Alloy solidification: Assessment and improvement of an easy-to-apply model

doi:10.1016/j.jmst.2022.03.038

Abstract

Abstract:

It has been a central task of solidification research to predict solute microsegregation. Apart from the Lever rule and the Scheil-Gulliver equation, which concern two extreme cases, a long list of microsegregation models has been proposed. However, the use of these models often requires essential experimental input information, e.g., the secondary dendrite arm spacing (λ), cooling rate ($\dot{T}$) or actual solidification range (ΔT). This requirement disables these models for alloy solidification with no measured values for λ, $\dot{T}$ and ΔT. Furthermore, not all of these required experimental data are easily obtainable. It is therefore highly desirable to have an easy-to-apply predictive model that is independent of experimental input, akin to the Lever rule or Scheil-Gulliver model. Gong, Chen, and co-workers have recently proposed such a model, referred to as the Gong-Chen model, by averaging the solid fractions (f_s) of the Lever rule and Scheil-Gulliver model as the actual solid fraction. We provide a systematic assessment of this model versus established solidification microsegregation models and address a latent deficiency of the model, i.e., it allows the Lever rule solid fraction f_s to be greater than one (f_s > 1). It is shown that the Gong-Chen model can serve as a generic model for alloy solidification until f_s reaches about 0.9, beyond which (f_s > 0.9) its applicability is dictated by both the equilibrium solute partition coefficient (k) and the solute diffusion coefficient in the solid (D_s), which has been tabulated in detail.

Key words: Solidification, Microsegregation, Solute, Back diffusion, Eutectic formation

H. Liu, Y. Liu, S.L. Lu, Y. Zhang, H. Chen, Y. Chen, M. Qian. Alloy solidification: Assessment and improvement of an easy-to-apply model[J]. J. Mater. Sci. Technol., 2022, 130: 1-11.

Figures/Tables 13

References 54

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Model	Expression	Empty Cell	Refs.
Brody-Flemings (1966)	$\begin{array}{l}X_{\mathrm{L}}=X_{0}\left[1-(1-\beta k) f_{\mathrm{s}}\right]^{\frac{k-1}{1-\beta k}} \\\alpha=\frac{4 D_{s} \Delta T}{\lambda^{2} T}\end{array}$	β=2α	[13]
Clyne-Kurz (1981)		$\beta=2 \alpha\left[1-\exp \left(-\frac{1}{\alpha}\right)\right]-\exp \left(-\frac{1}{2 \alpha}\right)$	[14]
Ohnaka (1986)		$\beta=\frac{2 \alpha}{1+2 \alpha} \text { or } \frac{4 \alpha}{1+4 \alpha}$	[15]
Voller (1999)		$\beta=\frac{2 \alpha}{\left(1-f_{\text {eutectic }}\right)^{2}+2 \alpha}$	[16]
Voller-Beckermann (1999)		$ f_{\text {eutectic }}=0 leads to the Ohnaka model$ $\beta=\frac{2 \alpha^{+}}{\left(1-f_{\text {eutectic }}\right)^{2}+2 \alpha^{+}}$	[17]
Won-Thomas (2001)		$where \alpha^{+}=\alpha+0.1 $ $\beta=2 \alpha^{+}\left[1-\exp \left(-\frac{1}{\alpha^{+}}\right)\right]-\exp \left(-\frac{1}{2 \alpha^{+}}\right)$	[18]

Model	Expression	Empty Cell	Refs.
Brody-Flemings (1966)	$\begin{array}{l}X_{\mathrm{L}}=X_{0}\left[1-(1-\beta k) f_{\mathrm{s}}\right]^{\frac{k-1}{1-\beta k}} \\\alpha=\frac{4 D_{s} \Delta T}{\lambda^{2} T}\end{array}$	β=2α	[13]
Clyne-Kurz (1981)		$\beta=2 \alpha\left[1-\exp \left(-\frac{1}{\alpha}\right)\right]-\exp \left(-\frac{1}{2 \alpha}\right)$	[14]
Ohnaka (1986)		$\beta=\frac{2 \alpha}{1+2 \alpha} \text { or } \frac{4 \alpha}{1+4 \alpha}$	[15]
Voller (1999)		$\beta=\frac{2 \alpha}{\left(1-f_{\text {eutectic }}\right)^{2}+2 \alpha}$	[16]
Voller-Beckermann (1999)		$ f_{\text {eutectic }}=0 leads to the Ohnaka model$ $\beta=\frac{2 \alpha^{+}}{\left(1-f_{\text {eutectic }}\right)^{2}+2 \alpha^{+}}$	[17]
Won-Thomas (2001)		$where \alpha^{+}=\alpha+0.1 $ $\beta=2 \alpha^{+}\left[1-\exp \left(-\frac{1}{\alpha^{+}}\right)\right]-\exp \left(-\frac{1}{2 \alpha^{+}}\right)$	[18]

Alloys	Empty Cell	D_s (m²/s)	Solidus temperature at 0.3 at.% solute (K)	Refs.
Fe-based	C in δ-Fe S in δ-Fe P in δ-Fe Mn in δ-Fe Si in δ-Fe	5.7×10−9 2.9×10−10 6.5×10−11 2.5×10−11 5.2×10−11	1805 1806 1802 1807 1805	[18]
Al-based	Al-Cu Al-Mg Al-Si Al-Ni Al-Gd Al-Ge Al-Co Al-Fe Al-Mn Al-Cr Al-Ti Al-V	1.4×10−12 2.7×10−12 3.6×10−12 3.3×10−12 6.9×10−12 2.2×10−12 7.2×10−12 3.8×10−13 1.9×10−14 1.5×10−15 3.1×10−16 1.9×10−17	931 930 930 913 923 897 930 931 931 930 933 933	[25]
Mg-based	Mg -Al Mg-Zn Mg-Ca Mg-Ag Mg-Sn Mg-Li Mg-Mn Mg-Nd Mg-Y	1.4×10−12 7.3×10−12 4.0×10−12 7.2×10−12 1.2×10−12 4.1 ×10−12 2.6×10−13 5.5×10−13 5.5×10−13	921 921 877 909 921 922 923 923 902	[26]
	Mg-Sr Mg-Si Mg-Zr	1.7×10−11 5.1×10−13 3.3 ×10−16	856 911 923	[27]
Ti-based	Ti-Cr Ti-Al Ti-Zr Ti-V Ti-Mo Ti-Sn Ti-Fe Ti-Co Ti-Mn Ti-Ni Ti-P	6.8×10−11 2.1×10−11 3.7×10−11 5.6×10−11 1.9×10−11 2.5×10−11 3.9×10−10 7.1×10−10 2.0×10−10 7.9×10−10 9.1×10−10	1941 1944 1940 1943 1944 1943 1935 1930 1938 1921 1895	[28]
Cu-based	Cu-Sn Cu-Zn Cu-Sb Cu-Ni	5.5×10−12 1.6×10−12 6.0×10−12 2.2×10−13	1347 1356 1335 1359	[28]
Ni-based	Ni-Al Ni-Co Ni-Cr Ni-Cu Ni-Ti	1.5×10−12 6.8×10−13 9.6×10−13 1.2×10−12 1.5×10−12	1725 1728 1728 1727 1724	[28]
Zr-based	Zr-Ag Zr-Co Zr-Mo Zr-Nb	2.3×10−10 2.0×10−9 2.8×10−11 4.0×10−11	2117 2076 2128 2120	[28]

Alloys	Empty Cell	D_s (m²/s)	Solidus temperature at 0.3 at.% solute (K)	Refs.
Fe-based	C in δ-Fe S in δ-Fe P in δ-Fe Mn in δ-Fe Si in δ-Fe	5.7×10−9 2.9×10−10 6.5×10−11 2.5×10−11 5.2×10−11	1805 1806 1802 1807 1805	[18]
Al-based	Al-Cu Al-Mg Al-Si Al-Ni Al-Gd Al-Ge Al-Co Al-Fe Al-Mn Al-Cr Al-Ti Al-V	1.4×10−12 2.7×10−12 3.6×10−12 3.3×10−12 6.9×10−12 2.2×10−12 7.2×10−12 3.8×10−13 1.9×10−14 1.5×10−15 3.1×10−16 1.9×10−17	931 930 930 913 923 897 930 931 931 930 933 933	[25]
Mg-based	Mg -Al Mg-Zn Mg-Ca Mg-Ag Mg-Sn Mg-Li Mg-Mn Mg-Nd Mg-Y	1.4×10−12 7.3×10−12 4.0×10−12 7.2×10−12 1.2×10−12 4.1 ×10−12 2.6×10−13 5.5×10−13 5.5×10−13	921 921 877 909 921 922 923 923 902	[26]
	Mg-Sr Mg-Si Mg-Zr	1.7×10−11 5.1×10−13 3.3 ×10−16	856 911 923	[27]
Ti-based	Ti-Cr Ti-Al Ti-Zr Ti-V Ti-Mo Ti-Sn Ti-Fe Ti-Co Ti-Mn Ti-Ni Ti-P	6.8×10−11 2.1×10−11 3.7×10−11 5.6×10−11 1.9×10−11 2.5×10−11 3.9×10−10 7.1×10−10 2.0×10−10 7.9×10−10 9.1×10−10	1941 1944 1940 1943 1944 1943 1935 1930 1938 1921 1895	[28]
Cu-based	Cu-Sn Cu-Zn Cu-Sb Cu-Ni	5.5×10−12 1.6×10−12 6.0×10−12 2.2×10−13	1347 1356 1335 1359	[28]
Ni-based	Ni-Al Ni-Co Ni-Cr Ni-Cu Ni-Ti	1.5×10−12 6.8×10−13 9.6×10−13 1.2×10−12 1.5×10−12	1725 1728 1728 1727 1724	[28]
Zr-based	Zr-Ag Zr-Co Zr-Mo Zr-Nb	2.3×10−10 2.0×10−9 2.8×10−11 4.0×10−11	2117 2076 2128 2120	[28]

Alloy system	k	ml	ΔT0	Ds	SDAS λ	Cooling rate,	Refs.
X0 (at.%)	Empty Cell	(K/at.%)	(K)	(m²/s)	(μm)	T˙ (K/s)	Ds	λ,T˙
Mg-0.3Al Mg-0.3Sn Al-0.3Cu Fe-0.3C Fe-0.3P	0.36 0.31 0.15 0.16 0.29	-6.63 -7.38 -7.03 -15.03 -22.27	3.54 4.93 11.95 24.01 21.15	1.2×10⁻⁴exp(−141814/RT) 1.1×10⁻⁴exp(−140925/RT) 4.4×10⁻⁵exp(−133900/RT) 1.3×10⁻⁶exp(−81398/RT) 2.9exp(−230175/RT)	45 45 35 14 100	8 8 10 65 1.5	[25][26] [26] [18] [18]	[29]* * [18] [30]