Skip to main content Accessibility help

Thermodynamically consistent phase-field modelling of contact angle hysteresis

Published online by Cambridge University Press:  20 July 2020

Pengtao Yue
Department of Mathematics, Virginia Tech, Blacksburg, VA24061, USA
E-mail address:


In the phase-field description of moving contact line problems, the two-phase system can be described by free energies, and the constitutive relations can be derived based on the assumption of energy dissipation. In this work we propose a novel boundary condition for contact angle hysteresis by exploring wall energy relaxation, which allows the system to be in non-equilibrium at the contact line. Our method captures pinning, advancing and receding automatically without the explicit knowledge of contact line velocity and contact angle. The microscopic dynamic contact angle is computed as part of the solution instead of being imposed. Furthermore, the formulation satisfies a dissipative energy law, where the dissipation terms all have their physical origin. Based on the energy law, we develop an implicit finite element method that is second order in time. The numerical scheme is proven to be unconditionally energy stable for matched density and zero contact angle hysteresis, and is numerically verified to be energy dissipative for a broader range of parameters. We benchmark our method by computing pinned drops and moving interfaces in the plane Poiseuille flow. When the contact line moves, its dynamics agrees with the Cox theory. In the test case of oscillating drops, the contact line transitions smoothly between pinning, advancing and receding. Our method can be directly applied to three-dimensional problems as demonstrated by the test case of sliding drops on an inclined wall.

JFM Papers
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below.


Abels, H., Garcke, H. & Grün, G. 2012 Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Meth. Appl. Sci. 22 (03), 1150013.CrossRefGoogle Scholar
Afkhami, S., Buongiorno, J., Guion, A., Popinet, S., Saade, Y., Scardovelli, R. & Zaleski, S. 2018 Transition in a numerical model of contact line dynamics and forced dewetting. J. Comput. Phys. 374, 10611093.CrossRefGoogle Scholar
Afkhami, S., Zaleski, S. & Bussmann, M. 2009 A mesh-dependent model for applying dynamic contact angles to VOF simulations. J. Comput. Phys. 228, 53705389.CrossRefGoogle Scholar
Alzetta, G., Arndt, D., Bangerth, W., Boddu, V., Brands, B., Davydov, D., Gassmoeller, R., Heister, T., Heltai, L., Kormann, K., et al. 2018 The deal.II library, version 9.0. J. Numer. Maths 26 (4), 173183.CrossRefGoogle Scholar
Ba, Y., Liu, H., Sun, J. & Zheng, R. 2013 Color-gradient lattice Boltzmann model for simulating droplet motion with contact-angle hysteresis. Phys. Rev. E 88, 043306.CrossRefGoogle ScholarPubMed
Bangerth, W. 2000 Using modern features of C++ for adaptive finite element methods: dimension-independent programming in deal.II. In Proceedings of the 16th IMACS World Congress 2000, Lausanne, Switzerland (ed. M. Deville & R. Owens). International Association for Mathematics and Computers in Simulation.Google Scholar
Bangerth, W., Hartmann, R. & Kanschat, G. 2007 deal.II—a general purpose object oriented finite element library. ACM Trans. Math. Softw. 33 (4), 24.CrossRefGoogle Scholar
Cahn, J. W. 1977 Critical-point wetting. J. Chem. Phys. 66, 36673672.CrossRefGoogle Scholar
Cahn, J. W. & Hilliard, J. E. 1958 Free energy of a nonuniform system. Part 1. Interfacial free energy. J. Chem. Phys. 28, 258267.CrossRefGoogle Scholar
Carlson, A., Do-Quang, M. & Amberg, G. 2009 Modeling of dynamic wettting far from equilibrium. Phys. Fluids 21, 121701.CrossRefGoogle Scholar
Carlson, A., Do-Quang, M. & Amberg, G. 2011 Dissipation in rapid dynamic wetting. J. Fluid Mech. 682, 213240.CrossRefGoogle Scholar
Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169194.CrossRefGoogle Scholar
Davis, T. A. 2004 Algorithm 832: Umfpack v4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30 (2), 196199.CrossRefGoogle Scholar
Ding, H. & Spelt, P. D. M. 2008 Onset of motion of a three-dimensional droplet on a wall in shear flow at moderate Reynolds numbers. J. Fluid Mech. 599, 341362.CrossRefGoogle Scholar
Ding, H., Spelt, P. D. M. & Shu, C. 2007 Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 226 (2), 20782095.CrossRefGoogle Scholar
Dong, S. 2014 An efficient algorithm for incompressible N-phase flows. J. Comput. Phys. 276, 691728.CrossRefGoogle Scholar
Dong, S. 2015 Physical formulation and numerical algorithm for simulating n immiscible incompressible fluids involving general order parameters. J. Comput. Phys. 283, 98128.CrossRefGoogle Scholar
Du, Q. & Nicolaides, R. 1991 Numerical analysis of a continuum model of phase transition. SIAM J. Numer. Anal. 28 (5), 13101322.CrossRefGoogle Scholar
Dupont, J.-B. & Legendre, D. 2010 Numerical simulation of static and sliding drop with contact angle hysteresis. J. Comput. Phys. 229 (7), 24532478.CrossRefGoogle Scholar
Dussan V., E. B. 1979 On the spreading of liquids on solid surfaces: static and dynamic contact lines. Annu. Rev. Fluid Mech. 11, 371400.CrossRefGoogle Scholar
Dussan V., E. B. & Chow, R. T.-P. 1983 On the ability of drops or bubbles to stick to non-horizontal surfaces of solids. J. Fluid Mech. 137, 129.CrossRefGoogle Scholar
Eral, B., ’t Mannetje, D. & Oh, J. 2013 Contact angle hysteresis: a review of fundamentals and applications. Colloid Polym. Sci. 291 (2), 247260.CrossRefGoogle Scholar
Extrand, C. W. 2002 Model for contact angles and hysteresis on rough and ultraphobic surfaces. Langmuir 18 (21), 79917999.CrossRefGoogle Scholar
Extrand, C. W. 2016 Origins of wetting. Langmuir 32 (31), 76977706.CrossRefGoogle ScholarPubMed
Eyre, D. J. 1998 Unconditionally gradient stable time marching the Cahn–Hilliard equation. In Computational and Mathematical Models of Microstructural Evolution (San Francisco, CA, 1998), Materials Research Society Symposium Proceedings, vol. 529, pp. 39–46. Materials Research Society.CrossRefGoogle Scholar
Fernández-Toledano, J.-C., Blake, T., Limat, L. & De Coninck, J. 2019 A molecular-dynamics study of sliding liquid nanodrops: dynamic contact angles and the pearling transition. J. Colloid Interface Sci. 548, 6676.CrossRefGoogle ScholarPubMed
Furmidge, C. 1962 Studies at phase interfaces. I. The sliding of liquid drops on solid surfaces and a theory for spray retention. J. Colloid Sci. 17 (4), 309324.CrossRefGoogle Scholar
Gao, L. & McCarthy, T. J. 2006 Contact angle hysteresis explained. Langmuir 22 (14), 62346237.CrossRefGoogle ScholarPubMed
Gao, M. & Wang, X.-P. 2014 An efficient scheme for a phase field model for the moving contact line problem with variable density and viscosity. J. Comput. Phys. 272, 704718.CrossRefGoogle Scholar
de Gennes, P. G. & Prost, J. 1993 The Physics of Liquid Crystals, 2nd edn. Oxford University Press.Google Scholar
Gomez, H. & van der Zee, K. G. 2017 Computational phase-field modeling. In Encyclopedia of Computational Mechanics, 2nd edn, pp. 135. American Cancer Society.Google Scholar
Guermond, J.-L. & Quartapelle, L. 2000 A projection FEM for variable density incompressible flows. J. Comput. Phys. 165 (1), 167188.CrossRefGoogle Scholar
Guo, Z. & Lin, P. 2015 A thermodynamically consistent phase-field model for two-phase flows with thermocapillary effects. J. Fluid Mech. 766, 226271.CrossRefGoogle Scholar
Guo, Z., Lin, P. & Lowengrub, J. 2014 A numerical method for the quasi-incompressible Cahn–Hilliard-Navier–Stokes equations for variable density flows with a discrete energy law. J.Comput. Phys. 276, 486507.CrossRefGoogle Scholar
Heister, T. & Rapin, G. 2013 Efficient augmented lagrangian-type preconditioning for the Oseen problem using Grad-DIV stabilization. Intl J. Numer. Meth. Fluids 71 (1), 118134.CrossRefGoogle Scholar
Hoffman, R. L. 1975 A study of the advancing interface. J. Colloid Interface Sci. 50, 228241.CrossRefGoogle Scholar
Huang, J.-J., Huang, H. & Wang, X. 2014 Numerical study of drop motion on a surface with stepwise wettability gradient and contact angle hysteresis. Phys. Fluids 26 (6), 062101.CrossRefGoogle Scholar
Iwamatsu, M. 2006 Contact angle hysteresis of cylindrical drops on chemically heterogeneous striped surfaces. J. Colloid Interface Sci. 297 (2), 772777.CrossRefGoogle ScholarPubMed
Jacqmin, D. 1999 Calculation of two-phase Navier–Stokes flows using phase-field modeling. J. Comput. Phys. 155 (1), 96127.CrossRefGoogle Scholar
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 5788.CrossRefGoogle Scholar
Janardan, N. & Panchagnula, M. V. 2014 Effect of the initial conditions on the onset of motion in sessile drops on tilted plates. Colloids Surf. A 456, 238245.CrossRefGoogle Scholar
Khatavkar, V. V., Anderson, P. D., Duineveld, P. C. & Meijer, H. E. H. 2007 a Diffuse-interface modelling of droplet impact. J. Fluid Mech. 581, 97127.CrossRefGoogle Scholar
Khatavkar, V. V., Anderson, P. D. & Meijer, H. E. H. 2007 b Capillary spreading of a droplet in the partially wetting regime using a diffuse-interface model. J. Fluid Mech. 572, 367387.CrossRefGoogle Scholar
Kusumaatmaja, H. & Yeomans, J. M. 2007 Modeling contact angle hysteresis on chemically patterned and superhydrophobic surfaces. Langmuir 23 (11), 60196032.CrossRefGoogle ScholarPubMed
Linder, N., Criscione, A., Roisman, I. V., Marschall, H. & Tropea, C. 2015 3D computation of an incipient motion of a sessile drop on a rigid surface with contact angle hysteresis. Theor. Comput. Fluid Dyn. 29 (5), 373390.CrossRefGoogle Scholar
Liu, H., Ju, Y., Wang, N., Xi, G. & Zhang, Y. 2015 Lattice Boltzmann modeling of contact angle and its hysteresis in two-phase flow with large viscosity difference. Phys. Rev. E 92, 033306.CrossRefGoogle ScholarPubMed
Lowengrub, J. & Truskinovsky, L. 1998 Quasi-incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. Lond. A 454 (1978), 26172654.CrossRefGoogle Scholar
Luo, L., Wang, X.-P. & Cai, X.-C. 2017 An efficient finite element method for simulation of droplet spreading on a topologically rough surface. J. Comput. Phys. 349, 233252.CrossRefGoogle Scholar
Maglio, M. & Legendre, D. 2014 Numerical simulation of sliding drops on an inclined solid surface. In Computational and Experimental Fluid Mechanics with Applications to Physics, Engineering and the Environment (ed. Sigalotti, L. Di G., Klapp, J. & Sira, E.), pp. 4769. Springer International Publishing.CrossRefGoogle Scholar
Makkonen, L. 2017 A thermodynamic model of contact angle hysteresis. J. Chem. Phys. 147 (6), 064703.CrossRefGoogle ScholarPubMed
Metzger, S. 2019 On stable, dissipation reducing splitting schemes for two-phase flow of electrolyte solutions. Numer. Algorithms 80 (4), 13611390.CrossRefGoogle Scholar
Prabhala, B., Panchagnula, M. V. & Vedantam, S. 2013 Three-dimensional equilibrium shapes of drops on hysteretic surfaces. Colloid Polym. Sci. 291, 279289.CrossRefGoogle Scholar
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 Molecular hydrodynamics of the moving contact line in two-phase immiscible flows. Commun. Comput. Phys. 1, 152.Google Scholar
Qian, T., Wang, X.-P. & Sheng, P. 2006 b A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 564, 333360.CrossRefGoogle Scholar
Quéré, D. 2008 Wetting and roughness. Annu. Rev. Mater. Res. 38 (1), 7199.CrossRefGoogle Scholar
Rahimi, P. & Ward, C. A. 2005 Contact angle hysteresis on smooth and homogenous surfaces in gravitational fiel ds. Microgravity Sci. Technol. 16 (1), 231235.CrossRefGoogle Scholar
Ren, W. & E, W. 2007 Boundary conditions for the moving contact line problem. Phys. Fluids 19, 022101.CrossRefGoogle Scholar
Ren, W. & E, W. 2011 a Contact line dynamics on heterogeneous surfaces. Phys. Fluids 23, 072103.CrossRefGoogle Scholar
Ren, W. & E, W. 2011 b Derivation of continuum models for the moving contact line problem based on thermodynamic principles. Commun. Math. Sci. 9, 597606.CrossRefGoogle Scholar
Schleizer, A. D. & Bonnecaze, R. T. 1999 Displacement of a two-dimensional immiscible droplet adhering to a wall in shear and pressure-driven flows. J. Fluid Mech. 383, 2954.CrossRefGoogle Scholar
Shen, J., Xu, J. & Yang, J. 2018 The scalar auxiliary variable (SAV) approach for gradient flows. J.Comput. Phys. 353, 407416.CrossRefGoogle Scholar
Shen, J. & Yang, X. 2010 Numerical approximations of Allen–Cahn and Cahn–Hilliard equations. Discrete Continuous Dyn. Syst. A 28, 16691691.CrossRefGoogle Scholar
Shen, J. & Yang, X. 2015 Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows. SIAM J. Numer. Anal. 53 (1), 279296.CrossRefGoogle Scholar
Shen, J., Yang, X. & Yu, H. 2015 Efficient energy stable numerical schemes for a phase field moving contact line model. J. Comput. Phys. 284, 617630.CrossRefGoogle Scholar
Shin, S., Chergui, J. & Juric, D. 2018 Direct simulation of multiphase flows with modeling of dynamic interface contact angle. Theor. Comput. Fluid Dyn. 32 (5), 655687.CrossRefGoogle Scholar
Soutas-Little, R. W. 1999 Elasticity. Dover.Google Scholar
Spelt, P. D. 2005 A level-set approach for simulations of flows with multiple moving contact lines with hysteresis. J. Comput. Phys. 207 (2), 389404.CrossRefGoogle Scholar
Vedantam, S. & Panchagnula, M. V. 2007 Phase field modeling of hysteresis in sessile drops. Phys. Rev. Lett. 99, 176102.CrossRefGoogle ScholarPubMed
Villanueva, W. & Amberg, G. 2006 Some generic capillary-driven flows. Intl J. Multiphase Flow 32 (9), 10721086.CrossRefGoogle Scholar
Voinov, O. V. 1976 Hydrodynamics of wetting. J. Fluid Mech. 11 (5), 714721 (translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 76–84, September–October 1976).Google Scholar
Wang, L., Huang, H.-B. & Lu, X.-Y. 2013 Scheme for contact angle and its hysteresis in a multiphase lattice Boltzmann method. Phys. Rev. E 87, 013301.CrossRefGoogle Scholar
Whyman, G., Bormashenko, E. & Stein, T. 2008 The rigorous derivation of Young, Cassie–Baxter and Wenzel equations and the analysis of the contact angle hysteresis phenomenon. Chem. Phys. Lett. 450 (4), 355359.CrossRefGoogle Scholar
Wu, S. & Ma, M. 2017 A contact angle hysteresis model based on the fractal structure of contact line. J.Colloid Interface Sci. 505, 9951000.CrossRefGoogle ScholarPubMed
Wylock, C., Pradas, M., Haut, B., Colinet, P. & Kalliadasis, S. 2012 Disorder-induced hysteresis and nonlocality of contact line motion in chemically heterogeneous microchannels. Phys. Fluids 24 (3), 032108.CrossRefGoogle Scholar
Xu, X. & Wang, X. 2011 Analysis of wetting and contact angle hysteresis on chemically patterned surfaces. SIAM J. Appl. Maths 71 (5), 17531779.CrossRefGoogle Scholar
Xu, X. & Wang, X. 2013 The modified Cassie's equation and contact angle hysteresis. Colloid Polym. Sci. 291 (2), 299306.CrossRefGoogle Scholar
Yang, X. 2016 Linear, first and second order and unconditionally energy stable numerical schemes for the phase field model of homopolymer blends. J. Comput. Phys. 302, 509523.Google Scholar
Yu, H. & Yang, X. 2017 Numerical approximations for a phase-field moving contact line model with variable densities and viscosities. J. Comput. Phys. 334, 665686.CrossRefGoogle Scholar
Yue, P. & Feng, J. J. 2011 Wall energy relaxation in the Cahn–Hilliard model for moving contact lines. Phys. Fluids 23, 012106.CrossRefGoogle Scholar
Yue, P., Feng, J. J., Liu, C. & Shen, J. 2004 A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293317.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, 279294.CrossRefGoogle Scholar
Yue, P., Zhou, C., Feng, J. J., Ollivier-Gooch, C. F. & Hu, H. H. 2006 Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing. J. Comput. Phys. 219, 4767.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

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 45
Total number of PDF views: 367 *
View data table for this chart

* Views captured on Cambridge Core between 20th July 2020 - 25th January 2021. This data will be updated every 24 hours.

Hostname: page-component-898fc554b-2qp9q Total loading time: 0.437 Render date: 2021-01-25T21:27:50.022Z Query parameters: { "hasAccess": "0", "openAccess": "0", "isLogged": "0", "lang": "en" } Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false }

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Thermodynamically consistent phase-field modelling of contact angle hysteresis
Available formats

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Thermodynamically consistent phase-field modelling of contact angle hysteresis
Available formats

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Thermodynamically consistent phase-field modelling of contact angle hysteresis
Available formats

Reply to: Submit a response

Your details

Conflicting interests

Do you have any conflicting interests? *