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Coupled imbibition and evaporation of droplets deposited on a nanoporous layer

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.