Hostname: page-component-745bb68f8f-cphqk Total loading time: 0 Render date: 2025-01-12T15:35:19.407Z Has data issue: false hasContentIssue false
  • English
  • Français

The motion, stability and breakup of a stretching liquid bridge with a receding contact line

Published online by Cambridge University Press:  06 January 2011

BIAN QIAN  and
KENNETH S. BREUER
BIAN QIAN
Affiliation:
Division of Engineering, Brown University, Providence, RI 02915, USA
KENNETH S. BREUER*
Affiliation:
Division of Engineering, Brown University, Providence, RI 02915, USA
*
‡Email address for correspondence: kbreuer@brown.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The complex behaviour of drop deposition on a hydrophobic surface is considered by looking at a model problem in which the evolution of a constant-volume liquid bridge is studied as the bridge is stretched. The bridge is pinned with a fixed diameter at the upper contact point, but the contact line at the lower attachment point is free to move on a smooth substrate. Experiments indicate that initially, as the bridge is stretched, the lower contact line slowly retreats inward. However, at a critical radius, the bridge becomes unstable, and the contact line accelerates dramatically, moving inward very quickly. The bridge subsequently pinches off, and a small droplet is left on the substrate. A quasi-static analysis, using the Young–Laplace equation, is used to accurately predict the shape of the bridge during the initial bridge evolution, including the initial onset of the slow contact line retraction. A stability analysis is used to predict the onset of pinch-off, and a one-dimensional dynamical equation, coupled with a Tanner law for the dynamic contact angle, is used to model the rapid pinch-off behaviour. Excellent agreement between numerical predictions and experiments is found throughout the bridge evolution, and the importance of the dynamic contact line model is demonstrated.

JFM classification

Instability: Instability Interfacial Flows (free surface): Liquid bridges
Type
Papers
Information
Journal of Fluid Mechanics , Volume 666 , 10 January 2011 , pp. 554 - 572
Copyright
Copyright © Cambridge University Press 2011

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: Department of Chemistry, Frick Laboratory, Princeton University, Princeton, NJ 08544, USA.

References

REFERENCES

Aarts, D. G. A. L., Lekkerkerker, H. N. W., Guo, H., Wegdam, G. H. & Bonn, D. 2005 Hydrodynamics of droplet coalescence. Phys. Rev. Lett. 95, 164503.CrossRefGoogle ScholarPubMed
Bird, J. C., Mandre, S. & Stone, H. A. 2008 Short-time dynamics of partial wetting. Phys. Rev. Lett. 100, 234501.CrossRefGoogle ScholarPubMed
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739–805.CrossRefGoogle Scholar
Deladi, S., Tas, N. R., Berenschot, J. W., Krijnen, G. J. M., de Boer, M. J., de Boer, J. H., Peter, M. & Elwenspoek, M. C. 2004 Micromachined fountain pen for atomic force microscope-based nanopatterning. Appl. Phys. Lett. 85, 5361–5363.CrossRefGoogle Scholar
Dodds, S., da Silveira Carvalho, M. & Kumar, S. 2009 Stretching and slipping of liquid bridges near plates and cavities. Phys. Fluids 21, 092103.CrossRefGoogle Scholar
Dussan, V. E. B., 1979 On the spreading of liquid on the solid surfaces: static and dynamic contact lines. Annu. Rev. Fluid Mech. 11, 371–400.CrossRefGoogle Scholar
Eggers, J. 1993 Universal pinching of 3D axisymmetric free-surface flow. Phys. Rev. Lett. 71, 3458–3460.CrossRefGoogle ScholarPubMed
Eggers, J. & Dupont, T. F. 1994 Drop formation in a one-dimensional approximation of the Navier–Stokes equation. J. Fluid Mech. 262, 205–221.Google Scholar
Eggers, J., Lister, J. R. & Stone, H. A. 1999 Coalescence of liquid drops. J. Fluid Mech. 401, 293–310.CrossRefGoogle Scholar
Frankel, I. & Weihs, D. 1985 Stability of a capillary jet with linearly increasing axial velocity (with application to shaped charges). J. Fluid Mech. 155, 289–307.CrossRefGoogle Scholar
Gillette, R. D. & Dyson, D. C. 1971 Stability of fluid interfaces of revolution between equal solid circular plates. Chem. Engng J. 2, 44–54.CrossRefGoogle Scholar
Ginger, D. S., Zhang, H. & Mirkin, C. A. 2004 The evolution of dip-pen nanolithography. Angew. Chem. 43, 30–45.CrossRefGoogle ScholarPubMed
Liao, Y. C., Franses, E. I. & Basaran, O. A. 2006 Deformation and breakup of a stretching liquid bridge covered with an insoluble surfactant monolayer. Phys. Fluids 18, 022101.Google Scholar
Lide, D. R. (Ed.) 2010 CRC Handbook of Chemistry and Physics. Internet Version 2010. URL http://hbcpnetbase.com/Google Scholar
Mason, G. C. 1970 An experimental determination of the stable length of cylindrical liquid bubbles. J. Colloid Interface Sci. 32, 172–176.CrossRefGoogle Scholar
Meseguer, J. 1984 Stability of slender, axisymmetric liquid bridges between unequal disks. J. Cryst. Growth 67, 141–143.CrossRefGoogle Scholar
Meseguer, J. & Sanz, A. 1985 Numerical and experimental study of the dynamics of axisymmetric slender liquid bridges. J. Fluid Mech. 153, 83–101.Google Scholar
Moldovan, N., Kim, K.-H. & Espinosa, H. D. 2006 Design and fabrication of a novel microfluidic nanoprobe. J. Microelectromech. Syst. 15, 204–213.Google Scholar
Müller, U. R. & Nicolau, D. V. 2004 Microarray Technology and Its Application. Springer.Google Scholar
Myshkis, A. D., Babskii, V. G., Slobozhanin, N. D., Kopachevskii, L. A. & Tyuptsov, A. D. 1987 Low-Gravity Fluid Mechanics. Springer.CrossRefGoogle Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.CrossRefGoogle Scholar
Panditaratne, J. C. 2003 Deflection of microjets induced by asymmetric heating and related free surface flows with moving contact lines. PhD thesis, Purdue University.Google Scholar
Papageorgiou, D. T. 1995 Analytical description of the breakup of liquid jets. J. Fluid Mech. 301, 109–132.CrossRefGoogle Scholar
Perales, J. M., Meseguer, J. & Martinez, I. 1991 Minimum volume stability limits for axisymmetric liquid bridges subject to steady axial acceleration. J. Cryst. Growth 110, 855–861.Google Scholar
Plateau, J. 1863 Experimental and theoretical researches on the figures of equilibrium of a liquid mass withdrawn from the action of gravity. Annual Report of the Board of Regents of the Smithsonian Institution, p. 207. Washington: Smithsonian Institution.Google Scholar
Qian, B., Loureiro, M., Gagnon, D. A., Tripathi, A. & Breuer, K. S. 2009 Micron-scale droplet deposition on a hydrophobic surface using a retreating syringe. Phys. Rev. Lett. 102, 164502.CrossRefGoogle ScholarPubMed
Rayleigh, Lord 1878 On the instability of jets. Proc. Lond. Math. Soc. 10, 4–13.CrossRefGoogle Scholar
Russo, M. J. & Steen, P. H. 1986 Instability of rotund capillary bridges to general disturbances, experiment and theory. J. Colloid Interface Sci. 113, 154–163.CrossRefGoogle Scholar
Sanz, A. & Martinez, I. 1983 Minimum volume for a liquid bridge between equal disks. J. Colloid Interface Sci. 93, 235–240.CrossRefGoogle Scholar
Slobozhanin, L. A. & Alexander, J. I. D. 1998 Combined effect of disk inequality and axial gravity on axisymmetric liquid bridge stability. Phys. Fluids 10, 2473–2488.CrossRefGoogle Scholar
Slobozhanin, L. A., Alexander, J. I. D. & Resnick, A. H. 1997 Bifurcation of the equilibrium states of a weightless liquid bridge. Phys. Fluids 9, 1893–1905.Google Scholar
Slobozhanin, L. A. & Perales, J. M. 1993 Stability of liquid bridges between equal disks in an axial gravity field. Phys. Fluids A 5, 1305–1314.CrossRefGoogle Scholar
Tanner, L. H. 1979 The spreading of silicone oil drops on horizontal surfaces. J. Phys. D: Appl. Phys. 12, 1473–1484.CrossRefGoogle Scholar
Thoroddesen, 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
Villanueva, W., Sjödahl, J., Stjernström, M., Roeraade, J. & Amberg, G. 2007 Microdroplet deposition under a liquid medium. Langmuir 23, 1171–1177.Google Scholar
Wilson, S. K., Hunt, R. & Duffy, B. R. 2000 The rate of spreading in spin coating. J. Fluid Mech. 413, 65–88.CrossRefGoogle Scholar
Yildirim, O. E. & Basaran, O. A. 2001 Deformation and breakup of stretching bridges of Newtonian and shear-thinning liquids: comparison of one- and two-dimensional models. Chem. Engng Sci. 56, 211–233.Google Scholar
Zhang, X., Padgett, R. S. & Basaran, O. A. 1996 Nonlinear deformation and breakup of stretching liquid bridges. J. Fluid Mech. 329, 207–245.Google Scholar