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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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