Polygonal boundary gaps in multiple diffusion source precipitation systems in gel media.

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.

Simulation of geochemical banding: Theoretical modeling and fractal structure in acidization-diffusion-precipitation dynamics.

In an earlier work, we presented an experimental study wherein reaction-transport processes were forged in a real rock medium. Zonation of CaSO_{4}-rich and CaSO_{4}-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 H_{2}SO_{4}-acidization front propagating radially from a central source into a CaCO_{3} rock bed causes dissolution of the calcite mineral and precipitation of CaSO_{4} as either gypsum (CaSO_{4}·2H_{2}O) or anhydrite (anhydrous CaSO_{4}). The deposition of CaSO_{4} 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^{+}, Ca^{2+}, and SO_{4}^{2-}), the CaCO_{3} dissolution, and the deposition of CaSO_{4}, 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.

Revisited Chaos in a Diffusion-Precipitation-Redissolution Liesegang System.

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.

Random Spacing between Metal Tree Electrodeposits in Linear DLA Arrays.

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.

Ag fractal structures in electroless metal deposition systems with and without magnetic field.

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.

Routes to fractality and entropy in Liesegang systems.

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.

Fractal structures in two-metal electrodeposition systems II: Cu and Zn.

In this second part of our study on fractal co-electrochemical deposition, we investigate the Cu-Zn system. Macroscopic and microscopic inspection shows a sensitive dependence of the morphology of the final pattern on initial concentrations. The pattern is seen to undergo a transition from classical dendrites to randomly ramified deposits, with each slight increase in [Cu(2+)](0), while [Zn(2+)](0) is maintained constant. The variational trends in chemical composition, growth velocity, and fractal dimension with increasing [Cu(2+)](0) are analyzed. The latter is seen to generally increase with copper (II) ion concentration. In contrast, the growth rate of the deposits is seen to decrease with increasing concentration of Cu(2+) ions. A new probe of dense ramified morphology, the pattern density, is introduced and seen to increase with [Cu(2+)](0). XRD measurements reveal that the observed properties correlate with the birth of copper-rich nuclei, which disrupt the crystalline anisotropy of the two-metal alloy.

Band, target, and onion patterns in Co(OH)2 Liesegang systems.

The study of morphology and shape development has gained considerable interest in certain sciences, notably biology and geology. Liesegang experiments producing Co(OH)2 stratification are performed here, in one, two, and three dimensions for comparison of the pattern morphologies. We obtain well-resolved bands in one dimension, target patterns (rings) in two dimensions, and onion patterns (spherical shells) in three dimensions. The morphological characteristics of the various patterns (spacing coefficients, rate of growth of ring spacing with distance) were measured. The spacing ratio of the strata in the different spatial dimensions was found to be anticorrelated with the surface-to-volume ratio of the gel domain. Some studies featuring the importance of morphology in Liesegang systems are briefly surveyed.

Mechanism of revert spacing in a PbCrO4 Liesegang system.

Periodic precipitation of sparingly soluble salts yields parallel Liesegang bands in 1D whose spacings obey either one of two known trends. The overwhelming trend is an increase in spacing as we move away from the junction, while some systems display a decrease in spacing as the bands get further away from the interface. The latter trend is much less common and is known as the revert spacing law. Whereas the direct (normal) spacing law is generally well-understood, the revert spacing trend has not been explicitly and distinctly elucidated. In this paper, we propose a mechanism of revert spacing governed by the adsorption of the diffusing CrO4(2−) ions on the formed PbCrO4 Liesegang bands and carry out a set of experiments that support the suggested scenario. It is shown that this adsorption increases as the band number (n) increases in revert spacing systems, while it decreases as n increases in direct spacing systems. It is concluded that this correlation in opposite directions decisively reveals the role of adsorption in the mechanism. The attraction between the CrO4(2−) and Pb(2+) in the gel causes the bands to form gradually closer and closer. Secondary structure (thinner bands formed within the main ones) obtained under some conditions is discussed in view of the light sensitivity of the chromate ion and the stability of the lead chromate sol.

Fractal structures in two-metal electrodeposition systems I: Pb and Zn.

Pattern formation in two-metal electrochemical deposition has been scarcely explored in the chemical literature. In this paper, we report new experiments on zinc-lead fractal co-deposition. Electrodeposits are grown in special cells at a fixed large value of the zinc ion concentration, while that of the lead ion is increased gradually. A very wide diversity of morphologies are obtained and classified. Most of the deposited domains are almost exclusively Pb or Zn. But certain regions originating at the base cathode, ranging from a short grass alley to dense, grown-up bushes or shrubs, manifest a combined Pb-Zn composition. Composition is determined using scanning electron microscopy/energy dispersive x ray measurements as well atomic absorption spectroscopy. Pb domains are characterized by shiny leaf-like and dense deposits as well as flowers with round, balloon-like corollas. The Zn zones display a greater variety of morphologies such as thick trunks and thin and fine branching, in addition to minute "cigar flower" structures. The various morphologies are analyzed and classified from the viewpoint of fractal nature, characterized by the box-count fractal dimension. Finally, macroscopic spatial alternation between two different characteristic morphologies is observed under certain conditions.

Ring morphology and pH effects in 2D and 1D Co(OH)2 Liesegang systems.

We study the factors that affect the morphology of Co(OH)(2) Liesegang rings, in a way to obtain concentric rings with large spacing, upon an appropriate variation in the experimental conditions. Such well-resolved patterns are obtained under optimum conditions: decrease in the concentration of the outer electrolyte, increase in the concentration of both the inner electrolyte and the gelatin in the hosting gel medium, and increase in the strength of a constant radial electric field applied across the pattern domain. The effect of pH on the bands in a 1D Co(OH)(2) Liesegang pattern is also investigated. The initial pH of the diffusing solution plays a central role in altering the band morphology, because the outer electrolyte (NH(4)OH) is a base, strongly affected by the H(+) equilibrium. The number of bands decreases and the interband spacing increases with decreasing pH of the NH(4)OH solution. The pattern morphology in that case is controlled by the NH(4)Cl/NH(4)OH ratio.

Propagating fronts in thin tubes: concentration, electric, and pH effects in a two-dimensional precipitation pulse system.

In this paper, we studied the dynamics of a CaCO3 precipitate deposition pulse in a thin, long tube connecting two reservoir sinks of coprecipitates. The pulse profile, as well as the time t(c) and distance x(c) of the first appearance of precipitate, is studied as a function of the initial concentration of CO(3)(2-) in the right reservoir, [CO(3)(2-)](0), and later as a function of an applied external electric field at different voltages. The time variations of the pulse location and the pH at the center of the tube are determined. The distance from the calcium chloride sink (x) at any fixed time decreases as [CO(3)(2-)](0) increases. The time evolution of the front location exhibits a crossover between an early time regime and a late time regime. The pH-time curve shows a marked resemblance with a sigmoid shape. At any time, the pH consistently increases with [CO(3)(2-)](0). In the presence of a constant electric field applied across the tube (fixed voltage), t(c) decreased with the field strength, whereas x(c) exhibited a correlated increase. Irregularities in the variation of distance with the applied voltage (at a fixed time) were noted. The pH experiences a slight increase with the applied voltage. The pulse width exhibits a nonlinear time dependence, of the form w = a + bt(1/6). The shape of the deposition pulse deviates from a Gaussian distribution. This study is of special interest in the experimental simulation and modeling of precipitate deposition and potential clogging in microcapillary channels.

Front propagation in patterned precipitation. 3. Composition variations in two-precipitate stratum dynamics.

This paper is a study of the composition dynamics of Liesegang band strata of Co(OH)2 and Ni(OH)2 from NH4OH, with redissolution by complex formation with ammonia. At a fixed time, the cobalt hydroxide composition was found to exhibit a random variation with band number, yet within a general overall decrease. The decrease with band number becomes more pronounced as the initial concentrations of Co2+ and Ni2+ get closer to each other. At equal concentrations, periodic oscillations in Co(OH)2 composition appear over consecutive bands. The time evolution of the total Co(OH)2 mass percent (over the entire pattern of strata) passes through a maximum. The dynamics of this complex system has been simulated by theoretical calculations using the model of Müller and Polezhaev, modified by Al-Ghoul and Sultan in a series of two papers in J. Phys. Chem. A; the present paper is the third in the series. The simulations capture the essential features of the experimentally observed dynamics.

Patterns with high rhythmicity levels in multicomponent Liesegang systems.

Liesegang banding is the display of rhythmic strata of precipitate as co-precipitate ions interdiffuse in a gel medium. Complex periodic patterns as well as aperiodic structures could emerge, notably in systems where more than one salt is precipitated. The use of three cations (Co2+) Ni2+, and Mg2+) in the banded precipitation of their hydroxides resulted in an unusual pattern with a consistently increasing rhythmicity. A periodic structure marked by the succession of band multiplets with increasing number of bands (from singlets to doublets to triplets to quadruplets, consistently) was observed. Such rhythmic patterns are obtained as the initial Mg2+ concentration ([Mg2+]0), chosen as a control parameter, increases through a critical value. At high [Mg2+]0, the trend breaks after a long time elapses. Two types of bifurcation are therefore experienced by such a system: concentration bifurcation and diffusive (time/space) bifurcation. The dynamics is elucidated on the basis of an analysis of the bands in certain groups, and gel regions between these groups, as well as between group blocs (here, a bloc denotes a succession of multiplet groups, with repetitively the same number of bands). Finally, similarities between our system and naturally occurring rhythmic patterns are emphasized and discussed.

Effect of competitive complex formation on patterning and front propagation in periodic precipitation.

The Co(OH)2 Liesegang pattern (from Co2+ and NH4OH) propagates in space by periodic band formation at the head due to precipitation and band disappearance at the tail due to dissolution in excess NH4OH. Introduction of Ni2+, which competes with Co2+ for complex formation with ammonia, into the system led to several interesting observations: slower propagation, fewer bands, and increased spacing between them. Above a cutoff concentration of Ni2+ (0.13 M for [Co2+]0 = 0.100 M), only a uniform precipitation zone was observed. The quantity delta X = |XNi-Xfree|, namely, the difference in the position of last band in the system with nickel relative to that without nickel attained at a fixed time (93 h), was found to oscillate as the initial concentration of NH4OH ([NH4OH]0) was varied. It was shown that variations in the concentration of intermediate NH4+ act as precursors for these oscillations: [NH4+] was measured with an ammonia-specific electrode, and the difference delta [NH4+] = |[NH4+]Ni-[NH4+]free| at 93 h exhibited oscillations with [NH4OH]0 similar to those obtained in delta X. Theoretical calculations based on the model of Müller and Polezhaev agreed with the experimental results.