Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T21:48:28.473Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical simulations of self-propelled jumping upon drop coalescence on non-wetting surfaces

Published online by Cambridge University Press:  02 July 2014

Fangjie Liu ,
Giovanni Ghigliotti ,
James J. Feng Open the ORCID record for James J. Feng [Opens in a new window]  and
Chuan-Hua Chen
Fangjie Liu
Affiliation:
Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA
Giovanni Ghigliotti
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2
James J. Feng*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2 Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC, Canada V6T 1Z3
Chuan-Hua Chen*
Affiliation:
Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708, USA
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Coalescing drops spontaneously jump out of plane on a variety of biological and synthetic superhydrophobic surfaces, with potential applications ranging from self-cleaning materials to self-sustained condensers. To investigate the mechanism of self-propelled jumping, we report three-dimensional phase-field simulations of two identical spherical drops coalescing on a flat surface with a contact angle of 180°. The numerical simulations capture the spontaneous jumping process, which follows the capillary–inertial scaling. The out-of-plane directionality is shown to result from the counter-action of the substrate to the impingement of the liquid bridge between the coalescing drops. A viscous cutoff to the capillary–inertial velocity scaling is identified when the Ohnesorge number of the initial drops is around 0.1, but the corresponding viscous cutoff radius is too small to be tested experimentally. Compared to experiments on both superhydrophobic and Leidenfrost surfaces, our simulations accurately predict the nearly constant jumping velocity of around 0.2 when scaled by the capillary–inertial velocity. By comparing the simulated drop coalescence processes with and without the substrate, we attribute this low non-dimensional velocity to the substrate intercepting only a small fraction of the expanding liquid bridge.

JFM classification

Drops and Bubbles: Breakup/coalescence Mathematical Foundations: Computational methods Drops and Bubbles: Drops and Bubbles
Type
Papers
Information
Journal of Fluid Mechanics , Volume 752 , 10 August 2014 , pp. 39 - 65
Copyright
© 2014 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.)

Footnotes

‡

Present address: Laboratoire de Physique de la Matière Condensée, CNRS UMR 7336, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice CEDEX 2, France.

References

Andrieu, C., Beysens, D. A., Nikolayev, V. S. & Pomeau, Y. 2002 Coalescence of sessile drops. J. Fluid Mech. 453, 427–438.CrossRefGoogle Scholar
Basaran, O. A. 1992 Nonlinear oscillations of viscous liquid drops. J. Fluid Mech. 241, 169–198.Google Scholar
Benilov, E. S. & Vynnycky, M. 2013 Contact lines with a 180° contact angle. J. Fluid Mech. 718, 481–506.Google Scholar
Boreyko, J. B. & Chen, C. H. 2009 Self-propelled dropwise condensate on superhydrophobic surfaces. Phys. Rev. Lett. 103, 184501.CrossRefGoogle ScholarPubMed
Boreyko, J. B. & Chen, C. H. 2010 Self-propelled jumping drops on superhydrophobic surfaces. Phys. Fluids 22, 091110.Google Scholar
Boreyko, J. B. & Chen, C. H. 2013 Vapor chambers with jumping-drop liquid return from superhydrophobic condensers. Intl J. Heat Mass Transfer 61, 409–418.Google Scholar
Boreyko, J. B. & Collier, C. P. 2013 Delayed frost growth on jumping-drop superhydrophobic surfaces. ACS Nano 7, 1618–1627.Google Scholar
Boreyko, J. B., Zhao, Y. & Chen, C. H. 2011 Planar jumping-drop thermal diodes. Appl. Phys. Lett. 99, 234105.Google Scholar
Cahn, J. W. & Hilliard, J. E. 1958 Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267.Google Scholar
Celestini, F., Frisch, T. & Pomeau, Y. 2012 Take off of small Leidenfrost droplets. Phys. Rev. Lett. 109, 034501.CrossRefGoogle ScholarPubMed
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Chen, C. H., Cai, Q., Tsai, C., Chen, C. L., Xiong, G., Yu, Y. & Ren, Z. F. 2007 Dropwise condensation on superhydrophobic surfaces with two-tier roughness. Appl. Phys. Lett. 90, 173108.Google Scholar
Cheng, J., Vandadi, A. & Chen, C. L. 2012 Condensation heat transfer on two-tier superhydrophobic surfaces. Appl. Phys. Lett. 101, 131909.Google Scholar
Clanet, C., Béguin, C., Richard, D. & Quéré, D. 2004 Maximal deformation of an impacting drop. J. Fluid Mech. 517, 199–208.CrossRefGoogle Scholar
Dietz, C., Rykaczewski, K., Fedorov, A. G. & Joshi, Y. 2010 Visualization of droplet departure on a superhydrophobic surface and implications to heat transfer enhancement during dropwise condensation. Appl. Phys. Lett. 97, 033104.Google Scholar
Eggers, J., Lister, J. R. & Stone, H. A. 1999 Coalescence of liquid drops. J. Fluid Mech. 401, 293–310.Google Scholar
Enright, R., Miljkovic, N., Al-Obeidi, A., Thompson, C. V. & Wang, E. N. 2012 Condensation on superhydrophobic surfaces: the role of local energy barriers and structure length scale. Langmuir 28, 14424–14432.CrossRefGoogle ScholarPubMed
Feng, J., Pang, Y., Qin, Z., Ma, R. & Yao, S. 2012 Why condensate drops can spontaneously move away on some superhydrophobic surfaces but not on others. ACS Appl. Mater. Interfaces 4, 6618–6625.Google Scholar
Gao, P. & Feng, J. J. 2009 Enhanced slip on a patterned substrate due to depinning of contact line. Phys. Fluids 21, 102102.Google Scholar
Ghigliotti, G., Zhou, C. & Feng, J. J. 2013 Simulations of the breakup of liquid filaments on a partially wetting solid substrate. Phys. Fluids 25, 072102.CrossRefGoogle Scholar
He, M., Zhou, X., Zeng, X., Cui, D., Zhang, Q., Chen, J., Li, H., Wang, J., Cao, Z., Song, Y. & Jiang, L. 2012 Hierarchically structured porous aluminum surfaces for high-efficient removal of condensed water. Soft Matt. 8, 6680–6683.Google Scholar
Helbig, R., Nickerl, J., Neinhuis, C. & Werner, C. 2011 Smart skin patterns protect springtails. PLoS One 6, e25105.Google Scholar
Hernández-Sánchez, J. F., Lubbers, L. A., Eddi, A. & Snoeijer, J. H. 2012 Symmetric and asymmetric coalescence of drops on a substrate. Phys. Rev. Lett. 109, 184502.Google Scholar
Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85–101.CrossRefGoogle Scholar
Kapur, N. & Gaskell, P. H. 2007 Morphology and dynamics of droplet coalescence on a surface. Phys. Rev. E 75, 056315.CrossRefGoogle ScholarPubMed
Kollera, M. & Grigull, U. 1969 Über das Abspringen von Tropfen bei der Kondensation von Quecksilber (The bouncing off phenomenon of droplets with condensation of mercury). Wärme- und Stoffübertragung (Heat Mass Transfer) 2, 31–35.Google Scholar
Lee, M. W., Kang, D. K., Yoon, S. S. & Yarin, A. L. 2012 Coalescence of two drops on partially wettable substrates. Langmuir 28, 3791–3798.CrossRefGoogle ScholarPubMed
Leidenfrost, J. G. 1756 De Aquae Communis Nonnullis Qualitatibus Tractatus. Johann Straube, Duisburg (translation: 1966 Intl J. Heat Mass Transfer 9, 1153–1166).CrossRefGoogle Scholar
Liu, F., Ghigliotti, G., Feng, J. J. & Chen, C.-H. 2014a Self-propelled jumping upon drop coalescence on Leidenfrost surfaces. J. Fluid Mech. 752, 22–38.Google Scholar
Liu, T. Q., Sun, W., Sun, X. Y. & Ai, H. R. 2012 Mechanism study of condensed drops jumping on super-hydrophobic surfaces. Colloids Surf. A 414, 366–374.Google Scholar
Liu, X., Cheng, P. & Quan, X. 2014b Lattice Boltzmann simulations for self-propelled jumping of droplets after coalescence on a superhydrophobic surface. Intl J. Heat Mass Transfer 73, 195–200.CrossRefGoogle Scholar
Lv, C., Hao, P., Yao, Z., Song, Y., Zhang, X. & He, F. 2013 Condensation and jumping relay of droplets on lotus leaf. Appl. Phys. Lett. 103, 021601.Google Scholar
Menchaca-Rocha, A., Martínez-Dávalos, A., Núñez, R., Popinet, S. & Zaleski, S. 2001 Coalescence of liquid drops by surface tension. Phys. Rev. E 63, 046309.Google Scholar
Mertaniemi, H., Forchheimer, R., Ikkala, O. & Ras, R. H. A. 2012 Rebounding droplet–droplet collisions on superhydrophobic surfaces: from the phenomenon to droplet logic. Adv. Mater. 24, 5738–5743.CrossRefGoogle ScholarPubMed
Miljkovic, N., Enright, R., Nam, Y., Lopez, K., Dou, N., Sack, J. & Wang, E. N. 2013 Jumping-droplet-enhanced condensation on scalable superhydrophobic nanostructured surfaces. Nano Lett. 13, 179–187.Google Scholar
Miljkovic, N., Enright, R. & Wang, E. N. 2012 Effect of droplet morphology on growth dynamics and heat transfer during condensation on superhydrophobic nanostructured surfaces. ACS Nano 6, 1776–1785.CrossRefGoogle ScholarPubMed
Miljkovic, N. & Wang, E. N. 2013 Condensation heat transfer on superhydrophobic surfaces. MRS Bull. 38, 397–406.Google Scholar
Mognetti, B. M., Kusumaatmaja, H. & Yeomans, J. M. 2010 Drop dynamics on hydrophobic and superhydrophobic surfaces. Faraday Discuss. 146, 153–165.Google Scholar
Nam, Y., Kim, H. & Shin, S. 2013 Energy and hydrodynamic analyses of coalescence-induced jumping droplets. Appl. Phys. Lett. 103, 161601.Google Scholar
Nilsson, M. A. & Rothstein, J. P. 2011 The effect of contact angle hysteresis on droplet coalescence and mixing. J. Colloid Interface Sci. 363, 646–654.CrossRefGoogle ScholarPubMed
Orme, M. 1997 Experiments on droplet collisions, bounce, coalescence and disruption. Prog. Energy Combust. Sci. 23, 65–79.Google Scholar
Paulsen, J. D., Burton, J. C. & Nagel, S. R. 2011 Viscous to inertial crossover in liquid drop coalescence. Phys. Rev. Lett. 106, 114501.Google Scholar
Peng, B., Wang, S., Lan, Z., Xu, W., Wen, R. & Ma, X. 2013 Analysis of droplet jumping phenomenon with lattice Boltzmann simulation of droplet coalescence. Appl. Phys. Lett. 102, 151601.CrossRefGoogle Scholar
Qian, J. & Law, C. K. 1997 Regimes of coalescence and separation in droplet collision. J. Fluid Mech. 331, 59–80.CrossRefGoogle Scholar
Quéré, D. 2005 Non-sticking drops. Rep. Prog. Phys. 68, 2495–2532.Google Scholar
Quéré, D. 2013 Leidenfrost dynamics. Annu. Rev. Fluid Mech. 45, 197–215.Google Scholar
Rayleigh, Lord 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. 29, 71–97.Google Scholar
Reid, W. H. 1960 The oscillations of a viscous liquid drop. Q. Appl. Maths 18, 86–89.CrossRefGoogle Scholar
Reyssat, M., Richard, D., Clanet, C. & Quéré, D. 2010 Dynamical superhydrophobicity. Faraday Discuss. 146, 19–33.Google Scholar
Richard, D. & Quéré, D. 2000 Bouncing water drops. Europhys. Lett. 50, 769–775.CrossRefGoogle Scholar
Ristenpart, W. D., McCalla, P. M., Roy, R. V. & Stone, H. A. 2006 Coalescence of spreading droplets on a wettable substrate. Phys. Rev. Lett. 97, 064501.CrossRefGoogle ScholarPubMed
Rykaczewski, K., Osborn, W. A., Chinn, J., Walker, M. L., Scott, J. H. J., Jones, W., Hao, C., Yao, S. & Wang, Z. 2012a How nanorough is rough enough to make a surface superhydrophobic during water condensation? Soft Matt. 8, 8786–8794.Google Scholar
Rykaczewski, K., Paxson, A. T., Anand, S., Chen, X., Wang, Z. & Varanasi, K. K. 2012b Multimode multidrop serial coalescence effects during condensation on hierarchical superhydrophobic surfaces. Langmuir 29, 881–891.CrossRefGoogle Scholar
Sprittles, J. E. & Shikhmurzaev, Y. D. 2012 Coalescence of liquid drops: different models versus experiment. Phys. Fluids 24, 122105.Google Scholar
Thoroddsen, S. T., Takehara, K. & Etoh, T. G. 2005 The coalescence speed of a pendent and a sessile drop. J. Fluid Mech. 527, 85–114.Google Scholar
Trinh, E. & Wang, T. G. 1982 Large-amplitude free and driven drop-shape oscillations: experimental observations. J. Fluid Mech. 122, 315–338.Google Scholar
Tsai, P., Hendrix, M. H. W., Dijkstra, R. R. M., Shui, L. & Lohse, D. 2011 Microscopic structure influencing macroscopic splash at high weber number. Soft Matt. 7, 11325–11333.Google Scholar
Wang, F. C., Yang, F. & Zhao, Y. P. 2011 Size effect on the coalescence-induced self-propelled droplet. Appl. Phys. Lett. 98, 053112.Google Scholar
Watson, J. A., Cribb, B. W., Hu, H. M. & Watson, G. S. 2011 A dual layer hair array of the brown lacewing: repelling water at different length scales. Biophys. J. 100, 1149–1155.Google Scholar
Wisdom, K. M., Watson, J. A., Qu, X., Liu, F., Watson, G. S. & Chen, C. H. 2013 Self-cleaning of superhydrophobic surfaces by self-propelled jumping condensate. Proc. Natl Acad. Sci. USA 110, 7992–7997.CrossRefGoogle ScholarPubMed
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, 293–317.Google Scholar
Yue, P., Zhou, C. & Feng, J. J. 2006a A computational study of the coalescence between a drop and an interface in Newtonian and viscoelastic fluids. Phys. Fluids 18, 102102.Google Scholar
Yue, P., Zhou, C. & Feng, J. J. 2007 Spontaneous shrinkage of drops and mass conservation in phase-field simulations. J. Comput. Phys. 223, 1–9.CrossRefGoogle Scholar
Yue, P., Zhou, C., Feng, J. J., Ollivier-Gooch, C. F. & Hu, H. H. 2006b Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing. J. Comput. Phys. 219, 47–67.Google Scholar
Zhang, Q., He, M., Chen, J., Wang, J., Song, Y. & Jiang, L. 2013 Anti-icing surfaces based on enhanced self-propelled jumping of condensed water microdroplets. Chem. Commun. 49, 4516–4518.Google Scholar
Zhou, C., Yue, P., Feng, J. J., Ollivier-Gooch, C. F. & Hu, H. H. 2010 3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids. J. Comput. Phys. 229, 498–511.CrossRefGoogle Scholar

Liu et al. supplementary movie

Coalescence on the substrate (figure 3a): Oh=0.00375, xz view (duration T*=6).

Download Liu et al. supplementary movie(Video)
Video 1.5 MB

Liu et al. supplementary movie

Coalescence in the air (figure 3b): Oh=0.00375, xz/xy view (duration T*=6).

Download Liu et al. supplementary movie(Video)
Video 1.8 MB

Liu et al. supplementary movie

Coalescence in the air (figure 3b): Oh=0.00375, xz/xy view (duration T*=6).

Download Liu et al. supplementary movie(Video)
Video 3.3 MB

Liu et al. supplementary movie

Coalescence on the substrate (figure 4a): Oh=0.00375, yz view (duration T*=6).

Download Liu et al. supplementary movie(Video)
Video 1.5 MB

Liu et al. supplementary movie

Coalescence on the substrate (figure 4b): Oh=0.00375, xy view (duration T*=6).

Download Liu et al. supplementary movie(Video)
Video 1.5 MB

Liu et al. supplementary movie

Coalescence on the substrate (figure 9a): Oh=0.375, xz view (duration T*=8).

Download Liu et al. supplementary movie(Video)
Video 1.9 MB

Liu et al. supplementary movie

Coalescence in the air (figure 9b): Oh=0.375, xz/xy view (duration T*=8).

Download Liu et al. supplementary movie(Video)
Video 1.7 MB