Interaction Mechanisms between Sessile Droplets
Solid objects that float on the surface of a liquid are known to attract each other because they deform the surface of the liquid. This is known as the “Cheerios Effect” because it is frequently observed on milk with these cereals. But what about liquid objects (droplets) on top of a solid surface? At first sight, one would not expect any interaction because the solid merely serves as an inert support. But this interaction does exist in various scenarios that we investigate in the lab. Such interactions could be highly relevant in technological applications or in biological systems, think, for instance, of adherent cells at the surface of tissue or a biofilm.
Droplets actually pinch on solid surfaces due to capillary forces, analogous to focal adhesion of biological cells: They pull with their edge and press with their body (the net force must vanish according to Newton’s law). On glass, nothing much would happen because it is too rigid. But a soft material like a gel or a soft silicone rubber (or pudding) will deform significantly. That deformation is called the “wetting ridge”. Importantly, this ridge extends quite far beyond the droplet, which can be “sensed” by other droplets nearby [Karpitschka et al., PNAS 113 (2016) 7403]. Consider two such droplets. Will they attract or repel? That depends in a very subtle way on geometry and there are at least three length scales involved: The size of the drop, the size of the wetting ridge, and the thickness of the soft material. Tersely phrased: on thin soft layers, droplets repel, on bulky elastic blocks, they attract. Importantly, the mechanism of the interaction relies only on interfacial tensions and elasticity – it should be present for any kind of object (think about cells in our body) that pinches on soft solids. As compared to the “classical” Cheerios effect, the role of solid and liquid are exchanged, which is why we have named it the “Inverted Cheerios Effect”. But also in terms of the involved physics, it is quite different: A body force like gravitation is not required, it is a pure result of capillarity and elasticity. The resulting droplet motion is then governed by dissipation in the soft substrate, which can quantitatively be calculated and matches the observed droplet velocities well [Pandey et al., Soft Matter 13 (2017) 6000].
But is elasticity even necessary for such an effect? Couldn’t a thin liquid film on top of a rigid surface mediate a similar kind of interaction? Such scenario has recently gained great attention under the catchphrase “SLIPS”: these “Slippery Liquid-Infused Porous Surfaces” can almost fully suppress droplet pinning, and are believed to be useful in numerous technical applications. Droplets on top of such surfaces create a liquid meniscus in the film around them, very similar to the wetting ridge observed on soft surfaces [Tress et al., Soft Matter 13 (2017) 3760]. Micron-sized drops interact by a capillary mechanism when deposited on films of immiscible viscous liquids [Hack et al., Appl. Phys. Lett. 113 (2018) 183701]. This scenario is highly relevant in modern ink-jet printing, where the print substrate is typically coated with a primer liquid, e.g. to promote precipitation of color pigments. Each deposited droplet emits a capillary wave and, at the same time, interacts with the waves that are emitted by its neighbors. Consequentially, the interaction forces not only depend on geometry, but also on time. Waves are composed of a sequence of hills and valleys, for which reason the droplet interaction follows a complicated pattern of attractive and repulsive zones that in addition change over time. Despite such complicated behavior, there is an elegant description of these spatio-temporal interaction patterns. The capillary wave emitted by a droplet grows in a self-similar way. Thus the wave profiles is ‘stretched’ as time proceeds, with a power law ~t1/4, to be precise. The same scaling is recovered in the interaction pattern, and zones of repulsion and attraction are aligned with the sign of the curvature (upward or downward) of the wave profile emitted by a single droplet.