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Dynamics of fluid bridges between a rising capillary tube and a substrate

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Micro/nanolithography is an emerging technique to create micro-/nano-features on substrate. New capillary-based lithography method has been developed to overcome the limitations (e.g., the direct contact with the substrate) of existing lithography techniques including dip-pen nanolithography, nano-imprint lithography, and electron-beam lithography. The understanding of the behavior of the liquid bridge formed between a capillary tube and a substrate is essential for the recently developed capillary-based lithography method that is non-invasive to the substrate. A three-dimensional spectral boundary element method has been employed and modified to describe the dynamics of the liquid bridge. Starting with a steady-state liquid bridge shape, the transient bridge deformation is computed as the capillary tube is being lifted away from the substrate. The motion of the three-phase contact line on the substrate is taken into consideration. The computational results are validated with the experimental findings. The influences of the lifting speed of the capillary, liquid properties, and contact line slip conditions on the bridge dynamics are investigated. We conclude by computations that with a higher lifting speed of the capillary tube, the contact line radius on the substrate and the bridge neck radius reduce in a faster manner; a smaller residual droplet tends to form on the substrate if the viscosity ratio (the bridge liquid versus the surrounding medium) is larger or the substrate is more slippery to the fluids.

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  • Ahn BY, Duoss EB, Motala MJ, Guo X, Park SI, Xiong Y, Yoon J, Nuzzo RG, Rogers JA, Lewis JA (2009) Omnidirectional printing of flexible, stretchable, and spanning silver microelectrodes. Science 323:1590–1593

    Article  Google Scholar 

  • Ambravaneswaran B, Basaran OA (1999) Effects of insoluble surfactants on the nonlinear deformation and breakup of stretching liquid bridges. Phys Fluids 11:997–1015

    Article  MATH  Google Scholar 

  • Bird RB, Stewart WE, Lightfoot EN (2001) Transport phenomena. Wiley, New York

    Google Scholar 

  • Davis AMJ, Frenkel AL (1992) Cylindrical liquid bridges squeezed between parallel plates: exact Stokes flow solutions and hydrodynamic forces. Phys Fluids A 4:1105–1109

    Article  MATH  Google Scholar 

  • Dimitrakopoulos P, Higdon JJL (2003) On the displacement of fluid bridges from solid surfaces in viscous pressure-driven flow. Phys Fluids 15:3255–3258

    Article  Google Scholar 

  • Dodds S, Carvalho MD, Kumar S (2009) Stretching and slipping of liquid bridges near plates and cavities. Phys Fluids 21:092103

    Article  Google Scholar 

  • Eggers J, Dupont TF (1994) Drop formation in a one-dimensional approximation of the Navier–Stokes equation. J Fluid Mech 262:205–221

    Article  MATH  MathSciNet  Google Scholar 

  • Gaudet S, McKinley GH, Stone HA (1996) Extensional deformation of Newtonian liquid bridges. Phys Fluids 8:2568–2579

    Article  MATH  Google Scholar 

  • Gillette RD, Dyson DC (1971) Stability of fluid interfaces of revolution between equal solid circular plates. Chem Eng 2:44–54

    Article  Google Scholar 

  • Hanna G, Barnes WJP (2007) Adhesion and detachment of the toe pads of tree frogs. J Exp Biol 155:103–125

    Google Scholar 

  • Khan MA, Wang Y (2010) Droplet motion in a microconfined shear flow via a three-dimensional spectral boundary element method. Phys Fluids 22:123301

    Article  Google Scholar 

  • King BH, Dimos D, Yang P, Morissette SL (1999) Direct-write fabrication of integrated, multilayer ceramic components. J Electroceram 3:173–178

    Article  Google Scholar 

  • Liao YC, Franses EI, Basaran OA (2006) Deformation and breakup of a stretching liquid bridge covered with an insoluble surfactant monolayer. Phys Fluids 18:022101

    Article  Google Scholar 

  • Lowry BJ, Steen PH (1994) Stabilization of an axisymmetric liquid bridge by viscous flow. Int J Multiph Flow 20:439–443

    Article  MATH  Google Scholar 

  • Lutfurakhmanov A, Loken GK, Schulz DL, Akhatov IS (2010) Capillary-based liquid microdroplet deposition. Appl Phys Lett 97:124107

    Article  Google Scholar 

  • Mason G (1970) An experimental determination of the stable length of cylindrical liquid bubbles. J Colloid Interface Sci 32:172–176

    Article  Google Scholar 

  • Meseguer J, Sanz A (1985) Numerical and experimental study of the dynamics of axisymmetric slender liquid bridges. J Fluid Mech 153:83–101

    Article  Google Scholar 

  • Meseguer J, Espino JL, Perales JM, Laveron-Simavilla A (2003) On the breaking of long, axisymmetric liquid bridges between unequal supporting disks at minimum volume stability limit. Eur J Mech B/Fluids 22:355–368

    Article  MATH  Google Scholar 

  • Muldowney GP, Higdon JJL (1995) A spectral boundary element approach to three-dimensional Stokes flow. J Fluid Mech 298:167–192

    Article  MATH  Google Scholar 

  • Perales JM, Meseguer J, Martinez I (1991) Minimum volume stability limits for axisymmetric liquid bridges subject to steady axial acceleration. J Cryst Growth 110:855–861

    Article  Google Scholar 

  • Plateau J (1863) Annual report of the Board of Regents of the Smithsonian Institution. Government Printing Office, Washington

    Google Scholar 

  • Pozrikidis C (1992) Boundary integral and singularity methods for linearized viscous flow. Cambridge University Press, New York

    Book  MATH  Google Scholar 

  • Qian B, Breuer KS (2011) The motion, stability and breakup of a stretching liquid bridge with a receding contact line. J Fluid Mech 666:554–572

    Article  MATH  Google Scholar 

  • Qian B, Loureiro M, Gagnon DA, Tripathi A, Breuer KS (2009) Micron-scale droplet deposition on a hydrophobic surface using a retreating syringe. Phys Rev Lett 102:164502

    Article  Google Scholar 

  • Qu X, Wang Y (2012) Dynamics of concentric and eccentric compound droplets suspended in extensional flows. Phys Fluids 24:123302

    Article  Google Scholar 

  • Rayleigh L (1878) On the instability of jets. Proc Lond Math Soc 10:4–12

    Article  Google Scholar 

  • Sanz A (1985) The influence of the outer bath in the dynamics of axisymmetric liquid bridges. J Fluid Mech 156:101–140

    Article  MATH  Google Scholar 

  • Slobozhanin LA, Alexander JID (1998) Combined effect of disk inequality and axial gravity on axisymmetric liquid bridge stability. Phys Fluids 10:2473–2487

    Article  Google Scholar 

  • Villanueva W, Sjodahl J, Stjernstrom M, Roeraade J, Amberg G (2007) Microdroplet deposition under a liquid medium. Langmuir 23:1171–1177

    Article  Google Scholar 

  • Vozzi G, Previti A, de Rossi D, Ahluwalia A (2002) Microsyringe-based deposition of two-dimensional and three-dimensional polymer scaffolds with a well-defined geometry for application to tissue engineering. Tissue Eng 8:1089–1098

    Article  Google Scholar 

  • Wang Y, Dimitrakopoulos P (2006) A three-dimensional spectral boundary element algorithm for interfacial dynamics in Stokes flow. Phys Fluids 18:082106

    Article  MathSciNet  Google Scholar 

  • Wang Y, Dimitrakopoulos P (2012) Low-Reynolds-number droplet motion in a square microfluidic channel. Theor Comput Fluid Dyn 26:361–379

    Article  MATH  Google Scholar 

  • Yildirim OE, Basaran OA (2001) Deformation and breakup of stretching bridges of Newtonian and shear-thinning liquids: comparison of one- and two-dimensional models. Chem Eng Sci 56:211–233

    Article  Google Scholar 

  • Zhang X, Padgett RS, Basaran OA (1996) Nonlinear deformation and breakup of stretching liquid bridges. J Fluid Mech 329:207–245

    Article  MATH  Google Scholar 

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This work is supported in part by the Department of Energy under award #DE-FG52-08NA28921 and NDSU Advance FORWARD program sponsored by National Science Foundation HRD-0811239. I.S.A. was also supported by the Grants of the Ministry of Education and Science of the Russian Federation (11.G34.31.0040).

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Correspondence to Yechun Wang.

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Wang, Y., Lutfurakhmanov, A. & Akhatov, I.S. Dynamics of fluid bridges between a rising capillary tube and a substrate. Microfluid Nanofluid 18, 807–818 (2015).

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