Coupled imbibition and evaporation of droplets deposited on a nanoporous layer

TL;DR

This work tackles the coupled dynamics of droplet imbibition and evaporation on a thin nanoporous layer, revealing a halo around the droplet whose growth is controlled by RH and Kelvin effects. The authors develop a unified 1D and 2D theoretical framework incorporating mass conservation, Darcy flow, and phase-change thermodynamics, and validate it with experiments on oxidized porous silicon. Key findings include RH-driven divergence of halo size due to the Kelvin effect, an apparent RH-dependent imbibition coefficient $w$, and the proposal that lateral vapor transport along the surface significantly contributes to halo dynamics. The results highlight the need to account for nanoscale thermodynamics and surface-transport coupling when interpreting halo dynamics and suggest RH as a powerful control parameter for designing infiltration patterns in porous materials with potential applications in sensing, printing, and actuation.

Abstract

Liquids in nanoscale hydrophilic pores generate capillary pressures so large that they could theoretically climb kilometers against gravity. However, droplets on thin nanoporous layers form imbibition fronts stopping at millimeters or less due to evaporation competing with capillary flow. Such droplet infiltration dynamics is of growing interest for studying confined fluids and for applications such as water harvesting, printing, chemical delivery, actuation, and sensing. Here, we investigate theoretically and experimentally the spontaneous imbibition and evaporation of sessile droplets into thin mesoporous layers, focusing on their dependence on imposed relative humidity (RH). Theoretically, we provide a unified analytical approach for the dynamics of the wetted annulus ("halo") around the droplet, accounting for arbitrary halo dimensions and confinement-induced thermodynamic shifts (Kelvin effect). Experimentally, we study water droplets on oxidized porous silicon layers (pore diameter 3-4 nm, thickness 5 $μ$m), systematically investigating how halo and droplet dynamics depend on RH. We show that halo formation timescales diverge at a critical RH due to the Kelvin effect, as illustrated by comparing RH-dependent evaporation rates in the halo (confined liquid) and in the droplet (bulk liquid). Our analysis also reveals an apparent divergence of the imbibition coefficient, unexplained by standard capillary models, suggesting a key role for Kelvin-driven vapor transport along the porous surface. The complex couplings revealed by our study call for caution in interpreting halo dynamics data. Our results also highlight RH as a powerful control parameter for tuning droplet imbibition behavior and infiltration patterns.

Figures (11)

• Figure 1: Experimental setup. (a) In a chamber controlled in relative humidity ($\mathcal{H}$), we deposit a water droplet on a thin porous layer (oxidized porous silicon). Spontaneous imbibition results in a wetted zone in the pores (the halo) of extension $L(t)$; $R_\mathrm{drop}$ is the fixed radius of the pinned, sessile drop. The inset shows a schematic close-up view of the imbibition front at the edge of the halo. A camera with a macro lens and LED ring allows us to record top view images of the drop. (b) Pseudo-color, background-subtracted image of a 1 droplet at a relative humidity of $\mathcal{H} = 0.3$ (30RH), at a time, $t = \qty{53}{\second}$ after deposition on the sample. The inner dashed circle represents the droplet's contact line on the substrate, and the outer dashed circle shows the external limit of the halo. The bright circular shapes visible inside the drop are optical reflections of the LED illumination, which we use to estimate the droplet's shape and volume.
• Figure 2: Water sorption isotherm: liquid filling fraction of the pores, $f$, as a function of the imposed relative humidity, $\mathcal{H} = p / p_\mathrm{sat}$, following either condensation (increase of RH, gray dots) or evaporation (decrease of RH, black dots) branches. Five successive condensation/evaporation cycles are shown. The data was obtained with a method based on white light interferometry. The shaded, blue region corresponds to the typical equilibrium RH of the confined fluid, $\mathcal{H}_\mathrm{eq}$, see text.
• Figure 3: 1D model for halo dynamics. (a) Pressure field (Equation \ref{['eq : PressureFieldDimensionless']}) for different position of the imbibition front ($\tilde{L}=0.1$, $0.2$, $0.4$, $0.6$ and $1$ from left to right where $\tilde{L} = L / L^\ast$ ; $\tilde{L}=1$ correspond to the steady-state while $\tilde{L} < 1$ represent a growing halo). (b) Corresponding solution for the position of the front squared as a function of time (Equation \ref{['eq : FrontEquationSquared_1D']}, continuous line); limiting regimes (early time and steady-state) are also displayed as dashed lines.
• Figure 4: 2D model for halo dynamics. (a) Numerical solution for the square of the position of the front as a function of time $L^2(t) = (R(t) - R_\mathrm{drop})^2$ (Equation \ref{['eq : FrontDifferentialEquation_2D']}, continuous blue line, solved in the case $L^\ast / R_\mathrm{drop} = 2$). Analytical solutions for limiting regimes (early time, Equation \ref{['eq : FrontDynamics2D_Init']} and steady-state, Equation \ref{['eq : Rmax']}) are also displayed as dashed lines. The continuous pale gray line is the prediction from the 1D model (Equation \ref{['eq : FrontEquationSquared_1D']}) for comparison. (b) Data in (a) plotted in terms of the effective halo size, $L_\mathrm{eff}$ (Equations \ref{['eq : Leff']}-\ref{['eq : Leff_L']}): continuous blue line: numerical simulation, continuous light gray line (indistinguishable from the previous one): analytical solution (Equation \ref{['eq : FrontEquationSquared_2D_Leff']}), dashed lines: limiting cases (Equations \ref{['eq : FrontDynamics2D_Init_Leff']} and \ref{['eq : Leff_max']}). This graph is identical to Figure \ref{['fig : Theory_1D']}b, replacing $L$ by $L_\mathrm{eff}$. Inset: ratio between effective halo extension $L_\mathrm{eff}$ and actual halo extension $L$, as a function of relative size between halo extension and droplet radius, calculated using Equation \ref{['eq : Leff_L']}.
• Figure 5: Complete dynamics after deposition on the mesoporous substrate of a 1 water droplet, at $\mathcal{H} = \qty{30}{\percent}$RH. After droplet deposition with the micro-pipette (first image), we observe 4 stages in the dynamics: i) fast spreading, ii) halo growth, iii) steady-state halo and droplet evaporation, iv) recession of droplet and halo. Note that time intervals between images are not constant due to various time scales in the dynamics. Images are background-subtracted and displayed in pseudocolors as in Figure \ref{['fig : ExpSetup']}b.