Hostname: page-component-77c89778f8-fv566 Total loading time: 0 Render date: 2024-07-17T23:35:49.395Z Has data issue: false hasContentIssue false

Contact lines over random topographical substrates. Part 2. Dynamics

Published online by Cambridge University Press:  11 February 2011

Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
Email address for correspondence:


We examine the dynamics of a two-dimensional droplet spreading over a random topographical substrate. Our analysis is based on the formalism developed in Part 1 of this study, where a random substrate was modelled as band-limited white noise. The system of integrodifferential equations for the motion of the contact points over deterministic substrates derived by Savva and Kalliadasis (Phys. Fluids, vol. 21, 2009, 092102) is applicable to the case of random substrates as well. This system is linearized for small substrate amplitudes to obtain stochastic differential equations for the droplet shift and contact line fluctuations in the limit of shallow and slowly varying topographies. Our theoretical predictions for the time evolution of the statistical properties of these quantities are verified by numerical experiments. Considering the statistics of the dynamics allows us to fully address the influence of the substrate variations on wetting. For example, we demonstrate that the droplet wets the substrate less as the substrate roughness increases, illustrating also the possibility of a substrate-induced hysteresis effect. Finally, the analysis of the long-time limit of spreading dynamics for a substrate represented by a band-limited white noise is extended to arbitrary substrate representations. It is shown that the statistics of spreading is independent of the characteristic length scales that naturally arise from the statistical properties of a substrate representation.

Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover.Google Scholar
Breiman, L. 1992 Probability. SIAM.Google Scholar
Cazabat, A. M. & Cohen-Stuart, M. A. 1986 Dynamics of wetting: effects of surface roughness. J. Phys. Chem. 90, 58455849.Google Scholar
Chung, J. Y., Youngblood, J. P. & Stafford, C. M. 2007 Anisotropic wetting on tunable micro-wrinkled surfaces. Soft Matt. 3, 11631169.Google Scholar
Cox, R. G. 1983 The spreading of a liquid on a rough solid surface. J. Fluid Mech. 131, 126.Google Scholar
Cramér, H. 1962 Random Variables and Probability Distributions, 2nd edn. Cambridge University Press.Google Scholar
Ehrhard, P. & Davis, S. H. 1991 Non-isothermal spreading of liquid drops on horizontal plates. J. Fluid Mech. 229, 365388.Google Scholar
Gardiner, C. W. 1985 Handbook of Stochastic Methods. Springer.Google Scholar
Gaskell, P. H., Jimack, P. K., Sellier, M. & Thompson, H. M. 2004 Efficient and accurate time adaptive multigrid simulations of droplet spreading. Intl J. Numer. Meth. Fluids 45, 11611186.Google Scholar
Gramlich, C. M., Mazouchi, A. & Homsy, G. M. 2004 Time-dependent free surface Stokes flow with a moving contact line. II. Flow over wedges and trenches. Phys. Fluids 16, 16601667.CrossRefGoogle Scholar
Hocking, L. M. 1983 The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Math. 36, 5569.Google Scholar
Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35 (1), 85101.Google Scholar
Kalliadasis, S. 2000 Nonlinear instability of a contact line driven by gravity. J. Fluid Mech. 413, 355378.Google Scholar
Katzav, E., Adda-Bedia, M., Amar, M. Ben & Boudaoud, A. 2007 Roughness of moving elastic lines: crack and wetting fronts. Phys. Rev. E 76, 051601.CrossRefGoogle ScholarPubMed
Moulinet, S., Guthmann, C. & Rolley, E. 2002 Roughness and dynamics of a contact line of a viscous fluid on a disordered substrate. Eur. Phys. J. E 8, 437443.Google Scholar
Nikolayev, V. S. 2005 Dynamics and depinning of the triple contact line in the presence of periodic surface defects. J. Phys. Condens. Matter 17, 21112119.Google Scholar
Rednikov, A. Y., Rossomme, S. & Colinet, P. 2009 Steady microstructure of a contact line for a liquid on a heated surface overlaid with its pure vapor: parametric study for a classical model. Multiphase Sci. Technol. 21, 213248.CrossRefGoogle Scholar
Rice, S. O. 1939 The distribution of maxima of a random curve. Am. J. Math. 61, 409416.CrossRefGoogle Scholar
Rice, S. O. 1945 The mathematical analysis of random noise. Bell Syst. Tech. J. 24, 46156.CrossRefGoogle Scholar
Savva, N. & Kalliadasis, S. 2009 Two-dimensional droplet spreading over topographical substrates. Phys. Fluids 21, 092102.CrossRefGoogle Scholar
Savva, N., Pavliotis, G. A. & Kalliadasis, S. 2011 Contact lines over random topographical substrates. Part 1. Statics. J. Fluid Mech. 672, 358383.Google Scholar
Schwartz, L. W. 1998 Hysteretic effects in droplet motions on heterogeneous substrates: direct numerical simulation. Langmuir 14, 34403453.Google Scholar
Schwartz, L. W. & Eley, R. R. 1998 Simulation of droplet motion on low-energy and heterogeneous surfaces. J. Colloid Interface Sci. 202, 173188.Google Scholar
Sodtke, C., Ajaev, V. S. & Stephan, P. 2008 Dynamics of volatile liquid droplets on heated surfaces: theory versus experiment. J. Fluid Mech. 610, 343362.CrossRefGoogle Scholar
Tanguy, A. & Vettorel, T. 2004 From weak to strong pinning. I. A finite size study. Eur. Phys. J. B 38, 7182.Google Scholar
Troian, S., Herbolzheimer, S., Safran, S. & Joanny, J. 1989 Fingering instabilities of driven spreading films. Europhys. Lett. 10, 2539.CrossRefGoogle Scholar