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Moving contact line dynamics: from diffuse to sharp interfaces

  • H. Kusumaatmaja (a1), E. J. Hemingway (a1) and S. M. Fielding (a1)
Abstract

We reconcile two scaling laws that have been proposed in the literature for the slip length associated with a moving contact line in diffuse interface models, by demonstrating each to apply in a different regime of the ratio of the microscopic interfacial width $l$ and the macroscopic diffusive length $l_{D}=(M{\it\eta})^{1/2}$ , where ${\it\eta}$ is the fluid viscosity and $M$ the mobility governing intermolecular diffusion. For small $l_{D}/l$ we find a diffuse interface regime in which the slip length scales as ${\it\xi}\sim (l_{D}l)^{1/2}$ . For larger $l_{D}/l>1$ we find a sharp interface regime in which the slip length depends only on the diffusive length, ${\it\xi}\sim l_{D}\sim (M{\it\eta})^{1/2}$ , and therefore only on the macroscopic variables ${\it\eta}$ and $M$ , independent of the microscopic interfacial width $l$ . We also give evidence that modifying the microscopic interfacial terms in the model’s free energy functional appears to affect the value of the slip length only in the diffuse interface regime, consistent with the slip length depending only on macroscopic variables in the sharp interface regime. Finally, we demonstrate the dependence of the dynamic contact angle on the capillary number to be in excellent agreement with the theoretical prediction of Cox (J. Fluid Mech., vol. 168, 1986, p. 169), provided we allow the slip length to be rescaled by a dimensionless prefactor. This prefactor appears to converge to unity in the sharp interface limit, but is smaller in the diffuse interface limit. The excellent agreement of results obtained using three independent numerical methods, across several decades of the relevant dimensionless variables, demonstrates our findings to be free of numerical artefacts.

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Corresponding author
Email address for correspondence: halim.kusumaatmaja@durham.ac.uk
References
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Aarts, D. G. A. L., Dullens, R. P. A., Lekkerkerker, H. N. W., Bonn, D. & van Roij, R. 2004 Interfacial tension and wetting in colloid–polymer mixtures. J. Chem. Phys. 120, 19731980.
Anderson, D. M., McFadden, G. B. & Wheeler, A. A. 1998 Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139165.
Bray, A. J. 1994 Theory of phase-ordering kinetics. Adv. Phys. 43, 357459.
Briant, A. J., Wagner, A. J. & Yeomans, J. M. 2004 Lattice Boltzmann simulations of contact line motion. I. Liquid–gas systems. Phys. Rev. E 69, 031602.
Briant, A. J. & Yeomans, J. M. 2004 Lattice Boltzmann simulations of contact line motion. II. Binary fluids. Phys. Rev. E 69, 031603.
Cahn, J. W. 1977 Critical point wetting. J. Chem. Phys. 66, 36673672.
Chen, H.-Y., Jasnow, D. & Viñals, J. 2000 Interface and contact line motion in a two phase fluid under shear flow. Phys. Rev. Lett. 85, 16861689.
Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169.
Davidovitch, B., Moro, E. & Stone, H. A. 2005 Spreading of viscous fluid drops on a solid substrate assisted by thermal fluctuations. Phys. Rev. Lett. 95, 244505.
Diotallevi, F., Biferale, L., Chibbaro, S., Pontrelli, G., Toschi, F. & Succi, S. 2009 Lattice Boltzmann simulations of capillary filling: finite vapour density effects. Eur. Phys. J. Spec. Top. 171, 237243.
Eggers, J. 2004 Hydrodynamic theory of forced dewetting. Phys. Rev. Lett. 93, 094502.
Eyre, D. J.1998 An unconditionally stable one-step scheme for gradient systems. http://www.math.utah.edu/∼eyre/research/methods/stable.ps, pp. 1–15.
de Gennes, P. G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827863.
Gompper, G. & Zschocke, S. 1991 Elastic properties of interfaces in a Ginzburg–Landau theory of swollen micelles, droplet crystals and lamellar phases. Europhys. Lett. 16, 731736.
Gross, M. & Varnik, F. 2014 Spreading dynamics of nanodrops: a lattice Boltzmann study. Intl J. Mod. Phys. C 25, 1340019.
Guillén-González, F. & Tierra, G. 2013 On linear schemes for a Cahn–Hilliard diffuse interface model. J. Comput. Phys. 234, 140171.
Hu, D. L., Chan, B. & Bush, J. W. M. 2003 The hydrodynamics of water strider locomotion. Nature 424, 663666.
Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85101.
Jacqmin, D. 1999 Calculation of two-phase Navier–Stokes flows using phase-field modeling. J. Comput. Phys. 155, 96127.
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 5788.
Ladd, A. J. C. & Verberg, R. 2001 Lattice–Boltzmann simulations of particle–fluid suspensions. J. Stat. Phys. 104, 11911251.
Lai, M.-C. & Peskin, C. S. 2000 An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J. Comput. Phys. 160, 705719.
Lu, C.-Y. D., Olmsted, P. D. & Ball, R. C. 2000 Effects of nonlocal stress on the determination of shear banding flow. Phys. Rev. Lett. 84, 642645.
Mognetti, B. M., Kusumaatmaja, H. & Yeomans, J. M. 2010 Drop dynamics on hydrophobic and superhydrophobic surfaces. Faraday Discuss. 146, 153165.
Morrow, N. R. 1990 Wettability and its effect on oil recovery. J. Petrol. Tech. 42, 14761484.
Olmsted, P. D. & Lu, C.-Y. D. 1999 Phase coexistence of complex fluids in shear flow. Faraday Discuss. 112, 183194.
Parker, A. R. & Lawrence, C. R. 2001 Water capture by a desert beetle. Nature 414, 3334.
Peskin, C. S. 2002 The immersed boundary method. Acta Numerica 11, 479517.
Pooley, C. M., Kusumaatmaja, H. & Yeomans, J. M. 2008 Contact line dynamics in binary lattice Boltzmann simulations. Phys. Rev. E 78, 056709.
Pooley, C. M., Kusumaatmaja, H. & Yeomans, J. M. 2009 Modelling capillary filling dynamics using lattice Boltzmann simulations. Eur. Phys. J. Spec. Top. 171, 6371.
Pozrikidis, C. 2011 Introduction to Theoretical and Computational Fluid Dynamics. Oxford University Press.
Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. 1992 Numerical Recipes in C. Cambridge University Press.
Renardy, M., Renardy, Y. & Li, J. 2001 Numerical simulation of moving contact line problems using a volume-of-fluid method. J. Comput. Phys. 171, 243263.
Sbragaglia, M., Sugiyama, K. & Biferale, L. 2008 Wetting failure and contact line dynamics in a couette flow. J. Fluid Mech. 614, 471493.
Seppecher, P. 1996 Moving contact lines in the Cahn–Hilliard theory. Intl J. Engng Sci. 34, 977992.
Servantie, J. & Muller, M. 2008 Statics and dynamics of a cylindrical droplet under an external body force. J. Chem. Phys. 128, 014709.
Setu, S. A., Dullens, R. P. A., Hernndez-Machado, A., Pagonabarraga, I., Aarts, D. G. A. L. & Ledesma-Aguilar, R. 2015 Superconfinement tailors fluid flow at microscales. Nature Commun. 6, 7297.
Spelt, P. D.M. 2005 A level-set approach for simulations of flows with multiple moving contact lines with hysteresis. J. Comput. Phys. 207, 389404.
Tabeling, P. 2010 Introduction to Microfluidics. Oxford University Press.
Thampi, S. P., Adhikari, R. & Govindarajan, R. 2013 Do liquid drops roll or slide on inclined surfaces? Langmuir 29, 33393346.
Yue, P., Zhou, C. & Feng, J. J. 2010 Sharp-interface limit of the Cahn–Hilliard model for moving contact lines. J. Fluid Mech. 645, 279.
Zheng, Y., Gao, X. & Jiang, L. 2007 Directional adhesion of superhydrophobic butterfly wings. Soft Matt. 3, 178182.
Zhou, M.-Y. & Sheng, P. 1990 Dynamics of immiscible-fluid displacement in a capillary tube. Phys. Rev. Lett. 64, 882885.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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