arXiv subjects

Mark Frenkel

Publications and source records attributed to Mark Frenkel.

13 recordsLinked to original sources

arXiv2024

Voronoi Entropy and long-range order of 2D point sets

The Voronoi Entropy (VE) and the continuous measure of symmetry (CSM) characterize the orderliness of a set of points on a 2D plane. The Voronoi entropy is the Shannon entropy of the Voronoi tessellation of the plane into polygons, quantifying the diversity of polygons. The VE is widely used to study the self-assembly of colloidal, supramolecular, and other systems. The value of VE changes from S=0 for a completely ordered system, built of polygons with an equal number of sides, to S=1.690+/-0.001 for a random set of points. While the VE takes into account only neighboring polygons, covering the 2D plane imposes constraints on the number of polygons and the number of edges in polygons. Consequently, unlike the conventional Shannon Entropy, the VE captures some long-range order properties of the system. We calculate the VE for several hyperuniform sets of points and compare it with the values of exponents of collective density variables characterizing long-range correlations in the system. We show that the VE correlates with the latter up to a certain saturation level, after which the value of the VE falls to S=0, and we explain this phenomenon.

cond-mat.stat-mech↗

arXiv2023

Continuous Measure of Symmetry as a Dynamic Variable: a New Glance on the Three-Body Problem

The time evolution of the continuous measure symmetry for the system built of the three bodies interacting via the potential U(r)~1/r is reported. Gravitational and electrostatic interactions between the point bodies were addressed. In the case of the pure gravitational interaction the three-body-system deviated from its initial symmetrical location, described by the Lagrange equilateral triangle, comes to collapse, accompanied by the growth of the continuous measure of symmetry. When three point bodies interact via the Coulomb repulsive interaction, the time evolution of CMS is quite different. CMS calculated for all of studied initial configurations of the point charges and all of their charge-to-mass ratios always comes with time to its asymptotic value, evidencing the stabilization of the shape of the triangle, constituted by the interacting bodies.

physics.gen-ph↗

arXiv2022

Ramsey Theory and Geometry of Closed Loops

We apply the Ramsey theory to the analysis of geometrical properties of closed contours. Consider a set of six points placed on a closed contour. The straight lines connecting these points are y_ik (x)={\alpha}_ik x+\b{eta}_ik (i,k=1...6), {\alpha}_ik is not equal to 0. We color the edges connecting the points for which {\alpha}_ik>0 holds with red, and the edges for which {\alpha}_ik<0 with green with red. At least one monochromic triangle should necessarily appear within the curve (according to the Ramsey number R(3,3)=6). This result is immediately applicable for the analysis of dynamical billiards. The second theorem emerges from the combination of the Jordan curve and Ramsey theorem. The closed curve is considered. We connect the points located within the same region with green links and the points placed within the different regions with red links. In this case, transitivity/intransitivity of the relations between the points should be considered. Ramsey constructions arising from the differential geometry of closed contours are discussed.

math.DS↗

arXiv2020

Investigation of the Impact of Cold Plasma Treatment on the Wettability of Medical Grade Polyvinylchloride

The impact of the Corona, dielectric barrier discharge and low pressure radiofrequency air plasmas on the wettability of the medical grade polyvinylchloride was investigated. Corona plasma treatment exerted the most pronounced increase in the hydrophilization of polyvinylchloride. The specific energy of adhesion of the pristine and plasma treated PVC tubing is reported. The kinetics of hydrophobic recovery following the plasma treatment was explored. The time evolution of the apparent contact angle under the hydrophobic recovery is satisfactorily described by the exponential fitting. Energy-dispersive X-ray spectroscopy of the chemical composition of the near-surface layers of the plasma treated catheters revealed their oxidation. The effect of the hydrophobic recovery is hardly correlated with oxidation of the polymer surface, which is irreversible.

physics.med-ph↗

arXiv2020

Voronoi Diagrams Generated by the Archimedes Spiral

Voronoi mosaics inspired by the seed points placed on the Archimedes Spirals are reported. Voronoi entropy was calculated for these patterns. Equidistant and non-equidistant patterns are treated. Voronoi mosaics built from cells of equal size which are of primary importance for decorative arts are reported. The pronounced prevalence of hexagons is inherent for the patterns with an equidistant and non-equidistant distribution of points, when the distance between the seed points is of the same order of magnitude as the distance between the turns of the spiral. Penta- and heptagonal 'defected' cells appeared in the Voronoi diagrams due to the finite nature of the pattern. The ordered Voronoi tessellations demonstrating the Voronoi entropy larger than 1.71, reported for the random 2D distribution of points, were revealed. The dependence of the Voronoi entropy on the total number of the seed points located on the Archimedes Spirals is reported. The aesthetic attraction of the Voronoi mosaics arising from seed points placed on the Archimedes Spirals is discussed.

math.HO↗

arXiv2020

Spiral Thermal Waves Generated by Self-Propelled Camphor Boats

Spiral thermal surface waves arising from self-propulsion of the camphor-driven objects are reported. Spiral thermal waves were registered for the dissolution and evaporation-guided self-propulsion. Soluto-capillarity is accompanied by thermo-capillarity under self-propulsion of camphor boats. The jump in the surface tension due to the soluto-capillarity is much larger than that due to the thermo-capillarity. The observed thermal effect is related to the adsorption of camphor molecules at the water/vapor interface. The observed spirals are shaped as Archimedean ones.

cond-mat.soft↗

arXiv2017

Magnetic Field Inspired Contact Angle Hysteresis Drives Floating Polyolefin Rafts

Displacement of floating polymer (polyolefin) rafts by steady magnetic field is reported. The effect is due to the interplay of gravity deforming the water/vapor interface, contact angle hysteresis and diamagnetic properties of the liquid support. Magnetic field (ca. 0.06 T) deformed the water/vapor interface and impacted the interfacial apparent contact angle. This deformation gave rise to the propulsion force, displacing the polymer raft. The velocity of displacement of the center of mass of rafts was registered. The effect is related to the contact angle hysteresis affected by the magnetic field, enabling the change in the apparent interfacial contact angle. The semi-quantitative model of the process is suggested

physics.flu-dyn↗

arXiv2017

Paradoxical Coffee-Stain Effect Driven by the Marangoni Flow Observed on Oil-Infused Surfaces

Formation of coffee stain deposits under evaporation of droplets containing aqueous solution of salts placed on silicone-oil impregnated substrates was observed. The formation of ring-like deposits was registered for various molar concentrations of salts for the droplets of 5-300 microlitres in volume. The effect occurred when the contact line was de-pinned, and the evaporation from the edge of a droplet was stopped by the silicone oil. The formation of the coffee stain deposit is related to the soluto-capillary Marangoni flow; the influence of the thermo-capillary flow taking place in parallel is negligible.

cond-mat.soft↗

arXiv2017

Self-Propulsion of a Metallic Superoleophobic Micro-Boat

The self-propulsion of a heavy, superoleophobic, metallic micro-boat carrying a droplet of various aqueous alcohol solutions as a fuel tank is reported. The micro boat is driven by the solutocapillary Marangoni flow. The jump in the surface tension owing to the condensation of alcohols on the water surface was established experimentally. Maximal velocities of the self-propulsion were registered as high as 0.05 m/s. The maximal velocity of the center mass of the boat correlates with the maximal change in the surface tension, due to the condensation of alcohols. The mechanism of the self-locomotion is discussed. The phenomenological dynamic model describing the self-propulsion is reported.

cond-mat.soft↗

arXiv2017

Under-liquid Self-Assembly of Submerged Buoyant Polymer Particles

The self-assembly of submerged cold-plasma-treated polyethylene beads is reported. The plasma-treated immersed millimetrically-sized polyethylene beads formed well-ordered 2D quasi-crystalline structures. The submerged floating of light polyethylene beads is possible due to the energy gain achieved by the wetting of the high-energy plasma-treated polymer surface prevailing over the energy loss due to the upward climb of the liquid over the beads. The capillary immersion attraction force is responsible for the observed self-assembly. The observed 2D quasi-crystalline structures demonstrate dislocations and point defects. Mechanical vibration of self-assembled rafts built of polyethylene beads leads to the healing of point defects. The immersion capillary lateral force governs the self-assembly, whereas the elastic force is responsible for the repulsion of polymer beads.

cond-mat.soft↗

arXiv2017

Camphor-Engine-Driven Micro-Boat Guides Evolution of Chemical Gardens

A micro-boat self-propelled by a camphor engine, carrying seed crystals of FeCl3, promoted the evolution of chemical gardens when placed on the surface of aqueous solutions of potassium hexacyanoferrate. Inverse chemical gardens (growing from the top downward) were observed. The growth of the inverse chemical gardens was slowed down with an increase in the concentration of the potassium hexacyanoferrate. Heliciform precipitates were formed under the self-propulsion of the micro-boat. A phenomenological model, satisfactorily describing the self locomotion of the camphor-driven micro-boat, is introduced and checked.

cond-mat.soft↗

arXiv2017

Self-Propelling Rotator Driven by Soluto-Capillary Marangoni Flows

The self-propelled, longstanding rotation of the polymer tubing containing camphor continuing for dozens of hours is reported. The rotator is driven by the solutocapillary Marangoni flows owing to the dissolution of camphor. The phenomenological model of self-propulsion is suggested and verified. Scaling laws describing the quasi-stationary self-propulsion are proposed and tested experimentally. The change in the surface tension, arising from the dissolution of camphor and driving the rotator is estimated as 0.3 mN/m.

cond-mat.soft↗

arXiv2017

Friction, Free Axes of Rotation and Entropy

Friction forces acting on rotators may promote their alignment and therefore eliminate degrees of freedom in their movement. The alignment of rotators by friction force was demonstrated by experiments performed with different spinners, demonstrating how friction generates negentropy in a system of rotators. A gas of rigid rotators influenced by friction force is considered. The orientational negentropy generated by a friction force was estimated with the Sackur-Tetrode equation. The minimal change in total entropy of a system of rotators, corresponding to their eventual alignment, decreases with temperature. The reported effect may be of the primary importance for a phase equilibrium and motion of ubiquitous colloidal and granular systems.

cond-mat.soft↗