Droplet motions fill a periodic table

Abstract

Drawing parallels to the symmetry breaking of atomic orbitals used to explain the periodic table of chemical elements; here we introduce a periodic table of droplet motions, also based on symmetry breaking but guided by a recent droplet spectral theory. By this theory, higher droplet mode shapes are discovered and a wettability spectrometer is invented. Motions of a partially wetting liquid on a support have natural mode shapes, motions ordered by kinetic energy into the periodic table, each table characteristic of the spherical-cap drop volume and material parameters. For water on a support having a contact angle of about 60°, the first 35 predicted elements of the periodic table are discovered. Periodic tables are related one to another through symmetry breaking into a two-parameter family tree. droplet vibrations | sessile drop dynamics | meniscus motions | capillary ballistics | moving contact line D roplets and droplet motions surround us. Our harvests depend on rain drops. We sweat, we shower, and we drink. Our eyes make tears and our blood splats. Drops enable the protein content of our bodily fluids to be measured (1), our silicon chips to be fabricated (2), and complex parts to be additively sculpted by drop-on-demand processing (3,4). Water droplets in motion are shaped into objects of beauty by surface tension. Their images have become symbols of purity and cleanliness, selling beer, jewelry, clothing, and automobiles. However, despite more than a century of study, the motions of droplets on a support have resisted systematic classification. This paper introduces the periodic table classification of capillary-ballistic droplet motions.The capillary-ballistic model assumes an ideal fluid with surface tension acting on the deformable surface (SI Appendix). Capillary-ballistic motions are typical of thin liquids like water. Prototypical of dynamics of this kind are free drop vibrations, predicted by Rayleigh to have frequencies (5) λ kl as inwhere the corresponding deformation is Y l k ðθ, φÞ in spherical coordinates. Here, wavenumber k is the degree and l is the order of the spherical harmonic Y l k (6). Frequencies [1] and mode shapes Y l k constitute the so-called Rayleigh spectrum, predictions verified experimentally (7,8). Note that, in [1], different l′s share the same frequency. These degeneracies arise from the perfect symmetry of the spherical free drop. The introduction of a support typically breaks these degeneracies.Deformations of the supported drop (9), Fig. 1 (Bottom Row), break from the Y l k ðθ, φÞ shapes. The number of layers n (Top Row, in schematic) and of sectors l [Bottom Row: bold lines, rendered shape (Right)] characterize modal symmetry. Using k = l + 2ðn − 1Þ, symmetries are alternatively classified "mathematically" by wavenumber pairs ½k, l. Modes are either "symmetric" (short for axisymmetric), e.g., [6,0]leftmost, "star," e.g., [6,6]rightmost, or "layer" modes (short for layer sector), e.g., [6,2] and ½6,4middle two modes. Note that the Rayleigh spectrum [1] splits. That is, [...

Surface wave pattern formation in a cylindrical container

“…2015). For this case, a new organizing principle has been developed that culminates in the periodic table of droplet motions (Steen, Chang & Bostwick 2019). By exploiting our experimental approach to include a range of contact angles (as opposed to just the two explored herein), it may be possible to observe similar spectral reordering due to wetting effects and this should be explored further.…”

Drop impact on solids: contact-angle hysteresis filters impact energy into modal vibrations

“…During post-pinning, the lg vibrations can be decomposed into vertical and horizontal centre-of-mass motions, largely defined by the impact angle and wetting properties of the substrate. In this case, it may be useful to use the classification structure for sessile drop motions introduced by Steen et al (2019). This topic should be pursued in future studies.…”

“…For the sessile drop, symmetry considerations restrict mode numbers k + l − even and break the degeneracy of the free drop spectrum to the azimuthal mode number l, such that each sessile drop mode of vibration has a unique resonance frequency. This has been observed experimentally by Chang et al (2013Chang et al ( , 2015 and an organizing principle for the sessile drop spectrum has been developed resulting in the 'periodic table of droplet motions' (Steen, Chang & Bostwick 2019).…”

On a toroidal method to solve the sessile-drop oscillation problem

“…Although the former approach leads to a highly simplified physical model, the latter requires hybrid analytical-numerical schemes: neither is then suitably accurate and accessible for use by an experimentalist. , Steen, Chang & Bostwick (2016) have presented a detailed account of the underlying physics and mechanics of this problem. The contact angle θ c (and shape) of the drop is established at static equilibrium by balancing the liquid-gas, liquid-solid and solid-gas interfacial tensions.…”