Arrays of truncated silicon microcones fabricated by deep reactive ion etching with SU-8 photoresist tops exhibit three wetting regimes selected by cone height and pitch: the Cassie-Baxter state with trapped air, the fully penetrating Wenzel state, and an intermediate sunny-side-up state in which the liquid penetrates the texture beyond the droplet footprint while both the apparent contact angle and the contact angle hysteresis stay low. At specific height-pitch combinations the Cassie-Baxter state collapses into the Wenzel state within seconds without any external forcing. Follow-up work by the same group proposed a pinned-contact-line criterion that predicts the Cassie-Wenzel boundary on rigid conical substrates, but the microscopic origin and existence boundary of the sunny-side-up state and the stability limits and mechanism of the spontaneous collapse remain unexplained in the literature. The problem is to establish a predictive, physically grounded account of this wetting-state map.

Background

The apparent contact angle of a water droplet on a rough surface, and the contact angle hysteresis (the difference between advancing and receding angles), are conventionally described by two limiting models. In the Cassie-Baxter state the droplet sits on top of the texture with air trapped beneath it, giving high apparent angles and low hysteresis; in the Wenzel state the liquid fully wets the texture, and the apparent angle is set by roughness amplification of the chemistry. For posts with flat tops, increasing the pitch (center-to-center spacing) at fixed chemistry drives an abrupt Cassie-to-Wenzel transition, as observed by Barbieri et al. on perfluorinated microscale obstacles. Which state is realized, and how stable it is, is controlled by the solid area fraction (the ratio of pillar top area to projected area), the pillar height, and the shape of the lateral walls.

Park, Fernandino and Dorao (2017) fabricated arrays of truncated silicon microcones by photolithography and deep reactive ion etching (DRIE) through SU-8 photoresist masks, with the SU-8 deliberately left capping the cone tops as a local-wetting controller (control measurements showed less than 2 degrees of contrast between the SU-8-capped and etched reference surfaces). Across 64 samples (four total heights from 4 to 75 micrometers, sixteen pitches from 15 to 315 micrometers) they mapped three wetting regimes: Cassie-Baxter (only for the taller cones), Wenzel, and a regime they labeled sunny-side-up, observed for 35 micrometer tall cones at pitches of 20-40 micrometers, in which the liquid penetrates the texture beyond the droplet footprint while both the static angle and the hysteresis are low. They also reported abrupt transitions from Cassie-Baxter to Wenzel within a few seconds without external forcing near 55 micrometer height at 60 micrometer pitch and 75 micrometer height at 70 micrometer pitch, attributing the collapse to the decreased solid area fraction; the state boundaries on their map were drawn as hypothetical lines. The companion journal study by the same authors extended the map to 96 samples including hierarchical cones and showed superhydrophilic and impregnating Cassie regimes (a state in which the texture is flooded up to the cone tops and the droplet sits on the impregnated surface); using literature models it found that the sag transition (the liquid-air interface bending down until it touches the substrate between cones, per Patankar) was not the dominant collapse mechanism on these cones and suggested depinning (the contact line unpinning from the cone tops under the depinning critical pressure, per Blow and Yeomans) instead. Both studies kept the sidewall angle fixed near 65 degrees, dictated by the etch recipe.

The most direct later advance is the criterion study by Park, Ribe, Fernandino and Dorao (2023). For conical microstructures with the three-phase contact line pinned on the cone sidewalls, the criterion for the Cassie-Wenzel transition can be written as cos⁡θcr=(fs−1)/(r−fs)\cos\theta_{cr}=(f_s-1)/(r-f_s), where fsf_s is the fraction of the solid surface and rr is the solid-liquid interface roughness; the half apex angle of the cone enters through the contact angle relative to the horizon, and a penetration ratio near 0.8 reproduced the observed transition location on their silicon samples at several equilibrium contact angles. The same study showed that this rigid-substrate criterion fails for elastic replicas in PDMS (polydimethylsiloxane, a soft elastomer), whose deformation shifts the transition. The sunny-side-up state, however, is not named or analyzed in any of these follow-ups, and the term has not been adopted elsewhere in the wetting literature.

Problem Statement

For arrays of truncated silicon microcones of the type fabricated by Park, Fernandino and Dorao (silicon cones whose flat tops are capped with a layer of SU-8 photoresist, sidewall angle near 65 degrees, heights of order 4-75 micrometers, pitches of order 15-315 micrometers), establish a predictive, physically grounded account of the wetting-state map and its stability. Concretely: (1) explain the microscopic origin of the sunny-side-up state — the regime at low cone height and low pitch in which the liquid penetrates the texture beyond the droplet footprint while both the apparent contact angle and the contact angle hysteresis (the difference between the advancing and receding contact angles) remain low — and determine from geometry and equilibrium contact angle whether and where it exists on this platform, without a priori restricting its physical nature (equilibrium impregnating state, long-lived partially penetrating state, or metastable transient on the way to Wenzel); and (2) determine the mechanism and stability limits of the spontaneous, unforced collapse from the Cassie-Baxter state (the droplet resting on top of the texture with air trapped beneath it) to the Wenzel state (the liquid fully wetting the texture) observed near 55 micrometer height at about 60 micrometer pitch and 75 micrometer height at about 70 micrometer pitch, specifying for which height-pitch combinations the Cassie-Baxter state is stable, metastable, or absent. Both (1) and (2) are required, and the account must be predictive across the platform's height-pitch geometry — reproducing the regimes reported at the measured height-pitch combinations — rather than fitted to those combinations alone. The origin and mechanism required by (1) and (2) must likewise be part of that regime-reproducing account rather than a qualitative narrative on its own.

References

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