Numerical simulation for dendrite growth in directional solidification using LBM-CA (cellular automata) coupled method

doi:10.1016/j.jmst.2020.01.047

摘要/Abstract

Abstract:

To predict the dendrite morphology and microstructure evolution in the solidification of molten metal, numerically, lattice Boltzmann method (LBM) - cellular automata (CA) model has been developed by integrating the LBM to solve the mass transport by diffusion and convection during solidification and the CA to determine the phase transformation with respect to the solid fraction based on the local equilibrium theory. It is successfully validated with analytic solutions such as Lipton-Glicksman-Kurz (LGK) model in static melt, and Oseen-Ivantsov solution under the fluid flow conditions in terms of tip radius and velocity of the dendrite growth. The proposed LBM-CA model does not only describe different types of dendrite formations with respect to various solidification conditions such as temperature gradient and growth rate, but also predict the primary dendrite arm spacing (PDAS) and the secondary dendrite arm spacing (SDAS), quantitatively, in directional solidification (DS) experiment with Ni-based superalloy.

Key words: Cellular automata (CA), Lattice Boltzmann method (LBM), Dendritic growth, Directional solidification

. [J]. 材料科学与技术, 2020, 49(0): 15-24.

Wonjoo Lee, Yuhyeong Jeong, Jae-Wook Lee, Howon Lee, Seong-hoon Kang, Young-Min Kim, Jonghun Yoon. Numerical simulation for dendrite growth in directional solidification using LBM-CA (cellular automata) coupled method[J]. J. Mater. Sci. Technol., 2020, 49(0): 15-24.

图/表 18

Fig. 1. Lattice arrangement of D2Q9 model [46].

Fig. 2. Overall flowchart for LBM-CA analysis.

Table 1 Material properties of Al-4 wt% Cu alloy [35].

Property and symbol	Value	Units
Density, ρ	2.475×10³	kg m^-3
Solute diffusivity in liquid, D_l	3×10^-9	m² s^-1
Gibbs-Thomson coefficient, Γ	2.4×10^-7	mK
Partition coefficient, k₀	0.17
Liquidus slope, m_L	-2.6	K wt%^-1
Liquidus temperature, T_eq	921.15	K
Initial composition, C₀	4	wt%
Anisotropy coefficient, ε	0.0267

Fig. 3. Schematic representation of each phases based on cell structure in CA analysis: (a) single solid cell surrounded by interface cells, (b) growth of solid and interface cells.

Fig. 4. Comparison between LGK analytic solution and LBM-CA prediction in terms of dendrite tip velocity and radius: (a) dendrite tip velocity, (b) dendrite tip radius.

Table 2 Material properties of Fe-0.82 wt% C alloy [52].

Property and symbol	Value	Units
Density, ρ	7.020×10³	kg m^-3
Solute diffusivity in liquid, D_l	6.36×10^-9	m² s^-1
Gibbs-Thomson coefficient, Γ	1.9×10^-7	mK
Partition coefficient, k₀	0.34
Liquidus slope, m_L	-78.0	K wt%^-1
Liquidus temperature, T_eq	1809	K
Initial composition, C₀	0.82	wt%
Anisotropy coefficient, ε	0.04
Viscosity, μ	5.5×10^-3	kg m^-1 s^-1

Fig. 5. Analysis results with proposed LBM-CA model under convectional melt: (a) geometrical boundary condition, (b) convectional melt.

Fig. 6. Comparison of Ossen-Ivantsov solution and proposed LBM-CA result.

Fig. 7. Directional solidification process: (a) schematic diagram, (b) cylindrical DS specimen.

Table 3 Chemical composition of the Ni-based superalloy.

Elements	Cr	Al	Ti	Ta	W	Mo	Zr	Hf	Co	Ni
wt.%	7.94	5.22	0.67	3.07	9.07	0.60	0.02	1.23	9.21	Bal.

Fig. 8. As-casted microstructures in DS process along: (a) radial, (b) axial direction.

Fig. 9. Numerical results with ProCAST [62] on distribution of temperature and solid fraction for directional casing.

Fig. 10. Temperature and cooling rate with respect to solidification time in ProCAST analysis.

Fig. 11. Microstructure prediction along radial direction between experiment and numerical results: (a) experiment, (b) ProCAST, (c) LBM-CA.

Fig. 12. Microstructure prediction along axial direction between experiment and numerical results: (a) experiment, (b) ProCAST, (c) LBM-CA.

Table 4 Nucleation parameters for numerical simulations.

Property and symbol	Value	Units
Maximum density, n_max	5.0×10⁷	m^-3
Standard deviation, ΔT_σ	0.1	K
Mean undercooling, ΔT_m	10	K

Table 5 Material properties of Ni-based superalloy [53,54].

Property and symbol	Value	Units
Density, ρ	8.780×10³	kg m^-3
Solute diffusivity in liquid, D_l	3.6×10^-9	m² s^-1
Gibbs-Thomson coefficient, Γ	3.65×10^-7	mK
Partition coefficient, k₀	0.788
Liquidus slope, m_L	-3.95	K wt%^-1
Liquidus temperature, T_eq	1672	K
Initial composition, C₀	37.0	wt%
Anisotropy coefficient, ε	0.02
Viscosity, μ	8.5×10^-3	kg m^-1 s^-1

Fig. 13. Various types of dendrite formation with respect to the processing conditions (cell size = 4 μm, domain size = 4 mm): (a-c) $\dot{T}$ = 1.5 K/s, (d-f) $\dot{T}$ =3.0 K/s.

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