Pattern Formation in Precipitation Reactions: The Liesegang Phenomenon

Designing Complex Tapestries with Photography-Inspired Manipulation of Self-Organized Thin-Films

Thin-films patterned with complex motifs are of fundamental interest because of their advanced optical, mechanical and electronic properties, but fabrication of these materials remains challenging. Self-organization strategies, such as immersion controlled reaction-diffusion patterning, have shown great potential for production of patterned thin-films. However, the autonomous nature of such processes limits controllable pattern customizability and complexity. Here, it is demonstrated that photography inspired manipulation processes can overcome this limitation to create highly-complex tapestries of micropatterned films (MPF's). Inspired by classical photographic processes, MPF's are developed, bleached, exposed, fixed, and contoured into user-defined shapes and photographic toning reactions are used to convert the chemical composition MPF's, while preserving the original stripe patterns. By applying principles of composite photography, highly complex tapestries composed of multiple MPF layers are designed, where each layer can be individually manipulated into a specific shape and composition. By overcoming fundamental limitations, this synergistic approach broadens the design possibilities of reaction-diffusion processes, furthering the potential of self-organization strategies for the development of complex materials.

Design of chemobrionic and biochemobrionic scaffolds for bone tissue engineering

Chemobrionic systems have attracted great attention in material science for development of novel biomimetic materials. This study aims to design a new bioactive material by integrating biosilica into chemobrionic structure, which will be called biochemobrionic, and to comparatively investigate the use of both chemobrionic and biochemobrionic materials as bone scaffolds. Biosilica, isolated from Amphora sp. diatom, was integrated into chemobrionic structure, and a comprehensive set of analysis was conducted to evaluate their morphological, chemical, mechanical, thermal, and biodegradation properties. Then, the effects of both scaffolds on cell biocompatibility and osteogenic differentiation capacity were assessed. Cells attached to the scaffolds, spread out, and covered the entire surface, indicating the absence of cytotoxicity. Biochemobrionic scaffold exhibited a higher level of mineralization and bone formation than the chemobrionic structure due to the osteogenic activity of biosilica. These results present a comprehensive and pioneering understanding of the potential of (bio)chemobrionics for bone regeneration.

Periodic Stratification of Colloids in a Liquid Phase Produced by a Precipitation Reaction and Gel Swelling

Pattern formation is a frequent phenomenon occurring in animate and inanimate systems. The interplay between the mass transport of the chemical species and the underlying chemical reaction networks generates most patterns in chemical systems. Periodic precipitation is an emblematic example of reaction-diffusion patterns, in which the process generates a spatial periodic structure in porous media. Here, we use the dormant reagent method to produce colloidal particles of Prussian blue (PB) and PB analogues at the liquid-gel interface. The generated particles produced a stable periodic stratification pattern in time in the liquid phase placed on top of the solid hydrogel. The phenomenon is governed by periodic swelling of the gel driven by the osmotic stress and stability of the formed particles. To illustrate the phenomenon, we developed an extended reaction-diffusion model, which incorporated the gel swelling and sedimentation effect of the formed colloids and could qualitatively reproduce the pattern formation in the liquid phase.

Periodic Precipitation in a Confined Liquid Layer

Pattern formation is a ubiquitous phenomenon in animate and inanimate systems generated by mass transport and reaction of chemical species. The Liesegang phenomenon is a self-organized periodic precipitation pattern always studied in porous media such as hydrogels and aerogels for over a century. The primary consideration of applying the porous media is to prevent the disintegration of the precipitation structures due to the sedimentation of the precipitate and induced fluid flow. Here, we show that the periodic precipitation patterns can be engineered using a Hele-Shaw cell in a confined liquid phase, restricting hydrodynamic instability. The patterns generated in several precipitation reaction systems exhibit spatiotemporal properties consistent with patterns obtained in solid hydrogels. Furthermore, analysis considering the Rayleigh-Darcy number emphasizes the crucial role of fluidity in generating periodic precipitation structures in a thin liquid film. This exploration promises breakthroughs at the intersection of fundamental understanding and practical applications.

Anisotropic Liesegang pattern for the nonlinear elastic biomineral-hydrogel complex

The Liesegang pattern is a beautiful natural anisotropic patterning phenomenon observed in rocks and sandstones. This study reveals that the Liesegang pattern can induce nonlinear elasticity. Here, a Liesegang-patterned complex with biomineral-hydrogel repetitive layers is prepared. This Liesegang-patterned complex is obtained only when the biomineralization is performed under the supersaturated conditions. The Liesegang-patterned complex features a nonlinear elastic response, whereas a complex with a single biomineral shell shows a linear behavior, thus demonstrating that the Liesegang pattern is essential in achieving nonlinear elasticity. The stiff biomineral layers have buffered the concentrated energy on behalf of soft hydrogels, thereby exposing the hydrogel components to reduced stress and, in turn, enabling them to perform the elasticity continuously. Moreover, the nonlinear elastic Liesegang-patterned complex exhibits excellent stress relaxation to the external loading, which is the biomechanical characteristic of cartilage. This stress relaxation allows the bundle of fiber-type Liesegang-patterned complex to endure greater deformation.

Growth of Nanocrystal Superlattices from Liquid Crystals

The growth of superlattices (SLs) made from self-assembled nanocrystals (NCs) is a powerful method for creating new materials and gaining insight into fundamental molecular dynamics. Previous explorations of NCSL syntheses have mostly compared them to crystallization. However, NCSL synthesis has not broadly shown cooling crystallization from saturated solutions as a reversible crystallization-dissolution process. We demonstrate the reversible growth of NCSLs by dispersing NCs in liquid crystal (LC) "smart solvents," and harnessing the transitions between the isotropic and nematic phases of the LCs. The growth mode and morphology can be tuned. This process is a model platform for studying crystallization and demonstrates great potential in manufacturing NCSLs as colloidal crystals through liquid-phase epitaxy or colloidal synthesis.

Artificial Homeostasis Systems Based on Feedback Reaction Networks: Design Principles and Future Promises

Feedback-controlled chemical reaction networks (FCRNs) are indispensable for various biological processes, such as cellular mechanisms, patterns, and signaling pathways. Through the intricate interplay of many feedback loops (FLs), FCRNs maintain a stable internal cellular environment. Currently, creating minimalistic synthetic cells is the long-term objective of systems chemistry, which is motivated by such natural integrity. The design, kinetic optimization, and analysis of FCRNs to exhibit functions akin to those of a cell still pose significant challenges. Indeed, reaching synthetic homeostasis is essential for engineering synthetic cell components. However, maintaining homeostasis in artificial systems against various agitations is a difficult task. Several biological events can provide us with guidelines for a conceptual understanding of homeostasis, which can be further applicable in designing artificial synthetic systems. In this regard, we organize our review with artificial homeostasis systems driven by FCRNs at different length scales, including homogeneous, compartmentalized, and soft material systems. First, we stretch a quick overview of FCRNs in different molecular and supramolecular systems, which are the essential toolbox for engineering different nonlinear functions and homeostatic systems. Moreover, the existing history of synthetic homeostasis in chemical and material systems and their advanced functions with self-correcting, and regulating properties are also emphasized.

Study on Carbonation of Porcine Blood Hydrogel in the Composite Mortar of Ancient Chinese Architectural Painting

In the ancient Chinese recipe for composite mortar used in the construction of ground layers for architectural painting, the mixture of porcine blood and lime water is one of the constituent materials. Herein, according to the traditional recipe, the interaction between porcine blood and lime water was systematically and deeply investigated. The experimental investigation demonstrated that porcine blood mixed with lime water at the ratio found in the recipe can form a hydrogel with a hydrophobic surface. During air-drying, the lime water in porcine blood hydrogel can react with CO2 to form calcium carbonate. The crystal morphology of the formed calcium carbonate depends on the surrounding micro-environment of calcium ions in the porcine blood hydrogel. The formed morphology of calcium carbonate includes small calcite crystallites, small graininess calcite crystals with round features, calcite aggregates with layered ladder-like structures, and amorphous calcium carbonate (ACC). Interestingly, the calcium carbonate formed in the inner part of the porcine blood hydrogel exhibits lamellar distribution due to a Liesegang pattern formation. Based on the findings that the porcine blood hydrogel has surface hydrophobicity and brittleness, it can be predicted that in the preparation process of composite mortar for ancient building color painting base course, porcine blood used in the form of a hydrogel is not only easier to be dispersed in hydrophobic tung oil than in liquid porcine blood but also the affinity between porcine blood gel and tung oil is enhanced. As constituent material dispersed in the composite mortar, the layered distribution of calcium carbonate in the porcine blood hydrogel may presumably be beneficial to reduce the internal stress of the composite mortar material.

Coiling of Secondary Tubes Formed from the Colloidal Exhaust of Primary Chemical Gardens

Chemical gardens are self-organized precipitate structures such as thin-walled tubes and membrane-bound cells reminiscent of biological shapes. These usually inorganic precipitates compartmentalize the reaction system and allow the study of materials synthesis in very steep concentration gradients. We create such tubes by steadily injecting a mixture of MnCl2 and CuSO4 solutions into a large reservoir of sodium silicate solution. The growing tube is open at its tip and ejects a stream of colloidal particles that aggregate to form a secondary tube above the original one. This secondary tube can coil into a tightly wound nest-like structure, freely suspended underneath the solution-air interface. Using three-dimensional image reconstruction, we analyze the onset of coiling and show that the structure is helical with a helix radius that increases in the vertical direction. The height at which the coiling begins is lowered with each successive repeat of the growth experiment, suggesting that coiling is induced by small variations in the density of the silicate solution.

Patterns Lead the Way to Far-from-Equilibrium Materials

The universe is a complex fabric of repeating patterns that unfold their beauty in system-specific diversity. The periodic table, crystallography, and the genetic code are classic examples that illustrate how even a small number of rules generate a vast range of shapes and structures. Today, we are on the brink of an AI-driven revolution that will reveal an unprecedented number of novel patterns, many of which will escape human intuition and expertise. We suggest that in the second half of the 21st century, the challenge for Physical Chemistry will be to guide and interpret these advances in the broader context of physical sciences and materials-related engineering. If we succeed in this role, Physical Chemistry will be able to extend to new horizons. In this article, we will discuss examples that strike us as particularly promising, specifically the discovery of high-entropy and far-from-equilibrium materials as well as applications to origins-of-life research and the search for life on other planets.

Formation of Precipitation Ellipsoidal Disks and Spheres in the Wake of a Planar Diffusion Front

Pattern formation is one of the examples of self-organization. In the generation of patterns, the coupling between the mass transport of the chemical species and their chemical reactions plays an important role. Periodic precipitation (Liesegang phenomenon) is a type of pattern formation in which layered precipitation structures form in the wake of the diffusion front. Here, we show a new type of precipitation pattern formation in zeolitic imidazolate framework-67 in a solid hydrogel column in a test tube manifested in the generation of precipitation ellipsoidal disks and spheres in the wake of the planar diffusion front of the outer electrolyte (2-methylimidazole). To increase the probability of the emergence of ellipsoidal disks and spheres, the surfaces of the borosilicate test tubes were chemically treated and functionalized. To support the experimental findings, we developed a reaction-diffusion model that qualitatively describes the formation of precipitate ellipsoidal disks and spheres in a test tube.

Oppositely Charged Nanoparticles Precipitate Not Only at the Point of Overall Electroneutrality

Precipitation of oppositely charged entities is a common phenomenon in nature and laboratories. Precipitation and crystallization of oppositely charged ions are well-studied and understood processes in chemistry. However, much less is known about the precipitation properties of oppositely charged nanoparticles. Recently, it was demonstrated that oppositely charged gold nanoparticles (AuNPs), also called nanoions, decorated with positively or negatively charged thiol groups precipitate only at the point of electroneutrality of the sample (i.e., the charges on the particles are balanced). Here we demonstrate that the precipitation of oppositely AuNPs can occur not only at the point of electroneutrality. The width of the precipitation window depends on the size and concentration of the nanoparticles. This behavior can be explained by the aggregation of partially stabilized clusters reaching the critical size for their sedimentation in the gravitational field.

A Quasi Universal Matalon-Packter Law for a Periodically Precipitating System of Iron(II) Hydroxide Involving Volumes and Concentrations of the Invading Electrolyte

The Matalon-Packter (MP) empirical law of periodically precipitating (Liesegang phenomenon) systems under non-equilibrium conditions describes the dependence of the periodicity (spacing coefficient) on the initial concentration of the outer electrolyte. We aim to present the MP law in a more generalized form using a realistic approach wherein mass transfer in the gel column plays a role instead of the initial concentrations. This work is an attempt to make such progress. The Liesegang bands of Fe(OH)2 were studied by varying the reservoir concentrations (c) and volumes (V) of the outer electrolyte (NH4OH). The spacing coefficient was found to be a function of the volume and concentration of the outer electrolyte. It was observed that the amount of chemical substance (cV) and the average molar diffusion flux (Fdiff) of the ions of the outer electrolyte could be a unifying quantity for expressing the MP law instead of the initial electrolyte concentration. We demonstrated that a single model is possible for a system, irrespective of the V value. Three different volumes were employed, and the calculations were performed under small, intermediate, and larger reservoir volume regimes. Interestingly, a single model was observed for the diffusion coefficients for all of the Fdiff values.

Nonclassical Crystallization Causes Dendritic and Band-Like Microscale Patterns in Inorganic Precipitates

The self-organization of complex solids can create patterns extending hierarchically from the atomic to the macroscopic scale. A frequently studied model is the chemical garden system which consists of life-like precipitate shapes. In this study, we examine the thin walls of chemical gardens using microfluidic devices that yield linear Ni(OH)2 precipitate membranes. We observe distinct light-scattering patterns within the compositionally pure membranes, including disorganized spots, dendrites, and parallel bands. The bands are tilted with respect to the membrane axis and their spacing (20-100 μm) increases with increasing flow rates. Scanning electron microscopy reveals that the bands consist of submicron particles embedded in a denser material and these particles are also found in the reactant stream. We propose that dendrites and bands arise from the attachment of solution-borne nanoparticles. The bands are generated by particle-aggregation zones moving upstream along the slowly advancing membrane surface. The speed of the aggregation zones is proportional to the band distance and defines the system's dispersion relation. This speed-wavelength dependence and the flow-opposing motion of the aggregation zones are likely caused by low particle concentrations in the wake of the zones that only slowly recover due to Brownian motion and particle nucleation.

Patterning Complex Line Motifs in Thin Films Using Immersion-Controlled Reaction-Diffusion

The discovery of self-organization principles that enable scalable routes toward complex functional materials has proven to be a persistent challenge. Here, reaction-diffusion driven, immersion-controlled patterning (R-DIP) is introduced, a self-organization strategy using immersion-controlled reaction-diffusion for targeted line patterning in thin films. By modulating immersion speeds, the movement of a reaction-diffusion front over gel films is controlled, which induces precipitation of highly uniform lines at the reaction front. A balance between the immersion speed and diffusion provides both hands-on tunability of the line spacing (d = 10 - 300 μ m $d = 10-300 \; \umu \text{m}$ ) as well as error-correction against defects. This immersion-driven patterning strategy is widely applicable, which is demonstrated by producing line patterns of silver/silver oxide nanoparticles, silver chromate, silver dichromate, and lead carbonate. Through combinatorial stacking of different line patterns, hybrid materials with multi-dimensional patterns such as square-, diamond-, rectangle-, and triangle-shaped motifs are fabricated. The functionality potential and scalability is demonstrated by producing both wafer-scale diffraction gratings with user-defined features as well as an opto-mechanical sensor based on Moiré patterning.

Propagating wave merging in a precipitation reaction

Propagating precipitation waves are a remarkable form of spatiotemporal behavior that arise through the coupling of reaction, diffusion, and precipitation. We study a system with a sodium hydroxide outer electrolyte and an aluminum hydroxide inner electrolyte. In a redissolution Liesegang system, a single propagating precipitation band moves down through the gel, with precipitate formed at the band front and precipitate dissolved at the band back. Complex spatiotemporal waves occur within the propagating precipitation band, including counter-rotating spiral waves, target patterns, and annihilation of waves on collision. We have also carried out experiments in thin slices of gel, which have revealed propagating waves of a diagonal precipitation feature within the primary precipitation band. These waves display a wave merging phenomenon in which two horizontally propagating waves merge into a single wave. Computational modeling permits the development of a detailed understanding of the complex dynamical behavior.

Transition from continuous to microglobular shaped peptide assemblies through a Liesegang-like enzyme-assisted mechanism

Enzyme-assisted self-assembly confined within host materials leads to Liesegang-like spatial structuration when precursor peptides are diffusing through an enzyme-functionalized hydrogel. It is shown here that playing on peptide and enzyme concentrations results in a transition from continuous self-assembled peptide areas to individual microglobules. Their morphology, location, size and buildup mechanism are described. Additionally, it is also found that the enzymes adsorb onto the peptide self-assemblies leading to co-localization of peptide self-assembled microglobules and enzymes. Finally, we find that large microglobules grow at the expense of smaller ones present in their vicinity in a kind of Ostwald ripening process, illustrating the dynamic nature of the peptide self-assembly process within host hydrogels.

Cu(II)-Metallized Three-Layered Cu-8HQ Complex with Hierarchical Crystallite Morphologies Synthesized via Reaction-Diffusion

Self-organization of regular band patterns of the precipitate via a reaction-diffusion (RD) framework is called Liesegang phenomenon. This non-equilibrium system is emerging as an efficient method for synthesizing materials with unique morphologies that may have desired properties. The formation of continuous precipitation inside a band with poor control over the shape and size of sparingly soluble salts has been well documented. However, only a few reports on forming organic-inorganic bonds are available. In the present work, we demonstrate the formation of 2D frameworks of bis-(8-hydroxyquinoline) copper(II) in the agar gel via RD. The macroscopic particles were dumbbell-shaped, with aspect ratios ranging from 2.7 (inner bands) to 0.7 (outer bands). The particles were composed of ribbon-shaped crystallites at the microscopic level, each with three layers of parallelogram prismatic monoclinic sheets stacked over one another, which could easily be exfoliated. The powder X-ray diffraction patterns at low angles and the surface areas of the crystallites indicated the formation of metal-organic frameworks. It was observed that the sizes of the particles could be tuned by controlling the extent of diffusion using reactant concentrations. Since such heterostructures have energy storage capacity, the cyclic voltammograms of the unexfoliated and exfoliated materials showed that they fall in the pseudocapacitor category with potential application as the electrode material. The frameworks were further characterized by techniques such as optical and electron microscopy, X-ray diffraction, IR spectroscopy, and UV-visible spectrophotometry.

Liesegang Phenomenon of Liquid Metals on Au Film

Room temperature liquid metals (LM) such as gallium (Ga) own the potential to react with specific materials which would incubate new application categories. Here, diverse self-organized ring patterns due to nonequilibrium reaction-diffusion and spreading-limitation of Ga-based LM clusters on gold (Au) film are reported, among which diffusion is the controlling step and the self-limiting oxide layer plays the role of kinetic barrier. Such phenomena, classically known as the Liesegang rings, mainly occur in electrolyte media. Unlike existing systems, the present periodic crystallization mechanism enables highly symmetric spatiotemporal periodic Liesegang rings on a smaller scale under ambient conditions. Typically, the Ga-Au and eutectic gallium-indium alloy (EGaIn)-Au reaction-diffusion-spreading systems are constructed, obtaining the revert type and hybrid type concentric Liesegang patterns, respectively. The competitive patterning behavior of the intermediate phase products AuGa₂ and AuIn₂ in hybrid Liesegang patterns is further analyzed by altering the initial Ga/In mass ratio, first-principles calculations, and molecular dynamic simulations. When the mass ratio of In in GaIn alloy exceeds 15%, it will preferentially react with Au. The discovery of LM Liesegang phenomenon is expected to be a flashpoint for self-organized reaction-diffusion systems and offers promising rules for diverse areas such as materials synthesis and the jewelry design industry.

How the Addition of Chitosan Affects the Transport and Rheological Properties of Agarose Hydrogels

Agarose hydrogels enriched by chitosan were studied from a point of view diffusion and the immobilization of metal ions. Copper was used as a model metal with a high affinity to chitosan. The influence of interactions between copper and chitosan on transport properties was investigated. Effective diffusion coefficients were determined and compared with values obtained from pure agarose hydrogel. Their values increased with the amount of chitosan added to agarose hydrogel and the lowest addition caused the decrease in diffusivity in comparison with hydrogel without chitosan. Liesegang patterns were observed in the hydrogels with higher contents of chitosan. The patterns were more distinct if the chitosan content increased. The formation of Liesegang patterns caused a local decrease in the concentration of copper ions and concentration profiles were affected by this phenomenon. Thus, the values of effective diffusion coefficient covered the influences of pore structure of hydrogels and the interactions between chitosan and metal ions, including precipitation on observed Liesegang rings. From the point of view of rheology, the addition of chitosan resulted in changes in storage and loss moduli, which can show on a "more liquid" character of enriched hydrogels. It can contribute to the increase in the effective diffusion coefficients for hydrogels with higher content of chitosan.

Periodic aggregation patterns of oxide particles on corroding metals: chemical waves due to solution feedback processes

Chemical waves that produce periodic patterns are common occurrences in nature. The underlying processes involved have been studied in many disciplines of science, but rarely reported in the chemistry of corrosion. In this study of carbon steel corrosion, iron oxide crystals are observed to deposit in concentric wave patterns or in discrete bands, known as Liesegang patterns. We demonstrate that oxide growth in these patterns is preceded by the formation of a hydrogel network, which consists of a semi-stationary phase of loosely connected metal-hydroxide colloids and a mobile phase of solution saturated with metal cations. Once the hydrogel network covers the metal surface, a metal cation produced by corrosion reactions at the metal surface must diffuse through the layer into the bulk solution. While diffusing through the porous network, the metal cation undergoes adsorption-precipitation as metal-hydroxide colloids which later can dissolve back into the solution. When the kinetics of precipitation and dissolution of the metal cation can be effectively coupled with the transport flux of the dissolved metal cation, the precipitation-dissolution-diffusion cycles can be sustained over time which can lead to periodic aggregation patterns of metal-hydroxide colloids at a specific time. We also establish that for transition metal cations the precipitation-dissolution-diffusion process can couple with reversible redox reactions between the soluble and less soluble metal cations, which can affect the overall transport of banded aggregates of metal-hydroxide colloids and the growth and transformation of metal-hydroxides into crystalline oxides. If systemic feedback between different elementary processes is sustained over long durations, iron-oxide crystals of different chemical compositions and shapes aggregate in Liesegang patterns. This work demonstrates unequivocally that non-uniform deposition of metal oxides during corrosion can occur via strongly coupled solution reactions and transport processes, and not simply as a result of metallurgical non-uniformity and/or localized solution environments.

Role of Stochasticity in Helical Self-Organization during Precipitation

Spontaneous pattern formation with a well-defined periodicity is ubiquitous in nature. The Liesegang phenomenon is a chemical model of such a spontaneous pattern formation. In this study, we investigated the role of stochasticity in reaction-diffusion precipitation processes by demonstrating the temperature dependence of spontaneous symmetry breaking and helix formation in the Liesegang pattern with CuCrO₄ precipitates; experimental analysis and numerical simulations based on reaction-diffusion equations were used. At high temperatures, helices with no, single, and double branches appeared in addition to the discrete parallel band characteristic of the Liesegang phenomenon. The probability of helix formation increased drastically when the experimental temperature during the pattern formation exceeded 20 °C. Moreover, the spacing coefficient, quantitatively representing the periodicity of obtained patterns, increased at high temperatures. Numerical simulations were performed to investigate the temperature dependence of the probability of helix formation and spacing coefficients. The stochasticity of the initial chemical reaction, which can trigger consequent nucleation and crystal growth, critically affected the probability of helix formation and the spacing coefficient. These features were explained in the framework of the prenucleation model by considering the degree of stochasticity in the initial chemical reaction step.

Periodic Self-Assembly of Poly(ethyleneimine)-poly(4-styrenesulfonate) Complex Coacervate Membranes

Coacervation is a self-assembly strategy based on the complexation of polyelectrolytes, which is utilized in biomedicine and agriculture, as well as automotive and textile industries. In this paper, we developed a new approach to the on-demand periodic formation of polyelectrolyte complexes through a Liesegang-type hierarchical organization. Adjustment of reaction conditions allows us to assemble materials with a tunable spatiotemporal geometry and establish materials' production cycles with a regulated periodicity. The proposed methodology allows the membrane to self-assemble when striving to reach balance and self-heal after exposure to external stimuli, such as potential difference and high pH. Using chronopotentiometry, K⁺ ion permeability behavior of the PEI-PSS coacervate membranes was demonstrated. The periodically self-assembled polyelectrolyte nanomembranes could further be integrated into novel energy storage devices and intelligent biocompatible membranes for bionics, soft nanorobotics, biosensing, and biocomputing.

Effect of the Polarity of Solvents on Periodic Precipitation: Formation of Hierarchical Revert Liesegang Patterns

Liesegang pattern (LP) is one example of self-organized periodic precipitation patterns in nonequilibrium systems. Several studies have demonstrated that the LP morphology can track physicochemical environmental conditions (e.g., temperature); however, the polarity effect has not been explored to date. In this study, a copper chromate system is used to reveal the impact of solvent polarity on the evolving LP structure using water/organic solvent mixtures. In the typical case of using water/dimethyl sulfoxide (DMSO) mixtures, two drastic changes in LP morphology with increasing DMSO contents were found: (i) increasing frequency of the original structure and (ii) formation of a hierarchical pattern with the appearance of another, lower-frequency structure. Furthermore, the simulation model operating with a bimodal size distribution, allowing both homogeneous and heterogeneous precipitations showed good agreement with the experimental results. Therefore, this study demonstrated that LP can be tailored by solvent polarity and can be used for designing hierarchical precipitation patterns in a straightforward manner.

Rapid formation of uniformly layered materials by coupling reaction-diffusion processes with mechanical responsiveness

Straightforward manufacturing pathways toward large-scale, uniformly layered composites may enable the next generation of materials with advanced optical, thermal, and mechanical properties. Reaction-diffusion systems are attractive candidates to this aim, but while layered composites theoretically could spontaneously arise from reaction-diffusion, in practice randomly oriented patches separated by defects form, yielding nonuniformly patterned materials. A propagating reaction front can prevent such nonuniform patterning, as is the case for Liesegang processes, in which diffusion drives a reaction front to produce layered precipitation patterns. However, while diffusion is crucial to control patterning, it slows down transport of reactants to the front and results in a steady increase of the band spacing as the front advances. Here, we circumvent these diffusive limitations by embedding the Liesegang process in mechanically responsive hydrogels. The coupling between a moving reaction front and hydrogel contraction induces the formation of a self-regulated transport channel that ballistically carries reactants toward the area where patterning occurs. This ensures rapid and uniform patterning. Specifically, large-scale ([Formula: see text]5-cm) uniform banding patterns are produced with tunable band distance (d = 60 to 160 µm) of silver dichromate crystals inside responsive gelatin-alginate hydrogels. The generality and applicability of our mechanoreaction-diffusion strategy are demonstrated by forming patterns of precipitates in significantly smaller microscopic banding patterns (d = 10 to 30 µm) that act as self-organized diffraction gratings. By circumventing the inherent limitations of diffusion, our strategy unlocks the potential of reaction-diffusion processes for the manufacturing of uniformly layered materials.

Tunability of Self-Organized Structures Based on Thermodynamic Flux

Nature establishes structures and functions via self-organization of constituents, including ions, molecules, and particles. Understanding the selection rule that determines the self-organized structure formed from many possible alternatives is fundamentally and technologically important. In this study, the selection rule for the self-organization associated with a reaction-diffusion system was explored using the Liesegang phenomenon, by which a periodic precipitation pattern is formed as a model system. Experiments were conducted by systematically changing the mass flux. At low mass fluxes, a vertically periodic pattern was formed, whereas at high mass fluxes, a horizontally periodic pattern was formed. The results inferred that a structural vertical-to-horizontal periodicity transition occurred in the self-organized periodic structure at the crossover flux at which the entropy production rate reversed. Numerical analyses attributed the as-observed flux-dependent structural transition to the selection of the self-organized pattern with a higher entropy production rate. These findings contribute to our understanding of how nature controls self-organized structures and geometry, potentially facilitating the development of novel designs, syntheses, and fabrication processes for well-controlled organized functional structures.

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.

Liesegang rings in the setting of end‐stage renal disease

Introduction

Liesegang rings are acellular, lamellar, concentric rings of organic or inorganic material naturally formed in both biologic and environmental systems. Description in human tissue is scarce. Liesegang rings have exclusively been identified in association with pathologic disease processes and thus are not typically considered in differential diagnosis. They are usually described with cystic or inflammatory lesions. Histologically, Liesegang rings show features that are also seen in sections of parasitic ova, larvae, psammoma bodies, and by radiology as calcifications in cystic diseases of the breast and kidney.

Case presentation

We noted at autopsy of a 59‐year‐old diabetic woman multiple black “stones” in the renal medulla. Microscopic examination demonstrated these to contain Liesegang rings.

Conclusion

Liesegang rings formation should be considered in the differential diagnosis of atypical appearing deposits in the kidneys and other tissues. They may play a role in the pathogenesis of kidney stones.

Cs sorption of Mn-Fe based Prussian blue analogs with periodic precipitation banding in agarose gel

We report the understanding of Cs sorption/desorption properties of Mn-Fe based Prussian blue analogs (Mn-Fe PBA) and Prussian blue (PB) in agarose gel via X-ray fluorescence spectroscopy and scanning electron microscopy. After contacting a 2 mass% agarose gel containing 100 mmol dm^-3 [Fe(CN)₆]^3- ions (inner electrolyte gel) with a 1 mass% agarose gel containing 550 mmol dm^-3 Mn²⁺ or Fe²⁺ ions (outer electrolyte gel) in a plastic straw for 11 days, the clipped 10 mm-long gel columns of uniformly formed precipitates (named "Mn-Fe PBA gel" and "PB gel") were used to investigate their Cs sorption/desorption properties. The Mn-Fe PBA gel showed several interesting features that were not observed in the PB gel. The Cs sorption capacity of the Mn-Fe PBA gel increased over time, and after ∼1200 h reached a value comparable to the most optimal values previously reported. During Cs sorption, Mn²⁺ ions were constantly released from the Mn-Fe PBA gel, and several periodic precipitation bands were generated. The positions of the periodic bands agreed with those of the peak distributions of Cs, Mn, and Fe, suggesting that the concentration fluctuation of Mn^II-Fe^III PBA in the gel resulted in the periodic band formation. In these bands, large crystallites (>10 μm) were dominant, suggesting the contribution of Ostwald ripening. During Cs desorption, while Mn²⁺ ions were released from the Mn-Fe PBA gel, the release of Cs⁺ ions was considerably suppressed (∼1/3 for the PB gel). Based on these results, a model for Cs sorption by the Mn-Fe PBA gel was proposed, and its potential applications were discussed.

Bioinspired Control of Calcium Phosphate Liesegang Patterns Using Anionic Polyelectrolytes

The Liesegang phenomenon is a spontaneous pattern formation, which is a periodic distribution of the precipitate discovered in diffusion-limited systems. Over the past century, it has been experimentally attempted to control the periodicity of patterns and structures of precipitates by varying the concentration of the hydrogel or electrolytes, adding organic or inorganic impurities, and applying an electric or pH field. In this work, the periodic patterns of calcium phosphate were manipulated with an anionic macromolecular additive inspired by bone mineralization in which various noncollagenous proteins are involved in the formation of a polymer-induced liquid precursor. The periodic patterns were systematically controlled by adjusting the amount of poly(acrylic acid), and they were numerically simulated by adjusting the threshold concentration of nucleation. The change of the pattern is explained by improved stability and directional diffusion of the intermediate.

Fungal hyphae develop where titanomagnetite inclusions reach the surface of basalt grains

Nutrient foraging by fungi weathers rocks by mechanical and biochemical processes. Distinguishing fungal-driven transformation from abiotic mechanisms in soil remains a challenge due to complexities within natural field environments. We examined the role of fungal hyphae in the incipient weathering of granulated basalt from a three-year field experiment in a mixed hardwood-pine forest (S. Carolina) to identify alteration at the nanometer to micron scales based on microscopy-tomography analyses. Investigations of fungal-grain contacts revealed (i) a hypha-biofilm-basaltic glass interface coinciding with titanomagnetite inclusions exposed on the grain surface and embedded in the glass matrix and (ii) native dendritic and subhedral titanomagnetite inclusions in the upper 1–2 µm of the grain surface that spanned the length of the fungal-grain interface. We provide evidence of submicron basaltic glass dissolution occurring at a fungal-grain contact in a soil field setting. An example of how fungal-mediated weathering can be distinguished from abiotic mechanisms in the field was demonstrated by observing hyphal selective occupation and hydrolysis of glass-titanomagnetite surfaces. We hypothesize that the fungi were drawn to basaltic glass-titanomagnetite boundaries given that titanomagnetite exposed on or very near grain surfaces represents a source of iron to microbes. Furthermore, glass is energetically favorable to weathering in the presence of titanomagnetite. Our observations demonstrate that fungi interact with and transform basaltic substrates over a three-year time scale in field environments, which is central to understanding the rates and pathways of biogeochemical reactions related to nuclear waste disposal, geologic carbon storage, nutrient cycling, cultural artifact preservation, and soil-formation processes.

Chemical Tracking of Temperature by Concurrent Periodic Precipitation Pattern Formation in Polyacrylamide Gels

In nature, nonequilibrium systems reflect environmental changes, and these changes are often "recorded" in their solid body as they develop. Periodic precipitation patterns, aka Liesegang patterns (LPs), are visual sums of complex events in nonequilibrium reaction-diffusion processes. Here we aim to achieve an artificial system that "records" the temperature changes in the environment with the concurrent LP formation. We first illustrate the differences in 1-D LPs developing at different temperatures in terms of band spacings, which can demonstrate the time, ramp steepness, and extent of a temperature change. These results are discussed and augmented by a mathematical model. Using scanning electron microscopy, we show that the average size of the CuCrO₄ precipitate also reflects the temperature changes. Finally, we show that these changes can also be "recorded" in the 2-D and 3-D LPs, which can have applications in long-term temperature tracking and complex soft material design.

Phase separation mechanism for a unified understanding of dissipative pattern formation in a Liesegang system

Dissipative patterns with solid-phase transitions are ubiquitous in nature. Despite their ubiquitous nature, there is no unified understanding of the non-equilibrium self-assembly mechanisms of such pattern formation. The Liesegang pattern (LP) is a typical model that has the potential to describe dissipative pattern formation arising from the nonlinear coupling of directional mass transport of water-soluble substances into a porous media with their solid-phase transition processes. However, the conventional mechanism in a Liesegang system lacks practicality because most of the existing studies have focused only on the transition mechanism of nucleation from the molecular to the solid state. In this study, we demonstrate a novel experimental system based on a phase transition and separation mechanism that does not require nucleation, namely, the pH-induced aggregation of gold nanoparticles modified with 11-mercaptoundecanoic acid (MUA-Au NPs) by H⁺ diffusion in a solid hydrogel. Combined experiments and numerical simulations reveal that pattern formation is driven by the macroscopic phase-separation mechanism. Furthermore, the pattern periodicity obtained from both experiments and simulations follows the classical spacing law of LP, namely, the LP morphology is determined without the need for nucleation. Therefore, we can show that the formation of LPs can be described in a unified mechanism, regardless of whether nucleation occurs. This finding opens the possibility that the chemical Liesegang system can be applied as a practical model for proving the mechanisms of similar dissipative pattern formation.

Macroscopic Banding Pattern of Collagen Gel Formed by a Diffusion-Reaction Process

Shapes and patterns observed in internal organs and tissues are reproducibly and robustly produced over a long distance (up to millimeters in length). The most fundamental remaining question is how these long geometries of shape and pattern form arise from the genetic message. Recent studies have demonstrated that extracellular matrix (ECM) critically participates as a structural foundation on which cells can organize and communicate. ECMs may be a key to understanding the underlying mechanisms of long-distance patterning and morphogenesis. However, previous studies in this field mainly focused on the complexes and interaction of cells and ECM. This paper pays particular attention to ECM and demonstrates that collagen, a major ECM component, natively possesses the reproducible and definite patterning ability reaching centimeter-scale length. The macroscopic pattern consists of striped transparent layers. The observation under crossed Nicols demonstrates that the layers consist of alternately arranged polarized and unpolarized parts. Confocal fluorescence microscopy studies revealed that the polarized and unpolarized segments include collagen-rich and -poor regions, respectively. The patterning process was proposed based on the Liesegang banding formation, which are mineral precipitation bands formed in hydrogel matrixes. These findings will give hints to the questions about long-distance cell alignment and provide new clues to artificially control cell placement over micron size in the field of regenerative medicine.

Synthesis of zeolitic imidazolate framework-8 and gold nanoparticles in a sustained out-of-equilibrium state

The design and synthesis of crystalline materials are challenging due to the proper control over the size and polydispersity of the samples, which determine their physical and chemical properties and thus applicability. Metal − organic frameworks (MOFs) are promising materials in many applications due to their unique structure. MOFs have been predominantly synthesized by bulk methods, where the concentration of the reagents gradually decreased, which affected the further nucleation and crystal growth. Here we show an out-of-equilibrium method for the generation of zeolitic imidazolate framework-8 (ZIF-8) crystals, where the non-equilibrium crystal growth is maintained by a continuous two-side feed of the reagents in a hydrogel matrix. The size and the polydispersity of the crystals are controlled by the fixed and antagonistic constant mass fluxes of the reagents and by the reaction time. We also present that our approach can be extended to synthesize gold nanoparticles in a redox process.

Naturally Extracted Hydrophobic Solvent and Self-Assembly in Interfacial Polymerization

Pharmaceutical, chemical, and food industries are actively implementing membrane nanofiltration modules in their processes to separate valuable products and recover solvents. Interfacial polymerization (IP) is the most widely used method to produce thin-film composite membranes for nanofiltration and reverse osmosis processes. Although membrane processes are considered green and environmentally friendly, membrane fabrication has still to be further developed in such direction. For instance, the emission of volatile solvents during membrane production in the industry has to be carefully controlled for health reasons. Greener solvents are being proposed for phase-separation membrane manufacture. For the IP organic phase, the proposition of greener alternatives is in an early stage. In this work, we demonstrate the preparation of a high-performing composite membrane employing zero vapor pressure and naturally extracted oleic acid as the IP organic phase. Its long hydrophobic chain ensures intrinsic low volatility and acid monomer dissolution, while the polar head induces a unique self-assembly structure during the film formation. Membranes prepared by this technique were selective for small molecules with a molecular weight cutoff of 650 g mol^-1 and a high permeance of ∼57 L m^-2 h^-1 bar^-1.

Precise Evaluation of Spatial Characteristics of Periodically Precipitating Systems via Measurement of RGB (Red, Green, and Blue) Values of Pattern Images

In the present study, a method is described for precise determination of spatial characteristics of Liesegang bands formed by employing a classical 1D setup using a web-based free resource (https://www.ginifab.com/feeds/pms/color_picker_from_image.php). The method involves the compartmentalization of the information on each pixel into R (red), G (green), or B (blue) values from the pattern images obtained using a simple digital camera. The values can further be converted to absorbance values by using the system blank. Each trough (or peak) in the graph of RGB values (or absorbance values) corresponds to a band in the pattern. The method is employed to determine the spacing and width of the periodically precipitating AgCl, AgBr, and Co(OH)₂ in an agar gel. It is observed that AgCl shows revert banding, and AgBr shows revert banding at the top of the tube and then diverges to regular banding at the bottom of the tube, whereas the Co(OH)₂ patterns explicitly show regular banding under given experimental conditions. It is also observed that minute instabilities, such as the formation of secondary bands, can also be visualized by the present method.

Modification of the Matalon-Packter Law for Self-Organized Periodic Precipitation Patterns by Incorporating Time-Dependent Diffusion Flux

Spontaneous pattern formation is common in both inanimate and living systems. Although the Liesegang pattern (LP) is a well-studied chemical model for precipitation patterns, various recent LP systems based on artificial control could not be easily evaluated using classical tools. The Matalon-Packter (MP) law describes the effect of the initial electrolyte concentration, which governs the diffusion flux (Fdiff), on the spatial distribution of LP. Note that the classical MP law only considers Fdiff through the initial concentration of electrolytes, even though it should also depend on the volume of the reservoir used for the outer electrolyte because of the temporal change in the concentration therein due to diffusion. However, there has been no report on the relationship between the MP law, the reservoir volume, and Fdiff. Here, we experimentally demonstrated and evaluated the effect of the reservoir volume on LP periodicity according to the classical MP law. Numerical simulations revealed that the reservoir volume affects the temporal modulation of Fdiff. By expressing the MP law as a function of estimated Fdiff after a certain period of time, we provide a uniform description of the changes in periodicity for both small and large reservoir volumes. Such modification should make the MP law a more robust tool for studying LP systems.

Evolution of Micron-Spaced Patterns within Precipitating Patterns of In Situ Synthesized Silver Nanoparticles in a Nanodot-Embedded PVA/PVP Film

Periodic pattern formation beyond conventional precipitation reactions of electrolytes is of greater importance for gaining insights into the driving forces behind spontaneous spatiotemporal pattern formation in living matter. The Liesegang phenomenon is considered to be one of the important models for understanding well-defined periodic patterns. In this study, we have used biomolecule-derived photoluminescent carbon nanodots as reducing agents that were embedded in thin polymer films. The poor water content of polyvinyl alcohol/polyvinyl pyrrolidone films has been found to dictate the temporal scale of reaction-diffusion kinetics. Moreover, the precursors for the synthesis of nanodots have been varied to decipher the role of thiol groups present in glutathione in micron-spaced pattern formation of silver nanoparticles. A method to develop periodic patterns of plasmonic silver nanoparticles is of significant interest from technological aspects. Moreover, the formation of a micron-spaced pattern has been rationed by invoking a lowered nucleation threshold in terms of slow reaction-controlled aggregation. We expect that such an understanding of the chemical reaction-based pattern formation will help in resolving the formation of artistic spatiotemporal patterns in nature.

Self-organization of nanoparticles and molecules in periodic Liesegang-type structures

Chemical organization in reaction-diffusion systems offers a strategy for the generation of materials with ordered morphologies and structural hierarchy. Periodic structures are formed by either molecules or nanoparticles. On the premise of new directing factors and materials, an emerging frontier is the design of systems in which the precipitation partners are nanoparticles and molecules. We show that solvent evaporation from a suspension of cellulose nanocrystals (CNCs) and l-(+)-tartaric acid [l-(+)-TA] causes phase separation and precipitation, which, being coupled with a reaction/diffusion, results in rhythmic alternation of CNC-rich and l-(+)-TA-rich rings. The CNC-rich regions have a cholesteric structure, while the l-(+)-TA-rich bands are formed by radially aligned elongated bundles. The moving edge of the pattern propagates with a finite constant velocity, which enables control of periodicity by varying film preparation conditions. This work expands knowledge about self-organizing reaction-diffusion systems and offers a strategy for the design of self-organizing materials.

Regular-Type Liesegang Pattern of AgCl in a One-Dimensional System

The Liesegang phenomenon can be used for micro- and nanofabrication processes to yield materials with periodic precipitation of diverse types of materials. Although there have been several attempts to control the periodicity of the Liesegang patterns, it remains unclear whether the periodic precipitation of AgCl in gel medium causes regular- or revert-type patterns. To confirm the periodicity of the AgCl pattern, we conduct one-dimensional experiments under various ion concentration conditions. From microscopic observations, three different precipitation modes were observed, i.e., continuous precipitation with a sharp front, periodic precipitation and continuous precipitation with a gradual front. For these three modes, numerical analyses of the pattern geometry are performed for the periodic precipitation. It was confirmed that the regular-type pattern appeared for all concentration conditions conducted in the present experiments. Furthermore, the pattern was found to obey the spacing law and the Matalon–Packter law. From our experiments, we concluded that AgCl forms regular-type Liesegang patterns, regardless of the dimension of diffusion.

Ginzburg-Landau amplitude equation for nonlinear nonlocal models

Regular spatial structures emerge in a wide range of different dynamics characterized by local and/or nonlocal coupling terms. In several research fields this has spurred the study of many models, which can explain pattern formation. The modulations of patterns, occurring on long spatial and temporal scales, cannot be captured by linear approximation analysis. Here, we show that, starting from a general model with long range couplings displaying patterns, the spatiotemporal evolution of large-scale modulations at the onset of instability is ruled by the well-known Ginzburg-Landau equation, independently of the details of the dynamics. Hence, we demonstrate the validity of such equation in the description of the behavior of a wide class of systems. We introduce a mathematical framework that is also able to retrieve the analytical expressions of the coefficients appearing in the Ginzburg-Landau equation as functions of the model parameters. Such framework can include higher order nonlocal interactions and has much larger applicability than the model considered here, possibly including pattern formation in models with very different physical features.