Crystallographic and Morphological Evidence of Solid–Solid Interfacial Energy Anisotropy in the Sn-Zn Eutectic System

Abstract

In this paper, we explore the morphological evolution during two-phase growth in the Sn-Zn eutectic system, which has a particularly low volume fraction of the minority Zn phase. The reason for this choice is its exotic nature, as even with such a low volume fraction, the reported morphology is “broken-lamellar,” in contrast to the usually expected hexagonal arrangement of Zn rods in the Sn matrix. Thus, the main objective of the study is to investigate the reasons behind this phenomenon. We begin by presenting experimental results detailing the morphology and crystallography of the eutectic microstructures under various combinations of thermal gradients and velocities in directional solidification conditions. Based on the crystallography and further specially designed experiments we find that the solid–solid interface between the Sn and Zn crystal is anisotropic. On the basis of the results, we propose a hypothesis that the presence of solid–solid interfacial energy anisotropy leads to the formation of predominantly broken-lamellar structures, even when the minority fraction is significantly low.

Appendices

Appendix A. Calculation of Thermal Gradient at the Solid–Liquid Interface

The model of the directional solidification experimental setup is as shown in Figure A1a. We calculate the thermal profiles by solving heat transfer equations in the software OpenFOAM[65], which allows us to determine accurate thermal gradient at the solid–liquid interface. The material parameters reported by References [43, 64] for the alloy are used in the calculation. The computed thermal profile is shown in Figure A1b. The axial temperature profile (Figure A1c) in liquid shows a gradient close to 6 °C/mm.

Fig. A1

Modeling using OpenFOAM software, where (a) is directional solidification setup 2D view, (b) is thermal profile, and (c) is axial temperature profile in liquid

Appendix B. Calculation of Jackson–Hunt Parameters for Lamellar and Rod Morphology

Using Jackson and Hunt theory,[47] undercooling vs. spacing relation for lamellar (L) and rod morphology (R) writes,

$$\begin{aligned} \Delta T = K_1V \lambda + K_2/\lambda . \end{aligned}$$

(B1)

For lamellar morphology, $K_{1}^{L} = {\overline{m}} C_{o} P / f_{\alpha }f_{\beta } D$ and $K_2^{L}=2{\overline{m}} (\Gamma _{\alpha } \sin \theta _{\alpha }/f_{\alpha }m_{\alpha } + \Gamma _{\beta } \sin \theta _{\beta }/f_{\beta }m_{\beta })$, where $C_o=\left( C_\alpha - C_\beta \right)$ ( i.e., $C_o$ is the composition difference between the solid phases at the eutectic temperature), $f_\alpha$, $f_\beta$ are the volume fractions of the solid phases, $1/{\overline{m}} = \left( 1/m_{\alpha } + m_\beta \right)$, $m_\alpha$, $m_\beta$ are the liquidus slopes of the solid phases. $\Gamma _\alpha$ and $\Gamma _\beta$ are the Gibbs–Thomson coefficients of the solid and the liquid phases while the $\theta _\alpha$ and $\theta _\beta$ are angles the $\alpha$-liquid and the $\beta$-liquid tangents make with the horizontal. P is a function of the volume fraction of one of the phases that reads $\sum _{n=1}^{\infty } \dfrac{1}{n\pi ^{3}} \sin ^{2}\left( n\pi f_\alpha \right)$. Similarly for rod morphology, $K_1^{R}={\overline{m}} C_{o} M / f_{\beta } D$, where $M=\sum _{n=1}^{\infty }\dfrac{1}{\left( \gamma _n\right) ^{3}}\dfrac{J_1^{2}\left( \sqrt{f_\alpha }\gamma _n\right) }{J_0^{2}\left( \gamma _n\right) }$, where $\gamma _n$ is the $n$th zero of the first-order Bessel function $J_1$, while $J_o$ is the zeroth-order Bessel function. We calculated $K_1^{L}$ and $K_1^{R}$ values as 0.00836 and 0.0018, respectively, using material parameters reported[43,64] for the Sn-Zn alloy. Thereafter using the value of $\lambda ^{2}V$ (= $K_2^{L}/K_1^{L}$) that we find using our experiments, we assessed the value of $K_2^{L}$ as 0.226. Using the relation $K_2^{R}/K_2^{L} = \sqrt{f_\alpha }$ from Jackson–Hunt theory, we calculated $K_2^{R}$ value as 0.0678. This allows us to determine the $\Delta T$ vs. spacing $\lambda$ relationship for both morphologies.

Appendix C. Calculation of Interface Shape and Undercooling

Assumptions are

• Directional solidification conditions with gradient G and velocity V;

• Composition profile derived for a planar solidification interface; and

• All other Fourier coefficients are derived from the planar solution except the boundary layer composition $B_0$

We integrate the basic undercooling equation which writes

$$\begin{aligned} \Delta T\left( x\right) = m^{\nu }\left( C_E - c\left( x\right) \right) + a/r\left( x\right) , \end{aligned}$$

(C1)

where $\Delta T$ is the undercooling, m is the liquidus slope of the solid phase concerned, $C_E$ is the eutectic composition, $c\left( x\right)$ is the composition profile derived from the solution of the diffusion for a planar solidification front, a is the Gibbs–Thomson coefficient of the relevant solid–liquid interface, while $r\left( x\right)$ is its curvature. The composition $c\left( x\right)$ can be explicitly written as

$$\begin{aligned} c\left( x\right) = C_E + c_\infty + B_0 + \sum _{n=1}^{\infty } B_n \sin \left( \dfrac{n\pi x}{S_\alpha +S_\beta }\right) , \end{aligned}$$

(C2)

where $B_0$ is the boundary layer composition, $S_{\alpha ,\beta }$ are, respectively, the half widths of each of the lamellae. $c_\infty$ is the departure of the far-field concentration from the equilibrium value, $C_E$ is the eutectic composition, while each of the Fourier amplitudes $B_n$ can be written as

$$\begin{aligned} B_n = \dfrac{2}{\left( n\pi \right) ^{2}}\left( S_\alpha + S_\beta \right) \dfrac{V}{D}C_0\sin \left( \dfrac{n\pi S_\alpha }{S_\alpha + S_\beta }\right) , \end{aligned}$$

(C3)

where $C_0$ is $|C_E - C_\alpha | + |C_\beta - C_E|$ and D is the diffusivity in the liquid.

The undercooling $\Delta T\left( x\right)$ can in turn be expressed using the thermal gradient and the interface position as

$$\begin{aligned} \Delta T&= T_E - (T_0 + G\left( z - z_0\right) ) \nonumber \\&= (T_E - (T_0 + G z_0)) - Gz \nonumber \\&= \Delta T_0 - Gz \end{aligned}$$

(C4)

$T_0$ and $z_0$ being the temperature and z-ordinate of the triple point which is shared by both solid–liquid interfaces. The composite term $(T_E - (T_0 + G z_0))$ is expressed as $\Delta T_0$ which by definition is the same for both the phases. With this Eqn. C1 becomes,

$$\begin{aligned} \Delta T_0 - Gz&= m^{\nu }\left( C_E - c\left( x\right) \right) + a/r\left( x\right) . \end{aligned}$$

(C5)

Integrating the preceding equation once over each of the solid–liquid interfaces, with the boundary conditions, that the center of each of the phases is flat and the tangent at the triple point is given by the balance of the surface tensions, we derive two relations, respectively, for each of the solid–liquid interfaces, which can be written as

$$\begin{aligned}&\Delta T_0 - G {\bar{z}}^{\alpha } -m^{\alpha }\left( -c_\infty - B_0 - 2\dfrac{\lambda V}{D}C_0P\left( \eta _\alpha \right) \right) \nonumber \\&\quad =a^{\alpha } \dfrac{\sin \left( \theta _{\alpha l}\right) }{\eta _\alpha \lambda } \nonumber \\&\Delta T_0 - G {\bar{z}}^{\beta } -m^{\beta }\left( -c_\infty - B_0 + 2\dfrac{\lambda V}{D}C_0P\left( \eta _\alpha \right) \right) \nonumber \\&\quad = a^{\beta }\dfrac{\sin \left( \theta _{\beta l}\right) }{\eta _\alpha \lambda }, \end{aligned}$$

(C6)

where $\lambda = S_\alpha + S_\beta$, and $\eta _\alpha =S_\alpha /(S_\alpha + S_\beta )$, while ${\bar{z}}^{\alpha }$ and ${\bar{z}}^{\beta }$ are average interface positions of the $\alpha$ and the $\beta$ interfaces, respectively. $P\left( \eta _\alpha \right)$ is written as $\sum _{n=1}^{\infty }\dfrac{\sin ^{2}\left( n\pi \eta _\alpha \right) }{\left( n\pi \right) ^{3}}$

The previous expressions can now be simplified to derive $\Delta T_0$ as well as $B_0$ in terms of the average interface positions as

$$\begin{aligned} B_0&= G\dfrac{\left( {\bar{z}}^{\alpha } - {\bar{z}}^{\beta }\right) }{m^{\beta } - m^{\alpha }} \nonumber \\&\quad + \dfrac{1}{m^{\beta } - m^{\alpha }}\left( \dfrac{a^{\alpha }\sin \left( \theta _{\alpha l}\right) }{\eta _\alpha \lambda } - \dfrac{a^{\beta }\sin \left( \theta _{\beta l}\right) }{\left( 1-\eta _\alpha \right) \lambda }\right) \nonumber \\&\quad -\dfrac{2\lambda VC_0P}{D\left( m^{\beta } - m^{\alpha }\right) }\left( \dfrac{m^{\alpha }}{\eta _\alpha } - \dfrac{m^{\beta }}{1-\eta _\alpha }\right) , \end{aligned}$$

(C7)

while $\Delta T_0$ is written as

$$\begin{aligned} \Delta T_0&= G{\bar{z}}^{\alpha } + m^{\alpha }\left( -c_\infty - B_0 - \dfrac{2\lambda VC_0P}{D\eta _\alpha }\right) \nonumber \\&\quad +\dfrac{a^{\alpha }\sin \left( \theta _{\alpha l}\right) }{\eta _\alpha \lambda } \end{aligned}$$

(C8)

Thereafter we set out to integrate Eqn. C5 using a finite difference approach where we discretize the curvature using central differences and the slopes as one-sided forward differences. The boundary conditions again involve imposing a zero slope at the mid-points of the lamellae which are at $x=0$ and $x= S_\alpha + S_\beta$. Also, the tangents at the triple point $x= S_\alpha$ are given by the Young’s condition. In this discretized equation, the derived expressions for $B_0$ and $\Delta T_0$ which are expressed in Eqns. C7 and C8 are used. This gives us a set of N-2 non-linear equations (N being the points on a particular solid–liquid interface) for each of the solid–liquid interfaces, which are solved using the non-linear solution routine of “fsolve” which is in the optimization tool box of “octave-forge.” The result is the solution to the interface z-ordinates of each of the solid–liquid interface from which one can derive both the boundary layer composition $B_0$ and $\Delta T_0$ and thereby the temperature of the triple point $T_0$ (since $z_0$ is derived from the solution) along with the temperatures along each of the solid–liquid interface.