A Voronoi diagram is a mathematical tool for nesting as set of points on a 2D area, according to a specific mathematical rule. The outcome is a tiling of the whole surface, with the boundaries of every tile (or cell), being based on the distance to the points around it. Each point p(k) has its own corresponding cell R-k. In Chemistry, and in material science, Voronoi diagrams offer aid in crystal growth mechanisms, notably in relation to their topological distribution. A simple application of constructing such topographical maps resides in diffusion-precipitation dynamics. When the precipitation originates from different sources with random spatial distribution, void, precipitate free gaps delineate the various domains, just like the cell boundaries in a mathematical Voronoi diagram. We construct such tessellated sheets, using a variety of precipitate systems including cobalt hydroxide, cobalt phosphate, copper chromate, lead chromate; and precipitates involving the Prussian blue reagents, such as: silver ferrocyanide, silver ferricyanide, cobalt ferrocyanide and cobalt ferricyanide. Besides holes of equal concentration in the diffusing electrolyte (the traditional method), experiments were conducted while varying the concentrations and the hole sizes. Distance and angle measurements allowed a classification of the tessellation parameters from the viewpoint of kinetic laws, chaotic dynamics, and notably entropy calculations.

We carry out experiments in diffusion-precipitation systems in a framework with multiple diffusion sources. The advancing precipitation fronts reach a point where they stop, leaving a rift or gap between the precipitate zones. In a similar setting, fractal metal deposits emerging from various reduction centers were synthesized. They were shown to meander in the medium, but fail to meet or overlap, thus also leaving a gap or rift just like in the precipitation system. The obtained patterns were analyzed by calculating entropy and front velocity for the former system, and fractal dimension for the latter. The observed structures represent chemical analogs of Voronoi diagrams.

Liesegang patterns present a display of parallel stripes of precipitate that arise from the interdiffusion of coprecipitate ions in a gel medium. The bands observed in rocks are typical analogies of this phenomenon, and their composition is not restricted to the banded deposition of a single mineral. We here extend the study to a three-precipitate system, wherein Co2+, Ni2+, and Cd2+ cations are precipitated by the same anion (OH- from NH4OH) to form Co(OH)2, Ni(OH)2, and Cd(OH)2, respectively. The resulting pattern exhibits an alternation of compact mixed Co(OH)2 and Ni(OH)2 bands with granules of pure Cd(OH)2 between them. The obtained pattern confirms the generic type of (A + B)/C/(A + B)/C/(A + B)/C alternation conjectured analytically using the competitive particle growth (CPG) model and stability analysis.

We carry out an evolutionary study of the COVID-19 pandemic, focusing on the case of Lebanon. The disease spread exhibits four eruption phases or waves. Chaos theory tools point toward a correlation of events, essentially obeying a quasi-deterministic chaotic regime. The analysis of the time series yields a largest Lyapunov exponent of 0.263, indicative of a chaotic trend. The review of past and recent analyses and modeling of pandemics could assist in the predictabilty of their course of evolution, effective management and decision making for health authorities.

We carry out a chemical treatment (acidization or basification) of typical rock specimens in-situ, and characterize the emerging pattern resulting from the infiltration-precipitation scenario. Galena and limestone samples were treated with sulfuric acid, while pyrite was reacted with sodium hydroxide. Various infiltration techniques were employed, after selection of the most feasible method for each rock separately. The patterns of anglesite (PbSO 4 ), anhydrite (CaSO 4 ) and goethite (FeOOH) deposition presented different alteration modes of the bare rock textures. Among the three deposited minerals, only the anhydrite (CaSO 4 ) displayed a band stratification. The formation of a Liesegang pattern in the rock of highest porosity indicates a plausible correlation between the band formation and a minimum porosity requirement. A banded rock of compact texture could then be formed by a cementation mechanism, governing the long time evolution of the rock.

We investigate multiple reaction-diffusion processes that engender the formation of distinct precipitation zones. In this paper, we carry out various original precipitation reactions in a gel medium, wherein the interdiffusion of the co-precipitates occurs from various sources arranged in a symmetric framework in 2D Petri dishes. The distinct precipitation zones are separated by clear polygonal boundaries, in congruence with the spatial distribution of the diffusion holes hosting the outer electrolyte. We use scanning electron microscopy, energy dispersive x-ray diffraction spectrometry, and notably powder x-ray diffraction for the characterization of the differentiated precipitate patterning zones for each system studied. The obtained patterns find their application niche in the chemical analogs of Voronoi diagrams and the rift scenery in geological landscapes.

EDITORIAL article Front. Phys., 18 October 2022Sec. Physical Chemistry and Chemical Physics Volume 10 - 2022 | https://doi.org/10.3389/fphy.2022.1051493

We investigate the formation of CaCO3 zonation using a displacement reaction in the solid phase through the diffusion of aqueous carbonic acid into a slaked lime [Ca(OH)(2)] gel putty. The obtained calcium carbonate gives rise to a variety of mosaic patterns, and interesting zonation in 3D. The system also displays Liesegang banding. We use pH measurements, scanning electron microscopy (SEM), Fourier Transform Infrared Spectroscopy (FTIR), and powder X-ray diffraction (XRD), for the characterization of the various differentiated zones within the pattern, and the crystalline forms therein. (C) 2021 Elsevier B.V. All rights reserved.

A small toroidal reactor is filled with CoCl2 center dot 6H(2)O and 1% agar gel solution. Two outer electrolytes, of suitably chosen concentrations, are employed to diffuse into the gelled solution: from one end NH4OH to precipitate Co(OH)(2), and from the other end Na3PO4 to precipitate Co-3(PO4)(2). The resulting diffusion-precipitation (Liesegang) pattern exhibits unusual trends wherein gaps and formation of cobalt phosphate hydrates are observed. The different precipitates are characterized by IR, X-ray and SEM techniques. Using the well-known generic empirical laws, we compute the band locations, and represent them graphically within the torus, for two experiments with different sets of initial concentrations.

We study a variety of reaction-diffusion processes that lead to the formation of exotic patterns. 1. We carry out precipitation reactions in gel media, wherein the interdiffusion of the co-precipitates takes place from multiple diffusion sources arranged in a symmetric framework. The precipitation zones are delimited by clear polygonal boundaries in congruence with the spatial distribution of the diffusion pools. 2. A displacement reaction in a solid-gel medium is conducted as a carbonic acid diffusion front invades an agar-calcium hydroxide gel putty. The formation of calcium carbonate yields a diversity of patterns, ranging from mosaic structures to Liesegang bands. 3. A Liesegang experiment precipitating lead chromate from the interdiffusion of lead and chromate ions in 2D yields a pattern of rings exhibiting revert spacing. When the diffusion comes from a constantly fed unstirred source (or reactor, CFUR), the patterns transit to a chaotic regime which is sensitive to the concentrations used and the flow rate.

In an earlier work, we presented an experimental study wherein reaction-transport processes were forged in a real rock medium. Zonation of CaSO4-rich and CaSO4-depleted domains were obtained and characterized. In the present study, we present a theoretical model to simulate the reaction-diffusion processes underlying the dynamics of the system. An H2SO4-acidization front propagating radially from a central source into a CaCO3 rock bed causes dissolution of the calcite mineral and precipitation of CaSO4 as either gypsum (CaSO4 center dot 2H(2)O) or anhydrite (anhydrous CaSO4). The deposition of CaSO4 is shown to exhibit a banded texture (irregular concentric rings in two dimensions). The model involves reaction-diffusion evolution equations for three aqueous species (H+, Ca2+, and SO42-), the CaCO3 dissolution, and the deposition of CaSO4, which is taken to obey a scaled Cahn-Hilliard equation. The output captures the zonation observed experimentally. Fractal analysis of the experimental contour shapes of the deposits reveals an oscillation in the fractal dimension over successive band numbers. Such oscillation is interpreted in terms of the precipitation-depletion tug scenario, not observable in regular two-dimensional Liesegang systems with high circular symmetry.

We present a novel study of a PbCrO4 Liesegang pattern exhibiting revert spacing. Scanning Electron Microscopy (SEM), Atomic Absorption Spectrometry (AAS) and Energy Dispersive X-ray (EDX) spectroscopy measurements are carried out, and are shown to support the adsorption of CrO42- on the precipitate, which becomes more enhanced as we move farther from the gel interface. Such an ascending differential adsorption scenario favors revert spacing over the direct spacing trend.

When we examine the random growth of trees along a linear alley in a rural area, we wonder what governs the location of those trees, and hence the distance between adjacent ones. The same question arises when we observe the growth of metal electro-deposition trees along a linear cathode in a rectangular film of solution. We carry out different sets of experiments wherein zinc trees are grown by electrolysis from a linear graphite cathode in a 2D film of zinc sulfate solution toward a thick zinc metal anode. We measure the distance between adjacent trees, calculate the average for each set, and correlate the latter with probability and entropy. We also obtain a computational image of the grown trees as a function of parameters such as the cell size, number of particles, and sticking probability. The dependence of average distance on concentration is studied and assessed.

Co(OH)2 Liesegang periodic precipitation systems exhibit oscillations in the number of bands due to band redissolution in high NH4OH concentration. We revisit the problem previously considered (Nasreddine and Sultan, J. Phys. Chem. A 1999, 103, 2934-2940) by rigorously refining the experiments and the Chaos analysis. Chaos is established in this diffusion-precipitation-redissolution system, as is evident from the refined outputs of the Chaos analysis tools. A brief account of possible applications of Chaos in Liesegang systems is presented.

Liesegang patterns of silver dichromate (Ag2Cr2O7) are studied in two different gel media: agar and gelatin, based on the work of Lagzi and Ueyama (2009). Whereas in gelatin, standard Liesegang bands are obtained as a result of the interdiffusion of Ag+ and Cr2O72- random crystallites with dendritic ramifications are observed in agar. We revisit this phenomenon and demonstrate the proposed mechanism, wherein dense heterogeneous nucleation in gelatin leads to Liesegang bands, as opposed to surface nucleation in agar yielding crystallites. We use viscosity, pH measurements, and notably scanning electron microscopy (SEM) in this endeavor. (C) 2018 Elsevier B.V. All rights reserved.

Metal electrodeposition systems display tree-like structures with extensive ramification and a fractal character. Electrolysis is not a necessary route for the growth of such dendritic metal deposits. We can grow beautiful ramification patterns via a simple redox reaction. We present here a study of silver (Ag) deposits from the reduction of Ag+ in (AgNO3) solution by metallic copper. The experiments are carried out in discotic geometry, in a Petri dish hosting a thin AgNO3 solution film. A variety of deposited structures and patterns is obtained at different Ag+ concentrations, yet with essentially the same fractal dimension averaged at 1.64, typical of diffusion-limited aggregation (DLA). A linear magnetic field of low induction (0.50-1.0 T) applied across the medium causes a notable transformation in the morphology of the deposits. In both the field off and the field on cases, the effect of vertical (hence 3D) heaving seems to be dominant, perhaps explaining the nearly constant fractal dimension.

This paper is the first among two articles which aim at exploring the possible similarities between the well-known Liesegang banding phenomenon in precipitate systems, and the stripe formation observed in a large number of rocks. In the present (first) article, we review a comprehensive and long study wherein patterning experiments were performed in-situ (real rock systems), to simulate the band formation through the acidization of a ferruginous limestone rock, causing dissolution and precipitation reactions. The results are analyzed by microscopy, AA and XRD techniques. The correlation between the Liesegang gel experiment and the processes taking place inside the rock medium is established. In the second part (following paper), a theoretical model is set forth to support our experiments. The fractal nature of the contours of the various regions will be explored.

Liesegang bands are formed when solutions of co-precipitate ions interdiffuse in a 1D gel matrix. In a recent study [R. F. Sultan, Acta. Mech. Sin. 27, 119 (2011)], Liesegang patterns have been characterized as fractal structures. In addition to experimentally obtained patterns, geometric Liesegang patterns were constructed in conformity with the well-known empirical laws. Both mathematical fractal dimensions and box count dimensions for images of PbF2 and PbI2 Liesegang patterns have been calculated. Liesegang patterns can also be described by the entropy state function, and categorized as more or less ordered structures. We revisit the relation between entropy and fractal dimension, and apply it to simulated geometrical Liesegang patterns. We have resort to three different routes for the estimation of the entropy of a Liesegang pattern. The HarFA software enabled the calculation of the Hausdorff dimension and the topological entropy, then the information dimension and the Shannon entropy. In a third pathway, analytical calculations were carried out by estimating the probability of occurrence of a fractal element or coverage. The product of Shannon entropy and Boltzmann constant yields the thermodynamic entropy. The values for PbF2 and PbI2 Liesegang patterns attained the order of magnitude of the reported Third Law entropies, but yet remained lower, in conformity with the more ordered Liesegang structures.

Complex reaction-transport dynamics can lead to the formation of ordered structures. A constant dissipation of free energy is a requirement for sustaining macroscopic order, especially in solution. In the solid phase, the evolved pattern can be locked for days, months or even years. Liesegang bands are stratified stripes of precipitate that appear and persist, when co-precipitate ions interdiffuse in a gel medium. A host of interesting properties characterize such rich dynamical systems: band spacing laws (direct and revert), band splitting, rhythmic multiplicity, multiple precipitate formation and band redissolution are but a few manifested characteristics, emerging from a complex dynamics with a great diversity of scenarios. The familiar and well-known band formation in rocks could be the result of a complex coupled diffusion-percolation-chemical reaction mechanism. Similarities between geochemical self-organization and the Liesegang phenomenon are surveyed and analyzed. The simulation of band generation in a rock bed is realized and carried out in- situ, by injection and infusion of the reactant components into the rock medium. Ramified, tree-like structures (dendrites) are obtained during the electrodeposition or simple electroless redox deposition of metal systems. A great variety of morphologies