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Three-dimensional equilibrium shapes of drops on hysteretic surfaces

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Abstract

In this paper, we study equilibrium three-dimensional shapes of drops on hysteretic surfaces. We develop a function coupled with the publicly available surface energy minimization code Surface Evolver to handle contact angle hysteresis. The function incorporates a model for the mobility of the triple line into Surface Evolver. The only inputs to the model are the advancing and receding contact angles of the surface. We demonstrate this model’s versatility by studying three problems in which parts of the triple line advance while other parts either recede or remain stationary. The first problem focuses on the three-dimensional shape of a static pendant drop on a vertical surface. We predict the finite drop volume when impending sliding motion is observed. In the second problem, we examine the equilibrium shapes of coalescing sessile drops on hysteretic surfaces. Finally, we study coalescing puddles in which gravity plays a leading role in determining the equilibrium puddle shape along with hysteresis.

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Notes

  1. Triple line is defined as the set of points which are in contact with the liquid, solid, and vapor phases simultaneously.

  2. The local contact angle is defined as the angle between the liquid–vapor interface and the solid substrate.

References

  1. Brandon S, Wachs A, Marmur A (1997) Simulated contact angle hysteresis of a three-dimensional drop on a chemically heterogeneous surface: a numerical example. J Colloid Interface Sci 191:110

    Article  CAS  Google Scholar 

  2. Brandon S, Haimovich N, Yeger E, Marmur A (2003) Partial wetting of chemically patterned surfaces: the effect of drop size. J Colloid Interface Sci 263:237

    Article  CAS  Google Scholar 

  3. Vedantam S, Panchagnula MV (2008) Constitutive modeling of contact angle hysteresis. J Colloid Interface Sci 321:393

    Article  CAS  Google Scholar 

  4. Vedantam S, Panchagnula MV (2007) Phase field modeling of hysteresis in sessile drops. Phys Rev Lett 99:176102

    Article  Google Scholar 

  5. Prabhala BR, Panchagnula MV, Subramanian VR, Vedantam S (2010) Perturbation solution of the shape of a nonaxisymmetric sessile drop. Langmuir 26:10717

    Article  Google Scholar 

  6. Brown RA, Orr FM, Scriven LE (1980) Static drop on an inclined plate: analysis by the finite element method. J Colloid Interface Sci 73:76

    Article  CAS  Google Scholar 

  7. Iliev SD (1997) Static drops on an inclined plane: equilibrium modelling and numerical analysis. J Colloid Interface Sci 194:287

    CAS  Google Scholar 

  8. Iliev SD (1999) The effects of resistance to shift of the equilibrium state of a liquid droplet in contact with a solid. J Colloid Interface Sci 213:1

    Article  CAS  Google Scholar 

  9. Dimitrakopoulos P, Higdon JJL (1999) On the gravitational displacement of three-dimensional fluid droplets from inclined solid surfaces. J Fluid Mech 395:181

    Article  CAS  Google Scholar 

  10. Dimitrakopoulos P, Higdon JJL (2001) On the displacement of three-dimensional fluid droplets adhering to a plane wall in viscous pressure-driven flows. J Fluid Mech 435:327

    Article  Google Scholar 

  11. Das AK, Das PK (2009) Simulation of a sliding drop over an inclined surface using smoothed particle hydrodynamics. Langmuir 25:11459

    Article  CAS  Google Scholar 

  12. Hong S-J, Chang F-M, Chou T-H, Chan SH, Sheng Y-J, Tsao H-K (2011) Anomalous contact angle hysteresis of a captive bubble: advancing contact line pinning. Langmuir 27:6890

    Article  CAS  Google Scholar 

  13. Brakke KA (1992) The surface evolver. Exp Math 1:141

    Article  Google Scholar 

  14. Ruiz-Cabello FJM, Kusumaatmaja H, Rodríguez-Valverde MA, Yeomans J, Cabrerizo-Vílchez MA (2009) Modelling the corrugation of the three-phase contact line perpendicular to a chemically striped substrate. Langmuir 25:8357

    Article  Google Scholar 

  15. Santos MJ, White JA (2011) Theory and simulation of angular hysteresis on planar surfaces. Langmuir 27:14868

    Article  CAS  Google Scholar 

  16. Joanny JF, de Gennes PG (1984) A model for contact angle hysteresis. J Chem Phys 81:552

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  18. Blake TD, Haynes JM (1969) Kinetics of liquid/liquid displacement. J Colloid Interface Sci 30:421

    Article  CAS  Google Scholar 

  19. Pomeau Y (2000) Recent progress in the moving contact line problem: a review. CR Acad Sci, Ser IIb: Mec, Phys, Chim, Astron 238:411

    Google Scholar 

  20. van Mourik S, Veldman AEP, Dreyer ME (2005) Simulation of capillary flow with a dynamic contact angle. Microgravity Sci Technol 17:87

    Article  Google Scholar 

  21. Iliev D, Iliev SD (2009) Dumping of capillary-gravity waves in a channel: the wedge dissipation effect. In: Proceedings of the thirty eighth spring conference of the Union of Bulgarian Mathematicians, vol 38, p 178

  22. Iliev S, Pesheva N, Nikolayev V (2005) Quasistatic relaxation of arbitrarily shaped sessile drops. Phys Rev E 72:011606

    Article  Google Scholar 

  23. Tukey HB (1970) The leaching of substances from plants. Annu Rev Plant Physiol 21:305

    Article  CAS  Google Scholar 

  24. Knoche M (1994) Effect of droplet size and carrier volume on performance of foliage-applied herbicides. Crop Prot 13:163

    Article  Google Scholar 

  25. Stone HA, Stroock AD, Ajdari A (2004) Engineering flows in small devices. Annu Rev Fluid Mech 36:381

    Article  Google Scholar 

  26. Beysens D, Knobler CM (1986) Growth of breath figures. Phys Rev Lett 57:1433

    Article  Google Scholar 

  27. Beysens D, Steyer A, Guenoun P, Fritter D, Knobler CM (1991) How does dew form? Phase Transit 31:219

    Article  CAS  Google Scholar 

  28. Fritter D, Knobler CM, Beysens DA (1991) Experiments and simulation of the growth of droplets on a surface (breath figures). Phys Rev A 43:2558

    Article  Google Scholar 

  29. Murray PE (1996) Commun Numer Methods Eng 12:447

    Article  Google Scholar 

  30. Narhe R, Beysens D, Nikolayev VS (2004) Contact line dynamics in drop coalescence and spreading. Langmuir 20:1213

    Article  CAS  Google Scholar 

  31. Narhe R, Beysens D, Nikolayev VS (2005) Dynamics of drop coalescence on a surface: the role of initial conditions and surface properties. Int J Thermophys 26:1743

    Article  CAS  Google Scholar 

  32. Liao Q, Zhu X, Xing SM, Wang H (2008) Visualization study on coalescence between pair of water drops on inclined surfaces. Exp Therm Fluid Sci 32:1647

    Article  Google Scholar 

  33. Wang H, Zhu X, Liao Q, Sui PC (2010) Numerical simulation on coalescence between a pair of drops on homogeneous horizontal surface with volume-of-fluid method. J Supercond Nov Magn 23:1137

    Article  CAS  Google Scholar 

  34. Dussan EB (1979) On the spreading of liquids on solid surfaces: static and dynamic contact lines. Annu Rev Fluid Mech 11:371

    Article  Google Scholar 

  35. Tadmor R (2004) Line energy and the relation between advancing, receding, and young contact angles. Langmuir 20:7659

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  37. Anantharaju N, Panchagnula MV, Vedantam S (2009) Asymmetric wetting hysteresis on composite surfaces of intrinsically hysteretic materials. Langmuir 25:7410

    Article  CAS  Google Scholar 

  38. Amirfazli A, Kwok YD Gaydos J, Neumann AW (1998) Line tension measurements through drop size dependence of contact angles. J Colloid Interface Sci 205:1

    Article  CAS  Google Scholar 

  39. Dupont JB, Legendre D (2010) Numerical simulation of static and sliding drop with contact angle hysteresis. J Comput Phys 229:2453

    Article  CAS  Google Scholar 

  40. Thampi SP, Govindarajan R (2011) Minimum energy shapes of one-side-pinned static drops on inclined surfaces. Phys Rev E 84:046304

    Article  Google Scholar 

  41. ElSherbini AI, Jacobi AM (2004) Liquid drops on vertical and inclined surfaces I. An experimental study of drop geometry. J Colloid Interface Sci 273:556

    Article  CAS  Google Scholar 

  42. Montes Ruiz-Cabello FJ, Rodríguez-Valverde MA, Cabrerizo-Vílchez MA (2011) A new method for evaluating the most-stable contact angle using tilting plate experiments. Soft Mater 7:10457

    Article  Google Scholar 

  43. Pierce E, Carmona FJ, Amirfazli A (2008) Understanding of sliding and contact angle results in tilted plate experiments. Colloids Surf A 323:73

    Article  CAS  Google Scholar 

  44. Kapur N, Gaskell PH (2007) Morphology and dynamics of droplet coalescence on a surface. Phys Rev E 75:056315

    Article  Google Scholar 

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Acknowledgements

We thank Prof. K. A. Brakke for the yeoman service in making the Surface Evolver program publicly available.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Tennessee Technological University, Cookeville, TN, 38501, USA

    Bharadwaj R. Prabhala

  2. Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai, TN, 600036, India

    Mahesh V. Panchagnula

  3. Department of Engineering Design, Indian Institute, of Technology Madras, Chennai, TN, 600036, India

    Srikanth Vedantam

Authors
  1. Bharadwaj R. Prabhala
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  2. Mahesh V. Panchagnula
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  3. Srikanth Vedantam
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Corresponding author

Correspondence to Mahesh V. Panchagnula.

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This article is part of the Topical Collection on Contact Angle Hysteresis

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Prabhala, B.R., Panchagnula, M.V. & Vedantam, S. Three-dimensional equilibrium shapes of drops on hysteretic surfaces. Colloid Polym Sci 291, 279–289 (2013). https://doi.org/10.1007/s00396-012-2774-z

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  • DOI: https://doi.org/10.1007/s00396-012-2774-z

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