Helices in the wake of precipitation fronts

Abstract

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.

Spiraling eutectic dendrites

“…In this paper, we show that a minimal phase-field model of ternary freezing is able to describe the spiraling ternary eutectic dendrites, and perform a detailed numerical study of this exotic growth mode. We demonstrate that the multiarm eutectic spiral patterns are robust, so they should be experimentally accessible, and that analogously to the findings for Liesegang reactions [10], the number of spirals results from an interplay of stochastic effects and the competition of nonlinear modes.…”

“…The complex microstructure of some ternary alloys is suspected to originate from eutectic dendrites [9]. Remarkably, multiarm spiraling has been reported experimentally in excitable media [1], in binary eutectics [4], and in Liesegang reactions [10], and theoretically in the FitzHugh-Nagumo model, in which the multiarm spirals form due to the attraction of single spirals [11]. It is yet unclear how general this behavior is, in particular whether multiarm spiraling is possible for ternary eutectic dendrites, and what governs the number of spiraling eutectic arms.…”

Phase Separation Patterns from Directional Quenching

“…We are not aware of a systematic study of directional quenching processes in the mathematical literature. Our work here is motivated to a large extent by phase separation processes in recurrent precipitation [20,7,25,26], studies of patterning in LangmuirBlodgett transfer [5,18,28], and numerical studies in [9].…”

Hopf Bifurcation from Fronts in the Cahn–Hilliard Equation

“…For example, precipitation models such as [12] and [54] Left: Instability is stationary (convective in the moving frame), and is "eaten" by the trailing homogeneous state. Right: Instability is absolute and is sustainned as the front propagates through the medium.…”

“…Experiments studying such mechanisms date back to the time of Liesegang [26], who studied the formation of periodic rings, now named after him, precipitating in the wake of a circular reaction front traveling through a gel solution. These processes have since been found to create an incredible array of spatial patterns, ranging from regular structures such as periodic stripes and dot arrays, to more complex ones such as helices, chevrons, and fractals (see [12], [22], [49], [54] and references therein).…”