A novel computational model for isotropic interfacial energies in multicomponent alloys and its coupling with phase-field model with finite interface dissipation

doi:10.1016/j.jmst.2022.04.057

Abstract

Abstract:

In this work, a novel computational model for the description of the temperature- and composition-dependent isotropic interfacial energy in multicomponent alloys was first developed in the framework of the CALculation of PHAse Diagram (CALPHAD) approach and implemented in a home-made code. By linking to the open-source code for interfacial energy calculation in alloys, OpenIEC, the databases for isotropic γ/liquid and γ/γ' interfacial energies in Ni-Al, Ni-Cr, Al-Cr, and Ni-Al-Cr systems were then efficiently established. After that, a direct coupling strategy between the current CALPHAD interfacial energy database and the phase-field model with finite interface dissipation was proposed and applied to three-dimensional (3-D) phase-field simulations of the primary γ dendritic growth in both Ni-Al and Ni-Al-Cr alloys during isothermal solidification. The effect of the interfacial energy on the morphology, tip growth rate, and partitioning coefficients in primary γ dendrites of binary Ni-Al and ternary Ni-Al-Cr alloys was investigated by comprehensively comparing the phase-filed simulation results using the composition-/temperature-dependent interfacial energies with those using the constant value. It is anticipated that the presently developed CALPHAD model for interfacial energy is of general validity for different multicomponent alloys and should be integrated with the phase-field model for quantitative simulation of their microstructure evolution.

Key words: Interfacial energy, CALPHAD, OpenIEC, Phase-field simulation, Dendritic solidification

Shenglan Yang, Jing Zhong, Jiong Wang, Jianbao Gao, Qian Li, Lijun Zhang. A novel computational model for isotropic interfacial energies in multicomponent alloys and its coupling with phase-field model with finite interface dissipation[J]. J. Mater. Sci. Technol., 2023, 133: 111-122.

Figures/Tables 12

Fig. 1. (a) Schematic for the concept of atomic pairs in both ternary A-B-C and quaternary A-B-C-D systems. Schematics for the cross-sectional view and end-members in α/β interface of (b) ternary A-B-C system and (c) quaternary A-B-C-D system.

Fig. 2. A framework for programming the interface energy model and its coupling with the phase-field code and OpenIEC code.

Table 1. Model parameters for the γ/liquid interfacial energies of Ni-Al, Ni-Cr, Al-Cr, and Ni-Al-Cr systems.

Parameters (mJ m^-2)	Expression (T in K)
σ_Ni−Al	-536.62 + 8.99910 · T + 6.07970 × 10^-4 · T ² - 1.28251 · T · ln(T)
σ_Ni−Cr	89.34 + 0.25978 · T - 6.34033 × 10^-5 · T ²
σ_Al−Cr	-1972.65 + 25.13782 · T + 1.56533 × 10^-3 · T ² - 3.55575 · T · ln(T)
$\Delta\sigma_{\mathrm{Ni-Al,Ni-Cr}}^0$	-740.10 + 0.41001 · T
$\Delta\sigma_{\mathrm{Ni-Al,Ni-Cr}}^1$	1293.93 - 0.88470 · T
$\Delta\sigma_{\mathrm{Ni-Al,Al-Cr}}^0$	-175.24
$\Delta\sigma_{\mathrm{Ni-Al,Al-Cr}}^1$	-747.61 + 0.50526 · T
$\Delta\sigma_{\mathrm{Ni-Cr,Al-Cr}}^0$	-87.32
$\Delta\sigma_{\mathrm{Ni-Cr}}$	-1098.13
$\Delta\sigma_{\mathrm{Al-Cr}}$	2014.65

Fig. 3. Model-predicted temperature-dependent γ/liquid interfacial energies for the (a) Ni-Al, (b) Ni-Cr, and (c) Al-Cr systems (denoted by solid lines), compared with the original data by OpenIEC (denoted by hollow symbols).

Fig. 4. Model-predicted temperature- and composition-dependent γ/liquid interfacial energies for the Ni-Al-Cr system (denoted by solid lines), compared with the original data by OpenIEC (denoted by hollow symbols). The mole fractions of Cr are fixed as (a) 0.05, (b) 0.08, (c) 0.11, (d) 0.14, (e) 0.17, and (f) 0.20, respectively.

Table 2. Model parameters for the γ/γ' coherent interfacial energies of Ni-Al, Ni-Cr, Al-Cr, and Ni-Al-Cr systems.

Parameters (mJ m^-2)	Expression (T in K)
σ_Ni−Al=σ_Al−Cr	138.04 - 0.11345 · T + 2.02702 × 10^-5 · T ²
σ_Ni−Cr	147.92 - 0.15161 · T + 3.90215 × 10^-5 · T ²
$\Delta\sigma_{\mathrm{Ni-Al,Al-Cr}}^0$	-9144.57 + 6.60663 · T
$\Delta\sigma_{\mathrm{Ni-Al,Al-Cr}}^1$	21,512.46 - 15.99501 · T
$\Delta\sigma_{\mathrm{Ni-Al,Al-Cr}}^2$	-12,244.89 + 9.26806 · T
$\Delta\sigma_{\mathrm{Ni−Cr,Al−Cr}}^0$	-1759.62 + 1.33749 · T
$\Delta\sigma_{\mathrm{Ni−Cr}}$r	18,351.18 - 13.49255 · T

Fig. 5. Model-predicted temperature-dependent γ/γ' interfacial energies for the (a) Ni-Al, (b) Ni-Cr, and (c) Al-Cr systems (denoted by solid lines), compared with the original data of Ni-Al alloys by OpenIEC (denoted by hollow squares).

Fig. 6. Model-predicted temperature- and composition-dependent γ/γ' interfacial energies for the Ni-Al-Cr system (denoted by solid lines), compared with the original data by OpenIEC (denoted by hollow symbols). The mole fractions of Al are fixed as (a) 0.06, (b) 0.08, (c) 0.10, (d) 0.12, (e) 0.14, and (f) 0.16, respectively.

Fig. 7. Cutaway drawings for Al composition field of phase-field simulated γ dendrite in the Ni–Al alloy during the isothermal solidification with the constant (a1–a3, c1–c3, and e1–e3) and temperature-dependent (b1–b3, d1–d3, and f1–f3) γ/liquid interfacial energies: (a1–a3 and b1–b3) 1570 K, (c1–c3 and d1–d3) 1573 K, and (e1–e3 and f1–f3) 1576 K. Dendrite contours are plotted at the position of $ \phi =0.3 $. σ0 denotes the constant γ/liquid interfacial energy from literature, while σ(T) denotes the temperature-dependent γ/liquid interfacial energy of the Ni–Al alloy predicted in this work.

Fig. 8. Evolution of (a) growth velocities and (b) partition coefficients of Al at the dendrite tips of phase-field simulated primary dendrite in the Ni-Al alloy at different temperatures. The symbols with solid lines denote the phase-field simulated results with the temperature-dependent γ/liquid interfacial energies, while the symbols with dashed lines represent those with the constant interfacial energy. The dash-dotted lines signify the equilibrium partition coefficients of Al in Ni-Al alloys at different temperatures.

Fig. 9. Cutaway drawings for Al (a-f) and Cr (g-l) composition fields of phase-field simulated γ dendrites of different Ni-Al-Cr alloys during isothermal solidification at 1473 K with the constant (a1-a3, c1-c3, e1-e3, g1-g3, i1-i3, and k1-k3) and temperature- and composition-dependent (b1-b3, d1-d3, f1-f3, h1-h3, j1-j3, and l1-l3) γ/liquid interfacial energies. a1-a3, b1-b3, g1-g3, and h1-h3: Ni-28 at.% Al-13 at.% Cr; c1-c3, d1-d3, i1-i3, and j1-j3: Ni-28 at.% Al-15 at.% Cr; and e1-e3, f1-f3, k1-k3, and l1-l3: Ni-28 at.% Al-17 at.% Cr. Dendrite contours are plotted at the position of ?=0.3. σ0 denotes the constant γ/liquid interfacial energy from literature, while σ(T,x) denotes the temperature- and composition-dependent γ/liquid interfacial energy of the Ni-Al-Cr alloys predicted in this work.

Fig. 10. Evolution of (a) growth velocities, (b) partition coefficients of Al, and (c) partition coefficients of Cr at the tips of phase-field simulated primary dendrites in the Ni-Al-Cr alloys during the isothermal solidification at 1473 K. The symbols with solid lines represent the simulated results with the temperature- and composition-dependent γ/liquid interfacial energies of Ni-Al-Cr alloys, while those with dashed lines represent the simulated results with the constant interfacial energy. The dash-dotted lines signify the equilibrium partition coefficients of Al and Cr in different Ni-Al-Cr alloys at 1473 K.

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