domain: spi phys

text:
Tongzhao Gong

Yun Chen
email: chenyun@imr.ac.cn

Shanshan Li

Yanfei Cao

Dianzhong Li

Xing-Qiu Chen

Guillaume Reinhart

Henri Nguyen-Thi

Revisiting dynamics and models of microsegregation during polycrystalline solidification of binary alloy

Keywords: microsegregation, grain refinement, cooling rate, back diffusion, phasefield method

Microsegregation formed during solidification is of great importance to material properties. The conventional Lever rule and Scheil equation are widely used to predict solute segregation. However, these models always fail to predict the exact solute concentration at a high solid fraction because of theoretical assumptions. Here, the dynamics of microsegregation during polycrystalline solidification of refined Al-Cu alloy is studied via two-and three-dimensional quantitative phase-field simulations.

Simulations with different grain refinement level, cooling rate, and solid diffusion coefficient demonstrate that solute segregation at the end of solidification (i.e. when the solid fraction is close to unit) is not strongly correlated to the grain morphology and back diffusion. These independences are in accordance with the Scheil equation which only relates to the solid fraction, but the model predicts a much higher liquid concentration than simulations. Accordingly, based on the quantitative phase-field

Introduction

Microsegregation in as-cast alloys caused by solute partition during solidification is a composition variation at the grain scale (or within the grain), which influences the mechanical properties, corrosion resistance of cast products and formation of macrosegregation [START_REF] Schneider | A numerical study of the combined effects of microsegregation, mushy zone permeability and flow, caused by volume contraction and thermosolutal convection, on macrosegregation and eutectic formation in binary alloy solidification[END_REF][START_REF] Thevik | The influence of micro-scale solute diffusion and dendrite coarsening upon surface macrosegregation[END_REF]. Several microsegregation models with different assumptions and simplifications have been proposed to predict solute redistribution and related phenomena during binary alloy solidification (Table 1). The Lever rule [START_REF] Battle | Mathematical-modeling of solute segregation in solidifying materials[END_REF] describes equilibrium solidification with assuming complete solute diffusion in both liquid and solid phases, and the solute concentration in liquid, CL, is described as

( ) 0 L S 11
C C kf = -- (1)
where C0 is the nominal composition, k is the solute partition coefficient, and fS is the solid fraction. The Lever rule is simple to use but is usually inaccurate because solute diffusion in solid (back diffusion) is very slow for most metallic alloys. To take into account the non-equilibrium effects during alloy solidification, the Scheil equation [START_REF] Battle | Mathematical-modeling of solute segregation in solidifying materials[END_REF] can be employed to predict the solute concentration in liquid, which still assumes complete solute diffusion in liquid, but neglects diffusion in solid. The solute concentration in liquid is written as follows ( )

1 L 0 S 1 k C C f - =- (2)
According to Eq. ( 2) the Scheil equation is not adequate for predicting the solute concentration at the late solidification stage because CL → ∞ (for k < 1) or CL → 0 (for k > 1) as fS → 1. Moreover, the liquid concentrations predicted by these two simple models are independent of the microstructure details. In order to predict microsegregation more accurately, several models have been developed by considering the back diffusion, the size and morphology of grains, and coarsening [START_REF] Bower | Measurements of solute redistribution in dendritic solidification[END_REF][START_REF] Clyne | Solute redistribution during solidification with rapid solidstate diffusion[END_REF][START_REF] Ohnaka | Mathematical-analysis of solute redistribution during solidification with diffusion in solid-phase[END_REF][START_REF] Battle | Mathematical-modeling of microsegregation in binary metallic alloys[END_REF][START_REF] Voller | A semi-analytical model of microsegregation in a binary alloy[END_REF][START_REF] Voller | A semi-analytical model of microsegregation and coarsening in a binary alloy[END_REF][START_REF] Voller | A unified model of microsegregation and coarsening[END_REF][START_REF] Won | Simple model of microsegregation during solidification of steels[END_REF][START_REF] Xu | A unified microscale parameter approach to solidification-transport process-based macrosegregation modeling for dendritic solidification: Part II. Numerical example computations[END_REF]. The solute concentration in liquid predicted by those models could be reformulated in a generalized formula as follows ( )

1 1 L 0 S 11 k k C C k f - - = --   (3)
where Φ is a parameter linking to both the back diffusion and the grain morphology.

The expressions of parameter Φ in Eq. (3) of many previously developed microsegregation models are briefly overviewed in Table 1. Its value strongly depends on the solid diffusion coefficient (DS), local solidification time (tf), and the characteristic length of grains (λ), which are included into a parameter, α = 4DStf/λ 2 .

However, quantitative analysis in both experiments and numerical simulations has indicated that the effect of back diffusion on microsegregation is very weak for most of metallic alloys [START_REF] Bower | Measurements of solute redistribution in dendritic solidification[END_REF][START_REF] Ohno | Importance of microstructural evolution on prediction accuracy of microsegregation in Al-Cu and Fe-Mn alloys[END_REF], since the solute diffusion coefficient in solid is usually 3~4 orders of magnitude smaller than that in liquid. Another important issue is that the characteristic length of grains actually varies with the grain morphology, and therefore an accurate prediction using these derived models must access the dynamical evolution of the grain structure.

Table 1. The expressions of parameter Φ in Eq. ( 3) in different models.

Microsegregation model Parameter Φ References

Scheil equation Φ = 0 [START_REF] Battle | Mathematical-modeling of solute segregation in solidifying materials[END_REF] Lever rule Φ = 1 [START_REF] Battle | Mathematical-modeling of solute segregation in solidifying materials[END_REF] Brody-Flemings model Φ = 2α Regarding to the effect of grain morphology, it is well known that the solute distribution strongly relies on the solidification microstructure such as the grain size [START_REF] Glenn | The effect of grain refining on the microsegregation of aluminium-magnesium alloy 5182[END_REF][START_REF] Daloz | Microsegregation, macrosegregation and related phase transformations in TiAl alloys[END_REF] and the secondary dendritic arm spacing (SDAS) [START_REF] He | Microsegregation Formation in Al-Cu Alloy under Action of Steady Magnetic Field[END_REF], which depends on the process conditions (e.g. the growth rate of columnar grains during directional solidification [START_REF] Zhang | Characterization of the Microstructure Evolution and Microsegregation in a Ni-Based Superalloy under Super-High Thermal Gradient Directional Solidification[END_REF] and the cooling rate during equiaxed solidification [START_REF] Eskin | Experimental study of structure formation in binary Al-Cu alloys at different cooling rates[END_REF][START_REF] Paliwal | The evolution of the growth morphology in Mg-Al alloys depending on the cooling rate during solidification[END_REF]).

Accordingly, the grain refinement is conventionally viewed as a method to reduce solute segregation [START_REF] Flemings | Behavior of metal alloys in the semisolid state[END_REF]. However, this correlation is difficult to be verified directly by traditional experimental methods, since it is influenced by many factors such as melt flow, solidification history and diffusion process after solidification, etc. More significantly, measuring the solute distribution within a grain precisely is a real challenge for 3D dendrites with very complicated shape and branching. The traditional experimental characterization of solute concentration on a sectioned surface from a quenched sample is very likely to be not conserved since the solute transport occurs in 3D space during solidification. The obtained solute concentration that varies with solid fraction may mislead our understanding of the actual solute segregation dynamics. The developed numerical simulation techniques, like the cellular automaton (CA) method [START_REF] Gandin | A 3D Cellular Automaton algorithm for the prediction of dendritic grain growth[END_REF][START_REF] Gandin | A three-dimensional cellular automaton-finite element model for the prediction of solidification grain structures[END_REF][START_REF] Zhang | A three-dimensional cellular automaton model for dendritic growth in multi-component alloys[END_REF] and the phase-field (PF) [START_REF] Karma | Phase-field formulation for quantitative modeling of alloy solidification[END_REF][START_REF] Pan | A phase-field study on the peritectic phase transition in Fe-C alloys[END_REF][START_REF] Pan | Peritectic transformation with non-linear solute distribution in all three phases: Analytical solution, phase-field modeling and experiment comparison[END_REF]

The phase-field model and numerical implementations

The quantitative PF model of equiaxed dendritic growth of a binary alloy

The quantitative PF model presented in Ref. [START_REF] Echebarria | Quantitative phase-field model of alloy solidification[END_REF] is adopted to model slow solidification of binary alloys under continuous cooling condition, and the quantitative capability of this model to reproduce the time evolution of microstructure during alloy solidification has been demonstrated in many previous studies [START_REF] Chen | Quantitatively comparing phase-field modeling with direct real time observation by synchrotron X-ray radiography of the initial transient during directional solidification of an Al-Cu alloy[END_REF][START_REF] Chen | Tip-splitting instability and transition to seaweed growth during alloy solidification in anisotropically preferred growth direction[END_REF][START_REF] Clarke | Microstructure selection in thin-sample directional solidification of an Al-Cu alloy: In situ X-ray imaging and phase-field simulations[END_REF][START_REF] Boukellal | Scaling laws governing the growth and interaction of equiaxed Al-Cu dendrites: A study combining experiments with phase-field simulations[END_REF]. The governing equations of the phase field ϕ and the rescaled solute concentration U are respectively given as

( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 c ,, 0 2 3 2 c 0 1 1 1 i i x y z i W Rt WW m C t Rt U k m C          =      + =    +             + -- - -  -   n n n n (4) ( ) ( ) ( ) ( )
0 11 11 1 22 22 11 2 W k U kk U DU tt kU t        +-  + --  -     =    +       +-  +  (5)
with

( ) ( ) ( ) ( ) ( ) 0 2 / 1 1 1 1 1 C C k k U k k k   -+ --   = - + --   (6)
where C is the local solute concentration, C0 is the initial concentration, k is the solute partition coefficient, Rc is the cooling rate, and m is the liquidus slope. The solid and liquid phases are represented by ϕ = +1 and ϕ = -1, respectively. The spatial length and time are rescaled respectively by the interface width W0 and relaxation time τ0, and therefore the dimensionless value for the solute diffusion coefficient in liquid is 00 12 2 00 DW D a a Wd  == [START_REF] Battle | Mathematical-modeling of microsegregation in binary metallic alloys[END_REF] where d0 = Γ/[|m|C0(1-k)] is the chemical capillary length with Γ the Gibbs-Thomson coefficient, a1 = 5 2 / 8 and a2 = 47/75 [START_REF] Echebarria | Quantitative phase-field model of alloy solidification[END_REF]. The coupling coefficient λ in Eq. ( 4) is given as λ = a1W0/d0. The standard form of the fourfold symmetry of the surface energy anisotropy is considered, and thus W(n) = W0as(n) and τ(n) = τ0as(n) 2 where ( ) ( )

4 s 4 4 4 ,, 1 3
4 i i x y z a    =  = - +   n (8)
In order to expedite large-scale quantitative PF simulations, a nonlinear preconditioning [START_REF] Glasner | Nonlinear Preconditioning for Diffuse Interfaces[END_REF] of the phase field ϕ is employed to allow coarser grids to be used in the interface layer. The transformed phase field ψ is given as

tanh 2    =   (9)
And hence the standard PF model (Eqs. ( 4) and ( 5)) is transformed into the nonlinearly preconditioned formula, as follows

( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 c 0 s 0 s 0 2 s 2 0s ,, 2 c 0 1 2 tanh 2 2 tanh 2 2 1 tanh 1 2 i i x y z i Rt a W a m C t a Wa Rt U k m C          =     + =  -             +   +         - - -    -     nn n n (10)   2 0 2 (1 ) (1 ) tanh 2 2 1 tanh 1 tanh 1 22 1 (1 ) 2 2 2 1 tanh 1 (1 ) 2 2 2 kk U t D U W k U t kU t       + --   --  =    + + -  - + -  +                           (11)
The numerical accuracy and efficiency of the preconditioned PF formula has been examined carefully [START_REF] Gong | Fast simulations of a large number of crystals growth in centimeter-scale during alloy solidification via nonlinearly preconditioned quantitative phase-field formula[END_REF]. The detailed numerical and material parameters can be referred to our previous study [START_REF] Chen | Quantitative Phase-field Simulation of Dendritic Equiaxed Growth and Comparison with in Situ Observation on Al-4 wt.% Cu Alloy by Means of Synchrotron X-ray Radiography[END_REF], but C0 is set to be 1 wt.% to result in a larger capillary length. Large-scale simulations can be therefore realized quantitatively in a relatively wide range of cooling rates. The interface width parameter, ξ = W0/d0, is determined according to the convergence tests (Fig. S1 in the supplementary materials) to guarantee the results of simulations being quantitative.

The numerical implementations

The parallel adaptive finite element method, with multiple processors using distributed memory based on the MPI protocol, is employed to solve the nonlinearly preconditioned PF model. Numerical implementations are realized by the open source deal.II library [START_REF] Bangerth | The deal.II Library, Version 8.4[END_REF], and the adaptive mesh refinement/coarsening is achieved through a simple but efficient approach [START_REF] Gong | Fast simulations of a large number of crystals growth in centimeter-scale during alloy solidification via nonlinearly preconditioned quantitative phase-field formula[END_REF]. The front-tracking method [START_REF] Gong | Fast simulations of a large number of crystals growth in centimeter-scale during alloy solidification via nonlinearly preconditioned quantitative phase-field formula[END_REF] is used to capture crystallographic orientations to improve the computing efficiency dramatically. The computational domain size is 5 × 5 mm 2 and 1 × 1 × 1 mm 3 , respectively in 2D and 3D simulations. Grains nucleate simultaneously at the beginning of simulations, with the number of grains 36~324 in the 2D simulations and 8~125 in the much more timeconsuming 3D simulations. All the simulations were carried out on a supercomputer with an Intel ® Xeon E5-2683 v4 CPU (2.10 GHz and 32 cores) and 256 GB memory at each computing node. For the 2D simulation with the lowest cooling rate (0.01 K/s) and the largest grain number (324), it took about 14 days using 96 cores to finish the computation until the solid fraction fS = 0.99, and most of the regions which were not solidified formed grain boundaries eventually. Here, totally solidified zones are defined as ψ ≥ 0. However, the 3D simulations were ended at a relatively low solid fraction (fS = 0.25) because of the unbearable computing time when grains impinged. The equiaxed dendritic microstructure of refined Al-1 wt.% Cu alloy during isothermal solidification with a constant cooling rate Rc = 0.1 K/s in the 2D and 3D PF simulations is shown in Fig. 1 for various imposed number of grains. As the number of grains increases (by artificially adding more nucleation sites in simulations), the grain size becomes smaller and smaller, and the equiaxed crystal shape evolves from highly branched dendrite to nearly uniform globular, with the secondary arms less developed and even disappearing. Changes in the grain size and morphology can be attributed to the solute interaction between neighboring grains, which suppresses the growth of primary arms and the development of sidebranches, therefore bringing out a much denser equiaxed dendritic microstructure. The solute interaction between grains has been well characterized through the in situ and real-time observation of Al-Cu solidification experiments performed by Bogno et al. [START_REF] Bogno | Growth and interaction of dendritic equiaxed grains: In situ characterization by synchrotron X-ray radiography[END_REF]. They have found that equiaxed dendritic growth experiences two regimes, namely the free growth and the impinged growth. Impingement causes grain growth being slowed down or even stopped. The impinged growth took most of the computing time to complete solidification in current simulations, particularly for 3D computations. The average liquid concentration, CL, in both 2D and 3D simulations is plotted in Fig. 2(a). In general, the simulated CL in both 2D and 3D with different number of grains almost fall in one curve. Apart from a slight difference that branched dendrites will cause a slightly lower CL, the evolution of CL shows no strong dependence on the grain morphology, for either dendrites with developed branches or nearly globular grains with smooth interface. Furthermore, CL in the PF simulations, as expected, falls in between the predictions by the Scheil equation and Lever rule. At the early solidification stage (fS < 0.55) both analytical models predict CL identical to the simulations, indicating that the grain morphology is not a crucial factor to affect CL at a low solid fraction. As solidification proceeds, the grain impingement takes place (characterized by the slow-down increasing rate of fS in Fig. 2(b) at t ~ 30 s) and the analytical predictions deviate gradually from the PF simulations.

Dynamic evolution of microsegregation in polycrystalline alloy

Effects of the grain refinement on the solute segregation

Fig. 2(b)

shows the variation of solid fraction as a function of undercooling. Like the average solute concentration in liquid, the solid fraction has no significant correlation to the number of grains (or the grain morphology) too, and fS in the PF simulations is close to the predictions by the Scheil equation and Lever rule as fS < 0.55. However, as fS increases furtherly, it evolves gradually in between the predictions by these two microsegregation models. It is worthy noticing that, no visible difference in CL can be found here between 2D and 3D simulations (Fig. 2 (a)). Even though it has been quantitatively convinced that the crystal growth kinetics and solute transport are different in 2D and 3D PF simulations [START_REF] Gong | Quantitative comparison of dendritic growth under forced flow between 2D and 3D phase-field simulation[END_REF], the difference in solute transport in 2D planar and 3D cubic space could be offset because solute is rejected from solid into liquid through a curve in 2D while a surface in 3D. Unlike the traditional experimental measurement of solute distribution on a sectioned 2D surface where the conservation of measured solute cannot be ensured, the 2D simulations here like the 3D simulation always guarantee the solute conservation during the entire solidification process, and are therefore feasible to reveal the dynamic evolution of microsegregation.

Prediction for CL only gives the average solute concentration in the rest of liquid but does not describe the spatial distribution of solute. The segregation extent can be quantitatively evaluated using the solute segregation index, 0 I 0

1 i V CC S dV VC - = ò
with Ci the solute concentration at the ith quadrature point (in both solid and liquid phases in the computational domain of PF simulations) and V the total domain volume.

SI should be more appropriate to evaluate the segregation extent than the mean solid or liquid solute concentration, as SI spatially measures the deviation of the solute concentration from C0. The results in 2D and 3D PF simulations are plotted in Fig. 3(a).

Again, no apparent difference can be found for SI between 2D and 3D simulations.

During the early solidification stage, an increasing amount of solute is rejected into liquid and piles up ahead of the solid-liquid interface, thus leading to an increment in SI. As solidification proceeds, the difference from C0 for CL is larger and larger, while for the average solute concentration in solid, CS, becomes smaller and smaller. As a result, SI increases to a peak value at fS  0.8, and then drops down with fS → 1 because the area of solid weights more and more in the SI calculation. When fS is close to 1, though CL departs from C0 very obviously (Fig. 2(a)), the liquid fraction is so small that the solute distribution in solid plays a dominant role in determining SI. At the late solidification stage, the solute interaction between grains is going to suppress the growth of crystals, but CL continues increasing, which could be attributed to the fact that the crystal growth does not stop entirely when the solid fraction is high and the diffusion time is sufficient to homogenize the composition in the sample. The variation trend of SI observed in Fig. 3(a) is independent of the grain number or morphology, but the peak value of SI is apparently affected by the grain refinement. Interestingly, less grains with developed branches lead to a lower peak value of segregation index, which is consistent with the experimental study of He et al. [START_REF] He | Microsegregation Formation in Al-Cu Alloy under Action of Steady Magnetic Field[END_REF] who have indicated that the decreases in SDAS would reduce the microsegregation. However, values of SI for different number of grains almost overlap as fS → 1 (Fig. 3(a)). Again, it is revealed that the segregation extent after complete solidification has no strong correlation to the grain refinement. Another parameter quantifying the segregation is the maximum solute concentration in liquid, Cmax (Fig. 3(b)). For interdendritic segregation, Cmax locates at the root of dendritic branches and can be approximated by the liquidus equilibrium concentration in the phase diagram, CL E = C0 -ΔT/m with ΔT the applied undercooling.

Owing to the curvature effect of the concave solid-liquid interface, Cmax is in fact slightly larger than the equilibrium concentration CL E obtained from the phase diagram.

However, this difference is invisible in PF simulations as shown by Fig. 4(a to d) and the approximation Cmax ≈ CL E sounds reasonable. Another issue is that the grain shape should lead to a slight difference in Cmax, owing to the different CL E arising from the different undercooling required to approach the same solid fraction (Fig. 2 Nevertheless, the apparent enlargement of Cmax for N = 36 in the 2D PF simulation at a low solid fraction (fS < 0.5) is due to the higher undercooling (or solidification time) to reach the same fS (Fig. 2(b)). It means that liquid is solidified at a higher undercooling for the same solid fraction when grain number is reduced, thereby gaining a higher equilibrium concentration and thus a larger Cmax. However, as fS > 0.5, Cmax for N = 36 evolves into a slightly lower value than other cases owing to a shorter time taken to get the same fS (Fig. 2(b)). The slight difference in Cmax with various grain numbers can be seen more apparently in Fig. 4(c to f). At the same solid fraction (fS = 0.9), because of the lower undercooling for N = 36 (Fig. 2(b)), the solute concentration near the interface exhibits a smaller value than that for N = 324. To sum up, when the grain number is smaller (N = 36), the melt solidifies slower at a low solid fraction (fS < 0.5). In contrast, at the late solidification stage smaller number of grains solidifies a little faster because grains impinge with each other not so terribly as that for high numbers of grains, thus leading to a relatively lower CL E and Cmax, as well as smaller SI. Obviously, curves of CL with fS is always below CL E in Fig. 4(a andb).

Effects of the cooling rate on the solute segregation

Rapid solidification has been always considered as an efficient method to reduce segregation. Increasing the cooling rate not only shortens the solute transport time but also refines grains. To clarify the effect of cooling rate on microsegregation, here the solidification of Al-1 wt.% Cu alloy with different cooling rates are simulated and the results are shown in Fig. 5. Please notice that with a large cooling rate (e. g. ~1.0 K/s) the quantitative results in the present 2D PF simulations are not able to be ensured even though W0 has been reduced to 0.4 μm, in which the computational domain size and computing time will be augmented dramatically, making the large-scale quantitative PF simulations extremely difficult. Therefore, the maximum cooling rate in our PF simulation is limited to Rc = 0.2 K/s, at which the results can still be converged with the given available W0. Based on the limited range of Rc, the following discussions about the effects of cooling rates on microsegregation are restricted to the slow solidification regime where the local equilibrium at the solid-liquid interface is satisfied. Fig. 5 shows that when grain number is small (N = 36), a higher cooling rate causes a slightly lower CL, whereas an opposite behavior is observed when grain number is large (N = 324).

These opposite effects should be ascribed to the difference in grain morphology. For N = 36 (Fig. 5(g)), as the cooling rate increases the sidebranches have been well developed and SDAS is shortened. For N = 324 (Fig. 5(h)), the cooling rate has no obvious effect on grain morphology due to the grain impingement occurs very early, and the longer diffusion time at a lower cooling rate will thereby lead to a better solute homogenization in liquid. Regardless of the cooling rate or grain refinement, it can be drawn that higher undercooling required to reach the same fS results in higher CL (Fig. 4(c,d) and Fig. 5(a, b, e, andf)). However, the effect of cooling rate on CL is not apparent overall in all these cases and they all evolve along close curves that fall in between the predictions by the Scheil equation and Lever rule.

The effect of cooling rate on the dynamics evolution of Cmax and SI is shown in Fig.

6.

Increasing the cooling rate results in slightly larger Cmax when fS < 0.55 (Fig. 6(a,b)). This variation is in agreement with the fact that Cmax strongly depends on the undercooling of the melt so that it varies in the same manner as undercooling shown in Fig. 5(e,f). For fS > 0.55, no significant difference can be found except for the case with N = 324 and Rc = 0.01 K/s, which may be attributed to the well mixing solute in the liquid arising from the narrow spacing between grains and sufficient diffusion time.

As the cooling rate increases, SI is reduced in the simulation with a small grain number (N = 36 in Fig. 6(c)). Lower cooling rate causes a higher peak value of SI due to the increase in CL (Fig. 5(a)) and SDAS [START_REF] He | Microsegregation Formation in Al-Cu Alloy under Action of Steady Magnetic Field[END_REF] (Fig. 5(g andh)). However, analogous to CL, the influence of cooling rate on SI shows an opposite trend for a large grain number (N = 324 in Fig. 6(d)) with almost globular crystal shape (Fig. 5(h)). SI at the late solidification stage will be enlarged with the cooling rate increasing for N = 324 (Fig.

6(d))

and faster cooling causes a higher peak value. andh) 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).

These different evolution behaviors can be ascribed to that when the grain number is large enough to make the crystal shape almost globular, the sidebranches will not be well developed owing to the strong solute interaction between neighboring grains. In this case, the solute diffusion in liquid plays a dominant role to reduce SI, and the time for solute homogenization by diffusion at a slower cooling rate is longer, thus leading to a decline in SI. In contrast, with sufficient space to develop sidebranches, the grain morphology is the main factor affecting the segregation extent. A higher cooling rate promotes the sidebranch formation, thus leading to a lower SI arising from smaller CL and SDAS [START_REF] He | Microsegregation Formation in Al-Cu Alloy under Action of Steady Magnetic Field[END_REF]. Interestingly, similar to the effect of grain number, no matter what extent of the influence by the cooling rate exists during solidification, the curves of SI eventually drop down to a roughly same value when solidification nearly completes. This is because CS that occupies most of the computational domain dominates the calculation of SI at the late solidification stage. As solute should be kept conserved and meet the local equilibrium condition during polycrystalline solidification, CS approaches to C0 at the end of solidification for different conditions (Fig. 5(c,d)), leading to the coincidence feature of SI. and(c, d), respectively. The grain number in (a, c) is 36, while that in (b, d) is 324.

Effects of back diffusion on solute segregation

The back diffusion is commonly considered as an important factor that influences the solute microsegregation behavior. However, for most of metallic alloys, the effect of back diffusion on microsegregation is negligible since the solute diffusion coefficient in solid is usually 3~4 orders of magnitude smaller than that in liquid (e. g. DS/DL ≈ 10 -3 for Al-Cu alloys). Here, to ascertain how back diffusion influences microsegregation, the 2D PF simulations at the cooling rate Rc = 0.1 K/s and grain number N = 36 are performed with various solid diffusion coefficients (by artificially changing DS), and the simulated results including CL, Cmax, and SI are shown in Fig. 7.

No apparent differences in all these quantities are observed for fS < 0.6. However, as the solid fraction increases furtherly, it will take a relatively shorter time to reach the same fS with a larger DS (Fig. 7(a)), which causes a lower equilibrium concentration, CL E . As Cmax is equal to CL E approximately, Cmax will also be reduced with a larger DS (Fig. 7(b)), thus leading to a lower average solute concentration, CL (Fig.

7(c))

.Therefore, the solute segregation index, SI, will be lowered as DS increases (Fig.

7(d)).

The microsegregation behavior shown in Fig. 7 is still consistent with the experimental study by He et al. [START_REF] He | Microsegregation Formation in Al-Cu Alloy under Action of Steady Magnetic Field[END_REF], who have found that decreasing solid diffusion coefficient would give rise to the increase in microsegregation. However, the differences between the curves with no back diffusion (DS = 0) and a small DS (10 -3 DL) is not pronounced, which means that the back diffusion is not a crucial factor that influences microsegregation for most of the metallic alloys with a DS several orders of magnitude smaller than DL. Furthermore, it is also worth noticing that CL in the simulation with DS/DL = 10 -1 is very close to the prediction by the Lever rule, which demonstrated that the assumption of complete diffusion in both solid and liquid 21 employed in the Lever rule is well satisfied with such a large DS.

A new microsegregation model for binary alloys

In terms of above simulations and analysis, it has been confirmed that for relatively slow solidification rather than rapid solidification, the grain morphology, cooling rate, and back diffusion (if not relatively high), have some slight influences on microsegregation but not the crucial factors to change CL and the final segregation for substitutional alloys. The liquid concentration in simulations is between the predictions by the Scheil equation and Lever rule, which are two limiting cases for the real nonequilibrium solidification. As already recognized, these two models fail to predict the solute redistribution at the late solidification stage when the solid fraction is close to unit, in particular for Scheil equation which gives a highly unrealistic value (k < 1). To overcome this drawback, it is necessary to develop a new model to gain a more accurate prediction for CL. According to the Lever rule, the solute concentration in liquid during binary alloys solidification is given by Eq. ( 1). As shown in Fig. 2(a), CL Lever is in a good agreement with the PF simulations at the early solidification stage with fS < 0.55.

However, at the late solidification stage as fS → 1, CL is much larger than CL Lever and close to the maximum solute concentration in liquid, Cmax, owing to that the solute concentration in the very little residual liquid is nearly uniform. It should be noticed that although the Scheil equation describes the limiting case with considering nonequilibrium effects during solidification, the predicted CL is unrealistic at a high solid fraction, and hence Cmax is a better choice of the upper limit for CL. Therefore, CL could be interpolated between CL Lever and Cmax through a function of fS, as follows ( ) ( )

Lever L S L S max 1 C w f C w f C = - +   (12)
where w(fS) satisfies ( ) ( )

SS SS 0, 0 1, 1 f w f f w f == == (13)
which reflects that CL changes gradually from the predicted value by the Lever rule (the lower limit) to Cmax (the upper limit). Even though the geometric shape of the solidliquid interface leads to Cmax deviating from CL E (the equilibrium liquidus concentration at a given undercooling for a flat interface), Cmax still could be approximated by CL E , as

follows E max L 0 T C C C m   = - (14)
with m < 0 for k < 1. This approximation has been proved by Fig. 4(c,d) which shows that Cmax at the interdendritic region is very close to CL E for both developed dendrites and globular grains. Please notice that Cmax is larger than C0/k when solidification nearly completes because the temperature reached at the end of solidification is lower than the solidus temperature. Eqs. ( 1) and ( 14) are substituted into Eq. ( 12), yielding

( ) ( ) ( ) 0 L S S 0 S 1 11 C T C w f w f C k f m   = - + -    --  (15)
This relation of CL should be true for any binary alloys. But ΔT in Eq. ( 15) is also an unknown variable which makes the relationship between CL and fS implicit. According to the Lever rule and Scheil equation, the solid fraction could be expressed respectively as ( )

Lever S 0 1 T f k T k T  =  + - (16)
where ΔT0 = mC0(1 -1/k) is the solidification range. Interestingly, as shown in Fig. 8, the simulated solid fraction could be expressed as the mean of fS Scheil and fS Lever , namely ( ) Eq. ( 18) again arises from the fact that the rule and the Scheil equation describe two limiting cases and the real non-equilibrium solidification is in between the predictions by these two simple models. Using Eq. ( 18), an explicit relationship between fS and ΔT can be obtained naturally. However, the exact solution of Eq. ( 18) is difficult to be found because it is a high-order power equation. One way to solve the complicated Eq. ( 18) is to simplify it into a simple equation that can be solved easily.

( )( ) 1
Here three intermediate variables are defined as

0 S 1 1 1 1 21 X T mC p k qf   -  - - (19)
where X and q would be considered as the rescaled undercooling and solid fraction, respectively. Then Eq. ( 18) could be transformed into p q X pX p = --+ [START_REF] Flemings | Behavior of metal alloys in the semisolid state[END_REF] with 0 < X < 1 (for k < 1 and m < 0), X > 1 (for k > 1 and m > 0), and -1 ≤ q ≤ 1. It should be noticed that Eq. ( 20) is robust for different substitutional alloys with various k and m, because it is only a transformed from of Eq. ( 18) that depicts the relationship between the solid fraction and undercooling during solidification and no restriction of k and m is assumed in the derivation of Eq. [START_REF] Flemings | Behavior of metal alloys in the semisolid state[END_REF]. It is still quite difficult to solve Eq. [START_REF] Flemings | Behavior of metal alloys in the semisolid state[END_REF] analytically, but as shown in Fig. 9, a linear approximation between q and X could be found for different k as

q aX b =+ (21)
with a and b being two fitting parameters. It is easily to found that when X = 1 (ΔT = 0), q = a + b = -1, thus leading to that b = -(a + 1). The values of the fitting parameter a for different k are given in Fig. 10 and Table. S1 in the Supplementary Materials.

Since X and q are independent of k but Eq. ( 20) is equivalent to Eq. ( 21) which also relates to k, the parameter a must be a function of k. Indeed, from Fig. 10, a is proportional to 1/(k -1) which reflects an inverse relationship to the solubility of solute in solid. A higher solubility means a smaller parameter a and thus a lower CL. With a small solute partition coefficient (k < 0.3), an approximate linear relationship between a and k is found. For a larger k, the fitting parameter a could be obtained directly from Fig. 10 and Table. S1, or by solving Eq. ( 20) numerically using Newton iteration method or other methods. Numerical practice shows that the Newton iteration method converges to the solution very fast after only a few iterations.

Combining the simplified linear Eq. ( 21) and the fitted parameter a in Fig. 10 or Table. S1, an explicit relationship between CL and fS can be found readily. The variable X in Eq. ( 21) could be solved as

1 1 q X a + = + (22)
Replacing X in Eq. ( 19), the following equation could be obtained as

S 0 S 2 2 f T mC af = + (23)
Then CL in Eq. ( 15) could be expressed as a function of only three independent variables (C0, fS, and k) and an extra fitting parameter a (dependent on k), which is given as

( ) ( ) ( ) S L 0 S SS 1 1 1 2 wf a C C w f k f a f  - =+  -- +  (24)
CL predicted by Eq. ( 24) with w(fS) = fS is plotted in Fig. 11(a), and a much better agreement with the simulation has been achieved, compared with the Lever rule and Scheil equation (as well as the Clyne-Kurz model and the Won-Thomas model considering the back diffusion), particularly at the late solidification stage with a high solid fraction. Moreover, it could be seen from Fig. 11(b) that the choice of the interpolation function w(fS) has no apparent influence on CL predicted by Eq. ( 24), so it is reasonable to set a simple form of w(fS) = fS as shown in Fig. 11(a). Therefore, the final analytical model of CL is given as

( ) SS L0 SS 1 1 1 2 f af CC k f a f  - =+  -- +  (25)
where the value of a is given in Fig. 10 andTable. S1 for k in the range of [0, 1) and [START_REF] Schneider | A numerical study of the combined effects of microsegregation, mushy zone permeability and flow, caused by volume contraction and thermosolutal convection, on macrosegregation and eutectic formation in binary alloy solidification[END_REF][START_REF] Voller | A unified model of microsegregation and coarsening[END_REF] which covers most binary alloy systems. As the concentration in solid can be obtained by CS = kCL, based on the definition of the solute segregation index, SI also can be calculated accordingly as

S L I L S 00 S L I L S 00 1 1 (for 1) 1 1 (for 1) C C S f f k CC C C S f f k CC     = - + -              = - + -          (26)
with fL the liquid fraction. The predicted SI by Eq. ( 26) is plotted in Fig. 12, varying in the same manner as that in the PF simulation and showing a much better agreement with simulation than the Lever rule and Scheil equation before SI reaches the peak value as fS < 0.73.

Accordingly, using Eqs. ( 25) and ( 26) one can easily calculate the solid fraction at which the microsegregation is most severe for a given binary alloy. The proposed microsegregation model is easily applied to a different alloy. Once the solute partition coefficient is known, the parameter a could be obtained through Fig. 10 or Table. S1, and then CL could be calculated directly by Eq. ( 25). However, it needs to be emphasized that the proposed model in the present work is still limited into the solidification where the local equilibrium at the solid-liquid interface is satisfied, which is consistent with the Scheil equation. 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/λ 2 (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 + α C with α C = 0.1 [START_REF] Won | Simple model of microsegregation during solidification of steels[END_REF]. 26) and the classical models (the Lever rule and Scheil equation).

Conclusions

Microsegregation behavior during polycrystalline equiaxed solidification of Al-Cu alloy is studied through quantitative phase-field simulations. With solidification proceeding, although more and more solute is rejected into melt and the average solute concentration in liquid increases gradually, the solute segregation index that reflects solute distribution homogeneity in both liquid and solid, evolves in an anomalous manner as first increasing to a certain peak value and then decreasing. Interestingly, the segregation index when liquid solidifies completely is almost the same for all cases,

showing weak dependence on the grain refinement and cooling rate. This evolution behavior gives rise to the guidance of the best solid fraction for implementing rapid cooling or semisolid processing to reduce segregation. The grain size or morphology, and the cooling rate slightly influence the solute concentration in liquid, but they are not the crucial factors to change the microsegregation for the relatively slow solidification. For most of metallic alloys with substitutional solute elements, the effect of back diffusion on microsegregation is also not significant.

Moreover, it is revealed through PF simulations that the average solute concentration in liquid is between the Lever rule prediction and the maximum concentration in liquid that strongly relies on the applied undercooling. PF simulations also disclose that the solid fraction could be well estimated as the average of predictions by the Lever rule and Scheil equation with appropriate solute back diffusion in solid.

  model Φ = 2α + /[(1 -feut) 2 + 2α + ] [9, 10] Won-Thomas model Φ = 2α + [1 -exp(-1/α + )]exp[-1/(2α + )] [11]

Fig. 1 .

 1 Fig. 1. Equiaxed dendritic microstructure of Al-1 wt.% Cu alloy at a cooling rate Rc =

Fig. 2 .

 2 Fig. 2. The average solute concentration in liquid, CL (a), and the solid fraction, fS (b),

Fig. 3 .

 3 Fig. 3. Variation of SI (a) and Cmax (b) with fS in the 2D and 3D PF simulations of Al-1

Fig. 4 .

 4 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 CL E against fS; (c and

Fig. 5 .

 5 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

Fig. 6 .

 6 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.

1 Fig. 7 .

 17 Fig. 7. Variation of the undercooling (a), Cmax (b), CL (c), and SI (b) against fS in the 2D

2 Fig. 8 .

 28 Fig. 8. Comparison of fS predicted by two widely used microsegregation models (Lever

Fig. 9 .

 9 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 .f 3 SFig. 11 .

 10311 Fig. 10. Plotting of the fitting parameter, a, against k in Eq. (21). (a) k < 1; (b) k > 1.

Fig. 12 .

 12 Fig. 12. Comparison of SI in the 2D PF simulation at Rc = 0.1 K/s and N = 36 with

  Based on these findings, a new microsegregation model is deduced to accurately predict the solute concentration, particularly at the late solidification stage with a high solid fraction. The model is also as simple and easy-to-use as the Lever rule and Scheil equation, only relating to the sample nominal composition, solid fraction, partition coefficient, and a fitting parameter (also relying on the partition coefficient). The segregation behavior and analytical model presented in this work would provide a new insight into the control of solute segregation in alloys during relatively slow solidification, and help to more accurately calculate phases and transformation kinetics of substitutional alloys by the CALPHAD method, as well as being in favor of simulations of macrosegregation in castings and ingots. For rapid solidification with solute trapping at the interface, same as the Lever rule or Scheil equation, the proposed model fails to predict the liquid concentration. Further improvement should be considered in the present model for the metallic alloys with interstitial solute elements which have large solute diffusion coefficients in solid. In these alloys, such as carbon steels, the back diffusion plays an important role in microsegregation.

Acknowledgement

This work was supported by the Science Challenge Project (Grant No. TZ2016004), the Youth Innovation Promotion Association CAS, the Youth Talent program of Shenyang National Laboratory for Materials Science (L2019F08), the Special Scientific Projects of Inner Mongolia, and French National Space Center (CNES), Convention N°174800/00.