Parity Breaking at Faceted Crystal Growth Fronts during Ice Templating

Abstract

Directional solidification of water-based solutions has emerged as a versatile technique to template hierarchical porous materials, but this nonequilibrium process remains incompletely understood. Here we use phase-field simulations to shed light on the mechanism that selects the growth direction of the lamellar ice structure that templates those materials. Our results show that this selection can be understood within the general framework of spontaneous parity breaking, yielding quantitative predictions for the tilt angle of lamellae with respect to the thermal axis. The results provide a theoretical basis to interpret a wide range of experimental observations.

Figures (5)

• Figure 1: (a) Numerically calculated equilibrium shape. (b) The angular shape of the kinetic anisotropy in the form $\mu_k^{\left<0001\right>}/\mu_k(\mathbf{n})$. (c)-(d) Solid-liquid interfaces captured at $t = 17~\mathrm{s}$ (left), $t = 22~\mathrm{s}$ (center), and $t = 75~\mathrm{s}$ (right) in 3D PF simulations of the directional solidification of a 3 wt.% aqueous sucrose solution under growth conditions of pulling velocity $V_p = 15~\mathrm{\mu m/s}$ and temperature gradient $G = 12~\mathrm{K/cm}$. Panels (c) and (d) show results with free-energy anisotropy only and with both free-energy and kinetic anisotropies, respectively. In both cases, the $\left<11\bar{2}0\right>$ preferred growth direction is aligned with the temperature gradient $G$, which is parallel to the $x$-axis of the rectangular coordinates, while the $\left<0001\right>$ direction is parallel to the $z$-axis.
• Figure 2: The morphologies of ice crystals in 3D PF simulations with $\mu_k^{\left<0001\right>}=$ 12.1 (a), 41.1 (b), and 775.3 (c) $\mathrm{\mu m/s/K}$. (d) The measured drifting velocity as a function of $\mu_k^{\left<0001\right>}$ from 2D PF simulations. Simulations were performed for the directional solidification of a 3 wt.% aqueous sucrose solution under growth conditions of $V_p = 15~\mathrm{\mu m/s}$ and $G = 12~\mathrm{K/cm}$.
• Figure 3: (a) Illustration of the $\gamma_0$ angle between the $a$-axis and the temperature gradient $G$ within the plane containing both the $a$ and $c$ axes. (b) 3D PF simulation of a 3 wt.% aqueous sucrose solution with $V_p = 15~\mathrm{\mu m/s}$, $G = 12~\mathrm{K/cm}$, and $\gamma_0 = 15^{\circ}$. (c) Unilateral subfeatures on the ice-templated materials tilt towards the hot side of $G$.
• Figure 4: (a) Ice lamellae in two drifting modes at different $\gamma_0$ in 2D PF simulations of a 3 wt.% aqueous sucrose solution with $V_p = 15~\mathrm{\mu m/s}$ and $G = 12~\mathrm{K/cm}$. The arrows indicate the direction and magnitude of drifting. Drifting ceases at a critical angle $\gamma_c$ in Branch 2. (b) The measured drifting velocity (dots) as a function of $\gamma_0$, where drifting to the right is considered positive. The black dashed line represents the geometric drifting relation $V_d / V_p = \tan{(\gamma_0)}$, and the solid lines represent the relation in Eq. \ref{['Vd_pm']}. (c) The measured $\gamma_c$ (dots) agrees with the relation $\tan{(\gamma_c)} = V^0_d / V_p$ (dashed line), where $V^0_d$ is the magnitude of the kinetic drifting at $\gamma_0 = 0^{\circ}$. $\mu_k^{\left<0001\right>}$ is constant ($41.1\,\mathrm{\mu m/s/K}$) in (a)-(b) and varies from $19.4$ to $96.9~\mathrm{\mu m/s/K}$ in (c).
• Figure 5: Growth competition between ice lamellae in Branch 1 (green) and Branch 2 (blue) within a single crystal for $\gamma_0=$$3^{\circ}$ (a), $6.5^{\circ}$ (b), and $10^{\circ}$ (c). Ice lamellae in Branch 2 (red) cease drifting when $\gamma_0 = \gamma_c$. Arrows indicate the directions and magnitudes of drifting for branches of the corresponding color. The middle column shows the dynamically selected Branch 2. The right column presents the solidified length $L$ as a function of $\gamma_0$ when Branch 2 is dynamically selected. Simulations begin from a planar interface at the liquidus temperature during directional solidification of a 3 wt.% aqueous sucrose solution with $V_p = 15~\mathrm{\mu m/s}$ and $G = 12~\mathrm{K/cm}$.