Tilted surfaces prevent paint drips

Imagine you’re painting a room in your home, and you slather a thick layer of paint across the ceiling. Rather than form a uniform coating, the paint will develop drops that grow. Once they get large enough, they’ll fall to the floor or land on your body.

Pendant drops hanging from a horizontal surface can accelerate their growth by moving across the surface and sucking up yet more fluid. Then tilting the surface so that the drops move faster should further enhance their growth, right? It turns out that’s not always the case. A new study by Etienne Jambon-Puillet at Princeton University and coworkers found that beyond a critical tilt angle, sliding drops actually shrink.

In their experiments, the researchers applied silicone oil to the underside of a glass substrate. At around 100 μm, the layer of oil was thin enough that droplets wouldn’t spontaneously form. Next the researchers created individual drops, each a few millimeters across, by adding fluid with a micropipette.

The first figure shows the evolution of two droplets over time (from left to right). The droplet at a tilt angle, α, of 0.9° grew from its initial height, indicated by the dashed line. But the droplet at α = 3.4° shrank. The critical angle separating the two regimes was around 2°.

Jambon-Puillet and his coworkers turned to simulations to help them understand the counterintuitive behavior. The two regimes, they found, arose from the relative thickness of the film in front of the drop, h₀, and behind the drop, h_w. As a pendant drop slides, it absorbs fluid at its leading edge and releases fluid through its trailing edge. The rate of change of the drop’s volume depends on the difference between the film thicknesses, h₀ − h_w (see the second figure). For small inclinations, h_w was generally less than h₀, and the drops grew. But for inclinations greater than 2°, h_w was greater than h₀. The faster-moving drops left so much fluid in their wakes that they shrank.

Sloppy house painters aren’t the only ones concerned with controlling pendant-drop formation. Industrial printers and coaters design protocols that avoid the behavior. Drops can also form in more volatile situations: The instability that underlies the behavior can occur at the interface between any two fluids when the denser fluid is a thin film, including liquid metal–plasma interfaces in fusion reactors. (E. Jambon-Puillet et al., Phys. Rev. Lett. 127, 044503, 2021 .)