Quantitative Phase-field Modeling and Simulations of Solidification Microstructures

Abstract

This review presents the development of quantitative phase-field models for simulating the formation processes of solidification microstructures, with particular attention to the theoretical foundation and progress in modeling. The symmetry of interpolating functions required to reproduce the free-boundary problem in the thin-interface limit and the necessity of antitrapping current in the diffusion equation are discussed. In addition, new cross-coupling in the phase-field equation for two-sided asymmetric diffusion is briefly described. Recent achievements of large-scale simulations using high-performance computing techniques are explained. Furthermore, some important applications of quantitative phase-field simulations such as investigations of cellular and dendritic growth, microsegregation, and peritectic reaction in carbon steel are discussed.

1. Introduction

Alloy solidification is a multi-physics phenomenon involving thermal diffusion, solute diffusion, fluid dynamics, deformation of solid and so on. The characteristic length scale and/or time scale inherent in these phenomena often differ by orders of magnitude; therefore, understanding, predicting, and controlling the formation processes of solidification microstructures require observation, analysis, and modeling of the dynamics over a wide range of spatial and temporal scales. In-situ observation techniques have made substantial contributions to our understanding of the formation processes of solidification microstructures.^1,2,3,4,5) In particular, synchrotron radiation X-ray imaging techniques have enabled in-situ observation of cellular and dendritic growth in thin samples.^{6,7,8,9,10,11,12,13)} Although the growth processes analyzed by such techniques have been limited to the quasi two-dimensional (2D), recent X-ray tomographic microscopy has unveiled the time evolution of the three-dimensional (3D) morphology of solidification microstructures.^14,15,16,17) A more detailed investigation and a deeper understanding of solidification processes should be possible by improving the spatial and temporal resolutions of these techniques and by widening their range of applicability to a variety of alloy systems and solidification conditions.

In addition to in-situ experimental techniques, computational approaches currently offer an effective way to investigate the details of the formation processes of the solidification microstructures of alloys. There are a variety of computational methods. The rapid advancement in high-performance computing techniques has enabled investigations of the dynamics of moving solid-liquid interfaces at atomistic scales by means of molecular dynamics (MD) simulations.^18,19,20) For instance, the statistical aspect of the multiple nucleation behavior of crystals from the undercooled melt can be analyzed by MD simulations using parallel computing with multiple graphics processing units (GPUs).^21,22,23,24) The growth behavior of crystals at a larger scale, that is, microstructural evolution behavior, can be properly understood within the framework of the free-boundary problem (FBP) of the solid–liquid interface.²⁵⁾ The phase-field model,^{26,27,28,29,30,31)} the cellular automaton method,^{32,33,34,35,36,37)} and the level-set method^38,39,40) have been developed for solving the FBP; these methods have been applied to investigations of a variety of microstructural processes. A dendrite network model was recently developed for simulating the competitive growth processes of dendrite structures at a larger scale with reasonable computational cost.^41,42) In addition, the cellular automaton and Monte Carlo methods can be utilized to describe the growth of dendrite envelopes or solidifying grains for predicting structures at the ingot level.^{43,44,45,46,47)}

The focus of the present review is the phase-field model. The chief advantage of this diffuse interface method is that explicit tracking of the moving interface can be avoided and thus, the temporal evolution of the complex morphology of the microstructure can be computed using a simple algorithm. In addition, the shape of the curved interface can be smoothly and precisely expressed. Accordingly, the contribution of the interface curvature to the dynamics, that is, the Gibbs-Thomson effect can be computed with accuracy. Significant effort has been devoted to the development of this model. Excellent reviews on the phase-field model have already been reported.^{26,27,28,29,30,31)} Therefore, in this paper, particular attention is directed at “quantitative” phase-field models based on thin-interface asymptotics to reproduce the solution of the FBP. The theoretical foundations and progress of quantitative phase-field modeling will be explained. Furthermore, applications of quantitative phase-field simulations are discussed. The purpose of the current review is not to cover all related works. We shall discuss recent advances in high-performance computing techniques and some applications of quantitative phase-field simulations to cellular and dendritic growth, microsegregation, and peritectic reaction in carbon steel.

2. Fundamentals of the Quantitative Phase-field Model

The central issue in modeling the evolution of the microstructure during solidification is the description of the moving solid-liquid interface associated with heat and solute diffusion in the bulk phases, the conservation of energy and solute around the moving interface, and the Gibbs-Thomson effect. The FBP is the problem of tracking the position of the moving interface based on these laws. Many theoretical models such as Ivantsov’s solution,⁴⁸⁾ the theory of constitutional undercooling,⁴⁹⁾ Mullins-Sekerka perturbation theory,⁵⁰⁾ and the microsegregation models^51,52,53,54) can be regarded as approximate solutions of the FBP with simplifications and/or assumptions. The phase-field model has emerged as an effective numerical tool for directly solving the FBP.⁵⁵⁾ Significant effort has been devoted to the development of the method after the simulation of dendrite growth by Kobayashi.^56,57,58) The model for alloy solidification was developed by Wheeler, Boettinger, and McFadden (WBM model)^59,60,61) and was significantly improved by Kim, Kim, and Suzuki (KKS model).^62,63) The KKS model is a two-phase approach in which the solid and liquid concentration fields are separately defined, and are made mutually dependent on the condition of equal diffusion potentials. This ansatz enables the decoupling of the interface properties and the concentration field. Such decoupling was also archived in a different two-phase approach,⁶⁴⁾ and this issue was later addressed within the framework of the grand potential formulation.^65,66) Furthermore, models for multi-phase and/or polycrystalline solidifications have been developed^67,68,69,70) and have found widespread application in a variety of alloy systems. Importantly, Karma and Rappel proposed a model for solidification in pure substances with equal thermal diffusivities in liquid and solid (symmetric model).^71,72,73) This model is constructed based on the thin-interface limit and is called the quantitative phase-field method. This model has been extended to deal with single-phase solidification in binary^74,75) and multi-component alloys⁷⁶⁾ and to describe multi-phase solidification.^77,78) Although early quantitative models are applicable to systems without solid diffusion (one-sided model), models for systems with solid diffusion (asymmetric two-sided model) have been proposed.^79,80,81,82) Furthermore, quantitative models were developed for thermosolutal problems⁸³⁾ and are coupled with fluid dynamics based on the Navier-Stokes equations^{84,85,86,87,88)} and the lattice-Boltzmann equations.^89,90,91,92) Other important improvements have been tested and achieved by several groups.^{93,94,95,96,97,98,99,100,101)} As detailed later, quantitative phase-field simulations are increasingly utilized for investigating the formation processes of solidification microstructures and, in particular, its usefulness has significantly increased with the use of high-performance computing techniques. In the following, the essential points of the quantitative phase-field model are briefly explained.

The phase-field model is constructed to reproduce the FBP within the framework of the diffuse-interface. The early models reproduced the solution of the FBP in the sharp-interface limit where the interface thickness approaches zero. However, the interface thickness cannot be set to zero in the simulations. Hence, abnormal interface effects associated with surface diffusion, interface stretching, and discontinuity in the diffusion potential and/or temperature at the interface appear in the early models,^102,103) causing a deviation of the solution from that of the FBP. The quantitative phase-field model is constructed based on the thin-interface limit where the interface thickness is small but finite.⁷³⁾ Accordingly, the model is free from abnormal interface effects, and can convey quantitative meaning in the simulation. To exemplify some mathematical details of the quantitative model, let us consider isothermal solidification in a dilute binary alloy. The sharp-interface equations in the FBP are given as:   

$${\partial }_{t}c=D_i{\nabla }^{2}c$$(1)

  

$$(c_{int}-c_l^e)/(c_l^e-c_s^e)=-d_0\kappa -\beta v_n$$(2)

  

$$(1-k)c_{int}{v}_n=D_s{\partial }_{n}c{|}^{-}-D_l{\partial }_{n}c{|}^{+}$$(3)

where c is the concentration and D_i with i = s and l represents the solid and liquid diffusivity, respectively. Equation (1) represents the diffusion in the bulk. In Eq. (2), c_l^e (c_s^e) is the equilibrium concentration of the liquid (solid), d₀ is the capillary length, β is the linear kinetic coefficient, κ is the local curvature of the interface, v_n is the moving velocity normal to the interface, and c_int is the interface concentration on the liquid side of the interface. Equation (2) represents the Gibbs–Thomson effect. Here, the anisotropy of the interface properties is neglected for simplicity. In Eq. (3), k is the partition coefficient, ∂_ηc|⁻ and ∂_ηc|⁺ are the spatial derivatives of c in the direction normal to the interface on the solid and liquid sides, respectively. η represents the spatial coordinates in the normal direction of the interface. Equation (3) represents the conservation of the solute at the moving solid-liquid interface.

In the phase-field model, isothermal solidification in a binary alloy is described by the time evolution of the phase-field variable, ϕ, and the concentration field, c. In this study, ϕ is defined so that it is +1 in the solid and −1 in the liquid phase, and continuously changes from +1 to −1 in the solid–liquid interface. Here, we employ the two-phase approach^62,63,64,65) in which the local concentration is defined as c = c_s(1+ϕ)/2 + c_l(1−ϕ)/2. Here c_s and c_l are the solid and liquid concentration fields, respectively, and c_s depends on c_l according to the condition of equal diffusion potential.⁶²⁾ In particular, c_s is given as c_s = kc_l in a dilute solution. Consequently, the time evolution equations are given as:   

$$\tau {\partial }_t\varphi =W^2{\nabla }^2\varphi -f^{\prime }(\varphi )-\lambda g^{\prime }(\varphi )u$$(4)

  

$$\begin{array}{l}\left(\frac{1+k}{2}-\frac{1-k}{2}h(\varphi )\right){\partial }_{t}u=\nabla (D_{l}q(\varphi )\nabla u-J_{at})\\ \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\hspace{0.17em} }+(1+(1-k)u)\frac{1}{2}{\partial }_{t}h(\varphi )\end{array}$$(5)

where τ is the relaxation constant for ϕ, W is a measure of the interface thickness, λ is a constant given as λ = a₁W/d₀ with the constant, a₁, and u is the dimensionless supersaturation given as $u=(c_l-c_l^e)/(c_l^e-c_s^e)$. J_at in Eq. (5) is the antitrapping current necessary for quantitative modeling, as discussed later. In Eqs. (4) and (5), there are four interpolating functions: f(ϕ), g(ϕ), h(ϕ), and q(ϕ). f(ϕ) is the interpolating function representing the barrier potential between the solid and liquid; it has minima at ϕ = ±1. The commonly employed function is the double-well (DW) potential function given by f(ϕ) = (1 −ϕ²)²/4 or the double-obstacle (DO) potential function given by f(ϕ) = (1 − ϕ²)/2 with f(ϕ) = ∞ for ϕ<−1 and ϕ >+1. g(ϕ) is a monotonically increasing function of ϕ with g(±1) = ±1, smoothly interpolating the thermodynamic states of the solid and liquid. h(ϕ) and q(ϕ) are the interpolating functions for the concentration field and diffusivity, respectively, with h(±1) = ±1, q(+1) = kD_s/D_l, and q(−1) = 1. Note that the anisotropy of the interface property is neglected in Eq. (4).

The purpose of phase-field modeling is to reproduce the sharp-interface Eqs. (1), (2), (3). The relation between the phase-field model (Eqs. (4) and (5)) and the sharp-interface model (Eqs. (1), (2), (3)) is analyzed by reducing an interface thickness, W, that is much smaller than any physical length scale in the microstructural process. More specifically, the solutions of Eqs. (4) and (5) are analyzed perturbatively by expanding them in the inner and outer regions. The inner region is the diffuse interface region where ϕ varies rapidly and the outer region is the bulk phase region away from the interface. Matching of the solutions in both regions then provides the link between the sharp-interface (outer) and diffuse-interface (inner) models. This is called the matched asymptotic analysis.^104,105) ε = W/d₀ is usually employed as an expansion parameter. The early models can be mapped onto the FBP in the limit of ε → 0, which is called the sharp-interface limit.¹⁰⁴⁾ However, ε cannot be set to zero in phase-field simulations and, accordingly, simulations of the early models suffer from abnormal interface effects. On the other hand, the quantitative model is constructed based on the thin-interface limit, where ε is made small but finite. It yields all important correction terms at the second order in ε. According to the thin-interface asymptotics, the Gibbs-Thomson relation given by Eq. (2) can be reproduced when the following relations are satisfied:^80,102)   

$${\int }_0^{-1}\left(\frac{1}{q(\varphi )}-\frac{1}{q(-1)}\right)\frac{d\varphi }{{\partial }_{\eta }{\varphi }_0}={\int }_0^{+1}\left(\frac{1}{q(\varphi )}-\frac{1}{q(+1)}\right)\frac{d\varphi }{{\partial }_{\eta }{\varphi }_0}$$(6)

  

$${\int }_0^{-1}\left(\frac{h(\varphi )}{q(\varphi )}-\frac{h(-1)}{q(-1)}\right)\frac{d\varphi }{{\partial }_{\eta }{\varphi }_0}={\int }_0^{+1}\left(\frac{h(\varphi )}{q(\varphi )}-\frac{h(+1)}{q(+1)}\right)\frac{d\varphi }{{\partial }_{\eta }{\varphi }_0}$$(7)

  

$${\int }_0^{-1}\left(\frac{g(\varphi )}{q(\varphi )}-\frac{g(-1)}{q(-1)}\right)\frac{d\varphi }{{\partial }_{\eta }{\varphi }_0}={\int }_0^{+1}\left(\frac{g(\varphi )}{q(\varphi )}-\frac{g(+1)}{q(+1)}\right)\frac{d\varphi }{{\partial }_{\eta }{\varphi }_0}$$(8)

where ${\partial }_{\eta }{\varphi }_0=-\sqrt{f(\varphi )}$ and the contribution of J_at in Eq. (5) is neglected. The conservation law of the solute given by Eq. (3) can be reproduced by Eqs. (4) and (5) when the following relations are satisfied:   

$${\int }_0^{-1}(h(\varphi )-h(-1))\frac{d\varphi }{{\partial }_{\eta }{\varphi }_0}={\int }_0^{+1}(h(\varphi )-h(+1))\frac{d\varphi }{{\partial }_{\eta }{\varphi }_0}$$(9)

  

$${\int }_0^{-1}(q(\varphi )-q(-1))\frac{d\varphi }{{\partial }_{\eta }{\varphi }_0}={\int }_0^{+1}(q(\varphi )-q(+1))\frac{d\varphi }{{\partial }_{\eta }{\varphi }_0}$$(10)

Moreover, the relaxation time, τ, must be given as:   

$$\tau =\frac{\beta W^2}{d_0}+\frac{a_1{a}_2{W}^3(1+(1-k)u)}{D_l{d}_0}$$(11)

where a₂ is a constant depending on the forms of the interpolating functions. Equation (11) is strictly valid for the one-sided model^74,75) and is numerically effective for a two-sided case.⁸⁰⁾

The interpolating functions must satisfy the relations given by Eqs. (6), (7), (8), (9), (10). Otherwise, abnormal interface effects emerge. When symmetric diffusion with k = 1 and D_l = D_s (q(ϕ) =1) is considered, one needs to consider only Eqs. (8) and (9), and these are readily satisfied by choosing h(ϕ) and g(ϕ) as odd functions of ϕ. This model is equivalent to the quantitative phase-field model first proposed for solidification in pure substances.^71,73) In the case of k ≠ 1 and D_l ≠ D_s, it is not straightforward to satisfy all the relations simultaneously. When one-sided diffusion with D_s = 0 is considered, one needs to satisfy Eqs. (7), (9), and (10). However, it is difficult to find the forms of h(ϕ), q(ϕ), and g(ϕ) which meet these requirements simultaneously. Karma resolved this impasse by introducing the antitrapping current J_at, which is defined as follows:⁷⁴⁾   

$$J_{at}=-a_{at}(\varphi )(1+(1-k)u)W{\partial }_t\varphi \frac{\nabla \varphi }{\left|\nabla \varphi \right|}$$(12)

where a_at(ϕ) is the interpolating function. When f(ϕ) is the DW potential with h(ϕ) = ϕ and q(ϕ) = (1−ϕ)/2, a_at(ϕ) is given as $1/\left(2\sqrt{2}\right)$ for the one-sided case. The contribution of this current makes the relation of Eq. (7) equivalent to Eq. (9). Hence, all abnormal interface effects can be eliminated owing to the antitrapping current.

D_s is, in general, several orders of magnitude smaller than D_l and, hence, the one-sided model has been widely utilized for investigations of cellular and dendritic growth behavior, as described later. However, it is not possible to reproduce equilibrium solidification because D_s = 0. It inevitably describes the Scheil solidification at the extremely slow cooling rate and, accordingly, microsegregation cannot be examined by the one-sided model. Moreover, the one-sided model cannot be applied to the solidification of carbon steel because the ratio of D_s to D_l for carbon is of the order of 10⁻¹ to 10⁻² in carbon steel.¹⁰⁶⁾ In this regard, it is important to develop a model for D_s ≠ 0, that is, two-sided asymmetric diffusion. However, difficulties arise because one needs to consider all the requirements given by Eqs. (6), (7), (8), (9), (10). The antitrapping current is not helpful in satisfying Eqs. (6) and (8). Hence, an assumption of the diffusion flux near the interface was made in the early quantitative models for two-sided diffusion.⁸⁰⁾ On the other hand, a study of the solidification of a pure substance with asymmetric diffusion¹⁰⁷⁾ shows that all abnormal interface effects can be formally removed by introducing a coupling term between ϕ and c into Eq. (4). This coupling term can be regarded as the term originating from kinetic cross coupling.^107,108,109) On the other hand, a new method for the variational formulation of quantitative phase-field models was recently proposed for isothermal solidification in a dilute binary alloy¹¹⁰⁾ and nonisothermal solidification in multi-component alloys.¹¹¹⁾ The antitrapping current in Eq. (5) and the additional term in Eq. (4) automatically appears in this method, and the derived model is free from all abnormal interface effects. Additionally, the form of τ is automatically determined, and the diffusivity is derived as a tensor inside the interface. It is expected that such variational models will find widespread application. These important advancements were recently achieved for quantitative phase-field modeling.

3. Numerical Performance of the Quantitative Phase-field Model

As previously described, abnormal interface effects appear in conventional phase-field models. Since the magnitudes of such effects scale with W, the simulation results of the conventional models largely depend on W. Although the accuracy of the result is generally improved by decreasing W, the simulation time is proportional to W⁻⁵ when the standard finite difference method is employed.⁷⁴⁾ On the other hand, the quantitative phase-field model can reproduce the solution of the FBP with a finite thickness (W ≠ 0). The simulation result of the quantitative model converges to the solution of the FBP by decreasing W. Fast convergence of the results demonstrates high numerical performance.

Numerical testing of the convergence behavior of the quantitative models was conducted in several works.^{73,75,78,80,81,82,110,111,112,113,114)} An example of the convergence test is shown in Fig. 1,⁸⁰⁾ which indicates the time change of the dendrite tip velocity in a dilute binary alloy during isothermal solidification calculated for k = 0.15, D_s/D_l = 0.1, the initial value of the dimensionless supercooling, u₀ = −0.55, and the anisotropy parameter of the interfacial energy, ε₄ = 0.02. The anisotropy of the interfacial energy for a four-fold symmetry was introduced into Eq. (4). The simulations were carried out from a small solid seed in a 2D rectangular system with x/d₀ = 400 and y/d₀ =800. The dendrite tip growing in the y-direction was tracked in a computational frame moving in the y-direction. The results for different values of W were obtained using the quantitative model and the model without J_at. Abnormal interface effects appear in the latter model. In all cases, after the initial decrement, the dendrite tip velocity reached a constant value, which corresponds to the steady state value. The results of the quantitative model agree with each other, and therefore, the result for d₀/W = 0.24 is well converged. On the other hand, the result of the model without J_at for d₀/W =0.24 is different from that for d₀/W = 0.554, and does not converge. This example demonstrates that the convergence of the result of the quantitative model is faster than that of the conventional model.

Fig. 1.

Temporal changes in the dendrite tip velocity during isothermal solidification in a two-dimensional system calculated by the quantitative phase-field models with and without antitrapping current for d₀/W = 0.554 and 0.240.⁸⁰⁾ The results represented by the solid, dashed, and dotted lines are almost superimposed. (Online version in color.)

The convergence behavior of quantitative models with different sets of interpolating functions was investigated for isothermal solidification of a dilute binary alloy.¹¹²⁾ In the models with DW and DO potentials employed for f(ϕ), high numerical accuracy can be achieved by employing the fifth-order and first-order polynomials for g(ϕ) and h(ϕ), respectively. The results of several convergence tests of such models are summarized in Fig. 2.¹¹²⁾ The circle and square symbols represent the steady-state values of the tip velocity, V_n, and tip radius, ρ, respectively, of the dendrite calculated for different values of W. V_n and ρ are normalized by V_c and ρ_c, respectively. For each solidification condition and alloy system, V_c and ρ_c are the converged values of V_n and ρ, respectively. The horizontal axis is the interface thickness W divided by ρ_c. Note that the definition of W in the model with the DO potential is different from that with the DW potential.¹¹²⁾ The convergence starts to break down for W/ρ_c ~ 0.2, regardless of the type of quantitative model, solidification condition, and alloy system. The condition of W/ρ_c ~ 0.2 approximately represents the limitation of accurately describing the dendrite tip of r_c with the diffuse interface thickness specified by W. In other words, the breakdown of the convergence of V_n and ρ is ascribable to the limitation of the spatial resolution due to the diffuse interface. Figure 2 shows that all abnormal interface effects were successfully suppressed in these quantitative phase-field models.

Fig. 2.

Convergence behavior of V_n (circles) and ρ (squares) calculated with the quantitative phase-field models with DW and DO potentials for u₀ = −0.3 and −0.5.¹¹²⁾ Data [a] and [b] are the results of isothermal solidification in binary alloys for D_s/D_l = 0.1 and 0.5, respectively.⁸⁰⁾ Data [c] and [d] are those for non-isothermal solidification for D_s/D_l = 0.0 and 0.1, respectively.⁸²⁾ Data [e] and [f] are the results of isothermal and non-isothermal solidification, respectively, in a ternary alloy.⁸²⁾ (Online version in color.)

4. High-performance Computing

The quantitative models exhibit fast convergence of the result with respect to W. This is computationally advantageous because lowering W yields a significant reduction in the computational time and, thereby, allows for large-scale simulations. Meanwhile, the progress of high-performance computing techniques enables large-scale phase-field simulations. In particular, the parallel computational technique using GPUs has been attracting attention.^{115,116,117,118,119,120)} In 2011, a very large-scale phase-field simulation was carried out using the GPU supercomputer TSUBAME 2.0 (Tokyo Institute of Technology) to describe the competitive growth of columnar dendrites in the Al–Si alloy. The computational performance of this multi-GPU method reached approximately 2.0 petaflops using 4000 GPUs, which was awarded the ACM Gordon Bell Prize in 2011.¹²¹⁾ Since then, multi-GPU computations have been employed for very large-scale phase-field simulations of dendrite growth, for instance, a completive growth of more than 100 dendrites in an Al–Si alloy in a 3D system with a side length of approximately 3 mm, which was divided into approximately 4000³ finite difference grid points.¹²²⁾ Such large-scale simulation techniques have led to many achievements, including the elucidation of the validity of the selection rule in competitive dendrite growth of bi-crystals^123,124,125) and polycrystal systems^126,127) and the symmetry of the arrangement of primary dendrite arms during directional solidification¹²⁸⁾ as well as investigations of dendritic growth coupled with fluid dynamics.^{92,129,130,131,132,133,134,135,136,137,138,139)}

Recently, large-scale phase-field simulations have become possible even with laboratory-scale computational resources. An example is shown in Fig. 3, which shows the result of a quantitative phase-field simulation of columnar dendrite growth during the directional solidification of Al-5mass% Si alloy under the conditions of a temperature gradient of 10 K/mm and a pulling speed of 90 μm/s. The 3D system with dimensions, 0.7 × 2.8 × 2.8 mm³, was divided into 512 × 2048 × 2048 grid points. More than 60 dendrites grew from the bottom and competitive growth occurred between them. This large-scale computation was carried out using 10 GPUs (Tesla P100).

Fig. 3.

Result of a quantitative phase-field simulation for a columnar dendrite growth during directional solidification in an Al-5mass% Si alloy with a temperature gradient of 10 K/mm and a pulling speed of 90 μm/s. The 3D system with dimensions, 0.7 × 2.8 × 2.8 mm³, was divided into 512 × 2048 × 2048 grid points. (Online version in color.)

As exemplified above, the rapid development of GPU computing techniques has contributed to a widening of the application range and increased the usefulness of the quantitative phase-field models. In addition, CPU-based parallel computing techniques^{140,141,142,143,144)} and efficient numerical schemes and algorithms such as adaptive mesh techniques^{87,114,145,146,147,148,149,150)} and preconditioning¹⁵¹⁾ have been developed for improving numerical performance of the phase-field simulations.

5. Applications

5.1. Cellular and Dendritic Growth

Quantitative phase-field models have been utilized for investigations of the formation processes of solidification microstructure.^{152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174, 175,176,177,178,179,180,181,182,183,184,185)} For instance, the validity of the theories for the free dendritic growth of alloys (the so-called LKT and LGK models^186,187,188)) was tested using 2D quantitative phase-field simulations.¹⁵⁸⁾ Competitive growth of primary branches in cellular and dendritic growth during directional solidification in bi-crystal materials has been extensively investigated in several works.^{123,124,125,159,160,161,162,163)} A number of quantitative phase-field simulations were carried out to compute the spacing of the primary branches in cellular and dendritic growth during directional solidification.^{128,157,164,165,166,167,168,169,170,171,172)} In addition, the growth orientation of cells and columnar dendrites tilted from the direction of the temperature gradient was analyzed in detail.^{160,173,174,175,176)} Recently, morphological changes of dendrites associated with transitions in the preferred growth direction were investigated.^177,178,179) Although most of the studies have focused on materials with four-fold symmetry, quantitative phase-field simulations have been applied to Mg-based alloys.^{180,181,182,183)}

5.2. Microsegregation

Although many analytical and numerical models have been developed for predicting microsegregation,^{51,52,53,54,106)} the quantitative accuracy of these methods is not always high because the shape and size of the microstructure and their time changes are greatly simplified or neglected. Quantitative phase-field simulations have been applied to analyses of the microsegregation behavior.^{189,190,191,192,193)} Figure 4 shows a comparison of the microsegregation behavior in fcc-solid solutions obtained by experiments and simulations.¹⁹³⁾ These data show the solute composition (Cu in Al-rich primary phase or Mn in Fe-rich primary phase) with respect to the solid fraction. The plots in each figure represent the experimental data. The dot-dashed line is the result of the one-dimensional (1D) finite difference method with the assumption of plate-shaped solid. In all cases, the 1D finite difference method underestimates the solute composition at low and medium solid fractions, and accordingly overestimates the composition at high solid fractions. This is a typical disagreement often observed in early studies on microsegregation. The accuracy of the 2D phase-field simulations is much better than that of the 1D finite difference method. Importantly, the results of the 3D quantitative phase-field simulations exhibit good agreement with the experimental results in all cases. This demonstrates that the quantitative phase-field model for two-sided diffusion can predict microsegregation with a high degree of accuracy.

Fig. 4.

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.¹⁹³⁾ In each figure, the dot-dashed, dashed, and solid lines represent the results of the 1D finite difference method and 2D and 3D quantitative phase-field simulations, respectively, while the circle plots represent the experimental data. (Online version in color.)

Although the binary alloy was investigated in the above-mentioned study, the quantitative model can be applied to investigations of multi-component alloys. In addition, it is not limited to dilute solutions. Figure 5 represents a snapshot of equiaxed dendrites in Fe-0.15mass%C-1.5massMn-0.2mass%Si-1.0mass%Cr-0.5massMo alloy. The initial temperature was 1784 K and the cooling rate was 50 K/s. The computational domain was divided into a mesh size of 2048 × 1024. The multi-binary extrapolation scheme,¹¹¹⁾ which is advantageous in terms of computational cost, was employed for the driving force. This example demonstrates that quantitative phase-field simulations can be applied to the detailed investigation of microsegregation of each solute in practical steel.

Fig. 5.

Result of a quantitative phase-field simulation with constant cooling at 50 K/s from 1784 K in the Fe-0.15mass%C-0.2mass%Si-1.5mass%Mn-1.0mass%Cr-0.5mass%Mo alloy. The 2D computational system was divided into 2048 × 1024 grid points. (Online version in color.)

5.3. Peritectic Reaction in Carbon Steel

It is commonly believed that peritectic reaction occur during the solidification of peritectic-solidified carbon steel. However, according to a recent in-situ observation,¹⁹⁴⁾ it is common that the austenite phase appears during ferritic solidification in a massive-like transformation mode, and the peritectic reaction and transformation rarely occurs during usual solidification and casting conditions. The microstructural change in such a massive-like transformation mode is consistent with the formation process of the coarse columnar grain structure of austenite during casting processes.^{195,196,197,198,199)} Yet, understanding the kinetics of the peritectic relation is very important in the advancement of solidification science and engineering, especially in relation to steel.

The peritectic reaction rate in carbon steel, namely the growth rate of austenite (γ) along the ferrite (δ)/liquid (L) interface was investigated by means of in-situ observations using a confocal scanning laser microscope (CSLM).^3,200) The measured rate was much faster than the diffusion-controlled reaction rate predicted by a theoretical model. Therefore, it was speculated that the peritectic reaction in carbon steel is not a diffusion-controlled process.³⁾ However, the theoretical model ignores the effect of solid diffusion on the reaction, and it does not explicitly deal with the motion of the triple junction of the δ/L, δ/γ and L/γ interfaces. Hence, it is necessary to employ a more reliable theory or numerical simulation. Quantitative phase-field models for two-phase solidification have been applied to the simulation of the peritectic reaction and transformation in a carbon steel.^201,202,203) The results for the peritectic transformation is shown in Fig. 6.²⁰¹⁾ This result shows the temperature dependence of the parabolic rate constant for the time dependence of the moving distances of the γ-L and γ-δ interfaces below the peritectic reaction temperature, 1768 K. The results of the quantitative phase-field simulations agree well with the experimental data.^204,205) This model was subsequently applied to the calculation of the peritectic reaction rate during the steady state. The results are shown in Fig. 7. The experimentally measured values³⁾ fall within the calculated range of the diffusion-controlled reaction rate. In addition, the calculated results are consistent with the findings of another in-situ observations.²⁰⁰⁾ Therefore, the comparison shown in Fig. 7 suggests that the reported experimental findings can be explained by the diffusion-controlled mechanism. This is an example in which the quantitative model was successfully utilized to clarify the transformation mechanism in carbon steel.

Fig. 6.

Dependence of the parabolic rate constants for the δ→γ transformation and L→γ solidification on the holding temperature calculated by the quantitative phase-field model in a one-dimensional system initially consisting of the diffusion couple of the liquid and δ phases.²⁰¹⁾ Data [A] and [B] represent the experimental data.^204,205)

Fig. 7.

Dependence of the peritectic reaction rate on the degree of undercooling from the peritectic reaction temperature obtained from a quantitative phase-field simulation.²⁰¹⁾ The open symbols represent the experimental data obtained using CLSM.³⁾

6. Conclusions

The quantitative phase-field models based on thin-interface asymptotics enable us to simulate the formation processes of solidification microstructures with a high degree of accuracy. In this paper, the theoretical foundations and progress in quantitative phase-field modeling was briefly explained, and some important applications of quantitative phase-field simulations was discussed. Since the pioneering work by Karma and Rappel,^71,72,73) numerous efforts have been devoted to the development of quantitative phase-field models to enhance its versatility. Scale-up of the simulations has been achieved through improvements in computing techniques and resources. Quantitative phase-field modeling and simulations will play an increasingly important role in the field of solidification science and engineering.

Acknowledgements

The author expresses his gratitude to Professor Tomohiro Takaki at Kyoto Institute of Technology, Professor Yasushi Shibuta at The University of Tokyo, Mr. Geunwoo Kim and Professor Kiyotaka Matsuura at Hokkaido University and Emeritus Professor Toshio Suzuki at The University of Tokyo. Also, the author acknowledges support of his research by a Grant-in-Aid for Scientific Research (B) (JSPS KAKENHI Grant No. 16H04541) from Japan Society for the Promotion of Science (JSPS).

References

• 1)   R.  Trivedi and  K.  Somboonsuk: Acta Metall., 33 (1985), 1061.
• 2)   H.  Esaka and  W.  Kurz: J. Cryst. Growth, 72 (1985), 578.
• 3)   H.  Shibata,  Y.  Arai,  M.  Suzuki and  T.  Emi: Metall. Mater. Trans. B, 31 (2000), 981.
• 4)   M. E.  Glicksman: Metall. Mater. Trans. B, 43 (2012), 207.
• 5)   S.  Akamatsu and  H.  Nguyen-Thi: Acta Mater., 108 (2016), 325.
• 6)   R. H.  Mathiesen,  L.  Arnberg,  F.  Mo,  T.  Weitkamp and  A.  Snigirev: Phys. Rev. Lett., 83 (1999), 5062.
• 7)   H.  Yasuda,  I.  Ohnaka,  K.  Kawasaki,  A.  Sugiyama,  T.  Ohmichi,  J.  Iwane and  K.  Umetani: J. Cryst. Growth, 262 (2004), 645.
• 8)   B.  Billia,  H.  Nguyen-Thi,  N.  Mangelinck-Noel,  N.  Bergeon,  H.  Jung,  G.  Reinhart,  A.  Bogno,  A.  Buffet,  J.  Hartwig,  J.  Baruchel and  T.  Schenk: ISIJ Int., 50 (2010), 1929.
• 9)   T. M.  Wang,  J. J.  Xu,  T. Q.  Xiao,  H. L.  Xie,  J.  Li,  T. J.  Li and  Z. Q.  Cao: Phys. Rev. E, 81 (2010), 042601.
• 10)   H.  Yasuda,  T.  Nagira,  M.  Yoshiya,  N.  Nakatsuka,  A.  Sugiyama,  K.  Uesugi and  K.  Umetani: ISIJ Int., 51 (2011), 402.
• 11)   A.  Bogno,  H.  Nguyen-Thi,  G.  Reinhart,  B.  Billia and  J.  Baruchel: Acta Mater., 61 (2013), 1303.
• 12)   T.  Nagira,  H.  Yokota,  S.  Morita,  H.  Yasuda,  M.  Yoshiya,  C. M.  Gourlay,  A.  Sugiyama,  K.  Uesugi and  K.  Umetani: ISIJ Int., 53 (2013), 1195.
• 13)   H.  Yasuda,  K.  Morishita,  N.  Nakatsuka,  T.  Nishimura,  M.  Yoshiya,  A.  Sugiyama,  K.  Uesugi and  A.  Takeuchi: Nat. Commun., 10 (2019), 3183.
• 14)   N.  Limodin,  L.  Salvo,  E.  Boller,  M.  Suéry,  M.  Felberbaum,  S.  Gailliègue and  K.  Madi: Acta Mater., 57 (2009), 2300.
• 15)   J. L.  Fife,  J. W.  Gibbs,  E. B.  Gulsoy,  C.-L.  Park,  K.  Thornton and  P. W.  Voorhees: Acta Mater., 70 (2014), 66.
• 16)   J. W.  Gibbs,  K. A.  Mohan,  E. B.  Gulsoy,  A. J.  Shahani,  X.  Xiao,  C. A.  Bouman,  M.  De Graef and  P. W.  Voorhees: Sci. Rep., 5 (2015), 11824.
• 17)   B.  Cai,  J.  Wang,  A.  Kao,  K.  Pericleous,  A. B.  Phillion,  R. C.  Atwood and  P. D.  Lee: Acta Mater., 117 (2016), 160.
• 18)   J.-J.  Hoyt,  M.  Asta and  A.  Karma: Mater. Sci. Eng. R Rep., 41 (2003), 121.
• 19)   Y.  Shibuta,  K.  Oguchi and  M.  Ohno: Scr. Mater., 86 (2014), 20.
• 20)   M.  Guerdane and  M.  Berghoff: Phys. Rev. B, 97 (2018), 144105.
• 21)   Y.  Shibuta,  K.  Oguchi,  T.  Takaki and  M.  Ohno: Sci. Rep., 5 (2015), 13534.
• 22)   Y.  Shibuta,  S.  Sakane,  T.  Takaki and  M.  Ohno: Acta Mater., 105 (2016), 328.
• 23)   Y.  Shibuta,  S.  Sakane,  E.  Miyoshi,  S.  Okita,  T.  Takaki and  M.  Ohno: Nat. Commun., 8 (2017), 10.
• 24)   Y.  Shibuta,  S.  Sakane,  E.  Miyoshi,  T.  Takaki and  M.  Ohno: Model. Simul. Mater. Sci. Eng., 27 (2019), 054002.
• 25)   J. S.  Langer: Rev. Mod. Phys., 52 (1980), 1.
• 26)   W. J.  Boettinger,  J. A.  Warren,  C.  Beckermann and  A.  Karma: Annu. Rev. Mater. Res., 32 (2002), 163.
• 27)   L. Q.  Chen: Annu. Rev. Mater. Res., 32 (2002), 113.
• 28)   L.  Gránásy,  T.  Pusztai and  J. A.  Warren: J. Phys. Condens. Matter., 16 (2004), R1205.
• 29)   I.  Steinbach: Model. Simul. Mater. Sci. Eng., 17 (2009), 073001.
• 30)   T.  Takaki: ISIJ Int., 54 (2014), 437.
• 31)   A.  Karma and  D.  Tourret: Curr. Opin. Solid State Mater. Sci., 20 (2016), 25.
• 32)   L.  Nastac and  D. M.  Stefanescu: Model. Simul. Mater. Sci. Eng., 5 (1997), 391.
• 33)   L.  Nastac: Acta Mater., 47 (1999), 4253.
• 34)   M. F.  Zhu and  C. P.  Hong: ISIJ Int., 41 (2001), 436.
• 35)   W.  Wang,  P. D.  Lee and  M.  McLean: Acta Mater., 51 (2003), 2971.
• 36)   M. F.  Zhu,  S. Y.  Lee and  C. P.  Hong: Phys. Rev. E, 69 (2004), 061610.
• 37)   K.  Reuther and  M.  Rettenmayr: Comput. Mater. Sci., 95 (2014), 213.
• 38)   Y. T.  Kim,  N.  Goldenfeld and  J.  Dantzig: Phys. Rev. E, 62 (2000), 2471.
• 39)   F.  Gibou,  R.  Fedkiw,  R.  Caflisch and  S.  Osher: J. Sci. Comput., 19 (2003), 183.
• 40)   L.  Tan and  N.  Zabaras: J. Comput. Phys., 221 (2007), 9.
• 41)   D.  Tourret and  A.  Karma: Acta Mater., 120 (2016), 240.
• 42)   D.  Tourret,  M. M.  Francois and  A. J.  Clarke: Comput. Mater. Sci., 162 (2019), 206.
• 43)   M.  Rappaz and  Ch.-A.  Gandin: Acta Metall. Mater., 41 (1993), 345.
• 44)   Ch.-A.  Gandin and  M.  Rappaz: Acta Mater., 45 (1997), 2187.
• 45)   M.  Vandyoussefi and  A. L.  Greer: Acta Mater., 50 (2002), 1693.
• 46)   M.  Yamazaki,  Y.  Natsume,  H.  Harada and  K.  Ohsasa: ISIJ Int., 46 (2006), 903.
• 47)   J. A.  Spittle and  S. G.  Brown: Acta Metall., 37 (1989), 1803.
• 48)   G. P.  Ivantsov: Dokl. Akad. Nauk SSSR, 58 (1947), 567 (in Russian).
• 49)   W. A.  Tiller,  K. A.  Jackson,  J. W.  Rutter and  B.  Chalmers: Acta Metall., 1 (1953), 428.
• 50)   W. W.  Mullins and  R. F.  Sekerka: J. Appl. Phys., 35 (1964), 444.
• 51)   E.  Scheil: Z. Metallkd., 34 (1942), 70.
• 52)   H. D.  Brody and  M. C.  Flemings: Trans. Metall. Soc. AIME, 236 (1966), 615.
• 53)   T. W.  Clyne and  W.  Kurz: Metall. Trans. A, 12 (1981), 965.
• 54)   I.  Ohnaka: Trans. Iron Steel Inst. Jpn., 26 (1986), 1045.
• 55)   G. J.  Fix: Free Boundary Problems: Theory and Applications, ed. by A. Fasano and M. Primicerio, Pitman, Boston, (1983), 580.
• 56)   R.  Kobayashi: RIMS Kôkyûroku, 614 (1987), 39 (in Japanese).
• 57)   R.  Kobayashi: Phys. D, 63 (1993), 410.
• 58)   R.  Kobayashi: Exp. Math., 3 (1994), 59.
• 59)   A. A.  Wheeler,  W. J.  Boettinger and  G. B.  McFadden: Phys. Rev. A, 45 (1992), 7424.
• 60)   A. A.  Wheeler,  W. J.  Boettinger and  G. B.  McFadden: Phys. Rev. E, 47 (1993), 1893.
• 61)   J. A.  Warren and  W. J.  Boettinger: Acta Metall. Mater., 43 (1995), 689.
• 62)   S. G.  Kim,  W. T.  Kim and  T.  Suzuki: Phys. Rev. E, 60 (1999), 7186.
• 63)   P. R.  Cha,  D. H.  Yeon and  J. K.  Yoon: J. Cryst. Growth, 274 (2005), 281.
• 64)   J.  Tiaden,  B.  Nestler,  H. J.  Diepers and  I.  Steinbach: Phys. D, 115 (1998), 73.
• 65)   M.  Plapp: Phys. Rev. E, 84 (2011), 031601.
• 66)   A.  Choudhury and  B.  Nestler: Phys. Rev. E, 85 (2012), 021602.
• 67)   I.  Steinbach,  F.  Pezzolla,  B.  Nestler,  M.  Seeßelberg,  R.  Prieler,  G. J.  Schmitz and  J. L. L.  Rezende: Phys. D, 94 (1996), 135.
• 68)   I.  Steinbach and  F.  Pezzolla: Phys. D, 134 (1999), 385.
• 69)   S. G.  Kim,  W. T.  Kim,  T.  Suzuki and  M.  Ode: J. Cryst. Growth, 261 (2004), 135.
• 70)   J.  Eiken,  B.  Böttger and  I.  Steinbach: Phys. Rev. E, 73 (2006), 066122.
• 71)   A.  Karma and  W.-J.  Rappel: Phys. Rev. E, 53 (1996), R3017.
• 72)   A.  Karma and  W.-J.  Rappel: Phys. Rev. Lett., 77 (1996), 4050.
• 73)   A.  Karma and  W.-J.  Rappel: Phys. Rev. E, 57 (1998), 4323.
• 74)   A.  Karma: Phys. Rev. Lett., 87 (2001), 115701.
• 75)   B.  Echebarria,  R.  Folch,  A.  Karma and  M.  Plapp: Phys. Rev. E, 70 (2004), 061604.
• 76)   S. G.  Kim: Acta Mater., 55 (2007), 4391.
• 77)   R.  Folch and  M.  Plapp: Phys. Rev. E, 68 (2003), 010602(R).
• 78)   R.  Folch and  M.  Plapp: Phys. Rev. E, 72 (2005), 011602.
• 79)   A.  Gopinath,  R. C.  Armstrong and  R. A.  Brown: J. Cryst. Growth, 291 (2006), 272.
• 80)   M.  Ohno and  K.  Matsuura: Phys. Rev. E, 79 (2009), 031603.
• 81)   M.  Ohno and  K.  Matsuura: Acta Mater., 58 (2010), 5749.
• 82)   M.  Ohno: Phys. Rev. E, 86 (2012), 051603.
• 83)   J. C.  Ramirez,  C.  Beckermann,  A.  Karma and  H.-J.  Diepers: Phys. Rev. E, 69 (2004), 051607.
• 84)   C.  Beckermann,  H.-J.  Diepers,  I.  Steinbach,  A.  Karma and  X.  Tong: J. Comput. Phys., 154 (1999), 468.
• 85)   X.  Tong,  C.  Beckermann and  A.  Karma: Phys. Rev. E, 61 (2000), R49.
• 86)   X.  Tong,  C.  Beckermann,  A.  Karma and  Q.  Li: Phys. Rev. E, 63 (2001), 061601.
• 87)   C. W.  Lan and  C. J.  Shih: Phys. Rev. E, 69 (2004), 031601.
• 88)   Y.  Lu,  C.  Beckermann and  J. C.  Ramirez: J. Cryst. Growth, 280 (2005), 320.
• 89)   W.  Miller,  S.  Succi and  D.  Mansutti: Phys. Rev. Lett., 86 (2001), 3578.
• 90)   D.  Medvedev,  T.  Fischaleck and  K.  Kassner: J. Cryst. Growth, 303 (2007), 69.
• 91)   D.  Medvedev,  T.  Fischaleck and  K.  Kassner: Phys. Rev. E, 74 (2006), 031606.
• 92)   R.  Rojas,  T.  Takaki and  M.  Ohno: J. Comput. Phys., 298 (2015), 29.
• 93)   J.  Bragard,  A.  Karma,  Y. H.  Lee and  M.  Plapp: Interface Sci., 10 (2002), 121.
• 94)   C.  Tong,  M.  Greenwood and  N.  Provatas: Phys. Rev. B, 77 (2008), 064112.
• 95)   J.  Rosam,  P. K.  Jimack and  A. M.  Mullis: Phys. Rev. E, 79 (2009), 030601(R).
• 96)   N.  Ofori-Opoku and  N.  Provatas: Acta Mater., 58 (2010), 2155.
• 97)   M.  Nicoli,  M.  Plapp and  H.  Henry: Phys. Rev. E, 84 (2011), 046707.
• 98)   J.  Kundin and  R.  Siquieri: Phys. D, 240 (2011), 459.
• 99)   S. Y.  Pan and  M. F.  Zhu: IOP Conf. Ser. Mater. Sci. Eng., 33 (2012), 012096.
• 100)   A.  Carré,  B.  Böttger and  M.  Apel: J. Cryst. Growth, 380 (2013), 5.
• 101)   J.  Kundin,  R.  Siquieri and  H.  Emmerich: Phys. D, 243 (2013), 116.
• 102)   R. F.  Almgren: SIAM J. Appl. Math., 59 (1999), 2086.
• 103)   M.  Plapp: Philos. Mag., 91 (2011), 25.
• 104)   G.  Caginalp: Phys. Rev. A, 39 (1989), 5887.
• 105)   R.  Folch,  J.  Casademunt,  A.  Hernández-Machado and  L.  Ramírez-Piscina: Phys. Rev. E, 60 (1999), 1724.
• 106)   T.  Matsumiya,  H.  Kajioka,  S.  Mizoguchi,  Y.  Ueshima and  H.  Esaka: Trans. Iron Steel Inst. Jpn., 24 (1984), 873.
• 107)   G.  Boussinot and  E. A.  Brener: Phys. Rev. E, 89 (2014), 060402(R).
• 108)   E. A.  Brener and  G.  Boussinot: Phys. Rev. E, 86 (2012), 060601(R).
• 109)   A.  Fang and  Y.  Mi: Phys. Rev. E, 87 (2013), 012402.
• 110)   M.  Ohno,  T.  Takaki and  Y.  Shibuta: Phys. Rev. E, 93 (2016), 012802.
• 111)   M.  Ohno,  T.  Takaki and  Y.  Shibuta: Phys. Rev. E, 96 (2017), 033311.
• 112)   M.  Ohno,  T.  Takaki and  Y.  Shibuta: J. Comput. Phys., 335 (2017), 621.
• 113)   K.  Oguchi and  T.  Suzuki: ISIJ Int., 49 (2009), 1536.
• 114)   J.  Rosam,  P. K.  Jimack and  A. M.  Mullis: Acta Mater., 56 (2008), 4559.
• 115)   A.  Yamanaka,  T.  Aoki,  S.  Ogawa and  T.  Takaki: J. Cryst. Growth, 318 (2011), 40.
• 116)   C.  Zhu,  J.  Jia,  H.  Zhang,  R.  Xiao and  L.  Feng: Mater. Trans., 55 (2014), 1841.
• 117)   Y.  Shibuta,  M.  Ohno and  T.  Takaki: JOM, 67 (2015), 1793.
• 118)   C.  Yang,  Q.  Xu and  B.  Liu: Comput. Mater. Sci., 136 (2017), 133.
• 119)   Y.  Shibuta,  M.  Ohno and  T.  Takaki: Adv. Theory Simul., 1 (2018), 1800065.
• 120)   W.  Sun,  R.  Yan,  Y.  Zhang,  H.  Dong and  T.  Jing: Comput. Mater. Sci., 160 (2019), 149.
• 121)   T.  Shimokawabe,  T.  Aoki,  T.  Takaki,  A.  Yamanaka,  A.  Nukada,  T.  Endo,  N.  Maruyama and  S.  Matsuoka: Proc. 2011 Int. Conf. for High Performance Computing, Networking, Storage and Analysis (SC ’11), ACM, New York, (2011), 1.
• 122)   T.  Takaki,  T.  Shimokawabe,  M.  Ohno,  A.  Yamanaka and  T.  Aoki: J. Cryst. Growth, 382 (2013), 21.
• 123)   T.  Takaki,  M.  Ohno,  T.  Shimokawabe and  T.  Aoki: Acta Mater., 81 (2014), 272.
• 124)   S.  Sakane,  T.  Takaki,  M.  Ohno,  T.  Shimokawabe and  T.  Aoki: IOP Conf. Ser. Mater. Sci. Eng., 84 (2015), 012063.
• 125)   T.  Takaki,  S.  Sakane,  M.  Ohno,  Y.  Shibuta,  T.  Shimokawabe and  T.  Aoki: ISIJ Int., 56 (2016), 1427.
• 126)   T.  Takaki,  M.  Ohno,  Y.  Shibuta,  S.  Sakane,  T.  Shimokawabe and  T.  Aoki: J. Cryst. Growth, 442 (2016), 14.
• 127)   T.  Takaki,  S.  Sakane,  M.  Ohno,  Y.  Shibuta,  T.  Aoki and  Ch.-A.  Gandin: Materialia, 1 (2018), 104.
• 128)   T.  Takaki,  S.  Sakane,  M.  Ohno,  Y.  Shibuta,  T.  Shimokawabe and  T.  Aoki: Acta Mater., 118 (2016), 230.
• 129)   T.  Takaki,  R.  Rojas,  M.  Ohno,  T.  Shimokawabe and  T.  Aoki: IOP Conf. Ser. Mater. Sci. Eng., 84 (2015), 012066.
• 130)   S.  Sakane,  T.  Takaki,  R.  Rojas,  M.  Ohno,  Y.  Shibuta,  T.  Shimokawabe and  T.  Aoki: J. Cryst. Growth, 474 (2017), 154.
• 131)   T.  Takaki,  R.  Rojas,  S.  Sakane,  M.  Ohno,  Y.  Shibuta,  T.  Shimokawabe and  T.  Aoki: J. Cryst. Growth, 474 (2017), 146.
• 132)   S.  Sakane,  T.  Takaki,  M.  Ohno,  Y.  Shibuta,  T.  Shimokawabe and  T.  Aoki: J. Cryst. Growth, 483 (2018), 147.
• 133)   T.  Takaki,  R.  Sato,  R.  Rojas,  M.  Ohno and  Y.  Shibuta: Comput. Mater. Sci., 147 (2018), 124.
• 134)   S.  Sakane,  T.  Takaki,  M.  Ohno and  Y.  Shibuta: Comput. Mater. Sci., 164 (2019), 39.
• 135)   T.  Takaki,  S.  Sakane,  M.  Ohno,  Y.  Shibuta and  T.  Aoki: Acta Mater., 164 (2019), 237.
• 136)   T.  Takaki,  S.  Sakane,  M.  Ohno,  Y.  Shibuta and  T.  Aoki: Comput. Mater. Sci., 171 (2020), 109209.
• 137)   T.  Takaki,  S.  Sakane,  M.  Ohno and  Y.  Shibuta: Model. Simul. Mater. Sci. Eng., 27 (2019), 054001.
• 138)   Y.  Mitsuyama,  T.  Takaki,  S.  Sakane,  Y.  Shibuta and  M.  Ohno: Acta Mater., 188 (2020), 282.
• 139)   S.  Sakane,  T.  Takaki,  M.  Ohno,  Y.  Shibuta and  T.  Aoki: Comput. Mater. Sci., 178 (2020), 109639.
• 140)   W. L.  George and  J. A.  Warren: J. Comput. Phys., 177 (2002), 264.
• 141)   Z.  Guo,  J.  Mi and  P. S.  Grant: J. Comput. Phys., 231 (2012), 1781.
• 142)   Z.  Guo,  J.  Mi and  P. S.  Grant: IOP Conf. Ser. Mater. Sci. Eng., 33 (2012), 012101.
• 143)   Z.  Guo,  J.  Mi,  S.  Xiong and  P. S.  Grant: Metall. Mater. Trans. B, 44 (2013), 924.
• 144)   M.  Bauer,  J.  Hötzer,  D.  Ernst,  J.  Hammer,  M.  Seiz,  H.  Hierl,  J.  Hönig,  H.  Köstler,  G.  Wellein,  B.  Nestler and  U.  Rüde: Proc. Int. Conf. for High Performance Computing, Networking, Storage and Analysis (SC ’19), ACM, New York, (2019), 1.
• 145)   N.  Provatas,  N.  Goldenfeld and  J.  Dantzig: Phys. Rev. Lett., 80 (1998), 3308.
• 146)   R.  Tönhardt and  G.  Amberg: Phys. Rev. E, 62 (2000), 828.
• 147)   C. W.  Lan,  C. M.  Hsu and  C. C.  Liu: J. Cryst. Growth, 241 (2002), 379.
• 148)   N.  Provatas,  M.  Greenwood,  B.  Athreya,  N.  Goldenfeld and  J.  Dantzig: Int. J. Mod. Phys. B, 19 (2005), 4525.
• 149)   K.  Oguchi and  T.  Suzuki: Mater. Trans., 48 (2007), 2280.
• 150)   C. C.  Chen,  Y. L.  Tsai and  C. W.  Lan: Int. J. Heat Mass Transf., 52 (2009), 1158.
• 151)   K.  Glasner: J. Comput. Phys., 174 (2001), 695.
• 152)   H. M.  Singer,  I.  Singer-Loginova,  J. H.  Bilgram and  G.  Amberg: J. Cryst. Growth, 296 (2006), 58.
• 153)   B.  Echebarria,  A.  Karma and  S.  Gurevich: Phys. Rev. E, 81 (2010), 021608.
• 154)   E.  Wesner,  A.  Choudhury,  A.  August,  M.  Berghoff and  B.  Nestler: J. Cryst. Growth, 359 (2012), 107.
• 155)   H.  Xing,  X.  Dong,  H.  Wu,  G.  Hao,  J.  Wang,  C.  Chen and  K.  Jin: Sci. Rep., 6 (2016), 26625.
• 156)   H.  Neumann-Heyme,  K.  Eckert and  C.  Beckermann: Acta Mater., 140 (2017), 87.
• 157)   Y.  Song,  S.  Akamatsu,  S.  Bottin-Rousseau and  A.  Karma: Phys. Rev. Mater., 2 (2018), 053403.
• 158)   J. C.  Ramirez and  C.  Beckermann: Acta Mater., 53 (2005), 1721.
• 159)   J.  Li,  Z.  Wang,  Y.  Wang and  J.  Wang: Acta Mater., 60 (2012), 1478.
• 160)   D.  Tourret and  A.  Karma: Acta Mater., 82 (2015), 64.
• 161)   D.  Tourret,  Y.  Song,  A. J.  Clarke and  A.  Karma: Acta Mater., 122 (2017), 220.
• 162)   C.  Guo,  J.  Li,  H.  Yu,  Z.  Wang,  X.  Lin and  J.  Wang: Acta Mater., 136 (2017), 148.
• 163)   H.  Xing,  M.  Ji,  X.  Dong,  Y.  Wang,  L.  Zhang and  S.  Li: Mater. Des., 185 (2020), 108250.
• 164)   M.  Greenwood,  M.  Haataja and  N.  Provatas: Phys. Rev. Lett., 93 (2004), 246101.
• 165)   S.  Gurevich,  A.  Karma,  M.  Plapp and  R.  Trivedi: Phys. Rev. E, 81 (2010), 011603.
• 166)   S.  Gurevich,  M.  Amoorezaei and  N.  Provatas: Phys. Rev. E, 82 (2010), 051606.
• 167)   Z.  Wang,  J.  Wang,  J.  Li,  G.  Yang and  Y.  Zhou: Phys. Rev. E, 84 (2011), 041604.
• 168)   M.  Amoorezaei,  S.  Gurevich and  N.  Provatas: Acta Mater., 58 (2010), 6115.
• 169)   V.  Fallah,  M.  Amoorezaei,  N.  Provatas,  S. F.  Corbin and  A.  Khajepour: Acta Mater., 60 (2012), 1633.
• 170)   Z.  Wang,  J.  Li,  J.  Wang and  Y.  Zhou: Acta Mater., 60 (2012), 1957.
• 171)   A. J.  Clarke,  D.  Tourret,  Y.  Song,  S. D.  Imhoff,  P. J.  Gibbs,  J. W.  Gibbs,  K.  Fezzaa and  A.  Karma: Acta Mater., 129 (2017), 203.
• 172)   J.  Kundin,  A.  Ramazani,  U.  Prahl and  C.  Haase: Metall. Mater. Trans. A, 50 (2019), 2022.
• 173)   J.  Ghmadh,  J. M.  Debierre,  J.  Deschamps,  M.  Georgelin,  R.  Guérin and  A.  Pocheau: Acta Mater., 74 (2014), 255.
• 174)   H.  Xing,  X. L.  Dong,  C. L.  Chen,  J. Y.  Wang,  L. F.  Du and  K. X.  Jin: Int. J. Heat Mass Transf., 90 (2015), 911.
• 175)   H.  Xing,  K.  Ankit,  X. L.  Dong,  H. M.  Chen and  K. X.  Jin: Int. J. Heat Mass Transf., 117 (2018), 1107.
• 176)   H.  Xing,  X. L.  Dong,  J. Y.  Wang and  K. X.  Jin: Metall. Mater. Trans. B, 49 (2018), 1547.
• 177)   T.  Haxhimali,  A.  Karma,  F.  Gonzales and  M.  Rappaz: Nat. Mater., 5 (2006), 660.
• 178)   J. A.  Dantzig,  P.  Di Napoli,  J.  Friedli and  M.  Rappaz: Metall. Mater. Trans. A, 44 (2013), 5532.
• 179)   G.  Kim,  T.  Takaki,  Y.  Shibuta,  S.  Sakane,  K.  Matsuura and  M.  Ohno: Comput. Mater. Sci., 162 (2019), 76.
• 180)   M.  Amoorezaei,  S.  Gurevich and  N.  Provatas: Acta Mater., 60 (2012), 657.
• 181)   S.  Gurevich,  M.  Amoorezaei,  D.  Montiel and  N.  Provatas: Acta Mater., 60 (2012), 3287.
• 182)   J.  Du,  A.  Zhang,  Z.  Guo,  M.  Yang,  M.  Li,  F.  Liu and  S.  Xiong: Acta Mater., 161 (2018), 35.
• 183)   Y.  Wang,  S.  Jia,  M.  Wei,  L.  Peng,  Y.  Wu,  Y.  Ji,  L.-Q.  Chen and  X.  Liu: J. Alloy. Compd., 815 (2020), 152385.
• 184)   J.  Kundin,  A.  Ramazani,  U.  Prahl and  C.  Haase: Metall. Mater. Trans. A, 50 (2019), 2022.
• 185)   Y.  Zhao,  B.  Zhang,  H.  Hou,  W.  Chen and  M.  Wang: J. Mater. Sci. Technol., 35 (2019), 1044.
• 186)   J.  Lipton,  M. E.  Glicksman and  W.  Kurz: Mater. Sci. Eng., 65 (1984), 57.
• 187)   J.  Lipton,  M. E.  Glicksman and  W.  Kurz: Metall. Trans. A, 18 (1987), 341.
• 188)   J.  Lipton,  W.  Kurz and  R.  Trivedi: Acta Metall., 35 (1987), 957.
• 189)   M.  Ohno,  T.  Takaki and  Y.  Shibuta: IOP Conf. Ser. Mater. Sci. Eng., 84 (2015), 012075.
• 190)   M.  Ohno,  D.  Kimura and  K.  Matsuura: Tetsu-to-Hagané, 103 (2017), 711 (in Japanese).
• 191)   M.  Ohno,  D.  Kimura and  K.  Matsuura: Tetsu-to-Hagané, 103 (2017), 720 (in Japanese).
• 192)   G.  Boussinot and  M.  Apel: Acta Mater., 122 (2017), 310.
• 193)   M.  Ohno,  M.  Yamashita and  K.  Matsuura: Int. J. Heat Mass Transf., 132 (2019), 1004.
• 194)   T.  Nishimura,  K.  Morishita,  M.  Yoshiya,  T.  Nagira and  H.  Yasuda: Tetsu-to-Hagané, 105 (2019), 290 (in Japanese).
• 195)   S.  Tsuchiya,  M.  Ohno,  K.  Matsuura and  K.  Isobe: Acta Mater., 59 (2011), 3334.
• 196)   M.  Ohno,  S.  Tsuchiya and  K.  Matsuura: Acta Mater., 59 (2011), 5700.
• 197)   S.  Tsuchiya,  M.  Ohno and  K.  Matsuura: Acta Mater., 60 (2012), 2927.
• 198)   M.  Ohno,  M.  Maruyama and  K.  Matsuura: Acta Mater., 61 (2013), 7334.
• 199)   M.  Ohno,  S.  Tsuchiya and  K.  Matsuura: ISIJ Int., 55 (2015), 2374.
• 200)   D.  Phelan,  M.  Reid and  R.  Dippenaar: Mater. Sci. Eng. A, 477 (2008), 226.
• 201)   M.  Ohno and  K.  Matsuura: Acta Mater., 58 (2010), 6134.
• 202)   M.  Ohno and  K.  Matsuura: ISIJ Int., 50 (2010), 1879.
• 203)   S.  Pan,  M.  Zhu and  M.  Rettenmayr: Acta Mater., 132 (2017), 565.
• 204)   K.  Matsuura,  Y.  Itoh and  T.  Narita: ISIJ Int., 33 (1993), 583.
• 205)   K.  Matsuura,  H.  Maruyama,  Y.  Itoh,  M.  Kudoh and  K.  Ishii: ISIJ Int., 35 (1995), 183.