Voronoi Tessellations and the Shannon Entropy of the Pentagonal Tilings.

We used the complete set of convex pentagons to enable filing the plane without any overlaps or gaps (including the Marjorie Rice tiles) as generators of Voronoi tessellations. Shannon entropy of the tessellations was calculated. Some of the basic mosaics are flexible and give rise to a diversity of Voronoi tessellations. The Shannon entropy of these tessellations varied in a broad range. Voronoi tessellation, emerging from the basic pentagonal tiling built from hexagons only, was revealed (the Shannon entropy of this tiling is zero). Decagons and hendecagon did not appear in the studied Voronoi diagrams. The most abundant Voronoi tessellations are built from three different kinds of polygons. The most widespread is the combination of pentagons, hexagons, and heptagons. The most abundant polygons are pentagons and hexagons. No Voronoi tiling built only of pentagons was registered. Flexible basic pentagonal mosaics give rise to a diversity of Voronoi tessellations, which are characterized by the same symmetry group. However, the coordination number of the vertices is variable. These Voronoi tessellations may be useful for the interpretation of the iso-symmetrical phase transitions.

From Chaos to Ordering: New Studies in the Shannon Entropy of 2D Patterns.

Properties of the Voronoi tessellations arising from random 2D distribution points are reported. We applied an iterative procedure to the Voronoi diagrams generated by a set of points randomly placed on the plane. The procedure implied dividing the edges of Voronoi cells into equal or random parts. The dividing points were then used to construct the following Voronoi diagram. Repeating this procedure led to a surprising effect of the positional ordering of Voronoi cells, reminiscent of the formation of lamellae and spherulites in linear semi-crystalline polymers and metallic glasses. Thus, we can conclude that by applying even a simple set of rules to a random set of seeds, we can introduce order into an initially disordered system. At the same time, the Shannon (Voronoi) entropy showed a tendency to attain values that are typical for completely random patterns; thus, the Shannon (Voronoi) entropy does not distinguish the short-range ordering. The Shannon entropy and the continuous measure of symmetry of the patterns demonstrated the distinct asymptotic behavior, while approaching the close saturation values with the increase in the number of iteration steps. The Shannon entropy grew with the number of iterations, whereas the continuous measure of symmetry of the same patterns demonstrated the opposite asymptotic behavior. The Shannon (Voronoi) entropy is not an unambiguous measure of order in the 2D patterns. The more symmetrical patterns may demonstrate the higher values of the Shannon entropy.

Directional Droplet Transport Mediated by Circular Groove Arrays. Part II: Theory of Effect.

In the first part of this research, we reported the experimental study of the drop impact on the superhydrophobic circular groove arrays, which resulted in a directional droplet transport. In the second part, we further explored the influence of the Weber number (We), ridge height (H), and the deviation distance (r) between the impacting point and the center of curvature on the lateral offset distance (ΔL) of bouncing drops. The suggested theoretical analysis is in reasonable agreement with the experimental observations. We demonstrate that a Cassie-Wenzel wetting transition occurred within the microstructures of the relief under the threshold Weber number, for example, We ≅ 19-25, which switched the nature of drop bouncing. The dynamic pressure plays a decisive role in the directional droplet transport. The reported investigation may shed light on the solid-liquid interactions occurring on the patterned hierarchical surfaces and open up new opportunities for directional droplet transportation.

Directional Droplet Transport Mediated by Circular Groove Arrays. Part I: Experimental Findings.

Directional transport of liquid droplets is crucial for various applications including water harvesting, anti-icing, and condensation heat transfer. Here, bouncing of water droplets with patterned superhydrophobic surfaces composed of circular equidistant grooves was studied. The directional transport of droplets toward the pole of the grooves was observed. The impact of the Weber number, initial polar distance r, and geometrical parameters of the surface on the directional droplet bouncing was experimentally explored. The nature of bouncing was switched when the Weber numbers exceeded We ≅ 20-25. The rebouncing height was slightly dependent on the initial polar coordinate of the impact point for a fixed We, whereas it grew for We > 20. The weak dependence of the droplet spreading time on the Weber number was close to the dependence predicted by the Hertz bouncing, thus evidencing the negligible influence of viscosity in the process. Change in the scaling exponent describing the dependence of the normalized spreading time on the Weber number was registered for We ≅ 25. The universal dependence of the offset distance ΔL of the droplets on the Weber number ΔL/D0 ∼ We1.5 was established. The normalized offset distance decreased with the normalized initial polar distance as ΔL/D0 ∼ (r/S)-1, where D0 and S are the droplet diameter and groove width, respectively. This research may yield more insights into droplet bouncing on patterned surfaces and offer more options in directed droplet transportation.

Faceted and Circular Droplet Spreading on Hierarchical Superhydrophobic Surfaces.

Bouncing of water droplets on the post-built superhydrophobic surfaces was studied. The topography of the surfaces was constituted by PDMS conical posts decorated with ZnO nanoparticles. Droplet impact on surface topographies built of posts with varied configuration and separation was studied under different Weber numbers. Faceted spreading and retraction of droplets were observed. Square-, pentagon-, and hexagon-shaped droplets were registered. It was shown that the nature of droplet spreading depended on both the Weber number and the topography of the post arrays. Even bouncing under small Weber numbers We ≅ 6.5 resulted in the Cassie-Wenzel transitions, starting from the area adjacent to the axis of droplets, and the area exposed to the wetting transitions scaled as [Formula: see text]. During spreading, two main stages were recorded as the kinematic (inertial) stage and the viscous stage. The viscous stage, in turn, appeared as a consequence of two substages governed by various time scaling laws. The faceted triple line was observed for the Cassie-like retraction of droplets.

Self-Arranged Levitating Droplet Clusters: A Reversible Transition from Hexagonal to Chain Structure.

Water microdroplets condense over locally heated water-vapor interfaces and levitate in an ascending vapor-air flow forming self-assembled ordered monolayer clusters. The droplets do not coalesce due to complex aerodynamic interactions between them. The droplet cluster formation is governed by the condensation/evaporation balance and by coupling of heat flux and vapor flow with aerodynamic forces. Here, we report the observations of a reversible structural transition from the ordered hexagonal-structure cluster to the chain-like structure and provide an explanation of its mechanism and conditions under which the transition occurs. The phenomenon provides new insights on the fundamental physical and chemical processes with microdroplets including their role in reaction catalysis in nature and their potential for aerosol and microfluidic applications.

Is the Voronoi Entropy a True Entropy? Comments on "Entropy, Shannon's Measure of Information and Boltzmann's H-Theorem", Entropy 2017, 19, 48.

The goal of this comment note is to express our considerations about the recent paper by A. Ben Naim (Entropy 2017, 19, 48). We strongly support the distinguishing between the Shannon measure of information and the thermodynamic entropy, suggested in the paper. We demonstrate that the Voronoi entropy should also be clearly distinguished from the entropy of a two-dimensional gas. Actually, the Voronoi entropy being an intensive value is the averaged Shannon measure of ordering for a given pattern.

Symmetry and Shannon Measure of Ordering: Paradoxes of Voronoi Tessellation.

The Voronoi entropy for random patterns and patterns demonstrating various elements of symmetry was calculated. The symmetric patterns were characterized by the values of the Voronoi entropy being very close to those inherent to random ones. This contradicts the idea that the Voronoi entropy quantifies the ordering of the seed points constituting the pattern. Extension of the Shannon-like formula embracing symmetric patterns is suggested. Analysis of Voronoi diagrams enables the elements of symmetry of the patterns to be revealed.

Toward an Understanding of Magnetic Displacement of Floating Diamagnetic Bodies, I: Experimental Findings.

Diamagnetic objects (polymer and metallic plates and spheres, ceramic beads, and liquid marbles), floating on water, and a variety of organic liquids may be driven by a steady magnetic field of 0.1 T, registered at the water-vapor surface. Diamagnetic bodies are attracted to the magnet, when the apparent contact angle at the solid/liquid interface is obtuse and repelled from the magnet, when the angle is acute. Cold plasma-treated polyolefin rafts and spheres, demonstrating underwater floating, are repelled by a permanent magnet. Addition of a surfactant to the water, as well as cold plasma treatment of the polyolefin bodies, can turn the attraction into the repulsion. We conjecture that the observed effects are caused by the interplay of two main phenomena. The first is the gravity, which induces sliding of the particle on the deformed liquid/vapor interface (the Moses effect). The second cause is the hysteresis of the contact angle at the bodies' boundaries.

Characterization of Self-Assembled 2D Patterns with Voronoi Entropy.

The Voronoi entropy is a mathematical tool for quantitative characterization of the orderliness of points distributed on a surface. The tool is useful to study various surface self-assembly processes. We provide the historical background, from Kepler and Descartes to our days, and discuss topological properties of the Voronoi tessellation, upon which the entropy concept is based, and its scaling properties, known as the Lewis and Aboav-Weaire laws. The Voronoi entropy has been successfully applied to recently discovered self-assembled structures, such as patterned microporous polymer surfaces obtained by the breath figure method and levitating ordered water microdroplet clusters.

Robust icephobic coating based on the spiky fluorinated Al2O3 particles.

Omniphobic and icephobic twin-scale surfaces based on the "urchin"-like fluorinated Al2O3 particles are presented. Combined effect of hierarchical topography and fluorination supplied to the surfaces omniphobic and icephobic properties. The study of the stability of the Cassie wetting state is reported. High apparent contact angles were accompanied with the low contact angle hysteresis and high stability of the Cassie air trapping wetting state. Time delay of the ice crystallization as high as [Formula: see text] min was established when compared to the ice formation on flat aluminum and non-fluorinated "urchin"-like surfaces. Crystallized water droplets formed on the reported nano-structured surfaces were easily blown out by the air jet with the velocity of [Formula: see text] m/s, (which is markedly lower than that common for exploitation of aircrafts and turbines). Heated "urchin"-like surfaces completely restored their omniphobic and icephobic surfaces after thawing. Qualitative analysis of water freezing is supplied.

Osmotic evolution of composite liquid marbles.

HYPOTHESIS: We hypothesized that the reported evolution (growth) of composite water marbles filled with saline water and coated with lycopodium dispersed in a thin layer of silicone oil is due to the osmotic mass transfer. The hypothesis is supported by the semi-empirical model of osmotic growth of small liquid marbles floating on distilled water. EXPERIMENTS: Saline composite, silicone oil-coated marbles floating on distilled water grew with time; whereas, composite marbles filled with distilled water floating on aqueous solutions of NaCl lost mass with time and shrunk. However, composite liquid marbles filled with saline water and floating on aqueous solutions of NaCl remained stable during 25 h of the laboratory experiment. FINDINGS: The reported findings are reasonably attributed to osmotic mass transport through the thin silicon layer filled with lycopodium particles coating the marbles, acting as an osmotic membrane. This is supported by the suggested model for the osmotic growth of marbles.

Negative Effective Mass in Plasmonic Systems.

We report the negative effective mass (density) metamaterials based on the electro-mechanical coupling exploiting plasma oscillations of a free electron gas. The negative mass appears as a result of the vibration of a metallic particle with a frequency of ω, which is close the frequency of the plasma oscillations of the electron gas m 2 relative to the ionic lattice m 1 . The plasma oscillations are represented with the elastic spring k 2 = ω p 2 m 2 , where ω p is the plasma frequency. Thus, the metallic particle vibrated with the external frequency ω is described by the effective mass m e f f = m 1 + m 2 ω p 2 ω p 2 - ω 2 , which is negative when the frequency ω approaches ω p from above. The idea is exemplified with two conducting metals, namely Au and Li.

Negative Effective Mass in Plasmonic Systems II: Elucidating the Optical and Acoustical Branches of Vibrations and the Possibility of Anti-Resonance Propagation.

We report the negative effective mass metamaterials based on the electro-mechanical coupling exploiting plasma oscillations of free electron gas. The negative mass appears as a result of the vibration of a metallic particle with a frequency ω which is close to the frequency of the plasma oscillations of the electron gas m2, relative to the ionic lattice m1. The plasma oscillations are represented with the elastic spring constant k2=ωp2m2, where ωp is the plasma frequency. Thus, the metallic particle vibrating with the external frequency ω is described by the effective mass meff=m1+m2ωp2ωp2-ω2, which is negative when the frequency ω approaches ωp from above. The idea is exemplified with two conducting metals, namely Au and Li embedded in various matrices. We treated a one-dimensional lattice built from the metallic micro-elements meff connected by ideal springs with the elastic constant k1 representing various media such as polydimethylsiloxane and soda-lime glass. The optical and acoustical branches of longitudinal modes propagating through the lattice are elucidated for various ratios ω1ωp, where ω12=k1m1 and k1 represents the elastic properties of the medium. The 1D lattice, built from the thin metallic wires giving rise to low frequency plasmons, is treated. The possibility of the anti-resonant propagation, strengthening the effect of the negative mass occurring under ω = ωp = ω1, is addressed.

Study of wetting of the animal retinas by Water and organic liquids and its Implications for ophthalmology.

Interfacial properties of the animal retinas are reported. Wetting of the retina-retinal pigment epithelium-choroid-sclera tissues of cow, sheep, and pig eyes by water, silicone and castor oil was explored experimentally. Both water and silicone oils demonstrated complete wetting of the retina, regardless of the viscosity of the silicone oil, whereas the castor oil demonstrated a partial wetting regime. Similar wetting regimes were observed for sheep, cow and pig retinas. The intact surface of animal retina was found to be both hydrophilic and oleophilic. Wetting experiments with double sandwich oil/water layers were performed. Water demonstrated stronger affinity to the retina than silicone and castor oils, and eventually replaced the oils at the liquid/retina interface. We conclude that aqueous solutions continuously secreted in the living eye may displace silicone oil from the retinal surface and contribute to retinal re-detachment. Study of dynamics of wetting of the animal retina by water and organic oils is reported. The exponent describing the dynamics of spreading of the castor oil is lower than that predicted by the Tanner law. Castor oil may provide more effective tamponade than silicone oil.

Manufacture and properties of composite liquid marbles.

HYPOTHESIS: Liquid marbles are non-stick droplets coated with colloidal usually hydrophobic particles. We suggest that "composite" liquid marbles, i.e. bi-liquid droplets, may be prepared with water droplets coated by a thin silicone oil layer containing hydrophobic, colloidal particles. EXPERIMENTS: The process enabling manufacturing water marbles coated with silicone oil containing fumed fluorosilica particles is reported. The marbles remained stable when placed on solid and liquid supports. Bouncing and coalescence of the composite marbles was explored. FINDINGS: Non-coalescence prolonged (ca. 20 min) jumping of composite marbles above a vibrating water bath was observed. Composite marbles withstand coalescence better than colloidal particle-stabilized liquid marbles. The effective surface tension of the composite marbles is markedly lower than that of water marbles coated with fumed fluorosilica particles. The coefficient of restitution of the composite marbles bouncing on a hydrophobic solid substrate is lower than that established for water marbles. This observation is related to the viscous dissipation occurring within the silicone layer making up the composite marbles.

Mini-Generator of Electrical Power Exploiting the Marangoni Flow Inspired Self-Propulsion.

The mini-generator of electrical energy exploiting Marangoni soluto-capillary flows is reported. The interfacial flows are created by molecules of camphor emitted by the "camphor engines" placed on floating polymer rotors bearing permanent magnets. Camphor molecules adsorbed by the water/vapor interface decrease its surface tension and create the stresses resulting in the rotation of the system. The alternative magnetic flux in turn creates the current in the stationary coil. The long-lasting nature of rotation (approximately 10-20 h) should be emphasized. The brake-specific fuel consumption of the reported generator is better than that reported for the best reported electrical generators. Various engineering implementations of the mini-generator are reported.

Self-Propulsion of Water-Supported Liquid Marbles Filled with Sulfuric Acid.

Self-propulsion of liquid marbles filled with sulfuric acid and coated with hydrophobic fluorosilica powder on a water surface is reported. The prolonged self-propulsion of marbles occurs over a couple of minutes with a typical velocity of the center of mass of the marble being [Formula: see text]. The shell of the marble is not uniform, resulting in the asymmetric absorption of water by a marble, giving rise to the nonuniform thermal field within its volume. The maximum temperature reached at the liquid marble surface was 70 °C. The self-propelled marble increased its mass by one-third during the course of its motion. The increase in mass followed by the marbles' heating is due to the adsorption of water vapor by their surface, which is permeable to gases. This gives rise to an exothermic chemical reaction, which in turn gives rise to Marangoni thermo-capillary flow driving the marble. Thermo-physical analysis of the problem is presented. The role of soluto-capillary flow in self-propulsion is negligible.