Simultaneous evaporation and imbibition of a droplet on a flooded porous substrate

TL;DR

This paper develops a mathematical model for a particle-laden droplet on a flooded porous substrate undergoing simultaneous evaporation and imbibition, revealing insights into droplet evolution, flow, and particle deposition relevant to various applications.

Contribution

It provides analytical expressions for droplet evolution and particle transport during simultaneous evaporation and imbibition, highlighting qualitative and quantitative behaviors.

Findings

A droplet undergoing pure imbibition in constant contact radius mode never fully imbibes.

Evaporation shortens the droplet's lifetime regardless of the mode.

Final particle deposit is independent of the physical mechanisms driving mass loss.

Abstract

A mathematical model for the evolution of, and deposition from, a thin particle-laden droplet on an infinitely thick, isotropic, flooded, porous substrate with interconnected pores undergoing simultaneous evaporation and imbibition is formulated and analysed. In particular, analytical expressions for the evolution of the droplet, as well as for the flow within the droplet and the substrate, and for the transport and deposition onto the substrate of the particles are obtained for droplets evolving in four different modes. While the physical mechanisms driving evaporation and imbibition are rather different, perhaps rather unexpectedly, it is found that there are a number of qualitative and quantitative similarities as well as differences in the resulting behaviour of the droplet as it loses mass to its environment. For example, it is shown that a droplet undergoing pure imbibition in the constant contact radius mode never completely imbibes, but if it evaporates (either with or without imbibition also occurring) then it has a finite lifetime, and increasing the strength of evaporation and/or imbibition shortens its lifetime. It is also shown that in the regime in which diffusion of particles is faster than axial advection but slower than radial advection of particles, the final deposit of particles left behind on the substrate after the droplet has completely evaporated and/or imbibed is independent of both the nature and the strength of the physical mechanism(s) driving the mass loss from the droplet. Not only are these results of theoretical interest, but they are also relevant to a wide variety of practical applications, including ink-jet printing, DNA chip manufacturing, and disease diagnostics, that would benefit from an improved ability to predict and/or control the final deposit pattern from a droplet undergoing simultaneous evaporation and imbibition.