Spiraling eutectic dendrites

Abstract

Eutectic dendrites forming in a model ternary system have been studied using the phase-field theory. The eutectic and one-phase dendrites have similar forms, and the tip radius scales with the interface free energy as for one-phase dendrites. The steady-state eutectic patterns appearing on these two-phase dendrites include concentric rings, and single-to multiarm spirals, of which the fluctuations choose, a stochastic phenomenon characterized by a peaked probability distribution. The number of spiral arms correlates with tip radius and the kinetic anisotropy. [7]. In turn, the newly discovered spiraling ternary eutectic dendrites emerge from the interplay of two-phase solidification with the Mullins-Sekerka-type diffusional instability caused by the third component [5]. This spiraling/helical structure has been identified as of interest for creating chiral metamaterials for optical applications via eutectic self-organization [8]. 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.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 free energy of a minimal ternary generalization of the binary phase-field model (see e.g. Ref.[12]) reads as

Phase-field modeling of eutectic structures on the nanoscale: the effect of anisotropy

“…Using a ternary version of the present model for a model system, in which ideal solution thermo-dynamics is used in the liquid and the regular solution model in the solid phase, we were able to capture this morphology and pattern, however, assuming kinetic anisotropy (Fig. 16) [57]. A later analytical treatment though suggests that no anisotropy is needed for the formation of the two-phase dendrites [58].…”

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.…”

“…[2] (however, now without f ori ) and the respective EOMs suffice to capture the essential properties of this interesting bicrystalline solidification morphology. [106] Remarkably, Fig. 14-Disordered dendrites formed in clay-filled polymer layers (darker panels, courtesy of Vincent Ferreiro and Jack F. Douglas) and in the GPB model (lighter panels) supplemented with 18,000 orientation pinning centers (about the number of clay particles on similar area in the experiment) distributed randomly (Reproduced with permission from Gra´na´sy et al, [28] Ó 2004 IOP Publishing). The simulations differ in the initialization of the random number generator.…”

Multi‐Step Crystallization of Self‐Organized Spiral Eutectics

“…Thus far, spiral patterns have been observed in the Al‐Th, Al‐Ag‐Cu, and Zn‐Mg alloy systems, as well as a few non‐metallic systems. These reports offer competing proposals for the mechanism of spiral growth, including different growth rates of the eutectic phases, grain rotations along the eutectic growth direction, diffusional instabilities caused by a third component, osmotic flow‐driven fingering, tilted growth in directional solidification, and thermal fluctuations . Such phenomena may occur simultaneously or sequentially over the course of crystallization, and thus it is difficult to conclusively isolate the dominant factor for spiral formation.…”

“…These reports offer competing proposals for the mechanism of spiral growth, including different growth rates of the eutectic phases, [19] grain rotations along the eutectic growth direction, [23] diffusional instabilities caused by a third component, [24] osmotic flow-driven fingering, [25] tilted growth in directional solidification, [26] and thermal fluctuations. [27] Such phenomena may occur simultaneously or sequentially over the course of crystallization, and thus it is difficult to conclusively isolate the dominant factor for spiral formation. For these reasons, spiral growth is the least understood among all eutectic morphologies (including lamellar and rod [28] ), yet it produces quite dramatic effects.…”