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Slipping moving contact lines: critical roles of de Gennes’s ‘foot’ in dynamic wetting

  • Hsien-Hung Wei (a1), Heng-Kwong Tsao (a2) and Kang-Ching Chu (a2)


In the context of dynamic wetting, wall slip is often treated as a microscopic effect for removing viscous stress singularity at a moving contact line. In most drop spreading experiments, however, a considerable amount of slip may occur due to the use of polymer liquids such as silicone oils, which may cause significant deviations from the classical Tanner–de Gennes theory. Here we show that many classical results for complete wetting fluids may no longer hold due to wall slip, depending crucially on the extent of de Gennes’s slipping ‘foot’ to the relevant length scales at both the macroscopic and microscopic levels. At the macroscopic level, we find that for given liquid height $h$ and slip length $\unicode[STIX]{x1D706}$ , the apparent dynamic contact angle $\unicode[STIX]{x1D703}_{d}$ can change from Tanner’s law $\unicode[STIX]{x1D703}_{d}\sim Ca^{1/3}$ for $h\gg \unicode[STIX]{x1D706}$ to the strong-slip law $\unicode[STIX]{x1D703}_{d}\sim Ca^{1/2}\,(L/\unicode[STIX]{x1D706})^{1/2}$ for $h\ll \unicode[STIX]{x1D706}$ , where $Ca$ is the capillary number and $L$ is the macroscopic length scale. Such a no-slip-to-slip transition occurs at the critical capillary number $Ca^{\ast }\sim (\unicode[STIX]{x1D706}/L)^{3}$ , accompanied by the switch of the ‘foot’ of size $\ell _{F}\sim \unicode[STIX]{x1D706}Ca^{-1/3}$ from the inner scale to the outer scale with respect to $L$ . A more generalized dynamic contact angle relationship is also derived, capable of unifying Tanner’s law and the strong-slip law under $\unicode[STIX]{x1D706}\ll L/\unicode[STIX]{x1D703}_{d}$ . We not only confirm the two distinct wetting laws using many-body dissipative particle dynamics simulations, but also provide a rational account for anomalous departures from Tanner’s law seen in experiments (Chen, J. Colloid Interface Sci., vol. 122, 1988, pp. 60–72; Albrecht et al., Phys. Rev. Lett., vol. 68, 1992, pp. 3192–3195). We also show that even for a common spreading drop with small macroscopic slip, slip effects can still be microscopically strong enough to change the microstructure of the contact line. The structure is identified to consist of a strongly slipping precursor film of length $\ell \sim (a\unicode[STIX]{x1D706})^{1/2}Ca^{-1/2}$ followed by a mesoscopic ‘foot’ of width $\ell _{F}\sim \unicode[STIX]{x1D706}Ca^{-1/3}$ ahead of the macroscopic wedge, where $a$ is the molecular length. It thus turns out that it is the ‘foot’, rather than the film, contributing to the microscopic length in Tanner’s law, in accordance with the experimental data reported by Kavehpour et al. (Phys. Rev. Lett., vol. 91, 2003, 196104) and Ueno et al. (Trans. ASME J. Heat Transfer, vol. 134, 2012, 051008). The advancement of the microscopic contact line is still led by the film whose length can grow as the $1/3$ power of time due to $\ell$ , as supported by the experiments of Ueno et al. and Mate (Langmuir, vol. 28, 2012, pp. 16821–16827). The present work demonstrates that the behaviour of a moving contact line can be strongly influenced by wall slip. Such slip-mediated dynamic wetting might also provide an alternative means for probing slippery surfaces.


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Albrecht, U., Otto, A. & Leiderer, P. 1992 Two-dimensional liquid polymer diffusion: experiment and simulation. Phys. Rev. Lett. 68, 31923195.10.1103/PhysRevLett.68.3192
Anderson, D. M. & Davis, S. H. 1995 The spreading of volatile liquid droplets on heated surfaces. Phys. Fluids 7, 248265.10.1063/1.868623
Ausserré, D., Picard, A. M. & Léger, L. 1986 Existence and role of the precursor film in the spreading of polymer liquids. Phys. Rev. Lett. 57, 26712674.10.1103/PhysRevLett.57.2671
Barry, A. J. 1946 Viscometric investigation of dimethylsiloxane polymers. J. Appl. Phys. 17, 10201024.10.1063/1.1707670
Beaglehole, D. 1989 Profiles of the precursor of spreading drops of siloxane oil on glass, fused silica, and mica. J. Phys. Chem. 93, 893899.10.1021/j100339a067
Benintendi, S. W. & Smith, M. K. 1999 The spreading of a non-isothermal liquid droplet. Phys. Fluids 11, 982989.10.1063/1.869970
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739805.10.1103/RevModPhys.81.739
Brochard, F. & de Gennes, P. G. 1984 Spreading laws for liquid polymer droplets: interpretation of the ‘foot’. J. Phys. Lett. 45, 597602.10.1051/jphyslet:019840045012059700
Brochard-Wyart, F., de Gennes, P. G., Hervert, H. & Redon, C. 1994 Wetting and slippage of polymer melts on semi-ideal surfaces. Langmuir 10, 15661572.10.1021/la00017a040
Chan, K. Y. & Borhan, A. 2006 Spontaneous spreading of surfactant-bearing drops in the sorption-controlled limit. J. Colloid Interface Sci. 302, 374377.10.1016/j.jcis.2006.05.068
Chan, T. K., McGraw, J. D., Salez, T., Seemann, R. & Brinkmann, M. 2017 Morphological evolution of microscopic dewetting droplets with slip. J. Fluid Mech. 828, 271288.10.1017/jfm.2017.515
Chen, J.-D. 1988 Experiments on a spreading drop and its contact angle on a solid. J. Colloid Interface Sci. 122, 6072.10.1016/0021-9797(88)90287-1
Colinet, P. & Rednikov, A. 2011 On integrable singularities and apparent contact angles within a classical paradigm. Eur. Phys. J. Spec. Top. 197, 89113.10.1140/epjst/e2011-01443-x
Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169194.10.1017/S0022112086000332
Duffy, B. R. & Wilson, S. K. 1997 A third-order differential equation arising in thin-film flows and relevant to Tanner’s law. Appl. Math. Lett. 10, 6368.10.1016/S0893-9659(97)00036-0
Eggers, J. & Stone, H. A. 2004 Characteristic lengths at moving contact lines for a perfectly wetting fluid: the influence of speed on the dynamic contact angle. J. Fluid Mech. 505, 309321.10.1017/S0022112004008663
Eggers, J. 2004 Toward a description of contact line motion at higher capillary numbers. Phys. Fluids 16, 34913494.10.1063/1.1776071
Eggers, J. 2005a Contact line motion for partially wetting fluids. Phys. Rev. E 72, 061605.10.1103/PhysRevE.72.061605
Eggers, J. 2005b Existence of receding and advancing contact line. Phys. Fluids 17, 082106.10.1063/1.2009007
Eggers, J. & Fontelos, M. A. 2015 Singularities: Formation, Structure, and Propagation. Cambridge University Press.10.1017/CBO9781316161692
de Gennes, P. G. 1979 Écoulements viscométriques de polymères enchevêtrés. C. R. Acad. Sci. Paris Sér. B 288, 219220.
de Gennes, P. G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827863.10.1103/RevModPhys.57.827
de Gennes, P. G. 1986 Deposition of Langmuir–Blodgett layers. Colloid Polym. Sci. 264, 463465.10.1007/BF01419552
Haley, P. J. & Miksis, M. J. 1991 The effect of the contact line on droplet spreading. J. Fluid Mech. 223, 5781.10.1017/S0022112091001337
Halpern, D., Li, Y.-C. & Wei, H.-H. 2015 Slip-induced suppression of Marangoni film thickening in surfactant-retarded Landau–Levich–Bretherton flows. J. Fluid Mech. 781, 578594.10.1017/jfm.2015.508
Halpern, D. & Wei, H.-H. 2017 Slip-enhanced drop formation in liquid falling down a vertical fibre. J. Fluid Mech. 820, 4260.10.1017/jfm.2017.202
Hervet, H. & de Gennes, P. G. 1984 Dynamique du mouillage: films précurseurs sur solid ‘sec’. C. R. Acad. Sci. Paris II 299, 499503.
Hocking, L. M. 1977 A moving fluid interface. Part 2. The removal of the force singularity by a slip flow. J. Fluid Mech. 79, 209229.10.1017/S0022112077000123
Hocking, L. M. 1983 The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Maths 36, 5569.10.1093/qjmam/36.1.55
Hocking, L. M. 1992 Rival contact-angle models and the spreading of drops. J. Fluid Mech. 239, 671681.10.1017/S0022112092004579
Hocking, L. M. & Rivers, A. D. 1982 The spreading of a drop by capillary action. J. Fluid Mech. 121, 425442.10.1017/S0022112082001979
Hoffman, R. L. 1975 A study of the advancing interface. I. Interface shape in liquid-gas systems. J. Colloid Interface Sci. 50, 228241.10.1016/0021-9797(75)90225-8
Huh, C. & Mason, S. G. 1977 The steady movement of a liquid meniscus in a capillary tube. J. Fluid Mech. 81, 401419.10.1017/S0022112077002134
Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85101.10.1016/0021-9797(71)90188-3
Kalliadasis, S. & Chang, H.-C. 1996 Dynamics of liquid spreading on solid surfaces. Ind. Engng Chem. Res. 35, 28602874.10.1021/ie950670r
Karapetsas, G., Sahu, K. C. & Matar, O. K. 2013 Effect of contact line dynamics on the thermocapillary motion of a droplet on an inclined plate. Langmuir 29, 88928906.10.1021/la4014027
Kavehpour, H. P., Ovryn, B. & McKinley, G. H. 2003 Microscopic and macroscopic structure of the precursor layer in spreading viscous drops. Phys. Rev. Lett. 91, 196104.10.1103/PhysRevLett.91.196104
Lacey, A.-A. 1982 The motion with slip of a thin viscous droplet over a solid surface. Stud. Appl. Maths 67, 217230.10.1002/sapm1982673217
Léger, L., Erman, M., Guinet-Picard, A. M., Ausserré, D. & Strazielle, C. 1998 Precursor film profiles of spreading liquid drops. Phys. Rev. Lett. 60, 23902393.10.1103/PhysRevLett.60.2390
Levinson, P., Cazabat, A. M., Cohen-Stuart, M. A., Heslot, F. & Nicolet, S. 1988 The spreading of macroscopic droplets. Revue Phys. Appl. 23, 10091016.10.1051/rphysap:019880023060100900
Li, Y.-C., Liao, Y.-C., Wen, T. C. & Wei, H.-H. 2014 Breakdown of the Bretherton law due to wall slippage. J. Fluid Mech. 741, 200227.10.1017/jfm.2013.562
Liao, Y.-C., Li, Y.-C., Chang, Y.-C., Huang, C.-Y. & Wei, H.-H. 2014 Speeding up thermocapillary migration of a confined bubble by wall slip. J. Fluid Mech. 746, 3152.10.1017/jfm.2014.117
Liao, Y.-C., Li, Y.-C. & Wei, H.-H. 2013 Drastic changes in interfacial hydrodynamics due to wall slippage: slip-intensified film thinning, drop spreading, and capillary instability. Phys. Rev. Lett. 111, 136001.10.1103/PhysRevLett.111.136001
Marsh, J. A., Garoff, S. & Dussan V., E. B. 1993 Dynamic contact angles and hydrodynamics near a moving contact line. Phys. Rev. Lett. 70, 27782781.10.1103/PhysRevLett.70.2778
Mate, C. M. 2012 Anomalous diffusion kinetics of the precursor film that spreads from polymer droplets. Langmuir 28, 1682116827.10.1021/la3041117
McHale, G., Shirtcliffe, N. J., Aqil, S., Perry, C. C. & Newton, M. I. 2004 Topography driven spreading. Phys. Rev. Lett. 93, 036102.10.1103/PhysRevLett.93.036102
Münch, A., Wagner, B. & Witelski, T. P. 2005 Lubrication models with small to large slip lengths. J. Engng Maths 53, 359383.10.1007/s10665-005-9020-3
Navier, C. L. 1823 (appeared in 1827) Memoire sur les lois du mouvement des fluides. Mem. Acad. R. Sci. France 6, 389440.
Noble, B. A., Mate, C. M. & Raeymaekers, B. 2017 Spreading kinetics of ultrathin liquid films using molecular dynamics. Langmuir 33, 34763483.10.1021/acs.langmuir.7b00334
Pahlavan, A. A., Cueto-Felgueroso, L., McKinley, G. H. & Juanes, R. 2015 Thin films in partial wetting: internal selection of contact-line dynamics. Phys. Rev. Lett. 115, 034502.
Savva, N. & Kalliadasis, S. 2009 Two-dimensional droplet spreading over topographical substrates. Phys. Fluids 21, 092102.10.1063/1.3223628
Savva, N. & Kalliadasis, S. 2011 Dynamics of moving contact lines: a comparison between slip and precursor film models. Europhys. Lett. 94, 64004.10.1209/0295-5075/94/64004
Sibley, D. N., Nold, A. & Kalliadasis, S. 2015 The asymptotics of the moving contact line: cracking an old nut. J. Fluid Mech. 764, 445462.10.1017/jfm.2014.702
Snoeijer, J. H. 2006 Free-surface flows with large slopes: beyond lubrication theory. Phys. Fluids 18, 021701.10.1063/1.2171190
Snoeijer, J. H. & Andreotti, B. 2013 Moving contact lines: scales, regimes, and dynamical transitions. Annu. Rev. Fluid Mech. 45, 269292.10.1146/annurev-fluid-011212-140734
Ström, G., Fredriksson, M., Stenius, P. & Radoev, B. 1990 Kinetics of steady-state wetting. J. Colloid Interface Sci. 134, 107116.10.1016/0021-9797(90)90256-N
Tanner, L. H. 1979 The spreading of silicone oil drops on horizontal surfaces. J. Phys. D 12, 14731484.10.1088/0022-3727/12/9/009
Ueno, I., Hirose, K., Kizaki, Y., Kisara, Y. & Fukuhara, Y. 2012 Precursor film formation process ahead macroscopic contact line of spreading droplet on smooth substrate. Trans. ASME J. Heat Transfer 134, 051008.10.1115/1.4005638
Voinov, O. V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11, 714721.10.1007/BF01012963
Wei, H.-H. 2018 Marangoni-enhanced capillary wetting in surfactant-driven superspreading. J. Fluid Mech. 855, 181209.10.1017/jfm.2018.626
Weng, Y.-H., Wu, C.-J., Tsao, H.-K. & Sheng, Y.-J. 2017 Spreading dynamics of a precursor film of nanodrops on total wetting surfaces. Phys. Chem. Chem. Phys. 19, 2778627794.10.1039/C7CP04979J
Zhou, M.-Y. & Sheng, P. 1990 Dynamics of immiscible-fluid displacement in a capillary tube. Phys. Rev. Lett. 64, 882885.10.1103/PhysRevLett.64.882
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Slipping moving contact lines: critical roles of de Gennes’s ‘foot’ in dynamic wetting

  • Hsien-Hung Wei (a1), Heng-Kwong Tsao (a2) and Kang-Ching Chu (a2)


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