Phase-Field Model of Freeze Casting

Abstract

Directional solidification of water-based solutions has emerged as a versatile technique for templating hierarchical porous materials. However, the underlying mechanisms of pattern formation remain incompletely understood. In this work, we present a detailed derivation and analysis of a quantitative phase-field model for simulating this nonequilibrium process. The phase-field model extends the thin-interface formulation of dilute binary alloy solidification with anti-trapping to incorporate the highly anisotropic energetic and kinetic properties of the partially faceted ice-water interface. This interface is faceted in the basal plane normal to the <0001> directions and atomically rough in other directions within the basal plane. On the basal plane, the model reproduces a linear or nonlinear kinetic relationship that can be linked to experimental measurements. In both cases, spontaneous parity breaking of the solidification front is observed, leading to the formation of partially faceted ice lamellae that drift laterally in one of the <0001> directions. We demonstrate that the drifting velocity is controlled by the kinetics on the basal plane and converges as the thickness of the diffuse solid-liquid interface decreases. Furthermore, we examine the effect of the form of the kinetic anisotropy, which is chosen here such that the inverse of the kinetic coefficient varies linearly from a finite value in the <0001> directions to zero in all other directions within the basal plane. Our results indicate that the drifting velocity of ice lamellae is not affected by the slope of this linear relation, and the radius and undercooling at the tip of an ice lamella converge at relatively small slope values. Consequently, the phase-field simulations remain quantitative with computationally tractable choices of both the interface thickness and the slope assumed in the form of the kinetic anisotropy.

Figures (11)

• Figure 1: (a) Hexagonal ice crystal. (b) Ice phase in a 3D PF simulation of the directional solidification of a 3 wt.% aqueous sucrose solution with growth conditions $V_p = 15~\mathrm{\mu m/s}$ and $G = 12~\mathrm{K/cm}$. (c) and (d) Anisotropy function $a_n(\mathbf{n})$ for the excess interface free energy in different cross-sections.
• Figure 2: (a) The numerically calculated equilibrium shape. (b)–(d) Contours of the equilibrium shape in different cross-sections (dots) compared with the 3D Wulff shape estimated using the $\boldsymbol{\xi}$-vector (lines).
• Figure 3: The shape of the anisotropy function $\tilde{A}(\mathbf{n})$ for interface kinetics in the plane containing both the $a$ and $c$ axes. The three curves correspond to kinetic anisotropy slopes of $r = 1$, $r = 8$, and $r = 64$, respectively.
• Figure 4: Ice crystals in 3D PF simulations of the directional solidification of a 3 wt.% aqueous sucrose solution under growth conditions of $V_p = 15~\mathrm{\mu m/s}$ and $G = 12~\mathrm{K/cm}$: (a) with only free-energy anisotropy, (b) with only kinetic anisotropy, and (c) with both anisotropies. In all simulations, 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 coordinate system, while the $\left<0001\right>$ direction is parallel to the $z$-axis.
• Figure 5: Ice crystals at time $t=$ 25 s (a), 66 s (b), and 400 s (c), captured in a 2D PF simulation of the directional solidification of a 3 wt.% aqueous sucrose solution under growth conditions of $V_p = 15~\mathrm{\mu m/s}$ and $G = 12~\mathrm{K/cm}$. The solid-liquid interface is initially planar, in a steady-state at rest, and located at the liquidus temperature. (d) A zoomed-in image of the tip region, where the colormap represents the solute concentration. The blue arrow indicates the drifting direction.