Interphase anisotropy effects on lamellar eutectics: A numerical study

In directional solidification of binary eutectics, it is often observed that two-phase lamellar growth patterns grow tilted with respect to the direction z of the imposed temperature gradient. This crystallographic effect depends on the orientation of the two crystal phases $\alpha$ and $\beta$ with respect to z. Recently, an approximate theory was formulated that predicts the lamellar tilt angle as a function of the anisotropy of the free energy of the solid($\alpha$)-solid($\beta$) interphase boundary. We use two different numerical methods—phase field (PF) and dynamic boundary integral (BI)—to simulate the growth of steady periodic patterns in two dimensions as a function of the angle ${\theta }_R$ between z and a reference crystallographic axis for a fixed relative orientation of $\alpha$ and $\beta$ crystals, that is, for a given anisotropy function (Wulff plot) of the interphase boundary. For Wulff plots without unstable interphase-boundary orientations, the two simulation methods are in excellent agreement with each other and confirm the general validity of the previously proposed theory. In addition, a crystallographic “locking” of the lamellae onto a facet plane is well reproduced in the simulations. When unstable orientations are present in the Wulff plot, it is expected that two distinct values of the tilt angle can appear for the same crystal orientation over a finite ${\theta }_R$ range. This bistable behavior, which has been observed experimentally, is well reproduced by BI simulations but not by the PF model. Possible reasons for this discrepancy are discussed.

Figure 1

Lamellar eutectic patterns observed in situ during thin-sample directional solidification ($V=0.5\mu \mathrm{m}{\mathrm{s}}^{-1}$) of a eutectic transparent (${\mathrm{CBr}}_4-{\mathrm{C}}_2{\mathrm{Cl}}_6$) model alloy. (a) Symmetric steady state in a “floating” eutectic grain. (b) Tilted lamellae in a “locked” eutectic grain. The growth direction is vertical (liquid on top). Bar: 20 $\mu \mathrm{m}$.

Figure 2

Schematic repeat units of lamellar eutectic patterns. (a) Isotropic system. (b) System with an interfacial anisotropy of the interphase boundary in the symmetric-pattern (SP) approximation. $\alpha ,\beta$: solid phases; $L$: liquid; z: growth direction parallel to the thermal gradient; x: direction of the isotherms. ${\theta }_{\mathrm{sp}}$: SP-approximation lamellar tilt angle. The lateral drift velocity is given by $\tan {\theta }_{\mathrm{sp}}=V_d/V$, with $V$ the pulling velocity. Other symbols: see text.

Figure 3

(a) Illustration of the Young-Herring equilibrium condition at the trijunction. $\vec{n}$ and $\vec{t}$: normal and tangent unit vectors of the interphase boundary. (b) Definition of the tilt angle ${\theta }_t$ and the angle $\delta$ between $\vec{\sigma }$ and z.

Figure 4

Steady-state lamellar patterns (BI simulations). (a) Symmetric pattern without anisotropy. (b) Tilted pattern with anisotropic interphase boundaries [anisotropy function given by Eq. (14); ${\epsilon }_g=0.2$; $w_g=0.1$; ${\theta }_R=3\pi /48\approx 11.{25}^{\circ }$; $\mathrm{Pe}=0.024$; ${\theta }_t=11.0^{\circ }$]. (c) Spatiotemporal diagram showing the steady-state dynamics corresponding to (a) and (b), successively, and the brief transient after the anisotropy was turned on.

Figure 5

Drifting lamellar patterns in the PF model; the anisotropy function is given by Eq. (15) with ${\epsilon }_g=0.2,w_g=0.1,{\epsilon }_2=0.0854$, and ${\epsilon }_4=0.0221$ (the same function as in Fig. 10 below): (a) “locked” tilted state (${\theta }_t={29}^{\circ },{\theta }_R={30}^{\circ })$, (b) transition from a locked to an unlocked tilted state upon change of ${\theta }_R$ from ${35}^{\circ }$ to ${40}^{\circ }$, (c) unlocked tilted state (${\theta }_t=10.8^{\circ },{\theta }_R={50}^{\circ })$. Other parameters: $d_0/l_D=7.9\times {10}^{-3},l_T/l_D=3.167$.

Figure 6

Steady-state tilt angle ${\theta }_t$ obtained from BI simulations as a function of the reduced lamellar spacing $\lambda /{\lambda }_m$ at constant $G$ and $V$ ($d_0/l_D=1.9531\times {10}^{-5},l_T/l_D=4$). The angle $-\delta$ is also plotted ($\delta$ is the angle between $\vec{\sigma }$ and z). Anisotropy function $a_c\left(\theta \right)=1-0.05cos\left[2(\theta -{\theta }_R)\right]$; ${\theta }_R=\pi /3$ (${\theta }_{\mathrm{sp}}=5.3^{\circ }$).

Figure 7

Lamellar tilt angle vs Péclet number at constant $\lambda /d_0=20.25,d_0/l_T=2.496\times {10}^{-3}$; $a_c=1+0.04cos\left[4(\theta -{\theta }_R)\right],{\theta }_R={15}^{\circ }$ (${\theta }_{\mathrm{sp}}=9.{07}^{\circ }$).

Figure 8

Lamellar tilt angle as a function of the rotation angle ${\theta }_R$. Weak, twofold symmetry anisotropy of the interphase boundary ($a_c=1-0.05cos\left[2(\theta -{\theta }_R)\right]$). The SP-approximation angle ${\theta }_{\mathrm{sp}}$ and the steady-state angle ${\theta }_t$ obtained with the BI and PF simulations are both shown. In this graph, as well as in the following, the represented ${\theta }_R$ range is limited to $[0,\pi /2]$ for obvious symmetry reasons.

Figure 9

(a) Lamellar tilt angle as a function of the rotation angle ${\theta }_R$. Mild lamellar locking effect caused by a shallow, smooth local minimum in the Wulff plot of the interphase boundary (see text). Same symbols as in Fig. 8. Dashed line: slope 1. (b) Wulff shape (${\theta }_R=0$). (c) Partial view of the Wulff shape (dash-dotted line) and the shape reconstructed from the ${\theta }_t$ data of Fig. 9 under the SP approximation (thin line).

Figure 10

(a) Lamellar tilt angle as a function of the rotation angle ${\theta }_R$. Strong lamellar locking effect with a sharp local minimum in the Wulff plot (see text). Same symbols as in Fig. 8. Inset: Wulff plot (thin line) and Wulff shape (thick line). Dashed lines: Unstable branches. (b) Angle $\delta$ of $\vec{\sigma }$ with z as a function of ${\theta }_R$ (BI simulations). (c) Shape of the lamellar pattern with the largest ${\theta }_t$ value ($\approx {48}^{\circ }$) simulated with the BI code.

Figure 11

Experimental observation of a coexistence of eutectic-growth domains with two different tilt angles in a single eutectic grain. Rotating directional solidification of a thin (10-$\mu \mathrm{m}$-thick) sample of a eutectic ${\mathrm{CBr}}_4-{\mathrm{C}}_2{\mathrm{Cl}}_6$ alloy. Horizontal dimension: 450 $\mu \mathrm{m}$.