Instabilities of ring-rivulets: Impact of substrate wettability

Rivulets and droplets are naturally appearing shapes when small amounts of liquid are deposited on a partially wettable substrate. Here we study, by means of numerical simulations, the dewetting dynamics of a ring-rivulet on substrates with various contact angles and wettability patterns. In particular, we consider, beyond the homogeous case, an annular band of lower contact angle as compared to the background and a constant radial gradient of contact angle, pointing either inward or outward from the centre. We show that by tuning the parameters characterizing the patterns, it is possible to control not only the stability of the rivulet, i.e. its breakup/collapse dynamics and the associated time scales, but also the dewetting morphology, in terms of number and position of the formed droplets.

💡 Research Summary

This paper investigates the dewetting dynamics of a circular rivulet (ring‑rivulet) deposited on partially wettable substrates, using high‑resolution numerical simulations based on the thin‑film equation (TFE). The authors first establish the baseline behavior on homogeneous substrates, where the key dimensionless parameter is the aspect ratio ψ₀ = 2 r₀ sin θ₀ / R₀ (r₀: minor radius, R₀: major radius, θ₀: equilibrium contact angle). For ψ₀ < 0.2 the rivulet undergoes a Rayleigh‑Plateau‑type breakup, forming multiple droplets; for ψ₀ > 0.2 it retracts and collapses into a single central droplet. The number of droplets n_d follows the linear stability analysis (LSA) prediction n_d ≈ π/(2 ψ₀) especially at low θ₀, while the breakup time τ_b grows with ψ₀ for small contact angles but becomes almost independent of ψ₀ for θ₀ > 10°, indicating that disjoining‑pressure effects dominate. Collapse time τ_c scales as τ_c ∝ (θ₀ ψ₀)⁻², a result derived analytically from a simplified radial dynamics equation.

The core contribution lies in exploring two classes of wettability patterns and demonstrating how they can be used to steer the rivulet toward either breakup or collapse, and to control the final droplet count and spatial arrangement.

1. Annular Band Pattern – A circular band of lower contact angle θ_a (10°–40°) is embedded in a background of higher contact angle θ_b = 60°. This creates an energetic barrier that suppresses the collapse mode. Simulations show that, regardless of ψ₀, the rivulet always breaks up (n_d > 1). The dependence of n_d on ψ₀ deviates from the LSA line, especially for larger θ_a, and the breakup time τ_b becomes largely governed by the local θ_a rather than ψ₀. The band thus acts as a confinement that decouples radial dynamics from the intrinsic instability, allowing a larger number of droplets even for relatively wide rings.

2. Radial Linear Gradient – The contact angle varies linearly with radial distance ξ: θ(ξ) = θ_a + (θ_b − θ_a) ξ / R₀. Two scenarios are examined: (i) inward‑increasing wettability (θ_a < θ_b) and (ii) outward‑increasing wettability (θ_a > θ_b). In case (i) the rivulet radius R(t) shrinks rapidly; steep gradients can completely suppress breakup, leading to a pure collapse. Moderate gradients allow simultaneous shrinkage and breakup, producing droplets that migrate toward the centre before coalescing. In case (ii) the outward gradient stabilizes the ring: R(t) reaches a plateau, and for the strongest gradients a slight initial expansion is observed. Thus, the gradient provides a tunable knob for both the speed of retraction and the balance between breakup and collapse.

The authors systematically map the parameter space (ψ₀, θ_a, θ_b) and present quantitative phase diagrams indicating where each dewetting pathway dominates. They also validate the analytical scaling laws for τ_b and τ_c against the simulation data, finding excellent agreement in the appropriate regimes.

From an application perspective, the ability to program the final droplet pattern by simply engineering substrate wettability opens new avenues in micro‑fabrication, coating technologies, and lab‑on‑a‑chip fluid handling. For instance, electrowetting can dynamically modify θ_a and θ_b, enabling real‑time reconfiguration of droplet arrays. The findings also suggest strategies for preventing unwanted dewetting in coating processes (by designing annular low‑wettability zones) or for deliberately generating ordered droplet lattices (by imposing controlled radial gradients).

In summary, the paper provides a comprehensive theoretical and computational framework for understanding and controlling the instabilities of ring‑rivulets. It identifies the aspect ratio ψ₀ and the contact angle θ₀ as fundamental stability parameters on uniform substrates, and demonstrates that patterned wettability—either as a confined annular band or as a radial gradient—offers powerful, experimentally accessible means to tailor the dewetting outcome, including the number, size, and placement of droplets. This work bridges fundamental fluid‑dynamics insights with practical surface‑engineering techniques, paving the way for advanced fluidic patterning technologies.