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Energy dissipation and the contact-line region of a spreading bridge

Published online by Cambridge University Press:  07 June 2012

H. B. van Lengerich  and
P. H. Steen
H. B. van Lengerich*
Affiliation:
School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA
P. H. Steen
Affiliation:
School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, USA
*
†Email address for correspondence: hbv3@cornell.edu
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Abstract

A drop on a circular support spontaneously spreads upon contact with a substrate. The motion is driven by a loss of surface energy. The loss of recoverable energy can be expressed alternatively as work done at the liquid–gas interface or dissipation through viscosity and sliding friction. In this paper we require consistency with the energy lost by dissipation in order to infer details of the contact-line region through simulations. Simulations with the boundary integral method are used to compute the flow field of a corresponding experiment where polydimethylsiloxane spreads on a relatively hydrophobic surface. The flow field is used to calculate the energy dissipation, from which slip lengths for local slip and Navier slip boundary conditions are found. Velocities, shear rates and pressures along the interface as well as interface shapes in the microscopic region of the contact line are also reported. Angles, slip length and viscous bending length scale allow a test of the Voinov–Hocking–Cox model without free parameters.

JFM classification

Low-Reynolds-number flows: Boundary integral methods Interfacial Flows (free surface): Contact lines Drops and Bubbles: Drops and Bubbles
Type
Papers
Information
Journal of Fluid Mechanics , Volume 703 , 25 July 2012 , pp. 111 - 141
Copyright
Copyright © Cambridge University Press 2012

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References

1. Archer, L. A. 2005 Wall slip: measurement and modelling issues. In Polymer Processing Instabilities (ed. Hadzikiriakos, S. G. & Migler, K. B. ), p. 73. Marcel Dekker.Google Scholar
2. Aris, R. 1989 Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover.Google Scholar
3. Blake, T. D. 2006 The physics of moving wetting lines. J. Colloid Interface Sci. 299 (1), 1–13.CrossRefGoogle ScholarPubMed
4. Blake, T. D. & Haynes, J. M. 1969 Kinetics of liquid/liquid displacement. J. Colloid Interface Sci. 30 (3), 421–423.CrossRefGoogle Scholar
5. Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81 (2), 739.CrossRefGoogle Scholar
6. Brochard-Wyart, F. & de Gennes, P. G. 1992 Dynamics of partial wetting. Adv. Colloid Interface Sci. 39, 1–11.CrossRefGoogle Scholar
7. Callen, H. B. 1985 Thermodynamics and an Introduction to Thermostatistics. Wiley.Google Scholar
8. Chaudhury, M. K. & George, M. W. 1992 How to make water run uphill. Science 256 (5063), 1539–1541.CrossRefGoogle ScholarPubMed
9. Chen, Q., Ramé, E. & Garoff, S. 1995 The breakdown of asymptotic hydrodynamic models of liquid spreading at increasing capillary number. Phys. Fluids 7, 2631.CrossRefGoogle Scholar
10. Chen, Q., Ramé, E. & Garoff, S. 1997 The velocity field near moving contact lines. J. Fluid Mech. 337, 49–66.CrossRefGoogle Scholar
11. Choi, C. H., Westin, K. J. A. & Breuer, K. S. 2003 Apparent slip flows in hydrophilic and hydrophobic microchannels. Phys. Fluids 15, 2897.CrossRefGoogle Scholar
12. Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169–194.CrossRefGoogle Scholar
13. Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131.CrossRefGoogle Scholar
14. Davis, R. H. 1999 Buoyancy-driven viscous interaction of a rising drop with a smaller trailing drop. Phys. Fluids 11, 1016.Google Scholar
15. Davis, S. H. 1980 Moving contact lines and rivulet instabilities. Part 1. The static rivulet. J. Fluid Mech. 98 (2), 225–242.CrossRefGoogle Scholar
16. Debenedetti, P. G. 1996 Metastable Liquids: Concepts and Principles. Princeton University Press.Google Scholar
17. Deen, W. M. 1998 Analysis of Transport Phenomena. Oxford University Press.Google Scholar
18. Doedel, E. J., Paffenroth, R. C., Champneys, A. R., Fairgrieve, T. F., Kusnetsov, Y. A., Sandstede, B., Oldeman, B., Wang, X. J. & Zhang, C. 2007 AUTO 07P – Continuation and Bifurcation Software for Ordinary Differential Equations. Department of Computer Science, Concordia University.Google Scholar
19. Dussan V, E. B. 1975 Hydrodynamic stability and instability of fluid systems with interfaces. Arch. Rat. Mech. Anal. 57 (4), 363–379.CrossRefGoogle Scholar
20. Dussan V, E. B. & Davis, S. H. 1974 On the motion of a fluid–fluid interface along a solid surface. J. Fluid Mech. 65 (1), 71–95.CrossRefGoogle Scholar
21. Dussan V, E. B. & Davis, S. H. 1986 Stability in systems with moving contact lines. J. Fluid Mech. 173, 115–130.CrossRefGoogle Scholar
22. Dussan V, E. B., Ramé, E. & Garoff, S. 1991 On identifying the appropriate boundary conditions at a moving contact line: an experimental investigation. J. Fluid Mech. 230, 97–116.CrossRefGoogle Scholar
23. Fordham, S. 1948 On the calculation of surface tension from measurements of pendant drops. Proc. R. Soc. Lond. A 194 (1036), 1–16.Google Scholar
24. Fuentes, J. & Cerro, R. L. 2005 Flow patterns and interfacial velocities near a moving contact line. Exp. Fluids 38 (4), 503–510.CrossRefGoogle Scholar
25. de Gennes, P. G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57 (3), 827–863.CrossRefGoogle Scholar
26. de Gennes, P. G., Brochard-Wyart, F. & Quéré, D. 2004 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer.Google Scholar
27. Hadjiconstantinou, N. G. 1999 Hybrid atomistic–continuum formulations and the moving contact-line problem. J. Comput. Phys. 154 (2), 245–265.CrossRefGoogle Scholar
28. Haley, P. J. Jr. & Miksis, M. J. 1991 Dissipation and contact-line motion. Phys. Fluids A 3, 487.Google Scholar
29. Hocking, L. M. & Rivers, A. D. 1982 The spreading of a drop by capillary action. J. Fluid Mech. 121, 425–442.CrossRefGoogle Scholar
30. Huh, C. & Mason, S. G. 1977 The steady movement of a liquid meniscus in a capillary tube. J. Fluid Mech. 81 (3), 401–419.CrossRefGoogle Scholar
31. Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35 (1), 85–101.CrossRefGoogle Scholar
32. Joseph, D. D. 1976 Stability of Fluid Motions, II, Tracts in Natural Philosophy, vol. 28 , p. 274 Springer.Google Scholar
33. Kissi, N., Piau, J. M., Attané, P. & Turrel, G. 1993 Shear rheometry of polydimethylsiloxanes. Master curves and testing of Gleissle and Yamamoto relations. Rheol. Acta 32 (3), 293–310.CrossRefGoogle Scholar
34. Kreyszig, E. 1991 Differential Geometry. Dover.Google Scholar
35. Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flow. Gordon & Breach.Google Scholar
36. Lauga, E., Brenner, M. P. & Stone, H. A. 2007 Microfluidics: the no-slip boundary condition. In Springer Handbook of Experimental Fluid Mechanics (ed. Tropea, C., Yarin, A. L. & Foss, J. F. ). p. 1219. Springer.CrossRefGoogle Scholar
37. Leal, L. G. 1992 Laminar Flow and Convective Transport Processes: Scaling Principles and Asymptotic Analysis. Butterworth-Heinemann.Google Scholar
38. Lee, S. H. & Leal, L. G. 1982 The motion of a sphere in the presence of a deformable interface. II. A numerical study of the translation of a sphere normal to an interface. J. Colloid Interface Sci. 87 (1), 81–106.Google Scholar
39. Lowndes, J. 1980 The numerical simulation of the steady movement of a fluid meniscus in a capillary tube. J. Fluid Mech. 101 (3), 631–646.Google Scholar
40. Maddocks, J. H. 1987 Stability and folds. Arch. Rat. Mech. Anal. 99 (4), 301–328.CrossRefGoogle Scholar
41. McConnell, A. J. 1932 Applications of the absolute differential calculus. Bull. Am. Math. Soc. 38 (1), 615–616.Google Scholar
42. Mhetar, V. & Archer, L. A. 1998 Slip in entangled polymer solutions. Macromolecules 31 (19), 6639–6649.CrossRefGoogle Scholar
43. Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech 18 (1), 1–18.Google Scholar
44. Navier, C. 1823 Mémoire sur les lois du mouvement des fluids. Mém. Présentés par Divers Savants Acad. Sci. Inst. Fr. 6 (2), 389–440.Google Scholar
45. Neto, C., Craig, V. S. J. & Williams, D. R. M. 2003 Evidence of shear-dependent boundary slip in Newtonian liquids. Eur. Phys. J. E 12, 71–74.CrossRefGoogle ScholarPubMed
46. Nitsche, M., Ceniceros, H. D., Karniala, A. L. & Naderi, S. 2010 High order quadratures for the evaluation of interfacial velocities in axi-symmetric Stokes flows. J. Comput. Phys. 229 (18), 6318–6342.CrossRefGoogle Scholar
47. North, S. H., Lock, E. H., King, T. R., Franek, J. B., Walton, S. G. & Taitt, C. R. 2009 Effect of physicochemical anomalies of soda-lime silicate slides on biomolecule immobilization. Anal. Chem. 82 (1), 406–412.CrossRefGoogle Scholar
48. Odqvist, F. K. G. 1930 Über die Randwertaufgaben der Hydrodynamik zäher Flüssigkeiten. Math. Z. 32 (1), 329–375.Google Scholar
49. Qian, T., Wang, X. P. & Sheng, P. 2006 A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 564 (1), 333–360.CrossRefGoogle Scholar
50. Ramé, E. & Garoff, S. 1996 Microscopic and macroscopic dynamic interface shapes and the interpretation of dynamic contact angles. J. Colloid Interface Sci. 177 (1), 234–244.CrossRefGoogle ScholarPubMed
51. Ramé, E., Garoff, S. & Willson, K. R. 2004 Characterizing the microscopic physics near moving contact lines using dynamic contact angle data. Phys. Rev. E 70 (3), 31608.CrossRefGoogle ScholarPubMed
52. Sauer, B. B. & Dee, G. T. 1991 Molecular weight and temperature dependence of polymer surface tension: comparison of experiment with theory. Macromolecules 24 (8), 2124–2126.Google Scholar
53. Schmatko, T., Hervet, H. & Léger, L. 2005 Friction and slip at simple fluid–solid interfaces: the roles of the molecular shape and the solid–liquid interaction. Phys. Rev. Lett. 94 (24), 244501.CrossRefGoogle Scholar
54. Schmatko, T., Hervet, H. & Léger, L. 2006 Effect of nanometric-scale roughness on slip at the wall of simple fluids. Langmuir 22 (16), 6843–6850.CrossRefGoogle ScholarPubMed
55. Shikhmurzaev, Y. D. 1997 Moving contact lines in liquid/liquid/solid systems. J. Fluid Mech. 334, 211–249.Google Scholar
56. Slattery, J. C., Sagis, L. & Oh, E. S. 2007 Interfacial Transport Phenomena. Springer.Google Scholar
57. Suo, Y., Stoev, K., Garoff, S. & Ramé, E. 2001 Hydrodynamics and contact angle relaxation during unsteady spreading. Langmuir 17 (22), 6988–6994.CrossRefGoogle Scholar
58. Teletzke, G. F., Davis, H. T. & Scriven, L. E. 1988 Wetting hydrodynamics. Rev. Phys. Appl. 23 (6), 989–1007.CrossRefGoogle Scholar
59. Truesdell, C. 1991 A First Course in Rational Continuum Mechanics: General Concepts. Academic Press.Google Scholar
60. Vogel, M. J. & Steen, P. H. 2010 Capillarity-based switchable adhesion. Proc. Natl Acad. Sci. 107 (8), 3377.Google Scholar
61. Voinov, O. V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11 (5), 714–721.CrossRefGoogle Scholar
62. Voinov, O. V. 1995 Motion of line of contact of three phases on a solid: thermodynamics and asymptotic theory: 1. Intl J. Multiphase Flow 21 (5), 801–816.CrossRefGoogle Scholar
63. Voinov, O. V. 2000 Wetting: inverse dynamic problem and equations for microscopic parameters. J. Colloid Interface Sci. 226 (1), 5–15.CrossRefGoogle ScholarPubMed
64. Vorvolakos, K. & Chaudhury, M. K. 2003 The effects of molecular weight and temperature on the kinetic friction of silicone rubbers. Langmuir 19 (17), 6778–6787.CrossRefGoogle Scholar
65. Weidner, D. E. & Schwartz, L. W. 1994 Contact-line motion of shear-thinning liquids. Phys. Fluids 6, 3535.CrossRefGoogle Scholar
66. Young, T. 1805 An essay on the cohesion of fluids. Phil. Trans. R. Soc. Lond. 95, 65–87.Google Scholar
67. Youngren, G. K. & Acrivos, A. 1975 Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech. 69 (2), 377–403.Google Scholar
68. Zettner, C. & Yoda, M. 2003 Particle velocity field measurements in a near-wall flow using evanescent wave illumination. Exp. Fluids 34 (1), 115–121.Google Scholar