Writing on water: topology-controlled patterns on liquid droplets

Counterintuitive ordered fluorescent patterns controllably self-assemble on the surfaces of spherical liquid droplets. This new phenomenon is at the convergence point of 2D crystallography, thin-plate elasticity, interfacial energetics, and topology, and may play an important role in morphogenesis.
Writing on water: topology-controlled patterns on liquid droplets
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Controllably patterning solid surfaces has been a crucial step in the development of the art of painting and of writing, at the dawn of humanity. The patterns were typically either carved into the substrate, or deposited onto the surface, in the form of a dye. Therefore, no patterns could be formed on liquid surfaces: the thermal motion of the liquid’s constituent particles would rapidly smear any such pattern, giving rise to the Latin idiom “in aqua scribere (“written in water”), denoting a futile task or, figuratively, a frivolous oath or testimony.

Starkly contrasting this idiom, our recent Nature Physics paper, “Topology-driven surface patterning of liquid spheres”, presents counterintuitive self-patterning of liquid droplets’ surface by fluorescent dye (Fig. 1a-c). The patterns are temperature controllable, persistent, and precisely reproducible. While our droplets are micron-sized, similar mechanisms potentially allow decorating droplets as small as a fraction of a zeptoliter (10-21 L). Furthermore, our method enables attachment of ligands and other functional groups at precise positions on the droplets’ surface. These, in turn, can form bonds with pre-determined orientations and symmetry, mimicking the covalent bonds in chemistry and thus allowing controlled self-assembly of droplets into higher-hierarchy structures and smart materials. Similar mechanisms likely play a role in protein recruitment to precise biomembrane positions, with important implications for cell development, morphogenesis, and vital physiological processes. Notably, the surfaces of many common viruses are decorated by precisely-positioned infection-mediating spikes. Recent studies suggest that mimicking spikes’ positioning, which is readily achievable by means of the developed pattern-formation mechanisms, may be exploited in rational design of vaccines.

Figure 1. Self-assembled fluorescent patterns on spherical liquid droplets. (a) The low-temperature pattern, consists of 12 small patches, which are maximally separated on the surface of the spherical liquid droplet. (b,c) the patches increase in size upon heating, developing into  the patterns shown. (d) Precisely the same pattern self-assembles on all of the droplets. The dashes mark the boundaries of the individual droplets.

Too late for a lunch...

This study started over a cup of coffee as a bet between some of the authors, two physicists and a chemist. The chemist claimed that no stable pattern could ever form on a liquid droplet’s surface: “Why would the fluorescent molecules localize at some spot, rather than spread homogeneously by entropy?”, the physicists maintained otherwise.  Betting a lunch at one of the top on-campus restaurants, the parties moved to the lab to test the case by experiment. Much to the surprise of all, light microscopy observations clearly demonstrated that the dye, which uniformly covers the surfaces of these droplets at a high temperature, desorbs upon cooling, but leaves behind well-defined, separate, fluorescent patches. While in the first observations the fluorescent patches were highly irregular and their shapes differed from droplet to droplet, these observations were definitely enough to win the bet. However, it was too late in the evening for lunch. Therefore, instead of a joyful walk to the restaurant, we stayed at the lab, following the dye’s readsorption onto the droplets in a heating scan: “The sample is already under the microscope, shouldn’t we play with this sample a bit longer, before it is disposed of ?”

Any experimental scientist, who has ever experienced such an exciting moment in his research, probably remembers it for life. As we continued working into the night, we were amazed to see that upon heating from a non-fluorescent state at low temperature, all droplets underwent, together, self-patterning to the same pattern at some temperature  (Fig. 1d). The pattern starts out as 12 tiny patches, equally spaced on the surface of the liquid spherical droplet (Fig. 1a). On further heating, the patches increase in size (Fig. 2b-c), eventually coalescing to render the full surface of the droplet uniformly fluorescing.

What is the mechanism?

So, what drives the fluorescent molecules to segregate and form these patterns, rather than spread uniformly over the whole surface? The answer lies in  the  chemical composition of the droplets. They are oil droplets, stabilized in water by surfactant molecules residing at their interface. The oil consist of long, linear, chain-molecules called normal-alkanes. The surfactants also comprise linear alkane-like chains, but are terminated by a water-soluble polar headgroup. The lengths of the surfactants’ alkyl tail and the alkane are almost identical, promoting the formation of a cocrystallized alkane:surfactant structure. Since their polar headgroup renders the surfactants insoluble in the bulk of the alkane droplets, the cocrystallized structure can only form at droplets’ interfaces. Thus, below the “interfacial freezing temperature”, typically denoted as Ts, a mixed alkane:surfactant crystalline monolayer forms at the surface, while the droplet’s oil bulk remains liquid.

The crystalline monolayer has a two-dimensional (2D)  hexagonal structure, which is incompatible with the topology of droplets’ surfaces. Hexagonal tiles alone cannot fully cover a closed surface, unless exactly 12 pentagonal tiles are also included. This has been postulated by the eminent mathematician Leonhard Euler, back in 1758. A famous manifestation of Euler’s condition are  the 12 pentagonal surface patches of a soccer ball (Fig. 2a). For exactly the same reasons, Euler’s condition dictates the presence of 12 five-coordinated lattice sites, within the otherwise-hexagonal structure of the crystalline monolayer covering our droplets’ surfaces. These five-coordinated lattice sites, known as “disclination” defects, cause a large extensional stress in the lattice. The stress energy is minimized by the disclinations self-distancing to maximal separations on the droplet’s surface. The stress is further reduced by attracting surface-active  guest molecules, if available. Using fluorescent guest molecules thus creates 12 bright fluorescent patches, maximally separated from each other, on the spherical surface of the droplet (Fig. 2b).

Since interfacial freezing transitions have recently been observed for many dozens of oil:surfactant combinations, the observed self-patterning phenomena may have implications for a wide range of technologies, where emulsions or aerosols are employed. In some of these, the disclination’s elastic stress relaxes further by the disclinations’ buckling out of the spherical surface of the droplet. This, in turn, causes the droplets to adopt the shape of an icosahedron, a 12-vertex polyhedron, while their bulk remains liquid. Other faceted liquid shapes were also observed, driven by coalescence of the disclinations.

Finally, in addition to the above-mentioned disclinations, 2D crystals typically also exhibit “dislocations”, pairs of adjacent 5- and 7-coordinated defects. While the role of dislocations in 2D  crystals’ elasticity has been extensively studied, with the findings being central to the 2002/2003 Wolf prize in Physics and to the 2016 Nobel prize, the interplay between dislocations, surface curvature, and topology-dictated disclinations, is yet to be further explored. In particular, we observe that in some salt-doped systems, the fluorescent patches are highly elongated, rather than rounded. Labyrinth-like fluorescent patterns occur as well (see the droplet at the left side of this post's banner). A role for chains of dislocations in forming such patterns is highly plausible, but the complete mechanism is yet to be resolved.

Figure 2. Euler’s condition dictates the inclusion of exactly 12 pentagonal patches in a full tiling of a closed surface by hexagons. A prominent example of this condition is the  soccer ball, the negative image of which is shown in (a). For the same reason, 12 disclination defects exist in the crystalline monolayer covering the surface of the droplets (b). Fluorescent dye segregates to these 12 disclinations, yielding 12 fluorescent patches, located at the same positions as the pentagonal patches in (a).

To conclude, the highly-counterintuitive patterns self-assembling on the surfaces of liquid droplets turn out to be at the convergence point of such remote fields as 2D crystallography, thin-plate elasticity, interfacial energetics, and mathematical topology. The fundamental mechanism behind the formation of these patterns is general and may play an important role in virus capsid and bacterial microcompartment formation, in vital positioning of proteins on cell membranes, lipid droplets, lipoproteins, and beyond. This phenomenon may also have consequences for a wide range of technologies, where emulsions are employed: from processing of food, water purification, and methods of oil extraction, to targeted drug delivery within the living body, self-assembly and nanotechnology.

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