Role of Stochasticity in Helical Self-Organization during Precipitation

Abstract

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 Precipitation in a Confined Liquid Layer

“…It is also noted that dislocation in the periodic structure, such as branching and position displacement of bands, was observed in most of our experiments. Our findings align with a previous study reporting that a high reaction rate enhances the stochasticity of reaction and precipitation processes, leading to the dislocation of Liesegang-type structures . In other words, the high concentration of reactants in our study likely contributed to an increased reaction rate, thereby enhancing the randomness of precipitation in the reaction space.…”

“…Our findings align with a previous study reporting that a high reaction rate enhances the stochasticity of reaction and precipitation processes, leading to the dislocation of Liesegang-type structures. 71 In other words, the high concentration of reactants in our study likely contributed to an increased reaction rate, thereby enhancing the randomness of precipitation in the reaction space. For further experimental validation in HS cells, effects of the initial inner electrolyte ([K 2 CrO 4 ] 0 ) were investigated, varying from 0.1 to 0.5 M with the fixed initial outer electrolyte concentration ([CuCl 2 ] 0 ) at 1.0 M ( Figure 2 b).…”