Helices in the wake of precipitation fronts

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.

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.

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.

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.

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.

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.

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.

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.

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.