J Mater Sci (2013) 48:7142–7152 DOI 10.1007/s10853-013-7530-3
Cellular automata modeling of static recrystallization based on the curvature driven subgrain growth mechanism Fengbo Han • Bin Tang • Hongchao Kou Jinshan Li • Yong Feng
•
Received: 29 March 2013 / Accepted: 12 June 2013 / Published online: 21 June 2013 Ó Springer Science+Business Media New York 2013
Abstract A two-dimensional cellular automata model was developed to describe the static recrystallization (SRX) arising from the subgrain growth, the driving force of which is dependent on boundary energy and local curvature. At the same time, the subgrain boundary energy and mobility rely on the boundary misorientation angle. On the basis, a deterministic switch rule was adopted to simulate the subgrain growth and kinetics of recrystallization quantitatively to provide an insight into the grain boundary bulging nucleation mechanism. Microstructure evolutions during SRX in different cases were simulated by the developed model. At the beginning of the simulation, the initial polycrystalline microstructure which contains large number of uniformly distributed subgrains in every preexisting grain was prepared using simple assumption based on experimental observations. Then, both homogeneous and inhomogeneous subgrain growth phenomena were captured by the simulation with different inter-subgrain misorientation, which showed continuous and discontinuous recrystallization, respectively. The effects of initial mean subgrain radius, distribution of initial subgrains, distribution of inter-subgrain misorientations, and annealing temperature on the recrystallization kinetics were also investigated.
F. Han (&) B. Tang H. Kou J. Li Y. Feng State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China e-mail:
[email protected] Y. Feng Western Superconducting Technologies Co., Ltd., Xi’an 710018, People’s Republic of China
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Introduction It’s well known that static recrystallization (SRX) may occur when a hot or cold deformed material is subsequently annealed. The SRX process is greatly affected by the deformation microstructure [1], some of which can be characterized by deformed pre-existing grains with welldefined and equiaxed subgrains inside them. Due to the importance of deformation microstructure in SRX, the researches aiming at the SRX process with deformation microstructure have been conducted. Therein, the method of simulation has been widely applied into the researches of SRX process with the deformation microstructure consisting of subgrains, e.g., Monte Carlo [2–4], vertex [5, 6], and phase field [7–9] method, in which the unified subgrain growth theory was employed [10, 11]. In the unified subgrain growth theory, nonuniform grain boundary mobility, and energy lead to abnormal subgrain and/or grain growth, which gives rise to recrystallization. However, all the simulations of recrystallization considering subgrain structures by aforementioned approaches used simulation time instead of real time, which should be improved to simulate the real recrystallization process in both length and time scale. Cellular automata (CA) model, as one of the mesoscale simulation methods, is more feasible to characterize evolution kinetics and topology features due to its intrinsic advantages of ready calibration to time, and length scale compared with the aforementioned simulation methods. Cellular automata are algorithms that describe the discrete spatial and temporary evolution of complex systems by applying local (or sometimes long-range) deterministic or probabilistic transformation rules to the cells of a regular (or nonregular) lattice, and each cell is characterized in terms of several state variables [12]. Ascribing to its
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excellent features, cellular automata model has been used successfully in the simulation of materials and metallurgies, especially SRX [12–15]. The original work about the application of CA method into SRX simulation was done by Hesselbarth and Go¨bel [16], who firstly simulated SRX using two dimension CA method two decades ago. In the model of Hesselbarth et al., they assumed that the SRX process consists of two independent processes, namely nucleation (recrystallization) and grain growth (nonrecrystallization), and then CA cells can be in one of two states. These assumptions became the basis for many different SRX models developed with the CA method afterward. Thereafter, Davies [17] studied the effect of neighborhood on the kinetics of a cellular automaton recrystallization model, and proposed a new kind of neighborhood, i.e., the pseudo-hexagonal one. Further on, Davies [18] developed a 3D model with probabilistic rules of grain growth in a CA simulation of recrystallization. He compared the recrystallization fractions obtained from the model with experimental results and analytical solution, and found a good ability of the CA model in reproducing real material behavior. Besides, Goetz and Seetharaman [19] studied the SRX kinetics with homogeneous and heterogeneous nucleation in 2D and 3D space using a CA model. Marx et al. [20] proposed a modified 3D CA model of primary recrystallization, allowing for the simulation of an orientation dependent growth rate. Davies and Hong [21] presented a model that allowed to perform simulation of recrystallization considering texture evolution. During the last decade, the amount of works dealing with SRX and CA has increased greatly. The CA method was applied to model SRX in various alloys, such as steel [22–25], copper [26], and aluminum [27, 28]. In order to simulate microstructure evolution during SRX process more accurately, the modeling of preceding deformation microstructure before annealing is of great importance. Crystal plasticity finite element (CPFE) method is a fascinating approach to obtain comprehensive information of deformation microstructure under different deformation conditions, including deformed grain shape, stored energy, and texture. Raabe and Becker [27] used a method in which the results of a CPFE simulation serve as a starting microstructure for a subsequent probabilistic kinetic cellular automaton for primary SRX simulation. The values of the state variables (dislocation density, crystal orientation) given at the integration points of the finite element (FE) mesh were mapped on the cellular automaton lattice to characterize the deformation microstructure. Zambaldi et al. [29] used a similar approach of applying a cellular automaton model to CPFE simulation data of a deformed superalloy for the prediction of local recrystallization phenomena. Zheng et al. [23] used the coupled method to simulate austenite static recrystallization followed by hot deformation in a low
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carbon steel. In the overview of Roters et al. [30] more detailed information of CPFE and the coupled method will be found. Obviously, almost all the CA simulations of SRX as well as the coupled method used the nucleation and growth models, and the stored energy serves as the driven force of grain growth during SRX. And the SRX nucleation rate was set as a constant value or given by a phenomenological equation depending on temperature, activation energy, and stored energy, and was usually measured or estimated from the experimental data. Besides, the nucleation site is often set as a cell in the cellular automata model. In these simulations, it should be noted that the deformation substructures such as well-defined subgrains are not considered, and that the nucleation criteria for SRX is a little simple without explicitly adopting physical based model for nucleation, for instance, grain boundary bulging nucleation mechanism. To our knowledge, there is no CA simulation of SRX based on the deformation microstructure consisting of subgrains. In this paper, CA simulation of SRX based on the unified subgrain growth theory was performed, in which the simulated initial microstructures with large number of subgrains inside the pre-existing grains were used representing the deformation microstructure. The curvature driven subgrain growth mechanism was adopted in the present work. One may argue that only considering curvature driven interface motion, and not including stored energy, is a significant simplification. We have to admit that there are shortages to separate subgrain structure and stored energy in studying SRX, since the relationship between subgrain structure and stored energy is complicated, and the interaction between these two factors in the process of annealing is not very clear. Therefore we only consider the subgrain structure evolution by the curvature driven mechanism to lessen the difficulty of the investigation. In the present research, the effects of inter-subgrain misorientation, initial mean subgrain radius, their distributions, and annealing temperature on the microstructure evolution and the kinetics of SRX were investigated using the developed CA model.
Model description and numerical methods Theory of curvature driven grain/subgrain growth Curvature driven grain/subgrain growth mechanism has been confirmed by experiments and theoretical analysis [31–33]. On the basis, curvature driven grain growth has been successfully simulated quantitatively by CA method [34]. In the process of grain/subgrain growth, the grain/ subgrain boundaries tend to move toward the center of the
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curvature, which is a common feature for the grain/subgrain growth when the main driving force is the interfacial energy. The velocity of a grain/subgrain boundary segment driven by the curvature can be expressed by: v ¼ MP
ð1Þ
where M is the grain/subgrain boundary mobility and P is the driving force for grain/subgrain boundary mobility. The mobility M is dependent on the boundary misorientation angle h, and assumed to be as [10], 4 !! h M ðhÞ ¼ MHAG 1 exp 5 ð2Þ hm where, hm is the misorientation when the boundary becomes a high-angle boundary. MHAG is the mobility of a high-angle boundary, and was estimated by the following equation [35]: dDb Qb MHAG ¼ exp ð3Þ kT RT where d is the characteristic grain boundary thickness, Db is the boundary self-diffusion coefficient, Qb is the boundary diffusion activation energy, k is the Boltzmann’s constant, R is the gas constant, and T is the absolute temperature. The driving force for boundary motion P can be expressed by P ¼ cj
ð4Þ
where c is the boundary energy and j is the boundary curvature. The boundary energy is dependent on the misorientation angle h, and can be calculated as the Read–Shockley equation [36]: h h c ¼ cm 1 ln ð5Þ hm hm where cm is the high-angle boundary energy. The boundary curvature j is theoretically defined as j¼
1 1 þ r1 r2
ð6Þ
where r1 and r2 are the principle radii of the boundary segment. In the current simulation, an equivalent approach known from solidification [37] is adopted to calculate the boundary curvature, which is defined as j¼
A Kink Ni Ccell N þ 1
ð7Þ
where Ccell is the cell size in the CA model, A = 1.28 is a topological coefficient, N = 24 is the total number of the first and second nearest neighbors for a square lattice, Ni is the number of cells within the neighborhood belonging to
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grain or subgrain i, and Kink = 15 is the number of cells within the neighborhood belonging to grain or subgrain i for a flat interface (j = 0). The detail topological considerations of this model can be found in Figs. A1–A3 in Ref. [37]. The CA model In the present model, the CA simulation mesh is 2500 9 2500 square lattice and periodic boundary conditions are used. The eight nearest cells are chosen as the neighborhood of a cell. The cell size is 0.04 lm, and the total simulation domain is 100 9 100 lm. Based on simple assumptions from the experimental observations of deformed microstructure in literature [38], initial microstructures of the simulation system comprising 12 grains with large number of subgrains inside every grain are generated. Figure 1a, b give the initial microstructure with different average subgrain radius r0, in which the subgrains are uniformly distributed in all grains, and they are similar to that in Ref [8]. Distribution of subgrains may vary from grain to grain for the deformation degree in every grain may be different. The initial microstructure with varying average subgrain radius from grain to grain is also generated, as shown in Fig. 1c. The 12 big pre-existing grains were created by a periodic voronoi tessellation method and the small subgrains were created by simulating normal grain growth. The orientation of every grain/subgrain was marked by an integral number, naming orientation number. In this research, the pre-existing grain boundaries are set as high-angle grain boundaries, while others are subgrain boundaries with low angle; herein, the unified subgrain growth theory was used. Since, a considerable large mesh scale of CA was employed and consequently a mount of boundaries needed to be treated lead to a substantial computing amount, thus only 2D simulation was considered and carried out. For the boundary cell at the grain/subgrain boundary, the curvature is calculated by Eq. (7). The value of curvature may be positive, zeros, or negative. Zero curvature means that the cell is at a flat boundary and the state of that cell will not change. Negative curvature induces the boundary to move away from the cell at which it is evaluated, therefore the state of that cell will not change [14]. In the contrary, boundary moves toward the cell at which the curvature is positive so that the state of that cell will change. According to the grain orientations and curvatures of the eight neighboring cells, the motion direction of the boundary cell at which the curvature is positive can be determined. The velocity of the corresponding boundary cell can be calculated by other equations listed in ‘‘Theory of curvature driven grain/subgrain growth’’ section. A deterministic transformation rule used by Zheng et al. [39]
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Fig. 1 a, b Initial microstructure with uniformly distributed subgrains, the average subgrain radius r0 is 0.31 and 0.58 lm, respectively; c initial microstructure with nonuniformly distributed subgrains, the average subgrain radius r0 varying from grain to grain
is applied to determine the changing state of each CA cell. For the cell (i, j) with positive curvature belonging to the moving boundary, the moving distance of the boundary cell in a single time step, Dt, is described as: Z tþDt t li;j ¼ vdt ð8Þ t
the indices (i, j) denote the coordinate of the selected boundary cell. The transformation fraction in cell (i, j), fi;jt , is then calculated by: . ð9Þ fi;jt ¼ lti;j LCA where LCA is the distance between two neighboring cells. If the accumulated value of the transformation fraction variable is greater than 1.0, the boundary cell switches into the new state from the neighboring cells belonging to the corresponding growing grain/subgrain determined previously. The transformation fraction of cells at the fast interfaces will reach 1.0 more quickly than that at slow interfaces. Cells at triple junctions have more than one interface, which makes it difficult to deal with. Partially transformation rule for different interfaces on the same cell at triple junctions is not implemented in the current model. The motion directions of cells at triple junctions are determined, and the transformation fraction only in this direction is calculated. All the parameters used in the present CA model are listed in Table 1.
Results and discussion Effect of inter-subgrain misorientation h Generally, boundary energy and mobility are dependent on the misorientation angle, and a low boundary energy and a Table 1 Key parameters used in the present article Qb (KJ/mol) 107
dDb (m3/s) -14
5.4 9 10
b (m) -10
2.56 9 10
cm (Jm2)
hm
0.625
15°
high boundary mobility of recrystallization fronts favor abnormal growth and discontinuous recrystallization [8]. In the actual microstructure, the boundaries with different misorientation angles usually distribute non-uniformly. In order to investigate the effect of inter-subgrain misorientation h on the SRX process, two kinds of misorientation distribution schemes are made. The first scheme is that all the subgrain boundaries inside the pre-existing grains are with the same misorientation h. In this scheme, cases with different inter-subgrain misorientation h, values between 38 and 158, are considered. The second scheme is that subgrain misorientations vary in the range of 1°–10°. In both schemes, the misorientations of the pre-existing grain boundaries were set to 158. Figure 2 depicts the temporal microstructure at different time under the condition of temperature T = 973 K, initial mean subgrain size r0 = 0.31 lm, and inter-subgrain misorientation h = 4°. Subgrain at the pre-existing grain boundaries bulging into its neighboring grain as recrystallization nucleus is observed, and it shows a better demonstration of the grain boundary bulging nucleation mechanism. High-angle boundary segments at the preexisting grain boundaries grow faster than subgrain boundaries inside the pre-existing grain, and they gradually grow into the neighboring grain, producing the recrystallization nucleus. It indicates that the boundaries with high misorientation angle at the pre-existing grains tend to be recrystallization fronts. The recrystallized region can be clearly distinguished from the unrecrystallized region before completely recrystallized, which shows discontinuous recrystallization. It should be emphasized that continuous and discontinuous recrystallization are purely phenomenological description of the spatial and temporal heterogeneity of microstructural evolution, and imply no specific micro-mechanism of recrystallization [10]. Finally the pre-existing grains are all consumed by the recrystallized grains, and the microstructure is relatively homogeneous and with no subgrains. Every subgrain was marked by a unique orientation, the coalescence among subgrains with the same orientation was completely avoided during the simulation. It should be noted that inter-subgrain
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misorientation held constant all the time. Although intersubgrain misorientation may change due to subgrain rotation during annealing, the evolution of inter-subgrain misorientation with time was not considered in the present simulation. Because the inter-subgrain misorientation at all subgrain boundaries are the same, the boundary energy and mobility are uniform in all directions, leading to uniformly subgrain growth inside the pre-existing grains. Figure 3 depicts the temporal microstructure at different time with the same temperature and initial mean subgrain size in Fig. 2, while the inter-subgrain misorientation h is higher than that in Fig. 2, which is 7°. In comparison with Fig. 2, recrystallized grains at the pre-existing grain boundary in Fig. 3 are relatively small in the same time, while subgrains inside the pre-existing grains are relatively large. For boundary energy and mobility of subgrain boundaries with misorientation angle h = 7° is close to that of high angle grain boundary, the competitive growth between high angle grain boundaries at the pre-existing grains and subgrain boundaries inside the pre-existing grains are more intense. Subgrains inside the pre-existing grains are more stable and maintain for a longer time with higher inter-subgrain misorientation h, which leads to a relatively slow recrystallization process. A distribution of misorientation instead of constant misorientations is more close to the actual microstructure. Figure 4 shows the simulated temporal microstructure at different time with the same temperature and initial mean subgrain size, as that in Figs. 2 and 3, while the initial inter-subgrain misorientation h varies in the range of Fig. 2 Simulated temporal microstructure at different time a 0.5 s, b 1 s, c 2 s, and d 5 s with annealing temperature T = 973 K, initial mean subgrain size r0 = 0.31 lm, and inter-subgrain misorientation h = 4°
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1°–10°, and the distribution of misorientation including both the subgrain and grain boundaries of the initial microstructure is shown in Fig. 5. In this case, the local misorientation h of subgrain boundary was computed by: h¼ 1þ9
j S2 S1 j Smax Smin
ð10Þ
where S1 and S2 are the orientation numbers of the two neighboring cells located on each side of the subgrain boundary, Smax is the maximum orientation number of the subgrains, and Smin is the minimum. The microstructure evolution in Fig. 4 is similar to that in Fig. 2, although the subgrain misorientation distribution of the former is quite different from the latter. The reason lies in the large amount low-angle boundaries between 1° and 7°, with which the subgrain boundary have relatively low energy and mobility, and thus the subgrains inside the pre-existing grains grow slowly. Despite this, the ununiformity of subgrain distribution in Fig. 4 is significant. There are some subgrains much larger than others, resulting from the ununiformly boundary misorientation. The variations of mean subgrain radius and recrystallized fraction with time can be used to investigate the kinetics of recrystallization. In order to calculate the recrystallization fraction quantitatively, we assume that if a subgrain at the pre-existing grain boundaries has a radius larger than the prescribed threshold value, then the subgrain is considered to be a recrystallized grain. In this study, 2 lm has been set as the threshold value. In every cellular automata step, the radius of every subgrains at the
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Fig. 3 Simulated temporal microstructure at different time a 0.5 s, b 1 s, c 2 s, and d 5 s with annealing temperature T = 973 K, initial mean subgrain size r0 = 0.31 lm, and inter-subgrain misorientation h = 7°
Fig. 4 Simulated temporal microstructure at different time a 0.5 s, b 1 s, c 2 s, and d 5 s with annealing temperature T = 973 K, initial mean subgrain size r0 = 0.31 lm, and varying inter-subgrain misorientation h in the range of 1°–10°
pre-existing grain boundaries are calculated and compared to the threshold value, then the recrystallized grains can be determined. Although the threshold value is given arbitrarily in the present work, it has little effect on the recrystallization kinetics except at the very early stages. Maybe a physically based nucleation model with accurate critical size of the recrystallization nucleus used by Zurob et al. [40] can be adopted in the future. The transition from
recrystallized region to unrecrystallized region due to a grain shrinkage, which has little effect on the recrystallization fraction, is not considered. Figure 6 shows the evolutions of the mean subgrain radius (including recrystallized grains) with time for various values of uniform inter-subgrain misorientation h (3°, 4°, 7°, 8°, 10°, 15°, respectively), and a nonuniform intersubgrain misorientation h varying in the range of 1°–10°,
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Fig. 5 Misorientation distribution of the initial microstructure with varying inter-subgrain misorientation h
Fig. 6 Evolutions of mean subgrain radius (including recrystallized grains) with time for various values of uniform inter-subgrain misorientation h and a nonuniform inter-subgrain misorientation h varying in the range of 1°–10°, with constant annealing temperature T and initial mean subgrain radius r0
with constant annealing temperature T and initial mean subgrain radius r0. According to experimental observations, the mean misorientation angle between subgrains may not exceed 7° [1]. However, in order to demonstrate the transition from discontinuous to continuous recrystallization, misorientations from 3° up to 15° are considered in the current work. This figure indicates that the mean subgrain radius evolution is greatly dependent on the intersubgrain misorientation h. In the case of h = 3° and 4°, the microstructural evolutions are inhomogeneous, corresponding to discontinuous recrystallization. In these two cases, the variations of mean subgrain radius with time change significantly and can be divided into three stages: the incubation stage, recrystallization stage, and grain growth stage. The gradual increase in the value of mean
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subgrain radius at the beginning of the process corresponds to the incubation stage of discontinuous recrystallization. The subsequent rapid increase in the value of mean subgrain radius corresponds to the recrystallization stage. After fully recrystallized, grains continue growing and the mean subgrain radius increment speed slows down. In the case of h = 15°, the mean subgrain radius increases uniformly through the whole process and the microstructure evolution is homogeneous, corresponding to continuous recrystallization, or normal grain growth. In the case of h = 7°, 8°, and 10°, the variations of mean subgrain radius with time change slightly, indicating the microstructure evolution in these cases are between continuous and discontinuous recrystallization. In the case of nonuniform inter-subgrain misorientation h varying in the range of 1°–10°, the variation of mean subgrain radius with time is similar to that of the cases h = 3° and 4°, corresponding to discontinuous recrystallization. The slope of the rapid increase stage in this case is smaller than that of the cases h = 3° and 4°, for some amount of relatively high angle misorientations weakens the discontinuous recrystallization process. The transition from continuous to discontinuous recrystallization is successfully demonstrated by simulations, in which the inter-subgrain misorientation h changes. Figure 7 shows the evolutions of recrystallization fraction with time for all the cases in Fig. 6. All the plots show sigmoidal curves. In the cases of h = 3°, 4°, and nonuniform h, it achieves fully recrystallization quickly in a few seconds, while it takes longer time to achieve fully recrystallization in the cases of h = 7°, 8°, 10°, and 15°. The SRX transformation kinetics in the cases of h = 3°, 4°, and nonuniform h are very similar, and the SRX transformation kinetics in the three cases become slower in turn. The SRX transformation kinetics becomes slower with increasing inter-subgrain misorientation. It can also be deduced that the amount of different type subgrain boundary (distinguished by misorientation) greatly affect the SRX transformation kinetics. The larger amount of the low angle subgrain boundaries inside the pre-existing grain leads to a faster SRX transformation kinetics. Effect of initial mean subgrain radius r0 The effect of initial mean subgrain radius r0 on the recrystallization kinetics was investigated. Figure 8 depicts the temporal microstructure at different time under the condition of annealing temperature T = 973 K, initial mean subgrain size r0 = 0.58 lm and inter-subgrain misorientation h = 4°, in which case the initial mean subgrain radius is greater than that in Fig. 2. Compared to Fig. 2, subgrains inside the pre-existing grain in Fig. 8 are relatively large, and the degree of recrystallization is relatively small in the same time. In the case shown in Fig. 9, the
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Fig. 7 Evolutions of the recrystallization fraction with time for various values of uniform inter-subgrain misorientation h and a nonuniform inter-subgrain misorientation h varying in the range of 1°–10°, with constant annealing temperature T and initial mean subgrain radius r0
annealing temperature T and inter-subgrain misorientation h are the same with that in Figs. 2 and 8, while the subgrain distribution in the initial microstructure is nonuniform. Clearly we can see that pre-existing grains with finer subgrains inside which are consumed more quickly than that with coarser subgrains.
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Figure 10 shows the evolutions of mean subgrain radius with time for various values of initial mean subgrain radius r0 and for the nonuniformly distributed initial subgrains with constant annealing temperature T and inter-subgrain misorientation h. For the inter-subgrain misorientation h = 4°, the variations tendency of mean subgrain radius with time are consistent in all cases with different initial mean subgrain radius and a nonuniform initial subgrain size, showing the phenomenon of discontinuous recrystallization. But the rapid increment stage of the evolution of mean subgrain radius with time appears earlier with smaller initial mean subgrain radius, and the incubation times of recrystallization become shorter with decreasing initial mean subgrain radius r0. In the case of nonuniformly distributed initial subgrains, the mean subgrain size increment speed at the very early stage is relatively high, for the subgrains at the pre-existing grain boundaries with large subgrain size grows faster into grains with small subgrain size. After the grains with fine initial subgrains inside them been completely consumed, there are still large amount subgrains inside the grains with coarse initial subgrains, and it needs longer time to consume these subgrains, leading to a slowly SRX transformation kinetics. Figure 11 shows the evolutions of the recrystallization fraction with time for various values of initial mean subgrain radius r0, and for the nonuniformly distributed initial subgrains with constant annealing temperature T and inter-subgrain
Fig. 8 Simulated temporal microstructure at different time a 0.5 s, b 1 s, c 2 s, and d 5 s with temperature T = 973 K, initial mean subgrain size r0 = 0.58 lm, and intersubgrain misorientation h = 4°
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Fig. 9 Simulated temporal microstructure at different time a 0.5 s, b 1 s, c 2 s, and d 5 s with temperature T = 973 K, nonuniformly distributed initial subgrains (microstructure in Fig. 1c), and inter-subgrain misorientation h = 4°
Fig. 10 Evolutions of mean subgrain radius with time for various values of initial mean subgrain radius r0, and for the nonuniformly distributed initial subgrains with constant temperature T and intersubgrain misorientation h
Fig. 11 Evolutions of the recrystallization fraction with time for various values of initial mean subgrain radius r0, and for the nonuniformly distributed initial subgrains with constant annealing temperature T and inter-subgrain misorientation h
misorientation h. The sigmoidal curves indicate that both the initiation and the termination of the discontinuous recrystallization become faster with decreasing initial mean subgrain radius r0, which shows faster SRX transformation kinetics. In the case of nonuniformly distributed initial subgrains, the SRX transformation kinetics at the very early stage is relatively fast, but it slows down after some
time, for the grains with fine subgrains inside them have been completely consumed. Since subgrain boundaries can be described by well-defined dislocation walls, thus smaller subgrain radius means higher dislocation density and higher stored energy. The SRX transformation kinetics can be accelerated by higher driving force with smaller initial mean subgrain radius.
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Fig. 12 Simulated temporal microstructure at different time a 0.5 s, b 1 s, c 2 s, and d 5 s with annealing temperature T = 873 K, initial mean subgrain size r0 = 0.31 lm, and inter-subgrain misorientation h = 4°
Fig. 13 Evolutions of mean subgrain radius with time for various values of annealing temperature T with constant initial mean subgrain radius r0 and inter-subgrain misorientation h
Effect of annealing temperature T Annealing temperature affects the boundary mobility greatly, and further affects the microstructure evolution and kinetics during SRX. Figure 12 shows temporal microstructure at different time with annealing temperature T = 873 K, initial mean subgrain size r0 = 0.31 lm, and inter-subgrain misorientation h = 4°, in which case the
Fig. 14 Evolutions of recrystallization fraction with time for various values of annealing temperature T with constant initial mean subgrain radius r0 and inter-subgrain misorientation h
annealing temperature is lower than that in Fig. 2. Compared to Fig. 2, the recrystallized grains at the pre-existing grain boundaries and the degree of recrystallization in Fig. 12 are both relatively small in the same time. Figure 13 shows the evolutions of mean subgrain radius with time for various values of annealing temperature T with constant initial mean subgrain radius r0 and inter-subgrain misorientation h. The variation tendency of mean subgrain radius with time is
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consistent at different annealing temperature, but the rapid increment stage of the variations of mean subgrain radius with time appears earlier with higher annealing temperature, indicating the SRX transformation kinetics become faster with increasing annealing temperature. Figure 14 shows evolutions of recrystallization fraction with time for various values of annealing temperature T with constant initial mean subgrain radius r0 and inter-subgrain misorientation h. This figure also indicates that both the initiation and the termination of the discontinuous recrystallization become faster with increasing annealing temperature, showing faster SRX transformation kinetics. This is because the boundary mobility increases greatly with increasing annealing temperature, especially for the high-angle boundaries at the preexisting grain boundaries, thereby subgrains at the preexisting grain boundaries grow faster into the neighboring grain, resulting to faster SRX transformation kinetics. Conclusion Two-dimensional cellular automata modeling of static recrystallization based on the curvature driven subgrain growth mechanism was realized in the paper. Phenomenologically continuous and discontinuous recrystallization was successfully simulated with varied inter-subgrain misorientation, while the annealing temperature and initial mean subgrain radius were kept constant. The simulated results display that the boundaries with high misorientation angle at the pre-existing grains tend to be recrystallization fronts, and the grain boundary bulging nucleation mechanism was captured by the current model. Besides, it is found that the subgrain boundaries with lower misorientation angle are very unstable, while the subgrain boundaries with higher misorientation angle are more durable. Further, in order to explore the effects of initial subgrain radius and annealing temperature on static recrystallization, the simulations with fixed inter-subgrain misorientation and varied initial subgrain radius and annealing temperature were operated. Results show that the SRX kinetics can be promoted with reducing initial subgrain radius and increasing annealing temperature. Therefore, it is concluded that lower inter-subgrain misorientation inside the pre-existing grains, smaller initial mean subgrain radius and higher annealing temperature can accelerate the SRX kinetics. In addition, the simulations with nonuniformly distribution of inter-subgrain misorientations and that of initial subgrains were also done and results indicate that these two factors both have effects on the SRX kinetics. Acknowledgements This work was supported by the Project of Introducing Talents of Discipline to Universities (No. B08040) and the National Fundamental Research of China (Project No. 2011CB605502).
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