Controlled dripping from a grooved condensing plate

TL;DR

The paper addresses the poorly understood edge dripping of condensate on vertical surfaces and demonstrates that texture geometry can convert stochastic drainage into spatially organized, temporally regular release. By using forced condensation and patterns of parallel and convergent grooves, the authors show that groove spacing $s$, depth $d$, and width $w$ control the transition from gravity-driven to capillarity-organized dripping, including the localization of dripping to fixed points with convergent basins. A simple condensation–capillarity model, $\tau = \alpha\, \frac{m_{hd}}{c\,b\,(L-b/4)}$ with $\alpha \approx 0.25$, captures how basin width $b$ governs the dripping period, and experimental data collapse across basin widths confirms basins as independent capillary attractors. Overall, geometry provides a robust handle to control both where and when condensed water exits the substrate, with practical implications for dew harvesting, passive cooling, and millifluidic transport.

Abstract

Condensed water on vertical surfaces ultimately leaves the substrate at the lower edge, where accumulated liquid detaches as drops. While droplet growth and surface transport have been extensively studied, this final release step remains poorly understood and largely uncontrolled. Yet this boundary event determines how and when condensed water is removed. We ask whether geometry can replace randomness as the governing mechanism of edge dripping. By engraving vertical grooves upstream, we redirect water from surface flow into groove-guided drainage toward the boundary. This switch in transport mode changes how liquid accumulates and detaches at the edge. Using rapid forced condensation and high-resolution imaging, we systematically vary groove spacing s, aspect ratio d/w, and orientation. We then analyse how these geometric parameters influence the formation, stability, and spatial organization of droplets hanging below the edge. Smooth substrates exhibit irregular, impact-driven detachment. Grooved substrates produce localized and steady dripping points. When grooves converge, dripping occurs at fixed, geometry-defined locations. For convergent designs, a simple condensation-capillarity model captures the dependence of the dripping period on the area of the drainage basin. Together, these results demonstrate that geometry alone can transform stochastic edge dripping into spatially organized and temporally regular release, with implications for dew harvesting, passive cooling, and millifluidic transport.

Figures (8)

• Figure 1: (a) Experimental setup. Warm, humid air is generated by bubbling compressed air through a heated water reservoir and directed toward the vertically mounted substrate via four nozzles (right). Condensation forms on a square acrylic plate of side length $L = 80.00\,\mathrm{mm}$, either smooth ($s = 80.00\,\mathrm{mm}$) or patterned with parallel vertical grooves. The grooves have a depth $d$ and a width $w$, while the spacing $s$ between grooves ranges from $0.30$ to $10.00\,\mathrm{mm}$. (b) Steady-state condensation on a smooth substrate showing the characteristic band structure formed by successive sweep drops sliding down the surface. (c) Close-up view of the lower edge highlighting the three droplet types: (orange) droplets fully on the vertical face; (blue) flank droplets resting on the edge; (red) hanging droplets below the substrate.
• Figure 2: Spatio-temporal evolution of hanging droplets below the lower edge for decreasing groove spacing: (a) $s = 80.00\,\mathrm{mm}$, (b) $1.25\,\mathrm{mm}$, (c) $0.50\,\mathrm{mm}$, (d) $0.30\,\mathrm{mm}$. Each bar represents one droplet; its length corresponds to droplet width $\ell$ and its colour to volume $\Omega_{hd}$. The vertical axis is time $t$, while the horizontal axis corresponds to position $x$ along the edge. At large spacing (a), hanging droplets appear after a latency of $\sim500\,\mathrm{s}$ and form irregular, drifting bands produced by sporadic impacts of sliding droplets. As grooves are introduced (b–c), bands become thinner, straighter, and more numerous, reflecting increased positional stability and smoother volume evolution. For the most closely spaced grooves (d), the number of hanging droplets decreases, but their positions and volume variation periods become highly regular. The dripping pattern transitions from intermittent and impact-driven to structured and quasi-periodic as groove spacing decreases.
• Figure 3: Evolution of hanging droplets in the steady regime ($t > 1000\,\mathrm{s}$). (a) Mean number of hanging droplets $N_d$ as a function of groove spacing $s$. Two regimes appear: for large and intermediate spacings ($s > 1\,\rm{mm}$), the number of droplets increases as grooves are added; for narrow spacing ($s < 1\,\rm{mm}$), it decreases sharply as grooves become denser. This behavior closely mirrors the water-retention trend observed on the vertical face, with a transition near $s \approx R_c$. (b) Mean droplet width $\ell$ and mean inter-drop gap in the same steady regime. Droplet width decreases continuously as spacing narrows, from about $8\,\mathrm{mm}$ on the smooth surface to $6.5\,\mathrm{mm}$ on the most structured ones ($s < 1\,\mathrm{mm}$), corresponding roughly to twice the capillary length ($2\lambda$). The mean gap follows a similar trend for $s > R_c$ but shifted with a value around $\lambda$. Then it rises again for $s < R_c$, reaching values comparable to droplet size $2\lambda$. Together, these measurements reveal a transition from intermittent, gravity-driven dripping to a geometrically constrained, periodic regime as groove spacing decreases.
• Figure 4: Time evolution of flank and hanging droplets on a densely grooved substrate ($s = 0.30\,\mathrm{mm}$, edge length $80\,\mathrm{mm}$). Snapshots at four key moments illustrate how groove-fed drainage gives rise to a stable dripping pattern at the lower edge. ($t = 160$–$254\,\mathrm{s}$): rapid formation and lateral merging of groove-fed flank droplets, some spanning over $30\,\mathrm{mm}$ in width; ($t = 991\,\mathrm{s}$): local detachment begins as the last large flank droplet is about to break, marking the final stage before the system settles; ($t = 3000\,\mathrm{s}$): the configuration remains unchanged long after the last breakup, reflecting the persistent steady state established shortly after $t = 991\,\mathrm{s}$. Red and rose dashed lines outline "fragmentation basins", zones where successive generations of flank droplets have broken up and stabilized. The groove network pins the boundaries of these basins, locking the spatial organization of the dripping pattern in place.
• Figure 5: Snapshot of the bottom of the sample at $t = 2500\,\mathrm{s}$ for increasing groove spacing: $s = 0.30-0.50-1.25-80.00\,\mathrm{mm}$. Side-anchored (flank) droplets are highlighted in blue to aid visualization. As groove spacing increases, droplet organization evolves from an alternating arrangement of hanging and flank droplets to random deposition on the smooth substrate, reflecting the disappearance of groove-mediated capillary-driven drainage.