Fig. 1. Equiaxed dendritic microstructure of Al-1 wt.% Cu alloy at a cooling rate Rc = 0.1 K/s with different grain numbers, N, respectively at solid fraction fS = 0.9 in the 2D (a1 to a4) and fS = 0.2 in 3D (b1 to b4) PF simulations. (a1) N = 36; (a2) N = 100; (a3) N = 225; (a4) N = 324; (b1) N = 8; (b2) N = 27; (b3) N = 64; (b4) N = 125. The colors represent the crystal orientations.

Fig. 2. The average solute concentration in liquid, CL (a), and the solid fraction, fS (b), in the 2D and 3D PF simulations of Al-1 wt.% Cu alloy at Rc = 0.1 K/s with different grain numbers. The predictions by the Scheil equation and Lever rule are also plotted.

Fig. 3. Variation of SI (a) and Cmax (b) with fS in the 2D and 3D PF simulations of Al-1 wt.% Cu alloy at Rc = 0.1 K/s. The 3D simulations are ended at a relatively low solid fraction because of the unbearable computing time when grains impinge.

Fig. 4. Solute distribution in the 2D PF simulations of Al-1 wt.% Cu alloy at Rc = 0.1 K/s with N = 36 and 324. (a and b) Variation of CL, CS, Cmax, and CLE against fS; (c and b) The solute profile along the lines shown in (e and f) at fS = 0.9. (e and f) show the corresponding local grain morphology in the simulations with N = 36 and 324, respectively. Blue is solid and red is residual liquid.

Fig. 5. Effects of the cooling rate on CL (a and b), CS (c and d), the undercooling (e and f), and equiaxed dendritic microstructure (g and h) in the 2D PF simulations of Al-1 wt.% Cu alloy with N = 36 (a, c, e, g1 to g4) and N = 324 (b, d, f, h1 to h4). fS in (g and h) is about 0.9, and the colors represent the crystal orientations. The cooling rate is 0.01 K/s (g1, h1), 0.05 K/s (g2, h2), 0.1 K/s (g3, h3) and 0.2 K/s (g4, h4).

Fig. 6. Effects of cooling rates on the solute segregation behavior in the 2D PF simulations of Al-1 wt.% Cu alloy. Cmax and SI at various cooling rates are shown in (a, b) and (c, d), respectively. The grain number in (a, c) is 36, while that in (b, d) is 324.

Fig. 7. Variation of the undercooling (a), Cmax (b), CL (c), and SI (b) against fS in the 2D PF simulations of Al-1 wt.% Cu alloy at Rc = 0.1 K/s and N = 36, with various DS.

Fig. 8. Comparison of fS predicted by two widely used microsegregation models (Lever rule and Scheil equation) and the 2D PF simulation of Al-1 wt.% Cu alloy at Rc = 0.1 K/s and N = 36.

Fig. 9. Plotting of q against X with various solute partition coefficients. (a) k = 0.14 (for Al-Cu alloy); (b) k = 0.3; (c) k = 0.6; (d) k = 0.9; (e) k = 1.5; (f) k = 2.0.

Fig. 10. Plotting of the fitting parameter, a, against k in Eq. (21). (a) k < 1; (b) k > 1. The inset figure in (a) shows the values of parameter a with k < 0.3, which could be well fitted by a linear function.

Fig. 11. (a) Comparison between CL predicted by Eq. (24) and classical models, as well as the result in the 2D PF simulation with Rc = 0.1 K/s and N = 36. (b) CL predicted by Eq. (24) with different choice of w(fS). SDAS in the simulation with Rc = 0.1 K/s and N = 36 is about 50-100 μm at fS = 0.4-0.9, and for the other cases with N = 100-324, the average grain size is 150-500 μm. Therefore, the parameter α = 4DStf/λ2 (as shown in Table 1), a constant related to SDAS or the grain size (in the case of globular grain with no developed sidebranches) in the Clyne-Kurz model is estimated to be 0.01-0.10. For the Won-Thomas model, the Fourier number is α+ = α + αC with αC = 0.1 [11].

Fig. 12. Comparison of SI in the 2D PF simulation at Rc = 0.1 K/s and N = 36 with predictions by Eq. (26) and the classical models (the Lever rule and Scheil equation).