A Quantitative Study of Microsegregation in Aluminum–Copper Alloys

Abstract

This study quantitatively investigated the effects of a wide variety of solidification process conditions on the microsegregation behavior of Al–Cu binary and multi-component alloys. Quantitative measurements of microsegregation were conducted using electron probe micro-analysis (EPMA). The EPMA results systematically demonstrated the influences of solidification cooling rates and thermal gradients on the microsegregation behavior in these Al–Cu alloys. The effects of the cooling rate on the microstructure and volume fraction of eutectic phase were also quantitatively characterized. A comprehensive micro-model for microsegregation, which considers solid diffusion, non-equilibrium parameters, constitutional undercooling at the dendrite tip, and dendrite remelting during dendrite formation was applied to predict the microsegregation behavior over a wide range of cooling rates. The comprehensive micro-model is analytical, simple to implement and accurately predicts the microsegregation behavior in Al–Cu binary and multi-component alloys during both columnar and equiaxed dendritic solidification for a wide range of solidification conditions from laser surface remelting to near equilibrium conditions represented by equiaxed dendrites in thick cast sections.

Appendix: Derivation of the Comprehensive Microsegregation Model

The solute balance in a fixed-size domain should follows that the sum of the solute in the solid and liquid phase should be the total solute presents. Therefore, the solute balance equation can be written as

$${\int }_{0}^{{X}_{\mathrm{s}}}{C}_{\mathrm{s}}dX+{\int }_{{X}_{\mathrm{s}}}^{{{\lambda }_{\mathrm{f}}}/ {2}}{C}_{\mathrm{l}}dX={C}_{0},$$

(A1)

where ${C}_{\mathrm{s}}$, ${C}_{\mathrm{l}}$ and ${C}_{0}$ is the concentration in solid phase, liquid phase and total concentration respectively, ${X}_{\mathrm{s}}$ is the size of the solid phase, and ${\lambda }_{\mathrm{f}}$ is the final SDAS. If a perfect mixing in the liquid is assume, that is the ${C}_{\mathrm{l}}$ is constant in the liquid phase, Eq. [A1] can be written as

$${\int }_{0}^{{X}_{\mathrm{s}}}{C}_{\mathrm{s}}dX+\left(\frac{{\lambda }_{\mathrm{f}}}{2}-{X}_{\mathrm{s}}\right){C}_{\mathrm{l}}^{*}={C}_{0}.$$

(A2)

Differentiate Eq. [A2] with respect to time and

$${\int }_{0}^{{X}_{\mathrm{s}}}\frac{\partial {C}_{\mathrm{s}}}{\partial t}dX+{C}_{\mathrm{s}}^{*}\frac{\mathrm{d}{X}_{\mathrm{s}}}{\mathrm{d}t}+\left(\frac{{\lambda }_{\mathrm{f}}}{2}-{X}_{\mathrm{s}}\right)\frac{\mathrm{d}{C}_{\mathrm{l}}^{*}}{\mathrm{d}t}-{C}_{\mathrm{l}}^{*}\frac{\mathrm{d}{X}_{\mathrm{s}}}{\mathrm{d}t}=0.$$

(A3)

Assume local equilibrium at the interface and Fick’s diffusion is the mechanisms for solute diffusion, the Eq. [A3] can be written as

$${\int }_{0}^{{X}_{\mathrm{s}}}{D}_{\mathrm{s}}\frac{{\partial }^{2}{C}_{\mathrm{s}}}{\partial {X}^{2}}dX-(1-k){C}_{\mathrm{l}}^{*}\frac{\mathrm{d}{X}_{\mathrm{s}}}{\mathrm{d}t}+\left(\frac{{\lambda }_{\mathrm{f}}}{2}-{X}_{\mathrm{s}}\right)\frac{\mathrm{d}{C}_{\mathrm{l}}^{*}}{\mathrm{d}t}=0.$$

(A4)

Rearrange Eq. [A4] and get

$$\left(1-k\right){C}_{\mathrm{l}}^{*}\frac{\mathrm{d}{X}_{\mathrm{s}}}{\mathrm{d}t}={D}_{\mathrm{s}}\frac{\partial {C}_{\mathrm{s}}^{*}}{\partial X}+\left(\frac{{\lambda }_{\mathrm{f}}}{2}-{X}_{\mathrm{s}}\right)\frac{\mathrm{d}{C}_{\mathrm{l}}^{*}}{\mathrm{d}t}.$$

(A5)

Assume the chain rule and local equilibrium, and therefore

$$\frac{\partial {C}_{\mathrm{s}}^{*}}{\partial {X}_{\mathrm{s}}}\cong k\frac{\partial {C}_{\mathrm{l}}^{*}}{\partial {X}_{\mathrm{s}}}.$$

(A6)

Substitute Eq. [A6] in [A5] and get

$$\left(1-k\right){C}_{\mathrm{l}}^{*}\frac{\mathrm{d}{X}_{\mathrm{s}}}{\mathrm{d}t}={D}_{\mathrm{s}}k\frac{\partial {C}_{\mathrm{l}}^{*}}{\partial {X}_{\mathrm{s}}}+\left(\frac{{\lambda }_{\mathrm{f}}}{2}-{X}_{\mathrm{s}}\right)\frac{\mathrm{d}{C}_{\mathrm{l}}^{*}}{\mathrm{d}t}.$$

(A7)

Assume a parabolic growth of solid vs. time, which is

$${X}_{\mathrm{s}}=\frac{{\lambda }_{\mathrm{f}}}{2}\sqrt{\frac{t}{{t}_{\mathrm{f}}}} \underset{}{\Rightarrow } \frac{\mathrm{d}{X}_{\mathrm{s}}}{\mathrm{d}t}=\frac{{\lambda }_{\mathrm{f}}^{2}}{8{t}_{\mathrm{f}}{X}_{\mathrm{s}}}.$$

(A8)

Substitute Eq. [A8] in [A7] and get

$$\left(1-k\right){C}_{\mathrm{l}}^{*}\frac{\mathrm{d}{X}_{\mathrm{s}}}{\mathrm{d}t}={D}_{\mathrm{s}}k\frac{8{t}_{\mathrm{f}}{X}_{\mathrm{s}}}{{\lambda }_{\mathrm{f}}^{2}}\frac{\partial {C}_{\mathrm{l}}^{*}}{\partial t}+\left(\frac{{\lambda }_{\mathrm{f}}}{2}-{X}_{\mathrm{s}}\right)\frac{\mathrm{d}{C}_{\mathrm{l}}^{*}}{\mathrm{d}t}.$$

(A9)

Since ${f}_{\mathrm{s}}={{X}_{\mathrm{s}}}/ {\frac{{\lambda }_{\mathrm{f}}}{2}}$, therefore the differential equation can be written in

$$\left(1-k\right){C}_{\mathrm{l}}^{*}\frac{{\lambda }_{\mathrm{f}}}{2}\frac{\mathrm{d}{f}_{\mathrm{s}}}{\mathrm{d}t}=\left[\left(2k\frac{4{{D}_{\mathrm{s}}t}_{\mathrm{f}}}{{\lambda }_{\mathrm{f}}^{2}}-1\right)+\frac{{\lambda }_{\mathrm{f}}}{2}\right]\frac{\partial {C}_{\mathrm{l}}^{*}}{\partial t}.$$

(A10)

The back diffusion coefficient can be represented by $\alpha =\frac{4{{D}_{\mathrm{s}}t}_{\mathrm{f}}}{{\lambda }_{\mathrm{f}}^{2}}$, later the Clyne–Kurz proposed an empirical term that have a better solute conservation to replace $\alpha$. The term back diffusion term $\Omega$, which can be written as

$$\Omega =\alpha \left[1-\mathrm{exp}\left(-\frac{1}{\alpha }\right)\right]-\frac{1}{2}\mathrm{exp}\left(-\frac{1}{2\alpha }\right).$$

(A11)

The initial solid concentration in the Modified Clyne–Kurz should be

$${C}_{\mathrm{s},\mathrm{i}}^{*}={C}_{0}^{*}A=\frac{{C}_{0}^{*}}{\left[1-\left(1-{k}_{\mathrm{v}}\right)\mathrm{Iv}(Pe)\right]}.$$

(A12)

A partition coefficient is also considered, and therefore

$${k}_{\mathrm{v}}=\frac{{k}_{\mathrm{e}}+{a}_{0}v/{D}_{0}}{1+{a}_{0}v/{D}_{0}}.$$

(A13)

Integrate Eq. [A10] using results from [A11], [A12] and [A13], and the expression for Modified Clyne–Kurz is

$${C}_{\mathrm{s}}^{\boldsymbol{*}}=\frac{{C}_{0}^{*}{k}_{\mathrm{v}}}{\left[1-\left(1-{k}_{\mathrm{v}}\right)\mathrm{Iv}(Pe)\right]}{\left[1-\left(1-2{k}_{\mathrm{v}}\Omega \right){f}_{\mathrm{s}}\right]}^{{\left({k}_{\mathrm{v}}A-1\right)}/{\left(1-2{k}_{\mathrm{v}}\Omega \right)}}.$$

(A14)

In short, [A14] can be written as

$${C}_{\mathrm{s}}^{*}={k}_{\mathrm{v}}{AC}_{0}{\left(1-(1-{2{k}_{\mathrm{v}}A\Omega )f}_{\mathrm{s}}\right)}^{{{k}_{\mathrm{v}}A-1}/ {1-2{k}_{\mathrm{v}}A\Omega }}.$$

(A15)