Modeling of Ferrite-Austenite Phase Transformation Using a Cellular Automaton Model

, pp. 422 429 Modeling of Ferrite-Austenite Phase Transformation Using a Cellular Automaton Model Dong AN, 1) Shiyan PAN, 1) Li HUANG, 1) Ting DAI, 1) Bruce KRAKAUER 2) and Mingfang ZHU 1) * 1) Jiangsu Key Laboratory for Advanced Metallic Materials, School of Materials Science and Engineering, Southeast University, Nanjing, Jiangsu, 211189 China. 2) AO Smith Corporate Technology Center, Milwaukee, WI, 53224 USA. (Received on August 29, 2013; accepted on November 5, 2013) A two-dimensional (2D) cellular automaton (CA) model is proposed to simulate the ferrite-austenite transformation in binary low-carbon steels. In the model, the preferential nucleation sites of austenite, the driving force of phase transformation coupled with thermodynamic parameters, solute partition at the ferrite/austenite interface, and carbon diffusion in both the ferrite and austenite phases are taken into consideration. The proposed model is applied to simulate the ferrite-to-austenite transformation during isothermal heating at 760 C that is in the ferrite and austenite two-phase range, the austenite-to-ferrite transformation during continuous cooling, and carbon diffusion during tempering at different temperatures for an Fe-0.2969 mol.% C alloy. The results show that during the isothermal heating, austenite nucleates and grows. The austenite grains are mostly located at the boundaries of ferrite grains. The carbon concentration in austenite is higher than that in ferrite. The simulated microstructure agrees reasonably well with the experimental observation. During the continuous cooling process, the austenite-to-ferrite transformation occurs accompanied with carbon diffusion. After cooling from the heating temperature of 760 C to room temperature with a cooling rate of 2 C/s, the carbon concentration field is nearly uniform, while a higher cooling rate of 5 C/s results in a non-uniform carbon concentration field. After tempering at different temperatures for 20 min, the uniformity of carbon distribution increases with increasing tempering temperature. The simulation results are used to understand the mechanisms of the observed experimental phenomena that a cold-rolled low-carbon enameling steel presents different yield strengths after different heat treatment processes. KEY WORDS: modeling; cellular automaton; ferrite-austenite transformation; low-carbon steel. 1. Introduction * Corresponding author: E-mail: zhumf@seu.edu.cn DOI: http://dx.doi.org/10.2355/isijinternational.54.422 Ferrite-austenite transformations in low-carbon steels have attracted wide attention due to their fundamental role in phase transformation and industrial importance. Various experimental work 1 6) has been carried out to study the mechanisms of ferrite-austenite transformation and the relationship between microstructure and mechanical properties. However, the microstructure evolution could scarcely be observed in real-time, and some microstructural features cannot be examined with even the most sophisticated analytical tools. During recent several decades, various numerical models have been developed to simulate the microstructure evolution and solute diffusion during solid-solid phase transformation in steels. The phase field (PF) method 7 9) has been successfully applied to simulate the austenite-ferrite transformation. Yamanaka and co-workers 7) constructed a PF model to simulate the formation of Widmanstatten ferrite plates during the isothermal austenite-to-ferrite transformation. The PF model was also combined with the crystal plasticity finite element method to describe the austenite-toferrite transformation in the deformed austenite phase. 8) Tong et al. 10) applied a q-state Potts model-based Monte Carlo (MC) method to simulate the austenite-ferrite transformation during the isothermal austenite decomposition, in which the non-equilibrium austenite-ferrite interface is mixed diffusion/interface controlled. In addition, they combined the MC model with a crystal plasticity finite element method to study the influence of austenite deformation on the subsequent isothermal austenite-ferrite transformation. 11) Because of its simple structure and good computational efficiency, the cellular automaton (CA) method has also been adopted by many researchers 12 15) to study the mechanisms of solidsolid phase transformation. Kumar and co-workers 12) developed a CA model to simulate the competition between the nucleation and the early growth of ferrite from austenite. It is found that the competition determines the variation of ferrite grain size with the cooling rate and austenite grain size. Zhang et al. 13) employed a CA model to investigate the transformation of austenite to ferrite during continuous cooling. An hexagonal lattice was used in the model to reduce the anisotropy caused by the CA algorithm. The model incorporates the local concentration changes with a nucleation or growth function to obtain the final nucleation 2014 ISIJ 422

number, ferrite grain size and the kinetics of ferrite formation based on the cooling rate or the undercooled temperature. Taking both carbon diffusion and ferrite/austenite interface dynamics into consideration, Lan et al. 14) developed a two-dimensional CA model to predict the growth kinetics of ferrite grains. Moreover, Lan and Xiao et al. 15) combined the CA method with a crystal plasticity finite element model to simulate the austenite-ferrite transformation in a C Mn steel with an heterogeneously deformed microstructure. The results show the inhomogeneous microstructure evolution occurring in the deformed austenite decomposing process. However, limited work has so far been reported regarding applying simulations to analyze the processing-microstructure-property relationships of steels. In this paper, a two-dimensional CA model is proposed to simulate the ferrite (α) austenite (γ) transformation during isothermal heating and continuous cooling, as well as the carbon diffusion in the subsequent tempering process. The simulation results are applied to understand the mechanisms of the experimentally observed phenomena that a coldrolled low-carbon enameling steel exhibits different yield strengths after heat treatment with different cooling rates and then tempering at different temperatures. 2. Experimental Phenomena A cold-rolled enameling steel, containing 0.07 wt.% C, 0.43 wt.% Si, 1.22 wt.% Mn, 0.04 wt.% P and 0.001 wt.% S, was heated at 760 C for 5 min, and then cooled by air cooling (~5 C/s) and sand cooling (~2 C/s). After heated at 760 C for 5 min and then air cooled, the samples were then tempered at 200 C 500 C for 20 min. Table 1 shows the yield strengths of the cold rolled enameling steel before and after the different heat treatment processes. It is found that the yield strength of the sample by sand cooling is higher than that of the one by air cooling. After tempering, the yield strength increases with the tempering temperature increasing. Figures 1 and 2 present the stress-strain curves of the samples with different heat treatment processes. As shown in Fig. 1, the stress-strain curve of the sample by sand cooling has an evident yield platform, while the one obtained by air cooling does not have obvious yield platform. After the samples by air cooling are tempered, the yield platform becomes gradually obvious as tempering temperature increases, as shown in Fig. 2. It is known that yield platform is closely related to the Cottrell atmosphere which is associated with carbon diffusion and distribution. In the following sections, a CA model is proposed and applied to simulate ferrite (α) austenite (γ) transformation during different heat treatment processes to understand the mechanisms of the experimental phenomena exhibited in Figs. 1 and 2. 3. Model Description, Governing Equations, and Numerical Algorithms 3.1. Model Description The steel used in the experiment is a multi-component Fe alloy. For the sake of simplicity, a pseudo-binary Fe C alloy is adopted in the present simulations. Figure 3 shows the vertical section of the phase diagram calculated using Pandat, 16) Fig. 1. Fig. 2. The stress-strain curves of the samples cooled with different cooling rates after heated at 760 C for 300 s. (Online version in color.) The stress-strain curve of the samples before and after tempering at different temperatures. (Online version in color.) Table 1. Yield strengths of the samples before and after different heat treatment processes (MPa). 760 C for 5 min Air cooling + temping As rolled Sand cooling Air cooling 200 C 300 C 500 C 335.9 356.0 291.2 310.3 337.3 387.6 Fig. 3. The vertical section of the phase diagram calculated from the multi-component Fe alloy with 0.07 wt.% C, 0.43 wt.% Si, 1.22 wt.% Mn, 0.04 wt.% P and 0.001 wt.% S. (Online version in color.) 423 2014 ISIJ

a thermodynamic phase diagram calculation software, based on the composition of the experimental steel described in Section 2. The relevant thermodynamic parameters used in the simulations are taken from Fig. 3. The initial concentration of the pseudo-binary Fe C alloy used for simulations is taken as u 0=0.2969 mol.% C (indicated by the dotted line in Fig. 3). This composition will produce the equilibrium γ phase fraction of 0.12 at the temperature of 760 C, which is identical with that of the experimental steel. In the present work, the phase transformations of both α γ and γ α during heat treatment are simulated, while the coarsening of ferrite grains is ignored. The heat treatment processes used in the simulations are divided into three parts: (1) Isothermal heating process: The heating temperature is 760 C and holding time is 300 s. During isothermal heating, ferrite-to-austenite phase transformation, solute carbon partition at the α/γ interface, and carbon diffusion in both ferrite and austenite phases take place. Based on experimental observations, the nucleation sites of austenite are preferentially distributed at the ferrite grain boundaries. (2) Continuous cooling process: The continuous cooling processes are simulated from 760 C to room temperature with the cooling rates of 2 C/s and 5 C/s, corresponding to the cases of sand cooling and air cooling, respectively. During the cooling process, austeniteto-ferrite phase transformation, carbon partition at the α/γ interface, and carbon diffusion in both ferrite and austenite phases take place. For the sake of simplicity, the phase transformation of austenite to pearlite is not considered. As shown in Fig. 3, the eutectoid temperature is about 686 C. Considering the kinetic effect on phase transformation, the end temperatures of γ α phase transformation are taken as 650 C and 600 C for the cooling rates of 2 C/s and 5 C/s, respectively. When the temperature is cooled down below the end temperature of γ α phase transition, the retained austenite is assumed to be transformed to supersaturated ferrite, while carbon diffusion continues to the room temperature with temperature dependent diffusion coefficients. (3) Tempering process: The case cooled with 5 C/s is then isothermally heated at 200 C 500 C, respectively, and held for 20 min. During the tempering process, carbon diffusion takes place, but no phase transformation is considered. 3.2. Interface Migration According to the interface dynamic theory, 17) the velocity of interface migration, v n, is determined by Δgm / Vm = vn / Meff + σ K... (1) where Δg m is the driving force of phase transformation, V m is the mole volume, M eff is the interface mobility, σ is the interface energy, and K is the interface curvature. In the present model, the effect of interface curvature is neglected. The interface mobility, M eff, is calculated by 18) M = M exp Q RT V eff ( ) 0 /... (2) where M 0 is the pre-exponential factor, Q is the activation energy, and R is the gas constant. With the assumption of dilute solution, the following m equation can be derived: α α Δgm / Vm = RTM ( ke 1) u ( T) ue ( T) / Vm... (3) where T M is the allotropic transformation temperature, u α (T) is the actual concentration in ferrite at temperature T, k e is the equilibrium partition coefficient, defined as k e = γ α ue ( T) / ue ( T), where u α e ( T) and u γ e ( T) are the equilibrium concentrations in ferrite and austenite at temperature T, respectively. The equilibrium concentrations of ferrite and austenite, u α ( T ) and u γ T e e ( ), are determined by fitting the thermodynamic data in the phase diagram of the pseudo binary Fe C alloys shown in Fig. 3. The equilibrium concentrations of ferrite and austenite varying with temperature can be written in the following polynomial forms: u α ( T)= B T e 9 k= 1 u γ ( T)= FT e 9 k= 1... (4)... (5) where the coefficients B 1=1.9369 10 24, B 2=1.0019 10 21, B 3=4.4444 10 19, B 4=1.5343 10 16, B 5=3.1797 10 14, B 6= 1.2132 10 16, B 7=1.7459 10 19, B 8= 1.1201 10 22, B 9=2.6974 10 26 ; F 1=1.4970 10 22, F 2=8.1092 10 20, F 3=3.7673 10 17, F 4=1.3621 10 14, F 5=2.9566 10 12, F 6= 1.0041 10 14, F 7=1.2849 10 17, F 8= 7.3405 10 21, F 9=1.5789 10 24. When the temperature is above the eutectoid temperature of 686 C, the equilibrium concentrations of ferrite and austenite are temperature dependent calculated using Eqs. (4) and (5). While when the temperature is below the eutectoid temperature, the equilibrium concentrations of ferrite and austenite are taken as the constant values identical with those calculated at the eutectoid temperature. According to Eqs. (1) and (3), and ignoring the effect of interface curvature, the velocity of interface migration, v n, is determined by ( ) ( ) ( ) ( ) v = M Δ g / V n eff m m { } α α = M RT k 1 u T u T / V eff M e e m... (6) Therefore, the increment of α phase fraction can be evaluated by Δϕ = v nδx/δt, where ϕ is α phase fraction, Δx and Δt are the grid spacing and time step, respectively. Equation (6) can be used for the simulation of both α γ and γ α phase transformations. For the phase transformation of α γ, the value of v n is negative, and thus the fraction of ferrite is decreased, i.e., Δϕ <0. According to the lever law, the mean concentration u is calculated by α u = ϕu + 1 ϕ u... (7) Defining p(ϕ) = ϕ + k e(1 ϕ), Eq. (7) can be simplified as u = u α p(ϕ). Thus u α in Eq. (6) is calculated by... (8) 3.3. Carbon Diffusion During α γ phase transformation, solute partition between ferrite and austenite at the α/γ interface is considered according to u γ = k eu α. According to the Fick s second law, the governing equation of carbon diffusion in 2D can be written as k k ( ) α u u/ p ϕ = ( ) k k γ 2014 ISIJ 424

u = ( u/ p( ϕ)) + ( u/ p( ϕ)) D( ϕ) D( ϕ) t x x y y... (9) where D(ϕ) is the diffusion coefficient associated with the fraction of ferrite. Similar to the calculation of mean concentration u, D(ϕ) can be calculated by D( ϕ) = ϕd + k 1 ϕ D α... (10) where D α and D γ are carbon diffusivities in the α and γ phases, respectively. Equation (9) is derived based on the mass conservation and solute equilibrium at the α/γ interface. On the right-hand side of Eq. (9), an equivalent concentration u/p(ϕ) is used, which ensures the solute equilibrium at the α/γ interface. On the other hand, the derivative of mean concentration with respect to time, u/t, on the left-hand side of Eq. (9) includes the effect of solute partition due to α γ transition. Thus, Eq. (9) facilitates the problem of discontinuous carbon concentrations and solute partition at the α/γ interface in a straightforward manner, and the entire domain can be treated as a single phase for solute transport calculation. Equation (9) is solved using an explicit finite difference scheme with a time step determined by the carbon diffusivity in ferrite, and the zero-flux boundary condition is adopted. 3.4. Numerical Solution Sequence The CA algorithm used in the present work is described as follows. The simulation system is divided into a uniform orthogonal arrangement of cells. Each cell has the following variables: (1) grain orientation, I; (2) α phase fraction, ϕ (the cell represents α phase or γ phase, when ϕ =1 or ϕ =0, respectively); (3) the symbol ϕ int=1 indicates the cells at the ferrite/austenite (α/γ) interface; and (4) mean carbon concentration, u. The velocity of interface migration is calculated using Eq. (6). It is assumed that the kinetics of interface migration along grain boundaries is 2.5 times faster. Then, the increment of new phase fraction, Δf new, is evaluated by... (11) where G new is a geometrical factor that is introduced to eliminate the artificial anisotropy caused by the CA square cell. G new is related to the states of neighboring cells and defined by I II Gnew = min 1 1 4 1 4, Sm + S m= m m, 1 = 1 3 2... (12) f I II 0( new < 1) S, S = 1( f new = 1) where S I and S II indicate the state of the nearest neighbor cells and the second-nearest neighbor cells, respectively, and f new is the new phase fraction of neighbor cells. At the end of each time step, the fraction of the new phase of each interface cell is updated. When the fraction of new phase equals one, the interface cell transforms its state from interface to the new phase. This transformed new phase cell in turn captures a set of its neighbors of the parent phase to be the new α/γ interface cells. The phase transformation will thus continue in the next time step. Then, carbon redistribution and diffusion is calculated by solving Eq. (9). Since the simulations involve different heat treatment processes, the e ( ) Δf = G v Δt / Δx new new n γ solution sequences of the different processes are described below. I. The solution sequence for the simulation of isothermal heating process is as follows: (1) Initialize the simulation system with domain length, grid spacing, initial uniform carbon composition field, and ferrite grains with different orientations. (2) Set austenite nucleus at ferrite grain boundaries. The grain index, phase fraction ϕ, and mean concentration u of austenite are also initialized with the corresponding values. (3) Calculate the increment of γ phase fraction by solving Eq. (6) and Δϕ = G γv nδt/δx. When ϕ =0, the cell is transformed to austenite. (4) Calculate carbon diffusion in both α and γ phases by solving Eqs. (9) and (10). (5) Time step from Step (3) until the end of the simulation. II. The solution sequence for the simulation of continuous cooling process is as follows: (1) Initialize the simulation system with the microstructure and carbon concentration field calculated at the end of the isothermal heating process. (2) Calculate the increment of the α phase fraction by solving Eq. (6) and Δϕ = G αv nδt/δx. When ϕ =1, the cell is transformed to the ferrite phase. (3) Calculate carbon diffusion in both the α and γ phases by solving Eqs. (9) and (10) using the temperature dependent carbon diffusivities in α and γ phases. (4) Time step from Step (2) until the end of the simulation. III. The solution sequence for the simulation of tempering process is as follows: (1) Initialize the simulation system with the microstructure and carbon concentration field calculated at the end of the continuous cooling process. (2) Repeat calculating carbon diffusion in the domain by solving Eqs. (9) and (10) until the end of the simulation. The physical property parameters used in the simulations are listed in Table 2. 4. Results and Discussion 4.1. Isothermal Heating Process In the present work, the calculation domain consists of a 400 316 square grid with Δx=0.3 μm. The domain size equals the size of SEM micrographs. At the beginning of the simulation, the computational domain is initialized with a uniform carbon concentration u=u 0 and ferrite phase (ϕ =1, and ϕ int=0) with various grain orientations, as shown in Fig. 4. Table 2. Physical property parameters. Symbol Definition and Unit Value R Gas constant (J mol 1 K 1 ) 8.314 D α Carbon diffusivity in ferrite (m 2 s) 2.2 10 4 exp( 122 500/RT) 9) D γ Carbon diffusivity in austenite (m 2 s) 1.5 10 5 exp( 142 100/RT) 9) V m Mole volume (m 3 mol 1 ) 7.2 10 6 M eff Interface mobility (m 4 J 1 s 1 ) 0.035exp( 14 700/RT)V m 19) T M Allotropic transformation temperature, A 3(K) 1 168.64 425 2014 ISIJ

Since the isothermal holding temperature of 760 C is higher than the eutectoid temperature (686 C), during the isothermal heating process, ferrite-to-austenite phase transformation happens. According to the experimental observations, the nucleation of austenite mostly appears at the ferrite grain boundaries. The number of the austenite nuclei is also determined based on the experimental metallographs. With holding time increasing, the α phase transforms to γ phase accompanied with carbon diffusion, and austenite grains grow until the fraction of γ phase reaches the equilibrium value (about 0.12). Figure 5 presents a comparison of the simulated and experimental microstructures after isothermal holding at 760 C for 300 s. Figure 5(a) shows the simulated morphology in which the deep and light colors indicate ferrite and austenite phases, respectively, and the black lines represent the boundaries of different grains. Figure 5(b) shows the SEM image obtained by quenching the sample after heating at 760 C for 300 s to retain the austenite morphology. It is found that the simulated microstructure compared reasonably well with the experimental observation. Figure 6 shows the simulated austenite phase fraction varying with time during the isothermal heating at 760 C. As shown, at the early stage of heating, the increment of γ fraction is quite limited. Then the γ fraction increases evidently within 0.1 s 10 s, which implies that the α γ transformation might take place mainly in this period. After holding about 10 s, the γ fraction approaches gradually the equilibrium value of 0.12. The evolution of the carbon concentration field during isothermal heating at 760 C is shown in Fig. 7. Different colors represent different carbon concentration levels, and the black lines represent the boundaries of different grains. Since the carbon concentration in the γ phase is about two orders of magnitude larger than that in the α phases, the color levels in Figs. 7(a) 7(c) are adjusted to show clearly the evolution of the carbon concentration field in the α phase, and the maximum values of the color legends are lower than the real concentration in the γ phase. However, the color legend of Fig. 7(d) depicts the actual carbon concentration levels in both the γ and α phases. It can be seen that the carbon concentration in the α phase is much lower than that in the γ phase. In the early stage of isothermal heating, austenite nucleates at the ferrite grain boundaries. The area near the γ/α interface exhibits the lower carbon concentration, because of the solute partition between α and γ phases. As α γ phase transformation proceeds, more carbon atoms are absorbed by the growing austenite grains. Driven by the carbon concentration gradients, carbon diffusion happens. After the fraction of γ phase reaches the equilibrium value, the phase transformation is completed, while carbon diffu- Fig. 4. The initialized ferrite grains with different grain orientations. (Online version in color.) Fig. 6. Simulated austenite phase fraction varying with time during the isothermal heating at 760 C. (Online version in color.) Fig. 5. Comparison of microstructures after isothermal heating at 760 C for 300 s: (a) simulation and (b) SEM image obtained by quenching the sample after heating at 760 C for 300 s to retain the austenite morphology. (Online version in color.) 2014 ISIJ 426

Fig. 7. Simulated evolution of carbon concentration field during isothermal heating at 760 C for: (a) 0.1 s; (b) 1 s; (c) 10 s; and (d) 300 s. (Online version in color.) Fig. 8. Simulated carbon distribution after isothermal heating at 760 C for 300 s and cooled down to the room temperature with different cooling rates: (a) 2 C/s and (b) 5 C/s. (Online version in color.) sion continues, resulting in gradually uniform carbon concentration fields in both α and γ phases. The carbon concentrations in both phases tend to be stabilized with the new equilibrium values, as shown in Fig. 7(d). The minimum and maximum values in the legend of Fig. 7(d) correspond to the equilibrium concentrations in the α phase and γ phases, respectively. 4.2. Continuous Cooling Process During the continuous cooling process, the temperature of the calculation domain decreases from 760 C to room temperature with different cooling rates of 2 C/s and 5 C/s. During cooling, austenite-to-ferrite phase transformation, accompanied with carbon diffusion, takes place. As described in Section 3.1, the end temperatures of γ α phase 427 2014 ISIJ

Fig. 9. Simulated carbon distribution after tempering at different temperatures for 20 min: (a) 200 C; (b) 300 C; and (c) 500 C. (Online version in color.) transformation are chosen as 650 C and 600 C for the cooling rates of 2 C/s and 5 C/s, respectively. When the temperature is cooled down below the end temperature of γ α phase transition, the retained austenite is assumed to be transformed to supersaturated ferrite through allotropic transformation without considering the austenite to pearlite transformation. The diffusion of carbon continues to room temperature with the temperature dependent carbon diffusivity in the ferrite phase. Figure 8 shows the simulated carbon distribution after isothermal heating at 760 C for 300 s and cooled down to room temperature with different cooling rates of 2 C/s and 5 C/s. The legends in Fig. 8 are identical with that in Fig. 7(d) for comparison. As shown, the carbon concentration field for the case of 2 C/s is more uniform than that cooled with 5 C/s. For the latter case, some regions at the boundaries have relative higher carbon concentrations as shown in Fig. 8(b). Apparently, these regions should be the original austenite grains before cooling. The maximum and minimum carbon concentrations are 0.3001 mol.% and 0.2879 mol.% for the 2 C/s cooling rate (Fig. 8(a)), while they are 3.2367 mol.% and 0.0946 mol.% for the 5 C/s cooling rate (Fig. 8(b)), respectively. It is evident that the cooling process with lower cooling rate provides more sufficient time for carbon diffusion, which is beneficial for carbon diffusion to the locations of dislocations to form high concentration Cottrell atmospheres. It is well known that high concentration Cottrell atmospheres have a crucial importance on pinning dislocations, leading to a obvious yield platform in the stress-strain curve, and an increased yield strength. 20) Accordingly, it is understandable that the sample cooled by sand cooling (2 C/s) has a more obvious yield platform and a higher yield strength compared with the one by air cooling (5 C/s), as shown in Fig. 1 and Table 1. 4.3. Tempering Process The sample cooled by air cooling (5 C/s) undergoes an additional tempering process at different temperatures. Since the tempering temperatures are below the eutectoid temperature, there is no α-γ phase transformation. As shown in Fig. 8(b), at the end of the continuous cooling process with the cooling rate of 5 C/s, the carbon concentration field is still quite uneven. Therefore, carbon diffusion occurs during the tempering process. Figure 9 presents the simulated carbon concentration fields after tempering at different temperatures for 20 min. The color legends in Fig. 9 are also set to be identical with that of Fig. 7(d). It is noted that with tempering temperature increasing, the uniformity of carbon distribution increases gradually. The maximum and minimum carbon concentrations in Figs. 9(a) 9(c) are 3.2291 mol.% and 0.0966 mol.%, 2.9819 mol.% and 0.1794 mol.%, 0.2996 mol.% and 0.2960 mol.%, respectively. Comparing Figs. 9(a) with 8(b), it can be found that the tempering at 200 C for 20 min has less effect on the carbon concentration field. When the tempering temperature is increased to 500 C, the carbon distribution becomes much more uniform as shown in Fig. 9(c) that is very close to the one after isothermal heating at 760 C for 300 s and cooled with 2 C/s as shown in Fig. 8(a). This is due to the fact that the carbon diffusivity increases with temperature. The relative high carbon diffusivity at 500 C results in a more sufficient carbon diffusion. As discussed in the previous section, sufficient carbon diffusion is beneficial to forming high concentration Cottrell atmospheres, leading to a more obvious yield platform and increased yield strength. Based on this mechanism, it can be reasonably explained that the yield platform becomes gradually evident and yield strength increases with increasing tempering temperature as shown in Fig. 2 and Table 1. 5. Conclusions (1) A 2D CA model is proposed to simulate the ferriteto-austenite phase transformation. The model involves the preferential nucleation sites of austenite, the driving force of phase transformation, carbon solute partition at the ferrite/ austenite interface, and carbon diffusion in both the ferrite and austenite phases. The present model is able to simulate ferrite-to-austenite phase transition during isothermal heating in the temperature range of ferrite-austenite phase coexistence, austenite-to-ferrite phase transition during continuous cooling, and carbon diffusion during tempering processes of binary low-carbon steels. (2) During isothermal heating at 760 C that is within the temperature range of ferrite-austenite phase coexistence, ferrite-to-austenite phase transition takes place accompanied with carbon redistribution. Austenite nucleates at the boundaries of ferrite grains. The growing austenite absorbs carbon atoms from ferrite at the α/γ interface, resulting in carbon concentration gradients and then carbon diffusion in both phases. After holding for 300 s, the carbon concentrations in both phases are nearly uniform and stabilized with the 2014 ISIJ 428

corresponding equilibrium values. The simulated microstructure compared reasonably well with that obtained experimentally. (3) During the continuous cooling process, austenite-toferrite phase transition takes place accompanied with carbon redistribution. The cooling process with lower cooling rate results in a more uniform carbon concentration field, since the lower cooling rate provides more sufficient time for carbon diffusion. This indicates that the lower cooling rate is beneficial to forming the high concentration Cottrell atmospheres. This simulation can be used to understand the experimental phenomenon that the sample isothermally heated at 760 C and then cooled by sand cooling exhibits a more obvious yield platform and a higher yield strength than that by air cooling. (4) During the tempering process, carbon diffusion occurs. The carbon concentration distribution becomes more uniform as tempering temperature increases. This implies that the tempering process at higher temperature promotes carbon diffusion to pin dislocations and thus to form the high concentration Cottrell atmospheres. Consequently, the yield platform becomes more obvious and yield strength increases as tempering temperature increases. Acknowledgements The authors wish to thank CompuTherm Company, USA, for generously providing us with the license of Pandat software. This work was financially supported by the AO Smith Corporate Technology Center, USA, and NSFC (Grant No. 51371051). REFERENCES 1) C. P. Scott and J. Drillet: Scripta Mater., 56 (2007), 489. 2) F. L. G. Oliveira, M. S. Andrade and A. B. Cota: Mater. Charact., 58 (2007), 256. 3) T. Furuhara, K. Kikumoto, H. Saito, T. Sekine, T. Ogawa, S. Morito and T. Maki: ISIJ Int., 48 (2008), 1038. 4) Y. C. Liu, D. J. Wang, F. Sommer and E. J. Mittemeijer: Acta Mater., 56 (2008), 3833. 5) M. Jiang, X. F. Yang, S. Y. Pan, B. W. Krakauer and M. F. Zhu: J. Mater. Sci. Technol., 28 (2012), 737. 6) M. Ferry, M. Thompson and P. A. Manohar: ISIJ Int., 42 (2002), 86. 7) A. Yamanaka, T. Takaki and Y. Tomita: ISIJ Int., 52 (2012), 659. 8) A. Yamanaka, T. Takaki and Y. Tomita: Mater. Trans., 47 (2006), 2725. 9) M. Militzer, M. G. Mecozzi and J. Sietsma: Acta Mater., 54 (2006), 3961. 10) M. M. Tong, D. Z. Li and Y. Y. Li: Acta Mater., 53 (2005), 1485. 11) N. M. Xiao, M. M. Tong, Y. J. Lan, D. Z. Li and Y. Y. Li: Acta Mater., 54 (2006), 1265. 12) M. Kumar, R. Sasikumar and P. K. Nair: Acta Mater., 46 (1998), 6291. 13) L. Zhang, C. B. Zhang, Y. M. Wang, S. Q. Wang and H. Q. Ye: Acta Mater., 51 (2003), 5519. 14) Y. J. Lan, D. Z. Li and Y. Y. Li: Acta Mater., 52 (2004), 1721. 15) Y. J. Lan, N. M. Xiao, D. Z. Li and Y. Y. Li: Acta Mater., 53 (2005), 991. 16) S. L. Chen, S. Daniel, F. Zhang, Y. A. Chang, X. Y. Yan, F. Y. Xie, R. Schmid-Fetzer and W. A. Oates: Calphad., 26 (2002), 175. 17) C. Bos and J. Sietsma: Scripta Mater., 57 (2007), 1085. 18 G. P. Krielaart and S. van der Zwaag: Mater. Sci. Technol., 14 (1998), 10. 19) M. Hillert and L. Höglund: Scripta Mater., 54 (2006), 1259. 20) K. F. Ha: A Microscopic Theory of Mechanical Properties in Metal, Science Press, Beijing, (1983). 429 2014 ISIJ