Topography-induced symmetry transition of droplets on quasi-periodically patterned surfaces

Abstract Quasi-periodic structures of quasicrystals yield novel effects in diverse systems. However, there is little investigation on employing quasi-periodic structures in the morphology control. Here, we show the use of quasi-periodic surface structures in controlling the transition of liquid droplets. Although surface structures seem random-like, we find that on these surfaces, droplets spread to well-defined 5-fold symmetric shapes and the symmetry of droplet shapes spontaneously restore during spreading, hitherto unreported in the morphology control of droplets. To obtain physical insights into these symmetry transitions, we conduct energy analysis and perform systematic experiments by varying properties of both liquid droplet and patterned surface. The results show the dominant factors in determining droplet shapes to be surface topography and the self-similarity of the surface structure. Our findings significantly advance the control capability of the droplet morphology. Such a quasi-periodic patterning strategy can offer a new method to achieve complex patterns.

Quasi-periodic structures, being ordered while aperiodic, have brought excitements in diverse fields. For example, these structures have initiated the paradigm shift in crystallography (1), introduced special beauty to architectures (2), and recently extended to soft materials including colloids (3), 2 thin-film systems (4) and self-assembly of molecules (5). These structures were extended to a variety of applications and bring special effects, such as the acoustic and optical wave transport in quasicrystals (6, 7). Here, we employ quasi-periodic structures to morphology control by placing liquid droplets on solid surfaces decorated with these structures.
Wetting and spreading of liquid droplets on solid surfaces were extensively investigated in the past century (8-13) because of their ubiquitous existence in nature and applications. Recently, much research has focused on morphology control of droplets using micro-decorated surfaces (14-21). However, similar to the crystals with periodic structures, which have only 2, 3, 4 and 6-fold symmetries, surfaces patterned with regular micropillars in periodic arrangement can yield droplets with contact areas of corresponding symmetries. More specifically, droplets form contact areas of 2-fold symmetry on surfaces patterned with rectangle array (14, 15) and stretched hexagonal array of cylindrical micropillars (15), of 3-fold symmetry on surfaces with hexagonal array of equilateral triangle micropillars (16), of 4-fold symmetry on surfaces with square array of regular micropillars (14-22), and of 6-fold symmetry on surfaces with hexagonal array of cylindrical micropillars (14, 15). The challenging issue is whether it is possible to produce the wetted area of 5-fold symmetry by orderly arranged pillars. Motivated by the aperiodic but ordered structure of quasicrystals with 5-fold symmetric Bragg diffraction pattern and by such structure presents in nature (Fig. 1a), we use surfaces patterned with two-dimensional quasi-periodic micropillar arrays for droplet spreading. We expect to obtain novel topographic features of droplets on such surfaces in comparison with periodically patterned surfaces.
In this article, we find that droplets spread to 5- and 10-fold symmetric shapes during spreading on the surface structures of a Penrose tiling. Droplets experience remarkable shape transitions of 3 spontaneous loss and restoration of symmetry during spreading, for example, from circularity to 5-fold symmetry, and from 10-fold symmetry to circularity. Furthermore, we provide physical insights into how quasi-periodic structures control the shape formation of a droplet during the spreading process.
To begin with, a droplet of silicon oil, which is non-volatile and can completely wet (23) a quasi-periodic surface, was released at the local symmetrical center of the quasi-periodically patterned surface. The size of the liquid droplet is smaller than the capillary length, cal g   (γ, ρ and g are the liquid surface tension, density, and gravitational acceleration, respectively), such that the gravity is negligible. Figure 1b shows the surface, on which micropillars are arranged in the pattern of a Penrose tiling (24). This pattern has 5-fold symmetry and is self-similar (the same patterns appear at different scales) with many local symmetrical centers (Fig. 1c-d). The pillar density and roughness are uniform in a long-range area of the pattern.
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