Cellular Automata Modeling of Grain Coarsening and Refinement during the Dynamic Recrystallization of Pure Copper

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1 Materials Transactions, Vol. 51, No. 9 (1) pp to 16 #1 The Japan Institute of Metals Cellular Automata Modeling of Grain Coarsening and Refinement during the Dynamic Recrystallization of Pure Copper Ho Won Lee and Yong-Taek Im* National Research Laboratory for Computer Aided Materials Processing, Department of Mechanical Engineering, KAIST, Gusongdong, Yusonggu, Daejeon 35-71, Korea In this study, a cellular automata technique was developed to simulate a dynamic recrystallization process of pure copper. Moore s neighboring rule was applied with partial fraction and time step control in the current approach to represent the grain growth kinetics more accurately. The cellular automata model developed in this study was applied to a simulation of the dynamic recrystallization of pure copper during hot deformation and compared with the experimental flow stresses and grain sizes determined from hot compression tests for validation. The predicted results were in reasonably good agreement with the experimental results. The grain coarsening and refinement phenomena were also investigated in detail. Finally, the effects of the process parameters on the microstructure and flow stress were investigated from various simulation results. [doi:1.23/matertrans.m1116] (Received April 7, 1; Accepted June 28, 1; Published August 11, 1) Keywords: cellular automata, dynamic recrystallization, copper, grain coarsening, grain refinement, partial fraction 1. Introduction In the automobile industry, there is currently a strong driving force to produce high-strength metals owing to environmental and energy concerns. The basic strengthening mechanisms include substructural strengthening, solid solution strengthening, precipitation strengthening, grain boundary strengthening, and phase transformation strengthening. In particular, grain boundary strengthening through grain refinement is known to enhance both the strength and toughness of a material. 1) Abundant metallurgical phenomena are related to the evolution of the grain size. However, dynamic recrystallization (DRX) is one of the key processes that govern the grain size of materials during hot deformation. DRX generally takes place in metals such as copper and nickel which have low or medium levels of stacking fault energy. During the DRX process, new grains originate at high-angle grain boundaries such as prior grain boundaries, the boundaries of recrystallized grain, and deformation bands and twins. The new grains grow to the high dislocation density of the original grain due to the driving force, which is driven by the difference in the dislocation density. However, recrystallized grains will cease to grow as the material deforms further, which has the effect of reducing the dislocation density difference. The resulting microstructure, especially the grain size, varies with the deformation condition. Therefore, understanding the DRX is essential when attempting to control the mechanical properties of a material. Various models have been proposed thus far to predict microstructural changes during recrystallization. Typically, the Johnson-Mehl-Avrami-Kolmogrov (JMAK) theory 2) is widely used to model homogeneous recrystallization kinetics. However, the JMAK equation is not adequate for actual applications due to the heterogeneous nature of the recrystallization process. Recently, many modeling methods have been proposed to solve this problem, such as the vertex *Corresponding author, ytim@kaist.ac.kr model, 3) the Monte Carlo model, 4,5) and the phase field model. 6) Although these models successfully describe microstructural evolution during recrystallization, the CA model is used most often due to its straightforward time and length scale calibrations. Cellular automata model for recrystallization was firstly introduced by Hesselbarth and Gobel. 7) They successfully described recrystallization kinetics that was not predicted by the JMAK theory by introducing inhomogeneous characteristics of recrystallization. However, most studies have focused on static recrystallization and DRX with grain refinement phenomena. 7 1) Grain coarsening phenomenon has received less attention, although it occurs during hot deformation processes such as forging and rolling because of local irregularities in the process parameters. Therefore, in the present investigation, CA modeling of grain coarsening and refinement during the DRX of pure copper was conducted. To ensure isotropic grain growth, Moore s neighboring rule was applied with a partial fraction and controlled time step in the CA model. To verify the developed model, simulated microstructures and flow stresses were compared to the compression results. Grain coarsening and refinement phenomena related to the flow curves were subsequently investigated in detail. Finally, to investigate the effect of process parameters such as the temperature, strain, and strain rate on the microstructural evolution and flow stress, numerous numerical simulations were carried out under different conditions. 2. Experimental A set of compression tests was conducted to investigate the microstructural and flow stress changes during the DRX process. Commercially available pure copper was used in the experiments. Its chemical composition is given in Table 1. To ensure a homogeneous initial microstructure, the raw material was vacuum-annealed at 7 C for one hour and furnace-cooled to room temperature. The initial microstructure obtained from this process is given in Fig. 1, and the initial grain size was measured as approximately 64 mm.

2 Cellular Automata Modeling of Grain Coarsening and Refinement during the Dynamic Recrystallization of Pure Copper 1615 Table 1 Chemical composition of pure copper used in the current study. mass % Cu O Se S Pb Ag Sn Fe Ni Fig. 2 The growth aspects of grain using Moore s neighboring rule with partial fraction and time step control. Fig. 1 EBSD OIM map of prior microstructure after annealing. Cylindrical specimens with a height of 15 mm and a diameter of 1 mm were prepared from the processed material by machining. Hot compression tests were conducted using a Gleeble machine at a constant temperature and strain rate. The compression temperatures varied from 5 to 7 C and the strain rates varied from.2 to.1 s 1. First, the specimen was heated to the working temperature at a heating rate of 1 C/s. After a holding time of 3 s, the specimen was deformed up to a strain of.8. The compressed specimen was water-quenched immediately after the deformation for further microstructural investigation. The quenched specimen was cut in parallel along the compression axis and was polished using SiC and diamond papers. Finally, electrochemical polishing was conducted to prepare the electron backscattered diffraction (EBSD) specimen. An EBSD system (EDAX-TSL/Hikari) attached to field emission scanning electron microscope (FE-SEM, FEI/ Nova23) was used in the current study to investigate the microstructural changes that occurred during the DRX process. The acceleration voltage and working distance was kv and 11 mm, respectively. Due to the different grain sizes, the scanned area was varied from :3 :3 to 1 1 mm 2 and the step size was varied from 1.2 to 5. mm. Finally, while neglecting twin boundaries, an equivalent circle diameter (ECD) 2) was calculated using an orientation imaging microscopy (OIM) map. 3. CA Modeling of the DRX In the current investigation, two-dimensional square cells were employed for the CA modeling. Every cell had five state variables: a grain number variable that represented different grains; a variable representing recrystallized fraction; and three state variables representing whether or not the current cell is recrystallized, nucleated, and/or located on the boundary, respectively. Conventional Moore s and von Neumann s neighboring rules 11) with a deterministic transformation rule cannot demonstrate isotropic growth kinetics. Whereas the von Neumann s neighborhood contains besides the central cell itself only the four cells directly above, below and besides the central cell, the Moore s neighborhood contains the central cell and all eight cells adjacent to it. As a result, the final shape of the grains is either diagonal or square, both of which are unrealistic. Therefore, in the current study, Moore s neighboring rule was implemented by allowing a partial recrystallized fraction in a cell. The fraction increment F during the time increment t was calculated by summing the fraction from the interface cell during each time increment, as follows: F ¼ X v i t=leng i ð1þ where v i is the incoming velocity from the i-th neighboring cell, t is the time increment, and leng i represents the distance between the current and i-th neighboring cell. The grain shape using Moore s neighboring rule with a partial fraction and time step control can be nearly circular after a certain time step compared to those using Moore s and von Neumann s neighboring rules, as shown in Fig. 2. In addition, the time step was determined by finding the minimum time step to complete the grain growth of one partial cell. Cellular automata simulations were conducted with the following steps. The process is also shown in Fig. 3. Calculation of the minimum time step to ensure better results; calculation of dislocation density changes by work hardening at each time step; nucleation of the recrystallization embryo by comparing the dislocation densities in the grain boundaries with the critical dislocation density; simulation of the growth of the nucleus to the high dislocation density area.

3 1616 H. W. Lee and Y.-T. Im Start Calculate time step dt Calculate dislocation increment of every grain for dt N ρ Y End To model the dislocation density changes by work hardening, the one-parameter model by Kocks and Mecking 12) was used in the current study due to its simplicity and applicability in a hot deformation process. The Kocks- Mecking (KM) model is based on the assumption that the average dislocation density determines the kinetics of the plastic flow. In the kinetic equation for hot deformation, the flow stress is proportional to the square root of the dislocation density, as follows: p ¼ Gb ffiffiffi ð2þ In this equation, is a numerical constant, G denotes the shear modulus, b is the magnitude of Burger s vector of dislocation, and is the average dislocation density. The change in the dislocation density of a coarse-grained or single-phase material may be considered to consist of two components, as follows: pffiffiffi d=d" ¼ k 1 k2 ð3þ Here, k 1 ¼ s k 2 =Gb, k 2 ¼ f ðt; _"Þ, and s ¼fA 1 _" exp ðq def =RTÞg 1=n 1. The initial dislocation density was assumed to be 1 9 for all of the initial grains, and the increment of the dislocation density was calculated for every time step using eq. (3). Nucleation in the dynamic recrystallization process occurs when the dislocation density reaches a critical value. In the present investigation, the critical dislocation density proposed by Roberts and Ahlblom 13) was used. c ¼ð _"=3blM 2 Þ 1=3 ð4þ In this equation, l is the dislocation mean free path, M represents the grain boundary mobility, is the dislocation ρ i > c nucleation Nucleus growth t > t end Y Fig. 3 Schematics of the numerical procedure of cellular automata for calculating the DRX. N line energy (expressed as ¼ :5Gb 2 ), and is the grain boundary energy. In eq. (5), the grain boundary mobility (M) is given, as shown below. 14) M ¼ D b b=kt expð Q b =RTÞ Here, is the characteristic grain boundary thickness, D b denotes the boundary self-diffusion coefficient, Q b is the activation energy for boundary diffusion, k represents the Boltzmann s constant, and R is the gas constant. In eq. (4), the dislocation mean free path (l) was calculated from the following equation, as originally formulated by Takeuchi and Argon: 15) ð=gþðl=bþ n2 ¼ K 1 ð6þ In eq. (6), n 2 and K 1 are material constants. Nucleation in the dynamic recrystallization process occurs on the grain boundaries of prior grains. This assumption corresponds to the grain bulging nucleation mechanism of dynamic recrystallization. However, if the dislocation density of the recrystallized grain reaches a critical value, the grain boundaries of the recrystallized grain can also be considered as a possible position for nucleation. The dislocation density of the recrystallized grain was set close to zero and the orientation was randomly selected. The nucleation rate model suggested by Ding and Guo 1) was selected in the current study of nucleation. The model considers that the nucleation rate for dynamic recrystallization is a function of both the temperature and strain rate, as follows: _n ¼ C _" m 1 expð Q nucl =RTÞ ð7þ where C, m 1, and Q nucl are constants. The driving force for the growth of the nucleus in dynamically recrystallized grains is the stored strain energy difference between the recrystallized grain and the prior grains. A grain boundary moves with a velocity (v) in response to the net pressure (P) on the boundary. It is generally assumed that the velocity is directly proportional to the pressure and that the constant of proportionality is the mobility (M) of the boundary, as follows: 2) v ¼ MP ¼ Mð m d Þ ð8þ where d and m are the dislocation density of the dynamically recrystallized grain and that of the prior grain matrix, respectively. In the current investigation, the dynamic recrystallization of pure copper under hot compression was simulated with various temperatures and strain rates. The initial microstructure was created by the normal grain growth of the sitesaturated nucleus with randomly selected orientations, and the initial grain size was set as 64 mm, as initially measured. The simulation area was 1 1 mm 2, and 62,5 square cells with a periodic boundary condition were used. The parameters of the CA model simulated are given in Table 2. ð5þ Table 2 Parameters used in the CA model. G (N/m 2 ) b (m) (J/m 2 ) D b (m 3 /s) Q b (kj/mol) A 1 n 1 Q def (kj/mol) 4: : : :

4 Cellular Automata Modeling of Grain Coarsening and Refinement during the Dynamic Recrystallization of Pure Copper s -1 5 C C 7 C C Strain, ε Average Grain Diameter, D rx /µm Current [11] 1E11 1E12 1E13 1E14 1E15 1E16 1E17 1E18 1E19 Zener-Holomonn Parameter, Z Fig. 5 The relationship between the experimental grain size and Z. Fig s -1.1 s -1 1 C.5 s -1.2 s Results and Discussion Strain, Experimental flow stress curves for various deformation conditions. 4.1 Experimental result and validation of the CA model The flow stress curves obtained by the hot compression tests are shown in Fig. 4. The flow stress was higher for the large Zener-Hollomon parameter (Z) described as Z ¼ _" expðq act =RTÞ, in other words, at a high strain rate and low temperature. The critical strain, necessary to initiate the DRX, decreased as the Z value decreased. The flow stress curve at high Z values showed a broad single-peak (e.g., 5 C and.1 s 1 ). On the other hand, the flow stress curves at low Z values showed oscillating multi-peak curves under low strain and steady curves under high strain. It is known that single- and multi-peak curves are related to the prior and final grain sizes. If the prior grain size is greater than two times the recrystallized grain size, a single-peak curve may occur. 16) The relationship between the measured average grain size and Z is represented in Fig. 5. It was compared with the values from Blaz et al. 17) Considering different measurement techniques and materials, the current experimental result showed the general characteristics of the DRX process well. To verify the developed CA model, flow stress curves and final grain sizes at various temperatures and strain rates are compared with the experimental results, as shown in Fig. 6 and Fig. 7, respectively. The proposed CA model predicted the flow stress generally well, although it overpredicted the steady state stress at 5 C and.1 s 1 and underpredicted ε Experimental 5 C-.1s -1 C-.1s -1 C-.5s -1 C-.2s -1 7 C-.1s CA analysis 5 C-.1s -1 C-.1s -1 C-.5s -1 C-.2s -1 7 C-.1s Fig. 6 The comparison of flow stress curves between the experimental and CA model. the flow stress at C and.2 s 1. This occurred because the characteristics of the single- and multi-peak curves were rather different from each other. To represent both curves accurately, the CA model should be considered differently in some manner (e.g., nucleation). In spite of the errors that resulted, the CA results represented the single- and multi-peak characteristics well. The flow stress curve at 5 C and.1 s 1 showed only a single-peak curve because the ratio of the prior grain size and the recrystallized grain was greater than two, as shown in

5 1618 H. W. Lee and Y.-T. Im Average Grain Diameter, D rx /µm Prior grain size: 64µm Experimental CA model T5-.1 T-.1 T-.5 T-.2 Multi peaks Single peak T7-.1 Stress, σ/mpa 3 1. T=7 C, ε=.1 s -1 1st 2nd 3rd 4th 5th 6th 7th 8th 9th Average Grain Diameter, D rx /µm Fig. 9 Changes of the flow stress and average grain size during grain coarsening by the DRX. σ D rx 9 7 Fig. 7 The comparison of final grain diameter between the CA model and experimental observation Fig. 8 Changes of the microstructure during grain coarsening by the DRX. Fig. 7. The predicted grain sizes were generally in good agreement with the experimental results, although some errors existed in the results with small grains. 4.2 Grain coarsening and refinement As explained earlier, the shape of the flow stress curves is related to the ratio of the prior and recrystallized grain size. In other words, it is related to microstructural phenomena such as grain coarsening and refinement during the DRX process. Therefore, the grain coarsening, refinement phenomena, and the characteristics of the flow curves are investigated in detail in this study. The simulated microstructure during grain coarsening by the DRX process is given in Fig. 8, and the changes in the flow stress and average grain size are represented in Fig. 9. In Fig. 8, similar to the single-peak DRX, the nuclei originated at the prior grain boundary past the critical strain. However, most of the prior grain boundaries were consumed rapidly compared to the single-peak DRX due to the rapid grain boundary velocity and the larger recrystallized grain size. As a result, no more nucleation occurred after a certain strain (about.13), and the grain structure was maintained until new nuclei originated at the recrystallized grain site. Therefore, the grain structures at the strains of.14 and.16 were very similar. The flow stress also showed a curve similar to that of the single-peak DRX up to a strain of.1, as shown in Fig. 9. However, the flow stress increased again past a strain value of.12 owing to the lack of nucleation between the DRX cycles. The next DRX cycle was then initiated past a certain deformation, and the stress was decreased again by further nucleation. The average grain size was also oscillating owing to the several cycles of the DRX that were run. The average grain size initially decreased immediately after the initiation of the current cycle of the DRX as a consequence of the nucleation stage. It then increased rapidly due to the growth and remained constant until the next DRX cycle. This phenomenon was repeated for nine cycles for the current condition.

6 Cellular Automata Modeling of Grain Coarsening and Refinement during the Dynamic Recrystallization of Pure Copper T=5 C, ε=1 s -1. T= C, ε=.5 s σ Drx Drx.2 1stand 2nd cycle 1st cycle Fig. 11 Changes of the flow stress and average grain size during grain refinement by the single-peak DRX. The simulated microstructure during the grain refinement process with the single-peak DRX is given in Fig. 1, and the changes in the flow stress and average grain size are represented in Fig. 11. In Fig. 1, the nuclei for the DRX process originated at the prior grain boundaries (the black line in the figure) past the critical strain. The recrystallized grain then grew into the prior grain structure (the white area in the figure). The recrystallized grain underwent further deformation and reached the critical dislocation density again. Therefore, the nuclei also originated at the boundaries of the recrystallized grains before consuming all of the prior grain boundaries. This implies that more than one DRX cycle was taking place simultaneously in the grain for the singlepeak DRX. In Fig. 11, the slope of the flow stress was changed past the critical strain (about.2) and reached its peak at a strain of about.25. The flow curves then softened and reached a steady state at a strain value of.4. The average grain size curve changed rapidly twice due to the different cycles of the DRX process. The average grain size was held constant up to the critical strain, decreased rapidly σ Average Grain Diameter, Drx /µm Stress, σ/mpa Stress, σ/mpa Fig. 1 Changes of the microstructure during grain refinement by the single-peak DRX (5 C and.1 s 1 ). Fig. 12 Changes of the flow stress and average grain size during grain refinement by the multi-peaks DRX. directly past the critical strain, and then became saturated. However, it decreased rapidly again at a strain of.4 when the second DRX cycle occurred at the recrystallized grain boundaries. The grain refinement flow stress curve by the DRX process can also show a multi-peak result (e.g., C and.5 s 1 in Fig. 6). In this case, the mechanism is similar to that of grain coarsening. Although the recrystallized grain size is smaller than the prior grain size, it is large enough to consume the prior grain boundaries before the initiation of the next DRX cycle. The change in the average grain size is also observable in Fig Effect of process parameters on the DRX To investigate the effect of the process parameters on the microstructure, a histogram of the grain size is represented in Fig. 13. The final grain size decreased when the strain rate increased and the temperature decreased. This was due to the relationships among the nucleation rate, mobility, and the resulting grain size. If the strain rate increases, the nucleation rate will be increased. Therefore, the grain size decreases as

7 16 H. W. Lee and Y.-T. Im C,.1s C,.1s C,.5s Grain Diameter, D/µm Grain Diameter, D/µm Grain Diameter, D/µm.12.1 C,.2s C,.1s Grain Diameter, D/µm Grain Diameter, D/µm Fig. 13 Distributions of the grain size with different deformation conditions. the strain rate increases. If the temperature increases, the nucleation rate and mobility will increase together. However, the change in the growth rate is rather important compared to the change in the nucleation rate in the current condition. Therefore, the result is that the grain size increases as the temperature increases. In the upper three cases in the figure, the fraction of grains having smaller grain sizes than average is larger. The deviation from the average grain size was increased when the temperature increased and the strain rate decreased. The large deviation in the grain size can also be observed in Fig Conclusions A cellular automata model using Moore s neighboring rule with a partial fraction and controlled time step for the analysis of dynamic recrystallization was successfully developed in the present study. To validate the developed model, flow stresses and recrystallized grain sizes were determined experimentally and compared with the CA results. The predicted result was generally in good agreement with the experimental results, although the former showed some errors in particular conditions. The grain coarsening and refinement phenomena were also investigated. In terms of grain refinement, the flow stress curves showed both a single-peak and multiple-peaks up to the recrystallized grain size, although a multi-peak curve was noted during the grain coarsening phenomena. Finally, the effects of the process parameters were investigated in this study to derive results that matched those in earlier findings. Acknowledgements The authors wish to acknowledge the grant of National Research Laboratory program of the Ministry of Education, Science and Technology through National Research Foundation (No. RA ) and BK21. REFERENCES 1) N. J. Petch: J. Iron Steel Inst. 174 (1953) ) F. J. Humphreys and M. Hatherly: Recrystallization and Related Annealing Phenomena, (Elsevier, Oxford, UK, 4). 3) F. J. Humphreys: Scr. Metall. Mater. 27 (1992) ) A. D. Rollett, M. J. Luton and D. J. Srolovitz: Acta Metall. Mater. (1992) ) P. Peczak and M. J. Luton: Acta Metall. Mater. 41 (1993) ) T. Takaki, T. Hirouchi, Y. Hisakuni, A. Yamanaka and Y. Tomita: Mater. Trans. 49 (8) ) N. Yazdipour, C. H. J. Davies and P. D. Hodgson: Comp. Mater. Sci. 44 (8) ) N. Xiao, C. Zheng, D. Li and Y. Li: Comp. Mater. Sci. 41 (8) ) G. Kugler and R. Turk: Acta Mater. 52 (4) ) R. Ding and Z. X. Guo: Acta Mater. 49 (1) ) N. H. Packard and S. Wolfram: J. Statis. Phys. 38 (1985) ) H. Mecking and U. F. Kocks: Acta Metall. 29 (1981) ) W. Roberts and B. Ahlblom: Acta Metall. 26 (1978) ) H. P. Stüwe and B. Ortner: Met. Sci. 8 (1974) ) S. Takeuchi and A. S. Argon: J. Mater. Sci. 11 (1976) ) T. Sakai, M. Ohashi, K. Chiba and J. J. Jonas: Acta Metall. 36 (1988) ) L. Blaz, T. Sakai and J. J. Jonas: Met. Sci. 17 (1983)