Reaction-diffusion processes

This study investigated how cyclic switching between high and low voltages can generate Liesegang-like (but not conforming to the scaling laws) periodic precipitation bands of metal hydroxides in agarose gel columns, aiming to uncover the generality and underlying mechanism of this reaction-transport-reaction process. The process is generalizable to various combinations of metal anodes and cathodes, with the number of bands typically increasing with cycle number. The spatiotemporal evolution of the bands exhibits considerable anode-element dependence, but is less dependent on the cathode element, as well as the dimensions of the gel column, demonstrating the generality of this banding phenomenon. Analyses of the time-dependence of the electric current and the morphology and composition of the anode-derived deposits indicate that OH À ions react with the anode surface after their transport from the cathode. Numerical simulations based on extended Nernst-Planck equations suggest that periodic banding is possible if (i) the generation of reactant ions is periodically suppressed, (ii) the diffusion coefficients of the reactant ions differ considerably, and (iii) a quantitative imbalance exists between the produced and precipitated ions. A qualitative mechanism for the observed banding phenomenon is proposed, in which the tunneling effect of OH À ions, owing to the Grotthuss mechanism, is suggested as a possible cause of this quantitative imbalance. These findings offer new insights into electrochemically induced pattern formation and highlight the potential of reaction-transport-reaction systems for advancing our understanding of coupled reaction-transport phenomena.

We describe a new class of the periodic banding of Ag2O precipitates through reaction-transport-reaction processes in an agarose-hydrogel column; the gel contains NaNO3 as an additive and is inserted between Ag and Ti rods with an applied constant voltage of 4.0-7.0 V. Submillimeter-scale periodic dark-brown precipitation bands were transiently generated with high reproducibility in this system, in which (i) electrochemical reactions at the metal rods to generate Ag+ and OH− , (ii) transportation of the reactant ions by the electric field, and (iii) reactions to produce Ag O are expected to couple. The bands successively emerged mainly toward the cathode, accompanied by substantial changes in the electric current through the gel. The periodic banding depended on the applied voltage, duration of application, and concentration of NaNO 3 initially loaded in the gel. The banding was most clearly observed in a 2.0 mass% agarose gel column containing 0.005 M NaNO3 under a constant applied voltage of~5.5 V for 3 h. This optimal condition resulted in more than ten periodic bands, of which the bandwidth (w) and the distance between the adjacent bands (d) were randomly distributed around the average values: for the bands formed up to 2 h, w = 0.25 ± 0.04 mm and d = 0.42 ± 0.05 mm; for the bands formed after 2 h, w = 0.42 ± 0.11 mm and d = 0.68 ± 0.07 mm. The generated periodic bands were gradually painted out with time to form an almost continuous broad band, even after the applied voltage was discontinued, although immersion of the gel in deionized water for 3 h could suppress this uniformization. Observations using optical and scanning electron microscopes with an energy-dispersive X-ray detector suggested that the periodic bands were composed of gelatinous Ag2O and micrometer-scale (1-50 μm) clusters, which comprised Agabundant nanoparticles with considerable size dispersion (10-500 nm). The relationship between the periodic banding and inhomogeneous distribution of the reactant ions and potential applications of the observed banding phenomena are discussed.

Two main approaches have been considered for modelling the dynamics of the SIS model on complex metapopulations, i.e, networks of populations connected by migratory flows whose configurations are described in terms of the connectivity distribution of nodes (patches) and the conditional probabilities of connections among classes of nodes sharing the same degree. In the first approach migration and transmission/recovery process alternate sequentially, and, in the second one, both processes occur simultaneously. Here we follow the second approach and give a necessary and sufficient condition for the instability of the disease-free equilibrium in generic networks under the assumption of limited (or frequency-dependent) transmission. Moreover, for uncorrelated networks and under the assumption of non-limited (or density-dependent) transmission, we give a bounding interval for the dominant eigenvalue of the Jacobian matrix of the model equations around the disease-free equilibrium. Finally, for this latter case, we study numerically the prevalence of the infection across the metapopulation as a function of the patch connectivity.

The effect of magnetic fields on the precipitation patterns of Mn−Fe-based Prussian blue analogues in water−glass gels was studied using X-ray fluorescence and X-ray absorption near-edge structure spectroscopies. Three sets of two glass tubes, A, B, and C, were prepared using 1.20 M Mn 2+ /0.24 M [Fe(CN) 6 ] 3− , 0.60 M Mn 2+ /0.12 M [Fe(CN) 6 ] 3− , and 0.30 M Mn 2+ /0.06 M [Fe(CN) 6 ] 3− solutions, respectively. From each of these sets, one tube was subjected to a magnetic field of 0.5 T, whereas the other was not. The magnetic field barely affected the Liesegang bands in the tube from Set A, but there were noticeable differences in the tubes from sets B and C, where (1) the amounts of electrolytes were small, (2) the dominant Mn species was [Mn(H 2 O) 6 ] 2+ , and (3) there was stochasticity of the band formation. In these regions, the magnetic field painted out the spaces between the precipitation bands, even enhancing the formation of additional bands.

Two main approaches have been considered for modelling the dynamics of the SIS model on complex metapopulations, i.e, networks of populations connected by migratory flows whose configurations are described in terms of the connectivity distribution of nodes (patches) and the conditional probabilities of connections among classes of nodes sharing the same degree. In the first approach migration and transmission/recovery process alternate sequentially, and, in the second one, both processes occur simultaneously. Here we follow the second approach and give a necessary and sufficient condition for the instability of the disease-free equilibrium in generic networks under the assumption of limited (or frequency-dependent) transmission. Moreover, for uncorrelated networks and under the assumption of non-limited (or density-dependent) transmission, we give a bounding interval for the dominant eigenvalue of the Jacobian matrix of the model equations around the disease-free equilibrium. Finally, for this latter case, we study numerically the prevalence of the infection across the metapopulation as a function of the patch connectivity.

We propose a novel electrochemical system to form precipitation patterns of Cu-Fe-based Prussian blue analogues (Cu-Fe PBA) in agarose gels, using an applied voltage to produce reactant ions. The spatiotemporal evolution, spatial distribution, and crystallite morphologies of the precipitates were investigated by visual inspection, Fe Kα intensity distribution measurements, and optical and scanning electron microscope observations. The precipitation patterns and their evolution depended on the applied voltage. Multicolored periodic precipitation bands were stochastically formed under cyclic alternating voltage (4 V for 1 h and then 1 V for 4 h per cycle). The distances between adjacent bands were randomly distributed (0.30 ± 0.25 mm). The sizes and shapes of the crystallites generated in the gel were position-dependent. Almost cubic but fairly irregular crystallites (0.1–0.8 μm) were formed in the periodic bands, whereas definitely cube-shaped crystallites (1–3 μm) appeared close to ...

We propose a simple and novel system to form precipitation patterns of Cu-Fe-based Prussian blue analogues (Cu-Fe PBA) in agarose gel through coupled electrochemical reactions, reactant ion diffusion influenced by electric field, and precipitation reactions. The spatiotemporal evolution, spatial distribution, and crystallite morphologies of the precipitates were investigated by visual inspection, Fe Kα intensity distribution measurements, and optical and scanning electron microscope observations. The observed precipitation patterns and their evolution depended on the applied voltage. Multicolored periodic precipitation bands were stochastically formed under cyclic alternating voltage (4 V for 1 h and then 1 V for 4 h per cycle). The distances between adjacent bands were randomly distributed (0.30 ± 0.25 mm). The sizes and shapes of the crystallites generated in the gel were position-dependent. Cubic but fairly irregular crystallites (0.1–0.8 μm) were formed in the periodic bands, wh...

Precipitation patterns spontaneously formed by Mn–Fe PBAs in agarose gel were investigated over wide concentrations ranges for the outer (0.10 ≤ [Mn2+] ≤ 0.70 M) and inner electrolytes (0.01 ≤ [[Fe(CN)6]3−] ≤ 0.35 M) by visual/SEM observations.

Precipitation patterns spontaneously formed by Mn-Fe-based Prussian blue analogues (Mn-Fe PBAs) in agarose gel were investigated over wide concentration ranges for the outer (0.10 # [Mn 2+ ] # 0.70 M) and inner electrolytes (0.01 # [[Fe(CN) 6 ] 3À ] # 0.35 M). The precipitation patterns were classified into five types: continuous in the inner electrolyte gel, continuous in the outer electrolyte gel, a very short band close to the gel boundary, periodic bands in the inner electrolyte gel, and filament-like patterns in the inner electrolyte gel. The concentrations found to be suitable for applications as pulsatile 137 Cs adsorbent delivery materials, for which the periodic patterns were most distinct and frequently occurring, were [Mn 2+ ] z 0.55 M and [[Fe(CN) 6 ] 3À ] z 0.10 M. The filament-like pattern, reported here for the first time, was generated near the cylindrical gel surface in a local, stochastic, and twodimensional manner. Scanning electron microscopy (SEM) images of the periodic band patterns indicate that the Mn-Fe PBA precipitates in periodic patterns consist of cubic crystallites (3-10 mm, each side), suggesting significant contributions from Ostwald ripening to the precipitation process. In contrast, precipitates in the filament-like pattern consist of relatively smaller cubic crystallites ((1 mm, each side), implying an important role of rapid nucleation in the formation of this structure. SEM observations also suggest that, overall, agarose gels support crystallite ripening more effectively, and hence also better suppress the formation of definite periodic bands, than water-glass gels. These findings provide useful information for future applications of self-assembled Mn-Fe PBA patterns in gels.

A light source or an oven is used for several seconds to force a heat gradient along a buried channel in printed circuit board. Then, it is let to cool while we analyze the emitted heat through end points to expose the hidden defects. The heat diffusion equation is used numerically for a common defect to predict the electric current flow by means of thermal current flow. This patented method is a novel non contact optical inspection, to challenge the market demand for miniaturization with efficiency.

We propose a simple and novel system to form precipitation patterns of Cu-Fe-based Prussian blue analogues (Cu-Fe PBA) in agarose gel through coupled electrochemical reactions, reactant ion diffusion influenced by electric field, and precipitation reactions. The spatiotemporal evolution, spatial distribution, and crystallite morphologies of the precipitates were investigated by visual inspection, Fe Kα intensity distribution measurements, and optical and scanning electron microscope observations. The observed precipitation patterns and their evolution depended on the applied voltage. Multicolored periodic precipitation bands were stochastically formed under cyclic alternating voltage (4 V for 1 h and then 1 V for 4 h per cycle). The distances between adjacent bands were randomly distributed (0.30 ± 0.25 mm). The sizes and shapes of the crystallites generated in the gel were position-dependent. Cubic but fairly irregular crystallites (0.1-0.8 μm) were formed in the periodic bands, whereas definitely cube-shaped crystallites (1-3 μm) appeared close to the anode. These cube-like reddish-brown crystallites were assigned to Cu-Fe II PBA. In some periodic bands, plate-like blue crystallites (assigned to Cu(OH)2) were also present. Future issues for potential applications of the observed periodic banding for selective preparation of Cu-Fe PBA crystallites were discussed.