# Characterization and modeling of the influence of initial microstructure on
recrystallization of zircaloy-4 during hot forming

Victor Grand1,2 Baptiste Flipon1 Alexis Gaillac2 Marc Bernacki1

( 1Mines ParisTech, PSL - Research University, CNRS, CEMEF, UMR 7635
2Framatome - Components Research Center
)

###### Abstract

The present article proposes a detailed study of recrystallization of
zircaloy-4 under hot forming conditions by means of experimental and numerical
tools. Thermomechanical tests and characterization campaigns that have been
necessary for this work are described. Then, the different microstructure
evolution mechanisms are characterized, from the simplest one to the most
complex. Grain growth kinetics is quantified and the influence of second phase
particle population is analyzed. Then, a complete study of dynamic and post-
dynamic recrystallization is provided. The occurrence of a continuous
mechanism is confirmed and the influence of thermomechanical conditions upon
recrystallization is assessed. Later, the numerical framework used to simulate
grain growth, continuous and post-dynamic recrystallization is presented.
After having successfully reproduced the grain coarsening kinetics with and
without second phase particles, the model is used to describe continuous
dynamic recrystallization and post-dynamic recrystallization from an initial
equiaxed and fully recrystallized microstructure. The agreement between
experimental and numerical results is assessed in details. Finally, post-
dynamic recrystallization is simulated, starting from two deformed
microstructures characterized by electron back-scattered diffraction technique
and immersed into simulations. This allows to discuss and reproduce the
influence of initial microstructure. Keywords: Hot forming, Zircaloy-4,
Continuous recrystallization, Full field modeling, Level-Set,
Characterization, EBSD.

###### Contents

  1. 1 Introduction
  2. 2 Materials and methods
    1. 2.1 Materials
    2. 2.2 Thermomechanical testing
    3. 2.3 Characterization techniques
  3. 3 Characterization of microstructure evolutions
    1. 3.1 Grain growth
    2. 3.2 Recrystallization
      1. 3.2.1 Characterization of recrystallization mechanism
      2. 3.2.2 Influence of thermomechanical conditions
  4. 4 Numerical framework
    1. 4.1 Interface description
    2. 4.2 Subgrain formation and evolution
    3. 4.3 Material parameters
  5. 5 Modeling of grain growth and recrystallization
    1. 5.1 Modeling of grain growth
      1. 5.1.1 Identification of material parameters without SPP
      2. 5.1.2 Application to commercial material
    2. 5.2 Modeling of CDRX and PDRX
    3. 5.3 Modeling of PDRX using immersed EBSD data
  6. 6 Conclusion

##  1 Introduction

With the global technology advances, requirements for metallic materials have
also known a significant increase. Thus, a need for a better understanding of
metallurgy, microstructure evolutions and the link between microstructure
characteristics and macroscopic properties is required. This is especially
true for critical applications such as a aeronautics or nuclear industry.
Focusing on nuclear industry, the development of new reactors and the general
regulation reinforcement oblige to fully understand and master every single
processing step to ensure that each final product meets the high requirements.
The fabrication of zirconium alloy cladding fuel assembly follows this
principle. Thus, for several decades now, Framatome has been leading research
campaigns to improve knowledge about zirconium alloy properties [1, 2],
forming processes and microstructure evolutions [3, 4, 5].

Manufacturing route of tubes and grids consists in a succession of hot and
cold forming steps and heat treatments. Each of these stages is crucial since
it impacts both the following process and the final part. It influences the
piece geometry as well as some macroscopic properties such as ductility,
mechanical strength or even corrosion resistance. These characteristics being
intimately related to microstructural features such as grain size distribution
or second phase particle (SPP) properties. Consequently, understanding and
being able to predict how microstructure evolves during hot forming and heat
treatments, especially through recrystallization (ReX), is of first interest.
The particular case of hot extrusion used to produce cladding tubes
illustrates how this is critical. Indeed, a low recrystallized fraction after
hot forming is susceptible to induce crack formation during the subsequent
steps of cold pilgering [6, 7].

Recrystallization of zircaloy-4 (Zy-4) has been studied by the way of
experimental characterization campaigns in several studies in the past. Chauvy
et al. [3] have put in light that under common industrial hot deformation
conditions, Zy-4 exhibits continuous dynamic recrystallization (CDRX). This
mechanism is characterized by the formation of subgrains and their progressive
transformation into new recrystallized grains. The new recrystallized grains
are generally characterized by a low dislocation density and an equiaxed shape
[8]. During CDRX, they appear rather homogeneously throughout the whole
microstructure [9]. Additionally, Chauvy et al. have analyzed quantitatively
the formation of low angle grain boundaries (LAGB, i.e. grain boundaries (GB)
that have a disorientation lower than a given threshold value, generally taken
equal to $15^{\circ}$) and the influence of thermomechanical conditions on
their formation [3]. Logé et al. [10] have investigated in details the
influence of the lamellae thickness and of the precipitate distribution upon
texture evolution during hot deformation and its relation with
recrystallization. They exhibit how such initial microstructural features may
affect subsequent evolutions during processing.

As it has been shown by experimental studies, recrystallization of Zy-4 is
influenced by a high number of parameters [3, 4, 10]. Therefore, numerical
modeling appears as a suitable tool to assess separately the influence of one
or the other condition. Dunlop et al., in a series of two articles, have
developed models to predict the plastic deformation and the recrystallization
of zircaloy-4 [11, 12]. They propose a mean field model, i.e. a model based on
variables describing the average characteristics of the considered
microstructure, for both recovery and recrystallization, which predicts
successfully the recrystallization kinetics. Gaudout et al. [13] have adapted
the Gourdet-Montheillet model [14] to simulate the recrystallization of
zirconium alloys during and after hot deformation. This last study constitutes
at the moment the last and more detailed attempt to model CDRX of Zy-4.

Until now and to the authors knowledge, no study has been led to assess in
details the influence of initial microstructure on recrystallization of Zy-4.
All modeling efforts have been made to produce a consistent model of
continuous recrystallization. The present article intends to take advantage of
the recent advances made in full-field simulations in order to improve our
understanding of CDRX of Zy-4. It summarizes a complete and extensive study of
Zy-4 recrystallization during and after hot forming in conditions
representative of industrial processes [6]. First, it describes how
microstructure evolutions have been characterized and what are the main
takeaways about recrystallization mechanisms. It discusses to what extent
thermomechanical conditions and initial microstructure impact
recrystallization. Secondly, this paper presents the full-field simulation
framework used to model grain growth (GG), CDRX and post-dymamic
recrystallization (PDRX). Then, the results from the main simulation cases are
presented and a discussion upon the influence of several microstructure
characteristics is proposed.

##  2 Materials and methods

###  2.1 Materials

Two different materials corresponding to three characteristic microstructures
are used in the experimental campaigns presented in the following sections:

  * •

commercial zircaloy-4, with either equiaxed grains or a lamellar
microstructure. Samples with lamellar microstructure have been manufactured
for this study and exhibit mainly basket-weaved lamellae.

  * •

A zirconium alloy with a chemical composition as close as possible to the
specification of Zy-4, except for Iron and Chromium that are kept as low as
possible. Samples made out of this alloy are manufactured to obtain an alloy
with the same solid solution than Zy-4 but without SPP.

For the sake of clarity, those three different initial states will be
respectively called: equiaxed (eq.), basket-weaved (b-w.) and without SPP (no
SPP.). The manufacturing routes of each of those part are described in figure
1.

Figure 1: Presentation of the manufacturing routes of each material.

Due to the fact each of those three different initial states are not obtained
by the exact same processing route, it is not possible to manufacture
cylindrical samples for thermomechanical testing with the exact same geometry.
Nevertheless, the three geometries share the same cylindrical shape with a
ratio height over radius being equal to 1.4.

###  2.2 Thermomechanical testing

Both heat treatments and hot compression tests are made within this study.
Heat treatments are made either under vacuum or a protective atmosphere of
argon to limit surface oxidation. Hot compression experiments are made using a
hydraulic mechanical testing machine (MTS Landmark 370-25). A radiant furnace
is used to heat samples and dies at the temperature setpoint. The set-up
allows the furnace to move along compression direction in order to put sample
in place before experiment and to proceed quickly to the water quench at the
end of the experiment. Dies and samples are coated using a spray of boron
nitride. The whole test is filmed to determine the quenching time for each
experiment. Detailed tables presenting the conditions of thermomechanical
tests are available in table 1.

Strain | $\mathbf{0.01~{}s^{-1}}$ | $\mathbf{0.1~{}s^{-1}}$ | $\mathbf{1.0~{}s^{-1}}$
---|---|---|---
$\mathbf{450^{\circ}C}$ | 0.65 | 1.0 | 1.35 | 0.65 | 1.0 | 1.35 | 0.65 | 1.0 | 1.35
$\mathbf{550^{\circ}C}$ | 0.65 | 1.0 | 1.35 | 0.65 | 1.0 | 1.35 | 0.65 | 1.0 | 1.35
$\mathbf{650^{\circ}C}$ | 0.65 | 1.0 | 1.35 | 0.65 | 1.0 | 1.35 | 0.65 | 1.0 | 1.35
(a) Hot compression with equiaxed microstructures.

Strain | $\mathbf{0.01~{}s^{-1}}$ | $\mathbf{0.1~{}s^{-1}}$ | $\mathbf{1.0~{}s^{-1}}$
---|---|---|---
$\mathbf{450^{\circ}C}$ | 0.65 | 1.0 | 1.35 | - | - | 1.35 | - | - | -
$\mathbf{550^{\circ}C}$ | - | - | 1.35 | 0.65 | 1.0 | 1.35 | - | - | 1.35
$\mathbf{650^{\circ}C}$ | - | - | - | - | - | 1.35 | 0.65 | 1.0 | 1.35
(b) Hot compression with lamellar microstructures.

Holding time (s) | $\mathbf{0.1~{}s^{-1}}$ | $\mathbf{1.0~{}s^{-1}}$
---|---|---
$\mathbf{650^{\circ}C}$ | 0.65 | - | - | - | - | - | - | 7 | 12 | 25 | 50 | 100 | 200
1.35 | 7 | 12 | 25 | 50 | 100 | 200 | 7 | 12 | 25 | 50 | 100 | 200
(c) Hot compression and holding at temperature with equiaxed microstructures.

Holding time (s) | $\mathbf{0.1~{}s^{-1}}$ | $\mathbf{1.0~{}s^{-1}}$
---|---|---
$\mathbf{650^{\circ}C}$ | 0.65 | - | - | - | - | 12 | 25 | 50 | 100
1.35 | 12 | 25 | 50 | 100 | 12 | 25 | 50 | 100
(d) Hot compression and holding at temperature with lamellar microstructures.

Table 1: Description of the thermomechanical conditions of the hot compression
campaigns.

###  2.3 Characterization techniques

Two techniques are used to characterize microstructure evolutions: optical
microscopy under polarized light with image analysis [15] (OM-PL) and EBSD.
The first technique is used for characterizing fully recrystallized
microstructures such as the ones observed under GG condition. EBSD, on the
contrary, is required for studying strain hardened microstructures. EBSD
analysis are performed using a Carl Zeiss SUPRA 40 field emission gun scanning
electron microscope (FEG-SEM) equipped with a Bruker Quantax system (EBSD
e-Flash${}^{\textsc{hr}}$). Acceleration voltage is set equal to $20~{}kV$ and
step size to $0.1~{}\mu m$. The step size is kept constant for all samples and
is set sufficiently low to be able to capture local orientation
heterogeneities such as subgrains that are in general few hundred nanometers
wide. Each observation zone dimensions are: $130~{}\mu m$ in width and
$85~{}\mu m$ in height. This area contains between $500$ and $3000$ grains
which is enough to ensure minimum representativeness.

Samples observed using OM-PL technique are prepared according to the protocol
described in ref. [15]. Samples observed by EBSD are polished using first
grinding papers and a polishing solution that contains diamond particles with
a size of $15~{}nm$. Then, they are electropolished using a solution of $90\%$
methanol and $10\%$ perchloric acid. Voltage is set equal to $25~{}V$,
polishing is performed during $27~{}s$ at a temperature between 5 and
10${}^{\circ}C$.

EBSD data are post-processed using MTEX [16]. LAGB are considered to have a
disorientation within $\left[2^{\circ}~{};~{}15^{\circ}\right]$. LAGB that are
enclosing subgrains are designated as closed LAGB. All data have been filtered
using a half-quadratic filter to reduce noise [17]. Geometrically necessary
dislocation (GND) density is estimated following the method developed by
Pantleon [18]. Finally, a criterion based on average GND density per grain is
used for discrimination of recrystallized grains : $\rho_{\textsc{gnd}}\leq
1.0\times 10^{14}m^{-2}$. This choice is motivated to ensure thorough
comparison between experimental and numerical results. Indeed, the other
common variables used for identification of recrystallized grains, such as
grain orientation spread (GOS) or grain average kernel average misorientation
(GAKAM), are not computed in the numerical framework.

##  3 Characterization of microstructure evolutions

###  3.1 Grain growth

The numerical framework described below (section 4) requires to characterize
grain boundary mobility in the absence of SPP. However, there is no
temperature range within $\alpha$-phase stability domain where SPP are soluble
[2]. Consequently, some material have been manufactured with as little as
possible SPP fraction. For both eq. and no SPP. initial states, heat
treatments are conducted under three temperatures: $600$, $650$ and
$700^{\circ}C$. For the material without SPP, holding times are $1$, $3$ and
$8$ hours whereas for the material with SPP they are $1$, $2$, $4$ and $6$
hours. Figure 2 presents the arithmetic average of the grain equivalent circle
diameter ($\overline{ECD}$) as a function of the heat treatment duration.

(a) Evolution of $\overline{ECD}$ for the alloy without SPP.

(b) Evolution of $\overline{ECD}$ for Zy-4 with SPP.

Figure 2: Evolution of $\overline{ECD}$ as a function of holding time at
temperature.

First, it can be seen on figure 2 that the presence of SPP seems to be
responsible for the presence of a plateau for high durations. Those facts are
in agreement with the classical theory describing the Smith-Zener pinning
effect [19, 20, 21]. However, according to the standard Smith-Zener equation,
the limit grain size only depends on the SPP population properties that are
their volume fraction and their mean equivalent diameter. Consequently, the
different plateau values at the three temperatures seem to indicate that the
SPP population evolves differently during heat treatments executed at the
three considered temperatures. To evaluate this, SPP populations have been
characterized by transmission electron microscopy (TEM) for a sample
corresponding to the initial state and for three samples corresponding to the
final states after 6 hours of heat treatment. Figure 3 illustrates how SPP are
observed using TEM and the SPP distributions obtained.

(a) TEM image of SPP.

(b) $ECD$ surface distribution measured by TEM.

Figure 3: Characterization of SPP population by TEM.

One can note that the SPP size distribution does not change that much globally
from one heat treatment to the other. However, it is important to keep in mind
that some slight changes for low ECD values can greatly influence the number
of particles. Those results will be discussed in light of the simulation
results in section 5.1.

###  3.2 Recrystallization

####  3.2.1 Characterization of recrystallization mechanism

As said earlier, Zy-4 is supposed to undergo CDRX under the common industrial
thermomechanical conditions [3, 13]. However, previous studies have been
conducted only with Zy-4 presenting a lamellar microstructure at initial state
and do not provide enough in details microstructure characterization. This
study tackles those two remarks and provides a new insight.

(a) $T=450^{\circ}$, $\dot{\varepsilon}=0.01~{}s^{-1}$, $\varepsilon=1.35$,
$dt=1~{}s$.

(b) $T=650^{\circ}$, $\dot{\varepsilon}=1.0~{}s^{-1}$, $\varepsilon=0.65$,
$dt=1~{}s$.

(c) $T=650^{\circ}$, $\dot{\varepsilon}=1.0~{}s^{-1}$, $\varepsilon=1.35$,
$dt=1~{}s$.

(d) $T=650^{\circ}$, $\dot{\varepsilon}=1.0~{}s^{-1}$, $\varepsilon=1.35$,
$dt=50~{}s$.

(e) $T=650^{\circ}$, $\dot{\varepsilon}=0.01~{}s^{-1}$, $\varepsilon=1.35$,
$dt=1~{}s$.

Figure 4: Presentation of orientation maps with IPF Y color code for five
different microstructures after hot deformation.

Figure 4 presents four orientation maps obtained using EBSD. An inverse pole
figure Y (IPF Y) color coding is used (with Y axis being the compression
direction). Thermomechanical conditions are detailed for each map. It is
interesting to note that when the sample is quenched right after deformation,
the microstructure contains large grains that seem to present orientation
gradients and/or subgrains. Those grains are elongated along horizontal
direction which is perpendicular to the compression direction. The
microstructure also includes a large number of small grains dispersed rather
homogeneously. The orientation map in figure 4(d) corresponding to the
microstructural state after holding at temperature for $50s$ shows that a few
recrystallized grains seem to grow at the expense of the deformed area. This
illustrates that even if a large number of subgrains and small grains are
formed during hot deformation, only a limited amount fulfill the energetic
and/or kinetics criterion to expand. The low growth kinetics of recrystallized
grains is consistent with the observations made in previous section that are
supposing that Zy-4 has a low grain boundary mobility (see section 3.1).

To investigate more in details the mechanisms of DRX, some disorientation
profiles are plotted as well as some average LAGB properties in figure 5. The
disorientation profiles confirm the presence of orientation gradients, put in
light some well defined LAGB with disorientation lower than $5^{\circ}$ and
some LAGB with much higher disorientation (around $10^{\circ}$ and called
medium-low angle GB, MLAGB). The three graphs available in figures 5(c), 6(a)
and 6(b) illustrate how the substructure evolves globally in the whole
microstructure as the deformation occurs and the strain increases. The high
LAGB length fraction and the rather high average misorientation angle for all
strain levels prove that LAGB and MLAGB populations are significant. The
progressive increase of mean LAGB misorientation angle supports the hypothesis
that on average, LAGB know a progressive increase of disorientation. If CDRX
leads to significant evolution of the substructure, it appears that PDRX
impacts to a greater extent the grain structure (see figure 6).

(a) Evolution of disorientation along a given path.

(b) Evolution of disorientation along a given path.

(c) Evolution of average disorientation angle of LAGB with strain.

Figure 5: In details investigation of DRX mechanisms.

All those elements confirm that Zy-4 undergo CDRX [22, 23] by the progressive
formation of subgrains that, under subsequent deformation, will transform to
recrystallized grains either by migration through a zone with orientation
gradient or by stacking of dislocations [8]. They also put in light how PDRX
occurs by the growth of a few recrystallized grains that sweep the deformed
areas.

####  3.2.2 Influence of thermomechanical conditions

Orientation maps for samples quenched as fast as possible right after hot
compression available on figure 4 illustrate the wide range of observed
microstructures. Qualitatively, there is not so much difference observed for
the two samples deformed at $450^{\circ}C$ and $650^{\circ}C$. However, there
is a significant difference for the samples deformed at strain rates of
$0.01~{}s^{-1}$ and $1.0~{}s^{-1}$. Those qualitative observations are
confirmed by the figures 6 and 5(c). Indeed, it is clearly shown that:

  * •

grain size is refined with strain and this is more significant for higher
strain rates (figure 6(a)). It is interesting to note that this refinement
takes certainly place for low strain values and then reaches a rather steady
state value.

  * •

Average LAGB misorientation exhibits a slight increase and illustrates the
progressive disorientation of LAGB (figure 5(c)). This increase stays limited
since some new LAGB with rather low disorientation are created upon subsequent
deformation.

  * •

LAGB length fraction seems to decrease slightly with temperature and under
subsequent deformation (figure 6(b)). This can be explained by the fact that
LAGB starts to form quite rapidly under deformation and reaches a rather
steady-state. Then, subgrain to grain transition leads to the transformation
of LAGB to HAGB and to the reduction of LAGB length ratio.

(a) Evolution of $\overline{ECD}$ with strain.

(b) Evolution of LAGB length fraction with strain.

(c) Evolution of recrystallized fraction with holding time.

(d) Evolution of $\overline{ECD}$ with holding time.

Figure 6: Influence of thermomechanical conditions over some microstructure
characteristic values during CDRX and PDRX.

If the samples are maintained at high temperatures for a few minutes, the
influence of those thermomechanical conditions is even larger. The evolution
of recrystallized fraction and mean recrystallized grain size are plotted in
figure 6. It illustrates how recrystallized fraction, after holding at high
temperature, increases with prior deformation. The effect of strain rate upon
both recrystallized fraction and recrystallized mean grain size is also put in
light. Those characteristic values are not plotted for the samples quenched
immediately after deformation since recrystallized fraction is found to be
negligible.

##  4 Numerical framework

###  4.1 Interface description

The model used for this study relies on a level-set (LS) formulation within a
finite element (FE) framework. Initial digital microstructures are generated
either thanks to a Laguerre-Voronoï method [24] according to an experimental
ECD distribution or by direct immersion of an EBSD image. For this second
method, orientation and dislocation density are averaged inside each
grain/subgrain since the model only allows for constant values inside grains.
Then, interfaces are handled using LS functions initialized as signed
Euclidean distance functions [25]. Grain boundary migration is simulated by
solving transport equation for each LS function:

$\displaystyle\dfrac{\partial\phi}{\partial
t}+\overrightarrow{v}\cdot\overrightarrow{\nabla\phi}=0.$ (1)

The interface velocity is supposed to be equal to the sum of a term induced by
capillarity effects and of a second related to differences of stored energy
between neighboring grains due to gradient of GND density [26]:

$\displaystyle\overrightarrow{v}_{c}=-M\gamma\kappa\overrightarrow{n},$ (2)
$\displaystyle\overrightarrow{v}_{e}=M\llbracket
E\rrbracket\overrightarrow{n}.$ (3)

$M$, $\gamma$ and $\kappa$ are respectively the mobility, energy and curvature
of grain boundaries. $\llbracket E\rrbracket$ is the stored energy difference
between adjacent grains and $\overrightarrow{n}$ is the outward unitary normal
of grain boundaries.

Since SPP are known to reduce significantly grain boundary migration [27],
being able to take into account their effect is necessary. To do so, a
population of spherical SPP respecting the experimental size distribution and
surface fraction is generated. Then, mesh elements and nodes are deleted where
SPP are introduced and boundary conditions are applied to replicate the
interaction between grain boundaries and SPP interfaces [28]. In case some
stored energy due to deformation is considered in simulation, SPP will not be
included. Indeed, pinning effect is considered to be negligible in regards to
stored energy pressures. It allows to save a lot of computation resources,
especially since SPP in Zy-4 are particularly fine and therefore requires to
severely refine the mesh and the time step.

Different numerical improvements of this LS framework were proposed by
Scholtes et al. [29] and Maire et al. [30] and are used here.

###  4.2 Subgrain formation and evolution

To predict the formation of subgrains and their evolution under subsequent
deformation, equations from Gourdet-Montheillet model [14] have been adapted
and implemented.

Dislocation density of each grain evolves according to Yoshie-Laasraoui-Jonas
equation [31]:

$\displaystyle d\rho=\left(K_{1}-K_{2}\rho\right)d\varepsilon,$ (4)

where $K_{1}$ and $K_{2}$ are two material constants. They respectively
describe the strain hardening and the recovery.

Gourdet-Montheillet model assumes that dislocations created under deformation
can evolve:

  * •

through their absorption by HAGB which migrate. This is naturally captured by
setting to the areas swept a low dislocation density $\rho_{0}$ representing
dislocation density of the material free of dislocations.

  * •

By organization to form subgrains. The subgrains interface created for each
deformation increment ($dS^{+}$) by this reorganization follows next equation:

$\displaystyle dS^{+}=\dfrac{\alpha
bK_{2}\rho\dot{\varepsilon}dt}{\eta\theta_{0}},$ (5)

$\alpha=1-\exp{\left(\dfrac{D}{D_{0}}\right)^{m}}$ is a coefficient describing
the fraction of dislocations recovered to form new subgrains. $D$ is the grain
diameter, $D_{0}$ is a grain reference diameter and $m$ is a fixed
coefficient. $b$ is the Burgers vector, $\eta$ is the number of sets of
dislocations and $\theta_{0}$ the disorientation of newly formed subgrains
[14].

  * •

By stacking into preexisting LAGB. The increment of disorientation induced by
this phenomenon is modeled according to following equation [14]:

$\displaystyle d\theta=\dfrac{b}{2\eta}\left(1-\alpha\right)DK_{2}\rho
d\varepsilon.$ (6)

Since $\alpha$ depends on grain diameter, $dS^{+}$ is computed individually
for each grain. Thus, new subgrains naturally appear preferentially where
$dS^{+}$ is the highest. Subgrain diameter is initialized to respect the grain
size distribution measured experimentally. Initial subgrain orientation is set
by applying a small misorientation to the parent grain orientation.
Misorientation axis is taken to respect uniform distribution. Misorientation
angle is set to respect the disorientation measured experimentally at low
strain levels. Finally, once the misorientation axis is set at subgrain
formation, it is kept constant and subgrain misorientation is increased by
applying a rotation around this axis.

###  4.3 Material parameters

For GG simulations realized within this study, grain boundary energy and
mobility are considered isotropic. This assumption is justified by the fact
that the proportion of LAGB is very low in all analyzed samples and therefore
the influence of anisotropy of GB properties is very limited. On the contrary,
for CDRX and PDRX simulations, grain boundary energy $\gamma$ is defined by
Read-Schockley equation:

$\displaystyle\gamma(\theta)\begin{cases}\gamma_{max}\left(\dfrac{\theta}{\theta_{max}}\right)\left(1-\ln{\dfrac{\theta}{\theta_{max}}}\right),~{}\theta<\theta_{max},\\\
\gamma_{max},~{}\theta\geq\theta_{max},\end{cases}$ (7)

with $\theta_{max}=15^{\circ}$ the maximum LAGB disorientation (as for the
post-process of EBSD data described in section 2.3).

In all simulations, the mobility is considered to be isotropic and the
”Heterogeneous with gradient” formulation is employed [32]. A discussion about
misorientation dependence of mobility is discussed in [32, 33].

$K_{2}$ is supposed to be constant for all grains. To reproduce intergranular
deformation heterogeneity, $K_{1}$ varies from grain to grain according to a
distribution. Taylor equation is used to deduce flow stress from average
dislocation density. Thus, $\overline{K_{1}}$ and $K_{2}$ are identified based
on the stress-strain curves and on the GND density values measured using EBSD.
To determine $K_{1}$ distribution, the assumption is made that for strain
values around $1.35$, all grains have reached their saturation GND density
($\rho_{sat}$). Using equation 4, one can deduce that:

$\displaystyle\rho_{sat}=\lim_{\varepsilon\to\infty}rho=\dfrac{K_{1}}{K_{2}}.$
(8)

Accordingly, one can get from GND density estimated by EBSD the value of
$K_{1}$ for each individual grain and deduce its statistical distribution.

Finally, GND density is converted to energy using next equation:

$\displaystyle E=\tau\rho,$ (9)

where $\tau$ stands for dislocation line energy and is estimated through
relation: $\tau=\dfrac{\mu b^{2}}{2}$ with $\mu$ the shear modulus [34].

An extensive description of the numerical framework can be found in literature
[26, 32, 35, 36]. Developments dedicated to CDRX modeling will be detailed in
an upcoming article [33].

##  5 Modeling of grain growth and recrystallization

###  5.1 Modeling of grain growth

As it has been done regarding experimental characterization of GG, this
section first focuses on data obtained with the material having no SPP. Then,
identified parameters are used to simulate GG in the presence of SPP.

####  5.1.1 Identification of material parameters without SPP

In pure GG context, there is no stored energy, so the term $v_{e}$ is null and
the interface velocity is simply equal to the capillarity term $v_{c}$ (see
equations 2 and 3). Therefore, for being able to run GG simulations, one
should have values for the product $M\times\gamma$, also denoted as reduced
mobility. Using experimental data presented in section 3.1, it is possible to
get initial values for reduced mobility by fitting a Burke and Turnbull law
[37]. This allows to run a first set of simulation. Then, comparing simulation
results to experimental data, it is possible to adjust the reduced mobility
value by a given coefficient in order to minimize $L^{2}$ error. Finally, a
last set of simulations is launched using this adjusted reduced mobility
values. This succession of operations is of course repeated for all
temperatures, in our cases $600$, $650$ and $700^{\circ}C$. The last step of
the identification procedure consists in interpolating, and if required,
extrapolating, the reduced mobility parameters identified at different
temperatures. To do so, $\gamma$ is considered to be constant within the
temperature range and is taken from literature [12]. Then, an Arrhenius law is
fitted such as described by equation 10. This procedure is also applied and
described in details in ref. [38].

$\displaystyle M=M_{0}\times\exp{\dfrac{-Q}{RT}}.$ (10)

In equation 10, $M_{0}$ is considered as a constant, $Q$ is the activation
energy for GG and $R$ is the gas constant.

Figure 7(a) presents the evolution of $\overline{ECD}$ for the three
temperatures. One can note that the general agreement between experimental and
numerical results is satisfying. Few differences are still observable,
particularly for very short and very long heat treatments. Indeed, the
kinetics seems to be slightly underestimated for short durations and slightly
overestimated for long times. However, the simulation results still show about
less than $1~{}\mu m$ difference compared to experimental results which
remains precise enough.

####  5.1.2 Application to commercial material

After having identified reduced mobility using data representative of the
material presenting no SPP, it is particularly interesting to assess whether
these parameters are appropriate for modeling GG with a significant SPP
population. Simulation results detailed within this section follow the same
principles than the ones described in section 5.1.1. SPP populations are
included within this simulation according to the strategy described in section
4.1. SPP are considered to be static within the heat treatment. Nevertheless,
to ensure maximal representativeness, SPP populations are defined
independently for each heat treatment temperature according to the
experimental measurements done by TEM and describing the state of SPP at the
end of the heat treatment (see section 3.1). Figure 7 presents the evolution
of the $\overline{ECD}$ with time as well as $ECD$ histograms at different
instants.

(a) Evolution of $\overline{ECD}$ with time.

(b) Evolution of $\overline{ECD}$ with time.

(c) $ECD$ number histogram at $t=1~{}hour$.

(d) $ECD$ number histogram at $t=2~{}hours$.

(e) $ECD$ number histogram at $t=4~{}hours$.

Figure 7: Evolution of grain size population during GG simulation.

The description of mean grain size and of the general grain size distribution
is excellent for the lowest temperature. Moreover, the simulations are able to
reproduce the fact that $\overline{ECD}$ plateau at three distinct values.
According to classic Smith-Zener theory of GB pinning by SPP,
$\overline{ECD}_{sat}\propto R_{\textsc{spp}}/f_{\textsc{spp}}$. Therefore,
this increase of $\overline{ECD}_{sat}$ with temperature is consistent with a
phenomenon of Ostwald ripening happening during heat treatment. Nevertheless,
numerical results underestimate GG at $700^{\circ}C$. This could be explained
by the fact that TEM measurements of SPP population are made at room
temperature and so the SPP populations considered for these simulations are
potentially not fully representative due to some dissolution of SPP. Finally,
if one had access to the kinetics of SPP population evolution, it would be
possible to use the latest developments made in the used LS framework to
consider a population of SPP evolving with time [39].

###  5.2 Modeling of CDRX and PDRX

The implementations dedicated to CDRX simulation described in section 4.2 are
applied to propose a representative simulation of both CDRX and PDRX, starting
from a material with an equiaxed microstructure. Initial microstructure is
generated in order to respect experimental grain size and orientation
distribution representative of the initial state. Then, this representative
elementary volume (REV) is deformed until $\varepsilon=1.35$ at $650^{\circ}$
and is hold for $300s$. Some numerical and experimental GND density maps are
displayed in figure 8. The general qualitative agreement between experimental
and numerical microstructures is good. It is noteworthy to mention that the
model is able to capture microstructure heterogeneities with preferential
zones for the formation and growth of recrystallized grains. Indeed, as shown
in figure 8(a), the model is able to reproduce to some extent intergranular
deformation heterogeneities. Therefore, the grains having a higher dislocation
density are the first ones to be swept by recrystallized grains.

(a) GND density map from simulation data, averaged by subgrain.

(b) GND density map computed from experimental data, averaged by grain.

Figure 8: Comparison of experimental and digital GND density maps during CDRX
and PDRX.

Finally, a quantitative comparison of some characteristic values is presented
in figure 9. Recrystallized fraction and recrystallized $\overline{ECD}$ are
well predicted by the model. The model is able to evaluate correctly the
length of LAGB created during hot deformation and is in the right order of
magnitude during subsequent heat treatment. Nevertheless, it is not able to
describe properly the first instants of PDRX during which LAGB length ratio
stays constant.

(a) Evolution of recrystallized fraction with time.

(b) Evolution of RX grain $\overline{ECD}$ with time.

(c) Evolution of LAGB length fraction with time.

(d) Evolution of $\overline{\rho}$ with time.

Figure 9: Evolution of the main microstructural features during hot
deformation followed by heat treatment simulation.

###  5.3 Modeling of PDRX using immersed EBSD data

The most significant microstructure evolutions caused by recrystallization
take place during heat treatment after hot deformation (see section 3.2.1).
Therefore, to study in details the influence of the initial microstructure, it
appears interesting to use the EBSD data to define precisely the initial
microstructure. This is particularly relevant since with the actual framework,
deformation phenomena are simplified due to the fact that modeling deformation
using state of the art models such as crystal plasticity is too costly [40].

EBSD data from two samples are post-processed as described in section 2.3.
Those two samples have been deformed under the same thermomechanical
conditions:
$T=650^{\circ}C~{};~{}\dot{\varepsilon}=1.0~{}s^{-1}~{};~{}\varepsilon_{f}=1.35$.
Figure 10 illustrates how both digital and experimental microstructures evolve
during heat treatment. One can see that the exact same EBSD data, post-
processed either by averaging GND density into grains (figure 10(b)) or
subgrains (figure 10(c)) can influence drastically the general look of the
microstructure. Then, comparing experimental to simuation data, one can note
that the general agreement is good, especially if experimental data are
averaged by subgrain, which is consistent to what has been done with
simulation data. It is also interesting to notice that this difference between
grain and subgrain structure is much more significant than for an initial
equiaxed microstructure (as shown in previous section 5.2).

(a) GND density map from simulation data, averaged by subgrain.

(b) GND density map computed from experimental data, average by grain.

(c) GND density map computed from experimental data, averaged by
grain/subgrain.

Figure 10: Comparison of experimental and digital GND density maps during
PDRX.

Figure 11 presents the evolution of the main microstructure statistical
descriptors. One can observe that the numerical results for recrystallized
fraction are in good agreement with the experimental ones. It is especially
interesting to note that the model is able to reproduce the slower
recrystallization of basket-weaved microstructure. This could appear counter-
intuitive looking at figure 11(d) since dislocation density which constitutes
the main driving force for recrystallization is higher for basket-weaved
microstructure. However, two other phenomena could explain this difference.
First, it appears that the LAGB length ratio (see figure 11(c)) of the basket-
weaved microstructure exhibits the same evolution but delayed in time. This
could potentially be explained by the difference of spatial distribution of
SPP. Indeed, alignment of SPP at lamellae borders in quenched sample could
reduce the dislocation mean free path and limit their ability to form LAGB.
Another hypothesis is that the increase in deformation localization for
quenched microstructures could lead to preferential zones for the formation of
recrystallized grains. Those grains would then impinge themselves more quickly
and this would lead to a slower recrystallization kinetics.

One should also point out that the model ability to evaluate evolution of
other characteristics such as mean dislocation density and LAGB length ratio
is good but slightly less precise. Several hypothesis could explain those
differences:

  * •

the model used consider only averaged values by grain. Thus, it is not able to
capture the influence of local variations in dislocation density or
crystallographic orientation. It also only supports the formation of LAGB
which bounds perfectly closed and spherical subgrains which is not seen
experimentally. Consequently, this could reduce its ability to predict values
related to LAGB.

  * •

As it is used for producing the results presented here, no static recovery
mechanisms are considered. This, as well as the $\rho_{0}$ value associated to
areas swept by moving boundaries could influence significantly driving forces
for microstructure evolutions.

  * •

Finally, no nucleation during the PDRX stage is modeled. There are no clear
experimental evidences that this happens for the conditions studied but it
remains a probable phenomenon that could influence post-dynamic evolutions.

(a) Evolution of recrystallized fraction with time.

(b) Evolution of RX grain $\overline{ECD}$ with time.

(c) Evolution of closed LAGB length fraction with time.

(d) Evolution of $\overline{\rho}$ with time.

Figure 11: Evolution of the main microstructural features during heat
treatment after hot deformation simulation.

Except the hypothesis related to intragranular heterogeneities and
oversimplified deformation predictions, which are outside the scope of this
study, the other will be evaluated and discussed in an upcoming article.

Finally, the consistency of the results obtained either using the CDRX and
PDRX framework or only the PDRX one with immersed microstructure is noticeable
(see figures 9 and 11). This tends to confirm that the CDRX framework provides
precise enough results and that the general simulation environment used is
robust and consistent.

##  6 Conclusion

The present work has detailed some of the main outcomes obtained from an
extensive experimental and numerical study of recrystallization of Zy-4. The
main recrystallization mechanisms have been identified and described and the
influence of some material and thermomechanical parameters quantified.
Additionally, the mechanisms observed experimentally have been successfully
reproduced by the way of full-field simulations. Adaptation and implementation
of the Gourdet-Montheillet model into the LS framework has shown satisfying
results. Its ability to reproduce some intergranular heterogeneities and to
capture to some extent the evolution of some specific physical quantities
related to LAGB has been proven. Future works will be dedicated to model CDRX
with various initial microstructures and for different thermomechanical
conditions. A special attention will be paid to obtain an even better
agreement with experimental measurements by refining some parameters as
discussed in section 5.3. Finally, those numerical results will be analyzed in
details in order to assess the influence of some specific mechanisms such as
recovery and formation of subgrains in the post-dynamic regime.

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