Fe-C peritectic solidification of polycrystalline ferrite by phase-field method

Computational Materials Science 178 (2020) 109626

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Letter

Fe-C peritectic solidification of polycrystalline ferrite by phase-field method Chao Yang

a,d

, Xitao Wang

b,c,⁎

, Hasnain Mehdi Jafri

a,d

a,d

, Junsheng Wang

, Houbing Huang

a,d,⁎

T

a

School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, China Collaborative Innovation Center of Steel Technology, University of Science and Technology Beijing, Beijing 100083, China Shandong Provincial Key Laboratory for High Strength Lightweight Metallic Materials, Advanced Materials Institute, Qilu University of Technology (Shandong Academy of Science), Jinan, 250353, China d Advanced Research Institute of Multidisciplinary Science, Beijing Institute of Technology, Beijing 100081, China b c

A R T I C LE I N FO

A B S T R A C T

Keywords: Polycrystalline solidification Peritectic transformation Fe-C alloy Multi-phase-field model

A multi-phase-field model was applied to simulate the polycrystalline solidification of δ phase (δ-ferrite) and peritectic solidification of γ phase (γ-austenite) in Fe-C system. Four main modes of phase transformation are included in this simulation, which are solidification of δ phase L → δ, solidification of γ phase L → γ, peritectic transformation δ → γ and peritectic reaction L + δ → γ. The results reveal that the redistribution and diffusion of carbon not only controls the movement of δ-liquid, γ-liquid and γ-δ interfaces but also lead to the production of liquid channels and melting pools. In addition, the evolution of the volume fractions of each phase was analyzed, which shows the statistical law of the interaction of δ, γ and liquid phases at different stages. The present study therefore contributes to the understanding of polycrystalline simulation of Fe-C peritectic solidification and clarifies the formation mechanism of microstructure and micro-segregation.

1. Introduction The microstructure and micro-segregation produced by the peritectic solidification of the Fe-C system have a significant effect on the properties of the steel. As a classical peritectic process, the transformation from δ phase to γ phase in Fe-C system has been extensively studied. The microscopic process of this peritectic solidification includes the nucleation and growth of δ phase, redistribution and diffusion of carbon, peritectic growth of γ phase and interaction of grains, etc. Microstructure morphology and carbon distribution of the Fe-C peritectic solidification are determined by the polycrystalline multiphase transformation with interfacial interactions and carbon diffusion. Although a large number of experiments reveal the role of different factors in this process, it is still difficult to quantitatively predict and control the evolution of the microstructure and micro-segregation by the available theoretical or experimental methods. Therefore, the kinetic mechanism of Fe-C peritectic solidification can be further explored by simulation, which provides significant references for the process control of microstructure [1–4]. As an effective simulation method for microstructure evolution, the phase-field method can couple multi-physics to simulate the peritectic solidification of Fe-C system, which can consider multi-phase [5], polycrystal [6], multi-component [7] and other factors into the model

[8]. For the microscopic phenomenon of the Fe-C peritectic solidification, the nucleation of δ phase and γ phase, the formation of dendrites, the diffusion of carbon, and the collision and interaction of grains are all important parts in the formation of final microstructure. Therefore, a systematic simulation that includes all of these aspects requires a combination of multi-phase field, concentration field, and orientation field. In recent years, progress has been made in the study of Fe-C peritectic solidification by experiments and simulations. Shibata et al. [9] studied the rates of the peritectic reaction of Fe-C system by in situ dynamic observations, which focused on the moving speeds of γ-liquid and γ-δ interfaces and the influence of carbon diffusion. Pan et al. [10,11] applied both in-situ observation and multi-phase-field model to study the evolution of austenite platelet during the Fe-C peritectic solidification process, which presented the peritectic solidification mechanism near the coexistence region of ferrite, austenite and liquid phase. Alves et al. [12,13] studied the peritectic solidification of Fe-Mn alloys, highlighting the effect of grain anisotropy on peritectic reaction. However, these studies have not yet explored the peritectic transformation of polycrystalline grains. Especially, the mechanism of peritectic growth of equiaxed grains with different orientations is not fully understood. Therefore, our research focuses on Fe-C peritectic solidification of

⁎ Corresponding authors at: Collaborative Innovation Center of Steel Technology, University of Science and Technology Beijing, Beijing 100083, China (X. Wang). School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, China (H. Huang). E-mail addresses: [email protected] (X. Wang), [email protected] (H. Huang).

https://doi.org/10.1016/j.commatsci.2020.109626 Received 7 February 2020; Received in revised form 24 February 2020; Accepted 26 February 2020 0927-0256/ © 2020 Published by Elsevier B.V.

Computational Materials Science 178 (2020) 109626

C. Yang, et al.

simulated. For the two-dimensional growth simulation of anisotropic equiaxed grains, the anisotropy of each grain is based on a fixed orientation, which is a reference angle from 0 to 2π. Orientation field of Eq. (5) is introduced to provide different reference angles for phasefield to obtain a polycrystalline solidification system. According to the general method of polycrystalline evolution simulation [19–22], the following equation is developed as the dynamical equation of the orientation parameter,

polycrystalline ferrite by multi-phase-field model, which includes the equiaxed grains nucleation and growth of δ phase, heterogeneous nucleation and peritectic growth of γ phase grains along with the statistical analysis of volume fractions of each phase. At the scale of single grain, the simulation results not only explain how carbon diffusion affects the morphology of the dendrite but also clarify the causes of liquid channels and melting pools in the process of peritectic solidification. At the scale of multiple grains, the simulation presents the interaction between grains and the formation of grain boundary segregation. The final order parameter distribution and carbon distribution of the polycrystalline solidification reveal the microstructure and micro-segregation in the solidification of steel materials. Hence all these results can further expand the understanding of the kinetic mechanism of Fe-C peritectic solidification [14].

n

∂θ ⎛ = Mθ ∇ ·⎜ ∑ εθ2 h (ϕα ) ∇θ + ∂t ⎝ α=1

∂t

β≠α

εα2, β

⎣

n

n

∑

(∇2 ϕα − ∇2 ϕβ ) +

γ ≠ α, γ ≠ β

1 2 (εα, γ − εβ2, γ ) ∇2 ϕγ − Hα, β ϕα n

⎤ ϕβ (ϕβ − ϕα ) + 6ϕα ϕβ ΔGα → β⎥ + ηα ⎦

(1)

The peritectic solidification of γ phase was based on the appearance of the polycrystalline δ phase. In order to study the peritectic reaction L + δ → γ, peritectic transformation δ → γ and solidification L → γ in Fe-C binary system, the polycrystalline solidification L → δ in the first stage was simulated. The homogeneous nucleation of δ phase in the pure liquid phase was introduced by the fluctuation noise term in Eq. (1) [24–27]. As the iteration time increased, multiple grains of δ phase with different orientations begin to nucleate and grow up. The anisotropy of the δ phase crystal interacts with the concentration distribution of carbon, which results in the appearance of dendrites. Fig. 1 shows the results of the multi-grains solidification of δ phase. As shown in Fig. 1, the carbon diffusion plays an important role in the growth of δ phase grains. During the process of polycrystalline growth, the solute redistribution at the δ-liquid interface makes carbon continuously discharged from the δ phase to the liquid phase, which causes a high concentration gradient in front of the interface. At the tips of primary and secondary dendrites, the carbon concentration is smaller because of the faster diffusion rate of carbon, which makes the dendritic tips to grow up preferentially. On the contrary, in the gaps between the secondary dendrites, the concentration of carbon is higher because the local diffusion of carbon is usually restricted in the liquid channels. As a result, local higher carbon concentration derives a lower driving force of transformation from liquid to δ phase, which makes the δ-liquid interface movement in the gaps of secondary dendrites particularly slow. As the grains of δ phase grow up and collide, the carbon accumulated in the secondary dendrites gaps and the grain boundary forms high carbon melting pools. The complete solidification of these high carbon regions requires longer carbon diffusion time or lower temperatures. Therefore, the carbon redistribution and diffusion control the growth and morphology of equiaxed grains of δ phase. The heterogeneous nucleation of γ phase generally occurs on the δliquid interface. The phase-field method is limited by its characteristics and it is difficult to simulate this nucleation process. However, the transformation driving force ΔGL → γ and ΔGδ → γ was calculated from the local carbon concentration in front of δ-liquid interface, which can be used to evaluate the nucleation rate. The calculation shows that ΔGL → γ

(2)

n

∑ ϕα = 1

(3)

α=1

where

ϕ is the order parameter of phase-field; M is the kinetic coefficient of phase-field; ε is the gradient energy parameter; H is the barrier energy of double-well function; ΔG is the driving force of phase transformation; η is the noise term; n is the number of phases and α, β indicate the different phases. Concretely, n = 3 and α, β represent the δ phase, γ phase and liquid phase of the Fe-C system; C is the concentration of carbon; D is the diffusion coefficient. The ΔGα → β in Eq. (1) is the driving force of phase transformation from α to β phase [16], which can be expressed as follows,

ΔGα → β =

1 ⎛ ∂fα 2 ⎜⎜ ∂c ⎝

+ cα

∂fβ ∂c

cβ

⎞ ⎟⎟ (cβ − cα ) − (fβ (cβ ) − fα (cα )) ⎠

(5)

3. Results and discussion

n

∂C ⎛ ⎞ = ∇ ·⎜ ∑ ϕα Dα ∇cα ⎟ ∂t α = 1 ⎝ ⎠

⎠

Explicit finite difference method is used to solve the above dynamical equations (Eqs. (1), (2) and (5)) using Fortran. This program is based on the distributed memory parallel computing to improve the efficiency of the calculation. The size of the calculation grids was 5000 × 5000, and a set of periodic boundary conditions were used in the numerical simulations. In addition, the physical parameters of the simulation are adopted from our previous work [23].

∂ϕα n

∇θ

θ is the orientation parameter; Mθ is the kinetic coefficient of orientation field; εθ and S are the gradient energy coefficients of orientation field; h (ϕ) and p (ϕ) are the interpolation functions that activate the terms only in the solid phase.

In order to simulate the interaction of δ, γ and liquid phase in the Fe-C system, the order parameters of multi-phase field [15,16] was introduced to express the volume fractions of each phase and the concentration of diffusion equation to represent the carbon content. The dynamical equations of these parameters are given as follows.

∑ Mα,β ⎡⎢

α=1

where

2. Methods

=

n

∑ p (ϕα ) S |∇θ| ⎞⎟

(4)

where cα and cβ in Eqs. (2) and (4) are the fictitious concentrations derived from the KKS model [17,18]; fα and fβ are the free energy density functions of α and β phase. In addition, the order parameter ϕα was taken as the volume fraction of α phase, and therefore Eq. (3) is the constraint of the order parameters, which ensures that the sum of them is 1. Based on the multi-phase-field equation (Eq. (1)) and the carbon diffusion equation (Eq. (2)), peritectic solidification of Fe-C system was 2

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Fig. 1. The equiaxed solidification of δ phase in Fe-C liquid (the initial mole fraction of carbon is 0.01) at 1750 K as predicted by the phase-field model. The phasefield maps (a)–(d) are shown in the top row. The respective carbon concentration distribution maps (e)–(h) and orientation maps (i)–(l) are in the middle row and bottom row. (Coloring: In the phase-field maps, the red and blue regions represent the δ phase and liquid phase, respectively. In the carbon concentration distribution maps, continuous change of hue from blue to red indicates varying of carbon mole fraction from 0.0024 to 0.0380, respectively. In the orientation maps, different colors stand for different orientations of δ phase grains.) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

interface, the γ phase begins to wrap and engulf the δ phase grains by peritectic reaction and transformation, while continuously growing in the liquid phase by solidification. During this process, the γ phase grains inherit not only the equiaxed crystal morphology of δ phase grains but also the original micro-segregation of carbon. Some of the original liquid channels of δ phase grains are preserved by penetrating the γ phase, and some are blocked by the solid phase to become the internal melting pools. As shown in Fig. 2(m)–(p), the carbon concentration in δ phase, γ phase and liquid phase is from low to high. This microsegregation results from the redistribution of carbon in the peritectic solidification. For the γ-liquid interface, the first stage of evolution is similar to that of δ-liquid phase interface, mainly controlled by the carbon redistribution and diffusion. The carbon redistribution produces a higher concentration gradient in front of the δ-liquid interface. When the γ phase grains collide, the carbon accumulated around them forms melting pools on the grain boundary. These melting pools with higher carbon content can only be solidified by lower temperatures, which eventually brings about austenite grain boundary segregation. For the γ-δ interface, γ phase wraps δ phase grains at this interface by peritectic transformation δ → γ, whose moving speed is usually slower than the γliquid phase interface. Because the carbon diffusion in γ phase is relatively slow, the carbon concentration in the γ region transformed from δ phase is lower than the region transformed from liquid phase. In the process of δ phase reduction, some of the original melting pools distributed in the δ phase grains are retained, while others are transformed into γ phase, and whether a melting pool solidifies into δ phase depends on its carbon contents. For the δ-liquid interface, it is mainly distributed around the original liquid channels and melting pools. As the carbon slowly diffuses, the δ-liquid interface also moves slowly, which makes the liquid channels narrower and the melting pools smaller.

is usually bigger than ΔGδ → γ with same carbon concentration. Therefore, the way to simulate the γ phase nucleation on the δ-liquid interface can be achieved in two steps. First, the δ phase as γ phase were temporarily defined and the austenite solidification process was simulated in a very short time, during which the temporary γ phase grows slightly. Second, the initial solid-phase was set back as δ phase, and the newly grown part on the δ-liquid interface is regarded as γ phase film. Except for the Gibbs-Thomson effect caused by the interface curvature, the relative thickness of this γ phase film is proportional to the local driving force ΔGL → γ . This technique, regarded as an effective method to obtain the nucleation of γ phase, provides the initial condition for the simulation of γ-δ-liquid interaction. A set of simulations were performed with different thickness of γ phase film, which showed that the final microstructure is not sensitive to the effect of parameters. Due to the initial carbon distribution, this γ phase film always appears at the tips of the dendrites, and it rarely appears in the secondary dendrite gaps. At the time t = 29.9 s, a layer of γ phase film on the δ-liquid interface was placed as the heterogeneous nucleation of γ phase, whose maximum thickness is about six times the width of the phase-field interface. During the further evolution process, the γ phase film in the high carbon concentration region was rapidly swallowed by the liquid phase. Conversely, the γ phase film in low carbon concentration region begins to grow into the liquid phase. The setting of heterogeneous nucleation is actually based on the local driving force of transformation from liquid to γ phase and the local interface curvature. Taking the δ and liquid phases distribution and carbon distribution obtained in the first stage (t = 29.9 s) as the new starting point, the peritectic solidification was simulated with the δ, γ and liquid phases, including the interaction between each phase. As shown in Fig. 2, after the nucleation on the original δ-liquid 3

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Fig. 2. The peritectic solidification of Fe-C triple phases system at 1750 K as predicted by the multi-phase-field model. The triple-phases-field maps (a)–(h) are shown in the top row. The respective carbon concentration distribution maps (i)–(p) are in the bottom row. (e)–(h) and (m)–(p) show the partial enlargements of the dotted boxes in (a)–(d), (i)–(l). The initial phase field and carbon concentration field at t = 29.90 s inherit the final evolution results of the first step as shown in Fig. 1. (Coloring: In the phase-field maps, the blue, red and green regions represent the liquid, δ and γ phase, respectively. In the carbon concentration distribution maps, continuous change of hue from blue to red indicates the varying of carbon mole fraction from 0.005 to 0.038, respectively.) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

dotted line, located at t = 29.9 s, marks the dividing line of these two stages. Figs. 1 and 2 present the microstructure and micro-segregation evolution of the first stage and the second stage respectively. In the first stage, the entire evolution system only includes the polycrystalline nucleation and growth of δ phase grains in the liquid phase. For this stage, the solid fraction curve presents the basic characteristics of the Johnson-Mehl-Avrami-Kolmogorov equation, therefore the total interface area of all grains (the length of the interface indicates the interface area in this two-dimensional simulation) and the carbon diffusion are key factors for the volume fraction increase of δ phase. At the beginning of the first stage, the increase in δ phase volume fraction is limited by the interface area of δ-liquid phase, because the grains of δ phase are in the early stage of nucleation. As the grains grow up, the δ-liquid interface area becomes larger, carbon diffusion becomes a key factor limiting the increase of δ phase volume fraction. In addition, the liquid phase volume fraction of the first stage can be easily calculated because the sum of the liquid phase volume fraction and the δ phase volume fraction is 100% at this stage. In the second stage, through the nucleation at the δ-liquid phase interface, the γ phase begins to appear and grow up. The growth of γ phase is based on the heterogeneous nucleation at the initial δ-liquid interface, and the areas of the δ-liquid, γ-liquid, and γ-δ interface

Fig. 3. The volume fractions of liquid phase, δ phase and γ phase as a function of evolution time. The dotted line marks the start time of the γ phase nucleation at the δ-liquid phase interface.

As shown in Fig. 3, the statistical analysis of the multi-phase-field simulation can be used to calculate the volume fractions of each phase at different times. According to the nucleation time of γ phase, the evolution of this solidification process was divided into two stages. The 4

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determine the phase transformation rate of L → δ, L → γ and δ → γ, respectively. Therefore, the areas of the δ-liquid, γ-liquid, and γ-δ interfaces are one of the key factors. Through the peritectic transformation and solidification controlled by carbon diffusion, the γ phase continuously consumes the initial δ phase and liquid phase. At the beginning of the second stage, the γ phase volume fraction rapidly increases due to a large amount of nucleation area provided by the pregrown δ phase grains. With the growth and collision of γ phase peritectic grains, the δ and liquid phases are almost completely consumed and the increase of γ phase volume fraction gradually slows down. The volume fraction of δ phase was temporarily kept constant at the beginning of the second stage because the growth of δ phase in the liquid phase balances its consumption in peritectic transformation and peritectic reaction. Thereafter, as the peritectic transformation proceeds, the δ phase volume fraction gradually decreases. Furthermore, based on the basic constraint that the sum of the three-phase volume fractions is equal to 100%, the volume fraction of liquid phase in the second stage was calculated. At t = 60 s, the solid phase consisting of δ phase and γ phase occupies all the space, which makes the liquid fraction almost zero. However, the existence of melting pools makes the liquid fraction always greater than zero. Because the carbon concentration in these melting pools is very high, the solidification process at this temperature (1750 K) needs a long enough solute diffusion time to achieve.

Software, Validation, Visualization, Writing - original draft, Writing review & editing. Xitao Wang: Conceptualization, Investigation, Methodology, Writing - review & editing. Hasnain Mehdi Jafri: Writing - review & editing. Junsheng Wang: Writing - review & editing. Houbing Huang: Conceptualization, Investigation, Visualization, Methodology, Writing - review & editing. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement This work was sponsored by the National Science Foundation of China (51972028). References [1] J. Tiaden, J. Cryst. Growth 198 (1999) 1275–1280. [2] A. Choudhury, B. Nestler, A. Telang, M. Selzer, F. Wendler, Acta Mater. 58 (10) (2010) 3815–3823. [3] I. Loginova, J. Ågren, G. Amberg, Acta Mater. 52 (13) (2004) 4055–4063. [4] I. Loginova, J. Odqvist, G. Amberg, J. Ågren, Acta Mater. 51 (5) (2003) 1327–1339. [5] B. Nestler, A. Wheeler, Physica D 138 (1–2) (2000) 114–133. [6] M. Yamaguchi, C. Beckermann, Acta Mater. 61 (6) (2013) 2268–2280. [7] J. Eiken, B. Böttger, I. Steinbach, Phys. Rev. E 73 (6) (2006) 066122. [8] H. Xing, M. Ji, X. Dong, Y. Wang, L. Zhang, S. Li, Mater. Des. (2019) 108250. [9] H. Shibata, Y. Arai, M. Suzuki, T. Emi, Metall. Mater. Trans. B 31 (5) (2000) 981–991. [10] S. Pan, M. Zhu, Acta Mater. 146 (2018) 63–75. [11] S. Pan, M. Zhu, M. Rettenmayr, Acta Mater. 132 (2017) 565–575. [12] C.L.M. Alves, J.L.L. Rezende, D. Senk, J. Kundin, J. Mater. Res. Technol. 5 (1) (2016) 84–91. [13] C.L.M. Alves, J. Rezende, D. Senk, J. Kundin, J. Mater. Res. Technol. 8 (1) (2019) 233–242. [14] D. Phelan, M. Reid, R. Dippenaar, Mater. Sci. Eng., A 477 (1–2) (2008) 226–232. [15] B. Nestler, A.A. Wheeler, Phys. Rev. E 57 (3) (1998) 2602. [16] B. Böttger, J. Eiken, M. Apel, Comput. Mater. Sci. 108 (2015) 283–292. [17] M. Ode, T. Suzuki, S. Kim, W. Kim, Sci. Technol. Adv. Mater. 1 (1) (2000) 43–49. [18] S.G. Kim, W.T. Kim, T. Suzuki, Phys. Rev. E 60 (6) (1999) 7186. [19] M. Tang, W.C. Carter, R.M. Cannon, Phys. Rev. B 73 (2) (2006) 024102. [20] R. Kobayashi, Y. Giga, J. Stat. Phys. 95 (5–6) (1999) 1187–1220. [21] A.E. Lobkovsky, J.A. Warren, Phys. Rev. E 63 (5) (2001) 051605. [22] J.A. Warren, R. Kobayashi, A.E. Lobkovsky, W.C. Carter, Acta Mater. 51 (20) (2003) 6035–6058. [23] C. Yang, S. Li, X. Wang, J. Wang, H. Huang, Comput. Mater. Sci. 171 (2020) 109220. [24] L. Gránásy, T. Börzsönyi, T. Pusztai, Interface and Transport Dynamics, Springer, 2003, pp. 190–195. [25] D.T. Wu, L. Gránásy, F. Spaepen, MRS Bull. 29 (12) (2004) 945–950. [26] L. Gránásy, T. Pusztai, G. Tóth, Z. Jurek, M. Conti, B. Kvamme, J. Chem. Phys. 119 (19) (2003) 10376–10382. [27] L. Gránásy, T. Pusztai, T. Börzsönyi, J.A. Warren, B. Kvamme, P. James, Phys. Chem. Glasses 45 (2) (2004) 107–115.

4. Conclusions In short, a multi-phase-field model was developed to simulate the polycrystalline solidification of δ phase and peritectic solidification of γ phase in Fe-C system. At the stage of homogeneous nucleation and polycrystalline growth of δ phase grains, the diffusion of carbon controls the moving speed of the δ-liquid interface and induces the formation of dendrites. The accumulation of carbon in the secondary dendrites gaps and grains gaps leads to the production of liquid channels and melting pools. In the stage of heterogeneous nucleation and peritectic growth of γ phase, the γ phase wraps and engulfs the original δ phase grains and grows in the liquid phase, which is also controlled by carbon diffusion. The morphology of the original δ phase grains, the liquid channels and the melting pools evolves into the final microstructure of the γ phase grains. The carbon distribution of the original δ phase grains and liquid phase is inherited by the γ phase grains, which presents the micro-segregation of peritectic solidification. In these two stages, the changes in volume fractions of each phase with time indicate the growth or decrease rates of δ, γ, and liquid phases. The carbon diffusion rate and the areas of δ-liquid, γ-liquid and γ-δ interface are the key factors controlling the changes of volume fractions. CRediT authorship contribution statement Chao

Yang:

Conceptualization,

Investigation,

Methodology,

5