Modelling Microsegregation of Binary Alloy During Solidification

doi:10.1007/s40195-025-01884-4

Abstract

Abstract:

This work studies the impact of the carbon diffusion on the growth kinetics of austenite and the solute segregation, by utilizing the phase-field (PF) method to simulate the solidification of a Fe-C binary alloy. It is revealed that increasing the ratio of the carbon diffusion coefficient in solid to that in liquid is advantageous in reducing the solute segregation, and a novel microsegregation model is developed based on the quantitative analysis of the results from PF simulations. The simplified one-dimensional diffusion simulation is employed to analyse the quantitative relationship between the parameters of the proposed microsegregation model and the properties of materials. The universality and reliability of the new microsegregation model are then validated by comparing with the experimental data of various alloy systems. These findings contribute to our comprehension of the fundamental theory of solidification and also provide a potential and promising approach to controlling the solidification microstructure.

Key words: Binary alloy, Solidification, Microsegregation, Phase-field method, Solid back diffusion

Tongzhao Gong, Shuting Cao, Weiye Hao, Weiqi Fan, Yun Chen, Xing-Qiu Chen, Dianzhong Li. Modelling Microsegregation of Binary Alloy During Solidification[J]. Acta Metallurgica Sinica (English Letters), 2025, 38(9): 1628-1636.

Figures/Tables 10

Fig. 1 Numerical convergence test of the interface mobility. a Tip growth rate and concentration. b Maximum of the tip growth rate and concentration. The size of the computational domain is 200 μm × 200 μm. The cooling rate is Rc = 10 K/s and there is no solute diffusion in solid (DS = 0)

Fig. 2 Numerical convergence test of the interval for updating quasi-equilibrium thermodynamic data. a Tip growth rate and concentration. b Maximum of the tip growth rate and concentration. The size of the computational domain is 200 μm × 200 μm. The cooling rate is Rc = 10 K/s and there is no solute diffusion in solid (DS = 0)

Fig. 3 Effects of solute diffusivity on solidification of the Fe-1.0 wt% C alloy. a Variation of the solid fraction (fS) with undercooling (∆T). b Variation of the average solute composition in liquid (CL,avg) with the solid fraction. Data of the equilibrium solidification and Scheil solidification calculated by the Thermo-Calc® (TC) software and TCFE8 thermodynamic database are also plotted. It should be noted that cementite is considered in the TC calculation, so the abrupt change occurred as the cementite began to precipitate, when the solid fraction was about 0.95

Fig. 4 Relationship between the fitting parameter and the ratio of diffusivities. a Model 1: the hyperbola function. b Model 2: the logistic function. c Model 3: the exponential associate function. d Model 4: the Gaussian function

Table 1 Values of the fitting parameters Φ

D_S (μm²/s)	D_L (μm²/s)	D_S/D_L	Φ	Fitting accuracy, R²
0	10,000	0	0.005803	0.9912
10	10,000	0.001	0.1208	0.9968
50	10,000	0.005	0.2947	0.9960
100	10,000	0.01	0.4136	0.9958
500	10,000	0.05	0.7206	0.9963
1000	10,000	0.1	0.8388	0.9974
5000	10,000	0.5	0.9940	0.9980
10,000	10,000	1	0.9999	0.9977

Fig. 5 Validation of the microsegregation model. a Final fitting relationship between the parameter Φ and the diffusivity ratio DS/DL. b Comparison of the average solute concentration in liquid, CL,avg, predicted by the proposed model and the PF simulations at different solid fractions (fS = 0.3, 0.6, and 0.9)

Fig. 6 Solute segregation in the 1D diffusion simulations of the Fe-1.0 wt% C alloy. a Variation of the solid fraction against the undercooling. b Variation of the average solute concentration in liquid against the solid fraction. c Solute distribution with the same solid fraction (fS = 0.9). d Solute concentration on the solid and liquid sides at the interface with the same solid fraction (fS = 0.9)

Fig. 7 Segregation model parameters derived from the 1D diffusion simulations. a-c Parameter Φ against the diffusivity ratio at different partition coefficients (k = 0.1−0.8) and liquidus slopes (m = − 500 to − 2000 K/at.). d Values of parameter χ

Table 2 Values of the fitting parameters, a and b, in the approximate power function between χ and k

Liquidus slope, m (K/at.)	a	b	Fitting accuracy, R²
− 500	1.840 × 10⁻³	− 1.100	0.9956
− 1000	0.956 × 10⁻³	− 1.087	0.9842
− 2000	0.447 × 10⁻³	− 1.122	0.9893

Fig. 8 Comparison of the proposed microsegregation model with the experimental data and the classical models. a Al-2 wt% Cu alloy [17]. b and c Mg-x wt% Al (x = 3 and 6) alloy [32]. d Fe-0.016 wt% P alloy [31]

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