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Relaxation of a dewetting contact line. Part 1. A full-scale hydrodynamic calculation

Published online by Cambridge University Press:  02 May 2007

JACCO H. SNOEIJER ,
BRUNO ANDREOTTI ,
GILES DELON  and
MARC FERMIGIER
JACCO H. SNOEIJER
Affiliation:
Physique et Mécanique des Milieux Hétérogènes, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France
BRUNO ANDREOTTI
Affiliation:
Physique et Mécanique des Milieux Hétérogènes, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France
GILES DELON
Affiliation:
Physique et Mécanique des Milieux Hétérogènes, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France
MARC FERMIGIER
Affiliation:
Physique et Mécanique des Milieux Hétérogènes, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France
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Abstract

The relaxation of a dewetting contact line is investigated theoretically in the so-called ‘Landau–Levich’ geometry in which a vertical solid plate is withdrawn from a bath of partially wetting liquid. The study is performed in the framework of lubrication theory, in which the hydrodynamics is resolved at all length scales (from molecular to macroscopic). We investigate the bifurcation diagram for unperturbed contact lines, which turns out to be more complex than expected from simplified ‘quasi-static’ theories based upon an apparent contact angle. Linear stability analysis reveals that below the critical capillary number of entrainment, Cac, the contact line is linearly stable at all wavenumbers. Away from the critical point, the dispersion relation has an asymptotic behaviour σ∝|q| and compares well to a quasi-static approach. Approaching Cac, however, a different mechanism takes over and the dispersion evolves from ∼|q| to the more common ∼q2. These findings imply that contact lines cannot be described using a universal relation between speed and apparent contact angle, but viscous effects have to be treated explicitly.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 579 , 25 May 2007 , pp. 63 - 83
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Barrat, J.-L. & Bocquet, L. 1999 Large slip effect at a nonwetting fluid–solid interface. Phys. Rev. Lett. 82, 4671–4674.CrossRefGoogle Scholar
Blake, T. D. & Ruschak, K. J. 1979 A maximum speed of wetting. Nature 282, 489–491.Google Scholar
Blake, T. D., de Coninck, J. & D'Ortuna, U. 1995 Models of wetting: immiscible lattice Boltzmann automata versus molecular kinetic theory. Langmuir 11, 4588.CrossRefGoogle Scholar
Buckingham, R., Shearer, M. & Bertozzi, A. 2003 Thin film traveling waves and the Navier-slip condition. SIAM J. Appl. Maths 63, 722–744.Google Scholar
Cottin-Bizonne, C., Cross, B., Steinberger, A. & Charlaix, E. 2005 Boundary slip on smooth hydrophobic surfaces: intrinsic effects and possible artifacts. Phys. Rev. Lett. 94, 056102.CrossRefGoogle ScholarPubMed
Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. J. Fluid Mech. 168, 169–194.CrossRefGoogle Scholar
Delon, G., Fermigier, M., Snoeijer, J. H. & Andreotti, B. 2007 in preparation/qtoa Q1.Google Scholar
Dussan, V. E. B. & Davis, S. H. 1974 On the motion of a fluid–fluid interface along a solid surface. J. Fluid Mech. 65, 71–95.CrossRefGoogle Scholar
Eggers, J. 2004 Hydrodynamic theory of forced dewetting. Phys. Rev. Lett. 93, 094502.CrossRefGoogle ScholarPubMed
Eggers, J. 2005 Existence of receding and advancing contact lines. Phys. Fluids 17, 082106.CrossRefGoogle Scholar
de Gennes, P.-G. 1986 Deposition of Langmuir–Blodget layers. Colloid Polym. Sci. 264, 463–465.CrossRefGoogle Scholar
Golestanian, R. & Raphael, E. 2001 a Dissipation in dynamics of a moving contact line. Phys. Rev. E 64, 031601.Google ScholarPubMed
Golestanian, R. & Raphael, E. 2001 b Relaxation of a moving contact line and the Landau–Levich effect. Europhys. Lett. 55, 228–234.CrossRefGoogle Scholar
Golestanian, R. & Raphael, E. 2003 Roughening transition in a moving contact line. Phys. Rev. E 67, 031603.Google Scholar
Hocking, L. M. 2001 Meniscus draw-up and draining. Euro. J. Appl. Maths 12, 195–208.CrossRefGoogle Scholar
Hoffman, R. L. 1975 Dynamic contact angle. J. Colloid Interface Sci. 50, 228–241.CrossRefGoogle Scholar
Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85–101.CrossRefGoogle Scholar
Huppert, H. E. 1982 Flow and instability of a viscous gravity current down a slope. Nature 300, 427–429.Google Scholar
Israelachvili, J. 1992 Intermolecular and Surface Forces. Academic.Google Scholar
Joanny, J.-F. & de Gennes, P.-G. 1984 Model for contact angle hysteresis. J. Chem. Phys. 11, 552–562.CrossRefGoogle Scholar
Landau, L. D. & Levich, B. V. 1942 Dragging of a liquid by a moving plate. Acta Physicochim. URSS 17, 42–54.Google Scholar
Landau, L. D. & Lifschitz, E. M. 1959 Fluid Mechanics. Pergamon.Google Scholar
Le Grand, N., Daerr, A. & Limat, L. 2005 Shape and motion of drops sliding down an inclined plane. J. Fluid Mech. 541, 293–315.CrossRefGoogle Scholar
Münch, A. & Evans, P. L. 2005 Marangoni-driven liquid films rising out of a meniscus onto a nearly horizontal substrate. Physica D 209, 164–177.Google Scholar
Nikolayev, V. S. & Beysens, D. A. 2003 Equation of motion of the triple contact line along an inhomogeneous interface. Europhys. Lett. 64, 763–768.Google Scholar
Ondarçuhu, T. 1992 Relaxation modes of the contact line in situation of partial wetting. Mod. Phys. Lett. E 6, 901–916.CrossRefGoogle Scholar
Ondarçuhu, T. & Veyssié, M. 1991 Relaxation modes of the contact line of a liquid spreading on a surface. Nature 352, 418–420.CrossRefGoogle Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.CrossRefGoogle Scholar
Pismen, L. M. & Pomeau, Y. 2000 Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics. Phys. Rev. E 62, 2480–2492.Google ScholarPubMed
Pit, R., Hervet, H. & Léger, L. 2000 Interfacial properties on the submicron scale. Phys. Rev. Lett. 85, 980–983.CrossRefGoogle Scholar
Podgorski, T., Flesselles, J. M. & Limat, L. 2001 Corners, cusps and pearls in running drops. Phys. Rev. Lett. 87, 036102.CrossRefGoogle ScholarPubMed
Quéré, D. 1991 On the minimal velocity of forced spreading in partial wetting. C. R. Acad. Sci. Paris II 313, 313–318.Google Scholar
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, 0301608.Google ScholarPubMed
Rio, E., Daerr, A., Andreotti, B. & Limat, L. 2005 Boundary conditions in the vicinity of a dynamic contact line: experimental investigation of viscous drops sliding down an inclined plane. Phys. Rev. Lett. 94, 024503.Google Scholar
Sedev, R. V. & Petrov, J. G. 1991 The critical condition for transition from steady wetting to film entrainment. Colloids Surfaces 53, 147–156.CrossRefGoogle Scholar
Sekimoto, K., Oguma, R. & Kawasaki, K. 1987 Morphological stability analysis of partial wetting. Ann. Phys. 176, 359–392.Google Scholar
Snoeijer, J. H., Delon, G., Fermigier, M. & Andreotti, B. 2006 Avoided critical behavior in dynamically forced wetting. Phys. Rev. Lett. 96, 174504.Google Scholar
Thompson, P. A. & Robbins, M. O. 1989 Simulations of contact-line motion: slip and the dynamic contact angle. Phys. Rev. Lett. 63, 766–769.Google Scholar
Voinov, O. V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11, 714–721.CrossRefGoogle Scholar