Hierarchical structure formation by crystal growth-front instabilities during ice templating.

Directional solidification of aqueous solutions and slurries in a temperature gradient is widely used to produce cellular materials through a phase separation of solutes or suspended particles between growing ice lamellae. While this process has analogies to the directional solidification of metallurgical alloys, it forms very different hierarchical structures. The resulting honeycomb-like porosity of freeze-cast materials consists of regularly spaced, lamellar cell walls which frequently exhibit unilateral surface features of morphological complexity reminiscent of living forms, all of which are unknown in metallurgical structures. While the strong anisotropy of ice-crystal growth has been hypothesized to play a role in shaping those structures, the mechanism by which they form has remained elusive. By directionally freezing binary water mixtures containing small solutes obeying Fickian diffusion, and phase-field modeling of those experiments, we reveal how those structures form. We show that the flat side of lamellae forms because of slow faceted ice-crystal growth along the c-axis, while weakly anisotropic fast growth in other directions, including the basal plane, is responsible for the unilateral features. Diffusion-controlled morphological primary instabilities on the solid-liquid interface form a cellular structure on the atomically rough side of the lamellae, which template regularly spaced "ridges" while secondary instabilities of this structure are responsible for the more complex features. Collating the results, we obtain a scaling law for the lamellar spacing,  [Formula: see text] , where [Formula: see text] and [Formula: see text] are the local growth rate and temperature gradient, respectively.