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Journal of Engineering Mathematics

, Volume 101, Issue 1, pp 55–69 | Cite as

New analytical solutions for static two-dimensional droplets under the effects of long- and short-range molecular forces

  • J. R. Mac IntyreEmail author
  • J. M. Gomba
  • Carlos A. Perazzo
Article

Abstract

We report new analytical solutions for the thickness profile of partially wetting two-dimensional droplets. The model includes the effects of capillarity and both short- and long-range molecular forces. We analyze the dependence of the maximum thickness, the contact angle, and the cross-sectional area on the height of the nanometric precursor film that surrounds the droplet. We found asymptotic expressions for the thickness profile and for the contact angles for large and small droplets. The results are compared to those obtained previously for polar liquids. The analytical solutions found here are useful to assess the validity of the hypothesis and the semi-analytical solutions proposed in the literature. In addition, these solutions enable the inference of information about the molecular potential from the measured steady profiles.

Keywords

2D droplet Contact angle Disjoining/conjoining pressure Partial wetting 

Notes

Acknowledgments

The authors gratefully acknowledge the funding supports via the CONICET Grants PIP No. 356 and PIP No. 299, and the ANPCyP Grant No. 2012-1707.

References

  1. 1.
    Bonn D, Eggers J, Indekeu J, Meunier J, Rolley E (2009) Wetting and spreading. Rev Mod Phys 81:739–805ADSCrossRefGoogle Scholar
  2. 2.
    Lai YH, Yang JT, Shieh DB (2010) A microchip fabricated with a vapor-diffusion self-assembled-monolayer method to transport droplets across superhydrophobic to hydrophilic surfaces. Lab Chip 10:499–504CrossRefGoogle Scholar
  3. 3.
    Daunay B, Lambert P, Jalabert L, Kumemura M, Renaudot R, Agache V, Fujita H (2012) Effect of substrate wettability in liquid dielectrophoresis (ldep) based droplet generation: theoretical analysis and experimental confirmation. Lab Chip 12:361–368CrossRefGoogle Scholar
  4. 4.
    Arscott S, Descatoire C, Buchaillot L, Ashcroft AE (2012) A snapshot of electrified nanodroplets undergoing Coulomb fission. Appl Phys Lett 100(7):074103ADSCrossRefGoogle Scholar
  5. 5.
    Roberts CC, Rao RR, Loewenberg M, Brooks CF, Galambos P, Grillet AM, Nemer MB (2012) Comparison of monodisperse droplet generation in flow-focusing devices with hydrophilic and hydrophobic surfaces. Lab Chip 12:1540–1547CrossRefGoogle Scholar
  6. 6.
    Israelachvili JN (1992) Intermolecular and surface forces, 2nd edn. Academic Press, New YorkGoogle Scholar
  7. 7.
    Berim GO, Ruckenstein E (2004) On the shape and stability of a drop on a solid surface. J Phys Chem B 108:19330–19338CrossRefGoogle Scholar
  8. 8.
    Nold A, Sibley DN, Goddard BD, Kalliadasis S (2014) Fluid structure in the immediate vicinity of an equilibrium three-phase contact line and assessment of disjoining pressure models using density functional theory. Phys Fluids 26(7):072001ADSCrossRefGoogle Scholar
  9. 9.
    Derjaguin B, Kusakov M (1936) Contact-line dynamics of a diffuse fluid interface. Izv Akad Nauk SSSR Ser Khim 5:741Google Scholar
  10. 10.
    Gomba JM, Homsy GM (2009) Analytical solutions for partially wetting two-dimensional droplets. Langmuir 25(10):5684–5691CrossRefGoogle Scholar
  11. 11.
    Gomba JM, Perazzo CA (2012) Closed-form expression for the profile of partially wetting two-dimensional droplets under gravity. Phys Rev E 86(056):310Google Scholar
  12. 12.
    Gotkis Y, Ivanov I, Murisic N, Kondic L (2006) Dynamic structure formation at the fronts of volatile liquid drops. Phys Rev Lett 97(18):186101ADSCrossRefGoogle Scholar
  13. 13.
    Churaev NV, Sobolev VD (1995) Prediction of contact angles on the basis of the Frumkin–Derjaguin approach. Adv Colloid Interface Sci 61:1–16CrossRefGoogle Scholar
  14. 14.
    Teletzke GF, Davis HT, Scriven LE (1987) How liquids spread on solids. Chem Eng Commun 55:41–82CrossRefGoogle Scholar
  15. 15.
    Starov V, Velarde M, Radke C (2007) Wetting and spreading dynamics. Surfactant science series. CRC Press, Boca RatonGoogle Scholar
  16. 16.
    Derjaguin B, Churaev N (1974) Structural component of disjoining pressure. J Colloid Interface Sci 49(2):249–255CrossRefGoogle Scholar
  17. 17.
    Derjaguin BV, Rabinovich YI, Churaev NV (1978) Direct measurement of molecular forces. Nature 272:313–318ADSCrossRefGoogle Scholar
  18. 18.
    Oron A, Bankoff SG (2001) Dynamics of a condensing liquid film under conjoining/disjoining pressures. Phys Fluids 13:1107–1117ADSCrossRefzbMATHGoogle Scholar
  19. 19.
    Glasner KB, Witelski TP (2003) Coarsening dynamics of dewetting films. Phys Rev E 67:016302ADSCrossRefGoogle Scholar
  20. 20.
    Schwartz LW, Roy RV (2004) Theoretical and numerical results for spin coating of viscous liquids. Phys Fluid 16:569–584ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gratton MB, Witelski TP (2008) Coarsening of unstable thin films subject to gravity. Phys Rev E 77:016301ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Gratton MB, Witelski TP (2009) Transient and self-similar dynamics in thin film coarsening. Physica D 238(23–24):2380–2394ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    de Gennes PG (1985) Wetting: statics and dynamics. Rev Mod Phys 57:827–863ADSCrossRefGoogle Scholar
  24. 24.
    Oron A, Bankoff S (1999) Dewetting of a heated surface by an evaporating liquid film under conjoining/disjoining pressures. J Colloid Interface Sci 218(1):152–166CrossRefGoogle Scholar
  25. 25.
    Schwartz LW (1998) Hysteretic effects in droplet motions on heterogenous substrates: direct numerical simulations. Langmuir 14:3440–3453CrossRefGoogle Scholar
  26. 26.
    Derjaguin B, Churaev NV, Muller V (1987) Surface forces. Springer, New YorkCrossRefGoogle Scholar
  27. 27.
    Sur J, Witelski TP, Behringer RP (2004) Steady-profile fingering flows in marangoni driven thin films. Phys Rev Lett 93(24):247803ADSCrossRefGoogle Scholar
  28. 28.
    Thiele U, Neuffer K, Bestehorn M, Pomeau Y, Velarde MG (2002) Sliding drops on an inclined plane. Colloids Surf A 206:87–104CrossRefGoogle Scholar
  29. 29.
    Brochard-Wyart F, Di Meglio JM, Quere D, De Gennes PG (1991) Spreading of nonvolatile liquids in a continuum picture. Langmuir 7(2):335–338CrossRefGoogle Scholar
  30. 30.
    Zhang X, Neogi P, Ybarra R (2002) Stable drop shapes under disjoining pressure: I. A hierarchical approach and application. J Colloid Interface Sci 249(1):134–140CrossRefGoogle Scholar
  31. 31.
    Pismen LM, Eggers J (2008) Solvability condition for the moving contact line. Phys Rev E 78(056):304MathSciNetGoogle Scholar
  32. 32.
    Lubarda VA, Talke KA (2011) Analysis of the equilibrium droplet shape based on an ellipsoidal droplet model. Langmuir 27(17):10705–10713CrossRefGoogle Scholar
  33. 33.
    Diaz ME, Fuentes J, Cerro RL, Savage MD (2010) An analytical solution for a partially wetting puddle and the location of the static contact angle. J Colloid Interface Sci 348(1):232–239CrossRefGoogle Scholar
  34. 34.
    Gaskell PH, Jimack PK, Sellier M, Thompson HM (2004) Efficient and accurate time adaptive multigrid simulations of droplet spreading. Int J Numer Methods Fluids 45(11):1161–1186CrossRefzbMATHGoogle Scholar
  35. 35.
    Koh YY, Lee YC, Gaskell PH, Jimack PK, Thompson HM (2009) Droplet migration: quantitative comparisons with experiment. Eur Phys J Special Topics 166(1):117–120ADSCrossRefGoogle Scholar
  36. 36.
    Gomba JM, Homsy GM (2010) Regimes of thermocapillary migration of droplets under partial wetting conditions. J Fluid Mech 647:125–142ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Oron A, Davis SH, Bankoff SG (1997) Long-scale evolution of thin liquid films. Rev Mod Phys 69:931–980ADSCrossRefGoogle Scholar
  38. 38.
    Schwartz LW, Eley RR (1998) Simulation of droplet motion on low-energy and heterogeneous surfaces. J Colloid Interface Sci 202(1):173–188CrossRefGoogle Scholar
  39. 39.
    Mitlin VS (1994) On dewetting conditions. Colloid Surf A 89:97–101CrossRefGoogle Scholar
  40. 40.
    Bertozzi A, Grün G, Witelski TP (2001) Dewetting films: bifurcations and concentrations. Nonlinearity 14:1569ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Glasner K, Witelski T (2005) Collision versus collapse of droplets in coarsening of dewetting thin films. Physica D 209:80–104ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Teletzke GF, Davis H, Scriven LE (1988) Wetting hydrodynamics. Rev Phys Appl 23(6):989–1007CrossRefGoogle Scholar
  43. 43.
    Abramowitz M, Stegun IA (1964) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New YorkzbMATHGoogle Scholar
  44. 44.
    Eggers J (2005) Contact line motion for partially wetting fluids. Phys Rev E 72:061605ADSCrossRefGoogle Scholar
  45. 45.
    Savva N, Kalliadasis S (2011) Dynamics of moving contact lines: A comparison between slip and precursor film models. Europhys Lett 94(6):64004ADSCrossRefGoogle Scholar
  46. 46.
    Kavehpour HP, Ovryn B, McKinley GH (2003) Microscopic and macroscopic structure of the precursor layer in spreading viscous drops. Phys Rev Lett 91:196104ADSCrossRefGoogle Scholar
  47. 47.
    Solomentsev Y, White LR (1999) Microscopic drop profiles and the origins of line tension. J Colloid Interface Sci 218(1):122–136CrossRefGoogle Scholar
  48. 48.
    Dörfler F, Rauscher M, Dietrich S (2013) Stability of thin liquid films and sessile droplets under confinement. Phys Rev E 88:012402ADSCrossRefGoogle Scholar
  49. 49.
    Perazzo CA, Mac Intyre JR, Gomba JM (2014) Final state of a perturbed liquid film inside a container under the effect of solid–liquid molecular forces and gravity. Phys Rev E 89:043010ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • J. R. Mac Intyre
    • 1
    Email author
  • J. M. Gomba
    • 1
  • Carlos A. Perazzo
    • 2
  1. 1.Instituto de Física Arroyo Seco, CIFICEN, Universidad Nacional del Centro de la Provincia de Buenos AiresTandilArgentina
  2. 2.Dep. de Física y Química, FICENUniversidad FavaloroBuenos AiresArgentina

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