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The nascent coffee ring: how solute diffusion counters advection

Published online by Cambridge University Press:  18 June 2021

Madeleine Rose Moore Open the ORCID record for Madeleine Rose Moore [Opens in a new window] ,
D. Vella Open the ORCID record for D. Vella [Opens in a new window]  and
J.M. Oliver
Madeleine Rose Moore*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
D. Vella
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
J.M. Oliver
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, United Kingdom
*
†Email address for correspondence: moorem@maths.ox.ac.uk
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Abstract

We study the initial evolution of the coffee ring that is formed by the evaporation of a thin, axisymmetric, surface-tension-dominated droplet containing a dilute solute. When the solutal Péclet number is large, we show that diffusion close to the droplet contact line controls the coffee-ring structure in the initial stages of evaporation. We perform a systematic matched asymptotic analysis for two evaporation models – a simple, non-equilibrium, one-sided model (in which the evaporative flux is taken to be constant across the droplet surface) and a vapour-diffusion limited model (in which the evaporative flux is singular at the contact line) – valid during the early stages in which the solute remains dilute. We call this the ‘nascent coffee ring’ and describe the evolution of its features, including the size and location of the peak concentration and a measure of the width of the ring. Moreover, we use the asymptotic results to investigate when the assumption of a dilute solute breaks down and the effects of finite particle size and jamming are expected to become important. In particular, we illustrate the limited validity of this model in the diffusive evaporative flux regime.

JFM classification

Phase change: Condensation/evaporation Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 920 , 10 August 2021 , A54
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

Article last updated 07 March 2023

References

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