The nascent coffee ring with arbitrary droplet contact set: an asymptotic analysis

Abstract

We consider the effect of droplet geometry on the early-stages of coffee-ring formation during the evaporation of a thin droplet with an arbitrary simple, smooth, pinned contact line. We perform a systematic matched asymptotic analysis of the small capillary number, large solutal Péclet number limit for two different evaporative models: a kinetic model, in which the evaporative flux is effectively constant across the droplet, and a diffusive model, in which the flux is singular at the contact line. For both evaporative models, solute is transported to the contact line by a capillary flow in the droplet bulk, while local to the contact line, solute diffusion counters advection. The resulting interplay leads to the formation of the nascent coffee-ring profile. By exploiting a coordinate system embedded in the contact line, we solve explicitly the local leading-order problem, deriving a similarity profile (in the form of a gamma distribution) that describes the nascent coffee ring. Notably, for an arbitrary contact line geometry, the ring characteristics change due to the concomitant asymmetry in the shape of the droplet free surface, the evaporative flux (for diffusive evaporation) and the mass flux into the contact line. We utilize the asymptotic model to determine the effects of contact line geometry on the growth of the coffee ring for a droplet with an elliptical contact set. Our results offer mechanistic insight into the effect of contact line curvature on the development of the coffee ring from deposition up to jamming of the solute; moreover, our model predicts when finite concentration effects become relevant.

Evaporation of non-circular droplets

“…Moreover, a key factor in determining properties of the developing coffee ring in the diffusion-limited evaporative regime is the behaviour of the flux local to the contact line (Moore et al. 2021, 2022; Sáenz et al. 2017).…”

“…2017; Moore et al. 2021, 2022). However, in the present analysis, we shall focus on the final residue at the contact line – the ‘coffee ring’ – which simplifies the mathematics.…”

“…2021), diffusive effects (Moore et al. 2021; Moore, Vella & Oliver 2022), jamming effects (Popov 2005) and numerous attempts at control (Mampallil & Eral 2018). However, a critical avenue is the behaviour of non-circular droplets, with the only existing studies being numerical (Freed-Brown 2015; Sáenz et al.…”

“…2017) or considering the early stages of deposit formation (Moore et al. 2022). This is particularly surprising given the ubiquity of square/rectangular (Mai & Richerzhagen 2007) and hexagonal (Huo et al.…”

“…2017; Moore et al. 2022). Finally, given potential applications in, for example, printing OLEDs, we extend our analysis to consider regular polygonal droplets in § 5.1, and more complex shapes in § 5.2, presenting results for the evaporative flux for general droplets, as well as the internal flow and transport dynamics for large droplets.…”