Critical conditions and De-pinning/Sag crossover for evaporation-driven wetting transitions of droplets on three-dimensional micropillar arrays

When a droplet evaporates on a micropillar-textured superhydrophobic surface, the suspended Cassie-Baxter state eventually collapses into the fully wetting Wenzel state. Two canonical collapse mechanisms are known: De-pinning, where the contact line detaches from the pillar tops and slides down the sidewalls, and Sag, where the meniscus between pillars droops until it touches the substrate bottom. A 2020 two-dimensional lattice Boltzmann study (Dou and Hao) reproduced the De-pinning transition and confirmed its analytical critical radius. What remains open is a quantitative, three-dimensional account of the critical radii, prefactors, and De-pinning/Sag crossover in idealized three-dimensional micropillar arrays.

Background

Wetting on microtextured solid surfaces is governed by two canonical states. In the Wenzel state the liquid fully penetrates the texture, while in the Cassie-Baxter (or fakir) state the droplet rests on top of the microstructures with air trapped underneath; this suspended state produces the large apparent contact angles and low adhesion exploited by superhydrophobic self-cleaning, anti-icing, and drag-reducing surfaces (Bico et al. 2002; Quéré 2008). The intrinsic wettability of the flat material is set by Young's contact angle $\theta_Y$, and a micropillar array is described by its pillar width $d$, pitch (center-to-center spacing) $l$, and height $h$. The Cassie state on such arrays is frequently only metastable. As a droplet evaporates and shrinks, its Laplace pressure grows, and the liquid-air interface suspended between pillar tops can fail in two canonical ways: in the De-pinning mechanism the contact line pinned at the pillar edges detaches and slides down the pillar sidewalls, whereas in the Sag (touch-down) mechanism the meniscus bulges downward until it contacts the substrate bottom, after which the drop collapses immediately (Quéré 2008; Kusumaatmaja et al. 2008; Wang et al. 2022). Sharp-interface theory gives critical droplet radii for both routes: in two-dimensional geometry the De-pinning (free-energy-barrier) estimate takes the gap-based form $R^{*}=(l-d)/(2|\cos\theta_Y|)$, while the Sag estimate scales as $R^{*}_{\mathrm{sag}}\sim (l-d)^{2}/h$, so the dominant mechanism is expected to cross over as the geometry $(l,d,h)$ and $\theta_Y$ are varied.

The lattice Boltzmann (LB) method is a mesoscopic simulation technique that evolves discrete velocity distribution functions rather than directly discretizing the Navier-Stokes equations; in its pseudopotential multiphase variant, with a multiple-relaxation-time (MRT) collision operator and a coupled thermal model for liquid-vapor phase change, it can simulate evaporating droplets on textured substrates without explicit interface tracking. Dou and Hao (2020) applied such a coupled MRT pseudopotential-thermal LB model in two dimensions: they captured both Cassie and Wenzel states, verified the apparent contact angles against Cassie-Baxter/Wenzel theory, found coexistence of the two states when the substrate is moderately hydrophobic ($\theta_Y$ below a critical angle), observed the De-pinning Cassie-to-Wenzel transition once the evaporating droplet reached its critical radius, and obtained simulated critical radii in quantitative agreement with the analytical two-dimensional prediction, which in the sharp-interface lineage takes the gap-based form $R^{*}=(l-d)/(2|\cos\theta_Y|)$ (Kusumaatmaja et al. 2008). No Sag transition was reported in their two-dimensional simulations. The restriction to two dimensions is physically meaningful, not merely numerical: earlier quasi-three-dimensional simulations of evaporating drops on square post arrays (Kusumaatmaja et al. 2008) had already shown that three-dimensional geometry qualitatively changes the competition, since doubly curved menisci, the finite length of the pinned contact line, and the possibility of de-pinning from interior posts (leaving a bowl-shaped drop base suspended on its outer rim) make retraction and collapse interact in ways that two-dimensional theory does not capture. Experiments confirm that both mechanisms operate on real arrays and that simple analytical criteria misestimate the critical conditions there (Tsai et al. 2010; Papadopoulos et al. 2013; Shamim et al. 2023). What is still missing is a quantitative, three-dimensional account of the critical conditions and of the crossover between the two mechanisms.

Problem Statement

Determine the critical conditions under which the Cassie-Baxter state of an evaporating droplet on a three-dimensional micropillar array collapses into the Wenzel state, and how those conditions depend on the pillar geometry and wettability. Concretely:

1. Determine numerically in three dimensions the critical droplet radius (equivalently, critical Laplace pressure or volume) at which the Cassie state fails by the De-pinning mechanism and by the Sag mechanism, as a function of pillar geometry (pitch $l$, width $d$, height $h$) and Young's angle $\theta_Y$; test the gap-based analytical estimates $R^{*}=(l-d)/(2|\cos\theta_Y|)$ and $R^{*}_{\mathrm{sag}}\sim (l-d)^{2}/h$ (the two-dimensional sag prefactor derived in Kusumaatmaja et al. 2008 is $(l-d)^{2}/(8h)$) against the measured three-dimensional critical radii and report the measured prefactors; and establish the crossover conditions in the geometry-wettability parameter space at which the dominant collapse mechanism switches between De-pinning and Sag.

Two related directions remain open in the literature:

2. How these critical conditions depend on the thermal driving of the evaporation (the imposed superheat $\Delta T$, the temperature elevation of the heated substrate above the liquid's saturation temperature, and the associated evaporation rate), since transition thresholds measured under finite superheat may differ from their quasi-static values.
3. How realistic imperfections of the surface - geometric disorder of the pillar array, pillar edge defects, chemical heterogeneity, and contact-angle hysteresis - modify the metastability, the energy barriers, and the transition pathways of the Cassie state.

References

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