Probability of the Emergence of Helical Precipitation Patterns in the Wake of Reaction-Diffusion Fronts

Abstract

Helical and helicoidal precipitation patterns emerging in the wake of reaction-diffusion fronts are studied. In our experiments, these chiral structures arise with well defined probabilities PH controlled by conditions such as e.g., the initial concentration of the reagents. We develop a model which describes the observed experimental trends. The results suggest that PH is determined by a delicate interplay among the time-and length-scales related to the front and to the unstable precipitation modes and, furthermore, the noise amplitude also plays a quantifiable role.PACS numbers: 68.35.Ct Helices and helicoids are present from nano-to macroscale (ZnO nanohelices [1], macromolecules and inorganic crystals with a helical structure [2, 3], precipitation helices [4][5][6], fiber geometry of heart walls [7]). Formation of these fascinating structures generally follows two routes. First, templates with chiral symmetry (e.g., oragogel fibers) may exist in the system, and the symmetry is just transcribed to a structure (e.g., inorganic materials [8]) at a larger scale. Second, spontaneous symmetry breaking may occur through the self-assembly of achiral building blocks into a helical/helicoidal structure, as e.g. in case of crystals with chiral morphology [2,9].Theoretically, the symmetry-breaking route is more interesting. Universal aspects may emerge and the robust features of this self-organization process may be important for applications as well. Indeed, control over creating helical structures would make engineering (in particular, the bottom-up design of micro-patterns [10]) more flexible since chiral morphology of materials are known to affect their physical (electronic) properties [6,11].In order to develop insight into the genesis of helices/helicoids, we investigate an emblematic example of pattern formation, namely the formation of precipitation patterns in the wake of reaction-diffusion fronts [12,13]. The motivation for this choice comes from the observation that helicoidal structures have an axis, and the correlations are simple in the plane perpendicular to the axis. This suggests that building the perpendicular correlations in the wake of an advancing planar front may be a simple and natural mechanism of creating helices/helicoids. Additional motivation comes from the existence of a large body of knowledge in the related Liesegang phenomena [12,13]. It allows the use of well-established experimental and theoretical approaches, thus making it easier to develop a novel view on the formation of helical structures.Our main results concern the probabilistic aspects of the symmetry-breaking route. We determine the probability P H of the emergence of single helices/helicoids in Liesegang-type experiments as the conditions such as the initial concentration of inner or outer electrolytes, or the temperature are changed. P H is found to be well reproducible and large (P H > 0.5 for some parameter range). The results are understood by expanding and simulating a model of formation of precipitation patterns [14]...

Pattern‐forming fronts in a Swift–Hohenberg equation with directional quenching — parallel and oblique stripes

“…More precisely, we are interested in systems that exhibit stable or metastable ordered states, such as stripes, or spots arranged in hexagonal lattices. Examples of such systems arise for example in di‐block copolymers , phase‐field models , and other phase separative systems , as well as in phyllotaxis , and reaction–diffusion systems . Throughout, we will focus on a paradigmatic model, the Swift–Hohenberg equation where , , , subscripts denote partial derivatives, and .…”

Phase Separation Patterns from Directional Quenching

“…monolayer (right). Liesegang rings and helices formed through recurrent precipiation in tube-in-tube experiments Cu 2+ (aq) + CrO 2 4 (aq) → CuCrO4(s) in 1% agarose gel, schematic of relation to 2d-patterning, and numerical simuilations; reproduced with permission from [26], Copyright 2013, APS.…”

Phase-Field Modeling of Polycrystalline Solidification: From Needle Crystals to Spherulites—A Review

“…Apparently, the thermal fluctuations choose from the possible patterns, which display a peaked probability distribution, [106] a behavior analogous to the stochastic mode selection observed in helical Liesegang systems. [107] E.…”