Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-30T11:43:42.118Z Has data issue: false hasContentIssue false
  • English
  • Français

Droplet motion on inclined heterogeneous substrates

Published online by Cambridge University Press:  14 May 2013

Nikos Savva  and
Serafim Kalliadasis
Nikos Savva
Affiliation:
Cardiff School of Mathematics, Cardiff University, Cardiff CF24 4AG, United Kingdom Department of Chemical Engineering, Imperial College London, London SW7 2AZ, United Kingdom
Serafim Kalliadasis*
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, United Kingdom
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the static and dynamic behaviour of two-dimensional droplets on inclined heterogeneous substrates. We utilize an evolution equation for the droplet thickness based on the long-wave approximation of the Stokes equations in the presence of slip. Through a singular perturbation procedure, evolution equations for the location of the two moving fronts are obtained under the assumption of quasi-static dynamics. The deduced equations, which are verified by direct comparisons with numerical solutions to the governing equation, are scrutinized in a variety of dynamic and equilibrium settings. For example, we demonstrate the possibility for stick–slip dynamics, substrate-induced hysteresis, the uphill motion of the droplet for sufficiently strong chemical gradients and the existence of a critical inclination angle beyond which the droplet can no longer be supported at equilibrium. Where possible, analytical expressions are obtained for various quantities of interest, which are also verified by appropriate numerical experiments.

JFM classification

Interfacial Flows (free surface): Contact lines Drops and Bubbles: Drops Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 725 , 25 June 2013 , pp. 462 - 491
Copyright
©2013 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

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover.Google Scholar
Ben Amar, M., Cummings, L. J. & Pomeau, Y. 2003 Transition of a moving contact line from smooth to angular. Phys. Fluids 15 (10), 2949–2960.CrossRefGoogle Scholar
Berejnov, V. & Thorne, R. E. 2007 Effect of transient pinning on stability of drops sitting on an inclined plane. Phys. Rev. E 75, 066308.CrossRefGoogle Scholar
Berim, G. O. & Ruckenstein, E. 2008 Microscopic calculation of the sticking force for nanodrops on an inclined surface. J. Chem. Phys. 129, 114709.Google Scholar
Bertozzi, A. L. & Brenner, M. P. 1997 Linear stability and transient growth in driven contact lines. Phys. Fluids 9 (3), 530–539.CrossRefGoogle Scholar
Bikerman, J. J. 1950 Sliding drops from surfaces of different roughnesses. J. Colloid Sci. 5, 349–359.Google Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739–805.Google Scholar
Briscoe, B. J. & Galvin, K. P. 1991 The sliding of sessile and pendent droplets. The critical condition. Colloids Surf. 52, 219–229.CrossRefGoogle Scholar
Brown, R. A., Orr, F. M. Jr & Scriven, L. E. 1980 Static drop on an inclined plate: an analysis by the finite element method. J. Colloid Interface Sci. 73 (1), 76–87.CrossRefGoogle Scholar
Buckingham, R., Shearer, M. & Bertozzi, A. 2003 Thin films and the Navier slip condition. SIAM J. Appl. Maths 63, 722–744.Google Scholar
Chaudhury, M. K. & Whitesides, G. M. 1992 How to make water run uphill. Science 256 (5063), 1539–1541.Google Scholar
Das, A. K. & Das, P. K. 2009 Simulation of drop movement over an inclined surface using smoothed particle hydrodynamics. Langmuir 25 (19), 11459–11466.Google Scholar
Diez, J. A., González, A. G. & Kondic, L. 2012 Instability of a transverse liquid rivulet on an inclined plane. Phys. Fluids 24, 032104.Google Scholar
Diez, J. A. & Kondic, L. 2001 Contact line instabilities of thin liquid films. Phys. Rev. Lett. 86 (4), 632–635.CrossRefGoogle ScholarPubMed
Dimitrakopoulos, P. & Higdon, J. J. L. 1999 On the gravitational displacement of three-dimensional fluid droplets from inclined solid surfaces. J. Fluid Mech. 395, 181–209.CrossRefGoogle Scholar
Dupuis, A. & Yeomans, J. M. 2006 Dynamics of sliding drops on superhydrophobic surfaces. Europhys. Lett. 75 (1), 105–111.Google Scholar
Durbin, P. A. 1988 Considerations on the moving contact-line singularity, with application to frictional drag on a slender drop. J. Fluid Mech. 197, 157–169.Google Scholar
Dussan V, E. B. 1979 On the spreading of liquids on solid surfaces: static and dynamic contact lines. Annu. Rev. Fluid Mech. 11, 371–400.CrossRefGoogle Scholar
Dussan V, E. B. 1985 On the ability of drops or bubbles to stick to non-horizontal surfaces of solids. Part 2. Small drops or bubbles having contact angles of arbitrary size. J. Fluid Mech. 151, 1–20.CrossRefGoogle Scholar
Dussan V, E. B. & Chow, R. T.-P. 1983 On the ability of drops or bubbles to stick to non-horizontal surfaces or solids. J. Fluid Mech. 137, 1–29.Google Scholar
ElSherbini, A. I. & Jacobi, A. M. 2004a Critical contact angles for liquid drops on inclined surfaces. Prog. Colloid Polym. Sci. 128, 57–62.Google Scholar
ElSherbini, A. I. & Jacobi, A. M. 2004b Liquid drops on vertical and inclined surfaces I. An experimental study on drop geometry. J. Colloid Interface Sci. 273, 556–565.CrossRefGoogle Scholar
Extrand, C. W. & Kumagai, Y. 1995 Liquid drops on an inclined plane: the relation between contact angles, drop shape and retentive force. J. Colloid Interface Sci. 170, 515–521.Google Scholar
Finn, R. & Shinbrot, M. 1988 The capillary contact angle, II: the inclined plane. Math. Meth. Appl. Sci. 10, 165–196.Google Scholar
Frenkel, Y. I. 1948 On the behavior of liquid drops on a solid surface 1. The sliding of drops on an inclined surface. J. Expl Theor. Phys. 18, 659–668.Google Scholar
Furmidge, C. G. L. 1962 The studies at phase interfaces I. The sliding of liquid drops on solid surfaces and a theory for spray retention. J. Colloid Interface Sci. 17, 309–324.Google Scholar
Greenspan, H. P. 1978 On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84, 125–143.CrossRefGoogle Scholar
Hocking, L. M. 1981 Sliding and spreading of thin two-dimensional drops. Q. J. Mech. Appl. Maths 24, 37–55.CrossRefGoogle Scholar
Hocking, L. M. 1983 The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Maths 36 (1), 55–69.Google Scholar
Hocking, L. M. 1990 Spreading and instability of a viscous fluid sheet. J. Fluid Mech. 211, 373–392.Google Scholar
Hong, S. D., Ha, M. Y. & Balachandar, S. 2009 Static and dynamic contact angles of water droplet on a solid surface using molecular dynamics simulation. J. Colloid Interface Sci. 339, 187–195.CrossRefGoogle ScholarPubMed
Huppert, H. 1982 Flow and instability of a viscous current down a slope. Nature 300, 427–429.Google Scholar
Hyväluoma, J., Koponen, A., Raiskinmäki, P. & Timonen, J. 2007 Droplets on inclined rough surfaces. Eur. Phys. J. E 23, 289–293.CrossRefGoogle ScholarPubMed
Iliev, S. D. 1997 Static drops on an inclined plane: equilibrium modelling and numerical analysis. J. Colloid Interface Sci. 194, 287–300.CrossRefGoogle Scholar
Kalliadasis, S. 2000 Nonlinear instability of a contact line driven by gravity. J. Fluid Mech. 413, 355–378.CrossRefGoogle Scholar
Kim, H.-Y., Lee, H. J. & Kang, B. H. 2002 Sliding of liquid drops down an inclined solid surface. J. Colloid Interface Sci. 247, 372–380.CrossRefGoogle ScholarPubMed
Koh, Y. Y., Lee, Y. C., Gaskell, P. H., Jimack, P. K. & Thompson, H. M. 2009 Droplet migration: quantitative comparisons with experiment. Eur. Phys. J. Special Topics 166, 117–120.CrossRefGoogle Scholar
Kondic, L. 2003 Instabilities in gravity driven flow of thin liquid films. SIAM Rev. 45 (1), 95–115.Google Scholar
Krasovitski, B. & Marmur, A. 2005 Drops down the hill: theoretical study of limiting contact angles and the hysteresis range on a tilted plate. Langmuir 21, 3881–3885.Google Scholar
Larkin, B. K. 1967 Numerical solution of the equation of capillarity. J. Colloid Interface Sci. 23, 305–312.CrossRefGoogle Scholar
Lawal, A. & Brown, R. A. 1982 The stability of inclined sessile drops. J. Colloid Interface Sci. 89 (2), 346–352.CrossRefGoogle Scholar
Le Grand, N., Daerr, A. & Limat, L. 2005 Shape and motion of drops sliding down an inclined plane. J. Fluid Mech. 541, 293–315.CrossRefGoogle Scholar
Limat, L. & Stone, H. A. 2004 Three-dimensional lubrication model of a contact line corner singularity. Europhys. Lett. 65 (3), 365–371.Google Scholar
López, P. G. J., Miksis, M. & Bankoff, S. G. 1997 Inertial effects on contact line instability in the coating of a dry inclined plate. Phys. Fluids 9 (8), 2117–2183.Google Scholar
Macdougall, G. & Ockrent, C. 1942 Surface energy relations in liquid/solid systems. I. The adhension of liquids to solids and a new method of determining the surface tension of liquids. Proc. R. Soc. Lond. A 180, 151–173.Google Scholar
Mahadevan, L. & Pomeau, Y. 1999 Rolling droplets. Phys. Fluids 11 (9), 2449–2453.CrossRefGoogle Scholar
Miwa, M., Nakajima, A., Fujishima, A., Hashimoto, K. & Watanabe, T. 2000 Effects of the surface roughness on sliding angles of water droplets on superhydrophobic surfaces. Langmuir 16, 5754–5760.CrossRefGoogle Scholar
Morita, M., Koga, T., Otsuka, H. & Takahara, A. 2005 Macroscopic wetting anisotropy on the line-patterned surface of fluoroalkylsilane monolayers. Langmuir 21, 911–918.Google Scholar
Pereira, A. & Kalliadasis, S. 2012 Equilibrium gas–liquid–solid contact angle from density-functional theory. J. Fluid Mech. 692, 53–77.Google Scholar
Pierce, E., Carmona, F. J. & Amirfazli, A. 2008 Understanding of sliding and contact angle results in tilted plate experiments. Colloids Surf. A 323, 73–82.CrossRefGoogle Scholar
Podgorski, T., Flesselles, J.-M. & Limat, L. 2001 Corners, cusps and pearls in running drops. Phys. Rev. Lett. 87 (3), 036102.Google Scholar
Popova, L. N. 1983 Non-axisymmetric equilibrium shapes of a drop on a plane. Fluid Dyn. 18 (4), 634–637.Google Scholar
Quéré, D. 1998 Drops at rest on a tilted plane. Langmuir 14, 2213–2216.Google Scholar
Richard, D. & Quéré, D. 1999 Viscous drops rolling on a tilted non-wettable solid. Europhys. Lett. 48 (3), 286–291.CrossRefGoogle Scholar
Rotenberg, Y., Boruvka, L. & Neumann, A. W. 1984 The shape of non-axisymmetric drops on inclined planar surfaces. J. Colloid Interface Sci. 102 (2), 424–434.Google Scholar
Roura, P. & Fort, J. 2001 Equilibrium of drops on inclined hydrophilic surfaces. Phys. Rev. E 64, 011601.Google Scholar
Roura, P. & Fort, J. 2002 Comment on ‘effects of the surface roughness on sliding angles of water droplets on superhydrophobic surfaces’. Langmuir 18, 566–569.CrossRefGoogle Scholar
Savva, N. & Kalliadasis, S. 2009 Two-dimensional droplet spreading over topographical substrates. Phys. Fluids 21, 092102.Google Scholar
Savva, N. & Kalliadasis, S. 2011 Dynamics of moving contact lines: A comparison between slip and precursor film models. Europhys. Lett. 94, 64004.CrossRefGoogle Scholar
Savva, N. & Kalliadasis, S. 2012 Influence of gravity on the spreading of two-dimensional droplets over topographical substrates. J. Engng Maths 73, 3–16.Google Scholar
Savva, N., Kalliadasis, S. & Pavliotis, G. A. 2010 Two-dimensional droplet spreading over random topographical substrates. Phys. Rev. Lett. 104, 084501.CrossRefGoogle ScholarPubMed
Savva, N., Pavliotis, G. A. & Kalliadasis, S. 2011a Contact lines over random topographical substrates. Part 1. Statics. J. Fluid Mech. 672, 358–383.Google Scholar
Savva, N., Pavliotis, G. A. & Kalliadasis, S. 2011b Contact lines over random topographical substrates. Part 2. Dynamics. J. Fluid Mech. 672, 384–410.Google Scholar
Servantie, J. & Müller, M. 2008 Statics and dynamics of a cylindrical droplet under an external body force. J. Chem. Phys. 128, 014709.CrossRefGoogle ScholarPubMed
Sheng, X. & Zhang, J. 2011 Directional motion of water drop on ratchet-like superhydrophobic surfaces. Appl. Surf. Sci. 257, 6811–6816.CrossRefGoogle Scholar
Silvi, N. & Dussan V, E. B. 1985 On the rewetting of an inclined solid surface by a liquid. Phys. Fluids 28 (1), 5–7.Google Scholar
Snoeijer, J. H., Le Grand-Piteira, N., Limat, L., Stone, H. A. & Eggers, J. 2007 Cornered drops and rivulets. Phys. Fluids 19, 042104.CrossRefGoogle Scholar
Snoeijer, J. H., Rio, E., Le Grand, N. & Limat, L. 2005 Self-similar flow and contact line geometry at the rear of cornered drops. Phys. Fluids 17, 072101.Google Scholar
Sommers, A. D. & Jacobi, A. M. 2006 Creating micro-scale surface topology to achieve anisotropic wettability on an aluminum surface. J. Micromech. Microengng 16, 1571–1578.CrossRefGoogle Scholar
Sommers, A. D. & Jacobi, A. M. 2008 Wetting phenomena on micro-grooved aluminum surfaces and modeling of the critical droplet size. J. Colloid Interface Sci. 328, 402–411.Google Scholar
Suzuki, S., Nakajima, A., Tanaka, K., Sakai, M., Hashimoto, A., Yoshida, N., Kameshima, Y. & Okada, K. 2008 Sliding behaviour of water droplets on line-patterned hydrophobic surfaces. Appl. Surf. Sci. 254, 1797–1805.Google Scholar
Tadmor, R., Chaurasia, K., Yadav, P. S., Leh, A., Bahadur, P., Dang, L. & Hoffer, W. R. 2008 Drop retention force as a function of resting time. Langmuir 24, 9370–9374.Google Scholar
Thiele, U., Neuffer, K., Bestehorn, M., Pomeau, Y. & Velarde, M. G. 2002 Sliding drops on an inclined plane. Colloids Surf. A 206, 87–104.CrossRefGoogle Scholar
Thiele, U., Velarde, M. G., Neuffer, K., Bestehorn, M. & Pomeau, Y. 2001 Sliding drops in the diffuse interface model coupled to hydrodynamics. Phys. Rev. E 64, 061601.CrossRefGoogle ScholarPubMed
Troian, S. M., Herbolzheimer, E., Safran, S. A. & Joanny, J. F. 1989 Fingering instabilities of driven spreading films. Europhys. Lett. 19 (1), 25–30.CrossRefGoogle Scholar
Vellingiri, R., Savva, N. & Kalliadasis, S. 2011 Droplet spreading on chemically heterogeneous substrates. Phys. Rev. E 84, 036305.Google Scholar
Yadav, P. S., Bahadur, P., Tadmor, R., Chaurasia, K. & Leh, A. 2008 Drop retention force as a function of drop size. Langmuir 24, 3181–3184.CrossRefGoogle ScholarPubMed
Yoshimitsu, Z., Nakajima, A., Watanabe, T. & Hashimoto, K. 2002 Effects of surface structure on the hydrophobicity and sliding behavior of water droplets. Langmuir 18, 5818–5822.Google Scholar
Zheng, Y., Gao, X. & Jiang, L. 2007 Directional adhesion of superhydrophobic butterfly wings. Soft Matt. 3, 178–182.CrossRefGoogle ScholarPubMed