Karpitschka Group


Marangoni Contraction

Evaporation induces Marangoni flows in a complex liquid and changes the apparent contact angle Zoom Image
Evaporation induces Marangoni flows in a complex liquid and changes the apparent contact angle

Sessile droplets typically evaporate faster near their edge than at their center. Capillarity compensates these losses and causes a strong flow toward the edge of the droplet. This is the reason why coffee-stains look like rings: coffee particles move with the capillary flow and are deposited in a ring around the droplet. If a second liquid is added to the droplet, this liquid will enrich near the edge of the droplet. However, different liquids have different surface tensions, which causes another type of flow, known as the Marangoni effect. It can be directed toward the contact line, which is the case for salty droplets [Soulie et al., PCCP 17 (2015) 22296]. If this flow is directed the other way round, e.g. as for glycerol/water mixtures, it can lead to a contraction of the whole droplet [Karpitschka et al., Langmuir 33 (2017) 4682]. Why is this important? Virtually any micro-electronic circuit, as in phones or computers, is made by wet processing and has to be dried at the end. So far, the wetting behavior of liquids on solids was known to be determined by the material’s surface energies (that’s Young’s law) and surface topography (Wenzel’s law), and nothing much could be changed about this. Drying microelectronics was (and is) a challenge, because material and topography are prescribed by the requirements of the electronics, not the drying; droplets would simply stick to the circuit. However, these contracted droplets seemingly forget about Young’s  and Wenzel’s laws. Instead, their wetting behavior is governed by evaporation and can be controlled remotely by changing the atmospheric composition around the droplet.

Inverted Cheerios Effect

Droplets on soft substrates sense each other due to elastocapillary forces Zoom Image
Droplets on soft substrates sense each other due to elastocapillary forces

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? We would not expect a mechanical interaction because the solid is solid. But this interaction does exist! Droplets pinch on solid surfaces due to capillary forces: They pull with their edge and press with their body (the net force vanishes – that’s 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.

The equivalence of adhesion and wetting

What is a singularity? If we describe a physical phenomenon with a continuum theory, like the adhesion of an elastic material to a rigid sphere, we frequently observe that a physical quantity, in this case the stress in the solid, becomes infinitely large at some “singular point”. Here the singular point is the edge of the contact. In nature however, nothing becomes truly infinite, so there must be something wrong with the continuum theory. 

Adhesion of a sphere to a soft solid
  Zoom Image
Adhesion of a sphere to a soft solid

Obviously, the equations did not take into account the discrete (atomic) nature of materials. But fully describing all atoms is a formidable task, because there’s so many of them – how could we “save” continuum theory, and take microscopic physics into account? The concept of surface tension is one example: molecules in the bulk of a liquid interact with each other on all sides. The molecules at the interface to a gas are missing half of their interaction partners. This causes surface tension, which can be included in continuum theories. Scientists agree that surface tension is utterly important to describe liquids. For solids, surface tension was largely ignored in the past because there, the consequences are much smaller due to the rigidity of a solid. But approaching a singularity, it becomes important again. The point at which it becomes important can be calculated by comparing surface tension to stiffness, ɣ/E. This gives is a length, the elastocapillary length. Below it, surface tension is important if we approach the edge of an elastic contact. After a couple of lines of variational calculus, one can show that surface tension also provides a boundary condition for the angle of contact between the elastic material and the sphere [Karpitschka et al., Soft Matter 12 (2016) 4463]. This is exactly equivalent to what was known for liquids as Young’s contact angle. Close to the edge of a contact, adhesion and wetting are the same.

Interfacial Singularities

Localized indentations on soft solids have a universal shape Zoom Image
Localized indentations on soft solids have a universal shape

Compressing a slice of soft bread, one can observe an inward fold with a self-contacting surface of the bread. A very similar effect is responsible for the morphology of our brains, where an inward fold is termed “sulcus”. Sulci form because the outer part of the brain grows faster than the inner part. Despite its importance in biology and engineering, the morphology of such inward-folds, in mechanics called creases, remained unexplored. In a recent paper [Karpitschka et al., Phys. Rev. Lett. 119 (2017) 198001] we discovered a continuous precursor to a crease, a “furrow”: it can be generated by deforming a soft surface inward, for instance by deflating a liquid inclusion beneath the surface, or pulling a solid inclusion downward. The furrow bifurcates sub-critically into a crease, when the surface region becomes compressed beyond a critical point. In the reverse direction unfolding proceeds smoothly. Both, the furrow and the crease, are described by a simple mathematical concept: self-intersecting curves. The “simplest” set of curves that undergo self-intersection belongs to the universality class of “cusps”. When the radius of curvature reaches zero, the surface forms an interfacial singularity and folds. Of course, for a solid, self-intersection is replaced by a self-contact. This contact is locally similar to the edge of a Hertz contact but, due to the large deformation, for an anisotropic material. Both, cusp singularity and Hertz contact, cause a universal profile that follows a 2/3rd power law.

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