Skip to main content
SpringerLink
Log in

Surfactant-driven motion and splitting of droplets on a substrate

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

A theoretical and computational model is presented to predict the motion of a small sessile liquid droplet, lying on a solid substrate including surfactant effects. The model, as formulated, consists of coupled partial differential equations in space and time, and several auxilliary relationships. The validity of the long-wave, or ‘lubrication’ approximation is assumed. It is shown that there are circumstances where surfactant injection or production will cause the droplet to split into two daughter droplets. It is conjectured that the results are relevant to basic mechanisms involved in biological cell division (cytokinesis). It is also demonstrated that motion of a droplet, analogous to the motility of a cell, can be produced by surfactant addition. Computed examples are given here, in both two and three space dimensions. Approximate energy requirements are also calculated for these processes. These are found to be suitably small.

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

Similar content being viewed by others

References

  1. I.B. Ivanov (ed.), Thin liquid films; Fundamentals and Applications. New York: Marcel-Dekker (1988) 1126 pp.

    Google Scholar 

  2. B. Warburton, Interfacial rheology. Cur. Opinion in Colloid Interf. Sci. 1 (1996) 481–487.

    Article  Google Scholar 

  3. K. Mysels, K. Shinoda and S. Frankel. Soap Films, Studies of their Thinning. New York: Pergamon (1959) 116 pp.

    Google Scholar 

  4. P.L. Evans, L.W. Schwartz and R.V. Roy, A mathematical model for crater defect formation in a drying paint layer. J. Colloid Interf. Sci. 227 (2000) 191–205.

    Google Scholar 

  5. D.E. Weidner, L.W. Schwartz and R.R. Eley, Role of surface tension gradients in correcting coating defects in corners. J. Colloid Interf. Sci. 179 (1996) 66–75.

    Google Scholar 

  6. S.D. Howison, J.A. Moriarty, J.R. Ockendon, E.L. Terrill and S.K. Wilson, A mathematical model for drying paint layers. J. Engng. Math. 32 (1997) 377–394.

    Article  MathSciNet  MATH  Google Scholar 

  7. D.J. Benney, Long waves on liquid films. J. Math. Phys. 45 (1966) 150–155.

    MATH  MathSciNet  Google Scholar 

  8. R.W. Atherton and G.M. Homsy, On the derivation of evolution equations for interfacial waves. Chem. Engng. Comm. 2 (1976) 57–77.

    Google Scholar 

  9. E.O. Tuck and L.W. Schwartz, A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows. SIAM Rev. 32 (1990) 453–469.

    Article  MathSciNet  MATH  Google Scholar 

  10. L.W. Schwartz and H.M. Princen, A theory of extensional viscosity for flowing foam. J. Colloid Interf. Sci. 118 (1987) 201–211.

    Google Scholar 

  11. D.P. Gaver and J.B. Grotber, The dynamics of a localized surfactant on a thin film. J. Fluid Mech. 213 (1990) 127–148.

    ADS  MATH  Google Scholar 

  12. A. De Wit, D. Gallez and C.I. Christov, Nonlinear evolution equations for thin liquid films with insoluble surfactant. Phys. Fluids 6 (1994) 3256–3266.

    Article  ADS  MATH  Google Scholar 

  13. L.W. Schwartz, D.E. Weidner and R.R. Eley, An analysis of the effect of surfactant on the leveling behavior of a thin liquid coating layer. Langmuir 11 (1995) 3690–3693.

    Article  Google Scholar 

  14. J.B. Grotberg and D.P. Gaver, A synopsis of surfactant spreading research. J. Colloid Interf. Sci. 178 (1996) 377–378.

    Google Scholar 

  15. D.W. Peaceman and H.H. Rachford, Jr., The numerical solution of parabolic and elliptic differential equations. SIAM J. 3 (1955) 28–41.

    MathSciNet  MATH  Google Scholar 

  16. S.D. Conte and R.T. Dames, On an alternating direction method for solving the plate problem with mixed boundary conditions. J. Assoc. Comp. Mach. 7 (1960) 264–273.

    MathSciNet  MATH  Google Scholar 

  17. N.N. Yanenko, The Method of Fractional Steps. New York: Springer-Verlag (1971) 160 pp.

    MATH  Google Scholar 

  18. H.P. Greenspan, Dynamics of cell cleavage. J. Theor. Biol. 65 (1977) 79–99.

    Google Scholar 

  19. H.P. Greenspan, On the deformation of a viscous droplet caused by variable surface tension. Stud. Appl. Maths. 57 (1977) 45–58.

    MATH  MathSciNet  Google Scholar 

  20. H.P. Greenspan, On fluid-mechanical simulation of cell division and movement. J. Theor. Biol. 70 (1978) 125–134.

    Article  Google Scholar 

  21. H.P. Greenspan, On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84 (1978) 125–143.

    ADS  MATH  Google Scholar 

  22. R. Rappaport, Cytokinesis in Animal Cells. Cambridge: Cambridge Univ. Press (1996) 386 pp.

    Google Scholar 

  23. A.N. Frumkin, On the phenomena of wetting and sticking of bubbles. Zhurnal Fizicheskoi Khimii 12 (1938) 337 (In Russian).

    Google Scholar 

  24. B.V. Derjaguin. Theory of the capillary condensation and other capillary phenomena taking into account the disjoining effect of long-chain molecular liquid films. Zhurnal Fizicheskoi Khimii 14 (1940) 137 (in Russian).

    Google Scholar 

  25. N.V. Churaev and V.D. Sobolev, Prediction of contact angles on the basis of the Frumkin-Derjaguin approach. Adv. Colloid Interf. Sci. 61 (1995) 1–16.

    Google Scholar 

  26. V.S. Mitlin and N.V. Petviashvili. Nonlinear dynamics of dewetting: Kinetically stable structures. Physics Letters: Part A 192(5–6) (1994) 323–326.

    ADS  Google Scholar 

  27. L.W. Schwartz, Unsteady simulation of viscous thin-layer flows. In: P. Tyvand (ed.), Free Surface Flows with Viscosity. Southampton, 1997: Computational Mechanics Publications, pp. 203–233.

  28. L.W. Schwartz, Hysteretic effects in droplet motions on heterogeneous substrates: Direct nummerical simulation. Langmuir 14 (1998) 3440–3453.

    Article  Google Scholar 

  29. L.W. Schwartz and R.R. Eley, Simulation of droplet motion on low-energy and heterogeneous surfaces. J. Colloid Interf. Sci. 202 (1998) 173–188.

    Google Scholar 

  30. A. Sharma, Equilibrium and dynamics of evaporating or condensing thin fluid domains: thin film stability and heterogeneous nucleation. Langmuir 14 (1998) 4915–4928.

    Article  Google Scholar 

  31. C. Huh and L.E. Scriven, Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid. Interf. Sci. 35 (1971) 85–101.

    Google Scholar 

  32. J. Spek, Oberflachenspannungensdifferenzen als eine Ursache der Zellteilung. Arch. f. Entwick-lungs Mech. 44 (1918) 5–113.

    Google Scholar 

  33. X. He and M. Dembo, Numerical simulation of oil droplet cleavage by surfactant. J. Biomech. Engng. 118 (1996) 201–209.

    Google Scholar 

  34. D. Zimmerman and A. Nir, On the viscous deformation of biological cells under anisotropic surface tension. J. Fluid Mech. 193 (1988) 217–241.

    ADS  MathSciNet  Google Scholar 

  35. S.B. Carter, Haptotaxis and the mechanism of cell motility. Nature 213 (1967) 256–260.

    ADS  Google Scholar 

  36. D. Marsland and J.V. Landau, The mechanisms of cytokinesis. J. Exp. Zool. 125 (1954) 507–539.

    Article  Google Scholar 

  37. L.D. Landau and E.M. Lifshitz, Fluid Mechanics. Oxford: Pergamon Press (1959) 536 pp.

    Google Scholar 

  38. V.G. Levich, Physiochemical Hydrodynamics. Englewood Cliffs: Prentice-Hall, Inc. (1962) 700 pp.

    Google Scholar 

  39. H.A. Stone, A simple derivation of the time-dependent convection-diffusion equation for surfactant transport along a deforming interface. Phys. Fluids (A) 2 (1990) 111.

    ADS  Google Scholar 

  40. H. Wong, D. Rumschnitzki and C. Maldarelli, On the sufactant mass balance at a deforming fluid interface. Phys. Fluids 8 (1996) 3203–3204.

    Article  ADS  MATH  Google Scholar 

  41. L.W. Schwartz, R.A. Cairncross and D.E. Weidner, Anomalous behavior during leveling of thin coating layers with surfactant. Phys. of Fluids 8 (1996) 1693–1695.

    Article  ADS  MATH  Google Scholar 

  42. M.H. Eres, D.E. Weidner and L.W. Schwartz, Three-dimensional direct numerical simulation of surface-tension-gradient effects on the leveling of an evaporating multi-component fluid. Langmuir 15 (1999) 1859–1871.

    Article  Google Scholar 

  43. E.N. Harvey, Tension at the cell surface. Protoplasmatologia IIE5 (1954) 1–30.

    Google Scholar 

  44. R.M. Hochmuth, Micropipette aspiration of living cells. J. Biomech. 33 (2000) 15–22.

    Article  Google Scholar 

  45. L.W. Schwartz and R.V. Roy, Some results concerning the potential energy of interfaces with nonuniformly distributed surfactant. Phys. Fluids 13 (2001) 3089–3092.

    Article  ADS  Google Scholar 

  46. D. Branton and D.W. Deamer, Membrane structure. Protoplasmatologia II (1972) 1–70.

    Google Scholar 

  47. D.M. Cazabat, F. Heslot, S.M. Troian and P. Carles, Fingering instability of thin spreading films driven by temperature gradients. Nature 346 (1990) 824–826.

    Article  ADS  Google Scholar 

  48. M.K. Smith, Thermocapillary migration of a two-dimensional droplet on a solid surface. J. Fluids Mech. 294 (1995) 209–230.

    ADS  MATH  Google Scholar 

  49. S.W. Benintendi and M.K. Smith, The spreading of a nonisothermal liquid droplet. Phys. Fluids 11 (1999) 982–989.

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ICI Paints Research Center, 44136, Strongsville, OH, U.S.A.

    R. V. Roy & R. R. Eley

  2. Surfaces & Colloids, Flemington, NJ, U.S.A.

    H. M. Princen

  3. Department of Mechanical Engineering, University of Delaware, 19716, Newark, DE, U.S.A.

    L. W. Schwartz

Authors
  1. L. W. Schwartz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. V. Roy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. R. Eley
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. H. M. Princen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. W. Schwartz.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schwartz, L.W., Roy, R.V., Eley, R.R. et al. Surfactant-driven motion and splitting of droplets on a substrate. J Eng Math 50, 157–175 (2004). https://doi.org/10.1007/s10665-004-0959-2

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-004-0959-2

Key words

Search

Navigation