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Contact lines over random topographical substrates. Part 1. Statics

Published online by Cambridge University Press:  11 February 2011

NIKOS SAVVA ,
GRIGORIOS A. PAVLIOTIS  and
SERAFIM KALLIADASIS
NIKOS SAVVA
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
GRIGORIOS A. PAVLIOTIS
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
SERAFIM KALLIADASIS*
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: s.kalliadasis@imperial.ac.uk
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Abstract

We investigate theoretically the statistics of the equilibria of two-dimensional droplets over random topographical substrates. The substrates are appropriately represented as families of certain stationary random functions parametrized by a characteristic amplitude and wavenumber. In the limit of shallow topographies and small contact angles, a linearization about the flat-substrate equilibrium reveals that the droplet footprint is adequately approximated by a zero-mean, normally distributed random variable. The theoretical analysis of the statistics of droplet shift along the substrate is highly non-trivial. However, for weakly asymmetric substrates it can be shown analytically that the droplet shift approaches a Cauchy random variable; for fully asymmetric substrates its probability density is obtained via Padé approximants. Generalization to arbitrary stationary random functions does not change qualitatively the behaviour of the statistics with respect to the characteristic amplitude and wavenumber of the substrate. Our theoretical results are verified by numerical experiments, which also suggest that on average a random substrate neither enhances nor reduces droplet wetting. To address the question of the influence of substrate roughness on wetting, a stability analysis of the equilibria must be performed so that we can distinguish between stable and unstable equilibria, which in turn requires modelling the dynamics. This is the subject of Part 2 of this study.

JFM classification

Interfacial Flows (free surface): Contact lines Drops and Bubbles: Drops Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 672 , 10 April 2011 , pp. 358 - 383
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Baker, G. A. & Graves-Morris, P. R. 1996 Padé Approximants. Cambridge University Press.CrossRefGoogle Scholar
Barabási, A.-L. & Stanley, H. E. 1995 Fractal Concepts in Surface Growth. Cambridge University Press.CrossRefGoogle Scholar
Berry, M. V. & Lewis, Z. V. 1980 On the Weierstrass–Mandelbrot fractal function. Proc. R. Soc. Lond. A 370 (1743), 459–484.Google Scholar
Bhushan, B. 2000 Surface roughness analysis and measurement techniques. In Modern Tribology Handbook (ed. Bhushan, B.), vol. 1, chap. 2, pp. 49–119. CRC Press LLC.CrossRefGoogle Scholar
Bhushan, B. & Majumdar, A. 1992 Elastic–plastic model for bifractal surfaces. Wear 153, 53–64.CrossRefGoogle Scholar
Bico, J., Tordeux, C. & Quéré, D. 2001 Rough wetting. Europhys. Lett. 55 (2), 214–220.CrossRefGoogle Scholar
Blake, T. D. 1993 Dynamic contact angles and wetting kinetics. In Wettability (ed. Berg, J. C.), pp. 251–310. Marcel Dekker.Google Scholar
Blossey, R. 2003 Self-cleaning surfaces–virtual realities. Nature Mater. 2, 301–306.CrossRefGoogle ScholarPubMed
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739–805.CrossRefGoogle Scholar
Borgs, C., De Coninck, J., Kotecký, R. & Zinque, M. 1995 Does the roughness of the substrate enhance wetting? Phys. Rev. Lett. 74 (12), 2292–2294.CrossRefGoogle ScholarPubMed
Breiman, L. 1992 Probability. SIAM.CrossRefGoogle Scholar
Callies, M. & Quéré, D. 2005 On water repellency. Soft Matt. 1, 55–61.CrossRefGoogle Scholar
Chung, J. Y., Youngblood, J. P. & Stafford, C. M. 2007 Anisotropic wetting on tunable micro-wrinkled surfaces. Soft Matt. 3, 1163–1169.CrossRefGoogle ScholarPubMed
Cox, R. G. 1983 The spreading of a liquid on a rough solid surface. J. Fluid Mech. 131, 1–26.CrossRefGoogle Scholar
Dussan, V. E. B., 1979 On the spreading of liquids on solid surfaces: static and dynamic contact lines. Annu. Rev. Fluid Mech. 11, 371–400.CrossRefGoogle Scholar
Edwards, S. F. & Wilkinson, D. R. 1982 The surface statistics of a granular aggregate. Proc. R. Soc. Lond. A 381, 17–31.Google Scholar
Geary, R. C. 1930 The frequency distribution of the quotient of two normal variates. R. Stat. Soc. J. 93, 442–446.CrossRefGoogle Scholar
de Gennes, P.-G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57 (3), 827–863.Google Scholar
de Gennes, P.-G., Brochard-Wyart, F. & Quéré, D. 2003 Capillarity and Wetting Phenomena. Springer.Google Scholar
Good, R. J. 1952 A thermodynamic derivation of Wenzel's modification of Young's equation for contact angles; together with a theory of hysteresis. J. Am. Chem. Soc. 74, 5041–5042.CrossRefGoogle Scholar
Greenwood, J. A. & Williamson, J. B. P. 1966 Nominally flat surfaces. Proc. R. Soc. Lond. A 295 (1442), 310–319.Google Scholar
Hazlett, R. D. 1990 Fractal applications: wettability and contact angle. J. Colloid Interface Sci. 173 (2), 527–533.CrossRefGoogle Scholar
Hazlett, R. D. 1992 On surface roughness effects in wetting phenomena. J. Adhes. Sci. Technol. 6 (6), 625–633.CrossRefGoogle Scholar
Hitchcock, S. J., Carroll, N. T. & Nicholas, M. G. 1981 Some effects of substrate roughness on wettability. J. Mater. Sci. 16, 714–732.CrossRefGoogle Scholar
Huh, C. & Mason, S. G. 1977 Effects of surface roughness on wetting (theoretical). J. Colloid Interface Sci. 60 (1), 11–38.Google Scholar
Jansons, K. M. 1985 Moving contact lines on a two-dimensional rough surface. J. Fluid Mech. 154, 1–28.CrossRefGoogle Scholar
Joanny, J. & de Gennes, P.-G. 1984 A model for contact angle hysteresis. J. Chem. Phys. 81 (1), 552–562.CrossRefGoogle Scholar
Johnson, R. E. & Dettre, R. H. 1964 Contact angle hysteresis. I. Study of an idealized rough surface. In Advances in Chemistry Series, vol. 43, Contact Angle, Wettability and Adhesion (ed. Fowkes, F. M.), pp. 112–135. American Chemical Society.CrossRefGoogle Scholar
van Kampen, N. G. 2007 Stochastic Processes in Physics and Chemistry, 3rd edn. Elsevier.Google Scholar
Katzav, E., Adda-Bedia, M. & Derrida, B. 2007 Fracture surfaces of heterogeneous materials: a 2D solvable model. Europhys. Lett. 78, 46006.CrossRefGoogle Scholar
Krishnamoorthy, K. 2006 Handbook of Statistical Distributions with Applications. Chapman & Hall/CRC.CrossRefGoogle Scholar
Longuet-Higgins, M. S. 1958 On the intervals between successive zeros of a random function. Proc. R. Soc. Lond. A 246, 99–118.Google Scholar
Majumdar, A. & Tien, C. L. 1990 Fractal characterization and simulation of rough surfaces. Wear 136, 313–327.CrossRefGoogle Scholar
Mandelbrot, B. B. 1982 The Fractal Geometry of Nature. Freeman.Google Scholar
McHale, G. 2007 Cassie and Wenzel: were they really so wrong? Langmuir 23, 8200–8205.CrossRefGoogle ScholarPubMed
Mulvaney, D. J., Newland, D. E. & Gill, K. F. 1989 A complete description of surface texture profiles. Wear 132, 173–182.CrossRefGoogle Scholar
Øksendal, B. 2003 Stochastic Differential Equations, 6th edn. Springer.CrossRefGoogle Scholar
Oliver, J. F., Huh, C. & Mason, S. G. 1977 The apparent contact angle of liquids on finely-grooved solid surfaces – a SEM study. J. Adhes. 8, 223–234.CrossRefGoogle Scholar
Oliver, J. F., Huh, C. & Mason, S. G. 1980 An experimental study of some effects of solid surface roughness on wetting. Colloids Surf. 1, 79–104.CrossRefGoogle Scholar
Palasantzas, G. & De Hosson, J. Th. M. 2001 Wetting on rough surfaces. Acta Mater. 49, 3533–3538.CrossRefGoogle Scholar
Pomeau, Y. & Vannimenus, J. 1985 Contact angle on heterogeneous surfaces: weak heterogeneities. J. Colloid Interface Sci. 104 (2), 477–488.CrossRefGoogle Scholar
Poon, C. Y. & Bhushan, B. 1995 Comparison of surface roughness measurements by stylus profiler, AFM and non-contact optical profiler. Wear 190, 76–88.CrossRefGoogle Scholar
Rice, S. O. 1945 The mathematical analysis of random noise. Bell Syst. Tech. J. 24, 46–156.CrossRefGoogle Scholar
Robbins, M. O. & Joanny, J. F. 1987 Contact angle hysteresis on random surfaces. Europhys. Lett. 3 (6), 729–735.CrossRefGoogle Scholar
Roberts, A. P. & Torquato, S. 1999 Chord-distribution functions of three-dimensional random media: approximate first-passage times of Gaussian processes. Phys. Rev. E 59 (5), 4953–4963.Google 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 2. Dynamics. J. Fluid Mech. 672, 384–410.CrossRefGoogle Scholar
Sayles, R. & Thomas, T. R. 1978 Surface topography as a non-stationary random process. Nature 271, 431–434.CrossRefGoogle Scholar
Shibuichi, S., Onda, T., Satoh, N. & Tsujii, K. 1996 Super water-repellent surfaces resulting from fractal structure. J. Phys. Chem. 100, 19512–19517.CrossRefGoogle Scholar
Thomas, T. R. 1999 Rough Surfaces, 2nd edn. Imperial College Press.Google Scholar
VanMarcke, E. 1983 Random Fields: Analysis and Synthesis. MIT Press.Google Scholar
Wenzel, R. N. 1936 Resistance of solid surfaces to wetting by water. Ind. Engng Chem. 28, 988–994.CrossRefGoogle Scholar
Whitehouse, D. J. 2001 Fractal or fiction. Wear 249, 345–353.Google Scholar
Whitehouse, D. J. & Archard, J. F. 1970 The properties of random surfaces of significance in their contact. Proc. R. Soc. Lond. A 316 (1524), 97–121.Google Scholar
Zeitak, R. 1997 Short time expansion for first-passage distributions. Phys. Rev. E 56 (3), 2560–2567.CrossRefGoogle Scholar
Zhou, X. B. & De Hosson, J. Th. M. 1995 Influence of surface roughness on the wetting angle. J. Mater. Res. 10 (8), 1984–1992.Google Scholar