Numerical Simulations of Locked Lamellar Eutectic Growth Patterns

Abstract

We present two-dimensional numerical simulations of tilted lamellar growth patterns during directional solidification of nonfaceted binary eutectic alloys in the presence of an anisotropy of the free energy $\gamma$ of the interphase boundaries in the solid. We used a dynamic boundary-integral (BI) method. The physical parameters were those of the transparent eutectic $\mathrm CBr_4$-$\mathrm C_2Cl_6$ alloy. As in Ghosh et al. (Phys Rev E 91, 022407, 2015), the anisotropy of $\gamma$ was described by a model function with tunable parameters. The lamellar-locking effect in the vicinity of a deep minimum of the interfacial energy was reproduced. For a weak anisotropy, the lamellar tilt angle $\theta _t$ was shown to depend on the growth conditions. We systematically studied the influence of usual control parameters (pulling velocity, temperature gradient, lamellar spacing, alloy concentration) on the tilted-lamellar pattern. We identified experimentally accessible conditions under which $\theta _t$ falls close to the theoretical prediction based on the so-called symmetric-pattern approximation. We finally simulated locked and weakly locked lamellar patterns and found empirically a good morphological matching with experimental observations during directional solidification of thin $\mathrm CBr_4$-$\mathrm C_2Cl_6$ samples.

Appendices

Appendix A

In this Appendix, we propose a list of the symbols used in the text. In Table II, physical parameters are listed. Table III provides a list of dimensionless variables and symbols.

Table II Physical Variables and Parameters

Table III Dimensionless Variables, Parameters and Symbols

Appendix B

In this Appendix, we were aiming to show useful details related to the anisotropy effect that was simulated in Figures 9 (Figure 11) and 10 (Figure 12) in Section IV–E. We recall that the anisotropic surface energy of the interphase boundary is written in the form of $\gamma (\theta ) = \gamma _0 [1-a_c(\theta -\theta _R)]$, with $\theta _R$ setting the angular orientation of the eutectic grain (see Eq. [14]). The model anisotropy function $a_c$ is taken in the form $a_c(\theta ) = \epsilon _g \exp \left[ -(\theta /w_g)^2\right] - \epsilon _2\cos 2\theta - \epsilon _4\cos 4\theta$ (see Eq. [15]). The relevant values of the various coefficients are recalled in the captions of Figures 11 and 12 for convenience.

Fig. 11

(a) Wulff shape (green line) and $\gamma$ plot (blue line) of the anisotropy function used for the simulations shown in Fig. 9. The graph is rotated by an angle equal to the relevant $\theta _R$ value. (b) Tilt angle $\theta _{sp}$ in the sp approximation for the same anisotropy function. (c) Detail of the experimental microstructure. Red dots: locked (label L) and floating (label F) interphase boundaries. The L and F points are both close to an intersect in the Wulff shape. The arrow (label U) in (a) and (b) designates the unstable branch. Coefficients in the anisotropy function: $\epsilon _g=0.2$, $w_g=0.1$, $\epsilon _2=0.104$, $\epsilon _4=0.02208$, and $\theta _R=51^o$. Color online

Fig. 12

(a) Wulff shape (green line) and $\gamma$ plot (blue line) of the anisotropy function used for the simulations shown in Fig. 10. The graph is rotated by an angle equal to the relevant $\theta _R$ value. (b) Tilt angle $\theta _{sp}$ in the sp approximation for the same anisotropy function. (c) Details of the experimental microstructure. Red dots: locked (label L) and floating (label F) interphase boundaries. The arrow (label U) in (a) and (b) designates the unstable branch. Coefficients in the anisotropy function: $\epsilon _g=0.2$, $w_g=0.005$, $\epsilon _2=0.2$, $\epsilon _4=0.06$, and $\theta _R=35.0^o$. Color online