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Analysis of retarding effect on α to γ transformation in Fe–C alloy by addition of dispersed particles
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Original Research
Analysis of retarding effect on α to γ transformation in Fe–C alloy by addition of dispersed particles Liang Chena,n, Yun Hanb, Bin Zhouc, Jie Gonga a
Key Laboratory for Liquid–Solid Structural Evolution and Processing of Materials (Ministry of Education), Shandong University, Jinan 250061, China b Research Institute of Technology, Shougang Corporation, Beijing 100043, China c Schools of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 850287, USA Received 18 January 2015; accepted 27 March 2015
Abstract The effects of dispersed second phase particles on α-ferrite (α) to austenite (γ) transformation at 1140 K in Fe–C alloy were studied by means of phase field simulation. According to the simulated results, it was found that the particle could retard the migration of α/γ interface. Importantly, both the morphology of particles and the interfacial energy of particle/matrix (α or γ) interface affect the magnitude of the retarding effect. More specifically, the particles with smaller aspect ratio bring stronger retarding force, and when the interfacial energy of particle/γ interface is larger than that of particle/α interface, the retarding effect also becomes significant. These phenomena could be explained from the viewpoint of change in the total amount of the interfacial energy of the simulation system. & 2015 The Authors. Published by Elsevier GmbH. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: α⧸γ Interface; Retarding effect; Particle morphology; Interfacial energy
1. Introduction Before applying the hot forming processes such as hot rolling, forging and extrusion, the carbon steel is usually reheated above the eutectoid temperature from room temperature, during which the reverse transformation to austenite (γ) from α-ferrite (α), pearlite or martensite structures takes place inevitably. The reverse transformation has particular importance for the subsequent plastic deformation and cooling process, since the condition of reversely transformed γ significantly affects the final microstructures and mechanical properties [1–4]. In recent years, many researchers have transferred their attention from the austenite decomposition during cooling to the reverse transformation during reheating. Rudnizki [5] simulated the microstructure evolution during the formation of γ from the mixture of α and pearlite by means of the phase field method. n
Corresponding author. Tel./fax: þ 86 531 88392811. E-mail addresses: [email protected] (L. Chen), [email protected] (Y. Han). Peer review under responsibility of Chinese Materials Research Society.
Basabe [6] investigated the dynamic reverse transformation behavior of low carbon steel under different strains, strain rates and temperatures. Li [7] studied the effects of alloying elements on the reversion kinetics from pearlite to austenite, and proposed that the addition of different alloying elements can either accelerate or retard the kinetics. Kajihara [8] performed numerical analysis on the migration behavior of α/γ interface during isothermal carburization of Fe–C alloy. The migration distance shows parabolic relationship with the transformation time, which supports the carbon diffusion controlled mechanism. Schmidt [9] compared the kinetics of α/γ interface obtained from the numerical models and high temperature confocal scanning laser microscope. Son [10] compared the behavior of reverse transformation from ultrafine grains and coarse grains in low carbon steel. From the above literature, it should be noticed that these early efforts mainly focused on the effects of initial microstructures and alloying elements on the kinetics of reverse transformation, while little has been addressed the retarding effect of second phase particles. The reverse transformation is a process of nucleation and growth of γ phase. It is considered that if the γ growth could be retarded, there will be more positions and time for nucleation, and
http://dx.doi.org/10.1016/j.pnsc.2015.05.002 1002-0071/& 2015 The Authors. Published by Elsevier GmbH. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article as: L. Chen, et al., Analysis of retarding effect on α to γ transformation in Fe–C alloy by addition of dispersed particles, Progress in Natural Science: Materials International (2015), http://dx.doi.org/10.1016/j.pnsc.2015.05.002
L. Chen et al. / Progress in Natural Science: Materials International ] (]]]]) ]]]–]]]
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thus the γ grains after reverse transformation could be effectively refined [11]. On the other hand, it is well known that the second phase particles could strongly retard the grain growth, which is called Zener pinning [12–17]. Based on this point, the experimental work was carried out in our previous work to study the effects of insoluble ZrO2 and TiO2 particles on the kinetics of α to γ transformation recently, and it was indeed found that these particles could also retard the migration of α/γ interface [18], and the retarding effect becomes noticeable for particles with higher volume fraction or smaller size. However, there are some other factors that might influence the magnitude of the retarding effect, such as the particle morphology and the interfacial energy of particle/matrix interface, which are difficult to be clarified by performing experiments. Thus, in this paper, the isothermal α to γ transformation in Fe–C alloy was simulated by means of the phase field methods, which have been proved to be effective on providing qualitative evidence [18,19]. Importantly, the effects of particle morphology (circular and elliptical) and the interfacial energy of particle/matrix (α or γ) interface were investigated. The main objective of the present study is to study the retarding effect of particle on α to γ transformation. 2. Phase field modeling In recent decade, the phase field method has become a powerful tool to describe a variety of microstructure evolution. The diffused interface in phase field method allows us to describe the complex morphology of the microstructure without explicitly tracking the position of the interface. In the present study, the phase field model proposed by Steinbach and Pezzola [20,21] was utilized to investigate isothermal α to γ transformation in Fe–C alloy including dispersed particles. The detailed information of the modeling is summarized below. 2.1. Phase field variables The coexisting α, γ phases and insolvable particles are distinguished by the order parameters of ϕi, where the subscript i specifies the types of different phases. i and p represent α or γ phase and insoluble particles, respectively. More specifically, ϕα=1 and ϕγ=ϕp=0 in α phase, ϕγ=1 and ϕα=ϕp=0 in γ phase, ϕp=1 and ϕα=ϕγ=0 in particle region. ϕi varies from 0 to 1 smoothly across the thickness of phase interface. The carbon diffusion was coupled in Pthis model and the carbon concentration, c, is given by c ¼ ni¼1 ϕi ci , where ci is the carbon concentration associated with i phase. 2.2. Governing equation The time evolution of the order parameter is described by the following equation: " # n ∂φi 2X δF δF ¼ sij M ij ð1Þ N jai δφi δφj ∂t where t is the transformation time, N is the number of coexisting phases at a given spatial point, Mij is the mobility
of i/j interface, F is the free energy functional of the system, sij takes 1 when i and j phases coexist and it takes 0 otherwise. Then, the next step is the derivation of the free energy functional F. The free energy functional of a system of volume V is defined as: !# Z " X P T F¼ f þ f þ λL φi 1 dV ð2Þ V
i
where fP is the double well potential, fT is the thermodynamic potential and λL is the Lagrange multiplier. Thus, the functional derivation in Eq. (1) can be written as: " # " # n n X X ε2ij 2 δF ∂c j ∇ ϕj þ ωij ϕj þ f i ðci Þþ ¼ ϕj f jcj cj λL δϕi 2 ∂ϕi jai j ð3Þ where εij is the gradient energy coefficient and ωij is the height of the double well potential, fi(ci) is the free energy density of i phase with composition ci. Moreover, when i and j phases are in equilibrium condition, the following relationship is held: ( " # " #) n n X X ε2ij 2 ε2ij 2 δF δF ∇ φj þ ωij φj ∇ φi þ ωij φi ¼ δφi δφj 2 2 jai jai þ ðf ci f cj Þ ðci f c cj f c Þ
ð4Þ
where fc is the chemical potential. The condition of the equal chemical potential between the coexisting phases was introduced here [22]. In dilute alloy system, the following relation can be used: o RT n e;i f ci f cj ci f c cj f c ci cj ce;j i cj Vm ð5Þ where R is the gas constant, T is the temperature and Vm is molar volume. ce;j i represents the carbon concentration of i phase in equilibrium with j phase. Based on the above discussion, the final form of the evolution equation for order parameter ϕi is given as:
# " #9 8 n " 2 n X X εij ε2ij 2 > > > 2 > > ∇ φj þ ωij φj ∇ φi þωij φi > n < = ∂φi 2X 2 2 iaj ¼ sij M ij j a i h i > > ∂t N jai > > e;i > > : ; þ VRTm ðce;j i cj Þ ðci cj Þ
ð6Þ The parameters of εij and ωij could be calculated according to the following expressions: εij ¼
2 pffiffiffiffiffiffiffiffiffiffiffiffi 2σ ij W π
ð7Þ
ωij ¼
4σ ij W
ð8Þ
where σij is the interfacial energy between i and j phases, W is the interface thickness. The mobility, Mij was assumed as follows [23]:
Please cite this article as: L. Chen, et al., Analysis of retarding effect on α to γ transformation in Fe–C alloy by addition of dispersed particles, Progress in Natural Science: Materials International (2015), http://dx.doi.org/10.1016/j.pnsc.2015.05.002
L. Chen et al. / Progress in Natural Science: Materials International ] (]]]]) ]]]–]]]
M ij ¼
1 15 aεij 2 RT m;ij e;i 1 kij ce;j c j i 4 ωij Dij
ð9Þ
where a¼ 0.6276, kij is the partition coefficient defined as kij ¼ ci/cj and Tm,ij is the transition temperature between i and j phases in pure iron. The time evolution of carbon concentration, c, is described by n X X ∂c ¼∇ Di ϕi ∇ci ¼ ∇ Di ϕi kiγ ∇cγ ; ∂t i¼1
ð10Þ
where Di is diffusion coefficient in i phase, kiγ is the partition coefficient defined as kiγ ¼ ci/cγ. 2.3. Simulation condition The partial phase field simulation system is schematically shown in Fig. 1, including the α, γ phases and some dispersed particles. It can be seen that the initial α/γ interface was set to be planar shape. The Neumann boundary condition was applied along x direction and the periodic boundary condition was applied along y direction. The length of the initial γ and α phases in x direction was set to be 50 and 23 μm, respectively, and the length in y direction was set to be 3.5 μm for both of them. The added particles were assumed to be inert by setting the interface mobility of Mα/p, Mγ/p and diffusion coefficient of Dp to be 0. Thus, the particle shape, size and position would not change and the carbon cannot diffuse inside these particles during the whole transformation. Since one of the main objectives of this study is to clarify the particle morphology on retarding effects for α/γ interface, the particles with circular and elliptical shape were added in α phase as schematically shown in Fig. 1. For convenience, the particle aspect ratio was defined to be ε ¼ a/b, where a is the axis in x direction and b is the axis in y direction. Thus, the aspect ratios of the particles from Fig. 1(a) to (c) are 0.5, 1.0 and 2.0, respectively. For each case, 6 particles were applied and the distance between two particles was 3.5 μm. Moreover, for all cases, the area of each particle was kept to be identical, and thus the same area fraction of particles was obtained as 8.2%. The initial carbon concentration of γ phase was 0.45 wt%, while it was set to be 0 for α phase and particle region. The local equilibrium condition is assumed at the α/γ interface, where the carbon e;γ concentration changes continuously from ce;a γ to ca . According to the equilibrium Fe–C phase diagram, the simulation temperature was determined to be 1140 K to ensure the occurrence of α to γ transformation. Some important parameters used in the simulation are shown in Table 1.
Fig. 1. 2D phase field system including particles with varying aspect ratios.
3
The present phase field model was aimed to provide qualitative evidence for researching the effect of particle on α/γ interface migration due to the following simplifications and assumptions. The 2D simulation was carried out to save the computational time, since the kinetics of α to γ transformation is quite low and long transformation time is needed to be simulated. The nucleation stage of γ was not considered and only the growth of γ phase with transformation time was simulated. The added particles were assumed to be inert, which means that the shape, size and position would not change during the whole transformation. Moreover, the value of interfacial energy of matrix/particle interface was uncertain. Hence, in light of these facts, the present model will be used for qualitative comparison about the magnitude of retarding effect from particles with different morphologies and matrix/ particle interfacial energies. 3. Results and discussion 3.1. Evolution of α/γ interface In this section, the evolution of α/γ interface migration during isothermal α to γ transformation at 1140 K was discussed, as shown in Fig. 2. The system including dispersed particles with ε ¼ 0.5 was chosen as an example for explanation. It can be seen from Fig. 2 that α/γ interface continuously moved to α phase with the proceeding of transformation. The α/γ interface and particle/matrix interface met in two singular points in the case of 2D, and the planar α/γ interface was bent when it was passing one particle as shown in Fig. 2(b). When the α/γ interface escaped from the particles, the curved interface always moved towards its center of the curvature to release the redundant interfacial energy as shown in Fig. 2(c), and tried to be planar again. During these processes, the total amount of interfacial energy of this system involved in variety, which affected the interface kinetics. When α/γ interface interacted with particles, the α/γ interface associated with the contacted area vanishes. Then, when the α/γ interface was passing over particles, it caused the increase of the total amount of σα/γ. Therefore, the retarding force exerted by the particles was expected to be generated during this stage. On the other hand, when α/γ interface completely escaped from particles, the curved interface tried to be planar, and it involved in the reduction of the total amount of σα/γ. The change in the total amount of σp/m (the interfacial energy of particle/matrix interface) would be discussed later. Fig. 3 shows the evolution of carbon concentration at different transformation times corresponding to Fig. 2. The driving force for α to γ transformation originates from the difference in chemical potential between α and γ phases, and the transformation kinetics is carbon diffusion controlled in this study. However, as reported by Nakada [26,27], the diffusionless mechanism might also plays an important role especially in reverse transformation from martensite in highalloy at high heating rate. From Fig. 3, it can also be seen that the carbon diffused from γ phase with high concentration to α phase with low concentration during the whole transformation.
Please cite this article as: L. Chen, et al., Analysis of retarding effect on α to γ transformation in Fe–C alloy by addition of dispersed particles, Progress in Natural Science: Materials International (2015), http://dx.doi.org/10.1016/j.pnsc.2015.05.002
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4 Table 1 Parameters used in this study. Parameter System size Mesh size Time step Interface width Molar volume Interfacial energy [24] Diffusion coefficient [25]
i-j Phase transition temperature of pure iron
Symbol
Unit
Value (unit)
-
mm2 μm μm s μm m3 mol 1 J m2 m2 s 1 m2 s 1 m2 s 1 K
73 3.5 10 4 10 4 5.0 10 6 7.0 Δx 7.7 10 6 1.0 1.50 10 5exp( 14.20 104/RT) 2.20 10 4exp( 12.25 104/RT) 0 1185
Δx Δy Δt W Vm σα⧸γ Dγ Dα Dp Tm, aγ
Fig. 2. Evolution of α/γ interface (ε¼ 0.5) at different transformation times.
At the initial stage, the carbon concentration in α phase increased rapidly from 0 to the equilibrium condition. As the increase of transformation time, the carbon concentration in γ phase decreased gradually. And the carbon cannot diffuse inside particles as set in the phase field model. Although the existence of dispersed particles might have some effects on the carbon diffusion [18], it is out of the scope of the present study.
Fig. 3. Evolution of transformation times.
carbon
concentration
(ε¼ 0.5)
at
different
3.2. Effects of particle morphology on α/γ interface migration The relationship between migration distance of α/γ interface and transformation time in case of different particle morphologies are shown in Fig. 4. It should be emphasized that the initial microstructures, grain size and alloying elements should also have significant effects on the overall reverse transformation kinetics in steels [28,29], which were not taken into account for simplicity in the present model. From Fig. 4, it can be seen that the migration behavior for each case follows the parabolic law, which indicates that the α to γ transformation is diffusion controlled in the present study. By comparing with no particle case, it is obvious that the migration velocity of α/γ interface becomes smaller with the existence of particles, which indicates the emergence of retarding effect. Importantly, it can be seen that the magnitude of the retarding effect is
Fig. 4. Effects of particle morphology on the migration kinetics of α/γ interface.
significantly affected by the aspect ratio of particle, and smaller aspect ratio results in stronger retarding effect. For instance, when the migration distance of α/γ interface is 22 μm, the transformation time for particles with aspect ratio varying from 0.5 to 2.0 and no particle case are 192.5 s, 127.6 s, 116.2 s, 101.8 s, respectively.
Please cite this article as: L. Chen, et al., Analysis of retarding effect on α to γ transformation in Fe–C alloy by addition of dispersed particles, Progress in Natural Science: Materials International (2015), http://dx.doi.org/10.1016/j.pnsc.2015.05.002
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This phenomenon could be qualitatively explained based on Zener's theory, which was firstly proposed to explain the particle effect on grain growth. The following discussion would only consider the situation in 2D. As shown in Fig. 5, the α/γ interface meets the particle only at two distinct points, and the α/γ interface has dimple shape to balance the interfacial energy of σα/γ, σp/γ and σp/α at the triple junction point. It should be pointed out that both of σp/γ and σp/α were assumed to be 1.0 J m 2 in this section, and they counteracted each other. Hence, only the change in the total amount of σα/γ would contribute to the retarding force. The maximum retarding force FZ provided by single particle could be approximately calculated from the energy consideration. The increment of the total amount of σα/γ can be described as Fα/ γ ¼ 2bσα/γ, where b is vertical axis as shown in Fig. 5. The retarding force equal to Fα/γ/a was applied in the opposite direction of α/γ interface migration over a distance of a, which can be written as: FZ ¼
2bσ α=γ ¼ 2σ α=γ ε 1 : a
5
effect becomes much stronger as the increase of Δσ. In other word, if σp/γ 4 σp/α, α/γ interface is significantly retarded, while if σp/γ o σp/α, the retarding effect would disappear gradually. The transformation time when α/γ interface migrate 22 μm is shown in Fig. 7. One can see that for a given Δσ, the required time for ε ¼ 0.5 is always longer than that for ε ¼ 1.0, which is consistent with the discussion in the above section. Moreover, the required transformation time was obviously extended as the increase of Δσ. Especially when ε¼ 0.5 and Δσ ¼ 2.5 J m 2 it takes 265.0 s. Fig. 8 schematically shows the situation that α/γ interface is passing over one particle from position 1 to position 2. During this
ð11Þ
Thus, it is known that smaller aspect ratio, ε, brings stronger retarding force, which explains the phenomenon shown in Fig. 4. 3.3. Effects of σp/m on α/γ interface migration The above section focused on the effects of particle morphology, and the same value of σp/γ and σp/α was assumed to simplify the question. However, they should not be equal in general case, since the particle contacts with two different phases, viz., α and γ phases as shown in Fig. 5, which is an distinct difference with Zener pinning. Thus, the effects interfacial energy difference defined as Δσ ¼ σp/γ–σp/α on the magnitude of retarding effect was studied, and the system containing particles with aspect ratio of 0.5 and 1.0 as shown in Fig. 1(a) and (b) were used for this investigation. Fig. 6 shows the migration distance of α/γ interface with transformation time for different values of Δσ. Although each curve shows the similar tendency in Fig. 6(a) and (b), it is obvious that the kinetics of the transformation could be significantly affected by the value of Δσ, and the retarding
Fig. 6. Effects of Δσ on α/γ interface migration with particles: (a) ε ¼0.5 and (b) ε¼1.0.
Fig. 5. Schematic drawing of the interaction between α/γ interface and particle.
Fig. 7. Time consumption at migration distance of 22 μm for α/γ interface.
Please cite this article as: L. Chen, et al., Analysis of retarding effect on α to γ transformation in Fe–C alloy by addition of dispersed particles, Progress in Natural Science: Materials International (2015), http://dx.doi.org/10.1016/j.pnsc.2015.05.002
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magnitude of the retarding force. The increase of Δσ brings stronger retarding force.
Acknowledgments
Fig. 8. Schematic drawing of the α/γ interface migration.
process, the p/α interface is gradually transformed into p/γ interface. If σp/γ 4σp/α, the stronger retarding effect emerges, since the total amount of σp/m increases. If σp/γ oσp/α, the retarding effect becomes slight due to the reduction in the total amount of σp/m. Thus, the unequal values of σp/γ and σp/α would cause the change in the total amount of σp/m which was defined as Fp/m. Combining the discussion in Sections 3.1 and 3.2, the change in the total amount of the interfacial energy of the whole system can be described as: ΔF total ¼ ΔF α=γ þ ΔF p=m ¼ 2bσ α=γ þ L σ p=γ þ σ p=α ð12Þ where L is the perimeter of the particle. And the retarding force provided by single particle could be rewritten as: L σ p=γ σ p=α 1 F Z ¼ 2σ α=γ ε þ ð13Þ a The above equation considers the retarding force resulting from both of ΔFα/γ and ΔFp/m, and the phenomenon found from Figs. 4 and 6 could be well explained. 4. Conclusions In this paper, the isothermal α to γ transformation at 1140 K in Fe–C alloy with dispersed particles was simulated by means of the phase field method. From the simulated results, the following conclusions can be drawn: (1) the addition of particles retards the α to γ transformation, and the migration behavior of α/γ interface passing over one particle is similar with that of the Zener pinning. (2) with the decrease of the aspect ratio of particles, the retarding effect becomes much stronger. This is attributed to the fact that the particles with smaller aspect ratio bring more increment in the total amount of σα/γ when α/γ interface passed over one particle, and therefore stronger retarding force will be generated. (3) the difference in interfacial energy of particle/matrix interface, which was defined as Δσ ¼ σp/γ σp/α, is also an important aspect that should be considered for the
The authors would like to acknowledge the financial support from National Natural Science Foundation of China (51405268), Encouragement Foundation for Young Scholars of Shandong Province (BS2014ZZ001), Shandong Postdoctoral Creative Foundation (201402025), and Fundamental Research Funds of Shandong University (Grant no. 2014HW001). References [1] T. Furuhara, K. Kikumoto, H. Saito, T. Sekine, T. Ogawa, S. Morito, T. Maki, ISIJ Int. 48 (2008) 1038–1045. [2] F. García Caballero, C. Capdevila, C. García de Andrés, ISIJ Int. 41 (2001) 1093–1102. [3] T. Shirane, S Tsukamoto, K. Tsuzaki, Y. Adachi, T. Hanamura, M. Shimizu, F. Abe, Sci. Technol. Weld. Join. 14 (2009) 698–707. [4] K.T. Park, E.G. Lee, C.S. Lee, ISIJ Int. 47 (2007) 294–298. [5] J. Rudnizki, B. Böttger, U. Prahl, W. Bleck, Metall. Mater. Trans. A 42 (2011) 2516–2525. [6] V.V. Basabe, J.J. Jonas, H. Mahjoubi, ISIJ Int. 51 (2011) 612–618. [7] Z.D. Li, G. Miyamoto, Z.G. Yang, T. Furuhara, Metall. Mater. Trans. A 42 (2011) 1586–1596. [8] M. Kajihara, J. Mater. Sci. 44 (2009) 2109–2118. [9] E.D. Schmidt, E.B. Damm, S. Sridhar, Metall. Mater. Trans. A 38 (2007) 244–260. [10] Y.I. Son, Y.K. Lee, K.T. Park, Metall. Mater. Trans. A 37 (2006) 3161–3164. [11] S.E. Offerman, N.H. van Dijk, J. Sietsma, S. Grigull, E.M. Lauridsen, L. Margulies, H.F. Poulsen, M.Th. Rekveldt, S. van der Zwaag, Science 298 (2002) 1003–1005. [12] B.H. Hu, Q.W. Cai, H.B. Wu, J. Iron Steel Res. Int. 20 (2013) 69–77. [13] Y. Suwa, Y. Saitob, H. Onodera, Scr. Mater. 55 (2006) 407–410. [14] Y. Suwa, Y. Saitob, H. Onodera, Acta Mater. 55 (2007) 6881–6894. [15] T. Oikawa, J.J. Zhang, M. Enomoto, Y. Adachi, ISIJ Int. 53 (2013) 1245–1252. [16] A. Mallick, Comput. Mater. Sci. 67 (2013) 27–34. [17] V.Y. Novikov, Mater. Lett. 68 (2012) 413–415. [18] L. Chen, K. Matsuura, M. Ohno, D. Sato, ISIJ Int. 52 (2012) 1841–1847. [19] L. Chen, K. Matsuura, D. Sato, M. Ohno, ISIJ Int. 52 (2012) 434–440. [20] I. Steinbach, F. Pezzolla, Physica D 134 (1999) 385–393. [21] J. Tiaden, B. Nestler, H.J. Diepers, I. Steinbach, Physica D 115 (1998) 73–86. [22] S.G. Kim, W.T. Kim, T. Suzuki, Phys. Rev. E 60 (1999) 7186–7197. [23] M. Ohno, K. Matsuura, Acta Mater. 58 (2010) 5749–5758. [24] K. Nakajima, M. Apel, I. Steinbach, Acta Mater. 54 (2006) 3665–3672. [25] N. Moelans, B. Blanpain, P. Wollants, Acta Mater. 53 (2005) 1771–1781. [26] N. Nakada, R. Fukagawa, T. Tsuchiyama, S. Takaki, D. Ponge, D. Raabe, ISIJ Int. 53 (2013) 1286–1288. [27] N. Nakada, T. Tsuchiyama, S. Takaki, D. Ponge, D. Raabe, ISIJ Int. 53 (2013) 2275–2277. [28] H. Springer, M. Belde, D. Raabe, Mater. Sci. Eng. A 582 (2013) 235–244. [29] L. Yuan, D. Ponge, J. Wittig, P. Choi, J.A. Jiménez, D. Raabe, Acta Mater. 60 (2012) 2790–2804.
Please cite this article as: L. Chen, et al., Analysis of retarding effect on α to γ transformation in Fe–C alloy by addition of dispersed particles, Progress in Natural Science: Materials International (2015), http://dx.doi.org/10.1016/j.pnsc.2015.05.002
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Original Research
Analysis of retarding effect on α to γ transformation in Fe–C alloy by addition of dispersed particles Liang Chena,n, Yun Hanb, Bin Zhouc, Jie Gonga a
Key Laboratory for Liquid–Solid Structural Evolution and Processing of Materials (Ministry of Education), Shandong University, Jinan 250061, China b Research Institute of Technology, Shougang Corporation, Beijing 100043, China c Schools of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, AZ 850287, USA Received 18 January 2015; accepted 27 March 2015
Abstract The effects of dispersed second phase particles on α-ferrite (α) to austenite (γ) transformation at 1140 K in Fe–C alloy were studied by means of phase field simulation. According to the simulated results, it was found that the particle could retard the migration of α/γ interface. Importantly, both the morphology of particles and the interfacial energy of particle/matrix (α or γ) interface affect the magnitude of the retarding effect. More specifically, the particles with smaller aspect ratio bring stronger retarding force, and when the interfacial energy of particle/γ interface is larger than that of particle/α interface, the retarding effect also becomes significant. These phenomena could be explained from the viewpoint of change in the total amount of the interfacial energy of the simulation system. & 2015 The Authors. Published by Elsevier GmbH. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: α⧸γ Interface; Retarding effect; Particle morphology; Interfacial energy
1. Introduction Before applying the hot forming processes such as hot rolling, forging and extrusion, the carbon steel is usually reheated above the eutectoid temperature from room temperature, during which the reverse transformation to austenite (γ) from α-ferrite (α), pearlite or martensite structures takes place inevitably. The reverse transformation has particular importance for the subsequent plastic deformation and cooling process, since the condition of reversely transformed γ significantly affects the final microstructures and mechanical properties [1–4]. In recent years, many researchers have transferred their attention from the austenite decomposition during cooling to the reverse transformation during reheating. Rudnizki [5] simulated the microstructure evolution during the formation of γ from the mixture of α and pearlite by means of the phase field method. n
Corresponding author. Tel./fax: þ 86 531 88392811. E-mail addresses: [email protected] (L. Chen), [email protected] (Y. Han). Peer review under responsibility of Chinese Materials Research Society.
Basabe [6] investigated the dynamic reverse transformation behavior of low carbon steel under different strains, strain rates and temperatures. Li [7] studied the effects of alloying elements on the reversion kinetics from pearlite to austenite, and proposed that the addition of different alloying elements can either accelerate or retard the kinetics. Kajihara [8] performed numerical analysis on the migration behavior of α/γ interface during isothermal carburization of Fe–C alloy. The migration distance shows parabolic relationship with the transformation time, which supports the carbon diffusion controlled mechanism. Schmidt [9] compared the kinetics of α/γ interface obtained from the numerical models and high temperature confocal scanning laser microscope. Son [10] compared the behavior of reverse transformation from ultrafine grains and coarse grains in low carbon steel. From the above literature, it should be noticed that these early efforts mainly focused on the effects of initial microstructures and alloying elements on the kinetics of reverse transformation, while little has been addressed the retarding effect of second phase particles. The reverse transformation is a process of nucleation and growth of γ phase. It is considered that if the γ growth could be retarded, there will be more positions and time for nucleation, and
http://dx.doi.org/10.1016/j.pnsc.2015.05.002 1002-0071/& 2015 The Authors. Published by Elsevier GmbH. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article as: L. Chen, et al., Analysis of retarding effect on α to γ transformation in Fe–C alloy by addition of dispersed particles, Progress in Natural Science: Materials International (2015), http://dx.doi.org/10.1016/j.pnsc.2015.05.002
L. Chen et al. / Progress in Natural Science: Materials International ] (]]]]) ]]]–]]]
2
thus the γ grains after reverse transformation could be effectively refined [11]. On the other hand, it is well known that the second phase particles could strongly retard the grain growth, which is called Zener pinning [12–17]. Based on this point, the experimental work was carried out in our previous work to study the effects of insoluble ZrO2 and TiO2 particles on the kinetics of α to γ transformation recently, and it was indeed found that these particles could also retard the migration of α/γ interface [18], and the retarding effect becomes noticeable for particles with higher volume fraction or smaller size. However, there are some other factors that might influence the magnitude of the retarding effect, such as the particle morphology and the interfacial energy of particle/matrix interface, which are difficult to be clarified by performing experiments. Thus, in this paper, the isothermal α to γ transformation in Fe–C alloy was simulated by means of the phase field methods, which have been proved to be effective on providing qualitative evidence [18,19]. Importantly, the effects of particle morphology (circular and elliptical) and the interfacial energy of particle/matrix (α or γ) interface were investigated. The main objective of the present study is to study the retarding effect of particle on α to γ transformation. 2. Phase field modeling In recent decade, the phase field method has become a powerful tool to describe a variety of microstructure evolution. The diffused interface in phase field method allows us to describe the complex morphology of the microstructure without explicitly tracking the position of the interface. In the present study, the phase field model proposed by Steinbach and Pezzola [20,21] was utilized to investigate isothermal α to γ transformation in Fe–C alloy including dispersed particles. The detailed information of the modeling is summarized below. 2.1. Phase field variables The coexisting α, γ phases and insolvable particles are distinguished by the order parameters of ϕi, where the subscript i specifies the types of different phases. i and p represent α or γ phase and insoluble particles, respectively. More specifically, ϕα=1 and ϕγ=ϕp=0 in α phase, ϕγ=1 and ϕα=ϕp=0 in γ phase, ϕp=1 and ϕα=ϕγ=0 in particle region. ϕi varies from 0 to 1 smoothly across the thickness of phase interface. The carbon diffusion was coupled in Pthis model and the carbon concentration, c, is given by c ¼ ni¼1 ϕi ci , where ci is the carbon concentration associated with i phase. 2.2. Governing equation The time evolution of the order parameter is described by the following equation: " # n ∂φi 2X δF δF ¼ sij M ij ð1Þ N jai δφi δφj ∂t where t is the transformation time, N is the number of coexisting phases at a given spatial point, Mij is the mobility
of i/j interface, F is the free energy functional of the system, sij takes 1 when i and j phases coexist and it takes 0 otherwise. Then, the next step is the derivation of the free energy functional F. The free energy functional of a system of volume V is defined as: !# Z " X P T F¼ f þ f þ λL φi 1 dV ð2Þ V
i
where fP is the double well potential, fT is the thermodynamic potential and λL is the Lagrange multiplier. Thus, the functional derivation in Eq. (1) can be written as: " # " # n n X X ε2ij 2 δF ∂c j ∇ ϕj þ ωij ϕj þ f i ðci Þþ ¼ ϕj f jcj cj λL δϕi 2 ∂ϕi jai j ð3Þ where εij is the gradient energy coefficient and ωij is the height of the double well potential, fi(ci) is the free energy density of i phase with composition ci. Moreover, when i and j phases are in equilibrium condition, the following relationship is held: ( " # " #) n n X X ε2ij 2 ε2ij 2 δF δF ∇ φj þ ωij φj ∇ φi þ ωij φi ¼ δφi δφj 2 2 jai jai þ ðf ci f cj Þ ðci f c cj f c Þ
ð4Þ
where fc is the chemical potential. The condition of the equal chemical potential between the coexisting phases was introduced here [22]. In dilute alloy system, the following relation can be used: o RT n e;i f ci f cj ci f c cj f c ci cj ce;j i cj Vm ð5Þ where R is the gas constant, T is the temperature and Vm is molar volume. ce;j i represents the carbon concentration of i phase in equilibrium with j phase. Based on the above discussion, the final form of the evolution equation for order parameter ϕi is given as:
# " #9 8 n " 2 n X X εij ε2ij 2 > > > 2 > > ∇ φj þ ωij φj ∇ φi þωij φi > n < = ∂φi 2X 2 2 iaj ¼ sij M ij j a i h i > > ∂t N jai > > e;i > > : ; þ VRTm ðce;j i cj Þ ðci cj Þ
ð6Þ The parameters of εij and ωij could be calculated according to the following expressions: εij ¼
2 pffiffiffiffiffiffiffiffiffiffiffiffi 2σ ij W π
ð7Þ
ωij ¼
4σ ij W
ð8Þ
where σij is the interfacial energy between i and j phases, W is the interface thickness. The mobility, Mij was assumed as follows [23]:
Please cite this article as: L. Chen, et al., Analysis of retarding effect on α to γ transformation in Fe–C alloy by addition of dispersed particles, Progress in Natural Science: Materials International (2015), http://dx.doi.org/10.1016/j.pnsc.2015.05.002
L. Chen et al. / Progress in Natural Science: Materials International ] (]]]]) ]]]–]]]
M ij ¼
1 15 aεij 2 RT m;ij e;i 1 kij ce;j c j i 4 ωij Dij
ð9Þ
where a¼ 0.6276, kij is the partition coefficient defined as kij ¼ ci/cj and Tm,ij is the transition temperature between i and j phases in pure iron. The time evolution of carbon concentration, c, is described by n X X ∂c ¼∇ Di ϕi ∇ci ¼ ∇ Di ϕi kiγ ∇cγ ; ∂t i¼1
ð10Þ
where Di is diffusion coefficient in i phase, kiγ is the partition coefficient defined as kiγ ¼ ci/cγ. 2.3. Simulation condition The partial phase field simulation system is schematically shown in Fig. 1, including the α, γ phases and some dispersed particles. It can be seen that the initial α/γ interface was set to be planar shape. The Neumann boundary condition was applied along x direction and the periodic boundary condition was applied along y direction. The length of the initial γ and α phases in x direction was set to be 50 and 23 μm, respectively, and the length in y direction was set to be 3.5 μm for both of them. The added particles were assumed to be inert by setting the interface mobility of Mα/p, Mγ/p and diffusion coefficient of Dp to be 0. Thus, the particle shape, size and position would not change and the carbon cannot diffuse inside these particles during the whole transformation. Since one of the main objectives of this study is to clarify the particle morphology on retarding effects for α/γ interface, the particles with circular and elliptical shape were added in α phase as schematically shown in Fig. 1. For convenience, the particle aspect ratio was defined to be ε ¼ a/b, where a is the axis in x direction and b is the axis in y direction. Thus, the aspect ratios of the particles from Fig. 1(a) to (c) are 0.5, 1.0 and 2.0, respectively. For each case, 6 particles were applied and the distance between two particles was 3.5 μm. Moreover, for all cases, the area of each particle was kept to be identical, and thus the same area fraction of particles was obtained as 8.2%. The initial carbon concentration of γ phase was 0.45 wt%, while it was set to be 0 for α phase and particle region. The local equilibrium condition is assumed at the α/γ interface, where the carbon e;γ concentration changes continuously from ce;a γ to ca . According to the equilibrium Fe–C phase diagram, the simulation temperature was determined to be 1140 K to ensure the occurrence of α to γ transformation. Some important parameters used in the simulation are shown in Table 1.
Fig. 1. 2D phase field system including particles with varying aspect ratios.
3
The present phase field model was aimed to provide qualitative evidence for researching the effect of particle on α/γ interface migration due to the following simplifications and assumptions. The 2D simulation was carried out to save the computational time, since the kinetics of α to γ transformation is quite low and long transformation time is needed to be simulated. The nucleation stage of γ was not considered and only the growth of γ phase with transformation time was simulated. The added particles were assumed to be inert, which means that the shape, size and position would not change during the whole transformation. Moreover, the value of interfacial energy of matrix/particle interface was uncertain. Hence, in light of these facts, the present model will be used for qualitative comparison about the magnitude of retarding effect from particles with different morphologies and matrix/ particle interfacial energies. 3. Results and discussion 3.1. Evolution of α/γ interface In this section, the evolution of α/γ interface migration during isothermal α to γ transformation at 1140 K was discussed, as shown in Fig. 2. The system including dispersed particles with ε ¼ 0.5 was chosen as an example for explanation. It can be seen from Fig. 2 that α/γ interface continuously moved to α phase with the proceeding of transformation. The α/γ interface and particle/matrix interface met in two singular points in the case of 2D, and the planar α/γ interface was bent when it was passing one particle as shown in Fig. 2(b). When the α/γ interface escaped from the particles, the curved interface always moved towards its center of the curvature to release the redundant interfacial energy as shown in Fig. 2(c), and tried to be planar again. During these processes, the total amount of interfacial energy of this system involved in variety, which affected the interface kinetics. When α/γ interface interacted with particles, the α/γ interface associated with the contacted area vanishes. Then, when the α/γ interface was passing over particles, it caused the increase of the total amount of σα/γ. Therefore, the retarding force exerted by the particles was expected to be generated during this stage. On the other hand, when α/γ interface completely escaped from particles, the curved interface tried to be planar, and it involved in the reduction of the total amount of σα/γ. The change in the total amount of σp/m (the interfacial energy of particle/matrix interface) would be discussed later. Fig. 3 shows the evolution of carbon concentration at different transformation times corresponding to Fig. 2. The driving force for α to γ transformation originates from the difference in chemical potential between α and γ phases, and the transformation kinetics is carbon diffusion controlled in this study. However, as reported by Nakada [26,27], the diffusionless mechanism might also plays an important role especially in reverse transformation from martensite in highalloy at high heating rate. From Fig. 3, it can also be seen that the carbon diffused from γ phase with high concentration to α phase with low concentration during the whole transformation.
Please cite this article as: L. Chen, et al., Analysis of retarding effect on α to γ transformation in Fe–C alloy by addition of dispersed particles, Progress in Natural Science: Materials International (2015), http://dx.doi.org/10.1016/j.pnsc.2015.05.002
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4 Table 1 Parameters used in this study. Parameter System size Mesh size Time step Interface width Molar volume Interfacial energy [24] Diffusion coefficient [25]
i-j Phase transition temperature of pure iron
Symbol
Unit
Value (unit)
-
mm2 μm μm s μm m3 mol 1 J m2 m2 s 1 m2 s 1 m2 s 1 K
73 3.5 10 4 10 4 5.0 10 6 7.0 Δx 7.7 10 6 1.0 1.50 10 5exp( 14.20 104/RT) 2.20 10 4exp( 12.25 104/RT) 0 1185
Δx Δy Δt W Vm σα⧸γ Dγ Dα Dp Tm, aγ
Fig. 2. Evolution of α/γ interface (ε¼ 0.5) at different transformation times.
At the initial stage, the carbon concentration in α phase increased rapidly from 0 to the equilibrium condition. As the increase of transformation time, the carbon concentration in γ phase decreased gradually. And the carbon cannot diffuse inside particles as set in the phase field model. Although the existence of dispersed particles might have some effects on the carbon diffusion [18], it is out of the scope of the present study.
Fig. 3. Evolution of transformation times.
carbon
concentration
(ε¼ 0.5)
at
different
3.2. Effects of particle morphology on α/γ interface migration The relationship between migration distance of α/γ interface and transformation time in case of different particle morphologies are shown in Fig. 4. It should be emphasized that the initial microstructures, grain size and alloying elements should also have significant effects on the overall reverse transformation kinetics in steels [28,29], which were not taken into account for simplicity in the present model. From Fig. 4, it can be seen that the migration behavior for each case follows the parabolic law, which indicates that the α to γ transformation is diffusion controlled in the present study. By comparing with no particle case, it is obvious that the migration velocity of α/γ interface becomes smaller with the existence of particles, which indicates the emergence of retarding effect. Importantly, it can be seen that the magnitude of the retarding effect is
Fig. 4. Effects of particle morphology on the migration kinetics of α/γ interface.
significantly affected by the aspect ratio of particle, and smaller aspect ratio results in stronger retarding effect. For instance, when the migration distance of α/γ interface is 22 μm, the transformation time for particles with aspect ratio varying from 0.5 to 2.0 and no particle case are 192.5 s, 127.6 s, 116.2 s, 101.8 s, respectively.
Please cite this article as: L. Chen, et al., Analysis of retarding effect on α to γ transformation in Fe–C alloy by addition of dispersed particles, Progress in Natural Science: Materials International (2015), http://dx.doi.org/10.1016/j.pnsc.2015.05.002
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This phenomenon could be qualitatively explained based on Zener's theory, which was firstly proposed to explain the particle effect on grain growth. The following discussion would only consider the situation in 2D. As shown in Fig. 5, the α/γ interface meets the particle only at two distinct points, and the α/γ interface has dimple shape to balance the interfacial energy of σα/γ, σp/γ and σp/α at the triple junction point. It should be pointed out that both of σp/γ and σp/α were assumed to be 1.0 J m 2 in this section, and they counteracted each other. Hence, only the change in the total amount of σα/γ would contribute to the retarding force. The maximum retarding force FZ provided by single particle could be approximately calculated from the energy consideration. The increment of the total amount of σα/γ can be described as Fα/ γ ¼ 2bσα/γ, where b is vertical axis as shown in Fig. 5. The retarding force equal to Fα/γ/a was applied in the opposite direction of α/γ interface migration over a distance of a, which can be written as: FZ ¼
2bσ α=γ ¼ 2σ α=γ ε 1 : a
5
effect becomes much stronger as the increase of Δσ. In other word, if σp/γ 4 σp/α, α/γ interface is significantly retarded, while if σp/γ o σp/α, the retarding effect would disappear gradually. The transformation time when α/γ interface migrate 22 μm is shown in Fig. 7. One can see that for a given Δσ, the required time for ε ¼ 0.5 is always longer than that for ε ¼ 1.0, which is consistent with the discussion in the above section. Moreover, the required transformation time was obviously extended as the increase of Δσ. Especially when ε¼ 0.5 and Δσ ¼ 2.5 J m 2 it takes 265.0 s. Fig. 8 schematically shows the situation that α/γ interface is passing over one particle from position 1 to position 2. During this
ð11Þ
Thus, it is known that smaller aspect ratio, ε, brings stronger retarding force, which explains the phenomenon shown in Fig. 4. 3.3. Effects of σp/m on α/γ interface migration The above section focused on the effects of particle morphology, and the same value of σp/γ and σp/α was assumed to simplify the question. However, they should not be equal in general case, since the particle contacts with two different phases, viz., α and γ phases as shown in Fig. 5, which is an distinct difference with Zener pinning. Thus, the effects interfacial energy difference defined as Δσ ¼ σp/γ–σp/α on the magnitude of retarding effect was studied, and the system containing particles with aspect ratio of 0.5 and 1.0 as shown in Fig. 1(a) and (b) were used for this investigation. Fig. 6 shows the migration distance of α/γ interface with transformation time for different values of Δσ. Although each curve shows the similar tendency in Fig. 6(a) and (b), it is obvious that the kinetics of the transformation could be significantly affected by the value of Δσ, and the retarding
Fig. 6. Effects of Δσ on α/γ interface migration with particles: (a) ε ¼0.5 and (b) ε¼1.0.
Fig. 5. Schematic drawing of the interaction between α/γ interface and particle.
Fig. 7. Time consumption at migration distance of 22 μm for α/γ interface.
Please cite this article as: L. Chen, et al., Analysis of retarding effect on α to γ transformation in Fe–C alloy by addition of dispersed particles, Progress in Natural Science: Materials International (2015), http://dx.doi.org/10.1016/j.pnsc.2015.05.002
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magnitude of the retarding force. The increase of Δσ brings stronger retarding force.
Acknowledgments
Fig. 8. Schematic drawing of the α/γ interface migration.
process, the p/α interface is gradually transformed into p/γ interface. If σp/γ 4σp/α, the stronger retarding effect emerges, since the total amount of σp/m increases. If σp/γ oσp/α, the retarding effect becomes slight due to the reduction in the total amount of σp/m. Thus, the unequal values of σp/γ and σp/α would cause the change in the total amount of σp/m which was defined as Fp/m. Combining the discussion in Sections 3.1 and 3.2, the change in the total amount of the interfacial energy of the whole system can be described as: ΔF total ¼ ΔF α=γ þ ΔF p=m ¼ 2bσ α=γ þ L σ p=γ þ σ p=α ð12Þ where L is the perimeter of the particle. And the retarding force provided by single particle could be rewritten as: L σ p=γ σ p=α 1 F Z ¼ 2σ α=γ ε þ ð13Þ a The above equation considers the retarding force resulting from both of ΔFα/γ and ΔFp/m, and the phenomenon found from Figs. 4 and 6 could be well explained. 4. Conclusions In this paper, the isothermal α to γ transformation at 1140 K in Fe–C alloy with dispersed particles was simulated by means of the phase field method. From the simulated results, the following conclusions can be drawn: (1) the addition of particles retards the α to γ transformation, and the migration behavior of α/γ interface passing over one particle is similar with that of the Zener pinning. (2) with the decrease of the aspect ratio of particles, the retarding effect becomes much stronger. This is attributed to the fact that the particles with smaller aspect ratio bring more increment in the total amount of σα/γ when α/γ interface passed over one particle, and therefore stronger retarding force will be generated. (3) the difference in interfacial energy of particle/matrix interface, which was defined as Δσ ¼ σp/γ σp/α, is also an important aspect that should be considered for the
The authors would like to acknowledge the financial support from National Natural Science Foundation of China (51405268), Encouragement Foundation for Young Scholars of Shandong Province (BS2014ZZ001), Shandong Postdoctoral Creative Foundation (201402025), and Fundamental Research Funds of Shandong University (Grant no. 2014HW001). References [1] T. Furuhara, K. Kikumoto, H. Saito, T. Sekine, T. Ogawa, S. Morito, T. Maki, ISIJ Int. 48 (2008) 1038–1045. [2] F. García Caballero, C. Capdevila, C. García de Andrés, ISIJ Int. 41 (2001) 1093–1102. [3] T. Shirane, S Tsukamoto, K. Tsuzaki, Y. Adachi, T. Hanamura, M. Shimizu, F. Abe, Sci. Technol. Weld. Join. 14 (2009) 698–707. [4] K.T. Park, E.G. Lee, C.S. Lee, ISIJ Int. 47 (2007) 294–298. [5] J. Rudnizki, B. Böttger, U. Prahl, W. Bleck, Metall. Mater. Trans. A 42 (2011) 2516–2525. [6] V.V. Basabe, J.J. Jonas, H. Mahjoubi, ISIJ Int. 51 (2011) 612–618. [7] Z.D. Li, G. Miyamoto, Z.G. Yang, T. Furuhara, Metall. Mater. Trans. A 42 (2011) 1586–1596. [8] M. Kajihara, J. Mater. Sci. 44 (2009) 2109–2118. [9] E.D. Schmidt, E.B. Damm, S. Sridhar, Metall. Mater. Trans. A 38 (2007) 244–260. [10] Y.I. Son, Y.K. Lee, K.T. Park, Metall. Mater. Trans. A 37 (2006) 3161–3164. [11] S.E. Offerman, N.H. van Dijk, J. Sietsma, S. Grigull, E.M. Lauridsen, L. Margulies, H.F. Poulsen, M.Th. Rekveldt, S. van der Zwaag, Science 298 (2002) 1003–1005. [12] B.H. Hu, Q.W. Cai, H.B. Wu, J. Iron Steel Res. Int. 20 (2013) 69–77. [13] Y. Suwa, Y. Saitob, H. Onodera, Scr. Mater. 55 (2006) 407–410. [14] Y. Suwa, Y. Saitob, H. Onodera, Acta Mater. 55 (2007) 6881–6894. [15] T. Oikawa, J.J. Zhang, M. Enomoto, Y. Adachi, ISIJ Int. 53 (2013) 1245–1252. [16] A. Mallick, Comput. Mater. Sci. 67 (2013) 27–34. [17] V.Y. Novikov, Mater. Lett. 68 (2012) 413–415. [18] L. Chen, K. Matsuura, M. Ohno, D. Sato, ISIJ Int. 52 (2012) 1841–1847. [19] L. Chen, K. Matsuura, D. Sato, M. Ohno, ISIJ Int. 52 (2012) 434–440. [20] I. Steinbach, F. Pezzolla, Physica D 134 (1999) 385–393. [21] J. Tiaden, B. Nestler, H.J. Diepers, I. Steinbach, Physica D 115 (1998) 73–86. [22] S.G. Kim, W.T. Kim, T. Suzuki, Phys. Rev. E 60 (1999) 7186–7197. [23] M. Ohno, K. Matsuura, Acta Mater. 58 (2010) 5749–5758. [24] K. Nakajima, M. Apel, I. Steinbach, Acta Mater. 54 (2006) 3665–3672. [25] N. Moelans, B. Blanpain, P. Wollants, Acta Mater. 53 (2005) 1771–1781. [26] N. Nakada, R. Fukagawa, T. Tsuchiyama, S. Takaki, D. Ponge, D. Raabe, ISIJ Int. 53 (2013) 1286–1288. [27] N. Nakada, T. Tsuchiyama, S. Takaki, D. Ponge, D. Raabe, ISIJ Int. 53 (2013) 2275–2277. [28] H. Springer, M. Belde, D. Raabe, Mater. Sci. Eng. A 582 (2013) 235–244. [29] L. Yuan, D. Ponge, J. Wittig, P. Choi, J.A. Jiménez, D. Raabe, Acta Mater. 60 (2012) 2790–2804.
Please cite this article as: L. Chen, et al., Analysis of retarding effect on α to γ transformation in Fe–C alloy by addition of dispersed particles, Progress in Natural Science: Materials International (2015), http://dx.doi.org/10.1016/j.pnsc.2015.05.002