Experiments in Fluids

, Volume 48, Issue 5, pp 851–862 | Cite as

An empirically validated model of the pressure within a droplet confined between plates at equilibrium for low Bond numbers

  • M. J. Schertzer
  • S. I. Gubarenko
  • R. Ben Mrad
  • P. E. Sullivan
Research Article

Abstract

An analytical model is presented that describes the equilibrium pressure within a confined droplet for small Bond numbers without prior knowledge of the interface shape. An explicit equation for the pressure was developed as a function of the gap height, surface tension, and contact angle. This equation was verified empirically. The shape of the interface was found based on the pressure predicted by both the proposed model and a model commonly used in electrowetting on dielectric (EWOD) investigations. These shapes were compared against experimentally observed interfaces for aspect ratios between 3.5 and 18. The pressures and shapes predicted by the proposed model were at least an order of magnitude more accurate than those predicted with a more commonly used model. At an aspect ratio of 3.5, the average error in the predicted shape was almost 4%, but decreased below the experimental error at an aspect ratio of 6. An aspect ratio of 15 is required for an EWOD device to split water droplets in air. The error in the model pressure and its predicted interface in this case were approximately 0.3%. The analytical pressure model proposed here can be used to increase the accuracy of models of practical EWOD devices. Better accuracy can be attained for small aspect ratios by iteratively calculating pressure using the model proposed here.

List of symbols

\( g \)

Acceleration due to gravity, \([g] \) = m/s2

\( \bar{h}\,,\,h \)

Half height of the gap, \( [ {\bar{h}}] \) = m

\( N_{B} \)

Bond number

\(\user2{n} \)

Outward facing normal vector

\( \bar{p}\,,\,p \)

Pressure within the droplet, \([ {\bar{p}}] \) = Pa

\( \bar{R} \)

Droplet radius in the horizontal plane, \([ {\bar{R}}] \) = m

\( \bar{r}\,,\,r \)

Droplet radius in the vertical plane, \( \left[ {\bar{r}} \right] \) = m

\( \bar{z}\,,\,z \)

Position on the vertical axis, \([ {\bar{z}}] \) = m

\( \gamma \)

Surface tension between the droplet fluid and the surrounding medium, \([ \gamma] \) = N/m

\( \eta \)

Change in \( \xi \) with respect with \( z \)

\( \theta \)

Angle between tangent to interface and horizontal plane, \( \left[ \theta \right] \) = rad

\( \left. \theta \right|_{ - h} \equiv \theta_{ - } \)

Interface angle at \( z = - h \) (also \( \pi \) minus the contact angle)

\( \bar{\kappa }\,,\,\kappa \)

Total local curvature of the interface, \( \left[ {\bar{\kappa }} \right] \) = 1/m

\( \bar{\xi }\,,\,\xi \)

Horizontal distance between the interface and the contact line, \( \left[ {\bar{\xi }} \right] \) = m

\( \bar{\rho }\,,\,\rho \)

Circular-cylindrical coordinate, \( \left[ {\bar{\rho }} \right] \) = m

\( \rho_{F} \)

Fluid density, \( \left[ {\rho_{F} } \right] \) = kg/m3

\( \varphi \)

Circular-cylindrical coordinate, \( \left[ \varphi \right] \) = rad

\( \bar{\phi }\,,\,\phi \)

Level set function, \( \left[ {\bar{\phi }} \right] \) = m

\( \overline{\nabla }\,,\,\nabla \)

Gradient operator, \( \left[ {\overline{\nabla }} \right] \) = 1/m

Notes

Acknowledgments

The support of the Canadian Foundation for Innovation and Engineering Services Inc. (ESI) and National Sciences and Engineering Research Council (NSERC) of Canada through Discovery grants and PGS D scholarships is greatly appreciated.

References

  1. Acero FJ, Ferrera C, Cabezas MG, Montanero JM (2005) Liquid bridge equilibrium contours between non-circular supports. Micrograv Sci Technol 17:8–20CrossRefGoogle Scholar
  2. Berthier J (2008) Microdrops and digital microfluidics. William Andrew Pub Norwich, NYGoogle Scholar
  3. Brochard F (1989) Motions of droplets on solid surfaces induced by chemical or thermal gradients. Langmuir 5:432–438CrossRefGoogle Scholar
  4. Carter WC (1988) The forces and behavior of fluids constrained by solids. Acta Metall 36:2283–2292CrossRefGoogle Scholar
  5. Chatterjee D, Hetayothin B, Wheeler AR, King D, Garrell RL (2006) Droplet-based microfluidics with non-aqueous solvents and solutions. Lab on a Chip 6:199–206CrossRefGoogle Scholar
  6. Cho S, Moon H, Kim C (2003) Creating, Transporting, Cutting, and Merging liquid droplets by electrowetting-based actuation for digital microfluidic circuits. J Microelectromech Syst 12:70–79CrossRefGoogle Scholar
  7. do Carmo MP (1976) Differential geometry of curves and surfaces. Prentice-Hall, Inc, Englewood CliffsMATHGoogle Scholar
  8. Endo T, Okuyama A, Matsubara Y, Nishi K, Kobayashi M, Yamamura S, Morita Y, Takamura Y, Mizukambi H, Tamiya E (2005) Fluorescence-based assay with enzyme amplification on a micro-flow immunosensor chip for monitoring coplanar polychlorinated biphenyls. Analytica Chimica Acta 53:7–13CrossRefGoogle Scholar
  9. Fan SK, Hashi C, Kim CJ (2003) Manipulation of multiple droplets on NxM grid by cross-reference EWOD driving scheme and pressure-contact packing. In: Proceedings of the IEEE Conference on Micro Electro Mechanical Systems (MEMS ’03), Kyoto, Japan, Jan 2003, pp 694–697Google Scholar
  10. Giardino J, Hertzberg J, Bradley E (2008) A calibration procedure for millimeter-scale stereomicroscopic particle image velocimetry. Exp Fluids 45:1037–1045CrossRefGoogle Scholar
  11. Lai S, Wang S, Luo J, Lee L, Yang S, Madou M (2004) Design of a compact disk-like microfluidic platform for enzyme-lined immunosorbent assay. Anal Chem 76:1832–1837CrossRefGoogle Scholar
  12. Lee S, Lee S (2004) Micro total analysis system (μ-TAS) in biotechnology. Appl Microbiol Biotechnol 64:289–299CrossRefGoogle Scholar
  13. Montanero JM, Cabezas G, Acero J (2002) Theoretical and experimental analysis of the equilibrium contours of liquid bridges of arbitrary shape. Phys Fluids 14:682–693CrossRefMathSciNetGoogle Scholar
  14. Moon P, Spencer DE (1971) Field theory handbook (Including coordinate systems, Differential equations and their solutions), 2nd edn. Springer, New YorkGoogle Scholar
  15. Osher S, Fedkiw R (2003) Level set methods and dynamic implicit surfaces. Springer, New YorkMATHGoogle Scholar
  16. Otsu N (1979) A threshold selection method from grey-level histograms. IEEE Trans Syst Man Cybern SMC-9:62–66Google Scholar
  17. Pollack M (1999) Electrowetting-based microactuation of droplets for digital microfluidics, Ph.D. thesis, Duke University, North CarolinaGoogle Scholar
  18. Ren H, Fair R, Pollack M, Shaughnessy E (2002) Dynamics of electro-wetting droplet transport. Sensors and Actuators B 87:201–206CrossRefGoogle Scholar
  19. Verges MA, Larson MC, Bacou R (2001) Forces and shapes of liquid bridges between circular pads. Exp Mech 41:351–357CrossRefGoogle Scholar
  20. Walker SW, Shapiro B (2006) Modeling the fluid dynamics of electro-wetting on dielectric (EWOD). J Microelectromech Syst 15:986–1000CrossRefGoogle Scholar
  21. Wixforth A, Strobl C, Gauer C, Toegl A, Scriba J, Guttenberg ZV (2004) Acoustic manipulation of small droplets. Anal Bioanal Chem 379:289–991CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • M. J. Schertzer
    • 1
  • S. I. Gubarenko
    • 1
  • R. Ben Mrad
    • 1
  • P. E. Sullivan
    • 1
  1. 1.Department of Mechanical and Industrial EngineeringUniversity of TorontoTorontoCanada

Personalised recommendations