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

- 198 Downloads
- 3 Citations

## 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.Bonn D, Eggers J, Indekeu J, Meunier J, Rolley E (2009) Wetting and spreading. Rev Mod Phys 81:739–805ADSCrossRefGoogle Scholar
- 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.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.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.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.Israelachvili JN (1992) Intermolecular and surface forces, 2nd edn. Academic Press, New YorkGoogle Scholar
- 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.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.Derjaguin B, Kusakov M (1936) Contact-line dynamics of a diffuse fluid interface. Izv Akad Nauk SSSR Ser Khim 5:741Google Scholar
- 10.Gomba JM, Homsy GM (2009) Analytical solutions for partially wetting two-dimensional droplets. Langmuir 25(10):5684–5691CrossRefGoogle Scholar
- 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.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.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.Teletzke GF, Davis HT, Scriven LE (1987) How liquids spread on solids. Chem Eng Commun 55:41–82CrossRefGoogle Scholar
- 15.Starov V, Velarde M, Radke C (2007) Wetting and spreading dynamics. Surfactant science series. CRC Press, Boca RatonGoogle Scholar
- 16.Derjaguin B, Churaev N (1974) Structural component of disjoining pressure. J Colloid Interface Sci 49(2):249–255CrossRefGoogle Scholar
- 17.Derjaguin BV, Rabinovich YI, Churaev NV (1978) Direct measurement of molecular forces. Nature 272:313–318ADSCrossRefGoogle Scholar
- 18.Oron A, Bankoff SG (2001) Dynamics of a condensing liquid film under conjoining/disjoining pressures. Phys Fluids 13:1107–1117ADSCrossRefzbMATHGoogle Scholar
- 19.Glasner KB, Witelski TP (2003) Coarsening dynamics of dewetting films. Phys Rev E 67:016302ADSCrossRefGoogle Scholar
- 20.Schwartz LW, Roy RV (2004) Theoretical and numerical results for spin coating of viscous liquids. Phys Fluid 16:569–584ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 21.Gratton MB, Witelski TP (2008) Coarsening of unstable thin films subject to gravity. Phys Rev E 77:016301ADSMathSciNetCrossRefGoogle Scholar
- 22.Gratton MB, Witelski TP (2009) Transient and self-similar dynamics in thin film coarsening. Physica D 238(23–24):2380–2394ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 23.de Gennes PG (1985) Wetting: statics and dynamics. Rev Mod Phys 57:827–863ADSCrossRefGoogle Scholar
- 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.Schwartz LW (1998) Hysteretic effects in droplet motions on heterogenous substrates: direct numerical simulations. Langmuir 14:3440–3453CrossRefGoogle Scholar
- 26.Derjaguin B, Churaev NV, Muller V (1987) Surface forces. Springer, New YorkCrossRefGoogle Scholar
- 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.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.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.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.Pismen LM, Eggers J (2008) Solvability condition for the moving contact line. Phys Rev E 78(056):304MathSciNetGoogle Scholar
- 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.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.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.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.Gomba JM, Homsy GM (2010) Regimes of thermocapillary migration of droplets under partial wetting conditions. J Fluid Mech 647:125–142ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 37.Oron A, Davis SH, Bankoff SG (1997) Long-scale evolution of thin liquid films. Rev Mod Phys 69:931–980ADSCrossRefGoogle Scholar
- 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.Mitlin VS (1994) On dewetting conditions. Colloid Surf A 89:97–101CrossRefGoogle Scholar
- 40.Bertozzi A, Grün G, Witelski TP (2001) Dewetting films: bifurcations and concentrations. Nonlinearity 14:1569ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 41.Glasner K, Witelski T (2005) Collision versus collapse of droplets in coarsening of dewetting thin films. Physica D 209:80–104ADSMathSciNetCrossRefzbMATHGoogle Scholar
- 42.Teletzke GF, Davis H, Scriven LE (1988) Wetting hydrodynamics. Rev Phys Appl 23(6):989–1007CrossRefGoogle Scholar
- 43.Abramowitz M, Stegun IA (1964) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New YorkzbMATHGoogle Scholar
- 44.Eggers J (2005) Contact line motion for partially wetting fluids. Phys Rev E 72:061605ADSCrossRefGoogle Scholar
- 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.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.Solomentsev Y, White LR (1999) Microscopic drop profiles and the origins of line tension. J Colloid Interface Sci 218(1):122–136CrossRefGoogle Scholar
- 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.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