Improved multi-order parameter and multi-component model of polycrystalline solidification

doi:10.1016/j.jmst.2021.06.017

Abstract

Abstract:

In this paper, we present an improved multi-order parameter model for multi-component model of polycrystalline solidification. We introduce an interpolation function in the phase field dynamical equation to obtain controllable grain boundary energy at large undercooling. The same interpolation function is also employed in the kinetics coefficient to allow for better control of grain boundary migration. Temperature dependent phase field parameters and noise terms are consistently coupled into the dynamics of a binary system in a manner that allows for quantitative simulations in the thin interface limit. The model is applied to multi-phase solidification in Al-Cu alloy, where a parabolic fitting method is employed to model the free energy of Al-Cu phases and two-phase nucleation is demonstrated in directional solidification.

Key words: Phase field model, Multi-phase, Multi-component, Grain boundary, Consistent nucleation

Laishan Yang, Zhibo Dong, Lei Wang, Nikolas Provatas. Improved multi-order parameter and multi-component model of polycrystalline solidification[J]. J. Mater. Sci. Technol., 2022, 101: 217-225.

Figures/Tables 8

Fig. 1. Experimental free energy (purple and dark purple) overlaid with fitted parabolic free energies (other colors) at T=821 K.

Fig. 2. Phase diagram of Al-Cu alloy:(a) generated from database of literature and (b) reproduced after free energy parabolic fitting described in the text. Only the left part of Al-Cu system is considered here.

Fig. 3. Dimensionless growth velocity of a two-dimensional dendrite in the Al-Cu alloy at 880K, for different choices of the coupling constant λα.

Fig. 4. Grain boundary energy prediction with ωαβ. The overlap function interaction parameter used is Λ = 1000.

Table 1 Simulation and alloy parameters used for grain boundary coalescence simulations [26].

Parameter	Value
Melting temperature of Al, T_m	933.47 K
Liquidus slope, m	-2.6 K/wt%
Partition coefficient, k	0.14
Liquid diffusion coefficient, D_L	3.0 × 10^-⁹ m²/s
Solid diffusion coefficient, D_s	3.0 × 10^-¹³ m²/s
Capillary length of L-α surface, d0α	2.4 × 10^-⁷ Km
Average concentration of Cu, c∞	2.0 wt%
Interface width, W_α	1.0 × 10^-⁷ m

Fig. 5. Homogeneous nucleation rate with chemical driving force. $\bar{\omega}_{sl}$ is the dimensionless grand potential difference between solid phase and liquid phase.

Fig. 6. Concentration distribution after two-phase nucleation occurs in directional solidification. Low concentration of Cu (blue) is α phase, high concentration of Cu (red) is θ phase and the left is liquid.

Table 2 Simulation and alloy parameters used for two-solid phase solidification simulations [26,33].

Parameter	Value
Temperature gradient, G	2.0 × 10⁵ K/m
Pulling speed, Vp	0.01 m/s
Average concentration of Cu, c∞	8.09 at%
Liquid diffusion coefficient, D_L	3.0 × 10^-⁹ m²/s
Solid diffusion coefficient, D_s	3.0 × 10^-¹³ m²/s
L-α interfacial energy, γαl	0.16340 J/m²
L-θ interfacial energy, γθl	0.08778 J/m²
α-α grain boundary energy, γαα	0.52288 J/m²
θ-θ grain boundary energy, γθθ	0.29090 J/m²
α-θ phase boundary energy, γαθ	0.40189 J/m²
Interaction coefficient, ωαβ	200
Interface width, Wα,β0	5.28 × 10^-⁸ m

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