Skip to main content
SpringerLink
Log in

The spreading of a viscous microdrop on a solid surface

  • Research Paper
  • Published:
Microfluidics and Nanofluidics Aims and scope Submit manuscript

Abstract

The spreading of a liquid microdrop across a solid surface is examined using the interface formation model. This model allows for variable surface tension at constant temperature and a flow induced Maragoni effect, by incorporating irreversible thermodynamics into the continuum model. The model is solved for small Capillary number and small Reynolds number. This problem has been considered before for much larger drops in Shikhmurzaev (Phys Fluids 9:266, 1997a), which examined the spreading of a drop for ε = τ U CL/R ≪ 1, where U CL is the speed of the moving contact line across the solid surface, τ is the surface tension relaxation time of the viscous liquid, and R is a typical length scale for the size of the drop. This paper extends that work by examining ε = O(1), which will be shown to be the appropriate scaling for very small liquid drops, on the scale of micrometres or less.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  • Ahn K, Link D, Griffiths A, Weitz D (2005) Enzyme–inhibitor assay using microdroplets. Bull Am Phys Soc 50(1):1291

    Google Scholar 

  • Anagnostopoulos CN, Chwalek JM, Delametter CN, Hawkins GA, Jeanmaire DL, Lebens JA, Lopez A, Trauernicht DP (2003) Micro-jet nozzle array for precise droplet metering and steering having increased droplet deflection. In: Proceedings of the 12th international conference on solid state sensors, IEEE, Boston, pp 368–371

  • Anna S, Mayer H (2005) A microfluidic tensiometer. Bull Am Phys Soc 50(1):986

    Google Scholar 

  • Bae SC, Anthony S, Granick S (2005) Stick or slip? Measuring slip lengths with nm resolution. Bull Am Phys Soc 50(1):985

    Google Scholar 

  • Basaran OA (2002) Small-scale free surface flows with breakup: drop formation and emerging applications. AIChE J 48(9):1842–1848

    Article  Google Scholar 

  • Blake TD, Shikhmurzaev YD (2002) Dynamic wetting by liquids of different viscosity. J Colloid Interface Sci 253:196–202

    Article  Google Scholar 

  • Blake TD, Clarke A, Ruschak KJ (1994) Hydrodynamic assist of dynamic wetting. AIChE J 40(2):229–242

    Article  Google Scholar 

  • Blake TD, Bracke M, Shikhmurzaev YD (1999a) Experimental evidence of nonlocal hydrodynamic influence on the dynamic contact angle. Phys Fluids 11(8):1995–2007

    Article  Google Scholar 

  • Blake TD, Clarke A, De Coninck J, de Ruijter M, Voue M (1999b) Droplet spreading: a microscopic approach. Colloid Surf A Phys Chem Eng Asp 149:123–130

    Article  Google Scholar 

  • Blake TD, Decamps C, De Coninck J, de Ruijter M, Voue M (1999c) The dynamics of spreading at the microscopic scale. Colloid Surf A Phys Chem Eng Asp 154:5–11

    Article  Google Scholar 

  • Braun RJ, Murray BT, Boettinger WJ, McFadden GB (1995) Lubrication theory for reactive spreading of a thin drop. Phys Fluids 7:1797–1810

    Article  MATH  Google Scholar 

  • Brenner T, Glatzel T, Zengerle R, Ducree J (2005) Frequency-dependent transversal flow control in centrifugal microfluidics. Lab Chip 5(2):146–150

    Article  Google Scholar 

  • Chatterjee D, Hetayothin B, Wheeler A, King D, Garrell R (2005) Transport of nonconductive and conductive droplets in a parallel plate array. Bull Am Phys Soc 50(1):987

    Google Scholar 

  • Chwalek JM, Trauernicht DP, Delametter CN, Sharma R, Jeanmaire DL, Anagnostopoulos CN, Hawkins GA, Ambravaneswaran B, Panditaratne JC, Basaran OA (2002) A new method for deflecting liquid microjets. Phys Fluids 14(6):37–40

    Article  Google Scholar 

  • Cox RG (1986) The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J Fluid Mech 168:169–194

    Article  MATH  Google Scholar 

  • Daneshbod Y, Sterling JD, Nadim A (2005) Analysis of drop shapes during electrowetting on a dielectric. Bull Am Phys Soc 50(1):987

    Google Scholar 

  • Davies JT, Rideal EK (1963) Interfacial phenomena, 2nd edn. Academic, San diego, ISBN 0-12-206056-3

  • Delametter CN, Trauernicht DP, Chwalek JM (2002) Novel microfluidic jet deflection: significant modeling challenge with great application potential. In: Proceedings of international conference on modeling and simulation of microsystems, NSTI, Puerto Rico, pp 44–47

  • Dupont V, Thome JR (2005) Evaporation in microchannels: influence of the channel diameter on heat transfer. Microfluid Nanofluid 1:119–127

    Article  Google Scholar 

  • Dussan VEB, Davis SH (1974) On the motion of a fluid-fluid interface along a solid surface. J Fluid Mech 65:71–95

    Article  Google Scholar 

  • Eggers J, Evans R (2004) Comment on ’Dynamic wetting by liquids of different viscosity’ by T.D. Blake and Y.D. Shikhmurzaev. J Colloid Int Sci 280:537–538

  • Ehrhard P, Davis SH (1991) Non-isothermal spreading of liquid drops on horizontal plates. J Fluid Mech 229:365–388

    Article  MATH  Google Scholar 

  • Furlani EP (2005a) Thermal modulation and instability of Newtonian liquid microjets. In: Proceedings of NSTI Nanotech 2005, Anaheim, 8–12 May 2005

  • Furlani EP (2005b) Temporal instability of viscous liquid microjets with spatially varying surface tension. J Phys A Math Gen 38:263–276

    Article  MATH  Google Scholar 

  • Furlani EP, Delametter CN, Chwalek JM, Trauernicht D (2001) Surface tension induced instability of viscous liquid jets. In: 4th International conference modeling and simulation of microsystems, April 2001

  • de Gennes PG (1985) Wetting: statics and dynamics. Rev Mod Phys 57:827–863

    Article  Google Scholar 

  • Greenspan HP (1978) On the motion of a small viscous droplet that wets a surface. J Fluid Mech 84:125–143

    Article  MATH  Google Scholar 

  • Haley PJ, Miksis MJ (1991) The effect of the contact line on droplet spreading. J Fluid Mech 223:57–81

    Article  MATH  MathSciNet  Google Scholar 

  • Hardy WB (1919) The spreading of fluids on glass. Philos Mag 38:49–55

    Google Scholar 

  • Jang J, Wereley ST (2005) Pressure distributions of gaseous slip flow in straight and uniform rectangular microchannels. Microfluid Nanofluid 1:41–51

    Article  MATH  Google Scholar 

  • Kerbage C, Ahn K, Hunt T, Westervelt R, Weitz D (2005) Optical detection and magnetic manipulation of drops in microfluidic devices. Bull Am Phys Soc 50(1):1291

    Google Scholar 

  • Krupenkin T, Taylor A, Kolodner P, Pau S, Lyons A, Hodes M (2005) Manipulating liquids on the tunable nanostructured surfaces. Bull Am Phys Soc 50(1):986

    Google Scholar 

  • Loscertales IG, Barrero A, Guerrero I, Curtiju R, Marquez M, Ganan-Calvo AM (2002) Micro/Nano encapsulation via electrified coaxial liquid jets. Science 296:1695

    Article  Google Scholar 

  • Lukiyanov A, Shikhmurzaev YD, King AC (2005) Curtain coating at low Reynolds numbers. Phys Rev E (in press)

  • Morini GL, Lorenzini M, Spiga M (2005) A criterion for experimental validation of slip-flow models for incompressible rarefied gases through microchannels. Microfluid Nanofluid 1:190–196

    Article  Google Scholar 

  • Moseler M, Landman U (2000) Formation, stability and breakup of nanojets. Science 289:1165–1169

    Article  Google Scholar 

  • Mukherjee A, Kandlikar SG (2005) Numerical simulation of growth of a vapor bubble during flow boiling of water in a microchannel. Microfluid Nanofluid 1:137–145

    Article  Google Scholar 

  • Quake S (2005) Biological large scale integration. Bull Am Phys Soc 50(1):652

    Google Scholar 

  • de Ruijter M, Blake TD, De Coninck J (1999a) Dynamic wetting studied by molecular modeling simulations of droplet spreading. Langmuir 15:7836–7847

    Article  Google Scholar 

  • de Ruijter M, Blake TD, Clarke A, De Coninck J (1999b) Droplet spreading: a tool to characterize surfaces at the microscopic scale. J Petrol Sci Eng 24:189–198

    Article  Google Scholar 

  • Shikhmurzaev YD (1993) The moving contact line on a smooth solid surface. Int J Multi phase Flow 19:589–610

    Article  Google Scholar 

  • Shikhmurzaev YD (1994) Mathematical modelling of wetting hydrodynamics. Fluid Dyn Res 13:45–64

    Article  Google Scholar 

  • Shikhmurzaev YD (1996) Dynamic contact angles in gas–liquid–solid systems and flow in vicinity of moving contact line. AIChE J 42:601

    Article  Google Scholar 

  • Shikhmurzaev YD (1997a) Spreading of drops on solid surfaces in a quasi-static regime. Phys Fluids 9:266

    Article  Google Scholar 

  • Shikhmurzaev YD (1997b) Moving contact lines in liquid/liquid/solid systems. J Fluid Mech 334:211–249

    Article  MATH  MathSciNet  Google Scholar 

  • Shikhmurzaev YD (1998) On cusped interfaces. J Fluid Mech 359:313–328

    Article  MATH  MathSciNet  Google Scholar 

  • Shikhmurzaev YD (2000) Coalescence and capillary breakup of liquid volumes. Phys Fluids 12(10):2386–2396

    Article  MathSciNet  Google Scholar 

  • Shikhmurzaev YD (2001) Dynamic wetting: issues resolved and rasied. In: Proceedings of the IUTAM symposium. Free-Surface flows, pp 19–28

  • Shikhmurzaev YD (2002) On metastable regimes of dynamic wetting. J Phys Condens Matter 14:319–330

    Article  Google Scholar 

  • Shikhmurzaev YD (2005a) Capillary breakup of liquid threads: a singularity-free solution. IMA J Appl Math 70:880−907

    Article  MATH  MathSciNet  Google Scholar 

  • Shikhmurzaev YD (2005b) Macroscopic mechanism of rupture of free liquid films. C R Mec 333:205–210

    Google Scholar 

  • Shikhmurzaev YD (2005c) Fluid dynamics with transitions in the topology of the flow domain: breakup of jets and rupture of films. Doklady Phys 50:40–43

    Article  Google Scholar 

  • Shikhmurzaev YD, Blake TD (2004) Response to the comment on [J Colloid Interface Sci 253 (2002): 196] by J. Eggers and R. Evans. J Colloid Interface Sci 280:539–541

    Google Scholar 

  • Sirringhaus H, Kawase T, Friend RH, Shimoda T, Inbasekaran M, Wu W, Woo EP (2000) High resolution inkjet printing of all-polymer transistor circuits. Science 290:2123

    Article  Google Scholar 

  • Tabeling P, Willaime H, Barbier V, Menetrier L (2005) Nonlinear phenomena in two- phase flows in microfluidic systems. Bull Am Phys Soc 50(1):1291

    Google Scholar 

  • Tanner L (1979) The spreading of silicone oil drops on horizontal surfaces. J Phys D 12:1473–1484

    Article  Google Scholar 

  • Wang MR, Li ZX (2005) Numerical simulations on performance of MEMS-based nozzles at moderate or low temperatures. Microfluid Nanofluid 1:62–70

    Article  MATH  Google Scholar 

  • Wilson MCT, Summers JL, Shikhmurzaev YD (1999) Hydrodynamic assist of wetting: theoretical results. In: Durst F, Raszillie H (eds) Advances in coating and drying of thin films. Proceedings of the 3rd European Coating Symposium, Shaker-Verlag, Aachen, pp 75–80

  • Wilson MCT, Summers JL, Gaskell PH, Shikhmurzaev YD (2001) Moving contact- line models and the effect of hydrodynamic assist of dynamic wetting. In: Proceedings of the IUTAM symposium on free-surface flows, pp 345–352

  • Wu Z, Nguyen N-T (2005) Convective-diffusive transport in parallel lamination micromixers. Microfluid Nanofluid 1:208–217

    Article  Google Scholar 

  • Xuan XC, Li DQ (2005) Electroosmotic flow in microchannels with arbitrary geometry and arbitrary distribution of wall charge. J Colloid Interface Sci 289(1):291–303

    Article  Google Scholar 

  • Ye C, Xuan X, Li D (2005) Eccentric electrophoretic motion of a sphere in circular cylindrical microchannels. Microfluid Nanofluid 1:234–241

    Article  Google Scholar 

  • Zheng B, Tice J, Ismagilov R (2005) Formation of droplets of different compositions in microfluidic channels and applications to protein crystallization. Bull Am Phys Soc 50(1):1291

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK

    S. P. Decent

Authors
  1. S. P. Decent
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. P. Decent.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Decent, S.P. The spreading of a viscous microdrop on a solid surface. Microfluid Nanofluid 2, 537–549 (2006). https://doi.org/10.1007/s10404-006-0094-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10404-006-0094-7

Keywords

Search

Navigation