Importance of microstructural evolution on prediction accuracy of microsegregation in Al-Cu and Fe-Mn alloys

International Journal of Heat and Mass Transfer 132 (2019) 1004–1017

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Review

Importance of microstructural evolution on prediction accuracy of microsegregation in Al-Cu and Fe-Mn alloys Munekazu Ohno a,⇑, Masayoshi Yamashita b, Kiyotaka Matsuura a a b

Division of Materials Science and Engineering, Faculty of Engineering, Hokkaido University, Kita 13 Nishi 8, Kita-ku, Sapporo, Hokkaido 060-8628, Japan Graduate School of Engineering, Hokkaido University, Japan

a r t i c l e

i n f o

Article history: Received 28 September 2018 Received in revised form 14 November 2018 Accepted 10 December 2018

Keywords: Microsegregation Solidification Phase-field model Dendrite Binary alloy

a b s t r a c t Microsegregation in Al-Cu and Fe-Mn alloys is analyzed by conducting solidification experiments, twodimensional (2D) and three-dimensional (3D) quantitative phase-field simulations for two-sided asymmetric diffusion, and using the one-dimensional (1D) finite difference method (FDM). Comparisons of these results substantiate that the 3D quantitative phase-field simulation can predict microsegregation behaviors with a high degree of accuracy. The disagreement between the results of the 1D-FDM simulation and the experiment, which is similar to that often observed in early works, can be ascribed to the fact that 1D-FDM simulation does not consider the details of microstructural evolution. The main factor causing disagreements in microsegregation among the results of 1D, 2D and 3D simulations is discussed in terms of the solid-liquid interfacial area density and its relationship with overlapping of the solute diffusion layer. Ó 2018 Elsevier Ltd. All rights reserved.

Contents 1. 2.

3.

4.

5.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Al-Cu alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Fe-Mn alloy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Quantitative phase-field model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. One-dimensional finite difference method (1D-FDM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Comparison between the experimental and numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conflict of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Supplementary material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1004 1005 1005 1006 1006 1006 1007 1008 1008 1009 1010 1015 1015 1016 1016 1016

1. Introduction

⇑ Corresponding author. E-mail address: [email protected] (M. Ohno). https://doi.org/10.1016/j.ijheatmasstransfer.2018.12.055 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.

Microsegregation is a non-uniform distribution of alloying elements at the dendrite scale. Since it corresponds to traces of the solute diffusion field during solid growth, it provides a wealth of information about the solidification process such as size and

M. Ohno et al. / International Journal of Heat and Mass Transfer 132 (2019) 1004–1017

morphology of the growing solid and solidification path [1–3]. Furthermore, microsegregation affects the mechanical properties, corrosion resistance of cast products and formation behavior of macrosegregation [4,5]. Therefore, understanding and controlling microsegregation behavior in alloys are important issues from both the scientific and the engineering viewpoints. Many analytical models based on different assumptions and simplifications have been developed for predicting microsegregation [3–14]. Moreover, one-dimensional (1D) numerical approaches have been proposed based on the finite difference method (FDM) for explicitly solving diffusion equations of solid and liquid [15,16]. These models can be used practically by combining them with the thermodynamic calculations, that is, the CALPHAD approach [3]. These analytical and numerical models allow for reasonable prediction of microsegregation in some alloy systems [14]. However, their quantitative accuracy is not always adequate. Detailed comparisons of predicted and experimental results often exhibit noticeable disagreements between them [17–20]. Many studies have attempted to clarify the main factor causing such disagreements. For instance, in a recent study on Al-Mg alloys, a significant difference was observed between the experimental results and the numerical results of 1D-FDM [19]. The authors argued that good agreement can be achieved by doubling the reported solid diffusivity and they considered that such an increment of solid diffusivity may be explained by the existence of excess vacancies in the solid near the solid-liquid interface. Moreover, studies on Al-Cu alloy system have showed that the microsegregation behavior predicated by the numerical model is different from the experimental result [18,20]. The difference can be eliminated by modifying the reported solubility of Cu in Alrich solid solution [18] or by considering the possibility of an unusual partition coefficient that is considerably larger than the equilibrium partition coefficient [20]. Discussion on reasons for such disagreements has centered on accuracy of input parameters used in the calculation, as described above, and validity of the description of back diffusion. In this regard, one must focus on the fact that details of microstructural evolution are not considered explicitly in these models, and the microstructural features are greatly simplified like plate-like dendrite, columnar or equiaxed symmetry. The validity of such simplifications needs to be investigated. Although the importance of microstructural features in the prediction of microsegregation has been well recognized [10,13,21] and efforts have been devoted to incorporation of effects of microstructural features [14,17], it is not straightforward to consider all details of the time evolution of complex microstructures in the analytical and simplified (1D) numerical models. The phase-field model is a powerful tool for describing microstructural evolution processes during phase transformations, including alloy solidification [22–28]. Since the phase-field simulations, in general, are computationally demanding, most phase-field simulations of microsegregation were conducted using 1D or twodimensional (2D) systems [29–32]. However, recent advances in high-performance computing techniques have enabled largescale and long-time phase-field simulations of 3D dendritic growth [33–35] and it is now possible to carry out full 3D analysis of microsegregation. Here, it is important to point out that accurate analysis of microsegregation cannot easily be performed using early-developed phase-field models. This is because interface effects such as solute trapping are abnormally magnified in the early models, and accordingly, it is not always straightforward to obtain results in a quantitative manner [36]. The development of quantitative phase-field models has opened up the way for quantitative simulations of solidification microstructures [36–38]. These models were developed to reproduce the solution of the freeboundary problem without abnormal interface effects. The quantitative models developed in early works can be applied only to alloy

1005

solidification processes without diffusion in the solid (one-sided diffusion) [37,38], and therefore, they cannot be used for analysis of microsegregation. Importantly, the quantitative phase-field model was recently extended to analyze alloy systems with an arbitrary value of solid diffusivity (two-sided asymmetric diffusion) [39–43]. Hence, it is expected that microsegregation can be calculated accurately by means of 3D quantitative phase-field simulations for two-sided asymmetric diffusion. However, the ability of the quantitative phase-field model to predict microsegregation behavior has not been investigated yet. The purpose of this study is twofold: one is to evaluate the prediction accuracy of the quantitative phase-field simulation for microsegregation and the other is to gain insights into the reason for the disagreement between the results of the 1D numerical model and the experiment observed often in early works. To these ends, we carry out solidification experiments, 1D-FDM simulation, 2D and 3D quantitative phase-field simulations, and compare the results in detail. We focus on microsegregation in the columnar dendrite structure of Al-2.0 mass%Cu and Al-3.5 mass%Cu alloys. In addition, we investigate microsegregation in the columnar and equiaxed dendrite structures of Fe-14 mass%Mn alloy to check the generality of the discussion. The remainder of this paper is organized as follows. The experimental procedures are explained in the next section. 1D-FDM and the quantitative phase-field model are explained in Section 3. The results and discussion are given in Section 4, followed by the conclusions in Section 5.

2. Experimental procedures 2.1. Al-Cu alloys We focus on microsegregation in Al-2.0 mass%Cu and Al-3.5 mass%Cu alloys. Although these compositions are lower than the maximum solubility of Cu in Al-rich solid solution (5.7 mass%) [44], the eutectic structure generally appears in casting processes due to non-equilibrium solidification. Al with 99.99% purity and Cu with 99.99% purity were employed as raw materials. A sample of 200 g was melted in an electric furnace by holding it at 750 °C for 1 h. Then, the melt was cast into the rapid directional solidification mold. The mold consists of a steel plate at room temperature and an Al2O3 pipe with an outer diameter of 40 mm, an inner diameter of 30 mm and a height of 50 mm. The Al2O3 pipe preheated to 750 °C was placed on the steel plate and it served as a side wall of the cast. The sample solidified directionally from the bottom (steel plate) toward the top. A K-type thermocouple was inserted into the center of the melt to measure temperature during the solidification. The cooling rate during the primary crystallization was approximately 1.3 K/s. At 115 s after casting, the temperature decreased to 535 °C, which is lower than the eutectic temperature of 548 °C, and the sample was then quenched in water to suppress influence of solid diffusion on the microsegregation. The solidified sample was sectioned vertically along the center axis, polished and etched with 1% hydrofluoric acid solution. Microstructural observation was carried out using an optical microscope. It was observed that the columnar dendrites developed upward from the bottom. Then, the sample was sectioned horizontally to measure the primary dendrite arm spacing (PDAS) K1 on the horizontal section. Line analyses of Cu composition were carried out along a direction perpendicular to the growth direction of the columnar dendrites on the vertical section by using an electron probe micro analyzer (EPMA). The distance between the measurement points was set to 3 lm, and the analyses were conducted at three different positions. The measured data were required to be processed into a form suitable for comparison with the numerical results. In this study, we employed the Gungor’s method [1], which

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has been used frequently in experimental analyses of microsegregation [2,19]. In this method, the number of data in a prescribed composition range is divided by total data points, and the quotient is regarded as the solid fraction solidified in the prescribed composition range. Hence, one can obtain the relationship between the solid fraction and the solute composition in the solid. In this study, this solid fraction is called the converted solid fraction fcs to distinguish it from the true solid fraction fs. For fair comparison, also, numerical results are represented using the converted solid fraction fcs.

models with two-sided asymmetric diffusion were recently developed [39–43], and they can be used to analyze microsegregation [45]. However, the prediction accuracy of the quantitative phasefield simulation for the microsegregation behavior has not been evaluated yet. This important subject is tackled in the present paper. In this study, we employed the quantitative model developed in [41] in light of the balance between computational cost and numerical accuracy [46]. When non-isothermal solidification in a dilute binary alloy is considered, the microstructural evolution process is described by the following time evolution equations:

2.2. Fe-Mn alloy


2
2 ! s0 ð1 þ ð1  ke ÞuÞas ! n @ t / ¼ W 2 r as n r/

In this study, we investigated microsegregation in Fe-14 mass% Mn alloy as well. According to the binary phase diagram, solidification of this alloy occurs by crystallization of the austenite phase without the peritectic reaction. We employed pure Fe (Fe-0.0006 C-0.0007Si-0.0006Mn-0.0001S-0.0005P in mass%) and pure Mn (Mn-0.0003Fe-0.002Si-0.0001P-0.023S in mass%) as raw materials. A sample of 130 g was put in a cylindrical Al2O3 crucible and melted at 1540 °C in an SiC furnace filled with Ar gas of five-nine purity, followed by holding at this temperature for 1 h. Then, the crucible with the sample was removed from the furnace and cooled in air. The B-type thermocouple was quickly inserted into the melted sample to measure the temperature at the center of the sample. The cooling rate was measured to be about 0.9 K/s. When the temperature decreased to 1385 °C, the sample was quenched in iced water. The solidified sample was sectioned vertically along the central axis. The section was polished and etched with 3%-nital solution to observe the microstructure. As shown later, the microstructure consisted of an equiaxed structure at the sample center and a columnar structure near the sample bottom that developed upward from the bottom. In the region containing the columnar structure, the sample was sectioned horizontally to measure PDAS. Line analyses were carried out on the vertical sections of both the equiaxed and columnar regions by means of EPMA, where the distance between the measurement points was set to 3 lm. Four and three spatial profiles of Mn composition were obtained in the regions of equiaxed and columnar structures, respectively. Then, the spatial composition profiles of each region were converted to the composition at the converted solid fraction by using Gungor’s method [1]. 3. Numerical methods 3.1. Quantitative phase-field model In the phase-field model, the interface is not sharp but diffuse, having a finite thickness. The most appealing feature of this diffuse interface approach is that one can avoid explicitly tracking the positions of moving interfaces during the time evolution of complex patterns [22–28]. This model serves as an effective computational tool for solving the free-boundary problem of solid-liquid interfaces. However, when a finite value is assigned to the thickness, it is difficult to use early models in a quantitative manner because these models were constructed to reproduce the solution of the free-boundary problem at the limit of zero thickness (the sharp-interface limit). In actual simulations with finite thickness, these models are affected by unphysical magnification of the interface effects that cause an undesired dependence of the simulation result on thickness [36–43]. This serious problem was resolved in the quantitative phase-field model that is constructed to reproduce the solution of the free-boundary problem at the limit of non-zero thickness (the thin-interface limit) [36]. Quantitative phase-field






3
2 @as ! n ! 5 þW @ i 4jr/j as n @ ð@ i /Þ i¼x;y;z 2

X

2

2

 2 þ /  /3  k 1  /2 ðh þ uÞ ð1Þ 1 ð1 þ ke  ð1  ke Þ/Þ@ t u 2  1 ¼ r ðke DS þ DL þ ðke DS  DL Þ/Þru 2 

1 ke Ds r/ ð1 þ ð1  ke ÞuÞW@ t / þ pffiffiffi 1  DL jr/j 2 2 ! 1 þ þ ð1 þ ð1  ke ÞuÞ@ t /  r J noise 2

ð2Þ

where / is the phase-field that takes values of +1 in a solid and 1 in a liquid and it changes continuously from +1 to 1 inside the solid-liquid interface. u is the dimensionless diffusion field of solute, as defined by u = (cL  c0)/[(1  ke)c0], with the liquid concentration cL, average concentration c0 and partition coefficient ke. W is the interface thickness, and a1 is a constant given as a1 = 0.8839. DS and DL are the diffusivity in solid and liquid, respectively. k is the coupling constant given as k = a1W/r[(1  ke)2c0 RTm/vm], where r is the solid-liquid interfacial energy, R is gas constant, Tm is melting point of the solvent and vm is molar volume. h is the dimensionless undercooling defined as h = (T  T0)/[mLc0(1  k)] with the liquidus slope mL and reference temperature T0, which was set to the solidus temperature in this study. s0 is given as s0 = a2kW2/DL, where ! a2 = 0.6267. n ¼ rn=jrnj is the unit vector normal to the inter
 ! face, and as n is a function associated with anisotropy of the interfacial energy. In this study a fourfold crystalline anisotropy is considered as follows:


 X @4/ ! i as n ¼ 1  34 þ 44 4 i¼x;y;z jr/j

ð3Þ

where e4 is the anisotropic strength of the solid-liquid interfacial ! energy. J noise is the noise flux representing fluctuations in the solute diffusion field that triggers the side branching. The temperature was calculated within the frozen temperature approximation as follows,

T ðyÞ ¼ T 0  Rc t þ Gy

ð4Þ

where Rc is the cooling rate, G is temperature gradient, and y is spatial coordinate in y-direction. Therefore, the temperature gradient was applied in the y-direction in the simulations of columnar dendritic growth. On the other hand, G was set to 0 in the simulations of equiaxed structures. Eqs. (1) and (2) were discretized using a second-order finite difference scheme with grid spacing Dx and were integrated using a

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first-order Euler scheme. 2D and 3D simulations of columnar dendritic growth were carried out in Al-2.0 mass%Cu and Al-3.5 mass% Cu alloys. In the 2D simulations, the mirror boundary condition was applied at two sides in the x-direction, and the growth of a half of the columnar dendrite in the y-direction was calculated from the initial solid placed at the origin of system. In the 3D simulations, the mirror boundary condition was applied at two sides in both the x- and z-directions, and the growth of a quarter of the columnar dendrite in the y-direction was calculated from the initial solid placed at the origin of system. In both 2D and 3D cases, the zero Neumann boundary condition was applied at two edges of the yaxis. The computational system was divided into Nx  Ny  Nz gird points where Ni denotes the number of grid points in the idirection. The length of the computational system is represented by Lx = NxDx, Ly = NyDx and Lz = NzDx in the x-, y- and zdirections. The 2D simulations were carried out with Lx = 256Dx = K1/2 and Ly = 2048Dx. The 3D simulations were performed with Lx = Lz = 128Dx = K1/2 and Ly = 512Dx. Here, K1 is the PDAS measured experimentally. Using experimental data of K1 described later, specifically, we employed Dx = 0.365(0.730) lm and 0.355(0.711) lm in 2D (3D) simulations for Al-2mass%Cu and Al-3.5 mass%Cu alloys, respectively, and Dx = 0.455(0.910) lm in 2D (3D) simulations for Fe-Mn alloy. To check for influence of system size on the calculated microsegregation, we carried out 2D simulations of the system with Lx = 256Dx = K1/2 and Ly = 4096Dx (Ly is doubled) and the system with Lx = 512Dx = K1 and Ly = 2048Dx (Lx is doubled). In the latter case (system with doubled Lx), two solid seeds were placed at (x, y) = (0, 0) and (Lx, 0). In the same way, additional 3D simulations were carried out using the system with Lx = Lz = 128Dx = K1/2 and Ly = 1024Dx (Ly is doubled) and the system with Lx = 256Dx = K1, Ly = 512Dx and Lz = 128Dx (Lx is doubled), where initial solid seeds were placed at (x, y, z) = (0, 0, 0) and (Lx, 0, 0). All simulations were stopped when the temperature at y = Ly reached 535 °C which is the quenching temperature in the experiments. In reality, the eutectic structure appears, and the growth of primary phase does not take place below the eutectic temperature of 548 °C. Therefore, the motion of solid-liquid interface, that is, the time evolution of / (Eq. (1)) was not calculated below 548 °C and only the diffusion equation (Eq. (2)) was solved. The relationship between the composition in the solid and the converted solid fraction was obtained using Gungor’s method from the composition distribution at the final time step in the same manner as in the experimental analysis. In this study, the regions near the bottom and the top in the ydirection (y < 1/10Ly and y > 9/10Ly) were excluded from the 2D and 3D analyses of microsegregation to avoid the influence of microstructures at the beginning and end of solidification.

In addition, we carried out 2D and 3D simulations of the Fe-14 mass%Mn alloy. Columnar dendritic growth was simulated in the same manner as in case of the Al-Cu alloys. In case of the equiaxed structure, the mirror boundary condition was applied to all boundaries in the 2D and 3D systems. The initial solid seed was placed at the origin, and the growth of a quarter and one eighth of a dendrite was calculated in 2D and 3D systems, respectively. The 2D and 3D simulations were carried out in systems with Lx = Ly = 512Dx = Ke/2 and Lx = Ly = Lz = 128Dx = Ke/2, respectively. Here, Ke is the distance between the equiaxed dendrites measured in the experiment. The simulations were stopped when the temperature reached 1385 °C. In case of the equiaxed dendrite, the converted solid fraction was calculated from the Mn composition distribution over the entire region of the final microstructure. In all phase-field simulations, the time step Dt was set to Dt = Dx2/(aDDmax) where Dmax is the maximum value of diffusivities in the simulation and aD is a constant given as aD = 5 and 7 for 2D and 3D cases, respectively. A primary arm array of columnar dendrites should affect the microsegregation behavior. Some arrays such as quadrilateral and hexagonal arrays have been reported for columnar dendritic growth [47,48]. Our recent large-scale phase-field study showed that the hexagonal array is dominant in both dendritic and cellular growth in the long-time limit of directional solidification [49]. Also, it has been indicated that various types of arrays exist until the steady state is reached. In our experiments, as shown later, both quadrilateral and hexagonal arrays were observed, and it was difficult to determine which was dominant under the solidification conditions used in this study. In the 3D simulations, we assumed the quadrilateral array for the sake of simplicity, as described above. All input parameters used in the phase-field simulations of the Al-Cu and Fe-Mn alloys are summarized in Table 1[15,34,49–52]. The temperature gradient G was determined approximately from the dependence of K1 on Rc and G [47,49] by using the measured values of K1 and Rc. All simulations were accelerated using a TESLA P100 graphics processing unit. 3.2. One-dimensional finite difference method (1D-FDM) We carried out the 1D simulation by using the FDM model developed by Matsumiya et al. [15]. In this model, the diffusion equations in the solid and liquid are solved in an explicit manner based on a finite difference scheme with second-order accuracy in space and first-order accuracy in time. The temperature is uniform throughout the whole system and it is cooled at a given constant rate. The liquidus temperature of the alloy TL was calculated

Table 1 Input parameters of 1D-FDM and phase-field simulations [12,26,42–44]. Al-Cu

Fe-Mn

Cooling rate, Rc (K/s) Temperature gradient, G (K/m)

1.3 5000

Arm spacing, Ki (lm)

K1 = 187 (2.0 mass%) K1 = 182 (3.5 mass%) 620 0.14 1.06 ∙ 107  exp(2.41 ∙ 104/(RT)) 4.44 ∙ 105  exp(1.34 ∙ 105/(RT)) 1.0  105 0.16 0.02

0.9 5000 (Columnar) 0 (Equiaxed) K1 = 233 (Columnar) Ke = 151 (Equiaxed) 497* 0.785* 3.85 ∙ 107  exp(6.95 ∙ 104/(RT)) 5.5 ∙ 106  exp(2.49 ∙ 105/(RT)) 7.17  106 0.204 0.02

Liquidus slope, mL (K/mol) Partition coefficient, ke Liquid diffusivity, DL (m2/s) Solid diffusivity, DS (m2/s) Molar volume, vm (m3/mol) Interfacial energy, r (J/m2) Anisotropy strength, e4

(i) These values were obtained in the present experiments. (ii) These were employed not in 1D-FDM but in the phase-field simulations. * These values were obtained from the thermodynamic calculations (CALPHAD method) [44].

(i) (i), (ii) (i)

(ii) (ii) (ii)

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(a) Al-2.0 mass%Cu (vertical section)

500 μm (c) Al-3.5 mass%Cu (vertical section)

400 μm

(b) Al-2.0 mass%Cu (horizontal section)

500 μm (d) Al-3.5 mass%Cu (horizontal section)

400 μm

Fig. 1. Columnar dendrite structures observed on (a, c) vertical section and (b, d) horizontal section in (a, b) Al-2.0 mass%Cu alloy and (c, d) Al-3.5 mass%Cu alloy.

as TL = Tm + mLCL⁄, where Tm is the melting point of solvent, mL is the liquidus slope and CL⁄ is the solute composition in the liquid at the solid-liquid interface. The solid-liquid interface moves when T becomes lower than TL. Then, solute redistribution occurs based on the equilibrium partition coefficient ke. In this study, the plate-like dendrite was assumed in the 1D analysis. The 1D computational system was discretized with a uniform grid spacing Dx. The mirror boundary condition was employed at both boundaries. The system length was set to K1/2 in the case of columnar dendrites in both alloys, while it was set to Ke/2 in the case of equiaxed dendrite of the Fe-Mn alloy. In all calculations of 1D-FDM, Dx = 0.2 lm and Dt = Dx2/(2DL⁄), where DL⁄ is the liquid diffusivity at the initial temperature. The system was cooled at the experimentally measured cooling rate. The simulation was stopped at the quenching temperature, that is, 535 °C for Al-Cu alloys and 1385 °C for Fe-Mn alloy. Similar to the phasefield simulations, in the calculations of the Al-Cu alloys, the motion of the solid-liquid interface was stopped and only diffusion in bulk was calculated below the eutectic temperature of 548 °C. The input parameters of the 1D simulations are listed in Table 1. 4. Results and discussion 4.1. Experimental results Fig. 1 shows microstructures observed on (a, c) vertical and (b, d) horizontal sections in (a, b) Al-2.0 mass%Cu and (c, d) Al-3.0 mass%Cu alloys. The vertical direction in Fig. 1(a) and (c) corresponds to the vertical direction of the sample. The dark and bright regions represent the primary phase (Al-rich solid solution) and the eutectic region, respectively. In each alloy, the columnar dendrites developed upward. The secondary dendrite arm spacing

(SDAS) K2 was measured by the intercept method on the vertical section, while PDAS K1 was measured on the horizontal section. SDAS and PDAS were measured to be 45 lm and 187 lm, respectively, in Al-2.0 mass%Cu alloy and 49 lm and 182 lm, respectively, in Al-3.5 mass%Cu alloy. The PDAS were employed in the present numerical simulations, as already explained above. The primary arm array can be seen on the horizontal sections (Fig. 1 (b) and (d)). Both the quadrilateral and the hexagonal arrays are mixed and it is difficult to determine which is dominant in the present experiments. We assumed quadrilateral arrays in the numerical simulations for simplicity, as described in the previous section. The results of EPMA analysis are shown in Fig. 2, where figure (a) is an example of a line profile of Cu composition across the columnar dendrites in Al-2.0 mass%Cu alloy. Note that high values of Cu composition around the eutectic composition are omitted in this figure because our focus is Cu composition profile in the primary phase. These data were changed using Gungor’s method to determine the relationship between the Cu composition in the solid C⁄Cu and the converted solid fraction fcs, as shown in Fig. 2(b). The square, triangle, and circular plots represent the results obtained from single line-profile, two line-profiles and three lineprofiles, respectively. All results show gradual increment of the solid composition with increasing fcs. The result does not depend substantially on the number of line-profiles used in the data conversion. Therefore, we considered that the result obtained from the three line-profiles is sufficiently accurate for comparison with the numerical results. Fig. 2(c) shows the result of Al-3.5 mass%Cu alloy. Although the result from single line-profile obviously deviates from the result obtained using two line-profiles, the latter result is almost identical to that obtained using three lineprofiles. Hence, the result obtained using three line-profiles will be compared with the numerical results.

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(b) Al-2.0 mass%Cu alloy Cu composition in solid, CCu* (mass%)

(a) Al-2.0 mass%Cu alloy Cu composition, CCu (mass%)

8 7 6 5 4 3 2 1 0

0

200

400

600

800

8 7 6

1 line 2 lines 3 lines

5 4 3 2 1 0 0.0

0.2

0.4

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Converted solid fraction, fsc

Distance, x ( m)

Cu composition in solid, CCu* (mass%)

(c) Al-3.5 mass%Cu alloy 8

6

1 line 2 lines 3 lines

4

2

0 0.0

0.2

0.4

0.6

0.8

1.0

Converted solid fraction, fsc Fig. 2. Results of composition analysis of Al-Cu alloys. (a) Spatial profile of Cu composition in Al-2.0 mass%Cu alloy. (b, c) Cu composition in solid versus converted solid fraction obtained from single line-profile (square), two line-profiles (triangle), and three line-profiles (circle) of (b) Al-2.0 mass%Cu alloy and (c) Al-3.5 mass% Cu alloy.

The microstructures of the Fe-Mn sample are shown in Fig. 3. The columnar dendrite structure was observed on the vertical section near the bottom of sample as shown in Fig. 3(a). SDAS K2 and PDAS K1 were estimated to be 46 and 233 lm, respectively. The equiaxed structure was observed in the center region of the sample as shown in Fig. 3(c). The distance between the equiaxed grains Ke was measured to be 150 lm. These values of K1 and Ke were utilized for the numerical simulations. Fig. 4(a) shows an example of the line-profile in the columnar region. The relationships between Mn composition and the converted solid fraction in the columnar and equiaxed regions are shown in Fig. 4(b) and (c), respectively. In the columnar region (Fig. 4(b)), the result obtained using two line-profiles is almost identical to that obtained using three lineprofiles. Similarly, no substantial difference was found between the results obtained using the three line-profiles and the four line-profiles in the equiaxed region. Hence, we employed the results obtained using the three line-profiles in the columnar region and that obtained using four line-profiles in the equiaxed region for comparison with the numerical results. 4.2. Numerical results The results of phase-field simulations of Al-2.0 mass%Cu alloy are shown in Figs. 5 and 6. Fig. 5 shows the results of 2D phasefield simulation. Fig. 5(a) shows the snapshots of microstructures

visualized using the / profile (regions with / > 0.0 are indicated by blue), while Fig. 5(b) shows the composition map of Cu, where the maximum value of Cu composition was set to 8 mass% because our focus is the composition profile in the primary phase. The dendrite first grows upward from the bottom (t = 0.45), followed by the development of secondary arms (t = 0.9 s) and the coarsening of these arms (t = 9.0 s). Cu is enriched in the last solidifying liquid at t = 45 s and the microsegregation forms. Fig. 6 shows the results of 3D phase-field simulation, where figure (a) shows the microstructures visualized using / profile (regions with / > 0.0 are indicated by blue), while figure (b) shows the composition map of Cu on the x-y, y-z, and z-x planes. The growth of the primary arm is immediately followed by the development of secondary arms, as can be seen in the microstructures at t = 0.37 s and 0.74 s. In other words, growth of the secondary arm occurs just below the tip of the primary arm, which is different from the result of 2D simulation, where the secondary arms start to develop far below the tip of the primary arm (Fig. 5). The similar trend was observed for the columnar dendritic growth in Al-3.5 mass%Cu alloy and Fe-Mn alloy. Table 2 shows SDAS K2 obtained from the experiments, as well as the 2D and 3D simulations of the columnar growth. We estimated SDAS from the composition profile in the final microstructure for fair comparison with the experimental results. More specifically, the number of peaks in composition profiles along y-direction in the interdendritic region was counted to

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(a) Vertical section of columnar region

400 μm

(b) Horizontal section of columnar region

400 μm

(c) Equiaxed region

400 μm Fig. 3. Microstructures of Fe-Mn sample. (a, b) Columnar dendrite structures observed on (a) vertical section and (b) horizontal section near the bottom of sample. (c) Equiaxed dendrite structure observed near sample center.

estimate SDAS. The results of the 2D and 3D simulations agree well with the experimental results for both alloys. An example of results of 3D phase-field simulations for equiaxed structure in Fe-Mn alloy is shown in Fig. 7 where one can see growth of one eighth of equiaxed dendrite. The results shown in Figs. 5 and 6 represent the growth of a half (2D) and quarter (3D) of the columnar dendrite, respectively. To examine effects of system size on the prediction accuracy of microsegregation, we conducted simulations using larger systems. Fig. 8 show the results for columnar dendritic growth in Al-2.0 mass%Cu alloy simulated with (a) 128  1024  128 (Ly is doubled) and (b) 256  512  128 grid points (Lx is doubled). The growth shapes near the tip of the primary arm and the coarsening process of the secondary arms in these large systems are very similar to those shown in Fig. 6. The microsegregation behaviors obtained in these simulations are shown in Fig. 8(c). The result is almost independent of the system size, which holds true in all other cases focused in this study. Therefore, effects of system size can be neglected in the present analyses of microsegregation. The results of the 2D and 3D simulations, as well as the result of 1D-FDM, will be compared with the experimental results in the following subsection. 4.3. Comparison between the experimental and numerical results The microsegregation behaviors obtained from the 1D-FDM, 2D and 3D phase-field simulations are compared with the experimental results in Fig. 9. The results of the 3D phase-field simulation show excellent agreement with the experimental results in all cases. This is important evidence that the quantitative phasefield model for two-sided diffusion can predict the microsegregation with a high degree of accuracy. Note that the results of the 3D simulations are not completely identical to the experimental

results. More specifically, the 3D simulation overestimates Cu composition at high solid fractions (0.7–0.9) in Al-3.5 mass%Cu alloy, and it slightly overestimates Mn composition at low solid fractions (0.1–0.3) during the columnar dendritic growth of FeMn alloy. Some simplifications were introduced in the present phase-field simulations, such as constant partition coefficient and constant liquidus slope (i.e., dilute solution approximation), frozen temperature approximation, and no fluid flow. These simplifications may cause discrepancies between the results of the 3D simulation and the experiment. However, the discrepancies are not significantly large. By contrast, large discrepancies appear between the experimental result and the other numerical results in each case. The mean absolute percentage error between the experimental and the numerical results is summarized in Table 3. The 3D simulation provides the most accurate result, while the 1D simulation exhibits the lowest accuracy in all cases. The present comparison clearly and quantitatively highlights the importance of appropriately describing microstructural evolution for predicting microsegregation. In all cases (Fig. 9(a)–(d)), 1D-FDM underestimates the solute composition at low and medium solid fractions, and it accordingly overestimates the composition at high solid fractions. Although not shown owing to space limitations, we performed 1D-FDM simulations using K2 instead of K1 and found the similar disagreements between the numerical and the experimental results. Actually, very similar disagreements have often been observed in early works [19,20]. Technically, the disagreements can be eliminated by enhancing back diffusion or by increasing the amount of solute distributed into the solid at the solid-liquid interface. This is why the possibilities of enhanced solid diffusivity or unusual partition coefficient were argued in the early works [19,20]. However, the comparison in Fig. 9 demonstrates that the disagreements often observed in the early works originate essentially not from inaccu-

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(b) Columnar region

Mn composition, CMn (mass%)

24 22 20 18 16 14 12 10

0

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800

1000

Mn composition in solid, CMn* (mass%)

(a)

24 22 20

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18 16 14 12 10 0.0

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Converted solid fraction, fsc

Distance, x ( m)

Mn composition in solid, CMn* (mass%)

(c) Equiaxed region 24 22 20 18

1 line 2 lines 3 lines 4 lines

16 14 12 10 0.0

0.2

0.4

0.6

0.8

1.0

Converted solid fraction, fsc Fig. 4. Results of composition analysis of Fe-Mn alloy. (a) Spatial profile of Mn composition in columnar dendrite region. (b, c) Mn composition in solid versus converted solid fraction in (b) columnar dendrite region and (c) equiaxed dendrite region.

racies in the input parameters but from disregard of details of microstructural evolution processes. In fact, the accuracy of the 2D phase-field simulations is considerably better that of 1D-FDM. However, large discrepancy arises between the numerical and the experimental results of Al-Cu alloys when fcs is less than 0.4 (Fig. 9(a) and (b)) even in the 2D simulation. Moreover, slight overestimation occurs when fcs is high in the case of Fe-Mn alloys (Fig. 9 (c) and (d)). Fig. 9 shows that 3D quantitative phase-field simulation allows for the accurate prediction of microsegregation behaviors, which accordingly demonstrates the usefulness of the quantitative phase-field model. Furthermore, it is shown that not considering detailed microstructural evolution results in large disagreements between the experimental and numerical results. For deeper understanding of the microsegregation behavior and for further development of simple but accurate analytical or numerical model for microsegregation, it is important to reveal the main factor causing such disagreements. Because the results of 3D simulation are essentially consistent with those of the experiments, we focus on the reason for the disagreement between the 1D, 2D and 3D simulations in the following discussion. First, we focus on effects of dimensionality of diffusion process. The growth of plate shape solid proceeds via 1D diffusion process, while the growth of columnar and spherical solids proceed via 2D and 3D diffusion processes, respectively. The microsegregation

behavior depends on the dimensionality of diffusion process. Hence, we carried out additional simulations of 1D-FDM using cylinder and polar coordinates to describe the growth of columnar and spherical solids, respectively. The computational conditions are the same as those described in Section 3.2. The results are shown in Fig. 10. One can understand from Fig. 10 that the experimental data cannot be accurately reproduced even with assumptions of columnar and sphere solids. The differences are rather large in comparison with the case of plate shape solid. In the early and middle periods of solidification, the concentrations in columnar and spherical solid are lower than the one in plate shape solid, which is a general trend found in early theoretical works (e.g., Ref. [13]). Therefore, it is not sufficient for accurate prediction of the microsegregation to consider only dimensionality of diffusion process with a simplified shape of solid. Back diffusion is one of the important factors affecting the microsegregation. Improving the description of back diffusion has been a key issue in theoretical works on microsegregation [7– 14]. To understand the influence of back diffusion in the present cases, we carried out additional simulations with DS = 0 (no diffusion in the solid), focusing on columnar dendritic growth in the Al-2.0 mass%Cu and Fe-Mn alloys. The results are shown in Fig. 11 along with the experimental data. Back diffusion does not contribute to microsegregation in these numerical results because of the absence of diffusion in the solid. In case of the Al-Cu alloy

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(a)

t = 0.9 s

t = 9.0 s

t = 0.37 s

t = 45 s

t = 0.74 s

t = 9.2 s

t = 45 s

y z

x

y x

(b) t = 0.37 s

t = 9.2 s

t = 45 s

8.0

t = 0.9 s

t = 9.0 s

t = 45 s

CCu (mass%)

(b) t = 0.45 s

t = 0.74 s

8.0 CCu (mass%)

0.3

0.3

Fig. 6. Results of 3D phase-field simulation for columnar dendritic growth in Al-2.0 mass%Cu alloy. (a) Phase-field and (b) Cu composition. The solid fraction fs = 0.13 at t = 0.37 s, fs = 0.285 at t = 0.74 s, fs = 0.822 at t = 9.2 and fs = 0.985 at t = 45 s.

Table 2 Secondary dendrite arm spacing (SDAS), K2. Fig. 5. Results of 2D phase-field simulation of columnar dendritic growth in Al-2.0 mass%Cu alloy. (a) Phase-field and (b) Cu composition. The solid fraction fs = 0.19 at t = 0.45 s, fs = 0.38 at t = 0.9 s, fs = 0.795 at t = 9.0, and fs = 0.969 at t = 45 s.

(Fig. 11(a)), agreement between the results of the 3D simulation and the experiment is not as good as that in Fig. 9(a). This indicates the importance of back diffusion for achieving high prediction accuracy of microsegregation in this case. By contrast, the results of the Fe-Mn alloy are not obviously different from those in Fig. 9(c). Because the solid diffusivity of Mn is about four orders of magnitude smaller than its liquid diffusivity, the effect of back diffusion on the numerical results is negligible in case of the FeMn alloy. On the other hand, the ratio of solid to liquid diffusivity of Cu in Al-Cu alloy is about one order of magnitude larger than the one of Mn in Fe-Mn alloy during the solidification and, therefore, the influence of back diffusion appears in the case of Al-Cu alloy. Importantly, there remain obvious disagreements among the results of the 1D, 2D and 3D simulations of both alloys, as shown in Fig. 11. Therefore, the disagreements should not mainly be associated with the accuracy of the description of back diffusion. When back diffusion is negligible, the microsegregation behavior is entirely determined by solute redistribution at the solidliquid interface during solidification. In Fig. 9, noticeable disagreements between the results of numerical simulations appear from

Experiment 2D simulation 3D simulation

Al-2.0 mass%Cu

Al-3.5 mass%Cu

Columnar in Fe-Mn

45 41 43

49 47 50

46 43 42

Fig. 7. Results of 3D phase-field simulation for equiaxed structure at (a) t = 8.5 s and (b) t = 21.3 s in Fe-Mn alloy.

the beginning of solidification. Therefore, we focus on solute redistribution at the solid-liquid interface in the early stages of solidification. In alloy systems with ke < 1, such as the present Al-Cu and

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(a)

(b)

Cu composition in solid, CCu* (mass%)

(c) 8

Lx = 128Δx, Ly = 512Δx 6

Lx = 128Δx, Ly = 1024Δx Lx = 256Δx, Ly = 512Δx

4

2

0 0.0

0.2

0.4

0.6

0.8

1.0

Converted solid fraction, fsc

Fig. 8. Results of 3D phase-field simulations for columnar dendritic growth in Al-2.0 mass%Cu alloy for different system sizes. The computational system consists of (a) 128  1024  128 and (b) 256  512  128 grid points. These are the microstructures at (a) t = 1.8 s and (b) at t = 0.9 s. (c) Cu composition versus converted solid fraction obtained from 3D phase-field simulations for columnar dendritic growth in Al-2.0 mass% Cu alloy.

Fe-Mn alloys, the solute is rejected from the solid into the liquid at the solid-liquid interface. Then, a solute diffusion layer with high solute composition develops in the liquid ahead of the moving interface. When the solid-liquid interfaces approach each other, their solute diffusion layers start overlapping each other. Such overlap results in increased solute compositions at the interface on both liquid and solid sides. Hence, when overlapping of the solute diffusion layer occurs frequently, a large amount of solute is distributed into the solids at the interfaces. Note that the frequency of overlapping of the solute diffusion layer is closely related to the size and morphology of the microstructure. SDAS is a measure used to characterize the size of dendrite structures. A small value of SDAS can be associated with high frequency of the overlapping of solute diffusion layer. As shown in Table 2, SDAS in the 2D simulation is very close to the one in the 3D simulation. However, the SDAS value in Table 2 represents the spacing after

the coarsening process, which is not directly relevant to the solute redistribution in the early stages of solidification. The problem here is that SDAS cannot be well defined when the secondary arms do not develop in the early stages of solidification. As described in early works [53–56], the specific area of the solid-liquid interface Ss or the interfacial area density Sv is an integral measure that characterizes the overall morphology of the solidification microstructure. Ss is defined as the total interfacial area divided by the total volume of the solid, while Sv is defined as the total interfacial area divided by the total volume of the system. Hence, these quantities are related to each other as given by Sv = fsSs, where fs is the true solid fraction. These quantities can be well defined for any morphology. Moreover, Sv can be employed to describe the coarsening kinetics precisely [54]. It can accordingly be used to characterize the process of multiplication or branching of the dendrites [56], which is associated with overlapping of the solute diffusion layer

M. Ohno et al. / International Journal of Heat and Mass Transfer 132 (2019) 1004–1017

Cu composition in solid, CCu* (mass%)

(a) Al-2.0 mass%Cu alloy

(b) Al-3.5 mass%Cu alloy

8

Experimental data 1D simulation 2D simulation 3D simulation

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Mn composition in solid, CMn* (mass%)

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20

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Converted solid fraction, fsc

22

Experimental data 1D simulation 2D simulation 3D simulation

24 22 20

Experimental data 1D simulation 2D simulation 3D simulation

18 16 14 12 10 0.0

0.2

0.4

0.6

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Converted solid fraction, fsc

Fig. 9. Comparisons of microsegregation behaviors between experimental and simulation results for columnar dendrite structures in (a) Al-2.0 mass%Cu, (b) Al-3.5 mass%Cu, (c) Fe-Mn alloys and (d) equiaxed structure in Fe-Mn alloys. In each figure, dot-dashed, dashed and solid lines represent the results of 1D, 2D and 3D simulations, respectively, while the circle plots are the experimental data.

and the resulting microsegregation behavior. Fig. 12 shows the Sv value calculated for columnar dendritic growth in the Al-2.0 mass%Cu and the Fe-Mn alloys. The x-axis represents the true solid fraction fs. The Sv value obtained from the 1D and the 2D simulations are shown in Fig. 12 as well. Sv in 1D system corresponds to the reciprocal of PDAS in the present case. Sv in 2D system is the total length of the solid-liquid interface divided by total system area. Fig. 12 shows that as fs increases, Sv first increases and then starts to decrease at around fs = 0.5 in the 3D simulations for both alloys. The increase in Sv indicates refinement of the microstructure, that is, development of secondary arms due to branching, while the decrease in Sv represents coarsening of the arms. When Sv is high, that is, many solid-liquid interfaces exist in the microstructure, the diffusion layer overlaps readily, and thus, a

Table 3 Mean absolute percentage errors.

1D 2D 3D

Al-2.0 mass% Cu

Al-3.5 mass% Cu

Columnar in FeMn

Equiaxed in FeMn

51.8 18.8 7.1

68.2 50.0 10.0

7.1 5.4 4.1

7.5 5.5 2.1

large amount of solute is distributed into the solids at the solidliquid interface. In Fig. 12, Sv is always lower in the 2D system than in the 3D system in the early and middle stages of solidification. More precisely, Sv in the 2D system is less than or equal to about half of Sv in the 3D system until fs = 0.5. Therefore, the overlapping of the solute diffusion layer occurs more frequently in the 3D system than in the 2D system. This explains the higher values of solute composition in the 3D simulation than that in the 2D simulation in the early and middle stages of solidification in Fig. 9. In case of the 1D system, Sv is constant, and overlapping of the solute diffusion layer does not occur at low solid fractions, resulting in lower solute compositions in the early stages of solidification than those in the 2D and 3D systems in Fig. 9. As discussed above, the disagreements in microsegregation between 1D, 2D, and 3D simulations should be related closely to the differences in interfacial area density Sv. Note that the disagreements between the numerical results in the Al-Cu alloy are more remarkable than those in the Fe-Mn alloy in Fig. 9. Similarly, the differences in Sv between the results of numerical simulations of the Al-Cu alloy are larger than those of the Fe-Mn alloy in Fig. 12. Further discussion of this point requires an understanding of the relationship between Sv and the solute composition distributed in the solid at the interface, which will be tackled in a future work. In this regard, we would like to refer to a recent work

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(a) Al-2.0 mass%Cu alloy Cu composition in solid, CCu* (mass%)

Cu composition in solid, CCu* (mass%)

(a) 8 7 6 5

Experimental data 1D-FDM plate 1D-FDM columnar 1D-FDM sphere

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(b) Columnar structure in Fe-Mn alloy Mn composition in solid, CMn* (mass%)

(b) Mn composition in solid, CMn* (mass%)

Experimental data 1D simulation 2D simulation 3D simulation

24 22 20

Experimental data 1D-FDM plate 1D-FDM columnar 1D-FDM sphere

18 16 14 12 10 0.0

0.2

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0.6

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24 22 20 18 16 14 12 10 0.0

1.0

in which a general evolution equation of Sv was developed as a function of time and solid fraction [55].

5. Conclusions We analyzed microsegregation in Al-Cu and Fe-Mn alloys. The results obtained from one-dimensional (1D) finite difference method (FDM), two-dimensional (2D) and three-dimensional (3D) quantitative phase-field simulations for two-sided asymmetric diffusion were compared to those obtained from solidification experiments. It was shown that the 3D simulation provides the most accurate results, while the 1D simulation provides the least accurate results in all cases. Accordingly, the present comparison highlights the importance of appropriate description of microstructural evolution for predicting microsegregation accurately. In this study, it was shown that the disagreement between the 1D-FDM results and the experimental results, observed often in early works, can be ascribed to disregard for details of microstructural evolution in 1D-FDM. It was discussed that the main factor causing such disagreements is the frequency of overlapping of the solute diffusion layer in early stages of solidification, which is characterized by the solid-liquid interfacial area density Sv. Future

0.2

0.4

0.6

0.8

1.0

Converted solid fraction, fsc

Converted solid fraction, fsc Fig. 10. Comparisons of microsegregation behavior obtained by 1D-FDM for plate, columnar and sphere solid in (a) Al-2.0 mass%Cu and (b) Fe-Mn alloys.

Experimental data 1D simulation 2D simulation 3D simulation

Fig. 11. Comparisons of microsegregation behavior between experimental and simulation results obtained for columnar dendrite structures in (a) Al-2.0 mass%Cu and (b) Fe-Mn alloys. The solid diffusivity was set to 0 in these simulations.

studies should aim to investigate the relationship between Sv and solute composition at the solid-liquid interface during solidification in greater detail. It is important to extend the present investigation to multi-component alloys as well. In this study, it was shown that the result of 3D quantitative phase-field simulation agrees very well with the experimental result in all cases. This finding substantiates that 3D simulation by using the present quantitative phase-field model for twosided asymmetric diffusion allows for accurate prediction of microsegregation behaviors. In this regard, we would like to refer to recent works, in which a method for prediction of the microsegregation was developed based on machine learning (deep neural network) with results of quantitative phase-field simulations [57], and then, the method was coupled to a model for simulating macrosegregation [58]. In addition, a macrosegregation simulation was recently accelerated by a Lattice-Boltzmann method [59]. A combination of these works will allow for highly accurate and cost-effective prediction of macrosegregation.

Conflict of interest None.

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(a) Al-2.0 mass%Cu Interface area density, Sv ( m-1)

0.10 0.08

1D 2D 3D

0.06 0.04 0.02 0.00 0.0

0.2

0.4

0.6

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Solid fraction, fs

(b) Columnar structure in Fe-Mn system Interface area density, Sv ( m-1)

0.10 0.08

1D 2D 3D

0.06 0.04 0.02 0.00 0.0

0.2

0.4

0.6

0.8

1.0

Solid fraction, fs Fig. 12. Dependences of interface area density on solid fraction during columnar dendritic growth in (a) Al-2.0 mass%Cu alloy and (b) Fe-Mn alloy.

Acknowledgments This research was partly supported by a Grant-in-Aid for Scientific Research (B) (JSPS KAKENHI Grant No. 16H04541) from Japan Society for the Promotion of Science (JSPS).

Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.ijheatmasstransfer. 2018.12.055.

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