Probability of the emergence of helical precipitation patterns in the wake of reaction-diffusion fronts

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.

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.

Diffusion-controlled annihilation A+B→0: Coalescence, fragmentation, and collapse of nonidentical A-particle islands submerged in the B-particle sea

We present a systematic analysis of diffusion-controlled interaction and collapse of two nonidentical spatially separated d-dimensional A-particle islands in the B-particle sea at propagation of the sharp reaction front A+B→0 at equal species diffusivities. We show that at a sufficiently large initial distance between the centers of islands 2ℓ and a relatively large initial ratio of island-to-sea concentrations, the evolution dynamics of the island-sea-island system demonstrates remarkable universality and, depending on the system dimension, is determined unambiguously by two dimensionless parameters Λ=N_{0}^{+}/N_{Ω} and q=N_{0}^{-}/N_{0}^{+}, where N_{0}^{+} and N_{0}^{-} are the initial particle numbers in the larger and smaller of the islands, respectively, and N_{Ω} is the initial number of sea particles in the volume Ω=(2ℓ)^{d}. We find that at each fixed 0<q≤1, there are threshold values Λ_{★}(q) and Λ_{s}(q)≥Λ_{★}(q) that depend on the dimension and separate the domains of individual death of each of the islands Λ<Λ_{★}(q), coalescence and subsequent fragmentation (division) of a two-centered island Λ_{★}(q)<Λ<Λ_{s}(q), and collapse of a single-centered island formed by coalescence Λ>Λ_{s}(q). We demonstrate that regardless of d, the trajectories of the island centers are determined unambiguously by the parameter q, and we reveal a detailed picture of the evolution of islands and front trajectories with an increase in Λ, focusing on the scaling laws of evolution at the final collapse stage and in the vicinity of starting coalescence and fragmentation points.

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.

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-Field Modeling of Biomineralization in Mollusks and Corals: Microstructure vs Formation Mechanism

While biological crystallization processes have been studied on the microscale extensively, there is a general lack of models addressing the mesoscale aspects of such phenomena. In this work, we investigate whether the phase-field theory developed in materials' science for describing complex polycrystalline structures on the mesoscale can be meaningfully adapted to model crystallization in biological systems. We demonstrate the abilities of the phase-field technique by modeling a range of microstructures observed in mollusk shells and coral skeletons, including granular, prismatic, sheet/columnar nacre, and sprinkled spherulitic structures. We also compare two possible micromechanisms of calcification: the classical route, via ion-by-ion addition from a fluid state, and a nonclassical route, crystallization of an amorphous precursor deposited at the solidification front. We show that with an appropriate choice of the model parameters, microstructures similar to those found in biomineralized systems can be obtained along both routes, though the time-scale of the nonclassical route appears to be more realistic. The resemblance of the simulated and natural biominerals suggests that, underneath the immense biological complexity observed in living organisms, the underlying design principles for biological structures may be understood with simple math and simulated by phase-field theory.

Fine tuning of pattern selection in the cadmium-hydroxide-system

Controlling self-organization in precipitation reactions has received growing attention in the efforts of engineering highly ordered spatial structures. Experiments have been successful in regulating the band patterns of the Liesegang phenomenon on various scales. Herein, we show that by adjusting the composition of the hydrogel medium, we can switch the final pattern between the classical band structure and the rare precipitate spots with hexagonal symmetry. The accompanying modeling study reveals that besides the modification of gel property, tuning of the time scale of diffusional spreading of hydroxide ions with respect to that of the phase separation drives the mode selection between one-dimensional band and two-dimensional spot patterns.

Diffusion-controlled coalescence, fragmentation, and collapse of d-dimensional A-particle islands in the B-particle sea

We present a systematic analysis of diffusion-controlled interaction and collapse of two identical spatially separated d-dimensional A-particle islands in the B-particle sea at propagation of the sharp reaction front A+B→0 at equal species diffusivities. We show that at a sufficiently large initial distance between the centers of islands 2ℓ compared to their characteristic initial size and a relatively large initial ratio of island to sea concentrations, the evolution dynamics of the island-sea-island system is determined unambiguously by the dimensionless parameter Λ=N_{0}/N_{Ω}, where N_{0} is the initial particle number in the island and N_{Ω} is the initial number of sea particles in the volume Ω=(2ℓ)^{d}. It is established that (a) there is a d-dependent critical value Λ_{★} above which island coalescence occurs; (b) regardless of d the centers of each of the islands move toward each other along a universal trajectory merging in a united center at the d-dependent critical value Λ_{s}≥Λ_{★}; (c) in one-dimensional systems Λ_{★}=Λ_{s}, therefore, at Λ<Λ_{★} each of the islands dies individually, whereas at Λ>Λ_{★} coalescence is completed by collapse of a single-centered island in the system center; (d) in two- and three-dimensional systems in the range Λ_{★}<Λ<Λ_{s} coalescence is accompanied by subsequent fragmentation of a two-centered island and is completed by individual collapse of each of the islands. We discuss a detailed picture of coalescence, fragmentation, and collapse of islands focusing on evolution of their shape and on behavior of the relative width of the reaction front at the final collapse stage and in the vicinity of starting coalescence and fragmentation points. We demonstrate that in a wide range of parameters, the front remains sharp up to a narrow vicinity of the coalescence, fragmentation, and collapse points.

Pattern Formation in Precipitation Reactions: The Liesegang Phenomenon

Pattern formation is a frequent phenomenon in physics, chemistry, biology, and materials science. Bottom-up pattern formation usually occurs in the interaction of the transport phenomena of chemical species with their chemical reaction. The oldest pattern formation is the Liesegang phenomenon (or periodic precipitation), which was discovered and described in 1896 by Raphael Edward Liesegang, who was a German chemist and photographer who was born 150 years ago. The purpose of this feature article is to provide a comprehensive overview of this type of pattern formation. Liesegang banding occurs because of the coupling of the diffusion process of the reagents with their chemical reactions in solid hydrogels. We will discuss several phenomena observed and discovered in the past century, including reverse patterns, precipitation patterns with dissolution (due to complex formation), helicoidal patterns, and precipitation waves. Additionally, we will review all existing models of the Liesegang phenomenon including pre- and postnucleation scenarios. Finally, we will highlight several applications of periodic precipitation.

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.

Role of Electrolyte in Liesegang Pattern Formation

Pattern formation based on the Liesegang phenomenon is considered one of the useful models for gaining a mechanistic understanding of spontaneous spatiotemporal pattern formations in nature. However, for more than a century, the Liesegang phenomenon in chemical systems has been investigated by using electrolytes as both the reaction substrate and aggregation promoter, which has obfuscated the role of the electrolyte. Here, we distinguish the electrolyte (Na₂SO₄) from the reaction substrates (Ag⁺ ion and citrate), where Na₂SO₄ does not participate in the reaction step and acts as an aggregation promoter. The addition of Na₂SO₄ in Ag⁺-citrate-type Liesegang rings gave well-resolvable clear bands with a larger spacing coefficient. The observed changes were discussed by using the classical DLVO (Derjaguin-Landau-Verwey-Overbeek) theory, where the role of the electrolyte is to shield the electrostatic repulsive interaction among the reaction products. Furthermore, the numerical simulation of the reaction-diffusion equation with different aggregation thresholds reproduced the salt-dependent change in the spacing coefficient. We expect that an understanding of the exact role of the electrolyte as the aggregation promoter reported here will offer novel insight into how nature spontaneously forms beautiful spatiotemporal patterns.

Diffusion-controlled formation and collapse of a d-dimensional A-particle island in the B-particle sea

We consider diffusion-controlled evolution of a d-dimensional A-particle island in the B-particle sea at propagation of the sharp reaction front A+B→0 at equal species diffusivities. The A-particle island is formed by a localized (point) A-source with a strength λ that acts for a finite time T. We reveal the conditions under which the island collapse time t_{c} becomes much longer than the injection period T (long-living island) and demonstrate that regardless of d the evolution of the long-living island radius r_{f}(t) is described by the universal law ζ_{f}=r_{f}/r_{f}^{M}=sqrt[eτ|lnτ|], where τ=t/t_{c} and r_{f}^{M} is the maximal island expansion radius at the front turning point t_{M}=t_{c}/e. We find that in the long-living island regime the ratio t_{c}/T changes with the increase of the injection period T by the law ∝(λ^{2}T^{2-d})^{1/d}, i.e., increases with the increase of T in the one-dimensional (1D) case, does not change with the increase of T in the 2D case and decreases with the increase of T in the 3D case. We derive the scaling laws for particles death in the long-living island and determine the limits of their applicability. We demonstrate also that these laws describe asymptotically the evolution of the d-dimensional spherical island with a uniform initial particle distribution generalizing the results obtained earlier for the quasi-one-dimensional geometry. As striking results, we present a systematic analysis of the front relative width evolution for fluctuation, logarithmically modified, and mean-field regimes, and we demonstrate that in a wide range of parameters the front remains sharp up to a narrow vicinity of the collapse point.

Probing the mystery of Liesegang band formation: revealing the origin of self-organized dual-frequency micro and nanoparticle arrays

Periodic precipitation processes in gels can result in impressive micro- and nanostructured patterns known as periodic precipitation (or Liesegang bands). Under certain conditions, the silver nitrate-chromium(vi) system exhibits the coexistence of two kinds of Liesegang bands with different frequencies. We now present that the two kinds of bands form independently on different time scales and the pH-dependent chromate(vi)-dichromate(vi) equilibrium controls the formation of the precipitates. We determined the spatial distribution and constitution of the particles in the bands using focused ion beam-scanning electron microscopy (FIB-SEM) and scanning transmission X-ray spectromicroscopy (STXM) measurements. This provided the necessary empirical input data to formulate a model for the pattern formation; a model that quantitatively reproduces the experimental observations. Understanding the pattern-forming process at the molecular level enables us to tailor the size and the shape of the bands, which, in turn, can lead to new functional architectures for a range of applications.

Self-assembly of like-charged nanoparticles into Voronoi diagrams

The self-assembly of nanoscopic building blocks into higher order macroscopic patterns is one possible approach for the bottom-up fabrication of complex functional systems. Macroscopic pattern formation, in general, is determined by the reaction and diffusion of ions and molecules. In some cases macroscopic patterns emerge from diffusion and interactions existing between nanoscopic or microscopic building blocks. In systems where the distribution of the interaction-determining species is influenced by the presence of a diffusion barrier, the evolving macroscopic patterns will be determined by the spatiotemporal evolution of the building blocks. Here we show that a macroscopic pattern can be generated by the spatiotemporally controlled aggregation of like-charged carboxyl-terminated gold nanoparticles in a hydrogel, where clustering is induced by the screening effect of the sodium ions that diffuse in a hydrogel. Diffusion fronts of the sodium ions and the induced nanoparticle aggregation generate Voronoi diagrams, where the Voronoi cells consist of aggregated nanoparticles and their edges are aggregation-free and nanoparticle-free zones. We also developed a simple aggregation-diffusion model to adequately describe the evolution of the experimentally observed Voronoi patterns.

Self-organization in precipitation reactions far from the equilibrium

Far from the thermodynamic equilibrium, many precipitation reactions create complex product structures with fascinating features caused by their unusual origins. Unlike the dissipative patterns in other self-organizing reactions, these features can be permanent, suggesting potential applications in materials science and engineering. We review four distinct classes of precipitation reactions, describe similarities and differences, and discuss related challenges for theoretical studies. These classes are hollow micro- and macrotubes in chemical gardens, polycrystalline silica carbonate aggregates (biomorphs), Liesegang bands, and propagating precipitation-dissolution fronts. In many cases, these systems show intricate structural hierarchies that span from the nanometer scale into the macroscopic world. We summarize recent experimental progress that often involves growth under tightly regulated conditions by means of wet stamping, holographic heating, and controlled electric, magnetic, or pH perturbations. In this research field, progress requires mechanistic insights that cannot be derived from experiments alone. We discuss how mesoscopic aspects of the product structures can be modeled by reaction-transport equations and suggest important targets for future studies that should also include materials features at the nanoscale.

Self-organized target and spiral patterns through the "coffee ring" effect

We studied the precipitation pattern of fullerene C60 nanocrystals generated through the evaporation of a confined liquid bridge. In contrast to the usual "coffee ring" pattern, both target and spiral patterns were observed. The characteristics of the pattern critically depended on the concentration of the solution, the temperature, and the level of vacuum. In addition, the morphology of the microscopic precipitates varied greatly as a function of these experimental parameters. This pattern formation can be interpreted as a two-step rhythmic nucleation/precipitation of fullerene crystals during receding motion of the contact line. Symmetric motion of the contact line produces a target pattern, and the propagation of distortion of the liquid interface caused by a disturbance generates a spiral pattern.

Ternary eutectic dendrites: Pattern formation and scaling properties

Extending previous work [Pusztai et al., Phys. Rev. E 87, 032401 (2013)], we have studied the formation of eutectic dendrites in a model ternary system within the framework of the phase-field theory. We have mapped out the domain in which two-phase dendritic structures grow. With increasing pulling velocity, the following sequence of growth morphologies is observed: flat front lamellae → eutectic colonies → eutectic dendrites → dendrites with target pattern → partitionless dendrites → partitionless flat front. We confirm that the two-phase and one-phase dendrites have similar forms and display a similar scaling of the dendrite tip radius with the interface free energy. It is also found that the possible eutectic patterns include the target pattern, and single- and multiarm spirals, of which the thermal fluctuations choose. The most probable number of spiral arms increases with increasing tip radius and with decreasing kinetic anisotropy. Our numerical simulations confirm that in agreement with the assumptions of a recent analysis of two-phase dendrites [Akamatsu et al., Phys. Rev. Lett. 112, 105502 (2014)], the Jackson-Hunt scaling of the eutectic wavelength with pulling velocity is obeyed in the parameter domain explored, and that the natural eutectic wavelength is proportional to the tip radius of the two-phase dendrites. Finally, we find that it is very difficult/virtually impossible to form spiraling two-phase dendrites without anisotropy, an observation that seems to contradict the expectations of Akamatsu et al. Yet, it cannot be excluded that in isotropic systems, two-phase dendrites are rare events difficult to observe in simulations.

Helical Turing patterns in the Lengyel-Epstein model in thin cylindrical layers

The formation of Turing patterns was investigated in thin cylindrical layers using the Lengyel-Epstein model of the chlorine dioxide-iodine-malonic acid reaction. The influence of the width of the layer W and the diameter D of the inner cylinder on the pattern with intrinsic wavelength l were determined in simulations with initial random noise perturbations to the uniform state for W < l/2 and D ∼ l or lower. We show that the geometric constraints of the reaction domain may result in the formation of helical Turing patterns with parameters that give stripes (b = 0.2) or spots (b = 0.37) in two dimensions. For b = 0.2, the helices were composed of lamellae and defects were likely as the diameter of the cylinder increased. With b = 0.37, the helices consisted of semi-cylinders and the orientation of stripes on the outer surface (and hence winding number) increased with increasing diameter until a new stripe appeared.

Propagating fronts and morphological instabilities in a precipitation reaction

Precipitation processes are essential in many natural systems, especially in biomineralization and in geological pattern formation. We observe temporal oscillations in the total mass of the precipitate, the formation of propagating and annihilating waves, and morphological instabilities in a thin precipitation layer in a two-side-fed gel reactor containing the AlCl3/NaOH reaction-diffusion system. Contrary to the standard Liesegang patterns, these structures form in the lateral direction at the meeting of the counterpropagating diffusion fronts of the electrolytes. The two main ingredients of the system are the amphoteric precipitate and the cross gradient of the chemicals due to the fixed boundary conditions. Simulations with a four-variable precipitation/redissolution model qualitatively reproduce the oscillations in the total mass of the precipitate and point out the stratified three-dimensional structure of the precipitate.

Critical onset of layering in sedimenting suspensions of nanoparticles

We quantitatively study the critical onset of layering in suspensions of nanoparticles in a solvent, where an initially homogeneous suspension, subject to an effective gravity a in a centrifuge, spontaneously forms well-defined layers of constant particle density, so that the density changes in a staircaselike manner along the axis of gravity. This phenomenon is well known; yet, it has never been quantitatively studied under reproducible conditions: therefore, its physical mechanism remained controversial and the role of thermal diffusion in this phenomenon was never explored. We demonstrate that the number of layers forming in the sample exhibits a critical scaling as a function of a; a critical dependence on sample height and transverse temperature gradient is established as well. We reproduce our experiments by theoretical calculations, which attribute the layering to a diffusion-limited convective instability, fully elucidating the physical mechanism of layering.

Self-organization of calcium oxalate by flow-driven precipitation

In the calcium-oxalate system fast kinetics with flow leads to the enrichment of the thermodynamically unstable dihydrate crystalline form.

Transition from rings to spots in a precipitation reaction–diffusion system

We report for the first time the transition from rings to spots with squared/hexagonal symmetry in a periodic precipitation system, which consists of sulfide/hydroxide ions diffusing into a gel matrix containing dissolved cadmium(ii) ions.

Magnetic-field-induced growth of silver dendrite-crystalline Liesegang rings

An external magnetic field can control the chirality of Liesegang rings and induce a change in the silver crystal structure.

Propagating precipitation waves: experiments and modeling

Traveling precipitation waves, including counterrotating spiral waves, are observed in the precipitation reaction of AlCl3 with NaOH [Volford, A.; et al. Langmuir 2007, 23, 961 - 964]. Experimental and computational studies are carried out to characterize the wave behavior in cross-section configurations. A modified sol-coagulation model is developed that is based on models of Liesegang band and redissolution systems. The dynamics of the propagating waves is characterized in terms of growth and redissolution of a precipitation feature that travels through a migrating band of colloidal precipitate.

Helices in the wake of precipitation fronts

A theoretical study of the emergence of helices in the wake of precipitation fronts is presented. The precipitation dynamics is described by the Cahn-Hilliard equation and the fronts are obtained by quenching the system into a linearly unstable state. Confining the process onto the surface of a cylinder and using the pulled-front formalism, our analytical calculations show that there are front solutions that propagate into the unstable state and leave behind a helical structure. We find that helical patterns emerge only if the radius of the cylinder R is larger than a critical value R>R(c), in agreement with recent experiments.