The peritectic transformation, where δ (ferrite) and L (liquid)

Transcription

1 Cellular automaton simulation of peritectic solidification of a C-Mn steel *Su Bin, Han Zhiqiang, and Liu Baicheng (Key Laboratory for Advanced Materials Processing Technology (Ministry of Education), Department of Mechanical Engineering, Tsinghua University, Beijing , China) Abstract: A cellular automaton model has been developed to simulate the microstructure evolution of a C-Mn steel during the peritectic solidification. In the model, the thermodynamics and solute diffusion of multi-component systems were taken into account by using Thermo-Calc and Dictra software package. Scheil model was used to predict the relationship between the solid fraction and the temperature, which was used to calculate the movement velocity of the L/δ and the L/γ interfaces. A mixed-mode model in multi-component systems was adopted to calculate the movement velocity of the δ/γ interface. To validate the cellular automaton model, the variation of manganese distribution was studied. The simulated results showed a good agreement with experimental results reported in literatures. Meanwhile, the simulated growth kinetics of peritectic solidification agreed well with the experimental results obtained using confocal scanning laser microscopy (CSLM). The model can simulate the growth kinetics of the peritectic solidification and the distribution of concentrations of all components in grains. Key words: cellular automaton; peritectic solidification; C-Mn steel; multi-component; numerical simulation CLC numbers: TP391.9/TG Document code: A Article ID: (2012) The peritectic transformation, where δ (ferrite) and L (liquid) transform into γ (austenite), is a topic of importance in cast steel containing 0.10wt.% to 0.51wt.%C [1-3]. Ueshima et al. [3] conducted directional solidification experiment and made a calculation via predicting the solute redistribution on the basis of assumption that the peritectic transformation is controlled by the diffusion of carbon. Shibata et al. [2] used confocal scanning laser microscopy (CSLM) in directional solidification experiments of Fe-C binary alloys, and proposed that this is caused by massive δ to γ transformation or by direct precipitation of γ from the melt during peritectic reaction. They also showed that the growth of γ in the peritectic transformation in Fe-0.42C can be predicted by carbon diffusion models. In recent years, a variety of computer models, such as cellular automaton (CA) [4 5], monte carlo [6] and phase field [7-11], have been proposed to simulate the microstructure evolution, for example, the processes of solid-state transformation and dendritic growth. However, there was little study on the simulation of the microstructure evolution during the peritectic solidification. *Su Bin Male, born in 1985, Ph.D candidate. His research interests mainly focus on the microstructure simulation during casting and heat treatment processes. Corresponding author: Han Zhiqiang zqhan@mail.tsinghua.edu.cn Received: ; Accepted: In this paper, a CA model has been developed to simulate the structure transformation in a low carbon C-Mn steel during peritectic solidification. Meantime, the distribution of manganese and the growth kinetics of the peritectic solidification were also simulated, which agreed well with the experimental results reported in literatures. The model can simulate the growth kinetics of the peritectic solidification and the distributions of concentrations of all components in grains. 1 Model description 1.1 L δ When T P <T<T L (T L is liquidus temperature, T P is peritectic temperature), molten metal transforms into δ-phase. In this study, Scheil model [17] was employed. The major assumptions in this model are that back diffusion is considered in the solid phase; diffusion in the liquid phase is so fast that the liquid phase always has a uniform composition; and the phase equilibrium between the liquid phase and the solid phase at the local interface can be reached. It is assumed that equiaxed grains grow spherically and the velocity of the L/δ interface can be obtained from the relationship between the solid fraction and the temperature. In this paper, the relationship between solid fraction and temperature can be calculated by Thermo-Calc software [12-13]. 1.2 Peritectic transformation When the temperature reaches T P, the peritectic reaction, 221

2 CHINA FOUNDRY L+δ γ, occurs. The peritectic reaction is followed by the growth of the intervening γ as a consequence of the transformations of δ to γ and L to γ [2]. In this process, we suppose that a γ grain covers the surface of a δ grain immediately. The growth kinetics of the L/γ interface can also be obtained from the relationship between the solid fraction and the temperature mentioned above. As for δ/γ interface, a mixed-mode model [14] was adopted to calculate the movement velocity of the interface. In the mixed-mode model, both the finite interface mobility and the finite diffusivity of alloying elements were taken into account. The basis of the mixed-mode model is the assumption that the interface's moving velocity v can be described as: where M δ /γ is the interface mobility, which includes the structural influence on the interface mobility, such as incoherence of δ/γ interface, pinning effect, set-up of stresses or solute drags. ΔG δ γ is the chemical driving force that can be calculated by: where N is the number of components in the system; and are chemical potential per mole of component i in δ and γ at the interface. x is the mole fraction of component i in γ at the interface. In this model, the interface mobility can be described as: (1) (2) Vol.9 No.3 energy for atom motion at the interface; R is the gas constant. In order to use Eq. (1), the concentrations of all components in δ and γ at the interface are used to determine and. In this article, it is assumed that the forming γ -phase is always formed with the equilibrium concentration. Atoms continuously transfer into γ from δ with δ/γ interface moving. The new concentrations of all components in δ and γ at the interface are calculated from the mass balance. 2 Cellular automaton algorithm A cellular automaton model was developed to simulate the phase transformation during the peritectic solidification. The size of a square cellular automaton cell is Δh, as shown in Fig. 1. In order to simulate phase transformation, every cell has seven variables: (1) a crystal orientation variable that represents the crystal orientation; (2) a phase state variable that implies each cell has a state: L, δ, γ or δ/γ interface; (3) four concentration variables that represent carbon and manganese concentration (the concentrations of all components in δ and γ at the interface are all considered); (4) a distance variable l, that represents migration distance at the δ/γ interface, for the δ/γ interface cell, 0<l<Δh, for the rest cell, l = 0 or Δh. The flow chart of calculation is shown in Fig. 2. (3) where is the pre-exponential factor; E δ/γ is the activation Fig. 1: An illustration of CA cells Fig. 2: Flow chart of calculation 222

3 3 Thermodynamic calculations and kinetic simulations in multicomponent systems Thermo-Calc is a powerful and flexible software and database package for all kinds of phase equilibrium, phase diagram and phase transformation calculations and thermodynamic assessments. With its application-oriented interface, many types of process simulations can also be performed. In the present model, the phase transformation temperature, the relationship between the solid fraction and the temperature, the chemical potential, as mentioned above, are taken directly from Thermo- Calc. Some calculated results are plotted in Fig. 3 and Fig. 4. Solid Fig. 3: Relationship between solid fraction and temperature during solidification of 0.13C-1.52Mn steel (a) (b) Fig. 4: Chemical potential of iron atom in δ-phase and γ-phase In a multi-component system a large number of diffusion coefficients have to be evaluated, and moreover, they are generally functions of alloy composition and are interrelated. The diffusion coefficient of solute B in one phase D B, is directly related to the mobility coefficient M B by means of the Einstein relation: M B can be divided into a frequency factor activation enthalpy Q B : Both (4) and an (5) and Q B depend upon the composition, temperature, and pressure. In the present model, the diffusion coefficient of alloying elements in different phases at different temperatures, D, can be calculated using Dictra software [15]. Some calculated results are plotted in Fig Model validation To validate the cellular automaton model, the distribution of manganese and the growth kinetics of the peritectic solidification were simulated to compare with the experimental results reported in literatures. Ueshima et al. [3] conducted solidification experiment and studied the solute redistribution during the peritectic transformation. According to the experimental conditions, we (a) (b) Fig. 5: Diffusion coefficients of carbon and manganese 223

4 CHINA FOUNDRY simulated the variation of manganese distribution to validate the present model. The experimental material is a low-carbon steel with the chemical compositions shown in Table 1. The samples were heated until complete melting was achieved; then some were cooled at a rate of 2.7 min -1, others were cooled at 15 min -1 at temperatures higher than 1,480 and at 27 min -1 at temperatures lower than 1,480, respectively. Then samples were dropped into a water tank at different temperatures for quenching, and the solute distribution was maintained. Carbon could not be sufficiently frozen by the water quenching employed in this experiment due to its large diffusion coefficient in solid. The distribution of manganese were studied by using a computer aided EPMA (computer aided X-ray microanalyzer). The isoconcentration lines of Table 2: Parameters used in simulation Parameters Values T L ( ) 1,515 T P ( ) 1,493.5 E δ/γ (kj mol -1 ) 140 Vol.9 No.3 manganese were obtained on the transverse cross section of primary dendrite by EPMA measurements. These lines are used to validate the model. The cell size is 2 μm in the simulated area with cell grids. It is assumed that there is one δ grain in the calculation domain. The parameters used in the simulation are shown in Table 2. M 0 δ/γ (mol m J -1 s -1 ) (Cooling rate: 2.7 min -1 ) 0.1 (Cooling rate: 27 min -1 ) Table 1: Chemical composition of low carbon steel (wt. %) C Si Mn S P (a) Manganese (b) Carbon Fig. 6: Simulated concentration field: T = 1,480, cooling rate: 2.7 min -1 Figure 6 is the simulated results of carbon and manganese concentration distribution at 1, 480 under 2.7 min -1. The distributions of manganese at different temperatures are shown in Fig.7 and Fig. 8. f A is area fraction which can be calculated by area from the center of the grain to the isoconcentration line of manganese divided by the total area. The simulated results agree well with the experimental results. Shibata et al. [2] studied the growth kinetics of the peritectic transformation by using CSLM. According to the experimental conditions, we simulated the changes of areas of δ and γ during the peritectic transformation to validate (a) (b) (c) Fig. 7: Variation of manganese distribution: cooling rate: 2.7 min -1 (a) (b) (c) Fig. 8: Variation of manganese distribution: cooling rate: 27 min

5 the present model. The composition of the steel is listed in Table 3. After melting, the liquid was cooled to 1,492 and held at this temperature. The rate of the transformation can be obtained. The cell size is 2 μm in the simulated area with cell grids. In calculations, several δ grains are distributed randomly in the calculation domain. E δ/γ is 140 kj mol -1, is mol m J -1 s -1. Figure 9 shows the simulated results of the peritectic transformation during isothermal holding at 1,492. Figure10 shows the changes in the areas of δ and γ during isothermal holding at 1,492 according to simulated and experimental results. Table 3: Chemical composition of steel (wt. %) C Si Mn S P Fig. 9: Peritectic transformation of 0.42C-0.01Mn steel (a) 2 s during isothermal holding at (b) 1, s Fig. 10: Changes in areas of δ and γ during isothermal holding at 1,492 5 Conclusions (1) A cellular automaton model has been developed to simulate the microstructure evolution of a C-Mn steel during the peritectic solidification. In the model, the thermodynamics and solute diffusion of multicomponent systems were taken into account by using Thermo- Calc and DICTRA software. Scheil model was used to predict the relationship between the solid fraction and the temperature, which was used to calculate the movement velocity of the L/δ and the L/γ interface; a mixed-mode model in multi-component systems was adopted to calculate the movement velocity of the δ/γ interface. (2) To validate the model, the variation of manganese distribution in grains was studied. The simulated results showed a good agreement with experimental results reported in literatures. Meanwhile, the simulated growth kinetics of the peritectic solidification agrees well with the experimental results obtained by using CSLM. References [1] Nassar H, Fredriksson H. On peritectic reactions and transformations in low-alloy steels. Metall Mater Trans, 2010, 41A: [2] Shibata H, Arai Y, Suzuki M, et al. Kinetics of peritectic reaction and transformation in Fe-C alloys. Metall Mater Trans, 2000, 31B: [3] Ueshima Y, Mizoguchi S, Matsumiya T, et al. Analysis of solute distribution in dendrites of carbon steel with δ/γ transformation during solidification. Metall Mater Trans, 1986, 17B: [4] Zhang L, Zhang C B, Wang Y M, et al. Cellular automaton modelling of the transformation from austenite to ferrite in low carbon steels during continuous cooling. Acta Metall Sin, 2004, 40: [5] Lan Y J, Li D Z, Li Y Y. Modeling austenite decomposition into ferrite at different cooling rate in low-carbon steel with cellular automaton method. Acta Mater, 2004, 52: [6] Tong M M, Li D Z, Li Y Y. Modeling the austenite-ferrite diffusive transformation during continuous cooling on a mesoscale using Monte Carlo method. Acta Mater, 2004, 52: [7] Ohno M, Matsuura K. Diffusion-controlled peritectic reaction process in carbon steel analyzed by quantitative phase-field simulation. Acta Mater, 2010, 58: [8] Choudhury A, Nestler B, Telang A, Selzer M and Wendler F. Growth morphologies in peritectic solidification of Fe-C: A phase-field study. Acta Mater, 2010, 58: [9] Militzer M, Mecozzi M G, Sietsma J, Zwaag S V. Three-dimensional phase field modelling of the austenite-to-ferrite transformation. Acta Mater, 2006, 54: [10] Huang C J, Browne D J. Phase-field model prediction of nucleation and coarsening during austenite/ferrite transformation in steels. Metall Mater Trans, 2006, 37A: [11] Loginova I, Odqvist J, Amberg G, et al. The phase-field approach and solute drag modeling of the transition to massive γ α transformation in binary Fe-C alloys. Acta Mater, 2003, 51: [12] Bos C, Sietsma J. Application of the maximum driving force concept for solid-state partitioning phase transformations in multi-component systems. Acta Mater., 2009, 57: [13] Andersson J, Helander T, Höglund L, et al. THERMO-CALC & DICTRA, computational tools for materials science. Calphad, 2002, 26 (2): [14] Chem S L, Yang Y, Chen S W, et al. Solidification simulation using Scheil model in multi-component systems. Journal of Phase Equilibria and Diffusion, 2009, 30: [15] Borgenstam A, Engstrom A, Hoghund L, et al. DICTRA, a tool for simulation of diffusional transformations in alloys. J Phase Equilibria, 2000, 21(3): This work was funded by the National Science and Technology Major Project of China (No.2011ZX ). 225