Phase-field Simulation of Recrystallization in Cold Rolling and Subsequent Annealing of Pure Iron Exploiting EBSD Data of Cold-rolled Sheet

Abstract

A unified theory for continuous and discontinuous annealing phenomena based on the subgrain growth mechanism was proposed by Humphreys around twenty years ago. With the developments in the unified subgrain growth theory, a number of Monte Carlo, vertex, and phase-field (PF) simulations have been carried out to investigate the nucleation and growth mechanisms of recrystallization by considering the local alignment of the subgrain structure.

In this study, the effects of the microstructural inhomogeneities created in the deformed state on recrystallization kinetics and texture development were investigated. Numerical simulations of static recrystallization were performed in three-dimensional polycrystalline structures by coupling the unified subgrain growth theory with PF methodology. To prepare the initial microstructures, two-dimensional electron back scattering diffraction (EBSD) measurements were carried out on 90% and 99.8% cold-rolled pure iron. Our previous experimental study has shown that there are large differences in the texture formation processes during the recrystallization of cold-rolled iron samples.

In cold-rolled iron with 90% reduction, the simulated texture exhibited nucleation and growth of γ-fiber (ND//<111>) grains at the cost of α-fiber (RD//<011>) components, where ND and RD denote normal direction and rolling direction, respectively. In contrast, the simulation results for cold-rolled iron with 99.8% reduction reproduced the high stability of the rolling texture during recrystallization. As a result, we conclude that the simulation results agreed with the experimentally observed textures in both the samples.

1. Introduction

When a metallic material is plastically deformed, numerous lattice defects are introduced inside the crystal, the internal energy increases, and the crystal hardens. Moreover, when the crystal is heated, the lattice defect density decreases and leads to lattice softening. The energy-release processes in such cases are recovery, recrystallization, and grain growth, and they play a very important role in controlling the structure of polycrystalline materials.¹⁾ The experimental optimization of cold-rolling and annealing conditions, which vary depending on the type of steel used, is an expensive process. Furthermore, there is a limit to the level of optimization that can be achieved using relational expressions obtained via experimental means. Therefore, there is an urgent need to develop models based on the physical metallurgy. In this study, the phase-field (PF) method²⁾ is used as a microstructure calculation method which employs the “subgrain (SG) growth model” devised by Humphreys.³⁾ This method can be applied to model recrystallization and the subsequent grain growth in a uniform manner. The ultimate goal is to consistently predict recrystallization and grain growth. There have been numerous reports on the numerical analysis results obtained by combining unified SG theory with microstructural simulations, such as, Monte Carlo (MC),^4,5) PF,^6,7,8) or Vertex models;^9,10) these are outlined in the reference.¹¹⁾

Discontinuous recrystallization behavior can be divided into nucleation or growth processes. The nucleation of recrystallization cannot be explained using the classical theory that “a region with low dislocation density is formed from a homogeneous deformed matrix due to thermal fluctuations”. Furthermore, it is known that the heterogeneous deformed structure plays a crucial role in nucleation.¹²⁾ As a result, the nucleation process needs to be modeled to reproduce the recrystallization behavior. Therefore, a model that clearly distinguishes nucleation and growth, and considers nucleation by treating the accumulated strain energy and the misorientation of boundaries, including the SG boundaries, as the threshold for nucleation, is required.¹³⁾ In the subsequent growth process, the strain energy difference is used as the main driving force for the growth of the recrystallizing grains. In this study, nucleation and growth are treated without distinction, according to the SG growth model described above. Specifically, the non-uniform deformed structure is converted into SG structures with various sizes and crystal orientations, and the nucleation of recrystallization is expressed as abnormal grain growth of a specific SG. Herein, the measurements obtained for the cold-rolled samples, which possess a deformed structure, using an SEM-EBSD (scanning electron microscope-electron back scatter diffraction) method are employed to convert it to an SG structure.⁸⁾ The possibility of abnormal grain growth of a specific SG, that is, “nucleation of discontinuous recrystallization” is expressed using the boundary energy and mobility, depending on the crystal orientation difference between the SGs. One must not always assume the initial configuration for the recrystallization process to be an SG structure; however, it is possible to express the spontaneous nucleation behavior of recrystallization, including the orientation of the recrystallizing nucleus, with the aid of such modeling. Using the above method, we explore the dependence of the cold-rolling rate on texture change in the recrystallization behavior of the strongly cold-rolled pure iron during annealing. Our previous experimental study¹⁴⁾ has shown that the texture formation processes during recrystallization are highly dependent on the cold-rolling rate.

2. Numerical Simulation Model and Calculation Method

2.1. PF Model for Recrystallization and Grain Growth

In this study, we assume that the SG boundary energy (boundary curvature) is the primary driving force of recrystallization. Although this assumption is controversial,^15,16) the possibility of an SG structure is not negligible, considering the high stacking fault energy and low solubility of the metal.³⁾ Holm et al.⁴⁾ also pointed out the following: the hypothesis that the driving force for SG motion is the reduction in the interfacial energy is controversial. Nevertheless, recent comparisons of experimental- and simulation-based results for the curvature-driven motion of SG coarsening in aluminum foils are in excellent agreement.¹⁷⁾ Herein, numerical simulation is carried out using the multi-phase-field (MPF) model developed by Steinbach et al.,²⁾ which can reproduce the boundary motion controlled by local mean curvature. In the MPF model, a set of continuous order parameters ϕ_i (i = 1, 2, ..., N) are defined to distinguish the orientation of the SGs, where ϕ_i (r, t) represents the existence ratio of each orientation at position r and time t and N is the total number of order parameters.

In this model, the sum of each order parameter at any position in the system is conserved.   

$${\sum }_{i=1}^N{\varphi }_i\left(r,t\right)=1.$$(1)

The time–evolution equations of the order parameter ϕ_i (i = 1, 2, ..., N) are given by   

$$\frac{\partial {\varphi }_i}{\partial t}=-\frac{2}{n\left(r,t\right)}{\sum }_j^N{s}_i{s}_j{M}_{ij}\left[f_i^C-f_j^C-\frac{8}{\pi }\sqrt{{\varphi }_i{\varphi }_j}\left(f_i^E-f_j^E\right)\right],$$(2)

where n is the number of non-zero order parameters at each spatial point. f_i^E is the excess free energy for the i-th SG, assumed to be 0 for i > 1 and M_ij is the phase-field mobility of the boundary between the i-th and j-th SGs. According to Hirouchi et al.¹⁸⁾ the phase-field mobility of multiple junctions is defined as follows:   

$$M_{ave}=\left({\sum }_{i,j}^N{M}_{ij}{\varphi }_i{\varphi }_j\right)/\left({\sum }_{i,j}^N{\varphi }_i{\varphi }_j\right).$$(3)

At the spatial point where three or more non-zero order parameters exist, M_ave is used as the phase-field mobility instead of M_ij in Eq. (2). Furthermore, f_i^c in Eq. (2) is expressed as:   

$$\begin{array}{l}{f}_i^C=\\ {\sum }_{k=1}^N\left[{\displaystyle \frac{{\epsilon }_{ik}^2}{2}}{\nabla }^2{\varphi }_k+W_{ik}{\varphi }_k+{\sum }_{l=1}^N\left(W_{ikl}{\varphi }_k{\varphi }_l+{\sum }_{m=1}^N{W}_{iklm}{\varphi }_k{\varphi }_l{\varphi }_m\right)\right].\end{array}$$(4)

W_ik, and ε_ik are related to the boundary energy between the i-th and k-th SGs. To improve the numerical accuracy and stability of the MPF simulations, the higher-order terms devised by Miyoshi and Takaki are incorporated into Eq. (4). The values of the W_ikl and W_iklm parameters are determined according to previous studies.^19,20) Although a restriction, such as j ≠ i, is required below the sigma symbol in Eqs. (2), (3) and (4), it is omitted to simplify the notation of the equation. Values of zero are assigned, instead: W_ii = 0; ε_ii = 0; and M_ii = 0. The value of n can be written as:   

$$n\left(r,t\right)={\sum }_{i=1}^N{s}_i\left(r,t\right),$$(5)

where s_i (r, t) is a step function that satisfies s_i (r, t) = 1, if ϕ_i (r, t) > 0 and s_i (r, t) = 0 otherwise. W_ij, ε_ij, and M_ij exhibit the following relationship with the boundary width δ, interface energy σ_ij, and the physical mobility M_ij^phys in the PF analysis.   

$$W_{ij}=\frac{4{\sigma }_{ij}}{\delta },{\epsilon }_{ij}=\frac{2}{\pi }\sqrt{2\delta {\sigma }_{ij}},M_{ij}=\frac{{\pi }^2}{8\delta }{M}_{ij}^{phys}$$(6)

For the purpose of numerical simulation, the set of PF Eq. (2) has to be solved numerically by discretizing them in space and time. We applied both parallel coding techniques²¹⁾ and the active parameter tracking (APT) algorithm^22,23,24) to accelerate computations and embody large-scale calculations.

2.2. Implementation of the Crystal Orientation Represented by Euler Angles

In this study, the recrystallization is reproduced using the SG growth model.³⁾ In this model, SG interfacial energy and mobility are related to the misorientation angle between the SGs, and the nucleation of the recrystallization is expressed by the abnormal grain growth of a specific SG. Conventionally, crystal orientation has been implemented into the phase-field models by connecting a crystal orientation g_i to an order parameter, ϕ_i. In these cases, the relation table with dimensions N × N had to be predetermined before simulation runs (where N is the total number of order parameters). This is a general practice, and is often employed in the MC method.²⁵⁾ As it is necessary to store the table in multiple CPU cores, execution of the numerical analysis when the value of N exceeded 10⁴ was challenging in some cases.

In addition, if we calculate the orientation relation between neighboring grains at each time step in the simulation runs without storing the table, the computational cost significantly increases. This is because twenty-four matrix operations are required to obtain an orientation relationship in the case of a cubic crystal. Therefore, to avoid such an increase in computational cost, the boundary energy and mobility related to the misorientation matrix Δg = g_i · g_j⁻¹ between SG_i and SG_j are stored in 1° increments in the memory space. (Hereafter, these matrices are referred to as E-table and M-table, respectively.) In this method, the computational cost can be significantly reduced by considering the crystal symmetry when creating the E- and M-tables. Therefore, a matrix operation is required to obtain an orientation relationship in the PF simulation runs. Although the total number of elements in each E- and M-table equals 360 × 180 × 360 ≒ 2.3 × 10⁷, the total number became smaller compared to that of N² if N > 4830 is satisfied. In this paper, the E- and M-tables are solely controlled by the misorientation angle θ; however, it is possible to also consider the effect of the rotation axis on misorientation.

2.3. Conversion of the Measured Orientation Map to an SG Structure using EBSD as the Initial Configuration for PF Simulations

Using the same cold-rolled samples previously reported,¹⁴⁾ 90% and 99.8% cold-rolled pure iron was polished and measured at 50 nm intervals. The measured cross-section is perpendicular to the TD (transverse direction) and is parallel to the RD (rolling direction) and ND (normal direction). The measured grain size of the hot-rolled sheet was 200–500 μm. As the final sheet thickness after cold-rolling was set to 0.1 mm, the number of grains along ND was highly dependent on the rolling reduction rate. In particular, the grain number was as low as five in the case of the 90% cold-rolled samples. Next, we discuss the conversion of the measured orientation map to an SG structure using EBSD. Ideally, it is preferable to use an unmodified measured orientation map as the initial configuration for the PF simulations of recrystallization. However, because a complete SG structure is required as the input data, the initial configuration was created using the following procedure. It should be noted that the sequence of operation performed in this section is a type of conversion from a deformed structure to an SG structure, that is, a model of the recovery process. To ensure numerical accuracy while calculating the curvature during the PF analysis, one measured grid point was correlated with multiple finite-difference grid points. Hereafter, a finite-difference grid point in the PF analysis is referred to as a PF grid point. Specifically, for a 90% cold-rolled sample, the results of which are discussed in section 3.1, one measured grid point was converted to 2 × 2 PF grid points. Moreover, for a 99.8% cold-rolled sample, as discussed in section 3.2, it was converted to 4 × 4 PF grid points. The measured intervals during the EBSD operation are the same; however, different PF grid points were assigned for the purpose of presenting the fineness of the measured structure,¹⁴⁾ which is dependent on the cold-rolling reduction rate. Furthermore, when carrying out the three-dimensional calculations, it was assumed that measurement points with the same crystal orientation and the same image quality value (hereafter abbreviated as IQ value) are arranged along the TD axis, that is, perpendicular to the measurement surface. As a result, the two-dimensional EBSD measurement results have been extended to three dimensions. In addition, regarding the length along the TD axis, preliminary studies suggested that three-dimensional constraints arising from the surrounding SGs had a significant effect on the nucleation frequency. The length along the TD axis was set to three times the average SG diameter, as a compromise between computational cost and accuracy. This will be explained using a relevant example in section 3.1.2.

The crystal orientation g (Euler angle in the Bunge expression) and the IQ value (corresponding to the accumulated strain, the value decreases as the strain increases²⁶⁾) are used as the outputs at the measured grid point for SG conversion. The conversion procedure is as follows:⁸⁾

(0) The IQ value is normalized so that the maximum value in the system is 1 and the minimum value is 0; this normalized value is called IQ’. In addition, the order parameter ϕ₁ with f₁E > 0 is set to 1 throughout the system for the purpose of creating an SG structure. Using this operation, the SG nucleus rapidly grows into a region with an order parameter value of ϕ₁ = 1. Thus, an SG structure is obtained. The crystal orientation of the SG nucleus reflects the crystal orientation of the measured orientation, and the crystal orientations at the PF grid points in a SG are identical. Furthermore, we assume that the dislocation density inside the SGs is sufficiently small, which is assigned the value of f_i^E = 0 (for i > 1).

(1) A point, P (x, y, z), is randomly selected from the entire system.

(2) The SG radius R (x, y, z) is defined as R = R_min × {1 + IQ’ × (L_es−1)} using the IQ’ value at coordinate P. Here, R_min is the minimum SG radius, and L_es is the parameter determined when L_es×R_min corresponds to the maximum SG radius. At this time, the SG radius takes the maximum value L_es×R_min when IQ’ = 1 and the minimum value R_min when IQ’ = 0.

(3) Let R_i be the radius of an SG with a nucleus coordinate P_i, where i takes a value of 1 to n_n and n_n is the total number of SGs adjacent to the SG with the nucleus P. If the distance between P and P_i, defined as | P − P_i |, satisfies the condition | P − P_i |> (R + R_i) for all i, the SG with nucleus P is accepted as the newest SG.

(4) Steps (1) to (3) are repeated until a new SG nucleus cannot be placed. As the total number of SGs is N_SG and the order parameter 1 is assigned as the region being invaded by the SGs, we can write N = N_SG + 1.

(5) By carrying out a short-time PF simulation, the region with an order parameter value of ϕ₁ = 1 completely disappears and an SG structure is obtained.

In the above mentioned procedure, one SG is equiaxed; however, SGs with similar crystal orientations are compressed into the ND and elongated into the RD. In addition, it is possible to determine the SG size according to the IQ value using this procedure. However, because the relationship between the strain accumulation via deformation and SG size is not completely clear, L_es is treated as an adjustable parameter in the PF simulations. Furthermore, in this procedure, it is implicitly assumed that all SGs are formed at the same time. In actual materials, the deformation state differs, depending on the crystal orientation and the surrounding constraints; as a result, the SG formation time (or recovery speed) is different.¹⁾ Finally, in this conversion procedure, the SGs formed from different nuclei are assigned different order parameters. Therefore, a boundary with zero misorientation is assumed to exist between SGs with the same orientation.

2.4. Calculation Conditions

In this paper, the boundary properties are only controlled by the misorientation across the boundary. In a simple dislocation model, the energy of a low-angle tilt boundary between two cubic crystals has been obtained as a function of misorientation.²⁷⁾   

$${\sigma }_{ij}\left(\theta \right)={\sigma }_0\frac{\theta }{{\theta }_m}\left(1-ln\frac{\theta }{{\theta }_m}\right),\hspace{1em}\theta \le {\theta }_m,$$(7)

where σ₀ and θ_m are the boundary energy and the misorientation when the boundary becomes a high-angle boundary.

Similarly, the mobility M^phys (θ) is constant at high-angle grain boundaries. We consider the M^phys versus θ curve to be sigmoidal with the following form:²⁾   

$$M_{ij}^{phys}=M^{phys}\left(\theta \right)=M_0^{phys}\left\{1-exp\left(-5{\left(\frac{\theta }{{\theta }_m}\right)}^4\right)\right\},\mathrm{\hspace{0.17em} }\theta \le {\theta }_m,$$(8)

where M₀^phys is the mobility of a high-angle boundary. Following the previous study, θ_m is assumed to have a value of 15°. Furthermore, considering the stability of the numerical calculation, conditions of 0.01 M₀^phys < M^phys (θ) and 0.52 σ₀ < σ (θ) are given. Here, the value of 0.52 is assigned to prevent a wetting phenomenon at a triple junction. That is, it was assumed that all SG boundaries with θ < 3° had the same boundary energy. This wetting phenomenon²⁸⁾ is also called ‘solid-state wetting phenomenon’, in which one high-energy boundary is divided into multiple low-energy interfaces. In this paper, the above assumptions are used to suppress the instability of numerical calculations because of the occurrence of wetting; however, it is necessary to verify the role of wetting in primary recrystallization in future studies. We used the following conditions in the current study:

• boundary width δ in the PF method: 4 PF lattice spacing;

• the mobility of a high-angle boundary: M₀^phys = 2.0 × 10⁻¹² m⁴ J⁻¹ s⁻¹;

• the energy of a high-angle boundary: σ₀ = 1.0 J m⁻²;

• boundary conditions: periodic boundary conditions were applied along the three axes;

• calculation time: 80000 steps (In this paper, calculation time will be expressed in time steps, although conversion to real time is possible);

• time required for numerical analysis was 9.5 h, using part of a shared memory parallel computer equipped with a CPU with a reference operating frequency of 3.2 GHz (16 cores × 16 nodes = 256 cores). This machine time is obtained from a simulation run using the 99.8% cold-rolled material (SG050-2-a) discussed in section 4.2. Moreover, the machine time is dependent not only on the total number of grid points in the system, but also on the total volume of the SG boundary region. Therefore, the time is dependent on the recrystallization rate.

3. Simulation Results

3.1. Recrystallization of 90% Cold-rolled Pure Iron

3.1.1. Calculation Conditions

The number of calculation grid points was set to 48 (TD) × 3744 (ND) × 960 (RD). As the interval of the grid spacing is 2.5 × 10⁻⁸ m, it corresponds to 1.2 μm (TD) × 94 μm (ND) × 24 μm (RD). The interval of time integration was set to 4.4 × 10⁻⁵.

3.1.2. Recrystallization Behavior Simulation Results

Table 1 shows the conversion conditions to SG structure and the number of SGs (N_{sg_init}) at the completion of step (5) in section 2.3. To evaluate the effects of the SG size distribution function on the recrystallization behavior, the size distributions were varied while introducing minimal changes in N_{sg_init} and the average SG radius <R>_init. We prepared three types of SG size distributions by changing R_min and L_es, as indicated in Table 1. In addition, the random number sequence for obtaining the position P was altered thrice for each type of SG size distribution. Therefore, nine simulation runs were carried out in total. Here, each random number sequence was designated as a, b, and c. Figure 1 shows the initial SG size distribution function normalized by <R>_init for all nine conditions. Note that under all the conditions, <R>_init was desirably equal to 0.17 μm. Furthermore, Fig. 2 shows the orientation distribution function (hereafter abbreviated as ODF) obtained from the initial microstructure. Here, because the TD size is 3.5 times the average SG diameter, which is significantly smaller than that in other directions, the ODF is created based on a TD surface. All the ODF diagrams shown in this paper are φ_{2, Euler} = 45° cross-sections created with a TD surface, and an orthotropic symmetry was selected for the samples. The α-fiber (RD // <011>) and γ-fiber (ND // <111>), which are the main orientations obtained when a metal with a bcc structure is cold-rolled, are also shown in Fig. 2. In addition, when the allowable misorientation was set to 15°, each of the fractions in the initial microstructure were α; 0.56 and γ; 0.23. Both the α-fiber and γ-fiber exhibited a (111) [1–10] orientation and were counted twice. Note that the macroscopic texture is completely preserved in the conversion. Figure 3(a) shows the EBSD measurements, and (b) and (c) show the initial SG microstructure of the TD surface converted under the SG100-2-b conditions. In this figure, the color map is assigned IPF diagrams for an ND surface. To explain the effect of extending the two-dimensional EBSD measurement results to three dimensions, an ND–TD cross-sectional view taken at the black line in Fig. 3(b) is shown in Fig. 3(c). From this figure, we also confirm the SGs are arranged along the TD axis. The constraints from the surrounding SGs had a significant effect on the nucleation of recrystallization using the SG growth model. However, the EBSD measurements were limited to two dimensions initially. Therefore, we extended the two-dimensional EBSD measurements to three-dimensional measurements, as described above. A comparison of these results with theoretical results using three-dimensional measurements will be the subject of future studies.

Table 1. SG structure conversion conditions for 90% cold-rolled iron and the properties of the converted SG structures. Regarding the conversion conditions, R_min is the minimum radius and R_min × L_es corresponds to the maximum radius. In the converted structures, N_{sg_init} and <R>_init are the total number of SGs and the mean radius of the SGs, respectively.

Name	Conversion conditions		Converted properties
	Rmin [μm]	Les	Nsg_init	<R>init [μm]
SG075-3-a	0.075	3	126533	0.17
SG075-3-b	0.075	3	126758	0.17
SG075-3-c	0.075	3	126629	0.17
SG100-2-a	0.1	2	134899	0.17
SG100-2-b	0.1	2	134785	0.17
SG100-2-c	0.1	2	134785	0.17
SG125-1.2-a	0.125	1.2	134662	0.17
SG125-1.2-b	0.125	1.2	134915	0.17
SG125-1.2-c	0.125	1.2	134818	0.17

Fig. 1.

Normalized SG size distributions in the initial stage for 90% cold-rolled iron. (Online version in color.)

Fig. 2.

ODF section (φ_{2, Euler = 45°}) of the converted SG structure under SG100-2-b conditions. (Online version in color.)

Fig. 3.

(a) ND-orientation map measured using an EBSD technique and (b), (c) those of the converted SG structure under SG100-2-b conditions. In (b) and (c), the white lines indicate boundaries with θ values of 15° or more, and the black lines indicate boundaries with θ values less than 15°. The horizontal black line in (b) represents the ND–TD cross-sectional view taken in (c). (Online version in color.)

Next, the results of the numerical analysis of recrystallization behavior are described. Figure 4 shows the time evolution of the recrystallization rate. The threshold value of the radius for judging recrystallized grains was R_th = 20 [g.p.] = 0.5 [μm]. This value is equivalent to three times of <R>_init. In Fig. 4, nearly equivalent recrystallization behavior is obtained, independent of the initial size distribution. However, upon closer inspection, a slight difference depending on the calculation conditions can be observed. In contrast, no clear dependence on the SG size distribution function in the initial configuration can be confirmed for the change in recrystallization behavior. Figures 5(a)–5(i) shows ODFs of the microstructure obtained immediately after the completion of recrystallization (i.e., recrystallization rate of 99%) for all nine simulation runs. In most cases, a decrease in α-fibers is confirmed with the development of γ-fibers; this is consistent with experimental results.¹⁴⁾ Although a direct comparison is difficult because the measurement method and measured area are different, the ODF obtained for SG100-2-c after recrystallization is in good agreement with the experimental results. It should be noted that, in the reference, the annealing process was performed at a heating rate of 10°C/min, starting from room temperature.¹⁴⁾ However, the calculations in this paper assumed a fixed M₀^phys value, corresponding to isothermal annealing conditions. Although the calculation starts from the same EBSD measurements, the results differ depending on the initial conditions, especially for texture formation; the reasons for this difference are discussed in section 4.1. Before carrying out the numerical analysis, we predicted that a clear tendency in the texture development would be obtained according to L_es, a parameter that determines the size distribution of SGs in the initial condition. From Fig. 5, γ-fiber development can be seen with an increasing L_es value. An increasing L_es value corresponds to a widening size distribution of the SGs; however, this trend is not obvious.

Fig. 4.

Temporal evolutions of the recrystallized fraction of the SG structures converted from 90% cold-rolled iron. (Online version in color.)

Fig. 5.

(a–i) ODF sections (φ_{2, Euler = 45°}) of 90% cold-rolled iron calculated from the microstructures just after recrystallization, showing the influence of the different SG conversion conditions. (Online version in color.)

3.2. Recrystallization of 99.8% Cold-rolled Pure Iron

3.2.1. Calculation Conditions

The number of calculation grid points was set to 48 (TD) × 3584 (ND) × 1440 (RD). As the interval of the grid spacing is 1.25 × 10⁻⁸ m, it corresponds to 0.6 μm (TD) × 45 μm (ND) × 18 μm (RD). The interval of time integration was set to 1.1 × 10⁻⁵.

3.2.2. Recrystallization Behavior Simulation Results

Table 2 shows the conditions used for conversion to an SG structure and the number of SGs (N_{sg_init}) at the completion of step (5) in section 2.3. As in section 3.1.2, we attempted to control the size distribution function solely in the absence of any changes in N_{sg_init} and the average SG radius, <R>_init. We prepared two types of SG size settings by changing R_min and L_es, as indicated in Table 2. The random number sequence was changed thrice under each condition. Therefore, six simulation runs were carried out in total. As in section 3.1.2, the different random number sequences were designated as a, b, and c. The data presented in Table 2 confirmed that both N_{sg_init} and <R>_init are kept at approximately constant values. Figure 6 shows the initial SG size distribution function normalized by <R>_init for all six conditions. As was intended, the spread of the distribution changes depending on the conversion conditions, while the random number sequence does not affect the distribution function. Figure 7 shows the ODF calculated for the initial configuration. For the as-deformed sample, the α-fiber texture developed more strongly than that of the 90% cold-rolled samples, and its orientation was confirmed as {100} <011> to {113} <011>.¹⁴⁾ Here, the ODF before conversion (i.e., the as measured ODF) is not shown; however, it should be noted that the macro texture is completely maintained in the conversion. Figures 8(a), 8(b) shows the IPF (ND) maps obtained using EBSD measurement and the initial SG structure converted under SG050-2-a conditions. Figures 8(a), 8(b) confirmed that the lamellar structure with a width of 0.2 μm or less,¹⁴⁾ which is characteristic of cold-rolled pure iron, has been successfully converted to an SG structure. Moreover, the simulation volume was narrowed compared to the 90% cold-rolled material. This is because a high rolling ratio of 99.8% renders the microstructure unit following rolling smaller, and the number of grains (in terms of hot-rolled sheets) present along the ND axis is approximately 50 times higher. Comparing the ODF obtained from the calculation region shown in Fig. 7 with the ODF obtained from the as-measured microstructure shown in Fig. 8(a), the γ-fiber strength was slightly different; however, overall, both were nearly equal. Moreover, it was evident that the measurement area could be represented by the calculation area.

Table 2. SG structure conversion conditions for 99.8% cold-rolled iron and properties of the converted SG structures. Regarding the conversion conditions, R_min is the minimum radius and R_min × L_es corresponds to the maximum radius. In the converted structures, N_{sg_init} and <R>_init are the total number of SGs and the mean radius of the SGs, respectively.

Name	Conversion conditions		Converted properties
	Rmin [μm]	Les	Nsg_init	<R>init [μm]
SG050-2-a	0.05	2	178050	0.085
SG050-2-b	0.05	2	177811	0.085
SG050-2-c	0.05	2	178001	0.085
SG075-1-a	0.075	1	179030	0.086
SG075-1-b	0.075	1	179000	0.086
SG075-1-c	0.075	1	179029	0.086

Fig. 6.

Normalized SG size distributions in the initial stage for 99.8% cold-rolled iron. (Online version in color.)

Fig. 7.

ODF section (φ_{2, Euler = 45°}) of the converted SG structure with under SG050-2-a conditions. (Online version in color.)

Fig. 8.

(a) ND-orientation map measured using an EBSD technique and (b) that of the converted SG structure under SG050-2-a conditions. In (b), the white lines indicate boundaries with θ values of 15° or more, and the black lines indicate boundaries with θ values less than 15°. (Online version in color.)

The numerical analysis results of the recrystallization behavior are described below. Figure 9 shows the time evolution of the recrystallization rate. Regarding the judgment of recrystallized grains, the threshold radius was set to R_th = 20 [g.p.] = 0.25 [μm] in consideration of the initial SG size. This value is equivalent to three times of <R>_init. The recrystallization behavior in the 90% cold-rolled samples is also shown in Fig. 9 for comparison. The graph is drawn considering that one calculation step for the 90% cold-rolled samples corresponds to four steps for the 99.8% cold-rolled samples, owing to the difference in the PF calculation grid size. Concerning the curves for the 99.8% cold-rolled samples, an approximately equivalent recrystallization behavior was obtained without a dependence on the initial size distribution. In addition, the subtle change in the recrystallization behavior depending on the random number sequence that was observed in the simulation results for the 90% cold-rolled material, was not observed in this case. Figures 10(a)–10(f) shows the ODFs of the microstructure obtained immediately after the completion of recrystallization (i.e., recrystallization rate of 99%) for all six simulation runs. Although a slight difference can be observed, an appreciable reproduction of the experimental characteristics was observed. Hardly any change in the texture before and after recrystallization was evident under all the conditions. The robustness of the calculation results for the 99.8% cold-rolled materials is considered to be closely related to the fineness of the above-mentioned microstructure units and will be discussed in section 4.2. As information on the adjacent recrystallized grains cannot be included in the ODF, the misorientation distribution function (here, the function was calculated from the entire three-dimensional calculation region) elucidated at the same time as the ODF shown in Fig. 10, is shown in Fig. 11. In Fig. 11, a decrease in the low-angle grain boundary fraction and an increase in the high-angle grain boundary fraction because of the progress in recrystallization behavior was confirmed. However, no difference ascribed to the difference in the initial microstructure was confirmed.

Fig. 9.

Temporal evolutions of the recrystallized fraction of the SG structures converted from 99.8% cold-rolled iron. For comparison, recrystallization kinetics obtained for 90% cold-rolled iron are plotted in the figure (black line). (Online version in color.)

Fig. 10.

(a–f) ODF sections (φ_{2, Euler = 45°}) of 99.8% cold-rolled iron calculated from the microstructures just after recrystallization showing the influence of the different SG conversion conditions. (Online version in color.)

Fig. 11.

Misorientation distribution functions of 99.8% cold-rolled iron calculated from the microstructures just after recrystallization. (Online version in color.)

4. Discussion

4.1. Dependence of Recrystallized Texture on Initial SG Structure for 90% Cold-rolled Steel

To investigate the textural change behavior during recrystallization for the 90% cold-rolled material in detail, an IPF (ND) diagram, used for creating the ODF shown in Fig. 5, is presented in Figs. 12(a)–12(i). It is evident that there is no significant difference in the grain size during recrystallization, which is also the case for the above-mentioned change in the recrystallization rate with time. However, when we consider the texture development, the development of γ-fiber seems to be determined by whether the γ-fibers encroach the area where nucleation hardly occurs (i.e., α-fiber, typically the purple area at the left end in Fig. 3). To verify this hypothesis, we compared the microstructure development obtained under SG075-3-b conditions with the most developed γ-fiber and under SG125-1.2-c conditions with the least developed γ-fiber. Figures 13(a)–13(f) shows the time evolution of the IPF (ND) map under the above two conditions. From Figs. 13(a)–13(f), it is evident that in both the cases, the nucleation of recrystallization occurs preferentially in: (1) the γ-fiber region and its periphery; and, (2) the region where a mixture of orientations exist (circled in the figure) because of the high number of boundaries with large misorientation. As time proceeds, recrystallized grains from these nuclei invade the α-fiber region where there is hardly any nucleation, owing to the lack of boundaries with large misorientation. This evolution sequence did not change in either case, and agrees well with experimental results.¹⁴⁾ However, there was a major difference in crystal orientation of the surviving nuclei for both cases. In the case of SG075-3-b, in which the SG size distribution is broad, the γ-fiber SGs survived and grew as recrystallized grains in the simulation run, as shown in Figs. 13(a)–13(c). In the case of SG125-1.2-c, in which the SG size distribution is narrow, the surviving ratio of the γ-fiber SGs was lower, as shown in Figs. 13(d)–13(f). Next, we consider this point in more detail. In this study, the SG formation process is modeled by converting the measured values to SG structure, as described in section 2.3. As the crystal orientation of the SG nucleus reflected the crystal orientation of the measured orientation in the conversion, the macro texture was fully maintained. However, unfortunately, information like the detailed morphology of each SG is lost because of the conversion. Furthermore, in the region where the nucleation frequency is high, the orientation gradient is large. Therefore, the orientation of the nuclei may highly depend on the conversion conditions. Initially, we expected to be able to control γ-fiber development by changing the initial SG size distribution. However, as described in section 3.1.2, no clear relationship between the development of γ-fibers and the SG size distribution was obtained. At the end of this section, we consider techniques that are considered effective in improving prediction accuracy.

Fig. 12.

(a–i) ND-orientation maps of 90% cold-rolled iron calculated from the microstructures just after recrystallization. (Online version in color.)

Fig. 13.

Simulated microstructural evolutions of 90% cold-rolled iron during recrystallization under SG075-3-b (a–c) and SG125-1.2-c (d–f) conditions. These figures show the ND-orientation maps. The dotted circle indicates a region with a variety of orientations in the initial SG structure. (Online version in color.)

(1) [Measurement] Improvement of resolution in the EBSD measurements.

(2) [Measurement, model] Enlargement of the measurement area, including three-dimensional measurement, i.e., enlargement of simulation volume.

(3) [Model] Detailed description of the SG boundary characteristics.

(4) [Model] Improvement of the SG conversion procedure.

We believe that the most effective strategy for reducing the instability in the recrystallized texture obtained by calculation is technique (2), which promotes the average calculation results. For technique (1), the effect is limited because, in this paper, only the crystal orientation of the representative point of each SG is used for SG conversion. Therefore, simultaneous execution with technique (4) is essential. For technique (3), a change in the way of providing the SG boundary energy and mobility described in section 2.4 is required. For example, considering the cusp of the misorientation boundary energy curve may be effective. Technique (4) requires a detailed modeling of the recovery process. For example, according to the crystal orientation, it may be conceivable to change the time taken to form SGs. However, because no clear improvement guidelines for technique (4) have been obtained up to now, we believe it is necessary to first focus on confirming an improvement in the prediction accuracy by employing technique (2) and (3).

4.2. Texture Change during Recrystallization in 99.8% Cold-rolled Pure Iron

As mentioned in section 3.2.2, the effect of conversion to the initial SG structure on the recrystallization behavior is considerably small. Therefore, in this section, we select a simulation run, in which the microstructural development details are investigated. In this simulation, SG050-2-a was selected as the target condition. Figures 14(a)–14(d) presents the time evolution of the microstructure (i.e. IPF diagrams for an ND surface.) and the time evolution of the ODF in the case of SG050-2-a. In this model, high-angle grain boundaries with high mobility are required for the nucleation of recrystallization. In the case of 90% cold-rolled samples, nucleation occurred preferentially in the γ-fiber region and its periphery. Nucleation was also observed in the regions where there was a variety of orientations because of the high number of boundaries with large misorientation. In contrast, the 99.8% cold-rolled samples exhibit numerous high-angle grain boundaries, owing to the extremely fine-layered structure¹⁴⁾ that is generated under the exceedingly large ratios during the reduction; nucleation is present across the entire sample. Therefore, it is probable that there was no clear selectivity for the orientation of the recrystallized grains and, similar to the experimental results,¹⁴⁾ no significant change was obtained in the ODF following recrystallization. Furthermore, this is strongly related to the robustness of the calculation results. In other words, compared to the 90% cold-rolled material, the initial structure is finer; moreover, because the movement of the recrystallization tip during recrystallization covers a short distance, it is possible to represent the recrystallization behavior in a narrower region. If we focus on, for example, the 20000-step technique presented in Fig. 14(c), the structure is bimodal, where unrecrystallized SGs and recrystallized grains coexist. It can be confirmed that discontinuous recrystallization has occurred locally. Finally, the formation of “island grains” often observed in our simulation results are indicated using arrows in Fig. 14. When SGs with low misorientation exist adjacently (5000 steps), and if, remarkably, only one grows (10000 steps), it surrounds the others (20000 steps). In this case, low-angle boundaries (low mobility, low energy) play a crucial role; as time proceeds, the “island grain” shrinks (30000 steps); however, the rate of shrinkage is exceedingly low because it is surrounded by low-angle boundaries.

Fig. 14.

(a–d) Simulated microstructural evolutions of 99.8% cold-rolled iron during recrystallization under SG050-2-a conditions and corresponding ODF (φ_{2, Euler = 45°}) sections calculated from the microstructures. These figures show the ND-orientation maps. Arrows indicate the formation process for “island” grain structures. (Online version in color.)

5. Conclusions

We developed a three-dimensional PF simulation system that can be applied to model nucleation of recrystallization and to the subsequent grain growth after recrystallization, in the same framework. In this study, we prepared cold-rolled pure iron samples with reduction rates of 90% and 99.8%. Our previous experimental study showed that there are large differences in the texture-formation processes during the recrystallization of these samples. To establish a method for converting EBSD measurements to an SG structure, the effects of the conversion parameters on the recrystallization behavior were investigated. The obtained results are summarized below.

Simulation results for 90% cold-rolled samples

(1) The influence of the difference in initial SG size distribution on the recrystallization rate is negligible. The reasons for this are as follows: nucleation of recrystallization preferentially occurs in regions with a high number of boundaries with large misorientation. Following this, the recrystallized grains from these nuclei invade the α-fiber region, where there is hardly any nucleation (due to a lack of boundaries with large misorientation); this evolution sequence is highly reproducible.

(2) Although the texture just after the completion of recrystallization is qualitatively consistent with the experimental results, it changes quantitatively according to the conversion conditions. Currently, the mechanism by which SGs are selected as recrystallized grains is not completely clear. Prior to cold-rolling, the number of grains in our sample was around five. Therefore, it is necessary to run simulations using a wider range of measured values, including the possibility of three-dimensional measurements.

Simulation results for 99.8% cold-rolled samples

(3) The influence of the initial SG size distribution change on the recrystallization rate is negligible.

(4) The texture evolution was in good agreement with the experimental results. That is, there is hardly any change in the ODF of the entire system before and after recrystallization.

(5) In addition, the prediction reproducibility of the recrystallization texture was better than that of the 90% cold-rolled samples. This is because the severely cold-rolled sample is composed of a very fine layered structure due to which nuclei can be formed everywhere. Therefore, the calculation area is considered to be sufficiently large compared to the representative volume required for determining the recrystallized structure.

Simulation results for both samples

(6) The recrystallization behavior of the 90% and 99.8% cold-rolled samples with different texture formation mechanisms was successfully reproduced in a unified manner by employing an SG growth model capable of expressing spontaneous nucleation behavior. In addition, three-dimensional analysis was carried out in a practical timeframe of 10 h using an efficient algorithm and a parallel coding technique.

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