2007.CrossRefGoogle Scholar
Blake, T.D. & De Coninck, J. 2011 Dynamics of wetting and Kramers's theory. Eur. Phys. J. Spec. Top. 197 (1), 249–264.CrossRefGoogle Scholar
Blake, T.D. & Haynes, J.M. 1969 Kinetics of liquid/liquid displacement. J. Colloid Interface Sci. 30 (3), 421–423.CrossRefGoogle Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81 (2), 739–805.CrossRefGoogle Scholar
Cassie, A. & Baxter, S. 1944 Wettability of porous surfaces. Trans. Faraday Soc. 40, 546–551.CrossRefGoogle Scholar
Chan, T.S., Yang, F. & Carlson, A. 2020 Directional spreading of a viscous droplet on a conical fibre. J. Fluid Mech. 894, A26.CrossRefGoogle Scholar
Choi, W., Tuteja, A., Mabry, J.M., Cohen, R.E. & McKinley, G.H. 2009 A modified Cassie–Baxter relationship to explain contact angle hysteresis and anisotropy on non-wetting textured surfaces. J. Colloid Interface Sci. 339, 208–216.CrossRefGoogle ScholarPubMed
Clarke, A. & Stattersfield, E. 2006 Direct evidence supporting nonlocal hydrodynamic influence on the dynamic contact angle. Phys. Fluids 18 (4), 048106.CrossRefGoogle Scholar
Cox, R.G. 1983 The spreading of a liquid on a rough solid surface. J. Fluid Mech. 131, 1–26.CrossRefGoogle Scholar
Cox, R. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169–194.CrossRefGoogle Scholar
Cox, R.G. 1998 Inertial and viscous effects on dynamic contact angles. J. Fluid Mech. 357, 249–278.CrossRefGoogle Scholar
Davis, M.J. & Davis, S.H. 2013 Droplet spreading: theory and experiments. C.R. Phys. 14 (7), 629–635.CrossRefGoogle Scholar
Di, Y., Xu, X. & Doi, M. 2016 Theoretical analysis for meniscus rise of a liquid contained between a flexible film and a solid wall. Europhys. Lett. 113 (3), 36001.CrossRefGoogle Scholar
Doi, M. 2013 Soft Matter Physics. Oxford University Press.CrossRefGoogle Scholar
Doi, M. 2015 Onsager priciple as a tool for approximation. Chin. Phys. B 24, 020505.CrossRefGoogle Scholar
Doi, M., Zhou, J., Di, Y. & Xu, X. 2019 Application of the Onsager–Machlup integral in solving dynamic equations in nonequilibrium systems. Phys. Rev. E 99 (6), 063303.CrossRefGoogle ScholarPubMed
Dupont, J.-B. & Legendre, D. 2010 Numerical simulation of static and sliding drop with contact angle hysteresis. J. Comput. Phys. 229 (7), 2453–2478.CrossRefGoogle Scholar
Dussan, E.B. 1979 On the spreading of liquids on solid surfaces: static and dynamic contact lines. Annu. Rev. Fluid Mech. 11 (1), 371–400.CrossRefGoogle Scholar
Duvivier, D., Blake, T.D. & De Coninck, J. 2013 Toward a predictive theory of wetting dynamics. Langmuir 29 (32), 10132–10140.CrossRefGoogle Scholar
Eggers, J. 2005 Contact line motion for partially wetting fluids. Phys. Rev. E 72 (6), 061605.CrossRefGoogle ScholarPubMed
Eggers, J. & Evans, R. 2004 Comment on “Dynamic wetting by liquids of different viscosity,” by T.D. Blake and Y.D. Shikhmurzaev. J. Colloid Interface Sci. 280 (2), 537–538.CrossRefGoogle Scholar
Extrand, C.W. 2002 Model for contact angles and hysteresis on rough and ultraphobic surfaces. Langmuir 18, 7991–7999.CrossRefGoogle Scholar
Gao, L. & McCarthy, T.J. 2007 How Wenzel and Cassie were wrong. Langmuir 23, 3762–3765.CrossRefGoogle ScholarPubMed
Gao, M. & Wang, X.-P. 2012 A gradient stable scheme for a phase field model for the moving contact line problem. J. Comput. Phys. 231 (4), 1372–1386.CrossRefGoogle Scholar
de Gennes, P.-G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57 (3), 827–863.CrossRefGoogle Scholar
de Gennes, P.G., Brochard-Wyart, F. & Quere, D. 2003 Capillarity and Wetting Phenomena. Springer.Google Scholar
Golestanian, R. 2004 Moving contact lines on heterogeneous substrates. Phil. Trans. R. Soc. Lond. A 362 (1821), 1613–1623.CrossRefGoogle ScholarPubMed
Golestanian, R. & Raphaël, E. 2003 Roughening transition in a moving contact line. Phys. Rev. E 67 (3), 031603.CrossRefGoogle Scholar
Guan, D., Wang, Y., Charlaix, E. & Tong, P. 2016 a Asymmetric and speed-dependent capillary force hysteresis and relaxation of a suddenly stopped moving contact line. Phys. Rev. Lett. 116 (6), 066102.CrossRefGoogle ScholarPubMed
Guan, D., Wang, Y., Charlaix, E. & Tong, P. 2016 b Simultaneous observation of asymmetric speed-dependent capillary force hysteresis and slow relaxation of a suddenly stopped moving contact line. Phys. Rev. E 94 (4), 042802.CrossRefGoogle ScholarPubMed
Guo, S., Gao, M., Xiong, X., Wang, Y.J., Wang, X., Sheng, P. & Tong, P. 2013 Direct measurement of friction of a fluctuating contact line. Phys. Rev. Lett. 111 (2), 026101.CrossRefGoogle ScholarPubMed
Guo, S., Xu, X., Qian, T., Di, Y., Doi, M. & Tong, P. 2019 Onset of thin film meniscus along a fibre. J. Fluid Mech. 865, 650–680.CrossRefGoogle Scholar
Gurtin, M.E., Polignone, D. & Vinals, J. 1996 Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Meth. Appl. Sci. 6 (06), 815–831.CrossRefGoogle Scholar
Haley, P.J. & Miksis, M.J. 1991 The effect of the contact line on droplet spreading. J. Fluid Mech. 223, 57–81.CrossRefGoogle Scholar
Huh, C. & Scriven, L.E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35 (1), 85–101.CrossRefGoogle Scholar
Iliev, S., Pesheva, N. & Iliev, P. 2018 Contact angle hysteresis on doubly periodic smooth rough surfaces in Wenzel's regime: the role of the contact line depinning mechanism. Phys. Rev. E 97 (4), 042801.CrossRefGoogle ScholarPubMed
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 57–88.CrossRefGoogle Scholar
Joanny, J.F. & De Gennes, P.-G. 1984 A model for contact angle hysteresis. J. Chem. Phys. 81 (1), 552–562.CrossRefGoogle Scholar
Joanny, J. & Robbins, M. 1990 Motion of a contact line on a heterogeneous surface. J. Chem. Phys. 92 (2), 32063212.CrossRefGoogle Scholar
Johansson, P. & Hess, B. 2018 Molecular origin of contact line friction in dynamic wetting. Phys. Rev. Fluids 3 (7), 074201.CrossRefGoogle Scholar
Johnson, R.E. Jr. & Dettre, R.H. 1964 Contact angle hysteresis. III. Study of an idealized heterogeneous surfaces. J. Phys. Chem. 68, 1744–1750.CrossRefGoogle Scholar
Lindner-Silwester, T. & Schneider, W. 2005 The moving contact line with weak viscosity effects–an application and evaluation of Shikhmurzaev's model. Acta Mech. 176 (3), 245–258.CrossRefGoogle Scholar
Man, X. & Doi, M. 2016 Ring to mountain transition in deposition pattern of drying droplets. Phys. Rev. Lett. 116 (6), 066101.CrossRefGoogle ScholarPubMed
Marmottant, P. & Villermaux, E. 2004 On spray formation. J. Fluid Mech. 498, 73–111.CrossRefGoogle Scholar
Marmur, A. & Bittoun, E. 2009 When Wenzel and Cassie are right: reconciling local and global considerations. Langmuir 25, 1277–1281.CrossRefGoogle ScholarPubMed
Miskis, M.J. & Davis, S.H. 1994 Slip over rough and coated surfaces. J. Fluid Mech. 273, 125–139.Google Scholar
Mohammad Karim, A., Davis, S.H. & Kavehpour, H.P. 2016 Forced versus spontaneous spreading of liquids. Langmuir 32 (40), 10153–10158.CrossRefGoogle ScholarPubMed
Neumann, A.W. & Good, R.J. 1972 Thermodynamics of contact angles: I. Heterogeneous solid surfaces. J. Colloid Interface Sci. 38, 341–358.CrossRefGoogle Scholar
Nold, A., Sibley, D.N., Goddard, B.D. & Kalliadasis, S. 2014 Fluid structure in the immediate vicinity of an equilibrium three-phase contact line and assessment of disjoining pressure models using density functional theory. Phys. Fluids 26 (7), 072001.CrossRefGoogle Scholar
Pavliotis, G.A. & Stuart, A.M. 2008 Multiscale Methods: Averaging and Homogenization. Springer.Google Scholar
Pismen, L.M. 2002 Mesoscopic hydrodynamics of contact line motion. Colloids Surf. A 206 (1), 11–30.CrossRefGoogle Scholar
Pismen, L.M. & Pomeau, Y. 2000 Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics. Phys. Rev. E 62 (2), 2480–2492.CrossRefGoogle ScholarPubMed
Prabhala, B., Panchagnula, M.V. & Vedantam, S. 2013 Three-dimensional equilibrium shapes of drops on hysteretic surfaces. Colloid Polym. Sci. 291, 279–289.CrossRefGoogle Scholar
Priest, C., Sedev, R. & Ralston, J. 2007 Asymmetric wetting hysteresis on chemical defects. Phys. Rev. Lett. 99 (2), 026103.CrossRefGoogle ScholarPubMed
Qian, T., Wang, X.P. & Sheng, P. 2003 Molecular scale contact line hydrodynamics of immiscible flows. Phys. Rev. E 68, 016306.CrossRefGoogle ScholarPubMed
Qian, T., Wang, X.P. & Sheng, P. 2006 A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 564, 333–360.CrossRefGoogle Scholar
Raj, R., Enright, R., Zhu, Y., Adera, S. & Wang, E.N. 2012 Unified model for contact angle hysteresis on heterogeneous and superhydrophobic surfaces. Langmuir 28, 15777–15788.CrossRefGoogle ScholarPubMed
Raphael, E. & de Gennes, P.-G. 1989 Dynamics of wetting with nonideal surfaces. The single defect problem. J. Chem. Phys. 90 (12), 7577–7584.CrossRefGoogle Scholar
Ren, W., Trinh, P.H. & E, W. 2015 On the distinguished limits of the Navier slip model of the moving contact line problem. J. Fluid Mech. 772, 107–126.CrossRefGoogle Scholar
Ren, W. & E, W. 2007 Boundary conditions for the moving contact line problem. Phys. Fluids 19 (2), 022101.CrossRefGoogle Scholar
Ren, W. & E, W. 2011 Contact line dynamics on heterogeneous surfaces. Phys. Fluids 23, 072103.CrossRefGoogle Scholar
Renardy, M., Renardy, Y. & Li, J. 2001 Numerical simulation of moving contact line problems using a volume-of-fluid method. J. Comput. Phys. 171 (1), 243–263.CrossRefGoogle Scholar
Savva, N. & Kalliadasis, S. 2011 Dynamics of moving contact lines: a comparison between slip and precursor film models. Europhys. Lett. 94 (6), 64004.CrossRefGoogle Scholar
Schwartz, L.W. & Eley, R. 1998 Simulation of droplet motion on low-energy and heterogeneous surfaces. J. Colloid Interface Sci. 202 (1), 173–188.CrossRefGoogle Scholar
Schwartz, L.W. & Garoff, S. 1985 Contact angle hysteresis on heterogeneous surfaces. Langmuir 1, 219–230.CrossRefGoogle Scholar
Seppecher, P. 1996 Moving contact lines in the Cahn–Hilliard theory. Intl J. Engng Sci. 34 (9), 977–992.CrossRefGoogle Scholar
Seveno, D., Vaillant, A., Rioboo, R., Adao, H., Conti, J. & De Coninck, J. 2009 Dynamics of wetting revisited. Langmuir 25 (22), 13034–13044.CrossRefGoogle ScholarPubMed
Sharma, A. 1993 Relationship of thin film stability and morphology to macroscopic parameters of wetting in the apolar and polar systems. Langmuir 9 (3), 861–869.CrossRefGoogle Scholar
Shikhmurzaev, Y.D. 1993 The moving contact line on a smooth solid surface. Intl J. Multiphase Flow 19 (4), 589–610.CrossRefGoogle Scholar
Shikhmurzaev, Y.D. 2006 Remark on the moving contact line with weak viscosity effects–an application and evaluation of Shikhmurzaev's model by T. Lindner-Silwester and W. Schneider (Acta Mech. 176, 245–258, 2005). Acta Mech. 182 (1), 141–143.CrossRefGoogle Scholar
Shikhmurzaev, Y.D. 2007 Capillary Flows with Forming Interfaces. Chapman and Hall/CRC.CrossRefGoogle Scholar
Shikhmurzaev, Y.D. 2020 Moving contact lines and dynamic contact angles: a litmus test for mathematical models, accomplishments and new challenges. Eur. Phys. J. Spec. Top. 229 (10), 1945–1977.CrossRefGoogle Scholar
Shikhmurzaev, Y.D. & Blake, T.D. 2004 Response to the comment on [J. Colloid Interface Sci. 253 (2002) 196] by J. Eggers and R. Evans. J. Colloid Interface Sci. 280 (2), 539–541.CrossRefGoogle Scholar
Sibley, D.N., Nold, A. & Kalliadasis, S. 2015 a The asymptotics of the moving contact line:cracking an old nut. J. Fluid Mech. 764, 445–462.CrossRefGoogle Scholar
Sibley, D.N., Nold, A., Savva, N. & Kalliadasis, S. 2013 The contact line behaviour of solid–liquid–gas diffuse-interface models. Phys. Fluids 25 (9), 092111.CrossRefGoogle Scholar
Sibley, D.N., Nold, A., Savva, N. & Kalliadasis, S. 2015 b A comparison of slip, disjoining pressure, and interface formation models for contact line motion through asymptotic analysis of thin two-dimensional droplet spreading. J. Engng Maths 94 (1), 19–41.CrossRefGoogle Scholar
Sibley, D.N., Savva, N. & Kalliadasis, S. 2012 Slip or not slip? A methodical examination of the interface formation model using two-dimensional droplet spreading on a horizontal planar substrate as a prototype system. Phys. Fluids 24 (8), 082105.CrossRefGoogle Scholar
Snoeijer, J.H. & Andreotti, B. 2013 Moving contact lines: scales, regimes, and dynamical transitions. Annu. Rev. Fluid Mech. 45, 269–292.CrossRefGoogle Scholar
Spelt, P.D.M. 2005 A level-set approach for simulations of flows with multiple moving contact lines with hysteresis. J. Comput. Phys. 207 (2), 389–404.CrossRefGoogle Scholar
Stoev, K., Ramé, E. & Garoff, S. 1999 Effects of inertia on the hydrodynamics near moving contact lines. Phys. Fluids 11 (11), 3209–3216.CrossRefGoogle Scholar
Sui, Y., Ding, H. & Spelt, P. 2014 Numerical simulations of flows with moving contact lines. Annu. Rev. Fluid Mech. 46, 97–119.CrossRefGoogle Scholar
Sui, Y. & Spelt, P.D.M. 2013 a An efficient computational model for macroscale simulations of moving contact lines. J. Comput. Phys. 242, 37–52.CrossRefGoogle Scholar
Sui, Y. & Spelt, P.D.M. 2013 b Validation and modification of asymptotic analysis of slow and rapid droplet spreading by numerical simulation. J. Fluid Mech. 715, 283–313.CrossRefGoogle Scholar
Tanner, L.H. 1979 The spreading of silicone oil drops on horizontal surfaces. J. Phys. D: Appl. Phys. 12 (9), 1473.CrossRefGoogle Scholar
Voinov, O.V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11 (5), 714–721.CrossRefGoogle Scholar
Wang, X.P., Qian, T. & Sheng, P. 2008 Moving contact line on chemically patterned surfaces. J. Fluid Mech. 605, 59–78.CrossRefGoogle Scholar
E, W. 2011 Principle of Multiscale Modeling. Cambridge University Press.Google Scholar
Wenzel, R.N. 1936 Resistance of solid surfaces to wetting by water. Ind. Engng Chem. 28, 988–994.CrossRefGoogle Scholar
Whyman, G., Bormashenko, E. & Stein, T. 2008 The rigorous derivative of Young, Cassie–Baxter and Wenzel equations and the analysis of the contact angle hysteresis phenomenon. Chem. Phys. Lett. 450, 355–359.CrossRefGoogle Scholar
Wu, C., Lei, S., Qian, T. & Wang, X. 2010 Stick-slip motion of moving contact line on chemically patterned surfaces. Commun. Comput. Phys. 7 (3), 403–422.CrossRefGoogle Scholar
Xu, X. 2016 Modified Wenzel and Cassie equations for wetting on rough surfaces. SIAM J. Appl. Maths 76 (6), 2353–2374.CrossRefGoogle Scholar
Xu, X., Di, Y. & Doi, M. 2016 Variational method for contact line problems in sliding liquids. Phys. Fluids 28, 087101.CrossRefGoogle Scholar
Xu, X., Di, Y. & Yu, H. 2018 Sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines. J. Fluid Mech. 849, 805–833.CrossRefGoogle Scholar
Xu, X. & Wang, X.P. 2010 Derivation of the Wenzel and Cassie equations from a phase field model for two phase flow on rough surface. SIAM J. Appl. Maths 70, 2929–2941.CrossRefGoogle Scholar
Xu, X. & Wang, X.P. 2011 Analysis of wetting and contact angle hysteresis on chemically patterned surfaces. SIAM J. Appl. Maths 71, 1753–1779.CrossRefGoogle Scholar
Xu, X. & Wang, X.P. 2013 The modified Cassie's equation and contact angle hysteresis. Colloid Polym. Sci. 291, 299–306.CrossRefGoogle Scholar
Xu, X. & Wang, X.-P. 2020 Theoretical analysis for dynamic contact angle hysteresis on chemically patterned surfaces. Phys. Fluids 32 (11), 112102.CrossRefGoogle Scholar
Xu, X., Zhao, Y. & Wang, X.-P. 2019 Analysis for contact angle hysteresis on rough surfaces by a phase field model with a relaxed boundary condition. SIAM J. Appl. Maths 79, 2551–2568.CrossRefGoogle Scholar
Yatsyshin, P., Savva, N. & Kalliadasis, S. 2015 Wetting of prototypical one-and two-dimensional systems: thermodynamics and density functional theory. J. Chem. Phys. 142 (3), 034708.CrossRefGoogle ScholarPubMed
Young, T. 1805 An essay on the cohesion of fluids. Phil. Trans. R. Soc. Lond. 95, 65–87.Google Scholar
Yue, P. 2020 Thermodynamically consistent phase-field modelling of contact angle hysteresis. J. Fluid Mech. 899, A15.CrossRefGoogle Scholar
Yue, P. & Feng, J.J. 2011 Can diffuse-interface models quantitatively describe moving contact lines? Eur. Phys. J. Spec. Top. 197 (1), 37–46.CrossRefGoogle Scholar
Yue, P., Zhou, C. & Feng, J.J. 2010 Sharp-interface limit of the Cahn–Hilliard model for moving contact lines. J. Fluid Mech. 645, 279–294.CrossRefGoogle Scholar
Zhang, Z. & Ren, W. 2019 Distinguished limits of the Navier slip model for moving contact lines in stokes flow. SIAM J. Appl. Maths 79, 1654–1674.CrossRefGoogle Scholar
Zhang, J. & Yue, P. 2020 A level-set method for moving contact lines with contact angle hysteresis. J. Comput. Phys. 418, 109636.CrossRefGoogle Scholar
Zhou, J. & Doi, M. 2018 Dynamics of viscoelastic filaments based on Onsager principle. Phys. Rev. Fluids 3 (8), 084004.CrossRefGoogle Scholar
Zhou, M.-Y. & Sheng, P. 1990 Dynamics of immiscible-fluid displacement in a capillary tube. Phys. Rev. Lett. 64 (8), 882–885.CrossRefGoogle Scholar