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

TL;DR

A lubrication-theory framework is developed to model a thin, particle-laden droplet on a flooded porous substrate undergoing simultaneous evaporation and imbibition. The authors derive coupled governing equations for the free surface, droplet and substrate flows, and particle-transport/deposition, introducing evaporation and imbibition numbers $\mathcal{E}$ and $\mathcal{I}$ and detailing four evolution modes: constant radius (CR), constant angle (CA), stick–slide (SS), and stick–jump (SJ). They obtain analytical expressions for droplet evolution, flow fields, particle transport, and final deposit patterns (ring, distributed, both, or multiple rings) with regime-dependent lifetimes and critical angles; in particular, a watershed boundary emerges in the mixed-transport regime separating surface- and substrate-deposited particles. The results provide predictive insight into control of final deposition patterns in applications such as ink-jet printing and diagnostics, and highlight how evaporation and imbibition shape flow and deposition even under similar mass-loss dynamics.

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.

Figures (20)

• Figure 1: Sketch of a thin, axisymmetric, particle-laden droplet on an infinitely thick, isotropic, flooded, porous substrate with interconnected pores undergoing simultaneous evaporation and imbibition.
• Figure 2: Sketches of the evolutions of the free surface $\hat{h}(\hat{r},\hat{t})$, the contact radius $\hat{R}(\hat{t})$, and the contact angle $\hat{\theta}(\hat{t})$, respectively. Each row of the figure corresponds to a different mode of evolution: (a) the CR mode, (b) the CA mode, (c) the SS mode, and (d) the SJ mode. The vertical dashed lines correspond to the lifetime of the droplet, while the additional vertical dashed lines in (c) correspond to the instant $\hat{t}=\hat{t}^*$ at which the droplet contact line de-pins, and the additional vertical dashed lines in (d) correspond to the instants $\hat{t}=\hat{t}_n$ ($n=1,2,3,\ldots$) at which the droplet contact line de-pins and jumps inwards from $\hat{R}_n$ to $\hat{R}_{n+1} (<\hat{R}_n)$. The arrows indicate the directions of increasing $\hat{t}$.
• Figure 3: Sketches of the final deposit patterns arising from particle-laden droplets evolving in the CR, CA, SS and SJ modes, respectively, namely (a) a ring deposit, (b) a distributed deposit, (c) a combined ring and distributed deposit, and (d) multiple ring deposits.
• Figure 4: The evolutions of (a) $\theta$ for a droplet evolving in the CR mode given by \ref{['eq:CR_evolution']} plotted as a function of $t/t_{\textrm{CR}}$ for ${\cal{E}}=1$ and ${\cal{I}}=2^n$ ($n=-15,-14,\ldots,15$) and (b) $R$ for a droplet evolving in the CA mode given by \ref{['eq:CA_evolution']} plotted as a function of $t/t_{\textrm{CA}}$ also for ${\cal{E}}=1$ and ${\cal{I}}=2^n$ ($n=-15,-14,\ldots,15$). The arrows indicate the directions of increasing ${\cal{I}}$.
• Figure 5: The evolutions of (a) $R$ and (b) $\theta$ for a droplet evolving in the SS mode given by \ref{['eq:SS_evolution_CA_phase']} and \ref{['eq:CR_evolution']} with \ref{['eq:tstar']} for pure evaporation (dashed line), simultaneous evaporation and imbibition with ${\cal{E}}={\cal{I}}=1$ (solid line), and pure imbibition (dotted line) plotted as functions of $t/t_{\textrm{SS}}$ when $\theta^*=1/2$.