Available via license: CC BY-NC-ND 4.0
Content may be subject to copyright.
ISIJ International, Vol. 61 (2021), No. 1
© 2021 ISIJ 350
ISIJ International, Vol. 61 (2021), No. 1, pp. 350–360
https://doi.org/10.2355/isijinternational.ISIJINT-2020-226
* Corresponding author: E-mail: suwa.fm2.yoshihiro@jp.nipponsteel.com
Phase-field Simulation of Recrystallization in Cold Rolling and
Subsequent Annealing of Pure Iron Exploiting EBSD Data of Cold-
rolled Sheet
Yoshihiro SUWA,1)* Miho TOMITA,2) Yasuaki TANAKA1) and Kohsaku USHIODA3)
1) Advanced Technology Research Labs., Technical Research & Development Bureau, Nippon Steel Corporation, 1-8 Fuso-
Cho, Amagasaki, Hyogo, 660-0891 Japan.
2) Setouchi Works, Nippon Steel Corporation, 1 Fuji-cho, Hirohata-ku, Hyogo, 671-1188 Japan.
3) Nippon Steel Research Institute Corporation, Kokusai Bldg., 3-1-1 Marunouchi, Chiyoda-ku, Tokyo, 100-0005 Japan.
(Received on April 21, 2020; accepted on September 8, 2020; originally published in Tetsu-to-Hagané,
Vol. 105, 2019, No. 5, pp. 540–549)
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 car-
ried 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 recrys-
tallization kinetics and texture development were investigated. Numerical simulations of static recrystalliza-
tion 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.
KEY WORDS: computer simulation; recrystallization; texture; sub-grain; phase-field model; electron back
scattering diffraction (EBSD).
1. Introduction
When a metallic material is plastically deformed, numer-
ous lattice defects are introduced inside the crystal, the
internal energy increases, and the crystal hardens. More-
over, 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-eld (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 uni-
form manner. The ultimate goal is to consistently predict
recrystallization and grain growth. There have been numer-
ous reports on the numerical analysis results obtained by
combining unied SG theory with microstructural simu-
lations, such as, Monte Carlo (MC),4,5) PF,6–8) 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
© 2021 The Iron and Steel Institute of Japan. This is an open access article under the terms of the Creative Commons
Attribution-NonCommercial-NoDerivs license (https://creativecommons.org/licenses/by-nc-nd/4.0/).
ISIJ International, Vol. 61 (2021), No. 1
© 2021 ISIJ351
theory that “a region with low dislocation density is formed
from a homogeneous deformed matrix due to thermal uc-
tuations”. 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 repro-
duce 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 subse-
quent growth process, the strain energy dierence is used
as the main driving force for the growth of the recrystalliz-
ing grains. In this study, nucleation and growth are treated
without distinction, according to the SG growth model
described above. Specically, the non-uniform deformed
structure is converted into SG structures with various sizes
and crystal orientations, and the nucleation of recrystalliza-
tion is expressed as abnormal grain growth of a specic
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 diraction) method are employed to convert it to an
SG structure.8) The possibility of abnormal grain growth of
a specic SG, that is, “nucleation of discontinuous recrystal-
lization” is expressed using the boundary energy and mobil-
ity, depending on the crystal orientation dierence between
the SGs. One must not always assume the initial congura-
tion 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 recrystalliza-
tion 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 recrys-
tallization. Although this assumption is controversial,15,16)
the possibility of an SG structure is not negligible, consid-
ering 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 reduc-
tion 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-
eld (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 dened to distin-
guish 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.
i
i
Nrt
,.
1
1 ............................. (1)
The time–evolution equations of the order parameter
ϕ
i (i = 1, 2, ..., N) are given by
i
ij ij i
C
j
C
ij i
E
j
E
j
N
tnrt ss Mf
ff
f
28
,,
.......................................... (2)
where n is the number of non-zero order parameters at each
spatial point. fiE is the excess free energy for the i-th SG,
assumed to be 0 for i > 1 and Mij is the phase-eld mobility
of the boundary between the i-th and j-th SGs. According to
Hirouchi et al.18) the phase-eld mobility of multiple junc-
tions is dened as follows:
MM
aveiji j
ij
N
ij
ij
N
,,
/.
.............. (3)
At the spatial point where three or more non-zero order
parameters exist, Mave is used as the phase-eld mobil-
ity instead of Mij in Eq. (2). Furthermore, fic in Eq. (2) is
expressed as:
f
WW W
i
C
ik
kikk iklkliklmklm
m
N
l
N
2
2
11
2
kk
N
1.
.......................................... (4)
Wik, 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 Wikl and Wiklm parameters are deter-
mined according to previous studies.19,20) Although a restric-
tion, 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: Wii =
0;
ε
ii = 0; and Mii = 0. The value of n can be written as:
nr
ts
rt
i
i
N
,,
,
1......................... (5)
where si (r, t) is a step function that satises si (r, t) = 1,
if
ϕ
i (r, t) > 0 and si (r, t) = 0 otherwise. Wij,
ε
ij, and Mij
exhibit the following relationship with the boundary width
δ
, interface energy
σ
ij, and the physical mobility Mijphys in
the PF analysis.
WM
M
ij
ij
ij ij ij ij
phys
4228
2
,, ....... (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) algo-
rithm22–24) to accelerate computations and embody large-
scale calculations.
2.2. Implementation of the Crystal Orientation Repre-
sented by Euler Angles
In this study, the recrystallization is reproduced using the
SG growth model.3) In this model, SG interfacial energy and
ISIJ International, Vol. 61 (2021), No. 1
© 2021 ISIJ 352
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 specic SG. Conven-
tionally, crystal orientation has been implemented into the
phase-eld models by connecting a crystal orientation gi to
an order parameter,
ϕ
i. In these cases, the relation table with
dimensions N × N had to be predetermined before simula-
tion 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 simula-
tion runs without storing the table, the computational cost
signicantly 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 = gi · gj−1
between SGi and SGj 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 com-
putational cost can be signicantly reduced by considering
the crystal symmetry when creating the E- and M-tables.
Therefore, a matrix operation is required to obtain an ori-
entation 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 > 4 830 is satised.
In this paper, the E- and M-tables are solely controlled by
the misorientation angle
θ
; however, it is possible to also
consider the eect of the rotation axis on misorientation.
2.3. Conversion of the Measured Orientation Map to
an SG Structure using EBSD as the Initial Cong-
uration 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 direc-
tion). The measured grain size of the hot-rolled sheet was
200–500
μ
m. As the nal 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 ve 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 unmodied measured ori-
entation map as the initial conguration for the PF simula-
tions of recrystallization. However, because a complete SG
structure is required as the input data, the initial congura-
tion 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 cor-
related with multiple nite-dierence grid points. Hereafter,
a nite-dierence grid point in the PF analysis is referred
to as a PF grid point. Specically, 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, dierent PF grid points were assigned for the pur-
pose of presenting the neness of the measured structure,14)
which is dependent on the cold-rolling reduction rate.
Furthermore, when carrying out the three-dimensional cal-
culations, 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, regard-
ing the length along the TD axis, preliminary studies sug-
gested that three-dimensional constraints arising from the
surrounding SGs had a signicant eect 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 accu-
mulated 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
ϕ
1 with f1E > 0 is set to 1 throughout
the system for the purpose of creating an SG struc-
ture. Using this operation, the SG nucleus rapidly
grows into a region with an order parameter value of
ϕ
1 = 1. Thus, an SG structure is obtained. The crystal
orientation of the SG nucleus reects the crystal ori-
entation 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 suciently small, which is assigned
the value of fiE = 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 dened as R = Rmin × {1 +
IQ’ × (Les− 1)} using the IQ’ value at coordinate P.
Here, Rmin is the minimum SG radius, and Les is the
parameter determined when Les ×Rmin corresponds to
the maximum SG radius. At this time, the SG radius
takes the maximum value Les ×Rmin when IQ’ = 1 and
the minimum value Rmin when IQ’ = 0.
(3) Let Ri be the radius of an SG with a nucleus coor-
dinate Pi, where i takes a value of 1 to nn and nn is
the total number of SGs adjacent to the SG with the
nucleus P. If the distance between P and Pi, dened
as | P − Pi |, satises the condition | P − Pi | > (R +
Ri) 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 NSG
and the order parameter 1 is assigned as the region
ISIJ International, Vol. 61 (2021), No. 1
© 2021 ISIJ353
being invaded by the SGs, we can write N = NSG +
1.
(5) By carrying out a short-time PF simulation, the region
with an order parameter value of
ϕ
1 = 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 com-
pressed 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 relation-
ship between the strain accumulation via deformation and
SG size is not completely clear, Les is treated as an adjust-
able 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
diers, depending on the crystal orientation and the sur-
rounding constraints; as a result, the SG formation time (or
recovery speed) is dierent.1) Finally, in this conversion pro-
cedure, the SGs formed from dierent nuclei are assigned
dierent 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)
ij
mm
m
ln
01,, ............ (7)
where
σ
0 and
θ
m are the boundary energy and the misorien-
tation when the boundary becomes a high-angle boundary.
Similarly, the mobility Mphys (
θ
) is constant at high-angle
grain boundaries. We consider the Mphys versus
θ
curve to
be sigmoidal with the following form:2)
MM Mexp
ij
phys phys phys
m
0
4
15 ,, ,
m
.......................................... (8)
where M0phys is the mobility of a high-angle boundary. Fol-
lowing 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 M0phys < Mphys (
θ
) and 0.52
σ
0 <
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 cur-
rent study:
• boundary width
δ
in the PF method: 4 PF lattice spac-
ing;
• the mobility of a high-angle boundary: M0phys = 2.0 ×
10 − 12 m4 J −1
s −1
;
• the energy of a high-angle boundary:
σ
0 = 1.0 J m −2
;
• boundary conditions: periodic boundary conditions were
applied along the three axes;
• calculation time: 80 000 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) × 3 744 (ND) × 960 (RD). As the interval of the grid
spacing is 2.5 × 10 −8
m, it corresponds to 1.2
μ
m (TD) ×
94
μ
m (ND) × 24
μ
m (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 (Nsg_init) at the completion of step
(5) in section 2.3. To evaluate the eects of the SG size
distribution function on the recrystallization behavior, the
size distributions were varied while introducing minimal
changes in Nsg_init and the average SG radius < R>init. We
prepared three types of SG size distributions by changing
Rmin and Les, 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
Tab le 1. SG structure conversion conditions for 90% cold-rolled
iron and the properties of the converted SG structures.
Regarding the conversion conditions, Rmin is the m ini-
mum radius and Rmin × Les cor responds to the maximu m
radius. In the converted structures, Nsg _init and < R>init
are the total number of SGs and the mean radius of the
SGs, respectively.
Name
Conversion conditions Converted properties
Rmin [
μ
m] Les Nsg_in it <R > init [
μ
m]
SG075 -3- a 0.075 3126 533 0.17
SG075 -3- b 0.075 3126 758 0 .17
SG075 -3- c 0.075 3126 629 0.17
SG10 0-2- a 0.1 213 4 899 0.17
SG10 0-2- b 0.1 213 4 785 0 .17
SG10 0-2- c 0.1 2134 785 0.17
SG125-1.2-a 0.12 5 1.2 13 4 662 0.17
SG125-1.2-b 0.12 5 1.2 134 915 0.17
SG125-1.2-c 0.125 1.2 134 818 0.17
ISIJ International, Vol. 61 (2021), No. 1
© 2021 ISIJ 354
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 condi-
tions, < R>init was desirably equal to 0.17
μ
m. Furthermore,
Fig. 2 shows the orientation distribution function (hereafter
abbreviated as ODF) obtained from the initial microstruc-
ture. Here, because the TD size is 3.5 times the average SG
diameter, which is signicantly 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
-ber (RD //
< 011 > ) and
γ
-ber (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 frac-
tions in the initial microstructure were
α
; 0.56 and
γ
; 0.23.
Both the
α
-ber and
γ
-ber exhibited a (111) [1–10] orien-
tation 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 gure, the color
map is assigned IPF diagrams for an ND surface. To explain
the eect of extending the two-dimensional EBSD measure-
ment 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 gure, we also conrm the SGs are arranged
along the TD axis. The constraints from the surrounding
SGs had a signicant eect on the nucleation of recrystal-
lization using the SG growth model. However, the EBSD
measurements were limited to two dimensions initially.
Therefore, we extended the two-dimensional EBSD mea-
surements 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.
Next, the results of the numerical analysis of recrystalli-
zation behavior are described. Figure 4 shows the time evo-
lution of the recrystallization rate. The threshold value of the
radius for judging recrystallized grains was Rth = 20 [g.p.] =
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 st ructure
under SG100-2-b conditions. (Online version in color.)
Fig. 3. (a) ND- orientation map measured using an EBSD techn ique and (b), (c) those of the converted SG str uctu re
under SG100-2-b conditions. In ( b) and (c), the white lines ind icate boundar ies with
θ
values of 15° or more, and
the black lines indicate boundaries with
θ
values less t han 15°. The horizontal black line in (b) represents the
ND–TD cross-sectional view taken in (c). (Online version in color.)
ISIJ International, Vol. 61 (2021), No. 1
© 2021 ISIJ355
0.5 [
μ
m]. 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. How-
ever, upon closer inspection, a slight dierence depending
on the calculation conditions can be observed. In contrast,
no clear dependence on the SG size distribution function in
the initial conguration can be conrmed for the change in
recrystallization behavior. Figures 5(a)–5(i) shows ODFs of
the microstructure obtained immediately after the comple-
tion of recrystallization (i.e., recrystallization rate of 99%)
for all nine simulation runs. In most cases, a decrease in
α
-bers is conrmed with the development of
γ
-bers;
this is consistent with experimental results.14) Although
a direct comparison is dicult because the measurement
method and measured area are dierent, 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) How-
ever, the calculations in this paper assumed a xed M0phys
value, corresponding to isothermal annealing conditions.
Although the calculation starts from the same EBSD mea-
surements, the results dier depending on the initial condi-
tions, especially for texture formation; the reasons for this
dierence 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
Les, a parameter that determines the size distribution of SGs
in the initial condition. From Fig. 5,
γ
-ber development
can be seen with an increasing Les value. An increasing Les
value corresponds to a widening size distribution of the SGs;
however, this trend is not obvious.
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) × 3 584 (ND) × 1 440 (RD). As the interval of the
grid spacing is 1.25 × 10 −8
m, it corresponds to 0.6
μ
m
(TD) × 45
μ
m (ND) × 18
μ
m (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 (Nsg_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 Nsg_init and the average
SG radius, < R>init. We prepared two types of SG size set-
tings by changing Rmin and Les, 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 dierent random number
sequences were designated as a, b, and c. The data presented
Fig. 4. Temporal evolut ions of the recrystalli zed fr action of the
SG str uctures converted from 90% cold-rolled iron.
(Online ver sion in color.)
Fig. 5. (a–i) ODF sections (
φ
2, Euler = 45°) of 90% cold-rolled iron calculated from t he microstructures just after recrystal-
lization, showing the inuence of the d ierent SG conversion cond itions. (Online version in color.)
ISIJ International, Vol. 61 (2021), No. 1
© 2021 ISIJ 356
Tab le 2 . SG st ructure conversion condit ions for 99.8% cold-rolled
iron and proper ties of the converted SG str uctures.
Regarding the conversion conditions, Rmin is the m ini-
mum radius and Rmin × Les cor responds to the maximu m
radius. In the converted structures, Nsg _init and < R>init
are the total number of SGs and the mean radius of the
SGs, respectively.
Name
Conversion conditions Converted properties
Rmin [
μ
m] Les Nsg_in it <R > init [
μ
m]
SG050-2-a 0.05 217 8 0 50 0.085
SG050-2-b 0.05 217 7 811 0.085
SG050-2-c 0.05 217 8 001 0.085
SG075-1-a 0.075 1179 0 30 0.086
SG075-1-b 0.075 1179 000 0.086
SG075-1-c 0.075 1179 02 9 0.086
Fig. 6. Normalized SG size distributions in the initial stage for
99.8% cold-rolled iron. (Online version in color.)
Fi g. 7. ODF section (
φ
2, Euler = 45°) of the converted SG str uct ure
with under SG050-2-a conditions. (Online version in
co lo r.)
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), t he white lines indicate bound aries w ith
θ
values of 15° or more, and the black
lines indicate boundaries wit h
θ
values less t han 15°. (Online version in color.)
in Table 2 conrmed that both Nsg_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 distribu-
tion changes depending on the conversion conditions, while
the random number sequence does not aect the distribution
function. Figure 7 shows the ODF calculated for the initial
conguration. For the as-deformed sample, the
α
-ber
texture developed more strongly than that of the 90% cold-
rolled samples, and its orientation was conrmed as {100}
< 011 > to {113} < 011 >.14) Here, the ODF before conver-
sion (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
ISIJ International, Vol. 61 (2021), No. 1
© 2021 ISIJ357
IPF (ND) maps obtained using EBSD measurement and the
initial SG structure converted under SG050-2-a conditions.
Figures 8(a), 8(b) conrmed that the lamellar structure
with a width of 0.2
μ
m 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 micro-
structure 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
-ber strength was slightly dierent;
however, overall, both were nearly equal. Moreover, it was
evident that the measurement area could be represented by
the calculation area.
The numerical analysis results of the recrystallization
behavior are described below. Figure 9 shows the time
evolution of the recrystallization rate. Regarding the judg-
ment of recrystallized grains, the threshold radius was set
to Rth = 20 [g.p.] = 0.25 [
μ
m] in consideration of the
Fig. 10. (a–f ) ODF sections (
φ
2, Euler = 45°) of 99.8% cold-rolled iron calculated from the microstr uctu res just after recrys-
tallization showing the i nuence of the dierent SG conversion conditions. (Online version in color.)
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 dierence in the PF
calculation grid size. Concerning the curves for the 99.8%
cold-rolled samples, an approximately equivalent recrystal-
lization 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 recrystalliza-
tion (i.e., recrystallization rate of 99%) for all six simula-
tion runs. Although a slight dierence 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
neness of the above-mentioned microstructure units and
will be discussed in section 4.2. As information on the adja-
cent 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 conrmed. However, no
dierence ascribed to the dierence in the initial microstruc-
ture was conrmed.
4. Discussion
4.1. Dependence of Recrystallized Texture on Initial SG
Structure for 90% Cold-rolled Steel
To investigate the textural change behavior during recrys-
Fig. 9. Temporal evolutions of the recrystallized fraction of the
SG struct ures converted from 99.8% cold-rolled iron. For
comparison, recr ystallization kinet ics obtained for 90%
cold-rolled iron are plotted in the gure (black line).
(Online ver sion in color.)
ISIJ International, Vol. 61 (2021), No. 1
© 2021 ISIJ 358
tallization 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 signicant dierence in the grain size during recrystal-
lization, 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
-ber seems to be determined by whether the
γ
-bers
encroach the area where nucleation hardly occurs (i.e.,
α
-ber, 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
-ber and under SG125-1.2-c conditions
with the least developed
γ
-ber. 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 prefer-
entially in: (1) the
γ
-ber region and its periphery; and, (2)
the region where a mixture of orientations exist (circled in
the gure) because of the high number of boundaries with
large misorientation. As time proceeds, recrystallized grains
from these nuclei invade the
α
-ber 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 dierence 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
-ber 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
-ber 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 reected 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 con-
version. Furthermore, in the region where the nucleation
frequency is high, the orientation gradient is large. There-
fore, the orientation of the nuclei may highly depend on the
conversion conditions. Initially, we expected to be able to
Fig. 12. (a–i) ND-orientation maps of 90% cold-rolled iron calcu-
lated from the microstructures just after recrystalliza-
tion. (Onli ne version in color.)
Fig. 11. Misorientation distribution fu nctions of 99.8% cold-
rolled iron calculated from the microstructures just af ter
recrystallization. (Online version in color.)
ISIJ International, Vol. 61 (2021), No. 1
© 2021 ISIJ359
control
γ
-ber development by changing the initial SG size
distribution. However, as described in section 3.1.2, no clear
relationship between the development of
γ
-bers and the SG
size distribution was obtained. At the end of this section, we
consider techniques that are considered eective in improv-
ing prediction accuracy.
(1) [Measurement] Improvement of resolution in the
EBSD measurements.
(2) [Measurement, model] Enlargement of the measure-
ment 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 proce-
dure.
We believe that the most eective strategy for reduc-
ing the instability in the recrystallized texture obtained by
calculation is technique (2), which promotes the average
calculation results. For technique (1), the eect 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 provid-
ing the SG boundary energy and mobility described in sec-
tion 2.4 is required. For example, considering the cusp of
the misorientation boundary energy curve may be eective.
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 rst focus on conrming 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 eect 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 develop-
ment 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
-ber region and its periphery. Nucleation was also
observed in the regions where there was a variety of orienta-
tions 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 ne-layered structure14) that is generated under
Fig. 13. Simulated microstruct ural evolutions of 90% cold-rolled
iron dur ing recrystallization under SG 075-3-b (a– c) and
SG125-1.2-c (d–f ) cond itions. These gures 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.)
Fi g. 14. (a–d) Simulated microstr uctural evolutions of 99.8%
cold-rolled iron during recr ystallization under SG050-
2-a conditions a nd cor responding ODF (
φ
2, Euler = 45°) sec-
tions calculated from the microstruct ures. T hese gures
show the ND-orientation maps. Ar rows indicate the for-
mation process for “island” grain st ructures. (Online
version in color.)
ISIJ International, Vol. 61 (2021), No. 1
© 2021 ISIJ 360
the exceedingly large ratios during the reduction; nucleation
is present across the entire sample. Therefore, it is prob-
able that there was no clear selectivity for the orientation
of the recrystallized grains and, similar to the experimental
results,14) no signicant 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 ner; 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 20 000-step technique presented in Fig. 14(c), the struc-
ture is bimodal, where unrecrystallized SGs and recrystal-
lized grains coexist. It can be conrmed 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 (5 000 steps), and if, remark-
ably, only one grows (10 000 steps), it surrounds the oth-
ers (20 000 steps). In this case, low-angle boundaries (low
mobility, low energy) play a crucial role; as time proceeds,
the “island grain” shrinks (30 000 steps); however, the rate
of shrinkage is exceedingly low because it is surrounded by
low-angle boundaries.
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
dierences in the texture-formation processes during the
recrystallization of these samples. To establish a method
for converting EBSD measurements to an SG structure, the
eects of the conversion parameters on the recrystallization
behavior were investigated. The obtained results are sum-
marized below.
Simulation results for 90% cold-rolled samples
(1) The inuence of the dierence in initial SG size
distribution on the recrystallization rate is negligible. The
reasons for this are as follows: nucleation of recrystalliza-
tion preferentially occurs in regions with a high number of
boundaries with large misorientation. Following this, the
recrystallized grains from these nuclei invade the
α
-ber
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 experi-
mental 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 ve. Therefore, it is necessary to run
simulations using a wider range of measured values, includ-
ing the possibility of three-dimensional measurements.
Simulation results for 99.8% cold-rolled samples
(3) The inuence 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 recrystal-
lization.
(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 ne layered structure due to
which nuclei can be formed everywhere. Therefore, the cal-
culation area is considered to be suciently 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 dierent texture formation mecha-
nisms was successfully reproduced in a unied manner by
employing an SG growth model capable of expressing spon-
taneous nucleation behavior. In addition, three-dimensional
analysis was carried out in a practical timeframe of 10 h
using an ecient algorithm and a parallel coding technique.
REFERENCES
1) F. J. Humphreys, G. S. Rohrer and A. Rollett: Recrystallization and
Related Annealing Phenomena, 3rd ed., Elsevier, Amsterdam, (2017),
3.
2) I. Steinbach and F. Pezzolla: Physica D, 134 (1999), 385.
3) F. J. Humphreys: Acta Mater., 45 (1997), 4231.
4) E. A. Holm, M. A. Miodownik and A. D. Rollett: Acta Mater., 51
(2003), 2701.
5) B. Radhakrishnan, G. Sarma and T. Zacharia: Scr. Mater., 39 (1998),
1639.
6) Y. Suwa, Y. Saito and H. Onodera: Mater. Sci. Eng. A, 457 (2007),
132.
7) Y. Suwa, Y. Saito and H. Onodera: Comput. Mater. Sci., 44 (2008),
286.
8) T. Takaki and Y. Tomita: Int. J. Mech. Sci., 52 (2010), 320.
9) F. J. Humphreys: Scr. Metall. Mater., 27 (1992), 1557.
10) D. Weygand, Y. Brechet and J. Lepinoux: Philos. Mag. B, 80 (2000),
1987.
11) Y. Suwa: Tetsu-to-Hagané, 97 (2011), 173 (in Japanese).
12) F. J. Humphreys: Mater. Sci. Forum, 467–470 (2004), 107.
13) D. Raabe and R. C. Becker: Model. Simul. Mater. Sci. Eng., 8 (2000),
445.
14) M. Tomita, T. Inaguma, H. Sakamoto and K. Ushioda: Tetsu-to-
Hagané, 101 (2015), 204 (in Japanese).
15) M. Winning, G. Gottstein and L. S. Shvindlerman: Acta Mater., 49
(2001), 211.
16) G. Gottstein, A. H. King and L. S. Shvindlerman: Acta Mater., 48
(2000), 397.
17) M. C. Demirel, A. P. Kuprat, D. C. George and A. D. Rollett: Phys.
Rev. Lett., 90 (2003), 016106.
18) T. Hirouchi, T. Tsuru and Y. Shibutani: Comput. Mater. Sci., 53
(2012), 474.
19) E. Miyoshi and T. Takaki: Comput. Mater. Sci., 112 (2016), 44.
20) E. Miyoshi and T. Takaki: Comput. Mater. Sci., 120 (2016), 77.
21) Y. Suwa, Y. Saito and H. Onodera: Mater. Trans., 49 (2008), 704.
22) S. Vedantam and B. S. V. Patnaik: Phys. Rev. E, 73 (2006), 016703.
23) J. Gruber, N. Ma, Y. Wang, A. D. Rollett and G. S. Rohrer: Model.
Simul. Mater. Sci. Eng., 14 (2006), 1189.
24) S. G. Kim, D. I. Kim, W. T. Kim and Y. B. Park: Phys. Rev. E, 74
(2006), 061605.
25) E. A. Holm, G. N. Hassold and M. A. Miodownik: Acta Mater., 49
(2001), 2981.
26) S. I. Wright, D. P. Field and D. J. Dingley: Electron Backscatter Dif-
fraction in Materials Science, ed. by A. J. Schwartz et al., Springer
Science +Business Media, New York, (2000), 141.
27) W. T. Read and W. Shockley: Phys. Rev., 78 (1950), 275.
28) A. D. Rollett, D. J. Srolovitz and M. P. Anderson: Acta Metall., 37
(1989), 1227.
