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

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When a spilled droplet of coffee dries on a tabletop, it is well known that it leaves behind a stain that is darker towards its edges. This is called the 'coffee-ring' effect and is not unique to coffee, but occurs in a multitude of situations involving an evaporating liquid droplet that contains an inert (non-evaporating) solute. The edge of the droplet - called the contact line - becomes pinned on the solid surface so that, as the liquid dries, to replace fluid lost at the contact line, a flow develops in the droplet that takes liquid from the droplet interior to the contact line. This flow carries solute along with it. The advection of the solute is then counteracted by diffusion near the contact line, which drives the development of the coffee ring. The coffee-ring effect can be exploited in many industrial problems, for example in printing microscale circuits or colloidal patterning, or aligning DNA using the outward flow. Engineers may also seek to counter the effect in applications where a uniform deposit may be desired, such as in spray coating or inkjet printing.The coffee-ring effect has therefore seen a significant amount of attention since its discovery. However, a less well-known phenomenon is the possibility of enhanced internal deposits developing as part of the same process. At later stages of drying, the droplet shape can alter significantly so that the liquid surface begins to dip in the centre, getting very close to the solid surface. This may lead to solute becoming trapped between the liquid surface and the solid. Moreover, the change in the droplet shape alters the flow pattern, further increasing the movement of solute to the droplet interior. These internal deposits are called 'coffee ridges' or 'coffee eyes' and have been seen previously in experiments involving droplets containing a polymer. However, coffee-ridge formation is comparatively poorly understood compared to its ring counterpart, despite its key role in the final residual patterns. In fact, in several applications, coffee ridges may be more problematic than coffee rings, for example in the printing of QLED screens. A better understanding of the physics behind coffee ridges alongside a means to accurately predict and understand their formation is therefore an important mathematical and engineering challenge.This project seeks to address this challenge by deriving a mathematical model for coffee-ridge formation. I will begin by considering a problem where a droplet evaporates in a shallow well. This configuration has two advantages. First, it will inhibit coffee-ring formation, so as to allow me to focus on the flow dynamics in the droplet interior and, hence, the coffee ridge. Second, such a configuration is used in the printing of OLED/QLED screens, so the model has direct industrial relevance. I will systematically derive a reduced model by exploiting the shallowness of the well and the dominant effect of surface tension in the droplet. I will then explore the evolution of the solute distribution within the droplet using a hybrid approach that combines matched asymptotic analysis with numerical simulations. Of particular interest are key characteristics of the evolving deposit such as the size and location of the coffee ridge. These results will then be compared to existing experimental data in the literature. I will then build upon these results to consider the more common configuration in which a droplet evaporates on a flat surface. By carefully analysing the concurrent formation of the coffee ring and the coffee ridge, I will discover how the flow patterns evolve in time and investigate the interplay between the two features as they grow. The model will be used to inform future applications of droplet drying in industry and engineering.