Coffee rings and ridges: predicting late-time deposit profiles in drying droplets

Project Description

A spilled droplet of coffee that evaporates from a surface leaves behind a stain that is significantly stronger towards the edge of the deposit --- the famous `coffee-ring effect'. This phenomenon is not restricted to coffee and is in fact ubiquitous whenever a droplet containing a solute evaporates into the surrounding air. The mechanism behind coffee-ring formation is as follows: the droplet contact line becomes pinned on surface inhomogeneities so that, as it dries, liquid flows from the droplet bulk towards the edge to conserve mass, carrying solute with it. This effect has been studied extensively over recent decades due to its relevance in a variety of different industrial and engineering applications. The coffee-ring may be inhibited to promote an even deposit in, for example, the drying of ink droplets, while on the other hand it may also be encouraged, for example in printing micro-circuitry, colloidal patterning or in aligning DNA molecules for mapping purposes.

The majority of previous studies on the coffee-ring effect concentrate on the amount of solute mass transported to the pinned contact line, with a focus on the role of droplet geometry, substrate characteristics and different fluid mixtures (e.g. binary fluids) in the shape of the resulting coffee ring. However, at later stages of the process, the liquid free surface begins lose its concavity and may approach touchdown on the substrate in the droplet interior. As the liquid profile thins, the solute concentration begins to build up at these interior locations, which may lead to solute jamming, leaving an internal solute deposit at dryout - the `coffee ridge'.

In this project, we will investigate the formation of the coffee ridge in detail by developing a mathematical model for solute transport in the physically-relevant regime in which surface tension dominates the fluid dynamics, while advection dominates diffusion in the solute transport, except in the vicinity of the coffee ring and ridge. Beginning with the simplest possible example of an axisymmetric droplet evaporating diffusively into the surround air, we will use a combination of matched asymptotic analysis and numerical simulations to predict where the deposit grows as a function of the evaporation time. With this in hand, we will move on to consider more general droplet geometries and investigate the role the droplet profile plays in the internal accumulation of deposit. The model will shed insight onto the underlying physical mechanisms of ridge formation, as well as predicting the resulting deposit profiles, leading to the possibility of developing control mechanisms to dynamically alter the deposit in an industrial or engineering setting.