Phase-field Modeling and Simulation of Solid-state Phase Transformations in Steels

Abstract

The phase-field method is used as a powerful and versatile computational method to simulate the microstructural evolution taking place during solid-state phase transformations in iron and steel. This review presents the basic theory of the phase-field method and reviews recent advances in the phase-field modeling and simulation of solid-state phase transformations in iron and steel, with particular attention being paid to the modeling of the austenite-to-ferrite, pearlitic, bainitic, and martensitic transformations. This review elucidates that the phase-field method is a promising computational approach to investigate the microstructural evolutions (e.g., interface migration, solute diffusion, and stress/strain evolutions) that take place during the phase transformations. It also indicates that further improvements are required to enhance the predictive accuracy of the phase-field models developed to date. Finally, this review discusses the critical challenges and perspectives for the further improvement of the phase-field modeling of solid-state phase transformations in steel, i.e., the modeling of heterogeneous nucleation, the abnormal effect of the diffusion interface, and material parameter identification.

1. Introduction

Steel is widely used in industrial products owing to its excellent balance between toughness and ductility. To improve this balance, it is important to control the microstructure of steel by utilizing various types of solid-state phase transformations. In addition to experimental methods, numerical modeling and simulations have been used to understand and predict microstructural evolution during such phase transformations. For example, various numerical modeling methodologies, including the phase-field method, have been employed. For example, the Monte Carlo (MC) and cellular automaton (CA) methods have been actively used to simulate microstructural evolutions such as the austenite-to-ferrite transformation in Fe–C alloys and steels.^1,2,3) However, the MC and CA methods are generally unable to simulate such microstructure evolutions in real time and length scales. Moreover, consideration of the effects of interfacial curvature on the microstructural evolutions is challenging when MC and CA simulations are employed.

In the current review, the phase-field method of interest was originally developed as a numerical simulation approach to investigate dendritic solidification in pure materials and alloys.^4,5,6,7) This method led to a new era of numerical simulations for the microstructural evolution processes taking place in materials since it can easily simulate complex microstructural evolutions by introducing an order parameter to implicitly track a moving interface. From the viewpoint of applied mathematics, the phase-field method is also a powerful solver for free boundary problems, and it overcomes the aforementioned issues associated with the MC and CA methods. Moreover, the phase-field method has been actively developed in the field of computational materials engineering since it can be easily coupled with the thermodynamic database of alloys using the calculation of phase diagram (CALPHAD) method. In addition, various physical phenomena, such as elasticity, plasticity, magnetism, and electrochemistry, can be easily incorporated into such phase-field models. Owing to these advantages, a commercial phase-field simulation software named MICRESSS (MICRostructure Evolution Simulation Software)⁸⁾ and many open-source codes have already been released,⁹⁾ including OpenPhase,¹⁰⁾ PRISMS-PF,¹¹⁾ FiPy,¹²⁾ and MOOSE.¹³⁾ However, it should be noted that there are several issues in the phase-field modeling and simulation of the solid-state phase transformations in steel, as will be discussed later.

Thus, for the purpose of this review, recent advances in phase-field modeling and the simulation of solid-state phase transformations in iron, ferrous alloys, and steel are reviewed. More specifically, we focus on the phase-field modeling of the basic phase transformations taking place in steel, i.e., the austenite-to-ferrite, pearlitic, bainitic, and martensitic transformations that occur during continuous cooling. Furthermore, future challenges and perspectives in the phase-field modeling of these phase transformations are discussed. It should also be noted that many review papers focusing on the phase-field method have been published in recent decades,^{14,15,16,17,18)} and excellent review papers on the phase-field modeling of solid-state phase transformations in steel, such as the austenite-to-ferrite transformation,¹⁹⁾ the pearlitic transformation,²⁰⁾ and the martensitic transformation,²¹⁾ have been published. The author recommends that the reader refers to these excellent review papers.

2. Phase-field Models

Prior to reviewing recent advances in the phase-field modeling of solid-state phase transformations in steel (see Section 3), the basic theory of the phase-field method is described in the current section. More specifically, the phase-field method is explained by assuming the simulation of an isothermal austenite-to-ferrite transformation in an Fe–C binary alloy. Initially, the phase-field model, which uses a single order parameter and so is referred to as the single-phase-field (SPF) model, is explained. Subsequently, the multi-phase-field (MPF) model, where multiple order parameters are used to simulate polycrystalline microstructural evolution, is explained.

2.1. Single-phase-field Model

In the SPF method, an order parameter is defined as a function of coordinates r and time t to implicitly track a moving interface and/or analyze physical phenomena, such as the diffusion of solute atoms associated with the movement of the interface. The order parameter can be classified into two categories, namely non-conserved and conserved. If one wishes to model the isothermal austenite-to-ferrite transformation in an Fe–C alloy using the SPF method, a non-conserved order parameter is used to distinguish between the austenite and ferrite phases and to track the migration of the austenite/ferrite interface. In this review, the symbol φ(r, t) is used to represent the non-conserved order parameter. Although the definition of the order parameter value is arbitrary, φ(r, t) is defined as φ(r, t) = 1 in the ferrite (α) phase and φ(r, t) = 0 in the austenite (γ) phase. In the phase-field method, an interface is assumed to have a finite width larger than that in reality, which is known as a diffuse interface. For the SPF model described here, φ(r, t) is defined as 0 < φ(r, t) < 1 at the diffuse interface. Moreover, a conserved order parameter is defined to describe the evolution of the carbon concentration during the transformation, which is denoted by c(r, t) herein. If one wishes to consider the diffusion of other solute atoms (e.g., Mn and Si), a further conserved parameter is defined for the solute atom. Hereafter, for simplicity, (r, t) is omitted unless otherwise stated.

In the phase-field method, the time-evolution equations that represent the time evolutions of the order parameters are derived by assuming that the total free energy of a system decreases gradually over time. For SPF modeling of the isothermal austenite-to-ferrite transformation in an Fe–C alloy, the total free energy functional is defined as:   

$$G={\int }_V\left\{g_{\text{chem}}\left(\varphi ,c\right)+g_{\text{doub}}\left(\varphi \right)+g_{\text{grad}}\left(\nabla \varphi \right)+g_{\text{elast}}\left(\varphi ,c\right)\right\}dV$$(1)

where g_chem(φ, c) denotes the chemical free energy and, for example, is given as:^22,23)   

$$g_{\text{chem}}\left(\varphi ,c\right)=p\left(\varphi \right)g^{\alpha }\left(c,T\right)+\left\{1-p\left(\varphi \right)\right\}{g}^{\gamma }\left(c,T\right)$$(2)

where gⁱ(c, T) is the chemical free energy density of the i phase (i = α and γ), which can be calculated using the thermodynamic theory approach. In this case, the chemical free energy function based on the sublattice model can be employed, along with the parameters derived from the CALPHAD method.^15,24) In addition, in Eq. (2), p(φ) is an interpolation function, wherein p(φ) = φ³(6φ² – 15φ + 10) is often employed; g_doub(φ) is a double-well potential function, which is commonly represented as follows:   

$$g_{\text{doub}}\left(\varphi \right)=Wq\left(\varphi \right)$$(3)

where W represents the height of the energy barrier between the two phases (i.e., the α and γ phases), q(φ) is the double-well potential function (e.g., q(φ) = φ²(1 – φ)²), and g_grad(∇φ) denotes the gradient energy density, which is given by:   

$$g_{\text{grad}}\left(\nabla \varphi \right)=\frac{a^2}{2}{\left|\nabla \varphi \right|}^2$$(4)

In Eq. (4), a is the gradient energy coefficient, and g_elast(φ, c) is the elastic strain energy density that is essential for the modeling martensitic transformation, as reviewed in Section 5. In this case, g_elast(φ, c) can be calculated as follows:   

$$g_{elast}\left(\varphi ,c\right)=\frac{1}{2}{C}_{ijkl}(c)\left({ϵ}_{kl}-{ϵ}_{kl}^0(c)\right)\left({ϵ}_{ij}-{ϵ}_{ij}^0(c)\right)$$(5)

where C_ijkl(c) is the elastic coefficient matrix which is assumed to depend on c in this paper, and ε_ij is the total strain tensor. In addition, ${ϵ}_{ij}^0$ is the eigenstrain tensor, which corresponds to a lattice-misfit strain, a transformation strain, or a plastic strain, for example. As discussed later, it is important to incorporate the temperature- and solute concentration-dependences of the elastic modulus and the eigenstrain tensor into the phase-field model to allow quantitative prediction of the microstructural evolution. To calculate the stress and strain evolutions taking place during phase transformations, the stress equilibrium equation can be solved through use of the finite element method and the micromechanics method based on the phase-field microelasticity theory.²⁵⁾ It should be noted here that the phase-field microelasticity theory solves the stress equilibrium equation by means of a fast Fourier transform (FFT), and is therefore computationally efficient.

Assuming that the microstructure evolves with the gradual reduction of the total free energy (G) expressed by Eq. (1), the time evolution equation for the non-conserved order parameter is given by the Allen-Cahn equation:²⁶⁾   

$$\frac{\partial \varphi }{\partial t}=-M_{\varphi }\frac{\delta G}{\delta \varphi }$$(6)

where δG/δφ represents the functional derivative of the total free energy functional with respect to the non-conserved order parameter,²⁷⁾ and M_φ is the phase-field mobility. The time evolution equation for the conserved order parameter is given by the Cahn-Hilliard equation:²⁸⁾   

$$\frac{\partial c}{\partial t}=\nabla \cdot \left(M_c\nabla \frac{\delta G}{\delta c}\right)$$(7)

where M_c is the diffusion mobility of the solute atoms. The time evolution equations, given by Eqs. (6) and (7) are discretized and numerically solved using numerical calculation methods, such as the finite element method, the finite difference method, and the FFT-based spectral method.

The parameters included in the phase-field model shown above (i.e., W, a, M_φ) are derived based on the sharp-interface limit analysis,²⁹⁾ and are obtained by the functions of various physical properties, as follows:   

$$W=\frac{6\gamma b}{\delta }$$(8)

  

$$a=\sqrt{\frac{3\gamma \delta }{b}}$$(9)

  

$$M_{\varphi }=\frac{\sqrt{2W}}{6a}M$$(10)

Here, b is a constant given by b = 2tanh⁻¹(1 − 2ω), where ω is a constant in the order of 0.1. As shown in Eqs. (8), (9), (10), the parameters of the phase field model depend on the diffuse interfacial width δ, the interfacial energy γ, and the interfacial mobility M. It should therefore be noted that the result of a phase-field simulation depends on the width of the diffuse interface and the physical properties employed. As will be discussed in Section 7, accurate identification and calibration of the physical property values (i.e., γ and M) used in the phase-field model are crucial for improving the predictive accuracy of the phase-field simulation.

2.2. Multi-phase-field Model

The MPF models proposed by Kobayashi et al.,³⁰⁾ Fan and Chen,³¹⁾ and Steinbach and Pezzolla³²⁾ have often been used to simulate the evolution of polycrystalline microstructures in various materials. As an example, this review explains the MPF model of the isothermal austenite-to-ferrite transformation in an Fe–C–Mn alloy,³³⁾ which is based on the MPF model proposed by Steinbach and Pezzolla.³²⁾

In the MPF model, multiple non-conserved order parameters are used to describe the spatiotemporal evolution of multiple crystal grains. Assuming a material is composed of N crystal grains, the non-conserved order parameters are denoted by φ_i (i = 1, 2, 3, …, N), where φ_i = 1 is defined inside the ith grain, and φ_i = 0 in other grains. The variation in φ_i at a diffuse interface is defined as 0 < φ_i < 1. The non-conserved order parameters must satisfy the following condition at an arbitrary coordinate:   

$$\sum _{i=1}^N{\varphi }_i=1$$(11)

The concentration of the alloying element, k (here, k = C and Mn) is defined as the conserved order parameter c^k, which is given by   

$$c^k=\sum _{i=1}^N{\varphi }_i{c}_i^k$$(12)

where c_i^k is the local concentration of k atoms (k = C or Mn) in the ith grain. When a diffusional phase transformation in alloys is modeled using the MPF model, two different methods for calculating the local concentration are often used. The first method is based on the linearized phase diagram and partition coefficient,^34,35) while another method is based on the parallel tangent law, which is commonly referred to as the KKS model.⁷⁾ The MPF model presented in this section is based on the KKS model. It should be noted that the KKS model is not appropriate for phase transformations that occur far from equilibrium. Instead, for a non-equilibrium phase transformation, non-equilibrium MPF models^36,37) that do not employ the KKS model are used.

The total free energy functional for the MPF model is given by:   

$$\begin{array}{l}G={\int }_V[\sum _{i=1}^N\left({\varphi }_i{f}_i\right)+\sum _{i=1}^N\sum _{j=i+1}^N\left(W_{ij}{\varphi }_i{\varphi }_j\right)\\ \hspace{1em}+\sum _{i=1}^N\sum _{j=i+1}^N\left(-\frac{a_{ij}^2}{2}\nabla {\varphi }_i\cdot \nabla {\varphi }_j\right)+\lambda \sum _{i=1}^N\left({\varphi }_i-1\right)]dV\end{array}$$(13)

where the first, second and third terms on the right-hand side of Eq. (13) are the chemical free energy density, the double-obstacle potential, and the gradient energy density, respectively. In addition, f_i represents the chemical free energy of ith crystal grain to which the chemical free energy function based on the CALPHAD method can be coupled.

The time evolution equations of the non-conserved order parameters are given by the Allen-Cahn equation,²⁶⁾ which is defined as:   

$$\frac{\partial {\varphi }_i}{\partial t}=-\sum _{j=1}^N\frac{2M_{ij}}{N}\left(\frac{\delta G}{\delta {\varphi }_i}-\frac{\delta G}{\delta {\varphi }_j}\right)$$(14)

Substituting Eq. (13) into Eq. (14), the following equation is obtained:   

$$\begin{array}{l}\frac{\partial {\varphi }_i}{\partial t}=\\ -\frac{2}{n_{\varphi }}\sum _{j=1}^{n_{\varphi }}{M}_{ij}^{\varphi }\left[\sum _{k=1}^{n_{\varphi }}\left\{\left(W_{ik}-W_{jk}\right){\varphi }_k+\frac{1}{2}\left(a_{ik}^2-a_{jk}^2\right){\nabla }^2{\varphi }_k\right\}-\frac{8}{\pi }\sqrt{{\varphi }_i{\varphi }_j}\mathrm{\Delta }{E}_{ij}\right]\end{array}$$(15)

where n_φ is the number of non-zero non-conserved order parameters at arbitrary coordinates, and ΔE_ij denotes the driving force of the phase transformation. In the right-hand side of Eq. (15), $-8/\pi \sqrt{{\varphi }_i{\varphi }_j}$ is added for the stable computation. When the KKS model is employed, the driving force is expressed as follows:   

$$\mathrm{\Delta }{E}_{ij}=f_j-f_i-\sum _{k=1}^n{\tilde{\mu }}^k\left(c_j^k-c_i^k\right)$$(16)

where n denotes the total number of solute species. When an Fe–C–Mn alloy is considered, n is equal to 2. In addition, ${\tilde{\mu }}^k$ is given by the following equation to satisfy the parallel tangent law:   

$${\tilde{\mu }}^k\equiv \frac{\partial f_i}{\partial c_i^k}=\frac{\partial f_j}{\partial c_j^k}$$(17)

The time evolution equations of the conserved order parameter are given by the Cahn-Hilliard equation²⁸⁾ as follows:   

$$\frac{\partial c^k}{\partial t}=\nabla \cdot \left(\sum _{l=1}^n{M}_{kl}^c\nabla \frac{\delta G}{\delta c^l}\right)$$(18)

where M_kl^c is the diffusion mobility of the solute element k with respect to the potential gradient of solute element l and is given by:   

$$M_{kl}^c=\sum _{j=1}^N{\varphi }_j{M}_{kl}^{cj}$$(19)

where M_kl^cl is the diffusion mobility of the solute atom k with respect to the potential gradient of solute element l in the jth crystal grain.

As described in the previous section for the SPF model, the parameters included in the MPF model are given as functions of various physical properties:   

$$W_{ij}=\frac{4{\gamma }_{ij}}{\delta }$$(20)

  

$$a_{ij}=\frac{2}{\pi }\sqrt{2\delta {\gamma }_{ij}}$$(21)

  

$$M_{ij}^{\varphi }=\frac{{\pi }^2}{8\delta }{M}_{ij}$$(22)

where γ_ij, δ, and M_ij are the interfacial energy between the ith and jth grains, the width of the diffuse interface, and the mobility of the interface between the ith and jth grains, respectively. As indicated by Eqs. (20), (21), (22), the result of the MPF simulation depends on the selection of the diffuse interface width δ and the values of the physical properties (i.e., γ_ij and M_ij).

3. Austenite-to-ferrite Transformation

3.1. Formation of Allotriomorph Ferrite

During a continuous cooling process from the high-temperature range in which the austenite phase is stable, the austenite-to-ferrite transformation takes place to form polygonal ferrite at the austenite grain boundaries. This morphology of the ferrite phase is referred to as allotriomorphic ferrite. The phase-field modeling of allotriomorphic ferrite formation was reviewed previously,¹⁹⁾ and so in the current review, we focus on recent progress and future challenges in the phase-field modeling of allotriomorphic ferrite formation.

One of the pioneering studies on the SPF modeling of the austenite-to-ferrite transformation was conducted by Yeon et al.,³⁸⁾ who investigated the growth of the ferrite phase in an Fe–C–Mn alloy under local equilibrium (LE) and para-equilibrium (PE) conditions. Subsequently, Loginova et al.³⁹⁾ presented excellent work on the phase-field modeling of the austenite-to-ferrite transformation in Fe–C alloys, in which the phase-field model was developed by extending the phase-field model of solidification in a binary alloy. A notable feature of Loginova’s phase-field model is that both diffusion-controlled growth and massive growth of the ferrite phase were successfully simulated by introducing the CALPHAD-based chemical free energy into the phase-field model. Later, pioneering work on the MPF modeling of allotriomorphic ferrite formation in polycrystalline austenite was presented by Huang et al.,⁴⁰⁾ who employed an MPF model based on the Kobayashi-Warren-Cater (KWC) approach.³⁰⁾ Subsequently, two- and three-dimensional MPF simulations of allotriomorphic ferrite formation were extensively studied,^41,42,43) wherein the nucleation conditions of the ferrite phase were investigated, along with the ferrite/austenite interface mobility.⁴³⁾ It is notable that MPF models for examination of the austenite-to-ferrite transformation in Fe–C–Mn alloys had already been proposed in the early 2000s.⁴²⁾ A review by Militzer¹⁹⁾ provides further details regarding the above-mentioned MPF modeling, should the reader wish to consult further background in this area.

To apply MPF models to practical steel-making processes, such as hot rolling and heat treatment processes, further advances in the MPF modeling of ferrite formation in a deformed austenite phase are also important. In this context, Yamanaka et al.⁴⁴⁾ proposed the numerical modeling of a deformed austenite-to-ferrite transformation by combining the crystal plasticity finite element and MPF methods. They calculated the nucleation rate of the ferrite phase in the deformed austenite phase based on the classical nucleation theory,⁴⁵⁾ and analyzed the growth of the ferrite phase using the MPF model. Furthermore, the rate of ferrite nucleation at the austenite grain boundary corner and edge was set higher than that at the grain boundary face. However, establishing a theory and calculation method to determine the priority of ferrite nucleation sites in polycrystalline austenite remains a challenge. It is well known that the ferrite/austenite interface, which deviates from the Kurdjumov-Sachs orientation relationship (KS OR) with the austenite phase, is more mobile than that of the near-KS OR.^46,47) As a result, the allotriomorphic ferrite formed at the austenite grain boundary often exhibits one-sided growth into either of the neighboring austenite grains. However, no MPF model has been developed that can quantitatively reproduce such a one-sided growth of the ferrite phase based on the KS OR. Thus, to accurately simulate the formation of allotriomorphic ferrite along austenite grain boundaries, the effect of the KS OR on the interfacial energy and mobility should be introduced in future MPF models.

MPF modeling of the austenite-to-ferrite transformation, which includes the diffusion of substitutional alloying elements in multi-component alloys, such as Fe–C–Mn ternary alloys and Fe–C–Mn–Si quaternary alloys, has advanced in recent years. In this context, Chen et al.⁴⁸⁾ performed an MPF simulation of the cyclic austenite-to-ferrite transformation to investigate the interactions between the migration of the ferrite/austenite interface and the solute diffusion in multi-component alloys. Their MPF simulation revealed that a spike in the content of the substitutional alloying element at the ferrite/austenite interface triggers the stagnation stage where the austenite-to-ferrite transformation is suppressed despite a temperature decrease.⁴⁹⁾ Later, as shown in Fig. 1, Segawa et al. also investigated the stagnant stage in the cyclic transformation taking place in Fe–C–Mn–Si alloys using the non-equilibrium MPF mode.³³⁾ Moreover, Kohtake et al. analyzed the transition of the transformation mode from the PE mode to the partitionless local equilibrium (PLE) mode during the austenite-to-ferrite transformation in an Fe–C–Mn alloy using the MPF model.⁵⁰⁾ As discussed above, the modeling substitutional alloying element redistribution and the alloying element spike at the ferrite/austenite interface strongly affect the MPF simulation results for the diffusional austenite-to-ferrite transformation. Therefore, further advances in phase-field modeling of the austenite-to-ferrite transformation, including the effects of solute drag and alloying element segregation on the moving ferrite/austenite interface, would be expected to contribute to advances in the quantitative phase-field simulations of allotriomorph ferrite formation.

Fig. 1.

Volume fraction of the ferrite phase during the cyclic austenite-to-ferrite and ferrite-to-austenite transformations in (a) Fe–C–Mn, (b) Fe–C–Mn–Si, and (c) Fe–C–Mn–Si alloys calculated using the non-equilibrium MPF model after Segawa et al.,³³⁾ reprinted with permission from Elsevier. The stagnant stage is highlighted by a bold line.

3.2. Formation of Ferrite Sideplates

During continuous cooling, the morphology of the ferrite phase changes from polygonal to plate-like; these morphologies are more commonly known as acicular ferrite and Widmanstätten ferrite, and they are important microstructures for tailoring the mechanical properties of the heat-affected zone in welded steel. For the purpose of the current review, the acicular and Widmanstätten ferrite morphologies are referred to as ferrite sideplates.

To model the formation of ferrite sideplates using the phase-field method, the formation mechanism should be clarified. To date, two key mechanisms have been proposed, namely the diffusion-controlled mechanism that depends on the interface instability, and the interface-controlled mechanism that is governed by the elastic strain energy. This section reviews the phase-field modeling of ferrite sideplate formation using the SPF and MPF models, wherein both of the above mechanisms are examined.

The key pioneering work on the phase-field modeling of ferrite sideplates was performed by Loginova et al.,⁵¹⁾ who modeled the formation of Widmanstätten ferrite in Fe–C alloys based on the diffusion-controlled mechanism. The formation of Widmanstätten ferrite was reproduced by introducing a strong anisotropy of the ferrite/austenite interfacial energy into their developed phase-field model. Subsequently, Yamanaka et al.^22,23) analyzed the Widmanstätten ferrite growth process using different approaches to incorporate a strong interfacial energy anisotropy using the facet interface model developed by Eggleston et al.⁵²⁾ Furthermore, Yan et al.⁵³⁾ analyzed the growth rate of Widmanstätten ferrite by introducing the KKS model into Loginova’s PF model, and compared the calculated growth rate with the experimental results. More recently, Bhattacharya et al. proposed the MPF model of Widmanstätten ferrite formation and reported that the MPF model reproduces the experimental observations for the plate growth rate.⁵⁴⁾

Phase-field modeling of ferrite sideplate formation based on an interface-controlled mechanism has also advanced in recent years. As shown in Fig. 2, Cottura et al.⁵⁵⁾ reproduced the formation of a Widmanstätten plate driven by the stress field generated by anisotropic transformation strain. In their later work, they revealed that stress relaxation by plastic deformation (i.e., plastic accommodation) reduces the growth rate of the Widmanstätten plate.⁵⁶⁾ Furthermore, Amos et al.⁵⁷⁾ proposed a chemo-elastic phase-field model and quantitatively analyzed the growth rate of the ferrite sideplates.

Fig. 2.

Lengthening of a Widmanstätten plate as a function of time and distribution of the von Mises equivalent stress in the matrix phase after Cottura et al.,⁵⁶⁾ reprinted with permission from Elsevier.

Since the pioneering work of Loginova et al., PF modeling of ferrite sideplate growth has progressed. However, to quantitatively predict ferrite sideplate formation, it is necessary to obtain an accurate interface mobility and interface energy, which depend on several physical factors, such as the orientation relationship between the ferrite and austenite phases and the temperature. Moreover, because the interfacial curvature strongly affects the migration rate of the ferrite/austenite interface, a three-dimensional phase-field simulation should be performed to quantitatively predict the experimental growth rate of the ferrite sideplates.

4. Pearlitic Transformation

During the continuous cooling process, when the temperature decreases below the onset temperature of the pearlite transformation, a pearlitic lamella consisting of ferrite and cementite two-phase layered structures is formed. Previous phase-field modeling of the pearlitic transformation has mainly focused on the growth rate and morphology of the pearlitic lamellae. Thus, the current section reviews recent advances in the phase-field modeling of pearlitic transformations.

The key pioneering work on the phase-field modeling of pearlitic transformation was carried out by Nakajima et al.,⁵⁸⁾ who investigated the effect of the diffusion path of carbon atoms in the pearlite lamellae on the pearlite growth rate in Fe–C alloys. Their results showed that the obtained growth rate was comparable to previously reported experimental data based on carbon diffusion through both the austenite bulk phase and the ferrite phase. Moreover, Steinbach and Apel⁵⁹⁾ investigated the effect of the lattice strain-induced stress field on the growth rate of a pearlitic lamella. They found that the stress field suppressed cooperative growth of the ferrite and cementite phases, resulting in staggered growth of the cementite phase. As shown in Fig. 3, they found that the calculated growth rate of the pearlitic lamellae was close to the experimental data when considering the staggered growth of the cementite phase.

Fig. 3.

Comparison of the growth rate of pearlite lamella versus temperature. The results obtained using the Zener-Hillert model, the phase-field simulations, and previous experimental data are compared after Steinbach and Apel,⁵⁹⁾ reprinted with permission from Elsevier.

Phase-field analysis of the growth rate of pearlitic lamellae was further performed by Mouri et al.⁶⁰⁾ As shown in Fig. 4, they investigated the effects of the interfacial diffusion of carbon atoms on the lamellar growth rate in Fe–C alloys. In previous studies on the phase-field modeling of the pearlite transformation, the chemical free energy of the cementite phase was calculated assuming a cementite phase with a stoichiometric chemical composition. In contrast, Mouri et al. used the sublattice model based on the CALPHAD method to calculate the chemical free energy of the cementite phase to accurately determine the growth rate of the pearlitic lamellae. Their simulation results showed that as in the above cases, the calculated growth rate of the pearlitic lamellae was still smaller than the experimentally derived rates, even when the interfacial diffusion of carbon atoms was taken into account.

Fig. 4.

Growth of a single pearlite lamella and evolution of the carbon concentration as simulated by the phase-field method considering carbon diffusion (a) only through the austenite phase and (b) through both the austenite phase and the interphase boundary.⁶⁰⁾

More recent work on the phase-field modeling of the pearlitic transformation focused on the divergence of pearlitic lamellae in Fe–C–Mn alloys.⁶¹⁾ It was reported that the Mn concentration distribution at the growth tip of a pearlitic lamella affects the growth rate and morphology of the diverged pearlite. Furthermore, the same research group analyzed the non-steady-state growth of diverged pearlite using the phase-field model.⁶²⁾

The annealing of pearlite is often used to control the mechanical properties of pearlitic microstructures, since spheroidization of the cementite phase occurs during the annealing process. In this context, Amos et al. reported a detailed three-dimensional MPF simulation of cementite spheronization,⁶³⁾ wherein they clearly demonstrated that the phase-field method enables investigation of the spatiotemporal process for plate-like cementite spheronization.

However, the phase-field modeling of pearlitic transformations remains a challenge, since it is necessary to quantitatively investigate the lamellar growth rate. As a result, a three-dimensional MPF simulation of the pearlite lamellar growth is required to accurately consider the grain boundary curvature. Furthermore, as discussed by Mouri et al.,⁶⁰⁾ it is necessary to develop an accurate database of physical properties, such as the diffusion coefficient of carbon, the interfacial energy, and the interfacial mobility, to reproduce the experimentally-observed growth rate of pearlite lamellae using the phase-field method.

5. Martensitic Transformation

The martensite phase, which is formed by quenching, exhibits various morphologies depending on the chemical alloying composition and temperature. This microstructure is used to increase the strength of steel. Therefore, understanding the martensitic phase formation process is important for the design and development of high-strength steel. The typical morphologies of the martensite phase are referred to as the lens, the lath, and the plate morphologies.⁶⁴⁾ To date, the majority of studies on the phase-field modeling of the martensitic transformation have focused on the formation of lath martensite since this microstructure is known to improve the strength–toughness balance of high-strength steel. Since Mamivand et al.²¹⁾ reviewed phase-field models for simulating martensitic transformation, the current review focuses only on recent progress in this area.

The phase-field modeling of martensitic transformation in alloys can be categorized into various types, as suggested by Mamivand et al.²¹⁾ Among the previous phase-field models of martensitic transformation, the most pioneering work involved the development of MPF models based on the phase-field microelasticity theory.^25,65) In particular, the first phase-field model of martensitic transformation was developed by Wang and Khachatruyan for the improper cubic-tetragonal transformation.⁶⁶⁾ Subsequently, the same research groups developed additional phase-field models and simulated martensitic microstructure formation in both ferrous and non-ferrous alloys.^67,68,69) The major contribution of these phase-field models is that the self-organized multivariant martensitic microstructure was successfully reproduced using phase-field models, thereby opening a new era of numerical modeling for the martensitic transformations taking place in alloys.

The morphology of the martensitic microstructure is characterized by the relaxation of the stress field generated by the transformation strain. In previous studies, two types of stress relaxation mechanisms have been reported, namely self-accommodation, in which multiple-variant structures are formed to relax the stress field, and plastic accommodation, which relaxes the stress field by plastic deformation in the austenite and martensite phases. In this context, Yamanaka proposed the first phase-field model of martensitic transformation considering both stress accommodation mechanisms,^70,71) wherein an elastoplastic phase-field model was employed.⁷²⁾ Subsequently, Malik et al.⁷³⁾ extended the elastoplastic phase-field model and conducted a three-dimensional simulation of martensite microstructure formation. Moreover, they used the phase-field model to investigate the effects of external stress on the transformation kinetics and microstructural morphology,⁷⁴⁾ the nucleation of the martensite phase,^75,76,77) the reverse transformation from the martensite to the austenitic phase,⁷⁸⁾ and the austenite grain size dependency on the martensite morphology.⁷⁹⁾

Recently, Shchyglo et al. proposed a new phase-field model to simulate lath martensite formation.⁸⁰⁾ To reproduce the lath martensite microstructure with 24 KS variants,⁸¹⁾ they considered 24 rotational components of the deformation gradient tensor related to the martensitic transformation, and they also proposed solving the time evolution equations for 24 different non-conserved order parameters for the 24 KS variants. It was found that the phase-field model can reproduce microstructure formation, giving similar results to those obtained experimentally for lath martensite formation. Using a similar approach, Ahlwalia et al. developed an elasto-plastic phase-field model of martensitic transformation using 24 different non-conserved order parameters.⁸²⁾ Their phase-field models, in which the 24 order parameters were used to describe the formation of the KS variants, are based on a phenomenological concept. As discussed below, to model the lath martensite formation, it might be natural to consider that formation of the 24 KS variants is a result of the plastic accommodation of the stress field generated by the Bain strain in the martensite phase.

Later, the phase-field modeling of lath martensite formation in low-carbon steels, which is different from the above-mentioned models,^80,82) was developed by Murata and co-workers.^{83,84,85,86,87)} They proposed a new theory of plastic accommodation in the martensitic transformation by considering the slip deformation in the martensite phase, which was named the “two types of slip deformation” (TTSD) model. Using the TTSD model, it was shown theoretically for the first time that the hierarchical structures of lath martensite and the KS variants are formed as a result of slip deformation in the martensite phase.^83,84) Furthermore, by combining the TTSD model with the MPF model that was originally developed to simulate dislocation dynamics,⁸⁸⁾ the elastoplastic phase-field model was developed to clearly explain the formation mechanism of lath martensite.⁸⁴⁾ This phase-field model does not use the 24 different order parameters but defines a new non-conserved order parameter to describe dislocation slip in the martensite phase. Thus, the plastic accommodation during lath martensite microstructure formation was analyzed by solving the time evolution equations of the non-order parameters. As shown in Fig. 5, Tsukada et al. demonstrated that the elastoplastic phase-field model based on the TTSD model reproduces the formation of a (111) habit plane during lath martensite formation.⁸⁵⁾ Subsequently, the phase-field model was also applied to the modeling of lath martensite recovery.⁸⁷⁾ In the future, further development of the phase-field model is expected for the quantitative prediction of not only lath martensite formation, but also other martensite morphologies in polycrystalline austenite.

Fig. 5.

Habit plane of the martensitic cluster simulated using the elastoplastic phase-field model based on the TTSD model. The blue-colored plane shown in the left-hand image represents the (111)γ plane.⁸⁵⁾

6. Bainitic Transformation

The bainitic microstructure, as well as the martensitic phase, is increasingly essential for the design and development of advanced high-strength steels. The bainitic microstructure is formed by the intermediate transformation mechanism between the diffusional and the displacive (martensitic) transformations. Similar to the martensite phase, the bainitic microstructure exhibits various microstructural morphologies, such as upper and lower bainitic microstructures, depending on the chemical composition of the alloys and the temperature. The competition between the formation of bainitic ferrite (i.e., the migration of the ferrite/austenite interface) and carbon diffusion is a key phenomenon in understanding the mechanism of bainitic microstructure formation in steels. According to the traditional theory of bainitic transformation kinetics, the stress field around the bainitic ferrite generated by the transformation strain is relaxed by the plastic accommodation over a relatively low-temperature range, similar to in the case of the martensitic transformation. This plastic accommodation introduces dislocation defects into the bainitic microstructure. Therefore, to model the formation of the bainitic microstructure using the phase-field method, it is important to correctly analyze not only the chemical driving force for the bainitic transformation, but also to consider the elastic and plastic deformations in the austenite and bainitic ferrite phases.

The work of Arif and Qin in 2013 was a pioneering work on the phase-field modeling of the bainitic transformation in steel.⁸⁹⁾ More specifically, they proposed the first SPF modeling of the formation of bainitic ferrite in Fe–C alloys. This SPF model was coupled with the CALPHAD-based chemical free energy function, and the elastic field was calculated based on the inclusion theory. As shown in Fig. 6, they performed three-dimensional phase-field simulations of bainitic ferrite formation and successfully simulated the formation of a bainite sheaf by autocatalytic nucleation.⁹⁰⁾ However, it should be noted that the formation of carbides in the bainitic microstructure was not considered in this work.

Fig. 6.

A sheaf of bainitic ferrite simulated by the phase-field model after Arif and Qin,⁸⁹⁾ reprinted with permission from Elsevier.

In a series of studies by Düsing and Mahnken,^91,92) MPF models were proposed to analyze the formation of lower bainitic microstructures and investigate carbide formation inside the bainitic ferrite. In a subsequent study, Düsing and Mahnken extended their MPF model⁹²⁾ to investigate upper and lower bainite formations,⁹³⁾ focusing on the locations of carbide formation. Furthermore, in a recent study, the influence of an elastic field on the formation of bainitic microstructures was examined, wherein the formation of upper and lower bainitic microstructures were evaluated based on the competition between the diffusional and the displacive transformations.⁹⁴⁾

Subsequently, Toloui et al. proposed the MPF modeling of carbide-free bainitic microstructure formation without cementite precipitation using the MICRESS software.⁹⁵⁾ In their study, the nucleation and interfacial anisotropies of bainitic ferrite were chosen phenomenologically to fit the volume fractions of the bainitic microstructure measured by their experiments. This MPF model was also recently used to model the microstructural evolution in TRIP steel.⁹⁶⁾

In the last few years, a number of articles have been published related to the phase-field modeling of bainite microstructure formation, wherein both diffusional and displacive transformations were considered. More specifically, Elhigazi and Artemev⁹⁷⁾ proposed an MPF model for modeling bainitic-type transformations, considering both diffusional and non-diffusional transformation kinetics, and reported that the model represented the change in the microstructural morphology with variation in the transformation rate. In addition, Schoof et al. evaluated the elastic field around the bainitic ferrite formed by the fully-displacive transformation using the MPF method. As shown in Fig. 7, their results provided interesting details regarding the elastic field and the elastically preferred region (i.e., the well-favored region shown in Fig. 7) for autocatalytic nucleation around the initially formed bainitic ferrite.⁹⁸⁾ To accurately calculate the chemical driving force for the bainitic transformation, it would be desirable to calculate the competition between carbon diffusion and migration of the ferrite/austenite interface, as discussed by Düsing et al.⁹⁴⁾

Fig. 7.

The elastically preferred (favored) region for the autocatalytic nucleation and growth of bainitic ferrite and its temporal evolution with the growth of the nucleus after Schoof et al.,⁹⁸⁾ reprinted with permission from Elsevier.

Although there are many phase-field models for the bainitic transformation, few phase-field models reported so far can reproduce the realistic bainitic microstructure morphology observed in experiments. This may not be due to the inability of the phase-field method to represent the physical phenomenon related to bainitic transformation. Further understanding of the bainitic transformation mechanism is still needed to improve the phase-field modeling capability of the bainitic transformation proposed in previous studies. Furthermore, because the bainitic transformation is the intermediate transformation between the diffusional transformations such as the ferritic and pearlitic transformations and the displacive (martensitic) transformation, the issues on the phase-field models described in Sections 3 to 5 must be addressed for further improvement. In particular, experimental data on temperature-dependent physical property values and parameters, such as the transformation strain, interfacial energy, and interfacial mobility, are important for further development of phase-field models of bainitic transformation.

7. Challenges and Perspectives

As outlined above, the phase-field modeling of solid-state phase transformations in steels and ferrous alloys has progressed significantly in recent years, with new phase-field models for the ferrite, pearlitic, martensitic, and bainitic transformations have been proposed. Further development of these phase-field models would therefore be expected to lead to quantitative simulations and the prediction of realistic microstructural morphologies. The challenges and future perspectives for the further development of phase-field models are discussed below.

7.1. Quantitative Phase-field Modeling

As described in Section 2, due to the fact that the parameters of the phase-field models (e.g., W and a for the SPF model) are functions of the width of the diffuse interface, the result of the phase-field simulation also depends on the width of the diffuse interface. This is a vital issue, especially when considering the phase-field modeling of diffusional phase transformations in multicomponent alloys. Furthermore, conventional phase-field models with a diffuse interface, whose parameters are derived based on sharp-interface limit analysis,⁶⁾ suffer from abnormal interfacial effects associated with surface diffusion, interface stretching, and discontinuity of the diffusion potential. The issue of abnormal interfacial effects can be solved using a quantitative phase-field model based on thin-interface limit analysis.^{100,101,102,103,104)} The development of quantitative phase-field models has preceded the field of solidification modeling,⁹⁹⁾ and so to further improve the phase-field models reviewed herein, the development of a quantitative phase-field model for solid-state phase transformations would be desirable.

7.2. Nucleation

It is difficult for the phase-field method to describe the nucleation behavior of the newly produced phase since it is based on the principle of total free energy minimization. However, nucleation modeling, which can be incorporated into the phase-field method, is key to quantitatively predicting realistic microstructural evolutions in the solid-state phase transformations of steel.

As is well known, two types of nucleation behaviors exist, namely homogeneous and heterogeneous nucleation. Heterogeneous nucleation is particularly important for predicting microstructural evolution during solid-state phase transformations, such as in the case of the heterogeneous nucleation of the ferrite phase at austenite grain boundaries. Due to its importance, nucleation modeling for phase-field simulations has been studied for a number of years. For example, as reviewed by Heo and Chen,¹⁰⁵⁾ the Langevin noise method¹⁰⁶⁾ and the explicit nucleation method¹⁰⁷⁾ have been employed to simulate heterogeneous nucleation behavior in a phase-field simulation. Although the explicit nucleation method may be a practical methodology to simulate heterogeneous nucleation, in which the size of the nucleus and the nucleation rate are determined based on other theories, such as the classical nucleation theory, it should be noted that the classical nucleation theory assumes the spherical shape of a nucleus. On the other hand, the non-classical approach to predict the morphology of a nucleus has also been actively studied. More specifically, Zhang et al.^108,109) investigated the critical nucleus morphology using the total free energy functional given by the phase-field approach and the minimax method, which is a mathematical variational method. They showed that the morphology of the critical nuclei formed during the solid-state phase transformation is not always spherical, and that it strongly depends on the contribution from the elastic strain energy. In addition, Song et al.¹¹⁰⁾ investigated the morphology of the ferrite phase nucleated at the austenite grain boundary in pure iron using a combination of the MPF method and the nudged elastic band (NEB) method, which is a mathematical algorithm to search the saddle point on the minimum energy path of the free energy functional space.¹¹¹⁾ Using their combined MPF-NEB method, they successfully demonstrated that the facetted morphology of the ferrite phase nucleated at the austenite grain boundary was in good agreement with that predicted by the molecular dynamics simulation of the heterogeneous ferrite nucleation.^112,113,114) The experimental observation of heterogeneous nucleation during the solid-state transformation in iron and steel is difficult even when state-of-the art experimental observation techniques are used. Thus, further exploration using a multi-scale simulations based on first principles calculations, molecular dynamics simulations, and the phase-field method are crucial for understanding and modeling heterogeneous nucleation, and would be expected to lead to quantitative prediction of the transformation kinetics and a realistic microstructural morphology.

7.3. Parameter Identification

As described in Section 2, the phase-field method is a phenomenological continuum model; thus, the empirical parameters and physical property values must be known a priori. Accurate identification of these parameters and physical properties is therefore required to realize the quantitative prediction of microstructural evolution using phase-field methods. To address this issue, multiscale simulations could be employed, in which the physical properties and parameters are obtained from first-principles and molecular dynamics calculations. Another effective solution may involve a combination of the phase-field method with first-principles calculations (i.e., the first-principles phase-field (FPPF) method).^{115,116,117,118)} This would be desirable since it is not necessary to couple the FPPF method with the CALPHAD method because the first-principles free-energy calculations are performed simultaneously with the phase-field simulations. Furthermore, because the FPPF method does not require empirical parameters, it clearly represents one of the next-generation phase-field methods, and so in the future, FPPF models are expected to be applied to solid-state phase transformations in steel.

Moreover, data-driven phase-field modeling with experimental data obtained from advanced experimental observation techniques will also be key to further progress in the development of phase-field methods. In the area of aluminum alloy solidification, a pioneering study for material parameter identification was performed by combining phase-field simulations with in-situ observations of dendrite solidification achieved using synchrotron radiation.¹¹⁹⁾ Furthermore, Zhang et al. combined the multi-phase-field simulation with the in-situ observation of grain growth in pure iron to perform the inverse identification of the grain boundary properties.¹²⁰⁾

Finally, data-driven phase-field modeling using the data assimilation technique based on the Bayesian inference is a promising method for estimating unknown material parameters and improving the predictive accuracy of phase-field simulations.^121,122,123) Indeed, multiple studies have reported the inverse estimation of physical property values and parameters from experimental results by implementing various Bayesian data assimilation algorithms into phase-field models. For example, in the case of steel, Ensemble Kalman Filter-based data assimilation methods have been applied to the phase-field simulations of the austenite-to-ferrite transformation,¹²⁴⁾ the polycrystalline grain growth,¹²⁵⁾ the solidification,^126,127) and the dendrite solidification in Fe–C–Mn alloys.¹²⁸⁾ The advantage of Bayesian data assimilation is that it enables not only the inverse estimation of physical property values and parameters from experimental data but also the evaluation of the uncertainty of the estimated parameters.

8. Conclusions

The current review focused on recent advances in the phase-field modeling of solid-state phase transformations in iron and steel, and a number of advantages associated with the phase-field modeling approach were highlighted. Firstly, the phase-field method is a powerful numerical modeling tool that can be applied to various solid-state phase transformations in iron and steel because it can simulate multi-physics phenomena, such as the migration of interfaces, the diffusion of solute atoms, and evolution of the stress–strain fields during phase transformations. Owing to the recent development of the calculation phase diagram (CALPHAD) method, which can be easily coupled with the phase-field method, a number of phase-field models have been developed for modeling basic solid-state phase transformations in steel, such as the ferrite, pearlitic, bainitic, and martensitic transformations. Although the advantages of the phase-field method are indisputable, the solid-state phase transformations taking place in iron and steel are extremely complex, and several issues must be solved in future research. For example, current phase-field models are generally unable to easily provide quantitative simulation results for the solid-state phase transformations taking place in multi-component steels containing substitutional alloying elements. In particular, it is difficult to eliminate the influence of the diffuse interface width on the results of the phase-field simulation (i.e., the abnormal interfacial effect described in Section 7.1). Quantitative phase-field models should therefore be developed for solid-state transformations in future studies. In addition, in the phase-field modeling of the martensitic transformation, the introduction of plasticity calculations into the phase-field model should be considered. As in the phase-field model of lath martensite formation developed by Murata and co-workers, the incorporation of the crystal plasticity concept into the phase-field model is crucial to the quantitative simulation of martensitic microstructure formation. Furthermore, due to the fact that solid-state phase transformations in steel often occur continuously during the continuous cooling process, integration of the developed phase-field models for basic phase transformations is crucial to extend their utility. Moreover, the theoretical understanding of heterogeneous nucleation and its modeling methodology, which is suitable for combination with the phase-field method, should be deepened further. Finally, to further develop the phase-field models, additional experimental data are required relating to the temperature and chemical composition dependences of the physical property values and parameters, such as the interfacial energy and interfacial mobility. To solve this issue, data-driven phase-field modeling as well as multi-scale modeling coupled with first-principles calculations and molecular dynamics simulations should be considered in future research.

Acknowledgement

The author acknowledges the support of this research by a Grant-in-Aid for Scientific Research (B) (JSPS KAKENHI Grant No. 20H02476) from the Japan Society for the Promotion of Science (JSPS).

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